Remarks on analyticity and unitarity in the presence of a Strongly Interacting Light Higgs

Applying the three axiomatic criteria of Lorentz invariance, analyticity and unitarity to scattering amplitudes involving the Goldstone bosons and the Higgs boson, we derive a general sum rule for the Strongly Interacting Light Higgs Lagrangian. This sum rule connects the IR coefficient $c_H$ to the UV properties of the theory, and can be used, for instance, to capture the role of resonances in processes like $V_{\rm L}V_{\rm L}\to hh$ and $V_{\rm L}V_{\rm L}\to V_{\rm L}V_{\rm L}$, with $V=W^{\pm},Z$.


I. INTRODUCTION
The Higgs boson [1][2][3] was found in July 2012 after a near half-century search [4,5]. As predicted by the Standard Model (SM), the Higgs boson is a scalar particle, and all the experimental data collected so far at the LHC seem to point towards a positive parity [6,7]. The value of the mass is m h = 125.5 ± 0.2 stat +0.5 −0.4 sys GeV measured by ATLAS [8], and m h = 125.7±0.3 stat ±0.3 sys GeV measured by CMS [9]. Despite this astonishing discovery, however, none of the mysteries related to the existence of the Higgs boson have been solved. Light scalars are unnatural in quantum field theory, unless a specific mechanism keeps their mass safe from large radiative corrections. The most elegant way to solve this problem is to protect the Higgs boson using a symmetry. This particular theoretical framework is realized in the context of Composite Higgs models, where the Higgs boson arises as pseudo Nambu-Goldstone boson of a spontaneously broken global symmetry [10][11][12][13][14][15]. 1 This scenario * Electronic address: alfredo.urbano@sissa.it 1 A more precise formulation of the hierarchy problem in the context of Composite Higgs models is the following. On a general ground, it is well known that the Higgs mass -via quantum corrections -is quadratically sensitive to the existence of any new physics beyond the SM, i.e. ∆m 2 h ∼ (g 2 /16π 2 )Λ 2 where g is a coupling constant and Λ is the scale of new physics. The so-called "big hierarchy problem" refers to the question why is m h Λ. The most straightforward solution to this problem relies on the following arguments. First, the scale of new physics has to be relatively low, Λ TeV; second, the new physics, irrespective from its nature, has to render the Higgs boson insensitive to further quantum corrections above the TeV scale. In Composite Higgs model the occurrence of the latter condition is ensured by compositeness itself -i.e. by the fact that in the fundamental theory above the TeV scale there exists no scalar operators of dimension less than 4 that can be added to the Lagrangian -while the condition m h TeV follows from the pseudo Nambu-Goldstone nature of the Higgs scalar. The "little hierarchy problem", on the contrary, refers to the lack of direct and/or indirect evidences of this new physics at the TeV scale. In Composite Higgs models has profound phenomenological implications; it predicts potentially large deviations in the couplings of the Higgs boson with the SM gauge bosons and fermions, as well as the existence of new resonances, with a mass around the TeV scale, coupled to the Higgs doublet. Unambiguous fingerprints of compositeness, therefore, could be present in sizable deformations of the Higgs couplings, and significantly enhanced cross sections describing scattering processes between the Higgs boson and/or the longitudinal gauge bosons W ± L , Z L . The former are under scrutiny at the LHC [16,17], and the current experimental bounds are still compatible with the presence of deviations from the SM predictions, especially considering loop-induced couplings (see, for instance, Refs. [18][19][20]). As far as the latter is concerned, processes like V L V L → hh, V = W ± , Z have a distinctive signature at the LHC: the production of two Higgses in association with two forward jets, well separated in pseudorapidity, related to the primary partons that radiate the V L V L pair. The possibility to detect these processes at the LHC is extremely challenging, given the tiny value of the corresponding SM cross sections [21,22]; on the other hand, this also implies that new physics effects -in particular due to the s-channel exchange of a new resonance in V L V L → hh, V L V L → V L V L -are more likely to be seen [23].
Actually, apart from experimental complications, there exists also a nontrivial theoretical obstruction. Composite Higgs models postulate the existence of a new strongly-coupled sector to which these resonances belong, thus making the usual perturbative approach completely this problem is mitigated by the fact that the Higgs mass correction is actually of the form ∆m 2 h ∼ (g 2 /16π 2 )f 2 and is set by the global symmetry breaking scale f rather by the new physics scale Λ. Following the previous discussion, as a consequence, one would expect f TeV while the cut-off scale of the theory will lie at higher values Λ ∼ 4πf , thus alleviating the little hierarchy problem.
useless. The possibility to make model-independent predictions without any knowledge of the underlying UVcompletion of the theory may seem, as a consequence, completely hopeless.
