1 Introduction

CERN-PH-TH/2012-176

KA-TP-26-2012

SFB/CPP-12-43

ZU-TH 09/12

LPN12-062

Higgs Low-Energy Theorem (and its corrections)

[3mm] in Composite Models

M. Gillioz, R. Gröber, C. Grojean, M.Mühlleitner and E. Salvioni

Institut für Theoretische Physik, Universität Zürich, CH-8051 Zürich, Switzerland

Institut für Theoretische Physik, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany

Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland

Dipartimento di Fisica e Astronomia, Università di Padova and INFN, Via Marzolo 8,

Abstract

The Higgs low-energy theorem gives a simple and elegant way to estimate the couplings of the Higgs boson to massless gluons and photons induced by loops of heavy particles. We extend this theorem to take into account possible nonlinear Higgs interactions as well as new states resulting from a strong dynamics at the origin of the breaking of the electroweak symmetry. We show that, while it approximates with an accuracy of order a few percents single Higgs production, it receives corrections of order for double Higgs production. A full one-loop computation of the cross section is explicitly performed in MCHM5, the minimal composite Higgs model based on the coset with the Standard Model fermions embedded into the fundamental representation of . In particular we take into account the contributions of all fermionic resonances, which give sizeable (negative) corrections to the result obtained considering only the Higgs nonlinearities. Constraints from electroweak precision and flavor data on the top partners are analyzed in detail, as well as direct searches at the LHC for these new fermions called to play a crucial role in the electroweak symmetry breaking dynamics.

1 Introduction

There is growing evidence that a Higgs boson is the agent of electroweak symmetry breaking (EWSB) [1]. And one of the most pressing questions is to uncover the true nature of this Higgs boson: is it the elementary scalar field of the Standard Model? Is it part of a supermultiplet? Is it a composite scalar emerging as a bound state from a strongly coupled sector? Theoretical arguments based on naturalness considerations tend to favor one of the two latter scenarios. But the LHC experiments have now opened a new data-driven era and the first experimental indications might come from possible deviations in the measurements of the Higgs couplings [2] compared to the ones predicted by the Standard Model (SM) that are unambiguously fixed by the value of the Higgs mass itself. However, both supersymmetric and composite Higgs bosons in the decoupling limits can be arbitrarily close to the SM Higgs at the energy scale currently probed by the LHC and deciphering the different scenarios from one another might require more luminosity than the one currently accumulated. A more unambiguous answer would come from a direct observation of additional particles: supersymmetric partners of the SM particles or additional resonances of the strong sector. In both scenarios, new particles in the top sector have a special status: in supersymmetric models, the stops cannot be too heavy without destabilizing the weak scale [3]; the top partners in composite models are responsible for generating the potential of the would-be Goldstone Higgs boson and, as recently noticed in Refs. [4], they have to be lighter than about 700 GeV to naturally accommodate a Higgs boson as light as 125 GeV. Even if the actual numbers are model-dependent, this conclusion is rather generic and certainly calls for improving the ongoing dedicated direct searches for the top partners [5].

In principle, indirect information on these top partners could also be obtained in Higgs physics. It is indeed quite ironic that, while the Higgs boson is supposed to be at the origin of the masses of all elementary particles, the currently most sensitive channels are the ones that involve massless particles, i.e. particles with no direct coupling to the Higgs boson: the gluons for its production and the photons for its decay. Clearly these processes appear only at the loop level in the SM and are therefore potentially sensitive to new states circulating in the loops. The structure of the Higgs couplings to photons and gluons are beautifully captured by the background field method known as Low-Energy Theorem (LET) [6, 7] and the effects of the top partners, or any new particles, are encoded by their sole contributions to the QED and QCD beta functions. In composite Higgs models, the corrections due to the Higgs compositeness to the SM gluon fusion production cross section, , and to the SM decay width into two photons, , were estimated in Ref. [8] to be generically of the order of , where  246 GeV is the weak scale and is the characteristic scale of the strong sector (the equivalent of in QCD). From dimensional arguments, the additional corrections due to the top partners should be of the order of , being respectively the mass of the top and the typical mass scale of his partners, i.e. a correction potentially as big as the one originating from the strong dynamics itself when the top partners are around 700 GeV. However, it was quickly realized [9, 10, 11] that these leading-order corrections from the top partners actually cancel and that they only give a contribution that we shall estimate (see Section 3.3) to scale like , i.e. one order of magnitude smaller than the strong dynamics contribution, sweeping any hope to learn anything about the top partners from the measurement of the Higgs production by gluon fusion and leaving only the Higgs production in association with a top-antitop pair [12] as a place to indirectly look for new physics in the first LHC run. A proper effective description of the top partners/fermionic resonances would be needed to study this promising channel.

