PPrepared for submission to JHEP
A More Natural Composite Higgs Model
Hsin-Chia Cheng and Yi Chung
Center for Quantum Mathematics and Physics (QMAP), Department of Physics,University of California, Davis, CA 95616, U.S.A.
E-mail: [email protected] , [email protected] Abstract:
Composite Higgs models provide an attractive solution to the hierarchy prob-lem. However, many realistic models suffer from tuning problems in the Higgs potential.There are often large contributions from the UV dynamics of the composite resonances tothe Higgs potential, and tuning between the quadratic term and the quartic term is re-quired to separate the electroweak breaking scale and the compositeness scale. We considera composite Higgs model based on the SU (6) /Sp (6) coset, where an enhanced symmetry onthe fermion resonances can minimize the Higgs quadratic term. Moreover, a Higgs quarticterm from the collective symmetry breaking of the little Higgs mechanism can be realizedby the partial compositeness couplings between elementary Standard Model fermions andthe composite operators, without introducing new elementary fields beyond the StandardModel and the composite sector. The model contains two Higgs doublets, as well as severaladditional pseudo-Nambu-Goldstone bosons. To avoid tuning, the extra Higgs bosons areexpected to be relatively light and may be probed in the future LHC runs. The deviationsof the Higgs couplings and the weak gauge boson couplings also provide important tests asthey are expected to be close to the current limits in this model. a r X i v : . [ h e p - ph ] J u l ontents SU (6) /Sp (6) Composite Higgs Model 6 SU (6) /Sp (6) Zf ¯ f Couplings 28 SU (5) /SO (5) Composite Higgs Model 31B Couplings between SM Fermions and Composite Operators, and TheirPeccei-Quinn Charges 32 – 1 –
Introduction
The Standard Model (SM) of particle physics successfully describes all known elementaryparticles and their interactions. At the center of SM is the mechanism of electroweaksymmetry breaking (EWSB), which is responsible for the masses of gauge bosons andfermions. The discovery of Higgs bosons in 2012 [1, 2] filled in the last missing piece ofthe SM. However, the Higgs boson itself brings new questions and puzzles that need to beanswered. As a minimal model to realize EWSB, the Higgs field is characterized by thepotential V ( H ) = − µ | H | + λ | H | (1.1)with just two parameters. The two parameters are now fixed by the observed Higgs vacuumexpectation value (VEV) v (cid:39) GeV and Higgs boson mass M h (cid:39) GeV as µ (cid:39) (88 GeV ) , λ (cid:39) . . (1.2)However, SM does not address the UV-sensitive nature of scalar bosons. The Higgsmass-squared receives quadratically divergent radiative corrections from the interactionswith SM fields, which leads to the well-known hierarchy problem. To avoid the largequadratic corrections, the most natural way is to invoke some new symmetry such that thequadratic contributions cancel in the symmetric limit. This requires the presence of newparticles related to SM particles by the new symmetry, such as top partners, in order tocut off the divergent loop contributions.One such appealing solution to the hierarchy problem is the composite Higgs model(CHM), where the Higgs doublet is the pseudo-Nambu-Goldstone boson (pNGB) of a spon-taneously broken global symmetry of the underlying strong dynamics [3, 4]. Through theanalogy of the chiral symmetry breaking in quantum chromodynamics (QCD), which nat-urally introduces light scalar fields, i.e., pions, we can construct models with light Higgsbosons in a similar way. In a CHM, an approximate global symmetry G is spontaneouslybroken by some strong dynamics down to a subgroup H with a symmetry breaking scale f .The heavy resonances of the strong dynamics are expected to be around the compositenessscale ∼ πf generically. The pNGBs of the symmetry breaking, on the other hand, can nat-urally be light with masses < f as they are protected by the shift symmetry. The potentialof the Higgs field arises from the explicit symmetry breaking effects, such as the interactionswith other SM fields. The largest coupling of the Higgs field in SM is to the top quark. Asa result, for naturalness, the top partners which regulate the top loop contribution to theHiggs potential should not be too heavy. The top loop contribution to the Higgs mass termcan be estimated as ∆ µ ∼ N c π y t M T ∼ (220 GeV ) (cid:18) M T . TeV (cid:19) , (1.3)where M T is the top partner mass. On the other hand, the bounds on the SM coloredtop partners have reached beyond 1 TeV from the collider searches [5, 6]. Compared withEq.(1.2), we see that the models with colored top partners (including both the minimal– 2 –upersymmetric standard model (MSSM) and the CHM) already require some unavoidable O (10%) tuning, albeit not unimaginable.In most CHMs, however, the tuning is much worse than that is shown in Eq. (1.3).Depending on the coset G/H and the representations of composite operators that coupleto the top quarks, the strongly interacting resonances of the top sector in the UV oftengive a bigger contribution to the Higgs potential than Eq. (1.3), which requires more tuningto cancel. Another problem is that, unlike the pions, the Higgs field needs to develop anonzero VEV v . The current experimental constraints require v < f / . On the other hand,for a generic pNGB potential, the natural VEV for the pNGB is either 0 or f . To obtaina VEV much less than f , a significant quartic Higgs potential compared to the quadraticterm is needed. In little Higgs models [7–9], a Higgs quartic term can be generated withoutinducing a large quadratic term from the collective symmetry breaking. Such a mechanismis not present in most CHMs, which is another cause of the fine-tuning issue.In this study, our goal is to find a more natural CHM by removing the additional tuningbeyond Eq. (1.3). We first identify the cosets and the composite operator representationsthat couple to the top quarks, which can preserve a larger symmetry for the resonances tosuppress the UV contribution to the Higgs potential. Next, we implement the collectivesymmetry breaking to generate a Higgs quartic potential while keeping the quadratic termat the level of Eq. (1.3). In this way we can naturally separate the scales of v and f ,resulting in a more natural CHM.This paper is organized as follows. In section 2, we review the tuning problems inCHMs and identify the sources of the extra tuning, using the SO (5) /SO (4) CHMs as anexample. In section 3, we introduce the SU (6) /Sp (6) CHM, including the interactions thatproduce the SM Yukawa couplings, and show how the large UV contribution to the Higgspotential is avoided. We then move on to the next step to generate an independent Higgsquartic term from collective symmetry breaking in section 4. The resulting Higgs potentialof the 2HDM is discussed in section 5. The complete potential and spectrum of all thepNGBs in our model are summarized in section 6 with numerical estimation. Section 7and Section 8 are devoted to the phenomenology of this model. Section 7 focuses on thecollider searches and constraints. The analyses of the indirect constraints from the precisionexperimental measurements are presented in Section 8. Section 9 contains our summariesand conclusions. In Appendix A we briefly discuss the possibility of constructing a similarmodel based on the SU (5) /SO (5) coset. We point out the differences and some drawbacksof such a model. Appendix B contains the details of the interactions between elementaryfermions and composite operators for a realistic implementation of the SU (6) /Sp (6) CHMmodel.
We first give a brief review of the tuning problem of the Higgs potential in general CHMs,which was comprehensively discussed in Ref. [10, 11]. This will help to motivate pos-sible solutions. As an illustration, we consider the Minimal Composite Higgs Models(MCHMs) [12] with the symmetry breaking SO (5) → SO (4) . The four pNGBs are iden-– 3 –ified as the SM Higgs doublet. The SM gauge group SU (2) W × U (1) Y is embedded in SO (5) × U (1) X , with the extra U (1) X accounting for the hypercharges of SM fermions.The explicit breaking of the global symmetry introduces a pNGB potential such that atthe minimum the SO (5) breaking VEV f is slightly rotated away from the direction thatpreserves the SU (2) W × U (1) Y gauge group. The misalignment leads to the EWSB at ascale v (cid:28) f .The explicit global symmetry breaking comes from SM gauge interactions and Yukawainteractions. The SM Yukawa couplings arise from the partial compositeness mechanism [13]:elementary fermions mix with composite operators of the same SM quantum numbers fromthe strong dynamics, L = λ L ¯ q L O R + λ R ¯ q R O L , (2.1)where q L , q R are elementary fermions and O L , O R are composite operators of some repre-sentations of G ( = SO (5) in MCHMs). The values of couplings λ L , λ R depend on the UVtheory of these interactions and are treated as free parameters to produce viable models.With these interactions, the observed SM fermions will be mixtures of elementary fermionsand composite resonances. The SM fermions can then couple to the Higgs field through theportion of the strong sector with couplings given by y (cid:39) λ L λ R g ψ (cid:39) (cid:15) L · g ψ · (cid:15) R , (2.2)where g ψ is a coupling of the strong resonances and is expected to be (cid:29) , (cid:15) L,R are ratios λ L,R /g ψ , which are expected to be small. The resonances created by O L,R have masses ∼ g ψ f , and play the roles of SM fermion partners. They cut off the divergent contributionsto the Higgs potential and make it finite. Notice that the operators belong to representationsof the global symmetry G , but the resonances are divided into representations of H afterthe symmetry breaking. Because the elementary fermions in general do not fill the wholerepresentations of G , the partial compositeness couplings λ L , λ R explicitly break the globalsymmetry G and generate a nontrivial Higgs potential.The pNGB Higgs field parametrizes the coset G/H so the potential is periodic in theHiggs field. The Higgs potential can be expanded in sin(