In the sixties, the ambitious goal of the "S-matrix theory" was to compute the elements of the S-matrix by requiring them to respect three general properties that ought to be valid independently of the actual existence of a Lagrangian description: Lorentz invariance, analyticity and unitarity [24].
The S-matrix theory was developed in order to describe the transition amplitudes in the presence of the strong interaction responsible for nuclear forces, like for instance the pion-nucleon scattering. Mutatis mutandis, we can try to apply the same basic principles in the presence of the strong dynamics of a Composite Higgs model. In this way, one can pursue the possibility to study the structure of scattering amplitudes through an elegant union between the analytical approach and the exploitation of the underlying symmetries.
Keeping this aim in mind, in this paper we apply the principles of the S-matrix theory to study the scattering amplitudes in presence of a Strongly Interacting Light Higgs (SILH), and in particular we focus on the processes involving the Goldstone bosons and the Higgs boson. Thanks to the Equivalence Theorem [25], in fact, at high energies these processes formally take the place of This work is organized as follows. In Section II we describe the theoretical setup of our computation, provided by the SILH effective Lagrangian [26]. In Section III, using a dispersion relation, we derive a sum rule that is the main result of this paper. In Section IV we discuss some phenomenological implications. Finally, we conclude in Section V. In Appendix A we summarize the basic principles of the S-matrix theory. In Appendix B we construct the scattering amplitudes used in Section III. In Appendix C we generalize the Froissart-Martin bound to inelastic scattering amplitudes. In Appendix D we construct the non-linear σ-model Lagrangian describing the coset SO(4, 1)/SO(4).

II. SETUP: THE SILH EFFECTIVE LAGRANGIAN
The scalar sector of the SM is described by the following Lagrangian with µ 2 < 0; H is the usual Higgs doublet with vacuum expectation value (vev) H = (0, v) T / √ 2, and is the covariant derivative related to the gauging SU (2) L ⊗ U (1) Y , being σ a=1,2,3 the usual Pauli matrices. In Eq. (2) π ± ≡ (π 1 ∓ iπ 2 )/ √ 2, π 0 are the Goldstone bosons while h is the Higgs boson. The minimum of the potential occurs for v 2 = −2µ 2 /λ, and after electroweak symmetry breaking the Higgs boson acquires the mass m 2 h = λv 2 . The Higgs Lagrangian L H possesses, in the limit g Y → 0, the larger global symmetry SO(4) ≈ SU (2) L ⊗SU (2) R , spontaneously broken by the Higgs vev into the diagonal custodial subgroup SO(3) ≈ SU (2) C . On the one hand -in the unbroken phase -the Goldstone bosons and the Higgs boson transform under the action of the global symmetry SO(4) according to its fundamental representation or, equivalently, according to the bi-doublet representation of SU (2) L ⊗SU (2) R ; on the other one -after electroweak symmetry breaking -under the action of the custodial group the Higgs boson transforms as a singlet, while the Goldstone bosons transform as a triplet, parametrizing the coset SU (2) L ⊗ SU (2) R /SU (2) C .
Given this setup, one may wonder if the global symmetry SO(4) ≈ SU (2) L ⊗ SU (2) R is just an accidental property encoded in the Higgs Lagrangian of the SM or if its presence is rooted in a more profound theoretical ground.
The latter scenario is realized in the context of Composite Higgs models, in which the Higgs is a pseudo Nambu-Goldstone boson, and -in analogy with the pions in QCD -it originates from the spontaneous breaking of a global symmetry. In more detail, the picture to bear in mind is the following. In addition to the elementary sector, formed by all the SM fields with the exception of the Higgs doublet, 2 there exists a composite sectoraround the TeV scale -described by a new fundamental strongly-coupled theory, and characterized by the global symmetry G. At some new scale f , this global symmetry is spontaneously broken by a dynamical condensate into the subgroup H ⊃ SU (2) L ⊗ U (1) Y . The crucial assumption is that, in the limit in which all the SM couplings are zero, the Higgs doublet H is an exact Nambu-Goldstone boson doublet living in the coset G/H. Assuming that the strong sector preserves the custodial symmetry, the minimal choice turns out to be G/H = SO(5)/SO (4). The picture is completed by the SM gauge and Yukawa couplings; they break explicitly the global symmetry, thus making the Higgs a pseudo Nambu-Goldstone boson, and generating radiatively the electroweak potential.
