Higgs doublet as a Goldstone boson in perturbative extensions of the standard model

Higgs doublet as a Goldstone boson in perturbative extensions of the Standard Model arXiv:0805.2107v2 [hep-ph] 7 Nov 2008 Brando Bellazzinia, Stefan Pokorskib, Vyacheslav S. Rychkova,b, Alvise Varagnoloa,c a b Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy Institute of Theoretical Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland c Dip. di Fisica, Univ. di Roma La Sapienza, P.le A. Moro, 2, I-00185 Rome, Italy Abstract We investigate the idea of the Higgs doublet as a pseudo-Goldstone boson in perturbative extensions of the Standard Model, motivated by the desire to ameliorate its hierarchy problem without conflict with the electroweak precision data. Two realistic supersymmetric models with global SU (3) symmetry are proposed, one for large and another for small values of tan β. The two models demonstrate two different mechanisms for EWSB and the Higgs mass generation. Their experimental signatures are quite different. Our constructions show that a pseudo-Goldstone Higgs doublet in perturbative extensions is just as plausible as in non-perturbative ones. 1 Introduction The idea of the Higgs doublet as a pseudo-Goldstone boson of some extended global symmetry has been proposed to ameliorate the hierarchy problem of the Standard Model (SM) [1]. Usually, it is linked to a new strongly interacting sector, responsible for spontaneous breaking of the global symmetry [2],[3],[4]. There are interesting signatures of this idea, among others related to the unitarization procedure in WW scattering [5],[6]. However, there are strong constraints on the scale f of the spontaneous breaking of global symmetry of the strong sector. Low values of f , say f . 500 GeV, cannot be easily reconciled with electroweak precision tests and B-physics data [7], while larger f reintroduces the hierarchy problem with the required finetuning growing as1 (f /v)2 . So, in practice, models of this kind do not avoid certain tension. There is some room for the idea of the Higgs doublet as a pseudo-Goldstone boson in perturbative extensions of the SM as well, with global symmetry broken in the perturbative regime. In general, one may expect such perturbative models to avoid excessive finetuning in the ElectroWeak Symmetry Breaking (EWSB) sector with no conflict with the electroweak precision data, generic for non-perturbative models. This possibility has been discussed in non-supersymmetric [7],[8] and supersymmetric [9],[10],[11],[12],[14],[13] models, however for various reasons those models are not fully satisfactory. In the present paper we explore it further in supersymmetric (SUSY) models with extended global symmetry of the Higgs sector. We discuss two models which differ in various respects and illustrate various aspects of the general approach. As global symmetry we take SU(3), the minimal one that can give Higgs doublet as a Goldstone boson in SUSY. The first model (Model I) remains perturbative up to the GUT scale. The global symmetry and the electroweak symmetry are broken by radiative corrections to the mass parameters, generated by a large Yukawa coupling, similarly to the Minimal Supersymmetric Standard Model (MSSM) (for earlier attempts, see [11]). Stabilization of the global symmetry breaking scale f can be achieved by quartic scalar coupling in large tan β regime. The model relies on the “double protection” mechanism, where the interplay between supersymmetry and an approximate global symmetry forbids the quadratic higgs term to receive a large logarithmic contribution from the UV cutoff Λ ∼ MGUT which is actually replaced by the scale of global symmetry breaking f . Thus one may hope to get, for the same values of stop mass, much less finetuning than in the MSSM. The values f & 2 TeV minimize finetuning of the model, while at the same time allowing the physical Higgs boson mass above the experimental bound of 115 GeV. Phenomenology of the Higgs sector of the model is very similar to the decoupling regime of the MSSM. In particular, for f & 2 TeV the WW scattering is unitarized almost completely by the lightest Higgs boson. The model is however distinguished by the presence of a relatively light doubly-charged Higgsino. In our second example (Model II) supersymmetry provides a consistent framework for stabilizing the minimum of the global symmetry breaking. It is a supersymmetric version of the approach to the EWSB proposed in Ref. [7], with the breaking driven by a tadpole of the SU(2) × U(1) singlet component of the full scalar multiplet. An interesting point about this mechanism is that this tadpole, while being linear in the fundamental field, generates the Higgs quartic when the σ-model structure is taken into account. This quartic dominates the usual D-term quartic at low tan β, so that the physical Higgs mass is determined by the soft SUSY breaking terms only. Modell II has a very different phenomenology with respect to MSSM since it allows for low f . However, it needs 1 Here and throughout the paper v is the electroweak scale in v ≃ 174 GeV normalization. 