Effective Lagrangian for a light Higgs-like scalar

We reconsider the effective Lagrangian that describes a light Higgs-like boson and better clarify a few issues which were not exhaustively addressed in the previous literature. In particular we highlight the strategy to determine whether the dynamics responsible for the electroweak symmetry breaking is weakly or strongly interacting. We also discuss how the effective Lagrangian can be implemented into automatic tools for the calculation of Higgs decay rates and production cross sections.


Introduction
The exploration of the weak scale has marked an important step forward with the discovery by the ATLAS [1] and CMS [2] collaborations of a boson with mass m h 125 GeV, whose production cross section and decay rates are compatible with those predicted for the Higgs boson of the Standard Model (SM). At the same time, no hint of the existence of additional new particles has emerged yet, which might shed light on the origin of the electroweak symmetry breaking (EWSB). One is thus faced with the problem of which is the best strategy to describe the properties and investigate the nature of the new boson h, beyond the framework of the Standard Model. In absence of a direct observation of new states, our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian for the light boson.
Such an effective description is valid as long as New Physics (NP) states appear at a scale M m h , and is based on an expansion in the number of fields and derivatives [3]. The detailed form of the effective Lagrangian depends on which assumptions are made. Considering that the observation made by the LHC experiments is in remarkable agreement with the SM prediction, although within the current limited experimental precision, it is reasonable to assume that h is a CP-even scalar that forms an SU (2) L doublet together with the longitudinal polarizations of the W and Z, so that the SU (2) L × U (1) Y electroweak symmetry is linearly realized at high energies. Under these assumptions the effective Lagrangian can be expanded into a sum of operators with increasing dimensionality, where the leading NP effects are given by dimension-6 operators.
The parametrization of the deviations of the Higgs couplings in terms of higher-dimension operators started more than two decades ago. The experimental observation of the Higgs boson, however, calls for a more detailed analysis. First, a compilation of a complete and updated list of constraints on the various Wilson coefficients is in need. Second, the rather precise estimation of the Higgs mass below the gauge boson thresholds necessitates a careful computation including off-shell effects that have not been incorporated up-to-now when the SM Lagrangian is supplemented by higher-dimensional operators. It is the purpose of this paper to perform such an updated analysis. We will also discuss in detail the implications of the custodial symmetry on the generalized Higgs couplings and clarify a few other issues which were not exhaustively addressed in the previous literature, like for example the connec-tion with the effective Lagrangian for a non-linearly realized electroweak symmetry. Finally, a precise comparison of the Higgs couplings with the SM predictions can only be done when higher-order effects are included in a consistent way, and we will develop a strategy to this end.
The paper is structured as follows. In section 2 we review the construction of the effective Lagrangian for a light Higgs doublet. By means of a naive power counting we estimate the coefficients of the various operators and review the most important bounds set on them by present experimental results on electroweak (EW) and flavor observables. Focusing on Higgs physics, we then discuss in Section 3 the relative effect of the various operators on physical observables. Such an analysis, first proposed in Ref. [4], will allow us to identify which operators can probe the Higgs coupling strength to the new states and which instead are sensitive only to the mass scale M . This is of key importance to distinguish between weaklycoupled UV completions of the Standard Model, like Supersymmetric (SUSY) theories, and theories where the EW symmetry is broken by a new strongly-interacting dynamics which forms the Higgs boson as a bound state [5][6][7]4]. These are the two most compelling scenarios put forward to solve the hierarchy problem of the Standard Model. We conclude the section by discussing how the assumption of a Higgs doublet and linearly-realized SU (2) L × U (1) Y can be relaxed. We illustrate the non-linear effective Lagrangian valid for the case of a generic CP-even scalar h and discuss the implications of custodial invariance. Section 4 is devoted to clarify a few issues related to the use of the effective Lagrangian beyond the tree level.
We present our concluding discussion in Section 5. In the Appendices A-C we collect useful formulas and give further details on the construction of the effective Lagrangian. The details of how we derived the bounds on the dimension-6 operators are reported in Appendix D.
As an illustration of our analysis and to better demonstrate how the effective Lagrangian can be implemented into automatic tools for the computation of physical quantities like Higgs production cross sections and decay rates, we have written eHDECAY 1 , a modified version of the program HDECAY [8], which includes the full list of leading bosonic operators.
We will describe the program in a separate companion paper [9].

