arXiv:hep-ph/0203079v3 4 Mar 2003
FERMI–PUB–02/045-T
BUHEP-01-09
March, 2003
Strong Dynamics and Electroweak
Symmetry Breaking
Christopher T. Hill1
and
Elizabeth H. Simmons2,3
1
Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, IL, 60510
2
Dept. of Physics, Boston University
590 Commonwealth Avenue, Boston, MA, 02215
3
Radcliffe Institute for Advanced Study and
Department of Physics, Harvard University
Cambridge, MA, 02138
Abstract
The breaking of electroweak
symmetry, and origin of the associated “weak
q √
scale,” vweak = 1/ 2 2GF = 175 GeV, may be due to a new strong interaction. Theoretical developments over the past decade have led to viable models
and mechanisms that are consistent with current experimental data. Many of
these schemes feature a privileged role for the top quark, and third generation, and are natural in the context of theories of extra space dimensions at
the weak scale. We review various models and their phenomenological implications which will be subject to definitive tests in future collider runs at the
Tevatron, and the LHC, and future linear e+ e− colliders, as well as sensitive
studies of rare processes.
Contents
1 Introduction
1.1 Lessons from QCD . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Weak Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Superconductors, Chiral Symmetries, and Nambu-Goldstone Bosons
1.4 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Purpose and Synopsis of the Review . . . . . . . . . . . . . . . . .
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2 Technicolor
2.1 Dynamics of Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The TC ↔ QCD Analogy . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Estimating in TC by Rescaling QCD; fπ , FT , vweak . . . . . . . . .
2.2 The Minimal TC Model of Susskind and Weinberg . . . . . . . . . . . . . .
2.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Spectroscopy of the Minimal Model . . . . . . . . . . . . . . . . . .
2.2.3 Non-Resonant Production and Longitudinal Gauge Boson Scattering
2.2.4 Techni-Vector Meson Production and Vector Meson Dominance . .
2.3 Farhi–Susskind Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Production and Detection at Hadron Colliders . . . . . . . . . . . .
2.3.4 Production and Detection at e+ e− Colliders . . . . . . . . . . . . .
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3 Extended Technicolor
3.1 The General Structure of ETC . . . . . . . . . . . . . .
3.1.1 Master Gauge Group GET C . . . . . . . . . . . .
3.1.2 Low Energy Relic Interactions . . . . . . . . . .
3.1.3 The α-terms: Techniaxion Masses. . . . . . . .
3.1.4 The β terms: Quark and Lepton Masses . . . .
3.1.5 The γ terms: Flavor-Changing Neutral Currents
3.2 Oblique Radiative Corrections . . . . . . . . . . . . . .
3.3 Some Explicit ETC Models . . . . . . . . . . . . . . .
3.3.1 Techni-GIM . . . . . . . . . . . . . . . . . . . .
3.3.2 Non-Commuting ETC Models . . . . . . . . . .
3.3.3 Tumbling and Triggering . . . . . . . . . . . . .
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3.4
3.5
3.6
3.7
3.3.4 Grand Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Walking Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Schematic Walking . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Schwinger-Dyson Analysis . . . . . . . . . . . . . . . . . . . . . . .
Multi-Scale and Low-Scale TC . . . . . . . . . . . . . . . . . . . . . . . . .
Direct Experimental Limits and Constraints on TC . . . . . . . . . . . . .
3.6.1 Searches for Low-Scale Color-singlet Techni–ρ’s and Techni–ω’s (and
associated Technipions) . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Separate Searches for color-singlet P 0 , P 0′ . . . . . . . . . . . . . .
3.6.3 Separate searches for color-singlet PT± . . . . . . . . . . . . . . . . .
3.6.4 Searches for Low-Scale Color-octet Techni–ρ’s (and associated Leptoquark Technipionss) . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.5 Searches for W ′ and Z ′ bosons from SU(2) × SU(2) . . . . . . . . .
Supersymmetric and Bosonic Technicolor . . . . . . . . . . . . . . . . . . .
3.7.1 Supersymmetry and Technicolor . . . . . . . . . . . . . . . . . . .
3.7.2 Scalars and Technicolor: Bosonic Technicolor . . . . . . . . . . . .
4 Top Quark Condensation and Topcolor
4.1 Top Quark Condensation in NJL Approximation . . . . . . . .
4.1.1 The Top Yukawa Quasi-Infrared Fixed Point . . . . . .
4.1.2 The NJL Approximation . . . . . . . . . . . . . . . . .
4.2 Topcolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Gauging Top Condensation . . . . . . . . . . . . . . .
4.2.2 Gauge Groups and the Tilting Mechanism . . . . . . .
4.2.3 Top-pion Masses; Instantons; The b-quark mass . . . .
4.2.4 Flavor Physics: Mass Matrices, CKM and CP-violation
4.3 Topcolor Phenomenology . . . . . . . . . . . . . . . . . . . . .
4.3.1 Top-pions . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Colorons: New Colored Gauge Bosons . . . . . . . . .
4.3.3 New Z ′ Bosons . . . . . . . . . . . . . . . . . . . . . .
4.4 Top Seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 The Minimal Model . . . . . . . . . . . . . . . . . . . .
4.4.2 Dynamical Issues . . . . . . . . . . . . . . . . . . . . .
4.4.3 Including the b-quark . . . . . . . . . . . . . . . . . . .
4.5 Top Seesaw Phenomenology . . . . . . . . . . . . . . . . . . .
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4.5.1
4.5.2
4.6 Extra
4.6.1
4.6.2
Seesaw Quarks . . . . . . .
Flavorons . . . . . . . . . .
Dimensions at the TeV Scale
Deconstruction . . . . . . .
Little Higgs Theories . . . .
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143
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5 Outlook and Conclusions
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6 Acknowledgements
160
Appendix A: The Standard Model
161
Appendix B: The Nambu-Jona-Lasinio Model
168
Bibliography
172
3
1
Introduction
1.1
Lessons from QCD
The early days of accelerator-based particle physics were largely explorations of the strong
interaction scale, associated roughly with the proton mass, of order 1 GeV. Key elements
were the elaboration of the hadron spectroscopy; the measurements of cross-sections;
the elucidation of spontaneously broken chiral symmetry, with the pion as a NambuGoldstone phenomenon [1, 2, 3]; the evolution of the flavor symmetry SU(3) and the
quark model [4, 5, 6]; the discovery of scaling behavior in electroproduction; [7, 8, 9];
and the observation of quarks and gluons as partons. Eventually, this work culminated
in Quantum Chromodynamics (QCD): a description of the strong scale based upon the
elegant symmetry principle of local gauge invariance, and the discovery of the Yang-Mills
gauge group SU(3) of color [10, 11]. Today we recognize that the strong scale is a welldefined quantity, ΛQCD ∼ 100 MeV, the infrared scale at which the perturbatively defined
running coupling constant of QCD blows up, and we are beginning to understand how to
perform detailed nonperturbative numerical computations of the associated phenomena.
There are several lessons in the discovery of QCD which illuminate our present perspective on nature. First, it took many years to get from the proton, neutron, and pions,
originally thought to be elementary systems, to the underlying theory of QCD. From the
perspective of a physicist of the 1930’s, knowing only the lowest-lying states, or one of the
1950’s seeing the unfolding of the resonance spectroscopy and new quantum numbers, such
as strangeness, it would have seemed astonishing that those disparate elements came from
a single underlying Yang-Mills gauge theory [12]. Second, the process of solving the problem of the strong interactions involved a far-ranging circumnavigation of all of the ideas
that we use in theoretical physics today. For example, the elaboration of the resonance
spectrum and Regge behavior of QCD led to the discovery of string theory1 ! QCD embodies a rich list of phenomena, including confinement, perturbative asymptotic freedom
[13, 14], topological fluctuations, and, perhaps most relevant for our present purposes, a
BCS-like [15, 16] mechanism leading to chiral symmetry breaking [1, 2, 3]. Finally, the
strong interactions are readily visible in nature for what might be termed “contingent”
reasons: the nucleon is stable, and atomic nuclei are abundant. If a process like p → e+ +γ
occured with a large rate, so that protons were short-lived, then
√ the strong interactions
would be essentially decoupled from low energy physics, for s < 2mπ . A new strong
dynamics, if it exists, presumably does not have an analogous stable sector (else we would
have seen it), and this dynamics must be largely decoupled below threshold.
QCD provides direct guidance for this review because of the light it may shed on the
scale of electroweak symmetry breaking (EWSB). The origin of the scale ΛQCD , and the
“hierarchy” between the strong and gravitational scales is, in principle, understood within
QCD in a remarkable and compelling way. If a perturbative input for αs is specified at
1
Proponents of concepts such as nuclear democracy, duality, and Veneziano models (which contain an
underlying string theory spectrum) believed this perspective, at the time, to be fundamental.
4
some high energy scale, e.g., the Planck scale, then the logarithmic renormalization group
running of αs naturally produces a strong-interaction scale ΛQCD which lies far below
MP lanck . The value of scale ΛQCD does not derive directly from the Planck scale, e.g.
through multiplication by some ratio of coupling constants. Rather ΛQCD arises naturally
and elegantly from quantum mechanics itself, through dimensional transmutation [17].
Hence, QCD produces the large hierarchy of scales without fine-tuning. The philosophy
underlying theories of dynamical EWSB is that the “weak scale” has a similar dynamical
and natural origin.
1.2
The Weak Scale
We have mentioned two of the fundamental mass √
scales in nature, the strong-interaction
scale ΛQCD and the scale of gravity, MP lanck = ( GN )−1 ≈ 1019 GeV. The mass scale,
which
q √ will figure most directly in our discussion, is the scale of weak physics, vweak =
( 2 2GF )−1 = 175 GeV. The weak scale first entered physics, approximately 70 years
ago, when Enrico Fermi constructed the current-current interaction description of βdecay and introduced the constant, GF , into modern physics [18]. The Standard Model
[19, 20, 21] identifies vweak with the vacuum expectation value (VEV) of a fundamental,
isodoublet, “Higgs” scalar field. Both the gravitational and weak scales are associated
with valid low energy effective theories. In the case of MP lanck we have classical General
Relativity; In the case of vweak we have the Standard Model.
The Standard Model is predictive and enjoys spectacular success in almost all applications to the analysis of existing experimental data. This hinges upon its renormalizability:
it is a valid quantum theory. Now that current experiments are sensitive to electroweak
loop effects, these two aspects have become closely entwined. In almost all channels, experiment continues simply to confirm the Standard Model’s predictions; those processes
which are sensitive to loop contributions are starting to constrain the mass of the Higgs
Boson. Combining all experimental data sensitive to electroweak loop corrections yields
an upper bound on the Standard Model Higgs boson’s mass of order 200 GeV [22].2
Beyond this fact, however, we know nothing in detail about the Higgs boson, or
whether or not it actually exists as a fundamental particle in nature. If new dynamics
were assumed to be present at the TeV scale, the Higgs boson could be a bound state
and the upper bound on the composite Higgs mass would rise to ∼ 1 TeV [24]. Certain
models we will describe in this review, such as the Top Quark Seesaw scheme, predict a
composite Higgs boson with a mass mH ∼ 1 TeV and are otherwise in complete agreement
with electroweak constraints (see Section 4).
In considering the Higgs sector, the foremost question is that of motive: “why should
2
At this writing there are internal inconsistencies in the precision electroweak data; the Z 0 -pole data
of leptons alone predicts a Higgs boson mass that is ∼ 20 − 40 GeV and is directly ruled out, while the
hadronic AbF B predicts a very heavy Higgs mass ∼ 300 GeV; these discrepancies are significant, at the
3.5σ level, and it is not clear that the Higgs mass bound obtained by combining all data is meaningful
[23].
5
nature provide a unique elementary particle simply for the purpose of breaking a symmetry?” Other issues involve naturalness, i.e., the degree of fine-tuning required to provide
the scale of the mass of the putative Higgs boson [25, 26, 27]. Certainly, when compared
to the completely natural origin of ΛQCD relative to MP lanck , the origin of a Higgs boson
mass, mH ∼ 200 GeV in the Standard Model is a complete mystery. These questions hint
at the need for a more general mechanism or some enveloping symmetries that do the
equivalent job of, or provide a rationalization for, the Higgs Boson. Therefore, it is fair
to say that the true mechanism of EWSB in nature is unknown.
The paradigm we will explore in the present review is that the EWSB physics, in analogy to the strong interactions of QCD, arises from novel strong dynamics. We emphasize
at the outset that new strong dynamics (NSD) is incompatable with a completely perturbative view of physics near the electroweak scale – but does not exclude the possibility of
a low-scale Supersymmetry (SUSY). For the most part in this review, our discussion will
focus on the nature and implications of the new strong dynamics itself, and not on the
additional possible presence of SUSY.
In focusing this review in this direction, we should consider what we hope to learn by
looking beyond the more popular supersymmetric theories. Certainly, SUSY is an elegant
extension of the Lorentz group that fits naturally into string theory, our best candidate
for a quantum theory of gravity. SUSY, moreover, gives us an intriguing raison d’etre
for the existence of fundamental scalar particles: If we look at the fermionic content of a
model, such as the Minimal Supersymmetric Standard Model (MSSM) we see that Higgs
bosons can be viewed as superpartners of new vector-like leptons (i.e., a pair of lefthanded leptonic isodoublets, one with Y = 1 and the other with Y = −1 in the MSSM).
As additional rewards we find that: (i) the hierarchy, while not explained, is protected
by the chiral symmetries of the fermions; (ii) the resulting theory can be weakly coupled
and amenable to perturbative studies; and (iii) there is reasonably precise unification of
the gauge coupling constants.
On the other hand, SUSY offers fairly limited insight into why there occurs EWSB,
since the Higgs sector is essentially added by hand, just as the original Higgs boson was
added to the Standard Model, to accomodate the phenomenon. The significance of the
successful unification of the gauge coupling constants [28, 29, 30], which is certainly one of
the more tantalizing aspects of the MSSM, is nevertheless inconclusive because the unification condition that obtains in nature remains unknown. For example, higher dimension
operators associated with the Planck or GUT scales can modify the naive unification condition, thus permitting unification in models that might otherwise be rejected [31, 32].
Moreover, new Strong Dynamical Models (NSD’s) of EWSB can in principle unify. However, because the primary dynamical issues in models of new strong dynamics arise at
the weak scale and because complete NSD models are few in number, unification of NSD
models has not been developed very far.
We believe that the central problem facing particle physics today is to explain the
origin of the electroweak mass scale, or equivalently, EWSB: What causes the scale
vweak ∝ (GF )−1/2 in nature? Theories of new strong dynamics offer new insights into
6
possible mechanisms of electroweak symmetry breaking. We recall that before QCD was
understood to be a local gauge theory with its intrinsic rich dynamics, speculation about
what lay beyond the strong interaction scale was systematically flawed. Thus, while
many of the fundamental symmetries controlling the known forces in nature are understood, speculation as to what lies on energy scales well above vweak should be viewed as
tentative. Let us therefore focus on physics at the weak scale.
We will begin by providing an introductory tutorial survey of the elementary physical
ingredients of dynamical symmetry breaking. In Section 1.3, we will consider a sequence of
“five easy pieces,” or illustrative models which successively incorporate the key elements
of the electroweak Standard Model. This is complemented by a discussion of the full
Standard Model, including oblique radiative corrections, in 1.4, 3.2, and Appendix A,
and by a discussion of a toy model of strong dynamics, the Nambu–Jona-Lasinio model,
in Appendix B. Issues related to naturalness in the QCD and electroweak sectors of the
Standard Model are discussed in Section 1.4. These are intended to provide newcomers
or non-specialists with the collected ideas and a common language to make the rest
of the review accessible. The arrangement of our discussion of modern theories and
phenomenology of dynamical EWSB is given in Section 1.5.
A reader who may wish to “cut to the chase” is advised to skip directly to Section 1.5.
1.3
Superconductors, Chiral Symmetries, and Nambu-Goldstone
Bosons
A particle physicist’s definition of an ordinary electromagnetic superconductor is a “vacuum” or groundstate in which the photon becomes massive. For example, when a block of
lead (Pb) is cooled to 3o K in the laboratory, photons impinging on the material acquire
a mass of about 1 eV, and the associated phenomena of superconductivity arise (e.g. expulsion of magnetic field lines, low-resistance flow of electric currents, etc.). The vacuum
of our Universe, the groundstate of the Standard Model, is likewise an “electroweak superconductor” in which the masses of the W ± and Z 0 gauge bosons are nonzero, while
the photon remains massless. Moreover there occurs in QCD the phenomenon of “chiral
symmetry breaking,” [1, 2, 3], analogous to BCS superconductivity [15, 16] , in which the
very light up, down and strange quarks develop condensates in the vacuum from which
they acquire larger “constituent quark masses,” in analogy to the “mass gap” of a BCS
superconductor.3
In what follows, we will develop an understanding of chiral symmetry breaking and dynamical mass generation in the more familiar context of electrodynamics and the LandauGinzburg model of superconductivity. To accomplish this, we examine the following sequence of toy models: (i) the free superconductor in which the longitudinal photon is a
massless spin-0 field and manifest gauge invariance is preserved; (ii) a massless fermion;
3
In a superconductor the mass gap is actually a small Majorana-mass, ∼ ψψ + h.c. for an electron, an
operator which carries net charge ±2
7
(iii) the simple U(1)L × U(1)R fermionic chiral Lagrangian in which the fermion acquires
mass spontaneously and a massless Nambu-Goldstone boson appears; (iv) the Abelian
Higgs model (also known as the Landau-Ginzburg superconductor); and putting it all
together, (v) the Abelian Higgs model together with the fermionic chiral Lagrangian in
which the Nambu-Goldstone boson has become the longitudinal photon. We will then
indicate how the discussion generalizes to our main subject of interest in Section 1.4, the
electroweak interactions as described by the Standard Model.
1.3(i) Superconductor ↔ A Massive Photon
The defining principle of electrodynamics is local U(1) gauge invariance. Can a massive
photon be consistent with the gauge symmetry? After all, a photon mass term like
1
M 2 Aµ Aµ appears superficially not to be invariant under the gauge transformation Aµ →
2
Aµ + ∂µ χ. It can be made manifestly gauge invariant, however, if we provide an additional
ingredient in the spectrum of the theory: a massless spinless (scalar) mode, φ, coupled
longitudinally to the photon. The Lagrangian of pure QED together with such a mode
may then be written:
1
L = − Fµν F µν + 12 (∂µ φ)2 + 21 e2 f 2 (Aµ )2 − ef Aµ ∂ µ φ
4
1
1
= − Fµν F µν + 12 e2 f 2 (Aµ − ∂µ φ)2
4
ef
(1.1)
In the second line we have completed the square of the scalar terms and we see that
this Lagrangian is gauge invariant, i.e., if the transformation Aµ → Aµ + ∂µ χ/ef is
accompanied by φ → φ + χ then L is invariant. The quantity f , is called the “decay
constant” of φ. It is the direct analogue of fπ for the pion of QCD (for a discussion of
normalization conventions for fπ see Section 2.1).
We see that the photon vector potential and the massless mode have combined to form
a new field: Bµ = Aµ − ∂µ φ/ef . Physically, Bµ corresponds to a “gauge-invariant massive
photon” of mass mγ = ef . Thus, the Lagrangian can be written directly in terms of Bµ
as
1
L = − FBµν FBµν + 21 m2γ (Bµ )2
4
(1.2)
where FBµν ≡ ∂µ Bν − ∂ν Bµ = Fµν . The φ field has now blended with Aµ to form the
heavy photon field; we say that the field φ has been “eaten” by the gauge field to give it
mass.
More generally, in order for this mechanism to produce a superconductor, the Lagrangian for the mode φ must possess the symmetry φ → φ + ξ(x) where ξ(x) can be
any function of space-time (φ → φ + ξ where ξ is a constant is the corresponding global
symmetry in the absence of the gauge fields) . This essentially requires a massless field φ,
with derivative couplings to conserved currents ∼ jµ ∂ µ φ. The shift will then change the
φ action by at most a total divergence, and we can eliminate surface terms by requiring
that all fields be well behaved at infinity.
8
Fields like φ, called Nambu-Goldstone bosons (NGB’s), always arise when continuous
symmetries are spontaneously broken.
1.3(ii) A Massless Fermion → Chiral Symmetry
Consider now a fermion ψ(x), described by a four component complex Dirac spinor.
We define the “Left-handed” and “Right-handed” projected fields as follows:
ψL = 12 (1 − γ5 )ψ ;
ψR = 12 (1 + γ5 )ψ
(1.3)
The (1 ± γ5 )/2 operators are projections, and the reduced fields are equivalent to two
independent two-component complex spinors, each, by itself, forming an irreducible representation of the Lorentz group. The Lagrangian of a massless Dirac spinor decomposes
into two independent fields’ kinetic terms as:
L = ψ̄i∂/ ψ = ψ̄L i∂/ ψL + ψ̄R i∂/ ψR
(1.4)
This Lagrangian is invariant under two independent global symmetry transformations,
which we call the “chiral symmetry” U(1)L × U(1)R :
ψL → exp(−iθ)ψL ;
ψR → exp(−iω)ψR ;
(1.5)
The symmetry transformation corresponding to the conserved fermion number has (θ = ω)
while an axial, or γ 5 , symmetry transformation has (θ = −ω). The corresponding Noether
currents are:
jµL ≡
δL
= 21 ψ̄γµ (1 − γ5 )ψ;
δ∂µ θ(x)
jµR ≡
δL
= 12 ψ̄γµ (1 + γ5 )ψ;
δ∂µ ω(x)
(1.6)
We can form the vector current, jµ = jµR + jµL = ψ̄γµ ψ and the axial vector current,
jµ5 = jµR − jµL = ψ̄γµ γ5 ψ̄.
If we add a mass term to our Lagrangian we couple together the two independent Land R-handed fields and thus break the chiral symmetry:
L = ψ̄i∂/ ψ − mψ̄ψ = ψ̄L i∂/ ψL + ψ̄R i∂/ ψR − m(ψ̄L ψR + ψ̄R ψL )
(1.7)
The original U(1)L × U(1)R chiral symmetry of the massless theory has now broken to a
residual U(1)L+R , which is the vectorial symmetry of fermion number conservation. We
can see explicitly that the vector current is conserved since the transformation eq.(1.5)
with θ = ω is still a symmetry of the Lagrangian eq.(1.7). The axial current, on the other
hand, is no longer conserved:
←
→
∂µ ψ̄γ µ γ5 ψ = ψ̄ ∂/ γ5 ψ + ψ̄γ5 ∂/ ψ
= −2imψ̄γ5 ψ
The Dirac mass term has spoiled the axial symmetry (θ = −ω).
9
(1.8)
1.3(iii) Spontaneously Massive Fermion → Nambu-Goldstone Boson
Through a sleight of hand, however, we can preserve the full U(1)L × U(1)R chiral
symmetry, and still give the fermion a mass! We introduce a complex scalar field Φ
with a Yukawa coupling (g) to the fermion. We assume that Φ transforms under the
U(1)L × U(1)R chiral symmetry as:
Φ → exp[−i(θ − ω)]Φ
(1.9)
that is, Φ has nonzero charges under both the U(1)L and U(1)R symmetry groups. Then,
we write the Lagrangian of the system as:
L = ψ̄L i∂/ ψL + ψ̄R i∂/ ψR − g(ψ̄L ψR Φ + ψ̄R ψL Φ∗ ) + LΦ
(1.10)
LΦ = |∂Φ|2 − V (|Φ|)
(1.11)
where
Unlike the previous case where we added the fermion mass term and broke the symmetry
of the Lagrangian, L remains invariant under the full U(1)L × U(1)R chiral symmetry
transformations. The vector current remains the same as in the pure fermion case, but
the axial current is now changed to:
→
←
jµ5 = ψ̄γµ γ5 ψ + 2iΦ∗ ( ∂ µ − ∂ µ )Φ
(1.12)
We can now arrange to have a “spontaneous breaking of the chiral symmetry” to give
mass to the fermion. Assume the potential for the field Φ is:
V (Φ) = −M 2 |Φ|2 + 12 λ|Φ|4
(1.13)
The vacuum built √
around the field configuration hΦi = 0 is unstable. Therefore, let us
ask that hΦi = v/ 2, and without loss of generality we can take v real. The potential
energy is minimized for:
v
M
√ =√
(1.14)
2
λ
We can parameterize the “small oscillations” around the vacuum state by writing:
1
Φ = √ (v + h(x)) exp(iφ(x)/f )
2
(1.15)
where φ(x) and h(x) are real fields. Substituting this anzatz into the scalar Lagrangian
(1.11) we obtain:
LΦ =
s
λ
1
Mh3 − λh4
2
8
√
2
2M
1
v
h(∂φ)2 + Λ
+ 2 (∂φ)2 + 2 h2 (∂φ)2 +
2
2f
2f
λf
1
(∂h)2
2
2 2
−M h −
10
(1.16)
where we have a negative vacuum energy density, or cosmological constant, Λ = −M 4 /2λ
(of course, we can always add a bare cosmological constant to have any arbitrary vacuum
energy we wish).
We see that φ(x) is a massless field (a Nambu–Goldstone mode). It couples only
derivatively to other fields because of the symmetry φ → φ + ξ.4 The field h(x), on
the other hand, has a positive mass-squared of m2 = 2M 2 . The proper normalization of
the kinetic term, for φ, i.e., (v 2 /2f 2 )(∂φ)2 , requires that f = v. Again, f is the decay
constant of the pion–like object φ. The decay constant f is always equivalent
to the
√
vacuum expectation value (apart from a possible conventional factor like 2).
Notice that the mass of h(x) can be formally taken to be arbitrarily large, i.e., by taking
the limit M → ∞, and λ → ∞ we can hold v 2 = f 2 = 2M 2 /λ fixed. This completely
suppresses fluctuations in the h field, and leaves us with a nonlinear σ model [3]. In
this case only the Nambu-Goldstone φ field is relevant
at low energies. In the nonlinear
√
σ model we can directly parameterize Φ = (f / 2) exp(iφ/f ). The axial current then
becomes:
jµ5 = ψγµ γ 5 ψ − 2f ∂µ φ
(1.17)
where the factor of 2 in the last term stems from the axial charge 2 of Φ (eq.(1.9)). Let
us substitute this into the Lagrangian eq.(1.10) containing the fermions:
√
(1.18)
L = ψ̄L i∂/ ψL + ψ̄R i∂/ ψR + 21 (∂φ)2 − (gf / 2)(ψ̄L ψR eiφ/v + ψ̄R ψL e−iφ/f )
If we expand in powers of φ/f we obtain:
√
√
L = ψ̄i∂/ ψ + ψ̄i∂/ ψ + 21 (∂φ)2 − (gf / 2)ψ̄ψ − i(g/ 2)φψ̄γ 5 ψ + ...
(1.19)
√
We see that this Lagrangian describes a Dirac fermion of mass m = gf / 2, and a massless pseudoscalar
√ Nambu-Goldstone boson φ, which is coupled to iψ̄γ5 ψ with coupling
strength g = 2m/f . This last result is the “unrenormalized Goldberger-Treiman relation” [33]. The Goldberger-Treiman relation holds experimentally in QCD for the axial
coupling constant of the pion gA and the nucleon, with m = mN , f = fπ , and is one of
the indications that the pion is a Nambu-Goldstone boson. The Nambu-Goldstone phenomenon is ubiquitous throughout the physical world, including spin-waves, water-waves,
and waves on an infinite stretched rope.
1.3(iv) Massive Photon → Eaten Nambu-Goldstone Boson
We now consider what happens if Φ is a charged scalar field, with charge e, coupled
in a gauge invariant way to a vector potential. Let us “switch off” the fermions for the
present. We construct the following Lagrangian:
1
L′Φ = − Fµν F µν + |(i∂µ − eAµ )Φ|2 − V (Φ)
4
4
(1.20)
This is a general feature of a Nambu–Goldstone mode, and implies “Adler decoupling”: any NGB
emission amplitude tends to zero as the NGB four–momentum is taken to zero.
11
This is gauge invariant in the usual way, since with Aµ → Aµ + ∂µ χ we can rephase Φ
as Φ → e−ieχ Φ. The scalar potential
V (Φ) is as given in eq.(1.13), hence, Φ will again
q
develop a constant VEV v = 2/λM. We can parameterize the oscillations around the
minimum as in eq.(1.15) and introduce the new vector potential,
1
Bµ = Aµ − ∂µ φ .
e
(1.21)
The Lagrangian (1.20) in this reparameterized form becomes:
1
L′Φ = − Fµν F µν + 12 e2 v 2 Bµ B µ + 12 (∂µ h)2
4
s
√
!
λ
1 4 1 2 2
2M
2 2
3
M h −
Mh − λh + 2 e h +
h Bµ B µ
2
8
λ
(1.22)
where M is defined as in eq.(1.13).
Hence, we have recovered the massive photon Bµ together with an electrically
neu√
tral field h, which we call the “Higgs boson.” The Higgs boson has a mass 2M and
has both cubic and quartic self-interactions, as well as linear and bilinear couplings to
pairs of the massive photon. This model is essentially a (manifestly Lorentz invariant)
Landau-Ginzburg model of superconductivity, also known as the “abelian Higgs model.”
We emphasize that it is manifestly gauge invariant because the gauge field and NambuGoldstone mode occur in the linear combination of eq.(1.21), as in eq.(1.1). We say that
the Nambu-Goldstone boson has been “eaten” to become the longitudinal spin degree of
freedom of the photon.
1.3(v) Massive Photon and Massive Fermions Come Together
Finally, we can put all of these ingredients together in one grand scheme. For example,
we can combine, e.g., a left-handed, fermion, ψL , of electric charge e and a neutral righthanded fermion, ψR , with an Abelian Higgs model:
L = L′Φ + ψ̄L (i∂/ − eA
/ )ψL + ψ̄R i∂/ ψR − g(ψ̄L ψR Φ + ψ̄R ψL Φ† )
(1.23)
The Lagrangian is completely invariant under the electromagnetic gauge transformation.
Also, the theory is intrinsically “chiral” in that the left-handed fermion has a different
gauge charge than the right-handed one. Now we see that, upon writing Φ as in eq.(1.15),
and performing a field redefinition, ψL → exp(iφ/v)ψL , we obtain:
1
L = − Fµν F µν + 12 e2 v 2 Bµ B µ + 12 (∂µ h)2
4
1
−M 2 h2 − Mh3 − λh4 + 21 e2 h2 Bµ B µ
8
1
+ψ̄i∂/ ψ − eB µ ψ̄L γµ ψL − mψ̄ψ − √ ghψ̄ψ
2
12
(1.24)
Thus we have generated: (1)√ a dynamical gauge boson mass, ev, and (2) a dynamical
fermion Dirac mass m = gv/ 2. The Dirac mass mixes chiral fermions carrying different
gauge charges, and would superficially appear to violate the gauge symmetry and electric
charge conservation. However, electric charge conservation is spontaneously broken by
the VEV of Φ. There remains a characteristic coupling of the fermion to the Higgs field
h proportional to g.5
1.4
The Standard Model
1.4(i) Ingredients
Analogues of all of the above described ingredients are incorporated into the Standard Model of the electroweak interactions. In the Standard Model electroweak sector,
the gauge group is SU(2) × U(1). The scalar Φ is replaced by a spin-0, weak isospin1/2, field, known as the Higgs doublet. An arbitrary component of Φ develops a VEV
which defines the neutral direction in isospin space (it is usually chosen to be the upper
component of Φ without loss of generality). Three of the four components of the Higgs
isodoublet then become Nambu-Goldstone bosons, and combine with the gauge fields to
make them massive. The W ± and Z bosons acquire masses, while the photon remains
massless. The fourth component of Φ is a left-over, physical, massive object called the
Higgs boson, the analogue of the h field in our toy models above. Because the Standard Model weak interactions are governed by the non-abelian group SU(2), tree-level
(h, h2 ) × (W ± W ∓ , ZZ) interactions occur; the analogous tree-level hγγ coupling is absent
in the abelian Higgs model (and in the Standard Model) since the QED U(1) symmetry
is unbroken. The electroweak theory is thus, essentially, a mathematical generalization
of a (Lorentz invariant) Landau–Ginzburg superconductor to a nonabelian gauge group.
We give the explicit construction of the Standard Model in Appendix A.
We also see that there are striking parallels between the dynamics of spontaneous
symmetry breaking with an explicit Higgs field, such as Φ, and the dynamical behavior of
QCD near the scale ΛQCD . Consider QCD with two flavors of massless quarks (Ψ ≡ (u, d))
LQCD = Ψ̄L iD/ ΨL + Ψ̄L iD/ ΨL
(1.25)
where Dµ is the QCD covariant derivative. The Lagrangian possesses an SU(2)L ×SU(2)R
chiral symmetry. When the running QCD coupling constant becomes large at the QCD
scale, the strong interactions bind quark anti-quark pairs into a composite 0+ field Ψ̄Ψ.
This develops a non-zero vacuum expectation value hΨ̄Ψi ≈ Λ3QCD , in analogy to the
Higgs mechanism. This, in turn, spontaneously breaks the chiral symmetry SU(2)L ×
SU(2)R down to SU(2) of isospin. The light quarks then become heavy, developing
their “constituent quark mass” of order mN ucleon /3. The pions, the lightest pseudoscalar
5
Unfortunately, this simple model is not consistent at quantum loop level, since an axial anomaly
occurs in the gauge fermionic current. This can be remedied by, e.g., introducing a second pair of chiral
fermions with opposite charges.
13
mesons, are the Nambu-Goldstone bosons associated with the spontaneous symmetry
breaking and are massless at this level (the pions are not identically massless because of
the fundamental quark masses, mu ∼ 5 MeV, md ∼ 10 MeV). The essence of this dynamics
is captured in a toy model of QCD chiral dynamics known as the Nambu-Jona-Lasinio
(NJL) model [34, 35]. The NJL model is essentially a transcription to a particle physics
setting of the BCS theory of superconductivity. In Appendix B we give a treatment of
the NJL model.
If we follow these lines a step further and switch off the Higgs mechanism of the
electroweak interactions, then we would have unbroken electroweak gauge fields coupled
to identically massless quarks and leptons. However, it is apparent that the QCD-driven
condensate hΨ̄Ψi =
6 0 will then spontaneously break the electroweak interactions at a
scale of order ΛQCD . The resulting Nambu-Goldstone bosons (the pions) will then be
eaten by the gauge fields to become the longitudinal modes of the W ± and Z bosons.
The chiral condensate characterized by a quantity ΛQCD ∼ fπ would then provide the
scale of the W ± and Z masses, i.e., the weak scale in a theory of this kind is given
by vweak ∼ fπ . Because fπ ≈ 93 MeV is so small compared to vweak ∼ 175 GeV, the
familiar hadronic strong interactions cannot be the source of EWSB in nature. However,
it is clear that EWSB could well involve a new strong dynamics similar to QCD, with a
higher-energy-scale, Λ ∼ vweak , with chiral symmetry breaking, and “pions” that become
the longitudinal W ± and Z modes. This kind of hypothetical new dynamics, known as
Technicolor, was proposed in 1979 (Section 2).
1.4(ii) Naturalness
Various scientific definitions of “naturalness” emerged in the early days of the Standard
Model. “Strong naturalness” is associated with the dynamical origin of a very small physical parameter in a theory in which no initial small input parameters occur. The foremost
example is the mechanism that generates the tiny ratio, ΛQCD /MP lanck ∼ 10−20 in QCD.
This is also the premiere example of the phenomenon of “dimensional transmutation,” in
which a dimensionless quantity (αs ) becomes a dimensional one (ΛQCD ) by purely quantum effects. Here, the input parameter at very high energies is αs = gs2 /4π, the gauge coupling constant of QCD, which is a dimensionless number, of order O(10−1 ) − O(10−2 ), not
unreasonably far from unity. The renormalization group and asymptotic freedom of QCD
(effects of order h̄) then determine ΛQCD as the low energy scale at which αs (Λ) → ∞.6
More introspectively, the scale ΛQCD arises from the explicit scale breaking in QCD
that is encoded into the “trace anomaly,” the divergence of the scale current Sµ :
∂ µ Sµ = Tµµ = −
β(gs ) µν
G Gµν
2gs
(1.26)
where β(αs ) is the QCD-β function, arising from quantum loops, and is of order h̄ (we
neglect quark masses and, indeed, this has nothing to do with the quarks; the phenomenon
6
Perhaps an enterprising string theorist will one day compute αs (MP lanck ), obtaining a plausible result
such as αs (MP lanck ) ∼ 1/4π 2 thus completely explaining the detailed origin of ΛQCD .
14
happens in pure QCD). The smallness of αs at high energies implies that scale invariance
is approximately valid there. Asymptotic freedom implies that as we descend to lower
energy scales, αs slowly increases, until the scale breaking becomes large, finally selfconsistently generating the dynamical scale ΛQCD and the hierarchy ΛQCD /MP lanck . Note
that most of the mass of the nucleon (or constituent quarks) derives from the nucleon
matrix element of the RHS of eq.(1.26). In a sense, the “custodial symmetry” of this
enormous hierarchy is the approximate scale invariance of the theory at high energies,
in the “desert” where ∂ µ Sµ ≈ 0. Indeed, if we were to arrange for β(g) = 0, either
by cancellations in the functional form of β or by having a nontrivial fixed point, then
the coupling would not run and ΛQCD → 0! Strong naturalness thus underlies one large
hierarchy we see in nature, i.e., how the ratio ΛQCD /MP lanck ∼ 10−20 can be generated in
principle is more-or-less understood!
A parameter in a physical theory that must be tuned to a particular tiny value is
said to be “technically natural” if radiative corrections to this quantity are multiplicative.
Thus, the small parameter stays small under radiative corrections. This happens if setting the parameter to zero leads the theory to exhibit a symmetry which forbids radiative
corrections from inducing a nonzero value of the parameter. We then say that the symmetry “protects” the small value of the parameter; this symmetry is called a “custodial
symmetry.”
For example, if the electron mass, me , is set to zero in QED, we have an associated
chiral symmetry U(1)L × U(1)R which forbids the electron mass from being regenerated
by perturbative radiative corrections. The chiral U(1)L × U(1)R in the me = 0 limit is the
custodial symmetry of a small electron mass. Radiative corrections in QED to the electron
mass are a perturbative power series in α, and they multiply a nonzero bare electron mass.
Multiplicative radiative corrections insure that the electron mass, once set small, remains
small to all orders in perturbation theory. Now, clearly, technical naturalness begs a
deeper, strongly natural, explanation of the origin of the parameter me , but no apparent
conflict with any particular small value of me .
Typically scalar particles, such as the Higgs boson, have no custodial symmetry, such
as chiral symmetry, protecting their mass scales. This makes fundamental scalars, such
as the Higgs boson, unappealing and unnatural. The scalar boson mass is typically
subject to large additive renormalizations, i.e., radiative corrections generally induce a
mass even if the mass is ab initio set to zero.7 The important exceptions to this are
(i) Nambu-Goldstone bosons which can have technically natural low masses due to their
spontaneously broken chiral symmetry; (ii) composite scalars which only form at a strong
scale such as ΛQCD and could receive only additive renormalizations of order ΛQCD ; (iii)
a technically natural mechanism for having fundamental low mass scalars is also provided
by SUSY because the scalars are then associated with fermionic superpartners. The chiral
symmetries of these superpartner fermions then protect the mass scale of the scalars so
long as SUSY is intact. Hence, to use SUSY to technically protect the electroweak mass
7
This point is actually somewhat more subtle; scale symmetry can in principle act as a custodial
symmetry if there are no larger mass scales in the problem; see [36].
15
scale in this way requires that SUSY be a nearly exact symmetry on scales not far above
the weak scale.
In attempting to address the question of naturalness of the Standard Model, we are
thus led to exploit these several exceptional possibilities in model building to construct
a natural symmetry breaking (Higgs) sector. In SUSY, the Higgs boson(s) are truly
fundamental and the theory is perturbatively coupled. The SUSY technical naturalness
protects the mass scales of the scalar fields, and one hopes that the strong natural explanation of the weak scale will be discovered eventually, perhaps in the origin of SUSY
breaking, perhaps incorporating a trigger mechanism involving the heavy top quark. Here
one takes the point of view that there are, indeed, fundamental scalar fields in nature,
and they are governed by the organizing principle of SUSY that mandates their existence.
This leads to the MSSM in which all of the Standard Model fields are placed in N = 1
supermultiplets, and are thus associated with superpartners. SUSY and the electroweak
symmetry must be broken at similar energy scales to avoid unnaturally fine-tuning the
scalar masses.
In Technicolor, the scalars are composites produced by new strong dynamics at the
strong scale. Pure Technicolor, like QCD, is an effective nonlinear–σ model [3],8 and the
longitudinal W and Z are composite NGB’s (technipions). More recent models, such as
the Top Quark Seesaw, feature an observable composite heavy Higgs boson. In theories
with composite scalar bosons, one hopes to imitate the beautiful strong naturalness of
QCD. This strategy, first introduced by Weinberg [37] and Susskind [38] in the late 1970’s
seems a priori compelling. Leaving aside the problem of the quark and lepton masses,
one can immediately write down a theory in which there are new quarks (techniquarks),
coupled to the W and Z bosons, and bound together by new gluons (technigluons) to
make technipions. If the chiral symmetries of the techniquarks are exact, some of the
technipions become exactly massless, have decay constants, f ∼ vweak , and are then
“eaten” by the W and Z to provide their longitudinal modes. We will call this “pure
Technicolor.” Pure TC can be considered to be a limit of the Standard Model in which
all quarks and leptons are approximately massless and the EWSB is manifested mainly
in the W and Z boson masses. In this limit the longitudinal W and Z are the original
massless NGB’s, or pions, of Technicolor, and the scale of the new strong dynamics (i.e.,
the analogue of ΛQCD ) is essentially ∼ vweak . Again, the scale is set by quantum mechanics
itself; one need only specify αtechni at some very high scale, such as the Planck scale, to
be some reasonable number of O(10−2), and the renormalization group produces the scale
vweak automatically.
A third possibility, is that the Higgs boson is a naturally low-mass pseudo-NambuGoldstone boson [39], like the pion in QCD. This idea, dubbed “Little Higgs Models,” has
recently come back into vogue in the context of deconstructed space-time dimensions [40,
41, 42] (see Section 4). The renaissance of this idea is so recent that, unfortunately, we will
not be able to give it an adequate review. It is currently being examined for consistency
8
The Higgs boson is then the analogue of the σ-meson in QCD, which is a very wide state, difficult
to observe experimentally, and can be decoupled in the nonlinear σ-model limit.
16
with electroweak constraints and the jury is still out as to how much available parameter
space, and how little fine-tuning, will remain in Little Higgs Models. Nonetheless, the
basic idea is compelling and may lead ultimately to a viable scenario.
1.5
Purpose and Synopsis of the Review
This review will show the interplay between theory and experiment that has guided the
development of strong dynamical models of EWSB, particularly during the last decade.
Because they invoke new strongly-interacting fields at an (increasingly) accessible energy
scale of order one TeV, dynamical models are eminently testable and excludable. To the
extent that they attempt to delve into the origins of flavor physics, they become vulnerable
to a plethora of low-energy precision measurements. This has forced model-builders to
be creative, to seek out a greater understanding of phase transitions in strongly-coupled
systems, to seek out connections with other model-building trends such as SUSY, and
to re-examine ideas about flavor physics. Because experiment has played such a key role
in guiding the development of these theories, we choose to present the phenomenological
analysis in parallel with the theoretical. Each set of experimental issues is introduced at
the point in the theoretical story where it has had the greatest intellectual impact.
Chapter 2 explores the development of pure Technicolor theories. As already introduced in this chapter and further discussed in 2.1, pure Technicolor (an asymptotically
free gauge theory which spontaneously breaks the chiral symmetries of the new fermions
to which it couples) can explain the origins of EWSB and the masses of the W and
Z bosons. Section 2.2 discusses the mathematical implementation of these ideas in the
minimal two-flavor model and the resulting spectrum of strongly-coupled techni-hadron
resonances. The phenomenology of these resonances and the prospects for discovering
new strong dynamics in studies of vector boson scattering at future colliders are are also
explored. The one-family TC model and its rich phenomenology are the subject of section
2.3.
A more realistic Technicolor model must include a mechanism for transmitting EWSB
to the ordinary quarks and leptons, thereby generating their masses and mixing angles.
The original suggestion of an Extended Technicolor (ETC) gauge interaction involving
both ordinary and techni-fermions alike is the classic physical realization of that mechanism. As discussed in sections 3.1 and 3.2, the extended interactions can cause the strong
Technicolor dynamics to affect well-studied quantities such as oblique electroweak corrections or the rates of flavor-changing neutral curent processes. Moreover, the extended
interactions require more symmetry breaking at higher energy scales, so that the merits of the weak-scale theory are, as with SUSY, entwined with mechanisms operating at
higher energies. These issues have had a profound influence on model-building. Section
3.3 describes some of the explicit ETC scenarios designed to address questions of flavor
physics, further symmetry breaking, and unification.
As one moves beyond the minimal TC and ETC theories, the conflict inherent in
a theory of flavor dynamics become sharper: creating large quark masses requires a low
17
ETC scale, while avoiding large flavor-changing neutral currents mandates a high one. An
intriguing resolution is provided by “Walking” Technicolor dynamics (section 3.4). This
departs radically from the QCD analogy: the dynamics remains strong far above the TC
scale, up to the ETC scale, because the β-function is approximately zero. This, in turn,
has led to multi-scale and low-scale theories of Technicolor (section 3.5), which predict
many low-lying resonances with striking experimental signatures at LEP, the Fermilab
Tevatron and LHC. As discussed in section 3.6, first searches for these resonances have
been made and extensive explorations are planned for Run II. Finally, while the initial
motivation for TC theories was the avoidance of fundamental scalars, several variants
of model-building have led to low-energy effective theories that incorporate light scalars
along with TC; these are the subject of section 3.7.
Chapter 4 explores an idea that has taken hold as it became clear that the top quark’s
mass is of order the EWSB scale (vweak ∼ 175 GeV): it is likely that the top quark plays
a special role in any complete model of strong electroweak symmetry breaking. In some
sense, the top-quark may be a bona-fide techniquark with dynamical mass generation of
its own. The first attempts at models along these lines, known as top-quark condensation
(section 4.1), demonstrate the idea in principle, but are ultimately unacceptably finetuned theories. However, by generalizing the idea of top-quark condensation, and building
realistic models of the new “Topcolor” forces that underpin the dynamics, one is led back
to acceptable schemes under the rubric of Topcolor-Assisted Technicolor (TC2). The TC2
models incorporate the best features of the TC and Topcolor ideas in order to explain the
full spectrum of fermion masses, while avoiding the classic isospin violation and FCNC
dilemmas that plague traditional ETC models. The Topcolor theory, its relationship to
TC, and associated phenomenology are the focus of sections 4.2 and 4.3. Further insights
into the dynamics of mass generation have arisen in the context of Top-Seesaw models
(section 4.4), in which the top quark’s large mass arises partly through mixing with
strongly-coupled exotic quarks.
Most recently, as discussed in section 4.5, Topcolor is a forerunner of and has a natural
setting in latticized or “deconstructed” extra dimensions [41, 42]. Topcolor may represent
a connection between the phenomenlogy of EWSB and the possible presence of extradimensions of space-time at the ∼ TeV scale. All in all, new information about the top
quark and new ideas about the structure of space-time have fostered a mini–renaissance
in the arena of new strong dynamics and EWSB.
18
2
Technicolor
Motivations underlying Technicolor have been described for the reader in Section 1. We
wish to mention a few of the many detailed earlier reviews. The review of Farhi and
Susskind [43] and vintage lectures by various authors [44, 45, 46, 47, 48, 49, 50] remain
useful introductions. There also exists a collection of reprints [51] tracing the early developments. To our knowledge there is no comprehensive review of the “medieval” period of
TC, ca. late 1980’s to early 1990’s. For more recent surveys, the reader should consult
the reviews of K. Lane, [52, 53] and S. Chivukula [54, 55, 56]. S. King has also written
a more recent review [57] which develops some specific models, particularly of ETC (see
Section 3).
Certain aspects of TC model-building will not be addressed in the present discussion
and we refer the interested reader to the literature. We will not discuss gauge coupling
unification (see, e.g., [58, 59, 60, 61, 62, 63]), nor will we discuss cosmological implications
(see, e.g., [64]). We will only briefly mention the idea of Supersymmetric TC [65, 66, 67],
in the section on SUSY and EWSB in Section 3.7. While TC has largely evolved in
directions somewhat orthogonal to Supersymmetry, the overlap of these approaches may
blossom in coming years should evidence for NSD should emerge at the weak scale.
We presently begin with a description of the essential elements of TC theories, addressing the problem of generating the W and Z masses. We postpone to Section 3 the
more involved details of ETC and the problems and constraints associated with creating
fermion masses. Accordingly, we will discuss the core phenomenology of TC models in
this section, and additional phenomenological discussion will appear in Section 3 as the
more detailed schemes unfold.
2.1
Dynamics of Technicolor
Technicolor (TC) was introduced by Weinberg [37] and Susskind [38] in the late 1970’s.
The heaviest known fermion at that time was the b-quark, with a mass of ∼ 5 GeV and
the top quark was widely expected to weigh in around 15 GeV. The predicted Standard
Model gauge sector, on the other hand, was composed of the massless photon and gluon,
and the anticipated, heavy gauge bosons, W and Z, with MW ∼ 80 GeV and MZ ∼ 90
GeV. Since the matter sector appeared to contain only relatively light fermions, it was
useful to contemplate a limit in which all of the elementary fermions are approximately
massless, and seek a mechanism to provide only the heavy gauge boson masses. TC was
a natural solution to this problem.
