# Electroweak bounds on Higgs pseudo-observables and \(h\rightarrow 4\ell \) decays

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## Abstract

We analyze the bounds on the Higgs pseudo-observables following from electroweak constraints, under the assumption that the Higgs particle is the massive excitation of an \(\text {SU}(2)_L\) doublet. Using such bounds, detailed predictions for \(h\rightarrow 4\ell \) decay rates, dilepton spectra, and lepton-universality ratios are presented.

## Keywords

Effective Field Theory Contact Term Higgs Particle Higgs Data Wilson Coefficient## 1 Introduction

The decays of the Higgs particle, *h*(125), can be characterized by a set of pseudo-observables (PO) that describe, in great generality, possible deviations from the Standard Model (SM) in the limit of heavy New Physics (NP) [1]. These PO should be considered as independent variables in the absence of specific symmetry or dynamical assumptions; however, relations among themselves and also between Higgs and non-Higgs PO arise in specific NP frameworks. Testing if such relations are verified by data provides a systematic way to investigate the nature of the Higgs particle and, more generally, to test the underlying symmetries of physics beyond the SM.

The constraints on the Higgs PO following from the hypotheses of CP invariance, flavor universality and custodial symmetry have been discussed in Ref. [1]. These symmetries lead to a series of relations among PO that can be tested using Higgs data only. In this paper we analyze the constraints following from the hypothesis that the Higgs particle is the massive excitation of a pure \(\text {SU}(2)_L\) doublet, i.e. constraints and relations among the PO that hold in the so-called linear effective field theory (EFT) regime.

Under this assumption, the *h* field appears in the effective SM+NP Lagrangian through the combination \((v + h)^n\), where \(v \approx 246 \text { GeV}\) is the \(\text {SU}(2)_L\)-breaking vacuum expectation value. This implies that processes involving the Higgs particle can be related to electroweak (EW) precision observables that do not involve the physical Higgs boson. As pointed out in Ref. [2], testing if such relations are satisfied represents a very powerful tool to discriminate linear and non-linear EFT approaches to Higgs physics. In particular, sizable deviations from the SM in the \(h\rightarrow 4\ell \) spectra are allowed, in general, in the non-linear EFT (i.e. if *v* and *h* are decoupled) [3], while they are significantly constrained by EW precision observables in the linear EFT [4, 5, 6]. Observing sizable deviations from the SM in the \(h\rightarrow 4\ell \) spectra could therefore allow to exclude that *h* is the massive excitation of a pure \(\text {SU}(2)_L\) doublet [2] (for additional tests about the \(\text {SU}(2)_L\) properties of the *h* boson see Ref. [7]).

In order to precisely quantify the above statement, in this paper we present a systematic evaluation of the bounds on the Higgs PO following the EW constraints in the linear EFT regime, with particular attention to the PO entering both \(h\rightarrow 4\ell \) and \(h\rightarrow 2\ell 2\nu \) decays. Using such bounds, we derive predictions for \(h\rightarrow 4\ell \) decay rates, dilepton invariant-mass spectra, and lepton-universality ratios.

## 2 Relating Higgs pseudo-observables to EW observables

Given the present and near-future level of precision in Higgs physics, and the absence of any significant deviation from the SM, it is a good approximation to work at the tree level in the linear EFT, as far as NP effects are concerned. In this limit, the Higgs PO (\(\kappa _i\) and \(\epsilon _i \)) can be expressed as linear combinations of the Wilson coefficients of the EFT Lagrangian. Analogously, the EW PO, such as the *Z*- and *W*-pole effective couplings, the *W* mass, and the effective triple gauge boson couplings (TGC), can be expressed as linear combinations of the same Wilson coefficients. By inverting these relations it is possible to get rid of some of the (basis-dependent) Wilson coefficients and derive basis-independent relations between Higgs and EW PO.

*Z*and

*W*effective on-shell couplings to fermions, and the effective anomalous TGC:

*Z*and

*W*couplings to the fermion

*f*, again in an implicit \(3\times 3\) notation, \(\mathbf{1}_3\) is the identity matrix, and \(\delta g_{1,z}\) and \(\delta \kappa _\gamma \) are the effective anomalous TGC extracted from \(e^+ e^- \rightarrow W^+ W^-\) and single

*W*production [10] (see “Appendix” for the definition of the various terms).

^{1}The parameters \(\{c_\theta , s_\theta , t_\theta \}\) denote the cosine, sine and tangent of the Weinberg angle, defined as in Ref. [11].