Effective Lagrangian for a light Higgs doublet
The most general SU (3) C × SU (2) L × U (1) Y -invariant Lagrangian for a weak doublet H at the level of dimension-6 operators was first classified in a systematic way in Refs. [10].
Subsequent analyses [11,12] pointed out the presence of some redundant operators, and a minimal and complete list of operators was finally provided in Ref. [13]. As recently discussed in Ref. [4], a convenient basis of operators relevant for Higgs physics, assuming that the Higgs is a CP-even weak doublet (this assumption will be relaxed in Appendix C) and the baryon and lepton numbers are conserved, is the following:  we will denote as O i the dimension-6 operator whose coefficient is proportional toc i .
Our higher-dimensional Lagrangian, which is supposed to capture the leading New Physics effects, counts 12 (∆L SILH ) + 8 (∆L F 1 ) + 8 (∆L F 2 ) = 28 operators. Five extra bosonic oper- which affect the gauge-boson propagators and self-interactions but with no effect on Higgs physics, should also be added to complete the operator basis, as well as 22 four-Fermi baryonnumber-conserving operators. 2 A comparison with Ref. [13] shows that two of our operators 2 Notice that the last three operators in Eq. (2.6) can be rewritten in favor of three additional independent four-Fermi operators, as in the basis of Ref. [13]. The coefficientsc 2W ,c 2B contribute respectively to the W and Y parameters defined in Ref. [14].
are actually redundant. As we shall explain in more detail in Section 3 (see Eqs. (3.28), (3.29)), it is well known [12,15] that two particular linear combinations of the fermionic operators in ∆L F 1 are equivalent to pure oblique corrections parametrized by the operators where the sum runs over all fermion representations, ψ = q L , u R , d R , L L , l R , whose hypercharge has been denoted as Y ψ . These two linear combinations have then to be excluded from ∆L F 1 , and we end up with exactly 53 linearly-independent operators as in Ref. [13]. 3 Any other dimension-six operator can be obtained from these 53 operators by using the equations of motion, or equivalently by performing appropriate field redefinitions. 4 Even though our basis (2.2)-(2.4) is equivalent to the one proposed in Ref. [13], we advocate that it is more appropriate for Higgs physics for at least three reasons [4]: i) Generic models of New Physics generate a contribution to the obliqueŜ parameter [16,14] at treelevel, which in the basis of Ref. [13] would have to be encoded in the two fermionic operators O Y Hψ and O Hq + O HL even in the absence of direct couplings between the SM fermions and the New Physics sector. There is an advantage in describing the oblique corrections in terms of the operators in (2.2) rather than in terms of the operators with fermionic currents, which generate vertex corrections and modify the Fermi constant. ii) The basis (2.1) isolates the contributions to the decays h → γγ (from O γ ) and h → γZ (from O γ and O HW − O HB ) that occur only at the radiative level in minimally coupled theories. iii) Our basis of operators is more appropriate to establish the nature of the Higgs boson and determine the strength of its interactions. For example, as we shall explain momentarily, if the Higgs boson is a pseudo Nambu-Goldstone boson (pNGB) the coefficient of the operator O γ , hence the rate 3 For completeness we collect in Appendix C also the extra 6 bosonic operators of dimension-six that are CP-odd. 4 In particular, the following identities hold: where λ ψ denotes the coupling of a generic SM fermion ψ to the new dynamics. It should be stressed that these estimates are valid at the UV scale M , at which the effective Lagrangian is matched onto explicit models. Renormalization effects between M and the EW scale mix operators with the same quantum numbers, and give in general subdominant corrections to the coefficients. We shall comment on these renormalization effects in Section 4. Notice that the estimates ofc W,B ,c Hψ ,c Hψ andc T apply when these coefficients are generated at 5 Notice that our normalization differs from the one of Ref. [4], and it is more convenient than the latter for a model-independent implementation of Eq. (2.2) in a computer program. The factor multiplying each operator in the effective Lagrangian has been conveniently defined such that the dependence on M and g * is fully encoded in the dimensionless coefficientsc i . A special and phenomenologically motivated case is represented by theories where the Higgs doublet is a composite Nambu-Goldstone (NG) boson of a spontaneously-broken symmetry G → H of the strong dynamics [5][6][7]4]. For these models the scale f must be identified with the decay constant associated with the spontaneous breaking, and the naive estimate is included as part of the G/H transformations. This means that they cannot be generated in absence of an explicit breaking of the global symmetry. It follows, in particular, that the naive estimates of the operators O γ and O g carry in this case an additional suppression factor [4],c (2.10) where g G denotes any weak coupling that breaks the Goldstone symmetry (one of the SM weak couplings in minimal models, i.e. the SM gauge couplings or the Yukawa couplings).  [17] for a natural way to obtain this alignment). This implies one coefficient for the up-type quarks (c u ), one for down-type quarks (c d ), and one for the charged leptons (c l ), i.e. thec u,d,l are proportional to the identity matrix in flavor space.