2.1.1
The TC ↔ QCD Analogy
TC is a gauge theory with properties similar to those of QCD. For concreteness, consider
a TC gauge group GT = SU(NT ), having NT2 − 1 gauge bosons, called “technigluons.”
We introduce identically massless chiral “techniquarks” subject to this new gauge force:
19
ai
Qai
L and QR , where a refers to TC and i is a flavor index. We will assume that the
Q’s fall into the fundamental, NT representation of SU(NT ). We further assume that
we have NT f flavors of the Q’s. This then implies that we have an overall global chiral
symmetry: SU(NT f )L × SU(NT f )R × [U(1)A ] × U(1)Q (where the U(1)A is broken by the
axial anomaly and is thus written in the square brackets [...]). We will call this the “chiral
group” of the TC theory.
TC, like QCD, is assumed to be a confining theory9 and has an intrinsic (confinement)
mass scale ΛT , which must be of order the weak scale vweak . Hence, the physical spectrum
will consist of TC singlets that are either technimesons, composed of QQ, or technibaryons
composed of NT techniquarks. We expect various resonances above the lowest lying states,
showing Regge behavior, precocious scaling, and ultimately even technijet phenomena [69].
Since these objects are not found as stable states in nature, the complete theory must
provide for the decay of techniquarks into the light observed quarks and leptons. This is
part of the function of a necessary extension of the theory, called Extended Technicolor
(ETC), which will serve the role of giving the light quarks and leptons their masses as
well. We postpone a detailed discussion of ETC until Section 3.
In complete analogy to QCD, the theory produces a dynamical chiral condensate of
fermion bilinears in the vacuum, i.e., if QiL,R are techniquarks of flavor i, then TC yields:
D
E
QiL QjR ≈ Λ3T C δij
(2.27)
This phenomenon occurs in QCD, in analogy to the Cooper pair condensate in a BCS
superconductor, and gives rise to the large nucleon and/or constituent quark masses. The
general implication of eq.(2.27) is the occurence of a “mass gap”: the techniquarks acquire
constituent masses of order m0 ∼ ΛT . Technibaryons composed of NT techniquarks will
be heavy with masses of order ∼ NT ΛT . There must also occur NT2 f − 1 massless NambuGoldstone bosons, with a common decay constant FT ∼ ΛT .
By analogy, in QCD, if we consider the two flavors of up and down quarks to be
massless, then there is a global chiral symmetry of the Lagrangian of the form SU(2)L ×
SU(2)R × [U(1)A ] × U(1)B , (where “A” stands for axial, and “B” for baryon number).
Within an approximation to the chiral dynamics of QCD, known as the “chiral constituent
quark model” (e.g., see [70]), based upon the Nambu–Jona-Lasinio (NJL) model, (Appendix B) we can give a description of the dynamical chiral symmetry breaking in QCD
or TC (we’ll refer to this as the NJL model below). In this approximation, we obtain
the Georgi-Manohar [71] chiral Lagrangian for constituents quarks and mesons as the low
energy solution to the model. In this NJL approximation there is a cut-off scale M, which
is of order ∼ mρ , and we can relate fπ to the dynamically generated “mass gap” of the
theory, i.e., the “constituent quark mass” of QCD. If Nf fermion flavors condense, each
having Nc colors, then the quarks will have a common dynamically generated constituent
mass m0 and produce a common decay constant, fπ for the (Nf2 − 1) Nambu-Goldstone
9
Note that this is not the case in other schemes, such as Topcolor, and a spontaneously broken
nonconfining TC has been considered [68].
20
bosons given by the Pagels-Stokar relation [72] (see Appendix B; in the next section we
discuss normalization conventions for fπ ):
Nc 2
m ln(M 2 /m20 ) .
4π 2 0
fπ2 =
(2.28)
In the NJL approximation we also obtain an explicit formula for the quark condensate
bilinear:
D
E
Nc
QiL QjR = δij 2 m0 M 2
(2.29)
8π
Improvements can be made to the NJL model by softening the four-fermion interaction
and treating technigluon exchange in the ladder approximation (see, e.g., [73, 74, 75]).
Alternatively, lattice gauge theory techniques can be brought to bear upon TC as well
(see, e.g., [76, 77]).
The TC condensate is diagonal in an arbitrary basis of techniquarks, Qi ’s, where the
chiral subgroup, SU(NT f )L × SU(NT f )R × U(1)Q , is an exact symmetry (U(1)A is broken
by instantons, and the Techni-η’ is heavy like the η’ of QCD). The Standard Model gauge
interactions, SU(3) × SU(2)L × U(1)Y will be a gauged subgroup of this exact TC chiral
subgroup. Indeed, since we want to dynamically break electroweak symmetries, then
SU(2)L × U(1)Y must always be a subgroup of the chiral group. In the minimal model
described in the next section, QCD is not a subgroup of the chiral group, while in the
Farhi-Susskind model, both QCD and electroweak gauge groups are subgroups of the
chiral group.
When the SU(3)×SU(2)L ×U(1)Y interactions are turned on, a particular basis for the
ai
QL,R has thus been selected (the general models of Q’s contains various SU(3)×SU(2)L ×
U(1)Y representations). Thus, an “alignment” occurs in the dynamical condensate pairing
i
of QL with QiR . In general it is not obvious ab initio that this alignment preserves the
exact gauge symmetries (like electromagnism and QCD; i.e., the electric charge and color
generators must commute with the condensate to preserve these symmetries), and breaks
yet other symmetries (electroweak) in the desired way. This is one example of the “vacuum
alignment” problem [78, 79, 80, 81, 82, 83, 84, 85]. In the simplest TC representations
the desired vacuum alignment is manifest.
2.1.2
Estimating in TC by Rescaling QCD; fπ , FT , vweak
Since TC is based upon an analogy with the dynamics of QCD, we can use QCD as
an “analogue computer” to determine, by appropriate rescalings, the properties of the
pure TC theory . A convenient set of scaling rules due originally to ‘t Hooft [86, 87, 88]
(see, furthermore, e.g., [71, 89]), characterize the behavior of QCD. These rules have been
extensively applied to TC [90]. The main scaling rules are:
fπ ∼
q
Nc ΛQCD
D
E
Qi Qj ∼ δij Nc Λ3QCD
21
m0 ∼ ΛQCD
(2.30)
These rules follow from the NJL approximation with the identification m0 ∼ M ∼ ΛQCD .
When we discuss TC models we will use the notation, FT , to refer to the corresponding
NGB, or technipion, decay constant.
−3/4 −1/2
We typically refer to the weak scale
GF = 175 GeV, which is related to
√ vweak = 2
the usual Higgs VEV as vweak = v0 / 2, and v0 = 246 GeV. Hence, in the spontaneously
broken phase of the Standard Model we can parameterize the Higgs field with its VEV
as:
√ !
√
v0 / 2 + h0 / 2
a a
.
(2.31)
H = exp(iπ τ /v0 )
0
This gives the kinetic terms for the π a (and h0 ) the proper canonical normalizations in the
limit of switching off the gauge fields. ¿From the Higgs boson’s kinetic terms we extract,
where the electroweak covariant derivative Dµ is defined in eq.(A.1):
Dµ H † D µ H →
g2
g2
g2
g1
v0 Wµ+ ∂ µ π − + v0 Wµ− ∂ µ π + + v0 ( Wµ0 + Bµ )∂ µ π 0 + ...
2
2
2
2
(2.32)
Now, in QCD fπ is defined by:
< 0|jµa5 |π b >= ifπ pµ δab
Fπ ≈ 93 MeV
(2.33)
a
where jµa5 = ψγµ γ 5 τ2 ψ where ψ = (u, d) in QCD (Note: another definition in common
√
use involves
√ the matrix elements of the charged currents and differs by a factor of 2,
i.e., Fπ = 2fπ ). When pions (Nambu-Goldstone bosons) or technipions are introduced
through chiral Lagrangians, we have typically a nonlinear-σ model field U that transforms
under GL × GR as U → LUR† , and its kinetic term is of the form:
U = exp(iπ a τ a /f )
L=
f2
Tr(∂ µ U † ∂µ U)
4
(2.34)
Then the normalization is f = fπ = 93 MeV, which can be seen by working out the axial
current, jµ5 = δL/δ∂µ π a and comparing with eq.(2.33).
We will similarly define FT as the techni-pion, π̃, to vacuum matrix element for the
corresponding techniquark axial current, involving a single doublet of techniquarks, in TC
a
models, i.e., j̃µa5 = Qγµ γ 5 τ2 Q where Q = (T, B) are techniquarks:
< 0|j̃µa5 |π̃ b >= iFT pµ δab
FT ∝ vweak
(2.35)
Including electroweak gauge interactions the techniquark kinetic terms take the form:
QL iD/ QL + QR iD/ QR −→
FT2
Tr((D µ U)† (Dµ U))
4
(2.36)
where Dµ is defined in eq.(A.1). We have also written the corresponding chiral Lagrangian
describing the technipions with a nonlinear-σ model, or chiral field U = exp(iπ a τ a /FT )
(in the chiral Lagrangian the left-handed electroweak generators act on the left side of U,
while vectorial generators act on both left and right, and are commutators with U). We
22
can thus form matrix elements of the techniquark kinetic terms between vacuum and technipion states, or expand the chiral Lagrangian to first order in π a . ¿From eqs.(2.35,2.36)
we obtain the effective Lagrangian describing the longitudinal coupling of (W ± , Z) to
(π ± , π 0 ):
g2
g2
g1
g2
FT Wµ+ ∂ µ π − + FT Wµ− ∂ µ π + + FT ( Wµ0 + Bµ )∂ µ π 0
(2.37)
2
2
2
2
Hence, comparing eq.(2.32) to eq.(2.37) we see that the Higgs VEV v0 = FT when we
have a single doublet of technipions. If ND doublets carry weak charges then
√ eq.(2.36)
contains ND terms. FT remains the same, but the weak scale becomes v0 = ND FT .10
Consider a TC gauge group SU(NT ) with ND electroweak left-handed doublets and
2ND singlets of right-handed techniquarks, each in in the NT representation. The strong
SU(NT ) gauge group will form a chiral condensate pairing the left-handed fermions with
the right-handed fermions. This produces (2ND )2 − 1 Nambu-Goldstone bosons (technipions, πT , and the singlet ηT′ ) each with decay constant FT . Hence, we can estimate FT
from the QCD analogue fπ using the scaling rules:
FT ∼
s
NT
3
!
ΛT
fπ ;
ΛQCD
v0 =
q
ND FT ∼
s
ND NT
3
!
ΛT
fπ
ΛQCD
(2.38)
As stated above, v02 receives contributions from ND copies of the electroweak condensate.
Now, in the above example
of scaling, we have taken the point of view that ΛT is fixed
√
and, e.g., v0 varies as ∼ NT ND as we vary NT and ND . This is an unnatural way to
define the TC theory, since v0 is an input parameter whose value is fixed by GF . Hence,
in discussing TC models from now on we will use the “TC Scaling” scheme in which we
hold v0 fixed, and vary ΛT together with NT and ND . Hence, the same example rewritten
in the language of TC Scaling is:
s
√
1
v0 3
FT ∼ v0
;
ΛT = ΛQCD √
;
v0 = 246 GeV.
(2.39)
ND
fπ ND NT
With these scaling rules in hand, we turn to the key properties of the main classes of TC
models.
10
If we could switch off nature’s EWSB, the preceding discussion indicates precisely how QCD itself
would then break the electroweak interactions. Indeed, (see e.g. [52]) if there is no EWSB (no Higgs
boson), then the up and down quarks are identically massless. Then we would have the QCD chiral condensate, generating constituent quark masses for up and down of order 300 MeV, and massless composite
pions in the absence of electroweak gauge interactions. In the presence of electroweak gauge interactions
the pions are mathematically “eaten” to become WL and ZL . Now, however the W and Z masses given
′
by MW
= fπ MW /v0 ∼ 29 MeV and MZ′ = fπ MZ /v0 ∼ 33 MeV. The longitudinal W and Z would thus
be the ordinary π’s of QCD. Thus, QCD misses the observed masses by ∼ 4 orders of magnitude, but it
gets the ratio of MW /MZ correct!
23
2.2
The Minimal TC Model of Susskind and Weinberg
2.2.1
Structure
In the minimal TC scheme, introduced by Weinberg [37] and Susskind [38], the gauge
group is SU(NT ) × SU(3) × SU(2)L × U(1)Y . In addition to the ordinary Standard Model
fermions, we include at least one flavor doublet of color singlet technifermions, (T, B).
These form two chiral weak doublets, (T, B)L and (T, B)R , and the chiral group is therefore
SU(2)L × SU(2)R × [U(1)A ] × U(1)B . The left-handed weak doublet (T, B)L will have
I = 21 electroweak SU(2)L gauge couplings, while the (T, B)R form a pair of singlets. The
gauge anomalies of the Standard Model and TC vanish if we take “vectorlike” assignments
under the weak hypercharge Y :
QaL =
T
B
!a
(Y = 0);
QaR = (TR , BR )a
L
1 1
Y
= ( ,− ) .
2
2 2
(2.40)
As “a” is the TC index, we have NT TC copies of these objects. Anomalies involving
Y are absent because we have introduced a vector-like pair (TR , BR ), each element with
opposite Y (the Witten global SU(2) anomaly [91] vanishes provided NT is even for any
ND ). We can readily generalize the model to include arbitrary flavors, ND > 1 doublets,
of the same color-singlet technifermions. With these assignments the techniquarks have
electric charges as defined by Q = I3 + Y /2, of +1/2 for T and −1/2 for B.
Without developing a detailed treatment of grand unification, we can see that the
scale ΛT plausibly corresponds to the electroweak scale, and can be generated naturally
by choosing the contents of the model appropriately. If we assume that the TC gauge
coupling constant is given by a high energy theory (e.g., a GUT or string theory) with
the unification condition,
αT (MGU T ) = α3 (MGU T ),
(2.41)
then evolving down by the renormalization group, we obtain for the TC scale ΛT to
one-loop precision:
#
"
ΛT
2π(b′0 − b0 )
(2.42)
= exp
ΛQCD
b0 b′0 α3 (MGU T )
where,
11NT
4
2
and
b′0 =
− ND
(2.43)
b0 = 11 − nf
3
3
3
b0 and b′0 are the one-loop β-function coefficients of QCD and TC respectively. Putting
in some “typical” numbers, we find that NT = 4, nf = 6, ND = 4 and α3−1 (MGU T ) ≈ 30
implies that ΛT /ΛQCD ≈ 8.2 × 102 (for smaller ND the ratio rapidly increases). Hence,
using ΛQCD = 200 MeV, we find ΛT ≈ 165 GeV, and FT ≈ 95 GeV, and we predict
vweak ≈ 190 GeV, not far from the known 175 GeV. The minimal model thus crudely
exhibits how TC, in principle, can generate the electroweak hierarchy, v0 /MP lanck ∼ 10−17 ,
thus generate a TC mass scale ΛT ∼ 102 − 103 GeV, in a natural way from the scale
anomaly (renormalization group running) and grand unification.
24
Let us turn this around and assume that we have naturally obtained the hierarchy
ΛT /MP lanck ∼ 10−17 in some particular and complete TC theory. We will not bother
further with unknown details of unification, and take, rather, a bottom-up approach. The
QCD/TC scaling laws allow us to make estimates of the chiral dynamics if we know ΛT ,
and to conversely derive ΛT from the fixed electroweak parameter v0 . In a minimal TC
model with NT = 4 and ND = 1, ΛQCD ∼ 200 MeV, fπ ∼ 93 MeV, we obtain the result
from eq.(2.39) that ΛT ∼ 458 GeV (whereas we obtain ΛT ∼ 205 GeV with ND = 5,
which is closer to consistency with the top-down estimate given above).
2.2.2
Spectroscopy of the Minimal Model
2.2.2(i) Techniquarks
The spectrum of the minimal model follows QCD as well. The mass gap of the theory
can be estimated by scaling from QCD. From eq.(2.39)
the techniquarks will acquire
√
√
dynamical constituent masses of order: mT Q ∼ m0 v0 3/fπ NT ND where m0 ∼ mN /3 ∼
300 MeV is the constituent quark mass in QCD. This gives mT Q ∼ 690 GeV for the
minimal model with NT = 4, ND = 1. Hence, there
√ will be a spectrum of “baryons”
composed of QQQQ with a mass scale of order ∼ 3/ ND TeV. At the pre-ETC level the
lightest of these objects is stable, and cosmologically undesireable [64]. However, in the
presence of the requisite ETC interactions there will be Q → q +X, Q → ℓ+X transitions
and the baryons will become unstable to decay into high multiplicity final states of light
quarks and leptons, e.g., final states containing 12 top– or bottom– quarks! Note that at
higher energies the minimal model is asymptotically free. At very high energies, E >> 10
TeV we would expect the formation of “techni-jets” [69].
2.2.2.(ii) Nambu-Goldstone Bosons
The spectrum of the lowest lying mesons is predictable in the QCD-like TC models
by analogy with QCD. It is controlled by the chiral group of the model. With ND = 1 we
have a chiral group of SU(2)L ×SU(2)R ×[U(1)A ]×U(1)V , and (switching off the gauge interactions) there are three identically massless isovector pseudoscalar Nambu-Goldstone
bosons, πT± , π√T0 , which are √dubbed technipions. The decay constant of the technipions
is FT = v0 / ND = 246/ ND GeV, so that the pion decay constant and v0 coincide
in models with a single weak-doublet of left-handed techniquarks. When we gauge the
SU(2)L × U(1)Y subgroup of the chiral group, the technipions become the longitudinal
weak gauge bosons WL± and ZL . following the conventional Higgs/superconductor mechanism as described in Section 1.3. Indeed, it is a general aspect of all dynamical symmetry
breaking schemes that the WL± and ZL0 are composite NGB’s.
With ND = 1 the only remnant Nambu-Goldstone boson, corresponding to the spontaneous breaking of the ungauged U(1)A subgroup of the chiral group, is an isosinglet
Techni-η ′ , or ηT′ . This is the analogue of the η ′ in QCD. The ηT′ acquires mass through
TC instantons. The original analysis of this state in TC is due to deVecchia and Veneziano
25
[92], and a very nice and more recent treatment, which we follow presently, is given by
Tandean [93].
The symmetry, U(1)A , is broken by the axial vector anomaly, i.e. the triangle diagrams
with the emission of two technigluons. The anomaly is actually suppressed in the large
NT limit, asq∼ 3/NT , [94, 95, 96], and an estimate of the ηT′ mass is therefore11 mη′ T C ∼
√
(3 6/2NT ) 3/NT ND (v0 /fπ )mη′ ∼ 2 TeV for NT = 4 and ND = 1. Thus the ηT′ is a
relatively heavy state; in the minimal ND = 1 model, it TC is fairly well hidden from
direct searches at low energies E <
∼ 1 TeV.
The decay of ηT′ parallels, in part, that of the η ′ in QCD and involves anomalies. Note
that while the U(1)A anomaly is suppressed by 1/NT , the flavor anomalies relevant to
these decays (these are the analogues of the π 0 → 2γ anomaly in QCD) are generally
proportional to NT . We expect the principle decay modes ηT′ → ZZ, γZ, γγ and ηT′ →
W + W − Z, ZZZ as analogues to the multi-pion and photonic decays of the η ′ in QCD.
However, more novel decays directly into gg (g ∼ gluon) or into top quark pairs can
also be possible. These decay modes depend on the existence and details of a coupling
to ordinary quarks and leptons, which is the subject of ETC, discussed in Section 3
(statements about ETC are more model dependent, especially regarding the top quark).
Typical estimates for decay widths can be gleaned from [93]:
q
3
107λ2/ ND MeV.
Γ(ηT → tt) ∼
NT
q
3
Γ(ηT → gg) ∼
56/ ND MeV.
NT
q
3
Γ(ηT → W + W − ) ∼
26/ ND MeV.
NT
(2.44)
where the parameter λ describes the ETC coupling of ηT to the top quark. Note that the
2
tt mode, though model dependent, can dominate provided that λ ∼ MT2 /MET
C is not too
small. See [93] for more details.
Next, let us consider increasing the number of technidoublets. For general ND > 1,
we first turn off the Standard Model gauge interactions. Then our SU(NT ) theory has a
global chiral group of SU(2ND )L × SU(2ND )R × [U(1)A ] × U(1). This leads to (2ND )2 − 1
Nambu-Goldstone bosons, πTa in the adjoint representation of SU(2ND ), and the singlet,
ηT′ described above. These are analogues of the pseudoscalar octet, π, K, η in QCD, and
the singlet η ′ . Note that because our techniquarks carry no QCD color, the NGB’s will
likewise be colorless. In the absence of the electroweak interactions all of the remaining
NGB’s are massless (except ηT′ as described above). As such low mass objects are not seen
in the spectrum, sources of NGB masses will be required to avoid conflict with experiment.
Switching on the gauge interactions, the SU(2)L × U(1)Y becomes a gauged subgroup
of the full SU(2ND )L × SU(2ND )R × [U(1)A ] × U(1) chiral group. When we restore the
√
An overall 6/2 factor is associated with the present techni-flavor group SU (2) relative to the flavor
group SU (3) of QCD in which the η ′ lives [93].
11
26
electroweak interactions, specific linear combinations of triplets of the NGB’s are eaten to
give masses to W and Z. Of the NGB’s we thus have several classes of objects. Some of
the remaining πT ’s carry electroweak charges, but form linear representations of SU(2)L .
These are the NGB’s that correspond to generators of SU(2ND )L × SU(2ND )R that do
not commute with the SU(2)L × U(1) subgroup. They are analogues of K mesons in
SU(3) that form linear isodoublets. In addition there are generalizations of the η in
QCD, which is an isosinglet. These objects correspond to generators that commute with
SU(2)L × U(1)Y , and do not acquire masses via gauge couplings.
To count and to classify the various technipions, it is useful to use matrix direct
product notation. First ignore U(1)Y . Our particular model has an SU(2ND ) vector flavor
subgroup of the chiral group. This group has 4ND2 − 1 generators. We can arrange all of
the ND doublets into a column vector with respect to this group, i.e., the fundamental
2ND representation of SU(2ND ). On the other hand, the horizontal flavor subgroup,
which acts on weak doublets, is SU(ND ).
Hence, we can classify the generators associated with the NGBs according to their
transformation properties under the direct product of the SU(2)L and SU(ND ) as follows:
• The three generators of SU(2)L can be written as the direct product τ a ⊗ Id where
Id is the d-dimensional unit matrix acting on the left-handed doublets. Therefore,
the three NGBs corresponding to the generators τ a ⊗ Id are the states that become
the WL± and ZL0 .
• There are 3ND2 −3 matrices of the form τ a ⊗λA , where λA are the ND2 −1 generators of
SU(ND ). Since these matrices do not commute with the τ a ⊗Id SU(2)L charges, the
corresponding 3ND2 − 3 PNGB’s carry SU(2)L charge. These non-neutral PNGB’s
acquire mass when we switch on the gauge interactions, just as the π ± acquire a
mass splitting relative to π 0 in QCD due to the coupling to electromagnetism.
• The matrices of the form I2 ⊗ λA commute with SU(2)L , so there are ND2 − 1 objects
which are sterile under SU(2)L . These objects are PNGB’s that remain identically
massless (modulo electroweak instanton effects), and are dubbed “techni-axions.”
• The final element I2 ⊗ Id corresponds to the ηT′ .
We illustrate the above decomposition explicitly in Fig.(1) for the special case of ND = 2.
A similar analysis in the presence of U(1)Y adds a little more information. The
PNGB’s that can carry charges under U(1)Y are given by the generators that do not
commute with Y ⊗ Id . These are the electrically charged technipions – hence, it is easier
to examine the properties under U(1)EM . In contrast, the techniaxions are neutral under
U(1)Y , rendering them sterile under all of the gauge interactions.
To estimate the masses of the charged PNGB’s, we again use QCD as an analogue
computer, and rescale the π ± − π 0 mass-squared difference. The mass-squared’s of the
non-neutral weak-charged PNGB’s are therefore estimated to be of order m2πT ∼ α2 Λ2T
27
τa
0
0
τa
T1
B1
T2
B2
+ - 0
- 0
π+, π , π = WL, WL, Z L
a
0
0
τ
0
- τa
a
τ
0
-i τ
I
0
0
I
0
iI
I
0
-iI
0
τ
a
0
-I
0
a
iτ
a
9 PNGB’s
0
3 Techniaxions
Figure 1: The (2ND )2 − 1 Nambu-Goldstone bosons corresponding to an ND = 2 extension of
the minimal model. The ηT′ corresponds to the unit matrix.
or mπT ∼ 6 GeV. [97]) There are further small corrections when we switch on U(1)Y .
Unfortunately, charged scalars with such low masses are ruled out experimentally (see
section 3), another difficulty for the minimal model.
All electrically neutral PNGB’s remain massless at this level of model-building. In
particularly, the techniaxions remain perturbatively massless, since they correspond to a
residual exact global symmetry (spontaneously broken). Their associated axial currents
have electroweak anomalies like the π 0 , but no QCD anomaly, and this can lead to a
miniscule electroweak instanton source of mass, ∝ exp(−4π 2 /g22 )v0 . These objects behave
like axions, with decay constant FT . They would be problematic, since axion-like objects
have restricted FT ’s, typically ≥ 108 to 1010 GeV by the usual astrophysical arguments.
As we will see in Section 3, these problems are ameliorated, in part, by the effects of ETC,
which provide a stronger source of gauge mass for technipions.
Incidently, there is expected to be a θ-angle in TC, and it is ultimately interesting as
a potential novel source of CP-violation (see e.g., [97]).
28
2.2.2(iii) Vector Mesons
In TC there will generally occur isovector and isosinglet s-wave vector mesons, the
0
0
analogues of ρ(770) and ω(782) in QCD, which we denote: ρ±
T , ρT and ωT . The vector
mesons are particularly important phenomenologically, because of their decays to weak
gauge bosons and technipions (e.g., ρT → W W is the direct analogue of the QCD process ρ → ππ). The vector mesons provide potentially visible resonance structures in
processes like pp or e+ e− → W + W − and, more generally, make large contributions to
technipion pair production. The masses of vector
q mesons can be estimated by TC scaling, mρ ∼ mωT C ∼ mρ (FT /fπ ) ∼ mρ (v0 /fπ ) 3/NT ND which yields the approximate
√
value of mρ,ωT C ∼ 1.8/ ND TeV for NT = 4.
Let us follow the conventional discussion of vector mesons in QCD by introducing the
dimensionless phenomenological “decay constants” fρT C and fωT C :
< ρaT |jµb |0 >= ǫµ δ ab
m2ρT C
fρT C
< ωT |jµ0 |0 >= ǫµ
m2ωT C
fρT C
(2.45)
√
where jµ0 = Qγµ Q/ 2, jµ0,a = Qγµ (τ a /2)Q, and Q is a techniquark doublet. It is difficult
to extract fω from QCD due to ω − φ mixing, so one typically assumes “nonet” symmetry,
fρ = fω . We then determine the decay constants from the partial width, Γ(ρ0 → e+ e− ) =
4παmρ /3fρ2; this implies fρ ≈ fω ≈ 5.0.
We must determine how fρ and fω undergo TC scaling. Note that the current matrix
elements of eq.(2.45) involve a TC singlet combination of techniquarks,
q and a normalized
initial state, and might therefore be expected to scale from QCD as ∼ NT /3(ΛT /ΛQCD )2 .
q
However, for fixed v0 we write this as ∼ (v02 /fπ2 ) 3/NT ND2 . so the amplitude m2ρT C /fρT C
must scale as ∼
q
3/NT ND2 . Since mρT C ∼
fρT C ∼
q
q
3/NT ND we see that [98]:
3/NT fρ ∼ 4.3
for NT = 4
(2.46)
This result makes intuitive sense only if one keeps in mind that, in TC scaling, we are
holding v0 fixed!
The decay modes of the vector mesons have been considered in the literature [99,
100, 101, 98]. Some can be treated by scaling from the principle QCD decay modes
ρ± → π ± π 0 , ρ0 → 2π 0 , ρ0 → π + π − , and ω → π + π 0 π − , ω → π 0 γ. Note that the
ω decay modes are associated with anomalies in a chiral Lagrangian description, and
have relatively tricky scaling properties [98]. In TC, by invoking the “equivalent NambuGoldstone boson rule” for the longitudinal gauge bosons, one finds the analogue modes
± 0
+
−
+ 0
−
0
0
ρ±
T → WL ZL , ρT → WL WL , 2ZL , and ωT → WL ZL WL , ωT → Zγ, ZZ, W W . Scaling
from QCD yields:
Γ(ρ0T
+
−
0
→ W W + 2Z ) ∼
3k1
NT
!
mρT
3
Γ(ρ → π + π − ) ∼
mρ
NT
29
3/2
280
√
GeV.
ND
Γ(ρ±
T
Γ(ωT →
±
3k1
NT
0
→W Z ) ∼
WL+ ZL0 WL− )
∼ k2
!
mρT
3
Γ(ρ → π + π − ) ∼
mρ
NT
3
NT
5/2
3/2
280
√
GeV.
ND
mωT
35
GeV. (2.47)
Γ(ω → π + π 0 π − ) ∼ √
mω
ND
where k1 ∼ 1.2 and k2 ∼ 4 are compensation factors arising from the ρ and ω decay
having phase-space suppression owing to the finite pion mass [98]. Other decays of the ω
are treated in [98]:
q
3 1/2
2.3/ ND GeV,
Γ(ωT → Z γ) ∼
NT
q
3 1/2
0 0
Γ(ωT → Z Z ) ∼
1.1/ ND GeV′
NT
q
3 1/2
+
−
5.2/ ND GeV.
Γ(ωT → W W ) ∼
NT
0
(2.48)
(iv) Higher p-wave Resonances
The parity partners of the ρT and ωT are the the p-wave axial vector mesons, a1T and
f1T . If the chiral symmetry breaking of QCD or TC were somehow switched off 12 states
and their parity partners would be degenerate. Given the presence of spontaneous chiral
symmetry breaking, we must estimate the masses
mesons by scaling
q of the axial vector
√
from QCD: ma1T ≈ mf1T ∼ ma1 ,f1 (1260)(v0 /fπ ) 3/NT ND ∼ 2.9/ ND TeV.
The spectrum should also include p-wave parity partners of the πT and ηT : the
0
isotriplet and isosinglet multiplets of 0+ mesons which analogues of the QCD q
states a±
0 , a0 ,
and f0 . Their masses are roughly given by ma0T ≈ mf0T ∼ ma0 ,f0 (980)(v0 /fπ ) 3/NT ND ∼
√
2.2/ ND TeV. A chiral Lagrangian approach to estimating the masses of the 0+ states
more carefully is beyond the scope of this discussion. Note that instanton effects can also
be substantial in a chiral quark model for these states. Typically the a0 nonet in the NJL
approximation to QCD has a low mass of order 2mQ ∼ 600 MeV, while contributions
from the ’t Hooft determinant can roughly double this result, bringing it into consistency
with the experimental values.
(iv) Summary of Static Properties
Table 1 summarizes the spectroscopy and decays of the main components of the minimal model of Susskind and Weinberg. These static properties, as well as production
cross-sections and observable processes, are extensively discussed in EHLQ [101], [103],
and [99, 100, 98]. We briefly review production and detection of the techni-vector mesons
of the minimal model in the next section.
12
There really is no conceptual limit of the true theory that can do this. It would be analogous to
taking the coupling constant on the NJL model to be sub-critical; see [102]
30
state
± 0
πT , πT → WL± , ZL
ηT0 (a)
0
ρ±
T , ρT
0
ωT
±
a0T , a0T 0
0 ∼σ
f0T
T
a1T ± , a1T 0
f1T
I(J P C )
1(0−+ )
0(0−+ )
1(1−− )
0(1−− )
1(0++ )
0(0++ )
1(1++ )
0(1++ )
mass (TeV)
MW , MZ (eaten)
∼ rmη ∼ 0.4 → 0.8
qmρ /s ∼ 1.2
∼ qmω /s ∼ 1.2
rma0 ∼ 1.5
qf0 /s ∼ 2
qma1 /s ∼ 2
qmf 1 /s ∼ 2
decay width (GeV)
ΓW , ΓZ
Γtt ∼ 8.0 − 64.0, Γgg ∼ 0.3 − 3.0
ΓρT (W W ) ∼ Γρ (ππ)/sq 2 ∼ 350
ΓωT (W W Z) ∼ Γω (πππ)/sq 2 ∼ 80
ηT W ± , ηT Z 0 ; Γ ∼ Γa0 /s ∼ 100
Γ ∼ Γf0 /s ∼ 1000
Γ(W W ) ∼ Γa1 /s ∼ 700
Γ(4W ) ∼ Γf1 /s ∼ 100
Table 1:
Estimated properties of lowest-lying (pseudo-) scalar and (axial-) vector mesons in
the minimal
p TC model with a single electroweak doublet of techniquarks ND = 1, NT = 4,
and q = 3/NT = 0.86. We take r ≡ ΛT /ΛQCD = 1.5 × 103 , and ΛQCD = 200 MeV, and
s ≡ fπ /FT = 5.7 × 10−4 , where fπ = 100MeV, FT = 175 GeV. The combination r 3 s2 = 1.1 × 103
frequently occurs. (a) These are estimates from the discussion of [104] .
2.2.3
Non-Resonant Production and Longitudinal Gauge Boson Scattering
Since the longitudinal W and Z are technipions, the minimal TC model predicts that high
energy WL −WL , ZL −ZL or WL −ZL scattering will be a strong-interaction phenomenon.
Studying the pair-production and scattering of the longitudinal W and Z states thus
provides a potential window on new strong dynamics. As Nambu–Goldstone bosons, the
longitudinal W and Z are described by a nonlinear σ-model chiral Lagrangians, [105, 106,
107, 108, 109, 110, 111, 112, 113, 114, 115, 116]. This is called “the equivalence theorem,”
[117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133], and often
this is viewed as an abstract approach, without specific reference to TC. However, to the
extent that we can use QCD as an analogue computer for TC, we expect that many of the
familiar low energy π − π theorems of QCD transcribe into the “low energy” TC regime,
<
∼ 1 TeV. Therefore, in models with a strongly–coupled EWSB sector, certain “lowenergy-theorem” or “non-resonant” contributions to the production and scattering of the
Nambu–Goldstone bosons are present. In theories where the EWSB sector also includes
resonances, such as the techni-ρ, that couple strongly to the Nambu–Goldstone bosons,
the scattering contributions from the resonances may also be present and even dominate.
The longitudinal W − W scattering processes are therefore a minimal requirement of new
strong dynamics.
There are several important mechanisms for producing vector boson pairs at future
hadronic or leptonic colliders. The first is annihilation of a light fermion/anti-fermion
pair. This process yields vector boson pairs that are mostly transversely polarized and
will usually be a background to the processes of interest here. A key exception is the
production of longitudinally polarized vector bosons in a J = 1 state (see [134] and
references therein), which renders this production channel sensitive to new physics with
a vector resonance like a techni-ρ.
A second mechanism applicable to hadronic colliders is gluon fusion [135, 136, 137,
31
138], in which the initial gluons turn into two vector bosons via an intermediate state
(e.g. top quarks, colored techihadrons) that couples to both gluons and electroweak
gauge bosons. In this case, only chargeless VL VL pairs can be produced, and thus this
channel is particularly sensitive to new physics with a scalar resonance like a heavy Higgs
boson. Finally, the vector-boson fusion processes [139, 140, 141, 142, 143], VL VL → VL VL ,
are important because they involve all possible spin and isospin channels simultaneously,
with scalar and vector resonances as well as non-resonant channels.
The review article of Golden, Han and Valencia [134] examines the possibilities of
making relatively model-independent searches for the physics of EWSB in VL VL scattering at the LC (in e+ e− or γγ modes) and LHC. Their discussion compares three basic
scenarios: (i) No resonances present in the experimentally accessible region (∼ 1.0 − 1.5
TeV) so that PNGB production is dominated by the nonresonant low energy theorems;
(ii) Production physics dominated by a spin-zero isospin-zero resonance like the Higgs
boson or a techni-sigma; (iii) physics dominated by a new spin-one isospin-one resonance
like a techni-ρ. We summarize their results, along with updates from other sources (see,
e.g. [144, 145, 146, 147, 148]), here and in Table 2.
Figure 2: Event yields at the LHC for ρT → W ± Z 0 → ℓ± νℓ ℓ+ ℓ− for MρT = 1.0, 2.5 TeV; from
Ref.[134]. A conventional techni-ρ resonance of mass much above 1 TeV would be invisible in
the channel ρT → W ± Z 0 → ℓ± νℓ ℓ+ ℓ− ,.
The LHC can detect strongly interacting EWSB physics in di–boson production.
Large Standard Model (SM) backgrounds for pair-production of leptonic W ’s can be
suppressed if one imposes stringent leptonic cuts, forward-jet-tagging and central-jetvetoing. Refs. [134, 118] explored complementarity for W ± Z and W ± W ± channels in
studying a vector-dominance model or a non-resonant model. A systematic comparison
of the different final states allows one to distinguish between different models to a degree.
A statistically significant signal can be obtained for every model (scalar, vector, or nonresonant) in at least one channel with a few years of running at an annual luminosity of
100 fb−1 . Detector simulations demonstrate that the semileptonic decays of a heavy Higgs
boson, H → W + W − → lνjj and H → ZZ → l+ l− jj, can provide statistically significant
signals for mH = 1 TeV, after several years of running at the high luminosity.
The LHC’s power to use di-boson production to see vector resonances associated with
32
Model
Channel
SM Scal O(2N) V1.0 V2.5 CG
ZZ(4ℓ)
1.0 2.5
3.2
1.0
3.7
4.2 3.5
ZZ(2ℓ2ν) 0.5 0.75
+
−
W W
0.75 1.5
2.5
8.5
9.5
±
W Z
7.5
W ±W ±
4.5 3.0
4.2
1.5
1.5 1.2
LET-K
Dly-K
4.0
5.7
1.2
2.2
Number of years at LHC with annual luminosity 100 fb−1 required for a 99%
confidence level signal. The models considered are: the Standard Model (SM), strongly-coupled
models with new scalar resonances (Scal, O(2N)), strongly-coupled models with new vector
resonances of mass 1 TeV (V1.0) or 2.5 TeV (V2.5), and strongly-coupled models with nonresonant scattering following the low-energy theorems (CG, LET-K, Dly-K). From ref. [134].
Table 2:
a strong EWSB sector has limited reach in mass [134, 118]. For example, a conventional
techni-ρ resonance of mass much above 1 TeV would be invisible in the channel ρT →
W ± Z 0 → ℓ± νℓ ℓ+ ℓ− , as shown in Fig.(2). A heavier techni-ρ would, instead, make itself
felt in the complementary W ± W ± channel [134, 118]. Models of “low-scale” TC with
lighter vector resonances more visible in the W Z channel at the LHC will be discussed in
Section 3.5 and 3.6.
2.2.4
Techni-Vector Meson Production and Vector Meson Dominance
In the minimal model with ND = 1 the lowest-lying resonances that can provide an
obvious signal of physics beyond the Standard Model are the techni-vector mesons. The
first accessible process involves the annihilation (at scales of order a few TeV) of a fermion
and antifermion into a techni-vector meson such as ρT and its subsequent decay into gauge
boson pairs, known as the “Susskind process” [38]. The vector boson pairs produced in this
way are mostly transversely polarized (the case of longitudinal polarizaion was discussed
in the previous section). Production and detection of techni-ρ states in f f¯ → ρT → V V
processes at various present and future colliders have been discussed extensively in the
literature, beginning with EHLQ [101, 103], and [122, 98, 149, 150, 151, 152, 153, 154,
155, 156]. Examples of the calculated cross-sections for ρT producion and decay at an
LHC or VLHC are given in Figures 3 and 4. Detailed search strategies and limits for the
“low-scale” variants of the minimal model are discussed in Section 3.5.
Absent a direct coupling between ordinary and techni-fermions such as ETC can provide (Section 3), how are the techni-vector mesons of the pure Minimal TC model to
be produced in q q̄ or e+ e− annihilation? The answer is that Vector Meson Dominance
(VMD) enables the techni-vector mesons to couple to currents of ordinary fermions. Most
relevant to a discussion of ρT production are vector dominance mixing of the ρ0T with γ
±
and Z and the ρ±
T with W , which we discuss below. The production and detection of
the techni-ω is considered in [98].
33
d σ /dM [nb/(GeV/c 2)]
−6
+ −
pp −> W W + anything
10
−7
10
100 TeV
40 TeV
−8
20 TeV
10
|y| < 1.5
−9
10
1.4
1.6
1.8
+
−
2
2.0 W W mass (TeV/c )
Figure 3: Vector Meson Dominance production of techni-ρ with subsequent decay to W + Z in
pp collider with center-of-mass energies, 20, 40 and 100 TeV (from EHLQ [157]).
d σ /dM [nb/(GeV/c 2)]
−6
−
+
pp −> ZW + ZW + anything
10
−7
10
100 TeV
40 TeV
−8
10
20 TeV
|y| < 1.5
−9
10
1.4
1.6
1.8
2
2.0 WZ mass (TeV/c )
Figure 4: Vector Meson Dominance production of techni-ρ with subsequent decay to W ± Z in
pp collider with center-of-mass energies, 20, 40 and 100 TeV (from EHLQ [157]).
34
Let us briefly review the theory of VMD. Consider a schematic effective Lagrangian
in which we introduce the photon, Aµ together with a single neutral vector meson13 ρµ :
1
1
µ
L = − (FAµν )2 − (Fρµν )2 − 21 ǫFAµν Fρ µν + 12 m2ρ (ρµ )2 − eAµ J µ − κρµ Jhad
4
4
(2.49)
µ
J µ is the ordinary electromagnetic current, and we define FXµν = ∂ µ X ν − ∂ ν X µ . Jhad
is
µ
a vector hadronic current which describes the strong interactions of the ρ; Jhad
contains,
for example, iπ + ∂µ π − , and we would typically fit the parameter κ to describe the strong
interaction decay ρ → π + π − . The ǫ term represents mixing between the photon and
ν
ρ, and can be viewed as arising from the nonzero amplitude < 0|T J µ (0) Jhad
(x)|0 >;
we will work to order ǫ. Note that the vector meson, ρ, can be viewed as a gauge field
that has acquired mass through spontaneous symmetry breaking [160] (Indeed, Bando,
Kugo, Yamawaki and others [161, 162, 163, 164] have argued that vector meson effective
Lagrangians always contain a hidden local symmetry). This is why we choose the ρ kinetic
term to be in the form of the photon kinetic term, and it implies that we are always free
to choose a gauge, such as ∂µ ρµ = 0.
Upon integrating by parts, we can rewrite the ǫ term as ǫAν ∂µ Fρ µν . Using equations
of motion for the ρ to order ǫ we obtain:
1
1
µ
ν
+ 21 m2ρ (ρµ )2 − eAµ J µ − κρµ Jhad
(2.50)
L = − (FAµν )2 − (Fρµν )2 + ǫm2ρ Aν ρν + ǫκAν Jhad
4
4
The first ǫm2ρ Aν ρν term can be viewed as arising from the matrix element of the electromagnetic interaction, eAµ < ρ|J µ |0 >, and by comparison with eq.(2.45) we identify
ǫ = e/fρ . In this form we have an induced mass mixing between the ρ and photon.14
The ρ and photon can now redefined as ρ → ρ − ǫA, A → A + ǫρµ . Thus we obtain:
1
1
µ
L = − (FAµν )2 − (Fρµν )2 + ǫρµ J µ + 12 m2ρ (ρµ )2 − eAµ J µ − κρµ Jhad
4
4
(2.51)
ν
This removes the mass mixing term, and the ǫAν Jhad
term, but leads to an induced
µ
ǫρµ J term. Thus, we can view the ρ as having an induced direct coupling to the full
electromagnetic current of strength ǫ!
Alternatively, upon integrating by parts, we could have written the ǫ term as ǫρν ∂µ FAµν ,
and using equations of motion for the ρ to order ǫ we have ǫeρν J ν = e2 ρν J ν /fρ . Thus,
we can view the effect of the ǫ term as directly inducing the coupling of the ρ to any
electromagnetic current with strength e2 /fρ , e.g., the ρ will couple directly to the electron’s
electromagnetic current. While this is a small coupling, the ρ is generally a narrow state.
On-resonance the production rate can be substantial.
This can be directly generalized to electroweak gauge fields and an isotriplet ρa , or to gluons and a
color octet ρa8 , but see [158, 159].
14
This does not violate gauge invariance. While a shift Aµ → Aµ + ∂µ θ leads to ǫm2ρ ∂ν θρν , we can
integrate by parts −ǫm2ρ θ∂ν ρν , and the gauge invariance of the ρ allows this term to be set to zero. That
is, ρµ behaves like a conserved current if it is a gauge field with hidden local symmetry [164].)
13
35
An equivalent non-Lagrangian description of this treats the ǫ term as a mixing effect in
the propagator of the photon. The propagator becomes a matrix, allowing the photon to
mix with the ρ and couple directly to Jhad . The propagator connecting an electromagnetic
current to the hadronic current becomes:
−i
−i
2
×
−iǫ
q
×
q2
q 2 − m2ρ
(2.52)
In the context of VMD, we can summarize our expectations for the minimal model
with ND = 1. We have already estimated the the techni-ρ mass to be of order ∼ 1.8 TeV
in this case. The dominant expected production and decay modes of the ρT are then:
and
f f¯ → (γ, Z 0 ) → ρ0T → W + W −
(2.53)
± 0
f f¯ → (W ± ) → ρ±
T → W Z
(2.54)
The annihilation of a fermion and anti–fermion a ρT and its subsequent decay into technipion + gauge boson, or technipions can also be significant. With ND = 1 we would
expect f f¯ → ρT → ηT′ + Z to be of interest.15 .
In models with more doublets, ND > 1, there are more vector meson states which
can be enumerated and classified according to the scheme discussed above for technipions
(see Fig.(1)). However, the only vector mesons which participate in the VMD mixing are
those transforming like the Standard Model gauge fields, i.e. as τ a ⊗ Id . These vector
mesons can, in turn, decay to any pair of technipions for which the V and technipions
together form an overall I ⊗ I combination16 . Hence, decays to gauge boson pairs are
allowed since (τ a ⊗ Id ) × (τ b ⊗ Id ) × (τ c ⊗ Id ) contains the singlet ǫabc (I ⊗ I). Moreover,
we see that the decays to the 3(ND2 − 1) PNGB’s of the form τ a ⊗ λa are also possible.
Decays to the techniaxions of the form I ⊗ λa , however, are disallowed.
Because we expect the techni-ρ to have MρT > 2MπT in the minimal model with
ND > 1, the dominant decay of the techni-ρ should be to a pair of technipions. The
technipions will, in turn, decay to W W , W Z, and so forth, through direct couplings or
anomalies. Variant models may include decays to gluon pairs; for models incorporating
ETC interactions, final states with fermion pairs are also expected. We defer further
discussion of searches and phenomenology for now, and move on to sketch out a more
general TC model.
15
Tandean [93] has also considered the signature and detectability of the ηT′ produced in a TeV scale
γγ collider.
16
Recall that under multiplcation of direct product representations, we have (X ⊗ Y ) × (W ⊗ U ) =
XW ⊗ Y U
36
2.3
2.3.1
Farhi–Susskind Model
Structure
The minimal model is neither a unique prescription for the construction of a TC theory,
nor is it likely to contain sufficient richness to ultimately allow the generation of the observed range of fermion masses and mixing angles. We anticipate that a more complete
model would need to include both a “quark” sector of color-triplet techniquarks, and a
“leptonic” sector of color-singlet technifermions (like those in the minimal model). These
states could ultimately act under extended TC interactions17 to give mass terms to the ordinary quarks and leptons. Toward this end, we turn now to describing the Farhi-Susskind
TC model [165] which contains a much richer spectrum of technifermions imitating the
anomaly free representations of the Standard Model.
The Farhi-Susskind model extends the flavor content of the minimal model to imitate
one full generation of quarks and leptons with the usual anomaly free isospin and Y
assignments:
T
B
QL =
QR = (TRi ,
!i
L
BRi
N
E
Y = y;
!
Y = −3y;
L
(2.55)
NR , LR ) Y = (y + 1, y − 1, −3y + 1, −3y − 1)
where the SU(3) color index i takes the values i = 1, 2, 3 for the techniquarks. This is an
anomaly free representation for any choice of the parameter y. For the particular standard
choice of y = 1/3, the techniquarks and technileptons have electric charges identical to
those of the quarks and leptons.
In the present model the techniquarks and technileptons couple to the full SU(3) ×
SU(2)L × U(1)Y gauge group in the usual way. Now we postulate that each Q multiplet
carries, in addition, an SU(NT C ) quantum number in the NT C representation of the
strong TC gauge group. Note that the existence of a right-handed technineutrino, NR , is
required to provide anomaly cancellation for the TC gauge interaction. The SU(NT C ) is
essentially a gauged horizontal generation symmetry.
Unlike the minimal model with ND = 1, in which all three NGB’s are absorbed into
the longitudinal modes of the electroweak gauge bosons, the Farhi-Susskind model has a
low energy spectrum containing numerous pseudo-Nambu-Goldstone bosons (PNGB’s).
