Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 877 (2013) 792–806

www.elsevier.com/locate/nuclphysb

Higgs-boson couplings beyond the Standard Model

Martin B. Einhorn a,b, José Wudka c

a Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, United States 1

b Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, United Statesc Department of Physics and Astronomy, University of California, Riverside, CA 92521-0413, United States

Received 27 August 2013; received in revised form 29 October 2013; accepted 4 November 2013

Available online 8 November 2013

Abstract

The implications for Higgs decays of potential new physics beyond the Standard Model (BSM) areconsidered in the context of effective field theory, assuming perturbative decoupling. Using existing datato restrict which dimension-six operators can arise, it is shown that, given the existing experimentalconstraints, only a small number of operators can affect the decays of the Higgs: those that may bepotentially-tree-generated (PTG) and modify the Higgs–fermion couplings, or those that may be loop-generated (LG) that modify the Higgs couplings to γ γ , Zγ and GG. Implications for specific branchingratios are given in terms of the coefficients of various dimension-six operators. In such a scenario, the ratiosΓ (H → WW∗)/Γ (H → ZZ∗) and Γ (H → W�ν)/Γ (H → Z��) equal to their Standard Model valuesto an accuracy of O(1%) or less.© 2013 Elsevier B.V. All rights reserved.

Keywords: Higgs-boson couplings; Beyond Standard Model; Effective field theory

1. Introduction

The observation of a new particle by two detectors [1,2] at the LHC has offered a candidate forthe long-sought Higgs boson of the Standard Model (SM). Within errors, the observed propertiesare consistent with the Higgs boson of the SM insofar as its production rate and branching ratiosare concerned, coming primarily from data in the WW ∗, ZZ∗, and γ γ decay modes. There issome supporting evidence from enhancements in TeVatron data [3], primarily from bb decays.So far, the evidence is consistent with SM expectations, with the exception of the rate in theγ γ -channel, which apparently exceeds SM estimates in one of the experiments [1].

1 Current address.

0550-3213/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysb.2013.11.004

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 793

Naturally, a primary goal of further experiments is to determine whether the couplings of theHiggs to weak bosons gHWW , gHZZ and the couplings to fermions agree with the SM expec-tations. These studies may be informed by theoretical expectations, and many papers have beenwritten (for recent summaries see e.g. [4]) about the implications of models that include particlesbeyond the SM (BSM). With a mass mH ≈ 125 GeV, the Higgs would appear to have a smallenough self-coupling for perturbation theory to be reliable. In that case, there are two possibilitiesfor the additional particles in such models: either (1) they involve new “light” particles of a masscomparable to or lighter than the Higgs boson, as, for example, in models having two Higgsdoublets, including some supersymmetric models,2 in which most or all of their mass derivesfrom the electroweak scale as do SM particles, or (2) all new particles are more massive thanmH , deriving their mass from some new underlying scale. An example would be softly-brokensupersymmetric models with super-renormalizable couplings large compared to the weak scale.In the absence of the observation of a new particle, it can be difficult to decide in which situationwe find ourselves.

The question is, what can be inferred from deviations of experimental data from SM expec-tations of the properties of the observed scalar? In the former case, there tend to be rather largedeviations from the SM, arising from mixing angles between two or more multiplets. Typically,couplings of a Higgs boson differ already at tree level by factors such as tanβ , the ratio of vacuumexpectation values of different doublets. In the latter case, one may perform a model-independentanalysis using a generic effective Lagrangian approach, taking into account that the first cor-rections to the SM can be described in terms of higher-dimensional operators (HDO). A largenumber of publications have appeared recently that discuss various aspects of this approach; itwould require a lengthy review to cite all the papers that have been written on this, and such alist would be out of date by the time of this publication; for a representative sample see [7,8].

A related question is, if there are no other particles discovered and no deviations from the SMobserved, what conclusions can be drawn, given the level of accuracy of the experiments? Asmuch as possible, one would like to draw model-independent conclusions, although that may bevery difficult in the near term.

Some years ago, the authors [9] and others [10,11] performed such analyses, both for weakly-interacting, decoupling scenarios and for strongly interacting models [12,13]. In this paper, weshall assume the underlying physics is decoupling and weakly coupled, at least to a good ap-proximation. What this means in practice is that the particles that we call “light,” such as the topquark, predominantly get their masses as a result of spontaneous breaking of SU(2) ⊗ U(1). Weassume that the “heavy” particles get their masses primarily via some other mechanism, althoughthey may also receive electroweak contributions. For example, this would be the case if the scaleof supersymmetry-breaking were large compared to the weak scale, giving some superpartnersparametrically large masses.

In phenomenological studies of deviations from the SM, it has been advocated3 that, to fitexperimental events involving the production and decay of a single Higgs boson, one employ aneffective Lagrangian of the form

Leff = H

v

[(2cWM2

WW−μ W+

μ + cZM2ZZ2

μ

) + ctmt t t + cbmbbb + cτmτ τ τ]

(1)

+ H

3πv

[cγ

2α

3F 2

μν + cg

αS

4G2

μν

]. (2)

2 Some recent fits of such models to LHC data are [5,6].3 See, e.g., the reviews in Refs. [6,14].

794 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

This effective Lagrangian is presumed to describe interactions in the so-called unitary gauge,where the Higgs doublet is of the form φ = ((v +H)/

√2 )(0,1)T (and v = √

2 〈φ〉 246 GeV.)In the SM, the interactions described in Eq. (2) arise at the one-loop level, whereas those inEq. (1) arise already in tree approximation. To leading order, all the coefficients ck = 1, which,given present experimental accuracies, is sufficient, although further radiative corrections can beincluded if necessary.