1 an UV completion at a scale O(20 TeV), where the SUSY model becomes strongly interacting. In both models finetuning is O(10%). 2 Model I In the MSSM, the lightest Higgs boson mass is determined by the effective quartic coupling, which depends logarithmically on the stop mass. Large tan β is then favored, to minimize the value of the stop mass consistent with the experimental bound mh > 115 GeV, and the finetuning in the Higgs potential. The latter is proportional to m2t̃ log Λ and for Λ ∼ MGUT remains, unfortunately, of order of 1%. The model we propose in this section retains the MSSM correlation between the stop mass and the Higgs boson mass, thus also requiring large tan β for reasonable values of mt̃ . However, it is based on the idea of the double protection of the Higgs potential [10],[11] and gives, for the same values of mt̃ , factor 10 less finetuning than MSSM. We begin with the effective model below certain scale F based on the symmetry SU(2)L ×U(1)y , where SU(2)L is gauged subgroup of a global SU(3). Its UV completion (above F ) can be similar to that of [11], and we will return to it below. The SU(3)-symmetric Higgs sector consists of a triplet Hd and an antitriplet Hu , while the chiral fermion multiplets of the top sector are the triplet Ψ = (Q, T )T and quark singlets tc and T c . Under SU(2)L × U(1)y the triplets split into doublets Hu,d and singlets Su,d :2 HdT = (HdT , Sd ), Hu = (Hu , Su ). We look for a model in which the global SU(3) is spontaneously broken by vacuum expectation values (VEVs) aligned so that SU(2)L × U(1)y gauge symmetry remains unbroken: HdT = (0, 0, fd ), Hu = (0, 0, fu ), and tan β ≡ fu /fd is large. We shall assume that the soft term mass scale msoft ∼ f ≪ F . The minimum of the SU(3)-symmetric scalar potential at large tan β generically requires negative Hu mass squared, which leads to runaway directions, unless there is a stabilization mechanism. Stabilization by D-terms of some, e.g. U(1), gauge interactions is not a satisfactory mechanism [10], while stabilization by the quartic coupling |Hu |4 is constrained by the holomorphicity of the superpotential. With two Higgs triplet chiral superfields, the minimal field content leading to stabilization by the quartic consists of two symmetric tensors Z1 and Z2 , √   Ti√ Hi / 2 Zi = . HiT / 2 zi Here the Ti ’s are SU(2) triplets with Y1,2 = ±1, Hi ’s are doublets with Y1,2 = ±1/2, and zi ’s are singlets. The superpotential of our model reads (in the following we anticipate large tan β solution) W = λZ2 Hu Hu + µHu Hd + µZ Z1 Z2 + yHu Ψ tc + mT c T. (2.1) 2 It’s useful to keep in mind that our doublet Hu is related to the MSSM doublet Hu via Hu.our = ε · Hu,MSSM . 2 The last term breaks the global SU(3) explicitly. It can originate from a UV completion as in [11]. The scalar potential reads3 V = |λHu Hu + µZ Z1 |2 + |2λZ2 Hu + µHd |2 + µ2Z |Z2 |2 + µ2 |Hu |2 + Vsoft , Vsoft = m2d |Hd |2 + m2u |Hu |2 + m2Z1 |Z1 |2 + m2Z2 |Z2 |2 − (m23 Hd Hu + H.c.) . (2.2) The soft terms in (2.2) depend on their initial values at the GUT scale and on the renormalization group (RG) running in the SU(3)-symmetric theory. We expect that the stop contribution will drive m2u to negative values (while m2d , m2Z1,Z2 > 0), and global SU(3) is spontaneously broken. Minimizing the potential for small m23 (see Appendix A for the running of m23 ) and assuming SU(3) to be broken in the SU(2) singlet direction, we get µ2u 2λ2eff h|Hu |i µ̃2 + m2d ≡ tan β ≃ ≫1 h|Hd |i m23 λµZ fu2 h|Z1 |i ≡ fZ1 ≃ − 2 µZ + m2Z1 2λµfu fd h|Z2 |i ≡ fZ2 ≃ − 2 µZ + m2Z2 + 4λ2 fu2 h|Hu |2 i ≡ fu2 ≃ − (2.3) where µ2u = m2u + µ2 < 0 (by assumption) m2 λ2eff = λ2 2 Z1 2 µZ + mZ1 m2Z2 + µ2Z µ̃2 = µ2 2 µZ + m2Z2 + 4λ2 fu2 The SU(3) is broken dominantly by fu and fZ1 , with fd and fZ2 suppressed by large tan β. Relative contribution of fZ1 versus fu decreases for smaller λ and µZ . The maximal value of λ at the Fermi scale is constrained by the requirement of remaining perturbative up to the GUT scale (see Appendix A for the discussion of λ running); we choose λ = 0.2 in the following. The mass parameter m2u gets SU(3)-symmetric negative contributions proportional to the Yukawa coupling y and the coupling λ (see Appendix A). In the following we will discuss the constraints on the parameter range following from the demand of no excessive finetuning in the potential for the SU(3) breaking. Spontaneous global SU(3) breaking leads to five Goldstone bosons: an SU(2)L doublet H and a real singlet η. The H plays the role of the SM Higgs doublet. The singlet η will not play any role in the following discussion; we will comment on its parametrization and physical effects below. For large tan β the Goldstones reside to a good approximation in the Hu and Z1 . Up to terms of 3 The omitted soft terms Z1 Z2 and Z2 Hu Hd will in general be produced by running from the GUT scale; we have checked that their typical generated values are small and do not affect the dynamics. 