The parameters \(\delta g^{Zf}\) relevant to this work are the leptonic *Z* couplings, which are tightly constrained by LEP-I data [6, 10, 12]. In particular, lepton flavor non-universal effects in the *Z* couplings are strongly suppressed, leading to per-mil constraints on the \(\delta g^{Z\ell }\) even in the most generic flavor scenario [13]. This is why we set these couplings to their SM values in the following. It must be stressed that the analyses of Refs. [6, 10, 13] neglect the impact of off-shell *Z* effects [14] and that of NLO terms in the EFT expansion, implicitly assuming a high EFT cutoff scale. A consistent inclusion of the latter effects may lead to a significant (relative) enlargement of the error on the \(\delta g^{Zf}\) couplings, especially in the case of a low cut-off scale for the EFT (see e.g. Ref. [15]). As we will comment in the following, we have explicitly checked that relaxing the bounds on \(\delta g^{Zf}\) to \(\mathcal {O}(1\,\%)\) does not lead to appreciable differences in our numerical analysis. The leptonic *W* couplings \(\delta g^{W\ell }\) are instead constrained only at the few percent level. In the following we consider the bounds from the non-universal fit of Ref. [13], reported also in “Appendix”.

In general, the parameters describing anomalous TGC in the effective Lagrangian are not PO [16]. Here we follow the approach of Ref. [10] where the \(e^+ e^- \rightarrow W^+ W^-\) cross-section is parameterized in terms of the effective on-shell *Z* and *W* couplings to fermions plus the three parameters \(\{\delta g_{1,z}, \delta \kappa _\gamma , \lambda _Z\}\), which therefore represent a consistent TGC PO set. The constraints on this set obtained in Ref. [10] are collected in “Appendix”. It should be stressed that a flat direction is present when all three TGC PO are included at the linear level [17] (see also [18]), which reflects into a very loose bound on \(\delta g_{1,z}\) when \(\lambda _Z\) is marginalized. In the following we will present results both for this case and for the case where \(\lambda _Z\) is fixed to zero, which is a common condition in many interesting explicit UV models.^{2}

As anticipated, given the strong bounds on \(\delta g^{Z \ell }\), and the expected experimental sensitivity on the Higgs contact terms, we can fix the former parameters to their SM values in Eq. (1) and study the allowed range of \(\epsilon _{Z f}\) and \(\epsilon _{W \ell }\) as determined by the TGC (whose constraints are obtained in the same limit) and *W* couplings only.

Since no LEP bound is available on the CP-violating TGC coupling \(\delta \tilde{\kappa }_\gamma \), at present \(\epsilon ^\mathrm{CP}_{ZZ}\) is an independent variable. However, in the future significant constraints on \(\delta \tilde{\kappa }_\gamma \) could be obtained from LHC data [19]. All in all, we are left with 3 CP-conserving couplings, \(\kappa _{ZZ}\) and \(\epsilon _{\gamma \gamma ,Z\gamma }\), and 3 CP-violating ones, \(\epsilon ^\mathrm{CP}_{\gamma \gamma ,Z\gamma ,{ZZ}}\).

*W*couplings to leptons, we find the following constraints on the Higgs contact terms

*Z*couplings to leptons and in \(\delta m\), the following relations among Higgs PO are satisfied:

*e*, \(4\mu \) decays) both in the general case (\(\lambda _Z\not =0\), marginalised) and for \(\lambda _Z=0\). It is interesting to notice that, even in the general case, only the direction

## 3 \(h\rightarrow 4\ell \) phenomenology under EW constraints

### 3.1 Partial decay rates

Some comments on these expressions are in order. First, it is easy to see that the contributions from the CP-violating terms \(\epsilon _{ZZ,Z\gamma ,\gamma \gamma }^\mathrm{CP}\) are negligible once the constraints from \(\Gamma (h \rightarrow \gamma \gamma , Z\gamma )\) are taken into account. We stress that this conclusion holds even for \(m_{\ell \ell }^\mathrm{min}\) as low as \(\mathrm{1~GeV}\). Indeed, despite the lack of bounds on \(\epsilon _{ZZ}^\mathrm{CP}\), its contribution to the total rate is below 4 % even for *O*(1) values. Thus, \(\Gamma (h\rightarrow 4\ell )\) can be expressed in terms of \(\kappa _{ZZ}\) and the five pseudo-observables \(\epsilon _{Ze_R,Ze_L,{ZZ},Z\gamma ,\gamma \gamma }\) bounded by Eqs. (5), (6) and (8).

The global fit of Ref. [20], that allows approximately for 30 % non-standard contributions in \(\Gamma (h\rightarrow 4\ell )\), can in principle be used to obtain a bound on \(\kappa _{ZZ}\) via Eq. (14). However, the error in the contact terms gets significantly enhanced when propagated to the total rate, which makes difficult to set a meaningful bound on \(\kappa _{ZZ}\) at this point. Moreover, the fit of Ref. [20] assumes SM-like differential spectra in \(h\rightarrow 4\ell \), that is not necessarily a safe assumption in the presence of sizable contact terms. We will come back to the combined bounds on \(\kappa _{ZZ}\) and the contact terms from the partial rates at the end of this section, after addressing the possible non-standard effects on the dilepton invariant-mass spectra.