The general features of this framework, and in particular the predicted deviations from the SM, can be captured in a model-independent way by using the SILH effective Lagrangian [26]. In this paper we are interested in the operators of the SILH Lagrangian that involve only the Higgs doublet, and therefore -at dim-6 -we have Notice that in the following, in order to simplify the notation, we shall refer to the generic component of the doublet H using the symbol π a , i.e. π a = π ± , π 0 , h. Furthermore, we focus only on the scattering processes π a π b → π c π d whose amplitude grows with the energy. As a consequence, we concentrate on the derivative of the Goldstone doublet described by the operator O H . In the next Section we shall derive a general sum rule for the SILH Lagrangian studying the scattering processes π a π b → π c π d . The underlying assumption is that the UV-completion of the SILH Lagrangian respects the postulates of Lorentz invariance, analyticity and unitarity (see Ref. [24], and Appendix A for a review of the basic definitions). This is a fundamental requirement that we expect to be true in any string-inspired UV-completion. 3

III. ANALYTICITY AND UNITARITY: IR/UV CONNECTION
Classifying the Goldstone bosons π ± , π 0 and the Higgs boson h according to the representation (2, 2) of SU (2) L ⊗SU (2) R , it is straightforward to realize that the scattering π ⊗ π has the following structure 3 See Ref. [27] for a recent discussion about the UV-completion of Composite Higgs models with partial compositeness.
This simply means, as a consequence, that the scattering amplitude A π a π b →π c π d (s, t) describing the generic process with s = (p 1 + p 2 ) 2 , t = (p 1 − p 3 ) 2 , can always be decomposed in terms of its projections with definite SU (2) L ⊗ SU (2) R quantum numbers. In full generality, therefore, instead of a single specific amplitude we focus on the combination A(s, t = 0) ≡ IJ=00,10,01,11 where κ IJ are arbitrary constants, and A IJ (s, t) are the projected amplitudes according to the decomposition in Eq. (6). In Eq. (8), moreover, we have explicitly considered the forward limit t = 0. In Appendix B we compute in detail all the scattering amplitudes A π a π b →π c π d (s, t) as a function of the projections A IJ (s, t) [see Eqs. (B25-B32)]. Following Refs. [28][29][30][31] we compute the integral where the contour of integration C is displayed in Fig. 1, and can be decomposed into two contributions: the contribution from the parts (denoted as I-IV in Fig. 1) surrounding the unitarity cuts, 4 and the contribution from the big circle at infinity, C ∞ . 5 Notice that the scattering amplitude A(s, t = 0) in Eq. (9) has been promoted to an analytic function of the complex variable s defined in the complex plane. The integral I can be computed in two different ways, providing a connection between the IR and UV behavior of the theory.
1. We compute the residual value of I at s = 0, where the scattering amplitudes A π a π b →π c π d (s, t) can be written explicitly extracting the interactions encoded in the operator O H . This approach, relying on the lowest order of the effective field theory description provided by the SILH Lagrangian, captures the IR limit of the theory. By direct computation we find [26,29] with equal amplitudes obtained substituting π 0 with h, i.e. A π ± π ∓ →π 0 π 0 (s, t) = A π ± π ∓ →hh (s, t). Evaluating the corresponding projections according to Eqs. (B33-B36), and taking the forward limit we obtain and, as a consequence 2. The second method is based on the explicit computation of the integral following the contour C in which we have separated the contribution from the cuts and the contribution from the big circle at infinity. Let us consider first the contribution from the cuts; dropping the t-dependence we have where the first (second) term represents the discontinuity of the scattering amplitude across the right (left) cut (see Fig. 1). As customary in this kind of computation, analyticity allows us to apply the crossing symmetry transformation that relates the amplitude in the u-channel and the amplitude in the s-channel; in terms of the projection A IJ in Eq. (8) we have the following matrix equation where A ≡ (A 00 , A 10 , A 01 , A 11 ) T . In Appendix B 2 we compute explicitly this transformation, and the matrix C su is given in Eq. (B45). All in all, using the decomposition in Eq. (8), the crossing symmetry transformation in Eq. (18), and remembering that the imaginary part of the physical scattering amplitude A IJ (s) in the s-channel is defined according to the contribution from the cuts in Eq. (17) can be rewritten as follows Assuming left-right symmetry, i.e. taking Im A 10 (s) = Im A 01 (s) ≡ Im A LR (s), we find Notice that this symmetry is formally defined as the invariance under the exchange of the generators of SU (2) L and SU (2) R , and it was introduced in Ref. [47] to prevent the presence of large corrections affecting the Zbb vertex.