Their quantum numbers may be ascertained by observing that, in the limit of vanishing
SU(3) × SU(2)L × U(1) couplings, there is an SU(8)L × SU(8)R × U(1)A × U(1) global
chiral group. The full Standard Model SU(3) × SU(2)L × U(1)Y interactions are a gauged
subgroup of this chiral group. At the scale ΛT C , the TC gauge coupling is strong, and
causes a degenerate chiral condensate to form:
D
E
D
E
D
E
D
E
T Li TRi = B Li BRi = N L NR = E L ER ∼ Λ3T C
17
See section 3.
37
(2.56)
where i is a (unsummed) color index ranging from i = 1, 2, 3. The chiral group is thus
broken spontaneously to an approximate SU(8) × U(1) vectorial symmetry, producing
63 + 1 NBB’s.
¿From the previous remarks about condensates, we see that there exist four composite
electroweak doublets. This is similar to the structure of a 4-Higgs-doublet model in which
each Higgs boson gets a common VEV FT . The electroweak scale is thus related to the
common VEV’s of the four Higgs bosons, FT , as v02 = 4FT2 , and thus FT = 123 GeV. One
combination of the NGB’s will become the longitudinal W and Z, while the orthogonal
states remain in the spectrum, as we will describe in the next subsection.
2.3.2
Spectroscopy
(i) Color {1}, {3}, and {8}, Pseudo-Nambu-Goldstone Bosons
Let us examine the content of the low lying (8 × 8) PNGB’s of the Farhi-Susskind
model. As might be expected, the enhanced variety of technifermions yields a larger
selection of PNGB states. Their properties are summarized in Table 3.
We will begin with the eight color-singlet states. By symmetry, three linear combinations are identically massless and become the longitudinal W ± and Z 0 :
T i B i + NE ∼ π −
B i T i + EN ∼ π +
T i T i − B i B i + N N − EE ∼ πL0
(2.57)
There remain 5 orthogonal color singlet objects, two with non-zero electric charge (we
follow the nomenclature and normalization conventions of Eichten, Hinchliffe, Lane and
Quigg (EHLQ) [103] [101]):
T i B i − 3NE ∼ P −
B i T i − 3EN ∼ P +
(2.58)
and three which are electrically neutral:
T i T i − B i B i − 3(N N − EE) ∼ P 0
T i T i + B i B i − 3(NN + EE) ∼ P 0′
(2.59)
and
T i T i + B i B i + N N + EE ∼ ηT′
(2.60)
The PNGB which is neutral under all gauge interactions receive mass via instantons.
The ηT′ , receives mass from the instantons of TC, and is expected to be heavy, as in
the case of the Weinberg-Susskind
model. From the discussion of section 2.2.2(ii) we
q
√
estimate mη′ T C ∼ ( 6/NT ) 3/NT ND (v0 /fπ )mη′ ∼ 700 GeV for NT = 4 (where ND = 4
in the Farhi-Susskind model). The P 0′ receives a mass only of order 1 GeV from QCD
instantons, but will receive a larger contribution from ETC effects (see Section 3). The
P 0 , likewise, receives its mass from ETC.
The PNGB’s with electroweak gauge charges, but no color, P0Q , receive masses from
the gauge interactions in analogy to the electromagnetic mass splitting of the ordinary
38
state
−
πT ∼ (T i B i + NE)
πT0 ∼ (T i T i − B i B i + NN − EE)
ηT′ ∼ (T i T i + B i B i + N N + EE)
P + ∼ (B i T i − 3EN)
P 0 ∼ (T i T i − B i B i − 3(NN − EE))
P 0′ ∼ (T i T i + B i B i − 3(NN + EE))
P31 ∼ ET
P30 ∼ NT − EB
P3−1 ∼ NB
P3 ′ ∼ NT + EB
P8+ ∼ BT
P80 ∼ T T − BB
P80 ′ ∼ T T + BB
I(J P ), color, [Q]
1[0− , 1− ] 0[−1]
1[0− , 1− ] 0 [0]
1[0− , 1− ] 0 [0]
1[0− , 1− ] 0 [1]
1[0− , 1− ] 0 [0]
0[0− , 1− ] 1 [0]
1(0− ) 3 [5/3]
1(0− ) 3 [2/3]
1(0− ) 3 [−1/3]
0(0− ) 3 [2/3]
1(0− ) 8 [1]
1(0− ) 8 [0]
0(0− ) 8 [0]
mass (GeV)
MW
MZ
∼ 103
∼ 100(4/NT C )1/2
∼ 100[ETC]
∼ 100[ETC]
∼ 160 (4/NT C )1/2
∼ 160 (4/NT C )1/2
∼ 160 (4/NT C )1/2
∼ 160 (4/NT C )1/2
∼ 240 (4/NT C )1/2
∼ 240 (4/NT C )1/2
∼ 240 (4/NT C )1/2
Table 3: Properties of scalar states in the Farhi-Susskind TC model following [103], [101];
see also [90]. To each listed spin-0 state there is a corresponding s-wave spin-1 analogue; there
will also be p-wave analogue resonance states. Nonself-conjugate states have corresponding
(unlisted) antiparticles. Quoted masses are crude estimates, quoted in GeV; their theoretical
values are very model dependent, modulo walking ETC, etc. For analogue vector and resonance
masses see the discussion of the text.
π + and π 0 of QCD. In QCD we have δm2π = m2π+ − m2π0 ≈ (35 MeV )2 . In the chiral
limit (current masses mu = md = 0) we have m2π0 = 0, and thus δm2π roughly reflects the
electromagnetic mass contribution of the π ± . Scaling from this, one finds that the full
electroweak induced mass for a technipion is of order m2P ± ∼ FT2 δm2π /fπ2 sin2 θW . Hence
0
numerically, mP ± ∼ 100(4/NT C )1/2 GeV. The P00 and P0 ′ masses arise from ETC alone.
0
More detailed estimates for the PNGB mass splittings in TC are discussed in Dimopoulos
[90] and Eichten and Lane [166].
The spectrum of the Farhi-Susskind model also includes technipions with non-zero
color charge: technileptoquark PNGB’s, P3 ∼ LQ, and color octet PNGB’s, P8 ∼
Q(λA /2)Q. These states acquire masses principally through the SU(3)QCD interactions
[167] [168] [79]. If the technipion carries net color in the R representation, then the QCD
contribution to the mass is of order m2πT ∼ (C2 (R)α3 (FT )/α(FT ))FT2 δm2π /fπ2 , giving us:
M(P3 ) ∼ 160 (4/NT C )1/2 GeV.
(2.61)
M(P8 ) ∼ 240 (4/NT C )1/2 GeV.
(2.62)
(here we use α3 (FT ) = 0.1, α(FT ) = 1/128, C2 (3) = 4/3, C2 (8) = 3). More generally, the
various electroweak and strong contributions to the mass of a colored PNGB are added in
quadrature to form the full mass, m2 = m2c + m2EW . In a more complete model, there can
also be Extended TC contributions to the masses, but these are expected to be smaller
than the QCD masses given above.
39
The reader is advised to consider the spectrum of states, and reasonable decay and
production estimates (which follow), but not take literally the model estimates of masses
at this stage. Any TC model we review from the 1980’s is, at best, incomplete, and
can only serve as a guide to what may be contained in more modern reincarnations of
the models. Moreover, in subsequent sections on ETC, Walking TC and hybrid models
like TC2 many mechanisms will surface that can rearrange the masses of states in these
models.
Color-triplet (leptoquark) technipions decay via Extended TC as P3 → qℓ (without
the standard choice of y = 1/3 these objects would be stable). The color octet technipions
decay into P8 → qq + ... final states. The rates for these PNGB decay processes depend
3
2
upon the scale and details of ETC, but are expected typically to be of order Γ <
∼ M /ΛT C
(e.g, in older ETC models ΛT C ∼ 1 TeV in this estimate is replaced by ΛET C ∼ 100
TeV, while in “Walking ETC” (Section 3.4) we expect ΛT C ∼ 1 TeV is replaced by
√
ΛT C ΛET C ∼ 10 TeV). We review the phenomenology of the colored technipions further
in Sections 2.3.3 and 2.3.4.
(ii) Vector Mesons
As was the case in the minimal TC models, the Farhi-Susskind model includes s-wave
vector (J=1) states which we refer to collectively as ρT (the literature also refers to these
states as VT ). Their masses follow from the estimate in section 2.2.2(iii) if we set ND = 4.
Neglecting the QCD corrections, which are expected to be less than ∼ 15%, we find
M(ρT ≡ VT ) ∼ 700 (4/NT C )1/2 GeV
(2.63)
Note that there now exist color-octet V8 states which have the quantum numbers of the
gluon, and act like a multiplet of heavy degenerate gluons18 . These will exhibit vectordominance-like mixing with gluons in processes like qq → G → V8 → PA PB .
There are also, as in the minimal model, p-wave parity partners of the PNGB’s (the
techni-a0 ’s and techni-f0 ’s) and parity partners of the vector mesons, the axial-vector
mesons a1T ’s and f1T . Following the discussion of section 2.4, we find them to have
masses of order:
M(a1T , f1T ) ∼ 1700 (4/NT C )1/2 GeV
(2.64)
M(a0T , f0T ) ∼ 1300 (4/NT C )1/2 GeV
(2.65)
Generally speaking, the parity-partner states form identical representations of the symmetry groups and have identical charges, but are significantly heavier.
2.3.3
Production and Detection at Hadron Colliders
(i) Color-singlet PNGB: P 0 and P 0′
18
Similar objects will crop up as gauge particles (colorons) in Topcolor models (Section 4.2) or as KK
modes in extra-dimensional models (Section 4.6)
40
dσ/dy (y=0) (nb)
1
10 −
−
100
40
20
0
10 −
−
−
−
−−
−
−
−1 −
−
10 −
−
−
−
−
−
10
2
+
p− p −> P
0
+ anything
−2
10
15
25
35
45
55
2
Mass (GeV/c )
Figure 5: Differential cross section for production of P 0′ at y = 0 in pp and pp̄ collisions with
indicated center-of-mass energy in TeV (reproduced from EHLQ [157]).
EHLQ [101], [103], provided the original discussion of production and detection of
neutral PNGBs in hadron colliders at various energies. The differential cross-section for
production of P 0′ is shown in Figure 5. The relevant decay widths for for P 0′ in the
Farhi-Susskind model are (including possible ETC contributions) [169]:
(3) m2l(q)
2
2 3/2
¯
Γ(P → ll(q q̄)) =
m
1
−
4m
/m
P
l(q)
P
8π F 2
2 3
2
αs N mP
Γ(P 0′ → gg) =
6π 3 4
F2
2 3
2
α
N mP
Γ(P 0′ → γγ) =
.
3
27π
4
F2
0′
(2.66)
(2.67)
(2.68)
If mP 0′ < mt /2, then the best hope of finding P 0′ at a hadron collider is through the
decay modes P 0′ → γγ, P 0′ → bb, τ + τ − , since the dijet decay modes will be invisible
against the large QCD background. The signal in the two-photon channel will resemble
that of an intermediate mass Higgs boson; the small branching ratio (of order 0.001)
is compensated by the large production rate. The signal in the τ + τ − final state has as
background from the corresponding Drell-Yan process. According to ref. [103] the effective
integrated luminosity (i.e. luminosity times identification efficiency) required to find the
P 0′ in this channel would be in the range 3 × 1035 – 5 × 1036 cm−2 for colliders with
center-of-mass energies in the range 2 – 20 TeV.
Recently, Casalbuoni et al. [170] (see also [171]) have looked in detail at the possibility
41
Projected gg → P 0 → γγ signal rate S and irreducible background rate for an
integrated luminosity of 2 f b−1 at Tevatron Run II; from [170].
Figure 6:
of finding the P 0 state at the Tevatron and LHC. The P 0 could be visible in the process
gg → P 0 → γγ at Run II of the Tevatron in the mass range 60 GeV ≤ M ≤ 200 GeV
and at the LHC in the mass range 30 GeV ≤ M ≤ 200 GeV. Moreover, a very precise
measurement of the signal rate would be possible at the LHC, enabling the determination
of some model parameters. Figure 6 shows the projected signal and irreducible background
rates at Tevatron Run II.
(ii) Color-triplet PNGB’s: the P3 Leptoquarks
As we have seen, the spectrum of the Farhi-Susskind TC model includes color-triplet
technipions, with leptoquark quantum numbers [43, 101, 156]. These objects, P3 ’s, would
be predominantly resonantly pair-produced through a color-octet techni-ρ (V8 ) coupled
either to gluons, or through octet vector dominance mixing (VMD) with the ordinary
gluon. The latter case assures a signal in q q̄ annihilation provided the V8 can be excited.
P3 ’s decay (via ETC [172] [173]) preferentially to third-generation quarks and leptons.
At leading order, the leptoquark pair production cross section depends only on the masses
(MV8 , MP3 ) and the V8 decay width (ΓV8 ). The latter depends, in turn, on masses, on the
size of the TC group NT C and on the mass-splitting between the color-octet and colortriplet technipions (∆M). While we would expect that V8 VMD with the gluon would
provide the largest contribution to the hadronic production of colored PNGB’s, provided
the V8 pole is within reach of the machine energy, at this writing there is no study of this
contribution and its effects.
42
EHLQ [101], and more recently W. Skiba [174], have studied of hadron collider signatures of colored PNGB’s, without the V8 resonant enhancement. Possible processes
in which colored PNGBs can be produced in hadron colliders are: (i) Gluon-gluon and
quark–anti-quark annihilations as sources of PNGB pair production; (ii) Quark-gluon
fusion producing single PNGB’s; (iii) anomalous couplings to two gluons producing single PNGB’s. Direct PNGB couplings to fermions (through ETC effects) are typically too
small to give significant cross sections. The V8 VMD is neglected in the present discussion,
and we know of no reference in the literature in which its effects are included.
The cross section for the pair production of colored PNGB’s has been calculated in Ref.
[103] for the general case of pseudo(scalar) particles in any representation of SU(3)color .
Quark anti-quark annihilation yields
dσ
2παs2
kd β 2 (1 − z 2 )
(q q̄ → P P ) =
9ŝ2
dt̂
and g − g annihilation gives
!
dσ
3
kd
2πα2
− (1 − β 2 z 2 ) 1 − 2V + 2V 2 .
(gg → P P ) = 2 s kd
ŝ
d
32
dt̂
(2.69)
(2.70)
where kd is the “Dynkin index” of the d-dimensional representation (k3 = 21 , k8 = 3), z
the cosine of parton scattering angle in the center of mass system,
V =1−
4m2P
1 − β2
2
and
β
=
1
−
,
1 − β 2z2
ŝ
ŝ and t̂ are Mandelstam variables at the parton level. Using these formulae and parton
distributions from Ref. [175] (set 1), Skiba obtains [174] production rates for leptoquarks,
P3 , and octets, P8 , which agree with the results of Refs. [101] (see also Hewett et al. , [176]).
As the cross-section for pair production of leptoquarks at LHC energies is sizeable (see
Fig.(7)), the LHC can potentially observe leptoquarks with masses up to approximately
∼ 1 TeV.
(iii) Color-octet PNGB’s, P8 , and vector mesons ρT 8
Due to color factors, the production cross sections for color octet P8 ’s are an order
of magnitude larger than for leptoquarks, as a comparison of Figures 7 and 8 reveals.
The detection of P8 ’s is, on the other hand, far more difficult. P8 ’s typically decay into
two hadronic jets, so that pair-produced P8 ’s yield four-jet final states. QCD four-jet
production is the main source of background, and it is overwhelmingly large. We mention
here two suggestions from the literature as to how P8 states might be detected. One
involves using special cuts and kinematic variables to bring out the P8 signal in four-jet
final states. The other turns to a single-production of P8 and a rarer P8 decay mode with
lower backgrounds.
Chivukula, Golden and Simmons Ref. [177, 178] have evaluated the possibility of
detecting new colored particles at the Tevatron and LHC in multi-jet final states. The
43
σ(pp->TT) [pb]
104
pp -> TT
103
102
101
100
16
10-1
4
1.8
200
400
600
mT [GeV]
800
1000
Figure 7: The cross section for pair production of leptoquarks in pp collisions, for
√
s = 1.8,
4 and 16 TeV, (from W. Skiba [174]; T = P3 in Skiba’s notation).
analysis begins with estimating the QCD multi-jet background [179, 180, 181, 182, 183,
184] and calculating the signal from heavy particle decays. Isolation and centrality cuts
must be applied to ensure all jets are detectable. Then appropriate kinematic variables
must be chosen to make the signal stand out cleanly above background.
The following strategy allows one to pull a P8 signal out of the large QCD background
[177], as illustrated in Figure 9. For each four-jet event, consider all possible pairwise
partitions of the jets. Choose the partition for which the two pairs are closest in invariant
mass, and define the “balanced pair mass,” mbal , as the average of the two masses. The
signal cross-section will cluster about mbal = mnew particle while the background will not.
Imposing a large minimum-pT cut on the jets further enhances the signal; the background
is peaked at low pT due to infrared QCD singularities.
Analyses of this kind indicate that real scalar color-octet particles with masses as high
as 325 GeV should be visible at the LHC if a pT cut of about ∼ 170 GeV is employed
[169]. The lower end of the visible mass range depends strongly on just how energetic
(pmin
T ) and well-separated (∆R) jets must be in order for an event to be identified as
containing four distinct jets. Discovery of color-octet scalars at the Tevatron is likely to
be possible, albeit difficult, in a reduced mass range [177].
An alternative strategy is to use processes with smaller backgrounds, such as singleproduction followed by a rarer decay mode. At hadron colliders, the widths relevant for
44
σ(pp->θθ) [pb]
105
10
pp -> θθ
4
103
16 TeV
102
4 TeV
101
1.8 TeV
100
100
200
300
mθ [GeV]
400
500
Figure 8: The cross section for pair production of color-octet PNGBs in pp collisions, for
√
s=
1.8, 4 and 16 TeV, (from W. Skiba [174]; θ = P8 in Skiba’s notation).
single production and decay of the P80′ of the Farhi-Susskind model are [169, 185] :
3 m2q
2
2 3/2
m
1
−
4m
/m
P
q
P
16π F 2
5αs2 N 2 m3P
0′
Γ(P8 → gg) =
24π 3 4
F2
2
m3P
Negs
Γ(P80′ → gγ) =
.
4πF
576π
Γ(P80′ → q q̄) =
(2.71)
Figure 10 gives the decay widths of the P80′ in the Farhi-Susskind model. Figure 11
illustrates gluon fusion production at the LHC and we see that the rate of single P80′
production is high at a hadron collider. However, the signal in the primary decay channels
(gluon or b-quark pairs) is swamped by QCD background. If the PNGB mass is above
the tt̄ threshold, the decay mode P → tt̄ becomes dominant and may alter the standard
QCD value of the tt̄ cross section and tt̄ spin correlations. The decay channel P → gγ
[186, 185] holds some promise for the models of “low-scale” TC discussed in Section 3.5;
however the PNGB’s of the Farhi-Susskind TC model are not likely to be visible in this
mode.
The ρ0′
T 8 coupling to gg and q q̄ can be estimated by assuming that it mixes with the
gluon under a generalized vector meson dominance (VMD). VMD applies to the gg → ρ0′
T8
0′
process as well as the q q̄ → ρ0′
process
ref.
[159].
The
partial
widths
relevant
for
ρ
T8
T8
45
4-jet cross-section (pb/25GeV)
Mbal (GeV)
dσ
Four-jet rate dm
at
a
17
TeV
collider with pmin
of 100 GeV. QCD background
T
bal
(hatched) and 240 GeV technipion signal are shown. No resolution effects included (from ref.
[177]).
Figure 9:
production at hadron colliders in the narrow width approximation are [169]:
αs2
mρ
2αρ
5αs2
→ q q̄) =
mρ
6αρ
Γ(ρ0′
T 8 → gg) =
(2.72)
Γ(ρ0′
T8
(2.73)
0′
Figure 11 compares the single production cross section of the ρ0′
T 8 and P8 at the LHC.
If kinematically allowed the dominant decay mode of ρ0′
8T is into two colored PNGB
[169]:
3/2
αρT
Γ(ρ0′
mρ 1 − 4m2P /m2ρ
.
(2.74)
8T → P8 P8 ) =
4
In this case the ρ0′
T 8 contributes strongly to the cross section for color-octet PNGB pair
production discussed in above and improves the signal.
If the ρT 8 are light, e.g. in low-scale models (Section 3.5), there can be a sizeable cross
section for their pair production. Näively, well above the pair-production threshold one
46
Γ (GeV)
M P (GeV)
Figure 10: Partial widths for the decay of P80′ into gluon-gluon (solid line), b̄b (dashed line),
t̄t (dotted line) and Z-gluon (dot-dashed line) in the Farhi-Susskind model [169].
expects: [169] :
σ(pp → ρT 8 + X)
1
1
≃ 2 ≃ .
σ(pp → ρT 8 ρT 8 + X)
gρT
40
(2.75)
Note that all types of colored vector resonances can be pair-produced, whereas the isosinglet ρ0′
T 8 dominates single production. Hence, pair production, unlike single production,
can result in interesting decays to longitudinal electroweak gauge bosons. One may also
expect spectacular 8–jet events whose kinematics distinguish them from the QCD background, as in the case of color-octet PNGB pair production.
The ρT 8 can significantly affect the invariant mass distribution of t̄t final states [104].
Through gluonic vector dominance processes like q̄q → g + ρT 8 → t̄t the gluon and ρT 8
add coherently. Significant mass imits can be placed from the Tevatron in Run II (this
process has a direct analogue in Topcolor, where ρT 8 becomes the coloron [187]).
(iv) Non-Resonant Production and Longitudinal Gauge Boson Scattering
At the next level, below resonance production, we expect gauge boson fusion through
techniquark loops (including anomalous loops) into techni-vector mesons and/or technipions. This can include gluon fusion [135], [136], [137], [138] at machines such as the
LHC, or W − W fusion at an LHC or LC (see section 2.3.4) [139], [140], [141], [142],
47
dσ/dy|y=0 (fb)
108
107
106
105
100
200
300
400
500
600
700
800
900
1000
MP,ρ (GeV)
0′
Figure 11: Differential cross section at y = 0 for single production of ρ0′
T 8 (solid line) and P8
(dashed line) at the LHC in femtobarns [169].
[143]. Scattering of the longitudinal electroweak modes, VL VL → VL VL , is especially important (see Section 2.2.3). These processes involve all possible spin and isospin channels
simultaneously, and can proceed through scalar and vector resonances or non-resonant
channels.
(v) Rescattering
An alternative way of finding color-octet technipions is to detect them after they have
re-scattered into pairs of W or Z particles. For example, in the Farhi-Susskind TC model
the production of gauge boson pairs through gluon fusion includes a potentially sizeable
[188] contribution from loops of colored technipions through the process P80,± P80,± → W W
or ZZ. This may prove to be a very useful diagonostic. If colored technipions are first
detected in 4-jet final states, their association with electroweak symmetry breaking will
not be obvious. However, the combination of their discovery with the observation of a
large number of gauge-boson pairs may permit us to deduce that the colored scalars are
PNGB’s of the symmetry breaking sector [189].
The contribution of loops of colored technipions to the production of gauge-boson
pairs through gluon fusion was calculated to leading order in chiral perturbation theory
in ref. [188]. Unfortunately, general considerations [190, 191] show that in theories with
many Nambu–Goldstone bosons, chiral perturbation theory breaks down at very low
energies. In the Farhi-Susskind model, for example, chiral perturbation theory breaks
down at a scale of order 440 GeV! Hence only a qualitative estimate of the signal size is
possible. Figure 12 shows the ZZ differential cross section as a function of ZZ invariant
mass at the LHC in a toy O(N) scalar-model [189]; model parameters were chosen so the
48
dσ/dM (nb/GeV)
M (GeV)
Figure 12: The ZZ differential cross section in nb/GeV vs. MZZ in a toy O(N ) scalar-model
with three color-octet PNGBs (upper curve), and the continuum q q̄ annihilation background
(lower curve). A pseudo-rapidity cut |η| < 2.5 is imposed on the final state Z’s. From [189]
signal size is representative of Farhi-Susskind TC. Note that there are almost an order of
magnitude more events due to gluon fusion than due to the continuum q q̄ annihilation for
ZZ invariant masses between 300 GeV and 1 TeV. The observation of such a large two
gauge-boson pair rate at a hadron collider would be compelling evidence that the EWSB
sector couples to color.
2.3.4
Production and Detection at e+ e− Colliders
(i) pair-production of EW bosons
The s-channel process e+ e− → W W is an effective probe of strong electroweak symmetry breaking, especially for physics with a vector resonance [192, 193, 134, 194]. A 500
GeV linear collider with only 80 fb−1 of integrated luminosity would already be sensitive
to radiative corrections induced by vector resonances with masses up to about 2 TeV,
but would not be able to observe off-resonance contributions to W W production (see
Figure 13). With 500 fb−1 of integrated luminosity, even the non-resonant or low-energy49
theorem contributions would become distinguishable from Standard Model expectations,
as illustrated in Figure
14. A higher-energy e+ e− collider would have even greater search
√
potential. With s=1.5 TeV and an integrated luminosity of 190 fb−1 , it should be
possible to distinguish the radiative effects of a very heavy techni-ρ or a non-resonant
amplitude from those of the standard model with a light Higgs boson; the 4 TeV (6 TeV)
techni-ρ corresponds to a 6.5σ (4.8 σ) signal. At a slightly higher integrated luminosity of
225 fb−1 , it would be possible to obtain 7.1σ, 5.3σ and 5.0σ signals for a 4 TeV techni-ρ,
a 6 TeV techni-ρ, and non-resonant contributions, respectively.
Figure 13: Sensitivity of a 500 GeV LC with 80 f b−1 of data to MρT via the W –boson form
factor; from Ref.[193]. The predicted values for a 3-TeV ρT or non-resonant scattering (LET)
lie within the 95% c.l. curve for the prediction of the Standard Model; these cases cannot, then
be reliably distinguished. Lighter ρT give predictions which differ significantly from those of the
Standard Model.
The W W fusion processes are complementary to the s-channel W + W − mode since
they involve√more spin-isospin channels (the I = 0 and I = 2 channels). For an e+ e−
collider at s = 1.5 TeV with 190 fb−1 of data, the W + W − /ZZ event ratio can be
a sensitive probe of a strongly-interacting electroweak sector [134, 195, 196, 197, 198] as
illustrated in Table 4. Statistically significant signals are found for a 1 TeV scalar or vector
particle. There is also about a 6σ signal for non-resonant (low-energy-theorem) amplitudes
via the W + W − → ZZ channel alone. For an e− e− collider with the same energy and
luminosity, the non-resonant-amplitude signal rate for the ννW − W − (I = 2) channel [197]
is similar to that for e+ e− → ν̄νZZ, as anticipated from symmetry arguments, while the
background rate is higher [134]. The signals are doubled for an e−
L polarized beam (or
quadrupled for two e−
beams),
whereas
the
backgrounds
increase
by
smaller factors. A 2
L
50
0.2
LET
Mρ=2.5 TeV
1.6 TeV 1.2 TeV
Im(FT)
0.1
5 5
5555555555
555 55
55
5
55
5
5
5
5
5
5
5
5
5 5 5
5
5
5
55
5
55
5
5
555 5
5
555555555
0.0
-0.1
5
5
95% C.L.
-0.2
0.9
1
1.1
Re(FT)
1.2
1.3
1.4
Figure 14: Sensitivity of a 500 GeV LC with 500 f b−1 of data to MρT via the W –boson form
factor; from Ref.[194]. The predicted values for a 2.5-TeV ρT or non-resonant scattering (LET)
now lie outside 95% c.l. curve for the prediction of the Standard Model.
TeV e+ e− linear collider would increase the signal rates by roughly a factor of 2–2.5.
When both the both the pair production and gauge boson fusion processes are taken
into account, the direct signal of a strongly-coupled symmetry-breaking sector is generally
stronger at a high-energy linear collider than at the LHC; Figure 15.
A strongly–coupled electroweak symmetry breaking sector can also be investigated
[134] at photon–photon colliders in the modes γγ → ZZ, γγ → W + W − , γγ → W + W − W + W − ,
and γγ → W + W − ZZ. While irreducible backgrounds in the first two cases are severe,
the four-boson final states seem more promising [199]. To be sensitive to a heavy scalar
resonance (e.g. heavy Higgs) with a mass up to 1 TeV would require a 2 TeV e+ e− collider with luminosity of 200 fb−1 [200], [201]. A 1.5 TeV linear collider running in the
γγ mode is found to be roughly equivalent to a 2 TeV e+ e− collider for the purpose of
studying strongly-interacting ESB physics [200, 201] (the correspondence scales roughly
√
√
as sγγ ∼ 0.8 see ).
(ii) colorless neutral PNGBs, P 0 , P 0′
A light enough neutral PNGB can be singly produced at an e+ e− collider via the axialvector anomaly [202, 203] which couple the PNGB to pairs of electroweak gauge bosons.
51
channels
SM
mH = 1 TeV
S(e+ e− → ν̄νW + W − )
160
B(backgrounds)
170
√
S/ B
12
S(e+ e− → ν̄νZZ)
120
B(backgrounds)
63
√
S/ B
15
− −
−
−
S(e e → ννW W )
27
B(backgrounds)
230
√
S/ B
1.8
Scalar
MS = 1 TeV
160
170
12
130
63
17
35
230
2.3
Vector
MV = 1 TeV
46
4.5
22
36
63
4.5
36
230
2.4
LET
31
170
2.4
45
63
5.7
42
230
2.8
Table 4: Total numbers of W + W − → 4-jet and ZZ → 4-jet signal S and background B events
calculated for a 1.5 TeV LC with integrated luminosity 200 fb−1 . Results are shown for the
SM, for strongly-coupled models with scalar or vector resonances and for low-energy-theorem
(LET) scattering. Events are summed over the mass range 0.5 < MW W < 1.5 TeV except for
the W + W − channel with√a narrow vector resonance in which 0.9 < MW W < 1.1 TeV. The
statistical significance S/ B is also given. For comparison, results for e− e− → ννW − W − are
also presented, for the same energy and luminosity. The hadronic branching fractions of W W
decays and the W ± /Z identification/misidentification are included. From ref. [134].
At LEP I the PNGB, P a , would have been primarily produced [204, 205, 206] through an
anomalous coupling to the Z boson and either a photon (Z → γP a ) or a second, off-shell
Z boson (Z → Z ∗ P a ). At the higher center of mass energies of LEP II, PNGBs over a
wider range of masses could have been produced through s-channel γ ∗ /Z ∗ exchange and
through a 2 → 3 production mechanism [207, 170].
If the TC group is SU(NT ), the anomalous coupling between the PNGB and the gauge
bosons G1 and G2 is given, as for the QCD pion, by [99, 208, 209]
NT AG1 G2
g1 g2
ǫµνλσ k1µ k2ν ελ1 εσ2 ,
2
2π FT
(2.76)
where NT is the number of technicolors, AG1 G2 is the anomaly factor (see below), the gi
are the gauge couplings of the gauge bosons, and the ki and εi are the four-momenta and
polarizations of the gauge bosons.
The model-dependent value of the anomaly factor is calculated in [99], [208], [209].
The dominant production mode for a PNGB at LEP I was generally the Z → γP a process
[204] which has a branching ratio of order 10−5
ΓZ 0 →γP a
123GeV
= 2.3 × 10 GeV
FT
−5
2
(NT AγZ 0 )
2
MZ2 0 − MP2 a
MZ2 0
!3
.
(2.77)
The final states contain a hard, mono-energetic photon, along with the decay products
of P a . Production in combination with an off-shell Z 0 is harder to observe. An upper
52
Signal Significance
M = 1240 GeV
ρ
10 2
10 2
10
10
1
10 2
Signal Significance
M = 1600 GeV
ρ
LHC
LC
500
LC
1000
1
LC
1500
M = 2500 GeV
ρ
LC
500
LC
1000
LC
1500
LET
10 2
10
1
LHC
10
LHC
LC
LC
LC
500
1000
1500
1
LHC
LC
LC
LC
500
1000
1500
Figure 15: Direct strong symmetry breaking signal significance in σ’s for various masses Mρ
of a vector resonance in WL WL scattering [194]. The numbers below the “LC” labels refer to
the center-of-mass energy of the linear collider in GeV. The luminosity of the LHC is assumed
to be 300 f b−1√
, while the luminosities of the linear colliders are assumed to be 500, 1000, and
1000 f b−1 for s=500, 1000, and 1500 GeV respectively. The lower right hand plot “LET”
refers to the case where no vector resonance exists at any mass in strong WL WL scattering.
bound on the decay width of the process Z 0 → Z 0∗ P a → P a f f¯ is given by [205, 210].
One expects branching ratios of order 10−7 to 10−6 , depending on the process of interest.
At LEP II and an NLC, production modes include e+ e− → γ ∗ , Z ∗ → γ (∗) P a , Z (∗) P a and
the 2-to-3 process, e+ e− → e+ e− P a [207].
In the minimal models, the PNGB’s decay back into electroweak bosons. Which decays
are allowed or dominant depends on the values of the anomaly factors. For example, a
neutral colorless PNGB produced by Z-decay can certainly decay to an off-shell Z plus
another electroweak gauge boson (photon or Z). It may also be able to decay to a pair of
photons; if allowed, this mode dominates over decays via off-shell Z’s. In any given nonminimal TC model, the dominant decay mode of the neutral PNGB’s depends both on the
gauge couplings of the technifermions and on any interactions coupling technifermions to
ordinary fermions. If some technifermions are colored, the PNGB may have an anomaly
53
coupling allowing it to decay to gluons. If the PNGB gets its mass from effective fourfermion interactions (e.g. due to extended TC), then it will be able to decay to an f f¯
pair. Finally, in some models, the PNGB may decay dominantly to particles in an invisible
sector.
Searches for the neutral PNGB’s at LEP I and LEP II were sensitive only to PNGB’s
with small technipion decay constants and large anomaly factors. The P a of the FarhiSusskind model are invisible at these colliders. However, data from the LEP experiments
is able to test the properties of the PNGB’s of various extended models, such as the
Appelquist-Terning model [211], the Manohar-Randall weak isotriplet model [204], Lane’s
low-scale TC model [212, 213], and the Lane-Ramana multiscale model [156]. Details
about all of the constraints are available in Ref. [210]; a summary for a few walking TC
models will be given in Section 3. For now, we turn to the possibility of direct searches
at higher-energy linear colliders.
′
The possibility of searching at a 1 TeV Linear Collider for a generic neutral P 0( )
decaying predominantly to the heaviest available fermions
e+ e− → PT0 γ → tt̄γ → bb̄W + W − γ
MP > 2mt
→ bb̄γ
MΠ < 2mt .
(2.78)
was studied in ref. [214]. The signal events will stand out from the background due
to the hard monochromatic photon recoiling against the PNGB decay products. Based
on a Monte Carlo simulation of signal and background, ref. [214] finds that for models
with NT = 3 and fT = 123, the P 0′ can be excluded at the 95% c.l. up to a mass of
646 GeV and could be discovered at the 5-sigma level up to a mass of 452 GeV. The
P 0 production rate appears too small to allow discovery in the models examined. These
results are illustrated in Figures 16 and 17.
0(′ )
The electrically neutral color-octet P8 will be singly produced in association with
a gluon through the anomalous coupling of the PNGB to the gluon and a photon or
Z boson. Provided that the PNGB decays to the heaviest available fermion pair, the
production and decay chain will look like [214]
e+ e− → PT08 g → tt̄g → bb̄W + W − g
MP 8 > 2mt
→ bb̄g
MP 8 < 2mt .
(2.79)
Characteristics that will help distinguish the signal events from those of the q q̄g background are monochromaticity of the gluons in the signal events are monochromatic and
their large spatial separation from the PNGB decay products. The left-right asymmetries
and tt̄ spin correlations will also differ from those of the background distributions. Based
on a Monte Carlo simulation of signal and background, ref. [214] asserts that the electrically neutral color-octet PNGB can be excluded or discovered for any realistic values
′
of NT C provided that MP < 2mt . For higher-mass PNGB, finding the PT08 will require
including events with hadronic W decays for NT C < 7. For the specific case NT C = 3, it
′
will be possible for a 1 TeV LC to exclude a PT08 (PT08 ) of mass less than 761 (404) GeV
54
Figure 16: Projected 95% c.l. exclusion reach for electrically neutral PNGB as a function of
mass and number of technicolors at a 1 TeV LC [214].
at 95% c.l. or to discover a PT08 with five-sigma significance for M < 524 GeV [214]. The
exclusion and discovery limits are shown in Figures 16 and 17.
The ability of an LC to study the lightest PNGB of a particular model [171] has been
probed in ref.[170]. They find that the process e+ e− → P 0 γ would allow discovery so
long as the PNGB mass was not close to MZ and for NT ≥ 3. Moreover, if running in
γγ mode, the process γγ → P√0 → bb̄ could not only allow detection of a PNGB with
mass between 10% and 70% of s, but would also provide more precise information about
model parameters. A muon collider could in principle help determine the couplings of a
PNGB in greater detail.
The production and detection of the ηT′ state has been studied in [92] and [93], the
latter with emphasis on the laser back-scattering technique at an e+ e− LC [215].
(iii) colorless charged PNGBs, P ±
Electrically charged PNGB’s would be abundantly pair-produced at a high-energy,
high-luminosity linear electron-positron collider. In terms of the electromagnetic point
cross section σo = 4πα2 (s)/3s, the continuum cross-section for producing P0+ P0− pairs at
an NLC is 0.30β 3σo , where β is the c.m. frame velocity of P0± [214]. Assuming the P0±
decay dominantly to tb pairs, the final state contains a pair of on-shell W bosons and
four b quarks. The leading background comes from tt̄g ∗ events where the off-shell gluon
materializes as a bb̄ pair; the cross-section for this process is less than a tenth of the tt̄
production cross-section of 1.9σo . Given a detector with good angular coverage and high
55
Figure 17: Projected 5-sigma discovery reach for electrically neutral PNGB as a function of
mass and number of Technicolors at a 1 TeV LC [214].
±
efficiency for
√ detection of the b-jets, it should be possible to detect P0 with a mass below
about 0.45 s at the five-sigma significance level [214]. Similarly, the large electromagnetic
±
pair-production cross-section and moderate backgrounds for pair-production of PT√
8 should
make 5-sigma significance discovery possible out to a mass of approximately 0.49 s [214].
As seen from the discussion of section 2.2.4, the ρ0T will couple via VMD to any
electromagnetic current (or to the weak neutral current, which can include ωT ). VMD
production on the narrow resonances will produce dramatic signatures in the conventional
final states as in e+ e− → (ρT , ωT ) → (W W + ZZ, W π + ZP, P P ). Such processes have
been discussed in the context of muon colliders, where the better beam energy resolution
is an advantage, in ref.[157].
In contrast, the work of ref [216] suggests that it will be difficult to detect the effects
of charged technipions on single top production at an eγ collider in the process eγ → tbν.
56
Figure 18: Projected P 0 production rate (solid), rate including√PNGB decay to bb̄ (dashed),
and irreducible γγ → bb̄ background (dotted) for γγ collisions at
luminosity of 20 f b−1 [170].
3
3.1
s = 500 GeV with integrated
Extended Technicolor
The General Structure of ETC
A realistic Technicolor model must address the flavor problem. That is, it must incorporate a mechanism for generating quark and lepton masses and the various weak mixing
angles, together with CP-violation. This implies that the ordinary quarks and leptons
of the Standard Model need to couple to the technifermion condensate that breaks the
electroweak symmetry. There must also be a mechanism for violating the conserved technibaryon quantum number: techniquarks must be able to decay, since stable techni-baryonic
states are cosmologically problematic [64].
The classic way to fulfill these requirements is to extend the Technicolor gauge interactions to include additional gauge bosons coupling both to ordinary and Technicolored
fermions. The extended interactions are part of a large gauge group which necessarily breaks down to its Technicolor subgroup at an energy above the scale at which the
Technicolor coupling becomes strong, ΛT C .
The construction of the first theories of “Extended Technicolor” (ETC) is due to
Eichten & Lane [173] and Dimopoulos & Susskind [172]. Their work, and the papers
which followed in the early 1980’s, especially [208], [100], [217] [99], [78], [218], flesh
out the general scheme and identify the challenges. We will summarize the major issues,
57
indicate trends in modern model-building, and refer the reader to the literature for further
detail.
3.1.1
Master Gauge Group GET C
The kinds of new interactions that are required for ETC involve transition couplings of
technifermions QL,R into the ordinary quarks and leptons, ψL,R . Currents of the form
Q̄L,R γµ ψL,R are therefore required, coupling to the new ETC gauge bosons. In a full
theory we assume a large gauge group GET C which contains all of the desired currents of
the form Q̄Q, Q̄ψ and ψ̄ψ. A simple generalization is a blockwise imbedding of SU(NT C )
into SU(NET C ) with NET C > NT C . Clearly NET C must be large enough to accomodate
representations containing both ψ’s and Q’s.
For example, consider how this imbedding might work in the minimal model. The
minimal model includes one weak doublet of technifermions (ND = 1) and a typical choice
of Technicolor group NT C = 4. ETC must couple the technifermions to all 12 left-handed
electroweak doublets of ordinary quarks and leptons and all 12 singlets. Then we should
think of the ordinary fermions of each generation as falling into 3 doublets of quarks and
one of leptons under the gauged SU(2)L and an analogous set of doublets under a global
SU(2)R . This yields an ETC scheme with a gauge group GET C = SU(16) × SU(2)L ×
SU(2)R ×U(1). Both SU(NT C ) and SU(3)c are imbedded into SU(16); hypercharge arises
as Y /2 = I3R + (B − L)/2, where B − L is a diagonal generator in SU(16). The fermions
form two fundamental 16 multiplets; one of them:
1
2
3
(Qc , Qk , Qm , Qy , ψr1 , ψg1 , ψb1 , ψr2 , ψg2 , ψb2 , ψr3 , ψg3 , ψb3 , ψlep
, ψlep
, ψlep
)L
c, k, m, y = Technicolors; r, g, b = colors
(3.80)
is a doublet under the electroweak group SU(2)L , and a singlet under SU(2)R , a (2, 1);
the other is a (1, 2). The Technicolor condensate will therefore break both the SU(2)L
and the SU(2)R . For the Farhi-Susskind model, an analogous ETC extension might be to
a gigantic gauge group placing all of the quarks and leptons and techniquarks (including
each Technicolor copy) into a single multiplet of a compact gauge group, e.g., SU(56).
Here the full Standard Model gauge interactions themselves would be subgroups of the
master group SU(56). Clearly a large variety of models are possible, and we will describe
several general approaches to building consistent models in this Section (several variant
schemes are also reviewed in [57]).
Starting from a high-energy theory based on a master gauge group GET C , it is necessary
to arrive at a low-energy theory in which the only surviving gauge groups are those of
Technicolor and the Standard Model. The group GET C has generators, T a , which form a
Lie Group,
[T a , T b ] = if abc T c
(3.81)
The Technicolor gauge group GT C must be a subgroup of GET C , since the ψ’s do not carry
Technicolor, while the Q’s do. Then GET C must undergo symmetry breaking at a scale
58
ΛET C down to its subgroup GT C :
GET C → GT C × ...
at ΛET C .
(3.82)
where the ellipsis denotes other factor groups, including perhaps the full Standard Model
SU(3) × SU(2) × U(1). This leaves the Technicolor gauge bosons (generators denoted
T̃ a ) massless, and elevates all of the coset ETC gauge bosons to masses of order ΛET C .
Indeed, the breaking may proceed in a more complicated way, e.g., it may occur in n
steps GET C → G1 → G2 → ...Gn−1 → GT C . A class of models in which this symmetrybreaking sequence is achieved dynamically is known as “Tumbling Gauge Theories” [219,
220, 221, 222, 223, 224, 225]. Typically, at each step, the subgroup will evolve and become
strong, permitting new condensates to form, which further break the theory into the next
subgroup. This occurs repetitively at different scales, and may in principle be a way
of generating the mass hierarchy for quarks and leptons of different flavors. Another
possibility is that there are fundamental scalars associated with supersymmetry at some
high energy scale, and the breaking of GET C (whether single or sequential) can be driven
by a Higgs mechanism (see Section 3.7).
3.1.2
Low Energy Relic Interactions
While the only elements of the original ETC gauge group that survive at low energies
are the generators of the Technicolor and Standard Model gauge groups, the low-energy
phenomenology of these models includes additional effects caused by the broken ETC
generators. On energy scales µ <
∼ ΛET C exchange of the heavy ETC bosons corresponding
to those broken generators produces three types of effective contact interactions among
the ordinary and technifermions:
ᾱab
ψ̄γµ T̄ a ψ ψ̄γ µ T̄ b ψ
Q̄γµ T̄ a ψ ψ̄γ µ T̄ b Q
Q̄γµ T̄ a QQ̄γ µ T̄ b Q
+
β̄
+
γ̄
ab
ab
Λ2ET C
Λ2ET C
Λ2ET C
(3.83)
Here the α, β and γ are coefficents that are contracted with generator indices and their
structure depends upon the details of the parent ETC theory. In the T̄ a we include chiral
factors such as (1 ± γ 5 )/2 (i.e., the theory is flavor-chiral and the ETC generators have
different actions on left- and right-handed fermions). We can now Fierz rearrange these
operators to bring them into the form of products of scalar and pseudo–scalar densities.
Upon Fierz rearrangement, we can pick out the generic terms of greatest phenomenological
relevance:
αab
Q̄T a QQ̄T b Q
ψ̄L T a ψR ψ̄R T b ψL
Q̄L T a QR ψ̄R T b ψL
+
β
+
γ
+ ...
ab
ab
Λ2ET C
Λ2ET C
Λ2ET C
(3.84)
Note that after Fierzing we must include the identity matrix among the generators, which
we do by extending the range of the generator indices to include zero: 1 ≡ T0 .
As a consequence of these generic terms we see that the physical effects of ETC go
beyond generating the quark and lepton masses and mixings. On the positive side, ETC
59
Q
Q
q
q
Figure 19:
The exchange of an ETC gauge Dboson
E allows a standard quark (or lepton) to
communicate with the techniquark condensate, QQ .
interactions, such as the α-terms, can elevate the masses of some of the light NambuGoldstone bosons, such as the techniaxions, to finite nonzero values, essential to rendering
models consistent with experiment. On the negative side, Extended Technicolor produces
the γ-terms, four-fermion contact interactions amongst ordinary fermions of the same
Standard Model gauge charges. This leads generally to flavor-changing neutral current
effects in low energy hadronic systems, e.g., it can lead to dangerously large contributions
to the KL KS mass difference. Lepton number violation will also generally occur, leading
to enhanced rates for µ → e + γ and related processes. It is a difficult model-building
challenge to limit these dangerous effects while generating adequately large quark and
lepton masses to accomodate the heavier fermions, e.g., charm, τ , b, and especially t. It
is, moreover, important that the oblique electroweak corrections involving TC and ETC
effects be under control [226], as will be discussed in Section 3.2. These and other19
phenomenological considerations have had a large influence on ETC model-building. We
will look at explicit examples of models constructed to address such questions in Section
3.3.
3.1.3
The α-terms: Techniaxion Masses.
The four-technifermion terms (with coefficients α) can potentially solve a problem we encountered in the previous discussion of the minimal and Farhi-Susskind models in Section
2. Loops involving α term insertions, which represent ETC gauge boson exchange across
a technifermion loop, with external PNGB’s (Fig.(3.1.3)) generally induce masses for the
PNGB’s. This mechanism can elevate the masses of the undesireably light PNGB’s to
larger values more consistent with experiment.
To see this explicitly, we can make use of an NJL or Georgi-Manohar model of the
19
There has recently been a great deal of interest in the possibility of a small departure of the observed
g − 2 of the muon from the Standard Model expectation [227, 228]. Some authors claim that it is difficult
for TC (or Topcolor) to yield comparable effects [229], while others claim that it is not [230].
60
π
π
Figure 20: ETC gauge boson exchange across a fermion loop. This is an α term insertion
leading to PNGB mass.
techniquark condensate, which instructs us to replace the techniquark bilinears in eq.(3.84)
with the corresponding chiral fields (a discussion of the chiral dynamics of ETC may be
found in [168], [231] ). Specifically one makes the replacement, motivated from the NJL
approximation or QCD:
Σḃa ≡ exp(iπ c T̃ c /FT )ḃa
Q̄aR QḃL → cNT C Λ3T C Σḃa
(3.85)
where dotted indices are for SU(2)L and undotted are for SU(2)R and the T̃ a are generators of the Technicolor subgroup of GET C . This leads, through the diagram of Fig.(20),
to a PNGB interaction term in the effective Lagrangian:
∼
αab c2 NT2 C Λ6T C
Tr(ΣT a Σ† T b )
Λ2ET C
(3.86)
Expanding eq.(3.86) in the π a T˜a we see that the induced technipion mass terms are:
∼−
αab c2 NT2 C Λ6T C
Tr([π c T̃ c , T a ][T b , π d T̃ d ]
Λ2ET C FT2
(3.87)
Those technipions associated with the Technicolor generators, T̃ a that commute with
ETC, i.e., for which:
[T˜a , T b ] = 0 ,
(3.88)
will have vanishing mass contributions from ETC. Technipions associated with the noncommuting generators, on the other hand, will generally receive nonzero ETC contributions to their masses of order ∼ NT C Λ2T C /ΛET C .
