arXiv:1104.1255v1 [hep-ph] 7 Apr 2011
Discovering Technicolor
First Black Report
2011
Composite Higgs physics @ LHC
Discovering Technicolor
J.R. Andersenr O. Antipinr G. AzuelosN L. Del Debbio♣ E. Del Nobiler
S. Di Chiarar T. Hapolar M. Järvinen P.J. Lowdon♣ Y. Maravin♠
I. MasinaHr M. Nardecchiar C. Picar F. Sanninor
CP3 -Origins, University of Southern Denmark, Odense, Denmark
N
Universite de Montréal, Montréal, Canada and TRIUMF, Vancouver, Canada
♣
Tait Institute, University of Edinburgh, Edinburgh, Scotland, UK
Crete Center for Theoretical Physics, University of Crete, Heraklion, Greece
♠
Kansas State University, Manhattan, KS, USA
H
Università degli Studi di Ferrara and INFN Sez. di Ferrara, Italy
r
Abstract
We provide a pedagogical introduction to extensions of the Standard Model in
which the Higgs is composite. These extensions are known as models of dynamical
electroweak symmetry breaking or, in brief, Technicolor. Material covered includes:
motivations for Technicolor, the construction of underlying gauge theories leading
to minimal models of Technicolor, the comparison with electroweak precision data,
the low energy effective theory, the spectrum of the states common to most of the
Technicolor models, the decays of the composite particles and the experimental
signals at the Large Hadron Collider. The level of the presentation is aimed at
readers familiar with the Standard Model but who have little or no prior exposure
to Technicolor. Several extensions of the Standard Model featuring a composite
Higgs can be reduced to the effective Lagrangian introduced in the text.
We establish the relevant experimental benchmarks for Vanilla, Running, Walking, and Custodial Technicolor, and a natural fourth family of leptons, by laying
out the framework to discover these models at the Large Hadron Collider.
CP3 -Origins-2011-13
CCTP-2011-11
Contents
1
The need to go beyond
1.1 The Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Riddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Dynamical Electroweak Symmetry Breaking
2.1 Superconductivity versus electroweak symmetry breaking
2.2 From color to Technicolor . . . . . . . . . . . . . . . . . . .
2.3 Constraints from electroweak precision data . . . . . . . .
2.4 Standard Model fermion masses . . . . . . . . . . . . . . .
2.5 Walking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Ideal walking . . . . . . . . . . . . . . . . . . . . . . . . . .
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Phenomenology of Minimal Technicolor
3.1 Low energy theory for MWT . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Scalar sector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Vector bosons . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Fermions and Yukawa interactions . . . . . . . . . . . . . . .
3.1.4 Weinberg Sum Rules . . . . . . . . . . . . . . . . . . . . . . .
3.1.5 Passing the electroweak precision tests . . . . . . . . . . . .
3.1.6 The Next to Minimal Walking Technicolor Theory (NMWT)
3.2 Beyond MWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Partially Gauged Technicolor . . . . . . . . . . . . . . . . . .
3.2.2 Split Technicolor . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Vanilla Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 WW - Scattering in Technicolor and Unitarity . . . . . . . . . . . . .
3.4.1 Spin zero + spin one . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Spin zero + spin one + spin two . . . . . . . . . . . . . . . .
Phenomenological benchmarks
4.1 Spin one processes: Decay widths and branching ratios
4.1.1 pp → R → `` . . . . . . . . . . . . . . . . . . . . .
4.1.2 pp → R → WZ → ```ν . . . . . . . . . . . . . . .
4.1.3 pp → R → `ν . . . . . . . . . . . . . . . . . . . .
4.1.4 pp → R → jj . . . . . . . . . . . . . . . . . . . . .
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4.2
4.3
4.1.5 pp → R → γV and pp → R → ZZ . . . .
Ultimate LHC reach for heavy spin resonances
4.2.1 pp → R → 2`, `ν, 3`ν at 14 TeV . . . . . .
4.2.2 pp → Rjj at 14 TeV . . . . . . . . . . . .
Composite Higgs phenomenology . . . . . . .
4.3.1 pp → WH and pp → ZH . . . . . . . . .
4.3.2 Higgs vector boson fusion, pp → H jj . .
4.3.3 H → γγ and H → gg . . . . . . . . . . .
4.3.4 Higgs production via gluon fusion . . .
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5
A Natural fourth family of leptons at the TeV scale
5.1 The Standard Model leptons: a Mini-review . . . . . . . . . . . . . . . . .
5.2 Adding a fourth lepton family . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Heavy leptons not Mixing with Standard Model neutrinos . . . .
5.2.2 Promiscuous heavy leptons . . . . . . . . . . . . . . . . . . . . . .
5.3 LHC phenomenology for the natural heavy lepton family . . . . . . . . .
5.3.1 Production and decay of the new leptons . . . . . . . . . . . . . .
5.3.2 Collider signatures of heavy leptons with an exact flavor symmetry
5.4 Collider signatures of promiscuous heavy leptons . . . . . . . . . . . . .
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Conclusion
82
A Realization of the generators for MWT and the Standard Model embedding 84
B Technicolor on Event Generators
B.1 Ruling Technicolor with FeynRules . . . . . . . . . . . . . . . . . . . . . .
B.2 Madding Technicolor via MadGraph/MadEvent v.4 . . . . . . . . . . . .
B.3 Calculating Technicolor with CalcHEP . . . . . . . . . . . . . . . . . . . .
2
85
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1
The need to go beyond
The energy scale at which the Large Hadron Collider experiment (LHC) operates is
determined by the need to complete the Standard Model (SM) of particle interactions
and, in particular, to understand the origin of mass of the elementary particles. Together
with classical general relativity the SM constitutes one of the most successful models of
nature. We shall, however, argue that experimental results and theoretical arguments
call for a more fundamental description of nature.
In Fig. 1, we schematically represent, in green, the known forces of nature. The SM
of particle physics describes the strong, weak and electromagnetic forces. The yellow
region represents the energy scale around the TeV scale and is being explored at the
LHC, while the red part of the diagram is speculative.
Standard Model
Fermi Scale
Figure 1: Cartoon representing the various forces of nature. At very high energies one
may imagine that all the low-energy forces unify in a single force.
All of the known elementary particles constituting the SM fit on the postage stamp
shown in Fig. 2. Interactions among quarks and leptons are carried by gauge bosons.
Massless gluons mediate the strong force among quarks while the massive gauge
bosons, i.e. the Z and W, mediate the weak force and interact with both quarks and
leptons. Finally, the massless photon, the quantum of light, interacts with all of the
electrically charged particles. The SM Higgs is introduced to provide mass to the
elementary particles and in its minimal version does not feel strong interactions. The
interactions emerge naturally by invoking a gauge principle which is intimately linked
3
to the underlying symmetries relating the various particles of the SM. The asterisk
Low Energy Effective Theory
U(1)
SU(2)
SM
SU(3)
Figure 2: Postage stamp representing all of the elementary particles which constitute the SM. The
forces are mandated with the SU(3)×
SU(2) × U(1) gauge group.
Figure 3: The SM can be viewed as a lowenergy theory valid up to a high energy scale
Λ.
on the Higgs boson in the postage stamp indicates that it has not yet been observed.
Intriguingly the Higgs is the only fundamental scalar of the SM.
The SM can be viewed as a low-energy effective theory valid up to an energy scale Λ,
as schematically represented in Fig. 3. Above this scale new interactions, symmetries,
extra dimensional worlds or any other extension could emerge. At sufficiently low
energies with respect to this scale one expresses the existence of new physics via
effective operators. The success of the SM is due to the fact that most of the corrections
to its physical observables depend only logarithmically on this scale Λ. In fact, in the
SM there exists only one operator which acquires corrections quadratic in Λ. This is
the squared mass operator of the Higgs boson. Since Λ is expected to be the highest
possible scale, in four dimensions the Planck scale (assuming that we have only the SM
and gravity), it is hard to explain naturally why the mass of the Higgs is of the order
of the Electroweak (EW) scale. This is the hierarchy problem. Due to the occurrence
of quadratic corrections in the cutoff this SM sector is most sensitive to the existence of
new physics.
1.1
The Higgs
It is a fact that the Higgs allows for a direct and economical way of spontaneously
breaking the electroweak symmetry. It generates simultaneously the masses of the
4
quarks and leptons without introducing Flavor Changing Neutral Currents (FCNC)s
at the tree level. The Higgs sector of the SM possesses, when the gauge couplings are
switched off, an SU(2)L × SU(2)R symmetry. The full symmetry group can be made
explicit when re-writing the Higgs doublet field
!
1
π2 + i π1
(1.1)
H= √
2 σ − i π3
as the right column of the following two by two matrix:
1
~ ≡M.
√ σ + i ~τ · π
2
(1.2)
The first column can be identified with the column vector iτ2 H∗ while the second with
H. τ2 is the second Pauli matrix. The SU(2)L ×SU(2)R group acts linearly on M according
to:
M → gL Mg†R
gL/R ∈ SU(2)L/R .
(1.3)
1 + τ3
M
= (i τ2 H∗ , 0) .
2
(1.4)
and
One can verify that:
1 − τ3
M
= (0 , H) .
2
The SU(2)L symmetry is gauged by introducing the weak gauge bosons W a with a =
1, 2, 3. The hypercharge generator is taken to be the third generator of SU(2)R . The
ordinary covariant derivative acting on the Higgs, in the present notation, is:
Dµ M = ∂µ M − i g Wµ M + i g0 M Bµ ,
with
Wµ = Wµa
τa
,
2
Bµ = Bµ
τ3
.
2
(1.5)
The Higgs Lagrangian is
i m2M h
i λ h
i2
1 h
† µ
L =
Tr Dµ M D M −
Tr M† M − Tr M† M .
2
2
4
(1.6)
At this point one assumes that the mass squared of the Higgs field is negative and this
leads to the electroweak symmetry breaking. Except for the Higgs mass term the other
SM operators have dimensionless couplings meaning that the natural scale for the SM
is encoded in the Higgs mass1 . We recall that the Higgs Lagrangian has a familiar
form since it is identical to the linear σ Lagrangian which was introduced long ago to
describe chiral symmetry breaking in QCD with two light flavors.
1
The mass of the proton is due mainly to strong interactions, however its value cannot be determined
within QCD since the associated renormalization group invariant scale must be fixed to an hadronic
observable.
5
At the tree level, when taking m2M negative and the self-coupling λ positive, one
determines:
|m2 |
hσi2 ≡ v2 = M
and
σ=v+h,
(1.7)
λ
where h is the Higgs field. The global symmetry breaks to its diagonal subgroup:
SU(2)L × SU(2)R → SU(2)V .
(1.8)
To be more precise the SU(2)R symmetry is already broken explicitly by our choice of
gauging only an U(1)Y subgroup of it and hence the actual symmetry breaking pattern
is:
SU(2)L × U(1)Y → U(1)Q ,
(1.9)
with U(1)Q the electromagnetic abelian gauge symmetry. According to the NambuGoldstone’s theorem three massless degrees of freedom appear, i.e. π± and π0 . In
the unitary gauge these Goldstones become the longitudinal degree of freedom of the
massive elecetroweak gauge bosons. Substituting the vacuum value for σ in the Higgs
Lagrangian the gauge bosons quadratic terms read:
2
v2 2 1 µ,1
2
µ,2
3
0
g Wµ W + Wµ W
+ g Wµ − g Bµ
.
8
(1.10)
The Zµ and the photon Aµ gauge bosons are:
Zµ = cos θW Wµ3 − sin θW Bµ ,
Aµ = cos θW Bµ + sin θW Wµ3 ,
(1.11)
√
with tan θW = g0 /g while the charged massive vector bosons are Wµ± = (W 1 ± i Wµ2 )/ 2.
The bosons masses M2W = g2 v2 /4 due to the custodial symmetry satisfy the tree level
relation M2Z = M2W / cos2 θW . Holding fixed the EW scale v the mass squared of the
Higgs boson is 2λv2EW and hence it increases with λ.
Besides breaking the electroweak symmetry dynamically the ordinary Higgs serves
also the purpose to provide mass to all of the SM particles via the Yukawa terms of the
type:
ij
j
ij
j
− Yd Q̄iL HdR − Yu Q̄iL (iτ2 H∗ )uR + h.c. ,
(1.12)
where Yq is the Yukawa coupling constant, QL is the left-handed Dirac spinor of quarks,
H the Higgs doublet and q the right-handed Weyl spinor for the quark and i, j the flavor
indices. The SU(2)L weak and spinor indices are suppressed.
When considering quantum corrections the Higgs mass acquires large quantum
corrections proportional to the scale of the cutoff squared.
MH 2ren
−
M2H
6
kg2 Λ2
.
=
16π2
(1.13)
Here g is and electroweak constant and k a numerical factor depending on the specific
model, expected to be O(1). Λ is the highest energy above which the SM is no longer a
valid description of Nature and a large fine tuning of the parameters of the Lagrangian
is needed to offset the effects of the cutoff. This large fine tuning is needed because there
are no symmetries protecting the Higgs mass operator from large corrections which
would hence destabilize the Fermi scale (i.e. the electroweak scale). This problem is
the one we referred above as the hierarchy problem of the SM.
The constant value of the Higgs field evaluated on the ground state is determined by
the measured mass of the W boson. On the other hand, the value of the SM Higgs mass
(MH ) is constrained only indirectly by the electroweak precision data. The preferred
value of the Higgs mass (obtained by the standard fit which excludes direct Higgs
searches at LEP and Tevatron) is MH = 95.7+30.6
GeV at 68% confidence level (CL) with
−24.2
a 95% CL upper limit MH < 171.5 GeV, as given by the generic fitting package Gfitter
[1]. The corresponding results obtained by a fit including the direct Higgs searches
produces MH = 120.6+17.9
GeV at 68% confidence level (CL) with a 95% CL upper limit
−5.2
MH < 155.3 GeV, as reported on http://gfitter.desy.de/GSM/ by the Gfitter Group 2 .
The final result of the average of all of the measures, however, has a Pearson’s chisquare (χ2 ) test of 17.5 for 14 degrees of freedom. A Higgs heavier than 155.3 GeV is
compatible with precision tests if we allow simultaneously new physics to compensate
for the effects of the heavier value of the mass. The precision measurements of direct
interest for the Higgs sector are often reported using the S and T parameters as shown
in Fig. 5. From this graph one deduces that a heavy Higgs is compatible with data
at the expense of a large value of the T parameter. Actually, even the lower direct
experimental limit on the Higgs mass can be evaded with suitable extensions of the SM
Higgs sector.
Many more questions need an answer if the Higgs is found at the LHC: Is it composite? How many Higgs fields are there in nature? Are there hidden sectors?
1.2
Riddles
Why do we expect that there is new physics awaiting to be discovered? Of course,
we still have to observe the Higgs, but this cannot be everything. Even with the Higgs
discovered, the SM has both conceptual problems and phenomenological shortcomings.
In fact, theoretical arguments indicate that the SM is not the ultimate description of
nature:
• Hierarchy Problem: The Higgs sector is highly fine-tuned. We have no natural
explanation of the large hierarchy between the Planck and the electroweak scales.
2
All the plots and numerical results we use in this section are reported by the Gfitter Group and can
be found at the web-address: http://gfitter.desy.de/GSM/.
7
SM
Nov 10
G fitter
MZ
49
61 +- 26
32
97 +- 25
m0had
31
97 +- 25
R 0lep
30
100 +- 25
AFB
0,l
32
100 +- 26
Al(LEP)
31
92 +- 24
A l(SLD)
40
122 +- 32
0.4
sin2 Olept
(Q )
eff
30
94 +- 24
0.3
30
94 +- 24
0.2
0.5
G fitter
B
SM
Aug 10
FB
0,c
A FB
0,b
A FB
T
KZ
preliminary
+ 33
68 - 25
Ac
31
96 +- 24
Ab
31
96 +- 24
R 0c
31
96 +- 24
R 0b
31
96 +- 24
(5)
+ 62
6_had(M2)
43 - 25
MW
75
116 +- 34
KW
+ 31
96 - 24
mc
31
96 +- 24
+ 32
99 - 25
71
142 +- 75
Z
mb
mt
20
40
60
80 100 120 140 160
MH [GeV]
Figure 4: Values of the Higgs mass
from the standard fit (which does
not take into account direct Higgs
searches) obtained by excluding different electroweak observables. The
green band represent the 1σ error
range around the best fit value of MH .
0.1
0
MH D [114,1000] GeV
mt = 173.3± 1.1 GeV
-0.1
MH
-0.2
-0.3
68%, 95%, 99% CL fit contours
(M =120 GeV, U=0)
H
-0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
S
Figure 5: The 68%, 95%, and 99% CL countours of the electroweak parameters S and
T determined from different observables derived from a fit to the electroweak precision
data. The gray area gives the SM prediction
with mt and MH varied as shown. MH = 120
GeV and mt = 173.1 GeV defines the reference
point at which all oblique parameters vanish.
• Strong CP problem: There is no natural explanation for the smallness of the
electric dipole moment of the neutron within the SM. This problem is also known
as the strong CP problem.
• Origin of patterns: The SM can fit, but cannot explain the number of matter
generations and their mass spectrum.
• Unification of the forces: Why do we have so many different interactions? It is
appealing to imagine that the SM forces could unify into a single Grand Unified
Theory (GUT). We could imagine that at very high energy scales gravity also
becomes part of a unified description of nature.
8
There is no doubt that the SM is incomplete since we cannot even account for a number
of basic observations:
• Neutrino physics: Only recently it has been possible to have some definite answers about properties of neutrinos. We now know that they have a tiny mass,
which can be naturally accommodated in extensions of the SM, featuring for example a see-saw mechanism. We do not yet know if the neutrinos have a Dirac
or a Majorana nature.
• Origin of bright and dark mass: Leptons, quarks and the gauge bosons mediating the weak interactions possess a rest mass. Within the SM this mass can be
accounted for by the Higgs mechanism, which constitutes the electroweak symmetry breaking sector of the SM. However, the associated Higgs particle has not
yet been discovered. Besides, the SM cannot account for the observed large fraction of dark mass of the universe. What is interesting is that in the universe the
dark matter is about five times more abundant than the known baryonic matter,
i.e. bright matter. We do not know why the ratio of dark to bright matter is of
order unity.
• Matter-antimatter asymmetry: From our everyday experience we know that
there is very little bright antimatter in the universe. The SM fails to predict the
observed excess of matter.
These arguments do not imply that the SM is necessarily incorrect, but it must be
extended to answer any of the questions raised above. The truth is that we do not have
an answer to the basic question: What lies beneath the SM?
A number of possible generalizations have been conceived (see [2, 3, 4, 5, 6, 7] for
reviews). Such extensions are introduced on the base of one or more guiding principles
or prejudices. Two technical reviews are [8, 9].
In the models we will consider here the electroweak symmetry breaks via a fermion
bilinear condensate. The Higgs sector of the SM becomes an effective description of a
more fundamental fermionic theory. This is similar to the Ginzburg-Landau theory of
superconductivity. If the force underlying the fermion condensate driving electroweak
symmetry breaking is due to a strongly interacting gauge theory these models are
termed Technicolor (TC).
TC, in brief, is an additional non-abelian and strongly interacting gauge theory
augmented with (techni)fermions transforming under a given representation of the
gauge group. The Higgs Lagrangian is replaced by a suitable new fermion sector
interacting strongly via a new gauge interaction (technicolor). Schematically:
1
LHiggs → − Fµν Fµν + iQ̄γµ Dµ Q + . . . ,
4
(1.14)
where, to be as general as possible, we have left unspecified the underlying nonabelian
gauge group and the associated technifermion (Q) representation. The dots represent
9
new sectors which may even be needed to avoid, for example, anomalies introduced by
the technifermions. The intrinsic scale of the new theory is expected to be less or of the
order of a few TeV. The chiral-flavor symmetries of this theory, as for ordinary QCD,
break spontaneously when the technifermion condensate Q̄Q forms. It is possible to
choose the fermion charges in such a way that there is, at least, a weak left-handed
doublet of technifermions and the associated right-handed one which is a weak singlet.
The covariant derivative contains the new gauge field as well as the electroweak ones.
The condensate spontaneously breaks the electroweak symmetry down to the electromagnetic and weak interactions. The Higgs is now interpreted as the lightest scalar
field with the same quantum numbers of the fermion-antifermion composite field. The
Lagrangian part responsible for the mass-generation of the ordinary fermions will also
be modified since the Higgs particle is no longer an elementary object.
Models of electroweak symmetry breaking via new strongly interacting theories of
TC type [10, 11] are a mature subject. The two uptodate reviews on which this work is
based are [12, 13]. For older nice reviews, updated till 2002, see [14, 15].
One of the main difficulties in constructing such extensions of the SM is the very
limited knowledge about generic strongly interacting theories. This has led theorists to
consider specific models of TC which resemble ordinary QCD and for which the large
body of experimental data at low energies can be directly exported to make predictions
at high energies. To reduce the tension with experimental constraints new strongly
coupled theories with dynamics different from the one featured by a scaled up version
of QCD are needed [16].
We will review models of dynamical electroweak symmetry breaking making use of
new type of four dimensional gauge theories [16, 17, 18] and their low energy effective
description [19] useful for collider phenomenology. The phase structure of a large
number of strongly interacting nonsupersymmetric theories, as function of number of
underlying colors has been uncovered via traditional nonperturbative methods [20] as
well as novel ones [21, 22].
The theoretical part of this report should be integrated with earlier reviews [13, 14,
15, 23, 24, 25, 26, 27, 28] on the various subjects treated here.
10
2
Dynamical Electroweak Symmetry Breaking
It is a fact that the SM does not fail, when experimentally tested, to describe all of
the known forces to a very high degree of experimental accuracy. This is true even if
we include gravity. Why is it so successful?
The SM is a low energy effective theory valid up to a scale Λ above which new interactions, symmetries, extra dimensional worlds or any possible extension can emerge.
At sufficiently low energies with respect to the cutoff scale Λ one expresses the existence
of new physics via effective operators. The success of the SM is due to the fact that
most of the corrections to its physical observable depend only logarithmically on the
cutoff scale Λ.
Superrenormalizable operators are very sensitive to the cut off scale. In the SM there
exists only one operator with naive mass dimension two which acquires corrections
quadratic in Λ. This is the squared mass operator of the Higgs boson. Since Λ is
expected to be the highest possible scale, in four dimensions the Planck scale, it is hard
to explain naturally why the mass of the Higgs is of the order of the electroweak scale.
The Higgs is also the only particle predicted in the SM yet to be directly produced in
experiments. Due to the occurrence of quadratic corrections in the cutoff this is the SM
sector highly sensitve to the existence of new physics.
In Nature we have already observed Higgs-type mechanisms. Ordinary superconductivity and chiral symmetry breaking in QCD are two time-honored examples. In
both cases the mechanism has an underlying dynamical origin with the Higgs-like
particle being a composite object of fermionic fields.