3 higher order in H, we have the following parametrization for the Goldstone bosons: Hu ≃ αu H † Hd ≃ αd H, H1 ≃ αZ1 H † , H2 ≃ αZ2 H fZ2 fZ1 T1 ≃ 2 H † H † , T2 ≃ 2 H H f f Su,d ≃ fu,d , z1,2 ≃ fZ1,Z2, (2.4) (2.5) where √ αu,d = fu,d /f, αZi = 2fZi /f, 2 2 f 2 = fu2 + fd2 + 2fZ1 + 2fZ2 . In this parametrization H has canonical kinetic term. As we will see later, experimental limit on the Higgs mass requires fZ1,Z2 ≪ fu ∼ f. The global SU(3) is explicitly broken by the last term in the superpotential (2.1) and by the D-terms of SU(2) × U(1). Both terms contribute to the potential for the Goldstone boson H: V = δm2H |Hu |2 + (λ0 + δλ)|Hu |4 + . . . , where the δm2H and δλ are obtained from the one-loop effective potential and λ0 comes from the D-terms. We first discuss the effective potential contribution as it is responsible for the VEV of H and the EWSB by the top-stop loops. Diagonalizing the top mass matrix, for large tan β we find (we introduce dimensionless coupling ỹ, m = ỹh|Su |i): mt = yt h|Hu |i, mT = h|Su |i p yt ≡ p y ỹ y 2 + ỹ 2 , y 2 + ỹ 2 . For the couplings y and λ to remain perturbative up to the GUT scale (see Appendix), we need y . 1.2. Since yt ≃ 1 (for h|Hu |i ≃ v) we get ỹ & 1.8 and mT & 2.2h|S √ u |i, somewhat stronger than the theoretical lower bound mT = 2yt h|Su |i realized for y ≃ ỹ ≃ 2. To realize the double protection mechanism, we assume that soft stop masses are SU(3)symmetric at the scale F . To compute the effective potential, we assume for simplicity that these masses are universal: m2Q (|Q̃|2 + |T̃ c |2 + |t̃c |2 ), and we also neglect the possible left-right stop mixing. As in ref. [11] we get the following result: " !  # m2Q 3 2 m2T 2 2 2 δmH = − 2 yt mQ ln 1 + 2 + mT ln 1 + 2 + ∆, (2.6) 8π mQ mT where 3g22 M22 + gy2 My2 F , ln ∆⊃ 2 8π Msoft 4 (2.7) the contribution due to SU(2) × U(1)y gauginos with soft masses M2 ,My . At the same time the dominant contribution to δλ is given by4   m2Q 3 4 − 2x ln(1 + 1/x) , x = m2Q /m2T . (2.8) y ln 2 δλ ≃ 16π 2 t mt (1 + x) Notice that this correction is smaller than the corresponding MSSM correction for the same value of the stop mass, which is due to the negative contribution of the heavy T quark. In the mT ≫ mQ limit, which will turn out to be relevant below, we recover the standard MSSM equations, with the important difference that the scale of the logarithm in (2.6) is given by mT instead of MGUT . The D-term potential reads:
g 2 + gy2 2 g 2 + gy2 2 2 2 2 2 2 2 2 |Hu | − |Hd| + |H1 | − |H2| = (αu − αd2 + αZ1 − αZ2 ) |H|4. VD = 8 8 For the Higgs boson mass we get the following result:
  2 2 2 m2h ≃ 1 − v 2 /f 2 MZ2 (αu2 − αd2 + αZ1 − αZ2 ) + 4δλ αu4 v 2 . (2.9) The overall suppression factor is due to the σ-model correction to the wavefunction normalization of the Higgs doublet; it can be derived by keeping track of terms higher order in H which were omitted in (2.4). Considering the large tan β suppression resulting in αd and αZ2 going to zero, we see that for a given δλ the Higgs boson mass is maximized for tan β → ∞, αZ1 → 0, mmax = MZ2 + 4δλ v 2 h f → ∞,
1/2 . (2.10) (2.11) Expanding in the small negative corrections appearing when these parameters deviate from their optimal values, (2.9) can be numerically parametrized as follows # " 2  2  1.3 TeV αZ1 2 12 max , (2.12) + + mh ≃ mh − 1 GeV tan β f 0.15 where the first and the second corrections come from finite values of tan β and f respectively, and the third from a nonzero αZ1 . Since mmax cannot be much above 115 GeV without a significant h increase in finetuning (see Figs 1, 2 below), we should not allow the total loss in (2.12) to exceed 1 ÷ 2 GeV, which implies obvious constraints on the relevant parameters. We now discuss the results for the Higgs boson mass and estimate the level of finetuning. In Fig. 1 we plot the Higgs boson mass (2.11) (i.e. without negative corrections given in (2.12)) as a function of mQ and mT , using δλ from eq. (2.8) with mt = 172 GeV. Similarly to the MSSM, the Higgs boson mass increases with mQ . For a fixed mQ , the correction is maximized in the mT ≫ mQ limit. In Fig. 2 we plot the finetuning in the Higgs mass term m2 |H|2 which is needed to compensate the top-stop contribution (2.6): δm2H |∆=0 FT1 = . (2.13) m2h /2 5 4.0 mh max @GeVD 3.5 mT @TeVD 3.0 125 2.5 120 2.0 117 115 1.5 110 1.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 mQ @TeVD Figure 1: The maximal Higgs boson mass (2.11) as a function of mQ and mT , see the text. 4.0 FT1 3.5 40 mT @TeVD 3.0 2.5 30 2.0 1.5 20 15 10 1.0 5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 mQ @TeVD Figure 2: Finetuning (2.13) in the Higgs mass parameter needed to compensate for the top-stop loop contribution (2.6). 