*h*(125) belongs to a pure \(\text {SU}(2)_L\) doublet. In the right plot we consider the \(\lambda _Z=0\) case: in this limit the overall modifications are much reduced but still visible, while the correlation between \(h\rightarrow 2e2\mu \) and \(h\rightarrow 4e (4\mu )\) remains very strong, with possible deviations below any future realistic resolution.

The different effect of the photon pole in the two channels, discussed in Ref. [23], manifests itself in Eq. (12) as a \(\sim \) \(25\) times larger interference term of \(\epsilon _{\gamma \gamma }\) with \(\kappa _{ZZ}\) in \(h \rightarrow 4e(4\mu )\) vs. \(h \rightarrow 2e2\mu \). However, we stress that this is a tiny effect on the partial rates (below \(1\,\%\) with present cuts) once the LHC bound on \(\epsilon _{\gamma \gamma }\) is taken into account. This is why this effect is not visible in Fig. 2. The smallness of this effect also implies that improving the bounds on \(\epsilon _{\gamma \gamma }\) from \(h\rightarrow 4e (4\mu )\) is extremely challenging, especially in the general case where the SM deviations from all the PO are considered at the same time.

The strict correlation between \(h\rightarrow 2e2\mu \) and \(h\rightarrow 4e (4\mu )\) rates represents a firm prediction of the linear EFT frameworks that is worth to test with future data: any violation of the correlation would not only imply the existence of NP, but would also imply that (i) NP does not respect lepton universality, (ii) the Higgs particle has a non-\(\text {SU}(2)_L\) component.^{3}

### 3.2 Single dilepton invariant-mass spectra

In addition to the partial widths, the rich kinematics of the \(h\rightarrow 4\ell \) processes offers additional handles to probe the relevant pseudo-observables. Since the contact terms \(\epsilon _{Ze_L,Ze_R}\) have the same Lorentz structure as the SM term, angular distributions are not modified and the only effect is on the differential distributions in the dilepton invariant masses. On the other hand, the other pseudo-observables, \(\epsilon _{ZZ,Z\gamma ,\gamma \gamma }^{(\mathrm{CP})}\), modify also angular distributions and thus a complete study of the full kinematics of the events is necessary in order to extract them as efficiently as possible (see in particular Refs. [23, 24, 25, 26, 27] for a recent discussion). In this work we focus only on the invariant-mass distributions, both because the effect of the contact terms in \(h\rightarrow 4\ell \) is the one less studied in the previous literature and because, as shown above, these PO are the less constrained at the moment (at least in the general TGC case).

Since the effects on the partial widths have already been discussed, here we focus on the shapes, i.e. the normalized differential distributions. To this purpose, we have sampled sets of PO inside the 68 and \(95\,\%\) CL bounds, keeping into account their correlations. For each set we have determined the normalized dilepton invariant-mass spectrum and its ratio to the one obtained in the SM at tree level, and we have finally built the envelopes of such spectra.

For these reasons, we show the differential decay rate in Fig. 4 with a \(m_{\ell \ell }^\mathrm{min}=\mathrm{4~GeV}\) cut. One observes that also in this case the effect of the pseudo-observables \(\epsilon _{ZZ,Z\gamma ,\gamma \gamma }\) on the shape is at most of \(\mathcal {O}(10\,\%)\), except for the low \(m_{\ell \ell }\) region, where the sensitivity is significantly enhanced. We further stress that lowering the infrared cut on both dilepton invariant masses also leads to an enhancement of the rate in the 30–80\(\text { GeV}\) region, since then the other lepton pair is allowed to be near the photon pole.

The difference between Fig. 5 (left) and (right) could be used, in the future, to indirectly improve the current bounds on \(\lambda _Z\). Notice that \(\kappa _{ZZ}\) has a negligible impact on the normalized shape in Fig. 5 (right). This is because varying \(\kappa _{ZZ}\) corresponds to an overall rescaling of the amplitude assuming a SM-like kinematical dependence, and thus this effect cancels at linear order (in \(\delta \)PO\(_i\)). In the \(\lambda _Z \not =0\) case a residual effect survives because of quadratic corrections; in the \(\lambda _Z = 0\) case all the PO are constrained to be close to their SM values and the effect vanishes, as expected by a consistent EFT expansion.