2.1
Superconductivity versus electroweak symmetry breaking
The breaking of the electroweak theory is a relativistic screening effect. It is useful to
parallel it to ordinary superconductivity which is also a screening phenomenon albeit
non-relativistic. The two phenomena happen at a temperature lower than a critical one.
In the case of superconductivity one defines a density of superconductive electrons ns
and to it one associates a macroscopic wave function ψ such that its modulus squared
|ψ|2 = nC =
ns
,
2
(2.15)
is the density of Cooper’s pairs. That we are describing a nonrelativistic system is
manifest in the fact that the macroscopic wave function squared, in natural units, has
mass dimension three while the modulus squared of the Higgs wave function evaluated
at the minimum is equal to h|H|2 i = v2 /2 and has mass dimension two, i.e. is a relativistic
wave function. One can adjust the units by considering, instead of the wave functions,
the Meissner-Mass of the photon in the superconductor which is
M2 = q2 ns /(4me ) ,
11
(2.16)
with q = −2e and 2me the charge and the mass of a Cooper pair which is constituted by
two electrons. In the electroweak theory the Meissner-Mass of the photon is compared
with the relativistic mass of the W gauge boson
M2W = g2 v2 /4 ,
(2.17)
with g the weak coupling constant and v the electroweak scale. In a superconductor
the relevant scale is given by the density of superconductive electrons typically of the
order of ns ∼ 4 × 1028 m−3 yielding a screening length of the order of ξ = 1/M ∼ 10−6 cm.
In the weak interaction case we measure directly the mass of the weak gauge boson
which is of the order of 80 GeV yielding a weak screening length ξW = 1/MW ∼ 10−15 cm.
For a superconductive system it is clear from the outset that the wave function ψ is
not a fundamental degree of freedom, however for the Higgs we are not yet sure about
its origin. The Ginzburg-Landau effective theory in terms of ψ and the photon degree
of freedom describes the spontaneous breaking of the U(1)Q electric symmetry and it is
the equivalent of the Higgs Lagrangian.
If the Higgs is due to a macroscopic relativistic screening phenomenon we expect it
to be an effective description of a more fundamental system with possibly an underlying
new strong gauge dynamics replacing the role of the phonons in the superconductive
case. A dynamically generated Higgs system solves the problem of the quadratic
divergences by replacing the cutoff Λ with the weak energy scale itself, i.e. the scale of
compositness. An underlying strongly coupled asymptotically free gauge theory, a la
QCD, is an example.
2.2
From color to Technicolor
In fact even in complete absence of the Higgs sector in the SM the electroweak
symmetry breaks [26] due to the condensation of the following quark bilinear in QCD:
hūL uR + d¯L dR i , 0 .
(2.18)
This mechanism, however, cannot account for the whole contribution to the weak gauge
bosons masses. If QCD was the only source contributing to the spontaneous breaking
of the electroweak symmetry one would have
MW =
gFπ
∼ 29 MeV ,
2
(2.19)
with Fπ ' 93 MeV the pion decay constant. This contribution is very small with respect
to the actual value of the W mass that one typically neglects it.
According to the original idea of TC [10, 11] one augments the SM with another
gauge interaction similar to QCD but with a new dynamical scale of the order of the
electroweak one. It is sufficient that the new gauge theory is asymptotically free and has
global symmetry able to contain the SM SU(2)L × U(1)Y symmetries. It is also required
12
that the new global symmetries break dynamically in such a way that the embedded
SU(2)L × U(1)Y breaks to the electromagnetic abelian charge U(1)Q . The dynamically
generated scale will then be fit to the electroweak one.
Note that, except in certain cases, dynamical behaviors are typically nonuniversal
which means that different gauge groups and/or matter representations will, in general,
possesses very different dynamics.
The simplest example of TC theory is the scaled up version of QCD, i.e. an SU(NTC )
nonabelian gauge theory with two Dirac Fermions transforming according to the fundamental representation or the gauge group. We need at least two Dirac flavors to
realize the SU(2)L × SU(2)R symmetry of the SM discussed in the SM Higgs section.
One simply chooses the scale of the theory to be such that the new pion decaying
constant is:
FTC
(2.20)
π = v ' 246 GeV .
The flavor symmetries, for any NTC larger than 2 are SU(2)L × SU(2)R × U(1)V which
spontaneously break to SU(2)V × U(1)V . It is natural to embed the electroweak symmetries within the present TC model in a way that the hypercharge corresponds to
the third generator of SU(2)R . This simple dynamical model correctly accounts for the
electroweak symmetry breaking. The new technibaryon number U(1)V can break due
to not yet specified new interactions. In order to get some indication on the dynamics
and spectrum of this theory one can use the ’t Hooft large N limit [29, 30, 31]. For
example the intrinsic scale of the theory is related to the QCD one via:
r
3 FTC
π
ΛTC ∼
ΛQCD .
(2.21)
NTC Fπ
At this point it is straightforward to use the QCD phenomenology for describing the
experimental signatures and dynamics of a composite Higgs.
2.3
Constraints from electroweak precision data
The relevant corrections due to the presence of new physics trying to modify the
electroweak breaking sector of the SM appear in the vacuum polarizations of the
electroweak gauge bosons. These can be parameterized in terms of the three quantities
S, T, and U (the oblique parameters) [32, 33, 34, 35], and confronted with the electroweak
precision data. Recently, due to the increase precision of the measurements reported
by LEP II, the list of interesting parameters to compute has been extended [36, 37]. We
show below also the relation with the traditional one [32]. Defining with Q2 ≡ −q2
the Euclidean transferred momentum entering in a generic two point function vacuum
polarization associated to the electroweak gauge bosons, and denoting derivatives with
13
respect to −Q2 with a prime we have [37]:
Ŝ ≡ g2 Π0W3 B (0) ,
T̂ ≡
W ≡
Y ≡
Û ≡
V ≡
X ≡
g
2
M2W
(2.22)
[ΠW3 W3 (0) − ΠW+ W− (0)] ,
i
g2 M2W h 00
ΠW3 W3 (0) ,
2
i
g02 M2W h 00
ΠBB (0) ,
2h
i
2
−g Π0W3 W3 (0) − Π0W+ W− (0) ,
i
g2 M2W h 00
ΠW3 W3 (0) − Π00W+ W− (0) ,
2
0
gg M2W 00
ΠW3 B (0) .
2
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
Here ΠV (Q2 ) with V = {W 3 B, W 3 W 3 , W + W − , BB} represents the self-energy of the vector
bosons. The electroweak couplings are the ones associated to the physical electroweak
gauge bosons:
1
≡ Π0W+ W− (0) ,
g2
1
≡ Π0BB (0) ,
g02
(2.29)
while GF is
1
= −4ΠW+ W− (0) ,
√
2GF
(2.30)
as in [38]. Ŝ and T̂ lend their name from the well known Peskin-Takeuchi parameters
S and T which are related to the new ones via [37, 38]:
αS
= Ŝ − Y − W ,
4s2W
αT = T̂ −
s2W
1 − s2W
Y.
(2.31)
Here α is the electromagnetic structure constant and sW = sin θW is the weak mixing
angle. Therefore in the case where W = Y = 0 we have the simple relation
Ŝ =
αS
,
4s2W
T̂ = αT .
(2.32)
The result of the the fit is shown in Fig. 5. If the value of the Higgs mass increases the
central value of the S parameter moves to the left towards negative values.
In TC it is easy to have a vanishing T parameter while typically S is positive. Besides,
the composite Higgs is typically heavy with respect to the Fermi scale, at least for
14
0.5
0.4
T
0.3
0.2
0.1
0.0
-0.1
-0.2
0.0
0.2
0.4
S
Figure 6: T versus S for SU(3) TC with one technifermion doublet (the full disc) versus
precision data for a one TeV composite Higgs mass.
technifermions in the fundamental representation of the gauge group and for a small
number of techniflavors. The oldest TC models featuring QCD dynamics with three
technicolors and a doublet of electroweak gauged techniflavors deviate a few sigma
from the current precision tests as summarized in Fig. 6. Clearly it is desirable to
reduce the tension between the precision data and a possible dynamical mechanism
underlying the electroweak symmetry breaking. It is possible to imagine different ways
to achieve this goal and some of the earlier attempts have been summarized in [39].
The computation of the S parameter in TC theories requires the knowledge of
nonperturbative dynamics making difficult the precise knowledge of the contribution
to S. For example, it is not clear what is the exact value of the composite Higgs mass
relative to the Fermi scale and, to be on the safe side, one typically takes it to be quite
large, of the order at least of the TeV. However in certain models it may be substantially
lighter due to the intrinsic dynamics. We will discuss the electroweak parameters later
in this chapter.
It is, however, instructive to provide a simple estimate of the contribution to S
which allows to guide model builders. Consider a one-loop exchange of ND doublets
of techniquarks transforming according to the representation RTC of the underlying TC
gauge theory and with dynamically generated mass Σ(0) assumed to be larger than the
weak intermediate gauge bosons masses. Indicating with d(RTC ) the dimension of the
techniquark representation, and to leading order in MW /Σ(0) one finds:
Snaive = ND
d(RTC )
.
6π
(2.33)
This naive value provides, in general, only a rough estimate of the exact value of S.
However, it is clear from the formula above that, the more TC matter is gauged under
15
the electroweak theory the larger is the S parameter and that the final S parameter is
expected to be positive.
Attention must be paid to the fact that the specific model-estimate of the whole S
parameter, to compare with the experimental value, receives contributions also from
other sectors. Such a contribution can be taken sufficiently large and negative to
compensate for the positive value from the composite Higgs dynamics. To be concrete:
Consider an extension of the SM in which the Higgs is composite but we also have new
heavy (with a mass of the order of the electroweak) fourth family of Dirac leptons. In
this case a sufficiently large splitting of the new lepton masses can strongly reduce and
even offset the positive value of S. We will discuss this case in detail when presenting
the Minimal Walking Technicolor (MWT) model. The contribution of the new sector
(SNS ) above, and also in many other cases, is perturbatively under control and the total
S can be written as:
S = STC + SNS .
(2.34)
The parameter T will be, in general, modified and one has to make sure that the corrections do not spoil the agreement with this parameter. From the discussion above it
is clear that TC models can be constrained, via precision measurements, only model
by model and the effects of possible new sectors must be properly included. We presented the constraints coming from S using the underlying gauge theory information.
However, in practice, these constraints apply directly to the physical spectrum. To be
concrete we will present in Section 3.1.5 a model of walking TC passing the precision
tests.
2.4
Standard Model fermion masses
Since in a purely TC model the Higgs is a composite particle the Yukawa terms, when
written in terms of the underlying TC fields, amount to four-fermion operators. The
latter can be naturally interpreted as a low energy operator induced by a new strongly
coupled gauge interaction emerging at energies higher than the electroweak theory.
These type of theories have been termed Extended Technicolor (ETC) interactions
[40, 41].
In the literature various extensions have been considered and we will mention
them later in the text. Here we will describe the simplest ETC model in which the
ETC interactions connect the chiral symmetries of the techniquarks to those of the SM
fermions (see left panel of Fig. 7).
When TC chiral symmetry breaking occurs it leads to the diagram in the right panel
of Fig. 7. Let’s start with the case in which the ETC dynamics is represented by a
SU(NETC ) gauge group with:
NETC = NTC + N g ,
(2.35)
and N g is the number of SM generations. In order to give masses to all of the SM
fermions, in this scheme, one needs a condensate for each SM fermion. This can be
16
QL
ψR
ψR
QR
ETC
ETC
QL
QR
ψL
ψL
Figure 7: Left panel: ETC gauge boson interaction involving techniquarks and SM
fermions. Right panel: Diagram contribution to the mass to the SM fermions.
achieved by using as technifermion matter a complete generation of quarks and leptons
(including a neutrino right) but now gauged with respect to the TC interactions.
The ETC gauge group is assumed to spontaneously break N g times down to SU(NTC )
permitting three different mass scales, one for each SM family. This type of TC with
associated ETC is termed the one family model [42]. The heavy masses are provided by
the breaking at low energy and the light masses are provided by breaking at higher
energy scales. This model does not, per se, explain how the gauge group is broken
several times, neither is the breaking of weak isospin symmetry accounted for. For
example we cannot explain why the neutrino have masses much smaller than the
associated electrons. See, however, [43] for progress on these issues. Schematically
one has SU(NTC + 3) which breaks to SU(NTC + 2) at the scale Λ1 providing the first
3
2
generation of fermions with a typical mass m1 ∼ 4π(FTC
π ) /Λ1 at this point the gauge
group breaks to SU(NTC + 1) with dynamical scale Λ2 leading to a second generation
3
2
mass of the order of m2 ∼ 4π(FTC
π ) /Λ2 finally the last breaking SU(NTC ) at scale Λ3
3
2
leading to the last generation mass m3 ∼ 4π(FTC
π ) /Λ3 .
Without specifying an ETC one can write down the most general type of fourfermion operators involving TC particles Q and ordinary fermionic fields ψ. Following
the notation of Hill and Simmons [14] we write:
αab
Q̄γµ Ta Qψ̄γµ Tb ψ
Λ2ETC
+ βab
Q̄γµ Ta QQ̄γµ Tb Q
Λ2ETC
+ γab
ψ̄γµ Ta ψψ̄γµ Tb ψ
Λ2ETC
,
(2.36)
where the Ts are unspecified ETC generators. After performing a Fierz rearrangement
one has:
Q̄Ta Qψ̄Tb ψ
ψ̄Ta ψψ̄Tb ψ
Q̄Ta QQ̄Tb Q
αab
+
β
+
γ
+ ... ,
(2.37)
ab
ab
Λ2ETC
Λ2ETC
Λ2ETC
The coefficients parametrize the ignorance on the specific ETC physics. To be more
specific, the α-terms, after the TC particles have condensed, lead to mass terms for the
SM fermions
g2
mq ≈ ETC
hQ̄QiETC ,
(2.38)
M2ETC
17
where mq is the mass of e.g. a SM quark, gETC is the ETC gauge coupling constant
evaluated at the ETC scale, METC is the mass of an ETC gauge boson and hQ̄QiETC is
the TC condensate where the operator is evaluated at the ETC scale. Note that we have
not explicitly considered the different scales for the different generations of ordinary
fermions but this should be taken into account for any realistic model.
The β-terms of Eq. (2.37) provide masses for pseudo Goldstone bosons and also
provide masses for techniaxions [14], see Fig. 8. The last class of terms, namely the
γ-terms of Eq. (2.37) induce FCNCs. For example it may generate the following terms:
1
Λ2ETC
(s̄γ5 d)(s̄γ5 d) +
1
Λ2ETC
(µ̄γ5 e)(ēγ5 e) + . . . ,
(2.39)
where s, d, µ, e denote the strange and down quark, the muon and the electron, respectively. The first term is a ∆S = 2 flavor-changing neutral current interaction affecting
the KL − KS mass difference which is measured accurately. The experimental bounds on
these type of operators together with the very naive assumption that ETC will generate
these operators with γ of order one leads to a constraint on the ETC scale to be of the
order of or larger than 103 TeV [40]. This should be the lightest ETC scale which in
turn puts an upper limit on how large the ordinary fermionic masses can be. The naive
estimate is that one can account up to around 100 MeV mass for a QCD-like TC theory,
implying that the top quark mass value cannot be achieved.
The second term of Eq. (2.39) induces flavor changing processes in the leptonic
sector such as µ → eēe, eγ which are not observed. It is clear that, both for the precision
Π
ETC
Π
Figure 8: Leading contribution to the mass of the TC pseudo Goldstone bosons via an
exchange of an ETC gauge boson.
measurements and the fermion masses, a better theory of the flavor is needed. For the
ETC dynamics interesting developments recently appeared in the literature [44, 45, 46,
47]. We note that nonperturbative chiral gauge theories dynamics is expected to play
a relevant role in models of ETC since it allows, at least in principle, the self breaking
of the gauge symmetry. Recent progress on the phase diagrams of these theories has
appeared in [12].
18
In Fig. 9 we show the ordering of the relevant scales involved in the generation
of the ordinary fermion masses via ETC dynamics, and the generation of the fermion
masses (for a single generation and focussing on the top quark) assuming QCD-like
dynamics for TC.
ΛET C
ΛT C
mf ≈
2
gET
C
< Q̄Q >ET C
Λ2ET C
Electroweak breaks
< Q̄Q >ET C ≈ < Q̄Q >T C ∼ Λ3T C
mf ≈
2
gET
C
< Q̄Q >ET C
2
ΛET C
� mTop
Figure 9: Cartoon of the expected ETC dynamics starting at high energies with a more
fundamental gauge interaction and the generation of the fermion masses assuming
QCD-like dynamics.
2.5
Walking
To better understand in which direction one should go to modify the QCD dynamics,
we analyze the TC condensate. The value of the TC condensate used when giving mass
to the ordinary fermions should be evaluated not at the TC scale but at the ETC one.
Via the renormalization group one can relate the condensate at the two scales via:
!
Z ΛETC
hQ̄QiETC = exp
d(ln µ)γm (α(µ)) hQ̄QiTC ,
(2.40)
ΛTC
19
where γm is the anomalous dimension of the techniquark mass-operator. The boundaries of the integral are at the ETC scale and the TC one. For TC theories with a running
of the coupling constant similar to the one in QCD, i.e.
α(µ) ∝
1
,
ln µ
for µ > ΛTC ,
(2.41)
this implies that the anomalous dimension of the techniquark masses γm ∝ α(µ). When
computing the integral one gets
ΛETC γm
hQ̄QiTC ,
(2.42)
hQ̄QiETC ∼ ln
ΛTC
which is a logarithmic enhancement of the operator. We can hence neglect this correction and use directly the value of the condensate at the TC scale when estimating the
generated fermionic mass:
mq ≈
g2ETC
M2ETC
Λ3TC ,
hQ̄QiTC ∼ Λ3TC .
(2.43)
The tension between having to reduce the FCNCs and at the same time provide
a sufficiently large mass for the heavy fermions in the SM as well as the pseudoGoldstones can be reduced if the dynamics of the underlying TC theory is different
from the one of QCD. The computation of the TC condensate at different scales shows
that if the dynamics is such that the TC coupling does not run to the UV fixed point but
rather slowly reduces to zero one achieves a net enhancement of the condensate itself
with respect to the value estimated earlier. This can be achieved if the theory has a
near conformal fixed point. This kind of dynamics has been denoted as of walking type.
In Fig. 10 the comparison between a running and walking behavior of the coupling is
qualitatively represented.
In the walking regime:
∗
ΛETC γm (α )
hQ̄QiETC ∼
hQ̄QiTC ,
(2.44)
ΛTC
which is a much larger contribution than in QCD dynamics [48, 49, 50, 51]. Here γm is
evaluated at the would be fixed point value α∗ . Walking can help resolving the problem
of FCNCs in TC models since with a large enhancement of the hQ̄Qi condensate the
four-Fermi operators involving SM fermions and technifermions and the ones involving
technifermions are enhanced by a factor of ΛETC /ΛTC to the γm power while the one
involving only SM fermions is not enhanced.
We note that walking is not a fundamental property for a successful model of the
origin of mass of the elementary fermions featuring TC. In fact several alternative ideas
already exist in the literature (see [52, 53] and references therein). However, a near
conformal theory would still be useful to reduce the contributions to the precision data
and, possibly, provide a light composite Higgs of much interest to LHC physics [17].
20
Figure 10: Top left panel: QCD-like behavior of the coupling constant as function of
the momentum (Running). Top right panel: walking-like behavior of the coupling
constant as function of the momentum (Walking). Bottom right panel: cartoon of the
beta function associated to a generic walking theory.
2.6
Ideal walking
There are several issues associated with the original idea of walking:
• Since the number of flavors cannot be changed continuously it is not possible to
get arbitrarily close to the lower end of the conformal window. This applies to
the TC theory in isolation i.e. before coupling it to the SM and without taking into
account the ETC interactions.
• It is hard to achieve large anomalous dimensions of the fermion mass operator
even near the lower end of the conformal window for ordinary gauge theories.
• It is not always possible to neglect the interplay of the four fermion interactions
on the TC dynamics.
In [54] it has been argued that it is possible to solve simultaneously all the problems
above by consistently taking into account the effects of the four-fermion interactions
on the phase diagram of strongly interacting theories for any matter representation as
function of the number of colors and flavors. A positive effect is that the anomalous
dimension of the mass increases beyond the unity value at the lower boundary of the
new conformal window and can get sufficiently large to yield the correct mass for the
top quark. It has also been shown that the conformal window, for any representation,
shrinks with respect to the case in which the four-fermion interactions are neglected.
21
This analysis derives from the study of the gauged Nambu-Jona-Lasinio phase diagram
[55].
It has been made the further unexpected discovery that when the extended TC sector,
responsible for giving masses to the SM fermions, is sufficiently strongly coupled, the
TC theory, in isolation, must feature an infrared fixed point in order for the full model
to be phenomenologically viable and correctly break the electroweak symmetry [56].
22
3
Phenomenology of Minimal Technicolor
The existence of a new weak doublet of technifermions amounting to, at least, a
global SU(2)L × SU(2)R symmetry later opportunely gauged under the electroweak
interactions is the bedrock on which models of TC are built on.
It is therefore natural to construct first minimal models of TC passing precision tests
while also reducing the FCNC problem by featuring near conformal dynamics. By
minimal we mean with the smallest fermionic matter content. These models were put
forward recently in[16, 17]. To be concrete we describe here the (N)MWT [13] extension
of the SM.
The extended SM gauge group is now SU(2)TC ×SU(3)C ×SU(2)L ×U(1)Y and the field
content of the TC sector is constituted by four techni-fermions and one techni-gluon
all in the adjoint representation of SU(2)TC . The model features also a pair of Dirac
leptons, whose left-handed components are assembled in a weak doublet, necessary
to cancel the Witten anomaly [57] arising when gauging the new technifermions with
respect to the weak interactions. Summarizing, the fermionic particle content of the
MWT is given explicitly by
!
Ua
a
QL =
,
URa , DaR ,
a = 1, 2, 3 ,
(3.45)
Da L
with a being the adjoint color index of SU(2). The left handed fields are arranged in
three doublets of the SU(2)L weak interactions in the standard fashion. The condensate
is hŪU + D̄Di which correctly breaks the electroweak symmetry as already argued for
ordinary QCD in Eq. (2.18).
The model as described so far suffers from the Witten topological anomaly [57].
However, this can easily be solved by adding a new weakly charged fermionic doublet
which is a TC singlet [17]. Schematically:
!
N
LL =
,
NR , ER .
