Pseudoobservables in Higgs decays
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Abstract
We define a set of pseudoobservables characterizing the properties of Higgs decays in generic extensions of the Standard Model with no new particles below the Higgs mass. The pseudoobservables can be determined from experimental data, providing a systematic generalization of the “\(\kappa \)framework” so far adopted by the LHC experiments. The pseudoobservables are defined from onshell decay amplitudes, allow for a systematic inclusion of higherorder QED and QCD corrections, and can be computed in any Effective Field Theory (EFT) approach to Higgs physics. We analyze the reduction of the number of independent pseudoobservables following from the hypotheses of lepton universality, CP invariance, custodial symmetry, and linearly realized electroweak symmetry breaking. We outline the importance of kinematical studies of \(h\rightarrow 4\ell \) decays for the extraction of such parameters and present their predictions in the linear EFT framework.
1 Introduction
After the discovery phase [1, 2], Higgs physics is entering the era of precision measurements. Characterizing the properties of this particle with high precision, and possibly with the least theoretical bias, is of the utmost importance in order to investigate the nature of physics beyond the Standard Model (SM).
Several phenomenological analyses about the effective couplings of the Higgs boson to SM fields have appeared after its discovery in 2012 (see e.g. Ref. [3, 4, 5, 6, 7, 8, 9]). These analyses were mainly based on the socalled “\(\kappa \)framework” [10] or “signalstrength” results reported by ATLAS and CMS [11, 12]: the experimental determination of a single parameter, for each production or decay channel, characterizing the ratio between the observed rates and those expected within the SM. While this approach has been quite useful for a first characterization of the properties of the newly discovered particle, and it was appropriate given the low statistics so far available, it is insufficient in view of more precise studies, especially for channels with nontrivial kinematical distributions. The purpose of the present paper is to provide a systematic generalization of the “\(\kappa \)framework” suitable for highprecision studies of onshell Higgs decays.
Motivated by present Higgs data, we work under the hypothesis that \(h(125)\) is a spin zero particle. We also assume that there is no new particle with mass below (or around) \(m_h \simeq 125 ~ \text {GeV}\) able to provide significant kinematical distortions of the Higgs decays to SM particles. In other words, we assume to be in a regime where the Effective Field Theory (EFT) approach to Higgs physics is applicable. However, contrary to existing EFT studies, we keep our analysis as general as possible, without specifying many details about the underlying EFT. In particular, we do not specify if the \(h(125)\) state is part of an \(\text {SU}(2)_{ L}\) doublet (socalled linear EFT approach), or if \(h(125)\) is the mass eigenstate resulting from a more complicated symmetrybreaking sector, allowing an effective decoupling of \(h\) from the Goldstoneboson components of the \(\text {SU}(2)_{ L} \times \text {U}(1)_{ Y}/ \text {U}(1)_\mathrm{em}\) symmetry breaking (socalled nonlinear EFT approach). We also do not impose global symmetry hypotheses such as lepton universality, CP invariance, and custodial symmetry. Rather, we discuss how such hypotheses can be tested from Higgs data. The only key assumption we make is to neglect terms in the decay amplitudes that receive nonvanishing treelevel contributions from local operators with dimension greater than six (\(D>6\)), as specified in detail in the following.
Under such general assumptions it is possible to define a limited set of pseudoobservables that can be directly determined from experimental data on Higgs physics and that encode all possible New Physics (NP) effects. These pseudoobservables are the natural generalization of the “\(\kappa \)framework” so far adopted by the LHC experiments [10], and an extension of the pseudoobservables employed to characterize NP effects in \(Z\) physics at LEP [13, 14]. The pseudoobservables are indeed defined at the amplitude level, allowing for a systematic inclusion of higherorder QED and QCD corrections: this leads to an accurate theoretical description of Higgsdecay amplitudes that recovers the best uptodate SM predictions in absence of NP effects. The pseudoobservables thus determined from Higgsphysics data can be computed in specific EFT approaches and, depending on the EFT employed, can possibly be correlated with nonHiggsphysics observables for specific tests of the EFT approach.
The paper is organized as follows: in Sect. 2 we present a general discussion of Higgsdecay amplitudes and pseudoobservables. In Sect. 3 we define the pseudoobservables characterizing Higgs decays mediated by electroweak gauge bosons. In Sects. 4 and 5 we discuss the SM limit, the parameter counting, and the reduction of the number of independent pseudoobservables following from the hypotheses of lepton universality, CP invariance and custodial symmetry. In Sect. 6 we present a phenomenological analysis of the \(h\rightarrow 2e2\mu \) channel, focusing on the impact and the possible determination of the \(h\rightarrow Z \bar{\ell }\ell \) contact terms. The results are summarized in the Conclusions. Appendix A contains the mapping between the pseudoobservables introduced in Sect. 3 and the Wilson coefficients of \(D=6\) operators in the linear EFT approach. Appendix B contains an extended discussion of the constraints following from custodial symmetry.