6 4.0 2 FT2 3.5 10 f @TeVD 3.0 2.5 20 2.0 1.5 5 1.0 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 mQ @TeVD Figure 3: Finetuning (2.14) in the SU(3)-symmetric mass parameter m2u , see the text. This finetuning increases quadratically with mQ , but grows only logarithmically with mT . Comparing Fig. 1 with Fig. 2, we see that mQ ∼ 800 GeV and mT & 2.5 TeV give mh > 115 GeV with about 10% finetuning (FT1 = 10). Soft stop masses as small as mQ ∼ 600 GeV are possible provided that mT is raised up to 4 TeV. Another source of finetuning in Model I is the SU(3)-symmetric top-stop contribution to the m2u parameter of the scalar potential (2.2), which by (2.3) should not exceed 2λ2 fu2 . The corresponding finetuning parameter FT2 = δm2u,stop /(2λ2f 2 ) (2.14) is plotted in the mQ − f plane in Fig. 3, where we assume y ≃ 1, λ = 0.2. We see that this finetuning is less than 20% (FT2 < 5) for mQ ∼ 800 GeV and f & 2 TeV, which however translates into mT above 4 TeV. In general raising f (and mT ≃ 2f ) we eliminate FT2 while FT1 grows only logarithmically. Unfortunately, in this limit the heavy top quark becomes undiscoverable at the LHC, and the scalar spectrum of the model resembles the standard MSSM at large tan β in the decoupling limit. In this case the only significant difference from the MSSM is the presence in the low-energy spectrum of states described by the tensors Z1 and Z2 , i.e. triplets Ti , doublets Hi and singlets zi . The triplets contain doubly (T̃1++ or T̃2−− ) and singly (T̃1+ or T̃2− ) charged and neutral (T̃10 or T̃20 ) higgsinos and their scalar partners. Doublets Hi have the same composition as the Higgs chiral superfields Hu,d. All those fields have common supersymmetric mass parameter µZ . It enters in the equation (2.3) for the VEV of z1 and is constrained by the requirement that no large negative αZ1 -correction to the Higgs boson mass should be present in Eq. (2.12). Assuming that all scalar soft masses are of the same order: √ MSUSY ∼ mu ∼ 2λf ∼ 0.3f , µZ is bounded by µZ . αZ1 msoft . f /20. 4 The formula given in [11] contains an extra +3/2 term in square brackets. Our formula is correct provided that δλ is defined as the coefficient in the Higgs mass correction formula, Eq. (2.9). 7 This estimate, while being subject to significant uncertainty, does indicate that the masses of new fermions, in particular of the doubly charged Higgsinos, are expected to be below 200 ÷ 300 GeV even for f as high as 2 TeV. We shall return to the phenomenological issues at the end of this section. Finally, we need to comment on several other issues which are important for the consistency of our model. First, we note that the model can be UV completed as in Ref. [11], with the gauge group SU(3) × U(1)x broken to the electroweak SU(2) × U(1)y at the scale F . An extra pair of triplets ΦU,D is responsible for this breaking, so that the full global symmetry of the scalar potential is SU(3) × SU(3). The SU(3)-breaking term mT c T in the Model I superpotential (2.1) originates naturally from an SU(3)-symmetric term y1 ΦU ΨT c , y1 ∼ m/F = ỹf /F, (2.15) of the UV completed superpotential. As explained in [11], the soft mass terms of the ΦU and ΦD fields have to be very nearly universal at F , since their difference m2D − m2U contributes to the mass term of the Higgs doublet via D-terms. Even assuming that these masses are universal at the GUT scale, superpotential interaction (2.15) will contribute to the running of m2U , so that at F the masses will be split by m2D − m2U ≃ MGUT 3y12 2 (mQ + m2T c ) ln 2 8π F This contribution must be . v 2 which can be achieved by choosing F & 10f as can be seen from the second eq. in (2.15). On the other hand the F scale cannot be too high because of the gaugino contributions to the Higgs mass, Eq. (2.7). For completeness we have to say a few words about the 5th Goldstone boson η, a gauge singlet axion, which appears in addition to the Higgs doublet when SU(3) is broken spontaneously to SU(2), as already mentioned above. This Goldstone is associated with a global U(1) under which the gauge singlet components have charges Su (+1/2), Sd(−1/2), z1 (+1), z2 (−1), equal to the hypercharge of the upper components of the same SU(3) multiplets; it resides mostly in the phase of Su whose VEV dominates the spontaneous symmetry breaking: √ Su ≃ fu exp(iη/ 2fu ) . The η does not get mass from the SU(3)-breaking terms which we so far considered, since they preserve the above U(1); it can however get mass if we add a small SU(3)-breaking tadpole ∆Vsoft = −m3S Su + H.c., (2.16) which gives m2η = m3S /f. This term breaks Su → −Su symmetry and can be generated radiatively by adding to the superpotential a small term ∆W = m′ T tc breaking the same symmetry: m3S ≃ − 3y ′ 2 MGUT m mQ ln . 2π 2 F It should be noticed however that the discussed axion, even if exactly massless, would not be in conflict with experiment [11], since it couples very weakly to the ordinary matter (such a coupling 8 could for example proceed via mixing with heavy fermions which are needed to implement the SU(3) symmetry in the first and second generations). Our last comment concerns the impact of the triplets T1 , T2 on the precision electroweak 2 observables. According to Eq. (2.4) they get VEVs, Ti11 ≃ fZi fv 2 . Since according to our discussion below Eq. (2.12) we need fZ1 < 0.1f (and fZ2 is even smaller), the contribution of the triplets to the ρ parameter is sufficiently suppressed: 2  1.7 TeV (T111 )2 −4 . δρ = −2 2 , |δρ| < 10 v f We conclude that Model I is a fully consistent example of a supersymmetric model with a Higgs doublet as a Goldstone boson of extended global symmetry, perturbative up to the GUT scale and with large tan β. Its phenomenology is similar to