(3.46)
E L
In general, the gauge anomalies cancel using the following generic hypercharge assignment
!
y+1 y−1
y
Y(UR , DR ) =
,
,
(3.47)
Y(QL ) = ,
2
2
2
!
y
−3y + 1 −3y − 1
Y(LL ) = − 3 ,
Y(NR , ER ) =
,
,
(3.48)
2
2
2
where the parameter y can take any real value [17]. In our notation the electric charge
is Q = T3 + Y, where T3 is the weak isospin generator. One recovers the SM hypercharge assignment for y = 1/3. To discuss the symmetry properties of the theory it is
23
Minimal Walking Technicolor
ons in the SU (2)T C ad: a = 1, 2, 3;
� a �
UL
a
, URa , DR
a
DL
U (1)Y
N
extra
neutrino
SU (2)L
E
extra
electron
ptons to cancel Witten
�
NL
EL
�
SU (3)C
U
TC-up
D
, N R , ER
G
TC-gluon
SU (2)T C
TC-down
Figure 11: Cartoon of the Minimal Walking Technicolor Model extension of the SM.
convenient to use the Weyl basis for the fermions and arrange them in the following
vector transforming according to the fundamental representation
of SU(4)
Planck 2010
UL
D
Q = 2L ∗ ,
(3.49)
−iσ UR
2 ∗
−iσ DR
Sannino, Tuominen ‘04; Dietrich, Sannino, Tuominen ‘05
where UL and DL are the left handed techniup and technidown, respectively and UR
and DR are the corresponding right handed particles. Assuming the standard breaking
to the maximal diagonal subgroup, the SU(4) symmetry spontaneously breaks to SO(4).
Such a breaking is driven by the following condensate
β
hQαi Q j αβ Ei j i = −2 hUR UL + DR DL i ,
(3.50)
where the indices i, j = 1, . . . , 4 denote the components of the tetraplet of Q, and the
Greek indices indicate the ordinary spin. The matrix E is a 4 × 4 matrix defined in terms
of the 2-dimensional unit matrix as
!
0 1
E=
(3.51)
1 0 .
Here αβ = −iσ2αβ and hULα UR ∗ β αβ i = −hUR UL i. A similar expression holds for the
D techniquark. The above condensate is invariant under an SO(4) symmetry. This
24
leaves us with nine broken generators with associated Goldstone bosons, of which
three become the longitudinal degrees of freedom of the weak gauge bosons.
Replacing the Higgs sector of the SM with the MWT the Lagrangian now reads:
1 a aµν
LH → − Fµν
F + iQ̄L γµ Dµ QL + iŪR γµ Dµ UR + iD̄R γµ Dµ DR
4
+iL̄L γµ Dµ LL + iĒR γµ Dµ ER + iN̄R γµ Dµ NR
(3.52)
a
with the TC field strength Fµν
= ∂µ Aaν − ∂ν Aaµ + gTC abc Abµ Acν , a, b, c = 1, . . . , 3. For the
left handed techniquarks the covariant derivative is:
Dµ QaL
=
δ ∂µ +
ac
gTC Abµ abc
y
g
ac
0
ac
~
− i Wµ · ~τδ − ig Bµ δ QcL .
2
2
(3.53)
Aµ are the techni gauge bosons, Wµ are the gauge bosons associated to SU(2)L and Bµ is
the gauge boson associated to the hypercharge. τa are the Pauli matrices and abc is the
fully antisymmetric symbol. In the case of right handed techniquarks the third term
containing the weak interactions disappears and the hypercharge y/2 has to be replaced
accordingly to Eq. (3.47). For the left-handed leptons the second term containing the
TC interactions disappears and y/2 changes to −3y/2. Only the last term is present for
the right handed leptons with an appropriate hypercharge assignment.
3.1
Low energy theory for MWT
We construct the effective theory for MWT including composite scalars and vector
bosons, their self interactions, and their interactions with the electroweak gauge fields
and the SM fermions.
3.1.1
Scalar sector
The relevant effective theory for the Higgs sector at the electroweak scale consists,
in our model, of a composite Higgs σ and its pseudoscalar partner Θ, as well as nine
pseudoscalar Goldstone bosons and their scalar partners. These can be assembled in
the matrix
σ + iΘ √
a
a
a
e
+ 2(iΠ + Π ) X E ,
(3.54)
M=
2
which transforms under the full SU(4) group according to
M → uMuT ,
with
u ∈ SU(4) .
(3.55)
The Xa ’s, a = 1, . . . , 9 are the generators of the SU(4) group which do not leave the
Vacuum Expectation Value (VEV) of M invariant
v
hMi = E .
(3.56)
2
25
Note that the notation used is such that σ is a scalar while the Πa ’s are pseudoscalars.
It is convenient to separate the fifteen generators of SU(4) into the six that leave the
vacuum invariant, Sa , and the remaining nine that do not, Xa . Then the Sa generators
of the SO(4) subgroup satisfy the relation
Sa E + E Sa T = 0 ,
a = 1, . . . , 6 ,
with
(3.57)
so that uEuT = E, for u ∈ SO(4). The explicit realization of the generators and the
embedding of the electroweak generators in the SU(4) algebra are shown in Appendix
A. With the tilde fields included, the matrix M is invariant in form under U(4) ≡
SU(4)×U(1)A , rather than just SU(4). However the U(1)A axial symmetry is anomalous,
and is therefore broken at the quantum level.
The connection between the composite scalars and the underlying techniquarks
can be derived from the transformation properties under SU(4), by observing that the
elements of the matrix M transform like techniquark bilinears:
β
Mi j ∼ Qαi Q j εαβ
with i, j = 1 . . . 4.
(3.58)
Using this expression, and the basis matrices given in Appendix A, the scalar fields
can be related to the wavefunctions of the techniquark bound states. This gives the
following charge eigenstates:
,
Θ ∼ i Uγ5 U + Dγ5 D ,
v + H ≡ σ ∼ UU + DD
e 3 ∼ UU − DD ,
A0 ≡ Π
Π0 ≡ Π3 ∼ i Uγ5 U − Dγ5 D ,
e 1 − iΠ
e2
Π
∼ DU ,
√
2
e 1 + iΠ
e2
Π
≡
∼ UD ,
√
2
Π1 − iΠ2
∼ iDγ5 U ,
√
2
1
Π + iΠ2
≡
∼ iUγ5 D ,
√
2
A+ ≡
Π+ ≡
A−
Π−
(3.59)
for the technimesons, and
ΠUU ≡
ΠDD ≡
ΠUD ≡
e UU ≡
Π
e DD ≡
Π
e UD ≡
Π
Π4 + iΠ5 + Π6 + iΠ7
∼ UT CU ,
2
Π4 + iΠ5 − Π6 − iΠ7
∼ DT CD ,
2
Π8 + iΠ9
∼ UT CD ,
√
2
4
e
e5 + Π
e 6 + iΠ
e7
Π + iΠ
∼ iUT Cγ5 U ,
2
e 4 + iΠ
e5 − Π
e 6 − iΠ
e7
Π
∼ iDT Cγ5 D ,
2
e 8 + iΠ
e9
Π
∼ iUT Cγ5 D ,
√
2
26
(3.60)
for the technibaryons, where U ≡ (UL , UR )T and D ≡ (DL , DR )T are Dirac technifermions,
and C is the charge conjugation matrix, needed to form Lorentz-invariant objects. To
these technibaryon charge eigenstates we must add the corresponding charge conjugate
states (e.g. ΠUU → ΠUU ).
Three of the nine Goldstone bosons (Π± , Π0 ) associated with the relative broken
generators become the longitudinal degrees of freedom of the massive weak gauge
bosons, while the extra six Goldstone bosons will acquire a mass due to ETC interactions
as well as the electroweak interactions per se. Using a bottom up approach we will not
commit to a specific ETC theory but limit ourself to introduce the minimal low energy
operators needed to construct a phenomenologically viable theory. The new Higgs
Lagrangian is
i
1 h
Tr Dµ MDµ M† − V(M) + LETC ,
2
LHiggs =
(3.61)
where the potential reads
m2M
i
i2
h
λ h
Tr MM† + λ0 Tr MM† MM†
2h
4
i
00
− 2λ Det(M) + Det(M† ) ,
V(M) = −
Tr[MM† ] +
(3.62)
and LETC contains all terms which are generated by the ETC interactions, and not by
the chiral symmetry breaking sector. Notice that the determinant terms (which are
renormalizable) explicitly break the U(1)A symmetry, and give mass to Θ, which would
otherwise be a massless Goldstone boson.
In order to give masses to the remaining uneaten Goldstone boson we add this term
which is generated in the ETC sector:
LETC ⊃
m2ETC
4
i
h
Tr MBM† B + MM† ,
(3.63)
√
and B ≡ 2 2S4 is a specific generator in the SU(4) algebra.
The potential V(M) produces a VEV which parameterizes the techniquark condensate, and spontaneously breaks SU(4) to SO(4). In terms of the model parameters the
VEV is
v = hσi =
2
2
m2M
λ + λ0 − λ00
,
(3.64)
while the Higgs mass is
M2H = 2 m2M .
(3.65)
The linear combination λ + λ0 − λ00 corresponds to the Higgs self coupling in the SM.
The three pseudoscalar mesons Π± , Π0 correspond to the three massless Goldstone
27
bosons which are absorbed by the longitudinal degrees of freedom of the W ± and Z
boson. The remaining six uneaten Goldstone bosons are technibaryons, and all acquire
tree-level degenerate mass through the ETC interaction in (3.63):
M2ΠUU = M2ΠUD = M2ΠDD = m2ETC .
(3.66)
The remaining scalar and pseudoscalar masses are
M2A±
M2Θ = 4v2 λ00
= M2A0 = 2v2 (λ0 + λ00 )
(3.67)
for the technimesons, and
M2Π
e
UU
= M2Π
e
UD
= M2Π
e
DD
= m2ETC + 2v2 (λ0 + λ00 ) ,
(3.68)
for the technibaryons. To gain insight on some of the mass relations one can use [58].
3.1.2
Vector bosons
The composite vector bosons of a theory with a global SU(4) symmetry are conveniently described by the four-dimensional traceless Hermitian matrix
Aµ = Aaµ Ta ,
(3.69)
where Ta are the SU(4) generators: Ta = Sa , for a = 1, . . . , 6, and Ta+6 = Xa , for a = 1, . . . , 9.
Under an arbitrary SU(4) transformation, Aµ transforms like
Aµ → u Aµ u† , where u ∈ SU(4) .
(3.70)
Eq. (3.70), together with the tracelessness of the matrix Aµ , gives the connection with
the techniquark bilinears:
1 j
µ, j
µ
µ
Ai ∼ Qαi σαβ̇ Q̄β̇,j − δi Qαk σαβ̇ Q̄β̇,k .
4
(3.71)
Then we find the following relations between the charge eigenstates and the wavefunctions of the composite objects:
v0µ ≡ A3µ ∼ Ūγµ U − D̄γµ D
A1µ − iA2µ
+µ
v
≡
∼ D̄γµ U
√
2
1µ
A
+
iA2µ
v−µ ≡
∼ Ūγµ D
√
2
v4µ ≡ A4µ ∼ Ūγµ U + D̄γµ D
, a0µ ≡ A9µ ∼ Ūγµ γ5 U − D̄γµ γ5 D
A7µ − iA8µ
+µ
, a
≡
∼ D̄γµ γ5 U
√
2
7µ
A
+
iA8µ
, a−µ ≡
∼ Ūγµ γ5 D
√
2
,
28
(3.72)
for the vector mesons, and
µ
xUU
≡
µ
xDD ≡
µ
xUD ≡
µ
sUD ≡
A10µ + iA11µ + A12µ + iA13µ
∼ UT Cγµ γ5 U ,
2
A10µ + iA11µ − A12µ − iA13µ
∼ DT Cγµ γ5 D ,
2
A14µ + iA15µ
∼ DT Cγµ γ5 U ,
√
2
A6µ − iA5µ
∼ UT Cγµ D ,
√
2
(3.73)
for the vector baryons.
There are different approaches on how to introduce vector mesons at the effective
Lagrangian level. At the tree level they are all equivalent.
Based on this premise, the minimal kinetic Lagrangian is:
i
h
i
1 h e e µν i 1
1 h
Lkinetic = − Tr W
− Bµν Bµν − Tr Fµν Fµν + m2 Tr Cµ Cµ ,
µν W
2
4
2
(3.74)
e µν and Bµν are the ordinary field strength tensors for the electroweak gauge
where W
fields. Strictly speaking the terms above are not only kinetic ones since the Lagrangian
contains a mass term as well as self interactions. The tilde on W a indicates that the
associated states are not yet the SM weak triplets: in fact these states mix with the
composite vectors to form mass eigenstates corresponding to the ordinary W and Z
bosons. Fµν is the field strength tensor for the new SU(4) vector bosons,
h
i
Fµν = ∂µ Aν − ∂ν Aµ − i g̃ Aµ , Aν ,
(3.75)
and the vector field Cµ is defined by
Cµ ≡ Aµ −
and Gµ is given by
g
Gµ .
g̃
Gµ = g Wµa La + g0 Bµ Y ,
(3.76)
(3.77)
where La and Y are the generators of the left-handed and hypercharge transformations,
as defined in Appendix A, with Y. The parameter g̃ represents the coupling among
g
the vectors and the ratio g̃ is phenomenologically very important because it sets the
mixing among gauge eigenstates and composite vectors eigenstates. The mass term
in Eq. (3.74) is gauge invariant, and gives a degenerate mass to all composite vector
bosons, while leaving the actual gauge bosons massless. (The latter acquires mass as
usual from the covariant derivative term of the scalar matrix M, after spontaneous
symmetry breaking.)
29
The Cµ fields couple with M via gauge invariant operators. Up to dimension four
operators the Lagrangian is
h
i
h
i
LM−C = g̃2 r1 Tr Cµ Cµ MM† + g̃2 r2 Tr Cµ MCµ T M†
i
h
i h
i
r3 h
+ i g̃ Tr Cµ M(Dµ M)† − (Dµ M)M† + g̃2 s Tr Cµ Cµ Tr MM† .
2
(3.78)
The dimensionless parameters r1 , r2 , r3 , s parameterize the strength of the interactions
between the composite scalars and vectors in units of g̃, and are therefore naturally
expected to be of order one. However, notice that for r1 = r2 = r3 = 0 the overall
Lagrangian possesses two independent SU(2)L × U(1)R × U(1)V global symmetries. One
for the terms involving M and one for the terms involving Cµ 3 . The Higgs potential
only breaks the symmetry associated with M, while leaving the symmetry in the vector
sector unbroken. This enhanced symmetry guarantees that all r-terms are still zero after
loop corrections. Moreover if one chooses r1 , r2 , r3 to be small the near enhanced
symmetry will protect these values against large corrections [59, 60].
3.1.3
Fermions and Yukawa interactions
The fermionic content of the effective theory consists of the SM quarks and leptons, the new lepton doublet L = (N, E) introduced to cure the Witten anomaly, and a
composite techniquark-technigluon doublet.
We now consider the limit according to which the SU(4) symmetry is, at first, extended to ordinary quarks and leptons. Of course, we will need to break this symmetry
to accommodate the SM phenomenology. We start by arranging the SU(2) doublets in
SU(4) multiplets as we did for the techniquarks in Eq. (3.49). We therefore introduce
the four component vectors qi and li ,
νiL
uiL
diL
eiL
i
i
,
∗
∗
(3.79)
q = 2 i , l =
−iσ uR
−iσ2 νiR
∗
∗
−iσ2 diR
−iσ2 eiR
where i is the generation index. Note that such an extended SU(4) symmetry automatically predicts the presence of a right handed neutrino for each generation. In addition
to the SM fields there is an SU(4) multiplet for the new leptons,
NL
E
L
L =
(3.80)
,
−iσ2 NR ∗
−iσ2 ER ∗
3
The gauge fields explicitly break the original SU(4) global symmetry to SU(2)L ×U(1)R ×U(1)V , where
U(1)R is the T3 part of SU(2)R , in the SU(2)L × SU(2)R × U(1)V subgroup of SU(4).
30
and a multiplet for the techniquark-technigluon bound state,
eL
U
D
e
L
.
e =
Q
∗
−iσ2 U
e
2 e∗R
−iσ DR
(3.81)
The techniquark-technigluon states, Q̃, being bound states of the underlying MWT
model, have a dynamical mass.
With this arrangement, the electroweak covariant derivative for the fermion fields
can be written
Dµ = ∂µ − i g Gµ (YV ) ,
(3.82)
where YV = 1/3 for the quarks, YV = −1 for the leptons, YV = −3y for the new lepton
doublet, and YV = y for the techniquark-technigluon bound state. Based on this matter
content, we write the following gauge part of the fermion Lagrangian:
i
µ,α̇β
Lfermion = i qα̇ σ
i
µ,α̇β
Dµ qiβ + i lα̇ σ
µ,α̇β
Dµ liβ + i Lα̇ σ
e σµ,α̇β Cµ Q
eβ .
+ xQ
α̇
e σµ,α̇β Dµ Q
eβ
Dµ Lβ + i Q
α̇
(3.83)
We now turn to the issue of providing masses to the SM fermions. In the first chapter
the simplest ETC model has been briefly reviewed. Many extensions of TC have
been suggested in the literature to address this problem. Some of the extensions
use another strongly coupled gauge dynamics, others introduce fundamental scalars.
Many variants of the schemes presented above exist and a review of the major models
is the one by Hill and Simmons [14]. At the moment there is not yet a consensus on
which is the correct ETC. In our phenomenological approach will we parameterize our
ignorance about a complete ETC theory by simply coupling the fermions to our low
energy effective Higgs throughout the ordinary effective SM Yukawa interactions and
we assume that any dangerous FCNC operator is strongly suppressed and therefore
negligible.
A discussion regarding the implications of having a natural fourth family of leptons
is presented in detail in Section 5 of this report.
3.1.4
Weinberg Sum Rules
In order to make contact with the underlying gauge theory, and discriminate between different classes of models, we make use of the Weinberg Sum Rules (WSR)s. In
[61] it was argued that the zeroth WSR – which is nothing but the definition of the S
parameter –
#
" 2
F2A
FV
−
,
(3.84)
S = 4π
M2V M2A
31
and the first WSR,
F2V − F2A = F2π ,
(3.85)
do not receive significant contributions from the near conformal region, and are therefore unaffected. In these equations MV (MA ) and FV (FA ) are mass and decay constant
of the vector-vector (axial-vector) meson, respectively, in the limit of zero electroweak
gauge couplings. Fπ is the decay constant of the pions: since this is a model of dynamical electroweak symmetry breaking, Fπ = 246 GeV. The heavy vector boson masses
are:
g̃2 (s − r2 ) v2
,
4
g̃2 (s + r2 ) v2
2
,
= m +
4
M2V = m2 +
M2A
and
FV
FA
F2π
(3.86)
√
2MV
=
,
g̃
√
2MA
=
χ,
g̃
= (1 + 2ω) F2V − F2A ,
(3.87)
where
ω≡
v2 g̃2
4M2V
(1 + r2 − r3 ) ,
χ≡1−
v2 g̃2 r3
4M2A
.
(3.88)
Then Eqs. (3.84) and (3.85) give
8π
2
1
−
χ
,
g̃2
r2 = r3 − 1 .
S=
(3.89)
(3.90)
The second WSR, corresponding to a zero on the right hand side of the following
equation, does receive important contributions from the near conformal region, and is
modified to
F2V M2V − F2A M2A = a
8π2 4
F ,
d(R) π
(3.91)
where a is expected to be positive and O(1), and d(R) is the dimension of the representation of the underlying fermions [61]. For each of these sum rules a more general
spectrum would involve a sum over all the vector and axial states.
In the effective Lagrangian we codify the walking behavior in a being positive and
O(1), and the minimality of the theory in S being small. A small S is both due to the
small number of flavors in the underlying theory and to the near conformal dynamics,
which reduces the contribution to S relative to a running theory [61, 62, 63].
32
Figure 12: The ellipses represent the 90% confidence region for the S and T parameters.
The ellipses, from lower to higher, are obtained for a reference Higgs mass of 117 GeV,
300 GeV, and 1 TeV, respectively. The contribution from the TC sector of the MWT
theory per se and from the new leptons is expressed by the green region. The left panel
has been obtained using a SM type hypercharge assignment while the right one is for
y = 1.
3.1.5
Passing the electroweak precision tests
We have studied the effects of the lepton family on the electroweak parameters
in [17], we summarize here the main results in Fig. 12, where we have used the updated
experimental values for S and T given in [64]. The ellipses represent the 90% confidence
region for the S and T parameters. The ellipses, from lower to higher, are obtained for
a reference Higgs mass of 117 GeV, 300 GeV, and 1 TeV, respectively. The contribution
from the MWT theory per se and of the new leptons [65] is expressed by the green
region. The left panel has been obtained using a SM type hypercharge assignment
while the right one is for y = 1. In both pictures the regions of overlap between the
theory and the precision contours are achieved when the upper component of the weak
isospin doublet is lighter than the lower component. The opposite case leads to a total
S which is larger than the one predicted within the new strongly coupled dynamics per
se. This is due to the sign of the hypercharge for the new leptons. The mass range used
in the plots is MZ 6 mE,N 6 10 MZ . The plots have been obtained assuming a Dirac
mass for the new neutral lepton (in the case of a SM hypercharge assignment).
The analysis for the Majorana mass case has been performed in [66] where one can
again show that it is possible to be within the 90% contours.
33
3.1.6
The Next to Minimal Walking Technicolor Theory (NMWT)
The theory with three technicolors contains an even number of electroweak doublets, and hence it is not subject to a Witten anomaly. The doublet of technifermions, is
then represented again as:
!
U{C1 ,C2 }
{C1 ,C2 }
{C1 ,C2 }
{C1 ,C2 }
1 ,C2 }
QL
,
Q{C
=
=
U
,
D
.
{C1 ,C2 }
R
R
R
D
L
(3.92)
Here Ci = 1, 2, 3 is the technicolor index and QL(R) is a doublet (singlet) with respect
to the weak interactions. Since the two-index symmetric representation of SU(3) is
complex the flavor symmetry is SU(2)L × SU(2)R × U(1). Only three Goldstones emerge
and are absorbed in the longitudinal components of the weak vector bosons.
Gauge anomalies are absent with the choice Y = 0 for the hypercharge of the lefthanded technifermions:
!
U(+1/2)
(Q)
QL =
.
(3.93)
D(−1/2) L
Consistency requires for the right-handed technifermions (isospin singlets):
(+1/2)
−1/2
,
Q(Q)
=
U
,
D
R
R
R
Y =
+1/2, −1/2 .
(3.94)
All of these states will be bound into hadrons. There is no need for an associated fourth
family of leptons, and hence it is not expected to be observed in the experiments.
Here the low-lying technibaryons are fermions constructed with three techniquarks
in the following way:
3 ,C4 }
5 ,C6 } βγ
1 ,C2 }
B f1 , f2 , f3 ;α = Q{C
Q{C
Q{C
C1 C3 C5 C2 C4 C6 .