The second LHC run will certainly increase the sensitivity of the direct searches for top partners. But it will also open new possibilities to indirectly probe the top partner sector in Higgs physics by exploring multi-Higgs production. A generalization of the LET exists for double Higgs production in the SM, and it is known to give a reasonably good estimate of the total rate within 20% accuracy for a light Higgs [13, 14] but it badly fails to reproduce the differential distributions [15]. The extension of the LET to strong EWSB models is not totally trivial since, in the SM, there is a strong destructive interference between two contributions and therefore strong dynamics that gives rise to a third new contribution can have order-one effects on the double Higgs production by gluon fusion, , as already noticed in Refs. [16, 17, 18]. In an explicit composite Higgs model, we are going to compare the LET results to an explicit one-loop computation taking into account all the contributions from the fermionic resonances in the top sector. We will find that the LET is less accurate than in the SM, and underestimates the full result by up to . On the other hand the pure strong dynamics effects arising due to the sole Higgs nonlinearities are overestimating the production rate which receives corrections from the top partners.

The outline of our paper is as follows. In Section 2, we first present two equivalent effective Lagrangians, the linearly realized strongly-interacting light Higgs Lagrangian [8] as well as a chiral Lagrangian where the SM gauge symmetry is nonlinearly realized [19], to describe Higgs physics at the LHC and we relate these effective Lagrangians to explicit composite Higgs models. In Section 3, we recall the LET in the SM and extend it to composite Higgs models, reproducing the often heard results that the gluon fusion production cross sections can be significantly reduced by up to 50%70% compared to the SM production. We also re-discuss within the LET approximation the cancellation of the top partner contributions to single Higgs production by gluon fusion and we estimate the size of the corrections to the LET results. We finally derive the LET for double Higgs production by gluon fusion and show that a factor 23 enhancement over the SM can be obtained in specific examples. This LET enhancement factor is nonetheless smaller than the one obtained considering only the pure Higgs nonlinearities effects. Section 4 is devoted to the top partners in an explicit minimal composite Higgs model, MCHM5, based on the coset space with SM fermions embedded into fundamental representations of . We first discuss in detail the constraints on the masses and couplings of these new fermions coming from electroweak (EW) precision measurements and from flavor physics. We then present exhaustive constraints from direct searches for these top partners using the current Tevatron and LHC data. In Section 5, a detailed computation of the contribution of the top partners to is presented, confirming that the LET gives a good approximation and confirming the scaling behavior of the corrections obtained in Section 3. Finally, Section 6 is devoted to double Higgs production. We demonstrate explicitly that the full loop computation with the top partners can easily give negative corrections of order 30% to the pure strong dynamics computation where only the Higgs nonlinearities were taken into account [17]. Contrary to the single Higgs production case, the LET does not capture well these contributions of the top partners, and this discrepancy is further amplified when looking at particular regions of the phase space selected by kinematical cuts. Various technical details are collected in a series of appendices.

2 Low-energy effective Lagrangian for a composite Higgs boson

An interesting solution to the hierarchy problem is given by the Higgs boson being a composite bound state emerging from a new strongly-interacting sector, broadly characterized by a mass scale and a coupling . If in addition the Higgs emerges as a pseudo-Goldstone boson of a spontaneous symmetry breaking at the scale , then it can be naturally lighter than the other resonances of the strong sector and can be accommodated. A low-energy, model-independent description of this idea is given by the strongly-interacting light Higgs (SILH) Lagrangian [8], which applies to the general scenario where the Higgs is a light pseudo-Goldstone boson, including Little Higgs and Holographic composite Higgs models. At scales much smaller than , deviations from the SM are parameterized in terms of a set of dimension-six operators. The subset that will be relevant in our discussion is