Although the decay rates are unambiguous in the Standard Model, unless we know the form ofthe BSM Lagrangian, one cannot blithely use Eqs. (1) and (2) as an effective Lagrangian withoutacknowledging other implications of such a choice. Because of the equivalence theorem, whichwe will review in the next section, the form of an effective Lagrangian involves a certain de-gree of arbitrariness. The particular choice of operators does not affect expectations for S-matrixelements, but the associated Green’s functions may be very different. At the very least, theseadditional assumptions need to be spelled out in detail. The expressions Eqs. (1), (2), althoughnot gauge-invariant, are intended to be used in unitary gauge. Some decays can be well approx-imated by treating all particles on-shell, so that, e.g., cb, cτ , cγ may be defined with all threeparticles on-mass-shell, other parameters such as cW , cZ , ct represent coupling constants that,for kinematical reasons, cannot be determined experimentally. As an example, the H → WW ∗mode represents H → W�ν� (where � = e,μ), which receives a contribution from a virtual W

exchange, but others as well, and these depend on the operator basis being used. Extracting limitson coefficients such as cW requires a complete calculation that includes all relevant contributionsto insure the results are independent of the operator basis and fully gauge-invariant.

In fact, among other results below, we shall show that modifications to the SM couplingsfrom new physics cW , cZ are negligible within foreseeable experimental errors (except for a pos-sible common normalization effect that does not contribute to the branching ratios), unless thereare other light particles whose masses arise primarily, if not solely, from electroweak symmetry-breaking. In that case, the form of Eq. (1) is unsuitable as a starting point for fitting or interpretingexperimental data. No conclusions can be drawn from it without making presumptions aboutother operators and processes involving the Higgs boson.

2. Some features of effective Lagrangians

By now, the language of effective field theory has become familiar [15–18], especially toresearchers studying physics BSM. We will review it here only to the extent that we need toestablish notation and to summarize some features of the approach; the details of our generalapproach are provided in a companion paper [19].

We imagine a theory where the heavy scale Λ is assumed too large for the correspondingexcitations to be directly produced; their virtual effects, however, may be observable. Assumingalso that the heavy physics decouples then implies that at scales below Λ the effective action canbe expanded in a power series in Λ (multiplied by logarithmic corrections) where all each powermultiplies a local operator, and those terms that grow with Λ can be absorbed in a renormalizationof the low-energy parameters. After this renormalization the effective action takes the form

Seff =∫

d4xLeff, Leff = L0 +∑ 1

Λn

∑f

(n)i O(n)

i , (3)

n�5 i

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 795

where we assumed the underlying physics is weakly coupled.4 The coefficients fi encode allthe details of the underlying model and can therefore be used to parameterize all possible typesof heavy physics; in general they can also depend logarithmically on Λ. For the case beingconsidered here, L0 corresponds to the full SM Lagrangian, in which case there is a singledimension-five [20] operator that violates lepton number by two units and generates a neutrinoMajorana mass. Aside from this, dimension-six operators then represent the leading virtual newphysics effects resulting from any weakly-coupled, decoupling heavy particles.

The number of dimension-six operators is large (∼100), but not all need be included in cal-culations since all low-energy effects can be parametrized by the coefficients of a reduced set ofoperators we refer to as a basis. This is a set of (dimension-six) operators {Oa} (henceforth wedrop the superscript (6)) with the property that any other operator O obeys the relation

O −∑a

κaOa =∑φ

Uφ

δS0

δφ, (4)

where the κa are appropriately chosen constants; φ, a generic light field; δS0/δφ, the classicalequations of motion; and Uφ , local operators. Each term in Eq. (4) is gauge- and Lorentz-invariant. We will say that the combination O − ∑

κaOa vanishes “on-shell,” and that O isequivalent to

∑κaOa . In addition, we demand that no linear combination of basis elements

vanishes on shell.As an historical aside, it was only recently shown [21] that the original basis given in the

classic paper by Buchmuller and Wyler [25] contained redundant operators. This illustrates that,when one has so many fields and couplings, it isn’t so easy to establish the size of the basis, butredundancy is obviously better than false constraints.

To establish some terminology, note that the HDO form a vector space. An equivalence re-lation produces a unique partition of a vector space into equivalence classes. A basis will haveone operator from each equivalence class. A minimal basis choice for dimension-six operatorsis presented in [21]. While the preceding is a familiar construction, in [19], we put forward animproved method of choosing basis operators with reference how these operators may arise inextensions of the SM. An extension of L0 is any model containing heavy particles that reduces toL0 for operators of dimension four or less at scales below some threshold Λ. An extended modelmay also be referred to as an embedding of L0.

In general, HDO may be identified as either potentially-tree-generated (PTG) or loop-generated (LG) [9,19]. A PTG operator OPTG is one for which an extension can be found inwhich it arises from a tree-diagram. An LG operator OLG is one that (1) cannot emerge from atree-graph in any embedding of L0 and (2) can arise from loop-graphs.5 This is a useful distinc-tion because LG operators have coefficients that are typically suppressed by a factor ∼1/(4π)2n

relative to PTG operators, where n is the number of loops. Having potentially larger coefficients,the PTG operators may be more sensitive to new physics effects.

4 Although Leff must be Hermitian, it is not always most expedient to make each term in the sum Hermitian; e.g., inthe SM, the Yukawa couplings are an illustration. In such cases, each term is implicitly accompanied by its Hermitianconjugate.

5 This classification can be made either without restriction on the embeddings of L0 or, if one is interested in a restrictedset of embeddings, from extended models respecting some additional local or global symmetry. For example, the limitson violation of baryon- or lepton-number suggest that such operators may be ignored for analyzing LHC data, regardlessof whether they are PTG or LG.

