Constraining Composite Higgs Models using LHC data

We systematically study the modifications in the couplings of the Higgs boson, when identified as a pseudo Nambu-Goldstone boson of a strong sector, in the light of LHC Run 1 and Run 2 data. For the minimal coset SO(5)/SO(4) of the strong sector, we focus on scenarios where the standard model left- and right-handed fermions (specifically, the top and bottom quarks) are either in 5 or in the symmetric 14 representation of SO(5). Going beyond the minimal 5L-5R representation, to what we call here the 'extended' models, we observe that it is possible to construct more than one invariant in the Yukawa sector. In such models, the Yukawa couplings of the 125 GeV Higgs boson undergo nontrivial modifications. The pattern of such modifications can be encoded in a generic phenomenological Lagrangian which applies to a wide class of such models. We show that the presence of more than one Yukawa invariant allows the gauge and Yukawa coupling modifiers to be decorrelated in the 'extended' models, and this decorrelation leads to a relaxation of the bound on the compositeness scale (f>640 GeV at 95% CL, as compared to f>1 TeV for the minimal 5L-5R representation model). We also study the Yukawa coupling modifications in the context of the next-to-minimal strong sector coset SO(6)/SO(5) for fermion-embedding up to representations of dimension 20. While quantifying our observations, we have performed a detailed chi-square fit using the ATLAS and CMS combined Run 1 and available Run 2 data.


Introduction
With increasing precision in measurements of the Higgs boson properties at the Large Hadron Collider (LHC), the possibility that the Higgs may be a composite object [1][2][3][4][5] can be put to stringent tests. In this context, the scenarios where the Higgs is identified as a pseudo Nambu-Goldstone boson (pNGB) of a strongly interacting sector are of special interest. This has received considerable attention following its identification as a holographic dual of 5d gauge-Higgs unification models [6][7][8][9]. In this paper, however, we stick to an effective 4d scenario, and do not comment on possible UV completion of such models. The approximate shift-symmetry of the pNGBs can screen the weak scale from physics beyond the compositeness scale (f ∼ O(TeV)). This provides a well-motivated framework for natural electroweak symmetry breaking.
The direct signatures of these models at the LHC could be the appearance of additional resonances of the strong sector [10][11][12][13][14][15][16][17][18]. However taking cue from non-observation of these resonances, attempts have been made to push up the resonance masses while keeping the theory still natural [19][20][21][22][23]. The other inevitable and testable features of these models are deviations of the Higgs couplings compared to their standard model (SM) predictions. One of the consequences of compositeness is that the couplings are replaced by form factors which are momentum dependent. However, it is difficult to test this momentum dependence at the LHC. Nevertheless, the nonlinearity of the pNGB dynamics provides a finite shift in the Higgs couplings measurable in the precision era of the LHC. In this paper we make a systematic study of the pattern and constraints on such modifications that arise in a general class of composite Higgs models.
The couplings of the pNGB Higgs with the weak gauge bosons (V V h) are usually suppressed in a general class of composite models. The parameter ξ ≡ v 2 /f 2 1, where v = 246 GeV is the electroweak vacuum expectation value (vev), controls this suppression. The Yukawa couplings are generated through a mixing between the elementary fermions and the operators of the strong sector. Once the strong sector is integrated out the effective Higgs-fermion interaction term looks like [45,46], where F(H † H/f 2 ) is a function of the SU(2) L doublet Higgs field (H). The contributions from the higher dimensional operators with independent coefficients, added to the SM dimension-4 Yukawa term, give rise to a modification in the couplings of the Higgs with the fermions (ff h), see also [47,48] in a different context. In the minimal model, the SM fermions couple to only one operator of the strong sector. As a result the modification of the couplings depends on only one free parameter ξ. The other parameters in the effective Lagrangian are fixed from the requirement of reproducing the corresponding SM fermion mass. Therefore, ff h and V V h couplings get highly correlated, and stringent constraints on f emerge [49,50] from the increasingly precise measurements of Higgs production and decays at LHC. In the extended models, owing to the presence of more than one invariant in the Yukawa sector with different coefficients, the correlation between ff h and V V h modifiers is weakened, and we observe a possible relaxation of the bound on f . This happens in certain regions of the parameter space where a possible enhancement in ff h vertex can partially offset the suppression in V V h coupling. Additionally, the extended models, carrying more than one invariant in the Yukawa sector, have the distinct advantage of being free from 'double tuning' [30] 5 .
In this paper, we first concentrate on a systematic and comparative study of various possibilities of Higgs coupling modifications in the context of the extended models 6 . For each such possibilities, we 5 ∆ = 1/ξ is a measure of minimal tuning in any composite Higgs model. On top of this, an additional tuning, dubbed 'double tuning', arises in scenarios (e.g. MCHM5 L −5 R ) where the coefficients of the quadratic and quartic terms in the potential are not in the same order of the elementary-composite mixing parameter. This can be avoided when either the fermion kinetic and/or the Yukawa terms contain at least two invariants. 6 We do not consider representation 10 of SO(5) because it does not lead to more than one Yukawa invariant keeping a discrete parity that protects the Zbb vertex [29,51]. Note that the choice MCHM14 L −1 R , where tR can be fully composite, involves minimal tuning as compared to the double tuned MCHM5 L −5 R [30]. However, we do not consider this choice because it contains a single Yukawa invariant. construct one-loop Coleman-Weinberg Higgs potential [52], and identify regions of parameters space where the top mass, Higgs mass and the electroweak vev are reproduced. Next we consider the nextto-minimal model which contains a SM scalar-singlet (η) apart from the Higgs doublet. Their mixing can significantly modify the observed Higgs boson couplings. In this context also, we survey different fermionic representations and calculate the corresponding modifications to Yukawa couplings.
We then construct an effective phenomenological Lagrangian whose parameters capture the coupling modifications of a general class of models mentioned earlier. The explicit connection between the coefficients of the Lagrangian and the parameters of the specific models is specified on a case-by-case basis. We perform a χ 2 analysis with the ATLAS and CMS combined Run 1 [53] and available Run 2 data [54][55][56][57][58][59][60][61][62][63][64][65][66] to estimate a bound on f in the extended models and compare it with that of the minimal model. In the context of the next-to-minimal model, we provide an estimate of the amount of doublet-singlet scalar mixing allowed by the current data.
The rest of the paper is organized as follows. In Section 2 we review the consequences of various fermionic embedding for SO(5)/SO(4) as well as SO(6)/SO(5) cosets. In Section 3 we present a phenomenological Lagrangian that captures the generic features of a wide class of models in terms of the Higgs coupling modifiers. Following this parametrization we perform a fit to the existing data using the χ 2 minimization technique in Section 4. Finally, we draw our conclusions in Section 5.

