Effective Lagrangian for a light Higgs-like scalar
Roberto Contino, Margherita Ghezzi, Christophe Grojean, Margarete Muhlleitner, Michael Spira
CCERN-PH-TH/2013-047KA-TP-06-2013PSI-PR-13-04
Effective Lagrangian for a light Higgs-like scalar
Roberto Contino a , Margherita Ghezzi a , Christophe Grojean b,c ,Margarete M¨uhlleitner d and Michael Spira e a Dipartimento di Fisica, Universit`a di Roma La Sapienza and INFN, Roma, Italy b ICREA at IFAE, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain c Theory Division, Physics Department, CERN, Geneva, Switzerland d Institute for Theoretical Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany e Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
Abstract
We reconsider the effective Lagrangian that describes a light Higgs-like boson and bet-ter clarify a few issues which were not exhaustively addressed in the previous literature.In particular we highlight the strategy to determine whether the dynamics responsi-ble for the electroweak symmetry breaking is weakly or strongly interacting. We alsodiscuss how the effective Lagrangian can be implemented into automatic tools for thecalculation of Higgs decay rates and production cross sections. a r X i v : . [ h e p - ph ] M a r Introduction
The exploration of the weak scale has marked an important step forward with the discoveryby the ATLAS [1] and CMS [2] collaborations of a boson with mass m h (cid:39)
125 GeV, whoseproduction cross section and decay rates are compatible with those predicted for the Higgsboson of the Standard Model (SM). At the same time, no hint of the existence of additionalnew particles has emerged yet, which might shed light on the origin of the electroweak sym-metry breaking (EWSB). One is thus faced with the problem of which is the best strategy todescribe the properties and investigate the nature of the new boson h , beyond the frameworkof the Standard Model. In absence of a direct observation of new states, our ignorance of theEWSB 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 (cid:29) m h , and is based on an expansion in the number of fields and derivatives [3]. Thedetailed form of the effective Lagrangian depends on which assumptions are made. Consider-ing that the observation made by the LHC experiments is in remarkable agreement with theSM prediction, although within the current limited experimental precision, it is reasonableto assume that h is a CP-even scalar that forms an SU (2) L doublet together with the lon-gitudinal polarizations of the W and Z , so that the SU (2) L × U (1) Y electroweak symmetryis linearly realized at high energies. Under these assumptions the effective Lagrangian canbe expanded into a sum of operators with increasing dimensionality, where the leading NPeffects are given by dimension-6 operators.The parametrization of the deviations of the Higgs couplings in terms of higher-dimensionoperators started more than two decades ago. The experimental observation of the Higgsboson, however, calls for a more detailed analysis. First, a compilation of a complete andupdated list of constraints on the various Wilson coefficients is in need. Second, the ratherprecise estimation of the Higgs mass below the gauge boson thresholds necessitates a carefulcomputation including off-shell effects that have not been incorporated up-to-now when theSM Lagrangian is supplemented by higher-dimensional operators. It is the purpose of thispaper to perform such an updated analysis. We will also discuss in detail the implicationsof the custodial symmetry on the generalized Higgs couplings and clarify a few other issueswhich were not exhaustively addressed in the previous literature, like for example the connec-1ion 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 whenhigher-order effects are included in a consistent way, and we will develop a strategy to thisend.The paper is structured as follows. In Section 2 we review the construction of the effectiveLagrangian for a light Higgs doublet. By means of a naive power counting we estimate thecoefficients of the various operators and review the most important bounds set on them bypresent experimental results on electroweak (EW) and flavor observables. Focusing on Higgsphysics, we then discuss in Section 3 the relative effect of the various operators on physicalobservables. Such an analysis, first proposed in Ref. [4], will allow us to identify whichoperators can probe the Higgs coupling strength to the new states and which instead aresensitive only to the mass scale M . This is of key importance to distinguish between weakly-coupled UV completions of the Standard Model, like Supersymmetric (SUSY) theories, andtheories where the EW symmetry is broken by a new strongly-interacting dynamics whichforms the Higgs boson as a bound state [5–7,4]. These are the two most compelling scenariosput forward to solve the hierarchy problem of the Standard Model. We conclude the sectionby 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 genericCP-even scalar h and discuss the implications of custodial invariance. Section 4 is devotedto 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 usefulformulas and give further details on the construction of the effective Lagrangian. The detailsof 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 Lagrangiancan be implemented into automatic tools for the computation of physical quantities likeHiggs production cross sections and decay rates, we have written eHDECAY , a modifiedversion 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]. eHDECAY is available at the following URL: 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 aminimal and complete list of operators was finally provided in Ref. [13]. As recently discussedin Ref. [4], a convenient basis of operators relevant for Higgs physics, assuming that the Higgsis a CP-even weak doublet (this assumption will be relaxed in Appendix C) and the baryonand lepton numbers are conserved, is the following: L = L SM + (cid:88) i ¯ c i O i ≡ L SM + ∆ L SILH + ∆ L F + ∆ L F (2.1)with∆ L SILH = ¯ c H v ∂ µ (cid:0) H † H (cid:1) ∂ µ (cid:0) H † H (cid:1) + ¯ c T v (cid:16) H † ←→ D µ H (cid:17)(cid:16) H † ←→ D µ H (cid:17) − ¯ c λv (cid:0) H † H (cid:1) + (cid:16)(cid:16) ¯ c u v y u H † H ¯ q L H c u R + ¯ c d v y d H † H ¯ q L Hd R + ¯ c l v y l H † H ¯ L L Hl R (cid:17) + h . c . (cid:17) + i ¯ c W g m W (cid:16) H † σ i ←→ D µ H (cid:17) ( D ν W µν ) i + i ¯ c B g (cid:48) m W (cid:16) H † ←→ D µ H (cid:17) ( ∂ ν B µν )+ i ¯ c HW gm W ( D µ H ) † σ i ( D ν H ) W iµν + i ¯ c HB g (cid:48) m W ( D µ H ) † ( D ν H ) B µν + ¯ c γ g (cid:48) m W H † HB µν B µν + ¯ c g g S m W H † HG aµν G aµν , (2.2)∆ L F = i ¯ c Hq v (¯ q L γ µ q L ) (cid:0) H † ←→ D µ H (cid:1) + i ¯ c (cid:48) Hq v (cid:0) ¯ q L γ µ σ i q L (cid:1) (cid:0) H † σ i ←→ D µ H (cid:1) + i ¯ c Hu v (¯ u R γ µ u R ) (cid:0) H † ←→ D µ H (cid:1) + i ¯ c Hd v (cid:0) ¯ d R γ µ d R (cid:1) (cid:0) H † ←→ D µ H (cid:1) + (cid:18) i ¯ c Hud v (¯ u R γ µ d R ) (cid:0) H c † ←→ D µ H (cid:1) + h . c . (cid:19) + i ¯ c HL v (cid:0) ¯ L L γ µ L L (cid:1) (cid:0) H † ←→ D µ H (cid:1) + i ¯ c (cid:48) HL v (cid:0) ¯ L L γ µ σ i L L (cid:1) (cid:0) H † σ i ←→ D µ H (cid:1) + i ¯ c Hl v (cid:0) ¯ l R γ µ l R (cid:1) (cid:0) H † ←→ D µ H (cid:1) , (2.3)3 L F = ¯ c uB g (cid:48) m W y u ¯ q L H c σ µν u R B µν + ¯ c uW gm W y u ¯ q L σ i H c σ µν u R W iµν + ¯ c uG g S m W y u ¯ q L H c σ µν λ a u R G aµν + ¯ c dB g (cid:48) m W y d ¯ q L Hσ µν d R B µν + ¯ c dW gm W y d ¯ q L σ i Hσ µν d R W iµν + ¯ c dG g S m W y d ¯ q L Hσ µν λ a d R G aµν + ¯ c lB g (cid:48) m W y l ¯ L L Hσ µν l R B µν + ¯ c lW gm W y l ¯ L L σ i Hσ µν l R W iµν + h . c . (2.4)The SM Lagrangian L SM and our convention for the covariant derivatives and the gaugefield strengths are reported for completeness in Appendix A. In particular, λ is the Higgsquartic coupling and the weak scale at tree level is defined to be v ≡ √ G F ) / = 246 GeV . (2.5)By iH † ←→ D µ H we denote the Hermitian derivative iH † ( D µ H ) − i ( D µ H ) † H , and σ µν ≡ i [ γ µ , γ ν ] / y u,d,l and the Wilson coefficients ¯ c i in Eq. (2.3) are matrices in flavorspace, and a sum over flavors has been left understood. Note that the assumption of aCP-even Higgs implies that the coefficients ¯ c u , ¯ c d and ¯ c l are real. As specified in Eq. (2.1),we will denote as O i the dimension-6 operator whose coefficient is proportional to ¯ c i .Our higher-dimensional Lagrangian, which is supposed to capture the leading New Physicseffects, counts 12 (∆ L SILH ) + 8 (∆ L F ) + 8 (∆ L F ) = 28 operators. Five extra bosonic oper-ators, ¯ c W g m W (cid:15) ijk W i νµ W j ρν W k µρ , ¯ c G g S m W f abc G a νµ G b ρν G c µρ , ¯ c W m W ( D µ W µν ) i ( D ρ W ρν ) i , ¯ c B m W ( ∂ µ B µν ) ( ∂ ρ B ρν ) , ¯ c G m W ( D µ G µν ) a ( D ρ G ρν ) a , (2.6)which affect the gauge-boson propagators and self-interactions but with no effect on Higgsphysics, should also be added to complete the operator basis, as well as 22 four-Fermi baryon-number-conserving operators. A comparison with Ref. [13] shows that two of our operators Notice that the last three operators in Eq. (2.6) can be rewritten in favor of three additional independentfour-Fermi operators, as in the basis of Ref. [13]. The coefficients ¯ c W , ¯ c B contribute respectively to the W and Y parameters defined in Ref. [14]. L F are equivalent to pure oblique corrections parametrized by the operators O T , O W and O B : O YH Ψ ≡ (cid:88) ψ Y ψ O Hψ ∼ O T , O B and O (cid:48) Hq + O (cid:48) HL ∼ O W , (2.7)where the sum runs over all fermion representations, ψ = q L , u R , d R , L L , l R , whose hyper-charge has been denoted as Y ψ . These two linear combinations have then to be excludedfrom ∆ L F , and we end up with exactly 53 linearly-independent operators as in Ref. [13]. Any other dimension-six operator can be obtained from these 53 operators by using theequations of motion, or equivalently by performing appropriate field redefinitions. Even though our basis (2.2)–(2.4) is equivalent to the one proposed in Ref. [13], weadvocate that it is more appropriate for Higgs physics for at least three reasons [4]: i) Genericmodels of New Physics generate a contribution to the oblique ˆ S parameter [16, 14] at tree-level, which in the basis of Ref. [13] would have to be encoded in the two fermionic operators O YHψ and O (cid:48) Hq + O (cid:48) HL even in the absence of direct couplings between the SM fermions andthe New Physics sector. There is an advantage in describing the oblique corrections in termsof the operators in (2.2) rather than in terms of the operators with fermionic currents, whichgenerate vertex corrections and modify the Fermi constant. ii) The basis (2.1) isolates thecontributions to the decays h → γγ (from O γ ) and h → γZ (from O γ and O HW − O HB ) thatoccur only at the radiative level in minimally coupled theories. iii) Our basis of operatorsis more appropriate to establish the nature of the Higgs boson and determine the strengthof its interactions. For example, as we shall explain momentarily, if the Higgs boson is apseudo Nambu-Goldstone boson (pNGB) the coefficient of the operator O γ , hence the rate For completeness we collect in Appendix C also the extra 6 bosonic operators of dimension-six that areCP-odd. In particular, the following identities hold: g m W H † H W iµν W i µν ≡ O W W = O W − O B + O HB − O HW + 14 O γ gg (cid:48) m W H † σ i H W iµν B µν ≡ O W B = O B − O HB − O γ . (2.8) → γγ , is suppressed, while in the basis of Ref. [13] this reflects into a cancellation in thelinear combination 4¯ c γ + (¯ c W W − ¯ c W B ) (cf. footnote 4).While a complete classification of the operators is essential, having a power counting toestimate their impact on physical observables, hence their relative importance, is equallycrucial. In this sense a simple yet consequential observation was made in Ref. [4]: whenexpanding the effective Lagrangian in the number of fields and derivatives, any additionalpower of H is suppressed by a factor g ∗ /M ≡ /f , where g ∗ ≤ π denotes the couplingstrength of the Higgs boson to New Physics states and M is their overall mass scale; anyadditional derivative instead costs a factor 1 /M . If the light Higgs boson is a compositestate of the dynamics at the scale M , it is natural to expect g ∗ (cid:29)