2 General considerations
 I.
helicityviolating decays into a pair of onshell fermions (\(\bar{b} b \), \(\tau ^+\tau ^, \ldots \));
 II.
helicityconserving decays to four fermions, two fermions and a (hard) photon, and two photons (\(4\ell \), \(2\ell 2\nu \), \(\ell ^+\ell ^ \gamma \), \(\gamma \gamma \), ...).
An early attempt to provide a general EFTinspired description of \(h\rightarrow 4\ell \) decay amplitudes has been presented in Refs. [15, 16]. Our work provides a generalization of the parametrization proposed there, taking into account also the subleading effects of \(Z\gamma \) and \(\gamma \gamma \) intermediate states. We will also pay particular attention to a consistent separation of the pseudoobservables accessible in Higgs decays from those accessible via onshell \(Z\) or \(W\) decays, defined in Sect. 2.1. From this point of view, our approach has some similarities with the one recently proposed in Ref. [17] (see also Ref. [18]). However, we stress two conceptual differences with respect to Ref. [17]:
(1) our pseudoobservables are defined directly from the onshell decay amplitudes and, as such, are unambiguously related to observable distributions;
(2) we make no assumptions as regards custodial symmetry and \(\text {SU}(2)_{ L}\) properties of the \(h\) particle.
As anticipated, the only key hypothesis we employ is to neglect contributions to Higgsdecay amplitudes corresponding to local interaction terms of \(D>6\) after electroweak symmetry breaking. More precisely, we employ the following simple power counting for each interaction term, based on its canonical dimension: \(h\), gauge bosons, and derivatives (momenta) count as 1, while fermions count as \(3/2\). With this counting we systematically neglect interaction terms with dimension \(D>6\). This implies that our decomposition is able to accommodate all the effects generated, at tree level, by the \(D=6\) effective Lagrangian in the linear EFT framework (or the nexttoleading order terms in the expansion). Similarly, in the generic nonlinear EFT framework, our decomposition is able to accommodate all the nexttoleading order terms in the expansion (disregarding singleHiggs interactions with \(D\ge 7\)). Even if the predicted size of each pseudoobservable varies depending on the specific EFT and its UV completion, the fact that interaction terms corresponding to higherdimensional operators can be neglected is general (assuming no light NP). Note also that while the decomposition is able to describe the effects generated at a given order in the EFT expansion, the pseudoobservables are defined by the kinematical decomposition of the onshell decay amplitudes and, as such, they are well defined independently of the EFT expansion.
 III.
helicityviolating amplitudes resulting from effective dipole interactions of the Higgs field to (light) fermions and electroweak gauge bosons;
 IV.
fourquark final states resulting from the effective coupling of the Higgs to gluons.
The first category is expected to be suppressed by light fermion masses in most realistic models and, independently of that, it does not interfere with the leading SM amplitudes in the limit of vanishing fermion masses. More precisely, we can neglect such amplitudes in the limit where we assume an exact \(\text {U}(1)_f\) symmetry acting on each of the light fermion species.^{1} Note that such symmetry is a small subset of the full \(\text {U}(3)^5\) flavor symmetry often advocated in the EFT context: imposing such reduced symmetry group we can allow violations of lepton universality in the \(h\rightarrow 4\ell \) amplitudes (\(\ell = e,\mu \)), while consistently neglecting the helicityviolating dipole amplitudes and leptonflavor violating interactions.
The second category is hardly accessible from the experimental point of view: the \(hgg\) effective coupling is essential to determine the Higgs production cross section, but it cannot be identified via a wellmeasured Higgs partial decay width.
2.1 Pseudoobservables in \(Z\rightarrow f \bar{f} \) and \(W\rightarrow f \bar{f} \) decays
2.2 Pseudoobservables in \(h\rightarrow f \bar{f}\) decays
3 Higgs decays mediated by electroweak gauge bosons
In this section we provide a unified decomposition of the Higgs decay amplitudes into four fermions (\(h\rightarrow 4 f\)), a fermion–antifermion pair and one hard photon (\(h\rightarrow f\bar{f} \gamma \)), and two photons (\(h\rightarrow \gamma \gamma \)). The \(h\rightarrow 4 f\) amplitudes are particularly interesting since they allow us to investigate the effective \(hW^+W^\) and \(hZZ\) interaction terms, which cannot be probed onshell. However, in order to extract such information in a modelindependent way, it is necessary to take into account also the possible additional contributions to \(h\rightarrow 4 f\) due to contact terms and the effective couplings of the Higgs to photons.