phenomenology of the MSSM in the decoupling limit, but finetuning in the Higgs potential is diminished by at least a factor 10 compared to MSSM. The lightest Higgs mass is expected to be just above 115 GeV, with stop around 800 GeV, and the new top quark above 3 ÷ 4 TeV, probably unreachable at the LHC. However, stabilization of the SU(3) breaking potential requires new states. The extended scalar sector of Model I is unlikely to manifest itself at the LHC, since these particles are expected to be quite heavy (apart from the decoupled axion η), while their couplings to W W and tt̄ are suppressed due to large tan β and f /v ratio. Among new fermions, there are those with the same quantum numbers of MSSM chargino and neutralino states. However, the mixing between MSSM states and these new fermions is small because fZi ≪ f and the mass eigenstates remain almost MSSM-like. The new mass eigenstates, including T̃ ±± and z̃i ’s, are almost degenerate in mass, with masses m ≈ µZ ≈ 200 GeV. The details of the mass spectrum depend on the details of the mixing. Thus we expect new light fermions doubling the MSSM states, but the best chance to see a trace of the SU(3) structure is probably the doubly charged Higgsino. At the LHC, this double-chargino T̃ ±± will be pairproduced via the Drell-Yan process. It will most likely undergo a chain decay into T̃ ± and W and finally into the lightest neutralino and two same sign W bosons. Because of an approximate degeneracy of the mass spectra the decay products are likely to be off-shell. The double-chargino is likely to be long-lived and the decay vertex may be displaced. In any case, we expect events with two opposite-hemisphere pairs of same-sign leptons and missing transverse energy in the final state. We are not aware of experimental studies for signals of this type5 . By analogy with DrellYan chargino-neutralino searches, we can optimistically estimate the LHC integrated luminosity required for the discovery of T̃ ±± to be O(100) fb−1 . The fermionic neutral singlets z̃i ’s may also be phenomenologically interesting. Depending on the details of the mass spectrum one of them may be the LSP and a potential candidate for dark matter particle. Our model provides then a concrete example of a spectrum going beyond the MSSM spectrum. More detailed phenomenological study of such and similar spectra are of experimental interest but beyond the scope of this paper. 5 In [18] doubly charged Higgsinos with a significant coupling to leptons were considered, so that a dominant decay mode is into slepton-lepton pairs, with the slepton subsequently decaying into a lepton plus neutralino. This gives rise to a practically background-free same-sign, same-flavor lepton pair and missing energy signature. Such couplings in our model would violate the lepton number conservation and are by no means necessary (if allowed at all). Our case is definitely more challenging experimentally. 9 3 Model II It has been emphasized in [7] (following an earlier observation in [2]), that in models with extended global symmetry there exists also a mechanism for EWSB and the Higgs boson mass generation based on a tadpole contribution of an SU(2) × U(1) singlet component of the full scalar multiplet. This mechanism necessarily requires a small value of f , to minimize finetuning in the Higgs potential. In turn, this implies large quartic coupling and low UV cutoff. On the other hand, high enough Higgs boson mass can be produced even for moderate tan β. The simplest model achieving stabilization of SU(3) breaking at small f includes two Higgs triplets Hd and Hu , same as in Model I, and an SU(3) singlet N, with the superpotential W = κNHu Hd . As we will see, it generically leads to low values of tan β. We must have κ ≤ 2 for the Landau pole to be above ΛUV = 20 − 30 TeV; we will choose κ = 2 in what follows. All RG runnings UV below are considered from ΛUV down to the Fermi scale, log MΛSUSY ≃ 3 for MSUSY ∼ 1 TeV. The scalar potential reads: V = κ2 [|Hu Hd |2 + |N|2 (|Hu |2 + |Hd |2 )] + Vsoft Vsoft = m2u |Hu |2 + m2d |Hd |2 + m2N |N|2 − (A3 N + m23 Hu Hd + H.c.) (3.1) The masses m2u,d , m2N need to be positive to avoid runaway. We use m23 to break SU(3) spontaneously, while A3 will give a VEV to N and generate an effective µ-term (chargino mass). It is consistent to assume that all terms which are not included at the tree level remain small or vanish. E.g. A′ NH1 H2 term is generated by gaugino masses, g 2 κM2 δA ∼ × 3 ∼ 0.02M2 . 16π 2 ′ Possible modifications of the model can be obtained by adding µHu Hd and/or κF 2 N terms to the superpotential, as well as NHu Hd term to Vsoft . These modifications lead to models of comparable “quality”, and we will not consider them. Minimization of the potential (3.1) in the gauge singlet direction gives h|Hu,d|i ≡ fu,d , f 2 ≡ fu2 + fd2 = µ2u,d ≡ m2u,d + µ2 , m23 − µu µd , κ2 sin β cos β µ ≡ κfN , h|N|i ≡ fN = tan β ≡ µd fu = , fd µu (3.2) A3 , m2N + κ2 f 2 where we continue using notation of section 2 for the components of Hd and Hu . As in Model I, spontaneous breaking of SU(3) to SU(2) generates five Goldstone bosons, a doublet H and an axion. SU(3) symmetry must then be broken also explicitly, to get a potential for H, which will break electroweak symmetry. In this model the presence of the soft term m23 results in nonvanishing VEVs fu,d,N breaking the global SU(3). To avoid any risk of destabilization of the SU(3) breaking potential by a negative m2u , we keep the top-stop sector as in MSSM. Thus, 10 SU(3) is explicitly broken by the top-stop sector. The standard RG running from ΛUV generates a negative contribution to the Higgs mass squared