L;α, f1
L;β, f2
L;γ, f3
(3.95)
where fi = 1, 2 corresponds to U and D flavors, and we are not specifying the flavor
symmetrization which in any event will have to be such that the full technibaryon
wave function is fully antisymmetrized in technicolor, flavor and spin. α, β, and γ
assume the values of one or two and represent the ordinary spin. Similarly we can
construct different technibaryons using only right-handed fields or a mixture of leftand right-handed ones.
3.2
Beyond MWT
When going beyond MWT one finds new and interesting theories able to break
the electroweak symmetry while featuring a walking dynamics and yet not at odds
with precision measurements, at least when comparing with the naive S parameter.
A compendium of these theories can be found in [20]. Here we will review only the
principal type of models one can construct.
34
3.2.1
Partially Gauged Technicolor
A small modification of the traditional TC approach, which neither involves additional particle species nor more complicated gauge groups, allows constructing several
other viable candidates. It consists in letting only one doublet of techniquarks transform non-trivially under the electroweak symmetries with the rest being electroweak
singlets, as first suggested in [17] and later also used in [67]. Still, all techniquarks transform under the TC gauge group. Thereby only one techniquark doublet contributes
directly4 to the oblique parameter which is thus kept to a minimum for theories which
need more than one family of techniquarks to be quasi-conformal. It is the condensation of that first electroweakly charged family that breaks the electroweak symmetry.
The techniquarks which are uncharged under the electroweak gauge group are natural
building blocks for components of dark matter.
3.2.2
Split Technicolor
We summarize here also another possibility [17] according to which we keep the
technifermions gauged under the electroweak theory in the fundamental representation
of the SU(N) TC group while still reducing the number of techniflavors needed to be
near the conformal window. Like for the partially gauged case described above this can
be achieved by adding matter uncharged under the weak interactions. The difference
to Section 3.2.1 is that this part of matter transforms under a different representation
of the TC gauge group than the part coupled directly to the electroweak sector. For
example, for definiteness let’s choose it to be a massless Weyl fermion in the adjoint
representation of the TC gauge group. The resulting theory has the same matter content
as N f -flavor super QCD but without the scalars; hence the name Split Technicolor. The
matter content of Split Technicolor lies between that of super QCD and QCD-like theories
with matter in the fundamental representation. We note that a split TC-like theory has
been used in [69], to investigate the strong CP problem.
Split Technicolor shares some features with theories of split supersymmetry advocated and studied in [70, 71] as possible extensions of the SM. Clearly, we have
introduced Split Technicolor—differently from split supersymmetry—to address the
hierarchy problem. This is why we do not expect new scalars to appear at energy scales
higher than the one of the electroweak theory unless one tries to supersymmetrize the
model at higher energies.
In [72] one can find an explicit example of (near) conformal TC with two types
of technifermions, i.e. transforming according to two different representations of the
underlying TC gauge group [20, 73]. The model possesses a number of interesting
properties to recommend it over the earlier models of dynamical electroweak symmetry
breaking:
4
Via TC interactions all of the matter content of the theory will affect physical observables associated
to the sector coupled to the electroweak symmetry.
35
• Features the lowest possible value of the naive S parameter [32, 33] while possessing a dynamics which is near conformal.
• Contains, overall, the lowest possible number of fermions.
• Yields natural DM candidates.
Due to the above properties we term this model Ultra Minimal near conformal Technicolor
(UMT). It is constituted by an SU(2) TC gauge group with two Dirac flavors in the fundamental representation also carrying electroweak charges, as well as, two additional
Weyl fermions in the adjoint representation but singlets under the SM gauge groups.
By arranging the additional fermions in higher dimensional representations, it is
possible to construct models which have a particle content smaller than the one of
partially gauged TC theories. In fact instead of considering additional fundamental
flavors we shall consider adjoint flavors. Note that for two colors there exists only one
distinct two-indexed representation.
3.3
Vanilla Technicolor
Despite the different envisioned underlying gauge dynamics it is a fact that the
SM structure alone requires the extensions to contain, at least, the following chiral
symmetry breaking pattern (insisting on keeping the custodial symmetry of the SM):
SU(2)L × SU(2)R → SU(2)V .
(3.96)
We will call this common sector of any TC extension of the SM, the vanilla sector.
The reason for such a name is that the vanilla sector is common to old models of
TC featuring running and walking dynamics. It is worth mentioning that the vanilla
sector is common not only to TC extensions but to several extensions, even of extradimensional type, in which the Higgs sector can be viewed as composite. In fact,
the effective Lagrangian we are about to introduce can be used for modeling several
extensions with a common vanilla sector respecting the same constraints spelled out
in [19]. The natural candidate for a walking TC model featuring exactly this global
symmetry is NMWT [16].
Based on the vanilla symmetry breaking pattern we describe the low energy spectrum
in terms of the lightest spin one vector and axial-vector iso-triplets V ±,0 , A±,0 as well as
the lightest iso-singlet scalar resonance H. In QCD the equivalent states are the ρ±,0 ,
a1±,0 and the f0 (600) [74]. It has been argued in [13, 58], using Large N arguments, and in
[17, 20], using the saturation of the trace of the energy momentum tensor, that models
of dynamical electroweak symmetry breaking featuring (near) conformal dynamics
contain a composite Higgs state which is light with respect to the new strongly coupled
scale (4 π v with v ' 246 GeV). These indications have led to the construction of models
of TC with a naturally light composite Higgs. Recent investigations using SchwingerDyson [75] and gauge-gravity dualities [76] also arrived to the conclusion that the
36
composite Higgs can be light 5 . The 3 technipions Π±,0 produced in the symmetry
breaking become the longitudinal components of the W and Z bosons.
The composite spin one and spin zero states and their interaction with the SM fields
are described via the following effective Lagrangian which we developed, first for
minimal models of walking TC [19, 60]:
i
1 h e e µν i 1 e eµν 1 h
µν
µν
− Bµν B − Tr FLµν FL + FRµν FR
Lboson = − Tr W
µν W
2
4
2
h
i 1 h
i
h
i
µ
+ m2 Tr C2Lµ + C2Rµ + Tr Dµ MDµ M† − g˜2 r2 Tr CLµ MCR M†
2
i
i g̃ r3 h
−
Tr CLµ MDµ M† − Dµ MM† + CRµ M† Dµ M − Dµ M† M
4
i µ2 h
i h
i λ h
i2
g̃2 s h 2
+
Tr CLµ + C2Rµ Tr MM† + Tr MM† − Tr MM† ,
4
2
4
(3.97)
e µν and e
where W
Bµν are the ordinary electroweak field strength tensors, FL/Rµν are the
field strength tensors associated to the vector meson fields AL/Rµ 6 , and the CLµ and CRµ
fields are
g
fµ ,
CLµ ≡ ALµ − W
g̃
g0
fµ .
B
g̃
(3.98)
a = 1, 2, 3
(3.99)
CRµ ≡ ARµ −
The 2×2 matrix M is
1
M = √ [v + H + 2 i πa Ta ] ,
2
a
where
√ π are the Goldstone bosons produced in the chiral symmetry breaking, v =
µ/ λ is the corresponding VEV, H is the composite Higgs, and Ta = σa /2, where σa are
the Pauli matrices. The covariant derivative is
e µa Ta M + i g0 M e
Dµ M = ∂µ M − i g W
Bµ T3 .
(3.100)
When M acquires a VEV, the Lagrangian of Eq. (3.97) contains mixing matrices for the
spin one fields. The mass eigenstates are the ordinary SM bosons, and two triplets of
heavy mesons, of which the lighter (heavier) ones are denoted by R±1 (R±2 ) and R01 (R02 ).
These heavy mesons are the only new particles, at low energy, relative to the SM.
Now we must couple the SM fermions. The interactions with the Higgs and the
spin one mesons are mediated by an unknown ETC sector, and can be parametrized at
The Higgs boson here is identified with the lightest 0++ state of the theory. Calling it a dilaton or a
meson makes no physical difference since these two states mix at the 100% level and both couple to the
trace of the stress energy momentum tensor of the theory. In the construction of the low energy effective
theory saturating the trace anomaly there is no way to distinguish these states.
6
In [19], where the chiral symmetry is SU(4), there is an additional term whose coefficient is labeled
r1 . With an SU(N) × SU(N) chiral symmetry this term is just identical to the s term.
5
37
low energy by Yukawa terms, and mixing terms with the CL and CR fields. Assuming
that the ETC interactions preserve parity and do not generate extra flavor violation
beyond the SM like Yukawa terms, the most general form for the quark Lagrangian is 7
/ iL + q̄iR iDq
/ iR
Lquark = q̄iL iDq
#
"
1 − τ3
1 + τ3
j
j
i
i
q jR + q̄L (Yd )i M
q jR + h.c. ,
− q̄L (Yu )i M
2
2
(3.101)
where i and j are generation indices, i = 1, 2, 3, qiL/R are electroweak doublets, Yu and
Yd are 3×3 complex matrices. The covariant derivatives are the ordinary electroweak
ones,
a
e
/ T a − i g0 e
/ L qiL ,
/ iL = ∂/ − i g W
Dq
BY
/ R qiR ,
/ iR = ∂/ − i g0e
Dq
BY
(3.102)
where YL = 1/6 and YR = diag(2/3, −1/3). One can exploit the global symmetries of
the kinetic terms to reduce the number of physical parameters in the Yukawa matrices.
Thus we can take
Yu = diag(yu , yc , yt ) ,
Yd = V diag(yd , ys , yb ) ,
(3.103)
and
qiL
=
uiL
j
Vi d jL
!
,
qiR
=
uiR
diR
!
,
(3.104)
where V is the CKM matrix.
It is possible to further reduce the number of independent couplings using the
WSRs discussed above. For example in NMWT, featuring technifermions with three
technicolors transforming according to the two-index symmetric representation of the
TC gauge group, the naive one-loop S parameter is S = 1/π ' 0.3: this is a reasonable
input for S in Eq. (3.84).
With S = 0.3 the remaining parameters are MA , g̃, s and MH , with s and MH having
a sizable effect in processes involving the composite Higgs 8 .
3.4
WW - Scattering in Technicolor and Unitarity
The simplest argument often used to predict the existence of yet undiscovered
particles at the TeV scale comes from unitarity of longitudinal gauge boson scattering
7
The lepton sector works out in a similar way, the only difference being the possible presence of
Majorana neutrinos.
8
The information on the spectrum alone is not sufficient to constrain s, but it can be measured
studying other physical processes.
38
amplitudes. If the electroweak symmetry breaking sector (EWSB) is weakly interacting,
unitarity implies that new particle states must show up below one TeV, being these spin
zero isosinglets (the Higgs boson) or spin one isotriplets (e.g. Kaluza-Klein modes). A
strongly interacting EWSB sector can however change this picture, because of the strong
coupling between the pions (eaten by the longitudinal components of the standard
model gauge bosons) and the other bound states of the strongly interacting sector. An
illuminating example comes from QCD. In [77] it was shown that for six colors or
more, the 770 GeV ρ meson is enough to delay the onset of unitarity violation of the
pion-pion scattering amplitude up to well beyond 1 GeV. Here the ’t Hooft large N
limit was used, however an even lower number of colors is needed to reach a similar
delay of unitarity violation when an alternative large N limit is used [78]. Scaling up
to the electroweak scale, this translates in a 1.5 TeV technivector being able to delay
unitarity violation of longitudinal gauge boson scattering amplitudes up to 4 TeV or
more. As we discussed in the previous sections such a model, however, would not be
realistic for other reasons: a large contribution to the S parameter [32], and large FCNC
if the ordinary fermions acquire mass via an old fashioned ETC, to mention the most
relevant ones. It is therefore interesting to analyze the pion-pion scattering in generic
models of Walking TC. We follow the analyses performed in [108, 107].
In the effective theory for TC the scattering amplitudes for the longitudinal SM gauge
bosons approach at large energies the scattering amplitudes for the corresponding eaten
pions. We mainly analyze the contribution to the ππ scattering amplitude from a spin
zero isosinglet and a spin one isotriplet, and consider the case in which a spin two
isosinglet contributes as well.
3.4.1
Spin zero + spin one
The isospin invariant amplitude for the pion-pion elastic scattering is [79]:
!
"
#
3g2Vππ
h2
s2
s−u
s−t
1
2
s− 2
− gVππ
+
.
(3.105)
A(s, t, u) = 2 −
Fπ
M2V
MH s − M2H
t − M2V u − M2V
Note that our normalization for√gVππ , which is the heavy vector to two-pions effective
coupling, differs by a factor of 2 from that of Ref. [79]. The scalar H contribution is
proportional to the coupling h. These couplings are simply related to the ones of the
Vanilla TC Lagrangian, but the specific relation is not relevant here.
The amplitude of Eq. (3.105) has an s-channel pole in the Higgs exchange. In the
vicinity of this pole the propagator should be modified to include the Higgs width. In
order to catch the essential features of the unitarization√process we will take the Higgs
to be a relatively narrow state, and consider values of s far away from MH , where the
finite width effects can be neglected. If the Higgs or any other state is not sufficiently
narrow to be treated at the tree level, it would be relevant to investigate the effects
due to unitarity corrections using specific unitarization schemes as done for example
in [80]. In order to study unitarity of the ππ scattering the most general amplitude
39
MV � 1 TeV
0.4
a00
0.2
0.0
�0.2
�0.4
0.5
1.0
1.5
2.0
s �TeV�
2.5
3.0
Figure 13: I = 0 J = 0 partial wave amplitude for the ππ scattering. Here a Higgs with
mass MH = 200 GeV, and a spin-one vector meson with mass MV = 1 TeV contribute to
the full amplitude. The different groups of curves correspond, from top to bottom, to
gVππ = 2, 2.5, 3, 3.5, 4. The different curves within each group correspond, from top to
bottom, to h = 0, 0.1, 0.15, 0.2. Nonzero values of gVππ and h give negative contributions
to the linear term in s in the amplitude, and may lead to a delay of unitarity violation.
40
should be expanded in its isospin I and spin J components, aIJ . However the I = 0 J = 0
component,
Z 1
1
0
a0 (s) =
d cos θ [3A(s, t, u) + A(t, s, u) + A(u, t, s)] ,
(3.106)
64π −1
has the worst high energy behavior, and is therefore sufficient for our analysis. Since
we are interested in testing unitarity at few TeVs in presence of a light Higgs, we set
MH = 200 GeV as a reference value, and study the regions in the (MV , gVππ ) plane in
which a00 is unitary up to 3 TeV, for different values of h. If the Higgs mass is larger
than 200 GeV but still smaller than or of the same size of MV , we expect our results to
be qualitatively similar, even though finite width effects might be important due to the
pole in the s-channel. If the Higgs mass is much larger than MV the theory is Higgsless
at low energies. This case was studied in Ref. [108], and applies also to the light Higgs
scenario if H is decoupled from the pions, i.e. h = 0.
In order to study the effect of the Higgs exchange on the scattering amplitude,
consider the high energy behavior of A(s, t, u),
!
3g2Vππ
h2
1
− 2 s.
(3.107)
A(s, t, u) ∼ 2 −
Fπ
M2V
MH
This shows that the Higgs exchange provides an additional negative contribution at
large energies, which, together with the vector meson, contributes to delay
unitarity
√
violation to higher energies. In Fig. 13 a00 is plotted as a function of s for MV = 1
TeV, MH = 200 GeV, and different values of gVππ and h. The different groups of curves
from top to bottom correspond to gVππ = 2, 2.5, 3, 3.5, and 4. For comparison, the QCD
value that follows from Γ(ρ → ππ) ' 150 MeV would be gVππ ' 5.6 9 . Within each
group, the top curve corresponds to the Higgsless case, h = 0, while the remaining ones
correspond, from top to bottom, to h = 0.1, 0.15, and 0.2. For small values of gVππ the
presence of a light Higgs delays unitarity violation to higher energies: If the partial
wave amplitude has a maximum near 0.5 the delay is dramatic.
For a given value of MV , the presence of a light Higgs enlarges the interval of values
of gVππ for which the theory is unitary, provided that |h| is not too large.
3.4.2
Spin zero + spin one + spin two
In addition to spin-zero and spin-one mesons, the low energy spectrum can contain
spin two mesons as well [79]. The contribution of a spin-two meson F2 to the invariant
amplitude is
" 2
#
g22 s3
g22
s
t2 + u2
A2 (s, t, u) =
−
+
−
,
(3.108)
3
2
2(M2F2 − s)
12M4F2
Fig. 13 does not reproduce a scaled up version of QCD ππ scattering. For the latter to occur, the
vector resonance should be as large as (246 GeV/93 MeV)×770 MeV ' 2 TeV. However in a theory with
walking dynamics the resonances are expected to be lighter than in a running setup.
9
41
MV � 1 TeV, MF2 � 3 TeV, g2 � 4 TeV�1
0.4
0.2
0.2
0.0
a00
a00 �spin�two�
0.4
0.0
�0.2
�0.2
�0.4
�0.4
1
2
3
4
s �TeV�
5
6
0.5
1.0
1.5
2.0
2.5
s �TeV�
3.0
3.5
4.0
Figure 14: Left: Contribution from the spin-two exchanges to the I = 0 J = 0 partial
wave amplitude of the ππ scattering. The different groups of curves correspond, from
left to right, to MF2 = 2, 3, 4 TeV. Within each group, the different curves correspond,
from smaller to wider, to g2 = 2, 2.5, 3, 3.5, 4 TeV−1 . Right: I = 0 J = 0 partial wave
amplitude with all channels included (spin-zero, -one, and -two). The dashed curves
reproduce Fig. 13, with just the spin-zero and the spin-one channels included. The
solid curves contain also the spin-two exchanges, for MF2 = 3 TeV, and g2 = 4 TeV−1 .
If unitarity is violated at negative values of a00 , the spin-two exchanges may lead to a
delay of unitarity violation.
where MF2 and g2 are mass and coupling with the pions, respectively. A reference
value for g2 can be obtained from QCD: m f2 ' 1275 MeV and Γ( f2 → ππ) ' 160 MeV
give |g2 | ' 13 GeV−1 so that |g2 |Fπ ' 1.2. Scaling up to the eletroweak scale results in
|g2 | ' 4 TeV−1 . The contribution of F2 to the I = 0 J = 0 partial wave amplitude is given
in Fig. 14 (left) for different values of MF2 and g2 . Notice that
√ the amplitude is initially
positive, and then becomes negative at large values of s. If MF2 is large enough,
the positive contribution can balance the negative contribution from the spin-zero and
spin-one channels, shown in Fig. 13. This can lead to a further delay of unitarity
violation, as shown in Fig. 14 (right). Here the curves of Fig. 13 are redrawn dashed,
while the full contribution from spin-zero, spin-one, and spin-two is shown by the solid
lines, for MF2 = 3 TeV and g2 = 4 TeV−1 . If unitarity is violated at negative values of a00 ,
then the spin-two contibution delays the violation to higher energies.
The unitarity analysis presented here is for generic Vanilla TC theories, or any
other model, featuring spin zero, one, and two resonances. The specialization to
running and walking TC is described in detail in [108, 107]. The bottom line is that it
is possible to delay the onset of unitarity violation, at the effective Lagrangian level,
for phenomenologically viable values of the couplings and masses of the composite
spectrum.
42
4
Phenomenological benchmarks
We introduce here a natural classification of the different types of dynamical models
which can be constrained at the LHC. We do this by analyzing the constraints that different types of dynamics imposes on the coefficients of the general effective Lagrangian
introduced in the earlier sections. We start with counting the number of parameters.
The effective Lagrangian (3.97) contains 9 parameters:
• g and g0 , the SU(2)L × U(1)Y gauge couplings;
• µ and λ, the parameters of the scalar potential;
• g̃, the strength of the spin one resonances interaction;
• m2 , the SU(2)L × SU(2)R invariant vector-axial mass squared;
• r2 , r3 , s, the couplings between the Higgs and the vector states;
Using the experimental values of GF , MZ and α we can eliminate 3 of them. The
remaining parameters can be rearranged according to the needs of the model builder.
For the Vanilla TC case, we found useful to choose as independent parameters
the vector and axial vectors masses MV , MA , since they dictate the scale of the new
resonances, as well as the physical mass of the composite Higgs MH , the value of the S
parameter of the theory and the two couplings g̃ and s.
The Lagrangian couplings r2 , r3 , and m can be expressed as functions of MA , MV
and the S parameter in the following way:
m2 = M2A + M2V − g̃2 v2 s /2 ,
(4.109)
r2 = 2(M2A − M2V )/ g̃2 v2 ,
q
r3 = 4M2A 1 ± 1 − g̃2 S/8π / g̃2 v2 .
(4.110)
(4.111)
The input parameters for the various models analyzed here as special cases of the
Vanilla TC are listed in Table 1. Under the last column we quote only the constraints
that can be used to reduce the number of input parameters.
Briefly we comment about the different models:
• Walking TC: using the first WSR we eliminate MV from the list of input parameters. In walking dynamics, the second WSR is modified according to [61]. We
can use this relation as a constraint on the parameter space requiring that a > 0 in
(3.91).
• Running TC: in this case both the first and the second WSRs can be used to reduce
the number of input parameters. We eliminate MV and S.
43
Model
Input parameters
Constraints
Vanilla
Walking
Running
NMWT
Custodial
D-BESS
MH , g̃, MA , MV , s, S
MH , g̃, MA , s, S
MH , g̃, MA , s
MH , g̃, MA , s
MH , g̃, MA , s
MH , g̃, MA
1st WSR
1st and 2nd WSR
1st WSR, S = 0.3
r2 = r3 = 0
r2 = r3 = s = 0
Table 1: List of models to be used as benchmarks at the LHC. We classified them in a
simple manner for the experiments to be able to easily identify the relevant parameter
space.
• NMWT: we apply the same constraints as in the case of the general Walking TC,
but now we take S = 0.3 estimated knowing the underlying TC theory.
• Custodial TC: this scenario requires r2 = r3 = 0. In this case S is automatically 0
and MV = MA [60, 61, 81, 82].
• D-BESS model: in this model [59] all couplings involving one or more vector
resonances and one or more scalar fields vanish. Effectively it is like the previous
case plus the condition s = 0.
As for ordinary QCD besides the simplest spin one and spin zero sectors discussed
here the underlying dynamics will produce several composite states of bosonic and
fermionic nature. For example, there will be states analog to the familiar baryons of
QCD. The physics of these new states is very interesting, it will help selecting the specific
underlying gauge dynamics, and needs to be explored in much detail. However, we
have chosen to limit ourselves to the most common sectors which have been studied
extensively in the literature for an initial analysis.
We present basic processes relevant for the collider phenomenology of sensible
models of dynamical electroweak symmetry breaking. We have divided the processes
according to the spin of the new relevant composite particle which can be potentially
discovered at collider experiments.
Before launching in the actual predictions we highlight key features of the model
which will help understanding the search strategy and the result presented below.