Within a generic EFT approach, the problem is simplified by the fact that a local interaction \(h J_f^\mu J_{f^\prime }^\nu g_{\mu \nu }\) has canonical dimension \(D = 7\). As a result, as long as we neglect operators of \(D> 6\), the correlation function in Eq. (7) is nonlocal at the electroweak scale, with at least one fermion pair generated by the propagation of one electroweak gauge boson. This allows us to decompose the \(h\rightarrow 4 f\) amplitude into a sum of neutral and chargedcurrent contributions, according to the charge of fermion current in Eq. (7), and to expand around the physical poles produced by the propagation of the SM gauge bosons (\(W, Z\), and \(\gamma \)). These two types of contributions are discussed separately in Sects. 3.1 and 3.2. The complete structure of a generic \(h\rightarrow 4 f\) amplitude is presented in Sect. 3.3.
3.1 \(h\rightarrow 4 f\) neutral currents
Note that the fact that the \(g_Z^f\) are defined from onshell \(Z\) amplitudes is essential for \(\kappa _X\) and \(\epsilon _X\) to be welldefined physical quantities (independent of the choice of the EFT basis). Indeed, the decomposition in Eqs. (8–11) contains a set of \(Z\)pole pseudoobservables \(\{ g_Z^f, m_Z, \Gamma _Z \}\), plus the lowenergy input observables \(\{G_F, \alpha _\mathrm{em}\}\), plus the set of Higgspole pseudoobservables \(\{\kappa _{ZZ},\epsilon _X \}\).
3.2 \(h\rightarrow 4f\) charged currents
3.3 \(h\rightarrow 4f\) complete decomposition
3.4 \(h\rightarrow \gamma \gamma \) and \(h\rightarrow f \bar{f} \gamma \)
4 SM values
The genuine electroweak corrections generate: (i) small corrections to the treelevel values of \(\kappa _X\) and \(\epsilon _X\) in Eq. (22); (ii) small nonlocal contributions to the form factors; (iii) further tiny corrections that cannot be cast into the general decomposition in Eqs. (8) and (12). These effects can be derived, in principle, by comparing our general decomposition with the expression of the full SM nexttoleading order \(h\rightarrow 4 f\) amplitude [20]. As noted in Ref. [21], such corrections are very small (below the 1 % level compared to the treelevel terms) and practically unobservable, except in a few notable kinematical points. In particular, the only case where such corrections are relevant is for onshell hardphoton amplitudes (given that they vanish at the tree level within the SM) or almost onshell photonexchange contributions in neutralcurrent amplitudes.
5 Parameter counting and symmetry limits
We are now ready to identify the number of independent pseudoobservables necessary to describe various sets of Higgsdecay amplitudes, under the main assumption that only terms arising at \(D\le 6\) in a generic EFT expansion are kept. We focus our attention on leptonic channels, which are more interesting from the experimental point of view.
5.1 Flavor universality
5.2 CP conservation
5.3 Custodial symmetry
Summary of the pseudoobservables relevant to describe Higgs leptonic (and \(\gamma \gamma \)) decay modes. In the second column (“Maximal Symmetry”) we show the independent pseudoobservables needed for a given set of decay modes, assuming both CP invariance and flavor universality. The additional variables needed if we relax these symmetry hypotheses are reported in the third and fourth columns. In the bottom row we show the independent pseudoobservables needed for a combined description of charged and neutral modes, under the hypothesis of custodial symmetry

5.4 Linear vs. nonlinear EFT
In the SM the Higgs boson is part of an \(\text {SU}(2)_L\) doublet \(H\) and the electroweak gauge symmetry is linearly realized. The linear effective theory is built following this assumption: higherdimensional operators are constructed in terms of the \(H\) field [25, 26, 27, 28, 29]. This implies that the physical Higgs (\(h\)) appears in operators contributing also to nonHiggs processes, and in particular to electroweak observables measured at LEP. In this context it is thus possible to provide strong bounds on some Higgs observables using LEP data [17, 30, 31, 32, 33, 34, 35, 36]. A complete modelindependent analysis of these (nonHiggs) constraints for the Higgs pseudoobservables is still missing, and we postpone it to a future work.
Assuming a linearly realized electroweak gauge symmetry provides also some relations among the Higgs pseudoobservables. These are due to an accidental custodial symmetry present in some of the \(D=6\) operators. In particular, by matching our pseudoobservables with the coefficients of the \(D=6\) operators, it turns out that the relations of Eqs. (33), (34) and (36) are always exactly satisfied [30]. This result implies that, independently of any symmetry assumption, the dynamical hypothesis of an underlying linear EFT reduces the number of relevant leptonic pseudoobservables from 20 to 14 (from 15 to 11 if flavor universality is further assumed). In Appendix A we derive these relations by an explicit matching with the operator basis of Ref. [31]. Since the relations derived involve only pseudoobservables, the result is independent of the operator basis adopted. The other two custodialsymmetry relations, Eqs. (35) and (37), are not satisfied in general in the linear EFT and turn out to be violated by nonvanishing coefficients of custodialsymmetry violating operators (see Appendix A.1).