− δm2H |Hu |2 ≡ −δm2H |Hu |2 − |Su |2 (3.3) Another source of the explicit SU(3) breaking is a tadpole contribution m3S Su , which, we assume, is generated by strong dynamics at ΛUV 6 . Making use of (3.3), we can write the full SU(3) breaking potential as a function of Su :
∆V = m2H |Su |2 − m3S Su + H.c. , (3.4) which has the clear advantage of completely decoupling the SU(3)-symmetric potential minimization and vacuum disalignment. In this parametrization, we view the |Hu |2 contribution in (3.3) as a renormalization of m2u parameter in (3.1). Minimizing the CP-even part of (3.4), we find the VEV of Su : m3 h|Su |i = 2S , mH where we have to assume that the found minimum satisfies |Su | < fu , otherwise the true minimum will be at Su ≃ ±fu with no EWSB. On the other hand, |Su | < fu means vacuum disalignment, with the Higgs VEV h|Hu |2 i ≡ vu2 = fu2 − h|Su |2 i (3.5) and the EWSB scale given by " v 2 = vu2 + vd2 = f 2 1 −  m3S fu m2H 2 # . (3.6) We see that v ≪ f can be obtained only at the price of finetuning the ratio m3S /fu m2H to 1. To illustrate p this finetuning more clearly, we can express (3.4) as a function of |Hu | by expanding Su = |fu |2 − |Hu |2 , which is a good approximation for v ≪ f : ∆V ≃ const − (m2H − m3S /fu )|Hu |2 + λ|Hu |4 + . . . , λ= m3S . 4fu3 (3.7) We now see the origin of finetuning: it appears since we p are canceling the O(f 2) Higgs mass term with the quadratic term appearing in the expansion of fu2 − |Hu |2 , taking advantage of the non-linear structure of the σ-model. The finetuning discussed above can be quantified by means of the usual logarithmic derivative, or by measuring the portion of the uniformly distributed parameter space satisfying v ≤ 174 GeV; we get [7]7 2f 2 FT ≃ 2 . (3.8) v 6 Perturbative origin of the tadpole could be engineered if desired by breaking Su → −Su symmetry in the superpotential, similarly to the generation of (2.16) in Model I. 7 This finetuning does not increase even if mH and mS are scaled up, because the Higgs quartic λ in (3.7) and, correspondingly, the Higgs mass (see below), increases in the same limit, “improving naturalness” in the sense of [15]. 11 As a reference value we fix f = 350 GeV, corresponding to O(10)% finetuning8 . Other potential sources of finetuning in Model II are related to the RG running of the SU(3)symmetric potential parameters. According to Eq. (3.2) to avoid large cancellations in the potential for SU(3) breaking, we must have µu µd . κ2 f 2 /2 (3.9) i.e. effectively µ2 , m2u , m2d . κ2 f 2 /4 . As mentioned above, the |Hu |2 in Eq. (3.3) effectively renormalizes m2u ; naturalness thus requires that m2t̃ 3 ΛUV . m2u . κ2 f 2 /4 . (3.10) δm2H = 2 yt2 m2t̃ log ∼ 2 4π MSUSY 4 sin β Thus we get mt̃ . sin β κf . (3.11) Finally, we discuss the Higgs boson mass. The same expansion of the square root which allows us to finetune v ≪ f in (3.7), also generates a Higgs quartic λ. For small tan β this quartic easily dominates the standard D-term quartic; as a result the Higgs boson mass is solely determined by the soft terms and the coupling κ. Taking into account the σ-model wavefunction suppression and also using the exact expression of the Higgs potential (3.4) instead of expanding in Hu , we find the Higgs boson mass mh = sin β(mH /f )v . If we assume that m2H is entirely generated by stop loops we get mh = v mt̃ , 2 f √ and thus mt̃ ≃ 2f for mh ≃ 120 GeV, consistently with the constraint (3.11) on mt̃ for low tan β. The model then predicts a light Higgs boson since larger values of mt̃ /f are inconsistent with the constraint (3.11). We will discuss phenomenology of Model II for f = 350 GeV, 1 ≤ tan β ≤ 2, and κ = 2.We choose the potential parameters as follows (t ≡ tan β): κ2 f 2 /2 2t2 − 1 2 2 2 µ = = , md = 2 κ f /2 , 1 + t2 t +1 2κ2 f 2 µ m23 = , m2N = 6m2u , A3 = (m2N + κ2 f 2 ) , −1 t+t κ 2 m2u which is consistent with naturalness and produces a minimum of the potential at the given values of f and tan β. We see that Higgsinos are expected to be light, µ ∼ 100 ÷ 200 GeV, for f = 350 GeV. √ Comparing with [7], notice a factor 2 difference in normalization of f resulting from the change from real to complex fields. Our f = 350 GeV gives the same finetuning as f ≃ 500 GeV in [7]. 