When turning off the electroweak interactions some of the interesting decay modes,
depending on the parameters of the effective Lagrangian, involving spin one massive
vector states and composite scalars are:
V→ΠΠ,
A→HΠ,
H→ΠΠ,
(4.112)
with the appropriate charge assignments. We assumed in the equation above that the
composite Higgs is lighter than the vector states, however our effective theory allows
44
for a more general spectrum. Once the electroweak interactions are turned on the
technipions become the longitudinal components of the W and Z bosons. Therefore the
processes in (4.112) allow to detect the spin one resonances at colliders. The massive
spin one states mix with the SM gauge bosons. After diagonalizing the spin one mixing
matrices (see [19, 74]) the lightest and heaviest of the composite spin one triplets are
termed R1±,0 and R2±,0 respectively. In the region of parameter space where R1 is mostly
an axial-like vector (for a mass less than or about one TeV) and R2 mostly a vector state
one observes the following qualitative dependence of the couplings to the SM fields to
heavy vector bosons as function of the electroweak gauge coupling g and the heavy
vector self interaction coupling g̃:
gR1,2 f f¯ ∼
g
,
g̃
gR2 WW ∼ g̃ ,
gR1 HZ ∼ g̃ .
(4.113)
Notice that, since the heavy spin one states do not couple directly to SM fermions, the
couplings gR1,2 f f¯ arises solely from the mixing with W and Z. This coupling is roughly
proportional to g/ g̃.
We implemented the effective Lagrangian on event generators such as MadGraph
[123] and CalcHEP [125] in order to compare the predictions of the model for different
choices of its parameters with the experimental data that are and will be produced
at LHC. The implementation can be freely downloaded from the page http://cp3origins.dk/research/tc-tools, together with the calculator needed to change the parameters. The details of the implementation and how to use the different computer packages
are provided in Appendix B.
The analyses we are going to present are at the parton-level study, not accounting
for efficiencies and for some systematic uncertainties arising from detector effects.
4.1
Spin one processes: Decay widths and branching ratios
The existence of new heavy spin one resonances has been postulated in different
extensions of the SM. It is not hard to convince oneself that exploring the entire range of
the couplings and masses of the effective Lagrangian presented here one is able to cover
virtually the entire spectrum of models presented in the literature. The link to a specific
TC model can be obtained using the WSRs or the knowledge of the spectrum deduced
via first principle lattice simulations. In this subsection we consider different processes
where the spin one resonances appear in the intermediate diagrams in the s-channel
at the parton level. The event generators are then used to embed the parton process
within the proton-proton collision. We have also checked that there is no appreciable
difference in using MadGraph or CalcHEP, which can be taken as a direct test of the
validity of the two implementations.
In traditional TC models of running dynamics the enforcement of the first and
second WSRs leads to a spin one vector lighter than the axial one. An intriguing
possibility for walking theories [19, 61] is the possible mass spectrum inversion of the
45
MR±-MR± (TeV)
S=0.3
0.3
g̃=2,5
Mass Splitting
0.5
0.45
1
0.4
2
MV-MA (TeV)
0.5
0.2
S=0.3
0.4
0.35
g̃=2,5
0.3
0.25
0.1
0.2
0.15
0
0.1
-0.1
0.05
-0.2
0
0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2
0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2
MR± (TeV)
MA (TeV)
1
Figure 15: Mass splittings MV − MA (left) and MR±2 − MR±1 (right). The dotted lines are
for g̃ = 5 while the solid lines are for g̃ = 2.
vector and axial spin one mesons. This occurs since the second WSR gets modified
[61]. In Fig. 15 (left) we plot MV − MA as a function of MA for two reference values of g̃
and S = 0.3. For generic values of S the inversion occurs for
r
4π
inv
Fπ .
(4.114)
M =
S
This gives Minv ' 1.6 TeV for S = 0.3, as clearly shown in the plot. Fig. 15 (right)
(R±,0
) are the lighter (heavier) vector
shows MR±2 − MR±1 as a function of MR±1 , where R±,0
2
1
resonances, with tree-level electroweak corrections included. This mass difference is
always positive by definition, and the mass inversion becomes a kink in the plot. Away
from Minv R1 (R2 ) is an axial (vector) meson for MA < Minv , and a vector (axial) meson
for MA > Minv . The mass difference in Fig. 15 is proportional to g̃2 , and becomes
relatively small for g̃ = 2. The effects of the electroweak corrections are larger for small
e
g couplings. For example, the minimum of MR±2 − MR±1 is shifted from Minv ' 1.59 TeV
to about 1 TeV for g̃ = 2. To help the reader we plot in Fig. 16 the actual spectrum for
the vector boson masses versus MA . Preliminary studies on the lattice of MWT and the
mass inversion issue appeared in [83].
The widths of the heavy vectors are displayed in Fig. 17. The lighter meson, R1 , is
very narrow. The heavier meson, R2 , is very narrow for small values of g̃. In fact in
this case MR2 ' MR1 , forbidding decays of R2 to R1 (+anything). For large g̃, R2 is very
narrow for large masses, but then becomes broader when the R2 → R1 , X channels open
up, where X is a SM gauge boson. It becomes very broad when the R2 → 2R1 decay
channel opens up. The former are only important below the inversion point, where R1
is not too heavy. The latter is only possible when R2 is essentially a spin one vector and
MR2 > 2MR1 .
46
Mass Spectrum (TeV)
Mass Spectrum (TeV)
2.2
2
1.8
1.6
1.4
R2±,0
1.2
1
0.8
2.2
2
1.8
1.6
1.4
R2±,0
1.2
1
0.8
R1±,0
0.6
S=0.3
g̃=2
R1±,0
0.6
0.4
S=0.3
g̃=5
0.4
0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2
0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
MA (TeV)
2
2.2
MA (TeV)
K (GeV)
K (GeV)
Figure 16: The mass spectrum of the MR±,0 vector mesons versus MA for g̃ = 2 (left)
1,2
and g̃ = 5 (right). The masses of the charged vector mesons are denoted by solid lines,
while the masses of the neutral mesons are denoted by dashed lines.
S=0.3
g̃=2
10
2
10
10
3
10
2
S=0.3
g̃=5
±
R2
10
±
R1
±
R2
R1±
1
10
1
-1
10
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-1
2.2
0.6
0.8
1
1.2
1.4
10
S=0.3
g̃=2
10
2
10
1.8
2
2.2
Mass (TeV)
K (GeV)
K (GeV)
Mass (TeV)
3
1.6
10
3
10
2
S=0.3
g̃=5
R02
R01
10
R02
1
10
1
R01
-1
10
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
-1
0.6
Mass (TeV)
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Mass (TeV)
Figure 17: Decay width of the charged (first row) and neutral (second row) vector
mesons for S = 0.3 and g̃ = 2, 5. We take MH = 0.2 TeV, s = 0.
47
The narrowness of R1 (and R2 , when the R2 → R1 , X channels are forbidden) is
essentially due to the small value of the S parameter. In fact for S = 0 the trilinear
couplings of the vector mesons to two scalar fields of the strongly interacting sector
vanish. This can be understood as follows: the trilinear couplings with a vector
resonance contain a derivative of either the Higgs or the technipion, and this can only
come from r3 in Eq. (3.97). Since r3 = 0 implies S = 0, as Eqs. (3.89) and (3.88) show
explicitly, it follows that the decay width of R1 and R2 to two scalar fields vanishes as
S → 0. As a consequence, for S = 0 the vector meson decays to the longitudinal SM
bosons are highly suppressed, because the latter are nothing but the eaten technipions.
(The couplings to the SM bosons do not vanish exactly because of the mixing with the
spin one resonances.) A known scenario in which the widths of R1 and R2 are highly
suppressed is provided by the D-BESS model [84], where the spin one and the spin zero
resonances do not interact. Therefore, in D-BESS all couplings involving one or more
vector resonances and one or more scalar fields vanish, not just the trilinear coupling
with one vector field. The former scenario requires r2 = r3 = s = 0, the latter only
requires r3 = 0. A somewhat intermediate scenario is provided by custodial TC, in
which r2 = r3 = 0 but s , 0. Narrow spin one resonances seem to be a common feature
in various models of dynamical electroweak symmetry breaking. (see for example
[85]). Within our effective Lagrangian (3.97) this property is linked to having a small S
parameter.
The R1 branching ratios are shown in Fig. 18. The wild variations observed in the
plots around 1.6 TeV reflect the mass inversion discussed earlier. Here the mixing
e a , with a = 0, ±1, vanishes, suppressing the decay to SM fermions.
between Ra1 and W
The other observed structure for the decays in ZH and WH, at low masses, is due to
the opposite and competing contribution coming from the TC and electroweak sectors.
This is technically possible since the coupling of the massive vectors to the longitudinal
component of the gauge bosons and the composite Higgs is suppressed by the small
value of S.
Now we consider the R2 branching ratios displayed in Fig. 19. Being R2 heavier
than R1 by definition, new channels like R2 → 2R1 and R2 → R1 X show up, where X
denotes a SM boson. Notice that there is a qualitative difference in the R2 decay modes
for small and large values of g̃. First, for small g̃ the R2 − R1 mass splitting is not large
enough to allow the decays R2 → 2R1 and R2 → R1 H, which are instead present for
large g̃. Second, for small g̃ there is a wide range of masses for which the decays to
R1 and a SM vector boson are not possible, because of the small mass splitting. The
e
branching ratios to fermions do not drop at the inversion point, because the R2 − W
mixing does not vanish.
48
±
BR(R1)
±
BR(R1)
1
ff
nl
10
-1
tb
1
10
W±H
-1
ff
nl
W±Z
10
-2
10
-2
tb
W±Z
10
10
10
-3
W±H
10
-4
10
-5
10
-3
-4
-5
S=0.3
S=0.3
g̃=2
10
g̃=5
-6
10
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-6
2.2
0.6
0.8
1
1.2
1.4
1.6
1
10
0
-1
ff
ZH
+
-
W W
10
10
10
10
ZH
10
-1
ff
ll
-2
10
-3
10
-4
10
-5
10
ll
-2
-3
W+W-
-4
-5
S=0.3
g̃=2
g̃=5
-6
10
0.6
0.8
1
1.2
1.4
1.6
2.2
1
S=0.3
10
2
Mass (TeV)
BR(R1)
0
BR(R1)
Mass (TeV)
1.8
1.8
2
2.2
-6
0.6
Mass (TeV)
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Mass (TeV)
Figure 18: Branching ratios of the charged (first row) and neutral (second row) R1
resonance for S = 0.3 and g̃ = 2, 5. We take MH = 0.2 TeV, s = 0.
49
10
10
10
10
±
BR(R2)
±
BR(R2)
1
ff
nl
-1
tb
1
10
W±Z
-2
10
-3
±
10
W H
-4
10
±
Z R1
10
-5
10
W± R0
1
-1
-2
±
R1
R0
1
ff
nl
tb
-3
±
R1 H
-4
-5
W±H
S=0.3
10
W± R0
1
-6
0.8
1
S=0.3
g̃=2
g̃=5
10
0.6
W±Z
±
Z R1
1.2
1.4
1.6
1.8
2
-6
2.2
0.6
0.8
1
1.2
1.4
1.6
1
0
ff
10
ll
-1
W+W-
1.8
2
2.2
Mass (TeV)
BR(R2)
0
BR(R2)
Mass (TeV)
1
10
-1
WR1
R+
1 R1
ff
10
-2
10
-2
ll
ZH
10
-3
10
WR1
-3
0
R1H
10
10
-4
10
-5
10
+
W W
-5
ZH
S=0.3
S=0.3
g̃=2
10
-6
10
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-
-4
g̃=5
-6
2.2
0.6
0.8
1
1.2
Mass (TeV)
1.4
1.6
1.8
2
2.2
Mass (TeV)
Figure 19: Branching ratios of the charged (first row) and neutral (second row) R2
resonance for S = 0.3 and g̃ = 2, 5 . We take MH = 0.2 TeV, s = 0.
`+
q
R01,2
`−
q̄
Figure 20: Feynman diagram of the signal processes for the dilepton production.
50
4.1.1 pp → R → ``
√
Dilepton production was discussed also in [74], with s = 14 TeV and 100 fb−1
integrated luminosity. The Feynman diagram of this Drell-Yan process is shown
√ in
Fig. 20. We updated that analysis for the near future LHC using the parameters s = 7
TeV and 10 fb−1 . The signal and the background are obviously reduced compared to
the earlier studies, but in the optimal region of the parameter space signals are still
clearly visible. Increasing the effective TC coupling g̃ quickly flattens out the signal. In
Fig. 21 we plot the number of events with respect to the invariant mass of the lepton
pair, using g̃ = 2, 3, 4 and MA = 0.5, 1, 1.5 TeV, where MA is the mass of axial eigenstate
before mixing with the SM gauge bosons. We have applied cuts of |η` | < 2.5 and p`T > 15
GeV on the rapidity and transverse momentum of the leptons. The peaks from the R1
and R2 clearly stand out with signal-to-background ratio S/B > 10 for several bins
over the parameter space under consideration. The background is considered to be
the contribution coming from the SM gauge bosons Z and γ. In Table 2 the signal and
background cross sections are reported, applying the cut
Table 2: pp → R1,2 → `+ `− . Signal and background cross sections for g̃ = 2, 3, 4 and
estimates for required luminosity for 3σ and 5σ signals. MR1,2 are the physical masses
for the vector resonances in GeV.
g̃
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
MA
500
500
1000
1000
1500
1500
500
500
1000
1000
1500
1500
500
500
1000
1000
1500
1500
MR1,2
M1 = 517
M2 = 623
M1 = 1027
M2 = 1083
M1 = 1526
M2 = 2103
M1 = 507
M2 = 715
M1 = 1013
M2 = 1097
M1 = 1514
M2 = 1541
M1 = 504
M2 = 836
M1 = 1007
M2 = 1148
M1 = 1509
M2 = 1533
σS (fb)
194
118
4.57
16.4
0.133
0.776
93.5
0.447
1.32
2.94
3.19 · 10−3
0.120
34.6
0.0
0.234
0.0
1.31 · 10−3
1.43 · 10−2
σB (fb)
3.43
1.34
9.17 · 10−2
5.60 · 10−2
5.91 · 10−3
2.81 · 10−3
3.71
0.649
8.81 · 10−2
5.15 · 10−2
5.63 · 10−3
3.94 · 10−3
3.85
0.649
8.98 · 10−2
5.15 · 10−2
3.94 · 10−3
3.94 · 10−3
51
L (fb−1 ) for 3σ
0.012
0.019
0.53
0.13
26
2.7
0.037
39
2.7
0.79
6300
29
0.12
30
25000
435
L (fb−1 ) for 5σ
0.038
0.056
1.8
0.39
67
8.2
0.090
81
7.4
2.5
14000
68
0.34
85
57000
1200
-1
10
2
Number of events/20 GeV @ 10 fb
3
10
1
10
10
-1
-2
500
1000
1500
Mll(GeV)
Number of events/20 GeV @ 10 fb-1
Number of events/20 GeV @ 10 fb-1
10
10
3
10
2
10
3
10
2
10
1
10
10
-1
-2
500
1000
1500
Mll(GeV)
10
1
10
10
-1
-2
500
1000
1500
Mll(GeV)
Figure 21: Dilepton invariant mass distribution M`` for pp → R1,2 → `+ `− signal and
background processes. We consider g̃ = 2, 3, 4 respectively from left to right and masses
MA = 0.5 TeV (purple), MA = 1 TeV (red) and MA = 1.5 TeV (green).
52
|M`` − MR | < 5 GeV
(4.115)
separately for the R1 and R2 peaks. The choice of the value 5 GeV is dictated by the
dilepton invariant mass resolution [86]. The invariant mass resolution drops when
the mass of the resonance increases, in any event, we use the same cut for all of the
different mass values coming from the one with worst resolution. Also estimates for
the required integrated luminosity for the 3σ and 5σ discoveries are given in the Table.
The significance is defined as the number of signal events divided by the square root
of the number of background events, when the number of events is large. The Poisson
distribution is used for the small event samples. The dilepton final state should be
clearly visible at the LHC in this particular region of the parameter space already with
1 fb−1 integrated luminosity.
4.1.2 pp → R → WZ → ```ν
`+
q
R+(−)
1,2
Z
W +(−)
`−
`+ (ν̄)
ν (`− )
q̄
Figure 22: Feynman diagram for the process pp → R± → WZ± → ```ν.
The final state signature with three leptons and missing energy arises from the
process pp → R → WZ → ```ν (see Fig. 22), where ` denotes a muon or an electron
√
and ν denotes the corresponding neutrino. This was also studied in [74], with s = 14
TeV and 100 fb−1 , where it was shown to be a promising signature for higher values
of g̃ and MA . The technivector-fermion couplings are suppressed for large g̃, which
makes the dilepton final state uninteresting in that region of the parameter space. In
contrast, the technivector coupling to SM vector bosons is enhanced for large values
g̃, balancing the suppression coming from the quark couplings. This can be seen from
Fig. 23, where the second peak begins to go down slowly with increasing g̃. Following
[74], we have used the transverse mass variable
(MT3` )2
q
= [ M2 (```) + p2T (```) + |p/T |]2 − |~
pT (```) + ~p/T |2 ,
(4.116)
where p/T denotes the missing transverse momentum. The cuts for the leptons are
applied as in the previous subsection and in addition we impose a cut on the missing
/ T > 15 GeV. As a background we consider the SM processes with
transverse energy E
±
R1,2 replaced by the W ± .
53
10
10
-1
-2
400
600
800
1000
1200
T
M3l
(GeV)
Number of events/20 GeV @ 10 fb-1
1
10
1
10
10
-1
-2
400
600
800
1000
1200
T
M3l
(GeV)
Number of events/20 GeV @ 10 fb
-1
Number of events/20 GeV @ 10 fb-1
10
10
1
10
10
-1
-2
400
600
800
1000
1200
T
M3l
(GeV)
Figure 23: Transverse mass distribution MT3` for pp → R±1,2 → ZW ± → ```ν signal and
background processes. We consider g̃ = 2, 3, 4 respectively from left to right and masses
MA = 0.5 TeV (green), MA = 1 TeV (red).
54
In Table 3 we present the signal and background cross sections and quantify the
possible observability of the signal.
Table 3: pp → R±1,2 → ZW ± → ```ν. Signal and background cross sections for g̃ = 2, 3, 4
and estimates for required luminosity for 3σ and 5σ signals. Cuts are applied to take
into account only the region of the visible peak. MR1,2 are the physical masses (GeV).
g̃
2
2
3
3
4
4
MA
500
1000
500
1000
500
1000
MR1
512
1012
505
1009
503
1005
4.1.3 pp → R → `ν
MR2
619
1081
713
1094
835
1145
σB (fb)
1.75
7.34 · 10−2
1.12
0.117
0.659
8.92 · 10−2
σS (fb)
1.12
0.545
0.340
1.10
2.15 · 10−2
0.297
L (fb−1 ) for 3σ
16
8.4
89
3.7
10000
21
L (fb−1 ) for 5σ
35
24
243
11
60
`+ (ν̄)
q
R+(−)
1,2
ν (`− )
q̄
Figure 24: Feynman diagram for the process pp → R± → `ν.
This is the third signature, with leptons in the final state, studied in [74]. The essential complication compared to the `+ `− final state is that the longitudinal component of
the neutrino momentum cannot be measured. The cuts applied for the charged lepton
/ T > 15 GeV. In Fig. 25, the
and on the missing energy are |η` | < 2.5, p`T > 15 GeV and E
number of events is plotted with respect to the transverse mass variable
q
T
/ T pT (`) (1 − cos ∆φ`,ν ) ,
M` = 2 E
(4.117)
using the parameter space points g̃ = 2, 3, 4 and MA = 0.5, 1, 1.5 TeV. This final state
signature behaves like the dilepton final state. Due to the suppression of the fermion
couplings, the signal is reduced when g̃ is increased.
55
10
2
Number of events/20 GeV @ 10 fb-1
3
10
1
10
-1
500
1000
1500
MTl(GeV)
Number of events/20 GeV @ 10 fb-1
Number of events/20 GeV @ 10 fb-1
10
10
3
10
2
10
3
10
2
10
1
10
-1
500
1000
1500
MTl(GeV)
10
1
10
-1
500
1000
1500
MTl(GeV)
Figure 25: Transverse mass distribution MT` for pp → R±1,2 → `ν signal and background
processes. We consider g̃ = 2, 3, 4 respectively from left to right and masses MA = 0.5
TeV (purple), MA = 1 TeV (red) and MA = 1.5 TeV (green).
56
4.1.4 pp → R → jj
The tree-level MWT contributions to this process is the s-channel exchange of a
composite vector boson (Drell-Yan process, Fig. 26).
q̄ f
qi
R1,2
qf
q̄i
Figure 26: Feynman diagram for the Drell-Yan process pp → R → jj.
The quark couplings to the heavy spin one states are generated by the mixing with
the SM gauge bosons. Their value can then be expanded perturbatively in powers of
g/ g̃, where g is the SU(2)L coupling constant, for large g̃. Therefore the cross section
for this process is reduced by, at least, a factor of g̃−4 , respect to the SM one. Therefore
the signal drowns in the large QCD background for this process.
4.1.5 pp → R → γV and pp → R → ZZ
Up to dimension four operators, and before including operators involving the
Lorentz µνρσ invariant tensor [12, 81], the photon coupling γR+1 W − is forbidden. This
implies that we do not have diagrams with technivectors decaying into photon and
vector boson. Higher order operators as well as operators involving the µνρσ in a
coherent manner will be investigated in the near future. The pp → R → ZZ processes
are also absent in the present effective description.
4.2
Ultimate LHC reach for heavy spin resonances
We conclude this subsection summarizing the results for the ultimate reach of LHC,
i.e. 14 TeV and 100 fb−1 [74]. We consider the representative parameter space points
g̃ = 2, 5 and MA = 0.5, 1, 1.5, 2 TeV for our plots and discussion.
4.2.1
pp → R → 2`, `ν, 3`ν at 14 TeV
The invariant mass and transverse mass distributions for signatures pp → R →
2`, `ν, 3`ν are shown in Figs. 27-29. In the left frames of Figs. 27 and 28, corresponding
to g̃ = 2, clear signals from the leptonic decays of R01,2 and R±1,2 are seen even for 2 TeV
resonances. Moreover Fig. 27 demonstrates that for g̃ = 2 both peaks from R01 and R02
may be resolved.
57
4
10
3
10
2
S=0.3
g̃=2
10
1
500
1000
1500
2000
Mll (GeV)
Number of events/20 GeV @ 100 fb-1
Number of events/20 GeV @ 100 fb-1
10
10
4
10
3
10
2
S=0.3
g̃=5
10
1
500
1000
1500
2000
Mll (GeV)
10
5
10
4
10
3
10
S=0.3
g̃=2
2
10
1
500
1000
1500
2000
MTl
(GeV)
Number of events/20 GeV @ 100 fb-1
Number of events/20 GeV @ 100 fb-1
Figure 27: Dilepton invariant mass distribution M`` for pp → R01,2 → `+ `− signal and
background processes. We consider g̃ = 2, 5 respectively and masses MA = 0.5 TeV
(purple), MA = 1 TeV (red), MA = 1.5 TeV (green) and MA = 2 TeV (blue).