1, hence f (cid:28) M , whichimplies that operators with extra powers of H give the leading corrections to low-energyobservables. On the other hand, in weakly-coupled completions of the Standard Modelwhere g ∗ ∼ g , all operators with the same dimension can be equally important. A properanalysis of the experimental results through the language of the effective Lagrangian can thusgive indication on whether the dynamics at the origin of electroweak symmetry breaking isweakly or strongly interacting. According to the power counting of Ref. [4], one naivelyestimates ( ψ = u, d, l, q, L ) ¯ c H , ¯ c T , ¯ c , ¯ c ψ ∼ O (cid:18) v f (cid:19) , ¯ c W , ¯ c B ∼ O (cid:18) m W M (cid:19) , ¯ c HW , ¯ c HB , ¯ c γ , ¯ c g ∼ O (cid:18) m W π f (cid:19) ¯ c Hψ , ¯ c (cid:48) Hψ ∼ O (cid:18) λ ψ g ∗ v f (cid:19) , ¯ c Hud ∼ O (cid:18) λ u λ d g ∗ v f (cid:19) , ¯ c ψW , ¯ c ψB , ¯ c ψG ∼ O (cid:18) m W π f (cid:19) , (2.9)where λ ψ denotes the coupling of a generic SM fermion ψ to the new dynamics. It should bestressed that these estimates are valid at the UV scale M , at which the effective Lagrangianis matched onto explicit models. Renormalization effects between M and the EW scale mixoperators with the same quantum numbers, and give in general subdominant corrections tothe coefficients. We shall comment on these renormalization effects in Section 4. Noticethat the estimates of ¯ c W,B , ¯ c Hψ , ¯ c (cid:48) Hψ and ¯ c T apply when these coefficients are generated at Notice that our normalization differs from the one of Ref. [4], and it is more convenient than the latterfor a model-independent implementation of Eq. (2.2) in a computer program. The factor multiplying eachoperator in the effective Lagrangian has been conveniently defined such that the dependence on M and g ∗ is fully encoded in the dimensionless coefficients ¯ c i . R -parity in SUSY theories, can force the leading corrections to arise at the1-loop level.Equation (2.9) suggests that in the case of a strongly-interacting light Higgs boson (SILH)the leading New Physics effects in Higgs observables are parametrized by the operators O H,T, ,ψ , and, if the SM fermions couple strongly to the new dynamics, by the fermionicoperators of Eq. (2.3) [4]. Notice that, compared to the naive counting, ¯ c HW,HB,g,γ aresuppressed by an additional factor ( g ∗ / π ). This is because the corresponding operatorscontribute to the coupling of on-shell photons and gluons to neutral particles and modify thegyromagnetic ratio of the W , and are thus generated only at the loop level in a minimallycoupled theory. Similarly, the dipole operators of Eq. (2.4) are generated at the loop-levelonly, hence their estimates have an extra loop factor.A special and phenomenologically motivated case is represented by theories where theHiggs doublet is a composite Nambu–Goldstone (NG) boson of a spontaneously-broken sym-metry G → H of the strong dynamics [5–7,4]. For these models the scale f must be identifiedwith the decay constant associated with the spontaneous breaking, and the naive estimateof the Wilson coefficients ¯ c i is modified by the request of invariance under G in the limit ofvanishing explicit breaking. At the level of dimension-6 operators, O γ , O g , O , O u,d,l and thedipole operators of Eq. (2.4) violate the shift symmetry H i → H i + ζ i ( ζ i = const. ) thatis included as part of the G / H transformations. This means that they cannot be generatedin absence of an explicit breaking of the global symmetry. It follows, in particular, thatthe naive estimates of the operators O γ and O g carry in this case an additional suppressionfactor [4], ¯ c γ , ¯ c g ∼ O (cid:18) m W π f (cid:19) × g (cid:54) G g ∗ , (2.10)where g (cid:54) G denotes any weak coupling that breaks the Goldstone symmetry (one of the SMweak couplings in minimal models, i.e. the SM gauge couplings or the Yukawa couplings).The operators O , O ψ , O ψG , O ψW , O ψB have been defined so that their prefactor alreadyincludes one spurion coupling, precisely the Higgs quartic coupling λ in O , and the Yukawacoupling y ψ in the other operators – indeed, both these couplings vanish for an exact NGboson. The estimates of the corresponding coefficients ¯ c , ¯ c ψ , ¯ c ψG , ¯ c ψW , ¯ c ψB are thus not7odified.In writing Eq. (2.2) we have assumed that each of the operators O u,d,l is flavor-aligned withthe corresponding fermion mass term, as required in order to avoid large Flavor-ChangingNeutral Currents (FCNC) mediated by the tree-level exchange of the Higgs boson (see forexample Ref. [17] for a natural way to obtain this alignment). This implies one coefficientfor the up-type quarks (¯ c u ), one for down-type quarks (¯ c d ), and one for the charged leptons(¯ c l ), i.e. the ¯ c u,d,l are proportional to the identity matrix in flavor space. It is useful to review some of the most important constraints on the coefficients ¯ c i that followfrom current experimental results, such as electroweak precision tests, flavor data and low-energy precision measurements. For simplicity, we focus on the bounds on flavor-conservingoperators, keeping in mind that they can come also from flavor-changing processes. For adiscussion of the bounds on flavor-violating operators see for example the recent review ofRef. [18] as well as Ref. [19].Among the strongest bounds are those on operators that modify the vector-boson self-energies. The operator O T , for example, violates the custodial symmetry [20] and contributesto the EW parameter (cid:15) [21]. From the EW fit performed in Ref. [22], it follows, with 95%probability, ∆ (cid:15) ≡ ∆ ρ = ¯ c T ( m Z ) , − . × − < ¯ c T ( m Z ) < . × − . (2.11)Such a stringent bound can be more naturally satisfied by assuming that the dynamics atthe 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 thecustodial symmetry, as due for example to the Yukawa or hypercharge couplings. We willdiscuss these renormalization effects in more detail in Section 4. Notice that all the otherdimension-6 operators in the effective Lagrangian are (formally) custodially symmetric andtheir coefficients will not be suppressed at the scale M . The electroweak precision tests More precisely, for all the other operators the only violation of the custodial symmetry comes from O W + O B [4], since this linear combination contributes to theparameter (cid:15) [21]. With 95% probability, one has [22]:∆ (cid:15) = ¯ c W ( m Z ) + ¯ c B ( m Z ) , − . × − < ¯ c W ( m Z ) + ¯ c B ( m Z ) < . × − . (2.12)From the tree-level estimate of ¯ c W,B reported in Eq. (2.9), and assuming an approximatecustodial invariance to suppress ¯ c T as explained above, it follows that Eqs. (2.11) and (2.12)set a lower bound M (cid:38) a few TeV. This bound is quite robust and can be avoided only inweakly-coupled UV completions where an extra symmetry protection suppresses the leadingcontribution to ¯ c W,B by an additional loop factor. Notable examples are SUSY theories with R -parity.The fermionic operators in Eq. (2.3) are strongly constrained by Z -pole measurements,as they modify the couplings of the Z to quarks and leptons: δg Lψ g Lψ = 12 ( − ¯ c H Ψ + 2 T L ¯ c (cid:48) H Ψ ) T L − Q sin θ W , δg Rψ g Rψ = 12 ¯ c Hψ Q sin θ W , (2.13)where T L and Q are respectively the SU (2) L and electric charges of the fermion ψ , andΨ = { L, q } is the SU (2) L doublet to which ψ L belongs. We used the results of Ref. [22] toperform a fit on the coefficients ¯ c Hψ , ¯ c H Ψ , ¯ c (cid:48) H Ψ . The details of our analysis can be found inAppendix D (see also Ref. [23]). In the case of light quarks ( u, d, s ) we find the followingbounds − . < ¯ c Hq < . , − . < ¯ c (cid:48) Hq < . , − . < ¯ c Hq < . , − . < ¯ c (cid:48) Hq < . , − . < ¯ c Hu < . , − . < ¯ c Hd < . , − . < ¯ c Hs < . , (2.14) the explicit breaking due to the gauging of hypercharge. As such, this breaking is external to the EWSBdynamics, since it comes from the weak gauging of its global symmetries. Formal invariance of the operatorscan be restored by uplifting the hypercharge gauge field to a whole triplet of SU (2) R . The top Yukawacoupling is another source of explicit custodial breaking. c, b ) gives − . < ¯ c HL + ¯ c (cid:48) HL < . , − . < ¯ c HL − ¯ c (cid:48) HL < . , − . < ¯ c Hl < . , − . < ¯ c Hq − ¯ c (cid:48) Hq < . , − . < ¯ c Hc < . , − . < ¯ c Hq + ¯ c (cid:48) Hq < . , − . < ¯ c Hb < − . . (2.15)All the above bounds have 95% probability and by the various coefficients we mean theirvalues at the scale m Z . The weakest constraint is that on the operator O Hb , which modifiesthe coupling of b R to the Z boson. The operator involving two right-handed top quarks, O Ht , is unconstrained by EW data, but it is also not relevant for the Higgs decays andwill be neglected in the following. The coefficient ¯ c 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: − . × − < ¯ c Htb ( m W ) < . × − . (2.16)For a given ( v/f ), the above bounds set a limit on the couplings of the SM fermions to thenew dynamics, see Eq. (2.9). Unless the scale of New Physics is very large, or some specificsymmetry 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 theexception of the top quark.The constraints on the dipole operators of Eq. (2.4) come from the current experimentallimits on electric dipole moments (EDMs) and anomalous magnetic moments. The boundson 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: − . × − < Im(¯ c uB + ¯ c uW ) < . × − , − . × − < Im(¯ c dB − ¯ c dW ) < . × − , − . × − < Im(¯ c uG ) < . × − , − . × − < Im(¯ c dG ) < . × − , (2.17)10here the coefficients are evaluated at the low-energy scale µ ∼ O (1) CP-violating phases these results imply a bound on ( v/f ) atthe level of 10 − . In natural extensions of the SM, such a strong limit clearly points to theneed of a symmetry protection mechanism. For a discussion, see for example Ref. [23] forthe 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 operatorswith top quarks [27]. These come from the experimental limit on the neutron EDM, − . × − < Im(¯ c tG ) < . × − , (2.18)the b → sγ and b → sl + l − rates, − . < Re(¯ c tW + ¯ c tB ) − .
65 Im(¯ c tW + ¯ c tB ) < . , (2.19)and the t ¯ t cross sections measured at the Tevatron and LHC, − . × − < Re(¯ c tG ) < . × − . (2.20)All these bounds have 95% probability and have been derived by making use of the formulasreported in Ref. [27]. It is worth noting that the bounds of Eqs. (2.19) and (2.20) are stillabout one order of magnitude weaker than the size of ¯ c tG , ¯ c tW and ¯ c tB expected from thenaive estimate (2.9) with ( v/f ) ∼ .
1. Additional weaker constraints arise from the limitson anomalous top interactions based on top decays and single top production. From theresults of Ref. [28] we find that, with 95% probability: − . < Re(¯ c bW ) < . , − . < Re(¯ c tW ) < . . (2.21)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] The coefficients are evaluated at the following scales: µ = m t (Eqs. (2.18) and (2.20)), µ = m W (Eq. (2.19)). − . × − < Re(¯ c eB − ¯ c eW ) < . × − , . × − < Re(¯ c µB − ¯ c µW ) < . × − , (2.22) − . × − < Im(¯ c eB − ¯ c eW ) < . × − , − . < Im(¯ c µB − ¯ c µW ) < . , (2.23)where the coefficients are evaluated at the relevant low-energy scale. Notice that the non-vanishing value of Re(¯ c µB − ¯ c µW ) follows from the known ∼ . σ anomaly in the ( g −
2) ofthe muon (see Ref. [29] for an updated review). Among the bounds of Eqs. (2.21), (2.22),(2.23) only those on Im(¯ c eB − ¯ c eW ) and Re(¯ c µB − ¯ c µW ) have the sensitivity to probe thevalues naively expected for these coefficients as reported in Eq. (2.9). In particular, the firstone sets an upper bound on ( v/f ) of order 10 − for an O (1) CP phase. While the Lagrangian ∆ L = ∆ L SILH + ∆ L F + ∆ L F is completely general, the basis ofoperators of Eqs. (2.2)–(2.4) is particularly useful to characterize the interactions of theHiggs sector. In fact, as already anticipated, one of the main results of Ref. [4] is thatof identifying which operators, hence which observables, are sensitive to the strength ofthe Higgs interactions, rather than merely to the value of the New Physics scale M . Inwhat follows we will discuss this point in greater detail and, starting from the analysis ofRefs. [4, 35], we will try to highlight a possible strategy to determine whether the dynamicsbehind the electroweak breaking is weak or strong. Our analysis will be based on the naiveestimates of the Wilson coefficients at the matching scale. In the next Section, we will discusshow the running from the matching scale to the weak scale affects these estimates. Let us start by considering the effects of the operators O H , O T , O u,d,l and O : they modifythe tree-level couplings of the Higgs boson to fermions, vector bosons and to itself. In the12nitary gauge and upon canonical normalization of the Higgs kinetic term, the Lagrangianreads [36] L = 12 ∂ µ h ∂ µ h − m h h − c (cid:18) m h v (cid:19) h + . . . + m W W + µ W − µ (cid:18) c W hv + . . . (cid:19) + 12 m Z Z µ Z µ (cid:18) c Z hv + . . . (cid:19) − (cid:88) ψ = u,d,l m ψ ( i ) ¯ ψ ( i ) ψ ( i ) (cid:18) c ψ hv + . . . (cid:19) + . . . (3.24)where the Higgs couplings c i = W,Z,ψ, , have been defined such that c i = 1 in the SM, and v is defined by Eq. (2.5). Their expressions as functions of the coefficients of the effectiveLagrangian (2.2) are given in Table 1. The shifts from the SM value are of order δc i ∼ g ∗ v M = v f . (3.25)Hence, measuring the Higgs couplings probes the strength of its interactions to the newdynamics. Notice that the effective description given by ∆ L neglects higher powers of ( H/f ),and is thus valid only if the shifts in the Higgs couplings are small: δc i ∼ ( v/f ) (cid:28)
1. If theHiggs doublet is the NG boson of a spontaneously broken symmetry
G → H , on the otherhand, it is possible to resum all powers of (
H/f ) by making use of the invariance under (non-linear) G transformations. Such an improved effective Lagrangian thus relies only on theexpansion in the number of derivatives. For example, in models based on the SO (5) /SO (4)coset [7, 37] the couplings of the Higgs boson to W and Z are predicted to be c W = c Z ≡ c V = √ − ξ , where ξ ≡ ( v/f ) . The couplings to fermions, on the other hand, are notuniquely fixed by the choice of the coset, but depend on how the SM fermions are coupled tothe strong dynamics. The last two columns of Table 1 report the predictions of the MinimalComposite Higgs Model MCHM4 [7] and MCHM5 [37], where the SM fermions couple linearlyto composite operators transforming as the spinorial and fundamental representations of SO (5), respectively. For simplicity, the predictions are derived by including only the effectsof the Higgs non-linearities, and neglecting those from the heavy resonances, hence onlythe coefficients c V , c ψ and c are non-vanishing. The models MCHM4 and MCHM5 will beconsidered as benchmarks in the rest of this work.13iggs couplings ∆ L SILH
MCHM4 MCHM5 c W − ¯ c H / √ − ξ √ − ξc Z − ¯ c H / − ¯ c T √ − ξ √ − ξc ψ ( ψ = u, d, l ) 1 − (¯ c H / c ψ ) √ − ξ − ξ √ − ξc c − c H / √ − ξ − ξ √ − ξc gg α s /α ) ¯ c g c γγ θ W ¯ c γ c Zγ (cid:0) ¯ c HB − ¯ c HW − c γ sin θ W (cid:1) tan θ W c W W − c HW c ZZ − (cid:0) ¯ c HW + ¯ c HB tan θ W − c γ tan θ W sin θ W (cid:1) c W ∂W − c W + ¯ c HW ) 0 0 c Z∂Z − c W + ¯ c HW ) − c B + ¯ c HB ) tan θ W c Z∂γ c B + ¯ c HB − ¯ c W − ¯ c HW ) tan θ W The second column reports the values of the Higgs couplings c i defined in Eq. (3.46) interms of the coefficients ¯ c i of the effective Lagrangian ∆ L SILH . The last two columns show thepredictions of the MCHM4 and MCHM5 models in terms of ξ = ( v/f ) ; the effects of the heavyresonances have been neglected for simplicity, so that only the couplings c W,Z,ψ, are non-vanishing.The auxiliary parameter α is defined by Eq. (3.43). Note that the previous version of this papercontains an erroneous factor 2 in the dependence of c Z on ¯ c T .