Let us now consider the contribution from the big circle at infinity in Eq. (16). The rule of thumb in this kind of computation is to show that the scattering amplitude falls to zero, or at least remains constant, as |s| → ∞; in this case, in fact, it is straightforward to see that the integral goes to zero as soon as the big circle is pushed to infinity. In order to evaluate the integral following this criterium, we can retrace the argument already used in Ref. [31], and based on the application of the Regge theory. Therefore, let us first try to recap in a nutshell the main prerogatives of this theory. Considering in full generality the process ab → cd, the Regge theory reconstructs the behavior of the corresponding scattering amplitude A ab→cd (s, t) in the kinematical region s |t| according to the following expression [24] A ab→cd (s, t) where Z is a complex constant; Eq. (22) can be interpreted considering the exchange of an object (the socalled Reggeon) with couplings γ ac (t), γ bd (t), and angular momentum α(t) = α(0) + α t; in the forward limit we have On the other hand analyticity and unitarity, by virtue of the Froissart-Martin bound [48][49][50], impose the constraint 6 |A ab→cd (s, 0)| const s(ln s) 2 .
Regge amplitudes saturating the Froissart bound, therefore, grow like s, and give non-zero contribution to the integral over C ∞ . According to Regge theory, this happens in correspondence of the exchange of a Reggeon with intercept α(t) = 1, and quantum number of the vacuum (i.e. without exchange of isospin and charge, γ ab = Cδ ab ). This trajectory is the Pomeron, and the corresponding amplitude reads From Eqs. (B33-B36) it follows that all the projections A IJ (s) can accommodate the Pomeron exchange, and in 6 Notice that the Froissart-Martin bound controls the high-energy behavior of elastic scattering amplitude, i.e. A ab→ab (s, t). However, using unitarity, it can be generalized also to inelastic scattering amplitude as in Eq. (24). We provide a proof of this generalization in Appendix C.
As in Ref. [31], we can exploit the freedom in the choice of the coefficients κ IJ in such a way that (κ 00 +κ 10 +κ 01 + κ 11 ) = 0, (3κ 00 + κ 10 + κ 01 − κ 11 ) = 0; in this case the contribution to the integral over the big circle at infinity originating from Eq. (26) where we made use of the optical theorem sσ tot IJ (s) = Im A IJ (s), being σ tot IJ (s) the total cross section for the process IJ → anything. The equality in Eq. (27) holds in the limit of vanishing gauge couplings g L , g Y → 0, and in the limit of unbroken electroweak symmetry, where the relevant global symmetry governing the scattering amplitude is SU (2) L ⊗ SU (2) R . Relaxing the left-right symmetric condition, i.e. considering A 10 = A 01 , we can choose the coefficient in such a way that κ 10 = κ 01 = 0, κ 00 + κ 11 = 0, 3κ 00 − κ 11 = 0. In this case we find the following generalization

IV. DISCUSSION AND OUTLOOK
Let us now discuss some consequences of the sum rule derived in the last Section. The sum rule connects the IR limit of the theory, represented by the coefficient c H of the effective SILH Lagrangian, with a combination of total cross sections that is valid, in principle, up to arbitrary high energy. This connection is completely general, because it does not rely on specific details -apart from the postulates of unitarity and analyticity -of the underlying strong dynamics that, as a consequence, remains unknown.
One of the most intriguing consequences of this kind of sum rule, as already emphasized in Refs. [28,29,31], is the possibility to investigate the positivity of c H . The sign of this coefficient is particularly important both from a phenomenological and a theoretical viewpoint. On the one hand c H contributes to the Higgs propagator; as a consequence it modifies universally all the SM Higgs couplings, thus providing a direct connection with the corresponding measurements under investigation at the LHC [16,17,51]. Considering the Higgs couplings with electroweak gauge bosons (g hV V ) and fermions (g hf f ) at the first order in ξ ≡ v 2 /f 2 one finds [26] in which we have defined the scaling factors being y f the SM Yukawa coupling y f = √ 2m f /v. The sign of c H , therefore, is crucial to understand if these deformations point towards a depletion or an enhancement of the Higgs couplings w.r.t. the SM predictions. In Fig. 2 we show the result of a chi-square fit of the Higgs data at the LHC (see Ref. [18] for technical details), performed using as free parameters ξc H , ξc f =t,b,τ in Eqs. (29,30). The blue solid line is obtained marginalizing over ξc f =t,b,τ ; we find at 95% C.L. ξc H = −0.07 +0. 53 −0.43 . For comparison, the one-dimensional fit obtained using ξc H as free parameter and setting ξc f =t,b,τ = 0 gives at 95% C.L. ξc H = 0.03 +0.20 −0.18 (red dashed line). The phenomenological relevance of c H immediately leads us to consider the theoretical implications of its sign. To be more concrete, let us give some example. In Composite Higgs models based on a compact global symmetry G, for instance, one always find a positive value for c H ; in the Holographic Higgs model [15], e.g., we have c H = 1. Little Higgs models also predicts a similar behavior; in the littlest Higgs model with custodial symmetry [52], e.g., one has c H = 1/2. Composite Higgs models based on a non-compact global symmetry group G, on the contrary, have negative c H ; in the minimal Composite Higgs model based on SO(4, 1)/SO(4), e.g., one finds c H = −1 (see Refs. [31,53,54] and Appendix D), thus enhancing the Higgs coupling with the electroweak gauge bosons.