 LSILH= cH2f2∂μ(H†H)∂μ(H†H)+cr2f2H†H(DμH)†(DμH)−c6λf2(H†H)3 + (cyyff2H†H¯fLHfR+h.c.)+cgg2s16π2f2y2tg2ρH†HGaμνGaμν+cγg′216π2f2g2g2ρH†HBμνBμν (1)

where are the SM gauge couplings, whereas and are the Higgs quartic and Yukawa coupling appearing in the SM Lagrangian, respectively. The first four operators are genuinely sensitive to the strong interaction, whereas the last two parameterize the effective couplings of the Higgs to gluons and to photons, respectively, mediated by loops of heavy particles. As the operators proportional to and do not respect the symmetry under which the Goldstone Higgs shifts they have to be suppressed by powers of the couplings which break this symmetry, thus explaining the extra factor appearing in front of them. These operators are important in the presence of relatively light resonances, i.e. when . Recent analyses [4] show that in a large class of models, a Higgs as light as implies the presence of one or more anomalously light (sub-TeV) fermionic resonances, which thus can contribute sizeably to and . Also notice that by choosing flavor-diagonal couplings Minimal Flavor Violation (MFV) is automatically implemented in the Lagrangian so that it complies with flavor bounds.

In Eq. (1) we have kept explicitly the operator proportional to , which can be eliminated at by a field redefinition

 H→H+a(H†H)H/f2, (2)

under which

 cH→cH+2a,cr→cr+4a,c6→c6+4a,cy→cy−a, (3)

while and do not change under this transformation. The choice corresponds to the ‘SILH basis’, which can be reached starting from a generic basis where by applying the transformation in Eq. (2) with . We choose to keep explicitly the operator proportional to as the ‘natural’ basis for nonlinear -models actually corresponds to a non-vanishing [10]. Furthermore, since physical amplitudes have to be invariant under field redefinitions, Eqs. (2) and (3) will be used as a consistency check of our results. In Eq. (1) we have also omitted the custodial breaking operator

 cT2f2(H†\lx@stackrel⟷DμH)2, (4)

where , which gives a contribution to the parameter and thus is strongly constrained by electroweak data. This operator does not contribute to the processes that we will be interested in. If the strong sector is invariant under custodial symmetry, as it happens for example in models based on the coset , then vanishes at tree-level. For a discussion of Higgs physics where the assumption of custodial invariance is relaxed, see Ref.[20].

The SILH Lagrangian represents an expansion in and can be used in the vicinity of the SM, which corresponds to . On the other hand, the technicolor limit () requires the resummation of the full series in . Such a resummation is possible in the Holographic Higgs models of Refs. [21, 22, 23]. These models are based on a five-dimensional gauge theory in Anti-de-Sitter (AdS) space-time. In the minimal realization, the bulk symmetry is broken to the SM group on the ultraviolet (UV) brane and to on the infrared (IR) brane. The coset provides four Goldstone bosons, one of which is the physical Higgs boson and the three remaining ones are eaten by the massive SM vector bosons. The Higgs couplings to gauge bosons and its self-interactions are modified compared to the SM, and the modification factors can be expressed in terms of the parameter . The Higgs Yukawa couplings and the form of the Higgs potential of the low-energy effective theory depend on the way the SM fermions are embedded into representations of the bulk symmetry. In the second part of this work we refer to the model MCHM5 [22] where the fermions transform in the fundamental representation of . An alternative realization of the composite Higgs, denoted by MCHM4, contains fermions embedded into the spinorial representation [21] (for more details see App. B.2). In this case, however, large corrections to the coupling are present and rule out an important part of the parameter space [24]. In contrast, if fermions are embedded into the fundamental or adjoint representation of , the custodial symmetry of the strong sector includes a left-right parity, which protects the coupling from receiving tree-level corrections [25].