### 3.3 The \(\kappa _{ZZ}\) vs. \(\epsilon _{Z\ell }\) bound from \(\Gamma (h\rightarrow 4 \ell )\)

From the above discussion we conclude that, even when sizable modifications in the rates occurs, namely in the \(\lambda _Z \ne 0\) case, the \(h\rightarrow 4\ell \) spectrum remains SM like compared to the present level of experimental precision. As a result, the only useful bound on the PO that can be set at present from \(h\rightarrow 4\ell \) data, within the linear EFT framework, is the one following from the partial decay rates. We also learned that, given the existing constraints on the PO, the structure of Eq. (14), and the present level of precision on \(\Gamma (h\rightarrow 4\ell )\), this bound is an effective constraint on \(\kappa _{ZZ}\) vs. \(\epsilon _{Z e_L}\), fixing \(\epsilon _{Z e_R} = 0.48 \times \epsilon _{Z e_L}\), as implied by Fig. 1.

## 4 Conclusions

In this paper we have presented a systematic evaluation of the bounds on the Higgs PO that follow from the EW constraints in the linear EFT regime, with particular attention to the PO appearing in \(h\rightarrow 4\ell \) and \(h\rightarrow 2\ell 2\nu \) decays. Using such bounds we have derived a series of predictions for \(h\rightarrow 4\ell \) decay rates and differential distributions. A dedicated analysis of the \(h\rightarrow 2\ell 2\nu \) modes will be presented in a separate publication.

The results of the EW bounds are summarized in Fig. 1 and Eqs. (6) and (7). We find that, because of the flat direction in TGC bounds (\(\lambda _Z \simeq - \delta g_{1,z}\) unconstrained), EW plus Higgs data leave open the room for sizable \(h\rightarrow 4\ell \) contact terms (\(\epsilon _{Z\ell }\)), provided they are flavor universal and with the specific L–R alignment shown in Fig. 1 (left). In principle, \(h\rightarrow 4\ell \) data can be used to remove the degeneracy in the EFT parameter space implied by the TGC flat direction; however, the present level of precision is not good enough. As a result, the uncertainty on the contact terms reflects into a poor knowledge of \(\kappa _{ZZ}\), as shown in Fig. 6 (left). If the TGC direction is closed (by model-dependent dynamical considerations suggesting \(\lambda _Z=0\)), the contact terms are bounded at the few percent level, as shown in Fig. 1 (right), and have a minor impact in the determination of \(\kappa _{ZZ}\), as shown in Fig. 6 (right).

The phenomenological implications for \(h\rightarrow 4\ell \) decays of the EW bounds on the Higgs PO are summarized in Figs. 2, 3 and 4. On the one hand, the uncertainty of the predictions thus obtained determine the level of precision necessary, in future \(h\rightarrow 4\ell \) analyses, to improve our constraints on the Higgs linear EFT. In this respect, we confirm the conclusion of Refs. [6, 27] that shape-modifications in \(h\rightarrow 4\ell \) are significantly constrained in the linear EFT regime, although we find that deviations from the SM as large as \(10\,\%\) (\(20\,\%\)) are still possible for \(\lambda _Z=0\) (\(\lambda _Z\not =0\)). These predictions are robust under relaxation of LEP-I bounds to the \(\sim 1\% \) level.

On the other hand, these predictions can be interpreted as a series of tests that, if falsified by future \(h\rightarrow 4\ell \) data, would allow us not only to establish the existence of NP but also to exclude that *h* is the massive excitation of a pure \(\text {SU}(2)_L\) doublet. In this respect, we stress the firm prediction on the lepton-flavor universality ratios in Fig. 2, and the bounds on the normalized dilepton invariant-mass spectrum in Fig. 5.

## Footnotes

- 1.
We stress that the pseudo-observables \(\epsilon _{W f}\) and \(\delta g^{Wf}\), which in general are complex, are real in the linear EFT scenario [1].

- 2.
The flat TGC direction is lifted if quadratic terms in the cross section are included. In principle, this procedure is not consistent with the EFT power counting, given the lack of inclusion of contributions from \(d=8\) operators, that are formally of the same order. However, in Ref. [10] it is argued that the result of the quadratic fit are consistent with the EFT expansion since higher-dimension operators contributing to the

*s*-channel give a suppressed contribution. For our purposes, we notice that the constraints on \(\{\delta g_{1,z}, \delta \kappa _\gamma \}\) obtained from the quadratic fit are essentially equivalent to those obtained setting \(\lambda _Z = 0\). - 3.
We stress that these two conditions are not sufficient to ensure large deviations from universality in \(h\rightarrow 4\ell \) decays, but are necessary conditions to observe it.

## Notes

### Acknowledgments

We thank Adam Falkowski, Francesco Riva, and Michael Trott for useful comments and discussions. This research was supported in part by the Swiss National Science Foundation (SNF) under Contract 200021-159720. M.G.-A. is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the program ANR-11-IDEX-0007 of the French government.

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