Current bounds on flavor-preserving operators
It is useful to review some of the most important constraints on the coefficientsc i that follow from current experimental results, such as electroweak precision tests, flavor data and lowenergy precision measurements. For simplicity, we focus on the bounds on flavor-conserving operators, keeping in mind that they can come also from flavor-changing processes. For a discussion of the bounds on flavor-violating operators see for example the recent review of Ref. [18] as well as Ref. [19].
Among the strongest bounds are those on operators that modify the vector-boson selfenergies. The operator O T , for example, violates the custodial symmetry [20] and contributes to the EW parameter 1 [21]. From the EW fit performed in Ref. [22], it follows, with 95% probability, Such a stringent bound can be more naturally satisfied by assuming that the dynamics at the scale M possesses an (at least approximate) SU (2) V custodial invariance. In this case c T (M ) = 0, and a non-vanishing value will be generated through the renormalization-group (RG) flow of this Wilson coefficient down to m Z in the presence of an explicit breaking of the custodial symmetry, as due for example to the Yukawa or hypercharge couplings. We will discuss these renormalization effects in more detail in Section 4. Notice that all the other dimension-6 operators in the effective Lagrangian are (formally) custodially symmetric and their coefficients will not be suppressed at the scale M . 6 The electroweak precision tests 6 More precisely, for all the other operators the only violation of the custodial symmetry comes from also imply a strong bound on O W + O B [4], since this linear combination contributes to the parameter 3 [21]. With 95% probability, one has [22]: where T 3L and Q are respectively the SU (2) L and electric charges of the fermion ψ, and Ψ = {L, q} is the SU (2) L doublet to which ψ L belongs. We used the results of Ref. [22] to perform a fit on the coefficientsc Hψ ,c HΨ ,c HΨ . The details of our analysis can be found in Appendix D (see also Ref. [23] O Ht , is unconstrained by EW data, but it is also not relevant for the Higgs decays and will be neglected in the following. The coefficientc Htb is severely constrained by the b → sγ rate. Indeed, the expansion of O Htb around the vacuum contains a vertex of the type W t R b R , which at 1-loop gives a chirally-enhanced contribution to the rate (see for example Ref. [24]).
We find, with 95% probability: For a given (v/f ), the above bounds set a limit on the couplings of the SM fermions to the new dynamics, see Eq. (2.9). Unless the scale of New Physics is very large, or some specific symmetry protection is at work in the UV theory (see for example the discussion in Ref. [23]), it follows that the SM fermions must be very weakly coupled to the new dynamics, with the exception of the top quark.
The constraints on the dipole operators of Eq. (2.4) come from the current experimental limits on electric dipole moments (EDMs) and anomalous magnetic moments. The bounds on the neutron and mercury EDMs for example strongly constrain the dipole operators with u and d quarks. By using the formulas of Ref. [25] we find, with 95% probability, that:  where the coefficients are evaluated at the low-energy scale µ ∼ 1 GeV. According to the naive estimate (2.9), for O(1) CP-violating phases these results imply a bound on (v/f ) 2 at the level of 10 −3 . In natural extensions of the SM, such a strong limit clearly points to the need of a symmetry protection mechanism. For a discussion, see for example Ref. [23] for the case of composite Higgs theories, and Ref. [26] for the case of SUSY theories.
Among the heavier quarks the most interesting bounds are those on dipole operators with top quarks [27]. These come from the experimental limit on the neutron EDM, 19) and the tt cross sections measured at the Tevatron and LHC, (2.20) All these bounds have 95% probability and have been derived by making use of the formulas reported in Ref. [27]. 7 It is worth noting that the bounds of Eqs. (2.19) and (2.20) are still about one order of magnitude weaker than the size ofc tG ,c tW andc tB expected from the naive estimate (2.9) with (v/f ) 2 ∼ 0.1. Additional weaker constraints arise from the limits on anomalous top interactions based on top decays and single top production. From the results of Ref. [28] we find that, with 95% probability: where the coefficients are evaluated at the scale µ ∼ m t .
In the lepton sector, the current measurements and SM predictions of the muon [29,30] and electron [31,32] anomalous magnetic moments and the limits on their EDMs [33,34] imply the following 95% probability bounds: where the coefficients are evaluated at the relevant low-energy scale. Notice that the nonvanishing value of Re(c µB −c µW ) follows from the known ∼ 3.5σ anomaly in the (g − 2) of the muon (see Ref. [29] for an updated review). Among the bounds of Eqs. Refs. [4,35], we will try to highlight a possible strategy to determine whether the dynamics behind the electroweak breaking is weak or strong. Our analysis will be based on the naive estimates of the Wilson coefficients at the matching scale. In the next Section, we will discuss how the running from the matching scale to the weak scale affects these estimates.   Table 1. The shifts from the SM value are of order

Operators sensitive to a strongly-interacting Higgs boson
The couplings to fermions, on the other hand, are not uniquely fixed by the choice of the coset, but depend on how the SM fermions are coupled to the strong dynamics. The last two columns of Table 1  Higgs couplings In general, a shift of the tree-level Higgs couplings of order (v/f ) 2 implies that the theory gets strongly coupled at energies ∼ 4πf , unless new weakly-coupled physics states set in to regulate the energy growth of the scattering amplitudes. The dominant effect comes from the energy growth of the V L V L → V L V L (V = W ± , Z 0 ) scattering amplitudes, which become non-perturbative at the scale Λ s = 4πv/ |c H |. A modified coupling to the top quark leads The scale of New Physics is thus required to lie below, or at, such ultimate range of validity of the effective theory: M Λ s .

