Phenomenology of a Composite Higgs Model
PPhenomenology of a Composite Higgs Model
Luigi Del Debbio ∗ Higgs Centre for Theoretical PhysicsThe University of EdinburghEdinburgh EH9 3FD, Scotland, UKE-mail: [email protected]
Christoph Englert
School of Physics and AstronomyThe University of GlasgowGlasgow, G12 8QQ, Scotland, UKE-mail: [email protected]
Roman Zwicky
Higgs Centre for Theoretical PhysicsThe University of EdinburghEdinburgh EH9 3FD, Scotland, UKE-mail: [email protected]
Several UV complete models of physics beyond the Standard Model are currently under scrutiny,their low-energy dynamics being compared with the experimental data from the LHC. Latticesimulations can play a role in these studies by providing a first principles computations of thelow-energy constants that describe this low-energy dynamics. In this work, we study in detaila specific model recently proposed by Ferretti [1], and discuss the potential impact of latticecalculations. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b omposite Higgs phenomenology Luigi Del Debbio
1. Introduction
Strongly-interacting dynamics is one of the possible scenarios describing the breaking of theelectro-weak (EW) symmetry observed in the Standard Model (SM): new physics beyond the Stan-dard Model (BSM) is the manifestation of a new theory in a nonperturbative regime, giving rise tomodels known as composite Higgs models. The dynamics of the low-lying states in the spectrum ofthe strongly-interacting theory is described by effective field theories (EFT), where the nonpertur-bative dynamics is encoded in a number of low-energy constants (LECs). EFTs are the preferredtool to compare strongly-interacting models with experimental data. Measurements translate di-rectly into constraints on the LECs, without ever having to specify the details of the underlyingtheory, which is often referred to as the UV completion. Traditionally, UV completions are sup-posed to be gauge theories coupled to matter in one, or several, representations of the gauge group.We shall refer to the charge under this new gauge group as hypercolor , while we will use the word color for the usual QCD sector of the SM.On the other hand, given a UV completion in the form of a hypercolor theory, first-principleinformation on the spectrum, and the LECs of the EFT, can be determined from lattice MonteCarlo simulations of the UV completion. These simulations are computationally expensive, and it isdesirable to determine the relevant observables, and their desired precision, from phenomenologicalstudies.In this work we address the question of the potential impact of lattice simulations within theparticular model presented by Ferretti in Ref. [1]. We write down the EFT that encodes the patternof spontaneous symmetry breaking in the UV completion, and analyse the bounds on the LECsfrom current LHC data. While similar studies have already been performed, our analysis focussesespecially on finding the experimental bounds on the quantities that can be computed in latticesimulations.
2. Low-energy spectrum
Let us briefly recall the low-energy description of the model in Ref. [1]. The effective ac-tion includes the nonlinear sigma model action describing the Nambu-Goldstone bosons (NGB)of the strong theory, and their self-interactions, and terms that describe their coupling to the SMparticles. The UV completion is an SU(4) gauge theory, with 5 Weyl fermions ( ψ Imn ) in the two-index antisymmetric representation of the hypercolor group, and 3 Dirac fermions (represented asa pair of Weyl fermions χ am , ¯ χ a (cid:48) m ) in the fundamental representation of the hypercolor group. Inthe expressions above, the flavor indices are I = , . . . , a = , . . . ,
3, and a (cid:48) = , . . . ,