Another useful description of the low-energy theory is given by an effective chiral Lagrangian where the symmetry is nonlinearly realized. The Goldstone bosons () providing the longitudinal degrees of freedom of the and bosons are introduced by means of the field

 Σ(x)=eiσaπa(x)/v, (5)

where and are the Pauli matrices. The field transforms linearly under . Introducing a scalar field , assumed to transform as a singlet under the custodial symmetry, leads to the following effective Lagrangian [19]

 L =12(∂μh)2−V(h)+v24Tr[(DμΣ)†DμΣ](1+2ahv+bh2v2+b3h3v3+⋯) V(h) =12m2hh2+d3(m2h2v)h3+d4(m2h8v2)h4+⋯, L(4) =g2s48π2GμνaGaμν(kghv+12k2gh2v2+…)+e232π2FμνFμν(kγhv+…), (6)

with the mass of the scalar given by . In Eq. (6) we have introduced the higher-dimensional couplings , which are mediated at loop level by strong sector resonances. The Higgs couplings to fermions, , are assumed to be flavor-diagonal, so that MFV is realized. In Table 1 the values of the couplings in the effective Lagrangian Eq. (6) are listed in the SILH approach and in the holographic Higgs model MCHM5 (for the latter, only Higgs nonlinearities are considered).

The SM with an elementary Higgs boson corresponds to , and vanishing higher order terms in .

3 Applying the Higgs Low-Energy Theorem

In this section we discuss applications of the Higgs low-energy theorem [6, 7] in composite models. The LET allows one to obtain the leading interactions of the Higgs boson with gluons and photons arising from loops of heavy particles. By heavy particles we mean here both SM states ( and top) and new states belonging to the composite sector. These couplings are needed in the computation of the cross sections of single and double Higgs production via gluon fusion at the LHC as well as of the partial width of the decay . We will adopt a model-independent approach and compute these quantities in terms of the parameters of the effective Lagrangians defined in Section 2, Eqs. (1) and (6), putting special emphasis on the former, namely the SILH description. Our analysis extends the results of Refs. [10, 27] to Higgs pair production in gluon fusion, and also includes a discussion of corrections to the LET approximation arising from higher order terms in the expansion, where is the mass of the generic heavy particle running in the loops. Notice that the LET can be extended to 2-loop order to include the leading QCD corrections, see for example Ref. [7] for applications in the SM. However, our discussion will be mainly limited to couplings at the leading 1-loop order.

3.1 Higgs interactions with gluons

According to the LET the interactions of the physical Higgs boson with gluons, mediated by loops of heavy coloured particles, can be obtained by treating the Higgs as a background field and taking the field-dependent mass of each heavy particle as a threshold for the running of the QCD gauge coupling.111Throughout the paper, we will denote by both the Higgs doublet and the scalar field with , as it will always be clear from the context which one we are referring to. On the other hand, denotes the physical Higgs scalar. Assuming the heavy particles to transform in the fundamental representation of one obtains the following effective Lagrangian

 Leff=g2s64π2GaμνGaμν∑piδbpilogm2pi(H), (7)

where if particle is a Dirac fermion, and if it is a complex scalar. In this paper we will focus only on the effects of the heavy fermion sector, which in composite Higgs models typically includes new states beyond the top quark. By expanding the field-dependent masses of the heavy particles around the vacuum expectation value (VEV) we obtain the couplings of the Higgs boson to gluons mediated by loops of heavy fermions

 Lhngg=g2s96π2GaμνGaμν(A1h+12A2h2+…), (8)

where we have defined

 An≡(∂n∂HnlogdetM2(H))⟨H⟩ (9)

with , and is the heavy fermion mass matrix. In the SM only the top quark contributes222The bottom contribution is non-negligible, but cannot be computed using the low-energy theorem, due to the smallness of the bottom quark mass. with , so that Eq. (8) can be rewritten at all orders in as (see for example Refs. [7, 28])

 Lhngg=g2s48π2GaμνGaμνlog(1+hv). (10)

The corresponding gauge invariant operator is , which is associated with a chiral fermion. The lowest-order operator arising from vector-like fermions is instead . The effects of these two operators on double Higgs production were discussed in Ref. [29].