A more general approach to Higgs physics is to build an EFT allowing an effective decoupling of \(h\) from the Goldstoneboson components of the \(\text {SU}(2)_L \,\times \, \text {U}(1)_Y/ \text {U}(1)_\mathrm{em}\) symmetry breaking. In this case the electroweak symmetry is nonlinearly realized and the effective theory is built as a derivative expansion over the cutoff [26, 27, 37, 38, 39, 40, 41, 42, 43, 44, 45]. Given that the Higgs and the symmetrybreaking vev are independent, in this EFT it is not possible to connect electroweak observables (Higgsless processes) with Higgs observables. Moreover, since the Goldstone bosons are encoded in a dimensionless field, it is possible to write many more independent \(D\le 6\) operators than in the linear case. It is easy to verify that in this context each pseudoobservable of our parameterization receives a nonvanishing treelevel contribution from an independent combination of effective operators. In particular, it is possible to build custodially violating operators [42, 43, 44, 45] that violate all the relations in Eqs. (33)–(37).
Even though electroweak precision tests and early Higgs data set strong constraints on the nonlinear construction, favoring the linearly realized EFT, it is still early to draw a definite conclusion about this point. As shown in Ref. [16], \(h\rightarrow 4\ell \) decays prove a very useful tool for settling this issue from data: if a violation of the electroweak bounds on the contact terms is measured, this will be a strong hint toward the nonlinear realization. Given the above discussion, a similar conclusion could be derived in presence of a violation of the custodialsymmetry relations in Eqs. (33), (34) and (36).
6 Differential distributions for \(h\rightarrow e^+e^ \mu ^+\mu ^\)
6.1 Analytic invariant mass distributions
The tensor structure associated with \(F_4\) of Eq. (11) does not interfere with the SM in the double differential distribution in \(q_1^2\) and \(q_2^2\). In the rest of the paper we focus on the effects due to \(\delta \kappa _{ZZ}\) and the contact terms \(\epsilon _{Z f}\), leaving a more detailed phenomenological study of the other coefficients, which should involve also the analysis of angular distributions, to a future work.
Further kinematical studies on the \(h\rightarrow 4\ell \) modes can be found in Ref. [46, 47, 48, 49, 50]. CMS performed a comprehensive study of \(h\rightarrow 4\ell \) decays with present data, in the context of \(hVV\) anomalous couplings [50]. The main differences of the latter approach with respect to our proposal is the fact that we consider as final states the onshell leptons, and we do not assume the effective interaction of these leptons to the Higgs and other SM fields to be necessarily mediated by the SM gauge bosons. This leads to a more general decomposition of the \(h\rightarrow 4\ell \) amplitude.
6.2 Higherorder SM corrections
We have validated the analytic formula for the treelevel SM prediction with the Prophecy4F Monte Carlo generator [20]. In Fig. 1, we present the normalized differential distribution in \(m_{12}\equiv \sqrt{q_1^2}\). The solid black line corresponds to the results obtained after integrating Eq. (42) over \(q_2^2\) for \(\kappa _{ZZ}=1\) and \(\epsilon _X=0\), while lowest order Prophecy4F predictions are shown with blue dots. The two predictions are in perfect agreement. Full \(\mathcal {O}(\alpha )\) electroweak corrections obtained with Prophecy4F are shown with red dots. Prophecy4F results are obtained after generating \(10^8\) weighted events using the dipole subtraction formalism for photon radiation and switching on the photon recombination which ensures sufficient inclusiveness [20]. More specifically, photons and leptons are recombined if their invariant mass is less than \(5\) GeV. We impose no cuts on the decay products.
6.3 Measuring contact terms
In order to probe the contact terms \(\epsilon _{Zf}\) and \(\epsilon _{Zf'}\) it is mandatory to exploit the differential decay distributions in \(q_1^2\) and \(q_2^2\). As an illustration, let us consider first the case of sizable deviations in one of the \(\epsilon _{Zf_R}\) and in \(\kappa _{ZZ}\), while keeping other couplings SMlike. The ratio of the total Higgsdecay rate to \(e^+e^ \mu ^+\mu ^\) with respect to the SM prediction as a function of the couplings is shown in Fig. 2a. As can be seen, a measurement of the total rate alone is not capable of resolving the contribution from the contact terms. On the contrary, in Fig. 2b we show the deviations from the SM in the normalized single differential distributions in \(m_{12}\equiv \sqrt{q_1^2}\) and \(m_{34}\equiv \sqrt{q_2^2}\) in solidblue line and dashedred line, respectively. These are obtained after fully integrating Eq. (42) over the corresponding invariant mass. As a benchmark, we set \((\kappa _{ZZ},\epsilon _{Zf_R})=(0.88,0.10)\), for which the total decay rate remains as in the SM. A good discriminating variable would be the difference between the two distributions. This measurement would mainly probe \(\epsilon _{Zf_R}\) and provide a complementary information to the one sketched in Fig. 2a. Finally, the ratio of the double differential distribution with the SM prediction is shown in Fig. 2c.