The sum rule in Eq. (28) can not fix the sign of c H . On the right side, in fact, we have two combinations of total cross sections that enter with opposite signs. Nevertheless, Eq. (28) can isolate the source of negative contributions: they come from the total cross section in the channel with (3, 3) quantum numbers under the global symmetry SU (2) L ⊗ SU (2) R .
This information contains some interesting phenomenological consequences. Following Ref. [23], in fact, one can assume that a resonance of the strong sector is accidentally lighter than the cut-off scale of the theory, m ρ Λ ∼ 4πf . If so, this resonance may have sizable effects in the scattering processes involving the Higgs boson and/or the longitudinal gauge bosons W ± L , Z L . Using the Equivalence Theorem [25] (for a recent discussion see Ref. [55]) we can investigate these effects looking directly at the π ⊗ π scattering. In this case the only resonances that can be exchanged are those that possess the correct quantum numbers according to the SU (2) L ⊗SU (2) R decomposition in Eq. (6) η ∼ (1, 1) , spin = 0 , custodial = 0 , ∆ ∼ (3, 3) , spin = 0 , custodial = 0 + 1 + 2 , (35) where the spin assignment is dictated by Bose symmetry [23], and where we have indicated the decomposition under the custodial SU (2) C group. The role of these resonances in the π ⊗ π scattering can be immediately understood considering the expressions of the scattering amplitudes in terms of their SU (2) L ⊗ SU (2) R projections, collected in Eqs. (B25-B31). More precisely, we find the following classification (see Ref. [23] for the corresponding description based on the CCWZ effective Lagrangian [56,57]).
From the point of view of our sum rule, the existence of this resonance leads to an enhancement in the total cross section σ tot 00 (s); its presence, therefore, is favored in models featuring a positive value of c H .
From the point of view of our sum rule, the existence of this resonance leads to an enhancement in the total cross section σ tot 10 (s); its presence, therefore, is favored in models featuring a positive value of c H . Notice, moreover, that this kind of vector resonance is predicted in the minimal Composite Higgs model based on SO(5)/SO(4) saturating the Weinberg sum rules [59][60][61][62][63][64].
From the point of view of our sum rule, the existence of this resonance leads to an enhancement in the total cross section σ tot 11 (s); its presence, therefore, is favored in models featuring a negative value of c H .
Finally, let us compare our sum rule with the existing literature. Starting from Eq. (28), using the optical theorem, and writing explicitly the scattering amplitudes in terms of the charge eigenstates [see Appendix B, Eqs. (B25,B31)] we find our sum rule, as a consequence, recovers the result obtained in Ref. [29]. Similarly, making use of the following SU (2) C custodial decompositions under which the pions transform as a triplet [31] A π ± π ± →π ± π ± (s) = T 2 (s) , where the eigenvalues T I=0,1,2 are the analogous of the A IJ in Eq. (8) but for the SU (2) C combination 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5, we find thus recovering the sum rule obtained in Ref. [31].
Before concluding, a final caveat is mandatory. 7 The final result of this paper, Eq. (28), surely provides useful indications about the relation between possible deviations of the Higgs couplings and the existence of strongly coupled resonances. However, the statement that a light resonance in a given channel IJ would enhance the corresponding contribution to c H has to be taken with a grain of salt. What matters in the computation of the integral in Eq. (28), in fact, is the ratio Γ ππ IJ /M 3 IJ ∼ g 2 * /M 2 IJ , where Γ ππ IJ is the width of the IJ resonance into ππ, M IJ its mass, and g * its coupling with the Nambu-Goldstone bosons. If a light resonance is more weakly coupled than a heavy one, then the contribution of the former does not dominate.

V. CONCLUSIONS
In this paper we have derived in the context of the SILH Lagrangian the following sum rule The derivation of the sum rule is based on the axiomatic properties of Lorentz invariance, analyticity and unitarity, and it relies on the underlying global symmetry SU (2) L ⊗SU (2) R . The sum rule connects the low-energy coefficient c H to the UV properties of the theory, encoded into the combination of total cross sections that appears on the right-hand side. The value of this coefficient is currently under experimental scrutiny at the LHC, and the possibility to extract from this measurement useful informations about the ultimate structure of the theory responsible for the electroweak symmetry breaking is of vital importance. For a given model featuring a SILH, the sum rule can give some useful insight about the corresponding UVcompletion. In particular, the role of the resonances in the scattering processes between longitudinal gauge bosons and/or the Higgs boson has been discussed.
