Flavourful SMEFT likelihood for Higgs and electroweak data

We perform an updated fit to LHC Higgs data and LEP electroweak precision tests in the framework of the Standard Model Effective Field Theory (SMEFT). We assume a generic structure of the SMEFT operators without imposing any flavour symmetries. The implementation is released as part of the public global SMEFT likelihood. This allows one to fit parameters of a broad class of new physics models to combined Higgs, electroweak, quark flavour, and lepton flavour observables.

from quark flavour physics, lepton flavour physics, low-energy parity violation, and other low-energy precision tests. It is desirable to include the Higgs observables into the same framework, as many dimension-6 operators generated by typical BSM models are currently probed only via LHC Higgs searches. As a part of this work, the combined Higgs and electroweak likelihood was incorporated into the global SMEFT likelihood package smelli, which allows users to perform combined fits of BSM models to Higgs, electroweak, flavour, and other low-energy data. This paper is organized as follows. In Section II we briefly review the SMEFT framework in order to fix our conventions and notation. In Section III we summarize the data used in this analysis and describe our methodology of constructing global likelihoods and parameter fitting. In Section IV we give the best fit and confidence intervals for the Wilson coefficients affecting the Higgs and/or electroweak observables at the level. Section V contains two simple example applications of our flavourful likelihood: one designed to connect and compare to previous work, and the other illustrating the relevance of allowing for a general flavour structure of dimension-6 operators.

II. SMEFT FRAMEWORK
We briefly review the SMEFT framework to fix our notation. We consider the extension of the SM by all independent dimension-6 operators Q i invariant under the SM gauge symmetries [26,27], We work in the Warsaw basis of operators [27], selecting the weak basis for fermions where the down-type quark and charged lepton mass matrices are diagonal (coinciding with the Warsaw basis as defined in the WCxf [20]). In our conventions the Wilson coefficients C i have dimensions [mass] −2 . Operators with dimensions higher than six, as well as dimension-5 operators (which only generate tiny neutrino masses and have no observable effect on Higgs or electroweak phenomenology) are ignored in this analysis. We also ignore CP violating opearators, which affect our observables only at the quadratic level in the corresponding Wilson coefficients. The electroweak parameters g L , g Y , v are defined in the α-m Z -G F scheme. We take into account only tree-level effects of dimension-6 operators, except in the observables where the SM contribution itself appears first at one loop. In this approximation, the operators affecting the Higgs and electroweak precision data are shown in Table I • The Z and W pole observables. These are mostly unchanged compared to the analysis of Ref. [28], see Table 1 therein for the original references. The only new pieces of experimental information are i) the PDG combination for the W mass [29], which includes the recent ATLAS measurement [30]; ii) The measurements of Γ(W →τ ν) in D0 [31] and of Γ(W →µν) Γ(W →eν) in LHCb [32]. On the other hand, we no longer use the V tb determination from Ref. [33] to constrain the W tb vertex, as other dimension-6 operators beyond those in Table I may affect that measurement.
• The total and differential cross sections for W W pair production measured in LEP-2 (Tables 5.3 and 5.6 of Ref. [34]). These are unchanged compared to the analysis of Ref. [35].
• Combination of Run-1 ATLAS and CMS measurements of the Higgs signal strength measurements in 21 different channels (Table 8 of Ref. [36] with the correlation matrix in Fig. 27, plus the µµ signal strength). In addition, we use ATLAS [37] and CMS [38] Run-1 measurements of the signal strength in the Zγ channel.
• Combined ATLAS measurement of the signal strength based on the 80 fb −1 of run-2 data ( Table 6 of Ref. [39] with the correlation matrix in Fig. 6). We also include signal strength measurements in the µµ [40] and Zγ [41] channels.
• Combined CMS measurement of signal strengths based on the 35.9 fb −1 of run-2 data (Table 3 of Ref. [42] with the correlation matrix in the auxiliary material). We also include signal strength measurements in the Zγ [43] and cc channels [44].
In order to take into account the correlations, we have to treat the experimental likelihoods as (multivariate) Gaussians.
To this end, we symmetrize asymmetric uncertainties. SMEFT corrections to electroweak observables are calculated analytically at tree level. For most of the Higgs observables they are determined numerically using the SMEFTsim model implementation [45] in Madgraph [46]. The exceptions are the gg → h and h → γγ processes where we use analytic formulas, taking into account 1-loop effects due to modified Yukawa and hV µ V µ couplings. Logarithmically enhanced one-loop corrections proportional to the dimension-6 Wilson coefficient are included in our code via the renormalization group running effects [47]. Finite loop corrections (see e.g. [48][49][50][51] other than that described above are ignored in our analysis. Our calculation of the Higgs and diboson observables include the effects due to modified Z and W couplings to fermions [52,53]. We take into account the interference effects in h → V ( * ) V * → 4f decays [54][55][56].
Having expressed all observables as linear functions of SMEFT Wilson coefficients, O th (C i ), we construct the likelihood function L(C i ) = e −χ 2 (Ci)/2 , with where S is the experimental covariance matrix and is the difference between the observables predicted in the SMEFT and measured experimentally. Given the precision of SM calculations of the observables in question, we can neglect all theory uncertainties and thus obtain a likelihood function that only depends on Wilson coefficients, and not on nuisance parameters.