10
4
10
3
10
2
S=0.3
g̃=5
10
1
500
1000
1500
2000
MTl
(GeV)
Figure 28: MT` mass distribution for pp → R±1,2 → `± ν signal and background processes.
We consider g̃ = 2, 5 respectively and masses MA = 0.5 TeV (purple), MA = 1 TeV (red),
MA = 1.5 TeV (green) and MA = 2 TeV (blue).
58
S=0.3
g̃=2
2
10
1
10
-1
500
1000
1500
2000
Number of events/20 GeV @ 100 fb-1
Number of events/20 GeV @ 100 fb-1
10
10
S=0.3
g̃=5
2
10
1
10
-1
500
T
M3l
(GeV)
1000
1500
2000
T
M3l
(GeV)
Figure 29: MT3` mass distribution for pp → R±1,2 → ZW ± → 3`ν signal and background
processes. We consider g̃ = 2, 5 respectively and masses MA = 0.5 TeV (purple), MA = 1
TeV (red), MA = 1.5 TeV (green) and MA = 2 TeV (blue).
Let us now turn to the case of g̃ = 5 in the right frames of Figs. 27 and 28. For
large g̃ the R f f couplings are suppressed, so observing signatures pp → R → 2` and
pp → R → `ν could be problematic. However, for large g̃, the triple-vector coupling
is enhanced, so one can observe a clear signal in the MT3` distribution presented in
Fig. 29. At low masses the decays of the heavy vector mesons to SM gauge bosons are
suppressed and the signal disappears. This mass range can, however, be covered with
signatures pp → R → 2` and pp → R → 3`ν.
4.2.2
pp → Rjj at 14 TeV
qf
qi
W
R1,2
Z, W
q0i
q0f
Figure 30: Feynman diagram for the vector boson fusion process pp → Rjj.
Vector Boson Fusion (VBF) is, in principle, an interesting channel for the vector
meson production (see Fig. 30), especially in theories in which the fermion couplings
of the vector resonances are suppressed. Unfortunately, as we shall see, it is not
an appealing channel to explore at the LHC due to a small production rate of R±,0
.
1,2
59
Although the coupling between three vector particles increases with g̃ it is hard to
produce R±,0
on-shell.
1,2
Following [74], to be specific, we consider VBF production of the charged R1 and
j
R2 vectors impose the following kinematical cuts on the jet transverse momentum pT ,
energy E j , and rapidity gap ∆η j j , as well as rapidity acceptance |η j | [87, 88]:
|η j | < 4.5 ,
j
pT > 30 GeV ,
E j > 300 GeV ,
∆η j j > 4 .
(4.118)
10
10
10
10
10
10
10
m(pb)
m(pb)
The VBF production cross section for the charged R1 and R2 vector resonances is
-1
-2
±
R2
10
-3
10
-4
R1±
10
-5
10
-6
10
S=0.3
-7
0.4
10
g̃=2
10
0.6
0.8
1
1.2
1.4
1.6
1.8
-1
-2
±
R2
-3
-4
-5
R1±
-6
S=0.3
-7
2
g̃=5
0.6
Mass (TeV)
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Mass (TeV)
Figure 31: Vector boson fusion production cross sections for the R±1,2 resonances, with
j
S = 0.3 and g̃ = 2, 5. The jet cuts are: |η j | < 4.5, pT > 30 GeV, E j > 300 GeV, ∆η j j > 4. See
text for details.
shown in Fig. 31 for the set of cuts given by Eq. (4.118). An interesting feature of
the VBF production is the observed crossover around the mass degeneracy point for
e
g = 5. This is a direct consequence of the fact that the R1,2 resonances switch their
vector/axial nature at the inversion point. For smal g̃ the crossover does not occur
due to the interplay between the electroweak and the TC corrections. In D-BESS VBF
processes are not very relevant, since there are no direct interactions between the heavy
mesons and the SM vectors. However in fermiophobic Higgsless models VBF is the
main production channel of the heavy resonances. Since the production rate of R±1,2 is
below 1 fb, VBF is not a promising channel at the LHC.
4.3
Composite Higgs phenomenology
The composite Higgs phenomenology is interesting due to its interactions with the
new massive vector bosons and their mixing with SM gauge bosons. We first analyze
60
1
BR(H A 2*X )
gHWW/gHWWSM
S=0.3
s=+1
s=0
s=-1
g̃=2
0.95
g̃=5
1
–
bb
ZZ
10
-1
oo
–
10
–
tt
-2
0.8
0.75
400
–
cc
0.9
0.85
W +W -
aa
10
600
800
1000
1200
1400
1600
1800
2000
2200
MA (GeV)
-3
100
200
300
400
MH(GeV)
Figure 32: Left: gHWW /gSM
ratio as a function of MA . The behavior of the gHZZ
HWW
coupling is identical to the gHWW one. Results are presented for S = 0.3, g̃ = 2, 5 (solid
and dashed lines respectively), and for s = (+1, 0, −1) (black, blue and green colors
respectively). Right: branching ratios of the composite Higgs as function MH .
the Higgs coupling to the W and Z gauge bosons. In Fig. 32 we present the gHWW /gSM
HWW
ratio as a function of MA . The behavior of the gHZZ and gHWW couplings are identical.
We keep fixed S = 0.3 and consider two values of g̃, 2 (solid line) and 5 (dashed line).
We repeat the plots for three choices of the s parameter (+1, 0, −1) depicted in black,
blue and green colors respectively. The deviation of gHWW from gSM
increases with
HWW
MA due to the fact that we hold the S parameter fixed. One reaches deviations of 20%
from the SM couplings when MA ' 2 TeV. This is reflected in the small deviations of
the Higgs branching ratios when compared with the SM ones.
4.3.1 pp → WH and pp → ZH
The presence of the heavy vectors is prominent in the associate production of the
composite Higgs with SM vector bosons, as first pointed out in [89]. Parton level
Feynman diagrams for the pp → WH and pp → ZH processes are shown in Fig. 33
(left) and Fig. 33 (right) respectively. The resonant production of heavy vectors can
enhance HW and ZH production by a factor 10 as one can see in Fig. 34 (right). This
enhancement occurs for low values of the vector meson mass and large values of g̃.
61
q
q
H
H
γ, Z, R01,2
W ± , R±1,2
q̄0
q̄
W±
Z
1
m(pp AWH)(pb)
m(pp AWH)(pb)
Figure 33: Feynman diagrams for the composite Higgs production in association with
SM gauge bosons.
~
MH=200 GeV, S=0.3, g=3
sm
s=-1
s=0
s=+1
10
-1
1
~
MH=200 GeV, S=0.3, g=6
sm
s=-1
s=0
s=+1
10
500
1000
1500
2000
-1
500
MA(GeV)
1000
1500
2000
MA(GeV)
Figure 34: The cross section for pp → WH production at 7 TeV in the center of mass
(W + H and W − H modes are summed up) versus MA for S = 0.3, s = (+1, 0, 1) and g̃ = 3
(left) and g̃ = 6 (right). The dotted line at the bottom indicates the SM cross section
level.
62
m(pp A Hjj) (pb)
m(pp A Hjj) (pb)
0.14
0.12
0.1
0.08
MH = 200 GeV
0.06
g̃ = 2
MH = 400 GeV
g̃ = 2
g̃ = 5
0.1
SM
0.08
0.04
SM
0.02
0
0.12
0.06
g̃ = 5
0.04
0.14
0.02
500
1000
1500
0
2000
500
MA (GeV)
1000
1500
2000
MA (GeV)
Figure 35: Composite Higgs production cross section via the VBF mechanism at 7 TeV.
4.3.2
Higgs vector boson fusion, pp → H jj
We have also analyzed the composite Higgs production in VBF processes pp → H jj.
We find that it is not enhanced with respect to the corresponding process in the SM as
it is clear from Fig. 35. The behavior of σ(pp → H jj) as function of MA traces the one of
the Higgs-gauge bosons coupling shown in Fig. 32.
We used the same kinematic cuts as for the pp → Rjj section except for the transverse
j
momenta and energy of the jets that we take to be pT > 15 GeV and E j > 150 GeV.
4.3.3 H → γγ and H → gg
γ, g
H
γ, g
Figure 36: Feynman diagram for the composite Higgs decay to γγ or gg via SM fermion
loop.
The one loop Higgs decay (see Fig. 36) width to two photons for any weakly
interacting elementary particle contributing to this process can be neatly summarized
via the formula [90]:
63
2
α2 GF M2H X
Γ(H → γγ) =
ni Q2i Fi ,
√
128 2π3 i
(4.119)
where i runs over the spins, ni is the multiplicity of each species with electric charge Qi
in units of e. The Fi functions are given by
F1 = 2 + 3τ + 3τ(2 − τ) f (τ) , F1/2 = −2τ[1 + (1 − τ) f (τ)] , F0 = τ 1 − τ f (τ) , (4.120)
where τ =
4m2i
M2H
and
q 2
arcsin τ1 ,
f (τ) =
2
√
− 41 log 1+ √1−τ − iπ ,
1− 1−τ
if τ ≥ 1
(4.121)
if τ < 1
The lower index of the function F indicates the spin of each particle contributing to the
process. It is clear from this formula that there can be strong interferences between
the different terms contributing. In particular we anticipate that there is an important
interplay between the SM gauge bosons and the TC contribution for this process.
The contribution to the Higgs decay to two-photon process in the SM is due to
the Ws and SM fermion loops contributions (with the leading one given by the top
quark). In fact it is an excellent approximation to use just the top in the fermion loop,
since the ratio between the corresponding result and the one obtained including all the
contributions is around 0.98, for the reference value of the Higgs mass of MH ' 120 GeV.
The approximation improves as the Higgs mass increases. We consider here the case in
which the Higgs is a composite state constituted of some more fundamental matter as
in TC. The SM Higgs sector Lagrangian is therefore interpreted simply as a low energy
effective theory. We assume the Vanilla TC theory and therefore the composite Higgs
(H) effective field coincides with the SM one. It is straightforward to generalize the
global symmetries of the TC theories to large symmetry groups.
The contribution to the sought process is modeled by re-coupling, in a minimal way,
the composite Higgs to the techniquarks Q via the following operator:
LQH
√ MQ t
t
∗
QL · HDRt + QL · (i τ2 H )URt + h.c. ,
= 2
v
(4.122)
where MQ is the dynamical mass of the techniquark and t = 1, . . . , d[r] is the technicolor index and d[r] the dimension of the representation under which the techniquarks
transform. This type of hybrid models have been employed many times in the QCD
literature to extra dynamical information and predictions for different phenomenologically relevant processes. A close example is the σ(600) decay into two photons [91]
investigated using, for example, hybrid type models in [92] and [93].
64
We assumed a single TC doublet with respect to the Weak interactions as predicted
by minimal models of TC [16]. We added a new Lepton family to cure the Witten
topological anomaly (with respect to the weak interactions) [57] when the TC sector
features an odd number of doublets charged under the SU(2)L Weak interactions. This
occurs when the dimension of the TC matter representation is an odd number. A
further Leptonic interaction term is then added:
LLH =
√ MN
√ ME
2
LL · HER + 2
LL · (i τ2 H∗ )NR + h.c. ,
v
v
(4.123)
with ME and MN the fermion masses expected to be of the order of the electroweak
scale [94]. The detailed spectrum is partially dictated by the precision electroweak
constraints.
The techniquark dynamical mass is intrinsically linked to the technipion decay
√ π . To achieve the correct values of
constant Fπ via the Pagels-Stokar formula MQ ≈ 2πF
d[r]
the weak gauge bosons the decay constant is related to the electroweak scale via:
q
v = N f Fπ ,
with
v = 246 GeV .
(4.124)
We have checked that the numerical value of the contribution of the ME/N states is
negligible when the masses are of the order of the electroweak scale. We have used as
a reference value for these lepton masses 500 GeV 10 .
At this point we determine the contribution of the techniquarks, not carrying ordinary color charge, and new leptons for the diphoton process by evaluating the naive
one loop triangle diagrams in which the composite Higgs to two techniquarks, as well
as the new leptons, vertex can be read off from the Lagrangian above. The contribution
at the amplitude level for each new particle is identical to the one of a massive SM
fermion provided one takes into account the correct electric charge and degeneracy
factor.
Although we adopted a simple model computation we now argue that the final
geometric dependence on the gauge structure, matter representation and number of
techniflavors of the composite theory is quite general.
We plot in Fig. 37 the intrinsic dependence on the dimension of the TC matter
representation d[r] raccording to the ratio:
R=
ΓSM (H → γγ) − ΓTC (H → γγ)
.
ΓSM (H → γγ)
(4.125)
For any odd representation we included the lepton contribution and therefore we
could not simply join the points. Interestingly when d[r] is around 7 the contribution
from the techniquarks adds to one of the SM fermion ones (i.e. mainly the top) and
cancels the one due to the W’s leading to ΓTC ≈ 0. The techniquark matter dominates
the process for d[r] larger than around seven. A similar behavior occurs when we
65
1.0
R
0.5
0.0
-0.5
MH =120 GeV
-1.0
0
5
10
d@rD
15
20
Figure 37: Difference of the Walking TC and SM decay widths divided by the SM width
plotted with respect to number of colors. MH = 120 GeV.
1
0
R
-1
-2
-3
MH =120 GeV
-4
-5
0
2
4
6
8
10
Nf
Figure 38: Difference of the Walking TC and SM decay widths divided by the SM width
plotted with respect to number of flavors. MH = 120 GeV.
66
increase the number of flavors as shown in Fig. 38. The dependence on the Higgs mass
with d[r] = 2 and N f = 2 is shown in Fig. 39. The WW threshold is evident in the plot
0.40
R
0.35
0.30
0.25
120
140
160
mH
180
200
Figure 39: Dependence of R as function of the Higgs mass MH for a reference value of
the techniquark matter representation d[r] = 2 and two number of Dirac techniflavors.
which shows how the ratio decreases with the Higgs mass.
To illustrate our results we plot in Fig. 40 the Higgs to two-photon branching ratio
as function of the Higgs mass for the MWT model and compare it to the SM one. It is
10-1
BRHH->ΓΓL
SM
MWT
10-2
10-3
10-4
10-5
100
120
140
160
180
200
mH
Figure 40: Higgs to two photons branching ratio in MWT as function of the composite
Higgs mass versus the SM result.
clear from the plot that we expect sizable corrections to the SM Higgs to two photons
process to be visible at the LHC.
10
For ME/N MH the contribution approaches a constant due to the functional form of F1/2 .
67
It is interesting to estimate the impact on the digluon process coming from a potentially composite Higgs made by technimatter charged under QCD interactions. This
contribution can be estimated, as we have done for the diphoton process, by considering
the one loop contribution of the techniquarks arriving at:
2
α2s GF M3H X SM X
Γ(H → gg) =
Fi +
2T[r̃ j ]d[r j ]FTC
,
√
j
3
64 2π
i
j
(4.126)
where T[r̃ j ] is the index of the representation of the techniquarks under the SM color
group SU(3) and d[r j ] is the dimension of the representation under the TC gauge group.
For techniquarks transforming according to the fundamental representation of QCD
120
100
Rg
80
60
40
MH =120 GeV
20
0
2.0
2.5
3.0
3.5
d@rD
4.0
4.5
5.0
Figure 41: Difference of the Walking TC and SM decay widths divided by the SM width
plotted with respect to number of flavors. MH = 120 GeV.
the two gluon decay width of the Higgs is strongly enhanced. This is presented in
Fig. 41 where we have plotted the ratio
Rg =
ΓTC (H → gg) − ΓSM (H → gg)
.
ΓSM (H → gg)
(4.127)
as a function of the dimension of the representation under the TC gauge group. The
composite Higgs to two-gluon decay width increases monotonically with respect to
the number of technicolors in contrast with the diphoton decay width. The difference
resides in the fact that gluons do not couple directly to the other gauge bosons of the
SM.
We now estimate the viability of the models with colored techniquarks by comparing
our results with the study performed in [95]. Here the authors considered the effects of
the fourth fermion SM generation to the process σ gg→H × BR(H → W + W − ). The fourth
68
2.4
30
2.2
d@rD
2.0
1.8
20
1.6
1.4
1.2
1.0
120
10
140
160
mH
180
200
Figure 42: Contours in the d[r] versus Higgs mass plane for fixed values of the enhancement of the gg → H production cross section compared to the SM one. Obviously, in
practice, the dimension of the representation cannot be taken to be continuous and
the plot simply shows that already for d[r] around 2 (which is the minimal nontrivial
matter group-theoretical dimension) the enhancement factor is around 30.
generation quarks enhances the gg → H production cross section by a factor of 9 to
7.5 in the Higgs mass range MH = 110 − 300 GeV with respect to the SM result. This
enhancement leads to a 95% confidence level exclusion of the Higgs in the mass range
MH = 131 − 208 GeV or of the presence of a fourth generation if the Higgs turns out
to be within this mass energy range. Colored techniquakrs are further penalized with
respect to the SM fourth generation of quarks since they carry also technicolor leading
to a further dramatic enhancement of this process. In Fig. 42 we have plotted contours
in the dimension of the techniquark representation versus the Higgs mass when the
gg → H production cross section is enhanced by factors 10, 20 and 30 respectively
compared to the SM one. Obviously, in practice, the dimension of the representation
cannot be taken to be continuous and the plot simply shows that already for d[r]
around two (which is the minimal nontrivial matter group-theoretical dimension) the
enhancement factor is around 30. Thus we conclude that this process alone strongly
indicates that viable TC models should feature colorless techniquarks as it is the case of
MWT or that a light composite Higgs made of colored techniquarks is excluded by the
Tevatron experiment. These results are in perfect agreement with the LEP electroweak
precision data which require a small S parameter.
69
In these two last subsections we introduced a simple framework to estimate the
composite Higgs boson coupling to two photons and gluons for generic TC extensions
of the SM. By comparing the decay rates with the SM ones for the diphoton case we
show that the corrections are typically of order one and therefore this process can
be efficiently used to disentangle a light composite Higgs from a SM one. We then
turned our attention to the composite Higgs to two gluon process and showed that the
Tevatron results for the gluon-gluon fusion production of the Higgs exclude either the
techniquarks to carry color charges up to the 95% confidence level or the existence of a
light composite Higgs made of colored techniquarks.
4.3.4
Higgs production via gluon fusion
As we have shown above if the techniquarks do not carry color this process occurs
as in the SM, provided that the ETC interactions do not strongly modify the Yukawa
sector.
5
A Natural fourth family of leptons at the TeV scale
If the LHC discovers a new fourth family of leptons but no associated quarks, what
would that imply? Either the associated quarks are much heavier than the electroweak
scale or a new set of fermions are needed to account for the 4th family lepton induced
gauge anomalies. The new fermions could address directly the big hierarchy problem
of the SM if their dynamics leads to a composite Higgs scenario. This is the setup we
are going to discuss in this chapter, using as a template MWT [16, 94].
Before investigating the phenomenology of such a theory, we will first review the
leptonic sector of the SM (see e.g. [96] for a recent review). Secondly, we will consider
a general mass structure for the fourth neutrino and both the case of mixing and no
mixing with the SM neutrinos. We will also summarize the (in)direct phenomenological
constraints. We then analyze the interplay of the composite Higgs sector with the new
lepton family at the LHC.
We will study the production and decay of the new leptons in proton - proton
collisions which is relevant to select the LHC signatures for the discovery of these new
leptons. We show that the composite Higgs structure can affect and differentiate the
final signatures with respect to the case in which the Higgs is elementary. The bottom
line is that one can experimentally determine if the fourth family is associated to a
composite Higgs sector.
70
5.1
The Standard Model leptons: a Mini-review
In the SM, the three lepton families, ` = e, µ, τ, belong to the following representations of the gauge group SU(3)C × SU(2)L × U(1)Y :
L` = (ν` L `L )T ∼ (1, 2, −1/2)
,
`R ∼ (1, 1, −1)
,
(5.128)
where the chirality projectors PL = (1 − γ5 )/2 and PR = (1 + γ5 )/2 have been introduced
and the relation Q = T3 + Y has been adopted in order to define the hypercharge with
respect to the electric charge. The neutral and charged current interactions of the SM
leptons are then respectively accounted for by the Lagrangian terms:
g
1¯ µ
1
2
µ
µ
¯
¯ µ ` Aµ
LNC =
ν̄L γ νL − `L γ `L + sin θW `γ ` Zµ + e `γ
cos θW 2
2
(5.129)
g
LCC = √ `¯L γµ νL Wµ− + h.c.
2
where ` = `L + `R and θW is the Weinberg angle.
In the SM, due to the absence of gauge singlet fermions with R-chirality, it is not
possible to endow neutrinos with a Dirac-type mass term. An effective Majoranatype mass term can however be identifyed among the non-renormalizable operators of
dimension 5. Symbolically:
(LH)T (LH)
(5.130)
Λ
where Λ represents the energy scale at which additional interactions and/or fields show
up, which must violate total lepton number (the effective mass operator above violates
it by two units). The associated effective neutrino masses would thus be suppressed
with respect to the electroweak scale, v, by a factor O(v/Λ). Experiments on neutrino
oscillations, which measure differences of squared masses and mixing angles, have
estabished that at least two of the SM neutrinos have a small mass, not larger than the
eV scale [64]. This would indeed suggest a very high scale for lepton number violation,
Λ ∼ 1011 v.
The neutrino mass matrix is diagonalised in general by a 3 × 3 unitary matrix U,
called the PMNS mixing matrix:
mν = U∗ mdiag U† .
(5.131)
Since U and mdiag are in general complex, neutrino masses can provide three CP violating phases: the one in U (which is the analog of the CP violating phase in the CKM
quark mixing matrix) plus the two phase differences in mdiag , which are denoted as
Majorana phases.
Two distinct oscillation frequencies have been first measured in solar and atmospheric neutrino oscillations and later confirmed by experiments on earth. Two well
71
separated differences need at least three different neutrino mass eigenstates involved
in oscillations so that the three known neutrino species can be sufficient. Then at least
two neutrinos must be massive while, in principle, the third one could still be massless.
The mass eigenstates involved in solar oscillations are m1 and m2 and, by definition,
|m2 | > |m1 |, so that ∆m2sun = |m2 |2 − |m1 |2 > 0. The atmospheric neutrino oscillations
involve m3 : ∆m2atm = |∆m231 | with ∆m231 = |m3 |2 − |m1 |2 either positive (normal hierarchy)
or negative (inverse hierarchy). The present data are still compatible with both cases.
The degenerate spectrum occurs when the average absolute value of the masses is
much larger than all mass squared differences. With the standard set of notations and
definitions the present data are summarised by [64]
∆m2sun = (7.05 − 8.34) × 10−5 eV2 , ∆m2atm = (2.07 − 2.75) × 10−3 eV2 ,
0.25 ≤ sin2 θ12 ≤ 0.37 , 0.36 ≤ sin2 θ23 ≤ 0.67 .