3; the in-dices m , n = , . . . , G F = SU ( ) × SU ( ) × SU ( ) (cid:48) × U ( ) X × U ( ) (cid:48) . (2.1)The charges of the matter fields under the various symmetry transformations are explained in detailin Ref. [1]. The expected spontaneous symmetry breaking pattern of the model takes the form G F / H F = (cid:18) SU ( ) SO ( ) (cid:19) × (cid:18) SU ( ) × SU ( ) (cid:48) SU ( ) c (cid:19) × (cid:18) U ( ) X × U ( ) (cid:48) U ( ) X (cid:19) , (2.2)1 omposite Higgs phenomenology Luigi Del Debbio induced by the condensates (cid:104) ε mnpq ψ Imn ψ Jpq (cid:105) ∝ δ IJ , and (cid:104) ¯ χ a (cid:48) m χ am (cid:105) ∝ δ a (cid:48) a .The unbroken subgroup H F = SO ( ) × SU ( ) c × U ( ) X must contain the SM group. The EWgauge group SU ( ) L × U ( ) Y is embedded in the unbroken SO ( ) by considering the subgroupSO ( ) (cid:39) SU ( ) L × SU ( ) R , and then identifying a U ( ) subgroup generated by T R , the third gen-erator of SU ( ) R , and setting Y = T R + X . The unbroken vector subgroup SU ( ) c is gauged, andidentified with the QCD subgroup of the SM.The 14 NGB in the coset SU ( ) / SO ( ) can be classified according to their SM SU ( ) L × U ( ) R charges: → + ± / + + ± = ( η , H , Φ , Φ ± ) . (2.3)In composite models the Higgs boson is the NGB denoted by H in the list above. The field H is adoublet under SU ( ) L , and can be written as a two-component complex field H = ( H + , H ) . If weconsider the hypercolor theory with massless fermions in isolation, the NGB are exactly massless.The potential for the H field, and more generally the mass of all NGBs, is generated by the couplingof the hypercolor theory to the SM fields. Note that the NGB in the SU ( ) × SU ( ) (cid:48) / SU ( ) c cosetare color charged massless states that characterize this particular hypercolor theory. We will brieflydiscuss their properties below.The spin 1/2, color triplet states of the hypercolor theory are natural candidates to play therole of top partners, and hence generate the mass of the heaviest quark. It is often assumed, whenbuilding EFTs, that there exists one such state of mass M , which is lighter than the typical scaleof the hypercolor theory, Λ HC . A lattice study of the spectrum of the hypercolor theory wouldyield unambiguous predictions about the spectrum of expected resonances. In the context of anEFT description of the low-energy dynamics, the top partner is introduced in the effective actionas a Dirac fermion field Ψ transforming in the ( , ) / representation of H F . The SM quantumnumbers of this field can be found by decomposing the ( , ) / representation into irreduciblerepresentations of G SM ⊂ H F : ( , ) / → ( , ) / + ( , ) / + ( , ) / , (2.4)where the numbers on the RHS denote the irreducible representations of SU ( ) c × SU ( ) L × U ( ) Y .As usual the NGB are combined in a field Σ = exp (cid:16) i Π f (cid:17) , where f is the NGB decay constant ofthe hypercolor theory, and hence one of the LEC in the effective action. Σ transforms non-linearlyunder global SU ( ) transformations. It is convenient to write the effective action in terms of thefield U = ΣΣ T , which transforms according to U (cid:55)→ gU g T for g ∈ SU ( ) . The coupling to the SMgauge bosons is obtained by promoting the ordinary derivatives in the usual chiral lagrangian tocovariant derivatives: L ⊃ f
16 tr (cid:104)(cid:0) D µ U (cid:1) † D µ U (cid:105) , (2.5)where D µ U = ∂ µ U − igW a µ [ T aL , U ] − ig (cid:48) B µ (cid:2) T R , U (cid:3) . (2.6)The mass term for the fermions and the coupling to the SM fermions is L ⊃ M ¯ ΨΨ + λ q f ¯ˆ q L ΣΨ R + λ t f ¯ˆ t R Σ ∗ Ψ L , (2.7)2 omposite Higgs phenomenology Luigi Del Debbio where ˆ q L and ˆ t R are spurionic embedding of the SM quarks in the and ¯5 representations of SU ( ) respectively. The mass of the hypercolor state M is another LEC that can be determined from latticestudies of the spectrum of the theory. In this preliminary study, we will simply scan over a sensiblerange for M . Likewise λ q and λ t are LECs that determine the mass of the top quark.The contributions of the SM particles to the Coleman-Weinberg potential of the NGBs areresponsible for the misalignment of the vacuum, which leads to EW symmetry breaking. In partic-ular only the fermionic couplings are responsible for negative contributions to the potential, whichare necessary to generate a non-vanishing vev for the H component. Following the notation inRef. [1], we set H = h / √