796 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

It is important to note that whether an operator is LG or PTG is a property of the heavyphysics, while an equivalence relation of the form Eq. (4) is a property of the light theory. Anequivalence class of operators may contain only PTG operators, only LG operators, or both kinds.It is helpful to identify this property of each equivalence class and, in cases when a class containsboth kinds of operators, to choose a basis operator from among the PTG operators. This providesthe most conservative approach to interpreting experimental data, whether providing limits onor evidence for BSM physics effects. The reason is that the parametrization covers the widestclass of heavy physics theories: those that generate the operators in question at tree level, aswell as those that may generate them only through loop corrections. In [19], we delineated theequivalence relations for the SM and analyzed the basis chosen in [21]. We identified those thatwere LG, those that were PTG, and in cases where an equivalence class contained both types,showed that the basis chosen in [21] were in fact PTG operators, as required. This then is goodbasis for studying physics BSM. In the remainder of this paper, we indicate how this may beapplied to Higgs production and decay.

3. New physics contributions to Higgs decay

Limiting our attention to baryon- and lepton-conserving operators involving only SM fields(e.g. no right-handed neutrinos), and adopting the basis {Oa} of dimension-six operators givenin Ref. [21], the effective Lagrangian takes the form6

Leff = LSM +∑a

fa

Λ2Oa + · · · , (5)

where the ellipsis denote operators of dimension >6. We need only PTG operators that contributeto the various Higgs decay channels, except for the γ γ , γZ, and GG final states which occurat one loop in the SM and in all extensions of the SM. These therefore, invite comparison withone-LG corrections to new physics. As discussed above, the operators included describe theleading deviation from the SM, whereas those we neglect will be too small to be observed, atleast in present experiments. In the following we will assume that the coefficients fa are real.7

We find that the PTG operators contributing to the Higgs decay channels measured at theLHC are also involved in other well-measured process, namely, Z and W lepton decays andcustodial symmetry violations (for a review, see, e.g., [26]) associated with the oblique T (or ρ)parameter. Current data indicate that deviation from the SM in these processes lie below the levelof 0.1%–1%, so that the contributions to the corresponding operators to Higgs decays can beneglected given the current precision in that decay.

The PTG operator basis contributing to the measured Higgs decays can be separated into 3classes:

1. Pure Higgs operators8:

O∂φ = 1

2

(∂μ|φ|2)2

, Oφ = |φ|6. (6)

6 Although we will use the basis of [21], we prefer to denote the operators as Oa rather than Qa . For convenience, wereproduce them in Appendix A.

7 For those operators Oa that are Hermitian, fa are necessarily real; for others, including the ones relevant in Higgsdecays, it is equivalent to assuming that CP violation is unimportant for these applications.

8 In [21], O∂φ is replaced by −Oφ� , which is the same after integration by parts.

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 797

Ignoring self-interactions of the scalar field, the effect of O∂φ will be to change the normal-ization of the Higgs field after symmetry breaking. Indeed, in unitary gauge,

φ = 1√2

(0

v + h

)(7)

(with v = √2〈φ〉 246 GeV), so we get

L = LSM + f∂φ

Λ2O∂φ + · · · ≈ 1

2(1 + εf∂φ)(∂h)2 + · · · , (8)

where ε ≡ v2/Λ2. Thus, the canonically normalized Higgs field will be

H = √1 + f∂φε h ≈

(1 + 1

2f∂φε

)h. (9)

This modifies all processes involving a single Higgs boson in the same way, replacing theSM Higgs field h by H .The operator Oϕ also modifies the parameters in the Higgs sector, such as the vacuum ex-pectation value v, but these effects can be absorbed in finite renormalizations of L0 with noobservable effects. In that case, Oϕ could only be distinguished through Higgs self-couplingeffects [22,23] which, at present, are experimentally out of reach. In contrast the rescaling(9) does have (potential) observable consequences.

2. Operators modifying Higgs couplings to W and Z:These include basis operators of the type X2ϕ2, called OϕX , OϕX , OϕWB and OϕWB (seeAppendix A). All of these are LG operators and will be neglected in first approximation. Thatmeans that the Higgs coupling to ZZ and WW may be assumed to be SM to within about0.1–1%. This important result is analogous to our earlier result concerning BSM correctionsto triple-gauge-boson couplings [19,9].The basis we employ also contains an operator of the type ϕ4D2: OφD ≡ |φ†Dμφ|2. Thisoperator would generate a mass shift for the Z and produce a change in the ρ-parameter [27]or, equivalently, the so-called oblique T parameter [28] from its SM value, specifically

δT = − 1

αεfφD, (10)

where, in obtaining this relation, we ignored the effective-operator contributions to the Fermiconstant [29]. Current experimental constraints9 give |δT | � 0.1, implying that, even thoughOφD may affect Higgs decays (specifically, the H → ZZ∗ mode10) these effects will be toosmall to be observed, given the experimental precision achievable at the LHC.

3. Higgs and gauge boson couplings to fermions:These include all the operators of the types [21] ψ2ϕ2D, called (Oϕψ)pr and (Oϕud)pr , andψ2Xϕ, called (OψX)pr , for any fermion ψ , where p, r are family indices. The former are allPTG operators, but the latter are all LG. Therefore, in first approximation, we will neglectOψX .Limits on flavor-changing neutral currents11 suggest that the thresholds for generation-changing operators (Oϕψ)pr in the coupling of the Z are very high, so we may assume

9 See the review by Erler and Langacker in [24].10 See comments on this decay mode in next section.11 See the review by Ceccucci, Ligeti, and Sakai in [24].