Qualitatively, the same discussion holds if both \(\epsilon _{Zf_L}\) and \(\epsilon _{Zf_R}\) are present, except for a trivial rescaling in the magnitude of the effects. For instance, if \(\epsilon _{Zf_R}=\epsilon _{Zf_L}\) the difference in the differential distributions is rescaled by the factor \(g^f_V/g^f_R\).
Somewhat different signatures are obtained if both \(\epsilon _{Zf_R}\) and \(\epsilon _{Zf'_R}\) are sizable. As an example, in Fig. 3 we consider the case \(\epsilon _{Zf_R}=\epsilon _{Zf'_R}\) which corresponds to the relation imposed by flavor universality (see Sect. 5). Similarly to case analyzed in Fig. 2, the deviations from the SM predictions are reported. As can be seen, the overall size of the effect is much smaller. As expected, in this case the single differential distributions in \(m_{12}\) and \(m_{34}\) are the same, and the double differential distribution is symmetric under \(m_{12}\leftrightarrow m_{34}\).
7 Conclusions
The experimental precision on the Higgsdecay distributions, especially those into four light leptons, is expected to significantly improve in the next few years. This will allow us to investigate in depth a wide class of possible extensions of the SM. However, to reach this goal, an accurate and sufficiently general parameterization of possible NP effects in such distributions is needed.
In this paper we have identified the complete set of pseudoobservables appearing in onshell Higgsdecay distributions in the limit of heavy NP. More precisely, we only assumed that contributions to the decay amplitudes generated by effective operators of \(D >6\) in a generic EFT approach can be neglected. The pseudoobservables we have introduced are defined by the momentum expansion of the onshell Higgsdecay amplitudes. As such, they are welldefined physical parameters that can be directly extracted from data, providing a natural generalization of the socalled “\(\kappa \)framework”. They indeed consist of four universal “\(\kappa \)like” pseudoobservables (\(\kappa _{ZZ}\), \(\kappa _{Z\gamma }\), \(\kappa _{\gamma \gamma }\), \(\kappa _{WW}\)), whose expectation is 1 within the SM, and a series of \(\epsilon _X\) parameters, whose SM expectation is zero for all practical purposes (i.e. it is well below the experimental sensitivity even in the HLLHC era). The “\(\kappa \)like” observables differ from the signal strength measurements currently reported by ATLAS and CMS, being associated to a welldefined (SMlike) kinematical distribution: they describe the (channelindependent) effective couplings of the Higgs boson to the SM gauge fields. The \(\epsilon _X\) terms encode possible nonSM effects in the kinematical distributions as well as violations of the accidental SM symmetries. The complete list of the pseudoobservables for the Higgs decays to four leptons is reported in Table 1: it ranges from a maximum of 20 independent terms, if no additional symmetry assumption is made, to a minimum of seven terms under the hypotheses of CP invariance, leptonflavor universality and custodial symmetry.
As outlined in Sect. 3, this formalism is well suited to describe all \(h\rightarrow 4 f\) decay modes: the only difference between leptonic, hadronic, and semileptonic modes (such as \(h\rightarrow 2\ell 2q\)), is the list of \(\epsilon _{V f}\) parameters (\(V=W,Z\)) contributing to the given set of decay channels. In principle, the same formalism (and the same set of pseudoobservables) can also describe in general terms NP effects (with nontrivial kinematical distortions) in the Higgs production crosssections controlled by the correlation function in Eq. (7), namely \(\sigma (pp\rightarrow hV)\) and the vectorboson fusion process. However, in this case more dynamical assumptions are needed due to the possible breakdown of the momentum expansion at large energies (see e.g. Ref. [52]). This problem is absent in the Higgsdecay amplitudes discussed in this work, where the energy scale is set by \(m_h\).