This situation is akin to that of QCD: the T a are analogues of the electric charge
operator, and T˜a are the generators of nuclear isospin. The neutral pion of QCD is
associated with a generator, I3 that commutes with the electric charge, Q = I3 + Y /2,
hence the neutral pion receives no contribution to its mass from electromagnetism. The
charged pions, associated with generators I1 ± iI2 , which do not commute with Q, receive
nonzero electromagnetic contributions to their masses [167].
61
Among the PNGB’s that arise in the Minimal and the Farhi-Susskind models, the
techniaxions can receive masses, at best, only from ETC. In classic ETC the prospects
for mass generation have been treated in detail in [99]; a typical result is:
2
Maxion
≈
1
(a few GeV)2 .
NT C
(3.89)
We will see subsequently that electrically charged PNGB’s receive masses from ETC
that are small compared to their electromagnetic contributions. Similarly the colored
states, P3 and P8 of the Farhi-Susskind model receive larger masses from QCD than from
conventional ETC.
We will discuss below, however, an additional dynamical ingredient of modern ETC
theories known as “Walking.” In Walking Technicolor, any effective operator involving
the techniquarks can be significantly amplified by the effects of Technicolor. This tends
to enhance the PNGB masses by factors of order ΛET C /ΛT C ∼ 103 . This in principle
remedies the problem of elevating the masses of the techniaxions to evade experimental
bounds.
3.1.4
The β terms: Quark and Lepton Masses
The terms in eq.(3.84), with coefficients βab , will generally give masses and mixing
D angles
E
to the ordinary quarks and leptons. Technicolor condenses the technifermions, Q̄Q ∼
NT C Λ3T C at the TC scale. The natural scale for the ETC-induced quark and lepton masses
is then of order:
NT C Λ3T C
(3.90)
mq,ℓ ∼ β
Λ2ET C
This would seem to liberally allow a fit of ordinary quark or lepton masses to a scale of
order ∼ ΛT C . For example, with mq ∼ mcharm ∼ 1 GeV and ΛT C ∼ 100 GeV, we have for
3
β ∼ 1, the result, ΛET C <
∼ (few)×10 GeV. Thus, the ordinary quark and lepton masses
place an upper bound on the effective value of ΛET C . The pattern and scale of masses
and mixing angles is sensitive to the pattern of breaking of ETC. It is not unreasonable
to expect to obtain hierarchical mass patterns between generations.
Higher-order effects of the β terms also yield observable consequences. The key example for ETC model building is Rb . In a classic ETC model, mt is generated by the
exchange of an electroweak-singlet ETC gauge boson of mass MET C ∼ gET C ΛET C coupling with strength gET C . At energies below MET C , ETC gauge boson exchange may be
approximated by local four-fermion operators, and mt arises from an operator coupling
the left- and right-handed currents:
−
2
gET
C
i µ iw
w
ψ̄
γ
T
Ū
γ
t
+ h.c.
µ
R
L
L
R
2
MET
C
(3.91)
where T = (U, D) are technifermions, and i and k are weak and Technicolor indices.
62
-b
L
T
ETC
Z
T
b
L
Figure 21:
Direct correction to the Zbb̄ vertex from exchange of the ETC gauge boson that
gives rise to the top quark mass. Technifermions are denoted by ‘T’.
Assuming there is only a single weak doublet of technifermions:
mt =
2
gET
Λ3T C
C
h
ŪUi
≈
.
2
MET
Λ2ET C
C
(3.92)
The ETC boson which produces mt through eq.(3.91) also necessarily induces the
related operator [232]:
−
2
gET
C
i µ iw
i µ iw
+ h.c.
ψ̄
γ
T
T̄
γ
ψ
L
L
L
L
2
MET
C
(3.93)
This affects the Zbb vertex as shown in Figure 21, which indicates the ETC boson being
exchanged between the left-handed fermion currents (with T ≡ DL since the ETC boson
is a weak singlet). This process alters the Z-boson’s tree-level coupling to left-handed
bottom quarks gL = (− 21 + 13 sin2 θW )(e/ sin θW cos θW ) by [232]:
δgLET C = −
ξ 2 ΛT C
e
1 mt
e
(I3 ) = ξ 2
·
2
2 ΛET C sin θ cos θ
4 ΛT C sin θ cos θ
(3.94)
where the right-most expression follows from applying eq. (3.92). Here ξ is a mixing
parameter angle between the W and Z and the ETC bosons of the theory.
Such shifts in gL alter the ratio of Z boson decay widths
Rb ≡
Γ(Z → bb̄)
.
Γ(Z → hadrons)
(3.95)
This is a convenient observable to work with since oblique [233] [114] [115] [234] [116]
[235] and QCD corrections largely cancel in the ratio, making it sensitive to the direct
vertex correction. One finds:
δRb
mt
≈ −5.1%ξ 2
.
Rb
175GeV
(3.96)
Such a large shift in Rb would be readily detectable in current electroweak data. In fact,
the experimental value of Rb = 0.2179 ± 0.0012 lies close enough to the Standard Model
63
prediction [236] [237] (.2158) that a 5% reduction in Rb is excluded at better than the
10σ level. Alternatively we require a suppression of ξ <
∼ 0.2 at the 2σ level.
It should be noted that there are discrepancies in the LEP data with the Standard
Model, in particular in the Forward-Backward asymmetry of the b-quark [23], though,
from the present discussion, it is unclear that they can be reconciled with a Technicolorlike dynamics. ETC models in which the ETC and weak gauge groups commute are
evidently ruled out. However, models which can separate third generation and first and
second generation dynamics, such as non-commuting ETC (Section 3.3.2), can produce
acceptable Rb corrections. Low-Scale Technicolor (Section 3.5) and Topcolor or TopcolorAssisted Technicolor models (Section 4) generally also predict values of Rb in accord with
the data [238, 239].
3.1.5
The γ terms: Flavor-Changing Neutral Currents
Severe constraints from flavor-changing neutral currents turn out to exclude the possibility
of generating large fermion masses in classic ETC models. The interactions of the third
class of terms in eq.(3.84), associated with coefficients γab , cause these problems. Because
ETC must couple differently to fermions of identical Standard Model gauge charges (e.g.
e, µ, τ ) in order to provide the observed range of fermion masses, flavor-changing neutral
current interactions amongst quarks and leptons generally result. Processes like:
(s̄γ 5 d) (s̄γ 5 d) (µ̄γ 5 e) (ēγ 5 e)
+
+ ...
Λ2ET C
Λ2ET C
(3.97)
are generally induced, and these give new contributions to experimentally well-constrained
quantities [240]. For example, the first term causes ∆S = 2 flavor-changing neutral current
interactions which give a contribution to the experimentally well-measured KL KS mass
difference. The matrix element of the operator between K 0 and K̄ 0 yields [173]:
δm2 /m2K ∼ γ
fK2 m2K < −14
10
Λ2ET C ∼
(3.98)
where we might expect γ ∼ sin2 θc ∼ 10−2 in any realistic model. Hence, we obtain:
3
ΛET C >
∼ 10 TeV
(3.99)
The second term of eq.(3.97) induces the lepton-flavor-changing process µ → eēe, ; eγ;
1
from this, we estimate a somewhat weaker bound, ΛET C >
∼ 10 TeV [173].
Applying the bound eq.(3.98) to our expression for ETC-generated fermion masses
(3.90), and assuming α ∼ β ∼ γ, yields an upper bound on the masses of ordinary quarks
and leptons that a generic ETC model can produce (we use ΛT C <
∼ 1 TeV, β <
∼ 10,
NT C <
10):
∼
Λ3T C <
<
100 MeV
(3.100)
mq,ℓ ∼ NT C 2
ΛET C ∼
64
Hence, producing the mass of the charm quark is already problematic for a classic ETC
model.
A remedy for production of the charm and, marginally, the b-quark masses is “Walking
Technicolor,” as we discuss below (Section 3.4). We remark that another way to suppress
the γab effects is to construct theories based upon SO(N) groups containing ETC and
Technicolor, with fermions in the real N representation. These can avoid flavor-changing
neutral current-like interactions in the tree approximation [241] [242], a consequence of the
representations that can be generated by products of the fundamental N representation
at that order. An operator such as eq.(3.97) will be generated in loops, and one still
obtains a limit of order 1 GeV on the allowed fermion masses. Ultimately addressing
the heavy top quark requires new dynamical mechanisms such as non-commuting ETC
(Section 3.3.2) or “Topcolor” (section 4.2).
These γ-term problems, it should be noted, have analogues in the Minimal Supersymmetric Standard Model, under the rubric of “The SUSY Flavor Problem.” The mass
matrices of the squarks and sleptons may in general require diagonalization by different
flavor rotations than those of the quarks and leptons. This leads to similar unwanted
FCNC and lepton number violating processes. The typical constraints upon mass differ−3
2
ences between first and second generations sfermions are δm2 /M 2 <
∼ 10 , where M is
the center-of-mass of a multiplet. The need to keep M ∼ 1 TeV in SUSY places constraints on the models that may be hard to realize. It is intriguing that both SUSY
models, and Technicolor models focus our attention upon nagging flavor physics issues at
the multi-TeV scale.
3.2
Oblique Radiative Corrections
Precision electroweak measurements have matured considerably throughout the LEP and
Tevatron era, with the copious data on the Z–pole, the discovery of the top quark, and
precise W mass measurements. Taken together with low energy data such as atomic parity violation and neutrino scattering, these experiments test the Standard Model at the
level of multi-loop radiative corrections. The resulting constraints are often troublesome
for classic Extended Technicolor. Yet these constraints can generally be avoided or ameliorated in the presence of Walking ETC (Section 3.4) or new strong top quark dynamics
(section 4).
The conventional parlance for discussing the “oblique” radiative corrections to the
Standard Model is to use the S, T , and U parameters. For definitions of these parameters,
we follow the formalism of Peskin and Takeuchi [234] [235]. The S, T , and U parameters
are defined and computed for a fermion bubble in Appendix A (see eqs.(A.31)).
Precision electroweak observables are linear functions of S and T . Thus, each measurement picks out an allowed band in the S − T plane, and measurement of several
processes restricts one to a bounded region in this plane, the S − T error ellipse. By
convention, offsets are added to S and T so the point S = T = 0 corresponds to the
prediction of the Standard Model for a set of fixed “reference values” of the top quark
65
Figure 22: Fit of the precision electroweak data to the Standard Model plus the S, T parameters described in the text (from Peskin and Wells [243]). The ellipse is the fit; the warped grid
is the Standard Model predictions. The figure uses only the values of the three best electroweak
observables, MW , the value of the weak mixing angle which appears in Z 0 decay asymmetries,
and Γℓ (the leptonic width of the Z 0 . All data is combined in a plot of M. Swartz at the 1999
Lepton-Photon Conference [244]).
and Higgs boson masses. A recent review by Peskin and Wells [243] addresses general
issues relevant to theories beyond the Standard Model, including Technicolor, new strong
top quark dynamics, and extra-dimensions. Peskin and Wells take the reference values
to be mt = 174.3 GeV and mH = 100 GeV. In Fig. 22, we present the 68% confidence
contour (1.51 σ) for a current S − T fit from Peskin and Wells [243]. The overlay of the
perturbative Standard Model prediction with mt = 174.3±5.1 GeV and mH running from
100 GeV to 1000 GeV, with mH = 200, 300, 500 GeV is indicated with vertical bands in
the warped grid in the Figure.
While Figure 22 favors a low mass Higgs boson, Mh <
∼ 200 GeV, we emphasize that
this does not imply that a low mass Higgs boson is compelled to exist. The warped grid is
only a perturbative Higgs boson mass plot. If the Higgs boson is indeed heavy, then there
must be associated new strong dynamics, and there will occur additional new physics
contributing to S and T . One cannot, therefore, rely upon the perturbative mass plot
overlay for large Higgs boson masses. An example of a model in which the Higgs boson
mass is ∼ 1 TeV, but is naturally consistent with the S-T error ellipse is the Top Quark
Seesaw theory [245] [246], described in Section 4.4.
In general, however, one may still use the S − T error ellipse itself to assess the
agreement of a given theory with the data. The contributions to S from classic Technicolor
66
are large and positive. As seen in Appendix A, the contribution of a single weak isodoublet
with NT Technicolors is, in a QCD-like dynamics:
∆S ≈
NT
6π
(3.101)
since the techniquarks are massless and have degenerate constituent quark masses. Thus,
with NT = 4 the minimal model with a single weak isodoublet will give a correction to
the S = T = 0 reference point of ∆S = +0.2 and corresponds roughly to the case of a ∼ 1
TeV Higgs boson mass with ∆T ∼ −0.3. The Farhi-Susskind model has a full generation
of 3 colors each of techniquarks and a “fourth color” of leptons, and thus yields ∆S ∼ 0.8
with ∆T ∼ −0.3, and would appear to be ruled out. In the minimal model we might
be able to pull ourselves from the point (0.2, −0.3) back into the experimentally favored
ellipse by adding a positive contribution to ∆T ∼ 0.4 from additional new physics. To
return to the ellipse in a Farhi–Susskind–like model would require large negative sources
of S as well.
There are possible physical sources of the additional contributions to S and T which
are required to make Technicolor consistent with the measured oblique parameters. First,
strong dynamics models that address the flavor problem must usually include new higherdimension operators from physics at higher scales that will affect the oblique parameters;
a general analysis has been made by a number of authors [114], [247], [248] [249]. The
possible existence of such operators obviates any claim that new dynamical models are
ruled out by the S-T constraints. Second, the presence of precocious (∼ TeV scale) extra
dimensions can play an important role, as recently discussed by Hall and Kolda, [250].
Third, Appelquist and Sannino have shown that the ordering pattern for vector-axial
hadronic states in SU(N) vector-like gauge theories close to a conformal transition need
not be the same as predicted in QCD-like theories. As a consequence, the S-parameter
in near-critical technicolor theories can be greatly reduced relative to QCD-like theories
[251]. Fourth, specific particle content that can produce overall modest contributions to
S and T has been identified. We will mention some of the proposed models shortly.
The contribution to the S parameter from a scalar field with exotic weak charge was
considered in [252], [253]. Both the ordinary weak charge under SU(2)L and the charge
under custodial20 SU(2)R play a role. Building on this, Peskin and Wells [243] label
a field’s weak isospin jL and assign it the appropriate quantum number jR under the
custodial SU(2); they further define: j+ = jL + jR and j− = |jL − jR |. If one assumes a
particle with J = j− has the lowest mass m, and all other particles in the multiplet have
a common mass M, then ∆S is roughly:
∆S ∼
κ′
M2
log 2
3π
m
20
(3.102)
The custodial SU (2) is essentially the global SU (2)R acting on right-handed fermions, and broken
by U (1)Y and the Higgs-Yukawa couplings. It insures that when the electroweak symmetries are broken,
there remains an approximate SU (2) global symmetry. This global SU (2) symmetry implies, in a chiral
Lagrangian, that fπ+ = fπ0 .
67
where:
1
(j+ + 1)
κ′ = −
12
(j− + 1)
!2
− 1 j− (j− + 1)(2j− + 1) .
(3.103)
If the particles with the smallest j− are the lightest, then the multiplet can yield negative
contributions to S. Large values of |∆S| can be obtained from multiplets with large weak
isospin.
Several avenues for producing small or even negative contributions to S have been discussed in the literature. Models with Majorana neutrino condensates naturally produce
negative contributions to S [254], which can in principle compensate large positive contributions. Luty and Sundrum [255] constructed models in which the PNGB’s give negative
contributions to S. To obtain ∆S ≃ −0.1 from this source, one needs technifermions
with jL = 2 and PNGB’s with masses of order 200 GeV. More generally, other models
may include arbitrary numbers of vectorlike pairs of electroweak charged particles [256]
(e.g., both a left-handed and right-handed gauged doublet with j− = 0 and a Dirac mass
coming from other external dynamics) with no cost in S. The Top–Seesaw model (Section
4.4) contains additional vectorlike fermions, allowing both S and T to be completely consistent with present data. Likewise, the Kaluzsa-Klein recurrences of ordinary quarks and
leptons in extra-dimensional models are vectorlike pairs, and are largely unconstrained
by S. These examples are discussed in Section 4.6.
It is straightforward to engineer large positive contributions to ∆T [257]. Particles
with masses much larger than ∼ 1 TeV can contribute to ∆T if their masses have an
up-down flavor asymmetry. The contribution is of order:
|∆T | ∼
m2U − m2D
.
m2U + m2D
(3.104)
|mU − mD | can typically be at of order 100 GeV. One class of theories that generate this
kind of contribution to ∆T are the Technicolor models in which ETC gauge interactions
are replaced by exchange of weak-doublet techni-signlet scalars [258, 259, 260] (see Section
3.5).
The effects of new Z ′ bosons on oblique corrections were studied by many authors
(see [243] and references therein). Such particles occur in new strong dynamical models,
e.g. as in Topcolor Models. Peskin and Wells investigate the region of parameters for any
Z 0′ model in which the shifts due to the Z 0′ compensate those of a heavy Higgs boson.
Models with extra dimensions can also exhibit “compensation” between effects of new
vector bosons and heavy Higgs bosons. Rizzo and Wells [261] have shown that the 95%
C.L. bound on the Higgs boson mass could reach ∼ 300 to ∼ 500 GeV for Kaluzsa-Klein
masses MKK in the range ∼ 3 to ∼ 5 TeV. The coupling of the new sector is typically
strong (see Section 4). In this vein, Chankowski et al. [262] has argued that two-Higgsdoublet models (which can be viewed as effective Lagrangians for composite Higgs models)
can be made consistent with the electroweak fits for a Higgs boson mass of ∼ 500 GeV.
Technicolor models can also be made consistent with the precision electroweak fits through
these kinds of effects.
68
A modification of the dynamics of the Technicolor gauge theory itself can produce
smaller oblique corrections than naively estimated above. In Walking Technicolor models
(Section 3.4), where the Technicolor gauge coupling remains strong between the TC and
ETC scales21 , the theory is intrinsically non-QCD-like and conventional chiral lagrangian,
or chiral constituent techniquark estimates are expected to fail. Models of this type would
not have a visible Higgs boson and may have no new particles below the first techi-vector
mesons, or open techniquark/anti-techniquark threshold: it is difficult to anticipate the
spectroscopy in a reliable way since QCD no longer functions as an an analogue computer.
There is calculational evidence that dangerous contributions to ∆S are suppressed by
walking. For example, models have been proposed in which the Technicolor enhancements
to ∆S are of order 0.1 [263], [264], [211]. It is possible to construct a Technicolor model
that is consistent with the electroweak data by including enough weak isospin breaking
to provide a small positive correction to ∆T . Such a model might have, for example
∆S ∼ 0.2, ∆T ∼ 0.2.
To summarize, the potential conflicts between models of new strong dynamics and
the tight experimental constraints on the size of the oblique electroweak parameters can
be averted in several ways. Whether it is particle content, the details of Technicolor
dynamics, or the behavior of higher-dimension operators from higher-energy physics which
comes to the rescue, it is clear that strong dynamics models are not a priori ruled out.
The challenge for model-builders is to produce concrete models that are consistent with
the constraints.
3.3
Some Explicit ETC Models
Having introduced the broad theoretical outlines and main experimental challenges of
ETC, we now present some explicit models. Our summary emphasizes strategies employed
in model-building and does not represent a complete list of all models. This discussion
will anticipate some of the ideas of Walking ETC which will be more fully treated in
Section 3.4.
3.3.1
Techni-GIM
An essential problem in ETC is to find a way to suppress the flavor changing neutral
processes arising from the dangerous γ terms. We seek a mechanism to naturally make
γ << β. One possibility is to include a GIM mechanism in the ETC sector (TC-GIM).
Here, the aim is to construct a detailed model in which the β coefficients are of order unity
(to make the fermion masses as large as possible) and a GIM cancellation causes γ << β,
protecting against the unwanted ∆S = 2 flavor-changing neutral current interactions.
With such a GIM mechanism we might hope to suppress the flavor-changing current21
This could be engineered, for instance, by including many vectorial techniquarks which would affect
the β function without adding to S.
69
+ 0
NLC e e -> γ P
LHC pp -> TT
LHC pp -> θθ
4 TeV
Tevatron pp -> TT
LEP2 e+e- -> TT
Tevatron pp -> TT
+ 0
LEP e e -> γ P
fS-1 [GeV]
500
100
Figure 23: The potential for probing the scale of TC-GIM interactions at present and future colliders (the high-scale models). The most important reaction(s) for each collider is(are)
indicated inside the bars (from W. Skiba [174]; T = P3 , θ = P8 in Skiba’s notation).
current interactions of eq.(3.97) by a further factor of ∼ Λ2T C /Λ2ET C . The lower bound on
ΛET C would then be weakened to ΛET C >
∼ 10 TeV, allowing ETC to generate a generic
quark mass as large as 10 GeV. This would accomodate the c and b-quark, but would
remain insufficient to produce a top quark mass of order ΛT C .
A candidate mechanism for achieving a GIM mechanism is to introduce separate ETC
gauge groups for each weak hypercharge fermion species [221] [265] [266] [267]. Hence
one has a separate ETC gauge group for each the left-handed electroweak doublets, righthanded up-type singlets, and down-type singlets. Weak SU(2)L commutes with all of
these gauge groups. The ETC groups are broken in such a way that approximate global
symmetries hold, broken only by terms necessary to generate quark and lepton masses,
and by gauge symmetries. This implies that the (ψψ)2 operators must be approximately
invariant under these global symmetries, and thus, in general no ∆S = 2 operators can
occur.
This is akin to the Glashow-Weinberg discrete flavor symmetry [268] that was introduced long ago into the multi-Higgs boson extensions of the Standard Model. Indeed
the TC-GIM models essentially have Glashow-Weinberg-natural multi-Higgs models as
low energy effective Lagrangians. These models are fairly complicated and suffer from
the usual limitations on the fermion masses that can be generated in even a Walking
TC scheme, e.g., the top quark mass requires an ETC boson mass of order ΛT C within
this framework. Nonetheless, models which achieve the natural GIM suppression may
be possible. The third generation is then clearly special, and part of a sort of inverted
hierarchy.
70
A noticeable feature of these models is the presence of leptoquark PNGB’s, akin to
those in the Farhi-Susskind model. Leptoquarks, in most models, typically decay into a
quark and a lepton of the same generation. In the TC-GIM models, the leptoquarks can
carry the lepton and quark numbers of any generation. This means, for example, that
existing limits for leptoquarks decaying only to first-generation or 2nd-generation fermions
can be applied directly to TC-GIM models. For example, there are limits on “down-type”
¯ and on leptoquarks decaying
first-generation leptoquarks [269] (those decaying to e + d)
to 2nd generation fermions[270]. These essentially rule out versions of TC-GIM models
in which the dynamical scale is relatively low. The color-octet PNGBs also expected in
these models may be detectable at the LHC (see Section 2.3.3).
Skiba has given a detailed survey of signatures of TC-GIM models [174], as summarized
in Figure 23. One of most promising TC-GIM signatures at an NLC is pair production of
leptoquarks: e+ e− → γ, Z → P P̄ , where P → ℓq̄. Each leptoquark decays into a hadronic
jet and an isolated lepton. The events would have two opposite-sign leptons, two hadronic
jets and no missing energy, a signature easily disentangled from the backgrounds. It is
expected that leptoquarks can be discovered provided that their masses are at least a few
GeV below the NLC’s kinematic limit.
3.3.2
Non-Commuting ETC Models
Another approach to ETC model building, which addresses the deviation in Rb caused
by the β terms discussed in 3.1.4, is to drop the assumption that the ETC and weak
gauge groups commute [271]. A heavy top quark must then receive its mass from ETC
dynamics at low energy scales; if the ETC bosons responsible for mt carry weak charges,
we can embed the weak group SU(2)heavy under which (t, b)L is a doublet into an ETC
group, or more precisely, a subgroup of ETC which is broken at low scales. To retain
small masses, the light quarks and leptons should not be charged under this low-scale
ETC group; hence the weak SU(2)light group for the light quarks and leptons must be
distinct from SU(2)heavy . The resulting symmetry-breaking pattern is:
ET C
T C × SU(2)heavy
TC
TC
×
×
×
×
SU(2)light × U(1)′ ⇒ (f )
SU(2)light × U(1)Y ⇒ (u)
SU(2)W × U(1)Y ⇒ (v0 )
U(1)EM ,
where f , u, and v0 = 246 GeV are the VEV’s of the order parameters for the three different symmetry breakings. Note that, since we are interested in the physics associated
with top-quark mass generation, only tL , bL and tR must transform non-trivially under
ET C. However, to ensure anomaly cancelation we take both (t, b)L and (ντ , τ ) to be doublets under SU(2)heavy and singlets under SU(2)light , while all other left-handed ordinary
fermions have the opposite SU(2) assignment.
The two simplest possibilities for the SU(2)heavy ×SU(2)light transformation properties
71
of the order parameters that mix and break the extended electroweak gauge groups are:
hϕi ∼ (2, 1)1/2 ,
hσi ∼ (2, 2)0 ,
“heavy case” ,
(3.105)
hϕi ∼ (1, 2)1/2 ,
hσi ∼ (2, 2)0 ,
“light case” .
(3.106)
The order parameter hϕi breaks SU(2)L while hσi mixes SU(2)heavy with SU(2)light . In
the “heavy” case [271], the technifermion condensation that provides mass for the third
generation of quarks and leptons is also responsible for the bulk of electroweak symmetry
breaking (as measured by the contribution to the W and Z masses). The “light” case is
just the opposite: the physics that provides mass for the third generation does not provide
the bulk of electroweak symmetry breaking. While this light case is counter-intuitive (after
all, the third generation is the heaviest!), it may in fact provide a resolution to the issue
of how large isospin breaking can exist in the fermion mass spectrum (and, hence, the
technifermion spectrum) without leaking into the W and Z masses. This is essentially
what happens in multiscale models [272, 273] and in Topcolor Assisted Technicolor [274].
Such hierarchies of technifermion masses are also useful for reducing the predicted value
of S in Technicolor models[211].
Below the scale of ETC breaking, ETC boson exchange yields four-fermion operators
among third-generation quarks and technifermions [271]:
L4f ∼ −
1
2
fET
C
1
ξ ψ̄L γ UL + t̄R γ µ TR
ξ
µ
!
1
ξ ŪLγµ ψL + T̄R γµ tR
ξ
!
,
(3.107)
where ξ is a model-dependent coefficient. When the techniquark condensate forms in
the LR cross-terms22 in these operators we obtain the top quark mass of order eq.(3.94)
[172, 173]. In the heavy case the technifermions responsible for giving rise to the thirdgeneration masses also provide the bulk of the W and Z masses, and we expect FT ≈ 125
GeV (which, for mt ≈ 175 GeV, implies fET C ≈ 375 GeV) [277]. Even in the light case
there must also be some SU(2)heavy breaking VEV, in order to give the top quark a mass.
The spectrum of non-commuting ETC models includes an extra set of W and Z bosons
which affect weak-interaction physics at accessible energies. Mixing between the two
sets of weak gauge bosons alters the Zf f couplings. In addition, the one-loop diagram
involving exchange of the top-mass-generating boson shifts the coupling of bL to the Z
as in eq.(3.94); here, the techniquarks in the loop are up-type, reversing the sign of the
final answer. The two physical effects shift Rb in opposite directions by similar amounts,
leaving Rb consistent with experiment [277]. More generally, a recent [278] fit of precision
electroweak data to the predictions of the NCETC models yields limits on the masses
and couplings of the heavy W and Z bosons. Essentially, if the only modifications to
the electroweak observables come from the extra electroweak and ETC bosons of noncommuting ETC, then the heavy W and Z bosons must weigh at least 2.4 TeV in the
22
The LR-interactions become enhanced in strong-ETC models, in which the ETC coupling is finetuned to be close to the critical value necessary for the ETC interactions to produce chiral symmetry
breaking. Physically, this is due to the presence of a composite scalar [275, 276] which is light compared
to ΛT C and communicates electroweak symmetry breaking to the top quark.
72
“light” case and at least 3.3 TeV in the “heavy” case. More general tests of the idea of an
extended weak gauge sector at current and future colliders are discussed in Section 3.6.
3.3.3
Tumbling and Triggering
Any successful ETC scheme must provide a dynamical explanation for the breaking of the
ETC group to its TC subgroup, as well as for the breaking of the electroweak symmetry.
Some interesting ideas fall under the rubrics of “tumbling” and “triggering”.
As mentioned earlier, some ETC models seek to explain the flavor hierarchy of fermion
masses by tying it to the sequential breaking of a large ETC gauge group into a succession
of subgroups containing, ultimately, the Technicolor interactions. This is called “Tumbling ” [219, 220, 222, 223, 224, 225]. The breaking of the full ETC group down to the
Technicolor group proceeds in n steps GET C → G1 → G2 → ...Gn−1 → GT C . At each step,
the subgroup’s gauge coupling will evolve and become strong, permitting new condensates
to form, which further break the theory into the subsequent subgroup. Eventually a low
energy theory described by the Technicolor and Standard Model gauge groups is achieved.
For discussions of the detailed dynamics of Tumbling, we refer the reader to [279] and
more recently [280, 281]. Ref. [282] analyzes all possible tumbling schemes based upon
SU(N).
While the tumbling mechanism mentioned above is based on the idea of a group’s
breaking itself, some interesting models involve the alternative idea of triggering, in which
one gauge theory produces breaking in another. Appelquist et al. [264], [283], [211] explore
Technicolor models featuring a non-trivial infrared fixed point of the Technicolor gauge
coupling, in which QCD is used to play a triggering role for EWSB. These models, thus,
predict a relation between the electroweak scale and the QCD confinement scale. They
also predict exotic leptoquarks with masses of ∼ 200 GeV. Since the oblique corrections
are a perennial problem in ETC theories, we mention that isospin splitting and techniquark/technilepton mass splitting in one-family models of this kind have been found
reduce the predicted value of the oblique parameter ∆S to acceptable levels, without
giving large contributions to ∆T [211].
3.3.4
Grand Unification
While the dynamics of pure TC models is tied strongly to the electroweak scale, ETC
models inevitably involve physics at higher energy scales. We have already seen that
avoiding FCNC in the neutral-K system requires pushing the symmetry-breaking scale of
classic ETC models up to at least 1000 TeV. As a result, it is natural for one to ask whether
ETC theories can be embedded into a grand-unified gauge theory at still higher energies. A
proof-of-principle for the possibility of ETC unification was mentioned earlier: the FarhiSusskind extension to an SU(56) under which all quarks, leptons, and technifermions
transform within one multiplet. We mention some additional unified models in which
some of the key theoretical and phenomenological challenges of ETC are also addressed.
73
Giudice and Raby [241] [242] have considered a unified model that starts with an
ETC group of SO(4) with a full standard model family of fermions in the 4 vector representation. This sector of the model is, in spirit, similar to the Farhi-Susskind model
with SO(4) replacing SU(NT C ) and containing the ETC gauge interactions. At the ETC
breaking scale λET C ∼ 103 TeV the group SO(4) breaks to SO(3) which is the true gauge
Technicolor group. As mentioned above, theories based upon SO(N) groups containing
ETC and Technicolor, with fermions in the N representation, can avoid flavor-changing
neutral current-like interactions (γ terms) in the tree approximation. This leads to gauge
bosons with masses of order 5 TeV for third generation,and ∼ 103 TeV for first and second, that mediate the ETC interactions. The model exhibits a fixed point in the strong
coupling of the Technicolor group, leading the coupling to run slowly (see Section 3.4 on
Walking TC) above the TC scale; a key side-effect of walking, the non-calculability of the
S-parameter, is discussed. The model also invokes four fermion effective operators, and
shares features with Top Condensation models (see Section 4).
The main idea is to imbed this ETC theory into a large gauge group such as SO(18)
and attempt to achieve a grand unified theory. The breaking chains required for this GUT,
curiously, can be generated by Majorana condensates of neutrinos (see e.g. [284]), thus
incorporating the Gell-Mann–Ramond–Slansky seesaw mechanism [285] more intimately
into the dynamics. Note that neutrino condensates have been invoked elsewhere to break
electroweak symmetries, or in concert with fourth generation Farhi-Susskind Technicolor,
[286, 284, 287, 288, 289]. The model of Giudice and Raby leaves open the question of
breaking the ETC gauge group at ΛET C . The exact nature of the Technicolor fixed point
(walking) with the additional four-fermion operators is also left unclear.
King and Mannan [290], [57] give a schematic model based upon the three sequential breakings SO(10) → SO(9) → SO(8) → SO(8), where the ETC bosons for nth
generation of fermions is produced at the nth breaking scale. This attempts to explain
the generational hierarchy of masses and mixings of the quarks and leptons. The model
incorporates grand unification and neutrino masses, but does not address the dynamics
behind the sequential breakings as in the Giudice–Raby model.
3.4
3.4.1
Walking Technicolor
Schematic Walking
Extended Technicolor (ETC) has difficulties in producing the observed heavy quark and
lepton masses. Even the charm quark is heavy enough to cause problems in building
models with sufficiently suppressed flavor-changing neutral-current interactions. In the
preceeding section, we surveyed various attempts to deal with this difficulty from a structural point of view. We turn now to an intriguing dynamical possibility which emerges
from a closer examination of the full TC strong–dynamics.
Let us consider the TC radiative corrections to the operators from ETC that generate
the quark and lepton masses. These operators appear as the “β” contact terms of eq.(3.83,
74
3.84) at the scales µ <
∼ ΛET C . Since ΛET C >> ΛT C , these operators are subject to
renormalization effects by TC,
D
E
Q̄QET C = exp
Z
ΛET C
ΛT C
!
d ln(µ) γm (α(µ))
D
Q̄QT C
E
(3.108)
where γm is the operator’s anomalous dimension.
If TC is QCD-like, then the TC coupling constant α(µ) is asymptotically free, and falls
logarithmically as α(µ) ∝ 1/ ln(µ) above the scale ΛT C . With the anomalous dimension
γm ∝ α(µ) we see that the radiative correction is proportional to exp[γm ln(ln(µ)] ∼
(ln(ΛET C /ΛT C ))γm . Hence the radiative corrections are power-logarithmic factors, similar
to the behavior of QCD radiative corrections to the nonleptonic weak interactions in the
Standard Model.
If, however, α(µ) is approximately constant, i.e., if the TC theory exists approximately
at a “conformal fixed point,” α(µ) = α⋆ 6= 0, where β(α⋆ ) = 0, then the radiative correc⋆
tion is converted into a power law, proportional to exp[γm (α⋆ ) ln(µ)] ∼ (ΛET C /ΛT C )γm (α ) ,
which is a substantially larger renormalization effect.
Such a theory is not QCD-like, but is an a priori possible behavior of a Yang-Mills
gauge theory. This behavior can provide significant amplification to both the α and β
terms of ETC which involve the technifermion bilinears, but does not alter the dangerous
γ terms, which involve only the Standard Model fermions. This holds out the possibility
of enhancing ETC-generated fermion and PNGB masses without increasing the rate of
neutral flavor-changing processes. A TC theory with an approximately constant coupling
α(µ) = α⋆ in the range ΛT C <
∼ ΛET C is said to be “Walking Technicolor” (WTC), an
∼µ<
idea which was first proposed by Holdom [291]; some early implications were discussed in
refs. [292], [293], [294], [295].
3.4.2
Schwinger-Dyson Analysis
As demonstrated by Yamawaki, Bando and Matumuto, [292], and further elaborated by
Appelquist, Karabali and Wijewardhana [294], for Walking TC it suffices to have an
approximate fixed point, β(α⋆) << 1 with α⋆ ∼ αc near ΛT C , for the relevant scales
⋆
ΛT C <
∼ µ <
∼ ΛET C . Here α is near the the critical coupling αc for the formation of
chiral condensates. An analysis of the Schwinger-Dyson equations for the mass-gap of the
theory [292], [73] then shows that the radiative correction of eq.(3.108) can enhance the
techniquark bilinear operator by a factor of order ΛET C /ΛT C . Essentially, the solution of
the Schwinger-Dyson equation with fixed coupling gives large anomalous dimensions ≃ 1
near the critical coupling, α∗ ∼ αc . This result obtains in the chiral broken phase as well
as the symmetric phase, [292] near critical coupling and hence resolves the difficulties in
TC (which is supposed to be in the chiral broken phase near αc ). We recapitulate the
argument below.
The Euclideanized Schwinger-Dyson equation for the self-energy of a fermion in Lan75
dau gauge is given by [296]:
2
Σ(p ) = 3C2 (R)
Z
d4 k α((k − p)2 )
Σ(k 2 )
(2π)4 (k − p)2 Z(k 2 )k 2 + Σ2 (k 2 )
(3.109)
Typically we approximate Z(k 2 ) ≈ 1, and linearize the equation by neglecting the Σ2 (k 2 )
denominator term. We assume α(µ) ≈ αc is slowly varying. Two solutions are then found:
µ2
Σ(p2 ) = Σ(µ) 2
p
!b±
b± = 21 (1 ± (1 − α(µ)/αc)1/2 )
(3.110)
where the critical coupling constant is αc = π/3C2 (R), and C2 (R) is the quadratic Casimir
of the complex technifermion representation R (recall C2 = (N 2 − 1)/2N for the fundamental representation).
The normal perturbative anomalous dimension of the QQ operator is
γm = 1 − (1 − α(µ)/αc )1/2 ∼
3C2 (R)α(µ)
2π
(3.111)
Hence, the solution with b− corresponds to the running of a normal mass term of nondynamical origin. The solution with b+ ∼ 1 corresponds to the high momentum tail of a
dynamically generated mass having the softer ∼ 1/p2 behavior at high energies 23 . Note
that, at the critical coupling α(µ) = αc , the two solutions coincide, which is believed to
be a generic phenomenon, [73], [298], [299]. Moreover, if we suppose that α(µ) = α∗ > αc
for 0 ≤ µ ≤ Λ∗ , then we find that a true dynamical symmetry breaking solution exists
where:
s
!
α∗
∗
−1
(3.112)
Σ(0) ∼ Λ exp −π/
αc
In the energy range ΛT C ≤ µ ≤ ΛET C , the large value of α(µ) ≈ α∗ corresponds to
an anomalous dimension of order 1, making the radiative correction factor for the technifermion bilinear (3.108) of order ΛET C /ΛT C .
What are the implications of Walking Extended TC for the ordinary fermion masses?
In classic ETC, we have seen that fermion masses typically scale as Λ3T C /Λ2ET C . Since
ΛT C ≈ 1 TeV, the phenomenological constraint on ΛET C >
∼ 100 TeV implies mq,ℓ <
∼
100 MeV. Walking ETC brings a large renormalization enhancement of the techniquark
bilinear by a factor of order ∼ ΛET C /ΛT C , so that we now have [292]:
mq,ℓ ∼ Λ2T C /ΛET C ∼ 1 GeV ,
(3.113)
which is large enough to accomodate the strange and charm quarks, and the τ lepton.
This is born out by more detailed studies which include the full ETC boson exchange in
the gap equations [291], [300], [292], [293] [294], [295] [301].
23
Jackiw and Johnson [297] showed long ago that this solution also forms the Nambu-Goldstone pole,
confirming it is a dynamically generated mass.
76
On the other hand, this is barely large enough to accomodate the bottom quark and,
certainly not the top quark masses24 . Consider that, if TC has QCD-like dynamics, the
value of ΛET C required to fit the top mass is given by (see [54]):
ΛET C ∼ 1 TeV
175GeV
mt
1/2
.
(3.114)
The measured value of the top quark mass therefore implies ΛET C ∼ ΛT C . This leaves too
little “distance” between energy scales for walking to make a difference [303]. It therefore
appears that ETC alone, even in the presence of walking, can only contribute a fraction of
the observed mt . We will address alternative mechanisms for producing the masses of the
t and b quarks in Section 4. Some of these include new strong gauge dynamics peculiar
to the third generation, while others invoke instantons [304] [305].
What is the origin of the small β function and what other effects may arise as a
consequence? Consider the one-loop β function of an SU(N)T C gauge theory with Nf
techniquarks in the fundamental NT C representation:
βT C
g3
= − T C2
16π
8
11
NT C − Nf + · · ·
3
3
(3.115)
Clearly, for αT C (µ) = gT2 C /4π to walk requires having many technifermions active between the scales ΛT C and ΛET C . These need not all be electroweak doublets, e.g., they
may be singlets or vectorlike doublets with respect to SU(2)L , which can help suppress
contributions to S. Higher tensor representations of the SU(NT C ) gauge group are also
possible [166]. In general, after the ETC breaking, the fermions in the lower energy theory
fall into subsets carrying [i] only (TC); [ii] (TC)×(color); [iii] (TC)×(flavor), and so on,
including both the fundamental and higher representations of TC. The technifermions in
different representations may condense at different scales, as will be discussed in the next
section.
Whether walking is caused by the presence of many technifermions in the fundamental
TC representation or technifermions in higher TC representations, the chiral symmetrybreaking sector is enlarged relative to that of minimal TC models. As a result, one
expects a proliferation of technipions and small technipion decay constants FT ≪ v0 . At
first glance, it appears that the models will suffer from unacceptably large contributions
to S (because of the large number of technifermions) and from the presence of many light
pseudo-Nambu-Goldstone bosons (PNGBs) which have not been observed. However, the
effects of the strong walking-TC dynamics on both issues must be taken into account.
As emphasized by Lane [53], the ingredients that enter conventional QCD-inspired
estimations of S are not applicable to a walking ETC theory. Given the altered pattern
of resonance masses (possibly a tower of ρT and ωT states), the proliferation of flavors,
and the presence of fermions in non-fundamental gauge representations [52], it is unclear
24
For an heroic attempt see, e.g., [302]. Such models lead either to gross violations of precision
electroweak constraints, or to excessive fine-tuning.
77
how to estimate S reliably in a walking model. We note that when estimated naively in
a QCD-like TC theory, the S-parameter comes out larger than anticipated, roughly twice
the naive result in the fermion bubble approximation. The fermion bubble approximation
appears, in spirit, to be closer to the situation in walking ETC. Indeed, existing estimates
of S in walking models [306] [307], approach the fermion loop estimate. However, in the
absence of compelling evaluations of S in walking gauge theories, S does not provide a
decisive test of these models.
In contrast, the effect of walking dynamics on the masses of the PNGB’s is unequivocal: the enhanced condensate raises these masses. Previously we found m2P ∼
D
E2
Q̄Q /Λ2ET C FT2 ∼ NT2 C Λ6T C /Λ2ET C FT2 , which could be dangerously small. With the enhancements on the pair of bilinear techniquark operators this becomes:
m2P ∼ NT2 C Λ4T C /FT2 ∼ NT2 C Λ2T C
(3.116)
and the PNGB’s are now safely elevated out of harm’s way from current experiments, but
left potentially accessible to the Tevatron and LHC.
More generally, Walking TC is actually an illustration of the physics of chiral dynamics
in the large Nf lavor limit. Recent interest in the phase structure of chiral gauge theories
has been inspired in part by duality arguments in SUSY theories where there exist exact
results for the phase structure of an SU(N) gauge theory with Nf flavors (see the review
[308]). The presence of infrared fixed points of the gauge coupling appears to be fairly
generic in theories with a large number of flavors.
The infrared fixed point of the strong SU(N)T C gauge theory can also arise from the
interplay of the first two terms of the β-function (3.115). The size of the fixed point
coupling constant, α∗ , can be controlled by adjusting Nf and NT C . By judicious choice
of Nf ∼ NT C one can make α∗ small and a perturbative analysis should be valid. The
weak coupling fixed point, truncating β on the first two terms, is called the BanksZaks fixed point [309]. The ladder-approximation gap equation can be used to probe
the chiral breaking transition with fixed point α∗ [310], which appears to occur for large
25
Nf <
∼ 4N ± 20%. .
3.5
Multi-Scale and Low-Scale TC
Eichten and Lane [272] [312] have suggested that, if TC dynamically generates the weak
scale, there may be distinct sectors of the full theory that contribute components to
the full electroweak scale, so that v02 = v12 + v22 + .... This general idea is known as
“Multi-Scale TC.” The sector of the theory with the smallest vi , the “Low-Scale TC”
sector, may produce visible phenomenological consequences at the lowest energy scales
in current colliders. As we have seen in the previous section, a “Low-Scale” sector is
generally expected in a walking TC theory.
25
See [311] for a discussion of the interplay of chiral breaking and confinement in the SUSY case.
78
Low-Scale TC produces a rich phenomenology, much of which is directly accessible to
the Tevatron in Run II or to a low energy LC; a plethora of new states should also be visible
¯
at the LHC. Processes of interest in a f f-collider
are then the familiar f f¯ → (ρT , ωT ) →
(W W + ZZ, W PT + ZPT , PT PT ) as in the Farhi-Susskind model, but transplanted now
to the lower scale vi . In this section, we will first discuss how the Low-Scale TC idea fits
within the general ETC framework and then sketch the phenomenological consequences.
Current limits on these theories are discussed in the next section.
The motivation for Low-Scale TC arises directly from the ideas of WTC discussed in
the previous section. We have already seen that a WTC coupling is desirable because
it offers some remedies for the questions of fermion and PNGB masses. Suppose we
follow the lead of Eichten and Lane [272] in constructing a walking model by including
technifermions in higher-dimensional TC representations. As the theory evolves downward
towards the TC scale, chiral condensates form. However, they do not all form at a single
scale: the higher representations will condense out at higher energies since their binding
is controlled by the Casimir coefficients, C2 (R) of their representations [313]. Moreover,
since we have seen how walking dynamics enhances technifermion bilinears, the expected
separation of the various condensate scales may be large.
In the most phenomenologically interesting cases of Low-Scale TC, the sector of the
theory with the smallest vi includes states light enough to be accessible to current experiments. One caveat is that a very light Low-Scale sector may include charged technipions
P ± with masses less than mtop . If top quark decays are to remain dominated by the conventional Standard Model channel t → W + b, as consistent with Tevatron Run I data,
it is necessary to suppress a fast t → PT + b, decay. One possibility is to require that
mP >
∼ 160 GeV [314]. Alternatively, one can decouple the top quark from the Low-Scale
sector to suppress the tb̄P vertex [104]. This is generally accomplished by using separate
strong top dynamics to generate the top quark (and b-quark) masses and Low-Scale TC
for the bulk of the EWSB [274] [315]. A scheme of this type called “Topcolor Assisted
TC,” or TC2, and the phenomenological implications of the new strong top dynamics will
be discussed in Section 4.2.
In general, the spectroscopy of a Low-Scale TC model can accomodate everything
from a minimal model through the Farhi-Susskind structure. The Low-Scale spectrum
will, at the very least, include light PNGB’s (technipions) and techni-vector mesons. For
example, with FT ∼ 60 GeV one expects MPT ∼ 100 GeV and MρT ∼ 200 GeV [52]. The
technipions will be resonantly produced via techni-ρ vector meson dominance (VMD) 26
with large rates at the Tevatron, LHC, and a linear collder [316]. The technivector mesons
are expected to be, in analogy to the minimal model, an isotriplet, color-singlet ρT , and
the isoscalar partner ωT . Isospin is likely to be a good approximate symmetry, so ρT and
ωT should be approximately degenerate in mass, as is the I = 1 multiplet of technipions.
The enhancement of technipion masses due to walking suggests that the decay channels
26
Per the discussion of Section 2.3.3, there is an induced coupling of the ρT to any electromagnetic
current, so this also applies at e+ e− and µ+ µ− colliders [157]). See [159, 158] about the caveats that
apply when estimating production of color-octet ρT by vector meson dominance.
79
ρT → PT PT and ωT → PT PT PT are probably closed. Thus, the decay modes ρT → WL PT
and ZL PT , where WL , ZL are longitudinal weak bosons, and ρT , ωT → γPT may dominate.
Because technipion couplings to fermions, like those of scalars, are proportional to mass,
one expects the most important decay modes to be
PT0 → bb̄
PT+ → cb̄ or cs̄, τ + ντ .
(3.117)
Heavy-quark jet tagging is, then, important in searches for Low-Scale TC.
Eichten, Lane and Womersley [312] have performed an extensive analysis of the phenomenological signatures of Low-Scale TC at the Tevatron. They present simulations of
± 0
0
p̄p → ρ±
T → WL PT and ωT → γPT for the Tevatron collider with an integrated luminosity
of 1 fb−1 . For MρT ≃ 200 GeV and MPT ≃ 100 GeV, the cross sections at the Tevatron
are expected to be of order a few picobarns. The narrowness of the ρT and ωT suggests
that appropriate cuts, e.g., on the final state invariant mass, can significantly enhance
the signal/background ratio. Furthermore, the final states include b-quark jets from technipion decay and either a photon or the isolated lepton from weak boson decay. This
results in some dramatic signals which stand out well above background when b-tagging
is required. Cross sections for the LHC are an order of magnitude larger than at the
Tevatron, so detection of the light technihadrons should be straightforward there as well.
More specifically, ref.[312] considers one light isotriplet and isoscalar of color-singlet
technihadrons and uses VDM for techni-ρ production and decay to determine the rates
for:
→ WL± ZL0 ; WL± PT0 , PT± ZL0 ; PT± PT0
q q̄ ′ → W ± + ρ±
T
(3.118)
+
−
± ∓
+ −
0
0
q q̄ → γ + Z + ρT → WL WL ; WL PT ; PT PT
For MρT ∼ 200 GeV > 2MPT and MPT ∼ 100 GeV, the dominant processes have cross
sections of 1 − 10 pb at the Tevatron and ∼ 10 − 100 pb at the LHC. The modes with
the best signal-to-background are ρT → WL PT or ZLPT and ρT ωT → γPT .