14n general, a shift of the tree-level Higgs couplings of order ( v/f ) implies that the theorygets strongly coupled at energies ∼ πf , unless new weakly-coupled physics states set in toregulate the energy growth of the scattering amplitudes. The dominant effect comes fromthe energy growth of the V L V L → V L V L ( V = W ± , Z ) scattering amplitudes, which becomenon-perturbative at the scale Λ s = 4 πv/ (cid:112) | ¯ c H | . A modified coupling to the top quark leadsinstead to strong V L V L → t ¯ t scattering at energies of order Λ s = 16 π v / ( m t (cid:112) | ¯ c u + ¯ c H | ).The scale of New Physics is thus required to lie below, or at, such ultimate range of validityof the effective theory: M (cid:46) Λ s . The operators O W , O B can be generated at tree-level by the exchange of heavy particles, forexample heavy spin-1 states. In the unitary gauge they are written in terms of the followingthree operators ( D µ W + µν ) W − ν h , ( ∂ µ Z µν ) Z ν h , ( ∂ µ γ µν ) Z ν h (3.26)plus terms with zero or two Higgs fields. The fact that there are three possible operatorsin the unitary gauge indicates that their coefficients are related by one identity if the Higgsboson belongs to an SU (2) doublet, see Eq. (3.48). We will discuss this point in greaterdetail in Section 3.6.It is easy to see that O W , O B give corrections to the tree-level Higgs couplings andgenerate quartic interactions with one vector boson and two SM fermions that contributeto the three-body decays h → V V ∗ → V ψ ¯ ψ . Indeed, by making use of the equations ofmotion, iD µ W iµν = g H † σ i ←→ D ν H − ig ¯ ψ σ i γ ν ψ , i∂ µ B µν = g (cid:48) H † ←→ D ν H − ig (cid:48) ¯ ψY γ ν ψ , (3.27) 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 sincethey break the U (1) em local symmetry. We thank Riccardo Rattazzi for pointing this out to us. For simplicity we have left a sum over all fermion representations ψ understood in Eq. (3.27). O W and O B as O W = − O H + 4 v ( H † H ) | D µ H | + O (cid:48) Hq + O (cid:48) HL (3.28) O B = 2 tan θ W (cid:0) − O T + O YH Ψ (cid:1) , (3.29)where the linear combination O YH Ψ has been defined in Eq. (2.7). Upon the field redefinition H → H − c W ( H † H ) H/v , the operator ( H † H ) | D µ H | can be rewritten in terms of thosein Eq. (2.2). Specifically, Eq. (3.28) becomes: O W = − O H + 2 (( O u + O d + O l ) + h.c. ) + − O + O (cid:48) Hq + O (cid:48) HL . (3.31)From the estimates of ¯ c W , ¯ c B and ¯ c H , ¯ c T , ¯ c ψ , ¯ c in Eq. (2.9) one can see that the shifts to thetree-level Higgs couplings due to O W , O B are of order ( m W /M ) , hence subdominant in thecase of a strongly interacting Higgs boson. Notice that the couplings of the Higgs boson to W and Z get different shifts from O B (since ∆¯ c T (cid:54) = 0). In practice, the constraint (2.12)bounds this custodial-symmetry breaking effect down to an unobservable level, unless somefine tuning is in place in the combination ¯ c W + ¯ c B so that ¯ c B can be large. Notice thatdespite the operator O T is generated after using the equations of motion, its contributionto ∆ (cid:15) (corresponding to a non-vanishing ˆ T parameter [16, 14]) is exactly canceled by thevertex correction implied by the linear combination of fermionic operators which is alsogenerated. This is of course expected, since O W , O B only contribute to (cid:15) , and not to (cid:15) .In general, the contribution of O W , O B to inclusive observables, in particular to the Higgsdecay rates, is of order ( m W /M ): δ Γ( h → V V )Γ( h → V V ) (cid:12)(cid:12)(cid:12)(cid:12) O W ,O B ∼ O (cid:18) m W M (cid:19) , (3.32) By means of Eqs. (3.29) and (3.31) it is thus always possible to remove O W and O B provided thecoefficients of the other operators are shifted as follows: ¯ c i → ¯ c i + ∆¯ c i , with∆¯ c H = − c W , ∆¯ c T = − θ W ¯ c B , ∆¯ c = − c W , ∆¯ c ψ = 2 ¯ c W ∆¯ c (cid:48) Hq = ∆¯ c (cid:48) HL = ¯ c W c Hq = 32 ∆¯ c Hu = − c Hd = − c HL = − ∆¯ c Hl = − θ W ¯ c B . (3.30) See for example Eq. (9.10) of Ref. [15].
V V = W ( ∗ ) W ∗ , Z ( ∗ ) Z ∗ , Z ( ∗ ) γ, γγ . This implies that these operators aresensitive only to the value of the scale of New Physics M , and do not probe the couplingstrength g ∗ . From the quantitative side, the constraint (2.12) suggests that their effects ininclusive 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: Γ( h → W ( ∗ ) W ∗ )Γ( h → W ( ∗ ) W ∗ ) SM (cid:39) . c W , Γ( h → Z ( ∗ ) Z ∗ )Γ( h → Z ( ∗ ) Z ∗ ) SM (cid:39) . (cid:0) ¯ c W + tan θ W ¯ c B (cid:1) . (3.33)Notice that despite its custodial invariance, the operator O W affects in a slightly differentway the decay of the Higgs boson into W W and ZZ , due to the fact that at least one of thetwo final vector bosons is off-shell. At the one-loop level O W also contributes to the Higgsdecays into Zγ and γγ (while O B does not). We find:Γ( h → Zγ )Γ( h → Zγ ) SM (cid:39) . c W , Γ( h → γγ )Γ( h → γγ ) SM (cid:39) . c W , (3.35)which agree with Eqs. (82) and (83) of Ref. [4]. For ¯ c W,B ∼ − the above approximateformulas imply corrections too small to be observed at the LHC. On the other hand, onecould try to take advantage of the different predictions in terms of angular and invariantmass distributions which are implied by the dimension-6 operators compared to the tree-levelSM prediction. The most promising strategy could be in fact that based on the analysis ofthe angular distributions of the final fermions [38–40]. In the ideal case in which one is ableto kill completely the SM tree-level contribution by means of appropriate kinematic cuts, 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 insteadnot included, since their effect on the numerical prefactor appearing in front of the coefficients ¯ c i is of order( v /f )( α / π ) and thus beyond the accuracy of our computation. See Ref. [9] for more details. It is easy to check that for m h > m Z and on-shell decays one has:Γ( h → W W )Γ( h → W W ) SM (cid:39) c W , Γ( h → ZZ )Γ( h → ZZ ) SM (cid:39) (cid:0) ¯ c W + tan θ W ¯ c B (cid:1) . (3.34)These formulas coincide with those of Eqs. (79)–(80) of Ref. [4], which are thus valid only for on-shell decays. The easiest way to compute the one-loop contribution of O W to the Zγ and γγ rates is by using Eq. (3.28)to rewrite this operator in terms of the others. Among the operators generated in this way, only O H givesa contribution. Notice that if Eq. (3.31) is used instead, one has to take into account also the contributionof ( O u + O d + O l ) and the shift to the Fermi constant induced by O (cid:48) Hq + O (cid:48) HL . d Γ( h → V V ) d Ω (cid:46) (cid:18) d Γ( h → V V ) d Ω (cid:19) SM (cid:46) c W,B π g , (3.36)which might leave room for observable effects even for ¯ c W,B ∼ O (10 − ). Clearly, a moreprecise assessment of the efficiency of such a strategy requires a dedicated analysis [41]. Let us now focus on the operators O HW , O HB , O γ and O g , which are generated at the one-looplevel. In the unitary gauge, O HW,HB,γ are rewritten in terms of W + µν W − µν h , Z µν Z µν h , γ µν γ µν h , Z µν γ µν h (3.37)plus other terms with zero or two Higgs fields. Since the coefficients of the above fouroperators are functions of ¯ c HW , ¯ c HB and ¯ c 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 ZZ inclusive rates is of order ( V V = W W, ZZ ) δ Γ( h → V V )Γ( h → V V ) (cid:12)(cid:12)(cid:12)(cid:12) O γ ,O HW ,O HB ∼ O (cid:18) m W π f (cid:19) . (3.38)Although such an effect depends on the Higgs interaction strength, it is suppressed comparedto Eq. (3.32) by a loop factor. We find that the following approximate formulas hold Γ( h → W ( ∗ ) W ∗ )Γ( h → W ( ∗ ) W ∗ ) SM (cid:39) . c HW , Γ( h → Z ( ∗ ) Z ∗ )Γ( h → Z ( ∗ ) Z ∗ ) SM (cid:39) . (cid:0) ¯ c HW + tan θ W ¯ c HB (cid:1) − .
26 ¯ c γ . (3.40) For m h > m Z and on-shell decays, we find insteadΓ( h → W W )Γ( h → W W ) SM (cid:39) c HW , Γ( h → ZZ )Γ( h → ZZ ) SM (cid:39) c HW + tan θ W ¯ c HB ) −
16 tan θ W sin θ W ¯ c γ . (3.39)Comparing with the analog formulas in Eqs. (79) and (80) of Ref. [4], we find that in these latter there is amissing factor 2 and the term proportional to ¯ c γ was not included either. Notice also that the effect of theoff-shellness of the gauge bosons is rather large, as one can see by comparing Eq. (3.39) with Eq. (3.40). c HB and ¯ c γ explicitly violates the custodial symmetry andthus differentiates W W from ZZ , the different numerical factor multiplying ¯ c HW in the twoformulas above is due to the off-shellness of at least one of the two vector bosons, similarly toEq. (3.33). Although there is currently no stringent bound on the coefficients ¯ c HW,HB,γ , theestimate (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 invariantmass distributions of these decays can potentially uncover the effect of New Physics. Inparticular, an estimate similar to that of Eq. (3.36) can be derived also for ¯ c HW,HB,γ .The processes h → γγ , h → Zγ and h → gg (or equivalently gg → h ) can in principletest the Higgs interaction strengths much more powerfully, since they arise at the one-looplevel in the SM. Naively one expects: δ Γ( h → gg, γγ, Zγ )Γ( h → gg, γγ, Zγ ) (cid:12)(cid:12)(cid:12)(cid:12) O g ,O γ ,O HW ,O HB ∼ O (cid:18) v f (cid:19) . (3.41)We find that the following approximate formulas hold to good accuracy for small ¯ c i ’s:Γ( h → gg )Γ( h → gg ) SM (cid:39) . c g πα Γ( h → γγ )Γ( h → γγ ) SM (cid:39) − .