Comparing to existing experimental and phenomenological analyses of \(h\rightarrow 4 \ell \) decays, the main difference due to the use of the complete set of pseudoobservables is related to the \(\epsilon _{Vf}\) terms, which encode the contributions generated by \(hVf\bar{f}\) effective contact interactions [15]. As pointed out in Ref. [16], such terms are particularly interesting in order to discriminate from data the hypotheses of linear vs. nonlinear EFT expansion. This is so because the linear approach predicts relations between electroweak observables and \(hVf\bar{f}\) contact terms, leading to strong (and potentially falsifiable) bounds on the latter. As we have shown by means of the explicit calculation of the Higgs pseudoobservables in terms of EFT Wilson coefficients, the linear EFT approach also predicts definite relations among Higgs pseudoobservables, so that not all of them are independent. An experimental check of these relations, which involves only Higgsphysics data, would therefore offer an independent tool to possibly discriminate between the linear and the nonlinear EFT expansions.
A further interesting aspect of the contact terms (or the \(\epsilon _{Vf}\) pseudoobservables) is their potential flavor nonuniversal nature. Their experimental determination is therefore an interesting way to test, from data, the assumption of flavoruniversality in the Higgs sector (which is often assumed to hold, up to small breaking terms related to fermion masses). As we have illustrated with a few examples in the \(h\rightarrow e^+e^ \mu ^+\mu ^\) case, the extraction of such terms from data require nontrivial kinematical studies, but significant bounds could be obtained in the future with highstatistics data.
Summarizing, the framework of Higgs pseudoobservables provided in this work can capture all the physics accessible in Higgs decays if no new light state is coupled to the \(h(125)\) boson; it can be systematically improved with higherorder QCD and QED corrections, recovering the best uptodate SM predictions in absence of new physics; it can be generalized in a simple way in order to describe any onshell Higgs decay; it can be efficiently used to test the symmetries of the newphysics sector without specifying the details of the underlying Lagrangian. We advocate the use of such formalism in the era of precise Higgsboson physics, in order to shed light in a systematic and unbiased way on the structure and symmetries of possible extensions of the SM.
Footnotes
 1.
In the lepton sector \(f=e_{ L}, e_{R}, \mu _{ L}, \mu _{R}\), where the \(\text {U}(1)_{\ell _{L}}\) symmetries \((\ell =e,\mu )\) act on the \(\text {SU}(2)_{ L}\) doublets \((\ell _{ L}, \nu ^\ell _{ L})\).
 2.
In general, one could also write a righthanded coupling of \(W\) boson to quarks; however, this is forbidden in the limit of unbroken \(U(1)_{u_{R}}\times U(1)_{d_{R}}\) flavor symmetry.
 3.
In particular, LEP measurements at the \(Z\) pole allow one to set very precise constraints on the \(Z\) couplings to each charged lepton, to neutrinos (summed over all possible light species), to the \(b\), \(c\), and \(u\) quarks [14], and a common coupling to the \(s\) and \(d\) quarks. Also the \(W\) couplings to each lepton flavor, and a combination of the couplings to the light quarks can be constrained with high precision [19].
 4.
Here we generically denote by \(\epsilon _X\) the parameters \(\epsilon _{ZZ,Z\gamma , \gamma \gamma , Zf}\), and \(\epsilon ^\mathrm{CP}_{ZZ, Z\gamma , \gamma \gamma }\).
 5.
The analysis of a process involving quarks is equivalent, with the only difference that the \(\epsilon _{Wf}\) coefficients are in this case nondiagonal matrices in flavor space, as the \(g_{ud}^W\) effective couplings.
 6.We introduce here the couplings \(c_{Z\gamma } \simeq  4.85\) and \(c_{\gamma \gamma } \simeq  6.49\) [24], defined from the effective Lagrangian$$\begin{aligned} \mathcal L^\mathrm{eff} = \frac{\alpha }{4\pi } \frac{h}{v} \left( \frac{c_{Z\gamma }}{s_w c_w} Z_{\mu \nu } F^{\mu \nu } + \frac{c_{\gamma \gamma }}{2} F_{\mu \nu } F^{\mu \nu } \right) , \end{aligned}$$(23)
 7.
These approximated numerical expressions are precise at the \(1\, \%\) level for \(q_1^2 \lesssim (95~\text {GeV})^2\) in the case of \(\Delta _{\gamma \gamma }^\mathrm{SM1L}\) and \( (30 \text {GeV})^2 \lesssim q_Z^2 \lesssim (120~\text {GeV})^2\) in the case of \(\Delta _{Z \gamma }^\mathrm{SM1L}\).
 8.
The \(\text {U}(1)_X\) factor is needed only to assign the correct hypercharge \(Y = T_{R}^3 + X\) to the SM fermions.
 9.
Here and in the following we label by the index \(i=1 \ldots 3\) the three lepton generations and we denote by \(L_L^i\) the lepton doublet \((e_L^i, \nu ^i_L)^T\).
 10.
One flavor component of each of these operators is redundant. We choose \([c_{L}^{(3)\ell }]_{ee} = [c_{L}^{\ell }]_{ee} = 0\).
 11.