Figure 24(d) shows invariant mass distributions for the W jj system after kinematic
and topological cuts and b-tagging have been imposed as discussed in Ref.[312]. A clear
peak is visible just below the mass of the ρT , and the peaks in the dijet mass and the W jj
mass are correlated. If the ρT and PT exist in the mass range favored by the Low-Scale TC
models they can be easily found in Run II of the Tevatron. The ωT is likewise produced
in hadron collisions via vector-meson-dominance coupling through γ and Z 0 . We expect
ωT → γPT0 , Z 0 PT0 will dominate ωT → PT PT .
Figure 25(a) shows the invariant mass distribution of the two highest-ET jets for
signal (black) and background (grey) events, that pass certain kinematic criteria, for an
integrated luminosity of 1 fb−1 . The effect of topological cuts is seen in Fig. 25(b). Tagging
one b-jet significantly improves the signal/background as in Fig. 25(c), and a peak below
the PT mass can be seen. Figure 25(d) shows the photon+dijet invariant mass after
various topological cuts and b-tagging are implemented [312].
80
Events/8 GeV
1500
Kinematic cuts
(a)
1000
500
Events/8 GeV
0
Events/8 GeV
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
Topological cuts
(b)
400
200
0
0
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
Single b-tag
(c)
20
10
0
Events/16 GeV
0
40
30
20
10
0
0
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
Single b-tag
(d)
0
50
100
150
200
250
300
W + dijet mass (GeV)
350
400
Figure 24: Predicted invariant mass distributions at Tevatron Run II for ρT signal (black) and
W jj background (grey); vertical scale is events per bin in 1 fb−1 of integrated luminosity from
Ref.([312]). Dijet mass distributions (a) with kinematic selections only, (b) with the addition of
topological selections, and (c) with the addition of single b-tagging; (d) W +dijet invariant mass
distribution for the same sample as (c).
81
Events/8 GeV
Events/8 GeV
2000
1500
1000
500
0
Events/8 GeV
0
600
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
Topological cuts
(b)
400
200
0
0
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
60
Single b-tag
(c)
40
20
0
Events/16 GeV
Kinematic cuts
(a)
80
60
40
20
0
0
20
40
60
80 100 120 140 160 180 200
Dijet mass (GeV)
Single b-tag
(d)
0
50
100
150
200
250
γ + dijet mass (GeV)
300
350
400
Figure 25: Predicted invariant mass distributions at Tevatron Run II for ωT signal (black)
and γjj background (grey); vertical scale is events per bin in 1 fb−1 of integrated luminosity.
Dijet mass distributions (a) with kinematic selections only, (b) with the addition of topological
selections, and (c) with the addition of single b-tagging; (d) γ+dijet invariant mass distribution
for the same sample as (c).
The ultimate virtue of Low-Scale TC is that the new strong dynamics, presumably the
key to other fundamental questions, such as those of fermion masses and flavor physics,
would commence at relatively lower accessible energies. As we have seen, Low-Scale TC
signatures ρT → W PT and ωT → γPT can be discovered easily in Run II of the Tevatron
for production rates as low as a few picobarns. In the next section, we will examine
existing experimental constraints on these models.
3.6
Direct Experimental Limits and Constraints on TC
Experiments performed in the past few years have had the large data samples, high
energies, and heavy-flavor-tagging capabilities required to begin direct searches for the
new phenomena predicted by models of new strong dynamics. We have just examined
some of the theoretical implications for preferred channels of Low-Scale TC models. We
now summarize the status of experimental limits on the accessible scalar mesons, vector
mesons, and gauge bosons, that are generically predicted by these theories. Our focus
here is on the direct searches performed largely by the LEP and Tevatron experiments,
and we present many of their original exclusion plots. Tables summarizing many of these
82
results may be found in refs. [317, 194]. As appropriate, we will also comment on more
indirect searches for new physics via measurements of precision electroweak observables
and on the prospects for the LHC and future lepton colliders. For further discussion of
technicolor searches at LHC and even higher-energy hadron colliders, see ref. [194].
3.6.1
Searches for Low-Scale Color-singlet Techni–ρ’s and Techni–ω’s (and
associated Technipions)
In light of the Low-Scale TC model, and the generic phenomenology of TC, it is useful to
examine the present-day constraints that exist, mostly from LEP and the Tevatron, on
the direct observables, i.e., the masses and production signatures of technivector mesons
and techni-π’s. Note that a number of the searches have taken advantage of the fact
that the ωT of Low-Scale models can be visible in collider experiments [272, 316, 312,
52]. Enhancement of technipion masses by WTC can quench the decay ωT → PT PT PT ,
resulting in the dominant mode ωT → γPT [272]. Techni-ω decays to q q̄, ℓ+ ℓ− and ν ν̄ can
also be significant, but the decay ωT → ZPT is phase-space suppressed.
CDF has published two searches for color-singlet techni-ρ’s [318, 319]. One study
assumes that the channel ρT → PT PT is closed and the other that it is open. If MρT <
MW + MPT then the techni-ρ (and its isoscalar partner, ωT ) decays to pairs of ordinary
quarks or leptons. D0 has looked for light ρT , ωT decaying to e+ e− [320]. The LEP
collaborations L3 [321], DELPHI [322], and OPAL [323] have released preliminary results
on multi-channel searches for light ρT and ωT .
CDF performed a counting experiment looking for ρT → W PT → ℓνbb̄, ℓνcb̄ in 109
−1
pb of Run I data [318, 319]. They selected candidate lepton plus two-jet events with
at least one jet b-tagged. The presence of peaks in the Mb,jet and MW,b,jet distributions
would signal the presence of the PT and ρT , respectively, as indicated in figure 26. No
deviation from Standard Model expectations was observed. CDF therefore set upper
limits on the techni-ρ production cross-section for specific pair of techni-ρ and technipion
masses as indicated in Figure 27. In Run II, with a larger data sample and a doubled
signal efficiency, CDF expects to explore a significantly larger range of ρT masses using
the same selection criteria, as indicated in Figure 28.
CDF also searched [318, 319] for technipions in the shape of the two b-jet mass distribution in ℓ + 2-jet and 4-jet events, using 91 pb−1 of Run I data. The former topologies
arise as described above; the latter can result from either ρT → W PT or ρT → PT PT
where the technipions decay to heavy flavors and the W decays hadronically. CDF notes
that the upper limits (∼ 100 pb) this search sets on production of ∼ 200 GeV techni-ρ decaying to ∼ 100 GeV technipions provides no immediate improvement in the constraints
on TC models. Run II should provide significantly improved sensitivity to these modes.
D0 has looked for the light ρT and ωT in WTC in which the vector mesons are unable to
decay to W +PT [320, 324]. To study the decays ρT , ωT → e+ e− , they selected events with
two isolated high-ET electrons, one required to be central, from ∼ 120 pb−1 of Run I data.
No excess above expected backgrounds was seen. For a model with NT C = 4, techniquark
83
2
[ events / (15 GeV/c ) ]
CDF Preliminary
10
8
6
4
2
0
10
After ∆φ(jj) and PT(j) Cuts
After ∆φ(jj) and PT(j) Cuts
CDF Data
WTC Signal
+Background
Background
8
6
4
2
0
0
100
200
100
Dijet mass
200
300
400
500
2
W + 2jet mass [GeV/c ]
Figure 26: Invariant mass of the dijet system and of the W + 2 jet system (with a leptonically
decaying W) in the CDF search for ρT → W ± PT [319]. The points are data; the solid histogram
is background; the dashed histogram shows background plus the signal from a walking TC model
with MρT = 180 GeV and MPT = 90 GeV. The topological cuts leading to the lower figures are
described in [157].
2
πT mass [GeV/c ]
115
110
-1
CDF 109 pb
ρ±T → W± + π0T and
ρ0T → W± + π±T processes
95 % C.L.
Excluded
Region
( π0T → bb , π±T → bc, cb)
–
–
–
es
ho
ld
105
ica
lt
hr
100
15pb
10pb
5pb
π
T
ki
ne
m
at
95
ρ
T→
W
90
85
MRSG p.d.f. multiplied by Kfactor=1.3
Technicolor model by E.Eichten and K.Lane
Phys. Lett. B388:803-807, 1996
80
160 165 170 175 180 185 190 195 200 205
ρT mass [GeV/c ]
2
Figure 27: Excluded region for the CDF search for color singlet techni-ρ search in the mode
ρT → W ± PT . [319]
84
πT mass [GeV/c2]
PYTHIA v6.139 : MV=MA=200 GeV
±
150 ρT± → W + πT0 and
±
ρT0 → W + πT± processes
140
al
th
re
sh
old
130
120
kin
em
at
ic
110
Exclusion(95% C.L.)
at CDF RUN2
ρ
90
T→
W
π
T
100
5 σ discovery at CDF RUN2
80
70
160
Excluded(95% C.L.) at CDF RUN1
180
200
220
240
260
ρT mass [GeV/c2]
Figure 28: Predicted reach of CDF in Run II for ρT → W πT assuming MV = MA = 200 GeV
[52] (note the notational variance, PT = πT ).
electric charges of QD = QU − 1 = −1/3 and MT = 100 GeV (this mass parameter is the
Low-Scale TC scale, of order the weak interaction scale) [312], ρT and ωT mesons with
masses below 200 GeV were ruled out at 95% c.l. provided that MρT − MPT < MW (see
Figure 29). Projected improvements in the reach of this search in Run II are shown in
Figure 30.
L3 used 176.4 pb−1 of data collected at an average center of mass energy of 188.6 GeV
to search for color-singlet ρT . [321]. The search took into account the four major techni-ρ
decay modes
(′ )
ρT → WL WL , WL PT , PT PT , γPT
(3.119)
for the following range of techni-ρ and technipion masses
50 GeV < MPT < 150 GeV
150 GeV < MρT < 250 GeV
(3.120)
In the W W decay channel, all decay modes of the W bosons are included. The result
is that an upper limit of 0.47 pb was set at 95% c.l. on the possible increase of the
e+ e− → W W cross-section due to contributions from TC.
In other decay channels, the technipions decay predominantly to bb̄ or bc̄ (the calculated branching ratios ranged from 50% to 90%) no statistically significant excess of
techni-ρ-like events was observed. L3 found the following approximate mass ranges for
technipions and techni-ρ’s were excluded at 95% c.l., for 150 < MρT < 200 GeV, technipion masses 50 < MPT < 150 GeV were excluded and for 50 < MPT < 80 GeV, techni-ρ
masses 150 < MρT < 230 GeV were excluded. Figure 31 shows the boundaries of the
excluded region.
85
B(ρT,ωT→ ee) x Cross Section (pb)
Theory (Eichten, Lane, Womersley)
Mρ,ω-Mπ = 100 GeV
MT = 100 GeV
MT = 200 GeV
MT = 300 GeV
MT = 400 GeV
1
10
-1
σ95 from D0 Data
10
-2
100
150
200
250
300
350
400 450 500
Mρ,ω (GeV)
Figure 29: Excluded regions for the D0 search for ρT , ωT → e+ e− [324].
The DELPHI collaboration searched for technipion production in the final states WL PT
and PT PT in four-jet final states [325]. No significant contribution of technipion production was observed, either in the total rate, or in the Mjj distribution. DELPHI thus
excludes at 95% c.l. technipions of mass between 70 and 130 GeV when the techni-ρ
mass is between 120 and 200 GeV. For heavier MρT , the range of excluded technipion
masses shrinks, as shown in Figure 32, e.g., for MρT = 250 GeV, 70 < MPT < 100 GeV is
ruled out.
The OPAL collaboration [323] searched for technihadrons √
in the channels e+ e− →
+ −
ρT , ωT → PT PT → bq̄ b̄q ′ and e+ e− → ρT , ωT → P 0 γ → bb̄γ at s ≈ 200 − 209 GeV. No
significant excess over SM background was observed in the total number of events or the
dijet mass distribution. Cross-section × branching ratio upper bounds of 60 - 200 fb were
set at 95% c.l., depending on the P ± + T mass. In the search for neutral technipions,
events with a pair of b-jets and an energetic isolated photon were chosen. After kinematic
and topological cuts, no excess over the SM background was observed. An upper limit of
approximately 50 fb (except when the technipion mass is nearly MZ ) on the cross-section
× branching ratio was established at 95% c.l. The combined results of these searches are
shown in Figure 33.
CDF searched for ωT → γPT , assuming that the PT → bb̄ [326]. About 200 events
with a photon, a b-jet and at least one additional jet were selected from 85 pb−1 of data.
This was found to be consistent with the Standard Model. The distributions of Mb,jet
and Mγ,b,jet − Mb,jet show no evidence of resonance production. Thus, CDF obtains an
upper limit on (cross-section)×(branching ratio), which it compared to the predictions of
the TC models of ref. [312]. The range of excluded masses of PT and ωT are shown in
86
Figure 30: Projected reach of the D0 detector in Tevatron Run IIa for ρT , ωT → e+ e− [52].
Figure 34 (the exclusion region boundary is ragged because of statistical fluctuations in
the data). CDF noted that, if the channels ωT → PT PT PT or ωT → ZPT were open, the
excluded region would be reduced as shown in the Figure [326]. Had their Monte Carlo
included ρT → γPT and ρT , ωT → γπT′ , their excluded region would increase.
Future experiments should provide definitive tests of Low-Scale TC [312]. The crosssection for ωT production is ∼ 1 − 10 pb for Tevatron, and an order of magnitude larger
at the LHC. Simulations by the ATLAS collaboration [327] show that the relatively light
techni–vector resonances of Low-Scale TC [212] are well within reach at LHC (Figure 35).
Experiments at an eγ collider have potential to discover and study an ωT with a mass
up to about ∼ 1 TeV in the processes e− γ → e− ωT → e− γZ, e− W + W − Z [328]. As shown
in Figure 36, at the stage where only detector acceptance cuts have been applied, the
cross-section for an ωT decaying to W W Z can be of order ∼ 10 − 100 fb, and as much
as an order of magnitude above background (ωT → Zγ is below background). Applying
various kinematic cuts, the ωT could be √
visible for a range of masses and decay widths.
For example, with 200 f b−1 collected at s = 1.5 TeV, an ωT of width 100 GeV can be
detected at the 3σ level up to a mass of about ∼ 1.3 TeV.
3.6.2
Separate Searches for color-singlet P 0 , P 0′
As discussed in Sections 3.4 and 4, many TC models require a large number ND of weak
doublets of technifermions. For a given TC gauge group SU(NT C ), the number of doublets
required to make the gauge coupling gT C “walk” is ND ≈ 10, as in the models of refs. [315,
330, 331]. Topcolor-Assisted TC models (see Section 4) also tend to require many doublets
87
L3
140
Mπ (GeV)
120
100
80
Excluded
at 95% CL
60
160
180
200
220
240
Mρ (GeV)
Figure 31: The MρT − MPT region excluded by L3 at 95%c.l. [321].
of technifermions [212, 315]. One phenomenologically interesting consequence of the large
number of√doublets is the presence of PNGB states with small technipion decay constants:
FT ≈ v0 / ND . Such states will be lighter, and have generically longer lifetimes.
As outlined in Section 2.3.4, data from LEP can place limits on single production of
light neutral PNGB’s from a variety of TC models. One benchmark example is the Lane’s
low-scale TC “Straw Man Model” (TCSM) [212, 213], in which the lightest technifermion
doublet, composed of technileptons TU and TD can be considered in isolation. The result
is two, nearly degenerate neutral mass eigenstates, whose generators are given by PT0 ∼
T̄U γ5 TU − T̄D γ5 TD and PT0 ′ ∼ T̄U γ5 TU + T̄D γ5 TD . As shown in ref. [329] LEP searches for
hadronically-decaying scalars produced√in association with a Z or γ may be used to place
an upper bound on the product NT C ND A (where A is the relevant anomaly factor).
√
Inserting the value of A appropriate to a particular PNGB yields an upper on NT C ND .
Table 3.6.2 gives these bounds as a function of PNGB mass for the cases where either
2-gluon or b̄b decays dominate.
Consider the case where MPT0 ′ ≤ 30 GeV and b-quark decays dominate; the limit
√
NT C ND ≤ 24 applies. As a result, for NT C = (4, 6, 8, 10, 12) the largest number of
electroweak doublets of technifermions allowed by the LEP data is, respectively, ND =
(36, 16, 9, 5, 4). The results are very similar if the two-gluon decays of the PNGB dominate
instead. How do these results accord with the requirements of Walking TC? Requiring
the one-loop TC beta function to satisfy βT C ≈ 0, implies that 11NT C /4 weak doublets of
technifermions are needed, according to eq. (3.115). The analysis of ref. [329] thus shows
that WTC and a very light PT0 ′ can coexist only in models with NT C = 4, 6. Similarly,
the size of the TC group is restricted to NT C ≤ 6[12] if the PNGB mass is 80 GeV [160
88
M(πT) [GeV/c2]
DELPHI
e+e-→πTπT;πTWL
e+e-→ρT(γ) :
ρT→hadrons
ρT→WL+WL-
120
100
ND=9
80
60
100
200
300
400
M(ρT) [GeV/c2]
Figure 32: MρT − MPT region excluded by DELPHI at 95%c.l. [325]
GeV]. The results are similar if the 2-loop β-function is used.
As a second example, we mention the results of ref. [329] for the WTC model of Lane
and Ramana [156] whose LEP–II and NLC phenomenology was studied by Lubicz and
Santorelli [207]. The model has ND = 9: one color-triplet of techniquarks (NQ = 1)
and six color-singlets of technileptons (NL = 6). Of the several neutral PNGBs in this
model, the one whose relatively large anomaly factors and small decay constant (FT ≈ 40
GeV) makes it easiest to produce is PL3 ∼ N̄ℓ γ5 Nℓ − Ēℓ γ5 Eℓ where the subscript implies
a sum over technilepton doublets. This PNGB is expected to have a mass in the range
∼ 100 − 350 GeV [207]. Depending on the value of the ETC coupling between the PNGB
and fermions, the dominant decay of this PNGB may be into a photon pair or b̄b. Ref.
[329] finds that if the two-photon decays dominate, the PNGB must have a mass in excess
of 160 GeV; if the b̄b decay is preferred, the mass range 80 GeV ≤ MP ≤ 120 GeV is
excluded.
3.6.3
Separate searches for color-singlet PT±
Models with more than the minimal two flavors of new fermions (e.g. TC with more than
one weak doublet) typically contain electrically charged PNGB states (PT± ). Experimental
limits on charged scalars are often phrased in the language of a two-higgs-doublet model,
i.e., in terms of the ratio of the vacuum expectation values of the two doublets (tan β)
and the mass of the charged scalar (MH+ ).
(i) LEP limits
89
mπ [GeV]
140
OPAL Preliminary
–
π T0γ→bbγ MV=200GeV
120
–
π T0γ→bbγ MV=100GeV
–– ,
π T+ π T- → bqbq
100
80
60
excluded
250 300 350 400 450 500 550 600
mρ/ω [GeV]
Figure 33: The 95% c.l. excluded region in the (MρT ,ωT ) plane from the combination of the
PT+ PT− → bq̄ b̄q ′ search and the PT0 γ → bb̄γ search. The dashed lines show the corresponding
median expected exclusions for the background only hypothesis. [323]
Color-singlet electrically-charged technipions PT± with the quantum numbers of the
charged scalars in two-higgs-doublet models are directly constrained by the limits on
pair-production of H ± derived from LEP data. When tan β is large, the charged scalars
of two-higgs-doublet models decay mostly to τ ν ; if tan β is small, light charged scalars
decay to cs̄, but for MH ± heavier than about 130 GeV, the channel H + → tb̄ → W bb̄
dominates [332, 333]. The LEP searches assume that H + → τ + ντ , cs̄ (as consistent with
the mass range they can probe), and derive limits on the rate of H + H − as a function of
the branching ratio B(H + → τ + ντ ). As shown in Figure 37, the lower limit is at least 77
GeV for any value of the branching ratio.
If the mass of the charged scalar is less than mt − mb , then the decay t → H + b can
compete with the standard top decay mode t → W b. Since the tbH ± coupling can be
parameterized in terms of tan β as [334]
gtbH + ∝ mt cot β(1 + γ5 ) + mb tan β(1 − γ5 ) ,
(3.121)
we see that the additional decay mode for the top is significant for either large or small
values of tan β. The charged scalar, in turn, decays as H ± → cs or H ± → t∗ b → W bb
if tan β is small and as H ± → τ ντ if tan β is large. In any case, the final state reached
through an intermediate H ± will cause the original tt̄ event to fail the usual cuts for the
lepton + jets channel. A reduced rate in this channel can therefore signal the presence of
a light charged scalar. As shown in Figure 38, D0 and CDF have each set a limit [317]
on MH ± as a function of tan β and σtt . In Run II the limits should span a wider range of
tan β and reach nearly to the kinematic limit as shown in Figure 39.
90
250
ld
200
σxBR (pb)
5
es
ho
T
th
r
πT Mass (GeV/c2)
1
Zπ
150 200 250 300 350
2
ωT Mass (GeV/c )
150
ld
sho
thre
3π T
100
50
0
Exclusion Region C.L.:
100
150
200
90%
95%
250
300
ωT Mass (GeV/c2)
99%
350
Figure 34:
The 90%, 95% and 99% c.l. exclusion regions for the CDF search for ω → γPT
[326]. The inset shows the limit on σ · B for MPT = 120 GeV; the circles indicate the limit and
the solid line shows the prediction from [312].
Limits on the mass and coupling of PT± are also implied by the experimental 95% c.l.
upper limit B(b → sγ) < 4.5 × 10−4 obtained by the CLEO Collaboration [335, 336].
Radiative corrections due to the PT tend to increase [337, 338] the branching ratio above
the Standard Model prediction of (3.28±0.33)×10−4 [339, 340], as would be true in a model
with an extended Higgs sector. To the extent that this is the only new physics contribution
to b → sγ, it implies an upper bound of order 300 GeV on the mass of the charged scalar.
However, the contributions of other new particles or non-standard gauge couplings can
also affect the branching ratio, making the exact limit quite model-dependent. Weaker,
and also model-dependent, bounds can also be derived from measurements of b → sγ and
b → τ ντ X and from τ -lepton decays at LEP [341, 342, 343, 344, 345].
3.6.4
Searches for Low-Scale Color-octet Techni–ρ’s (and associated Leptoquark Technipionss)
In models of Walking TC, in which enhancement of the technipion masses prevents the
decay V8 → P̄T PT (V8 is the color-octet techni–ρ), decay to dijets can dominate [291, 316].
The CDF Collaboration has used 87 pb−1 of Run I data to search for the effects of V8 ’s
91
Figure 35: Simulated event and background rates in the ATLAS detector for ρT → W ± Z →
ℓ± νℓ ℓ+ ℓ− for various MρT and MPT in low-scale TC models [212]; from Ref. [327].
92
Figure 36: Cross-section vs e+ e− CM energy for e− γ → e− ωT with ωT → W W Z and Zγ and
the SM backgrounds thereto. Solid (long-dashed) lines are for MωT = 0.8 (1.0) TeV. In each
case, the upper curve is for W W Z and the lower is for Zγ. The dash-dot (dotted) line is for
the SM e− W W Z (e− Zγ) background. In (a), 0.25ΓW W Z = ΓZγ = 5 GeV (solid) or 20 GeV
(long-dashed); in (b), 0.5ΓW W Z = ΓZγ = 15 GeV (solid) or 40 GeV (long-dashed). (From [328])
MP 0′ ≤
T
30 GeV
60 GeV
80 GeV
100 GeV
120 GeV
140 GeV
160 GeV
√
NT C ND ≤
′
′
PT0 → gg PT0 → b̄b
28(b)
24(b)
67(b)
70(b)
283(b)
25(a)
—
40(a)
—
42(a)
—
49(a)
—
68(a)
Table 5:
Limits (from ref. [329] on the number of technicolors, NT C , and weak doublets of
technifermions, ND , for hadronically decaying PNGBs in TCSM [212, 213] models as a function
of the upper bound on the PNGB mass. The superscripted labels indicate the data used to
calculate the limits: (a) means AγγP a ; (b) means AγZP a .
93
Figure 37: LEP lower bounds on MH ± as a function of B(H + → τ ν). From [317].
on the dijet and b-tagged dijet invariant mass spectra [346]. No deviations from the
Standard Model backgrounds were observed. A narrow V8 with mass 350 < MV8 < 440
GeV is excluded at 95% c.l. using the b-tagged distribution, as may be seen in figure 57.
The mass range 260 < MV8 < 470 GeV is ruled out using the untagged sample.
In searches for leptoquark techni-π’s P3 , CDF sets NT C = 4 and allows a relevant
parameter, ∆M, to take on the expected value of 50 GeV and the limiting values of 0
and ∞. More precisely, ∆M is the mass difference between P8 and P3 , and enters the
calculation of the partial width for V8 → P̄3 P3 [156]. CDF reports joint limits on the
masses of the P3 and V8 .
In the first search [348] CDF considered the decay path P3 → b̄τ − . The observed
yield was consistent with Standard Model backgrounds (dominated by Z → τ τ plus jets,
diboson, and t̄t production). CDF was able to exclude techni–π masses up to (MV8 /2) for
V8 masses up to about ∼ 450 (500, 620) GeV assuming ∆M = 0 (50, ∞), as illustrated in
Figure 41. This extends a continuum leptoquark analysis [349] which had previously set
the limit MP3 ≥ 99 GeV.
In a second set of searches [347], CDF considered the decay paths P3 → cν̄τ and
P3 → bν̄τ . No excess of observed events over Standard Model backgrounds (dominated
by W + jets) was found. For P3 decaying to charm, CDF rules out (Figure 40) technipion
masses up to mt for techni-ρ masses up to about ∼ 450, (500, 650) GeV assuming ∆M = 0,
(50, ∞); heavier P3 would decay to tντ . The lower bound from the continuum search in
this channel is MP3 ≥ 122GeV at 95% c.l. For P3 decaying to bν̄τ , CDF’s continuum
search set the 95% lower bound MP3 ≥ 149GeV and the technipion search excludes MP3
up to the kinematic limit (MV8 /2) for techni-V8 masses up to about ∼ 600 (650, 700) GeV
94
Figure 38: Limits on charged scalar mass as a funciton of tan β. The 95% exclusion bounds
from CDF and D0 studies of top decays are strong functions of tan β. LEP limits from figure
37 are also shown. From [317].
assuming ∆M = 0 (50, ∞), as shown in Figure 41.
Leptoquarks typically decay into a quark and a lepton of the same generation. In the
TC-GIM models, the P3 ’s carry lepton and quark numbers of any generation. A leptoquark decaying into an e and d–quark has the signature of a “first generation leptoquark.”
The first generation leptoquarks decays into e + d¯ with branching ratio ǫ, or into a ν + ū
with branching ratio 1 − ǫ, and experimental limits are sensitive to the unknown ratio
ǫ. In TC-GIM models we have down-type leptoquarks with ǫ = 1, and distinct up-types
have ǫ = 0. We note that the D0 Collaboration [269] has excluded down-type leptoquarks
up to ∼ 130 GeV, while second generation leptoquarks are excluded by CDF up to ∼ 133
GeV [270]. This essentially rules out low energy scale versions of the TC-GIM models.
Searches for color-octet technipions will present a greater challenge. It is difficult
to pull color-octet technipions, P8 ’s, out of the multijet backgrounds at LHC. However,
TC-GIM models can have many different kinds of P8 ’s; if these are nearly degenerate in
mass they may give a correspondingly stronger signal. Run II searches for P8 ’s should
also consider the rare decay channel decay channel P8 → gγ [186, 185]. Although the
rate for this mode is down from the two-gluon channel by a factor of about ∼ 100, a
smaller background makes the signal potentially visible, as illustrated in Figure 42. By
employing a PYTHIA-level simulation the authors of ref. [185] were able to identify
cuts on the transverse momentum and invariant mass of the photon + jet system that
significantly enhance the signal relative to the background. They find that the Tevatron
Run II can exclude P80′ in Low-Scale TC models where FT is reduced to about ∼ 40 GeV,
up to ∼ 350 GeV, or achieve a 5σ discovery up to ∼ 270 GeV. The larger value of FT in
the Farhi-Susskind TC model relative to Low-Scale models renders its PNGB invisible in
this mode.
95
±
Figure
R 39: Projected Run II reach of D0 charged scalar search in t → H b assuming
TeV,
3.6.5
√
s=2
Ldt = 2fb−1 , and σ(tt̄) = 7pb.
Searches for W ′ and Z ′ bosons from SU(2) × SU(2)
As discussed in Section 3.3.2, an integral part of some dynamical theories is an extended
SU(2)h × SU(2)ℓ structure for the weak interactions in which the first two generations of
fermions are charged under the weaker SU(2)ℓ and the third generation feels the stronger
SU(2)h gauge force. Examples include the non-commuting extended TC (NCETC) models [271] and the related topflavor models [350, 351, 352, 353]. The low-energy spectrum
of the models includes massive W’ and Z’ bosons that couple differently to the thirdgeneration fermions. The more strongly suppressed the new gauge bosons’ couplings to
first and second generation fermions, the less effective traditional searches for new electroweak weak bosons become. It is also notable that the new gauge bosons couple, at
leading order, only to left-handed fermions (through weak isospin). This section discusses
search techniques that exploit the flavor non-universal couplings of the W’ and Z’.
The LEP experiments have studied the possibility that new physics is contributing to
electron-positron scattering via four-fermion contact interactions. This allows them to set
a lower bound on the mass of a Z’ boson, because at energies well below the mass of the
Z’ boson its exchange may be approximated by a four-fermion contact interaction. Their
strongest limits on exchange of a Z’ boson arising from extended weak interactions come
−
+ −
from processes involving pair production of third-generation fermions: e+
L eL → τL τL and
−
e+
L eL → bL b̄L . As discussed in ref. [329], the LEP limits on contact-interaction scale Λ
96
Figure 40: CDF’s 95% c.l. exclusion region for ρ8 → P̄3 P3 → ccντ ντ [347] for several limiting
values of ∆M = M (P8 ) − M (P3 ) which affect the V8 decay partial width.
Figure 41: CDF’s 95% c.l. exclusion region for V8 → P3 P3 → bbντ ντ [347] for several values
of ∆M = M (P8 ) − M (P3 ).
97
Figure 42: Invariant mass distribution for the pp̄ → P80′ → gγ signal (dashed) and background
(solid) in multi-scale TC with MP = 250 GeV and F = 40 GeV [185].
translate into the following bounds from τ τ production
q
MZ ′ = Λ αem /4 sin2 θ >
365 GeV ALEPH
355 GeV OPAL
.
(3.122)
523 GeV ALEPH
325 GeV OPAL
.
(3.123)
and from bb production
q
2
MZ ′ = Λ αem /4 sin θ >
While these are weaker than the current limits on non-commuting ETC [278] or topflavor
[350, 353] Z’ bosons from precision electroweak data (section 3.3.2), they are complementary in the following sense. The limits from a fit to electroweak data assume that the only
new relevant physics comes from the extra weak and ETC gauge bosons (or Higgs bosons
in topflavor); the contact-interaction limits are lower bounds on the Z’ mass regardless of
the other particle content of the theory.
The Fermilab Tevatron experiments have, likewise, searched for new contact interactions contributing to dijet and dilepton production. Because their searches involve only
first and second generation fermions, the implied bounds on the mass of a Z’ primarily
coupled to the third generation are significantly weaker than those from the LEP data
[329].
98
Required Luminosity (fb−1 )
30
20
10
0
0.6
0.7
0.8
0.9
sin φ
Figure 43: Luminosity required to discover SU(2) Z’ bosons preferentially coupled to the third
generation [329]. Dashed curves are 3σ discovery curves for a fixed mass, while solid curves are
5σ discovery curves. From bottom to top, 3σ curves are displayed for Z’ masses of 550 GeV,
600 GeV, 650 GeV, and 700 GeV. From bottom to top, 5σ curves are displayed for Z’ masses of
550 GeV, 600 GeV, and 650 GeV. The horizontal lines indicate luminosity targets for Run II.
A W’ boson preferentially coupled to the third generation fermions should be detectable in single top-quark production at Run IIb [354] if its mass is less than about
1-1.5 TeV. The ratio of cross-sections Rσ ≡ σ(p̄p → tb)/σ(p̄p → lν) can be measured
(and calculated) to an accuracy [355, 356, 357] of at least ±8%. The extra weak gauge
bosons present in a non-commuting Extended TC model can change Rσ in several ways27 .
Mixing of the Wh and Wℓ bosons alters the light W ’s couplings to the final state fermions.
Exchange of both W and W ′ mass eigenstate bosons contributes to the cross-sections. As
a result, a visible increase in Rσ is predicted.
A direct search for Z’ bosons primarily coupled to the third generation can be made
by looking at heavy flavor production at the Tevatron. No searches for these bosons
have been made thus far. A recent study [329] (using PYTHIA and a simple model of
the D0 detector) indicates that the channel pp̄ → Z ′ → τ τ → eµν ν̄ is promising. By
requiring large leptonic transverse momenta, low jet multiplicity, and a sufficiently large
opening angle between the e and µ, the Run II experiments should be able to overcome
the Standard Model backgrounds from Z 0 , tt̄ and W W production. A Z’ boson with a
mass up to 700 GeV (depending on mixing angle) should be visible, as indicated in figure
43.
Various methods of finding W ′ and Z ′ at the LHC, NLC or FMC have been suggested
in the context of topflavor models [350, 351, 352, 353]; in principle these should work
equally well for dynamical models. Calculations of expected production rates indicate
27
Exchange of the ETC boson that generates mt does not modify the W tb vertex, because the boson
does not couple to all of the required fermions: (tR , bR , UR , DR ).
99
that a number of processes are worth further study. The presence of a W ′ could boost
single top production at the LHC to a rate rivaling that of tt̄; the distinctive final state
would include a pair of b-jets and a high-pT lepton. Both single-lepton and dilepton
production at the LHC could be visibly modified by new weak bosons. A Z ′ boson might
visibly alter the rate of di-muon production at an NLC or that of tt̄ production at an
NLC, FMC or LHC. Since only the left-handed couplings of the top quark would be
affected, top angular distributions might also be be altered. Finally, if flavor-changing
mixing between second and third-generation leptons were large, production of unlike-sign
µτ pairs might be seen. Because the VH VL VL triple gauge boson coupling vanishes to
leading order, di-boson production will not reveal the presence of a W ′ or Z ′ .
3.7
3.7.1
Supersymmetric and Bosonic Technicolor
Supersymmetry and Technicolor
Shortly after the introduction of Technicolor, Witten described the general aspects of
a Supersymmetric Technicolor theory [65], while Dine, Fischler, and Srednicki [66] [67]
constructed explicit models. Several interesting possibilities arise when the ideas of Supersymmetry (SUSY) and TC are combined. Notably, it becomes possible to raise the
scale of ETC [358], and thus suppress the dangerous γ terms. One might even be able to
achieve dynamical electroweak symmetry breaking in concert with Supersymmetry breaking. These models are unfashionable at present, relative to the MSSM which guarantees
a low-mass Higgs boson accessible to Run IIb or the LHC. Nonetheless, the general idea
has some theoretical merit and the models have interesting and accessible phenomenology
(including the potential for light fundamental scalars).
6 0 is dynamically generated
In ordinary QCD we know that a chiral condensate, hqqi =
when the gauge coupling becomes strong. In Supersymmetric QCD a quark q possesses
a superpartner, denoted q̃, the squark. The supercharge, Q, generates a transformation
on these fields of the form:
{Q, q̃ † q + q † q̃} = qq
(3.124)
6 0 implies that:
Hence, the existence of the fermionic condensate hqqi =
Q|0 >6= 0
(3.125)
Whenever Q does not annihilate the vacuum it means that SUSY is broken. The field
q̃ † q + q † q̃ then becomes a massless Goldstone fermion, a Goldstino. Thus, SUSY breaking
and dynamical EWSB can have an intimate connection through fermion condensation.
In addition to the key papers mentioned above, we refer the interested reader to futher
works on Supersymmetric Technicolor, refs.[359, 360, 361, 362, 67], and to ref.[363], which
discusses the combination of Topcolor with SUSY.
100
3.7.2
Scalars and Technicolor: Bosonic Technicolor
The key advantage of introducing fundamental scalars into TC is to provide an alternative
to ETC. Most of the constraints and problems of ETC can be dismissed by assuming
general masses and couplings of the fundamental scalar sector. We thus consider theories
in which the additional fields that communicate the TC condensates to the ordinary
quarks and leptons are fundamental scalars. We view these scalars as ultimately associated
with Supersymmetry, where their masses can be protected by fermionic chiral symmetries
[364, 365, 366, 367, 368, 369, 360]. Alternatively, these scalars may, in principle, be bound
states arising within a high energy strongly coupled theory [275, 276]. Conceivably, they
can also be viewed as relics from extra dimensions, such as Wilson lines (Section 4.6).
In what follows, we momentarily disregard the details of the higher-energy physics that
enables the scalars to have masses of order the weak scale and consider the key features
and phenomenology of TC models with fundamental or effective scalars. We classify the
models according to the weak and TC charges of the scalar states.
(i) Weak-doublet Techni-singlet Scalars
We begin by discussing TC models whose spectrum includes one or more weak-doublet
TC-singlet scalars. These correspond to a natural low-energy limit of strongly-coupled
ETC models [275]. Moreover, the presence of a weak-doublet techni-singlet scalar has a
sufficiently large effect on the vacuum alignment of the technifermion condensate to make
an SU(2) TC gauge group viable [370].
In the minimal model of this type [258], one adds to the Standard Model gauge and
fermion sectors a simple SU(N) TC sector, with two techniflavors that transform as a lefthanded doublet ΥL = (pL , mL ) and two right-handed singlets, pR and mR , under SU(2)W ,
with weak hypercharge assignments Y (ΥL ) = 0, Y (pR ) = 1/2, and Y (mR ) = −1/2. The
technifermions and ordinary fermions each couple to a weak scalar doublet φ = (φ+ φ0 )T
which has the quantum numbers of the Higgs doublet of the Standard Model. Unlike
the Standard Model Higgs doublet, φ has a nontachyonic mass (Mφ2 ≥ 0) and is not the
primary source of electroweak symmetry breaking.
The scalar has Yukawa couplings to the technifermions:
LφT = ῩL φ̃ λ+ pR + ῩL φ λ− mR + h.c.
(3.126)
When the technifermions condense, these couplings cause φ to acquire an effective VEV,
hφi = f ′ ≈ 4πλT f 3 /Mφ2 . Because the scalar couples to ordinary fermions as well as
technifermions, the ordinary fermions obtain masses from the diagram sketched in Figure
44
4πf 3
mf ≈ λf h 2 .
(3.127)
Mφ
where h is the scalar coupling to the ordinary fermions. The coupling matrices λf are
proportional to the mass matrices mf and drive flavor symmetry breaking. The quark
101
-
TL
fR
φ
fL
TR
Figure 44:
Fermion-technifermion interaction through scalar exchange. This diagram is responsible for fermion mass generation once the technifermions condense.
flavor symmetries are broken according to the same pattern as in the Standard Model
so that the quarks mix via the usual CKM matrix and the standard GIM mechanism
prevails.
Both the technipion decay constant and the scalar VEV contribute to the electroweak
scale: f 2 + f ′2 = v02 . The technipions and the isotriplet components of φ mix. One linear
combination becomes the longitudinal component of the W and Z; the orthogonal one
remains in the low-energy theory as an isotriplet of physical scalars [371]. The coupling of
the charged physical scalars to quarks has the same form as in a type-I two-Higgs doublet
model [371]. Imposing the requirement that the isoscalar component of φ have no VEV
enables one to eliminate f and f ′ and calculate observables in terms of Mφ , h and λ. This
model has been studied in the limits (i) in which λ is negligible and one works in terms
of (Mφ , h) [258, 259, 371], and (ii) in which Mφ is negligible and one works in terms of
(λ, h) [260, 371]. The leading Coleman-Weinberg corrections to the scalar potential have
been included in studies of both limits.
The phenomenology of TC models with weak-doublet scalars has been found to be
in agreement with experiment. Models of this kind do not produce unacceptably large
contributions to K 0 -K 0 or B 0 -B 0 mixing, nor to the electroweak S and T parameters
[258, 259, 260]. In addition, the new scalars in the model can be made heavy enough to
evade detection, even in the limit where the scalar doublet is assumed to have a vanishing
SU(2) × U(1) invariant mass [260]. There are negative corrections to Rb , but they never
exceed −1% in the regions of parameter space allowed by other constraints; likewise, the
rate for b → sγ is less than in the Standard Model, but not so altered as to conflict with
experiment [371]. Current bounds on the parameter space of the model [372] are shown
in Figures 45 and 46; in creating the plots, the parameters Mφ (limit i) and λφ (limit ii)
have been eliminated in favor of mass of the physical isoscalar state, mσ .
Supersymmetrized models with weak-doublet techni-singlet scalars were introduced
in refs.[364, 365] and aspects of flavor physics and renormalization group evolution were
studied in refs.[366, 367, 368, 369]. The minimal such model [364, 365] contains all the
fields of the Minimal Supersymmetric Standard Model (MSSM), a set of SU(N)T C gauge
bosons and their superpartners, and color-singlet technifermion superfields transforming
102
Figure 45: Constraints on technicolor with scalars in limit [i], where the scalar self-coupling is
negligible, plotted in the physical basis (mσ , h). The allowed region of parameter space (shaded)
is bounded by the contours mσ = 114 GeV (solid), Rb − RbSM = 0.26% (dashes) and hf ′ = 4πf
(dot-dash). Other contours of constant Rb are shown for reference. The current bound from
searches for charged scalars mπp± = 79 GeV is shown (long dashes) along with the reference
curve mπp± = mt − mb . The constraint from B 0 B̄ 0 mixing is labeled “B-line”. [372]
under SU(N)T C × SU(2)W × U(1)Y as
TUR ≡ (NT C , 1, 1/2),
TDR ≡ (NT C , 1, −1/2),
TL ≡ (NT C , 1, 0).
(3.128)
The superpotential includes WSSM from the MSSM and an additional part WHT C from
Yukawa couplings of the two Higgs superfields HU and HD to the technifermion superfields
T.
WHT C = gU HU TUR TL + gD HD TDR TL
(3.129)
R-parity is imposed, just as in the MSSM; the technifermions transform like matter.
The Higgs fields have positive squared masses in the perturbative vacuum. The superpotential terms WHT C , however, produce terms linear in the Higgs fields in the Lagrangian when the technifermions condense [364, 365]. This causes the Higgs fields to
acquire VEV’s:
hHU i = gU hTUR TUL i/m2HU ,
hHD i = gD hTDR TDL i/m2HD .
(3.130)
Thus for hT T i ∼ (600GeV)3 , gU ∼ 1 and mHU ∼ 1 TeV, one obtains hHU i ∼ 100 GeV,
yielding a realistic top quark mass. If mHD > mHU , the VEV of HD will be much smaller
than that of HU , producing the required top-bottom mass splitting. In variant models
103
Figure 46: Constraints on technicolor with scalars in limit [ii], where the scalar mass is
negligible, plotted in the physical basis (mσ , h). Curves labeled as in Figure 45. From [372].
with larger technifermion content, WHT C generates technifermion current-algebra masses,
yielding masses of order 200 GeV – 1 TeV [364, 365] for the technipions in the spectrum.
These supersymmetrized TC models minimize the FCNC problems that usually affect
SUSY and TC models [366, 367, 368, 369]. In the minimal version, scalar exchange
among quarks and leptons generates no tree-level FCNC. If additional Higgs multiplets are
introduced to explain the hierarchy of fermion masses and mixings, tree-level FCNC may
be re-introduced; scalar masses of order ∼ 10 TeV suffice to suppress them. Furthermore,
because the lightest Higgs bare mass mHU is of order ∼ 1 TeV, superpartner masses of
order ∼ 1 − 10 TeV become natural, reducing the degree of low-energy squark and slepton
mass degeneracy required to avoid undue FCNC from loops.
(ii) Weak-singlet Technicolored Scalars
Theories which include weak-singlet technicolored scalars address the intergenerational
fermion mass hierarchy more directly. In the early model of ref.[373], exchange of technicolored weak-singlet scalars induces four-fermion interactions between a trio of technifermions and one fermion. Ordinary fermions then mix with technibaryons and become
massive. Unfortunately, this model predicts unacceptably large FCNC and tree-level contributions to the T parameter. A more natural account of the mass hierarchy is provided
by models [369, 374, 360, 375] in which exchange of technicolored weak-singlet scalars induces four-fermion interactions between pairs of fermions and technifermions. The small
CKM elements associated with the third-generation quarks arise because the tL and bL
104
TU
h
Q
i
L
i
<T Ti >i
TU
x
ωc
x
UL
ω
R
~
g
i
h
x
~
g
ωc x ω
U
i
UL
UR
(a)
UL
x
<T Ti >i
UR
UR
(b)
Figure 47: (a) Tree-level mass for two generations of fermions due to technifermion condensation. (b) Radiative mass contribution from squark and gluino exchange. Analogous contributions
to charged lepton masses arise from exchange of ξ and electroweak gauginos.
mass eigenstates are automatically aligned. However, because the scalar can couple to
both tR and bR , the top-bottom mass ratio must be provided by a ratio of Yukawa couplings. We will discuss three models with interesting features in more detail.
A supersymmetric model with technicolored weak-singlet scalars and some interesting
mass phenomenology was introduced by Kagan [369]. The gauge group is SU(NT C ) ×
SU(3) × SU(2) × U(1); three families of quark and lepton superfields and two sets (T1 , T2 )
of the color-singlet technifermion superfields as in equation (3.128) are present. The
Higgs sector, consists of a vector-like pair of color triplet Higgs fields transforming under
SU(NT C ) × SU(3) × SU(2) × U(1) as ω c ≡ (NT C , 3̄, 1, −1/6) and ω ≡ (NT C , 3, 1, 1/6) and
a vector-like pair of color singlet fields ξ c ≡ (NT C , 1, 1, 1/2) and ξ ≡ (NT C , 1, 1, −1/2).
While only two generations of quarks receive masses at tree-level in this model, gaugino
and sfermion exchange enable the first generation fermions to receive radiative masses,
as indicated in Figure 47. SUSY breaking masses can be in the 1 − 10 TeV range and
little or no squark and slepton mass degeneracy is needed. For NT C = 3, adding a
pair of technivector, color, weak- and hypercharge singlet Higgs superfields induces fourtechnifermion interactions of the form TL TL TUR TDR . When technifermion condensation
occurs, this generates large current masses for the technifermions, usefully enhancing
the masses of the technipions. Phenomenology associated with techniscalar-generated
chromomagnetic moments for the quarks and several scenarios for quark mass generation
in non-supersymmetric versions of the model are discussed in [369, 376].
Dobrescu has studied a related supersymmetric TC model with only one doublet of
technifermions [360]. Only the third generation fermions acquire large masses and the
other two generations’ smaller masses are generated radiatively by sfermion and gaugino
exchange. The predicted rates of neutral K and B meson mixing, CP violation in KL
and B meson decays, and CP asymmetries in B decays or ∆S = 1 transitions, are all
consistent with experiment.
Dobrescu and Terning [375] have also shown that a TC model with technicolored
weak-singlet scalars can produce a negative contribution to the S parameter. This model
includes the Standard Model fields, an SU(NT C ) technicolor gauge group, two flavors of
105
technifermions ΨR ≡ (PR , NR ) ≡ (NT C , 1, 2, 0), PL ≡ (NT C , 1, 1, 1), and NL ≡ (NT C , 1, 1, −1)
and three technicolored scalars transforming as φ ≡ (NT C , 3̄, 1, −1/3), ωt ≡ (NT C , 3̄, 1, −7/3),
and ωb ≡ (NT C , 3̄, 1, 5/3). The couplings of the Z boson to the (t, b)L doublet (δgL ) and
the tR and bR quarks (δgRt , δgRb ) are found to be shifted by amounts depending on the
Yukawa couplings and the scalar masses. The model therefore predicts that the value of
the S parameter is reduced from the typical one-doublet techifermion contribution by an
amount depending on those couplings:
S ≈ 0.1NT C − 1.02 3δgL − 2δgRt + δgRb .
(3.131)
(iii) Weak-doublet Technicolored Scalars
Models incorporating weak-doublet technicolored scalars [370] can explain not only the
intergenerational fermion mass heirarchy, but also some elements of the intragenerational
mass hierarchies. These models [370] include the Standard Model gauge and fermion
sectors together with a minimal TC sector. This takes the form of an asymptotically free
SU(NTC ) gauge group, which becomes strong at a scale of order 1 TeV, and one doublet
of technfermions which transform under the SU(NTC ) × SU(3)C × SU(2)W × U(1)Y gauge
group as ΨL ≡ (PL , NL ) = (NT C , 1, 2)0 , PR = (NT C , 1, 1)+1 and NR = (NT C , 1, 1)−1.
The large top quark mass is generated by a scalar multiplet, χt , which transforms under
the SU(N)TC × SU(3)C × SU(2)W × U(1)Y gauge group as: (NTC , 3, 2)4/3 . Its Yukawa
interactions may be written without loss of generality as
Lt = Cq qL3 N R χt + Ct ΨL tR iσ2 χt† + h.c. ,
(3.132)
where qL3 ≡ (tL , bL )⊤ is the left-handed weak eigenstate t − b quark doublet, and the
Yukawa coupling constants, Cq and Ct , are defined to be positive. Below the scale of
technifermion condensation, scalar exchange generates a top quark mass. Because the
hypercharge of χt allows it to couple to tR but not to bR , the model as described gives
mass only to the top quark. If other fermion masses arise from physics above the TC
scale, it will be natural for the top to be the heaviest fermion. Such physics could be
a weak-doublet techni-singlet scalar like that described earlier, or a set of technicolored
scalars. In any case, the oblique corrections and Z-pole observables predicted by this class
of models are found to be consistent with experiment [370].