Operators sensitive to the scale of New Physics
The operators O W , O B can be generated at tree-level by the exchange of heavy particles, for example heavy spin-1 states. In the unitary gauge they are written in terms of the following three operators 8 Here and in the following, derivatives acting on operators in the unitary gauge are covariant under local U (1) em transformations. Operators like (∂ µ Z µν )γ ν h or (∂ µ γ µν )γ ν h obviously cannot be generated since they break the U (1) em local symmetry. 9 We thank Riccardo Rattazzi for pointing this out to us. 10    bounds this custodial-symmetry breaking effect down to an unobservable level, unless some fine tuning is in place in the combinationc W +c B so thatc B can be large. Notice that despite the operator O T is generated after using the equations of motion, its contribution to ∆ 1 (corresponding to a non-vanishingT parameter [16,14]) is exactly canceled by the vertex correction implied by the linear combination of fermionic operators which is also generated. 12 This is of course expected, since O W , O B only contribute to 3 , and not to 1 .
In general, the contribution of O W , O B to inclusive observables, in particular to the Higgs decay rates, is of order (m 2 W /M 2 ): (3.30) 12 See for example Eq. (9.10) of Ref. [15].
where in this case V V = W ( * ) W * , Z ( * ) Z * , Z ( * ) γ, γγ. This implies that these operators are sensitive only to the value of the scale of New Physics M , and do not probe the coupling strength g * . From the quantitative side, the constraint (2.12) suggests that their effects in inclusive Higgs decay rates is too small to be observable. For example, we find that for small c W,B the tree-level correction to the W W and ZZ partial rates is well approximated by: 13 Notice that despite its custodial invariance, the operator O W affects in a slightly different way the decay of the Higgs boson into W W and ZZ, due to the fact that at least one of the two final vector bosons is off-shell. 14 At the one-loop level O W also contributes to the Higgs decays into Zγ and γγ (while O B does not). We find: which agree with Eqs. (82) and (83) of Ref. [4]. 15 Forc W,B ∼ 10 −3 the above approximate formulas imply corrections too small to be observed at the LHC. On the other hand, one could try to take advantage of the different predictions in terms of angular and invariant mass distributions which are implied by the dimension-6 operators compared to the tree-level SM prediction. The most promising strategy could be in fact that based on the analysis of the angular distributions of the final fermions [38][39][40]. In the ideal case in which one is able to kill completely the SM tree-level contribution by means of appropriate kinematic cuts, 13 Here and in the following our approximated formulas have been obtained by using eHDECAY [9] with m h = 125 GeV. QCD corrections to the decay rates are fully included. Electroweak corrections are instead not included, since their effect on the numerical prefactor appearing in front of the coefficientsc i is of order ) and thus beyond the accuracy of our computation. See Ref. [9] for more details. 14 It is easy to check that for m h > 2m Z and on-shell decays one has: These formulas coincide with those of Eqs. (79)-(80) of Ref. [4], which are thus valid only for on-shell decays. 15 The easiest way to compute the one-loop contribution of O W to the Zγ and γγ rates is by using Eq. the relative effect of NP becomes of order plus other terms with zero or two Higgs fields. Since the coefficients of the above four operators are functions ofc HW ,c HB andc γ , they are related by one identity, see Eq. (3.47).
We will discuss this point in greater detail in Section 3.6.
As implied from the naive estimates (2.9), the contribution of O HW,HB,γ to the W W and Although such an effect depends on the Higgs interaction strength, it is suppressed compared to Eq. (3.32) by a loop factor. We find that the following approximate formulas hold 16 (3.40) 16 For m h > 2m Z and on-shell decays, we find instead Comparing with the analog formulas in Eqs. (79) and (80) of Ref. [4], we find that in these latter there is a missing factor 2 and the term proportional toc γ was not included either. Notice also that the effect of the off-shellness of the gauge bosons is rather large, as one can see by comparing Eq. While the contribution due toc HB andc γ explicitly violates the custodial symmetry and thus differentiates W W from ZZ, the different numerical factor multiplyingc HW in the two formulas above is due to the off-shellness of at least one of the two vector bosons, similarly to Eq. (3.33). Although there is currently no stringent bound on the coefficientsc HW,HB,γ , the estimate (2.9) suggests that their correction to inclusive rates is unobservable at the LHC.
As discussed in the previous section, on the other hand, a study of the angular and invariant mass distributions of these decays can potentially uncover the effect of New Physics. In particular, an estimate similar to that of Eq. (3.36) can be derived also forc HW,HB,γ .
The processes h → γγ, h → Zγ and h → gg (or equivalently gg → h) can in principle test the Higgs interaction strengths much more powerfully, since they arise at the one-loop level in the SM. Naively one expects: We find that the following approximate formulas hold to good accuracy for smallc i 's: where we have conveniently defined and by α em we indicate the value of the running electromagnetic coupling α em (q 2 = 0) in the Thomson limit. If the Higgs boson is a NG boson, the coefficientsc g andc γ are further suppressed by a factor (g G /g * ) 2 , see Eq. (2.10), where g G is a weak coupling. This implies that in this class of theories the corrections to Γ(h → γγ) and Γ(h → gg) depend only on the scale of New Physics and not on the Higgs interaction strength. In fact, in the case of minimal models with linear couplings, like for example the MCHM4 and MCHM5, the low energy theorem [42,43] implies that the leading contribution to the γγ and gg decay rates from the virtual exchange of heavy fermions is additionally suppressed [44][45][46][47] due to a cancellation between the effect parametrized byc g,γ and the one that follows from the shift in the top Yukawa coupling due toc u andc H (see Ref. [46] for an interesting exception). In general, in theories with a pNGB Higgs boson the local corrections to the rates Γ(h → γγ) and Γ(h → gg) from O γ and O g are expected to be small and subdominant compared to the effect from the modified tree-level Higgs couplings.

Fermionic operators
The fermionic operators in ∆L (3.44) Compared to the corrections from O W and O B , the effect of the fermionic operators is potentially enhanced by a factor (λ 2 ψ /g 2 ). In practice, the possibility of large fermionic couplings λ ψ is strongly constrained by LEP, see Eqs. (2.14)-(2.16). Scenarios in which a large degree of compositeness of either the left-or right-handed quarks is not ruled out are generically those in which the corresponding operators in ∆L F 1 are not generated as due to some protecting symmetry (see for example Refs. [23,48,49]). Large corrections to the inclusive rate of the three-body decays h → Vψψ from ∆L F 1 are thus excluded, while the possibility of detecting the effects of these operators through the analysis of differential distributions should be explored, similarly to what has been discussed for O W and O B .
Among the dipole operators in ∆L F 2 , those with light fermions are already strongly constrained by current precision data, but potentially sizable effects could still come from the operators involving the top quark. For example, the contribution of O tG to gg → h, where we have definedĉ tG ≡ Re(c tG ) (m 2 t /m 2 W ) ∼ m 2 t /(16π 2 f 2 ) 3 × 10 −3 (v 2 /f 2 ). Notice that the experimental limit on the neutron EDM puts an upper bound on the imaginary part ofĉ tG at the 10 −4 level, see Eq. (2.18), which indicates that this is currently the most sensitive experiment on Im(c tG ). Some mechanism is however required to suppress the imaginary parts of the dipole operators involving light fermions, in order to satisfy the stringent constraints of Eq. (2.17). By the same mechanism also Im(c tG ) could be suppressed, so that the processes of Eq. (3.45) are essential to probe the contribution of O tG due to Re(c tG ).
From Eq. (3.45) and the naive estimate ofĉ tG it follows that the most sensitive process is perhaps gg → tt, in particular the events at large invariant mass, although a precision larger than the one currently achieved is required to constrain (v/f ). To this aim, the analysis of differential distributions and spin correlations could be a successful strategy [50,27]. The NP contribution to the process gg → tth can in principle get the largest enhancement from a cut on √ s, but the small rate might limit the actual sensitivity achievable at the LHC [51].
Finally, additional information comes from the experimental limits on top anomalous couplings obtained at the Tevatron and the LHC, although their sensitivity on NP is expected to be much smaller by naive estimate. The operator O tW , in particular, gives the largest effect and generates the anomalous coupling g R (g/m W )b L σ µν W − µν t R [28]. Naively one ex- an effect too small to be observed even for f of order v.