2, and all other fields to zero. The couplings of the field h to the SMgauge bosons, and fermions are as follows:tr (cid:2) U ( h ) W µ U ( h ) † W µ (cid:3) = [ + cos ( h / f )] W c µ W c µ , (2.8)¯ˆ q L U ( h ) ˆ t R + ¯ˆ t R U ( h ) ∗ ˆ q L = √ ( h / f ) ( ¯ t L t R + ¯ t R t L ) . (2.9)The Coleman-Weinberg potential is parametrized by two LECs, α and β : V ( h ) ∝ α cos ( h / f ) − β sin ( h / f ) . (2.10)The LECs encode the contributions of the SM sector to the potential, in a way that is analogous tothe electromagnetic corrections to the pion mass [2]. They can be computed from field correlatorsas described in Ref. [3]: α = − C LR ( g + g (cid:48) ) < , (2.11)2 β = − y C top . (2.12)Clearly EW symmetry breaking can only occur if α + β >
0. Moreover the value of the Higgs vev in units of f is determined by these constants. The LECs in the equations above can be computedfrom first principles from correlators in the UV complete theory. For instance C LR = ( π ) (cid:90) ∞ dq q Π LR ( q ) + O ( g , g (cid:48) ) , (2.13)where Π LR is defined from the Lorentz decomposition of the current/current correlators: (cid:0) q g µν − q µ q ν (cid:1) Π LR ( q ) = (cid:90) d D x e ikx tr (cid:104) J R µ ( x ) J L ν ( ) (cid:105) , (2.14)and J R , L µ ( x ) = ¯ ψγ µ ± γ ψ ( x ) . Similarly, C top can be computed on the lattice, and therefore the issueof EW dynamical symmetry breaking can be resolved by numerical simulations.Finally note that diagonalising the mass term in the effective action yields the top mass, atleading order in v , m t = √ M f λ q λ t (cid:113) M + λ q f (cid:112) M + λ t f v , (2.15)or, equivalently, m t / v = √ ρ M λ q λ t (cid:113) + λ q ρ M (cid:113) + λ t ρ M , (2.16)3 omposite Higgs phenomenology Luigi Del Debbio where we have introduced the ratio ρ M = f / M , which can be extracted from numerical simulations.Introducing spurions ˆ q iL ∈ , ˆ u iR ∈ , and ˆ d iR ∈ , where i = , , L ⊃ √ µ b tr (cid:16) ¯ˆ q L U ˆ d R + h . c . (cid:17) ; (2.17)the new LEC, µ b , is determined by requiring that the correct value is recovered for the bottomquark mass.
3. Constraints from data
Expanding the exponential in U , the effective lagrangian for the Higgs field reduces to the so-called minimal composite Higgs model (MCHM) [4, 5, 6]. The Higgs coupling to the EW gaugebosons is rescaled by a factor (cid:112) − ξ compared to the SM values, where ξ = v / f . Bounds on ξ from current experimental data have been studied in detail in the context of MCHM, see e.g. [7].As discussed in the previous section, this specific model predicts additional PNGBs, i.e. ad-ditional charged Higgs particles, whose masses are set by the EW contributions to the Coleman-Weinberg potential. The color octet of hyper-pions complete the low-energy spectrum of the theory,the lower bound on their mass being currently in the multi-TeV regime [8, 9].Assuming e.g. a mass m ≈
200 GeV for the exotic Higgses, and taking into account the exper-imental constraints on ξ , the production cross sections for these particles can be computed. Evenfor the lightest values of m , we find production cross sections of the order of 100 fb, which decreaseexponentially for higher masses. This is reassuringly consistent with the fact that these states havenot been observed experimentally.The customary observable for studies of BSM phenomenology is the signal strength in agiven channel, which is defined as the ratio of the expected events in the BSM extended modelto the events predicted by the SM. The additional Higgs particles discussed above modify the h → WW , γγ , ZZ signal strengths. For this preliminary report we focus on these three channelsonly, as examples of possible applications. In each channel we compute the signal strength pre-dicted by the effective theory above. The results of our study are shown in Figs. 1, 2 and 3. In allplots the LHC data are displayed by the green band as a function of ξ , or, equivalently, f . The blueband is obtained