54 ¯ c γ πα em , Γ( h → Zγ )Γ( h → Zγ ) SM (cid:39) . (cid:0) ¯ c HW − ¯ c HB + 8 ¯ c γ sin θ W (cid:1) π √ α α em , (3.42)where we have conveniently defined α ≡ √ G F m W π , (3.43)and by α em we indicate the value of the running electromagnetic coupling α em ( q = 0) inthe Thomson limit. If the Higgs boson is a NG boson, the coefficients ¯ c g and ¯ c γ are furthersuppressed by a factor ( g (cid:54) G /g ∗ ) , see Eq. (2.10), where g (cid:54) G is a weak coupling. This impliesthat in this class of theories the corrections to Γ( h → γγ ) and Γ( h → gg ) depend only onthe scale of New Physics and not on the Higgs interaction strength. In fact, in the caseof minimal models with linear couplings, like for example the MCHM4 and MCHM5, thelow energy theorem [42, 43] implies that the leading contribution to the γγ and gg decay19ates from the virtual exchange of heavy fermions is additionally suppressed [44–47] due to acancellation between the effect parametrized by ¯ c g,γ and the one that follows from the shiftin the top Yukawa coupling due to ¯ c u and ¯ c H (see Ref. [46] for an interesting exception). Ingeneral, 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 theeffect from the modified tree-level Higgs couplings. The fermionic operators in ∆ L F are sensitive to the strength of the couplings of the Higgsboson and of the SM fermions to the new dynamics. They lead to contact corrections to thethree-body decays h → V V ∗ → V ψψ which are naively of order δ Γ( h → V ¯ ψψ )Γ( h → V ¯ ψψ ) ∼ O (cid:18) v f λ ψ g ∗ (cid:19) . (3.44)Compared to the corrections from O W and O B , the effect of the fermionic operators ispotentially enhanced by a factor ( λ ψ /g ). In practice, the possibility of large fermioniccouplings λ ψ is strongly constrained by LEP, see Eqs. (2.14)-(2.16). Scenarios in which alarge degree of compositeness of either the left- or right-handed quarks is not ruled outare generically those in which the corresponding operators in ∆ L F are not generated asdue to some protecting symmetry (see for example Refs. [23, 48, 49]). Large corrections tothe inclusive rate of the three-body decays h → V ¯ ψψ from ∆ L F are thus excluded, whilethe possibility of detecting the effects of these operators through the analysis of differentialdistributions should be explored, similarly to what has been discussed for O W and O B .Among the dipole operators in ∆ L F , those with light fermions are already stronglyconstrained by current precision data, but potentially sizable effects could still come fromthe operators involving the top quark. For example, the contribution of O tG to gg → h , gg → t ¯ t , gg → t ¯ th is of order E / (16 π f ), where E is the energy scale relevant in theprocess. More in detail δσ ( gg → h ) σ ( gg → h ) ∼ ˆ c tG , δσ ( gg → t ¯ t ) σ ( gg → t ¯ t ) ∼ ˆ c tG √ sm t , δσ ( gg → t ¯ th ) σ ( gg → t ¯ th ) ∼ ˆ c tG sm t , (3.45)where we have defined ˆ c tG ≡ Re(¯ c tG ) ( m t /m W ) ∼ m t / (16 π f ) (cid:39) × − ( v /f ). Noticethat the experimental limit on the neutron EDM puts an upper bound on the imaginary20art of ˆ c tG at the 10 − level, see Eq. (2.18), which indicates that this is currently the mostsensitive experiment on Im(¯ c tG ). Some mechanism is however required to suppress the imag-inary parts of the dipole operators involving light fermions, in order to satisfy the stringentconstraints of Eq. (2.17). By the same mechanism also Im(¯ c tG ) could be suppressed, so thatthe 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 of ˆ c tG it follows that the most sensitive process isperhaps gg → t ¯ t , in particular the events at large invariant mass, although a precision largerthan the one currently achieved is required to constrain ( v/f ). To this aim, the analysis ofdifferential distributions and spin correlations could be a successful strategy [50, 27]. TheNP contribution to the process gg → t ¯ th can in principle get the largest enhancement froma 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 cou-plings obtained at the Tevatron and the LHC, although their sensitivity on NP is expectedto be much smaller by naive estimate. The operator O tW , in particular, gives the largesteffect and generates the anomalous coupling g R ( g/m W )¯ b L σ µν W − µν t R [28]. Naively one ex-pects g R = (4 m t /m W ) ¯ c tW ∼ m t m W / (16 π f ) = 1 . × − ( v/f ) , an effect too small to beobserved even for f of order v . 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] L = 12 ∂ µ h ∂ µ h − m h h − c (cid:18) m h v (cid:19) h − (cid:88) ψ = u,d,l m ψ ( i ) ¯ ψ ( i ) ψ ( i ) (cid:18) c ψ hv + . . . (cid:19) + m W W + µ W − µ (cid:18) c W hv + . . . (cid:19) + 12 m Z Z µ Z µ (cid:18) c Z hv + . . . (cid:19) + . . . + (cid:16) c W W W + µν W − µν + c ZZ Z µν Z µν + c Zγ Z µν γ µν + c γγ γ µν γ µν + c gg G aµν G aµν (cid:17) hv + (cid:16) c W ∂W (cid:0) W − ν D µ W + µν + h.c. (cid:1) + c Z∂Z Z ν ∂ µ Z µν + c Z∂γ Z ν ∂ µ γ µν (cid:17) hv + . . . (3.46)where, we recall, v is defined in Eq. (2.5). We have shown only terms involving up tothree bosonic fields, and we have omitted in particular those involving fermions that follow21rom ∆ L F + ∆ L F . Their form can be easily derived from Eqs. (2.3) and (2.4). Therelations between the couplings appearing in Eq. (3.46) and the coefficients of the dimension-6 operators in Eq. (2.2) are reported in Table 1. It is worth noting that the same Lagrangian(3.46) applies also to the case in which the electroweak symmetry SU (2) L × U (1) Y is non-linearly realized and h is a generic CP-even scalar, singlet of the custodial symmetry, notnecessarily connected with the EW symmetry breaking. Indeed, each of the terms in (3.46),being invariant under local U (1) em transformations, can be dressed up with the Nambu-Goldstone bosons that are eaten to form the longitudinal W and Z polarizations and mademanifestly SU (2) L × U (1) Y gauge invariant [52] (see also Ref. [53]). The explicit expressionin such a basis has been given in Refs. [54,55] at the level of four-derivative operators. In thissense the effective Lagrangian (3.46) is a generic tool to understand the origin of the newlydiscovered boson and the role it plays in the electroweak symmetry breaking dynamics. Itis valid for arbitrary values of the couplings c i appearing in Eq. (3.46), and it can be usedto make computations of observable quantities at a given order in an expansion in E/M and in α SM / π , where by the latter we indicate the generic SM loop expansion parameter.That is in full analogy with other well-known effective theories, see Ref. [3]. It should bestressed that, according to a well established methodology and similarly to Eq. (2.2), inthis effective Lagrangian all quantum fluctuations associated to short-length modes (high-energy modes) have already been considered and are parametrized by local operators withan increasing number of derivatives, while quantum fluctuations (loop diagrams) involvingthe light modes still have to be taken into account. For instance, top loops will give anadditional contribution to the on-shell h -gluon-gluon coupling. While Eq. (3.46) is general,the effective Lagrangian (2.2) assumes that h is part of an SU (2) L doublet and further relieson the expansion in powers of H/f . As such, it is valid only in the limit of small deviationsof the Higgs couplings from their SM values and up to corrections of order O ( v /f ). Another difference between the non-linear Lagrangian (3.46) and the SILH Lagrangian (2.2)is that the first one contains two more free parameters. This means that there are tworelations among the couplings of Eq. (3.46) which hold at the level of dimension-6 operators22f the Higgs is part of a doublet. As noticed in Sections (3.2) and (3.3), the first identityrelates c W W , c ZZ , c Zγ and c γγ , while the second relates c W ∂W , c Z∂Z and c Z∂γ . They read: c W W − c ZZ cos θ W = c Zγ sin 2 θ W + c γγ sin θ W (3.47) c W ∂W − c Z∂Z cos θ W = c Z∂γ θ W . (3.48)In fact both identities are not special to the case in which the Higgs is a doublet, but are ageneral consequence of custodial symmetry. This latter is accidental in the SILH Lagrangianif one restricts to the operators that lead to derivative couplings of the Higgs to vector bosons.Starting at the dimension-8 order, it is possible to write custodial-breaking operators thatlead to couplings that violate the relations (3.47) and (3.48). For instance¯ c W W g m W v (cid:0) H † W aµν σ a H (cid:1) (cid:0) H † W b µν σ b H (cid:1) + i ¯ c W gv m W (cid:0) H † σ a H (cid:1) ( D µ W µν ) a (cid:16) H † ←→ D ν H (cid:17) (3.49)gives rise to c Z∂Z = − c W , c Z∂γ = − θ W ¯ c W ,c ZZ = 8 cos θ W ¯ c W W , c Zγ = 4 sin 2 θ W ¯ c W W , c γγ = 8 sin θ W ¯ c W W , (3.50)and the relations (3.47) and (3.48) are not fulfilled. A third relation holds on the non-derivative couplings c W and c Z if one assumes thatcustodial symmetry is an invariance of the Lagrangian (2.2), so that ¯ c T = 0; it reads: c W = c Z . (3.51)As said above, while all three identities (3.47), (3.48) and (3.51) are a consequence of custo-dial symmetry, the first two are accidental at the level of dimension-6 operators if the Higgsis part of a doublet.To show that Eqs. (3.47), (3.48) and (3.51) follow from custodial invariance, let us con-sider the case in which the EWSB dynamics has a global SU (2) L × SU (2) R symmetry, andimagine to fully gauge this group by enlarging the hypercharge to a whole triplet of SU (2) R .In this case the diagonal custodial SU (2) V is exact even though g (cid:48) (cid:54) = g . The left and right The two operators in (3.49) give rise to the oblique parameter ˆ U , see for instance Ref. [14]: ˆ U = − ¯ c W − c HW while ˆ S = ¯ c HW . SU (2) L × SU (2) R and the interactions amongtwo gauge fields and the Higgs boson are fully characterized in momentum space by threeform factors:(Γ LL ) µνij ( p , p ) L iµ L jν h + (Γ LR ) µνij ( p , p ) L iµ R jν h + (Γ RR ) µνij ( p , p ) R iµ R jν h . (3.52)Here p , p are the momenta of the gauge fields and each form factor can be computed interms of a Green function with two conserved currents, Γ µνik = (cid:104) J µi J νk | h (cid:105) . In addition tothe usual massive W and Z bosons, which form a triplet ˆ V iµ of the custodial group, in thiscase there is a whole triplet of massless SU (2) V gauge fields (the photon plus its chargedcompanion), V iµ . The mass eigenstates V µ and ˆ V µ are related to the left and right gauge fieldsthrough a rotation by an angle θ W , where tan θ W = g (cid:48) /g . Their cubic interactions with theHiggs boson are thus characterized by three form factors, which are linear combinations ofthose in Eq. (3.52):Γ V V = sin θ W Γ LL + sin 2 θ W LR + Γ RL ) + cos θ W Γ RR Γ ˆ V V = sin 2 θ W LL + (cid:0) cos θ Γ LR − sin θ Γ RL (cid:1) − sin 2 θ W RR Γ ˆ V ˆ V = cos θ W Γ LL − sin 2 θ W LR + Γ RL ) + sin θ W Γ RR , (3.53)where we have defined Γ µνRL ( p , p ) ≡ Γ νµLR ( p , p ). Notice, in particular, that in this case thesame form factor Γ ˆ V ˆ V 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 switchingoff the unphysical R , µ fields. The interactions of two neutral vector bosons to the Higgs arestill described by the relations of Eq. (3.53), where Γ ZZ = Γ ˆ V ˆ V , Γ γγ = Γ V V and Γ Zγ = Γ ˆ V V .In the charged sector, instead, the W corresponds to a pure left gauge field, since it has nomixing with right-handed ones. This implies that its form factor is given by the last formulaof Eq. (3.53) with θ W = 0, that is: Γ W W = Γ LL . The four physical form factors are linearcombinations of the three defined in Eq. (3.52), and are thus related by one identity:Γ µνW W ( p , p ) − Γ µνZZ ( p , p ) cos θ W = (cid:0) Γ µνZγ ( p , p ) + Γ νµZγ ( p , p ) (cid:1) sin 2 θ W µνγγ ( p , p ) sin θ W . (3.54)24otice that this relation is a consequence of our initial assumption of SU (2) L × SU (2) R invariance of the EWSB dynamics. The custodial SU (2) V is broken in this case only bythe gauging of hypercharge. For g (cid:48) = 0 the custodial symmetry is unbroken and Eq. (3.54)implies Γ W W = Γ ZZ . It is straightforward to derive the relations (3.47), (3.48) and (3.51)from Eq. (3.54). At quadratic order in the momenta, the form factors can be computed fromthe effective Lagrangian (3.46); one has:Γ µνW W ( p , p ) = 2 m W c W η µν − c W W P µν − c W ∂W ( P µν + P µν )Γ µνZZ ( p , p ) = 2 m Z c Z η µν − c ZZ P µν − c Z∂Z ( P µν + P µν )Γ µνZγ ( p , p ) = − c Zγ P µν − c Z∂γ P µν Γ µνγγ ( p , p ) = − c γγ P µν , (3.55)where we have defined P µν ≡ η µν p − p µ p ν , P µν ≡ η µν p − p µ p ν and P µν ≡ η µν p · p − p ν p µ .This is in fact the most general decomposition which follows at the O ( p ) level for an on-shellHiggs boson by assuming CP invariance and requiring that: i) the Γ W W , Γ ZZ and Γ γγ formfactors are symmetric under the exchange { p , µ } ↔ { p , ν } ; ii) the Γ γγ and Γ Zγ form factorssatisfy the Ward identities implied by U (1) em local invariance: p µ Γ µνγγ ( p , p ) = 0 = p ν Γ µνγγ ( p , p ) , p ν Γ µνZγ ( p , p ) = 0 . (3.56)Additional structures proportional to p µ and p ν can be omitted since they give vanishingcontributions both when the vector bosons are on-shell and when they decay into a pair offermions by coupling to the corresponding conserved current. Inserting Eq. (3.55) into (3.54)one then obtains the identities (3.47), (3.48) and (3.51).From the above discussion it follows that if custodial symmetry is an invariance of theEWSB dynamics, the effective Lagrangians (3.46) and (2.2) have the same number of freeparameters, in terms of which all observables can be computed. This is true also if oneconsiders the fermionic operators (for a Higgs doublet these are listed in Eqs. (2.3) and(2.4)), as long as one focuses on terms with one Higgs boson. This means that by usingsingle-Higgs processes alone, one cannot distinguish the case in which the Higgs boson ispart of a doublet from the more general situation. The only possible strategy to this aimis exploiting the connection among processes with zero, one and two Higgs bosons which25s implied by the Lagrangian (2.1) at O ( v /f ) and does not hold in the case of the moregeneral non-linear Lagrangian. As a consequence of such connection, the bounds that EWand flavor data set on operators with zero Higgs fields severely constrain the size of the NPeffects in Higgs processes, as discussed in Section (2.1). If one were to find that single-Higgsprocesses violate these constraints, this would be an indication that the Higgs is not part ofa doublet. Furthermore, processes with double Higgs boson production can be predicted toa certain extent in terms of single-Higgs couplings, and can thus be used to probe the natureof the Higgs boson [56]. 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 assumingthe validity of the Standard Model. While the methodology is well established and variousexamples of its application exist in several different contexts, we think that a dedicateddiscussion can be useful to better clarify some specific points (see also Ref. [57] for a recentdiscussion). As an illustrative though important example, we will consider the calculationof the Higgs partial decay widths, and show how the corrections from dimension-6 operatorscan be incorporated in a consistent way. As a by-product of our analysis and to betterdemonstrate its applicability, in a companion paper [9] we will present a modified versionof the program