8 12 Furthermore, since f is small, phenomenology of the model is strongly influenced by its nonlinear structure. We recall that in the σ-model approximation ! |H| H sin( ) |H| f Hu = fu (3.12) cos( |H| ) f and similarly for Hd (|H| = a √ H † H). We also can parametrize the Higgs doublet H nonlinearly:   0 √ H=Σ , (3.13) v̄ + h/ 2 a where Σ = eiT G /v is the pion field containing the Goldstone bosons eaten by the W and Z. The fields are canonically normalized. The true electroweak scale reads v = f sin(v̄/f ). Additional heavy scalar modes describe deviation from the σ-model structure (3.12). For instance, fluctuations in the radial directions can be introduced by replacing √ (3.14) fu,d → fu,d + su,d / 2. We are interested in the couplings of the scalars h, su , sd to the vector boson pairs. By the equivalence theorem, these couplings can be obtained from the kinetic part of the Lagrangian for Hu and Hd inserting (3.13) into (3.12). The couplings of su,d are found using (3.14). We get:   √ v 2 2 2 L = v |Dµ Σ| + 2v|Dµ Σ| cos(v̄/f )h + (cos β su + sin β sd ) (3.15) f 1/2 We see that the h coupling to pions, and hence to W W, is suppressed by cos(v̄/f ) = (1 − v 2 /f 2 ) , and h unitarizes W W amplitude only partially. Unitarization is completed by the exchange of the heavy scalars. The su,dW W couplings appear because the radial directions obtain nonzero projection on the first two components of Hu,d when expanded around v 6= 0. The fields su and sd are not mass eigenstates. In the mass matrix, they mix with each other and also with the radial excitation of N. Thus, three heavy mass eigenstates complete the unitarization of W W scattering. Denoting the mass eigenstates by Si , i = 1, 2, 3, e.g. for f = 350 GeV, κ = 2 and tan β = 2 we get mSi ≃ (290, 850, 1000) GeV. The cubic W W -scalar interaction Lagrangian in this case is given by SM 2 L = gW W h [cos(v̄/f )h + (v/f )ci Si ]Wµ , c ≃ (0.86, −0.13, −0.5), (3.16) c2i = 1, so that the W W scattering is fully unitarized above mS3 . Other important couplings are the Si t̄t couplings as they determine the production rate of these scalars via the gluon fusion. They originate from the term v su yt √ tt̄, f 2 yt ≡ mt , v sin β which appears similarly to the su,d W W couplings discussed above. Due to the v/f coupling suppression, production rates of the heavy scalars via the gluon fusion, as well as via the vector boson 13 fusion, will be suppressed by at least one order of magnitude with respect to the corresponding production rates for the SM Higgs boson of the same mass. Nevertheless, at least the lightest of these heavy scalars should be quite easy to discover at the LHC in the gold-plated decays S1 → ZZ → 4l. Apart from the radial modes (3.14), another interesting heavy mode is a longitudinal fluctuation orthogonal to the pseudo-Goldstone mode: Hu = cos β H1 , Hd = − sin βH1 . (3.17) The SU(2) doublet H1 does not get a VEV and is decoupled from the vector boson pairs; it is analogous to the heavy MSSM doublet in the decoupling limit. It describes a degenerate heavy multiplet (H ± , H 0 , A0 ) of mass µu µd , m2H1 ≃ sin β cos β which can be found substituting (3.17) into the scalar potential. By (3.9), we expect mH1 = O(κf ) ∼ 700 GeV. The neutral members of this multiplet couple to tt̄ with strength cot β times the SM Higgs coupling. They will be produced via gluon fusion and will be seen as narrow resonances decaying into tt̄ pairs (total width around 30 GeV). Using the model-independent analysis of [16], we can estimate that O(10) fb−1 of integrated luminosity could be enough for their discovery at the LHC. Finally, we discuss the effect of the heavy scalars on the electroweak observables. As pointed out in Ref.[7], the relevant parameter is the effective Higgs boson mass, which in our case is given by  v2  Y 2 m̄ f 2 mEWPT = mh , m̄ = (mSi )ci , mh where the ci are the parameters appearing in the W W -scalar interaction Lagrangian (3.16). In our numerical example we get m̄ = 400 GeV and mEWPT = 155 GeV for mh = 115 GeV. Thus, mEWPT is slightly above the 144 GeV 95% C.L. limit, but various other supersymmetric contributions can easily compensate its effect in the (S, T ) plane. 4 Conclusions We have presented two realistic supersymmetric models with Higgs doublet as Goldstone boson of a spontaneously broken extended global symmetry. Model I is perturbative up to the GUT scale and realizes large tan β scenario, while Model II requires a rather low UV cut-off (∼ 20 TeV) and generically gives low tan β. Both models avoid excessive finetuning in the Higgs potential and are in fact motivated by this requirement. Being perturbative up to much higher cut-off than so-called “strongly interacting” models, they do not lead to any serious tension with precision electroweak data. The two models illustrate two different mechanisms for EWSB and the Higgs mass generation. Their experimental signatures are quite different. Clearly, the price for a small finetuning is some complexity (e.g. compared to the MSSM). Our constructions supplement the list of previous proposals for ameliorating the supersymmetric little hierarchy problem. E.g. the Next-to-Minimal Supersymmetric Standard Model easily solves the little hierarchy if its SHu Hd 14 coupling λ is allowed to become strong below the GUT scale (a possibility recently taken to the extreme in [17]). Its predictions are different from Model II as, for instance, it does not predict non-linear effects in the scalar couplings. We will wait and see what experiment tells us. For the moment, the main lesson of our constructions is that the possibility of the Higgs boson as a Goldstone boson in perturbative theories looks equally plausible as in non-perturbative scenarios with low cut-off and actually more predictive. 