798 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

that p = r , but this still leaves the Z couplings to the 3 families, (Oϕψ)pp for each type offermion ψ = {�, e, q,u, d}. Many of these are already precisely determined [24], especiallyfor ψ = leptons for all 3 generations, from LEP measurements. The Z-lepton couplings aremeasured to at least 1% and agree with the SM predictions to that precision, so their potentialcontributions to Higgs decay widths will lie in this range. These effects are of order f ε andcorrespond to a scale Λ > 2.5 TeV when f ∼ 1. For quarks, they are similar experimentallyconstrained for all flavors except for the t -quark. Although these operators break custodialsymmetry and therefore would change |δT |, their lowest order contributions are in one-loopcorrections and so are not strongly constrained by the experimental value of |T |. To improveon existing limits significantly seems to be beyond the reach of a hadron collider such asLHC.The operator (Oϕud)pr would modify couplings of the W in ways that would affect bothfamily-changing couplings as well as Higgs decays. These couplings of the W to the firsttwo generations have been well-studied, and there are even constraints from t → bW . Onceagain, this would also contribute to |δT |, but only at one-loop order, and, given that thetop quark only makes a contribution of about 0.2, a bound on the order of 10% gives nosignificant constraint on the corresponding εfϕud . The operator O(3)

ϕ� potentially modifies theW�ν coupling, but since measurements agree with the Standard Model to a precision below1%, the effective operator modifications to these couplings should lie in this range; as withthe Z case their contributions to Higgs decays can be ignored.This case also includes corrections to the Yukawa couplings of the Higgs, which would affectthe fermion masses as well as Higgs decays

(Oeϕ)pr = |ϕ|2�perϕ, (Ouϕ)pr = |ϕ|2qpur ϕ, (Odϕ)pr = |ϕ|2qp drϕ, (11)

where the fermions carry generational indices. If one simply replaces the Higgs field by itsvacuum expectation value, the contributions of these operators cannot be distinguished fromthe SM contribution. Therefore, the GIM mechanism will continue to work, as well as testsof CKM unitarity.12 However, they affect the Higgs decays to fermion pairs differently thanSM Yukawa couplings, since these are cubic in the scalar doublet whereas Yukawa couplingsare linear. Thus, comparing the Higgs decay rates to the fermion masses may provide a goodtest for the presence of these operators. At present, it has not yet been determined whetherthe SM couplings to the observed Higgs are proportional to the mass, so there is no constrainton the operator coefficients fψϕ .Even with much increased precision, the projected limits on the effective operator coeffi-cients may be harder to interpret (e.g. extracting limits on the scale of new physics) than onemight think, depending on the nature of the underlying theory. In the SM, the vanishing of afermion mass give rise to an enhanced chiral symmetry. If that were a property of the HDOas well, as is commonly believed to be the case, then the coefficient of these operators oughtto be proportional to at least the first power of the corresponding Yukawa coupling. Thus,even fairly inaccurate measurements that limit the size of these kinds of corrections to Higgsdecay would be a good indicator of whether this hypothesis is correct.

4. Loop-generated operators:In most applications, operators that are necessarily generated by heavy-particle loops aredisregarded as being subdominant, or because their effects are expected to be difficult to

12 See the review by Blucher and Marciano in [24].

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 799

detect experimentally. The rare decays H → γ γ , Zγ , GG (where G represents a gluon)present exceptions since the SM contributions are themselves loop-generated [30,31]. Inparticular, the experimental precision for H → γ γ may eventually reach the level neededto detect or set limits on deviations generated by the heavy physics. These deviations aregenerated by (i) modifications of the vertices involved in the SM loops (such as the Htt

coupling) and/or (ii) contributions from operators generated by heavy-physics loops. Theformer are listed above, the latter are the following:

OφX = 1

2|φ|2XμνX

μν, X = {GA,WI ,B

}; OWB = (φ†τ Iφ

)BμνWI

μν. (12)

There are also operators involving the dual tensors Xμν , but these do not interfere with theSM amplitudes, and so generate contributions of order Λ−4 or smaller. The coefficients ofthese operators are suppressed by a loop factor 1/(16π2); each field X is also accompaniedby the corresponding gauge coupling constant since gauge fields couple universally.

Summarizing: in the following, given the precision anticipated for LHC experiments studyingHiggs decays, we will ignore potential effects of the operator OϕD and those of type ψ2ϕ2D,viz., Oϕψ and Oϕud . It is worth noting that these same operators generate “contact” vertices ofthe form HZee and HWeν that could contribute to H → WW ∗,ZZ∗ decays, so these too canbe ignored.

Thus, the only dimension-six operators to be retained for Higgs decays at LHC are O∂ϕ andOψϕ . Recall that in the SM, the same transformation that diagonalizes the fermion masses willdiagonalize the Higgs couplings, so there remain no (quark) flavor-changing couplings. Whenthe operators (Oψϕ)pr are introduced however, the diagonalization of the fermion mass matrixwill not, in general, remove the flavor-changing Higgs couplings. The corresponding amplitudes,however, will be suppressed by a factor of ε, and therefore, the decay rates by ε2. Concerningthe flavor-diagonal operators, we will assume that their coefficients are proportional to the corre-sponding fermion masses, so that quantities of the form fψϕ/mψ have a finite limit as mψ → 0.This naturality assumption also makes these effects much harder to observe.

4. Implications for LHC experiments

We now determine the manner in which new heavy, decoupling physics can affect these decayswhile taking into consideration the limitations from existing data, as described above.