106
4
Top Quark Condensation and Topcolor
The large top quark mass is suggestive of new dynamics, potentially associated with
electroweak symmetry breaking. The early papers attempt to identify all of the EWSB
with the formation of a top quark condensate, and thus a bound-state Higgs composed
of t̄t. From this perspective, in contrast to TC, the W and Z and the top quark are the
first order massive particles; all other quarks and leptons viewed as a priori massless. We
first discuss these pure top condensation scenarios and then review more realistic models
incorporating top condensation into a larger dynamical framework.
4.1
4.1.1
Top Quark Condensation in NJL Approximation
The Top Yukawa Quasi-Infrared Fixed Point
Top quark condensation is related to the quasi-infrared fixed point of the top quark
Higgs–Yukawa coupling, first discussed by Pendleton and Ross [377] and Hill, [378], [379],
with implications for the Higgs boson mass, [380]. At issue are particular properties of
the solutions to the Renormalization Group (RG) equations for the top Yukawa coupling
constant. The Pendleton-Ross fixed point corresponds to a fixed ratio gt /gQCD [377] and
is the basis of the idea of “reduction of coupling constants” schemes (e.g., see [381]). It
defines a critical trajectory above which gt has a Landau pole, and below which gt is
asymptotically free. It predicted mt ∼ 120 GeV.
On the other hand, the formation of a Higgs bound-state composed of t̄t implies a very
large top quark Higgs-Yukawa coupling constant at the scale of the binding interaction
M. If the top quark Higgs–Yukawa coupling is sufficiently large at a high energy scale M
(e.g., if gt (M) is at least of order unity) then one obtains a robust low energy prediction
of mt = gt (mt )vwk , e.g. for M ∼ 1015 GeV of order mt ∼ 220 GeV in the Standard Model
[378], [379]. (or mt ∼ 200 sin β GeV, in the MSSM [382]). For arbitrary but sufficiently
large initial gt (M) the low energy value of gt (vwk ) is determined by the RG equations
alone. It remains logarithmically sensitive to M, but becomes insensitive to the initial
gt (M) boundary condition. The term “quasi-infrared fixed point” refers to this solution
for the top-Yukawa coupling, and is the RG improved solution of top quark condensation
schemes [383].
4.1.2
The NJL Approximation
Nambu discussed the idea of a t̄t condensate in the context of generic chiral dynamics,
[384]. This was independently, and more concretely, developed by Miransky, Tanabashi,
Yamawaki, et al. [385, 386], who placed the idea firmly in the context of an NJL model
and derive the relationship between the top quark mass and electroweak scale through
the Pagel’s-Stokar formula. It was subsequently elaborated by Marciano [387], [388], and
Bardeen, Hill and Lindner et al. [383]. The latter paper provides a detailed analysis of
107
the scheme with the connection to, and improvement by, the full renormalization group.
The Higgs boson is shown to be a “deeply bound state” [389] composed of t̄t, where the
top and Higgs masses are predicted by the quasi-infrared fixed point [379, 380].
The NJL model for top quark condensation must be considered as an approximation
to some supposed new strong dynamics. Indeed, the discussion of models only becomes
complete when a concrete proposal for the new dynamics is given, e.g., a new gauge
interaction, “Topcolor,” [304] which we discuss in the next section. Technically, the
Standard Model can always be rewritten as an NJL model for any fermion, by a sufficiently
arbitrary and wide range of choices of higher dimension operators at the composite scale,
as emphasized by Hasenfratz et al. [390]. For example, combining certain d = 8 operators
with the d = 6 NJL interaction, restricting oneself to the fermion bubble approximation,
and choosing coefficients for these operators to be absurdly large, of order ∼ 106 , one can
argue that the Higgs is composed of the electron and positron! In doing this, however,
one is perversely tuning cancellations of the coupling of composite Higgs boson to its
constituents in the infrared, a phenomenon one does not expect in any natural or realistic
dynamics.
In Topcolor the additional operators are under control, and one can estimate the
coefficients to be small, <
∼ O(1). Indeed, Topcolor was invented [304] to address the
criticism raised in ref.[390]. There are proposals other than Topcolor, based upon strongly
coupled U(1) or non–gauge interactions, which also aim to provide a concrete basis for
the new strong interaction, see refs. [391], [392], and [393]. As these interactions are not
asymptotically free, it is difficult to understand how they become strong in the infrared or
unified at high energies. Note that U(1) interactions, moreover, are typically subleading in
Nc and the NJL model is a large-Nc approximation. Topcolor thus has an advantage; we
will have a new dynamics such as Topcolor in mind throughout the following discussion.
We can implement the notion of top quark condensation by adapting the Nambu–JonaLasinio (NJL) model following ref.[385, 386]. A new fundamental interaction associated
with a high energy scale, M, involving principally the top quark, is postulated as a fourfermion interaction potential:
V ∼−
g2 a
(ψ̄ tRa )i (t̄Rb ψ b )i + ...
M2 L
(4.133)
where (a, b) are color indices and (i) is an SU(2)L index. This is viewed as a cut-off theory
at the scale M, (i.e., the interaction presumeably softens due to topgluon exchange above
this scale). For any g 2 this interaction is attractive and will form a bound-state boson,
H ∼ ψ̄L tR . With sufficiently large (supercritical) g 2 > gc2 the interaction will trigger
the formation of a low energy condensate, hHi ∼ ht̄ti. The condensate has the requisite
I = 1/2 and Y = −1 quantum numbers of the Standard Model Higgs boson condensate,
enabling it to break SU(2) × U(1) → U(1) in the usual way.
The subsequent analysis is a straightforward application of the NJL model, as discussed
in Appendix B. For supercritical g 2 > gc2 the theory undergoes spontaneous symmetry
breaking. The associated Nambu–Goldstone modes become the longitudinal W and Z. In
108
the fermion–loop approximation, one obtains the Pagels-Stokar formula which connects
the Nambu–Goldstone boson decay constant, fπ = vwk , to the constituent quark mass,
[385, 386, 383]:
M2
Nc 2
2
m
(log
+ k).
(4.134)
fπ2 = vwk
=
16π 2 t
m2t
Here mt is the top quark mass, which is now the dynamical mass gap of the theory. The
constant k is associated with the precise matching onto, and definition of, the high energy
theory; it comes from the ellipsis of eq.(4.133), and in Topcolor models k <
∼ O(1).
Now, if we take the cut–off to be M ∼ 1 TeV and k ≈ 1 we predict too large a top
quark mass, mt ∼ 600 GeV. On the other hand, with very large M ∼ 1015 GeV and k ≈ 1
we find remarkably that mt ∼ 160 GeV. Moreover, note that for large M we become
systematically less sensitive to k.
Unfortunately, in the limit of very large M >> vwk the model is extremely fine-tuned.
This is seen from the quadratic running of the (unrenormalized) composite Higgs boson
mass in the NJL model, e.g., eq.(B.5):
m2H (µ) =
2Nc
M2
−
(M 2 − µ2 )
2
g
(4π)2
(4.135)
Taking µ → vwk , we see that, in order to have M >> vwk ∼ mH , the coupling constant
of the NJL theory must be extremely fine-tuned:
g 2 Nc
2
= 1 + O(vwk
/M 2 )
8π 2
(4.136)
This implies extreme proximity of g 2 to the critical value gc2 = 8π 2 /Nc . If M ∼ 1015 GeV
then the coupling must be fine-tuned to within ∼ 1 : 10−30 of its critical value.
Critical coupling corresponds to a scale invariant limit of the low energy theory, as
in the case of critical coupling in a second order phase transition in condensed matter
physics. Choosing a near-critical coupling essentially tunes the hierarchy between the
large scale ∼ M and the weak scale vweak by having approximate scale invariance over a
large “desert.”
In the limit of M >> vwk one can reliably use the renormalization group to improve
the predictions for the low-energy top and Higgs masses. The full QCD and Standard
Model contributions can be included to arbitrary loop order. For the top quark, the main
effect is already seen in the one-loop RG equation:
16π 2
3 3
∂gt
2
gt − (Nc2 − 1)gt gQCD
= Nc +
∂ ln µ
2
(4.137)
with the “compositeness boundary condition”:
Nc
1
→
ln(M 2 /µ2)
2
gt
(4π)2
109
µ→M
(4.138)
The boundary condition states that if the Higgs doublet is a pure t̄t bound state, then the
Higgs-Yukawa coupling to the top-quark must have a Landau pole at the composite scale
M. The low energy value of gt , as given by the solution to eq.(4.137), is the quasi-infrared
fixed point, given by the approximate vanishing of the rhs of eq.(4.137). The fixed point
is only ln(ln(M)) sensitive to the UV scale M. The low energy quasi-infrared fixed point
prediction in the Standard Model of mt ≈ 220 GeV with M ∼ 1015 GeV [378, 379] is large
in comparison to the observed mt = 176 ± 3 GeV. The result does, however, depend on
the exact structure (e.g., the particle content) of the high energy theory, and one comes
sufficiently close to the physical top mass that perhaps new dynamics can be introduced
into the Standard Model to fix the prediction.
In the MSSM, for example, one obtains mt ∼ 200 sin β GeV, which determines a predicted tan β when compared to the experimental mt [382], [394]. Top quark condensation
has been adapted to supersymmetric models [395], but a problem arises. Essentially, the
nonrenormalization of the superpotential implies that the effective bound state Higgs mass
does not run quadratically; it runs only logarithmically owing to the Kahler potential, or
kinetic term, renormalization effects. Hence, the requisite new strong interaction scale M
must be of order vweak , and cannot be placed in any sensible way at ∼ MGU T . This may be
acceptable in the context of top-seesaw models (Section 4.4), but the general viability of
Supersymmetry in these schemes has not been examined. While the quasi-infrared fixed
point does play an important role in the MSSM [382], [394], it appears that the naive
compositeness interpretation is difficult to maintain in SUSY.
There have been many applications of these ideas to other specific schemes; we mention
a few variants here. M. Luty demonstrated how to produce multi-Higgs boson models in
the context of top and bottom condensation [396] (see also [397], [398]). The problem of
generating the masses of all third generation fermions has been attacked in this context
by various authors [399], [400].
Yet another application of these ideas is to the neutrino spectrum, and the formation of
neutrino condensates; Hill, Luty and Paschos have considered the dynamical formation of
right-handed neutrino Majorana condensates and the see-saw mechanism, [284] [284] (see
also: [288]). Some of these models include additional exotic matter such as a sequential
fourth generation or leptoquark bound states [401].
Suzuki has discussed the reinterpretation of the compositeness condition as indicating
formation of a composite right-handed top quark [402]. Such composites in the NJL context are necessarily Dirac particles, and the Suzuki model affords an interesting variation
on the NJL model in which a composite Dirac fermion arises (this of course happens
automatically in the composite SUSY models). For related comprehensive reviews of top
condensation which describe other variant schema see: [403], [57] and [404].
The key problem with top condensation models is that either (i) the new dynamics
lies at a very high energy scale, and the top mass is predicted, but there is an enormous
amount of fine–tuning, or (ii) the new dynamics lies at the TeV scale, hence less finetuning, yet the predicted top quark mass is then too large. We turn presently to a
discussion of more natural models of class (ii), based upon Topcolor, which can provide
110
remedies for these problems.
4.2
4.2.1
Topcolor
Gauging Top Condensation
Our previous discussion of top condensation noted that the relevant interaction Lagrangian, eq.(4.133), must be viewed as an effective description of a more fundamental
theory. Let us presently consider what that theory might be. A key observation is that a
Fierz rearrangement of the interaction term leads to [304]:
−
A
g2
λA i
g2 a
b i
µλ
(
ψ̄
t
)
(
t̄
ψ
)
=
(
ψ̄
γ
ψ
)(
t̄
γ
tR ) + O(1/Nc )
Ra
i
Rb
iL
µ
R
M2 L
M2
2 L
2
(4.139)
where Nc = 3 is the number of colors. This is exactly the form (including the sign)
induced by a massive color octet vector boson exchange, and suggests a new gauge theory
with certain properties: (i) it must be spontaneously broken at a scale of order M; (ii)
it must be strongly coupled at the scale M to produce deeply bound composite Higgs
bosons and trigger chiral condensates; (iii) it must involve the color degrees of freedom
of the top quark, analogous to QCD. The relevant models therefore involve embedding of
QCD into some large group G which is sensitive to the flavor structure of the Standard
Model [304,274].
¿From this point of view the embedding of QCD into a minimal SU(3)1 ×SU(3)2 gauge
group at higher energies seems a plausible scenario28 . The second (weaker) SU(3)2 gauge
interactions act upon the first and second generation quarks while the first (stronger)
SU(3)1 interaction acts upon the third generation. This drives the formation of the top
condensate ht̄ti = 0. Clearly some additional dynamics is then required to suppress the
formation of the b-quark consdensate, hb̄bi = 0.
We note that a different class of models based upon a strong U(1) gauge interaction
has also been proposed by Bonische, [405], Giudice and Raby, [241], and Lindner and
Ross, [392] in which the top quark carries the extra, strong U(1) charge. The desired NJL
interaction term of eq.(4.133) occurs, but with a 1/Nc suppression.
Topcolor Assisted Technicolor (TC2) [304,274] postulates that the top quark mass
is large because it is a combination of (i) a small fundamental component, ǫmt << mt
generated by, e.g., ETC or a fundamental Higgs boson, plus (ii) a large dynamical quark
mass component, (1−ǫ)mt ≈ mt , generated by Topcolor dynamics at the scale M ∼ 1 TeV,
which is coupled preferentially to the third generation. In a pure top condensation model
we would have ǫ = 0, and we produce the usual three NGB’s that become longitudinal
WL± and ZL . With nonzero ǫ we are relaxing the requirement that the ht̄ti condensate
account for all of the electroweak symmetry breaking.
28
We will see in Section 4.5 that such a structure occurs in theories with extra dimensions, and
anticipates the idea of “deconstruction.”
111
Figure 48: Fierz rearrangement of the attractive Nambu-Jona-Lasinio interaction contains the
color current-current interaction to leading order in 1/Nc .
The ETC component of the top quark mass, the ǫmt term, is expected to be ∼ mb .
Hence we assume ǫ <
∼ 0.1. Furthermore, the b-quark can receive mass contributions from
instantons in SU(3)1 [406,304], so that the fundamental EWSB sector (e.g., ETC) needs
to provide only a very small, or possibly none of the, fundamental contribution to mb .
Remarkably the strong CP–θ1 angle in the Topcolor SU(3)1 can provide the origin of
CKM CP-violation [407].
After all of the dynamical symmetry breaking there are three NGB’s from the TC
sector, and three NGB’s from top condensation sector. One linear combination of these,
mostly favoring the TC NGB’s, will become the longitudinal WL± and ZL . The orthogonal
linear combination will appear in the spectrum as an isovector multiplet of PNGB’s, π̃ a .
These objects acquire mass as a consequence of the interference between the dynamical
and ETC masses of the top quark, i.e., the masses of the π̃ a will be proportional to ǫ.29 We
refer to the π̃ a as top–pions (note that in the minimal pairing of the ψL = (t, b)L doublet
with the tR singlet there is no singlet top-η ′). For ǫ <
∼ 0.05 − 0.10, we will find that the
top–pions have masses of order ∼ 200 GeV. They are phenomenologically forbidden from
occuring much below ∼ 165 GeV [314] due to the absence of the decay mode t → π̃ + + b.
Because Topcolor and low-scale TC address complementary issues, it is natural to
combine these schemes into TC2. [315,52]. Happily, each half of the model appears
to solve many difficulties experienced by the other. For example, since the top quark
couples only weakly to ETC in TC2 models, the problem of preventing t → P + + b in
Low-Scale TC is avoided. Moreover, since ETC need not now provide the full top or
bottom quark masses, the devastating FCNC constraints on ETC are alleviated (perhaps
including some “walking”), producing a viable Technicolor component of the TC2 model.
Even the S parameter is likely to be less problematic. We do, however, have to deal
with FCNC’s induced by mass mixing of t and b with light quarks, and off-diagonal light
PNGB couplings. These can evidently be controlled by systematic choices of fermion
29
Technicolor itself can be a spontaneously broken theory with NJL-like dynamics [68]. This has the
benefit of suppressing resonant contributions to the S and T parameters, and may be part of the TC2
scenario described below.
112
mass matrix textures, and the model again appears viable. For discussion of these flavor
physics questions, we refer the reader to Section 4.2.4, to [407], which elaborates on the
issues raised in [408], and also to [409] and [410].
We mention some variant applications and elaborations of Topcolor. “Bosonic Topcolor,” [411,412], is based on maintaining the fundamental sector as an elementary Higgs
boson, similar in spirit to Bosonic Technicolor (section 3.7.2) and is consistent with current
phenomenological constraints. We view Bosonic Topcolor or TC as a useful framework for
identifying acceptable model-building directions, with the eventual aim of replacing the
fundamental Higgs boson with something dynamical or Supersymmetric [413]. Bosonic
Topcolor anticipates a class of models in which SUSY arrives, not near the EWSB scale,
but rather closer to the multi-TeV scale. “Topcolor Assisted Supersymmetry,” provides
a mechanism for solving the flavor problems in the Sfermion sector [363]. Some authors
[414] have argued for a natural origin of Topcolor-like structures in grand-unified models,
and models with mirror fermions [415,416]. Applications to left-right symmetry breaking
have also been considered in [287]. Finally, we note that related models involving the idea
of a horizontal replication of SU(2)L → G, where G contains some strong interactions,
have been explored under the rubrics of non-commuting extended technicolor [271], (see
section 3.3.2) and top-flavor [350,351,417]. A potent constraint, often overlooked in these
schemes, is the presence of potentially dangerous large instanton effects that violate B +L
over some portion of the parameter space of the model [406,304]).
4.2.2
Gauge Groups and the Tilting Mechanism
In Topcolor models, a new strong dynamics occurs primarily in interactions that involve
tttt, ttbb, and bbbb. The dynamics must cause the top quark to condense, htti =
6 D 0,Ewhile
simultaneously suppressing the formation of a large bottom quark condensate, bb ≈ 0.
This requires “tilting” the vacuum.
′
Common solutions involve introducing additional
D E strong U(1) interactions that are
attractive in the tt channel and repulsive in the bb channel [304]. This can take the
form of imbedding the weak hypercharge, U(1)Y → U(1)Y 1 × U(1)Y 2 , since Y has the
desired properties. There are then constraints on the running of the strong U(1) couplings
and the demands of tilting. Effective field theory analysis of tilting indicates it is not an
unreasonable possibility, but may require some fine-tuning at the few percent level [418].
(i) Classic Topcolor
Topcolor with a U(1)′ tilting mechanism leads to the following gauge group structure
at high energies [304]:
SU(3)1 × SU(3)2 × U(1)Y 1 × U(1)Y 2 × SU(2)L → SU(3)QCD × U(1)EM
(4.140)
where SU(3)1 × U(1)Y 1 (SU(3)2 × U(1)Y 2 ) couples preferentially to the third (first and
second) generations. The U(1)Y i are just rescaled versions of electroweak U(1)Y The
113
fermions are then assigned to (SU(3)1 , SU(3)2 , Y1, Y2 ) as follows:
(t, b)L ∼ (3, 1, 1/3, 0)
(ντ , τ )L ∼ (1, 1, −1, 0)
(u, d)L, (c, s)L ∼ (1, 3, 0, 1/3)
(ν, ℓ)L ℓ = e, µ ∼ (1, 1, 0, −1)
(t, b)R ∼ (3, 1, (4/3, −2/3), 0)
τR ∼ (1, 1, −2, 0)
(4.141)
(u, d)R , (c, s)R ∼ (1, 3, 0, (4/3, −2/3))
ℓR ∼ (1, 1, 0, −2)
The extended color interactions must be broken to the diagonal subgroup which can be
identified with QCD. We assume this is accomplished through an (effective) scalar field:
Φ ∼ (3, 3̄, y, −y)
(4.142)
In fact, when Φ develops a VEV the theory spontaneously breaks down to ordinary
QCD ×U(1)Y at a scale assumed to be ∼ 1 TeV, before Topcolor becomes confining.
Nonetheless, the SU(3)1 is assumed to be strong enough to form chiral condensates. The
vacuum will be tilted by theD U(1)
E Y 1 couplings which permits the formation of a htti
condensate but disallows the bb condensate is due to the U(1)Y i couplings.
The symmetry breaking will gives rise to three top-pion states near the top mass scale.
If the Topcolor scale is of the order of 1 TeV, the top-pions will have a decay constant,
estimated from the Pagel’s-Stokar
√ relation to be fπ ≈ 60 GeV, and a strong coupling
constant given by, gtbπ ≈ mt / 2fπ ≈ 2.5. If mπ̃ > mt + mb , the top-pions may be
observable in π̃ + → t + b.
Let us define the coupling constants (gauge fields) of SU(3)1 × SU(3)2 to be, respecA
tively, h1 and h2 (AA
1µ and A2µ ) while for U(1)Y 1 × U(1)Y 2 they are respectively q1 and q2 ,
(B1µ , B2µ ). The U(1)Y i fermion couplings are then qi Y2i , where Y 1, Y 2 are the charges of
the fermions under U(1)Y 1 , U(1)Y 2 respectively. A techni–condensate or explicit Higgs can
break SU(3)1 ×SU(3)2 ×U(1)Y 1 ×U(1)Y 2 → SU(3)QCD ×U(1)Y at a scale M >
∼ 240 GeV,
or it can fully break SU(3)1 ×SU(3)2 ×U(1)Y 1 ×U(1)Y 2 ×SU(2)L → SU(3)QCD ×U(1)EM
at the scale MT C = 240 GeV. Either scenario typically leaves a residual global symmetry
implying a degenerate, massive color octet of “topgluons,” BµA , and a singlet heavy Zµ′ .
A
′
′
The gluon AA
µ and topgluon Bµ (the SM U(1)Y field Bµ and the U(1) field Zµ ), are then
defined by orthogonal rotations with mixing angle θ (θ′ ):
cot θ = h1 /h2 ;
1
1
1
=
+
g32
h21 h22
(4.143)
cot θ′ = q1 /q2 ;
1
1
1
= 2 + 2;
2
g1
q1 q2
(4.144)
and:
where g3 (g1 ) is the QCD (U(1)Y ) coupling constant at MT C . We demand cot θ ≫ 1
and cot θ′ ≫ 1 to tilt the strongest interactions and to select the top quark direction
114
for condensation. The masses of the degenerate octet of topgluons and Z ′ are given by
MB ≈ g3 M/ sin θ cos θ MZ ′ ≈ g1 M/ sin θ′ cos θ′ . The usual QCD (U(1)Y electroweak) interactions are obtained for any quarks that carry either SU(3)1 or SU(3)2 triplet quantum
numbers (or U(1)Y i charges).
Integrating out the heavy bosons Z ′ and B gives rise to effective low energy four
fermion interactions. The effective Topcolor interaction mediated by B is strongest in the
third generation and takes the form:
L′T opC
!
λA
2πκ
λA
= − 2 t̄γµ t + b̄γµ b
MB
2
2
t̄γ
µλ
A
2
t + b̄γ
µλ
A
2
!
b .
(4.145)
where:
g32 cot2 θ
(4.146)
κ=
4π
This interaction is attractive in the color-singlet t̄t and b̄b channels and invariant under
color SU(3) and SU(2)L ×SU(2)R ×U(1)×U(1) where SU(2)R is the custodial symmetry
of the electroweak interactions.
In addition we have the U(1)Y 1 interaction Lagrangian (which breaks custodial SU(2)R ):
L′Y 1
2πκ1
=− 2
MZ ′
2
1
1
1
ψ̄L γµ ψL + t̄R γµ tR − b̄R γµ bR − ℓ̄L γµ ℓL − τ̄R γµ τR
6
3
3
2
2
(4.147)
where:
g12 cot2 θ′
(4.148)
4π
and where ψL = (t, b)L , ℓL = (ντ , τ )L and κ1 is assumed to be O(1). Note that while too
small a value for κ1 signifies fine-tuning in the model, phenomenological constraints limit
how large κ1 can be.
D
E
For sufficiently large κ, we trigger the formation of a low energy condensate, tt + bb ,
which breaks SU(2)L × SU(2)R × U(1)Y → U(1) × SU(2)c , where SU(2)c is a global
custodial symmetry. The U(1)Y 1 force is attractive in the tt channel and repulsive in the
bb channel. We find, in concert, the critical and subcritical values of the combinations:
κ1 =
κ+
2 κ1
> κcrit ;
9Nc
κcrit > κ −
κ1
9Nc
(4.149)
by using the large-Nc NJL (fermion loop) approximation (similar criticality constraints
were discussed previously in the NJL model of top condensation of Miransky, Tanabashi,
Yamawaki, et al. [385,386]).
The phase diagram of the model is shown in Fig.(49) from [407]. The criticality
conditions (4.149) define the allowed region in the κ1 –κ plane in the form of the two
straight solid lines intersecting at (κ1 = 0, κ = κcrit ). To the left of these lines lies the
symmetric phase, in between them the region where
D E only a ht̄ti condensate forms and to
the right of them the phase where both ht̄ti and b̄b condensates arise. The horizontal line
115
+ <ττ>
κ1
6.0
(B)
<tt>
4.0
(A)
symmetric
2.0
+ <bb>
(C)
1.0
2.0
κ
Figure 49:
The full phase diagram of the Topcolor model. The top quark alone condenses
for values of κ and κ1 corresponding to hatched region. In other regions additional unwanted
condensates turn on (for still larger κ1 a hτ̄ bi condensate forms. (A) MB = 1.0 TeV; (B)
MB = 3.0 TeV; (C) upper bound from Z → τ̄ τ (figure from [407]).
marks the region above which κ1 makes the U(1)Y 1 interaction strong enough to produce
a hτ̄ τ i condensate. This line is meant only as a rough indication, as the fermion-bubble
(large-Nc ) approximation, which we have used, fails for leptons. There is an additional
constraint from the measurement of Γ(Z → τ + τ − ), confining the allowed region to the one
below the solid curve. This curve corresponds to a 2σ discrepancy between the Topcolor
prediction and the measured value of this width. In the allowed region a top condensate
alone forms. The constraints favor a strong SU(3)1 coupling and allow a range of U(1)Y 1
couplings.
How does Technicolor know to break Topcolor when the latter triggers chiral condensates? This may involve self-breaking schemes, as considered by Martin [225], [225],
[419]. Topcolor interactions may themselves produce the condensate of Φ as a dynamical
boundstate, leading to self-breaking at the Topcolor scale. Other such scenarios require
further study.
We can also construct a model with no strong U(1) tilting interaction [304,274] called
“Type II Topcolor.” The model requires additional heavy vectorlike b’ quarks, but there
is no Z’ required for tilting because of the particular gauge charge assignments under
116
Topcolor. These models are intriguing, but have received little attention in the literature,
and we refer the interested reader to the original references [304,274,407].
(ii) Flavor-Universal Topcolor
The flavor-universal variant of Topcolor [420,331,421] is based on the same gauge
group as above, but assigns all quarks to be triplets under the strongly-coupled SU(3)1
gauge group and singlets of SU(3)2 . As a result, the heavy color-octet of gauge bosons
that results from breaking of the extended color group to its QCD subgroup are “flavoruniversal colorons” coupling to all quark flavors with equal strength [420]:
LC = g3 cot θ(C A · JCA )
where
JCµ A =
X
q̄γ µ
q
λA
q
2
(4.150)
(4.151)
The weak and hypercharge assignments of the quarks and leptons are as in Classic Topcolor, so that the properties and couplings of the Z’ boson are unchanged.
Once again, the values of κi = gi2/4π are jointly constrained by several different pieces
of physics. Above all, the top quark should be the only fermion to condense and acquire
a large mass. Applying the gauged NJL analysis [422–425] to all the Standard Model
¯ i = 0 for f 6= t. Top condensation
fermions, one can seek solutions with h t̄t i =
6 0 and h ff
30
occurs provided that [421,407]
κ3 +
2π 4
4
2
κ1 ≥
− αs − αY
27
3
3
9
(4.152)
where we again take κcrit,N JL ≈ 2π/3 [34,35]. Quarks other than the top quark will
not condense provided that additional limits on κ1 and κ3 are satisfied. In Classic TC2
models, the limit comes from requiring h b̄b i = 0, while in flavor-universal TC2 models,
an even stronger limit comes from ensuring h c̄c i = 0 [421]:
κ3 +
2 αY2
2π 4
4
<
− αs − αY
27 κ1
3
3
9
(4.153)
As shown in Figure (50), the phase diagram of the flavor-universal model includes a region
in which only top condensation occurs; the region is similar to that of classic TC2. (see
also discussion of phase plane in [424,425]).
Several types of physics further constrain the allowed region of the κ1 − κ3 plane.
As already mentioned for Classic TC2, , mixing between the Z and Z’ bosons alters
the predicted value of the Z decay width to tau leptons from the standard model value
[407,421]. The ρ or T parameter is also sensitive both to Z − Z ′ mixing [277] and to single
30
Likewise, if one applies the fermion bubble approximation to leptons, The strong U(1) interaction
will not cause leptons to condense if [421,407] κ1 < 2π − 6αY . This approximate condition is superceded
by other constraints.
117
<ττ > =/ 0
κ1
(3)
6
5.0
(4)
5
expt.
excludes
<cc>
=/ 0
κ
1
4
3.0
(1)
3
(2)
<tt> = 0
2
(5)
1.0
1
(6a)
κ3
(6b)
(6c)
0
1.4
1.5
1.6
1.6
1.7
κ3
1.8
1.8
1.9
2
2.1
2.0
Figure 50: Joint constraints on coloron and Z’ couplings (from [421]). Curves (1), (2), (3)
outline the ‘gap triangle’ where only h t̄t i =
6 0 in flavor-universal TC2 models; in ordinary TC2
models, the triangle is roughly symmetric about the lowest point. The region above curve (4) is
excluded by data on ∆ρ∗ ; the region above curve (5) is excluded by data on Z → τ + τ − . Lines
(6a-6c) are upper bounds on κ1 that hold if the U(1) Landau pole lies one, two, or five orders
of magnitude above Λ. The right-hand bound is steep because it comes from the h c̄c i = 0
constraint, eq. (4.153).
coloron exchange across the top and bottom quark loops of W and Z vacuum polarization
diagrams [426,421]. Finally, if the Landau pole of the strong U(1) interaction is to lie
at least an order of magnitude above the symmetry-breaking scale Λ, then κ1 <
∼ 1 [421].
Figure 50 summarizes these constraints on the κ1 − κ3 plane for flavor-universal TC2
models. The constraints from ∆ρ and the Landau pole also apply to the phase diagram
in figure 49. In the end, both kinds of Topcolor models are confined to a region of the
κ1 − κ3 plane in which κ1 is small and κ3 ≈ κcrit,N JL ≈ 2π/3 [421].
4.2.3
Top-pion Masses; Instantons; The b-quark mass
In TC2 a multiplet of PNGB top-pions is naturally present in the spectrum. Top-pion
masses are estimated from the loop of Figure(4.2.3), which feels both the dynamical mass
∼ mt and the explicit mass ∼ ǫmt [304,274]. The top–pion decay constant is estimated
from the Pagels-Stokar formula; using M = MB ∼ 1.5 TeV and mt = 175 GeV, it is
fπ ≈ 60 GeV. The Lagrangian for the coupling of the top-pions to quarks takes the form:
"
#
i
i
(4.154)
gbtπ̃ itγ tπ̃ + √ t(1 − γ 5 )bπ̃ + + √ b(1 + γ 5 )tπ̃ −
2
2
√
and the coupling strength is gbtπ̃ ≈ mt / 2fπ [274]. This Lagrangian, written above in
the current basis, will in general contain generational mixing when one passes to the
mass-matrix eigenbasis.
5
0
118
ε
mt
Figure 51: The interference of the explicit breaking tewrm ∝ ǫ, and the dynamical mass ∼ mt
yields an estimate of the top-pion mass.
Estimating the induced top-pion mass from the fermion loop yields:
m2π̃ =
ǫMB2
Nǫm2t MB2
=
8π 2 fπ2
log(MB /mt )
(4.155)
where the Pagels-Stokar formula is used for fπ2 (with k = 0) in the last expression. For
ǫ = (0.03, 0.1), MB ≈ (1.5, 1.0) TeV, and mt = 175 GeV this predicts mπ̃ = (180, 240)
GeV. We would expect that ǫ is subject to large renormaliztion effects and, even a bare
value of ǫ0 ∼ 0.005 consistent with ETC, can produce sizeable mπ̃ > mt .
Charged top-pions as light as ∼ 165 GeV, would provide a detectable decay mode
for top quarks [314]. Burdman has discussed potentially dangerous effects in Z → bb̄
resulting from low mass top-pions and decay constants as small as ∼ 60 GeV [239]. A
comfortable phenomenological range is slightly larger than our estimates: mπ̃ >
∼ 300 GeV
and fπ >
100
GeV.
These
values
remain
subject
to
large
uncertainties.
∼
The b receives a mass of ∼ O(1) GeV from ETC. Remarkably, however, it also obtains
an induced mass from instantons in SU(3)1 . The instanton effective Lagrangian may
be approximated by the ‘t Hooft flavor determinant31 , which is the effective Lagrangian
generated by zero-modes when instantons are integrated out. The instanton induced
b-quark mass can then be estimated as:
m⋆b ≈
3kmt
∼ 6.6 k̂ GeV
8π 2
(4.156)
where we generally expect k̂ ∼ 1 to 10−1 as in QCD, from fitting the QCD ’t Hooft
determinant to the η ′ mass. This yields a reasonable estimate of the observed b quark
mass, and for k̂ ∼ 1 it comes very close to the observed value. There also occurs an induced
top–pion coupling to bR coming from instantons. Many of the features of the theory we
have just outlined imitate the successful chiral-constituent quark model approximation to
QCD (see, e.g., [70]), thus yielding a reliable picture of Topcolor dynamics.
31
Note that the SU (3)1 CP-angle, θ1 , cannot be eliminated from the full quark mass matrix because
of the ETC contribution to the t and b masses. Indeed, it can lead to induced scalar couplings of the
neutral top–pion, and an induced CKM CP–phase, [407].
119
4.2.4
Flavor Physics: Mass Matrices, CKM and CP-violation
Topcolor has an obvious challenge: it violates the GIM symmetry by treating the interactions of third generation differently than those of the first and second. Indeed, it does
so with a new strong interaction. This contrasts with, e.g., the flavor universal coloron
model in which GIM is not violated by the strong SU(3) spectator interaction [420,410].
Of course, the Higgs–Yukawa couplings of the Standard Model violate GIM as well, and
so too must ETC interactions in Technicolor if they are to provide the observed nondegenerate fermion masses. Clearly, when we seek a deeper theory of the origin of fermion
masses we must face the true dynamical origin of GIM violation. Topcolor faces this issue
head-on at accessible energy scales. In consequence, it confronts significant constraints
and potential observable violations of Standard Model predictions.
We can study these questions in the context of Topcolor by examining the textures
of the fermion mass matrices that arise in these models. The textures are controlled
by the breaking patterns of horizontal global (flavor) symmetries [407]. The models
sketched above include a Topcolor symmetry at energies above the weak scale, presumably
subsequent to some initial breaking of a larger group structure at a much higher energy
scale M0 . The third generation fermions have different Topcolor assignments than do
the second and first generation fermions. Thus the mass matrix texture, particularly
the mixing of the light quarks and leptons with the third generation, will depend quite
strongly on the way in which Topcolor is broken. Flavor physics phenomenology associated
with mixing of the third and first generations, particularly B 0 − B̄ 0 mixing, will be seen
to constrain model-building significantly [409,410], while K physics turns out to be less
significant a probe.
Let us parameterize the electroweak symmetry breaking in TC2 by a fundamental
Higgs boson, which ultimately breaks SU(2)L × U(1)Y . This is simply shorthand for a
techniquark bilinear operator which receives a VEV of order vweak . Similarly, we consider
an effective field Φ which breaks Topcolor. We specify the full Topcolor charges of these
fields, e.g., under SU(3)1 × SU(3)2 × U(1)Y 1 × U(1)Y 2 × SU(2)L we choose:
1 1
Φ ∼ (3, 3̄, , − , 0)
3 3
1
H ∼ (1, 1, 0, −1, )
2
(4.157)
Also, let Σ = exp(iπ a T a /fπ ) be the nonlinear chiral field composed of the (bare) toppions. Then, the effective Lagrangian couplings to fermions that generate mass terms in
the up quark sector are of the form:
LMU = m0 T̄L ΣP TR + c33 T̄L tR H
†
† det Φ
+c23 C̄L tR HΦ
M04
det Φ†
+c13 ŪL tR HΦ†
M04
Φ
det Φ†
Φ
+ c31 T̄L uR H
+ c32 T̄L cR H
3
M0
M0
M0
+ c22 C̄L cR H + c21 C̄L uR H
+ c12 ŪL cR H + c11 ŪL uR H + h.c.
120
(4.158)
Here T = (t, b), C = (c, s) and U = (u, d). The mass m0 is the dynamical Topcolor
condensate top mass. Furthermore det Φ is defined by
1
det Φ ≡ ǫijk ǫlmn Φil Φjm Φkn
6
(4.159)
where in Φrs the first(second) index refers to SU(3)1 (SU(3)2 ). The matrix elements
require these factors of Φ to connect the third with the first or second generation color
indices. The down quark and lepton mass matrices are generated by couplings analogous
to (4.158).
To explore what kinds of textures arise naturally, let us assume that the ratio Φ/M0
is small, O(ǫ). The field H acquires a VEV of v. Then the resulting mass matrix is:
c11 v
c12 v
∼0
c21 v
c22 v
∼0
3
c31 O(ǫ)v c32 O(ǫ)v ∼ m0 + O(ǫ )v
(4.160)
where we have kept only terms of O(ǫ) or larger.
This is a triangular matrix (up to the c12 term). When it is written in the form
UL DUR† with UL and UR unitary and D positive diagonal, restrictions on UL and UR may
be inferred. In the present case, the elements UL3,i and ULi,3 are vanishing for i 6= 3 , while
the elements of UR are not constrained by triangularity. Analogously, in the down quark
sector DLi,3 = DL3,i = 0 for i 6= 3 with DR unrestricted. The situation is reversed when the
opposite corner elements are small, which can be achieved by choosing H ∼ (1, 1, −1, 0, 21 ).
The full CKM matrix is, as usual, given by K = UL DL† .
Triangularity, thus implies that either the matrix UL has large off diagonal elements,
while UR has small off diagonal elements (we’ll denote this case (U : 10)), or vice versa,
(U : 01). Without triangularity (U : 11) is allowed; with exact flavor symmetry we
have (U : 00). These are not fine-tunings, but rather systematic choices we make in
implementing the symmetry. They will ultimately have a deeper dynamical origin. For
an example application, we can invoke triangularity to cause the CKM matrix K = UL DL†
to be generated by pure UL rotations, with no contribution from DL , (U : 10, D : 01) or
(U : 10, D : 00), or vice versa (U : 01, D : 10) or (U : 00, D : 10).
The restrictions on the quark mass rotation matrices owing to their triangular structure
have important phenomenological consequences [407]. For instance, in the process B 0 →
B 0 there are potentially large contributions from top-pion and coloron exchange.
In the top-pion graph of Fig.(4.2.4) we find a dangerously large contribution to δm2 /m2 ,
about two orders of magnitude above the experimental limit if normal CKM angles are
assumed in a (D : 11) solution. However, owing to chirality the graph is proportional
†
to the product DL,3,1 DR,3,1
. Hence, we can systematically suppress this effect if we allow
one of the three solutions (D : 10), or (D : 01), or (D : 00), while (D : 11) is disallowed.
However, the topgluon graph yields a result that is similarly large, and proportional to
3,1 2
|DL3,1|2 + |DR
| [409]. Hence, we conclude that only the symmetric solution, (D : 00),
121
d
b
π
d
b
Figure 52: Top-pion contribution to B 0 − B̄ 0 mixing.
d
b
b
d
Figure 53: Top-gluon contribution to B 0 − B̄ 0 mixing.
is allowed, i.e., the off-diagonal mass matrix elements mixing (1, 3) in the down quark
sector are all small. This suppresses Topcolor b → sγ effects as well.
Thus, the CKM matrix elements mixing to the third generation must be controlled by
UL , and we thus require a (U : 10) or (U : 11) solution. The same effects occur in D 0 − D̄ 0
mixing, where top-pion graphs are ∝ UL UR† off-diagonal elements, and topgluon graphs
yield an expression ∝ |UL |2 + |UR |2 . Therefore, the latter result implies that D 0 − D̄ 0
must be in excess of the Standard Model prediction by about two orders of magnitude.
This is close to the current experimental limit. Thus, D 0 − D̄ 0 mixing emerges as an
intriguing probe of topcolor. Data showing that the rate of D 0 − D̄ 0 mixing is close to
the SM prediction would pose a severe difficulty for topcolor models.
There are various other effects and limits on the models that can be obtained in mostly
B and D, and to a lesser degree in K physics. Many of these are sensitive to the Z ′ which is
more model dependent. We refer the interested reader to some of the relevant literature,
beginning with [407], [427] and [428].
122
The measurement of Rb in Z → bb̄ also yields an important constraint, [429], [239],
[430]. This is subject to larger uncertainties because subleading 1/Nc top-pion loops
dominate the leading large Nc top-gluon loops. The degree of uncertainty can be seen by
examining the analysis of [239] and noting that in their exclusion figure, with the Topcolor
fπ ∼ 60 GeV the model appears to be ruled out for all reasonable top-pion masses, while
for fπ ∼ 100 GeV there is no constraint. We take the difference of these two results to be
within the errors of the subleading Nc calculation. Similar uncertainties plague efforts to
evaluate the oblique electroweak constraints on these models [430]. Also, diagrams that
generate off-shell longitudinal mixing of the Z with π 0 have not been included in any of
these analyses.
Studying questions related to flavor physics in the context of Topcolor models has
proven to be quite instructive. We now turn to other phenomenological questions and
evaluate the status of limits on Topcolor scenarios.
4.3
4.3.1
Topcolor Phenomenology
Top-pions
If Topcolor produces a t̄t condensate to elevate the top quark mass, it necessarily produces
an isovector of PNGB top-pions ∼ tb̄, tt̄−bb̄ and bt̄. There may also be a heavier top-Higgs
∼ tt̄+bb̄. In TC2 this multiplet of top-pions is uneaten by gauge bosons and the masses of
these objects have been estimated in minimal schemes to be of order ∼ 200 GeV, though
this is fairly uncertain. The top-pions are strongly coupled to t̄t, b̄b and t̄b.
Top-pion vertex corrections therefore can noticeably decrease Rb [239,430], though the
estimates are very sensitive to choice of fπ (the effect becomes negligible with fπ ∼ 100
GeV). Additional physics can in principle cancel the effect [238] (e.g., topgluons, for
example, can increase Rb [429]), but ref. [239] surveyed a number of possible sources of
cancellation and found none producing an effect of the requisite size. Exchange of the light
composite pseudoscalars in top seesaw models likewise contributes to the T parameter.
The interplay of the resulting limits is discussed in section 4.5. Several searches for top–
pion and top–higgs (σ) production have been proposed; we now review these.
(i) Neutral Top–pions, Top–Higgs
A singly-produced neutral π̂ 0 that dominantly decays to t̄c, via flavor mixing, could
0
possibly be detected [431] at Run IIb or the LHC. The
q strength of the π̂ coupling to t̄c is
governed by the model-dependent parameter Utc ≡ |UtcL |2 + |UtcR |2 where U L and U R are
the matrices that diagonalize the up-quark mass matrix. In order for the flavor-changing
decay to dominate, the decays to tt̄ must be energetically disallowed. Decays to the
electroweak gauge bosons is more model-dependent. In TC2 models, where the Yukawa
coupling of scalars coupling to tR is enhanced by r 2 ≈ (mt /fπt )2 ≈ 10, the top-scalar
coupling to vector boson pairs is likewise suppressed by 1/r2 [432]. In top seesaw models
(Section 4.4), r = 1 so that the CP-even scalar decays mainly to V V ; however, there can
123
Figure 54: Top-higgs production cross-section via gluon fusion times branching ratio to t̄c at
√
s = 2 TeV. [431]
also be a light CP-odd scalar that is unable to decay to V V and therefore decays to t̄c
instead [246].
As shown in Figure 54, the production cross-section times flavor-changing branching
ratio at Run IIb can provide several hundred top-higgs events even for Utc ∼ 0.02. Since
the top-higgs is expected to be rather narrow, the signal peak should be visible for scalar
masses up to the top threshold (see figure 55). At LHC, the production rate is dramatically
enhanced since the primary production mode is gluon-gluon fusion; the cross-section times
branching ratio is about 100 times that shown in Figure 54 for the same values of Utc .
Recent studies are optimistic for this process [433], but detailed background studies are
required.
Recent work [434–437] also suggests that π̂ 0 ’s may be visible through their t̄c decays
at a LC running in e+ e− or γγ collision mode. The cross-section for e+ e− → t̄c or
γγ → t̄c at a linear collider receives a contribution of order 20 fb from neutral top-pions;
an integrated luminosity of 50 fb−1 might make the top-pion visible [435]. On the other
hand, the processes e+ e− → γπt , Zπt , with πt → t̄c, have cross-sections of order
1.5 pb
√
and 0.3 pb, respectively for top-pion masses below 350 GeV [434]. A LC with s = 1 TeV
would produce of order 400 (100) γ t̄c (Z t̄c) events due to top-pions, potentially making
both channels observable.
(ii) Charged Top–pions
In the presence of sizeable flavor-changing couplings t̄c of the π̂t0 , a large flavor-mixing
coupling for the charged top-pions c̄bπt+ can be induced [438]. This enables charged toppions to be singly produced at sizeable rates in the s-channel at hadron colliders [439]
or photon-photon linear colliders [440]. Ref.[438] has calculated the production crosssections for several charged top-pion masses at a variety of colliders using the following
L
L
R
R
benchmark
√ set of couplings (typical for topcolor models): Utb = Ucb = 0, Utb = 5Ucb ,
gπtb = 3 2mt /v. As shown in Figure 56, the cross-sections for Run II and the LHC
124
Figure 55: Mass distribution
for the t̄c jet system for mht = 200, 300 GeV and for the W jj
√
and W bb backgrounds at
s = 2 TeV [431].
are 4.0 pb and 0.55pb, respectively; those at photon-photon colliders are also sizeable.
Hence, the authors of [438] suggest that charged top-pions would be visible up to masses
of 300-350 GeV at Run II and 1 TeV at LHC, and that light enough top-pions would
also be detectable at a linear collider running in the photon-photon mode.
In contrast, the work of ref [216] suggests that it will be difficult to detect the effects
of charged top-pions on single top production at an eγ collider in the process eγ → t̄bν.
We mention that in titling scenarios and variant schemes, there will be analogous
bottom-pions which couple strongly to b̄b. For a discussion of the associated phenomenology of these objects see [441,442,417].
4.3.2
Colorons: New Colored Gauge Bosons
Several of the dynamical models we have discussed include extended strong interactions
SU(3)1 ×SU(3)2 , with coupling constants h2 ≫ h1 . A general prediction of such models is
the existence of a color-octet of massive gauge bosons (colorons). The best search strategy
for colorons depends on how they couple to the different quark flavors. We will discuss
the phenomenology resulting from the two most common choices for the quarks’ charge
assignments.
(i) Topgluons
In models such as Topcolor [304], TC2 [274,407], or Top Seesaw [245], only the thirdgeneration quarks transform principally under the stronger SU(3)2 group. As a result,
the massive topgluons (B µa ) couple predominantly to third-generation quarks [304]. The
125
Figure 56: s-channel charged top-pion production with the benchmark Yukawa couplings given
in the text. For reference, the dashed curves show the results for Yukawa couplings satisfying
the 3-sigma Rb bound [438].
topgluons are expected to be heavy (M 0.5 − 2.0 TeV) and broad (Γ/M 0.3 − 0.7) resonances. In production at, e.g., the Tevatron, in the dominant process q̄q → (g, B) → t̄t
the amplitude involves (g3 tan θ) × (−g3 cot θ) so that the factors involving θ cancel (and
there is characteristic destructive interference above threshold). Thus, θ affects the rates
only through the decay width of B µa .
Topgluons of moderate mass may be produced directly at the Tevatron [187,443]. CDF
has recently used their measured upper limit on cross-section time branching ratio of new
resonances decaying to bb̄ to place a limit on topgluons [346] . They exclude topgluons
of width ΓB = 0.3M in the mass range 280 < M < 670 GeV, of width 0.5 M in the
range 340 < M < 640 GeV, and of width ΓB = 0.7M in the range 375 < M < 560 GeV.
A simulation of topgluon production and decay combined with an extrapolation of the
CDF b-tagged dijet mass data from Run I [444] indicates that in Run IIb, the topgluon
discovery mass reach in bb̄ final states should be 0.77-0.95 TeV with 2 f b−1 of integrated
luminosity and 1.0 - 1.2 TeV with 30fb−1 .
Lower backgrounds make topgluons easier to find in their decays to top quarks [187,443].