(5.132)
In addition, sin2 θ13 < 0.035 at 90% C.L. The pattern of neutrino mixing is drastically
different from the pattern of quark mixing. At present no experimental information on
the Dirac and Majorana CP violation phases in the neutrino mixing matrix is available.
Oscillation experiments do not provide information about the absolute neutrino
mass scale. Limits on that are obtained from the endpoint of the tritium beta decay
spectrum, from cosmology and from neutrino-less double beta decay. From tritium we
have an absolute upper limit of 2.2 eV (at 95% C.L.) on the mass of electron antineutrino,
which, combined with the observed oscillation frequencies under the assumption of
three CPT-invariant light neutrinos, also amounts to an upper bound on the masses of
the other active neutrinos.
Oscillation experiments cannot distinguish between Dirac and Majorana neutrinos.
The detection of neutrino-less double beta decay would provide direct evidence of
lepton number violation, and the Majorana nature of neutrinos. It would also offer
a way to possibly disentangle the 3 cases of degenerate, normal or inverse hierachy
neutrino spectrum. The present limit from 0νββ (with large ambiguities from nuclear
matrix elements) is about |mee | < (0.3 − 0.8) eV [96].
5.2
Adding a fourth lepton family
In the following, we will account for the light neutrino masses and mixings by
means of an effective Majorana mass term, namely we add to the SM Lagrangian a
dimension-5 non-renormalizable operator.
Our aim here is to study the phenomenology of an additional heavy lepton family,
with masses about the TeV scale [94]. Thus, we add a 4th-family of leptons - for which
we introduce the ζ-flavor - composed by a lepton doublet, a charged lepton singlet and
a gauge singlet:
Lζ = (νζ L ζL )T ∼ (1, 2, −1/2) ,
ζR ∼ (1, 1, −1) ,
72
νζ R ∼ (1, 1, 0) .
(5.133)
The ζ-charged lepton, ζ = ζL + ζR , will have a Dirac mass term like the other three
charged leptons of the SM, but large enough to avoid conflict with the experimental
limits. We work in the basis in which the 4 × 4 charged lepton mass matrix is diagonal.
5.2.1
Heavy leptons not Mixing with Standard Model neutrinos
We forbid, in this section, the mixing between the ζ-neutrino and the three light
neutrinos of the SM. The Lagrangian can be split as L = LSM + Lζ .
The Lagrangian mass terms we take for the ζ-sector reads:
!
#
"
!
0 mD
1
(νζL )c
mass
c
+ h.c. ,
(5.134)
Lζ = −mζ ζ ζ −
νζL (νζR )
νζR
mD mR
2
Diagonalizing the neutrino mass matrix above, we obtain two independent Majorana
eigenstates, N1 and N2 , with real and positive masses, M1 and M2 (for convention
M1 ≤ M2 ),
s
s
2
2
m
m
mR
mR
D
D
M1 =
(5.135)
1 + 4 2 − 1 , M2 =
1 + 4 2 + 1 ,
2
2
mR
mR
which are related to the original Dirac and Majorana masses according to M1 M2 = m2D
and M2 − M1 = mR .
The ζ-neutrino chiral states will be an admixture of the two Majorana eigenstates
N1 and N2 :
!
!
!
2mD
νζL
i cos θ sin θ
PL N1
.
(5.136)
=
, tan 2θ =
c
(νζR )
−i sin θ cos θ
PL N2
mR
In the limit mD mR the seesaw mechanism would be at work (leading to M1 ∼ m2D /mR ,
M2 ∼ mR , νζ L ∼ i PL N1 , (νζR )c ∼ PL N2 ). Here however we are more interested in the
situation mD ∼ mR , in which both Majorana neutrinos have a mass about the TeV scale
and hence have a large SU(2)-active component.
The neutral current interaction of the ζ-leptons in terms of the heavy neutrino
Majorana mass eigenstates reads:
LNC
ζ =
g
1
1
νζ L γµ νζ L − ζL γµ ζL + sin2 θW ζ̄γµ ζ Zµ + e ζ̄γµ ζ Aµ ,
cos θW 2
2
(5.137)
cos2 θ
sin2 θ
N̄1 γµ γ5 N1 −
N̄2 γµ γ5 N2 + i cos θ sin θN̄2 γµ N1 .
2
2
(5.138)
where
νζ L γµ νζ L = −
73
The interaction of the Z with a couple of N1 or N2 is axial, while the one with two
different Ni is a vector interaction. As for the charged current:
g
−
µ
LCC
ζ = √ Wµ ζ̄L γ (i cos θ PL N1 + sin θ PL N2 ) + h.c. .
2
The Dirac mass can
√ be written in terms of the Yukawa coupling yζ and the Higgs VEV
v as mD = yζ v/ 2. Hence the interaction of the new neutrinos with the Higgs field
reads:
mD
LH
=
−
ν
ν
+
ν
ν
(5.139)
ζ
ζ
ζ
ζ
R
L
L
R H
ζ
v
mD
= −
cos θ sin θ N̄1 N1 + N̄2 N2 − i cos(2θ) N̄1 γ5 N2 H .
v
5.2.2
Promiscuous heavy leptons
In this section we consider the possibility that the new heavy leptons mix with the
SM leptons. For clarity of presentation we assume that the heavy neutrinos mix only
with one SM neutrino of flavor ` (` = e, µ, τ). We refer to [94] for the discussion of the
general case. The entries of the mass matrix are:
O(eV) O(eV) m0 (ν`L )c
1
(5.140)
− L = ( ν`L νζL (νζR )c ) O(eV) O(eV) mD (νζL )c + h.c. .
2
m0
mD
mR
νζR
The measured values of the light neutrino masses suggest the entries of the upper 2×2
block to be of O(eV) while the remaining entries are expected to be at least of the order
of the electroweak energy scale. Given such a hierarchical structure and up to small
corrections of O(eV/M1,2 ) . 10−11 , one obtains the following form for the unitary matrix
which diagonalises (5.140):
0
i cos θ sin θ0 sin θ sin θ0
ν`L
PL N0
cos θ
ν
(5.141)
ζL = V PL N1 , V = − sin θ0 i cos θ cos θ0 sin θ cos θ0 .
c
(νζR )
PL N2
0
−i sin θ
cos θ
N0,1,2 are the new (Majorana) mass eigenstates and
m0
,
tan θ =
mD
0
m0D
tan 2θ = 2
,
mR
The light neutrino N0 has a mass of O(eV). Up
neutrinos N1,2 have masses given by:
s
0 2
m
mR
mR
D
M1 =
1 + 4 2 − 1 , M2 =
2
2
mR
74
m0D 2 ≡ m2D + m02 .
(5.142)
to corrections of O(eV), the heavy
s
0 2
m
D
1 + 4 2 + 1 .
m
R
(5.143)
Notice that the smaller is mR , the more the neutrinos N1 and N2 become (the two Weyl
components of) a Dirac state.
Including the neutrino of flavor `, the neutral fermion current in Eq. (5.138) is replaced
with
1
cos2 θ
sin2 θ
ν¯` L γµ ν` L + ν¯ζ L γµ νζ L = − N̄0 γµ γ5 N0 −
N̄1 γµ γ5 N1 −
N̄2 γµ γ5 N2
2
2
2
+ i cos θ sin θN̄2 γµ N1 ,
(5.144)
while the charged current terms in the Lagrangian, Eq. (5.139), become
g
−
µ
0
0
0
LCC
ζ = √ Wµ ζ̄L γ (− sin θ PL N0 + i cos θ cos θ PL N1 + sin θ cos θ PL N2 ) + h.c. (5.145)
2
and
g
−¯ µ
0
0
0
LCC
` = √ Wµ `L γ (cos θ PL N0 + i cos θ sin θ PL N1 + sin θ sin θ PL N2 ) + h.c.
2
(5.146)
Notice that the neutral current remains flavor diagonal at tree-level 11 , hence the heavy
neutrinos couple to the SM ones only through the charged current interactions at this
order. This is a distinctive feature of our TeV neutrino physics. The SM like Yukawa
interactions lead to the following terms involving the Higgs:
LH
ζ = −
m02
mD
(1 + 2 ) cos θ0 [sin θ cos θ(N̄1 N1 + N̄2 N2 ) − i cos(2θ) N̄1 γ5 N2 ] H . (5.147)
v
mD
The low energy effective theory we will use for determining the interesting signals for
LHC phenomenology [19, 60] is the effective Lagrangian for Vanilla Technicolor.
5.3
LHC phenomenology for the natural heavy lepton family
In this section we investigate aspects of the phenomenology related to the interplay
between the new weekly coupled sector, i.e. the heavy leptons with its mixing with the
SM fermions, and the new strongly coupled sector breaking the electroweak symmetry
dynamically [94]. We consider only the MWT global symmetries relevant for the
electroweak sector, i.e. the subsector SU(2) × SU(2) spontaneously breaking to SU(2).
The low energy spectrum contains, besides the composite Higgs, two SU(2) triplets
of (axial-) vector spin one mesons. The effective Lagrangian has been introduced in
[19, 74]. The spin one massive eigenstates are indicated with R1 and R2 and are linear
combinations of the composite vector/axial mesons of MWT and the weak gauge boson
eigenstates. We have implemented the SU(2) × SU(2) TC sector in CalcHEP [131] using
the LanHEP module [129] to generate the Feynman rules in [74]. We have added the
new leptons to this implementation for the present study.
11
The neutral current is not flavor diagonal in models with TeV scale right handed neutrinos involved
in the see-saw mechanism for the light SM neutrino masses, see e.g. [98].
75
Ni , ζ±
q
q
γ, Z, R01,2
W ± , R±1,2
N j , ζ∓
q̄
Ni
q̄0
ζ±
Figure 43: Feynman diagrams for direct production of the heavy leptons in MWT with
i, j = 1, 2.
5.3.1
Production and decay of the new leptons
The heavy leptons may be directly produced through the charged- and neutral
current interactions:
pp → W ± /R±1,2 → ζ± Ni ,
pp → Z/γ/R01,2 → ζ+ ζ− ,
pp → Z/R01,2 → Ni N j ,
i, j = 1, 2.
(5.148)
The corresponding Feynman diagrams are given in Fig. 43. The direct production
cross sections for the heavy leptons in MWT are largely independent of the parameters
associated to the TC sector: in the direct production of ζ+ ζ− the only free parameter is
the mass of the charged lepton Mζ ; the direct production of Ni N j and ζ± Ni depends, in
addition to the masses of the leptons, on the V matrix entries of Eq. (5.141) as follows
from Eqs. (5.144) and (5.145). Plots of the LHC cross sections for pp → ζ+ ζ− , ζ± Ni , Ni Ni
can be found in [94].
The final state distributions arising from the direct production of the leptons depend
on the specific parameters of the TC sector. In particular R1 is a (mostly) axial-resonance
and R2 is a (mostly) vector-resonance, so R1 mixes mostly with the Z boson while R2
mixes significantly with the photon. Consequently, the invariant mass distribution of
the heavy neutral leptons Ni Ni will be relatively more dominated by the R1 resonance
compared to the charged leptons ζ+ ζ− . This is demonstrated in Fig. 44. The masses
and widths of R1,2 as a function of MA , g̃ and S are given in [74].
Production of the heavy leptons can also proceed via the Yukawa-type couplings to
the composite Higgs following from Eq. (5.147). We are following [19] for an effective
way to give masses to the SM fermions in the MWT setup. The composite Higgs may
itself be produced through either gluon fusion, vector boson fusion or in association
with a SM vector boson:
gg → H ,
pp → qq0 H ,
pp → HZ/HW .
(5.149)
The process gg → H → Ni Ni → WWµµ (within the SM framework) was recently
considered in [99, 100, 101] where also a 4th generation of quarks were included that
enhance the gg → H cross section [100] (for a recent review of the scenario in which the
76
3
Number of events/10 GeV @ 10 fb-1
Number of events/5 GeV @ 10 fb-1
10
~
10
MWT+4, MA=750, g=3
sm+4
2
10
1
10
10
-1
-2
600
800
1000
1200
M(jj)(GeV)
10
3
~
10
MWT+4, MA=750, g=3
sm+4
2
10
1
10
10
-1
-2
600
800
1000
1200
M(N1N1)(GeV)
Figure 44: Invariant mass distributions, M(ζ+ ζ− ) and M(N1 N1 ) in pair production of
ζ+ ζ− (left) and N1 N1 (right) in the MWT (purple/gray) and when the leptons are added
to the SM (black). The plots are for LHC at 7 TeV in the centre of mass and 10 fb−1 . For
the second plot we assumed unit mixing matrix element. In both frames the first peak
corresponds to the R1 resonance while the second corresponds to the R2 resonance.
The neutral current coupling of N1 N1 is axial and therefore more dominated by the
R1 resonance. We take the values g̃ = 3, MA = 750, S = 0.1 for the parameters of the
TC sector. The corresponding masses and widths of R1,2 are given in [74]. sm + 4 and
MWT+4 in the legend refer to respectively to the SM plus the fourth family and to the
MWT with the fourth family implementation on event generators.
new leptons are accompanied by a fourth generation of quarks, see [102]). In the MWT
class of models, this channel is not expected to be enhanced compared to the SM since
the techniquarks are not colored. The same applies for vector boson fusion production
of the composite Higgs [74].
On the other hand, the associate production of the composite Higgs can actually be
enhanced in MWT models [74, 89], in particular the pp → HZ/HW channel, due to the
presence of a light axial-vector resonance as shown in [74]. We will therefore focus on
the associate production of the Higgs.
The relevant expressions for the decay widths of the heavy leptons can be found
in [94]. Clearly, the decay patterns depend on the mass hierarchy and the mixings
between the leptons. Notice that, in the regime where Mζ > M1 and MH > M1 , N1 could
decay only via its mixing with SM leptons, N1 → `± W ∓ , whose vertex is proportional to
sin θ0 . The decay width and lifetime of N1 are displayed in [94]. We note that a value of
77
sin θ0 ∼ 10−6 would yield a decay length of ∼ 1 m, which is enough for a relativistically
boosted particle to escape detection at the LHC and be considered as missing energy
in the various processes [103].
5.3.2
Collider signatures of heavy leptons with an exact flavor symmetry
Let us first consider the limit in which the new leptons do not mix appreciably with
the SM ones. If Mζ > M1 , then N1 constitutes a long lived neutral particle and will give
/ T signals. In particular, the decay
rise to missing momentum p/T and missing energy E
mode H → N1 N1 gives rise to an invisible partial width of the composite Higgs.
As pointed out in [74, 89], the cross-section for ZH production can be enhanced in
MWT models because the axial-vector resonance can be light [19, 61]. Here the `+ `− + p/T
final state will receive contributions both from ZH and N1 , N2 production, as shown in
Fig. 45. We therefore study the proces pp → ZN1 N1 → `+ `− + p/T . We consider limiting
q
q
N1
γ, Z, R01,2 H
N1
Z
`+
γ, Z, R01,2
N1
N2
`−
q̄
N1
q̄
`+
Z
`−
Figure 45: Feynman diagrams for the `+ `− + p/T signal due to heavy leptons in the MWT
model.
values of the parameters such that either the Higgs or the N2 state is too heavy to
contribute significantly as well as parameters where both contribute in the process.
The acceptance cuts relevant for LHC are
|η` | < 2.5 ,
p`T > 10 GeV ,
∆R(``) > 0.4 .
(5.150)
Here ` is a charged lepton, η` and p`T are the pseudo-rapidity and transverse momentum
of a single lepton while ∆R measures the separation between two leptons in the detector.
∆R is defined viapthe difference in azimuthal angle ∆φ and rapidity ∆η between two
leptons as ∆R ≡ (∆η)2 + (∆φ)2 .
The main sources of background come from di-boson production followed by leptonic decays [104, 105, 106]
ZZ → `+ `− νν̄ , W + W − → `+ ν`− ν̄ , ZW → `+ `− `ν
where in the last process the lepton from the W decay is missed.
78
(5.151)
We impose the additional cuts
|M`` − MZ | < 10 GeV ,
and ∆φ(``) < 2.5 .
(5.152)
The first is meant to reduce the WW background by requiring the invariant mass of
the lepton pair to reproduce the Z boson mass. The second cut on the azimuthal angle
separation together with taking large p/T reduces potential backgrounds such as single
Z production + jets with fake p/T [104, 105].
The results are given in Fig. 46 assuming a fully invisibly decaying Higgs. On the
left panel we show the signal and background arising from the SM featuring the new
heavy leptons. On the right hand panel we show the same signal but in the MWT model.
For p/T > 100 GeV where the signal could potentially be observed, the Higgs production
channel dominates and in the MWT model a very distinct p/T distribution arises due
to the effect of the R1 resonance. While invisible decays of a SM model Higgs at most
appear as an excess of events compared to the background in e.g. p/T distributions, the
presence of a light axial-vector resonance results in a peaked distribution, different
from the shape of the background, making it a more striking signal. In general the peak
of the R1 resonance will increase for larger values of g̃ and decrease with the mass of
R1 . In addition, due to the smallness of the S parameter, the electroweak contributions
to the ZH production cross section cancel the contribution from the TC sector for small
g̃ at particular values of MR1 , causing a strong depletion of the signal. We show this in
the left panel in Fig. 47. However, for a light axial resonance a relatively large value of
g̃ is favored by unitarity arguments [107, 108] and electroweak precision observables
[74, 19, 82].
The true invisible branching fraction of the Higgs will depend on the mass of the
N1 particle, as shown in Fig. 47. An invisible partial width of the composite Higgs
has been searched for at LEP in the proces e+ e− → HZ with Z decaying hadronically
[109]. However, no limits were achieved for MH > 114 GeV. The same final state
from HZ/HW production in pp collisions has been considered by several authors, also
recently in the context of TC models (see for instance [110] and references therein). The
LHC discovery potential for an invisibly decaying Higgs in this final state at LHC has
also been investigated at detector level [106, 111]. It was found that the HW mode is
not promising [111] while the ZH mode remains challenging. With a SM production
cross section of HZ a significance of 3.43 σ was achieved at MH = 160 GeV, dropping to
a 2 σ excess at MH = 200 GeV [106].
Thanks to the possible increase in the HZ production cross section found in [74]
together with the resonance like structure in the p/T distribution displayed in Fig. 46,
we believe that this channel could be interesting to investigate the interplay between
new long-lived heavy neutrinos and composite vector states at LHC.
If instead Mζ < M1 , then ζ can be a long-lived CHAMP (Charged Massive Particle).
Collider signatures of long lived charged leptons could either be displaced vertices or, if
79
2
tan 2e=0.01, M1=50 GeV
tan 2e=1.5, M1=50 GeV
sm
10
1
10
10
-1
-2
100
200
300
400
Missing pT(GeV)
Number of events/5 GeV @ 10 fb-1
Number of events/5 GeV @ 10 fb-1
10
10
2
tan 2e=0.01, M1=50 GeV
tan 2e=1.5, M1=50 GeV
sm
10
1
10
-1
100
200
300
Missing pT(GeV)
Figure 46: `+ `− + p/T signal from pp → ZN1 N1 → `+ `− N1 N1 in the SM (left) and in the
MWT (right). In black the full SM background. The purple/gray line corresponds to
MH = 150 GeV, M1 = 50 GeV, tan 2θ=0.01 (to decouple the second neutrino we set M2 to
2000 TeV); the red (bottom) line corresponds to MH = 150 GeV, M1 = 50 GeV, tan 2θ=1.5
(M2 = 175 GeV).
the charged lepton decays outside the detector, a muon like signal for which the heavy
mass should be reconstructable. Such a long-lived CHAMP arises in several scenarios
and has been studied in some detail, for a review see e.g. [112].
In [113] a Herwig based study of the LHC reach for long-lived leptonic CHAMPs
was considered. With a discovery criterion of 5 pairs of reconstructed opposite charge
heavy leptons a reach of Mζ = 950 GeV at 100 fb−1 was found, reduced to 800 GeV
without specialized triggers. We can expect this reach to be improved in our model
by searching also for single ζ production channel. Additionally, it was found that
long-lived leptonic and scalar CHAMPS could be distinguished for masses up to 580
GeV.
The discovery potential for long-lived CHAMPS has also been studied at detector
level for LHC. The CMS and Atlas collaboration has considered various long-lived
CHAMPS [114]. From their results we infer that 3 signal events with less than one
background event could be observed in CMS with 1 fb−1 and Mζ ∼ 300 GeV and
similarly in Atlas. More precisely, in [114] 3 signal events could be seen in direct pair
production of 300 GeV KK taus with a pair production cross-section of 20 fb, very
similar to what we found. Fig. 44 shows that it is interesting to investigate the invariant
80
400
10
N1N1, tan 2e=0.01, MH=200 GeV
N1N1, tan 2e=1, MH=200 GeV
N1N1, tan 2e=0.01, MH=500 GeV
N1N1, tan 2e=1, MH=500 GeV
N1N2, tan 2e=1, MH=500 GeV
m(pp A R01 A ZH)(pb)
BR(H A N1N1(2) )
1
-1
1
10
10
10
10
10
10
10
-2
50
100
150
200
250
10
~
g
~ =2, S=0.1
g=5, S=0.1
~
g
=2, S=0.3
~
g=5, S=0.3
-1
-2
-3
-4
-5
-6
-7
600
800
1000
1200
1400
MR01(GeV)
M1(GeV)
Figure 47: The invisible branching fraction of the composite Higgs to either N1 N1 or
N1 N2 as a function of M1 (left). The production cross-section times branching ratio
σ(pp → R01 → ZH) as a function of MR1 for S = 0.1 (red) and S = 0.3 (black) as well as
for g̃ = 2 (dotted lines) and g̃ = 5 (solid lines).
mass distribution of the leptonic CHAMP.
5.4
Collider signatures of promiscuous heavy leptons
If the heavy leptons mix with the SM leptons, this will give rise to Lepton Number
Violating (LNV) processes with same sign leptons and jets in the final state, e.g.
pp → W ± /R±1,2 → µ± Ni → µ± µ± W ∓ → µ± µ± jj
pp → Z/R01,2 → Ni N j → µ± µ± W ∓ W ∓ → µ± µ± jjjj ,
i = 1, 2,
(5.153)
as in Fig. 48.
The potential for observing the µ± µ± jj final state has been extensively studied
in scenarios with heavy right-handed neutrino singlets, both in the SM with 3 light
neutrinos [98, 115, 116, 117] and in the presence of additional new physics [103, 118].
Same sign lepton final states have been searched for at the Tevatron in [119, 120]. The
pp → W ± → `± Ni → `± `0± jj process in the SM with 3 light neutrinos was studied in
[121] at the level of a fast detector simulation. While backgrounds for same sign lepton
production are smaller than for opposite sign lepton production, arising in the Dirac
limit, they were found to be significantly larger than previously estimated in parton
level processes, in particular for Mi < MW .