The sum rule favors the existence of a scalar resonance η ∼ (1, 1) or a vector resonance ρ L ∼ (3, 1) [or, equivalently, ρ R ∼ (1, 3)] in models with a positive value of c H , like in Composite Higgs models based on a compact global symmetry group. In presence of the scalar resonance η ∼ (1, 1), in particular, the process W ± L W ∓ L → W ± L W ∓ L and the double Higgs production W ± L W ∓ L → hh and Z L Z L → hh are supposed to be enhanced. In models featuring a negative value of c H , on the contrary, the presence of a large contribution from a scalar resonance ∆ ∼ (3, 3) is mandatory. Daniele Amati and Yuri Dokshitzer for enlightening discussions, and to Slava Rychkov for reading the manuscript and encouragments. The author is also grateful to Francesco Riva for important comments. This work is supported by the ERC Advanced Grant n • 267985, "Electroweak Symmetry Breaking, Flavour and Dark Matter: One Solution for Three Mysteries" (DaMeSyFla).

Appendix A: Scattering amplitudes and the S-matrix
The remarkable goal of the S-matrix program, developed during the sixties before the rise of QCD, was to construct and compute scattering amplitudes using only three postulates as guiding principles [24]. To be more concrete, let us consider in full generality the scattering i → f from an initial state i to a final state f ; the corresponding S-matrix element is where A i→f is the relativistic scattering amplitude. The aforementioned three principles are the following.
i) The S-matrix is Lorentz invariant. This means that the scattering amplitude can be written as a function of the Lorentz invariants -scalar products and rest masses -involved in the process. Considering for definiteness the two-to-two scattering ab → cd, these Lorentz invariants can be recast in terms of the usual Mandelstam variables related by s + t + u = i=a,...d m 2 i . We denote the corresponding scattering amplitude as bearing in mind, however, that u is not an independent variable. In the following, whenever it is not necessary, we will omit the u-dependence.
ii) The S-matrix is unitary, S † S = SS † = 1. This property is a consequence of the conservation of probability. In terms of the scattering amplitude in Eq. (A1) the unitarity condition reads where dΠ n is the n-particle phase-space measure.
iii) The S-matrix is an analytical function of the Lorentz invariants regarded as variables in the complex plane. The singularities of the S-matrix are only those dictated by unitarity. It can be proved that this property is intimately connected with causality [24]. Apart from the usual formalities, the analyticity of a scattering amplitude finds an operative definition in the Cauchy integral formula where C is a contour that does not enclose the singularities of A i→f (s, t). Eqs. (A6, A7) are the key equations of the S-matrix program: once the imaginary part of A i→f (s, t) is known, in fact, the Cauchy integral formula -rewritten in terms of a dispersion relation [24] -allows to fully reconstruct the scattering amplitude.
Appendix B: The scattering amplitude A π a π b →π c π d In this Appendix we construct explicitly the scattering amplitudes for the process π a (p 1 ) + π b (p 2 ) → π c (p 3 ) + π d (p 4 ), where π a = π ± , π 0 , h. In B 1 we show how the SU (2) L ⊗SU (2) R symmetry dictates the general structure of these amplitudes, while in B 2 we discuss the corresponding transformations of crossing symmetry.
1. The role of the SU (2)L ⊗ SU (2)R symmetry In order to construct the scattering amplitude for the process π a π b → π c π d we make use of the symmetry SU (2) L ⊗SU (2) R under which the Goldstone bosons and the Higgs boson transform according to the bi-doublet representation From the composition of angular momenta it follows that the π ⊗ π combination admits the following deconstruction this means that we can organize the initial and the final state of the scattering process according to their SU (2) L ⊗ SU (2) R quantum numbers. To this purpose, we start from Eq. (B1), labeling the states in the representation (2, 2) using the notation |t L , t 3 L ; t R , t 3 where the last definition is nothing but a shorthand notation. In the combination π ⊗ π the states coming from the sum T L = t L + t L , T R = t R + t R generate the representation (2, 2) ⊗ (2, 2) that we label as |T L , T 3 L ; T R , T 3 R . According to Eq. (B3) we find [65,66] • Singlet (1, 1), (T L = 0, T R = 0) • Right Triplet (1, 3), (T L = 0, T R = 1) After reversing the system formed by the eigenstates in Eqs. (B8-B23), it is possible to use the Wigner-Eckart theorem to rewrite the scattering amplitude π c π d |A|π a π b in terms of the following eigenamplitudes (B24) As a consequence, we can recast the amplitude π c π d |A|π a π b as a function of the