Initial measurements of the invariant mass (Mtt ) and transverse momentum (pT ) distributions of the produced top quarks have been made, as shown in Figure 58. While
a comparison with the measured Mjj distribution for QCD dijets [450] illustrates how
126
10 2
Technirho
/
Topcolor Z
/
Standard Z
Vector
Gluinonium
10
1
b) Topgluons
● CDF 95% CL Limit
10 2
400
600
Excluded: 280 < M < 670
200
10
400
600
M(gT) (GeV/c2)
3
d) Topgluons
● CDF 95% CL Limit’
-
-
c) Topgluons
● CDF 95% CL Limit
σ • Br{gT → bb} (pb)
New Particle Mass (GeV/c2)
3
10
Γ/M=0.3
10
1
200
σ • Br{gT → bb} (pb)
10 3
-
σ • Br{gT → bb} (pb)
a) Narrow Resonances
● CDF 95% CL Limit
-
σ • Br{X → bb} (pb)
10 3
10 2
Γ/M=0.5
10
Excluded: 340 < M < 640
1
200
400
600
M(gT) (GeV/c2)
10 2
Γ/M=0.7
10
Excluded: 375 < M < 560
1
200
400
600
M(gT) (GeV/c2)
Figure 57: CDF search for topgluons in bb̄ [346]
Figure 4: The 95% CL upper limit on the cross section times branching ratio (points)
for a) narrow resonances, and topgluons of width b) = 0 3M, c) = 0 5M, and d)
= 0 7M
is compared
theoreticalis,
predictions
statistics-limited the
Run
I toptosample
some(curves).
preliminary limits on new physics are being
:
:
:
extracted32 . The pT distribution for the hadronically-decaying top in fully-reconstructed
lepton + jets events (Figure 59) constrains
17 any non-SM physics which increases the num+0.024
ber of high-pT events. The fraction R4 = 0.000+0.031
−0.000 (stat)−0.000 (sys) of events in the
highest pT bin (225 ≤ pT ≤ 300 GeV) implies [449] a 95% c.l. upper bound R4 ≤ 0.16 as
compared with the SM prediction R4 = 0.025.
In Run IIb, the σtt measurement will be dominated by systematic uncertainties; the
collaborations will use the large data sample to reduce reliance on simulations [452].
Acceptance issues such as initial state radiation, the jet energy scale, and the b-tagging
efficiency will be studied directly in the data. It is anticipated [452] that an integrated
luminosity of 1 (10, 100) fb−1 will enable σtt to be measured to ± 11 (6, 5) %. Topgluons
of mass up to 1.0-1.1 TeV (1.3 − 1.4 TeV) would be visible in 2 fb−1 (30 fb−1 ) at Run IIb,
either in a search for a new broad resonance, or through their effects on the magnitude of
the tt̄ total cross-section. Projected Run IIb limits on σ ·B for new resonances decaying to
tt̄ are illustrated in Figure 64. For further details on current and future topgluon searches
at the Tevatron, see Table III of ref. [194]. The situation at the LHC and VLHC has
been recently studied in considerable in [443], conclude that t̄t is overwhelmed by QCD
backgrounds.
(ii) Flavor-universal Colorons
In theories in which all quarks carry only SU(3)2 charge [420,331,421], the massive
colorons couple with equal strength to all quark flavors (section 4.1.2(ii)). As a result,
32
It has been noted, e.g. that a narrow 500 GeV Z’ boson is inconsistent with the observed shape of
the high-mass end of CDF’s Mtt distribution.[451]
127
Events/25 GeV/c2
(a)
92.2%
(b)
10.8%
6
4
2
200
400
600
200
mtt (GeV/c2)
400
600
Figure 58:
Invariant mass distribution for top pairs: pairs: D0 data (histogram), simulated
background (triangles), simulated S+B (dots). In (a) mt unconstrained; in (b) mt = 173
GeV.[445–448]
Figure 59: PT distribution for hadronically-decaying tops in lepton+jets events from CDF.[449]
they will appear in dijet production and heavy flavor production at hadron colliders. A
comparison of these processes [453] has indicated that the limits on colorons from dijet
production should be the more stringent ones.
Existing limits on flavor-universal colorons from several sources are summaried in
Figure 60. Exchange of light strongly-coupled colorons across quark loops would cause
large contributions to the T parameter; accordingly, the region (M/ cot θ) < 450 GeV
is excluded.33 Light narrow colorons would have been seen by a Run I CDF search for
new particles decaying to dijets (see [455]); this excludes the cross-hatched region at low
cot2 θ. The light-shaded region (M/ cot θ < 759 GeV) is excluded by the shape of the dijet
angular distribution measured by D0 [456]. Finally, the shape of the dijet mass spectrum
measured by D0 [450] sets the strongest limit: M/ cot θ > 837 GeV [455].
33
A fit to the full set of electroweak data gives a slightly stronger bound at large coloron mass [454].
128
20
cot2θ
15
D0
excludes
∆ρ
excludes
10
Mc/cotθ
> 837 GeV
5
0
1
CDF
excludes
2
3
4
2
Mc (TeV/c )
Figure 60: Experimental limits on flavor-universal colorons from sources described in the text
[455]. The horizontally hatched region at large cot θ is not part of the Higgs phase of the model
[420].
In the context of flavor-universal TC2 models, the value of κ3 must be approximately
2 in order for the colorons to help cause the top quark to condense as discussed earlier
[421]; this is equivalent to cot θ ≈ 4. Hence, in these models, the limit set by D0 on the
coloron mass is Mc > 3.4 TeV.
Run IIb will, naturally, be sensitive to somewhat heavier colorons. According to the
TeV 2000 report the limit on σ · B will improve as shown in Figure 61. The predicted
σ · B for a coloron of cot θ = 1 is equivalent to that for an axigluon (shown); larger values
of cot θ increase the rate.
4.3.3
New Z ′ Bosons
As discussed in section 4.1, both Classic TC2 [274,407] and Flavor-Universal TC2 [331,421]
models include an extra U(1) group and predict the presence of a massive Z ′ boson. The
couplings of this Z ′ to fermions are generally not flavor-universal, and the more strongly
suppressed the Z ′ couplings to first and second generation fermions are, the less effective
traditional searches for Z ′ become. This section discusses experimental searches and limits
that exploit the flavor non-universal couplings of the Z ′ .
TC2 models include an extended electroweak sector of the form
SU(2)W × U(1)h × U(1)ℓ .
(4.161)
where the coupling of the first U(1) group is the stronger one, gh >> gℓ . At a scale above
the weak scale, the two hypercharge groups break to a subgroup identified as U(1)Y . As
a result, a Z ′ boson that is a linear combination of the original two hypercharge bosons
129
Figure 61: Anticipated 95% c.l. upper limit on σ · B for new particles decaying to dijet as a
function of the new particle mass for various integrated luminosities at the Tevatron. Axigluon
curve corresponds to a coloron of cot θ = 1[452].
becomes massive. This heavy Z ′ boson couples to a fermion as
e
−i
cos θ
sχ
cχ
Yℓ − Yh
cχ
sχ
!
(4.162)
where Yh (Yℓ ) is the fermion’s hypercharge under the U(1)h (U(1)ℓ ) group and χ is the
mixing angle between the two original hypercharge sectors cot χ = (gh /gℓ )2 . Compared
with the couplings of the Z ′ boson from an extended weak group (Section 3.6.5), the
significant physical differences are that the overall coupling is of hypercharge rather than
weak strength, and the Z ′ couples to both left-handed and right-handed fermions at
leading order.
(i) Z ′ and Precision Tests
Studies of precision electroweak limits on the Z ′ bosons [278,457] in these models have
obtained lower bounds on MZ ′ as a function of sin2 χ. The lower bound is found to depend
130
MHZ_hL
Optimal Z’
14000
12000
10000
8000
6000
4000
2000
sin^2 chi
0.2
MHZ_hL
5000
0.4
0.6
1
0.8
Optimal Z’
4000
3000
2000
1000
sin^2 chi
0.0250.050.075 0.10.1250.150.175 0.2
Figure 62:
Lower bounds at 95% c.l. on mass of “optimal” non-universal Z’ boson. [278].
Solid curve is from a global fit to precision electroweak data; dashed line is from LEP II contact
interaction limits. The bottom plot is a close-up of the region where the smallest Z’ mass is
allowed.
quite strongly on the hypercharge assignments of the fermions because the shift in the Z
boson’s coupling to fermions depends directly on Yh :
e sin θ
δg f ≈ −
x cos θ cos2 χ
!
!
i
h
ǫ
f
2
f
1+
Y
−
sin
χY
h
sin2 χ
(4.163)
where ǫ ≡ 2fπ2t /v 2 (YhtL − YhtR ). In models with a so-called “optimal” Z ′ , in which the
third generation fermions have Yh = Y, Yℓ = 0 and those of the other two generations
have Yh = 0, Yℓ = Y , the Z ′ can be as light as 630 GeV (see figure 62) at 95% c.l. for
2
<
sin2 χ ≈ 0.0784; a Z ′ mass less than a TeV is allowed for 0.0744 <
∼ sin χ ∼ 0.0844 [278].
In the TC2 model of Lane [330], where the fermions have considerably larger Yh charges
(see table), the lower bound on the Z ′ mass is correspondingly higher, about 20 TeV. The
strong constraint here comes from sensitivity to atomic parity violation in Cs [330]. A
variant of Lane’s model [458] in which the lepton’s U(1)h couplings are vectorial has a Z ′
limit much closer to that of the “optimal” scenario (though lacking the low-mass region
near sin2 χ = .0784) [457].
131
1st, 2nd
(u, d)L, (c, s)L
u R , cR
d R , sR
(νe , e)L , (νµ , µ)L
eR , µR
Yh
-10.5833
-5.78333
-6.78333
-1.54
2.26
3rd
(t, b)L
tR
bR
(ντ , τ )L
τR
Yh
8.7666
11.4166
10.4166
-1.54
2.26
Table 6: Fermion charges for Lane’s model [330]
The non-universal couplings of the Z ′ boson to fermions cause it to induce lepton
number violating processes. A study in ref. [459] has shown that µ − e conversion in
nuclei is about an order of magnitude better than the decay µ → 3e [460] for constraining
the magnitudes of the lepton mixing angles. The decay µ → eγ yields weaker bounds.
Present data allows the Z ′ mass to be as large as 1 TeV and the magnitudes of the lepton
mixing angles lie roughly between the analogous CKM entries and their square-roots [459].
Looking to the future, ref. [461] finds that the MECO and PRIME experiments will probe
lepton flavor violating Z ′ bosons out to a mass of order 10 TeV, far better than the MEG
(µ → eγ) experiment can do.
At energies well below the mass of the Z ′ boson, its exchange in the process e+ e− → f f¯
where f is a τ lepton or b quark may be approximated by four-fermion contact interactions.
As discussed in section 3.6.5, limits the ALEPH and OPAL experiments have given on
the scale Λ of new contact interactions may be translated into lower bounds on the mass
of the Z ′ boson. The strongest such limits on an “optimal” Z ′ boson are for the process
−
+ −
e+
R eR → τR τR [329]
q
MZ ′ = Λ αem
/ cos2
θ >
370 GeV ALEPH
370 GeV OPAL
.
(4.164)
which improves a bit on the limit from precision electroweak data [457].
Dijet and Drell-Yan data from CDF and D0 can, likewise, be interpreted as setting
limits on the mass of a new Z ′ boson. The lower bounds on an “optimal” Z ′ derived
from limits on quark-lepton contact interactions are significantly weaker than those from
the LEP data [329]. The constraints on the Z ′ of Lane’s TC2 model [330] from dijet and
Drell-Yan data are sufficient to indicate that the model requires significant fine-tuning
[462]; as mentioned earlier, however, the bounds from precision electroweak data are even
stronger.
(ii) Z ′ Production Searches
Evidence of a Z ′ boson coupled primarily to the third generation can more profitably
be sought in heavy flavor production at the Tevatron. The CDF collaboration has searched
in tt̄ events [451] for a narrow leptophobic Z ′ boson present in some models of topcolor132
σX • Br{X→tt} (pb)
20
CDF 95% C.L. Upper Limits for Γ = 0.012MX
CDF 95% C.L. Upper Limits for Γ = 0.04MX
Leptophobic Topcolor Z′, Γ = 0.012MZ′
10
9
8
7
6
Leptophobic Topcolor Z′, Γ = 0.04MZ′
5
4
3
2
1
0.9
0.8
0.7
0.6
0.5
400
500
600
700
800
900
1000
MX (GeV/c2)
Figure 63:
The 95% c.l. upper limits on σX · BRX → tt̄ as a function of mass (solid and
open points) compared to the cross section for a leptophobic topcolor Z ′ (thick solid and dashed
curves) for two resonance widths [451].
assisted technicolor [463]. Finding no evidence of a new narrow residence, CDF sets the
limits MZ ′ > 780 GeV for such a Z ′ of natural width ΓZ ′ = 0.04MZ ′ and MZ ′ > 480 GeV
for ΓZ ′ = 0.012MZ ′ as shown in figure 63. Simulations of Z ′ → tt̄ → lνbb̄jj for Run II
indicate that with 2f b−1 (30 fb−1) of data a narrow topcolor Z ′ of mass up to 0.92 TeV
(1.15 TeV) would be visible [452]; figure 64 summarizes the anticipated Run II sensitivity
to σ · B in the Mtt distribution, and the corresponding predictions for topcolor Z ′ bosons..
The search by the CDF Collaboration for new resonances decaying to bb̄ [346], which
was previously mentioned (section 4.3.2, figure 57) as setting limits on technirhos, falls
slightly short of setting a limit on topcolor Z ′ bosons. In addition to pursuing Z’ bosons
in these heavy quark channels, the Run IIb will also be able to look for Z ′ bosons decaying
to τ leptons as described in section 3.6.5. In the case with the lowest standard model
background, Z ′ → τ τ → eµν ν̄, a Z ′ boson with a mass up to about 600 GeV should be
accessible (depending on mixing angle) [329]. For further details on current and future
Topcolor Z ′ boson searches at the Tevatron, see Table IV of ref. [194].
A Z ′ boson would also be visible in the process e+ e− → τ τ at an NLC [421]. Assuming
a 50% efficiency for idenitfying τ –pairs and requiring an excess over standard model
133
(pb)
Min σ∗B(X→tt) for a resonance
to be observed at the 5σ level.
1 fb-1
σ∗B
1
10 fb-1
-1
-1
100 fb
10
-2
TopColor Z‘, Γ=1.2%
TopColor Z‘, Γ=10%
10
400
500
600
700
800
Mtt GeV/c2
900
1000
Figure 64: Anticipated[452] Run IIb limits on σ · B(X → tt̄) and predictions for a topcolor Z’.
q
ττ
ττ
backgrounds of (N τ τ − NSM
) ≥ 5 NSM
, it appears that the effects of a 2.7 TeV Z ′ boson
√
with α1 tan2 χ ≤ 1 would be visible in a 50 fb−1 sample taken at s = 500 GeV. A 1.5
TeV NLC with 200 fb−1 of data would be sensitive to Z ′ bosons as heavy as 6.6 TeV. The
other channels suggested earlier in the context of topflavor models might also be useful
for finding this Z ′ at a LC, LHC, or FMC.
4.4
Top Seesaw
The electroweak mass gap, i.e., the dynamical fermion mass associated with the generation
of vweak through the Pagels-Stokar relation eq.(4.134), is ∼ 600 GeV when the scale of
Topcolor is taken to be a natural scale of order ∼ 1 TeV. If the top quark mass were this
large, our problems would be solved, and EWSB would be identified naturally with a t̄t
condensate. By itself, however, the top quark is too light to produce the full electroweak
condensate in this way.
We can, however, write a viable model in which the I = 21 top quark mass term, the
term associated with electroweak symmetry breaking, is indeed ∼ 600 GeV, and exploit a
seesaw mechanism to tune the physical mass of the top quark to mt = 175 GeV [245,246].
This gives up the predictivity of the top quark mass, since the tuning is done through a
new mixing angle. Because the top quark is heavy, however, the mixing angle need not
be chosen artificially small.
The Top Seesaw mechanism can be implemented with the introduction of a pair of
iso-singlet, vectorlike quarks, χL and χR which both have Y = 4/3, analogues of the tR .
134
This model produces a bound-state Higgs boson, primarily composed of t̄L χR , with the
Higgs mass ∼ 1 TeV, saturating the unitarity bound of the Standard Model. Such a large
Higgs mass would seemingly be a priori ruled out by the S − T parameter constraints.
Remarkably, however, the Top Seesaw theory has the added bonus of supplying a rather
large T -parameter contribution associated with the presence of the χ fermions [245,464].
In 1998 when the Top Seesaw theory was proposed, it was in poor agreement with the
S − T bounds. With the most recent compilation of LEP and worldwide data, including
a refined initial state radiation and W -mass determination, the Top Seesaw theory lies
within the 95% c.l. error ellipse (see Section 4.5). Indeed the theory lies within the S − T
plot for expected natural values of the χ masses. Hence, the theory has already scored a
predictive success: the S − T ellipse has moved to accomodate it. We may, in fact, view
the measured error ellipse as a determination of the χ mass in this scheme; we obtain
roughly Mχ ∼ 4 TeV. In this picture, the worldwide electroweak precision measurements
are probing the mass of a heavy new particle, the χ, significantly above the electroweak
scale.
Note that the Top Seesaw model, unlike TC2, does not invoke Technicolor; rather, it
replaces Technicolor entirely with Topcolor. It offers novel model building possibilities
for attacking the flavor problem [465,466,417,467]. Extensions have been constructed in
which the W , Z, t and b all receive dynamical masses. Construction of an explicit model
of all quark and lepton masses seems plausible [465,466]. Because the χ quarks need not
carry weak-isospin quantum numbers, and enter in vectorlike pairs34 , the constraints on
the number of techniquarks from the S parameter are essentially irrelevant for the Top
Seesaw. The Top Seesaw also finds an attractive setting in extra dimensional models,
as discussed in Section 4.6. The χ quarks and Topcolor itself may be interpreted as
Kaluza-Klein modes. Because the mass-gap is 600 GeV rather than mt = 175 GeV, the
masses of all the colorons, and any additional heavy gauge bosons, are naturally moved to
slightly larger mass scales than in TC2. One then has more model-building elbow room:
phenomenological exploration of the Top Seesaw dynamics will rely on the VLHC rather
than the LHC.
4.4.1
The Minimal Model
In the minimal Top Seesaw scheme the full EWSB occurs via the condensation of the
left-handed top quark with a new, right-handed weak-singlet quark, which we refer to as
a χ-quark. The χR quark has hypercharge Y = 4/3 and is thus indistinguishable from
the tR . The dynamics which leads to this condensate is Topcolor, as discussed below,
and no tilting U(1)′ is required. The fermionic mass scale of this weak-isospin I = 1/2
condensate is ∼ 0.6 TeV. This corresponds to the formation of a dynamical bound-state
weak-doublet Higgs field, H ∼ (χR tL , χR bL ). To leading order in 1/Nc this yields a VEV
for the Higgs boson, via the Pagels-Stokar formula, of the appropriate electroweak scale,
vweak = 175 GeV, and the top quark acquires an I = 21 dynamical mass µ. The (I = 0,
34
See refs. [417,467] for models where the χ’s are weak isodoublets.
135
Y = 4/3) χ-quarks have an allowed Dirac mass Mχ and a mass term m0 that mixes tR
and χL . Thus, the full fermionic mass matrix takes the form:
tL
χL
!
0
µ
m0 Mχ
!
tR
χR
!
(4.165)
where the vanishing entry is a term forbidden by the Topcolor assignments (see 4.4.2
below). We diagonalize the mass matrix:
M = UL† (φL )MD UR (φR )
leading to eigenvalues:
MD =
m1 0
0 m2
(4.166)
!
(4.167)
where:
m21
"
r
2
1
m20 + Mχ2 + µ2 −
=
m20 + Mχ2 + µ2 − 4µ2 m20
2
m20 µ2
+ O(m40 µ4 /Mχ6 )
≈
Mχ2 + m20 + µ2
#
(4.168)
and
m22
"
r
2
1
=
m20 + Mχ2 + µ2 +
m20 + Mχ2 + µ2 − 4µ2 m20
2
≈ Mχ2 + m20 + µ2 + O(m20 µ2 /Mχ2 )
#
(4.169)
where limits for large Mχ are indicated.
The fermionic mass matrix thus admits a conventional seesaw mechanism, yielding
the physical top quark mass as an eigenvalue that is ∼ m0 µ/Mχ << µ ≈ 600 GeV. The
top quark mass can be adjusted to its experimental value. The diagonalization of the
fermionic mass matrix does not affect the VEV (vweak = 175 GeV) of the composite Higgs
doublet. Indeed, the Pagels-Stokar formula is now modified to read:
2
vweak
≡ fπ2 =
M2
Nc m2t
(log
2 + k)
16π 2 sin2 φL
M
(4.170)
where mt is the physical top mass, and φR the mass matrix diagonalizing mixing angle.
The Pagels-Stokar formula differs from that obtained (in large Nc approximation) for top
quark condensation models by the large enhancement factor 1/ sin2 φL . This is a direct
consequence of the seesaw mechanism. The Top Seesaw employs ψL = (tL , bL ) as the
source of the weak I = 1/2 quantum number of the composite Higgs boson, and thus
the origin of the EWSB vacuum condensate. This neatly separates [258] the problem of
EWSB from that of the new weak-isosinglet states in the χL,R and tR sector, a distinct
advantage since the electroweak constraints on new physics are not very restrictive of
isosinglets.
136
4.4.2
Dynamical Issues
How does Topcolor produce the µ mass term? We introduce an embedding of QCD
into the gauge groups SU(3)1 × SU(3)2 , with coupling constants h1 and h2 respectively.
These symmetry groups are broken down to SU(3)QCD at a high mass scale M. The
assignment of the elementary fermions to representations under the full set of gauge
groups SU(3)1 × SU(3)2 × SU(2)W × U(1)Y is as follows:
ψL : (3, 1, 2, + 1/3) ,
χR : (3, 1, 1, + 4/3) ,
tR , χL : (1, 3, 1, + 4/3) .
(4.171)
This set of fermions is incomplete: the representation specified has [SU(3)1 ]3 , [SU(3)2 ]3 ,
and U(1)Y [SU(3)1,2 ]2 gauge anomalies. These anomalies will be canceled by fermions
associated with either the dynamical breaking of SU(3)1 × SU(3)2 , or with the b-quark
mass generation (a specific example of the latter case is given at the end of this section).
Schematically things look like this:
SU(3)1
!
t
b L
(χ)R
...
SU(3)2
!
t
b R
(χ)L
...
The dynamics of EWSB and top-quark mass generation will not depend on the details of
the additional fermions.
We introduce a scalar Higgs field, Φ, transforming as (3, 3, 1, 0), which develops a
diagonal VEV, hΦij i = Vδji . This field is presumably yet another dynamical condensate,
or it can be interpreted as a relic of compactification of an extra dimension. Topcolor is
then broken to QCD,
SU(3)1 × SU(3)2 −→ SU(3)QCD
(4.172)
yielding massless gluons and an octet of degenerate colorons with mass M given by
M 2 = (h21 + h22 ) V 2 .
(4.173)
We can also exploit Φ to provide the requisite Mχ by introducing a Yukawa coupling
of the fermions χL,R of the form: −ξ χR Φ χL + h.c. −→ −Mχ χχ. We emphasize that
this is an electroweak singlet mass term. ξ can be a perturbative coupling constant so
V ≫ ξV = Mχ . Finally, since both tR and χL carry identical Topcolor and U(1)Y quantum
numbers we are free to include the explicit mass term, also an electroweak singlet, of the
form m0 χL tR + h.c..
The Lagrangian of the model at scales below the coloron mass is SU(3)C × SU(2)W ×
U(1) invariant and becomes:
L0 = Lkinetic − (Mχ χL χR + m0 χL tR + h.c.) + Lint
137
(4.174)
Lint contains the residual Topcolor interactions from the exchange of the massive colorons:
Lint
h2
MA
= − 12 ψL γ µ
ψL
M
2
!
!
MA
χR + LL + RR + ...
χR γµ
2
(4.175)
where LL (RR) refers to purely left-handed (right-handed) current-current interactions.
It suffices to retain in the low energy theory only the effects of the operators shown in
eq. (4.175) even though higher dimension operators may be present. To leading order in
1/Nc , upon performing the familiar Fierz rearrangement, we have:
Lint =
h21
(ψL χR ) (χR ψL ) .
M2
(4.176)
It is convenient to pass to a mass eigenbasis with the following redefinitions:
χR → cos φR χR − sin φR tR ;
tR → cos φR tR + sin φR χR
where
tan φR =
m0
Mχ
(4.177)
(4.178)
In this basis, the NJL Lagrangian takes the form:
L0 = Lkinetic − M χR χL + h.c.
+
i
h21 h
ψ
(cos
φ
χ
−
sin
φ
t
)
L
R R
R R [(cos φR χR − sin φR tR ) ψL ] (4.179)
M2
where
M=
q
Mχ2 + m20 .
(4.180)
At this stage we have the choice of using the renormalization group, or looking at
the mass gap equations for µ. A rationale for studying the gap equations is that they
in principle allow one to explore limits, such as M > M which are conceptually more
difficult with the renormalization group35 . For a more complete analysis of the dynamics
see [468].
Let us summarize the gap equation analysis. We assume two dynamical mass terms:
−µ1 tL χR − µ2 tL tR
(4.181)
and we compute the gap equations to order O(µ3 ). This will produce no IR divergences
in terms to order cos2 φR , but there is an IR log-divergence in the last term of Fig.(65)
of order sin2 φR , and we take this to be ln(m2t ). Fig.(65) produces the coupled system of
35
For instance, the d = 6 operator makes no sense above the scale M in the renormalization group,
but the cut-off theory can still be expressed in the gap equation language.
138
µ1
µ1
χ
µ
t
µ2
t
χ
+
µ2
(t, χ )
t
t
Χ
tL
t
µ2
t
ΧR tL
µ1
+
µ2
χ
+
t
(χ ,t)
t
t
µ1 µ1
ΧR tL
tL
ΧR
χ
1
+
tL
µ1
t
ΧR
t
µ2
µ2 µ2
+ t
µ2
t
ΧR tL
t
R L
t
ΧR
Figure 65: Gap equations for µ1 . A similar set of terms is obtained for µ2 [468].
gap equations for µ1 :
µ1
2
h21 N
M 2 + M
2
2
2
µ
cos
φ
M
−
M
ln
=
1
R
2
8π 2 M 2
M
2
2
h2 N
µ21 M 2
M 2 + M
M2 + M
2
+ 21 2 µ1 cos2 φR −µ21 ln
−
µ
ln
+
2
2
2
2
8π M
M
M2 + M
M
2
!
M2 + M
h2 N
M2
2
+ 21 2 µ2 cos φR sin φR M 2 − µ21 ln
−
µ
ln
2
2
8π M
m2t
M
(4.182)
and a similar set for µ2 . With the substitutions µ1 = µ cos φR and µ2 = µ sin φR the two
independent gap equations reduce to a single mass gap equation:
2
h21 N
M2 + M
M 2 − M 2 cos2 φR ln
µ =
µ
2
8π 2 M 2
M
2
2
µ2 M 2
M 2 (M 2 + M )
M2 + M
2
4
−
µ
sin
φ
ln
+
(4.183)
−µ2 ln
R
2
2
2
M
M2 + M
m2t M
The gap equation eq.(4.183) shows that we require supercritical coupling as the mass M
becomes large. Moreover, for fixed supercritical coupling, h21 /4π, as we raise the scale M
the condensate turns off like a second order phase transition.
We can also see that these reproduce normal top condensation in the decoupling limit.
139
eff
κc /κc
4
3.5
3
2.5
2
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M /Λ
2
Figure 66: Gap equation
√ with coupling constant κ = h1 /4π (scaled by constant κc ≡ 2π/3) as
a function of M /M , for
tan φR = (0, 0.1, 0.2, 0.3, 0.5, 0.8). from Hill, He, and Tait [468]
For example, choose M → ∞ for fixed M, and we find:
h21 N
M2
2
2
2
2
µ =
µ
M
(1
−
cos
φ
)
−
µ
sin
φ
ln
R
R
8π 2 M 2
m2t
!!
(4.184)
and redefining g̃ = h1 sin φR , and mt = µ sin φR we have:
mt
g̃ 2 N
M2
2
2
=
µ
M
−
m
ln
t
8π 2 M 2
m2t
!!
(4.185)
which is the top condensation gap equation. Here we have decoupled χL and χR with
M → ∞. We can also obtain top condensation by setting sin2 φR = 0 and M → 0, which
decouples χL and tR , and causes χR to play the role of tR .
A separate question of interest is the structure of the electroweak corrections in the
Top Seesaw theory. This is most easily determined from an effective Lagrangian for the
composite Higgs boson. The most complete study to date is that of Hill, He and Tait [468].
The composite Higgs boson mass satisfies the approximate NJL result, mHiggs ≈ 2µ ∼ 1.2
TeV. Experience with Topcolor suggests that radiative corrections from SU(2)2 will reduce
this. The composite Higgs mass is consistent with the unitarity limit of the Standard
Model. By itself, this pulls the theory to large negative T . However, there are χ − t
mixing corrections to the T parameter as well. We obtain for T:
4
Nm2t
2µ2 M
µ2
T =
+
ln
− 1
2
32π 2 vweak
αZ m20 M 2
µ2 m20
140
(4.186)
mtχ /Λ
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M /Λ
Figure 67: Behavior of gap equation for fixed supercritical
coupling h21 /4πκc = 2.0 as M /M
√
is increased. In the notation of the figure, mtχ = µ. tan φR = (0, 0.1, 0.2, 0.3, 0.5, 0.8). The
figure shows results for our gap equation (solid) and an approximation (dotted) that treats M
as a mass insertion (see [468]).
Putting in typical numbers, such as µ = 0.6 TeV, m0 = mt M/µ ≈ 1.2 TeV, αZ = 1/128,
one observes a large positive correction to T , and one recovers consistency with the error
ellipse constraint for M ≈ 4 TeV. Note that the S-parameter is vanishing, since the
additional χ quarks are weak isosinglets.
We note that, using the freedom to adjust sin φR , we can in principle dynamically
accommodate any fermion mass lighter than ∼ 600 GeV – at the price of some finetuning. This freedom may be useful in constructing more complete models involving all
three generations [466,465]. The top quark is unique, however, in that it is very difficult
to accommodate such a heavy quark in any other way, and there is less apparent finetuning. We therefore believe it is generic, in any model of this kind, that the top quark
receives the bulk of its mass through this seesaw mechanism. In a more general theory
that includes the seesaw mechanism there are more composite scalars, and one of the
neutral Higgs bosons may be as light as O(100 GeV).
4.4.3
Including the b-quark
Inclusion of the b-quark is straightforward, and two distinct schemes immediately suggest
themselves. We include additional fermionic fields of the form ωL , ωR , and bR with the
assignments:
bR , ωL : (1, 3, 1, − 2/3) , ωR : (3, 1, 1, − 2/3) .
(4.187)
141
These fermion gauge assignments cancel the anomalies noted above. We further allow
ωL ωR and ωL bR mass terms, in direct analogy to the χ and t mass terms:
L0 ⊃ −(Mω ωLωR + mω ωL bR + h.c.)
(4.188)
With the previous assignments for the χ quarks, schematically, this looks like the following:
Inclusion of b-quark I:
SU(3)1
!
t
I = 21
b !L
χ
I=0
ω R
SU(3)2
!
t
I =0
b !R
χ
I=0
ω L
Alternatively, we may define the χ and ω fields to form isodoublets, with assignments
as:
bR , : (1, 3, 1, − 2/3) , (χ, ω)R , : (3, 1, 2,1/3) ,
, (χ, ω)L : (3, 1, 2,1/3) .
(4.189)
Inclusion of b-quark II:
SU(3)1
!
t
I = 21
b !L
χ
I = 21
ω R
SU(3)2
!
t
I =0
b !R
χ
I = 21
ω L
The schematic model affords a simple way to suppress the formation of a b-quark
mass comparable to the top quark mass. We
q can suppress the formation of the ωL bR
condensate altogether by choosing M ω = µ2ωω + µ2ωb ∼ M. In this limit we do not
produce a b-quark mass. However, by allowing µωω ≤ M and µωb /µωω ≪ 1 we can form
an acceptable b-quark mass in the presence of a small ωL bR condensate.
Yet another possibility arises, one which seems to be phenomenologically favored [468],
which is to exploit instantons. If we suppress the formation of the ωL bR condensate by
choosing M ω ∼ M, there will be a ωL bR condensate induced via the ’t Hooft determinant
when the t and χ are integrated out. We then estimate the scale of the induced ωL bR
mass term to be about ∼ 20 GeV, and the b-quark mass then emerges as ∼ 20µωb /µωω
GeV. We will not further elaborate the b-quark mass in the present discussion, since its
precise origin depends critically upon the structure of the complete theory including all
light quarks and leptons. Including partners for the b-quark, the T parameter is given by
a more general formula [464,468].
142
The Top Seesaw Model offers new possibilities for a dynamical scheme explaining
both EWSB and the origin of flavor masses and mixing angles. We focused here in some
detail on the third generation and the EWSB dynamics. A fully extended model for
light quark and lepton masses has not yet been analyzed, but it would seem that the Top
Seesaw affords interesting new directions and possibilities that should be examined. Some
varying attempts in this direction can be found in [466,465]. Remarkably, the vectorlike
fermions introduced here to provide the seesaw can also help to remedy the discrepancies
between lepton and b-quark forward-backward and left-right asymmetries in the LEP data
[23].
4.5
4.5.1
Top Seesaw Phenomenology
Seesaw Quarks
In the Top Seesaw and related models, the third generation quarks [245,246,464], and
possibly all Standard Model fermions [466,258], acquire mass through seesaw mixing with
exotic, weak-singlet fermions. As a result, there are several new types of states which can
affect the phenomenology: the new fermions themselves, and, composite scalars formed at
least in part from the new fermions. We begin by surveying current bounds on the mixing
angles φf between ordinary and weak-singlet fermions. We then look at low-energy limits
on the masses of the heavy fermionic seesaw partner, f H , states and the composite scalars
(these results apply generally to Kaluza-Klein modes as well). We comment on variant
models in which the mixing is with weak-doublet fermions instead of weak-singlets. And
to conclude, we discuss Tevatron limits on the masses of the f H states.
Suppose the mass matrix of a Standard Model top quark mixing with a vectorial
isosinglet quark χ has the seesaw form of eq.(4.165). The matrix is diagonalized by
performing separate rotations on the left-handed and right-handed fermion fields (eq.
4.166). Most of the phenomenology is sensitive to the mixing among the left-handed
fermions. Of the two mass eigenstates, the physical top quark is the lighter one t̃L and is
mostly weak-isodoublet:
t̃L = cos φt tL − sin φt χL ,
(4.190)
and has a mass of order m0 µ/Mχ ; the heavier (χ̃) is mostly weak-isosinglet:
TLH ≡ χ̃L = sin φt tL + cos φt χL
(4.191)
with a mass of order ∼ Mχ . This stucture is readily generalized to models in which more
than one ordinary fermion mixes with weak singlets [469].
Recent limits on the mixing angles between ordinary and weak-isosinglet fermions
from precision electroweak data were obtained in [470]. Separate limits at 95% c.l. are
given for the case in which each fermion flavor has its own weak partner [466,258]
sin2 φe ≤ 0.0024 ,
sin2 φd ≤ 0.015 ,
sin2 φu ≤ 0.013 ,
sin2 φµ ≤ 0.0030 ,
sin2 φs ≤ 0.015 ,
sin2 φc ≤ 0.020 .
143
sin2 φτ ≤ 0.0030
sin2 φb ≤ 0.0025
(4.192)
(4.193)
(4.194)
1
6
5
10
(a)
4
10
2
Topseesaw Higgs Mass at Large-Nc:
exact Mh (solid) & Mh≈2mtχ (dotted)
3
Mh (TeV)
2
1
0.9
0.8
0.7
Bottom to top: κ/κc = 1.05, 1.2, 1.5, 2, 4
1.2
Higgs Mass by Improved RG
1
Curves from left to right:
κ/κc = 4, 2, 1.5, 1.2, 1.05
0.8
0.6
0.4
(b)
0.2
1
10
10
2
Λ (TeV)
Figure 68: The predicted mass spectrum of the Top Seesaw Higgs boson: (a) by the large-Nc
fermion-bubble calculation; and (b) by an improved RG analysis including the Higgs self-coupling
evolution.
and the case in which only the third-generation quarks have weak partners [245,246,464]
sin2 φb ≤ 0.0013 .
(4.195)
The most important precison electroweak constraints on the Top Seesaw come from the
S and T parameters [464,470,468]. It is remarkable that the minimal Top Seesaw model,
which typically includes a heavy composite Higgs boson around 1 TeV (see Fig. 68), is nontrivially compatible with the S − T bounds. The composite Higgs boson’s contributions
will drive T in the negative direction relative to a light SM Higgs. However, the Top Seesaw
sector has generic weak-isospin violation from the t-χ mixing which will significantly
contribute to T in the positive direction. Extended models with bottom seesaw are more
complex because of the b-ω mixing and the two composite Higgs doublets. Corrections to
the Z → b̄b vertex are also relevant in this case. We summarize the essentials of the S − T
analysis of Top Seesaw models and refer the interested reader to the detailed discussion
in ref. [468].
In Fig. 69, we give the complete S and T contributions from the minimal Top Seesaw
model, including corrections from both the composite Higgs boson and the seesaw quarks,
and compare them with the 95% C.L. contour for S − T . Each figure corresponds to a
different choice of critical coupling, κ/κc , and shows the trajectory in the S − T plane as
144
-0.4
-0.2
0
0.2
-0.4
-0.2
0
0.2
5.0
(b)
0.8
0.8
κ/κc = 1.05
0.6
0.1
0.2
0.4
7.5
13
0.2
6.5 = Mχ
0.3
13 = Mχ
0.5
SM
mh =1.0 TeV
0
6.3
✣
4.9
8.7
0.6
6.3
7.5
5.4
✣
∆T
95%C.L.
3.8
4.3
0.4
5.0 TeV
3.3 TeV
3.8
95%C.L.
κ/κc = 1.2
(a)
0
Right curve: Fermion-bubble
Left curve: Improved RG
0.8
0.8
10
9.8
(c)
(d)
24
0.6
10
95%C.L. 13
0.4
95%C.L.
13
16
✣
✣
26
0.2
22 TeV
24
0.4
36
30
48
60
16 TeV = Mχ
0.6
36 = Mχ
0
0.2
0
κ/κc = 2
κ/κc = 4
-0.2
-0.2
-0.4
-0.2
0
0.2
-0.4
∆S
-0.2
0
0.2
Figure 69: Top Seesaw contributions to S and T are compared with the 95% c.l. error ellipse
(with mref
h = 1 TeV) for κ/κc = 1.05, 1.2, 2, 4, shown as a function of Mχ . In each plot, the
curve on the right is derived from the large-Nc fermion bubble calculation, the curve on the left
is deduced by an improved RG approach. For reference, the SM Higgs corrections to (S, T ),
SM varying from 100 GeV up to 1.0 TeV in plot (a).
relative to mref
h = 1 TeV, are given for mh
(from Hill, He and Tait [468]).
the χ mass varies. The results are based on both the large-Nc fermion-bubble calculation
and an improved RG in ref.[468]. The improved RG approach gives lower Higgs mass
values (around 400 − 500 GeV) so that the curves are slightly shifted towards the upper
left (the reality is somewhere between these two trajectories). The figure clearly illustrates
that the Top Seesaw model is consistent with the electroweak precision data provided Mχ
is in the appropriate mass range. For instance, when the Topcolor force is slightly supercritical, we see that precision data are effectively probing Mχ ∼ 4 TeV. In Fig. 70, we
display the same S − T trajectories as in Fig. 69, but with the corresponding Higgs mass
(Mh ) values marked [468].
The inclusion of a bottom seesaw generates additional b-ω mixing (ω ≡ bH is the
seesaw partner of b) which makes nontrivial contributions to the S and T parameters and
also to the Z → b̄b vertex. Furthermore, the composite Higgs sector now contains two
doublets and thus provides additional corrections to the precision observables. The b − ω
145
-0.4
-0.2
0
0.2
-0.4
-0.2
0
0.2
.501
(b)
0.8
0.8
κ/κc = 1.05
0.6
0.4
95%C.L.
965
471
0.1
0.2
✣
955
0.4
1.09
.434
0.2
943 = Mh
0.3
1.05 = Mh
0.5
SM
mh =1.0 TeV
0
1.1
.468
427
0.6
.482
455
✣
∆T
1.11 TeV
972 GeV
482
95%C.L.
κ/κc = 1.2
(a)
0
Right curve: Fermion-bubble
Left curve: Improved RG
0.8
0.8
.489
1.324
(c)
(d)
.459
0.6
1.320
95%C.L. .471
0.4
95%C.L.
.455
0.4
✣
1.307
0.6
1.42
.434
✣
1.41
.418
.426
0.2
1.43 TeV
1.40 = Mh
.407
1.294 TeV = Mh
0.2
0
0
κ/κc = 2
κ/κc = 4
-0.2
-0.2
-0.4
-0.2
0
0.2
-0.4
∆S
-0.2
0
0.2
Figure 70: Same as Fig. 69, but with the corresponding Mh values marked on the S-T trajectories instead. (from Hill, He and Tait [468]).
mixing induces a positive shift in the left-handed Z-b-b̄ coupling,
δgLb = +
e
2 sinθW cosθW
(sin φb )2 ,
(4.196)
which results in a decrease of Rb = Γ[Z → bb̄]/Γ[Z → hadrons], i.e., Rb ≃ RbSM −0.39(sbL )2 ,
as obtained in Ref. [464,468]. This puts an upper bound on the b-seesaw angle,
mb /µω
mb
sin φb ≃ √
≃
√ ,
Mω rb
1 + rb
(4.197)
and correspondingly a lower bound on the mass Mω (≃ Mχ ), as summarized in Fig. 72.
The Rb bound will mainly constrain the low tan β region of the effective composite two
doublet model.
Variant models [417,467] in which vectorial weak-doublet partners exist for both top
and bottom give similar contributions to T as the models with a weak-singlet partner for
top, while the contributions to Rb are suppressed by a small mixing angle. The lower
bound on these exotic quarks is, then, of order a few TeV, assuming they are degenerate.
In principle, one could experimentally distinguish between the models with weak-singlet
146
-0.4
-0.2
0
0.2
-0.4
-0.2
0
0.2
0.8
0.8
(a)
κ/κc = 2, tanβ = 2
0.6
(b)
17
6.3
95%C.L.
20
0.4
✣
0.4
23
26 TeV = Mχ
25 TeV = Mχ
32
0.3
0.5
0
✣
19
0.1
0.2
∆T
0.6
95%C.L.
0.2
0
mSM
h = 1.0 TeV
tanβ = 5
0.8
0.8
13
(c)
(d)
12
0.6
95%C.L.
95%C.L.
16
0.4
✣
13
0.4
✣
20
23 TeV = Mχ
0.2
0.6
0.2
20
26TeV = Mχ
0
tanβ = 12
0
tanβ = 40
-0.2
-0.2
-0.4
-0.2
0
0.2
-0.4
∆S
-0.2
0
0.2
Figure 71: Top and bottom seesaw contributions to S and T are compared with the 95% C.L.
−3 with a variety of values of tan β.
error ellipse (with mref
h = 1 TeV) for κ/κc = 2 and χ = 3 × 10
The S-T trajectories (including both Higgs and quark contributions) are shown as a function of
Mχ . For reference, the SM Higgs corrections to (S, T ), relative to mref
h = 1 TeV, are depicted
SM
for mh varying from 100 GeV up to 1.0 TeV in plot (a).
and weak-doublet mixings by measuring AtLR at an NLC [467]. The predicted shifts
relative to the SM value would be of similar size but opposite sign, as the ZtL tL coupling
is altered in the weak-singlet models while the ZtR tR coupling is altered in the weakdoublet models.
As discussed in [470], it is possible to use existing Tevatron data to set limits on direct
production of the χ-like states. New, mostly-singlet, quarks decaying via mixing to an
ordinary quark plus a W boson would contribute to the dilepton events used by the CDF
[471] and D0 [472] experiments to measure the top quark production cross-section. Since
the weak-singlet quarks are color triplets, they would be produced with the same crosssection as sequential quarks of identical mass. However the weak-singlet quarks can decay
via neutral-currents (e.g. dH → ZdL ) as well as charged-currents (e.g. dH → W uL ), and
this lowers the branching fraction of the produced quarks to the final states to which the
search is sensitive. In fact, the decay width bH → cL W is so strongly suppressed both by
Cabbibo factors and the large rate of bH → bL Z, that the Tevatron data do not provide
a lower bound on MbH . For models in which all quarks have weak-singlet partners, the
147
1
10
10
2
1
κ/κc = 2
tanβ = 1
-1
3
sinθLb
10
⇓
10
5
Allowed
-2
12
40
(a)
10
2σ Rb bound
-3
10 2
min
Mχ,ω (TeV)
(tanβ, Mχ,ω ) = (1, 20)
(3, 9)
10
(5, 5.5)
(12, 2.3)
1
(b)
1
(Different tanβ curves largely overlap)
min
(Mχ,ω in TeV)
10
10
2
Λ (TeV)
Figure 72: The Rb limits are shown for the b seesaw angle sbL = sin θLb in plot (a) and for the
mass Mω (≃ Mχ ) in plot (b). Here, we choose κ/κc = 2 and a wide range of tan β values.
limits
H
Md,s
>
153 GeV CDF
143 GeV D0
(4.198)
Similarly, mass limits on new mostly-singlet leptons (for flavor-universal mixing models) can be extracted from the results of LEP II searches for new sequential lepton doublets. In the relevant searches, the new neutral lepton N is assumed to be heavier than
its charged partner L and L is assumed to decay only via charged-current mixing with
a Standard Model lepton (i.e. BR(L → νℓ W ∗ ) = 1.0). The OPAL [473] and DELPHI
[474] experiments have each set a 95% c.l. lower bound of order 80 GeV on the mass
of a sequential charged lepton. As detailed in [470], when one adjusts for the increased
production rate and decreased charged-current branching fraction of the mostly-singlet
ℓH states, the resulting limit is
MℓH ≥ 84.9GeV .
(4.199)
is slightly stronger.
Finally, CDF limits [475] on heavy bH quarks pair-produced through QCD processes
and decaying via neutral currents can be applied to the weak-singlet fermions so long as
the value of B(bH → bL Z 0 ) is included. CDF finds 95% c.l. upper limits on the product
148
Figure 73: The 95% c.l. upper limit on pp̄ → b′ b̄′ X production cross section times the b′ → bZ
branching ratio squared (solid). The dashed curve shows the value predicted in a theory where
the branching ratio is 100% [475].
of cross-section and squared branching fraction as shown in figure 73. For B(bH →
bL Z 0 ) = 100%, heavy quarks with masses between 100 and 199 GeV are excluded. But
if the neutral-current branching fraction decreases for heavier b′ quarks, as in the models
discussed in [470], b′ quarks as light as 160 GeV may still be allowed by the data.
Ultimately, at higher-energy hadron colliders, direct pair-production of the χ states
and their subsequent decay via χ → ht → tt̄t could yield [468], a spectacular 6t final
state. As shown in Fig. 74, the cross section for this process with mχ = 1 TeV would be
∼ 10 pb.
4.5.2
Flavorons
As an extension of the Top Seesaw mechanism, models with flavor universal dynamical
symmetry breaking have been proposed [466,476]. In these models, the dynamics are
driven by family or large flavor gauge symmetries; when these symmetries break, multiplets of heavy “flavoron” bosons remain in the specrum. Their masses, like the symmetry
breaking scale, are expected to be of order a few TeV. The flavorons’ couplings to fermions
must be strong enough to generate condensates that break the electroweak symmetry and
provide the ordinary fermions with masses. As a result, the flavorons tend to be readily
produced at hadron colliders like the Tevatron – but also to have large decay widths that
make their detection in dijet invariant mass spectra a challenge [466].
For the moment, some of the best limits on flavor gauge bosons come from precision
electroweak data [454]. Flavorons coupling to ordinary fermions can give rise to direct
corrections to Zf f vertices; generally speaking, if the Zf f coupling is gf , the correction
149
10 4
mχ = 1 TeV
10 3
σχχ (fb)
10
mχ = 3 TeV
2
10
mχ = 5 TeV
1
10
10
10
10
-1
MG = 10 TeV
-2
p pbar
-3
pp
-4
20
40
60
80
100
120
140
E (TeV)
Figure 74: Cross-section for pair production of χ states at high energy hadron colliders,
leading to 6-top events [194].
is [429]
GκF MZ2
MF2
∆gf = gf
ln
6π MF2
MZ2
!
(4.200)
where G, κ ≡ gF2 /4π, and MF are the group theory factor, coupling, and mass appropriate
to the particular flavouron. Flavoron exchange across t and b quark loops will generally
contribute to the T parameter an amount (in leading log)
T = Nc (G′LL + 2G′RR )
h
i
κm4t
2
2 2
ln(M
/m
)
F
t
32π 2 sin2 θW cos2 θW MZ2 MF2
(4.201)
where G′LL,RR are group theory factors for flavoron couplings to left-handed and righthanded quarks. In addition, in models in which some generator of the flavor group mixes
with hypercharge or the diagonal generator of SU(2)W , the associated Z − Z ′ mixing also
contributes to T .