The operators (Oψϕ)pp, ψ = ep,up, dp , together with O∂φ , modify the determination of theYukawa couplings, which are not yet much experimentally constrained, as well as the Higgsdecays into fermions as follows:

Γ (H → ψψ) = κ2ψΓSM(H → ψψ), κ2

ψ =(

1 − f∂φε +√

2v

mψ

fψϕε

). (13)

Before proceeding further, some comments are in order concerning the decays referred to bythe CMS and ATLAS Collaborations as H → ZZ∗ and H → WW ∗. These are a shorthand forthe average of the leptonic final states in which the virtual Z∗ decays into either e+e− or μ+μ−and in which the W ∗ decays into either eνe or μνμ. If we consider the actual S-matrix elementsfor these processes, we find, e.g., the decay to a Z plus a lepton pair receives contributions from3 diagrams:

800 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

The vertices here are intended to include the sum of the SM couplings and the correspondingPTG dimension-six operators. The first has a Z internal line and, in addition to the SM contribu-tions, is affected by O∂φ . The third graph involves a contact HZee or HZμμ interaction, which,we have argued in the previous section, can be neglected.

The second diagram has an e or μ internal line. As emphasized earlier, the first two diagramsare not gauge-invariant in general, not even in the SM. However, that gauge dependence is as-sociated with the nonzero fermion mass, and the Yukawa couplings of the e and μ make a tinycontribution to these decays. We further assumed that the coefficients fψϕ were proportional tothe fermion mass. If we set the mass zero, then the second diagram vanishes, and the first be-comes gauge-invariant. Stated otherwise, in the limit of vanishing fermion mass, both the vectorand the axial-vector currents are conserved (the latter in the Goldstone mode).

Whether the preceding arguments remain true for the τ -lepton remains to be seen, but thedecay Z → ττ agrees with the SM to the same accuracy as for decays to ee and μμ.

For whatever reasons, whether because of this approximate chiral symmetry or because theirthreshold Λ is very large, the HDOs that could give rise to δT give negligible corrections as well.

As a result, only the effects from O∂φ remain unconstrained, so we obtain

Γ(H → ZZ∗) = κ2

ZΓSM(H → ZZ∗), κ2

Z = (1 − f∂φε). (14)

A similar discussion applies to the H → WW ∗ mode: the effects generated by possible devi-ations form the SM in the W�ν couplings are well below the current experimental precision towhich this decay mode is measured. So we find

Γ(H → WW ∗) = κ2

WΓSM(H → WW ∗), κ2

W = (1 − f∂φε). (15)

The expected modification for both of these decay widths are the same because the contributionsfrom O∂φ respect custodial symmetry, so the ratio Γ (H → ZZ∗)/Γ (H → WW ∗) equals theirStandard Model value to an accuracy of 1% or less. If, on the contrary, this ratio is observedto differ markedly from 1, it is likely there will be other particles whose mass scale are alsogenerated primarily by electroweak symmetry-breaking, as occur in models with more than oneHiggs doublet and in supersymmetric models.

Finally, we consider briefly three rare but important decays.13

1. The H → γ γ mode receives contributions both from tree-level modifications to the Htt andHWW couplings, as well as from the loop-induced effective operators OφX for X = WI , B

in (12). In order to display explicitly the loop nature of these operators and including the factthat gauge bosons couple universally, we will write

fφW = g2

16π2fW , fφB = g′ 2

16π2fB, (16)

13 In obtaining the numbers below, we will substitute the following values for the top and Higgs masses and for the SMvacuum expectation value: mt = 173.5 GeV, mH = 125 GeV, v = 246.22 GeV.

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 801

so that these contributions to the effective Lagrangian become

L(γ γ )

eff-loop = 1

Λ2(fφWOφW + fφBOφB) = ε

v

α

4πfγ γ

1

2HFμνF

μν, (17)

where fγ γ = fW + fB . Using the standard expressions for the top and W loop contributions[32], we find

Γ (H → γ γ ) = κ2γ γ ΓSM(H → γ γ ),

κ2γ γ = 1 − f∂φε + 0.30fγ γ ε + 0.28ftϕε. (18)

2. The H → Zγ mode also receives contributions from possible effective operator modifica-tions of the Htt and HWW vertices, as well as from OφXX = WI , B and OWB in (12) thatgenerate

L(Zγ )

1 loop = eg

16π2

v

Λ2fZγ FμνZ

μν, (19)

where

fZγ = 16π2

eg

[1

2(fφW − fφB)s2w − fWBc2w

], (20)

where s2w (c2w) denotes the sine (cosine) of twice the weak-mixing angle. Using the knownexpressions for the loop factors we find

Γ (H → Zγ ) = κ2Zγ ΓSM(H → Zγ ),

κ2Zγ = 1 − f∂φε + 1.82fZγ ε + 1.46ftϕε. (21)

3. Finally, the H → GG mode receives contributions from Otϕ as well as from OφG in (12).Writing the coefficient of the latter as fφG = g2

s fGG/(16π2), where gs is the SU(3)colorgauge coupling constant,

Γ (H → GG) = κ2GGΓSM(H → GG),

κ2GG = 1 − f∂φε + 2.91fGGε + 4ftϕε. (22)

This mode can potentially receive significant radiative corrections, however, explicit evalu-ation show that these are large only for mH > 2mt [33], which is not the case. (Radiativecorrections to the lighter quark modes are large, however, all contributions to the width fromlight quarks are suppressed by a factor (mq/v)2, and can be ignored.)