Under our flavor symmetry assumptions, the coefficient \(c_{LL}^{3\ell }\) contains two allowed flavor structures. Instead, we will follow the usual convention of keeping both \(\mathcal{O}_{LL}^{3\ell }\) and \(\mathcal{O}_{LL}^\ell \) but allowing only for one flavor structure, namely \(c_{ijkl}=\alpha _{ik}\delta _{ij}\delta _{kl}\).
 12.
Notice that Ref. [31] use \(g_f^Z\) for a different quantity, namely \(g_f^{Z,SM}/2\) in our notation, and that \(\delta m_W^2\) is defined in that work with the opposite sign.
 13.
We define our flavor symmetry in the basis where the downquark and charged lepton Yukawa matrices are diagonal, whereas the upquark Yukawa matrix has the form \(Y_U=V^\dagger Y_U^\mathrm{diag}\). We neglect the breaking of the symmetry induced by the Yukawa matrices but for its effect on fermion masses. See Ref. [54, (Section 3)] for a more detailed discussion.
 14.
The embedding of \(L_L\) in the bidoublet can be explicitly realized in a basis of \(2\times 2\) matrices as \(E_L = \sigma ^\alpha E_L^\alpha \). In particular we have \(E_L = \sigma ^+ \nu _L + \sigma ^{0} e_L\), where \(\sigma ^{\pm } = (\sigma ^1 \pm \sigma ^2)/2\) and \(\sigma ^{0\pm } = (\mathbf{1}_2 \pm \sigma ^3)/2\), such that \(T^3_{L}(\nu _L) = T^3_{R}(\nu _L) = T^3_{L}(e_L) = T^3_{R}(e_L) = 1/2\).
 15.
In terms of EFT operators, this redundancy is a consequence of the fact that, by using the equations of motion, one can rewrite the \(h V^\mu D^\nu V_{\mu \nu }\) operators, responsible for the \(\epsilon _{V\partial V}\) terms, as a combination of \(h V_\mu \bar{f}\gamma ^\mu f^\prime \) contact interactions and \(m_V^2 h V^\mu V_\mu \) terms.
Notes
Acknowledgments
We thank Michael DuehrssenDebling, Adam Falkowski, Andre Tinoco Mendes, and Michael Trott for useful discussions. We also thank Ilaria Brivio for comments on the manuscript. This project was partially funded by the Lyon Institute of Origins, grant ANR10LABX66 (M.G.A.). Note Added. While this work was in its final stage, Ref. [56] appeared, where relations analogous to Eqs. (35), (36) have been reported.
References
 1.G. Aad et al., ATLAS Collaboration, Phys. Lett. B 716, 1 (2012)Google Scholar
 2.S. Chatrchyan et al., CMS Collaboration, Phys. Lett. B 716, 30 (2012)Google Scholar
 3.A. Azatov, R. Contino, J. Galloway, JHEP 1204, 127 (2012). arXiv:1202.3415
 4.M. Klute, R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, Phys. Rev. Lett. 109, 101801 (2012). arXiv:1205.2699
 5.M. Montull, F. Riva, JHEP 1211, 018 (2012). arXiv:1207.1716
 6.J.R. Espinosa, C. Grojean, M. Muhlleitner, JHEP 1212, 045 (2012). arXiv:1207.1717
 7.D. Carmi, A. Falkowski, E. Kuflik, T. Volansky, J. Zupan, JHEP 1210, 196 (2012). arXiv:1207.1718
 8.P.P. Giardino, K. Kannike, I. Masina, M. Raidal, A. Strumia, JHEP 1405, 046 (2014). arXiv:1303.3570
 9.J. Ellis, T. You, JHEP 1306, 103 (2013). arXiv:1303.3879
 10.A. David et al., LHC Higgs Cross Section Working Group Collaboration, arXiv:1209.0040
 11.ATLAS Collaboration, ATLASCONF2013034, ATLASCOMCONF2013035Google Scholar
 12.CMS Collaboration, CMSPASHIG13005Google Scholar
 13.D.Y. Bardin, M. Grunewald, G. Passarino, arXiv:hepph/9902452
 14.S. Schael et al., ALEPH and DELPHI and L3 and OPAL and SLD and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavour Group Collaborations, Phys. Rep. 427, 257 (2006). arXiv:hepex/0509008
 15.G. Isidori, A.V. Manohar, M. Trott, Phys. Lett. B 728, 131 (2014). arXiv:1305.0663