H/f ) and up to the quartic termit takes the form V ( H ) = − ˆ αf sin Hf + ˆ βf sin Hf , (2.3)where ˆ α and ˆ β have mass dimension two and ˆ α corresponds to the mass-squared parameterof the Higgs field while ˆ β/f will contribute to the quartic term. By expanding sin( H/f ) ,higher powers of H can be generated from each term, but for convenience, we will simplycall the first term quadratic term and the second term quartic term. The parameters ˆ α and ˆ β are model dependent and are generated by explicit breaking parameters, like λ L and λ R .Given the potential, we can get the VEV and Higgs mass parameterized as v = (cid:115) ˆ α β f, M h = 8 ˆ β v f (1 − v f ) . (2.4)– 4 –he misalignment of the minimum from the SM gauge symmetry preserving direction isparametrized by ξ ≡ v f = sin (cid:104) θ (cid:105) = ˆ α β (cid:28) , (2.5)where angle (cid:104) θ (cid:105) ≡ (cid:104) h (cid:105) /f . Therefore, for a realistic model, we need ˆ α (cid:28) ˆ β and at the sametime, the correct size of ˆ β to get the observed Higgs boson mass M h (cid:39) GeV.From the most explicit symmetry breaking effects of the composite Higgs models,one typically gets ˆ α > ˆ β , which is the source of the tuning problem. For example, inMCHM [11, 12], the SM fermions mix with composite operators O L , O R ∈ of SO (5) .After the symmetry breaking, the composite resonances split into and representationsof SO (4) . The mass difference between and resonances generates a Higgs potential atthe compositeness (UV) scale with ˆ α ∼ N c π λ L,R M ψ ∼ (cid:15) L,R N c g ψ π f , (2.6a) ˆ β ∼ N c π λ L,R f ∼ (cid:15) L,R N c g ψ π f . (2.6b)The quartic term coefficient ˆ β arises at a higher order in (cid:15) than ˆ α , so generically ˆ β (cid:28) ˆ α is expected instead. It is then required more fine-tuning to achieve the correct EWSB. Insome models, it is possible to have ˆ α ∼ ˆ β . For example, MCHM [10] with O L , O R ∈ of SO (5) can lead to the potential with ˆ α ∼ ˆ β ∼ N c π λ L,R M ψ ∼ (cid:15) L,R N c g ψ π f , (2.7)where ˆ β arises at the same order as ˆ α . It requires less tuning to achieve ξ (cid:28) . This hasbeen called “minimal tuning.” But even so, the UV contribution of Eq. (2.7) to ˆ α is largerthan the IR contribution from the top quark loop ∆ m IR ∼ N c π y t M T ∼ (cid:15) L,R N c g ψ π f , (2.8)which already requires some levels of fine-tuning as shown in Eq. (1.3). This additional UVcontribution actually worsens the condition and requires more tuning. A less-tuned scenariois to have a composite right-handed top quark (which is a singlet of G ). In this case, (cid:15) R ∼ but does not contribute to the Higgs potential. The Higgs potential is controlled by λ L ∼ y t ,which can be smaller.From the above discussion, one can see that to obtain a more natural Higgs potentialin CHM, it would be desirable to suppress the contribution from the composite top-partnerresonances to the quadratic term. For example, a maximal symmetry was proposed inRef. [14] to keep the degeneracy of the whole G representation of the top-partner reso-nances. However, the maximal symmetry is somewhat ad hoc within a simple model andits natural realization requires more complicated model constructions by doubling the globalsymmetry groups or invoking a holographic extra dimension [15, 16]. We will look for cosets– 5 – /H such that the representation of the top-partner resonances do not split even after thesymmetry breaking of G → H so that it preserves a global symmetry G in any singlepartial compositeness coupling to prevent unwanted large contributions to the Higgs poten-tial. Besides, we need some additional contribution to the quartic term without inducingthe corresponding quadratic term simultaneously to make ˆ β > ˆ α naturally. This may beachieved by the collective symmetry breaking of the little Higgs mechanism [7–9]. Previousattempts include adding exotic elementary fermions to an SU (5) /SO (5) CHM model [17]and a holographic model with double copies of the global symmetry [18]. Another way ofgenerating the quartic term without the quadratic term using the Higgs dependent kineticmixing requires both new elementary fermions and an enlarged global symmetry or an extradimension [19]. We will take a more economical approach by implementing the little Higgsmechanism without adding exotic elementary fermions or invoking multiple copies of theglobal symmetry, but simply using the couplings that mix SM fermions with compositeresonances. SU (6) /Sp (6) Composite Higgs Model
Among the possible cosets, the cosets SU (5) /SO (5) and SU (6) /Sp (6) are potential can-didates to realize the ideas discussed at the end of the previous section. If the compositeoperator O L,R ∈ ( ) of SU (5)( SU (6)) , the corresponding resonances do not split under theunbroken subgroup SO (5)( Sp (6)) . Since they are still complete multiplets of G , there isan enhanced symmetry for each mixing coupling λ L,R , which protects the pNGB potential.The cosets were also some earliest ones employed in little Higgs models [9, 20] where thecollective symmetry breaking for the quartic coupling was realized. In CHMs, it requiresdifferent explicit implementations if no extension of the SM gauge group or extra elementaryfermions are introduced. The SU (5) /SO (5) model has a general problem that an SU (2) triplet scalar VEV violates the custodial SU (2) symmetry, leading to strong experimentalconstraints. We will focus on the SU (6) /Sp (6) model here and leave a brief discussion ofthe SU (5) /SO (5) model in Appendix A. SU (6) /Sp (6) To parametrize the SU (6) /Sp (6) non-linear sigma model, we can use a sigma field Σ ij , whichtransforms as an anti-symmetric tensor representation of SU (6) , where i, j = 1 , . . . are SU (6) indices. The transformation can be expressed as Σ → g Σ g T with g ∈ SU (6) or as Σ ij → g ik g jl Σ kl with indices explicitly written out. The scalar field Σ has an anti-symmetricVEV (cid:104) Σ (cid:105) = Σ αβ (with α , β representing Sp (6) index), where Σ = (cid:32) − II (cid:33) , (3.1) Naïvely they can split into two real representations, but if they carry charges under the extra U (1) X gauge group which is required to obtain the correct hypercharge, they need to remain complex. A CHM with the SU (6) /Sp (6) coset were considered in Ref. [21], but for a different prospect. – 6 –nd I is the × identity matrix. The Σ VEV breaks SU (6) down to Sp (6) , producing 14Nambu-Goldstone bosons.The 35 SU (6) generators can be divided into the unbroken ones and broken ones witheach type satisfying (cid:40) unbroken generators T a : T a Σ + Σ T Ta = 0 , broken generators X a : X a Σ − Σ X Ta = 0 . (3.2)The Nambu-Goldstone fields can be written as a matrix with the broken generator: ξ ( x ) = ξ iα ( x ) ≡ e iπa ( x ) Xa f . (3.3)Under SU (6) , the ξ field transforms as ξ → gξh † where g ∈ SU (6) and h ∈ Sp (6) , so ξ carries one SU (6) index and one Sp (6) index. The relation between ξ and Σ field is givenby Σ( x ) = Σ ij ( x ) ≡ ξ Σ ξ T = e iπa ( x ) Xa f Σ e iπa ( x ) XTa f = e iπa ( x ) Xaf Σ . (3.4)The complex conjugation raises or lowers the indices. The fundamental representation of Sp (6) is (pseudo-)real and the Sp (6) index can be raised or lowered by Σ αβ or Σ ,αβ .The broken generators and the corresponding fields in the matrix can be organized asfollows ( (cid:15) = iσ ): π a X a = √ φ a σ a − η √ H (cid:15)s H H † η √ − H T (cid:15) T s ∗ − H ∗ √ φ a σ a ∗ − η √ H ∗ H † H T η √ . (3.5)In this matrix, there are 14 independent fields. They are (under SU (2) W ): a real triplet φ a , a real singlet η , a complex singlet s , and two Higgs (complex) doublets H and H .We effectively end up with a two-Higgs-doublet model (2HDM). The observed Higgs bosonwill correspond to a mixture of h and h inside two Higgs doublets H = H / ⊃ √ (cid:0) h (cid:1) and H = H − / ⊃ √ (cid:0) h (cid:1) . Using the Nambu-Goldstone matrix, we can construct the lowenergy effective Lagrangian for the Higgs fields and all the other pNGBs. The SM electroweak gauge group SU (2) W × U (1) Y is embedded in SU (6) × U (1) X withgenerators given by SU (2) W : 12 σ a − σ a ∗
00 0 0 0 , U (1) Y : 12 − + X I . (3.6)– 7 –he extra U (1) X factor accounts for the different hypercharges of the fermion representa-tions but is not relevant for the bosonic fields. These generators belong to Sp (6) × U (1) X and not broken by Σ . Using the Σ field, the Lagrangian for kinetic terms of Higgs bosoncomes from L h = f tr (cid:104) ( D µ Σ)( D µ Σ) † (cid:105) + · · · , (3.7)where D µ is the electroweak covariant derivative. Expanding this, we get L h = 12 ( ∂ µ h )( ∂ µ h ) + 12 ( ∂ µ h )( ∂ µ h ) + f g W (cid:32) sin (cid:112) h + h √ f (cid:33) (cid:20) W + µ W − µ + Z µ Z µ cos θ W (cid:21) . (3.8)The non-linear behavior of Higgs boson in CHM is apparent from the dependence of trigono-metric functions.The W boson acquires a mass when h and h obtain nonzero VEVs V and V of m W = f g W (cid:32) sin (cid:112) V + V √ f (cid:33) = 14 g W ( v + v ) = 14 g W v , (3.9)where v i ≡ √ f V i (cid:112) V + V sin (cid:112) V + V √ f ≈ V i = (cid:104) h i (cid:105) . (3.10)The parameter that parametrizes the nonlinearity of the CHM is given by ξ ≡ v f = 2 sin (cid:112) V + V √ f . (3.11) SM gauge interactions explicitly break the SU (6) global symmetry, so they contribute tothe potential of the Higgs fields as well as other pNGBs. SM gauge bosons couple to pNGBsthrough the mixing with composite resonances: L = gW µ,a J µ,aW + g (cid:48) B µ J µY . (3.12)The J W and J Y belong to the composite operators in an adjoint representation of SU (6) . After the symmetry breaking, the composite operators are decomposed into and of Sp (6) . The masses of composite resonances of different representations of Sp (6) arein general different and this will generate a potential for pNGBs at O ( g ) . For SU (2) W ,it only breaks the global symmetry partially and generates mass terms for the two Higgsdoublets and the scalar triplet φ : SU (2) W : (for H , H ) c w π g g ρ f ≈ c w π g M ρ , (3.13)(for φ ) c w π g g ρ f ≈ c w π g M ρ , (3.14)where g ρ f ∼ M ρ is the mass of the vector resonances ρ which act as the gauge bosonpartners to cut off the SU (2) W gauge loop contribution to the pNGB masses, and c w is– 8 – O (1) constant. Similarly, for U (1) Y , the interaction also breaks the global symmetrypartially. It only generates mass terms for H , H : U (1) Y : c (cid:48) π g (cid:48) g ρ f ≈ c (cid:48) π g (cid:48) M ρ , (3.15)where c (cid:48) is also an O (1) constant.Combining these two contributions, we get the mass terms of the pNGBs from thegauge contributions at the leading order as M η = M s = 0 , M φ = c w π g M ρ ,M H = M H = c w π g M ρ + c (cid:48) π g (cid:48) M ρ ≈ (cid:18) g + g (cid:48) ( c (cid:48) /c w )8 g (cid:19) M φ . (3.16)From the gauge contributions only, we expect that M φ > M H = M H and they are belowthe symmetry breaking scale f . The SU (2) W × U (1) Y singlets s and η do not receive massesfrom the gauge interactions at this order, but they will obtain masses elsewhere which willbe discussed later. For partial compositeness, the elementary quarks and leptons couple to composite operatorsof G = SU (6) . To be able to mix with the elementary fermions, the representations of thecomposite operators must contain states with the same SM quantum numbers as the SMfermions. For our purpose, we can consider and ¯6 of SU (6) as they don’t split underthe Sp (6) subgroup. To account for the correct hypercharge, e.g., q L = 2 / , q R = 1 / for up-type quarks and q R = 1 − / for down-type quarks, the composite operators needto carry additional charges under the U (1) X outside SU (6) and the SM hypercharge is alinear combination of the SU (6) generator diag (0 , , / , , , − / and X . The compositeoperator as a / of SU (6) (where the subscript / denotes its U (1) X charge) can bedecomposed under SM SU (2) W × U (1) Y gauge group as O iL,R ∼ ξ iα Q αL,R ∼ / = / ⊕ / ⊕ ¯2 / ⊕ − / , (3.17)where Q L,R are the corresponding composite resonances. The composite