4.1. Branching ratios and production cross section

For mH ∼ 125 GeV, the main decays of the SM Higgs are into the bb (58%) and WW ∗ (21%)channels, but the GG (9%), ττ (6%), cc (3%) and ZZ∗ (3%) are also significant

Γ (H) = κ2H ΓSM(H), κ2

H = 1 − f∂φε + βε, (23)

where, using the Standard Model values available at [34],

β =∑

ψ=b,c,τ

fψϕ

√2vB(H → ψψ)

mψ

+ (2.91fGG + 4ftϕ)B(H → GG)

= 43.115fbϕ + 7.947fcϕ + 12.385fτϕ + 0.343ftϕ + 0.249fGG. (24)

802 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

Then, for any final state ξ , the branching ratio B is related to the SM ratio BSM as

B(H → ξ) = κ2ξ

κ2H

BSM(H → ξ). (25)

Specifically,

B(H → ψψ) =(

1 +√

2v

mψ

fψϕε − βε

)BSM(H → ψψ),

B(H → Z��) = (1 − βε)BSM(H → Z��),

B(H → W�ν) = (1 − βε)BSM(H → W�ν),

B(H → γ γ ) = (1 + 0.30fγ γ ε + 0.28ftϕε − βε)BSM(H → γ γ ),

B(H → Zγ ) = (1 + 1.82fZγ ε + 1.46ftϕε − βε)BSM(H → Zγ ),

B(H → GG) = (1 + 2.91fGGε + 4ftϕε − βε)BSM(H → GG). (26)

Note that the ratio B(H → Z��)/B(H → W�ν) is expected to have deviations below 1% fromthe SM value. Should this prove not to be the case, it would provide another strong indication ofthe presence of other light particles that affect these decays.

Although the decay mode H → GG has not been measured, the main contributions to theproduction cross section is in fact the inverse process of gluon fusion. Hence, to a good approxi-mation we have

σ prod κ2GGσ

prodSM . (27)

That might be probed, although it suffers from the usual difficulties in determining the absolutenormalization of a cross section.

The above new physics corrections are of order ε, which for Λ > 1 TeV is smaller than 0.1.This provides a measure of the precision that LHC experiments will need to reach in order toprobe physics at this scale (though sometimes a precision of 10% might suffice, depending onhow large the coefficients fi and the numerical coefficients multiplying them are). None of therelevant experiments have (yet) reached this level, in fact, current experimental precision in thesedecays allows only the exclusion of new physics at scales that have already been probed directly.

For general types of heavy physics the current experimental data does not allow for a completedetermination of the effective Lagrangian parameters contributing to the various decay modes.There is, however, one specific scenario where the data is sufficiently restrictive, namely, forthose types of heavy physics where no effective operator is generated at tree level; we refer tothis possibility as the all loop-generated scenario or ALGS.14 In this scenario, coefficients fψφ

are all ∼1/(16π2) but fGG,γ γ,Zγ remain O(1), so that the former can be ignored:

σ prod∣∣ALGS = (1 + 2.91fGGε)σ

prodSM ,

B(H → V V ∗)∣∣

ALGS = (1 − 0.249fGGε)BSM(H → V V ∗), V = W,Z,

B(H → γ γ )∣∣ALGS = (1 + 0.3fγ γ ε − 0.249fGGε)BSM(H → γ γ ). (28)

14 There is a significant class of SM extensions that fall in this category. They include any model with a symmetryunder which all heavy excitations transform non-trivially but the SM particles are singlets; this is the case for a varietyof supersymmetric and higher-dimensional models.

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 803

Using the existing experimental limits [35]

σ

σSM=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

ATLAS CMS channel

1.6 ± 0.3, 0.77 ± 0.27, H → γ γ,

1.0 ± 0.3, 0.68 ± 0.20, H → WW ∗,1.5 ± 0.4, 0.92 ± 0.28, H → ZZ∗,

(29)

we find

fGG = +1 ⇒ ΛALGS >

{520 GeV, ATLAS,

1420 GeV, CMS,

fGG = −1 ⇒ ΛALGS >

{730 GeV, ATLAS,

500 GeV, CMS,

fγ γ = +1 ⇒ ΛALGS >

{110 GeV, ATLAS,

140 GeV, CMS,

fγ γ = −1 ⇒ ΛALGS >

{174 GeV, ATLAS,

150 GeV, CMS,(30)

where we used 2-standard deviation intervals and the fact that the new physics effects in the ZZ∗and WW ∗ decay modes (14), (15) are the same. These bounds are competitive with the onesderived from the LEP data (within the ALGS scenario).

For general scenarios the current Higgs data imposes only weak constraints. Some bounds canbe derived by making additional and, in general unjustified, assumptions (such as the absence ofcancellations among the various effective contributions to a given observable). Still, it is to beexpected that as more Higgs decay and production modes are measured, it will become possibleto bound the complete set of coefficients and obtain limits on the scale of new physics in moregeneral situations. It is nonetheless remarkable that useful bounds on Λ for an interesting classof models can be obtained using the limited data currently available.

5. Conclusion

The purpose of this paper was to extend the analysis of Ref. [9] to include couplings of theSM Higgs, especially those relevant to LHC measurements, in a model-independent fashion,taking into account the existing constraints on breaking of the custodial SU(2) symmetry ofthe Higgs sector of the SM. Implications for specific branching ratios were given in terms ofthe coefficients of various dimension-six operators. For the types of heavy physics that do notgenerate tree-level effective operators, the analysis provides unambiguous limits on the scale Λ;these are competitive with those obtained at other colliders (though they do not necessarily referto the same types of heavy physics). It is noteworthy that with the limited data available suchconstraints can be derived.

Since the Higgs discovery a large number of papers have appeared that discuss various aspectsof the effective Lagrangian terms involving this new particle [7,8]. Among these publications,there are various that do not separate PTG and LG operators [8], so the constraints obtained onthe effective Lagrangian parameters are difficult to map into physical quantities (such as Λ).Other publications [7] are concerned with chiral effective theories associated with a strongly-interacting new physics, or with a generalized analysis of all operators involved. For this last

804 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

set of investigations limits are obtained only after adopting additional assumptions (such as thatonly one effective Lagrangian operator is present at a time or that some coefficients are equal).