Non-linear Lagrangian for a Higgs-like scalar
Summarizing, by working in the unitary gauge and in the basis of fermion mass eigenstates, the effective Lagrangian relevant for Higgs physics reads as follows [36]

Implications of custodial symmetry
Another difference between the non-linear Lagrangian (3.46) gives rise to In this case the diagonal custodial SU (2) V is exact even though g = g. The left and right gauge fields couple to the conserved currents of SU (2) L ×SU (2) R and the interactions among two gauge fields and the Higgs boson are fully characterized in momentum space by three form factors: where we have defined Γ µν RL (p 1 , p 2 ) ≡ Γ νµ LR (p 2 , p 1 ). Notice, in particular, that in this case the same form factor ΓVV describes the interaction of two W 's and two Z's to the Higgs boson, as due to custodial invariance.
The physical limit where only SU (2) L × U (1) Y is gauged is obtained by simply switching off the unphysical R 1,2 µ fields. The interactions of two neutral vector bosons to the Higgs are still described by the relations of Eq. (3.53), where Γ ZZ = ΓVV , Γ γγ = Γ V V and Γ Zγ = ΓV V .
In the charged sector, instead, the W corresponds to a pure left gauge field, since it has no mixing with right-handed ones. This implies that its form factor is given by the last formula of Eq. (3.53) with θ W = 0, that is: Γ W W = Γ LL . The four physical form factors are linear combinations of the three defined in Eq. (3.52), and are thus related by one identity: where we have defined P µν 1 ≡ η µν p 2 1 − p µ 1 p ν 1 , P µν 2 ≡ η µν p 2 2 − p µ 2 p ν 2 and P µν 12 ≡ η µν p 1 ·p 2 − p ν 1 p µ 2 . This is in fact the most general decomposition which follows at the O(p 2 ) level for an on-shell (2.4)), as long as one focuses on terms with one Higgs boson. This means that by using single-Higgs processes alone, one cannot distinguish the case in which the Higgs boson is part of a doublet from the more general situation. The only possible strategy to this aim is exploiting the connection among processes with zero, one and two Higgs bosons which is implied by the Lagrangian (2.1) at O(v 2 /f 2 ) and does not hold in the case of the more general non-linear Lagrangian. As a consequence of such connection, the bounds that EW and flavor data set on operators with zero Higgs fields severely constrain the size of the NP effects in Higgs processes, as discussed in Section (2.1). If one were to find that single-Higgs processes violate these constraints, this would be an indication that the Higgs is not part of a doublet. Furthermore, processes with double Higgs boson production can be predicted to a certain extent in terms of single-Higgs couplings, and can thus be used to probe the nature of the Higgs boson [56].

Implementing the Higgs effective Lagrangian beyond the tree level
In this section we address a few issues related to the use of the effective Lagrangians (2.1) and (3.46) beyond the tree level, as required to make Higgs precision physics without assuming the validity of the Standard Model. While the methodology is well established and various examples of its application exist in several different contexts, we think that a dedicated discussion can be useful to better clarify some specific points (see also Ref. [57] for a recent discussion). As an illustrative though important example, we will consider the calculation of the Higgs partial decay widths, and show how the corrections from dimension-6 operators can be incorporated in a consistent way. As a by-product of our analysis and to better demonstrate its applicability, in a companion paper [9] we will present a modified version of the program HDECAY