Composite Models and Modified Yukawa Couplings
In this section we consider different representations for fermions in SO(5)/SO(4) and SO(6)/SO (5) cosets and work out the modifications in the top quark Yukawa coupling in a systematic manner.

SO(5)/SO(4) Coset
As long as the coset is SO(5)/SO(4), the modification in V V h coupling is solely determined by ξ, as The number of Yukawa invariants, on the other hand, depends on the representations in which t L and t R are embedded. We write down the relevant invariants using the pNGB representation Σ = in the unitary gauge : Above, Q t L and T t R contain t L and t R as incomplete SO(5) multiplets, respectively (see Appendix A.1). The most general Lagrangian involving the top quark can be written as The dependence on the strong sector dynamics is encoded inside the momentum dependent Πfunctions. In Table 2 In the second equality of Eq. (2.3), the factor 1 − 1 2 ξ arises due to canonical normalization of the Higgs field. As argued in [31,67], the top quark contribution to the effective gluon-gluon-Higgs (ggh) coupling in composite Higgs models is independent of the wave function renormalization effects of the top quark due to cancellation with resonance loops. This would imply a deviation in effective ggh coupling compared to the effective tth coupling. The modification of the effective ggh coupling can be expressed as The one-loop Coleman-Weinberg Higgs potential receives largest contribution from the top quark, as The coefficients α and β above are integrals over the form factors. A similar contribution to α arises from gauge boson loops with opposite sign (largest contribution from SU(2) L gauge bosons), parametrized as [68,69] α g −c g 1 16π 2 where c g is an O(1) positive constant absorbing the details of the integration, g is the SU(2) L gauge coupling, and g ρ corresponds to that of strong sector spin-1 resonances. The gauge contribution to β is numerically small. To calculate the top-induced contribution to α and β, we use certain parametrization of the momentum dependent form factors based on scaling arguments. The decay constants and the top-partner masses are parametrized as where λ L,R i are dimensionless constants and g i denote strong couplings. In the present analysis, we keep |λ i | /g i < 1. The strong sector coupling strengths are kept well within the perturbative limits, i.e. 1 < g i < 2π. Regarding the integrals over the form factors in fermionic sector, the loop factors and the dimensionful variables are shown explicitly. Some group theoretic factors also emerge due to the decomposition of the SO(5) resonances in terms of SO(4) multiplets. We assume that sufficient number of resonances with coupling strengths g i saturate the form factors, rendering the integrals finite. As an illustration, we show one of the integrals involved in the Higgs potential, as parametrized in [3,68,69] where c (n) 1,2 are O(1) numbers and the forms factors are displayed in Appendix B. Finally we use the following phenomenological constraints to generate the allowed parameter space: We present the results of our numerical analysis in Fig. 2