states Q L,R createdby these operators belong to the representations of Sp (6) and play the roles of SM fermioncomposite partners. For SU (2) , and ¯2 are equivalent and related by the (cid:15) tensor. Wemake the distinction to keep track of the order of the fermions in a doublet. We see thatthe composite states have the appropriate quantum numbers to mix with the SM quarks.The left-handed elementary top quark can mix with either the first two componentsor the 4th and 5th components of the sextet. If we assume that it couples to the first twocomponents, the mixing term can be expressed as λ L ¯ q La Λ ai O iR = λ L ¯ q La Λ ai (cid:0) ξ iα Q αR (cid:1) (3.18)where a represents an SU (2) W index, and (Λ) ai = Λ = (cid:32) (cid:33) (3.19)– 9 –s the spurion which keeps track of the symmetry breaking.To get the top Yukawa coupling, we couple the elementary right-handed quark to the ¯6 / , which decomposes under SU (2) W × U (1) Y as O (cid:48) L,Rj ∼ ξ ∗ j β Σ βα Q αL,R ∼ ¯6 / = ¯2 / ⊕ − / ⊕ / ⊕ / . (3.20)The right-handed top quark mixes with the last component of the ¯6 / , which can be writtenas λ t R ¯ t R Γ t R j O (cid:48) Lj = λ t R ¯ t R Γ t R j (cid:16) ξ ∗ j β Σ βα Q αL (cid:17) , (3.21)where Γ t R = (0 0 0 0 0 1) is the corresponding spurion.Combining λ L and λ t R couplings, we can generate the SM Yukawa coupling for the topquark (and similarly for other up-type quarks), ∼ λ L λ t R ¯ q La Λ ai ξ iα Σ αβ ξ Tβ j Γ † t R j t R = λ L λ t R ¯ q La Λ ai Σ ij Γ † t R j t R ⊃ λ L λ t R (¯ q L H t R ) . (3.22)Similarly, for the bottom quark (or in general down-type quarks), we can couple b R tothe third component of ¯6 / with the coupling λ b R and spurion Γ b R = (0 0 1 0 0 0) . Thisgenerates a bottom Yukawa coupling of ∼ λ L λ b R ¯ q La Λ ai ξ iα Σ αβ ξ Tβ j Γ † b R j b R = λ L λ b R ¯ q La Λ ai Σ ij Γ † b R j b R ⊃ λ L λ b R (¯ q L H b R ) . (3.23)Alternatively, we could also couple the left-handed elementary quarks to ¯6 / and right-handed elementary quarks to / , λ (cid:48) L ¯ q La (cid:15) ab Ω bi O (cid:48) Ri = λ (cid:48) L ¯ q La (cid:15) ab Ω bi (cid:16) ξ ∗ i β Σ βα Q αR (cid:17) , (3.24)where (Ω) ai = Ω = (cid:32) (cid:33) (3.25)and λ (cid:48) b R ¯ b R Γ (cid:48) b R j O jL = λ (cid:48) b R ¯ b R Γ (cid:48) b R j (cid:0) ξ jα Q αL (cid:1) , (3.26)where Γ (cid:48) b R = (0 0 0 0 0 1) . Combining λ (cid:48) L and λ (cid:48) b R coupling, we can generate the SMYukawa coupling for bottom quark as ∼ λ (cid:48) L λ (cid:48) b R ¯ q La (cid:15) ab Ω bi ξ ∗ i β Σ βα ξ † αj Γ (cid:48)∗ jb R b R = λ (cid:48) L λ (cid:48) b R ¯ q La (cid:15) ab Ω bi Σ † ij Γ (cid:48)∗ jb R b R ⊃ λ (cid:48) L λ (cid:48) b R (cid:16) ¯ q L ˜ H b R (cid:17) , (3.27) If we had coupled the left-handed quarks to the 4th and 5th components of O R , ˜ λ L ¯ q La (cid:15) ab Λ (cid:48) bi O iR = ˜ λ L ¯ q La (cid:15) ab Λ (cid:48) bi (cid:16) ξ iα Q αR (cid:17) + h.c. , with the spurion (Λ (cid:48) ) bi = Λ (cid:48) = (cid:32) (cid:33) . The combination of ˜ λ L and λ t R would generate an up-type Yukawa coupling with H , ∼ ˜ λ L λ t R (cid:16) ¯ q L ˜ H t R (cid:17) . – 10 –here ˜ H ≡ (cid:15)H ∗ . In this case, the bottom mass also comes from VEV of H . Note thatthe combination of λ L and λ (cid:48) b R (or λ (cid:48) L and λ b R ) does not generate the SM Yukawa couplingbecause it does not depend on Σ .The lepton Yukawa couplings can be similarly constructed by coupling elementaryleptons to and ¯6 with X = − / . In 2HDMs, if the SM quarks have general couplingsto both Higgs doublets, large tree-level flavor-changing effects can be induced. To avoidthem, it is favorable to impose the natural flavor conservation [22, 23] such that all up-type quarks couple to one Higgs doublet and all down-type quarks couple to either thesame Higgs doublet (Type-I) or the other Higgs doublet (Type-II or flipped dependingon the lepton assignment). We can obtain all different possibilities by choosing the partialcompositeness couplings. For Type-II and flipped models, the b → sγ put strong constraintson the charged Higgs boson mass ( (cid:38) GeV) [24] which would require more tuning inthe Higgs potential. Therefore, we will assume the Type-I 2HDM for the remaining of thepaper, with the top Yukawa coupling coming from λ L λ t R and the bottom Yukawa couplingcoming from λ (cid:48) L λ (cid:48) b R . The partial compositeness coupling λ L or λ R individually cannot generate a potential for thepNGBs by itself, because the coupling Eq. (3.18) [or (3.21)] preserves an SU (6) symmetryrepresented by the α index. Although α is an Sp (6) index, without Σ , it cannot distinguish Sp (6) from SU (6) . To generate a nontrivial Higgs potential, we need at least an insertionof Σ , which distinguishes Sp (6) from SU (6) . It first arises through the combination of λ L and λ R in Eq. (3.22), which is just the top Yukawa coupling. Therefore, the first nontrivialHiggs potential shows up at the next order, i.e., O ( λ L λ R ) , as ∼ λ L λ R (cid:12)(cid:12)(cid:12) (Λ) ai (Γ ∗ ) j Σ ij (cid:12)(cid:12)(cid:12) (3.28)It gives a contribution to the H squared-mass term of the order ∆ M H ∼ − N c π λ L λ R f ∼ − N c π y t M T , (3.29)which is the same as the IR contribution from the top loop estimated in Eq. (1.3). Therefore,in this model, we avoid the potentially large O ( λ ) UV contribution and achieve the minimaltuning for the quadratic part of the Higgs potential.
In the previous section, we show that in the SU (6) /Sp (6) CHM the UV contribution fromthe strong dynamics to the Higgs potential is suppressed, minimizing the tuning of thequadratic term. However, we need some additional quartic Higgs potential to further reducethe tuning and to obtain a 125 GeV Higgs boson, as the IR contribution from the top quarkloop to the Higgs quartic term is not enough. Generating a Higgs quartic coupling withoutinducing the corresponding quadratic term is the hallmark of the little Higgs mechanism.For example, in the original SU (6) /Sp (6) little Higgs model [20], a Higgs quartic term from– 11 –he collective symmetry breaking can be generated by gauging two copies of SU (2) , withgenerators given by Q a = 12 σ a ×
00 0 0 0 and Q a = − × σ a ∗
00 0 0 0 (4.1)and gauge couplings g and g . The two SU (2) ’s are broken down to the diagonal SU (2) W by the Σ VEV. The potential for the pNGBs generated by the two gauge couplings takesthe form g f (cid:12)(cid:12)(cid:12)(cid:12) s + i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) + g f (cid:12)(cid:12)(cid:12)(cid:12) s − i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) . (4.2)The g term preserves the SU (4) symmetry of the , , , entries which contains the shiftsymmetry of H and H . If only the first term of the potential exists, the ˜ H † H dependencecan be absorbed into s by a field redefinition and the term just corresponds to a mass termfor s . Similarly, the g term preserves the SU (4) symmetry of the , , , entries underwhich H and H remain as Nambu-Goldstone bosons, but with a different shift symmetry.The combination of both terms breaks either of the shift symmetries, and a quartic Higgspotential is generated after integrating out the s field, λ (cid:12)(cid:12)(cid:12) ˜ H † H (cid:12)(cid:12)(cid:12) with λ = g g g + g . (4.3)The possibility of gauging two copies of SU (2) gauge group is subject to the strongexperimental constraints on W (cid:48) and Z (cid:48) . We would like to generate the quartic Higgspotential without introducing additional elementary fields to the SU (6) /Sp (6) CHM, so wewill consider the collective symmetry breaking from the interactions between the elementaryfermions and the resonances of the strong dynamics.From the discussion of the previous section, we see that the elementary quark doubletsmay couple to composite operators of SU (6) representations and/or ¯6 , and each containstwo doublets of the same SM quantum numbers: / = / ⊕ / ⊕ ¯2 / ⊕ − / , (4.4a) ¯6 / = ¯2 / ⊕ − / ⊕ / ⊕ / . (4.4b)Both operators can create the same resonances which belong to of the Sp (6) group.Now consider two elementary quark doublets couple to the first two components of thecomposite operators of and ¯6 respectively, while both representations contain the sameresonances: λ L ¯ q La Λ ai O iR = λ L ¯ q La Λ ai (cid:0) ξ iα Q αR (cid:1) , (4.5)where (Λ) ai = Λ = (cid:32) (cid:33) , (4.6)– 12 –nd λ (cid:48) L ¯ q (cid:48) La (cid:15) ab Ω bi O (cid:48) Ri = λ (cid:48) L ¯ q (cid:48) La (cid:15) ab Ω bi (cid:16) ξ ∗ i β Σ βα Q αR (cid:17) , (4.7)where (Ω) ai = Ω = (cid:32) (cid:33) . (4.8)The combination of the two interactions breaks the SU (6) global symmetry explicitly butpreserves an SU (4) symmetry of the , , , entries. It leads to a potential for the pNGBsat O ( λ L λ (cid:48) L ) of the form [(Λ) ai (Ω ∗ ) bj Σ ij ][(Ω) bm (Λ ∗ ) an Σ ∗ mn ] , (4.9)which can easily be checked by drawing a one-loop diagram, with q L , q (cid:48) L , Q R running inthe loop. After expanding it we obtain ∼ λ L λ (cid:48) L (cid:12)(cid:12)(cid:12) (Λ) ai (Ω ∗ ) bj Σ ij (cid:12)(cid:12)(cid:12) → N c π λ L λ (cid:48) L f (cid:12)(cid:12)(cid:12)(cid:12) s + i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) . (4.10)This is one of the terms needed for the collective symmetry breaking. The coefficient isestimated from the dimensional analysis.Notice that we have chosen different (generations of) elementary quark doublets, q L and q (cid:48) L in the two couplings. If q L and q (cid:48) L were the same, the loop can be closed at O ( λ L λ (cid:48) L ) and a large s tadpole term and Higgs quadratic term will be generated, ∼ λ L λ (cid:48) L (cid:16) (cid:15) ab (Λ) ai (Ω ∗ ) bj Σ ij (cid:17) → N c π λ L λ (cid:48) L g ψ f (cid:18) s + i f ˜ H † H (cid:19) . (4.11)Such a term is actually needed for a realistic EWSB, but it would be too large if it weregenerated together with Eq. (4.10) that will produce the Higgs quartic term. It can begenerated of an appropriate size in a similar way involving some other different fermionsand composite operators with smaller couplings.The way that the mass term for s can be generated without the tadpole term can beunderstood from the symmetry point of view. In addition to the SU (2) W × U (1) Y , the Σ preserves a global U (1) Peccei-Quinn (PQ) [25] subgroup of Sp (6) . This global U (1) symmetry corresponds to the unbroken generator U (1) P Q : 12 − − , (4.12)under which s has charge 1, both H , H have charge 1/2, and the rest of pNGBs havecharge 0. The s mass term is invariant under U (1) P Q while the tadpole term has charge1 so it will not be induced if the interactions can preserve the U (1) P Q symmetry. On the– 13 –ther hand, the composite operators in Eqs. (4.5), (4.7) have the following PQ charges fortheir components (assuming that they don’t carry an additional overall charge), = / ⊕ ⊕ ¯2 − / ⊕ , (4.13a) ¯6 = ¯2 − / ⊕ ⊕ / ⊕ , (4.13b)where the subscript here denotes the PQ charge instead of the X charge. We see that q L and q (cid:48) L couple to components of different PQ charges. If q L and q (cid:48) L are different, it ispossible to assign PQ charges, i.e., / for q L and − / for q (cid:48) L , so that the interactionsEqs. (4.5), (4.7) preserve the PQ symmetry and the s tadpole term will not be generated.If q L and q (cid:48) L are the same, then there is no consistent charge assignment that can preservethe PQ symmetry, and hence the s tadpole term can be induced. Furthermore, if differentgenerations of quarks carry different PQ charges, The U (1) P Q preserving interactions willnot induce flavor-changing neutral currents (FCNC) as they violate the PQ symmetry.The second term required in realizing the collective symmetry breaking can be gener-ated similarly by a different set of quarks (or leptons). They should couple to the 4th and5th components of the and ¯6 operators through the spurions (Λ (cid:48) ) ai = (cid:32) (cid:33) and (Ω (cid:48) ) ai = (cid:32) (cid:33) , (4.14)which preserve the SU (4) symmetry of the 1,2,3,6 entries. The combination of Λ (cid:48) and Ω (cid:48) can then introduce the potential ∼ λ L λ (cid:48) L (cid:12)(cid:12)(cid:12) (Λ (cid:48) ) ai (Ω (cid:48)∗ ) bj Σ ij (cid:12)(cid:12)(cid:12) → N c π λ L λ (cid:48) L f (cid:12)(cid:12)(cid:12)(cid:12) s − i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) , (4.15)which provides the other term needed for the collective symmetry breaking.To generate all the terms required for the Higgs quartic potential from collective sym-metry breaking, we need to use several different quarks and/or leptons, with different PQcharge assignments. As we mentioned earlier, we also need some smaller PQ-violating cou-plings between the elementary fermions and the composite operators, in order to generatea proper-sized ˜ H † H term, m ∼ N c π λ L λ (cid:48)(cid:48) L g ψ f , (4.16)where λ (cid:48)(cid:48) L represents the smaller U (1) P Q violating coupling. A more detailed couplingassignment for a realistic model is presented in Appendix B.With all the collective symmetry breaking interactions discussed above, we obtain apNGB potential, N c π λ k L λ (cid:48) (cid:96) L f (cid:12)(cid:12)(cid:12)(cid:12) s + i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) + N (cid:48) c π λ m L λ (cid:48) n L f (cid:12)(cid:12)(cid:12)(cid:12) s − i f ˜ H † H (cid:12)(cid:12)(cid:12)(cid:12) (4.17)(where the indices k, (cid:96), m, n here label different fermions). After integrating out the massive s field, we obtain a quartic term for the Higgs doublets (take N c , N (cid:48) c = 3 ) as λ (cid:12)(cid:12)(cid:12) ˜ H † H (cid:12)(cid:12)(cid:12) with λ = 316 π λ k L λ (cid:48) (cid:96) L λ m L λ (cid:48) n L λ k L λ (cid:48) (cid:96) L + λ m L λ (cid:48) n L ≈ π λ k L λ (cid:48) (cid:96) L . (4.18)– 14 –ncluding this quartic term, the coefficients of the Higgs potential in this model are esti-mated to be ˆ α ∼ π λ t L λ t R f , ˆ β ∼ π λ k L λ (cid:48) (cid:96) L f . (4.19)Therefore we can further improve upon the minimal tuning ( ˆ α ∼ ˆ β ) case by requiring λ (cid:48) L > λ t R = ⇒ ˆ β > ˆ α . (4.20)Of course, however, ˆ β can not be arbitrarily large because it is determined by the Higgsboson mass from Eq. (2.4). The required numerical parameters will be discussed in thenext section. The SU (6) /Sp (6) model contains two Higgs doublets. To analyze the EWSB and the Higgsboson masses, we need to consider the Higgs potential in a 2HDM. A review of 2HDM canbe found in Ref. [26]. The other pNGBs do not affect the Higgs potential much (they eitherare heavy or couple mostly quadratically to the Higgs doublets), so we will postpone theirdiscussion to the next section. The Higgs potential in our model can be parameterized as V ( H , H ) = m H † H + m H † H − m (cid:16) ˜ H † H + h.c. (cid:17) + λ (cid:16) H † H (cid:17) + λ (cid:16) H † H (cid:17) + λ (cid:12)(cid:12)(cid:12) ˜ H † H (cid:12)(cid:12)(cid:12) . (5.1)Notice that, in CHMs, due to the non-linearity of pNGBs, the Higgs potential should includetrigonometric functions instead of polynomials. Also, to match the potential here to theSM Higgs potential, an additional factor of cos (cid:104) θ (cid:105) will appear. However, since the deviationis strongly constrained by Higgs coupling measurements, we will take (cid:104) θ (cid:105) (cid:28) and expandsin x ∼ x in the following discussion for simplicity.In the 2HDM potential (5.1), both Higgs doublets develop nonzero VEVs. Denote theVEVs of H and H to be v and v respectively, and their ratio is defined as tan β ≡ v /v .The total VEV v satisfies v = v + v = v cos β + v sin β = (246 GeV ) . (5.2) H couples to the top quark and gets a large negative loop-induced contribution to itsquadratic term, so it is natural to expect v > v . On the other hand, the main quarticterm coming from the collective symmetry breaking is λ . To have a large enough effectivequartic term for the 125 GeV Higgs boson, we do not want either sin β ( ≡ s β ) or cos β ( ≡ c β ) to be too small. The current constraints [27–29] have ruled out the region tan β near 1, sowe will consider a benchmark with a medium value,tan β ∼ . (5.3)Also, the light neutral eigenstate should be close to the SM Higgs boson, which imposessome conditions on the parameters in the Higgs potential (5.1). In Subsec. 5.1, we firstdiscuss the quadratic potential, which will determine the spectrum of additional Higgsbosons in this model. Then, we will discuss the alignment issue in Subsec. 5.2 and thecorresponding values of the quartic terms in the Higgs potential.– 15 – .1 Estimating the Mass Terms The experimental constraints require that the 2HDM should be close to the alignment limit( β − α = π/ ) [30–33], where α is the mixing angle between the mass eigenstates of thetwo CP-even Higgs boson and the corresponding components in H , H (after removing theVEVs), h = − h sin α + h cos α . (5.4)To simplify the discussion of the quadratic terms, we assume that the alignment holdsapproximately, h ≈ h cos β + h sin β = h SM , (5.5)then we can calculate the SM Higgs potential by the transformation (cid:32) H H (cid:33) = (cid:32) cos β − sin β sin β cos β (cid:33) (cid:32) H SM H heavy (cid:33) . (5.6)The potential of the light SM Higgs doublet becomes (keeping the terms with H SM onlyand rewriting H SM → H ) V ( H ) = (cid:0) m cos β + m sin β − m sin β cos β (cid:1) | H | + (cid:18) λ cos β + λ sin β + λ sin β cos β (cid:19) | H | . (5.7)Matching the quadratic term with the SM Higgs potential implies that − µ = m cos β + m sin β − m sin β cos β ≈ − (88 GeV ) . (5.8)As shown in the previous section, these mass terms get contributions from different sources: m comes from gauge contributions, m gets an additional large negative contributionfrom the top quark besides the gauge contributions, and m comes from the PQ-violatinginteractions. No natural cancellation among the three terms in Eq. (5.8) is warranted.Therefore, the absolute values of all three terms should be of the same order as µ toavoid tuning. For example, for tan β = 3 Eq. (5.8) can be satisfied by m ∼ (360 GeV ) , m ∼ (120 GeV ) , and m ∼ (210 GeV ) without strong cancellations among the threeterms. These numbers are based on the alignment approximation. More accurate valuesneed to include the whole 2HDM potential and will be given after the discussion of thequartic terms. There are three quartic couplings in the Higgs potential (5.1): λ , λ , and λ . The effectivequartic coupling for the light Higgs, which can be seen from Eq. (5.7), is a combination ofthe three quartic couplings and tan β . To obtain a 125 GeV Higgs boson we need λ cos β + λ sin β + λ sin β cos β ≈ . . (5.9)– 16 – is mainly induced by the SM gauge loops and is expected to be small. λ receives thetop quark loop contribution, λ ∼ y t π ln M T v ∼ . . (5.10)This implies that we need λ which comes from the collective symmetry breaking to satisfy λ s β c β ∼ . ⇒ λ ∼ for tan β = 3 . (5.11)If it arises from the collective quartic term obtained in Eq. (4.18), it corresponds to λ L λ (cid:48) L ∼ ⇒ (cid:113) λ L λ (cid:48) L ∼ . . (5.12)These couplings between the elementary states and composite operators are quite large.However, the smallness of SM Yukawa couplings can be obtained by small λ R couplings.There are other experimental constraints with these large λ L couplings, which will bediscussed in the following sections.We have been assuming that the 2HDM potential is approximately in the alignmentregime. Let us go back to check how well the alignment can be achieved. A simple wayto achieve the alignment is the decoupling limit where the extra Higgs bosons are heavy.However, this would require more tuning in the Higgs mass parameters. In our model λ >λ , λ . Under this condition, we need tan β ∼ to achieve the exact alignment if the extraHiggs bosons are not too heavy. This is not compatible with the experiment constraints.Therefore we expect some misalignment and need to check whether the misalignment canbe kept within the experimental constraints.Solving the eigenvalue equations, we can get the following equations for the factor c β − α , c β − α = 1 M A tan β (cid:18) λ v (cid:18) − s α c β (cid:19) + λ v (cid:18) c α s β (cid:19) − M h (cid:18) − s α c β (cid:19)(cid:19) , (5.13) = 1 M A cot β (cid:18) − λ v (cid:18) − s α c β (cid:19) − λ v (cid:18) c α s β (cid:19) + M h (cid:18) c α s β (cid:19)(cid:19) . (5.14)As the misalignment should be small, to estimate its size, we can assume that the masseigenstates of the 2HDM are near alignment, which satisfy ( − s α , c α ) ≈ ( c β , s β ) approxi-mately for the right-handed side. We then have c β − α ≈ M A tan β (cid:0) λ v + λ v − M h (cid:1) , (5.15) ≈ M A cot β (cid:0) − λ v − λ v + M h (cid:1) . (5.16)Consider the benchmark valuestan β ≈ , λ ≈ , and M A ≈ GeV , (5.17)where the M A value is chosen to keep the misalignment small and to evade the direct searchin the A → hZ decay channel at the LHC [27]. The equations for c β − α becomes c β − α ≈ . λ + 0 . ≈ . − . λ . (5.18)– 17 –ince λ in this model is small, we have c β − α ≈ . which parametrizes the deviationfrom the alignment. The misalignment will have a direct consequence on Higgs physics andwill be discussed in the following sections. The most relevant deviation, the ratio of Higgsto vector bosons coupling to SM coupling, is proportional to s β − α ≈ . and should stillbe safe.Eq. (5.18) also implies that λ needs to be ≈ . , which is consistent with the estimatefrom the top quark loop contribution Eq. (5.10). To sum up, the three quartic couplings inour 2HDM potential take values λ ≈ (cid:29) λ ≈ . (cid:29) λ . (5.19) So far, all numbers in the above discussion are estimations based on simplified approxima-tions. In a realistic benchmark model, the exact values can be solved by directly diago-nalizing the mass matrix. To reproduce the correct Higgs boson mass M h = 125 GeV andsmall enough c β − α with fixed tan β ≈ and λ ≈ , we choose the following values as areference for our study:tan β ≈ . , λ ≈ . , λ ≈ . , and M A ≈ GeV . (5.20) λ is irrelevant as long as it is small so we don’t set its value. The value of λ is set byproducing the correct Higgs boson mass.With these numbers, we can diagonalize the mass matrix and get the mixing angle α and the misalignment β − α as s α = − . , c α = 0 .