RG evolution of the Wilson coefficients
Let us discuss the issue of the renormalization and RG evolution of the Wilson coefficients first. As done in the previous sections, we will assume that the Higgs boson is part of an parameter α SM /4π. This is in analogy with the renormalization of the pion effective Lagrangian in the chiral limit, see Ref. [58]. It thus follows that the RG equation is linear and homogeneous in thec i , and different operators with the same quantum numbers will in general mix with each other. At leading order in α SM , with α SM = α em , α 2 , α s , respectively, in the case of electromagnetic, weak and QCD corrections, one has where γ ij is the leading-order coefficient of the anomalous dimension. Some elements of the anomalous dimension matrix γ where α 2 has been defined in Eq. (3.43). It is well known that this RG running is associated with the IR contribution to the 3 parameter, and the same coefficient γ where N c = 3 is a color factor. In this case the RG correction is of order (λ 2 ψ /16π 2 ) log(M/µ) compared to the UV value of the coefficients, as one can immediately verify by using the estimates (2.9).
Loops of EW gauge fields give corrections which are suppressed by a weak loop factor (g 2 /16π 2 ), and the associated RG evolution is therefore generically small. An important exception is the case in which the Wilson coefficient has a value suppressed at the scale M . of Fig. 2, renormalizesc T and gives Although small, such a low-energy value ofc T has a strong impact on the EW precision tests performed at LEP [61]. 18 On the other hand, it is too small to be observable through a measurement of the Higgs couplings at the LHC. A similar renormalization ofc T also follows from loops of SM fermions through the insertion ofc Hψ , as illustrated by diagram (b) of Fig. 2. The explicit calculation for the case of a composite right-and left-handed top quark was performed for example in Ref. [62]. Naively, the effect goes like and is of order (y ψ /g ) 2 (λ ψ /g * ) 2 compared to the one from loops of hypercharge. 19 18 For example,c T (m Z ) ∼ 10 −3 forc H (M ) ∼ 0.1. 19 Notice that in case of a sizable fermion coupling λ ψ , a numerically larger contribution toc T comes from fermionic loops with two insertions ofc Hψ . The corresponding diagram is quadratically divergent, so that it gives a threshold correction toc T at the scale M , but does not contribute to its running. An explicit calculation can be found in Ref. [62] for the case of a composite top quark. Naively the effect is of order ∆c T ∼ N c (v/f ) 2 (λ ψ /16π 2 )(λ ψ /g * ) 2 , and can be numerically large. For example, if both t L and t R couple with the same strength λ t L = λ t R ∼ √ g * y t to the new dynamics, then it follows ∆c T ∼ N c (v/f ) 2 (y 2 t /16π 2 ).
In general, although small, the RG evolution of the Wilson coefficients due to EW loops must be properly taken into account in order to precisely match the experimental results obtained at low energy with the theory predictions at high energy. This is even more true in the case of QCD loop corrections, which can be large and will affect the coefficients of the dimension-6 operators with quarks and gluon fields. 20 The effect of the running of the Wilson coefficients can be easily incorporated in programs for the automatic calculation of production cross sections and decay rates by using the effective Lagrangian (2.1) and identifying the coefficients appearing there as their values at the relevant low-energy scale.

Decay rates at the loop level with the effective Lagrangian
In addition to the short-distance effects discussed above, which are parametrized in terms of the evolution of the coefficients of local operators, one-loop diagrams also lead to longdistance corrections to the observables under consideration. Specifically, while short-distance effects are related to the divergent terms, the long-distance contributions correspond to the finite parts and are defined in a given renormalization scheme. In general, the decay amplitude can be expanded as follows: 21 higher-order operators. 20 Notice that g 2 scg is not renormalized at one-loop by QCD corrections. This follows from the RGinvariance of the operator (β(g s )/g s )G µν G µν which contributes to the trace of the energy-momentum tensor [63][64][65]. See also the recent discussion in Ref. [59]. 21  according to the discussion of Section 3, we can quantify the various effects encoded by ∆A 0 as follows: Wc i (m h ), i = HW, HB, γ, g, ψW, ψB, ψG , insertion of the effective vertices has not been computed yet, but we can easily estimate its size: where the dots denote the subleading terms due to the other operators. The terms shown in Eq. (4.65) arise from the same 1-loop diagrams that give the SM amplitude A SM 1 , where each of the Higgs couplings gets shifted byc H ,c u andc 6 . By neglecting the unknown ∆A 1 one is omitting terms of order (v 2 /f 2 )(α 2 /4π), that is, of the same size of the tree-level contribution due to the operator O HW , see Eq. (4.63), since E = m h ≈ m W . This latter contribution can be easily computed and it is included in the formula of the decay rate to W W (and similarly that of O HW and O HB to ZZ is also included) implemented in the program eHDECAY discussed in Ref. [9]. The addition of the tree-level correction from O HW where Γ SM 0 (W ( * ) W * ) denotes the tree-level SM decay rate. For simplicity, we have not shown terms involving powers of E 2 /M 2 among the neglected contributions, since for E = As mentioned, this formula incorporates while the one-loop EW diagrams featuring one effective vertex give a correction of order where Γ SM 0 (qq) is the SM tree-level rate and κ QCD encodes the QCD corrections. This formula includes the leading O(v 2 /f 2 ), O(α 2 /4π) and QCD corrections. Mixed electroweak and QCD corrections can also be included by assuming that they factorize, as the nonfactorizable terms are known to be small. Compared to the decay rate into W W , Eq. (4.69) apparently does not include corrections of order m 2 h /M 2 . While there is indeed no operator whose contribution starts at that order, such corrections can arise from subleading contributions toc H andc ψ . For example, the tree-level exchange of heavy fermions can lead to a 23 The same remark as in footnote 21 applies.
wave-function renormalization of the SM ones, which can be re-expressed in our notation as a contribution toc ψ of order λ 2 ψ v 2 /M 2 . A similar resummation of the QCD corrections also works for the decay h → gg. In this case the SM tree-level amplitude vanishes, A SM O g contributes at tree-level, As discussed in Section 2 (see Eq. (2.10)), the above estimate is suppressed by an additional factor (g 2 G /g 2 * ) in the case of a NG Higgs boson. At the one-loop level one has Thus, the one-loop effect of O H and O u is expected to be as important as the tree-level one from O g , and even larger if the Higgs is a NG boson, as discussed in Section 3.3. This is in fact not surprising, sincec g arises at the 1-loop level in minimally coupled theories, whilē c H andc u can be generated at tree level. The contribution from the dipole operator O tG is suppressed by a factor y 2 t /16π 2 compared to that from O H and O u , as expected from the fact thatc tG is generated at the 1-loop level in minimally coupled theories. For this reason it can be neglected. It should be noted that without a complete computation of the NLO EW corrections of order (α 2 /4π)(v 2 /f 2 ), the LHC data on Higgs physics are not sensitive to the range of values ofc tG expected using the naive estimate (2.9) with (v/f ) 2 ∼ 0.1.
Furthermore, we stress that in order to distinguish the effect of O tG from that of O g , the tth channel should be measured [51] (single top production in association with the Higgs could also provide complementary information [66]). Also in this case, there are no operators giving m 2 h /M 2 corrections, although these terms will in general appear as subleading contributions toc g ,c H andc u , as discussed above. It is well known that higher-order α s corrections are large, so they must be included consistently in our perturbative expansion. This can be done easily in the approximation m h 2m t , which is reasonably accurate for m h = 125 GeV. In such a limit one can integrate out the top quark and match its one-loop contribution to that of the local operator O g . Then it trivially follows that the QCD corrections associated to the virtual exchange and real emissions of gluons and light quarks below the scale m t factorize in the rate, the multiplicative factor being the same for both the top quark and New Physics terms. By approximating A A SM 1 +A SM 2 +∆A 0 +∆A 1 , one arrives at the following formula for the h → gg decay rate: where Γ SM 1 (gg) is the 1-loop SM decay width. The factor c ef f includes all the dependence on m t and accounts for virtual QCD corrections to A SM 1 above that scale, while κ sof t parametrizes the soft radiative effects. By using Eq. The contributions to the decay h → γγ follow a similar pattern as for h → gg. At tree level: At one loop: Finally, the estimate of the corrections to h → γZ is the following: (4.77) In this case the 1-loop electroweak corrections are not known in the SM, so that the formula for the decay rate reads: of the code is given in Ref. [9], where more explicit formulas for each of the Higgs partial widths are provided.