SO(6)/SO(5) Coset
The next-to-minimal model, with SO(6)/SO(5) coset includes a real singlet scalar (η) along with the usual Higgs doublet. Quite a few interesting features emerge in this case, depending on whether η acquires a vev [21,43,44] or not [70][71][72][73]. In the present section we discuss the effect of the η-vev and consequently the doublet-singlet scalar mixing on the Yukawa couplings. Here we follow the convention and notation as presented in [21].
In this case the structure of the Lagrangian involving the top quark is similar to Eq. (2.2), with the exception that the Π-functions are dependent both on h and η, as shown in Appendix C, for different representations. Although compared to SO(5)/SO(4) coset, more possibilities of embedding t L and t R in different SO(6) multiplets exist, we stick to the choices shown in Appendix A.2 only. The Lagrangian, in terms of the canonically normalized quantum fields (h n , η n ), upon electroweak symmetry breaking, can be written as Due to the doublet-singlet mixing, the state corresponding to the observed Higgs field is where θ mix denotes the amount of mixing and is constrained by the LHC Higgs data. For the case, where both m η m h and η h , the mixing angle can be simply parametrized as [44] θ mix ∼ h η m 2 η 1 . (2.12) We also observe that where χ = η 2 /f 2 . Because of an inherent Z 2 symmetry associated with our choice of embedding, η couples with the top quark as η 2 . When the Z 2 symmetry is spontaneously broken the dependence on χ appears. The appearance of ξ is a consequence of constructing an SU(2) invariant Yukawa-like term involving the η field. Finally the expression for the Yukawa coupling modifier involving the observed Higgs is obtained as k tth = cos θ mix k tthn − sin θ mix k ttηn . (2.14) We show some representative plots illustrating the impact of χ on the Yukawa coupling modifier. In Fig. 2.2 we present the variation of k tth with χ for NMCHM 6 L −6 R . Obviously extra model dependence appears in the case of symmetric representation (20), where more than one Yukawa invariant exist.

Effective Phenomenological Lagrangian
The modifications in the Higgs couplings as discussed in the previous section have two generic features: (i ) modification in V V h coupling, arising from the non-linearity of the pNGBs, is universal as long as the coset belongs to SO(N )/SO(N − 1) group (modulo the mixing with other states), and (ii ) modification of the Yukawa couplings depends on the choice of fermion embeddings. These can be captured in an effective Lagrangian as,  [74]. In the SO(5)/SO(4) models, additional contributions to dimension-6 operators emerge, given by In the above Lagrangian we have dropped terms which are highly constrained by the electroweak precision observables [45]. One can read off the Yukawa and ggh/γγh coupling modifiers as and the modifier for V V h coupling as While ξ represents the ratio of the weak scale to the effective scale of the theory, thus naturally controlling the coupling modifiers, a brief discussion of the other two parameters, namely ∆ and δ, in