Using Eq. (8) it is straightforward to derive the expression of the and couplings in the SILH formalism. We refer the reader to App. A for a derivation, and simply report here the results. We remark that from now on we will work in the unitary gauge, where the Higgs doublet reads . The effective coupling of the Higgs boson to two gluons reads (see App. A)

 Lhgg=g2s48π2GaμνGaμνhv[12(∂∂logHlogdetM2(H))H=v−cH2ξ]. (11)

This coupling governs the rate of single Higgs production via gluon fusion, and its expression was already obtained in Refs. [10, 27]. The production rate normalized to the SM one is given by the square of the expression in square brackets in Eq. (11). On the other hand the effective coupling of two Higgs bosons to two gluons, which contributes to Higgs pair production via gluon fusion, has the following expression

 Lhhgg=g2s96π2GaμνGaμνh2v2[12((∂2∂(logH)2−∂∂logH)logdetM2(H))H=v−cr4ξ]. (12)

In terms of the effective Lagrangian in Eq. (6), the couplings read

 Lhgg=g2s48π2GaμνGaμνhv(c+kg),Lhhgg=g2s96π2GaμνGaμνh2v2(2c2−c2+k2g). (13)

In the expression of the coupling in Eq. (13), the first term comes from the triangle top loop involving the vertex, whereas the second is the contribution of top box diagrams, see Fig. 2. On the other hand, and are parameterizing the contributions from integrated-out heavy particles.

3.2 Higgs interaction with photons

Although the main focus of this paper is on gluon fusion, we give here the expression for the coupling of the Higgs boson to photons, as it is another loop process of crucial relevance for Higgs phenomenology at the LHC. This coupling receives contributions both from loops of heavy fermions and from the boson loop. Application of the LET leads to the following effective Lagrangian [7]

 (14)

which is valid for , and where we have assumed that the heavy fermions transform in the fundamental representation of . Expanding around the VEV we obtain the interaction

 Lhγγ=e216π2FμνFμνh(Q2tA1−74(∂∂Hlogm2W(H))⟨H⟩), (15)

where we have assumed that all fermions have electric charge equal to that of the top quark333In all models considered in this paper, only top-like resonances contribute to the coupling. The extension to heavy states with different electric charge is straightforward., , and was defined in Eq. (9). By performing simple manipulations we obtain (see App. A)

 Lhγγ=e232π2FμνFμνhv −Jγ(4m2W/m2h)(1+ξ(cr4−cH2))], (16)

where we have replaced the LET approximation for the loop with the full result encoded by the function

 Jγ(x)=F1(x),F1(x)=2+3x[1+(2−x)f(x)],f(x)=arcsin2(x−1/2), (17)

which tends for large to , where the first term comes from the transverse polarizations of the and is precisely equal to the gauge contribution to the function of the coupling, while the second term arises from the eaten Goldstone bosons. The use of the full expression for the loop implies that the formal validity of Eq. (16) is extended to . The rescaling of the decay width is obtained by comparing the square of the expression multiplying in Eq. (16) in the two cases. In terms of the parameters of the effective Lagrangian in Eq. (6) the coupling reads

 Lhγγ=e232π2FμνFμνhv(4Q2tc+kγ−aJγ(4m2W/m2h)). (18)

3.3 Single Higgs production via gluon fusion

For a SM Higgs boson, the gluon fusion process [30] gives the dominant production cross section at the LHC, see Refs. [31] for reviews. At leading order (LO), the process proceeds via a top loop, with a subleading contribution from the bottom loop, see Fig. 1.

In composite Higgs models extra heavy, colored fermions with sizeable couplings to the Higgs boson are typically present, whose contributions to the gluon fusion process should in general be taken into account. It has been shown [9, 10, 11, 32] that in explicit constructions the cross section (computed in the LET approximation) is insensitive to the details of the heavy fermion spectrum, i.e. it does not depend on the couplings and masses of composites, but only on the ratio , where is the overall scale of the strong sector. This was found to be true both in models with partial compositeness and in Little Higgs theories. In fact, although the top Yukawa coupling receives a correction due to the mixing with resonances which depends on composite couplings, this contribution is exactly canceled by the loops of extra fermions, leading to a dependence of the rate only on . This also implies that the cross section can be obtained by simply multiplying the SM one by , where is the rescaling of the top Yukawa coming only from the nonlinearity of the -model, and neglecting corrections due to fermionic resonances.