four scattering eigenvalues A 00 , A 10 , A 01 , A 11 . Reintroducing the notation π c π d |A|π a π b = A π a π b →π c π d (s, t, u) we find A π ± π 0 →π ± π 0 (s, t, u) = 1 4 (A 10 + 2A 11 + A 01 ) , where on the right side the same kinematical dependence A IJ = A IJ (s, t, u) is understood. Moreover, Eqs. (B26-B29) hold true replacing π 0 with the Higgs boson, i.e. for instance A π ± π 0 →π ± π 0 (s, t, u) = A π ± h→π ± h (s, t, u) and A π 0 π 0 →π 0 π 0 (s, t, u) = A hh→hh (s, t, u). On the contrary, we find A hh→π 0 h (s, t, u) = A π 0 π 0 →hπ 0 (s, t, u) = 0. Finally, notice that we need one more amplitude in order to disentangle the combination A 10 + A 01 in Eqs. (B26,B27,B31); in particular we find This scattering amplitude is different from zero only breaking the left-right symmetry, A 10 = A 01 . Including Eq. (B32) the system in Eqs. (B25-B31) can be immediately reversed, and a trivial computation leads to the following expressions A 00 = 4A π ± π ∓ →π 0 π 0 (s, t, u) + A π ± π ± →π ± π ± (s, t, u) , A 01 = A π ± π ± →π ± π ± (s, t, u) −2 [A π ± π 0 →π 0 π ± (s, t, u) − iA π + π − →hπ 0 (s, t, u)] . (B36) 2. SU (2)L ⊗ SU (2)R crossing symmetry One of the most powerful consequence of analyticity is crossing symmetry. Starting from Eq. (A5), and defining the corresponding crossed scattering amplitudes t − channel : a +c →b + d , crossing symmetry is formally defined by the following relations A ab→cd (s, t, u) = A ad→cb (u, t, s) , and corresponds to the fact that, thanks to the analytical continuation, it is possible to describe all the processes in Eqs. (B37-B39) with the same analytical function but interchanging the role of the Mandelstam variables [24]. In our case, because of the SU (2) L ⊗ SU (2) R structure, the crossing relations in Eqs. (B40,B41) have the following matrix form where A ≡ (A 00 , A 10 , A 01 , A 11 ) T . Using Eqs. (B25-B31) and Eqs. (B40-B41) we find As a simple cross-check, these matrices satisfy the rela- (sin θ) 1 2 . (C2) Using the optical theorem the first inequality can be immediately translated into a bound on the high-energy behavior of the total cross section σ tot ab (s) describing the process ab → anything σ tot ab (s) const (ln s) 2 .
From a more general point of view, one can be interested in inelastic processes where initial and final state are different [67,68]. In the following we shall derive a generalization of the Froissart-Martin bound in Eq. (C1), and our proof goes as follows.
Let us start considering the inelastic scattering process ab → cd among scalar particles with, respectively, fourmomenta p 1 , p 2 , p 3 , p 4 . For simplicity we assume that all the masses are equal, s + t + u = 4m 2 , with √ s m. In this limit the differential cross section is where A ab→cd (s, cos θ) is the scattering amplitude. The total cross section for the process ab → cd is therefore σ ab→cd (s) = 1 32πs However, given that we are interested in the forward limit of the scattering amplitude, |A ab→cd (s, cos θ = 1)|, this result is not enough for our purposes. In order to put a bound on |A ab→cd (s, cos θ = 1)|, we proceed following three steps.

We introduce the partial wave expansion
A ab→cd (s, cos θ) = ∞ l=0 (2l + 1)a l (s)P l (cos θ) , where a l (s) are the partial wave amplitudes and P l (cos θ) the Legendre polynomials. Bearing in mind that P l (1) = (2l + 1)(2m + 1)a l (s)a * m (s) , (C9) and, using the Cauchy-Bunyakovskii inequality, (C10) On the other hand, considering the total cross section in Eq. (C5), we have where we made use of the orthogonality relation 2. The next step is to relate the partial wave amplitudes a l (s) in Eq. (C10) to the amplitude describing the corresponding elastic process ab → ab. To this purpose, we use the unitarity condition in Eq. (A6); for a scattering process 1 + 2 → n, with initial state |p 1 , p 2 and final state |p 1 , . . . , p n we have Considering the two-body elastic scattering process ab → ab, and writing explicitly the independent kinematical variables, Eq. (A6) becomes 9 2Im A ab→ab (s, cos θ = 1) = n dΠ n |A ab→n | 2 . (C14) The right-hand side is a sum of positive numbers. Extracting only the process ab → cd, we have the following inequality 9 Notice that A ab→ab is the transition amplitude for the scattering process ab → ab in which the direction of motion is unchanged (initial and final state are equal). In other words, we are dealing with the elastic amplitude describing the forward scattering.
The elastic amplitude A ab→ab (s, 1), in turn, can be expanded in partial waves leading to being |f l (s)| Im f l (s). This inequality connects the partial wave amplitudes describing the elastic process ab → ab, f l (s), to those describing the forward inelastic process ab → cd, a l (s), according to Eq. (C10).