Modifiers
Dependence on parameters the effective Lagrangian is in order. The origin of ∆ can be traced back to the nonlinear realization of the pNGB sector. In scenarios containing only one Yukawa invariant this is a numerical constant (e.g. in MCHM 5 L −5 R , ∆ −3/2), while in the extended models with several invariants this factor may deviate depending on the details of the strong sector resonances. In contrast, δ reflects the effect of partial composite nature of the top quark in these theories, contributing to the anomalous dimension of the top quark. In the effective ggh and γγh vertex, in fact, contributions from the wave function renormalization cancel against the resonance loop contributions [31,67]. In our phenomenological analysis, that follows in the next section, we will employ the effective Lagrangian (Eqs. (3.1) and (3.2)), to confront the LHC Higgs data. All fitting are done assuming ∆ t and δ t to be free parameters. Further we assume that the bottom and τ Yukawa couplings are modified only by the universal factor ∆ b = ∆ τ = −3/2, i.e. they are always suppressed compared to their SM values. We also make a reasonable approximation δ b = δ τ = 0. A complete list of all the coupling modifiers within the SO(5)/SO(4) model is given in Table 3.1 7 . The explicit expressions of ∆ t and δ t in terms of the form factors are defined in Table D.1 of Appendix D.
The main feature that gets added when one moves to the next-to-minimal model is the presence of an additional singlet scalar and its mixing with the Higgs doublet. A description of the composite models including a SM singlet in the context of a strongly interacting light Higgs can be found in [75]. Here we present a simplified effective Lagrangian keeping only the dominant terms. We add the following piece involving η to Eqs. (3.1) and (3.2), Note that the dimension-5 operators involving a single η field is not allowed in the presence of a Z 2 symmetry, as discussed in previous section. Due to doublet-singlet scalar mixing (θ mix ), the Yukawa modifier for the observed Higgs boson (h ) assumes the following form where ∆ η f is a function of (∆ η f ) and the η-vev. The expressions for ∆ η t for different representations are given in Table D.2 of Appendix D. In the following analysis we assume ∆ η t = ∆ η b for simplicity. 7 In [53], effective ggh and γγh coupling modifiers have been calculated keeping only the dominant terms: k ggh ggh and k γγh 0.07 (k (t) The V V h coupling modifier now picks up the additional factor cos θ mix compared to the minimal coset (see Eq. (3.4)) k V V h = cos θ mix 1 − ξ. (3.7)
We perform a χ 2 fit to assess the present constraints starting from the effective Lagrangian introduced in the previous section. We use the combined ATLAS+CMS Run 1 results for signal strengths, given by the 'six-parameter' fit as shown in Table 15 of [53]. The so far available Run 2 (13 TeV) results are summarized in   panels, respectively. Solid black line represents ∆χ 2 = χ 2 − χ 2 min for the extended models, while the red dashed lines represents the same for MCHM 5L−5R . Green and yellow regions denote the allowed range for ξ at 68% and 95% CL, respectively.
The effective Lagrangian given in Eq. (3.2), which corresponds to SO(5)/SO(4) coset, have three independent parameters ξ, ∆ t and δ t . Using this parametrization we calculate the Higgs signal strengths in various final states normalized to their SM values. These are then compared with the data using the χ 2 fit. The minimum value of the χ 2 and corresponding best-fit values of the parameters in the extended models are given below. This may be compared with a similar fit obtained for the MCHM 5 L −5 R with only one free parameter ξ. The best-fit values are given by • Run 1 : • Run 1 + Run 2 : χ 2 min = 19.72, ξ = 0.00 . , with all other parameters fixed to their best-fit values. For comparison, we also show the curve for MCHM 5 L −5 R , and our results agree with [44] wherever we overlap. For the extended models, we obtain a lower bound f ≥ 465 GeV at 95% CL from Run 1 data only. This should be compared with f ≥ 780 GeV at 95% CL for MCHM 5 L −5 R . The relaxation of the bounds on f for the extended models follows from the reduced correlation between k tth and k V V h in the effective Lagrangian in Eq. (3.2) as compared to the tight correlation in MCHM 5 L −5 R . We find that combined Run 1 and Run 2 data give significantly more stringent lower bound, namely, f ≥ 640 GeV at 95% CL for the extended models.
In Figs. 4.2 we check whether the extended models fit the data better (i.e. χ 2 /d.o.f. is lower) than MCHM 5 L −5 R . In the blue shaded region the extended models, as parametrically encoded in the effective Lagrangian in Eq. (3.2), fit relatively better for the entire range of ξ. On the same plot we also throw the actual model points, with the resonance masses and decay constants as the strong sector inputs, discussed on a case-by-case basis in Section 2.1, satisfying the constraints shown in Eq. (2.9). In Figs. 4.3, the experimentally preferred regions for the coupling modifiers are shown at 68% and 95% CL in the (k tth -ξ) and (k (t) ggh -k tth ) planes. The model points are observed to span over a large range of the preferred regions. It may be noted that present experimental precision is not sensitive to the value of δ t separately; what is in fact bounded is the combination (∆ t + δ t ). Future colliders may have sufficient precision to sense the different modifications in the top Yukawa coupling and the effective ggh coupling.
Moving to the next-to-minimal coset, we deal with a new feature that the Higgs doublet now mixes with a real singlet. The mixing results in a further suppression of the observed Higgs boson coupling to the massive gauge bosons. The Yukawa couplings are modified too because of the presence of a singlet. We perform a similar χ 2 analysis with the combined Run 1 and Run 2 results to impose an upper bound on the amount of mixing. Fig. 4.4 shows that the maximum amount of mixing allowed so far at 95% CL is θ mix ∼ 0.35. Future data would constrain it even further [21].  ggh − k tth planes allowed at 68% (green) and 95% (yellow) CL using combined Run 1 and available Run 2 data. In the left panel the red line corresponds to MCHM 5L−5R . On the right panel, the grey dashed line corresponds to δ t = 0. Valid 'extended model' points are observed to lie within the experimentally allowed regions.