The expressions presented show that any deviation of the couplings gHWW or gHZZ from theSM, at a level of accuracy observable by LHC experiments (∼10% or higher), can be explainedonly by having ε ∼ 1, which corresponds to new physics at a scale below 1 TeV. If the ATLASenhancement in the γ γ mode is verified, then this is strengthened considerably; it seems asif only new physics around the electroweak scale could account for this effect. Thus, we havesharpened the contrast between models that modify the SM by introduction of some higher massscale and those, for example, having more than one Higgs doublet contributing to the weak scalevacuum expectation value v; the current anomaly in the photon mode would then belong tothe second possibility. As the data improve, it will be exciting to observe how BSM physics isfirst manifested. We hope this analysis provides another tool by which this conclusion may behastened.

Acknowledgements

The research of one of us (M.B.E.) was supported in part by the National Science Foundationunder Grant No. NSF PHY11-25915.

Appendix A. Dimension-six basis operators for the SM

Tables 1 and 2 are taken from [21], by permission of the authors. We changed the operatornames from Q to O to conform to the present conventions.

Table 1Dimension-six operators other than the four-fermion ones.

X3 (LG) ϕ6 and ϕ4D2 (PTG) ψ2ϕ3 (PTG)

OG f ABCGAνμ G

Bρν G

Cμρ Oϕ (ϕ†ϕ)3 Oeϕ (ϕ†ϕ)(lperϕ)

OG f ABCGAνμ G

Bρν G

Cμρ Oϕ� (ϕ†ϕ)�(ϕ†ϕ) Ouϕ (ϕ†ϕ)(qpur ϕ)

OW εIJKWIνμ W

Jρν W

Kμρ OϕD (ϕ†Dμϕ)�(ϕ†Dμϕ) Odϕ (ϕ†ϕ)(qpdrϕ)

OW εIJKWIνμ W

Jρν W

Kμρ

X2ϕ2 (LG) ψ2Xϕ (LG) ψ2ϕ2D (PTG)

OϕG ϕ†ϕ GAμνGAμν OeW (lpσμνer )τ

I ϕWIμν O(1)

ϕl(ϕ†i

←→D μ ϕ)(lpγ μlr )

OϕG ϕ†ϕ GAμνGAμν OeB (lpσμνer )ϕBμν O(3)

ϕl(ϕ†i

←→D I

μ ϕ)(lpτ I γ μlr )

OϕW ϕ†ϕ WIμνWIμν OuG (qpσμνT Aur )ϕ GA

μν Oϕe (ϕ†i←→D μ ϕ)(epγ μer )

OϕW ϕ†ϕ WIμνWIμν OuW (qpσμνur )τ

I ϕ WIμν O(1)

ϕq (ϕ†i←→D μ ϕ)(qpγ μqr )

OϕB ϕ†ϕ BμνBμν OuB (qpσμνur )ϕ Bμν O(3)ϕq (ϕ†i

←→D I

μ ϕ)(qpτ I γ μqr )

OϕB ϕ†ϕ BμνBμν OdG (qpσμνT Adr )ϕ GAμν Oϕu (ϕ†i

←→D μ ϕ)(upγ μur )

OϕWB ϕ†τ I ϕ WIμνBμν OdW (qpσμνdr )τ

I ϕ WIμν Oϕd (ϕ†i

←→D μ ϕ)(dpγ μdr )

OϕWB ϕ†τ I ϕ W IμνBμν OdB (qpσμνdr )ϕ Bμν Oϕud i(ϕ†Dμϕ)(upγ μdr )

M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806 805

Table 2Four-fermion operators conserving baryon number. All are PTG.

(LL)(LL) (RR)(RR) (LL)(RR)

Oll (lpγμlr )(lsγμlt ) Oee (epγμer )(esγ

μet ) Ole (lpγμlr )(esγμet )

O(1)qq (qpγμqr )(qsγ

μqt ) Ouu (upγμur )(usγμut ) Olu (lpγμlr )(usγ

μut )

O(3)qq (qpγμτI qr )(qsγ

μτI qt ) Odd (dpγμdr )(dsγμdt ) Old (lpγμlr )(dsγ

μdt )

O(1)lq

(lpγμlr )(qsγμqt ) Oeu (epγμer )(usγ

μut ) Oqe (qpγμqr )(esγμet )

O(3)lq

(lpγμτI lr )(qsγμτI qt ) Oed (epγμer )(dsγ

μdt ) O(1)qu (qpγμqr )(usγ

μut )

O(1)ud

(upγμur )(dsγμdt ) O(8)

qu (qpγμT Aqr )(usγμT Aut )

O(8)ud

(upγμT Aur )(dsγμT Adt ) O(1)

qd(qpγμqr )(dsγ

μdt )

O(8)qd

(qpγμT Aqr )(dsγμT Adt )

(LR)(RL) and (LR)(LR)

Oledq (ljper )(dsq

jt )

O(1)quqd

(qjpur )εjk(qk

s dt )

O(8)quqd

(qjpT Aur )εjk(qk

s T Adt )

O(1)lequ

(ljper )εjk(qk

s ut )

O(3)lequ

(ljpσμνer )εjk(qk

s σμνut )

References

[1] G. Aad, et al., ATLAS Collaboration, Phys. Lett. B 716 (2012) 1, arXiv:1207.7214 [hep-ex].[2] S. Chatrchyan, et al., CMS Collaboration, Phys. Lett. B 716 (2012) 30, arXiv:1207.7235 [hep-ex].[3] CDF Collaboration, D0 Collaboration, Tevatron New Physics, Higgs Working Group, arXiv:1207.0449 [hep-ex].[4] M. Flechl, et al., CMS Collaboration, arXiv:1307.4589 [hep-ex];

M. Baak, et al., GFitter Group, Eur. Phys. J. C 72 (2012) 2003, arXiv:1107.0975 [hep-ph];A. Djouadi, J. Phys. Conf. Ser. 447 (2013) 012002.