HDECAY [8] that features a full implementation of the effective Lagrangian∆ L SILH , Eq. (2.2), as well as its generalization to the case of a non-linearly realized EWsymmetry, Eq. (3.46).A first difficulty which arises when using either Eq. (2.1) or (3.46) is the presence of mul-tiple expansion parameters. For generic values of the Higgs couplings c i , the validity of theeffective Lagrangian (3.46) is based on a double perturbative expansion in the SM couplings, α SM / π , and in powers of E/M . The effective Lagrangian (2.1) further assumes ( v/f ) (cid:28) c i = 1 + δc i , with δc i (cid:46) O ( v /f ). All theseexpansion parameters must be properly taken into account when performing calculations.26urthermore, the non-renormalizability of the effective theory implies the presence of addi-tional divergences compared to the SM case which must be absorbed by a renormalizationof the Wilson coefficients of local operators. Let us discuss the issue of the renormalization and RG evolution of the Wilson coefficientsfirst. As done in the previous sections, we will assume that the Higgs boson is part of an SU (2) L doublet and use the Lagrangian (2.1). Since we are only interested in the divergentstructure of the diagrams, it is convenient to work in the limit of unbroken SU (2) L × U (1) Y and compute the Green functions in terms of the Higgs doublet H . The only 1-loop dia-grams which generate additional logarithmic divergences are those featuring one insertionof the effective vertices from dimension-6 operators. By dimensional analysis, further inser-tions of the effective vertices lead to power-divergent contributions to dimension-6 opera-tors (which are irrelevant to determine the RG running) and log-divergent contributions tohigher-dimensional operators. The same counting holds also at higher loop level: the onlylog-divergent contribution to dimension-6 operators comes from diagrams with one inser-tion of the effective couplings, and is thus suppressed by extra powers of the SM expansionparameter α SM / π . This is in analogy with the renormalization of the pion effective La-grangian in the chiral limit, see Ref. [58]. It thus follows that the RG equation is linearand homogeneous in the ¯ c i , and different operators with the same quantum numbers will ingeneral mix with each other. At leading order in α SM , with α SM = α em , α , α s , respectively,in the case of electromagnetic, weak and QCD corrections, one has¯ c i ( µ ) = (cid:18) δ ij + γ (0) ij α SM ( µ )4 π log (cid:16) µM (cid:17)(cid:19) ¯ c j ( M ) , (4.57)where γ (0) ij is the leading-order coefficient of the anomalous dimension. Some elements of theanomalous dimension matrix γ (0) ij have been recently computed in Refs. [59, 60].In the case in which the Higgs boson and possibly the SM quarks (in particular the topand the bottom) are strongly coupled to the new dynamics, the leading RG running effectcomes from loops of these particles and can be as large as ∆¯ c i / ¯ c i ( M ) ∼ ( g ∗ / π ) log( M/µ )or ( λ ψ / π ) log( M/µ ). This must be compared to the effects of order ( g SM / π ) log( M/µ )27 a) (b) Figure 1:
One-loop diagrams relevant for the RG running of ¯ c W and ¯ c B . Dashed, continuousand wiggly lines denote, respectively, a weak doublet H , a fermion and a vector field V = W, B .The symbol ⊗ denotes the insertion of the effective vertex from O H (in diagram ( a )) or O Hψ (indiagram ( b )). from loops of gauge fields. For example, the insertion of ¯ c H in the diagram ( a ) of Fig. 1 leadsto a renormalization of O W + B ≡ O W + O B :¯ c W + B ( µ ) = ¯ c W + B ( M ) − α π log (cid:16) µM (cid:17) ¯ c H ( M ) , (4.58)where α has been defined in Eq. (3.43). It is well known that this RG running is associatedwith the IR contribution to the (cid:15) parameter, and the same coefficient γ (0) W + B,H = − / c H ( M ) ∼ O ( v /f ), ¯ c W,B ( M ) ∼ O ( m W /M ), it follows that the correction to ¯ c W + B fromits RG evolution down to the scale µ is of order ∆¯ c W + B / ¯ c W + B ( M ) ∼ ( g ∗ / π ) log( M/µ )as anticipated. Similarly, the insertion of ¯ c Hψ into a loop of fermions, like in diagram ( b ) ofFig. 1, leads to a renormalization of ¯ c W and ¯ c B :∆¯ c W,B ≈ N c α π log (cid:16) µM (cid:17) ¯ c Hψ ( M ) , (4.59)where N c = 3 is a color factor. In this case the RG correction is of order ( λ ψ / π ) log( M/µ )compared to the UV value of the coefficients, as one can immediately verify by using theestimates (2.9).Loops of EW gauge fields give corrections which are suppressed by a weak loop factor( g / π ), and the associated RG evolution is therefore generically small. An importantexception is the case in which the Wilson coefficient has a value suppressed at the scale M .For example, if the dynamics behind the EW symmetry breaking is custodially invariant, then¯ c T ( M ) = 0. The insertion of ¯ c H into a loop of hypercharge gauge bosons, as in diagram ( a )28 a) (b) Figure 2:
One-loop diagrams relevant for the RG running of ¯ c T . Dashed, continuous and wigglylines denote, respectively, a weak doublet H , a fermion and a hypercharge field B . The symbol ⊗ denotes the insertion of the effective vertex from O H (in diagram ( a )) or O Hψ (in diagram ( b )). of Fig. 2, renormalizes ¯ c T and gives¯ c T ( µ ) = 32 tan θ W α π log (cid:16) µM (cid:17) ¯ c H ( M ) . (4.60)Compared to the naive estimate of Eq. (2.9), ¯ c T ( M ) ∼ O ( v /f ), valid in absence of custo-dial symmetry, the above correction is further suppressed by a factor ( g (cid:48) / π ) log( M/µ ).Although small, such a low-energy value of ¯ c T has a strong impact on the EW precision testsperformed at LEP [61]. On the other hand, it is too small to be observable through ameasurement of the Higgs couplings at the LHC. A similar renormalization of ¯ c T also followsfrom loops of SM fermions through the insertion of ¯ c Hψ , as illustrated by diagram ( b ) ofFig. 2. The explicit calculation for the case of a composite right- and left-handed top quarkwas performed for example in Ref. [62]. Naively, the effect goes like∆¯ c T ≈ N c y ψ π log (cid:16) µM (cid:17) ¯ c Hψ ( M ) , (4.61)and is of order ( y ψ /g (cid:48) ) ( λ ψ /g ∗ ) compared to the one from loops of hypercharge. For example, ¯ c T ( m Z ) ∼ − for ¯ c H ( M ) ∼ . Notice that in case of a sizable fermion coupling λ ψ , a numerically larger contribution to ¯ c T comes fromfermionic loops with two insertions of ¯ c Hψ . The corresponding diagram is quadratically divergent, so thatit gives a threshold correction to ¯ c T at the scale M , but does not contribute to its running. An explicitcalculation 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 ) ( λ ψ / π )( λ ψ /g ∗ ) , and can be numerically large. For example, if both t L and t R couplewith 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 ) ( y t / π ).
29n general, although small, the RG evolution of the Wilson coefficients due to EW loopsmust be properly taken into account in order to precisely match the experimental resultsobtained at low energy with the theory predictions at high energy. This is even more truein the case of QCD loop corrections, which can be large and will affect the coefficients ofthe dimension-6 operators with quarks and gluon fields. The effect of the running ofthe Wilson coefficients can be easily incorporated in programs for the automatic calculationof production cross sections and decay rates by using the effective Lagrangian (2.1) andidentifying the coefficients appearing there as their values at the relevant low-energy scale.
In addition to the short-distance effects discussed above, which are parametrized in termsof the evolution of the coefficients of local operators, one-loop diagrams also lead to long-distance corrections to the observables under consideration. Specifically, while short-distanceeffects are related to the divergent terms, the long-distance contributions correspond tothe finite parts and are defined in a given renormalization scheme. In general, the decayamplitude can be expanded as follows: A = A SM + A SM + ∆ A + ∆ A + . . . (4.62)where A SM ( A SM ) is the tree-level (one-loop) SM amplitude, and ∆ A (∆ A ) is the tree-level (one-loop) contribution from the dimension-6 operators of the effective Lagrangian inEqs. (2.2)–(2.4). The dots denote higher-loop contributions as well as the corrections due tohigher-order operators. Notice that g s ¯ c g is not renormalized at one-loop by QCD corrections. This follows from the RG-invariance of the operator ( β ( g s ) /g s ) G µν G µν which contributes to the trace of the energy-momentum ten-sor [63–65]. See also the recent discussion in Ref. [59]. In the strict sense this equation is valid for the genuine EW corrections only, while for simplicity weinclude the (IR-divergent) virtual QED corrections to the SM amplitude in the same way. The correspondingreal photon radiation contributions to the decay rates are treated in terms of a linear novel contributionto the Higgs coupling for the squared amplitude in order to obtain an infrared finite result. Pure QEDcorrections factorize as QCD corrections in general so that their amplitudes scale with the modified Higgscouplings. However, they cannot be separated from the genuine EW corrections in a simple way. h → W ( ∗ ) W ∗ . In this case the operators that cancontribute at tree-level are O H , O W , O HW , O ψW , O (cid:48) Hψ , as well as O Hud in the case in whichthe off-shell W decays into a pair of quarks. Based on the naive estimates of Eq. (2.9) andaccording to the discussion of Section 3, we can quantify the various effects encoded by ∆ A as follows:∆ A A SM ( W ( ∗ ) W ∗ ) = ˆ c H × O (cid:18) v f (cid:19) + ˆ c W × O (cid:18) E M (cid:19) + ˆ c HW × O (cid:18) E π f (cid:19) + ˆ c Hud × O (cid:18) v f λ u λ d g ∗ (cid:19) + ˆ c (cid:48) Hψ × O (cid:18) v f λ ψ g ∗ (cid:19) + ˆ c ψW × O (cid:18) Em ψ π f (cid:19) . (4.63)Here E = m h is the relevant energy of the process and we have conveniently defined each ofthe O (1) parameters ˆ c i to be equal to ¯ c i ( m h ) divided by its naive estimate in Eq. (2.9):ˆ c i = f v ¯ c i ( m h ) , i = H, T, , ψ, ˆ c i = M m W ¯ c i ( m h ) , i = W, B , ˆ c i = 16 π f m W ¯ c i ( m h ) , i = HW, HB, γ, g, ψW, ψB, ψG , ˆ c i = g ∗ λ ψ f v ¯ c i ( m h ) , ˆ c (cid:48) i = g ∗ λ ψ f v ¯ c (cid:48) i ( m h ) , i = Hψ , ˆ c Hud = g ∗ λ u λ d f v ¯ c Hud ( m h ) . (4.64)When the Higgs boson is pNGB, the two parameters ˆ c g and ˆ c γ are not of order one but arefurther suppressed by a factor g (cid:54) G /g ∗ . From Eq. (4.63) one can see that the contribution ofthe dipole operators O ψW is suppressed by ( m ψ /m h ) compared to that of O HW , while thatof O Hud and O (cid:48) Hψ is expected to be small given the existing constraints on the couplings λ ψ (see the discussion in Section 2.1). The dominant NP contribution thus comes from theterms in the first line of Eq. (4.63), among which the one proportional to ¯ c H is the leadingeffect for g ∗ > g . The 1-loop electroweak amplitude A SM gives a contribution of order A SM /A SM ∼ ( α / π ). We thus see explicitly that ∆ A and A SM encode the NLO correctionsin the three expansion parameters which we are considering: α / π (electroweak expansion), E /M (derivative expansion) and v /f . The contribution due to 1-loop diagrams with oneinsertion of the effective vertices has not been computed yet, but we can easily estimate itssize:∆ A A SM ( W ( ∗ ) W ∗ ) = ˆ c H × O (cid:18) v f α π (cid:19) + ˆ c u × O (cid:18) v f α π (cid:19) + ˆ c × O (cid:18) v f α π (cid:19) + . . . (4.65)31here the dots denote the subleading terms due to the other operators. The terms shownin Eq. (4.65) arise from the same 1-loop diagrams that give the SM amplitude A SM , whereeach of the Higgs couplings gets shifted by ¯ c H , ¯ c u and ¯ c . By neglecting the unknown ∆ A one is omitting terms of order ( v /f )( α / π ), that is, of the same size of the tree-levelcontribution due to the operator O HW , see Eq. (4.63), since E = m h ≈ m W . This lattercontribution can be easily computed and it is included in the formula of the decay rateto W W (and similarly that of O HW and O HB to ZZ is also included) implemented in theprogram eHDECAY discussed in Ref. [9]. The addition of the tree-level correction from O HW is clearly the first step towards a full inclusion of the O [( v /f )( α / π )] corrections, wherethe missing part will have to be computed from 1-loop diagrams featuring one insertion of O H , O u and O . It is worth noting that these diagrams, in general, contain logarithmicdivergences which must be reabsorbed by a renormalization of the Wilson coefficients andcontribute to their RG evolution as explained in the previous section. The finite part is thecontribution to ∆ A which awaits to be computed.By approximating the amplitude as A (cid:39) A SM + A SM + ∆ A one obtains the followingformula for the decay rate: Γ( W ( ∗ ) W ∗ ) = Γ SM ( W ( ∗ ) W ∗ ) (cid:40) | A SM | Re (cid:2)(cid:0) A SM (cid:1) ∗ (cid:0) A SM + ∆ A (cid:1)(cid:3) + O (cid:32)(cid:18) v f (cid:19) , (cid:18) α π v f (cid:19) , (cid:16) α π (cid:17) (cid:33) (cid:41) , (4.66)where Γ SM ( W ( ∗ ) W ∗ ) denotes the tree-level SM decay rate. For simplicity, we have notshown terms involving powers of E /M among the neglected contributions, since for E = m h ≈ m W one has E /M (cid:46) v /f if g ∗ (cid:38) g . As mentioned, this formula incorporatesthe O ( v /f ), O ( α / π ) and O ( m h /M ) corrections (NLO in the perturbative expansion),and can be easily implemented in existing codes for the automatic computation of the decayrate. The inclusion of the O ( m h /M ) tree-level correction due to O W is justified as long as g ∗ < π , since it is parametrically larger than the neglected O [( v /f )( α / π )] terms by afactor (16 π /g ∗ ). Notice that in the limit of large deviations of the Higgs couplings from theirSM values, ( v/f ) ∼ O (1), the neglected terms of O [( v /f )( α / π )] become as important The same remark as in footnote 21 applies.