 16.G. Isidori, M. Trott, JHEP 1402, 082 (2014). arXiv:1307.4051
 17.R.S. Gupta, A. Pomarol, F. Riva, arXiv:1405.0181
 18.G. Passarino, Nucl. Phys. B 868, 416 (2013). arXiv:1209.5538
 19.K.A. Olive et al., Particle Data Group Collaboration, Chin. Phys. C 38, 090001 (2014)Google Scholar
 20.A. Bredenstein, A. Denner, S. Dittmaier, M.M. Weber, Phys. Rev. D 74, 013004 (2006). arXiv:hepph/0604011
 21.Y. Chen, R. Harnik, R. VegaMorales, arXiv:1404.1336
 22.L. Bergstrom, G. Hulth, Nucl. Phys. B 259, 137 (1985) (erratumibid. B 276, 744, 1986)Google Scholar
 23.A. Djouadi, Phys. Rep. 457, 1 (2008). arXiv:hepph/0503172
 24.B. Grinstein, C.W. Murphy, D. Pirtskhalava, JHEP 1310, 077 (2013). arXiv:1305.6938
 25.W. Buchmuller, D. Wyler, Nucl. Phys. B 268, 621 (1986)CrossRefADSGoogle Scholar
 26.Z. Han, W. Skiba, Phys. Rev. D 71, 075009 (2005). arXiv:hepph/0412166
 27.B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, JHEP 1010, 085 (2010). arXiv:1008.4884
 28.A.V. Manohar, M.B. Wise, Phys. Lett. B 636, 107 (2006). arXiv:hepph/0601212
 29.G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, JHEP 0706, 045 (2007). arXiv:hepph/0703164
 30.R. Contino, M. Ghezzi, C. Grojean, M. Muhlleitner, M. Spira, JHEP 1307, 035 (2013). arXiv:1303.3876
 31.A. Pomarol, F. Riva, JHEP 1401, 151 (2014). arXiv:1308.2803
 32.A. Falkowski, F. Riva, A. Urbano, JHEP 1311, 111 (2013). arXiv:1303.1812
 33.M. Ciuchini, E. Franco, S. Mishima, L. Silvestrini, JHEP 1308, 106 (2013). arXiv:1306.4644
 34.J. Ellis, V. Sanz, T. You, JHEP 1407, 036 (2014). arXiv:1404.3667
 35.J. Ellis, V. Sanz, T. You, arXiv:1410.7703
 36.M. Beneke, D. Boito, Y.M. Wang, JHEP 1411, 028 (2014). arXiv:1406.1361
 37.T. Appelquist, C.W. Bernard, Phys. Rev. D 22, 200 (1980)CrossRefADSGoogle Scholar
 38.A.C. Longhitano, Phys. Rev. D 22, 1166 (1980)CrossRefADSGoogle Scholar
 39.A.C. Longhitano, Nucl. Phys. B 188, 118 (1981)Google Scholar
 40.F. Feruglio, Int. J. Mod. Phys. A 8, 4937 (1993). arXiv:hepph/9301281
 41.B. Grinstein, M. Trott, Phys. Rev. D 76, 073002 (2007). arXiv:0704.1505
 42.R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Phys. Lett. B 722, 330 (2013) (erratumibid. B 726, 926, 2013). arXiv:1212.3305
 43.I. Brivio et al., JHEP 1403, 024 (2014). arXiv:1311.1823
 44.G. Buchalla, O. Catá, C. Krause, Nucl. Phys. B 880, 552 (2014). arXiv:1307.5017
 45.G. Buchalla, O. Catá, C. Krause, Phys. Lett. B 731, 80 (2014). arXiv:1312.5624
 46.J.C. Romao, S. Andringa, Eur. Phys. J. C 7, 631 (1999). arXiv:hepph/9807536
 47.A. De Rujula, J. Lykken, M. Pierini, C. Rogan, M. Spiropulu, Phys. Rev. D 82, 013003 (2010). arXiv:1001.5300
 48.Y. Chen, N. Tran, R. VegaMorales, JHEP 1301, 182 (2013). arXiv:1211.1959
 49.G. Buchalla, O. Cata, G. D’Ambrosio, Eur. Phys. J. C 74 (3), 2798 (2014). arXiv:1310.2574
 50.V. Khachatryan et al., CMS Collaboration, CMSHIG14018, CERNPHEP2014265. arXiv:1411.3441
 51.O. Nicrosini, L. Trentadue, Phys. Lett. B 196, 551 (1987)CrossRefADSGoogle Scholar
 52.A. Biekoetter, A. Knochel, M. Kraemer, D. Liu, F. Riva, arXiv:1406.7320
 53.J. EliasMiro, J.R. Espinosa, E. Masso, A. Pomarol, JHEP 1311, 066 (2013). arXiv:1308.1879
 54.V. Cirigliano, J. Jenkins, M. GonzalezAlonso, Nucl. Phys. B 830, 95 (2010). arXiv:0908.1754
 55.K. Agashe, R. Contino, L. Da Rold, A. Pomarol, Phys. Lett. B 641, 62 (2006). arXiv:hepph/0605341
 56.A. Pomarol, arXiv:1412.4410 [hepph]
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