[5] P.M. Ferreira, et al., arXiv:1211.3131 [hep-ph].[6] G. Belanger, et al., arXiv:1212.5244 [hep-ph].[7] A. Pomarol, F. Riva, arXiv:1308.2803 [hep-ph];

H. Mebane, et al., arXiv:1306.3380 [hep-ph];G. Isidori, A.V. Manohar, M. Trott, arXiv:1305.0663 [hep-ph];B. Dumont, S. Fichet, G. von Gersdorff, J. High Energy Phys. 1307 (2013) 065, arXiv:1304.3369 [hep-ph];R. Contino, et al., J. High Energy Phys. 1307 (2013) 035, arXiv:1303.3876 [hep-ph];J. Elias-Miro, et al., arXiv:1302.5661 [hep-ph];T. Corbett, et al., Phys. Rev. D 86 (2012) 075013, arXiv:1207.1344 [hep-ph];G. Cacciapaglia, et al., J. High Energy Phys. 1303 (2013) 029, arXiv:1210.8120 [hep-ph];G. Passarino, Nucl. Phys. B 868 (2013) 416, arXiv:1209.5538 [hep-ph].

[8] E. Masso, V. Sanz, Phys. Rev. D 87 (2013) 033001, arXiv:1211.1320 [hep-ph];A. Hayreter, G. Valencia, arXiv:1304.6976 [hep-ph];W.-F. Chang, W.-P. Pan, F. Xu, Phys. Rev. D 88 (2013) 033004, arXiv:1303.7035 [hep-ph];P. Agrawal, S. Mitra, A. Shivaji, arXiv:1211.4362 [hep-ph];S. Kanemura, K. Tsumura, Eur. Phys. J. C 63 (2009) 11, arXiv:0810.0433 [hep-ph];K. Nishiwaki, S. Niyogi, A. Shivaji, arXiv:1309.6907 [hep-ph];A. Senol, Nucl. Phys. B 873 (2013) 293, arXiv:1212.6869 [hep-ph];D. Choudhury, R. Islam, A. Kundu, arXiv:1212.4652 [hep-ph];D. Choudhury, P. Saha, J. High Energy Phys. 1208 (2012) 144, arXiv:1201.4130 [hep-ph].

806 M.B. Einhorn, J. Wudka / Nuclear Physics B 877 (2013) 792–806

[9] C. Arzt, M.B. Einhorn, J. Wudka, Nucl. Phys. B 433 (1995) 41, arXiv:hep-ph/9405214.[10] A. De Rujula, et al., Nucl. Phys. B 384 (1992) 3.[11] K. Hagiwara, et al., Phys. Rev. D 48 (1993) 2182.[12] For reviews, see J. Ellison, J. Wudka, Annu. Rev. Nucl. Part. Sci. 48 (1998) 33, arXiv:hep-ph/9804322;

A. Pich, Rep. Prog. Phys. 58 (1995) 563, arXiv:hep-ph/9502366.[13] A. Pich, I. Rosell, J.J. Sanz-Cillero, Phys. Rev. Lett. 110 (2013) 181801, arXiv:1212.6769 [hep-ph];

R. Alonso, et al., Phys. Lett. B 722 (2013) 330, arXiv:1212.3305 [hep-ph].[14] F. Zwirner, Theory summary, in: XLVIIIth Rencontres de Moriond, Electroweak Interactions and Unified Theories,

9 March 2013.[15] H. Georgi, Annu. Rev. Nucl. Part. Sci. 43 (1993) 209.[16] S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, Cambridge, UK, 1996, 511 p.[17] L. Alvarez-Gaume, M.A. Vazquez-Mozo, Lect. Notes Phys. 839 (2012) 1.[18] A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, Princeton, NJ, 2010, 576 p.[19] M.B. Einhorn, J. Wudka, arXiv:1307.0478 [hep-ph].[20] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566.[21] B. Grzadkowski, et al., J. High Energy Phys. 1010 (2010) 085, arXiv:1008.4884 [hep-ph].[22] F. Bonnet, et al., Phys. Rev. D 85 (2012) 035016, arXiv:1105.5140 [hep-ph].[23] F. Bonnet, et al., Phys. Rev. D 86 (2012) 093014, arXiv:1207.4599 [hep-ph].[24] J. Beringer, et al., Particle Data Group Collaboration, Phys. Rev. D 86 (2012) 010001.[25] W. Buchmuller, D. Wyler, Nucl. Phys. B 268 (1986) 621.[26] M.B. Einhorn, The Standard Model Higgs Boson, North-Holland, Amsterdam, 1991, 390 p.[27] See, e.g., M.B. Einhorn, D.R.T. Jones, M.J.G. Veltman, Nucl. Phys. B 191 (1981) 146, and references therein.[28] M.E. Peskin, T. Takeuchi, Phys. Rev. D 46 (1992) 381.[29] For details, see G. Sanchez-Colon, J. Wudka, Phys. Lett. B 432 (1998) 383, arXiv:hep-ph/9805366.[30] J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl. Phys. B 106 (1976) 292.[31] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304.[32] M.A. Shifman, et al., Sov. J. Nucl. Phys. 30 (1979) 711, Yad. Fiz. 30 (1979) 1368;

For a pedagogical review see: L.B. Okun, Leptons and Quarks, North-Holland, Amsterdam, 1982.[33] M. Spira, et al., Nucl. Phys. B 453 (1995) 17, arXiv:hep-ph/9504378.[34] https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CERNYellowReportPageBR2.[35] G. Aad, et al. ATLAS-CONF-2013-034;

https://twiki.cern.ch/twiki/pub/CMSPublic/PhysicsResultsHIG/sqr_mlz_ccc_mH125.7_decay_noc.png.