Let us review how this cancellation arises. It is due to the fact that the determinant of the heavy fermion mass matrix takes the form

 detM2(H)=F(H/f)×P(λi,Mi,f), (19)

where is a function satisfying since the top becomes massless in the limit of unbroken electroweak symmetry, and is a function of the composite couplings and masses , but independent of . It is then immediate to see that the coupling in Eq. (11) does not depend on the masses and couplings of the resonances.444The coefficient does not receive contributions from the heavy fermion sector. It is, however, generated by integrating out heavy scalars or vectors [27], see App. B.1 for an explicit example. The origin of the factorization in Eq. (19) was explained in the context of partial compositeness in Ref. [11], by means of a spurion analysis. There it was also pointed out that such a factorization can break down if the top mixes with more than one composite operator, leading to a dependence of the vertex on composite couplings. Nevertheless in many explicit constructions, including Little Higgs models, the factorization in Eq. (19) takes place. Still, the independence of the vertex on the composite couplings (collectively denoted by ) holds exactly only in the LET approximation, and corrections due to finite fermion mass effects are expected. We can estimate the residual dependence on the due to finite fermion mass effects in a simple way. Assuming for simplicity the presence of only one top partner , the mass eigenvalues can be written at as

 mt(H)=ytH√2⎛⎝1−c(t)y2H2f2⎞⎠,mT(H)=λTf(1+aTH2f2), (20)

where is a parameter dependent on the couplings as .555If the quadratic divergence in the Higgs mass due to the top is cancelled by , then the absence of an term in implies . See for example the explicit values of and in the Littlest Higgs model, reported in App. B.1, Eq. (92). On the other hand, we can write , where is a constant arising from the pure nonlinearity of the -model. The LET result for the coupling reads, taking the effect of into account,

 Lhgg=g2s48π2GaμνGaμνhv(1−(c(t)y−2aT+cH2)ξ). (21)

Notice that in the limit where is heavy, corresponding to large , the effects of the heavy resonance on the coupling vanish. In fact, goes to zero, whereas , implying that only the nonlinearity in the top Yukawa arising from the nonlinear -model is relevant.

By using the expression of the top Yukawa coupling we can compute explicitly the top loop diagram, retaining the first subleading term in the expansion. This is the leading correction to the LET coupling, given that . Thus Eq. (21) is improved to

 (22)

where we have used , and the ellipses stand for subleading corrections (including terms of order ). The independence of the LET vertex of the composite couplings is equivalent to the statement that is a constant, .666Notice that by using Eq. (20) one finds . So if the factorization in Eq. (19) holds then is a constant. If this is the case then the dependence on the of Eq. (22) is due to the last term, and we can estimate the sensitivity of the cross section to the to be, for a light top partner ,

 δσ(gg→h)σ(gg→h)SM∼760m2hm2tξ≃0.06ξ, (23)

where in the last equality we assumed . Thus corrections are expected to be very small even for large . This estimate will be confirmed in Section 5, where the cross section will be computed in MCHM5 retaining the full mass dependence. We note that in this model the cross section can be strongly suppressed compared to the SM, reaching for a low compositeness scale . However, the Higgs branching ratios into and are enhanced compared to the SM, so MCHM5 can still be compatible with the excesses observed by ATLAS and CMS at even for values of as large as those considered in this paper [26].

3.4 Double Higgs production via gluon fusion

Within the SM, double Higgs production via gluon fusion received interest mainly because it is sensitive to the trilinear Higgs self-coupling [33], see the first diagram in Fig. 2.

In composite Higgs models, the process is affected essentially in two main ways. First, the nonlinearity of the strong sector gives rise to a coupling (which vanishes in the SM) and thus to a genuinely new contribution to the amplitude, see the second diagram in Fig. 2. Second, one should take into account the effects of top partners, which include also new box diagrams involving off-diagonal Yukawa couplings777Note that these Yukawa couplings only involve the top quark and its charge heavy composite partners. (shown in the second line of Fig. 2). A first study of in composite Higgs models, neglecting top partners, was performed in Ref. [17], where it was found that a large enhancement of the cross section is possible due to the new coupling (see also Ref. [16] for an earlier study in the context of Little Higgs models). For example, in MCHM5 with , which corresponds to , the cross section was found to be about 3.6 times larger than in the SM. Recently, Ref. [18] performed a model-independent study of the process, making reference to the effective Lagrangian in Eq. (6) and again neglecting the effects of top partners, and found a large sensitivity of the cross section to the coefficient parameterizing the coupling.