5 Acknowledgements V.S.R. was supported by the EU under RTN contract MRTN-CT-2004-503369 and ToK contract MTKD-CT-2005-029466. A Technical details on Model I In this appendix we collect some technical details relevant for Model I. It is convenient to normalize all charges and couplings with the UV completion into SU(3) × U(1)x in mind. For instance Hu has SU(3) × U(1)x quantum numbers 31/3 , DHu = (W a T a + 1/3Bx ) Hu . The unbroken generator is Y = √13 T 8 + X. The SU(3) gauge coupling g coincides at the scale F with the SU(2) gauge coupling g2 . The U(1)y coupling gy at F is: ggx . gy = p g 2 + gx2 /3 Numerical values of g and gx are 0.65 and 0.37 respectively. The RG equation for m23 in Eq. (2.2) valid above F reads 16π 2 16 dm23 = − g 2Mµ + U(1)x -gaugino contribution. d log Λ 3 Here M is the SU(3)-symmetric gaugino mass. For the running from the GUT scale we get δm23 ∼ 0.4Mµ . From (2.3), the natural value of tan β is m2d /δm23 . We see that tan β = O(10) is naturally allowed, provided that md is a factor of a few larger than M, µ. Running from F down generates only Hu Hd coefficient which is much smaller since the running is very short. Other mass parameters whose running is of interest for the model are m2u and m2Z2 . For log MGUT /F ∼ 30 one gets

1 1 2 2 2 2 2 2 2 6y m + m 16λ m + m × 30 c + mu × 30 + Q T u Z2 16π 2 16π 2
1 2 2 2 4λ m + m × 30 = u Z2 16π 2 δm2u = δm2Z2 15 The dominant contribution comes from the term proportional to the Yukawa coupling. It makes m2u negative and breaks the global SU(3) symmetry radiatively. Finally, we discuss the perturbativity constraint up to the GUT scale on the couplings λ and y. The RG equations read:   dy 16 2 2 2 2 2 2 16π = y 7y + 8λ − (g3 + g ) − O(gx) d log Λ 3   4 2 2 2 2 2 dλ = λ 18λ + 6y − 12g − gx 16π d log Λ 3 where g3 is the strong coupling constant. One can check that the safe range for values of the couplings at the scale F is y . 1.2, λ . 0.3. References [1] D. B. Kaplan and H. Georgi, “SU(2) X U(1) Breaking By Vacuum Misalignment,” Phys. Lett. B 136, 183 (1984). D. B. Kaplan, H. Georgi and S. Dimopoulos, “Composite Higgs Scalars,” Phys. Lett. B 136, 187 (1984). [2] K. Agashe, R. Contino and A. Pomarol, “The minimal composite Higgs model,” Nucl. Phys. B 719, 165 (2005) [arXiv:hep-ph/0412089]. [3] R. Contino, L. Da Rold and A. Pomarol, “Light custodians in natural composite Higgs models,” Phys. Rev. D 75, 055014 (2007) [arXiv:hep-ph/0612048]. [4] R. Contino, T. Kramer, M. Son and R. Sundrum, “Warped/Composite Phenomenology Simplified,” JHEP 0705, 074 (2007) [arXiv:hep-ph/0612180]. [5] G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, “The strongly-interacting light Higgs,”JHEP 0706, 045 (2007), arXiv:hep-ph/0703164. [6] A. Falkowski, S. Pokorski and J. P. Roberts, “Modelling strong interactions and longitudinally polarized vector boson scattering,” JHEP 0712, 063 (2007) [arXiv:0705.4653 [hep-ph]]. [7] R. Barbieri, B. Bellazzini, V. S. Rychkov and A. Varagnolo, “The Higgs boson from an extended symmetry,” Phys. Rev. D 76, 115008 (2007) arXiv:0706.0432 [hep-ph]. [8] C. Csaki, J. Heinonen, M. Perelstein and C. Spethmann, “A Weakly Coupled Ultraviolet Completion of the Littlest Higgs with T-parity,” arXiv:0804.0622 [hep-ph]. [9] A. Birkedal, Z. Chacko and M. K. Gaillard, “Little supersymmetry and the supersymmetric little hierarchy problem,” JHEP 0410, 036 (2004) [arXiv:hep-ph/0404197]. [10] P. H. Chankowski, A. Falkowski, S. Pokorski and J. Wagner, “Electroweak symmetry breaking in supersymmetric models with heavy scalar superpartners,” Phys. Lett. B 598, 252 (2004) [arXiv:hep-ph/0407242]. [11] Z. Berezhiani, P. H. Chankowski, A. Falkowski and S. Pokorski, “Double protection of the Higgs potential,” Phys. Rev. Lett. 96, 031801 (2006) [arXiv:hep-ph/0509311]. [12] T. Roy and M. Schmaltz, “Naturally heavy superpartners and a little Higgs,” JHEP 0601, 149 (2006) [arXiv:hep-ph/0509357]. 16 [13] C. Csaki, G. Marandella, Y. Shirman and A. Strumia, “The super-little Higgs,” Phys. Rev. D 73, 035006 (2006) [arXiv:hep-ph/0510294]. [14] A. Falkowski, S. Pokorski and M. Schmaltz, “Twin SUSY,” Phys. Rev. D 74, 035003 (2006) [arXiv:hep-ph/0604066]. [15] R. Barbieri, L. J. Hall and V. S. Rychkov, “Improved naturalness with a heavy Higgs: An alternative road to LHC physics,” Phys. Rev. D 74, 015007 (2006) [arXiv:hep-ph/0603188]. [16] V. Barger, T. Han and D. G. E. Walker, “Top Quark Pairs at High Invariant Mass - A ModelIndependent Discriminator of New Physics at the LHC,” Phys. Rev. Lett. 100, 031801 (2008) [arXiv:hep-ph/0612016]. [17] R. Barbieri, L. J. Hall, Y. Nomura and V. S. Rychkov, “Supersymmetry without a light Higgs boson,” Phys. Rev. D 75, 035007 (2007) [arXiv:hep-ph/0607332]. [18] D. A. Demir, M. Frank, K. Huitu, S. K. Rai and I. Turan, “Signals of Doubly-Charged Higgsinos at the CERN Large Hadron Collider,” arXiv:0805.4202 [hep-ph]. 17