977 = ⇒ c β − α = 0 . , s β − α = 0 . . (5.21)The eigenvalues of the matrix give the masses of the CP-even neutral scalar bosons as M h ≈ GeV and M H ≈ GeV . (5.22)The complete spectrum will be discussed in the next section.After we obtain the quartic couplings, we can go back to determine the mass terms.The value of M A is chosen to satisfy the experimental constraint. It also gives the value of m based on the relation m = M A s β c β ∼ (210 GeV ) . (5.23)Given the values of all the quartic couplings and m , we can obtain the other mass terms m = 3 m − λ v − λ v ∼ (320 GeV ) , (5.24) m = 13 m − λ v − λ v ∼ (90 GeV ) . (5.25)These numbers will serve as a benchmark for our phenomenological studies.– 18 –ssuming that these masses arise dominantly from the loop contributions discussed inthe previous sections, we can also estimate the masses of the composite states in the CHM, m = 332 π g M ρ ∼ (320 GeV ) , (5.26) m = 332 π g M ρ − π y t M T ∼ (90 GeV ) , (5.27) m = N c π λ L λ (cid:48)(cid:48) L g ψ f ∼ (210 GeV ) , (5.28)where we have ignored the small U (1) gauge contribution and O (1) coefficients. The m equation gives the mass of the gauge boson partners M ρ ∼ TeV. In the m equation,the top loop contribution needs to cancel the positive gauge contribution (320 GeV ) toproduce a (90 GeV ) term. From that, the top partner is estimated to be around M T ∼ . TeV. This corresponds to an O (10%) tuning between the gauge contribution and the topcontribution, but it is hard to avoid given the experimental constraints on the top partnermass. The desired size of m can be achieved by a suitable choice of the PQ-violatingcoupling λ (cid:48)(cid:48) L which is a free parameter in this model. After discussing the Higgs potential from the naturalness consideration, we are ready toprovide the estimates of masses of all other pNGBs, based on the benchmark point alludedin the previous section.
The 2HDM potential has been discussed in the previous section. In addition to the SM-like125 GeV Higgs boson, there is one more CP-even neutral scalar H , a CP-odd neutral scalar A , and a complex charge scalar H ± . Their masses from the Higgs potential (5.1) are M A = m s β c β , M H ± = M A − λ v ,M h,H = 12 (cid:16) M A ± (cid:113) M A − M H ± λ v s β c β (cid:17) , (6.1)which results in a spectrum M A > M H > M H ± . This is different from the 2HDM spectrumof the MSSM because the dominant quartic term is λ . For the benchmark point of theprevious section, the three masses are estimated to be M A ∼ GeV , M H ∼ GeV , and M H ± ∼ GeV . (6.2) In addition to the two doublets, the pNGBs also include a real triplet φ , a real singlet η ,and a complex singlet s . The triplet obtains its mass from the gauge loop as shown inEq. (3.14). For M ρ ∼ TeV, it gives M φ = 14 π g M ρ ∼ (500 GeV ) . (6.3)– 19 –he singlets do not receive mass contributions from SM gauge interactions. The com-plex singlet s obtains its mass from the collective symmetry breaking mechanism (4.17), M s = N c π λ k L λ (cid:48) (cid:96) L f + N (cid:48) c π λ m L λ (cid:48) n L f ≥ λ f ≈ (2 f ) , (6.4)which is expected to be at the TeV scale. There is also a tadpole term from the PQ-violatingpotential, which will introduce a small VEV for s , (cid:104) s (cid:105) ∼ m fM s ≤ (210 GeV ) f ∼ O (10 GeV ) . (6.5)It will have little effect on the mass of the singlet.Finally, the real singlet η does not get a mass at the leading order but it couplesquadratically to the Higgs doublets (e.g., from Eq. (3.28)), so it can still become massiveafter the Higgs doublets develop nonzero VEVs. Through Eq. (3.28), η receives a mass M η ∼ π y t M T · (cid:18) vf (cid:19) = ⇒ M η ∼ (cid:18) M T f (cid:19) GeV . (6.6)For naturalness, a relatively light top partner is preferred. On the other hand, the exper-imental constraints require η to be heavier than half of Higgs boson mass to avoid largeHiggs decay rate to the ηη channel. We expect a light singlet scalar around 100 GeV, whichcan be the lightest composite state in the spectrum. In CHMs, there will be new composite states of scalars, fermions, and vectors near orbelow the compositeness scale. The detailed spectrum and quantum numbers depend onthe specific realizations of the CHMs. In this section, we study the collider searches of andconstraints on these new states in the SU (6) /Sp (6) model discussed in this paper. Under the requirement of naturalness, the second Higgs doublet is expected to be amongthe lightest states of the new resonances and could be the first sign of this model. Inthe Type-II 2HDM, the flavor-changing process b → sγ has put strong constraints on thecharged Higgs mass to be above 600 GeV, which would require more tuning in the Higgspotential. Therefore, we focus on the Type-I 2HDM scenario. As explained in the previoussection, we will consider a relatively small tan β ∼ with a small misalignment c β − α ∼ . .The direct searches can be divided into two categories – charged Higgs bosons H ± andneutral Higgs bosons H , A . In the Type-I 2HDM with a small misalignment, neutralHiggs bosons to fermion couplings are characterized by a factor − s α /s β ∼ / and thecharged Higgs boson to fermion couplings are characterized by c β /s β ∼ / . Comparing toneutral Higgs bosons, the charged Higgs boson searches give a more reliable constraint ontan β because it doesn’t depend on the mixing angle α .– 20 – igure 1 . Constraints on extra neutral Higgs bosons in a Type-I 2HDM with a small misalignment c β − α = 0 . . This summary plot is taken from Ref. [27]. The charged Higgs boson is searched by its decays to SM fermions. For M H ± (cid:46) m t , thestrongest constraint comes from decaying to τ ν [34, 35]. Interpreted in the Type-I model,it excludes tan β < for M H ± ∼ GeV and tan β < for M H ± up to 150 GeV [36].For a heavier charged Higgs boson, the main constraint comes from the decay to tb , whichrules out tan β (cid:46) for M H ± in the range of 200-400 GeV, and becomes weaker for larger M H ± [28, 29].For neutral Higgs bosons, there are multiple decay channels being searched. For lightstates below the t ¯ t threshold, they can be searched by H/A → τ τ [37, 38] and H → γγ [39, 40] decays. For heavier states, the decay to t ¯ t becomes accessible and dominant.The searches of H/A → t ¯ t has been done at CMS and ATLAS [41, 42]. These searchestypically constrain tan β (cid:38) − up to M H/A ∼ GeV. When there is misalignment asexpected in this model, there are also additional decay channels of these neutral scalarswhich give important constraints. These include
H/A → W W [43, 44] and ZZ [45, 46], H → hh [47, 48], and A → hZ [49, 50]. The A → hZ and H → hh turn out to bemost constraining for the region that we are interested in. The A → hZ can exclude tan β up to 10 below the t ¯ t threshold. Some higher mass ranges are also constrained due todata fluctuations. H → hh constrains tan β to be (cid:38) for a wide mass range. Variousconstraints on the neutral scalars for 2HDMs are summarized in Ref. [27], and the relevantplot is reproduced in Fig. 1. We can see that the benchmark point chosen in the previoussection, M A ∼ GeV , M H ∼ GeV , and M H ± ∼ GeV , (7.1)– 21 –ith tan β = 3 is sitting in the gap of the constraints. It is still allowed by but very closeto the current constraints, hence it will be tested in the near future.For future searches, the most relevant channels for the more natural mass range aredi-boson channels H/A → V V , H → hh , and A → hZ . The current bounds are expectedto be improved by ∼ times [51]. It will probe the parameter region that we are mostinterested in. If we can also find the charged Higgs with a slightly lighter mass, thisparticular spectrum can be an indication of the specific 2HDM Higgs potential (differentfrom that of the MSSM) that arises from this type of CHMs. Besides the second Higgs doublet, there are also several additional scalar bosons, whichinclude a real triplet φ , a complex singlet s , and a real singlet η . At the leading order,they don’t directly connect to the SM fermions. However, the couplings to SM fermionsare induced through the mixing with Higgs bosons after EWSB, with a suppression factorof v/ f ∼ . (for ξ ∼ . ). Scalar triplet φ : The scalar triplet has unsuppressed gauge interactions with W and Z bosons, but only through four-point vertices. They can be paired produced throughthe vector boson fusion but the production is highly suppressed due to the large energyrequired. Therefore, here we only consider the single production through the interactionwith SM fermions. The scalar triplet includes a complex charged scalar φ ± and a neutralscalar φ . The collider searches of the charged scalar are similar to those of H ± of thesecond Higgs doublet but with the suppressed couplings. It can be produced in associationwith a top and a bottom. However, due to the suppressed coupling and the larger mass,the charged scalar φ ± is less constrained.The neutral scalar φ is searched in the same ways as the neutral scalars in 2HDMs.Guided by the benchmark scenario, we consider a scalar with mass ∼ GeV, whichgives a cross section fb. The dominant decay mode will be φ → t ¯ t with a branchingratio ∼ . The current bound from the LHC searches [41, 42] on the cross section is σ × BR < pb, which is still loose for a neutral scalar with σ × BR ∼ fb. The di-bosonmodes are also important with branching ratios ∼ for W W and ∼ for ZZ . Themost stringent current upper bound comes from φ → ZZ channel, which ruled out σ × BR above fb [45, 46]. It is also much larger than ∼ fb for the benchmark point. In thefuture, around . × φ (at 500 GeV) would be produced in the HL-LHC era with anintegrated luminosity of 3 ab − . The bound can be improved by times [51]. And a 500GeV φ could be within reach in the HL-LHC era. Scalar singlets:
The complex scalar s is expected to be at TeV scale and the real singlet η is around GeV. They both act like the neutral scalar φ discussed above, but withoutthe gauge interactions. They can be produced through the gluon fusion but the productioncross sections will be suppressed by ξ/ ∼ . .For the heavy complex scalar s , The expectation of its mass in the benchmark pointis above . TeV. The dominant decay channel will be a pair of neutral Higgs bosons– 22 – → h h ( hh, hH, HH ) or charged Higgs bosons due to the large s ˜ H † H coupling. Italso connects to the fermions sector through the mixing with Higgs bosons. However, theproduction is suppressed due to the large mass. Although it is an essential element of thecollective Higgs quartic term, it is hard to detect even at the HL-LHC. It may be accessiblein the next generation hadron collider.The light real scalar η should be heavy enough so that h → ηη is forbidden due to theconstraint from the Higgs invisible decay measurement [52]. This requires M T /f (cid:38) . fora realistic model, but it should remain relatively light if the top partner is not too heavy forthe naturalness reason. Since the interactions between η and SM particles are all throughthe mixing with the Higgs boson, the search modes are similar but with the ξ/ suppressionon the production rate. The cross section is ∼ . pb for a 100 GeV η . The dominant decaymodes are b ¯ b (78 . , τ τ (8 . and gg (7 . , but they all suffer from large backgrounds.On the other hand, the clean channel γγ suffers from a low branching ratio ∼ . . Forthe benchmark point, the diphoton channel has σ × BR ∼ fb. The latest search fromCMS [53] still has an uncertainty ∼ fb for a diphoton invariant mass ∼ GeV,much bigger than the cross section that we expect. With more data and improvements inthe background determinations, it might be discoverable at future LHC runs.