In this paper we include for the first time the effects of top partners in double Higgs production via gluon fusion. This is especially interesting in the light of the results of Refs. [4], where a naturally light composite Higgs was shown to be tightly correlated with the presence of light top partners, as such light resonances can in principle affect the cross section in a sizeable way. Our analysis will confirm that this is indeed the case.

We start by discussing the cross section in the LET approximation, which greatly simplifies the computation. In this limit, the amplitude is simply the sum of two diagrams, one with the effective coupling followed by a trilinear Higgs coupling and the other involving the effective coupling. Adopting the SILH formalism, and recalling the expressions of the relevant Feynman rules

 hgg:iαs3πvδab(pν1pμ2−p1⋅p2gμν)[12(∂∂logHlogdetM2(H))H=v−cH2ξ],hhgg:iαs3πv2δab(pν1pμ2−p1⋅p2gμν)[12((∂2∂(logH)2−∂∂logH)logdetM2(H))H=v−cr4ξ],hhh:−i3m2hv[1+ξ(c6−32cH−14cr)] (24)

(where denote the momenta of the incoming gluons), we can write the amplitude as

 ALET(gg→hh)=αs3πv2δab(pν1pμ2−p1⋅p2gμν)CLET(^s), (25)

where

 CLET(^s)= 3m2h^s−m2h[12(∂∂logHlogdetM2(H))H=v+ξ(c6−2cH−cr4)] +12((∂2∂(logH)2−∂∂logH)logdetM2(H))H=v−cr4ξ (26) = 3m2h^s−m2h(1−ξ(c(t)y−c6+2cH+cr4−3cgy2tg2ρ))−(1+ξ(c(t)y+cr4−3cgy2tg2ρ)),

with denoting the partonic center-of-mass (c.m.) energy. To obtain the second equality in Eq. (26) we used Eqs. (70) and (71) contained in App. A. It is immediate to check that the combinations and are invariant under the reparameterization in Eqs. (2) and (3). For completeness, we also give the result in terms of the coefficients of the effective Lagrangian in Eq. (6):

 CLeffLET(^s)=3m2h^s−m2h(c+kg)d3+2c2−c2+k2g. (27)

 ^σgg→hh=G2Fα2s(μ)^s128(2π)319√1−4m2h^sC2%LET(^s). (28)

The hadronic cross section is obtained by convolution with the parton distribution function of the gluon in the proton,

 σ=∫14m2h/sdτ∫1τdxxfg/P(x,Q)fg/P(τ/x,Q)^σgg→hh(τs), (29)

with the collider c.m. energy related to by . The renormalization scale and the factorization scale are chosen equal to the invariant mass of the Higgs boson pair, . Throughout the paper, the parton distribution functions of MSTW2008 [34] are employed. For , Eq. (26) correctly reproduces the SM result in the limit of large top mass [13, 14]

 CSMLET(^s)=3m2h^s−m2h−1. (30)

In the SM the limit gives a cross section in agreement with the full result only within for (for we find and ) and moreover it produces incorrect kinematic distributions, as noticed in Ref. [15]. Thus we expect the LET to be in general less accurate in than in single Higgs production.

From Eq. (26) we read off that in models where the factorization Eq. (19) of holds, the LET cross section is insensitive to composite couplings, due to a cancellation completely analogous to the one that we discussed for single Higgs production. In the left panel of Fig. 3 we show for  GeV and a c.m. energy of 14 TeV the cross section normalized to the SM cross section (both were computed applying the LET) as a function of for some well-known models, in all of which the cancellation holds.

We note that in MCHM5 the enhancement of the cross section is striking. This can be traced back to the behavior of the function , which is proportional to the LET amplitude and is shown in the right panel of Fig. 3 for the three models under consideration and for the SM. The enhancement for MCHM5 is evident. As pointed out for the first time in Ref. [17], where the process was studied in MCHM5 considering only Higgs nonlinearities (or equivalently in the limit of heavy top partners) but keeping the full dependence on , the dramatic increase of the cross section compared to the SM is mostly due to the presence of a new coupling. The large enhancement of in MCHM5 is in contrast with the strong suppression in the same model of the single Higgs production cross section, which for equals of the SM value (see Section 5).