32s those included through A SM . In order words, a proper inclusion of the EW corrections inthe limit v ∼ f requires a complete 1-loop calculation where each of the diagrams is rescaledby the appropriate coupling factor.A similar discussion applies to the Higgs decay into a pair of fermions, h → ¯ ψψ . In thiscase only O H and O ψ ( ψ = u, d, l ) contribute at tree level,∆ A A SM ( ¯ ψψ ) = (cid:18) ˆ c H c ψ (cid:19) × O (cid:18) v f (cid:19) , (4.67)while the one-loop EW diagrams featuring one effective vertex give a correction of order∆ A A SM ( ¯ ψψ ) = ˆ c H × O (cid:18) v f α π (cid:19) + ˆ c ψ × O (cid:18) v f α π (cid:19) + ˆ c × O (cid:18) v f α π (cid:19) + . . . (4.68)where the dots indicate the subleading terms due to the other operators. The calculationof ∆ A has not been performed yet, while the 1-loop EW corrections are known in the SM, A SM . Their inclusion is thus possible as long as ( v/f ) (cid:28)
1, so that the neglected termsin ∆ A are subleading. The case of QCD radiative corrections is different, since at leadingorder they factorize with respect to the expansion in the number of derivative and fields andcan thus be resummed up to higher orders. In the case of the Higgs decay into a pair ofquarks one can for example approximate A (cid:39) A SM + A SM + ∆ A and obtain the followingformula for the decay rate: Γ(¯ qq ) = Γ SM (¯ qq ) κ QCD (cid:40) | A SM | Re (cid:2)(cid:0) A SM (cid:1) ∗ (cid:0) A SM + ∆ A (cid:1)(cid:3) + O (cid:32)(cid:18) v f (cid:19) , (cid:18) α π v f (cid:19) , (cid:16) α π (cid:17) (cid:33) (cid:41) , (4.69)where Γ SM (¯ qq ) is the SM tree-level rate and κ QCD encodes the QCD corrections. This for-mula includes the leading O ( v /f ), O ( α / π ) and QCD corrections. Mixed electroweakand QCD corrections can also be included by assuming that they factorize, as the non-factorizable 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 h /M . While there is indeed no operatorwhose contribution starts at that order, such corrections can arise from subleading contri-butions to ¯ c H and ¯ c ψ . For example, the tree-level exchange of heavy fermions can lead to a The same remark as in footnote 21 applies. c ψ of order λ ψ v /M .A similar resummation of the QCD corrections also works for the decay h → gg . In thiscase the SM tree-level amplitude vanishes, A SM = 0, while the leading contribution arisesfrom the 1-loop exchange of top quarks. The two-loop EW corrections are known in the SMand give a correction of order A SM /A SM ∼ α / π . Among the dimension-6 operators, only O g contributes at tree-level, ∆ A A SM ( gg ) = ˆ c g × O (cid:18) v f (cid:19) . (4.70)As discussed in Section 2 (see Eq. (2.10)), the above estimate is suppressed by an additionalfactor ( g (cid:54) G /g ∗ ) in the case of a NG Higgs boson. At the one-loop level one has∆ A A SM ( gg ) = (cid:18) ˆ c H c u (cid:19) × O (cid:18) v f (cid:19) + ˆ c tG × O (cid:18) v f y t π (cid:19) . (4.71)Thus, the one-loop effect of O H and O u is expected to be as important as the tree-level onefrom O g , and even larger if the Higgs is a NG boson, as discussed in Section 3.3. This is infact not surprising, since ¯ c g arises at the 1-loop level in minimally coupled theories, while¯ c H and ¯ c u can be generated at tree level. The contribution from the dipole operator O tG issuppressed by a factor y t / π compared to that from O H and O u , as expected from thefact that ¯ c tG is generated at the 1-loop level in minimally coupled theories. For this reasonit can be neglected. It should be noted that without a complete computation of the NLOEW corrections of order ( α / π )( v /f ), the LHC data on Higgs physics are not sensitiveto the range of values of ¯ c tG expected using the naive estimate (2.9) with ( v/f ) ∼ . O tG from that of O g , the t ¯ th channel should be measured [51] (single top production in association with the Higgs couldalso provide complementary information [66]). Also in this case, there are no operators giving m h /M corrections, although these terms will in general appear as subleading contributionsto ¯ c g , ¯ c H and ¯ c u , as discussed above. It is well known that higher-order α s corrections arelarge, so they must be included consistently in our perturbative expansion. This can be doneeasily in the approximation m h (cid:28) m t , which is reasonably accurate for m h = 125 GeV. Insuch a limit one can integrate out the top quark and match its one-loop contribution to thatof the local operator O g . Then it trivially follows that the QCD corrections associated to the34irtual exchange and real emissions of gluons and light quarks below the scale m t factorizein the rate, the multiplicative factor being the same for both the top quark and New Physicsterms. By approximating A (cid:39) A SM + A SM +∆ A +∆ A , one arrives at the following formulafor the h → gg decay rate:Γ( gg ) = Γ SM ( gg ) κ soft (cid:40) c eff + 2 c eff | A SM | Re (cid:2)(cid:0) A SM (cid:1) ∗ (cid:0) A SM c eff + ∆ A + ∆ A c eff (cid:1)(cid:3) + O (cid:32)(cid:18) v f (cid:19) , (cid:18) α π v f (cid:19) , (cid:16) α π (cid:17) (cid:33) (cid:41) , (4.72)where Γ SM ( gg ) is the 1-loop SM decay width. The factor c eff includes all the depen-dence on m t and accounts for virtual QCD corrections to A SM above that scale, while κ soft parametrizes the soft radiative effects. By using Eq. (4.72), the existing four-loopcalculations of c eff [67–70] and κ soft [71–73] allow one to include the QCD corrections upto N LO.The contributions to the decay h → γγ follow a similar pattern as for h → gg . At treelevel: ∆ A A SM ( γγ ) = ˆ c γ × O (cid:18) v f (cid:19) . (4.73)At one loop: ∆ A A SM ( γγ ) = ˆ c H × O (cid:18) v f (cid:19) + ˆ c u × O (cid:18) v f (cid:19) + ˆ c W × O (cid:18) m W M (cid:19) + ˆ c HW × O (cid:18) m W π f (cid:19) + (ˆ c tW + ˆ c tB ) × O (cid:18) v f y t π (cid:19) . (4.74)The 2-loop electroweak corrections have been computed in the SM and can be included for( v /f ) (cid:28)
1, so that unknown O [( v /f )( α / π )] effects arising from 2-loop diagrams withone effective vertex are negligible. From Eq. (4.74) one can see that the 1-loop contributiondue to O HW is of the same order as such neglected terms. The 1-loop correction from O W ,on the contrary, is parametrically larger by a factor (16 π /g ∗ ) and should be included for g ∗ < π . The easiest way to compute it is by rewriting O W in terms of the other operatorsthrough the equations of motions [4], see Eq. (3.31). The 1-loop correction due to thedipole operators is suppressed by a factor y t / π and can be neglected. Approximating35 (cid:39) A SM + A SM + ∆ A + ∆ A one finds:Γ( γγ ) = Γ SM ( γγ ) (cid:40) | A SM | Re (cid:2)(cid:0) A SM (cid:1) ∗ (cid:0) A SM + ∆ A + ∆ A (cid:1)(cid:3) + O (cid:32)(cid:18) v f (cid:19) , (cid:18) α π v f (cid:19) , (cid:16) α π (cid:17) (cid:33) (cid:41) , (4.75)Finally, the estimate of the corrections to h → γZ is the following:∆ A A SM ( Zγ ) = ˆ c γ × O (cid:18) v f (cid:19) + (ˆ c HW − ˆ c HB ) × O (cid:18) v f (cid:19) , (4.76)∆ A A SM ( Zγ ) = ˆ c H × O (cid:18) v f (cid:19) + ˆ c u × O (cid:18) v f (cid:19) + ˆ c W × O (cid:18) m W M (cid:19) + ˆ c HW × O (cid:18) m W π f (cid:19) + ˆ c tW × O (cid:18) v f y t π (cid:19) + ˆ c tB × O (cid:18) v f y t π (cid:19) . (4.77)In this case the 1-loop electroweak corrections are not known in the SM, so that the formulafor the decay rate reads:Γ( Zγ ) = Γ SM ( Zγ ) (cid:40) | A SM | Re (cid:2)(cid:0) A SM (cid:1) ∗ (∆ A + ∆ A ) (cid:3) + O (cid:32)(cid:18) v f (cid:19) , (cid:16) α π (cid:17)(cid:33)(cid:41) , (4.78)where only the contributions from O H , O u and O W should be retained in ∆ A for consistency.Through the above discussion we sketched how the effective Lagrangian can be imple-mented beyond the tree level in the calculation of physical quantities. In the case of theHiggs partial decay widths, in particular, we have seen how the EW and QCD correctionscan be included consistently with the expansion in the number of fields and derivatives. As amore concrete illustration of these considerations, we have written a modified version of theprogram HDECAY , which we dub eHDECAY , where the corrections from all the local operatorsof the effective Lagrangians (2.2) and (3.46) are included at NLO. A detailed descriptionof the code is given in Ref. [9], where more explicit formulas for each of the Higgs partialwidths are provided. 36
Discussion
The discovery of a resonance with a mass around 125 GeV similar to the long-sought StandardModel Higgs boson brings the exploration of the electroweak symmetry breaking sector underquantitative scrutiny. The LHC experiments, together with those at the Tevatron, reportthe signal strengths, i.e. the product of the Higgs production cross section times its decaybranching ratio, for various final state channels. The main task of the community is now tointerpret these data and understand the implications for the theory of New Physics that isexpected to lie beyond the weak scale.The EW oblique parameters provide a bound on the scale of New Physics but do notgive detailed information about the nature of the NP sector. In order to understand how theweak scale is stabilized at the quantum level, i.e. how the hierarchy problem is solved, onecrucial question is whether EW symmetry breaking proceeds by weak or strong dynamics.The direct observation of new degrees of freedom would provide a straightforward answer.But a glimpse of New Physics can also be caught from a dedicated study of the Higgs bosonitself, and in particular from a measurement of its couplings, if a departure from the SMpredictions is ever observed. It is useful to parametrize the deviations from the SM by theeffective Lagrangian of Eq. (2.1). By measuring its Wilson coefficients ¯ c i one can infer whatkind of UV theory completes the SM.If the coupling strength of the Higgs boson to the NP sector is of the order of the SM weakcouplings, g ∗ ≈ g , then our power counting (2.9) shows that the coefficients of the operatorsthat can be generated at tree-level, O H , O u,d,l , O W and O B , are expected to be all of thesame order, m W /M , where M is the typical mass scale of the NP spectrum, unless somespecial selection rule suppresses some of them. It is instructive to examine the predictionsof the archetypal example of weakly-coupled UV completions: the Minimal SupersymmetricStandard Model (MSSM). First, R -parity protects the EW oblique parameters from anytree-level contributions, hence ¯ c W and ¯ c B are of order ( m W /M )( α / π ) and thus small.Second, the couplings of the lightest Higgs boson to the massive gauge bosons are given by c V = sin( β − α ), where α is the rotation angle to diagonalize the CP-even mass matrix andtan β is the ratio of the vacuum expectation values of the two neutral CP-even Higgs bosons.In the decoupling limit, α → β − π/
2, one has c V = 1 + O ( m Z /m H ), where m H is the mass of37he heaviest CP-even scalar (for a general treatment of the decoupling limit see for exampleRef. [74]). This means that at tree-level the deviations of the Higgs-gauge boson couplingsare generated by dimension-8 operators [75], while ¯ c H arises only through loop effects and isnaively of order ( m W /M )( α / π ). At the same time, the couplings to up- and down-typequarks read, respectively, c u = + cos α sin β = 1 + 2 m Z m H cos β cos 2 β + O (cid:18) m Z m H (cid:19) c d = − sin α cos β = 1 − m Z m H sin β cos 2 β + O (cid:18) m Z m H (cid:19) . (5.79)For moderately large tan β this implies ¯ c d ∼ m Z /m H , while ¯ c u is further suppressed by afactor ∼ / tan β (see for example Refs. [76, 77] and the recent discussion in Ref. [78]). Apattern with small values of ¯ c H , ¯ c W , ¯ c B and ¯ c u but with a ∼
15% enhancement of the Higgscoupling to down-type quarks due to ¯ c d , for example, would be indicative of the MSSM withlarge tan β and the additional Higgs bosons around 300 GeV. Generic two-Higgs doubletmodels lead to a similar pattern of couplings, while models where the Higgs boson mixeswith a scalar that is singlet under the SM gauge group can generate ¯ c H at the tree level. In theMSSM, loops of light stops or staus as well as charginos can also give sizable contributions tothe effective couplings of the light Higgs boson to photons and gluons, with ¯ c g , ¯ c γ satisfyingthe naive estimates (2.9). For example, loops of stops lead to ¯ c g ∼ ( g ∗ / π )( m W /m t ),where g ∗ = y t or A t /m ˜ t .This situation has to be contrasted with the case of strongly coupled theories. There, ourpower counting (2.9) singles out ¯ c H , ¯ c u,d as the dominant Wilson coefficients (¯ c controls onlythe Higgs self-interaction and measuring it at the LHC will be challenging), while ¯ c W and ¯ c B are suppressed by the ratio ( g/g ∗ ) . Furthermore, a composite Higgs boson can be naturallylight if it is the pseudo Nambu-Goldstone boson associated to the dynamical breaking of aglobal symmetry of the strong dynamics. This implies that the coefficients ¯ c g and ¯ c γ will alsobe suppressed by a factor ( g (cid:54) G /g ∗ ) , where g (cid:54) G is some weak spurion breaking the Goldstonesymmetry. The modifications in the gluon-fusion production cross section and in the decayrate to photons are thus controlled by ¯ c H and ¯ c u .The harvest of data collected by the LHC certainly calls for a definite theoretical frame-work to describe the Higgs-like resonance and compute production and decay rates accurately38n perturbation theory without restricting to the SM hypothesis. Effective Lagrangians areone of the tools at our disposal to achieve this goal. Elaborating on the operator classifi-cation of Ref. [4], we estimated the present bounds on the Wilson coefficients and providedaccurate expressions for the Higgs decay rates including various effects that were previouslyomitted in the literature. Assuming that the observed Higgs-like resonance is a spin-0 andCP-even particle, we discussed two general formulations of the effective Lagrangian, one ofwhich relies on the linear realization of SU (2) L × U (1) Y at high energies. One of the ques-tions that can be addressed by considering these two parametrizations is whether the theoryof New Physics flows to the SM in the infrared, that is, whether the Higgs-like resonanceis part of an EW doublet. If all the Higgs signal strengths measured at the LHC convergetowards the SM prediction, it would be a very suggestive indication that indeed the Higgsboson combines together with the longitudinal components of the W and Z to form an EWdoublet, since any other alternative requires some tuning to fake the SM rates. On theother hand, the doublet nature of the Higgs boson would be less obvious to establish if thesignal strengths exhibit deviations from their SM predictions (but note that some deviationsin the signal strengths could unambiguously indicate that the Higgs boson is not part of adoublet, this is in particular the case if a large breaking of the custodial symmetry is ob-served in conflict with the strong bound already existing from EW precision data). We havepointed out that, if the EWSB dynamics is custodially symmetric, it is not possible to testwhether the Higgs boson is part of a doublet by means of single-Higgs processes alone. Adirect 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 natureof the Higgs boson is another question that also requires accurate computations. If there islittle doubt that the observed resonance has a large CP-even component, the possibility ofa small mixing with a CP-odd component remains alive, and dedicated analyses will haveto 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 thetheoretical framework where this question can be addressed quantitatively.The absence so far of direct signals of New Physics at the LHC indicates that the roadto unveil the origin of the electroweak symmetry breaking might be long and go throughprecision analyses rather than copious production of new particles. For such a task, the well39stablished technology of effective field theories is the most powerful and general tool we haveto analyze the Higgs data and put them into a coherent picture together with the existingexperimental information without assuming the validity of the Standard Model. There isstill time for the LHC to disprove this pessimistic eventuality by reporting the discovery ofnew light particles or large shifts in some of the Higgs couplings. It is clear, however, that ifthe New Physics continues to remain elusive, a precise investigation of the Higgs propertieswill become the most urgent programme in high-energy physics both for the experimentaland the theoretical community. Acknowledgments
We thank B. Gavela, A. De Rujula, J.R. Espinosa, A. Falkowski, E. Franco, L. Merlo,A. Pomarol, R. Rattazzi, F. Riva, L. Silvestrini, M. Trott for insightful discussions, andthe participants of the LHC Higgs XS Working Group, in particular A. David, A. Denner,M. D¨uhrssen, M. Grazzini, G. Passarino and G. Weiglein for discussions and comments. Wealso thank J.F. Kamenik for useful explanations on the results of Ref. [27] and we thankA. Pomarol and E. Masso for pointing out a sign error in Eq. (2.13) and we thank F. Maltonifor reminding us about the Bianchi identities. This research has been partly supported by theEuropean Commission under the ERC Advanced Grant 226371 MassTeV and the contractPITN-GA-2009-237920 UNILHC. C.G. is supported by the Spanish Ministry MICNN undercontract FPA2010-17747. The work of R.C. was partly supported by the ERC AdvancedGrant No. 267985
Electroweak Symmetry Breaking, Flavour and Dark Matter: One Solutionfor Three Mysteries (DaMeSyFla) . M.M. is supported by the DFG SFB/TR9 ComputationalParticle Physics. 40
SM Lagrangian: notations and conventions
In this Appendix, we collect the conventions used throughout this paper. The field contentdecomposes under SU (3) C × SU (2) L × U (1) Y as H = (1 , , / , L iL = (1 , , − / , l iR = (1 , , − , (A.80) q iL = (3 , , / , u iR = (3 , , / , d iR = (3 , , − / , (A.81)where the hypercharge is defined as Y = Q − T L , and i = 1 , , D µ q L = (cid:18) ∂ µ − i g S λ a g aµ − i gσ i W iµ − i g (cid:48) B µ (cid:19) q L (A.82)where λ a , a = 1 . . .