For each of the three representative flavoron models mentioned here, we assume that
flavoron exchange at low energies may be treated in the NJL approximation (with fourfermion coupling 4πκ/2MF2 ). The critical coupling required for chiral symmetry breaking
is
2Nπ
κcrit = 2
.
(4.202)
N −1
The minimal model [466,476,477,83] has a gauged SU(3) family symmetry acting on the
left-handed quark doublets. This family symmetry group does not mix with the Standard Model gauge groups and the model leaves the GIM mechanism intact. The critical
coupling is κcrit = 2.36. Models in which the SU(9) symmetry of color and family multiplicity of the left-handed quarks is gauged [466,265] also preserve GIM. A proto-color
150
group acting on the right-handed quarks is also present; obtaining the correct value for
αs in its presence requires
κ ≥ 3αs (2TeV) ≈ 0.3
(4.203)
Ignoring mixing between the flavor and proto-color group generators, the critical coupling
for chiral symmetry breaking is κcrit = 0.71. If the full SU(12) flavor symmetry of all
the left-handed quark and lepton doublets is gauged [466,265], both a proto-color group
and a proto-hypercharge group must be included for the right-handed fermions, and the
constraint (4.203) still applies. In this case, κcrit = 0.53 – not far above the lower limit.
The 95% c.l. lower bound on MF from precision electroweak data for κ at its critical
value is approximately 2 TeV for all three models [454].
Comparable limits on MF (for critical coupling κ) in the SU(3) and SU(9) models
have been obtained from Tevatron Run I data [478]. Run II should be sensitive to MF
up to 2.5 - 3 TeV with 2 f b−1 of data. For the SU(3) model, both dijet and anomalous
single top production are likely to yield large signals; in the SU(9) model, the dijet signal
is relatively enhanced and the single top signal, relatively reduced in size, enabling the
two models to be distinguished if both channels are studied.
A significantly stronger bound on the flavorons of the SU(12) model derives from
atomic parity violation: MF >
∼ 10 TeV [478]. Flavorons from the SU(12) model light
enough to produce visible signals at the Tevatron are already excluded by this limit.
4.6
Extra Dimensions at the TeV Scale
Recently there has been considerable interest in theories of extra dimensions of space
which emerge not far from the weak scale. We cannot give a comprehensive review of
this burgeoning field, but we will outline some key ideas relating to electroweak symmetry
breaking and the physics of new strong dynamics. The ideas largely stem from the view
that string theories admit an arbitrary hierarchy between the compactification scale of the
extra dimensions, and the fundamental string scale. Moreover, with gravity in a higher
dimensional bulk the “true” Planck scale can emerge as a ∼ 100 TeV scale, but is subject
to rapid power-law renormalization; grand unification remains viable in these theories,
and one is led to a long list of novel phenomena.
Some of the key early works are those of Antoniadis, et al. and Lykken (weak scale
superstrings [479,480],[481]);36 Arkani-Hamed, Dimopoulos, and Dvali, (millimeter scale
gravity [482,483]); Dienes, Dudas and Gherghetta, (gauge fields in the bulk and powerlaw unification [484,485]); Randall, Sundrum (DeSitter space in the bulk and natural
hierarchy [486,487]). It is also essential that chiral fermions emerge in the compactified
theory, and this is possible by implementing orbifold compactification ala Horava and
Witten, [488], or domain walls ala Arkani-Hamed and Schmaltz [489]. In the latter case
36
Possibly the earliest proposal for “weak scale extra dimensions” was an heroic effort by Darrell R.
Jackson to identify the W and Z as Kaluza-Klein modes, (private communication to CTH, Caltech, ca.
1975.)
151
dynamical mechanisms arise for the origin of generational hierarchies, and the CKM
matrix. [490,491] Many phenomenological constraints have been placed upon the scales
of the new extra dimension(s), and many variations and developments of this theme now
exist, too numerous to review here. A compactification scale lower limit of order ∼ 1 TeV
seems to be indicated phenomenologically.
Why not consider extra space-time dimensions at the weak scale? After all, supersymmetry is an extra-dimensional theory in which the extra dimensions are fermionic,
or Grassmanian, variables. Supersymmetry leads naturally, upon “integrating out” the
extra fermionic dimensions (i.e., descending from a superspace action to a space-time
action), to a perturbative extension of the Standard Model, e.g., the MSSM. In such a
scheme the Higgs sector is at least a two-doublet model, and the lightest Higgs boson is
expected to be in a range determined by the perturbative electroweak constraints, <
∼ 140
GeV. ¿From a “bottom-up” perspective the key lesson from Supersymmetry is that an
organizing principle for physics beyond the Standard Model emerges from hidden extra
dimensions which are then integrated out. Upon specifying the algebraic properties of
the extra dimensions one is led to a particular symmetry structure and class of dynamics
for EWSB. The particular extra dimensions in which we will be interested are ordinary
c-numbers, but one can contemplate other possibilities [492–494].
Extra dimensions do not show up as new “real estate,” but rather they appear in
accelerator experiments as new particles 37 . These are the excited modes of existing
fields, e.g., quarks, leptons, gauge bosons, the graviton, that propagate in the compact
extra dimensions, and are known as Kaluza-Klein (KK) excitations. The KK modes
typically appear as a ladder spectrum of discrete resonances with the same quantum
numbers as the zero-mode field itself. For example, if we consider the possibility that
QCD gauge fields (gluons) can propagate in the bulk then the KK modes show up as a
sequence of massive colorons.
The idea of extra c-number dimensions near the weak scale has led to the proposal of
dynamical schemes that can imitate the strong dynamics we have discussed in this review.
Dobrescu first observed that the main ingredients of Topcolor, i.e., the colorons, appear
naturally as KK modes of the gluon [496] and a number of detailed models have been
discussed [497–500]. These essentially follow the idea of top condensation as discussed
in Section 4. The other key ingredient of the Top Seesaw model, the seesaw partner χ
quarks, can emerge as KK modes as well.
4.6.1
Deconstruction
As KK-modes begin to appear in accelerator experiments, we can ask: “how are they
to be described phenomenologically, as if we have no knowledge of the extra dimensions
themselves?” or, equivalently, “What is the effective low energy Lagrangian of the KK
modes?” This has led to a new twist in the development of these models known as “deconstruction.” Independently, the “Harvard” group of Arkani-Hamed, Cohen and Georgi,
37
For a discussion of one possibility for extra dimensions of time, see [495].
152
[42,501,502] and the “Fermilab” group Hill, Pokorski, Wang and Cheng [41,503,504], have
proposed using a lattice to describe the extra dimensions. The 3 + 1 dimensions of spacetime are continuous, while the extra compact dimensions are latticized (this is known
as a “transverse lattice” [505]). By using the lattice technique, one can “integrate out”
the extra dimensions, preserving local gauge invariance, and arrive at a manifestly gauge
invariant effective Lagrangian including Kaluza-Klein modes (in a sense the KK modes
are analogues of superpartners).
To get a flavor for deconstruction, let us consider the first KK-mode excitation of
the gluon in a compactified theory of D = 4 + 1 dimensions. We can write its effective
Lagrangian knowing only the SU(3) symmetry of QCD. The lowest KK mode appears
as a “coloron,” a heavy linear octet representation of the gauge group SU(3); its mass is
the only parameter determined by compactification. A heavy vector meson which forms
a multiplet under a (local or global) symmetry group G, can always be described as a
gauge field [163], with a spontaneously broken gauge group that is an identical copy of
G. For the single coloron case we have G = SU(3). Hence, we may take our theory of a
gluon plus its first excitation to be based upon the gauge group G′ = SU(3)1 × SU(3)2
and break this to G = SU(3) with a Higgs field Φ transforming as a (3, 3̄).
By constructing a Higgs potential:
V (Φ) = −M 2 Tr(Φ2 ) + λ Tr(Φ4 ) + λ Tr(Φ2 )2 + M ′ det(Φ)
(4.204)
we can arrange for Φ to develop a vacuum expectation value V , by which all unwanted
Higgs degrees of freedom are elevated to the large mass scale ∝ M. Note that in trying
to discard the extra 10 degrees of freedom in Φ as a Higgs field, we encounter a unitarity
bound [24,506,507] which predicts the string scale from M.
The Lagrangian containing the gluon and the first KK mode is therefore:
1 A Aµν 1 A Aµν
L = − F1µν
F1 − F2µν F2 + |Dµ Φ|2 − V (Φ).
4
4
(4.205)
The field Φ, after developing a VEV, and decoupling the Higgs degrees of freedom, can
be parameterized by a chiral field:
√
(4.206)
Φ → V exp(iφA T A / 2V )
This is exactly the structure that would emerge from a Wilson lattice with two transverse
lattice points (which are now “branes, each carrying a 3 + 1 theory). We also see that
this is identical to the gauge sector of Topcolor models. With the breaking in place, the
theory has two mass eigenstates, the gluon and coloron:
A
A
GA
µ = cos θA1µ + sin θA2µ ;
A
BµA = − sin θAA
1µ + cos θA2µ ;
(4.207)
where Aiµ belongs to SU(3)i . If we allow SU(3)i to have a coupling constant gi then we
find:
g1
tan θ =
(4.208)
g2
153
1
2
...
n
...
n+1
N
ΨR
X
X
X
X
X
X
X
ΨL
Figure 75:
Dirac fermion corresponding to constant φ has both chiral modes on all branes.
The ×’s denote the φ couplings on each brane, and the links are the latticized fermion kinetic
terms which become Wilson links when gauge fields are present. (from [508]) The spectrum here
has a singlet lowest massive mode, and doubled KK modes; by adding a Wilson term one can
remove one of the two cross-bars between adjacent branes, and elminate second Brillouin zone
doubling in the spectrum.
The mass of the coloron is given by
MK = V
q
g12 + g22 =
1
R
(4.209)
where R is identified with the radius of the compactified extra dimension. We also see
that the low energy QCD-coupling is related to gi as:
1
2
gQCD
=
1
1
+ 2
2
g1 g2
(4.210)
2
With g1 = g2 we have g12 = 2gQCD
and this is the onset of “power-law running” of the
coupling constant. The model indeed represents a single KK mode, descending from
higher dimensions. Conversely, observation of a coloron in the particle spectrum could be
taken to suggest that a new extra dimension is opening up.
The full lattice description of QCD is a straightforward generalization of this model
[41]. If we consider the full lattice gauge theory in 4 + 1 dimensions with a Wilson action
for the extra dimension, we obtain the Lagrangian:
L=−
N
−1
N
X
1X
A
Fjµν
FjAµν +
Tr(Dµ Φk )2
4 j=1
k=1
(4.211)
Here there are N − 1 Φ fields, and the kth field transforms as an (3k , 3̄k+1) representation,
straddling the nearest neighbor SU(3)k and SU(3)k+1 gauge groups. Φk is a unitary
154
1
2
...
n
...
n+1
N
ΨR
X
X
X
X
X
X
ΨL
Figure 76: A chiral fermion occurs on brane n where φ(x5 ) swings rapidly through zero. The
chiral fermion has kinetic term (Wilson links) connecting to adjoining branes. (from [508])
matrix, and represents the Wilson line:
Φk = V exp ig
Z
xk+1
xk
dx5 Aa5 T a
(4.212)
Φk may be parameterized by:
√
Φk = V exp(iφak T a / 2V )
(4.213)
where T a = λa /2. This clearly matches our previous discussion of a single KK mode.
The Φk kinetic terms imply a gauge field mass term of the form:
−1
1 2 2 NX
g V
(Aakµ − Aa(k+1)µ )2
2
k=1
(4.214)
Performing the mass spectrum analysis of the model is equivalent to analyzing the phonon
spectrum of a one-dimensional crystal lattice. The mass term is readily diagonalized and
we obtain eigenvalues reflecting a ladder of states with:
√
Mn = 2gV sin(πn/2N)
(4.215)
the lowest energy masses given by:
√
1
Mn ≈ πgV n/N 2 ≡
R
(4.216)
This is the spectrum assuming free boundary conditions on the lattice and corresponds
to orbifolding. With periodic boundary conditions instead, one gets the Aa5 as low energy
modes as well, which act like pseudo-scalars, and the KK mode levels are doubled.
155
1
2
...
n
n+1
...
N
ΨR
X
X
X
X
X
X
X
X
ΨL
tR
X
X
X
X
tL
Figure 77:
Pure top quark condensation by Topcolor is obtained in the limit of critical
coupling on brane n and decoupling to the nearest neighbors. Decoupling corresponds to taking
the compactification mass scale large; the links are then denoted by dashed lines. (from [508])
The net result is that the effective Lagrangian for, e.g., QCD propagating in the extra
dimensional bulk, is a chain of gauge groups of the form: SU(3) × SU(3) × SU(3) × ...,
one gauge group
√ per lattice brane. The high energy coupling constant common to each
group is g ∼ N gQCD , consistent with the power-law running. ¿From a “bottom-up”
perspective, one writes down this gauge structure with arbitrary relevant operators with
arbitrary coefficients to describe the most general low energy effective theory for extradimensional physics.
Various additional dynamical issues in deconstructed theories are currently under investigation, such as chiral dynamics [509,510], electroweak observables [511,512], unification [502,513] and GUT breaking [514]. A number of authors have applied deconstruction
to supersymmetric schemes and SUSY breaking [515,516] [517,518]. There are interesting
and novel topological issues in the presence of many gauge factor groups, e.g., [519,520].
Duality and supersymmetry can have an intriguing interplay [521] in these schemes via
topology. Deconstructing gravity is in need of elaboration, but Bander has made an
interesting first attempt in the language of vierbeins [522].
Fermions can be accomodated, and chirality can be engineered. Chiral fermions can
be localized in the fifth dimension by background fields [489,523,524]. A free fermion has
the continuum and latticized action (we neglect the gauge interactions here, and do not
include Wilson terms):
Z
5
5
5
d x Ψ̄(i∂/ − ∂5 γ − φ(x ))Ψ →
N Z
X
n=1
d4 x Ψ̄n (i∂/ − φn )Ψn + V Ψ̄n γ5 Ψn+1 + h.c. (4.217)
If the background field is approximately constant both chiral components of the
fermion appear on each lattice brane, as depicted in Fig.(75). The cross bars are the
156
1
2
...
n
n+1
...
N
ΨR
X
X
X
X
X
X
X
X
ΨL
tR
X
X
X
X
tL
Figure 78: Top Seesaw Model arises when the effects of nearest neighbor vectorlike fermions
are retained, i.e., when these heavier states are only partially decoupled. Keeping more links
maintains the seesaw. Usually we denote tRn ∼ χR , tLn+1 ∼ χL , tRn+1 ∼ tR . (from [508])
latticized links allowing hopping of the fermion from n to n + 1 in eq.(4.217) If φ(x5 )
swings through zero rapidly in the vicinity of brane n, i.e., φn = 0 then only a single
chiral component is normalizeable in the vicinity of n and one gets a dislocation in the
lattice as shown in Fig.(76).
The coupling strength of SU(3)n on the n-th brane will generally be renormalized
by the dislocation and can become supercritical, Fig.(77). It would, therefore, not be
coincidental to expect this to happen; indeed a variety of effects are expected to occur
near the dislocation, e.g., the chiral fermions themselves can feed-back onto the gauge
fields to produce such renormalization effects. The result is a chiral condensate on brane
n forming between chiral fermions. Identify Ψ = (t, b)L and tR as the chiral zero-modes
on brane n of two independent Dirac fields in the bulk. In the limit that we take the
compact extra dimension very small, the nearest neighbor links decouple at low energies.
In this limit we recover a Topcolor model with pure top quark condensation.
In Fig.(78) we illustrate the case that some of the links to nearest neighbors are not
completely decoupled. Again, this can arise from renormalizations due to background
fields, or to warping [503,504]. Thus the mixing with heavy vectorlike fermions occurs in
addition to the chiral dynamics on brane n. In this limit we obtain the Top Quark Seesaw
Model.
4.6.2
Little Higgs Theories
Recently another approach to electroweak symmetry breaking has been revived, inspired
by deconstruction, in which the Higgs itself is a PNGB, [501]. This is actually an old
idea, due to Georgi, Kaplan, and others (see, e.g. [525][526][527]). The novelty presently
157
is that delocalization in an extra dimension (or the minimal deconstructed description
of such) leads to a potentially softer scheme in which dangerous gauge loop quadratic
divergences contributing to the Higgs boson (mass)2 can be cancelled in one loop order.
The resulting models bear some resemblance to the MSSM Higgs structure.
There are now a number of schemes, most notably the “minimal moose model,” [528]
and SU(5)/SO(5) model [529,530], and the SU(6)/SO(6) scheme [531]. We briefly describe the “minimal moose” scheme.
When we periodically compactify a higher dimension we find that there is, in the
absence of loops, a massless mode corresponding to the zero-mode of Aa5 . For QCD this
would appear in the effective lagrangian as a color octet of pseudoscalars. However, since
these modes carry color, they will acquire mass ∼ gV . Arkani-Hamed, Cohen and Georgi
show how this leads to a Coleman-Weinberg potential for the modes, and is generally
finite. The authors then construct a scheme [501] in which this mode can arise as an
electroweak isodoublet.
The idea, from a lattice point of view, is to latticize two compact dimensions with
toroidal boundary conditions (T2 , the surface of a doughnut). The minimal plaquette
action is then a generalized Eguchi-Kawai model [532], but with the additional links
associated with the periodicity. At three of the four sites of the plaquette one attaches
SU(3) gauge groups. On the fourth cite one attaches SU(2) × U(1). The Wilson links
(linking Higgs) which attach the SU(2) × U(1) gauge group to nearest neighbor SU(3)’s
then contain components which transform as I = 21 Higgs fields. The Coleman-Weinberg
potential produces an unstable vacuum. The theory leads to an effective two-Higgs doublet
model. The gauge couplings would be expected to raise the mass of the lightest Higgs to
order ∼ g22 Λ2 , but the discrete symmetries of the model, associated with the delocalized
structure, lead to a GIM-like cancellation of this contribution. The dangerous term does
occur at two-loop level ∼ g24 Λ2 . With Λ ∼ 10 TeV the scale of the “compactification
dynamics” the lightest 0+ Higgs can be of low mass, ∼ 120 GeV. Hence a “little” hierarchy,
10 TeV <
∼µ<
∼ vweak is protected in this scheme.
These models are receiving fuller elaboration at present. For a critical discussion of
this scheme see K. Lane [533]. Two recent works attack several aspects of the models from
the perspective of consistency with electroweak radiative corrections [534]. The models
contain ingredients, such as I = 0 PNGB’s, that develop tadpole VEV’s and may violate
T cosntraints. Also, the models such as SU(5)/SO(5) employ a top seesaw to give to the
top quark mass, and the extended fermion structure can be problematic in loops [535].
Some recent papers examine collider phenomenology of these schemes [536,537].
Much remains to be done with Little Higgs schemes, in particular in establishing their
phenomenological naturalness.
158
5
Outlook and Conclusions
The next decade will bring great discoveries in the field of elementary particle physics. The
Tevatron Run-II program and the LHC will begin to reveal the mechanism(s) responsible
for the origin of mass. The Large Hadron Collider in concert with a Linear Collider can
begin detailed studies of the fields involved in electroweak symmetry breaking. In the
longer term, a Very Large Hadron Collider will have the power to probe the corollary
mechanisms that underlie the physics of flavor.
This review has focused on the possibility that the origin of mass involves a new strong
dynamics near the TeV scale. As we have argued here, dynamical electroweak symmetry
breaking, whether it comes in the guise of Technicolor, Topcolor, or some variant scheme,
can provide a natural explanation for the weak scale. It can also provide tantalizing clues
about the magnitudes and origins of the fermion masses and mixings, and may be able
to explain CP violation. Indeed, many issues of flavor physics that are relegated to the
Planck or Gut scale in perturbative theories, such as the MSSM, become accessible to
sufficiently energetic accelerators in the dynamical framework.
Technicolor was created in the era in which the W and Z bosons are heavy and
all known fermions are comparatively light. In the first approximation, therefore, the
fermions are massless and Technicolor is essentially a pure QCD-like theory which naturally generates the weak scale, in analogy to the way the strong scale is generated by QCD
through its “running coupling constant.” By itself, Technicolor requires a new gauge group
and additional fermions, i.e., techniquarks. Technicolor is, however, an incomplete theory.
Extended Technicolor was introduced to accomodate light fermion masses, but even the
charm quark begins to push the limits on ETC from rare decay processes. Hence, one
is led to various schemes to accomodate heavier fermion masses, such as Walking Technicolor and Multi-Scale Technicolor. Combinations with Supersymmetry and Bosonic
Technicolor have been considered.
In the post–1990 era, in addition to the heavy W and Z, we have the very heavy top
quark. This leads to an alternate point of departure for dynamical models of EWSB, based
on the idea of Topcolor. Topcolor by itself is a fine-tuned theory in which the top quark
alone condenses, predicting mt ∼ 220 GeV, and this first–order theory can be ruled out.
However, in concert with Technicolor, we arrive at viable models in which a new Technidynamics can coexist and the top quark acquires a dynamical mass through Topcolor.
Such models, “Topcolor Assisted Technicolor” or TC2, predict a rich phenomenology
that may be accessible to the Tevatron, and certainly to the LHC. In this view, the top
quark is playing the role of the first techniquark.
Topcolor by itself, with a small extension, can naturally yield a completely viable
theory of dynamical EWSB: the “Top Quark Seesaw.” In this scheme the top quark condenses, through strong Topcolor, but there are additional vectorlike fermions that mix to
give the physical mass of the top quark through a seesaw. The Top Quark Seesaw, despite
an effective boundstate Higgs boson with a mass of ∼ 1 TeV, is in agreement with present
oblique radiative corrections (the S − T plot). It explains the W and Z masses, as well as
159
mt and mb , the latter through instantons or a simple extension. Moreover, it involves the
fairly robust gauge group imbedding of QCD to SU(3) × SU(3) and additional vectorlike fermions, and is therefore as economical as Technicolor. Finally, the SU(3) × SU(3)
structure, and the new vectorlike fermions, are completely natural ingredients expected
from extra dimensions of space near the weak scale. The model may be extendable to
generate all known fermion masses and mixing angles, though no complete theory has yet
been written down.
Thus, strong dynamical models of EWSB have evolved and remain viable and consistent with all known experimental limits. Of course, regardless of the theoretical beauty of
any given scenario, its validity will ultimately be ascertained through experiment. Should
the slight discrepancies in the electroweak precision tests, e.g., the forward-backward
asymmetries of leptons and hadrons, open into a new phenomenology, then novel TeVscale dynamics may be in the offing. Should the hadron colliders find a spectrum of new
states with unexpectedly large production cross-sections, or should the Higgs boson be
shown to be very heavy through exclusion, then new strong dynamics will be indicated.
We have seen that new strong dynamics can be consistent with the presence of (even
weak-scale) Supersymmetry. Another exciting candidate system of organizing principles
is the existence of c-number extra dimensions, which we have found to be naturally associated with several types of strong dynamics, such as the Top Seesaw or the Little Higgs
models. Hence, we will likely uncover new fundamental organizing principles in our quest
to unravel EWSB, whether the dynamics is perturbative or strongly coupled.
The task of theorists is to prepare to understand whatever results emerge from the
upcoming experiments aimed at the electroweak scale. In this review we have focused upon
a set of ideas complementary to the most popular and well-studied perturbative MSSM
scenarios. We have seen that, when considered in equivalent detail, these dynamical
theories are no less viable, given current experimental bounds. They hint at connections
to deeper ideas about the structure and interplay of space, time and matter. They remain,
in our opinion, exciting prospects for discovery in the near future.
6
Acknowledgements
We wish to thank the following individuals for useful input and commentary to this
review: Tom Appelquist, William Bardeen, Gustavo Burdman, Sekhar Chivukula, Bogdan
Dobrescu, Estia Eichten, Keith Ellis, Hong-Jian He, Ken Lane, Chris Quigg, and Koichi
Yamawaki.
160
Appendix A: The Standard Model
A(a) Summary of Standard Model Structure
The Standard Model is a unified description of the electromagnetic and weak interactions based upon the gauge group SU(2) × U(1). The fermions are chiral, i.e., the
left-handed fermions are doublets with respect to the SU(2) gauge group, while the righthanded fermions are singlets. Left– and right–handed fermions also carry different charges
under the U(1) gauge group.
The concept of spontaneous symmetry breaking, to provide both the gauge boson and
fermion masses, is necessarily imbedded into the Standard Model. As in the LandauGinzburg superconductor, a scalar field, the “Higgs field,” is introduced explicitly, “by
hand,” to provide the symmetry breaking condensate. Recalling the toy model example of
Section 1.3(iii), the fermions can only acquire Dirac masses (or neutrino Majorana masses)
through coupling to the condensing Higgs boson, since explicit fermion mass terms are
not singlets under the gauge group, owing to the fermionic chiral charge assignments. The
gauge fields also acquire their masses through the condensing Higgs boson, as per the toy
example of Section 1.3(iv).
We begin by considering an SU(2) × U(1) gauge theory with one scalar field that
transforms as an isodoublet under the SU(2) group. The covariant derivative for the
theory is given by:
iDµ = i∂µ − g2 Wµa Qa − g1 Bµ
Y
2
Y
= i∂µ − g2 Wµ+ Q− − g2 Wµ− Q+ − g2 Wµ3 Q3 − g1 Bµ
(A.1)
2
√
√
where the “charge basis” expressions are Q± = (Q1 ±iQ2 )/ 2, and Wµ± = (Wµ1 ±iWµ2 )/ 2.
Note there are two gauge coupling constants, g2 and g1 , one for each simple subgroup of
the theory. The Qa are the SU(2) weak charges and Y is the U(1) hypercharge. The Qa
satisfy the SU(2) Lie algebra:
[Qa , Qb ] = iǫabc Qc .
(A.2)
These charges are at this stage abstract operators, the generators of the associated groups.
We specialize to particular representations for the charges, choosing an I = 12 weak isospin
representation for the left-handed fermions, and Higgs boson, in which Qa = τ a /2 where
the τ a are Pauli matrices. The right-handed fermions are singlets, annihilated by the Qa .
We further define the weak hypercharge for any representation by its eigenvalue, Yr , or
the abstract operator Y .
In defining the Standard Model we choose a specific generator to correspond to the
electric charge operator:
Y
(A.3)
QEM = Q3 +
2
Note that the choice of Q3 is a choice of basis, i.e., we could use any normalized linear
combination of the Qa in place of Q3 , but by a gauge rotation we can always align that
161
linear combination with Q3 . Given the electric charges of particles, this defines the Yr for
any representation.
The Standard Model left-handed fermionic matter doublets are then assigned the
following structure and weak hypercharges:
u2/3
d−1/3
!
ν0
e−1
1
, Yr = ;
3
L
!
L
, Yr = −1;
c
H =
φ+
−φ̄0
!
Yr = +1;
(A.4)
where the last entry is the “charge–conjugated Higgs field” defined by H c = iτ 2 H ∗ . The
right–handed fermions are SU(2) singlets, i.e., [Qa , ψR ] = 0, hence,
4
2/3
uR , Yr = ;
3
−1/3
dR
2
, Yr = − ;
3
e−1
R , Yr = −2;
νR0 , Yr = 0.
(A.5)
A right–handed neutrino is sterile, and usually omitted in the definition of the “Standard
Model.”
Returning to the gauge bosons, if we write the linear combinations:
Wµ3 = Zµ0 cos θ + Aµ sin θ
Bµ = −Zµ0 sin θ + Aµ cos θ
(A.6)
where Zµ0 (Aµ ) is the physical Z-boson (photon), then we see,
g2 sin θ = e;
g1 cos θ = e;
(A.7)
The photon thus couples to eQEM with strength e where:
1
1
1
= 2+ 2
2
e
g2 g1
(A.8)
The weak mixing angle is defined by:
tan θ =
g1
g2
(A.9)
The gauge covariant field strengths are defined by commutators of the covariant derivative:
i
Tr (τ a [Dµ , Dν ]) = ∂µ Aaν − ∂ν Aaµ + ǫabc Abµ Acν
g22
i
= − 2 Yr−1 Tr ([Dµ , Dν ]) = ∂µ Bν − ∂ν Bµ
g1
a
Fµν
= −
Fµν
(A.10)
Then the gauge field kinetic terms are:
1 a a
1
LG.B. kinetic = − Fµν
Fµν − Fµν Fµν
4
4
162
(A.11)
Consider the complex doublet scalar Higgs-boson with Yr = −1:
H=
φ0
φ−
!
(A.12)
The Lagrangian for H takes the form:
L = (Dµ H)† (D µ H) − V (H)
(A.13)
The masses of the gauge bosons are generated by spontaneous symmetry breaking, in strict
analogy to the generation of mass in a superconductor as discussed in Section 1.3(iv). We
assume that V (H) has an unstable extremum for H = 0 and a nontrivial minimum, e.g.,
we can write:
λ
2
V (H) = (H † H − vweak
)2
(A.14)
2
The Higgs boson then develops a vacuum expectation value. This may, without loss of
generality be taken in the upper component (which is, again, the choice of orientation
within the internal SU(2) space that defines the electric charge to be that combination
of generators that annihilates hHi):
!
!
!
vweak + √h2
πaτ a
(A.15)
hHi =
or:
H = exp i √
0
2vweak
√
In the present normalization vweak = (2 2GF )−1/2 = 175 GeV. Note that QEM acting on
the Higgs VEV is zero, which implies that the photon remains a massless gauge boson γ.
When we substitute the anzatz (A.15) into the Higgs boson kinetic term of eq. (A.13),
the masses of the gauge bosons are generated (as in the Abelian Higgs model of Section
1.3(iv)):
√
λ
1 2 + µ− 1 2
1 2 2
1
1
2
µ
(∂h) + MW Wµ W + MZ Zµ Z − MH h −
MH h3 − λh4
L =
2
2
2
2
2
8
1 2 MH
h (g22 Wµ+ W µ− + (g12 + g22 )Zµ Z µ )
(A.16)
+ h +
2
λ
√
where MH = vweak 2λ. This Lagrangian also exhibits the coupling of the Higgs h field
to itself and to the physical W and Z fields. Inverting eq.(A.6):
vweak
0
Aµ = sin θ Wµ3 + cos θ Bµ
Zµ = cos θ Wµ3 − sin θ Bµ =
(g2 Wµ3 − g1 Bµ )
(A.17)
q
g12 + g22
allows us to rewrite the covariant derivative as:
iDµ = i∂µ −
g2 Wµ+ Q−
−
g2 Wµ− Q+
163
− eAµ
Y
Q +
2
3
e
− Zµ Q
(A.18)
where the Z-boson couples to the neutral current charge:
e
Q
q
Y
τ3
− sin2 θ
cos θ
+
=
2
2
!
τ3
Y
= e cot θ
− tan θ
2
2
g12
g22
2
!
(A.19)
We can also extract the masses, upon comparing eq.(A.16) with eq.(A.13) using the full
covariant derivative (A.18):
1
2
2
;
MW
= g22 vweak
2
1 2
MZ2 = vweak
(g12 + g22).
2
(A.20)
¿From this, it follows:
2
MW
= cos2 θW
MZ2
1
2
vweak
= √
2 2GF
(A.21)
The coupling of the matter fields to gauge fields is then determined through the gauge–
invariant kinetic terms. For example, let us consider the left–handed top and bottom
quark doublet ΨL = (t, b)L . The kinetic term is:
1
1
Ψ̄L iD/ ΨL = Ψ̄L i∂/ ΨL − √ t̄γµ LbW µ + − √ b̄γµ LtW µ −
2
2
e
2e
e Ψ Z
− t̄γµ LtAµ + b̄LbAµ − Ψ̄L Qγ
µ L µ
3
3
(A.22)
where L = 21 (1 − γ 5 ). Dirac mass terms would be of the form Ψ̄L ψR and are therefore
isospin− 12 ; we cannot directly add such nonsinglet terms into the Lagrangian. We can,
however, assume there are terms of the form:
gt Ψ̄L · HtR + gb Ψ̄L · H c bR
(A.23)
which couple the left- and right-handed fermions to the Higgs field, and which are invariant
under SU(2) × U(1). When H develops its VEV, we see that we obtain masses mt =
gt vweak and mb = gb vweak for the top and bottom quarks respectively. In general, of course,
there will occur mixing effects when we include the light generations; these lead to the
Cabibbo-Kobayashi-Maskawa (CKM) matrix relating the gauge and mass eigenbases for
left-handed quarks.
For completeness, we mention that to minimally generate a neutrino Majorana mass,
we would write for a leptonic doublet ΨL = (ν, ℓ)L :
gν
(Ψ̄L H)(H C ΨC
L)
M
(A.24)
where ΨC
L is the charge-conjugated spinor. This term produces a Majorana mass for νL
2
of gν vweak
/M.
164
The interplay between gauge symmetries, and chiral symmetries, both of which are
broken spontaneously, is fundamental to the Standard Model. The left-handed fermions
carry the electroweak SU(2) quantum numbers, while the right-handed do not. All of the
mathematical features of the symmetric Lagrangian remain intact, but the spectrum of the
theory does not retain the original obvious symmetry properties. When a massive gauge
boson was discovered, such as the W ± or Z of the Standard Model, we also discovered an
extra piece of physics: the longitudinal component, i.e., the NGB which comes from the
symmetry breaking sector.
A(b) Summary of One–Loop Oblique Radiative Corrections
We will now summarize a particular subclass of electroweak radiative corrections, the
so-called “oblique” corrections. To begin, we generalize slightly our definition of vweak in
the masses of the W and Z. Let us generalize eq.(A.20) and write:
1 2 2
2
MW
= vW
g2 ;
2
1
MZ2 = vZ2 (g12 + g22 )
2
(A.25)
where vW (vZ ) is the Higgs VEV “as seen by” the W -boson (Z=boson). In tree level
2
2
vW
= vZ2 = vweak
, and we want to include radiative effects that split these quantities
and lead to their q 2 eveolution. The radiative corrections to the spontaneously broken
Standard Model thus involve keeping track of four underlying functions of the momentum
2
scale, q 2 which are: g12 (q 2 ), g22 (q 2 ) and vW
(q 2 ) and vZ2 (q 2 ). For the running of the coupling
constants, to a good approximation, we really only need to know α(MZ ) ≈ α(MW ) in
most applications.
Let us be more precise about vW and vZ . First, we scale away the coupling constants
by defining Ãaµ = g2 Aaµ and B̃µ = g2 Bµ . Loop corrections to correlation functions (or
inverse propagators) for any two gauge fields can be written as:
B
T
F.T. < 0|T ÃA
µ (0)õ (x)|0 >= gµν ΠAB − qµ qν ΠAB
(A.26)
(subscript T stands for “transverse”). We then make the specific definitions of vW and vZ
1 2
1 2
vZ =
v
− Π3B
2
2 weak
1 2
v
− Π3Q + Π33
=
2 weak
1 2
1 2
vW =
v
+ ΠW W − Π3Q
2
2 weak
and of the couplings
1
1
= 2 − ΠT33 − ΠT3B
2
g2
g2un
1
= 2 − ΠT3Q
g2un
165
(A.27)
1
1
= 2 − ΠTBB − ΠT3B
2
g1
g1un
1
= 2 + ΠT3Q − ΠTQQ
g1un
(A.28)
where gi un is an unrenormalized coupling constant. The dominant Standard Model con2
tributions to vW
and vZ2 at q 2 = 0 come from the third generation and a putative Higgs
2
boson. The difference vW
− vZ2 is, upon computing the loops, finite:
2
vW
− vZ2
"
Nc
2m2t m2b
2
2
=
(m
+
m
)
−
log(m2t /m2b )
t
b
32π 2
(m2t − m2b )
#
2
MW
m2H
MZ2 m2H
2
2
2
2
+ 2
ln(mH /MW ) − 2
ln(mH /MZ )
2
mH − MW
mH − MZ2
(A.29)
The ρ parameter of Veltman is:
ρ =
2
vW
vZ2
"
2m2t m2b
Nc
2
2
(m
+
m
)
−
log(m2t /m2b )
= 1+
t
b
32v02 π 2
(m2t − m2b )
#
2
MW
m2H
MZ2 m2H
2
2
2
2
+ 2
ln(mH /MW ) − 2
ln(mH /MZ )
2
mH − MW
mH − MZ2
(A.30)
2
In general, vW
and vZ2 have small, nonzero, q 2 dependence. Moreover, the effect of
new physics at the weak scale, |q| ∼ 1 TeV will generally be to induce additional q 2
2
2
dependence into vW
and vZ2 and additional splitting into vW
− vZ2 beyond the Standard
Model contributions. For example, new sequentially heavier chiral fermions will contribute
through loops, and their contributions do not decouple as their masses are taken large.
The squared couplings g12 and g22 are less susceptible to effects from new physics We seek,
therefore, a convenient parameterization of the effects of new physics that can be used for
comparison with a variety of experiments. The resulting interpolation between different
experiments may involve nonstandard values of these new parameters and yield evidence
for new physics.
2
A simple set of parameters can be generated by expanding vW
and vZ2 in a Taylor
series in q 2 as follows:
2
2
vW
(q 2 ) = vweak
+ σq 2 + τ vr2 + ωq 2
2
vZ2 (q 2 ) = vweak
+ σq 2 − τ vr2 − ωq 2
(A.31)
(A.32)
2
Thus, vweak
is the average (I = 0) of the two decay constants and contains most of the
physics of symmetry breaking, while τ (I = 1) is just a rewriting of the splitting between
these at zero momentum (i.e., the ρ parameter). The parameters σ and ω are respectively
(I = 0) and (I = 1) measures of physics contributing to the q 2 evolution in the effective
low energy theory.
166
A similar parameterization exists in the literature and is due to Kennedy and Lynn,
[538], Altarelli and Barbieri [539] and Peskin and Takeuchi [235]. These authors consider
the coefficients of the full vacuum polarization tensors ΠXY . Peskin and Takeuchi define:
#
"
∂
∂
S = 16π
Π33 |q2 =0 − 2 Π3Q |q2 =0
2
∂q
∂q
i
h
4π
T =
Π
|
2 =0 − Π33 |q 2 =0
W
W
q
2
sin θ cos2 θMZ2
"
#
∂
∂
U = 16π
ΠW W |q2 =0 − 2 Π33 |q2 =0
∂q 2
∂q
(A.33)
(A.34)
(A.35)
2
Recalling our definitions of vW
and vZ2 :
and:
1 2
1 2
vZ = vweak
+ Π33 − ΠT3Q
2
2
(A.36)
1 2
1 2
vW = vweak
+ ΠW W − ΠT3Q
2
2
(A.37)
we see that:
σ=
2S + U
;
16π
ω=
U
;
16π
τ=
α
sin2 θ cos2 θMZ2 T
= T;
2
4πvr
2
(A.38)
It is straightforward to compute S, T and U. One finds for the S parameter:
S =
oi
n
Nc h
1 − Y log m2b /m2t
6π
(A.39)
T is given by the usual ρ–parameter expression, or the difference of the zero–momentum
2
expressions for vW
and vZ2 :
"
Nc
2m2t m2b
2
2
T =
log(m2t /m2b )
(m
+
m
)
−
t
b
(m2t − m2b )
4π sin2 θ cos2 θMZ2
#
2 2
2
M
m
MW
m2H
Z
H
2
ln(m2H /MW
)− 2
ln(m2H /MZ2 )
+ 2
2
mH − MW
mH − MZ2
(A.40)
Similarly, for U we can obtain:
"
Nc
5m4 − 22m2t m2b + 5m4b
U =
− t
6π
3(m2t − m2b )2
#
n
o
m6t − 3m4t m2b − 3m2t m4b + m6b
2
2
+
log mt /mb
(m2t − m2b )3
(A.41)
√
Combining the inputs from different measurements, such as GF = 1/2 2vweak (0)2 ,
2
2
2
2
2
MZ2 = 21 (g12 (MZ2 ) + g22(MZ2 ))vweak
(MZ2 ), MW
= 21 g22 (MW
)vweak
(MW
), sin2 θZ−pole =
167
g1 (MZ2 )/(g1 (MZ2 ) + g2 (MZ2 )), and mtop , one identifies (at some chosen confidence level) an
ellipse-shaped region of the S −T plane with which the data is most consistent. The S −T
coordinate system is chosen with arbitrary offsets so that some particular Standard Model
value of MH defines the origin. One can then overlay the trajectories corresponding to
varying the value of MH or the effects of other new physics. An example of an S − T
error ellipse plot with theoretical overlays is shown in Figure (69).
Appendix B: The Nambu-Jona-Lasinio Model
Much of our intuition about dynamical symmetry breaking comes from the BCS theory
of superconductivity [15,16]. This involves a fermion pairing dynamics, leading to a rearrangement of the vacuum (ground state) in the presence of a strong attractive interactions.
It was imported into elementary particle physics to provide a successful picture of mass
generation in strong interactions, also known as chiral symmetry breaking [1,3]. A toy
version of this, known as the Nambu–Jona-Lasinio (NJL) model, [34] provides a simple
physical picture of chiral symmetry breaking. We present a synopsis of it here.
Consider an effective four–fermion vertex with coefficient, G = g 2/Λ2 . The theory at
the “high energy physics” scale Λ is:
LΛ = ψ̄La i∂/ ψLa + ψ̄Ra i∂/ ψRa + G(ψ̄La ψRa )(ψ̄Rb ψLb )
(B.1)
where (a, b) are global SU(N) “color” indices. Note that the four–fermion form of the
interaction is contained in a Fierz rearrangement of a single–coloron (massive gluon)
exchange potential:
!
!
λA
1 2
g µν
λA
g ψ̄γµ ψ 2
(B.2)
ψ̄γν ψ
2
2
q − Λ2
2
thus G = g 2/Λ2 at q 2 = 0. Thus, in some sense we may view this an approximation
to what we believe is happening in QCD on scales M 2 ∼ Λ2QCD , ignoring the effects of
confinement.
Let us view eq. (B.1) as an effective Lagrangian at the scale Λ; the coupling constant
g is thus the renormalized effective coupling constant at that scale. Of course, in any
realistic strongly interacting theory this would seem to be a drastic approximation [390],
but it underlies the reasonably successful chiral constituent quark model (see [71,70] and
references therein). We consider the solution to the theory at lower energies based upon
the effects of the fermion loops, i.e., a fermion bubble approximation. This is equivalent
to a large–Ncolor expansion. We will present the renormalization group version of the
solution to the NJL model, which is quite physical and transparent. The widely-used
alternative older approach is discussed in refs.[383,540].
We can rewrite eq.(B.1), introducing an auxilliary field H, as:
L = Lkinetic + (gψL ψR H + h.c.) − Λ2 H † H
168
(B.3)
If we integrate out the field H we reproduce the four–fermion vertex as an induced interaction with G ≡ g 2 /Λ2 . Note that G > 0 implies an attractive interaction and permits
the factorization in this form. More specifically, eq.(B.3) is the effective Lagrangian on
a scale Λ. To obtain the effective Lagrangian on a scale µ < Λ in the fermion bubble
approximation we integrate out the fermion field components on scales µ ↔ Λ. The full
induced effective Lagrangian at the new scale µ will then take the form:
Lµ = Lkinetic + gψ L ψR H + h.c.
+ZH |∂ν H|2 − m2H H † H −
λ0 † 2
(H H) − ξ0 RH † H
2
(B.4)
Here R is the geometric scalar curvature, and we see there is an induced “nonminimal”
coupling of the Higgs field to gravity, ξ. A direct evaluation of the induced parameters
by computing the relevant Feynman loops gives:
g 2 Nc
log(Λ2 /µ2 );
(4π)2
2g 4Nc
log(Λ2 /µ2 );
=
(4π)2
ZH =
λ0
m2H = Λ2 −
ξ0 =
2g 2Nc 2
(Λ − µ2 )
(4π)2
1 g 2Nc
log(Λ2 /µ2 ).
6 (4π)2
(B.5)
(the parameter g is unrenormalized at this stage in fermion loop approximation). The
induced low energy parameters, ZH and λ0 , and ξ0 are determined in terms of Λ, and we
are interested in the µ ≈ 0 limit of the theory.
We emphasize that the effective theory applies in either the spontaneously broken or
unbroken phases. The broken phase is selected by demanding that m2H < 0 for scales
µ2 ≪ Λ2 , thus requiring that Λ2 (1 − g 2 Nc /8π 2 ) < 0; hence, g 2 > 8π 2 /Nc = gc2 defines
a critical coupling. On the other hand, for positive m2H as µ → 0 the theory remains
unbroken (this is equivalent to a subcritical four–fermion coupling constant, g 2 ≤ gc2 ).
Let us bring the effective
Lagrangian into a conventionally normalized form by rescal√
ing the field H → H/ ZH :
Where we find:
L = Lkinetic + g̃ψ L ψR H + h.c.
λ̃
f2H H † H − (H † H)2 − ξRH † H
+|∂ν H|2 − m
2
g̃
2
=
f2H =
m
λ̃ =
ξ =
16π 2
g /ZH =
Nc log(Λ2 /µ2 )
m2H /ZH
32π 2
2
λ0 /ZH
=
Nc log(Λ2 /µ2 )
ξ0 /ZH = 1/6
(B.6)
2
169
(B.7)
These are the physical renormalized coupling constants. We tune the low energy value of
f2H to the desired value, and the remaining predictions of the model are contained in g̃,
m
λ̃ (and ξ) as we will see below. The compositeness of the H boson is essentially contained
in the result that g and λ are singular as µ → Λ (while ξ is constant and equal to its
conformal value of 1/6). We will refer to these as the “compositeness conditions.”
These results are easily recovered directly from the conventional differential renormalization group equations, supplemented with “compositeness conditions” as high energy
boundary conditions. We utilize the approximate β-functions which reflect only the presence of fermion loops:
∂
g = Nc g 3
(B.8)
16π 2
∂ ln µ
∂
16π 2
λ = (−4Nc g 4 + 4Nc g 2λ)
(B.9)
∂ ln µ
Solving the first RG equation gives:
1
g 2 (µ)
−
1
g 2 (Λ)
=
Nc
ln(Λ2 /µ2 )
16π 2
(B.10)
If we now use the boundary condition, 1/g 2 (Λ) = 0 we see that we recover the above result
g̃ 2 = gc2(µ). The second RG equation may then be solved by hypothesizing an anzatz of
the form λ = cg 2 . Substituting one finds:
16π 2
∂
1
g = (4c − 4)Nc g 3
∂ ln µ
2c
(B.11)
and demand that this must be consistent with the other RG equation. Thus one finds:
c = 2 and:
1
1
Nc
−
=
ln(Λ2 /µ2 )
(B.12)
λ(µ) λ(Λ)
32π 2
and again λ(Λ)−1 = 0 leads to the above result of λ̃ = λ(µ).
To obtain the phenomenological results of the NJL model we examine the low energy
Higgs potential from the action with µ = m:
λ̃
f2H H † H + (H † H)2 − (g̃ψ L ψR H + h.c.)
V (H) = −m
2
(B.13)
f2H < 0 so the neutral Higgs field develops a VEV: Re(H 0 ) =
Let us assume
that m
√
(v + h)/ 2. Therefore we find the dynamical fermion mass:
√
m = g̃v/ 2;
(B.14)
and the h mass is:
m2h = v 2 λ̃
and so:
m2h /m2 = 2λ̃/g˜2 = 4
170
or:
(B.15)
mh = 2m
(B.16)
using eqs.(B.7). This is the familar NJL result, mh = 2m. We note that this result is
subject to renormalizion when effects of other interactions are included [383,540]. It also
does not imply that the Higgs is loosely bound (see below)! We also obtain presently the
Pagels-Stokar relation:
1 2
Nc
v = m2 /g̃ 2 = m2
ln(Λ2 /m2 )
2
16π 2
(B.17)
It is also amusing to study the result ξ = 1/6 in the differential renormalization group
[540]. One finds that the solution for the scalar coupling to curvature,
ξ(µ) = 1/6,
(B.18)
is a RG constant for all scales. More generally, as one descends toward the infrared,
ξ = 1/6 is an attractive infra-red fixed point. Therefore, no matter what is the initial
value for ξ at the large scale Λ, given enough RG running time ξ will eventually reach
1/6 for small µ. Of course, the RG running only occurs for scales µ > mH . We thus
find in the usual fermion bubble approximation that H is conformally coupled to gravity,
even though scale breaking dynamics exists at high energies Λ. Moreover, ξ = 1/6 is an
attractive renormalization group fixed point in the infrared in this approximation [540].
This physical particle H is a bound state of ψ̄ψ, arising by the attractive four–fermion
interaction at the scale Λ. One might think that this is a loosely bound state, since it lies
on top of the threshold for open ψ̄ψ and apparently has vanishing binding energy to this
order. However, this is not a nonrelativistic bound state, and normal intuition does not
apply. The state is built from fully relativistic Feynman loops with momenta that extend
from m to Λ. The prediction mh = 2m cannot be viewed as an exact one and is subject
to corrections from subleading N effects and other interactions.
The RG approach can also be used to go beyond the leading order in large Nc . Essentially, we keep the compositeness boundary conditions but include the full dynamics
into the RG equations [383]. In this case, g̃ and λ̃ develop nontrivial infra-red fixed
points [379,377,380,378]. The NJL model is also readily generalized with the incorporation of flavor to be a chiral constituent quark model (see [71,70] and references therein),
or techniquark model. As discussed in Section 4, it is also the basis of Topcolor models.
The NJL model is readily adapted to QCD, or more general strongly interacting theories with nontrivial cboundstate or chiral dynamics. In that case the field H becomes
a more general object, such as a linear σ-model field (see, e.g., [70] or the Appendix of
[102], which closely follows the present discussion).
171
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