81
q
q
W∓
W ± , R±1,2 N1
q̄0
`
±
`0 ±
W∓
γ, Z, R01,2 N1
`±
N1
`±
q̄
W∓
Figure 48: Same sign leptons from production of N1 . We will consider the case where
the W’s decay to jets. Such that the final states we consider are µ± µ± jj (left) and µ± µ± jjjj
(right).
Again the production cross-sections are largely unaffected by the presence of heavy
vectors. However, the shape of the distributions are affected by the presence of the
heavy vectors.
To study these processes we impose jet acceptance cuts in addition to the leptonic
acceptance cuts given in Eq. (5.150)
|η j | < 3 ,
j
pT > 20 GeV ,
∆R(` j) > 0.5 .
(5.154)
The resulting invariant mass distributions for µ− µ− jj (left) and µ− µ− jjjj (right) are
given below in Fig. 49. We have taken cos θ = 0.7, sin θ0 = 0.098 and N2 decoupled.
While sin θ0 determines the mixing between N1 and ` and therefore is constrained
by experiment, cos θ is not. This means that the production cross section of N1 N1
potentially is significantly larger than the N1 µ production cross-section. If at the same
time N1 only decays to Wµ, we find the result given in the right frame of Fig. 49.
The production cross section σ(pp → R1 ) scales roughly as g̃−2 and independently of
S for small S. The detailed behavior may be inferred from the branching ratios presented
in [74]. The above shows that the interplay of heavy neutrinos and composite vector
mesons can lead to striking signatures at the LHC.
6
Conclusion
We introduced extensions of the SM in which the Higgs emerges as a composite
state. In particular we motivated TC, constructed underlying gauge theories leading to
minimal models of TC, compared the different extensions with electroweak precision
data and constructed the low energy effective theory.
We have then classified the spectrum of the states common to most of the TC
models and investigated their decays and associated experimental signals for the LHC.
We have set up the effective description to allow for easy extensions to account for
82
Number of events/5 GeV @ 100 fb-1
Number of events/5 GeV @ 100 fb-1
10
gt=3, MA=750 GeV, S=0.1
MN1=100 GeV, MN2=1000 GeV
1
10
10
10
10
-1
-2
-3
-4
500
600
700
800
900
1000
M(lljj)(GeV)
10
gt=3, MA=750 GeV, S=0.1
MN1=100 GeV, MN2=1000 GeV
1
10
10
10
-1
-2
-3
600
700
800
900
1000
1100
M(lljjjj)(GeV)
Figure 49: Invariant mass distributions for LHC at 14 TeV: pp → N1 µ− → µ− µ− jj
(left) and pp → N1 N1 → µ− µ− jjjj (right). The parameters in the TC sector are fixed
to be MA = 750 GeV, S = 0.1, g̃ = 3 while the new lepton sector parameters are
M1 = 100GeV, cos θ = 0.7, sin θ0 = 0.098. N2 is decoupled for these parameters.
specific features of given underlying gauge dynamics or for generic models in which
the Higgs emerges as a composite state.
We have established important experimental benchmarks for Vanilla, Running,
Walking and Custodial Technicolor for the potential discovery of these models at the
LHC.
Acknowledgments
It is a pleasure for us to thank M. Antola, A. Belyaev, S. Catterall, D.D. Dietrich,
R. Foadi, M.T. Frandsen, H. Fukano-Sakuma, M. Heikinheimo, J. Giedt, S.B. Gudnason,
C. Kouvaris, A. Nisati, T.A. Ryttov, V. Sanz, J. Schechter, P. Schleper, R. Shrock and
K. Tuominen for having shared part of the work on which this work is based on, long
and fruitful collaborations, relevant discussions, criticism and/or careful reading of the
manuscript.
83
A
Realization of the generators for MWT and the Standard Model embedding
It is convenient to use the following representation of SU(4)
!
!
A
B
C
D
Sa = †
,
Xi =
,
B −AT
D† CT
(A.155)
where A is hermitian, C is hermitian and traceless, B = −BT and D = DT . The S are
also a representation of the SO(4) generators, and thus leave the vacuum invariant
Sa E + ESa T = 0 . Explicitly, the generators read
!
1 τa
0
a
a = 1, . . . , 4 ,
(A.156)
S = √
aT ,
2 2 0 −τ
where a = 1, 2, 3 are the Pauli matrices and τ4 = 1. These are the generators of
SU(2)V × U(1)V .
!
1
0 Ba
a
S = √
, a = 5, 6 ,
(A.157)
a†
0
2 2 B
with
B5 = τ2 ,
B6 = iτ2 .
(A.158)
The rest of the generators which do not leave the vacuum invariant are
!
1 τi 0
i
i = 1, 2, 3 ,
X = √
iT ,
2 2 0 τ
and
!
0 Di
,
X =
i†
0
2 2 D
i
with
1
√
D4 = 1 ,
D5 = i1 ,
(A.159)
i = 4, . . . , 9 ,
(A.160)
D8 = τ1 ,
D9 = iτ1 .
(A.161)
D6 = τ3 ,
D7 = iτ3 ,
The generators are normalized as follows
h
i 1
Tr Sa Sb = δab ,
2
h
i 1
, Tr Xi X j = δi j ,
2
h
i
Tr Xi Sa = 0 .
(A.162)
Having set the notation, it is instructive to split the scalar matrix into four two by
two blocks as follows:
!
X O
M= T
,
(A.163)
O Z
with X and Z two complex symmetric matrices accounting for six independent degrees
of freedom each and O is a generic complex two by two matrix featuring eight real
84
bosonic fields. O accounts for the SM like Higgs doublet and a second copy as well as
for the three Goldstones which upon electroweak gauging will become the longitudinal
components of the intermediate massive vector bosons. The electroweak subgroup can
be embedded in SU(4), as explained in detail in [60]. Here SO(4) is the subgroup to
which SU(4) is maximally broken. The Sa generators, with a = 1, . . . , 4, together with
the Xa generators, with a = 1, 2, 3, generate an SU(2)L × SU(2)R × U(1)V algebra. This is
easily seen by changing genarator basis from (Sa , Xa ) to (La , Ra ), where
!
!
a
a
a
a
τa
0
0
S
−
X
S
+
X
0
T
,
(A.164)
, −Ra ≡ √
= 2
=
La ≡ √
aT
0
0
0 − τ2
2
2
with a = 1, 2, 3. The electroweak gauge group is then obtained by gauging SU(2)L , and
the U(1)Y subgroup of SU(2)R × U(1)V , where
T
Y = −R3 +
√
2 YV S4 ,
(A.165)
and YV is the U(1)V charge. For example, from Eq. (3.47) and Eq. (3.48) we see that
YV = y for the techniquarks, and YV = −3y for the new leptons. As SU(4) spontaneously
breaks to SO(4), SU(2)L × SU(2)R breaks to SU(2)V . As a consequence, the electroweak
symmetry breaks to U(1)Q , where
√
√
(A.166)
Q = 2 S3 + 2 YV S4 .
Moreover the SU(2)V group, being entirely contained in the unbroken SO(4), acts as a
custodial isospin, which insures that the ρ parameter is equal to one at tree-level.
The electroweak covariant derivative for the M matrix is
h
i
Dµ M = ∂µ M − i g Gµ (y)M + MGTµ (y) ,
(A.167)
where
g Gµ (YV ) = g Wµa La + g0 Bµ Y
√
T
= g Wµa La + g0 Bµ −R3 + 2 YV S4 .
(A.168)
Notice that in the last equation Gµ (YV ) is written for a general U(1)V charge YV , while
in Eq. (A.167) we have to take the U(1)V charge of the techniquarks, YV = y, since these
are the constituents of the matrix M, as explicitly shown in Eq. (3.58).
B
Technicolor on Event Generators
We now implement the effective Lagrangian (3.97) on event generators such as
MadGraph [122, 123] and CalcHEP [124, 125].
85
To implement the models on Monte Carlo event generators we used the FeynRules
Mathematica package [126, 127, 128]12 . FeynRules offers interfaces to several Monte
Carlo generators. The interfaces to CalcHEP and MadGraph have been tested for this
model. The implementation supports unitary gauge. The FeynRules model file and
the precompiled model files for MadGraph and CalcHEP can be downloaded here
https://feynrules.phys.ucl.ac.be/wiki/TechniColor
and here
http://cp3-origins.dk/research/tc-tools.
Unless the user wants to modify the model to include some new features, it is advisable
to download the precompiled input files. These include some tweaks with respect to
the output of FeynRules which makes their use easier.
B.1
Ruling Technicolor with FeynRules
The model contains the SM fermions and gauge bosons. The naming convention of
the particles was chosen to follow that of the SM implementation in MadGraph. As the
implementation is aimed for studying the signals from new high energy interactions,
the SM Lagrangian was taken to be a very simple one. In particular, the electron, the
muon, and the neutrinos, as well as the up, down and strange quarks, are taken to be
massless. This increases significantly the speed of Monte Carlo computations for some
processes. We include only the Cabibbo mixing of the quarks. The input parameters
(and their corresponding default values) of the SM sector are:
• EE = 0.313429, the electron charge;
• GF = 0.0000116637, the Fermi coupling constant (in units of GeV−2 );
• MZ = 91.1876, the mass of the Z boson (in units of GeV);
• aS = 0.118, the Z pole value of the strong coupling constant αs ;
• cabi = 0.227736, the Cabibbo angle;
• The masses of the heavy quarks MC = 1.3, MB = 4.2, MT = 172, and the tau
lepton MTA = 1.777;
• The widths of the weak gauge bosons and the top quark: wZ = 2.4952, wW =
2.141, and wT = 1.508.
12
The FeynRules implementation is a complete rewrite of the LanHEP [129] implementation that was
used in [74, 130].
86
The TC sector of the model implements the Vanilla Technicolor Lagrangian (3.97)
with the mass of the heavy vector spin one state fixed by the first WSR, i.e. the case of
generic walking TC in Table 1. Thus the composite states are
• H: the composite Higgs H;
• R1N and R2N: the neutral heavy spin one states R01 and R02 ;
• R1+, R1-, R2+, and R2-: the charged heavy spin one states R±1 and R±2 .
For these states the effective theories of MWT and NMWT coincide.
The input parameters of the composite TC sector are those listed already in Table 1:
• MA, the mass MA of the axial spin one states. Its allowed range is from about 500
(GeV) to about 3000, depending on the values of gt and S.
• gt is the coupling strength g̃ of the TC interactions and its allowed value ranges
from about 1 to circa 10, depending slightly on the parameters MA and PS.
• PS indicates the S parameter of the new strongly coupled sector in isolation. This
value is expected to be positive and small (much less than one). Estimates give
PS ' 0.1 for MWT and PS ' 0.3 for NMWT. The full phenomenologically relevant
S parameter comes from different contributions such as the one from the new
lepton family.
• MH is the mass MH of the composite Higgs boson. As discussed in the main text,
the Higgs is expected to be lighter than the vector mesons for walking TC models.
• rs is the parameter s that controls the coupling between the composite Higgs and
the heavy spin one states which is expected to be O(1).
An important feature of the model is that the widths of the heavy vectors and the
composite Higgs depend strongly on the parameters MA, gt, and MH. This dependence
needs to be included in the Monte Carlo calculations in order to obtain reliable results.
The FeynRules implementation does not include the expressions for these widths, since
they are computed in different ways, for example, in CalcHEP and in MadGraph. It
is possible to set CalcHEP to calculate and update the widths on the fly automatically
whenever the values of the parameters are changed (see the CalcHEP manual [131]). In
MadGraph, the widths can be included in the calculator program which takes care of
changing the parameters. A calculator for the TC model, which includes the widths of
the composite particles, can be downloaded from the same webpages of the FeynRules
model implementation given above.
87
B.2
Madding Technicolor via MadGraph/MadEvent v.4
MadGraph/MadEvent (just MadGraph or MG/ME in brief) is a software that allows
to generate amplitudes and events in models for particle interactions. The SM as well
as some beyond SM models are already included in the MG/ME model files and are
ready to be used once installed. The (N)MWT implementation can be downloaded
from the links given at the beginning of this Section.
Here the aim is to provide a fast, practical and ready-to-go guide throughout the
installation of MadGraph version 4 13 and its use with the MWT package; for further
informations and a deeper insight refer to the MadGraph manual, downloadable from
http://cp3wks05.fynu.ucl.ac.be/twiki/bin/view/.
To download the MadGraph package a registration is needed on the MG/ME website
or one of its mirrors. To compute the matrix elements only the MadGraph StandAlone
package is needed, while to also generate events, the entire MadGraph Developer’s kit
is needed.
We refer to the MadGraph manual for installation instructions. Inside the MG/ME
folder one finds two useful directories: the Model directory is the one where the MWT
model folder needs to be placed, while the Template directory is the one from where the
simulations are run. Here the files with the events generated by MadGraph are created.
Fig. 50 shows the position of the directories and files used.
To use the new model the first step is to specify the process or the processes one
wants to study. This is done by modifying the proc card.dat file inside the Template/Cards
folder. As shown in the default file it is possible to specify more than one process at the
time, provided that the various processes are identified by an integer number following
the character @. The names of the particles and respective antiparticles are defined in
the particles.dat file inside the specified model folder, and are case insensitive; in the
SM default implementation for instance e-, mu-, ta- are the charged leptons (with
antiparticles e+, mu+, ta+), ve, vm, vt the neutrinos (with antiparticles denoted by a ∼
after the name), the quarks are u, d, c, s, t, b (again with antiparticles denoted by a ∼
after the name), h is the Higgs boson and g, a, z, w+ and w- are the gauge bosons. In
the MWT implementation the SM particles preserve their names, and in addition one
has the new neutral vector resonances r1, r2 plus the charged vector resonances r1+,
r1-, r2+, r2-.
The process is specified listing initial and final particles separated by the character
>. For example to study e+ e− → µ+ µ− one should type e+e->mu+mu-. At the bottom of
this file one can set the definitions for the ‘multiparticles’, i.e. labels that are given to
sets of particles rather than a single one. For instance, one can define the proton as
P uu˜dd˜g
13
No modifications for MadGraph v. 5 are needed.
88
Figure 50: Graph illustrating the usage of the MWT implementation in MG/ME. Circles
represent directories, rectangles represent text files while hexagons are executable files.
Blue arrows show placement of files and folders into the various directories, thick
yellow arrows indicate input for the executable files while thick red ones indicate
output.
89
meaning that the proton p is composed by u, ū, d, d¯ and the gluon g. The process
pp>e+e- will henceforth consist of all the processes with any combination of two of the
following partons u, ū, d, d¯ and g in the initial state. Besides modifying the present
definitions, the user can provide some new ones. Note that the antiproton is not defined
here, and if one wants pp̄ in the initial state he should just write pp: whether these are
protons and/or antiprotons it will be specified later, in the choice of the collider type in
the run card.dat file.
When specifying the process, use the form xy>z>abc to require the particle z to
appear in a s-channel intermediate state, and use xy>abc/z to exclude the particle z to
appear in any diagram.
Right below the process specification one can assign the maximum number of QEDand QCD-type vertices in each Feynman diagram that will be produced.
The model one wants to use for his simulations shall be specified as the name of
the corresponding folder inside the Model directory; for example it is MWT for the MWT
model (case insensitive).
To generate the Feynman diagrams for the processes specified, one has now to enter
the Template/bin folder and run the command ./newprocess from shell. The result
can be seen by opening the file Template/index.html and by clicking on the link ‘Process
Information’; the Feynman diagrams are provided both as jpg files (click on ‘html’) as
well as ps files (click on ‘postscript’).
We can now generate events to compute different cross-sections. The result will
depend on both the Feynman diagrams that have been generated in the previous step
and the value of the couplings and parameters of the model used. To assign different
values to the MWT parameters it is necessary to use the calculator, downloadable from
the webpages linked at the beginning of this Section; the readme.txt file coming along
with it contains instructions on how to use it. Once extracted the package one has to
compile it typing make from shell; the executable file MWT calculator is then created.
One can assign new values to some or all external parameters editing the param input
file, where all the parameters that can be modified are listed. The calculator will assign
a default value to all those parameters that are not requested to be modified in this file.
To run the calculator, type
./MWT_Calculator parameter_input > param_card.dat
from shell, so that the file parameter input is given as input and a file named param card.dat
is produced. Opening this last file one can see the values of all the external parameters
and all the masses of the model; the values of the other parameters will be automatically
calculated by MadGraph. Substituting now this file to the existing one in the MWT
model folder inside the Model directory will update the model with the new parameters
values.
The second step before generating events is editing the Template/Cards/run card.dat
file, where one can set the simulation and the collider parameters, choose the parton
90
distribution functions (PDF) and specify some cuts. The simplest entries one can edit
are:
• the tag name for the run, that is just one word providing a label to the run;
• the number of (unweighted) events one wants;
• the random seed; it should be always left to zero, unless the user aims to reproduce
a specific set of results. If left to zero, it is incremented automatically in each run
so that successive runs are statistically independent;
• the collider type, that means the type of particles in each beam and their PDFs:
use 1 to indicate the proton and −1 for the antiproton; setting 0, the PDFs are
switched off. The default setting is pp, as in the case of LHC;
• the beams energy;
• the PDF choice: to have the list of different possibilities, see the MG/ME manual.
Note that the PDF automatically fixes also αs and its evolution;
• the cuts.
To generate events and get the cross sections of the processes, run now from inside
the directory Template/bin the command ./generate_events from shell; type 0 to run
the simulation on a single machine, and then provide the run name, i.e. a label for
the run. Depending on the number of particles and the number of Feynman diagrams
involved the run can take from minutes to hours. The result of the simulation can be
seen once again opening the file Template/index.html and clicking on the link ‘Results
and Event Database’. The files containing the events (in the LHEF format [132]) are
stored in the Template/Events folder.
Summarising:
1. Install the MadGraph/MadEvent package from the website;
2. download the MWT implementation and the calculator from
http://cp3-origins.dk/research/tc-tools,
and place the MWT model folder inside the MadGraph Models directory and
compile the calculator;
3. specify the processes you want to study and the model in the Cards/proc card.dat
file;
4. run ./newprocess from the Template/bin directory to produce the Feynman diagrams. You can check the result in the file Template/index.html;
91
5. to change the parameters of the MWT model, edit the parameter input file in the
calculator folder and then run
./MWT_Calculator parameter_input > param_card.dat
to generate the param card.dat file, that has to be placed into the MWT model
folder in the MadGraph Models directory;
6. in order to set the run parameters (collider type, beams energy, PDFs and so on)
and impose cuts, edit the Template/Cards/run card.dat;
7. run ./generate_events from Template/bin to calculate the cross section and generate the events;
8. check the result with the file Template/index.html; the event files are stored into the
Template/Events folder.
B.3
Calculating Technicolor with CalcHEP
CalcHEP is a package that allows the user to go from the Lagrangian to the physical
observables calculated at the lowest order in perturbation theory. The package can be
downloaded from the CalcHEP website, where also the manual and information about
new features are available. In addition users need Fortran and C compilers14 and X
Window System (X11)15 to install the program.
The current CalcHEP version 2.5.7 includes by default the SM and large extra
dimensional model implementations. Other models can be also downloaded from the
CalcHEP web page. The (N)MWT implementation can be downloaded from the links
given at the beginning of this Section.
First of all one has to download the tar file that contains the source code from the
CalcHEP web page, and extract the package. One has then to open the file getFlags in
the new CalcHEP folder and make sure that the names of the Fortran and C compilers
correspond to those which are installed in the computer (and in case they do not,
one has to change the names to the corresponding ones). To install the program
one should move to the CalcHEP folder and run the command make. If all works
properly the message CalcHEP is installed successfully and can be started should appear.
It is convenient to create a working directory placed outside the CalcHEP folder by
using the command ./mkUsrDir ../WorkDir. At this point the program is ready to
use and a Graphical User Interface (GUI) session can be launched with the command
./calchep, to be entered preferably from the working directory. One can navigate
from menu to menu by using the arrow and esc keys. Instructions and help can be
found pressing the F1 and F2 keys.
To install a new model in the current CalcHEP installation one should download
the model files and copy them to the models folder inside the CalcHEP main folder.
14
15
For example g77 and gcc.
In some operating systems additional X11 development libraries are required.
92
Among those, the file prtcls.mdl contains description of the particles, vars.mdl defines
the model parameters, func.mdl contains additional parameters which depend on those
defined in the previous file, and lgrng.mdl determines the interactions. To install the
new model one has to launch CalcHEP, choose from the second menu Import models,
and type CALCHEP/models: in the resulting menu one can scroll the available models
and choose the one to import.
Figure 51: Menus of the symbolic part.
CalcHEP contains two parts: a symbolic and a numerical one. Menus of the symbolic
part are presented in Fig. 51. In the second menu in Fig. 51 the user can choose the
gauge (either Feynman-’t Hooft or unitary), or edit the model. Choosing Enter Process
the user can define the process. The number of initial state particles have to be either
one or two. While the number of the final state particles is not limited in principle, the
increasing amount of time required by the simulation sets a practical limit between four
and six on that number, for a typical machine and process. It is also possible to exclude
some particles from the Feynman diagrams used to calculate the process amplitude.
In the third menu the user can view the created diagrams and also save them in
a separate file. Choosing Squaring techniques CalcHEP squares the diagrams. The
numerical session can now be started directly by choosing Make&Launch n calchep or
93
Figure 52: Menus of the numerical part.
94
the user can ask CalcHEP to calculate directly the squared matrix elements by choosing
Symbolic calculations. The resulting analytic expressions for the squared matrix elements
can be saved in different formats. One can then move to to the numerical session by
choosing C-compiler from the fifth menu in Fig. 51.
In numerical sessions the user can define the momentum of the initial state particles
and choose if and which parton distribution functions (PDF’s) to use. Several cuts can
be imposed and regularization of the propagators can be done before the numerical
integration. Also the model parameters can be changed at this point. CalcHEP automatically re-calculates values of the related parameters if the model parameters are
changed. The structure of the menus in the numerical part is presented in Fig. 52.
CalcHEP uses Vegas to perform the numerical integrals: the user has to choose the
number of Vegas calls and iteration steps. The actual number depends on how much
time the user allows and how well the integrals converge. Some convergence problems
can be fixed by the a wise choice of the regularization scheme and appropriate cuts.
The program can also generate plots of the differential cross section associated with
a given process in function of one of the independent variables. The important part
of the program is the event generator which allows to make predictions for collider
phenomenology. The method for producing the events is explained in the CalcHEP
manual.
Though intuitive, the graphical interface is not the only way to use CalcHEP: the user
can write at once all the information relative to the process to be simulated in the batch
file and then execute it in the working directory. For example, if the name of the batch
file is ee-mumu-batch one has to use the command ./calchep_batch ee-mumu-batch
to run the batch.
For more information on CalcHEP usage, visit the CalcHEP homepage where the
official manual can be downloaded.
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