The top partners in the SU (6) /Sp (6) CHM are vector-like fermionic resonances whichform a sextet of the Sp (6) global symmetry. Their quantum numbers under the SM gaugesymmetry are (3 , , / × , (3 , , / , and (3 , , − / , which are identical to those ofSM quarks. There are no exotic states with higher or lower hypercharges. These states aredegenerate in the limit of unbroken Sp (6) global symmetry. (Small splittings arise from theexplicit symmetry breaking effects and EWSB.) Their mass M T plays the important roleof cutting off the quadratic contribution from the top quark loop to the Higgs potential.Naturalness prefers M T to be as low as possible allowed by the experimental constraints.The current bound on the top partner mass has reached ∼ . TeV [5, 6]. The HL-LHC canfurther constrain the mass up to ∼ . TeV [54]. The benchmark value of 1.6 TeV is closeto but probably still beyond the reach of HL-LHC. A future 100 TeV collider will cover theentire interesting mass range of the top partners if no severe tuning conspires. It may evenbe able to find the fermionic partners of the other SM quarks, which are expected to bemuch heavier.
Unlike the top partners, the partners of SM gauge bosons (spin-1 resonances) are notnecessarily light because of the smallness of SU (2) W , U (1) Y gauge couplings. In fact,their masses need to be large enough to give a sufficiently large mass to the second Higgsdoublet and to cancel in a large part the negative contribution from the top sector to thequadratic Higgs potential. The largest couplings of these composite spin-1 resonances areto the composite states, including the pNGBs. Their mixings with SM gauge bosons arestrongly suppressed by their multi-TeV masses, hence their couplings to SM light fermionsare also suppressed, resulting in a small production rate as well as small decay branching– 23 –atios to SM elementary particles [55, 56]. The leading decay modes will be through thecomposite states, such as top partners or pNGBs which include the longitudinal modes of W and Z . The current searches of heavy vector triplets decaying into SM gauge bosonsfinal states have reached a bound about 4 TeV [57–60]. The bound is relieved for larger g ρ > with more suppression on the production rate. Besides, the model contains a richersector of the pNGBs which will dilute the decay branching fractions to SM gauge bosons,further reducing the bound. If the vector resonances are heavier than twice the top partnermass, the decaying into top partners will dominate and it would require different searchstrategies. As the production rate quickly diminishes for heavier vector resonances, thetypically expected masses of the vector resonances as in our benchmark will be out of reacheven at the HL-LHC. A future higher energy machine will be needed to discover them. In this section, we discuss the indirect tests of this model from precision experimentalmeasurements.
The Higgs boson couplings to SM fields in the SU (6) /Sp (6) CHM are modified by twoeffects: the nonlinear effect due to the pNGB nature of the Higgs boson and the misalign-ment from the mixing of the 2HDM. The deviation of the Higgs coupling to vector bosonsis parameterized by κ V ≡ g hV V g SMhV V = sin ( β − α ) cos (cid:112) V + V √ f , (8.1)where the first factor comes from the misalignment of the 2HDM and the second factor isthe nonlinear effect of the pNGB. For the benchmark point in Sec. 5, sin( β − α ) ≈ . ,which gives κ V ≈ (0 . (cid:114) − ξ ≈ . − . ξ , (8.2)The deviation of the Higgs coupling to fermion is universal in Type-I 2HDMs becauseit couples to all fermions in the same way. The expression is somewhat more complicatedin CHM, and here we only expand to O ( ξ ) , κ f ≡ g hff g SMhff = 1 s β (cid:18) c α − ξ
112 (3 s β c α + c β c α − s β c β s α ) (cid:19) ≈ . − . ξ , (8.3)where the numerical value of the last expression is obtained for the benchmark point.The current best-fit values of κ V and κ F from ATLAS [61] with an integrated luminosityof 80 fb − are κ V = 1 . ± . , (8.4) κ F = 1 . ± . , (8.5)with a 45% correlation between the two quantities. The central values for both quantitiesare slightly above the SM value 1, but without significant deviations given the uncertainties.– 24 – igure 2 . The fit of the Higgs coupling strengths to the gauge bosons ( κ V ) and fermions ( κ f )obtained by the ATLAS [61] from the 13 TeV LHC data. The cross is the observed central value.The circles from inside out represent the 68%, 95%, and 99% CL respectively. The red star showsthe SM value (1 , . The blue star in the predicted value of the 2HDM benchmark of Sec. 5 with ξ = 0 . Along the line, we show the predictions for the same benchmark with different ξ from 0 to0.3. As shown in Fig. 2, within 95% CL level, ξ ≤ . is still allowed (for the benchmark point),which gives a lower bound on the scale f ∼
700 GeV.In the future, the uncertainties in κ V and κ F can be improved to 1% and 3% respectivelyat the HL-LHC, [62]. Assuming the central values of (1 , , it can bound ξ down to . at99% CL. The next generation Higgs factories, such as ILC, CEPC, and FCCee, will havegreat sensitivities to the hZZ coupling and can measure κ V with a precision ≈ f up to several TeV and hence cover the entire natural parameter regionfor the CHMs.Another decay mode worth mentioning is h → γγ . The branching ratio of this decaymode will receive an additional contribution from charge Higgs bosons. But the currentbound from this decay mode is still loose. It will improve at HL-LHC and future Higgsfactories. It may provide a sign of the heavy charged Higgs bosons if they exist. New physics appearing near the TeV scale may introduce dangerously large flavor changingneutral currents (FCNCs), so the flavor-changing processes put strong constraints on themodel constructions. The SU (6) /Sp (6) model contains two light Higgs doublets. If generalYukawa couplings are allowed between them and SM fermions, large FCNCs will be induced.Therefore, it is desirable to impose the natural flavor conservation such that each type ofYukawa couplings only comes from one of the two Higgs doublets. Even so, a light chargedHiggs boson can induce a significant contribution to the branching ratio BR ( B → X s γ ) [63–68]. In the Type-II or flipped 2HDM, this gives a lower bound on the charged Higgs boson– 25 – H ± > GeV [24, 69], which would introduce more tuning in the Higgs potential. Tohave a more natural model, we therefore focus on the construction of the Type-I 2HDM. Ina Type-I model, the B → X s γ constraint rule out the region below tan β < [24, 69].The partial compositeness couplings between the elementary fermions and the compos-ite operators can potentially induce FCNCs. In our construction, the largest such couplings(for the top Yukawa and the collective Higgs quartic term) preserve a Peccei-Quinn symme-try with different PQ charges for different generations (see Appendix B). As a result, thereis no FCNC induced by these large couplings in the leading order. Some FCNCs may beinduced by other (smaller) couplings which are responsible for generating the complete SMfermion masses and mixings, but they are suppressed by the small couplings and dependon the details of their pattern. The electroweak oblique corrections provide important tests of new physics near the weakscale. They are usually expressed in terms of S , T , and U parameters [70, 71]. The currentglobal fit gives [72] S = − . ± . , T = 0 . ± . , U = 0 . ± . . (8.6)For heavy new physics, U is typically small as it is suppressed by an additional factor M /m Z . If one fixes U = 0 , then S and T constraints improve to S = 0 . ± . , T = 0 . ± . , (8.7)with a strong positive correlation (92%) between them. At 95% CL, one obtains S < . and T < . .There are several contributions to the oblique parameters in our model, with similaritiesand differences compared to the MCHM discussed in the literature. First, our model hastwo Higgs doublets. Their contributions to S and T can be found in Ref. [73–75]. To satisfythe other experimental constraints, the Higgs potential needs to be close to the alignmentlimit and the heavy states are approximately degenerate. The contributions are expectedto be small and do not provide a significant constraint [76]. The other contributions arediscussed below. The S parameter The leading contribution to the S parameter comes from the mixing between the SM gaugebosons and the composite vector resonances. It is estimated to be [77–79] ∆ S ∼ c S π v M ρ ∼ c S . (cid:18) TeV M ρ (cid:19) , (8.8)where c S is an O (1) factor. It gives a lower bound of ∼ . TeV on M ρ for c S = 1 .In CHMs, there is a contribution from the nonlinear Higgs dynamics due to the de-viations of the Higgs couplings, which result in an incomplete cancellation of the elec-troweak loops [80, 81]. This contribution is proportional to ξ and depends logarithmicallyon M ρ /M h . For M ρ = 5 TeV, it gives ∆ S ∼ . ξ which is well within the uncertainty. A factor of 1/2 is included due to the normalization of f compared to Ref. [80, 81]. – 26 –n the MCHM, there is also a contribution due to loops of light fermionic resonances. It islogarithmically divergent and its coefficient depends on the UV physics [81]. This contribu-tion can be significant, depending on the UV-sensitive coefficient. However, in our model,the fermionic resonances are complete multiplets of SU (6) and their kinetic terms remain SU (6) symmetric, so this divergent contribution is absent. The T parameter The T parameter parametrizes the amount of custodial SU (2) breaking. There are alsoseveral potential contributions in our model. First, the pNGB spectrum contains a real SU (2) W triplet φ . If it obtains a VEV induced by the trilinear scalar couplings to a pairof Higgs doublets, H † φH , H † φH , or ( H φH + h.c.), it will give a tree-level contributionto ∆ T . Its VEV is bounded to be less than ∼ GeV, putting strong constraints onthese couplings. However, if all the large couplings are real and the CP symmetry is(approximately) preserved, the real scalars φ and η are CP odd and the interactions H † φH , H † φH , and ( H φH + h.c. ) are forbidden by the CP symmetry. The η and φ fields needto couple quadratically to the Higgs fields. This also justifies the Higgs potential analysisbased on the 2HDM potential. Of course, CP symmetry has to be broken in order to allowthe nonzero phase in the CKM matrix. We assume that this is achieved with the smallpartial compositeness couplings so that the induced trilinear scalar couplings are kept smallenough to satisfy the bound.Apart from the potential triplet VEV contribution, the leading contribution to ∆ T comes from fermion loops. For the partial compositeness couplings in this model, thecustodial symmetry breaking