3. Finally, combining Eq. (C10) and Eq. (C17), we are now in the position to use the original argument of the Froissart theorem. This argument relies on the fact that, in the large s, l limit, we have the asymptotic behavior where δ is an integer. This simply means that the partial waves with l c √ s ln s, where c is some constant, can be neglected. Because of unitarity all the remaining ones, moreover, are bounded according to |f l (s)| 16π. All in all we find In this Appendix, we construct the non-linear σ-model based on the coset SO(4, 1)/SO(4) following the CCWZ prescription, originally proposed in Refs. [56,57] considering compact, connected, semisimple Lie group G. The correspondent generalization to the case in which G is a non-compact group, and H is its maximal compact subgroup [as in SO(4, 1)/SO(4)] is known in the context of supergravity theories (see, e.g., Ref. [69][70][71]).
The de Sitter group SO(4, 1) [72] finds an intuitive realization as the ten-parameter group of transformation matrices that acting on the five variables w, x, y, z, t holds invariant the indefinite quadratic form The generators of the corresponding algebra L ab = −L ba satisfy the commutation relations [L ab , L cd ] = i (−g ac L bd + g ad L bc + g bc L ad − g bd L ac ) , (D2) where g = diag(−1, −1, −1, −1, +1) is the internal metric of the algebra. The explicit solution used throughout this paper is (D3) The maximal compact subgroup of SO(4, 1) is the special orthogonal group SO(4). In more detail, recasting the generators as follows the algebra of the isomorphism SO(4) ≈ SU (2) L ⊗ SU (2) R can be recovered defining while the remaining generators T a=1,···,4 c ≡ L a5 / √ 2 define the coset SO(4, 1)/SO (4). Notice that, using the explicit realization in Eq. (D3), the generators of the de Sitter group are normalized as follows The coset SO(4, 1)/SO(4) is the four-dimensional hyperbolic space H 4 . To describe this space, first we introduce the coordinates ϕ 1,···,4 to parametrize the left cosets, then we define the coset representative field The coset representative field is an element of the group SO(4, 1) which transform under global SO(4, 1) transformation from the left and local SO(4) transformation from the right. It satisfies the defining relation of SO(4, 1), namely U T g U = g, and its inverse can be build using the metric g as U −1 = g U g. In the unitary gauge ϕ = (0, 0, 0, h) the coset representative field takes the explicit matrix form (D10) Following Ref. [69][70][71], we introduce the decomposition where d µ ≡ d a µ T a C , E µ ≡ E a L,µ T a L + E a R,µ T a R . Gauging the SU (2) L ⊗ U (1) Y SM subgroup amounts to promoting the ordinary derivatives to covariant ones, ∂ µ → D µ = ∂ µ − i(g L W a µ T a L + g Y B µ T 3 R ). The gauged version of Eq. (D11) becomes U −1 (D µ U ) =d µ +Ê µ , withd µ ≡d a µ T a C ,Ê µ ≡ E a L,µ T a L +Ê a R,µ T a R . The leading order non linear σ-model Lagrangian for the SM gauge and Goldstone bosons is We find From the above Lagrangian we read the value of the electroweak gauge boson masses from which we obtain Following Ref. [26], the scaling factor describing the coupling of the Higgs with the electroweak gauge boson V is Notice that in Composite Higgs models based on a compact global symmetry group G one finds k hV V = √ 1 − ξ. Similarly, the scaling factor describing the Higgs quadratic coupling to the electroweak gauge boson V follows from where L hhW W = (f 2 /8)g 2 L sinh 2 (h/f ), L hhBB = (f 2 /8)g 2 Y sinh 2 (h/f ). In Composite Higgs models based on a compact global symmetry group G one finds k hhV V = 1 − 2ξ.
The flipped sign in Eq. (D17) has important phenomenological implications [73], summarized for the sake of clarity in Fig. 3, where we fit the LEP data in the plane defined by the oblique parameters S and T [74,75] (see Ref. [18,76] for a detailed discussion about the fit). The reference point at which S and T vanish is defined by the SM with m h = 126 GeV and m t = 173.5 GeV, and it lies on the boundary of the 68% confidence contour, in agreement with the experimental data. Deviations of the Higgs couplings with the electroweak gauge bosons W ± , Z w.r.t. their SM values generate logarithmic correction [77] to the S and T parameters in the directions shown by the representative arrows in the plot. The correction in Eq. (D17) points towards the favored region, thus alleviating the tension with the electroweak precision measurements that affects the Composite Higgs models based on a compact global symmetry [78]. It is important to keep in mind, however, that extra contributions coming from the strong sector can drastically modify this picture.