8, and σ i , i = 1 . . .
3, are the usual Gell-Mann and Pauli matrices. Ac-cordingly, the gauge-field strengths are defined as G aµν = ∂ µ g aν − ∂ ν g aµ + g S f abc g bµ g cν , (A.83)where f abc are the SU (3) structure constants.The Yukawa interactions of the up-type quarks involve the Higgs charge-conjugate dou-blet defined as H c = iσ H ∗ . (A.84)The renormalizable Lagrangian of the SM thus reads: L SM = − G aµν G a µν − W iµν W iµν − B µν B µν + ( D µ H ) † ( D µ H )+ i (cid:0) ¯ L L γ µ D µ L L + ¯ l R γ µ D µ l R + ¯ q L γ µ D µ q L + ¯ u R γ µ D µ u R + ¯ d R γ µ D µ d R (cid:1) + µ H H † H − λ ( H † H ) + ( y u ¯ q L H c u R + y d ¯ q L Hd R + y l ¯ L L Hl R + h . c . ) (A.85) B Electroweak Chiral Lagrangian in non-unitary gauge
We report here the expression of the EW chiral Lagrangian valid in a generic gauge and inthe 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. The41calar h is assumed to be CP-even and a singlet of the custodial symmetry, and does notnecessarily belong to an SU (2) L doublet. The Lagrangian can be expanded in terms withan increasing number of derivatives L = L + L EW SB , L EW SB = − V ( h ) + L (2) + L (4) + . . . (B.86)where L contains the kinetic terms of the SU (3) c × SU (2) L × U (1) Y gauge fields and of theSM fermions, L EW SB describes the sector responsible for EWSB, and V ( h ) is the potentialfor h [36]: V ( h ) = 12 m h h + c (cid:18) m h v (cid:19) h + . . . (B.87)Under the request of SU (2) V custodial symmetry, the longitudinal W and Z polarizationscorrespond to the NG bosons of the global coset SU (2) L × SU (2) R /SU (2) V and are describedby the 2 × x ) = exp ( iσ a χ a ( x ) /v ) , (B.88)where σ a are the Pauli matrices. SU (2) L × U (1) Y (local) transformations read asΣ( x ) → U L Σ( x ) U † Y , U L = exp( iα aL σ a ) , U Y = exp( iα Y σ ) (B.89)and the covariant derivative is defined by D µ Σ = ∂ µ Σ − i g W aµ σ a Σ + i g (cid:48) B µ Σ σ . (B.90)At the level of two derivatives one has [36]: L (2) = 12 ( ∂ µ h ) + v (cid:0) D µ Σ † D µ Σ (cid:1) (cid:18) c V hv + · · · (cid:19) − v √ λ uij (cid:0) ¯ u ( i ) L , ¯ d ( i ) L (cid:1) Σ (cid:0) u ( i ) R , (cid:1) T (cid:18) c u hv + · · · (cid:19) + h.c. − v √ λ dij (cid:0) ¯ u ( i ) L , ¯ d ( i ) L (cid:1) Σ (cid:0) , d ( i ) R (cid:1) T (cid:18) c d hv + · · · (cid:19) + h.c. − v √ λ lij (cid:0) ¯ ν ( i ) L , ¯ l ( i ) L (cid:1) Σ (cid:0) , l ( i ) R (cid:1) T (cid:18) c l hv + · · · (cid:19) + h.c. (B.91)where the dots stand for terms with two or more Higgs fields and an implicit sum over flavorindices i, j = 1 , , x ) = 1, the sum of (B.87) and (B.91) coincides with thefirst two lines of Eq. (3.46) with c W = c Z = c V .At the level of four derivatives, there are 6 independent bosonic operators which affectcubic vertices with one h field: L (4) = c (cid:48) W W W aµν W µν a hv + c (cid:48) W B Tr (cid:0) Σ † W aµν σ a Σ B µν σ (cid:1) hv + c (cid:48) BB B µν B µν hv + c (cid:48) W m W D µ W aµν Tr (cid:16) Σ † σ a i ←→ D ν Σ (cid:17) h − c (cid:48) B m W ∂ µ B µν Tr (cid:16) Σ † i ←→ D ν Σ σ (cid:17) h + c gg G aµν G a µν hv + . . . (B.93)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) . In the unitary gauge,Eq. (B.93) coincides with the last three lines of Eq. (3.46). More specifically, the coefficients c W W , c ZZ , c Zγ , c γγ can be written as linear combinations of c (cid:48) W W , c (cid:48) BB , c (cid:48) W B , c W W = 2 c (cid:48) W W c ZZ = 2(cos θ W c (cid:48) W W − θ W cos θ W c (cid:48) W B + sin θ W c (cid:48) BB ) c γγ = 2(sin θ W c (cid:48) W W + 2 sin θ W cos θ W c (cid:48) W B + cos θ W c (cid:48) BB ) c Zγ = 2(sin θ W cos θ W c (cid:48) W W + cos 2 θ W c (cid:48) W B − sin θ W cos θ W c (cid:48) BB ) , (B.94)while c W ∂W , c Z∂Z can be expressed in terms of c (cid:48) W , c (cid:48) B : c W ∂W = 4 c (cid:48) W c Z∂Z = 4 c (cid:48) W + 4 tan θ W c (cid:48) B c Z∂γ = 4 tan θ W c (cid:48) W − c (cid:48) B . (B.95) The operator O = ( v/m W ) Tr (cid:2) ( D µ Σ) † ( D ν Σ) (cid:3) ∂ µ ∂ ν h that appeared in Eq. (B.85) of the first versionof this paper is actually redundant and can be eliminated in terms of the operators O (cid:48) W W and O (cid:48) W B . Forexample, the shift c (cid:48) W W → c (cid:48) W W + c and c (cid:48) W B → c (cid:48) W B + tan θ W c can be used to remove c . Another convenient basis, which can be more easily compared to Eq. (3.46), is one in which the first twooperators of Eq. (B.93) are replaced by W aµν Tr (cid:104) Σ † σ a i ←→ D µ Σ (cid:105) ∂ ν h , B µν Tr (cid:104) Σ † i ←→ D µ Σ σ (cid:105) ∂ ν h . (B.92)This is in fact the basis adopted in Ref. [54]. SU (2) L × U (1) Y transforma-tions. The part of Eq. (B.86) which does not depend on the Higgs field h coincides with thenon-linear chiral Lagrangian for SU (2) L × U (1) Y [79], in the limit of exact custodial sym-metry. This latter assumption can be relaxed by specifying the sources of explicit breakingof the custodial symmetry, i.e. its spurions, in terms of which one can construct additionaloperators formally invariant under SU (2) L × U (1) Y local transformations. For example, thelist of operators that follows in the case in which custodial invariance is broken by a fieldwith 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 thestrongest effects are expected to arise from the breaking due to the top quark, rather thanhypercharge), we do not report here any particular list of operators, and prefer to refer tothe existing literature for further details. C Relaxing the CP-even hypothesis
If one relaxes the hypothesis that h is CP-even, there are six extra dimension-6 operatorsthat need to be added to the effective Lagrangian (2.2):∆ L CP = i ˜ c HW gm W ( D µ H ) † σ i ( D ν H ) ˜ W iµν + i ˜ c HB g (cid:48) m W ( D µ H ) † ( D ν H ) ˜ B µν + ˜ c γ g (cid:48) m W H † HB µν ˜ B µν + ˜ c g g S m W H † HG aµν ˜ G aµν + ˜ c W g m W (cid:15) ijk W i νµ W j ρν ˜ W k µρ + ˜ c G g S m W f abc G a νµ G b ρν ˜ G c µρ , (C.96) Notice that h is invariant under SU (2) L × SU (2) R (hence SU (2) L × U (1) Y ) transformations. In thecase in which h belongs to an SU (2) L doublet H , this follows from the fact that h parametrizes the norm ofthe doublet: H † H = ( v + h ) / F µν = (cid:15) µνρσ F ρσ for F = W, B, G ( (cid:15) is thetotally antisymmetric tensor normalized to (cid:15) = 1). Furthermore, the coefficients of theoperators involving fermions will be in general complex numbers.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), which can affectcubic vertices with one h field:∆ L (4) CP = ˜ c (cid:48) W W ˜ W aµν W µν a hv + ˜ c (cid:48) W B Tr (cid:104) Σ † ˜ W aµν σ a Σ B µν σ (cid:105) hv + ˜ c (cid:48) BB ˜ B µν B µν hv + ˜ c gg G aµν G aµν hv . (C.97)In the unitary gauge, both Lagrangians ∆ L CP and ∆ L (4) CP are matched onto:∆ L (4) CP = (cid:18) ˜ c W W W + µν ˜ W − µν + ˜ c ZZ Z µν ˜ Z µν + ˜ c Zγ Z µν ˜ γ µν + ˜ c γγ γ µν ˜ γ µν + ˜ c gg G aµν ˜ G aµν (cid:19) hv + . . . (C.98)When the EW symmetry is linearly realized, the coefficients of Eq. (C.98) are related to theWilson coefficients of Eq. (C.96) through the same relations reported in Table 1 with thesimple exchange c i → ˜ c i (and with c W = c B = 0). In the non-linear case, ˜ c W W , ˜ c ZZ , ˜ c γγ and˜ c Zγ are given in terms of the Wilson coefficients of Eq. (C.97) by relations identical to theones of Eq. (B.94) (with c i → ˜ c i and c = 0). Notice that the Bianchi identities ensure that D µ ˜ V µν = 0 and therefore there are no CP-odd analogues to the operators O V ∂V .In addition to the new operators of Eq. (C.97), an imaginary value of the coupling c W ∂W also breaks the CP-invariance : i Im( c W ∂W ) ( W − ν D µ W + µν − h.c. ) . (C.99)In the non-unitary-gauge, this coupling originates from the operator (cid:15) abc Tr (cid:0) Σ † σ a Σ σ (cid:1) D µ W bµν Tr (cid:16) Σ † σ c i ←→ D ν Σ (cid:17) h. (C.100)In the linear realization of the SU (2) L × U (1) Y gauge symmetry, this coupling cannot beobtained from a dimension-6 operator but it originates from the dimension-8 operator: i(cid:15) abc (cid:0) H † σ a H (cid:1) (cid:16) H † σ b ←→ D ν H (cid:17) ( D µ W c µν ) . (C.101) We thank A. Pomarol for helping us to understand the issue of this additional CP-odd coupling. hV V obtained from dimension-6 operators built with an EW doublet and the couplings hV V obtained at the order p when the SU (2) L × U (1) Y symmetry is non-linearly realized,provided that the gauge fields couple to conserved currents.Finally, it should also be noted that when the CP-invariance assumption in the Higgssector 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-6operators reported in Section 2.1. For a given observable we construct a likelihood for thecoefficients ¯ c i as follows: L (¯ c i ) ∝ exp (cid:2) − ( O SM + δO (¯ c i ) − O exp ) / (2 ∆ O exp ) (cid:3) , (D.102)where O exp ± ∆ O exp is the experimental value of the observable, O SM denotes its SM pre-diction and δO (¯ c i ) is the correction due to the effective operators. If several observablesconstrain the same coefficients ¯ c i , the global likelihood is constructed by multiplying thoseof each observable. We include the theoretical uncertainty on the SM prediction by integrat-ing over a nuisance parameter whose distribution is appropriately chosen. We then quotethe 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 usedthe EW fit performed in Ref. [22] by the GFitter collaboration, and constructed a likelihoodfor the various coefficients by computing their contributions to the Z -pole observables. Forthe latter, we used the SM predictions and experimental inputs reported in Table 1 ofRef. [22], treating the uncertainties on the SM predictions as normally distributed. Weperformed two separate fits: one on the coefficients of the operators involving the lightquarks ( u, d, s ), and one on those with charged leptons and heavy quarks ( c, b ). We thusneglected, for simplicity, the correlations between these two sets of coefficients. The relevantobservables in the first fit are Γ tot , σ had and R l . They depend on the Wilson coefficients only46hrough the following linear combination: l = (cid:18) −
14 + 13 sin θ W (cid:19) (¯ c Hq − ¯ c (cid:48) Hq ) + (cid:18) −
16 sin θ W (cid:19) (¯ c Hq + ¯ c (cid:48) Hq + ¯ c Hq + ¯ c (cid:48) Hq )+ 13 sin θ W ¯ c Hu −
16 sin θ W (¯ c Hd + ¯ c Hs ) , (D.103)which with 95% probability is constrained to lie in the interval − . × − < l < . × − . (D.104)Although there are no further observables at the Z -pole which can resolve the degener-acy implied by this result, we thought it useful to report the limits that one obtains fromEq. (D.104) by turning on one coefficient at the time. These are the bounds reported inEq. (2.14).The second fit, performed on the coefficients of the operators with leptons and heavyquarks, makes use of all the observables at the Z pole and counts 7 unknowns, specifically: x i = { (¯ c Hq − ¯ c (cid:48) Hq ) , ¯ c Hc , (¯ c Hq + ¯ c (cid:48) Hq ) , ¯ c Hb , ¯ c Hl , (¯ c HL + ¯ c (cid:48) HL ) , (¯ c HL − ¯ c (cid:48) HL ) } . For simplicity weassume lepton universality, and thus take the coefficients ¯ c Hl , ¯ c HL , ¯ c (cid:48) HL to be the same for allthe leptonic generations. In terms of the above variables, the result of the fit is summarizedby their central values ¯ x i , standard deviations σ i and by the correlation matrix ρ ij :¯ c Hq − ¯ c (cid:48) Hq = (5 . ± . × − ¯ c Hc = (5 . ± . × − ¯ c Hq + ¯ c (cid:48) Hq = ( − . ± . × − ¯ c Hb = ( − . ± . × − ¯ c Hl = (1 . ± . × − ¯ c HL + ¯ c (cid:48) HL = (7 . ± . × − ¯ c HL − ¯ c (cid:48) HL = (5 . ± × − (D.105)47 = . . − . − .