By comparison with Ref. [17] we find that when fermionic resonances are above the cutoff, the LET underestimates the ratio by about : for example for , application of the LET gives a cross section of 2.6 times the SM, whereas Ref. [17] found an enhancement factor of 3.6. This difference is due to the fact that in the former case is assumed, whereas in the latter the full dependence was retained. Notice that the best estimate of the cross section that can be obtained using the LET is , because part of the corrections due to the finite top mass should cancel in the ratio of LET cross sections. In fact, in terms of cross sections the disagreement between the LET and the result obtained taking into account only Higgs nonlinearities is larger. For we obtain , whereas Ref. [17] found , i.e. the difference is of order .

In order to understand this behavior we investigated in more detail the validity of the LET both for single and double Higgs production. In single production the expansion parameter is and the series converges very quickly. In double Higgs production on the other hand, one needs to expand in with , which is not small, so in general the expansion does not work as well as for the single Higgs case. In MCHM5, the validity of the expansion gets even worse. The reason why the LET is less accurate for MCHM5 than for the SM (where it underestimates the cross section by about ) is mainly the presence of the new triangle diagram containing the coupling, which contrarily to the triangle diagram involving the virtual Higgs exchange does not vanish at large . This is confirmed by taking into account the corrections of to the LET result, which are reported in App. E.5. Compared to the SM we have an additional contribution from the two-Higgs two-fermion coupling which goes like , and which in contrast to the triangle diagram with virtual Higgs exchange is not suppressed by the Higgs propagator, see Eqs. (148) and (149). Therefore in MCHM5 for large , where the coupling is sizeable (see Table 1), the corrections do not improve at all the LET result. For a model-independent study of the process including only modifications to the top couplings, see Ref. [18].

While in the LET approximation the contributions of loops of top partners to the amplitude exactly cancel out with that coming from the modification of the top Yukawa due to mixing with resonances, the sensitivity to composite couplings of the full double Higgs production cross section (computed retaining all dependence on masses) is expected to be much larger than for , where it was shown to be negligible. By direct computation in MCHM5, we will see in Section 6 that this is the case, i.e. the full cross section has a sizeable sensitivity to the details of the spectrum of the top partners. This effect is not captured by the simple LET result, which is completely determined by . Therefore, while the low-energy theorem provides a useful tool to obtain a rough estimate of the cross section, a complete loop computation is needed to describe correctly the effects of top partners in the process.

4 Composite Higgs model with extra fermionic resonances

We consider a composite Higgs scenario with the symmetry group of the strongly-interacting sector given by , which is spontaneously broken down to at the scale . In order to correctly reproduce the fermion charges an additional local symmetry is introduced, leading to the symmetry breaking pattern . This is the minimal realisation including custodial symmetry and which implies four Goldstone bosons (GBs) transforming as a of . The SM electroweak group is embedded into and the hypercharge is then given by [21, 22]. The GBs are parameterized in terms of the field

 Σ=Σ0eΠ/f,Π=−i√2T^ah^a,Σ0=(0,0,0,0,1), (31)

where are the generators of , and are the 4 real GBs. Using the explicit expressions of the and generators ()

 TaL,Rij= −i2[12ϵabc(δbiδcj−δbjδci)±(δaiδ4j−δajδ4i)],aL,R=1,2,3,a,b,c=1,2,3 (32) T^aij= −i√2(δ^aiδ5j−δ^ajδ5i),^a=1,…,4 (33)

one obtains

 Σ=sin(h/f)h(h1,h2,h3,h4,hcotan(h/f)),h=√∑^ah2^a. (34)

 Lkin=f22(DμΣ)(DμΣ)T,DμΣ=∂μΣ+igWaμΣTaL+ig′BμΣT3R. (35)

By performing an rotation, it is always possible to align the Higgs VEV to the direction, thus identifying , where is the Higgs field (with ). Then in the unitary gauge

 Σ= Σ0⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝11cos(H(x)/f)−sin(H(x)/f)1sin(H(x)/f)cos(H(x)/f)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠≡Σ0ζ(x) (36) = (0,0,sin(H/f),0,cos(H/f)), (37)

and therefore

 Lkin=12∂μH∂μH+g2f24sin2(Hf)[W+μW−μ+12cos2θWZμZμ] (38)

which fixes .

Fermionic resonances are described using the language of partial compositeness. We introduce vector-like fermions which have quantum numbers such that they can mix linearly with the SM fermions