Discussion
The Ref. [74]). This means that at tree-level the deviations of the Higgs-gauge boson couplings are generated by dimension-8 operators [75], whilec H arises only through loop effects and is naively of order (m 2 W /M 2 )(α 2 /4π). At the same time, the couplings to up-and down-type quarks read, respectively, For moderately large tan β this impliesc d ∼ m 2 Z /m 2 H , whilec u is further suppressed by a factor ∼ 1/ tan 2 β (see for example Refs. [76,77] and the recent discussion in Ref. [78]). A pattern with small values ofc H ,c W ,c B andc u but with a ∼ 15% enhancement of the Higgs coupling to down-type quarks due toc d , for example, would be indicative of the MSSM with large tan β and the additional Higgs bosons around 300 GeV. Generic two-Higgs doublet models lead to a similar pattern of couplings, while models where the Higgs boson mixes with a scalar that is singlet under the SM gauge group can generatec H at the tree level. In the MSSM, loops of light stops or staus as well as charginos can also give sizable contributions to the effective couplings of the light Higgs boson to photons and gluons, withc g ,c γ satisfying the naive estimates (2.9). For example, loops of stops lead toc g ∼ (g 2 * /16π 2 )(m 2 W /m 2 t ), where g * = y t or A t /mt. This situation has to be contrasted with the case of strongly coupled theories. There, our power counting (2.9) singles outc H ,c u,d as the dominant Wilson coefficients (c 6 controls only the Higgs self-interaction and measuring it at the LHC will be challenging), whilec W andc B are suppressed by the ratio (g/g * ) 2 . Furthermore, a composite Higgs boson can be naturally light if it is the pseudo Nambu-Goldstone boson associated to the dynamical breaking of a global symmetry of the strong dynamics. This implies that the coefficientsc g andc γ will also be suppressed by a factor (g G /g * ) 2 , where g G is some weak spurion breaking the Goldstone symmetry. The modifications in the gluon-fusion production cross section and in the decay rate to photons are thus controlled byc H andc u .
The harvest of data collected by the LHC certainly calls for a definite theoretical framework to describe the Higgs-like resonance and compute production and decay rates accurately in perturbation theory without restricting to the SM hypothesis. Effective Lagrangians are one of the tools at our disposal to achieve this goal. Elaborating on the operator classification of Ref. [4], we estimated the present bounds on the Wilson coefficients and provided accurate expressions for the Higgs decay rates including various effects that were previously omitted in the literature. Assuming that the observed Higgs-like resonance is a spin-0 and CP-even particle, we discussed two general formulations of the effective Lagrangian, one of which relies on the linear realization of SU (2) L × U (1) Y at high energies. One of the questions that can be addressed by considering these two parametrizations is whether the theory of New Physics flows to the SM in the infrared, that is, whether the Higgs-like resonance is part of an EW doublet. If all the Higgs signal strengths measured at the LHC converge towards the SM prediction, it would be a very suggestive indication that indeed the Higgs boson combines together with the longitudinal components of the W and Z to form an EW doublet, since any other alternative requires some tuning to fake the SM rates. On the other hand, the doublet nature of the Higgs boson would be less obvious to establish if the signal strengths exhibit deviations from their SM predictions (but note that some deviations in the signal strenghts could unambiguously indicate that the Higgs boson is not part of a doublet, this is in particular the case if a large breaking of the custiodial symmetry is observed in conflict with the strong bound already existing from EW precision data). We have pointed out that, if the EWSB dynamics is custodially symmetric, it is not possible to test whether the Higgs boson is part of a doublet by means of single-Higgs processes alone.
A direct proof can come only from processes with multi-Higgs bosons in the final states [56], which are however challenging to study at the LHC. Precisely establishing the CP nature of the Higgs boson is another question that also requires accurate computations. If there is little doubt that the observed resonance has a large CP-even component, the possibility of a small mixing with a CP-odd component remains alive, and dedicated analyses will have to be performed to bound the mixing angle between the two components. To this aim too, an effective Lagrangian including the CP-odd operators listed in Appendix C provides the theoretical framework where this question can be addressed quantitatively.
The absence so far of direct signals of New Physics at the LHC indicates that the road to unveil the origin of the electroweak symmetry breaking might be long and go through precision analyses rather than copious production of new particles. For such a task, the well established technology of effective field theories is the most powerful and general tool we have to analyze the Higgs data and put them into a coherent picture together with the existing experimental information without assuming the validity of the Standard Model. There is still time for the LHC to disprove this pessimistic eventuality by reporting the discovery of new light particles or large shifts in some of the Higgs couplings. It is clear, however, that if the New Physics continues to remain elusive, a precise investigation of the Higgs properties will become the most urgent programme in high-energy physics both for the experimental and the theoretical community.