072 0 . − . − . .
74 1 . − . − .
085 0 .
11 0 .
15 0 . − . − .
078 1 . . − . − .
21 0 . − . − .
085 0 .
85 1 . − . − . − . .
24 0 . − . − .
40 1 . .
11 0 . − .
057 0 . − . − .
33 0 .
11 1 . − . − .
14 0 .
030 0 . − . . − .
35 1 . (D.106)The limits of Eq. (2.15) have been obtained by making use of the above formulas andmarginalizing over all the coefficients except the one on which the bound is reported.For the limits of Eqs. (2.11) and (2.12) we have used the fit on S and T performed inRef. [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 neutronand 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 givenrespectively in Ref. [80] and Ref. [81]. We included the theoretical errors by assuming thatthey are uniformly distributed within the stated intervals. Only two linear combinations ofthe coefficients ¯ c i can be constrained in this way, since two are the observables at disposal: l = − m d m W [Im(¯ c dG ) + 1 . c dB − ¯ c dW )] − m u m W [Im(¯ c uG ) − .
64 Im(¯ c uB + ¯ c uW )] l = − m u m W Im(¯ c uG ) + 2 m d m W Im(¯ c dG ) . (D.107)Using m u = 2 . m d = 4 . − . × − GeV − < l < . × − GeV − − . × − GeV − < l < . × − GeV − . (D.108)From the above result, by turning on one coefficient at the time, one obtains the limitsgiven in Eq. (2.17). The bound on Im(¯ c tG ) of Eq. (2.18) has been similarly derived fromthe neutron and mercury EDMs by following Ref. [27] and making use of the formulas giventhere. 48he limits of Eq. (2.22) have been obtained from the experimental measurements of theelectron [31] and muon [29] anomalous magnetic moments and their SM predictions (takenrespectively from Ref. [32] and Refs. [29, 30]). In this case we have included the theoreticalerrors by assuming that they are normally distributed. All the remaining bounds reported inSection 2.1, namely those of Eqs. (2.19)-(2.21) and Eq. (2.23) have been obtained by simplytranslating into our notation the results given in the references quoted in the text. References [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B (2012) 1 [arXiv:1207.7214[hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B (2012) 30 [arXiv:1207.7235[hep-ex]].[3] S. Weinberg, “The quantum theory of fields.” Cambridge, UK: Univ. Pr. (1996) 489 p[4] G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, JHEP (2007) 045 [hep-ph/0703164].[5] D. B. Kaplan and H. Georgi, Phys. Lett. B (1984) 183. S. Dimopoulos andJ. Preskill, Nucl. Phys. B , 206 (1982). T. Banks, Nucl. Phys. B , 125 (1984).D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. B , 187 (1984). H. Georgi,D. B. Kaplan and P. Galison, Phys. Lett. B , 152 (1984). H. Georgi and D. B. Ka-plan, Phys. Lett. B , 216 (1984). M. J. Dugan, H. Georgi and D. B. Kaplan, Nucl.Phys. B , 299 (1985).[6] R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B (2003) 148 [hep-ph/0306259].[7] K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B (2005) 165 [hep-ph/0412089].[8] A. Djouadi, J. Kalinowski and M. Spira, Comput. Phys. Commun. (1998) 56 [hep-ph/9704448]. A. Djouadi, M. M. Muhlleitner and M. Spira, Acta Phys. Polon. B (2007) 635 [hep-ph/0609292]. 499] R. Contino, M. Ghezzi, C. Grojean, M. M. Muhlleitner and M. Spira, “eHDECAY: animplementation of the Higgs effective Lagrangian into HDECAY”, work in progress.[10] C. J. C. Burges and H. J. Schnitzer, Nucl. Phys. B (1983) 464; C. N. Leung,S. T. Love and S. Rao, Z. Phys. C (1986) 433; W. Buchmuller and D. Wyler, Nucl.Phys. B (1986) 621.[11] R. Rattazzi, Z. Phys. C (1988) 605; B. Grzadkowski, Z. Hioki, K. Ohkuma andJ. Wudka, Nucl. Phys. B (2004) 108 [hep-ph/0310159]; P. J. Fox, Z. Ligeti,M. Papucci, G. Perez and M. D. Schwartz, Phys. Rev. D (2008) 054008[arXiv:0704.1482 [hep-ph]]; J. A. Aguilar-Saavedra, Nucl. Phys. B (2009) 181[arXiv:0811.3842 [hep-ph]]; J. A. Aguilar-Saavedra, Nucl. Phys. B (2009) 215[arXiv:0904.2387 [hep-ph]].[12] C. Grojean, W. Skiba and J. Terning, Phys. Rev. D (2006) 075008 [hep-ph/0602154].[13] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, JHEP (2010) 085[arXiv:1008.4884 [hep-ph]].[14] R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B (2004) 127[hep-ph/0405040].[15] R. Barbieri, CERN-TH-6659-92.[16] M. E. Peskin and T. Takeuchi, Phys. Rev. D (1992) 381.[17] K. Agashe and R. Contino, Phys. Rev. D (2009) 075016 [arXiv:0906.1542 [hep-ph]].[18] G. Isidori, arXiv:1302.0661 [hep-ph].[19] G. Isidori, Y. Nir and G. Perez, Ann. Rev. Nucl. Part. Sci. (2010) 355[arXiv:1002.0900 [hep-ph]].[20] P. Sikivie, L. Susskind, M. B. Voloshin and V. I. Zakharov, Nucl. Phys. B (1980)189. 5021] G. Altarelli and R. Barbieri, Phys. Lett. B , 161 (1991); G. Altarelli, R. Barbieriand S. Jadach, Nucl. Phys. B , 3 (1992) [Erratum-ibid. B , 444 (1992)].[22] M. Baak, M. Goebel, J. Haller, A. Hoecker, D. Kennedy, R. Kogler, K. Moenig andM. Schott et al. , Eur. Phys. J. C (2012) 2205 [arXiv:1209.2716 [hep-ph]].[23] M. Redi and A. Weiler, JHEP (2011) 108 [arXiv:1106.6357 [hep-ph]].[24] N. Vignaroli, Phys. Rev. D (2012) 115011 [arXiv:1204.0478 [hep-ph]].[25] M. Pospelov and A. Ritz, Annals Phys. (2005) 119 [hep-ph/0504231].[26] P. Paradisi and D. M. Straub, Phys. Lett. B (2010) 147 [arXiv:0906.4551 [hep-ph]].[27] J. F. Kamenik, M. Papucci and A. Weiler, Phys. Rev. D (2012) 071501[arXiv:1107.3143 [hep-ph]].[28] J. A. Aguilar-Saavedra, N. F. Castro and A. Onofre, Phys. Rev. D (2011) 117301[arXiv:1105.0117 [hep-ph]].[29] A. Hoecker and W.J. Marciano, PDG review on “The Muon Anomalous Magnetic Mo-ment”, in: J. Beringer et al. [Particle Data Group Collaboration], Phys. Rev. D ,010001 (2012).[30] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C (2011) 1515[Erratum-ibid. C (2012) 1874] [arXiv:1010.4180 [hep-ph]].[31] D. Hanneke, S. F. Hoogerheide and G. Gabrielse, arXiv:1009.4831 [physics.atom-ph].[32] G. F. Giudice, P. Paradisi and M. Passera, JHEP (2012) 113 [arXiv:1208.6583[hep-ph]].[33] G. W. Bennett et al. [Muon (g-2) Collaboration], Phys. Rev. D (2009) 052008[arXiv:0811.1207 [hep-ex]].[34] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt and E. A. Hinds,Nature (2011) 493. D. M. Kara, I. J. Smallman, J. J. Hudson, B. E. Sauer,51. R. Tarbutt and E. A. Hinds, New J. Phys. (2012) 103051 [arXiv:1208.4507[physics.atom-ph]].[35] R. Rattazzi, talk at the workshop Physics at LHC: from Experiment to Theory , Prince-ton University, March 21-24 2007[36] R. Contino, C. Grojean, M. Moretti, F. Piccinini and R. Rattazzi, JHEP (2010)089 [arXiv:1002.1011 [hep-ph]].[37] R. Contino, L. Da Rold and A. Pomarol, Phys. Rev. D (2007) 055014 [hep-ph/0612048].[38] S. Y. Choi, D. J. Miller, 2, M. M. Muhlleitner and P. M. Zerwas, Phys. Lett. B (2003) 61 [hep-ph/0210077].[39] A. De Rujula, J. Lykken, M. Pierini, C. Rogan and M. Spiropulu, Phys. Rev. D (2010) 013003 [arXiv:1001.5300 [hep-ph]].[40] S. Bolognesi, Y. Gao, A. V. Gritsan, K. Melnikov, M. Schulze, N. V. Tran and A. Whit-beck, Phys. Rev. D (2012) 095031 [arXiv:1208.4018 [hep-ph]].[41] A. Azatov, A. Falkowski, C. Grojean and E. Kuflik, work in progress.[42] J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B (1976) 292;M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl.Phys. (1979) 711 [Yad. Fiz. (1979) 1368].[43] B. A. Kniehl and M. Spira, Z. Phys. C (1995) 77 [hep-ph/9505225].[44] A. Falkowski, Phys. Rev. D (2008) 055018 [arXiv:0711.0828 [hep-ph]].[45] I. Low and A. Vichi, Phys. Rev. D (2011) 045019 [arXiv:1010.2753 [hep-ph]].[46] A. Azatov and J. Galloway, Phys. Rev. D (2012) 055013 [arXiv:1110.5646 [hep-ph]].[47] M. Gillioz, R. Grober, C. Grojean, M. Muhlleitner and E. Salvioni, JHEP (2012)004 [arXiv:1206.7120 [hep-ph]]. 5248] M. Redi, Eur. Phys. J. C (2012) 2030 [arXiv:1203.4220 [hep-ph]].[49] R. Barbieri, D. Buttazzo, F. Sala, D. M. Straub and A. Tesi, arXiv:1211.5085 [hep-ph].[50] C. Degrande, J.–M. Gerard, C. Grojean, F. Maltoni and G. Servant, JHEP (2011)125 [arXiv:1010.6304 [hep-ph]]; H. Hesari and M. M. Najafabadi, arXiv:1207.0339 [hep-ph]; C. Englert, A. Freitas, M. Spira and P. M. Zerwas, arXiv:1210.2570 [hep-ph].[51] C. Degrande, J. M. Gerard, C. Grojean, F. Maltoni and G. Servant, JHEP (2012)036 [arXiv:1205.1065 [hep-ph]].[52] S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. (1969) 2239; C. G. Callan, Jr.,S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. (1969) 2247.[53] C. P. Burgess and D. London, hep-ph/9203215.[54] A. Azatov, R. Contino and J. Galloway, JHEP (2012) 127 [arXiv:1202.3415 [hep-ph]].[55] R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, arXiv:1212.3305 [hep-ph].[56] R. Contino, C. Grojean, D. Pappadopulo, R. Rattazzi, A. Thamm, work in progress.[57] G. Passarino, Nucl. Phys. B (2013) 416 [arXiv:1209.5538 [hep-ph]].[58] S. Weinberg, Physica A (1979) 327.[59] C. Grojean, E. E. Jenkins, A. V. Manohar and M. Trott, arXiv:1301.2588 [hep-ph].[60] J. Elias-Miro, J. R. Espinosa, E. Masso and A. Pomarol, arXiv:1302.5661 [hep-ph].[61] R. Barbieri, B. Bellazzini, V. S. Rychkov and A. Varagnolo, Phys. Rev. D (2007)115008 [arXiv:0706.0432 [hep-ph]].[62] A. Pomarol and J. Serra, Phys. Rev. D (2008) 074026 [arXiv:0806.3247 [hep-ph]].[63] S. L. Adler, J. C. Collins and A. Duncan, Phys. Rev. D (1977) 1712.[64] J. C. Collins, A. Duncan and S. D. Joglekar, Phys. Rev. D (1977) 438.5365] N. K. Nielsen, Nucl. Phys. B (1977) 212.[66] M. Farina, C. Grojean, F. Maltoni, E. Salvioni and A. Thamm, arXiv:1211.3736 [hep-ph].[67] K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser, Nucl. Phys. B (1998) 61 [hep-ph/9708255].[68] M. Kramer, E. Laenen and M. Spira, Nucl. Phys. B (1998) 523 [hep-ph/9611272].[69] Y. Schroder and M. Steinhauser, JHEP (2006) 051 [hep-ph/0512058].[70] K. G. Chetyrkin, J. H. Kuhn and C. Sturm, Nucl. Phys. B (2006) 121 [hep-ph/0512060].[71] T. Inami, T. Kubota and Y. Okada, Z. Phys. C (1983) 69; A. Djouadi, M. Spiraand P. M. Zerwas, Phys. Lett. B (1991) 440; K. G. Chetyrkin, B. A. Kniehl andM. Steinhauser, Phys. Rev. Lett. (1997) 353 [hep-ph/9705240].[72] M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, Nucl. Phys. B (1995) 17[hep-ph/9504378].[73] P. A. Baikov and K. G. Chetyrkin, Phys. Rev. Lett. (2006) 061803 [hep-ph/0604194].[74] J. F. Gunion and H. E. Haber, Phys. Rev. D (2003) 075019 [hep-ph/0207010].[75] L. Randall, JHEP (2008) 084 [arXiv:0711.4360 [hep-ph]].[76] J. F. Gunion and H. E. Haber, Nucl. Phys. B (1986) 1 [Erratum-ibid. B (1993)567].[77] K. Blum, R. T. D’Agnolo and J. Fan, JHEP (2013) 057 [arXiv:1206.5303 [hep-ph]].[78] A. Azatov and J. Galloway, Int. J. Mod. Phys. A (2013) 1330004 [arXiv:1212.1380[hep-ph]].[79] T. Appelquist and C. W. Bernard, Phys. Rev. D (1980) 200; A. C. Longhitano,Phys. Rev. D (1980) 1166; A. C. Longhitano, Nucl. Phys. B (1981) 118.5480] C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Har-ris, P. Iaydjiev and S. N. Ivanov et al. , Phys. Rev. Lett. (2006) 131801 [hep-ex/0602020].[81] W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V. Romalis, B. R. Heckel andE. N. Fortson, Phys. Rev. Lett.102