Conclusions
Non-linearity of pNGB dynamics modifies the Higgs boson couplings with the weak gauge bosons as well as with the fermions compared to their SM expectations. The ratio ξ, which parametrizes the hierarchy between the weak scale and the strong sector spontaneous symmetry breaking scale, controls this deformation. In MCHM 5 L −5 R , the Yukawa sector contains a single invariant. Here, the single parameter ξ appears in the modifications of both V V h and ff h couplings, leading to a rather strong lower limit f ≥ 1 TeV, as the data show increasing affinity towards the SM predictions. In the extended models, MCHM 14 L −14 R , MCHM 14 L −5 R and MCHM 5 L −14 R , owing to the presence of more than one invariant in the Yukawa sector, the ff h coupling modifier depends on other parameters of the strong sector in addition to ξ. This releases the tension leading to a new lower limit f ≥ 640 GeV, which is much relaxed compared to the limit in MCHM 5 L −5 R .
An important feature of these models is the emergence of a parametric difference in the top Yukawa and the effective gluon-gluon-Higgs vertices. This arises because of a cancellation between the toppartner resonance masses in the loop with the wave function renormalization of the top quark in the calculation of the effective ggh vertex. However, the present data is insensitive to smell this difference.
We have in fact constructed a phenomenological Lagrangian which captures the effects of a vast array of such models with different fermionic representations. We have constrained the parameters of this Lagrangian using LHC data and observed that the allowed regions are quite consistent with a reasonable choice of strong sector input parameters of the individual models which yield correct values of m t , v and m h .
We have extended our analysis to the next-to-minimal model as well. The appearance of a real singlet scalar adds a new twist to phenomenology, whose mixing with the Higgs doublet is constrained by the LHC data. Interestingly, the singlet scalar also contributes to the top Yukawa through an effective higher dimensional operator.
Our analysis shows that further precision, likely to be achieved in future colliders, would constrain these scenarios to the extent that individual models could be discriminated, and the proposition that the Higgs boson may have a spatial extension would be challenged with more ammunition.

A.1 SO(5)/SO(4) Coset
Fundamental 5 and symmetric 14 representations of SO(5) can be decomposed under the unbroken SO(4) ≡ SU(2) L × SU(2) R as follows: We embed t L into the (2, 2)'s so that the correction to Zbb vertex is under control, while t R is embedded into the (1, 1). The embeddings of the top quarks into incomplete multiplets of 5 and 14 are given below. and

B Details of Form Factors
The form factors appearing in

C Π-functions for Next-to-Minimal Models
The Π-functions for the next-to-minimal case are given for different representations in  In Table D.1 we list the expressions for ∆ t and δ t for the models MCHM 5 L −5 R , MCHM 14 L −14 R , MCHM 14 L −5 R and MCHM 5 L −14 R , respectively. We present the expressions for (∆ t + δ t ) and ∆ η t , as defined in Eq. (2.12), in terms of the form factors, for different SO(6) representations in Table D.2.

Models
Coupling Modifiers