A SM Lagrangian: notations and conventions
In this Appendix, we collect the conventions used throughout this paper. The field content where the hypercharge is defined as Y = Q − T 3L , and i = 1, 2, 3 is a flavor index. The action of the gauge group is fully characterized by the conventions used to define the covariant derivative. For instance, for the left-handed quark doublet, we have where λ a , a = 1 . . . 8, and σ i , i = 1 . . . 3, are the usual Gell-Mann and Pauli matrices. Accordingly, the gauge-field strengths are defined as where f abc are the SU (3) structure constants.
The Yukawa interactions of the up-type quarks involve the Higgs charge-conjugate doublet defined as The renormalizable Lagrangian of the SM thus reads:

B Electroweak Chiral Lagrangian in non-unitary gauge
We report here the expression of the EW chiral Lagrangian valid in a generic gauge and in the most general case in which the SU (2) L × U (1) Y is non-linearly realized. For simplicity, we will restrict to the case in which the EWSB dynamics has a custodial invariance. The scalar h is assumed to be CP-even and a singlet of the custodial symmetry, and does not necessarily belong to an SU (2) L doublet. The Lagrangian can be expanded in terms with an increasing number of derivatives where L 0 contains the kinetic terms of the SU (3) c × SU (2) L × U (1) Y gauge fields and of the SM fermions, L EW SB describes the sector responsible for EWSB, and V (h) is the potential for h [36]: where σ a are the Pauli matrices. SU (2) L × U (1) Y (local) transformations read as and the covariant derivative is defined by At the level of two derivatives one has [36]: The dots stand for terms which have two or more h fields or do not lead to cubic vertices, see Refs. [54,55] for the complete list of bosonic operators in L (4) (B.95) 24 The operator This is in fact the basis adopted in Ref. [54]. The above discussion shows explicitly that every operator in Eq. with the EW quantum numbers of hypercharge has been recently discussed in Ref. [55].
Since the choice of quantum numbers of the spurions is model-dependent (and in fact the strongest effects are expected to arise from the breaking due to the top quark, rather than hypercharge), we do not report here any particular list of operators, and prefer to refer to the existing literature for further details.  where the dual field strengths are defined asF µν = 1 2 µνρσ F ρσ for F = W, B, G ( is the totally antisymmetric tensor normalized to 0123 = 1). Furthermore, the coefficients of the operators involving fermions will be in general complex numbers.

C Relaxing the CP-even hypothesis
In the case of the effective chiral Lagrangian with SU (2) L × U (1) Y non-linearly realized, there are four additional operators, to be added to those of Eq. (B.93) Table 1  Finally, it should also be noted that when the CP-invariance assumption in the Higgs sector is relaxed, the couplings c u,d,l are allowed to take some complex values.

D Current bounds on dimension-6 operators
In this Appendix we explain how we derived the bounds on the coefficients of the dimension-6 operators reported in Section 2.1. For a given observable we construct a likelihood for the coefficientsc i as follows: where O exp ± ∆O exp is the experimental value of the observable, O SM denotes its SM prediction and δO(c i ) is the correction due to the effective operators. If several observables constrain the same coefficientsc i , the global likelihood is constructed by multiplying those of each observable. We include the theoretical uncertainty on the SM prediction by integrating over a nuisance parameter whose distribution is appropriately chosen. We then quote the bound on a given coefficient by marginalizing over the remaining ones.
Let us consider for example the bounds of Eqs. (2.14) and (2.15). To derive them we used the EW fit performed in Ref. [22] by the GFitter collaboration, and constructed a likelihood for the various coefficients by computing their contributions to the Z-pole observables. For the latter, we used the SM predictions and experimental inputs reported in Table 1 of Ref. [22], treating the uncertainties on the SM predictions as normally distributed. We For the limits of Eqs. (2.11) and (2.12) we have used the fit on S and T performed in Ref. [22], by marginalizing on one parameter to extract the bound on the other.
To derive Eq. (2.17) we have used the theoretical predictions of the EDM of the neutron and mercury given in Ref. [25] in terms of the dipole moments of the quarks (see Eqs. (2.12), (3.65) and (3.71) of Ref. [25]), and the experimental results for these observables given respectively in Ref. [80] and Ref. [81]. We included the theoretical errors by assuming that they are uniformly distributed within the stated intervals. Only two linear combinations of the coefficientsc i can be constrained in this way, since two are the observables at disposal: The limits of Eq. (2.22) have been obtained from the experimental measurements of the electron [31] and muon [29] anomalous magnetic moments and their SM predictions (taken respectively from Ref. [32] and Refs. [29,30]). In this case we have included the theoretical errors by assuming that they are normally distributed. All the remaining bounds reported in Section 2.1, namely those of Eqs. (2.19)-(2.21) and Eq. (2.23) have been obtained by simply translating into our notation the results given in the references quoted in the text.