comes from λ R . The dominant contribution comes from thelight top partners and the corresponding mixing coupling λ t R The deviation is estimatedto be [79] ∆ T ∼ N c π α λ t R v M T ∼ . (cid:18) λ t R . (cid:19) (cid:18) . TeV M T (cid:19) . (8.9)There is also a contribution from the modifications of the Higgs couplings to gaugebosons due to the nonlinear effects of the pNGB Higgs. The contribution to ∆ T from thenonlinear effects again depends on ξ and is logarithmically sensitive to M ρ . For M ρ = 5 TeV,it gives ∆ T ∼ − . ξ [80, 81]. It is significant and can partially cancel the light top partnercontribution. The contribution from the mixing of the hypercharge gauge boson and vectorresonances is small due to the custodial symmetry. The tree-level contribution vanishes andthe loop contribution is negligible. The overall ∆ T correction is expected to be positiveand could help to improve the electroweak precision fit in the presence of a positive ∆ S .In summary, among the various sources of the corrections to the electroweak observ-ables, the contributions from the composite resonances are expected to be dominant. Theygive strong constraints on the masses of heavy resonances M ρ and M T as well as the rele-vant coupling like λ t R . Nevertheless, for natural parameter values as our benchmark, the The custodial symmetry of our model corresponds to the Case B in Ref. [21] The partial compositeness couplings are related to the top Yukawa coupling by λ t L λ t R ∼ y t g T . For y t s β ∼ . at 2 TeV and assuming g T ∼ , we need (cid:112) λ t L λ t R ∼ . . – 27 –orrections on ( S, T ) can still lie safely within the current uncertainty region. A future Z factory can greatly improve the precisions of the electroweak observables, which can providea strong test of the model. Zf ¯ f Couplings
The partial compositeness couplings generate mixings between elementary fermions andcomposite resonances. They can modify the Zf ¯ f couplings in the SM. This is a well-knownproblem in CHMs for the Zb ¯ b coupling in implementing the top partial compositeness. Asolution based on an extended custodial symmetry SU (2) V × P LR on the top sector byembedding the left-handed top-bottom doublet into the (2 , representation of SU (2) L × SU (2) R was proposed in Ref. [82]. The top sector in our construction does not have thisextended custodial symmetry. Furthermore, to obtain the collective quartic Higgs term, weneed several large partial compositeness couplings involving other light SM fermions. Asa consequence, we may expect significant deviations of the Zf ¯ f couplings for all fermionsinvolved and they present important constraints on this model.The third generation left-handed quark’s partial compositeness couplings modify the Zb L ¯ b L coupling. Its deviation δg b L from the current experimental determination is con-strained within × − [83]. This deviation comes from mixings between the bottomquark b and the corresponding composite resonances B . Under our assignment in Ap-pendix B, there are two terms that will have large positive contributions to δg b L . Theyare λ t L ¯ q ,L H B R → ( λ t L ν )¯ b L B R , (8.10) λ (cid:48) b L ¯ q ,L ˜ H B (cid:48) R → ( λ (cid:48) b L ν )¯ b L B (cid:48) R . (8.11)The first one is responsible for generating the top Yukawa coupling and induces the mixingbetween b L and the bottom partner B with PQ charge 0. The second introduces the bottomYukawa coupling and the collective quartic term. It induces the mixing with another bottompartner B (cid:48) with PQ charge 1. The deviations that they bring can be estimated as δg b L ≈ λ t L c β M ( TeV ) × (30 × − ) , δg b L ≈ λ (cid:48) b L s β M ( TeV ) × (30 × − ) , (8.12)where M and M are the masses of the fermions resonances B and B (cid:48) respectively. Notethat M is also the top partner mass which is responsible to cut off the top loop contributionto the quadratic Higgs potential so it should not be too large for naturalness. On the otherhand M is the bottom partner mass which can be much larger because of the small bottomYukawa coupling. These corrections impose strong constraints on the couplings and massesof the composite fermion resonances. For the first term, taking λ t L ≈ . and c β ≈ . fromthe benchmark model, it requires M = M T (cid:38) . TeV, which is still in the range we expect.Compared to the other models without the SU (2) V × P LR custodial symmetry, such as theMCHM [12], we are saved by the c β factor to allow a relatively light top partner. For thesecond one, taking λ (cid:48) b L ≈ and s β ≈ would require M (cid:38) . . TeV for the bottompartner. The bound on M can be reduced for a smaller value of λ (cid:48) b L , but at the cost of– 28 – larger λ c L if their combination is responsible for the collective Higgs quartic term, whichincreases the deviations for δg c L and δg s L .The collective Higgs quartic term needs at least four large λ L , λ (cid:48) L couplings. Eachof them will induce two δg L deviations from SM Zf ¯ f couplings and all of them reducethe magnitudes from the SM predicted values. Since the Z decay width and branchingratios are all well measured at O (10 − ) precision, we also need to examine their observableconsequences and the corresponding constraints.It is harder to extract the constraints on individual couplings from the observables thatdepend on more complicated combinations of different couplings. Therefore we consider theconstraints from Γ (hadron) and Γ (charged lepton) because they are directly proportionalto the couplings instead of some ratios. We predict smaller values for both Γ (hadron) and Γ (charged lepton), but their observed central values are both larger than the SM predictionsso the allowed parameter space is strongly restricted. At the 95% CL level, the allowednegative deviations are [72] ∆Γ( had ) ∼ − . MeV , ∆Γ( (cid:96) + (cid:96) − ) ∼ − . MeV . (8.13)From these, we obtain the constraints on allowed negative deviations on the magnitude ofdifferent left-handed fermion couplings (assuming only one term dominates) as follow, | δg u L | < . × − for up-type quarks, (8.14a) | δg d L | < . × − for down-type quarks, (8.14b) | δg e L | < . × − for charged leptons. (8.14c)They strongly constrain the parameters of our model. To satisfy these constraints, thecorresponding fermion partners need to be over 10 TeV if their couplings to the elementaryfermions are large enough to be responsible for the collective Higgs quartic term.These constraints can be relaxed somewhat if we use the neutrino couplings for thecollective Higgs quartic term. The Γ (invisible) is smaller than the SM prediction. Theallowed negative deviation is 4 MeV at the 95% CL level, which corresponds to | δg ν L | < × − for neutrinos. (8.14d)The resulting constraints on the corresponding fermion resonances are milder.The precision measurements of the Z couplings put strong constraints on our modelbecause we predict a reduction of all Zf L ¯ f L couplings in the construction. A future Z factory may improve the coupling measurements by more than one order of magnitude.Consequently, it can either establish a deviation from the SM predictions which points tonew physics in the nearby scales, or further affirm the SM predictions which will severelychallenge this model or any other models with similar predictions. Nevertheless, we wouldlike to emphasize that these constraints are indirect so it is quite possible that one canextend the model to introduce new contributions to cancel the deviations, at the expenseof complexity and/or tuning. – 29 – Conclusions
Composite Higgs models remain an appealing solution to the hierarchy problem. However,in realistic models, some tuning in the Higgs potential is often required to obtain thecorrect EWSB and the observed Higgs boson mass. One source is from the mass splittingswithin the top partner multiplet of the composite resonances, which can generate a largequadratic Higgs potential through the partial compositeness couplings at the order λ L ( R ) .The other is to obtain the necessary relative size between the quartic term and the quadraticterm of the Higgs potential in order to separate the EWSB scale and the compositenessscale. In this paper, we look for models that can address both problems. We show that aCHM based on the coset SU (6) /Sp (6) can achieve the goals without introducing additionalelementary fields beyond the SM and the composite sector, which otherwise will introducea new coincidence problem that why the new elementary fields and the compositenessresonances are at the same mass scale.A key part of the setup is to couple the elementary SM fermions to the composite op-erators of the fundamental representation of SU (6) . The composite resonances do not splitafter the symmetry is broken to Sp (6) and hence do not induce any large potential fromthe UV dynamics for the pNGBs. The leading contribution to the Higgs quadratic term isreduced to the unavoidable top quark loop in the IR. In addition, the fundamental repre-sentation of SU (6) contains two electroweak doublets of the same SM quantum numbers.This allows us to write down different ways of coupling between the elementary fermionsand the composite resonances, each of which preserves a subset of the global symmetry. Inthis way, a quartic Higgs potential can be generated from the collective symmetry breakingof the little Higgs mechanism, without inducing the corresponding quadratic terms. Thisindependent quartic term enables us to naturally separate the EWSB scale and the SU (6) global symmetry breaking scale, reducing the tuning of the Higgs potential.This model contains many more pNGBs than one Higgs double of the minimal model.In particular, there are two Higgs doublets and the second Higgs doublet should not betoo heavy for naturalness considerations. The extra Higgs bosons are already subject tocollider constraints and are the most likely new particles to be probed in the future LHCruns beside the top partners. The other pNGBs, having smaller couplings to SM particles,are more difficult to find. Together with the heavy vector and fermion resonances, they needhigher energy machines with large integrated luminosities. The top partners in this modeldo not include new particles with exotic charges, e.g., 5/3, as in many other CHMs. Themodel also predicts deviations of the Higgs couplings and weak gauge boson couplings. Thecurrent experimental data already provide substantial constraints on the model parametersin the most natural region. The Higgs coupling measurements will be greatly improved atthe HL-LHC and future Higgs factories. A future Z factory can also further constrain theelectroweak observables. Either the agreements with SM predictions with higher precisionswill push the model completely out of the natural scale for the solution to the hierarchyproblem, or some deviations will be discovered to point to the possible new physics, and ifany of the CHMs can provide an explanation for them.– 30 – cknowledgments We thank Felix Kling, Shufang Su, and Wei Su for letting us use the figure in Ref. [27]. Wealso thank Da Liu, Matthew Low, and Ennio Salvioni for useful discussions. This work issupported by the Department of Energy Grant number DE-SC-0009999.