by scanning over the values of the LECs. More precisely, we vary M in the range [ . , . ] TeV, λ t in the range [ , π ] . The value of λ q is then fixed by the mass of the top quark,as shown in Eq. 2.16. Finally µ b is engineered to reproduce the correct bottom quark mass of 4.7GeV.The plots show a bit of tension between the signal strength meaurements by ATLAS and CMSand the prediction in the model. In particular, WW production is currently favoured slightly belowthe SM expectation. However, the presence of the newly charged scalar states introduces additionalfreedom to obtain a signal strength in the diphoton channel that is comparable to the observed limitsover a wide range of signal strengths. The width of the blue and red bands highlights the sensitivityof the Higgs branching to photon pairs. 4 omposite Higgs phenomenology Luigi Del Debbio ξ µ ( WW ) Figure 1:
Signal strength µ ( WW ) for the decay into h → WW gauge bosons. The green band is the error inthe LHC data, while the blue data results from the scan of the parameters of the EFT. ξ µ ( γγ ) Figure 2:
Signal strength µ ( γγ ) for the decay into h → γγ gauge bosons. The green band is the error inthe LHC data, while the blue data results from the scan of the parameters of the EFT. The red data includescharged Higgs contributions to the diphoton decay width.
4. Conclusions
The preliminary results of our study suggest that the UV theory considered here is compatiblewith the data analysed so far by the experiments at the LHC, for a sensible range of the couplings inthe effective action, despite showing some tension e.g. the h → WW . Currently, the main source ofuncertainty is the experimental error in the data, while the variations in the theoretical predictionsdue to the scan in the space of parameters seems to be rather smaller.We have identified a number of LECs that appear in the effective action, and have discussed thepossibility of measuring them in lattice simulations of the UV complete underlying theory. Someof the LEC like the condensates, C LR , C top allow us to check the pattern of symmetry breaking,and the generation of the correct Higgs potential. Note that a robust prediction of the sign of5 omposite Higgs phenomenology Luigi Del Debbio ξ µ ( ZZ ) Figure 3:
Signal strength µ ( ZZ ) for the decay into h → ZZ gauge bosons. The green band is the error in theLHC data, while the blue data results from the scan of the parameters of the EFT. the combination α + β would already yield valuable information. Lattice results would alsoconstrain the region that needs to be explored when scanning over the space of LECs. In particulara determination of ξ and ρ M would reduce the scan to a lower-dimensional subspace, and thereforecould constrain the model substantially.As the experimental errors decrease, lattice simulations can lead to stringent tests of BSMmodels with a UV completion. References [1] G. Ferretti, JHEP (2014) 142 doi:10.1007/JHEP06(2014)142 [arXiv:1404.7137 [hep-ph]].[2] T. Das, G. S. Guralnik, V. S. Mathur, F. E. Low and J. E. Young, Phys. Rev. Lett. (1967) 759.doi:10.1103/PhysRevLett.18.759[3] M. Golterman and Y. Shamir, Phys. Rev. D (2015) no.9, 094506doi:10.1103/PhysRevD.91.094506 [arXiv:1502.00390 [hep-ph]].[4] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B (2003) 148doi:10.1016/j.nuclphysb.2003.08.027 [hep-ph/0306259].[5] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B (2005) 165doi:10.1016/j.nuclphysb.2005.04.035 [hep-ph/0412089].[6] R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D (2007) 055014doi:10.1103/PhysRevD.75.055014 [hep-ph/0612048].[7] G. Aad et al. [ATLAS Collaboration], JHEP (2015) 206 doi:10.1007/JHEP11(2015)206[arXiv:1509.00672 [hep-ex]].[8] V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B (2015) 98doi:10.1016/j.physletb.2015.04.045 [arXiv:1412.7706 [hep-ex]].[9] The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2016-084.(2015) 98doi:10.1016/j.physletb.2015.04.045 [arXiv:1412.7706 [hep-ex]].[9] The ATLAS collaboration [ATLAS Collaboration], ATLAS-CONF-2016-084.