A The SU (5) /SO (5) Composite Higgs Model
The SU (5) /SO (5) is also a possible coset that can naturally avoid large UV contributionsto the Higgs potential. It was one of the cosets considered in early composite Higgs modelsof 1980s [84, 85]. It was also the coset of the littlest Higgs model [9] which was one of thepioneer models to realize the mechanism of the collective symmetry breaking for the Higgsquartic coupling. The symmetry breaking can be parametrized by a symmetric tensor fieldwith a VEV (cid:104) Σ (cid:105) = Σ = I I , where I is the × identity matrix. (A.1)The SM SU (2) W and U (1) Y generators are embedded as σ a − σ a ∗ , − I I + X I , (A.2)where the extra U (1) X charge X accounts for the correct hypercharges of SM fermions.There are 14 pNGBs, with a complex doublet (which is identified as the Higgs field H ), a complex triplet φ , a real triplet ω , and a real singlet η . The partial compositenesscouplings can go through the and ¯5 representations of SU (5) . They do not split under SO (5) and hence do not give large UV contributions to the Higgs potential, just as in the SU (6) /Sp (6) case. Under the SM SU (2) W × U (1) Y , they decompose as x = x − / ⊕ x ⊕ ¯2 x +1 / , (A.3a) ¯5 x = ¯2 x +1 / ⊕ x ⊕ x − / . (A.3b)To mix with elementary fermions, we need to choose x = 2 / for the up-type quarks and − / for the down-type quarks.The Higgs quartic term arising from the collective symmetry breaking takes the form, κ f (cid:12)(cid:12)(cid:12)(cid:12) φ ij + i f ( H i H j + H j H i ) (cid:12)(cid:12)(cid:12)(cid:12) + κ f (cid:12)(cid:12)(cid:12)(cid:12) φ ij − i f ( H i H j + H j H i ) (cid:12)(cid:12)(cid:12)(cid:12) . (A.4)A drawback of this potential is that a nonzero VEV of the SU (2) W triplet φ will be inducedafter EWSB unless κ = κ . The triplet VEV violates the custodial SU (2) symmetry andis subject to the strong constraint of the T (or ρ ) parameter. Even if we ignore that fora moment, it is also more challenging to generate the collective quartic potential (A.4) in– 31 –his model. The two doublets in or ¯5 have different hypercharges if x (cid:54) = 0 and henceare not equivalent. We cannot couple the elementary SM fermion doublets to both and ¯5 in a way that preserves an SU (3) global symmetry to protect the Higgs mass, sothe mechanism introduced for the SU (6) /Sp (6) model in Sec. 4 does not work here. Onecould add additional exotic vector-like elementary fermions (with hypercharge / or − / )to couple to these composite operators for the purpose of generating the quartic term, butthese exotic elementary fermions should have masses comparable to the compositeness scale,which requires some coincidence. Another possibility is to use the lepton partners that have x = 0 , then the two doublets in , ¯5 are equivalent representations. One can write downthe partial compositeness couplings to generate Eq. (A.4), analogous to the SU (6) /Sp (6) model. However, the same interactions will induce the Majorana mass terms for the left-handed neutrinos through the triplet φ VEV. The couplings need to be O (1) in order toproduce a large enough quartic term. It means that unless the triplet VEV is tiny (whichrequires κ and κ to be equal to a very high accuracy), the induced neutrino masses willbe too large. This constraint on the φ VEV is even much stronger than that from thecustodial SU (2) violation. B Couplings between SM Fermions and Composite Operators, and TheirPeccei-Quinn Charges
Both SM Yukawa couplings and the Higgs quartic potential from collective symmetry break-ing arise from the partial compositeness couplings between the elementary fermions andcomposite operators. The leading interactions (with O (1) coupling strength) should re-spect an approximate U (1) P Q symmetry to avoid a too large quadratic ˜ H † H term andlarge FCNCs, so it is convenient to assign the PQ charges to the fermions in classifying thecouplings. We will construct a Type-I 2HDM model because of the weaker constraint onthe heavy Higgs bosons, and produce both terms needed for the collective quartic Higgspotential.For the quark sector, we include eight composite operators in and ¯6 representationsof SU (6) with overall PQ charges r = 0 , , , , r = r +1 / ⊕ r ⊕ ¯2 r − / ⊕ r (B.1a) ¯6 r = ¯2 r − / ⊕ r ⊕ r +1 / ⊕ r (B.1b)Here the subscript denotes the PQ charge instead of the hypercharge. The and ¯6 of thesame PQ charges create the same resonances which become the quark partners of differentflavors. The U (1) P Q charges of the three generations of elementary quarks are shown inTable 1. The lepton sector can be similarly assigned.There are some requirements for producing a Type-I 2HDM. First, to generate SMYukawa couplings, we need to couple one of q L and q R to and the other to ¯6 of the samePQ charge. In addition, each q L needs to couple to the composite operators at least intwo ways in order to generate the up-type and down-type Yukawa couplings with the sameHiggs doublet. If q L had only one coupling to (or ¯6 ), the up- and down-type quarks would– 32 – (1) P Q U (1) P Q U (1) P Q q ,L = ( t L , b L ) T t R b R q ,L = ( c L , s L ) T c R s R q ,L = ( u L , d L ) T u R d R Table 1 . PQ charges of elementary quarks. The PQ charge of u R appears out of the pattern. Asdiscussed in the text, the up quark Yukawa coupling comes from the U (1) P Q violating coupling,which also generates the required ˜ H † H term. couple to different Higgs doublets as we discussed in Sec. 3.4. Once q L couplings are fixed,the right-handed quark couplings follow directly from the PQ charges (except for the upquark). To generate the Higgs quartic term by collective symmetry breaking, we need tointroduce two pairs of couplings between the elementary doublets and the ( , ¯6 ) pairs, witheach pair of couplings preserving a different SU (4) symmetry. Finally, we add a U (1) P Q violating λ (cid:48)(cid:48) u L which serves to generate the mixed Higgs quadratic term in Eq. (4.16), andalso the up quark Yukawa coupling.From these requirements, a possible set of couplings between elementary quarks and thecomposite operators is shown below (in the parentheses after the corresponding compositeoperators). = / ( λ t L ) ⊕ ⊕ ¯2 − / ⊕ (B.2a) ¯6 = ¯2 − / ⊕ ⊕ / ⊕ ( λ t R ) (B.2b) = / ( λ c L ) ⊕ ⊕ ¯2 / ⊕ ( λ (cid:48) b R ) (B.2c) ¯6 = ¯2 / ( λ (cid:48) b L ) ⊕ ⊕ / ⊕ ( λ c R ) (B.2d) = / ⊕ ⊕ ¯2 / (˜ λ (cid:48) s L ) ⊕ (B.2e) ¯6 = ¯2 / ⊕ (˜ λ (cid:48) s R ) ⊕ / (˜ λ u L ) ⊕ (B.2f) = / ( λ (cid:48)(cid:48) u L ) ⊕ ⊕ ¯2 / ⊕ ( λ (cid:48) d R ) (B.2g) ¯6 = ¯2 / ( λ (cid:48) d L ) ⊕ ⊕ / ⊕ ( λ u R ) (B.2h)where the subscript of the coupling tells which elementary quark it is coupled to. (The left-handed couplings couple to the whole doublets despite the quark labels.) The SM quarkYukawa couplings are given by y t ∼ λ t L λ t R g ψ , y b ∼ λ (cid:48) b L λ (cid:48) b R g ψ , (B.3) y c ∼ λ c L λ c R g ψ , y s ∼ ˜ λ (cid:48) s L ˜ λ (cid:48) s R g ψ (B.4) y u ∼ λ (cid:48)(cid:48) u L λ u R g ψ , y d ∼ λ (cid:48) d L λ (cid:48) d R g ψ , (B.5)where g ψ r is the coupling of the strong resonances in r , ¯6 r , with their masses given by ∼ g ψ r f . To have a relatively light top partner, we should have g ψ ∼ , while all other g ψ r ’sare expected to be large. The quark flavor mixings (CKM matrix) can be generated by– 33 –dditional U (1) P Q violating couplings which are not shown. These couplings are expectedto be small and will not significantly affect the Higgs potential.For the Higgs quartic term, the combination of λ c L and λ (cid:48) b L generates one term of thecollective symmetry breaking, while the combination of ˜ λ (cid:48) s L and ˜ λ u L generates the other.Alternatively, we could also use the lepton sector to generate one of the collective symmetrybreaking terms. The quartic coupling is estimated to be λ = 316 π λ c L λ (cid:48) b L ˜ λ u L ˜ λ (cid:48) s L λ c L λ (cid:48) b L + ˜ λ u L ˜ λ (cid:48) s L ≈ π λ c L λ (cid:48) b L ( if ˜ λ u L ˜ λ (cid:48) s L (cid:38) λ c L λ (cid:48) b L ) . (B.6)To get a large enough λ , these couplings should be quite large ( (cid:38) ). The correct SMYukawa couplings can still be obtained by suitable choices of λ R couplings and g ψ r .The λ (cid:48)(cid:48) u L coupling violates the U (1) P Q symmetry as it mixes the q ,L with charge / with the composite doublet of charge / . By combining with λ (cid:48) d L , it will generate a mixingmass term for the two Higgs doublets, m ∼ π λ (cid:48) d L λ (cid:48)(cid:48) u L g ψ f . (B.7)In this way, all terms required in the Higgs potential for a realistic model can be generatedwithout introducing additional elementary fermions. References [1]
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