IV. CONFIDENCE INTERVALS FOR WILSON COEFFICIENTS
In this section we present marginalized 1 σ confidence intervals for the Wilson coefficients entering into our combined likelihood. This exercise illustrates, in a model-independent fashion, the constraining power of the current Higgs and electroweak data.
The electroweak and Higgs constraints dramatically differ in accuracy: for the former the typical precision is O(0.1%), while for the latter it is O(10%). In the Warsaw basis the two sets of constraints probe an overlapping set of Wilson coefficients, which leads to large correlations. For the sake of illustration, it is more transparent work with certain linear combinations of Wilson coefficients, such that the strongly constrained combinations are isolated from the weakly constrained ones [7,57]. With this in mind, we define ‡ the following linear combinations of the Warsaw ‡ Note that our δg's differ by 1/v 2 from the ones defined in Refs. [22,28]. basis Wilson coefficients: where and In fact, v 2 δg V f are the vertex corrections parametrizing deviations from the SM prediction of the V = Z, W boson couplings to fermion f . Since they are probed by the LEP electroweak precision tests, almost all δg's are independently constrained with a good accuracy, except for one weakly constrained direction in the space of the light quark vertex corrections [28].
We also define another set of linear combinations of the Warsaw basis Wilson coefficients The important point is that, unlike the combinations δg, C i do not affect electroweak observables at tree level. Therefore they are only loosely constrained by the less precise Higgs data, and they are weakly correlated with δg's. We now present the best fit for the Wilson coefficients after a couple of simplifying but physically reasonable assumptions. First, we ignore the dependence of the likelihood on the Wilson coefficients affecting only the Yukawa couplings of the light fermions: [c uϕ ] ii , [c dϕ ] ii , i = 1, 2, and [c eϕ ] 11 . Currently, these are probed mostly via their contributions to the total Higgs width (thus affecting all observed branching fractions uniformly), and their effect is negligible unless the shift of the Yukawa couplings largely exceeds the SM Yukawa. Leaving these parameters in the fit would lead to flat directions. We also ignore the contributions to Higgs observables proportional to the Wilson coefficient c G , which enters into our likelihood via its contributions to the tth production. This parameter is much better constrained by dijet observables [58], and after taking that into account it cannot affect the tth signal strength significantly. With the above assumptions, 31 independent combinations of Wilson coefficients are left as free parameters in the fit.
Our results are shown in Table II. As advertised, the combinations δg are more strongly constrained, at the level |δg| 10 −2 -10 −3 , compared to the remaining ones that are probed only by the Higgs data. There are a few exceptions from this rule, however. First, the uncertainty for the combinations δg Zq corresponding to light quark vertex corrections are larger [28] . While LEP-1 measures the total hadronic width with a per mille precision, it does  not resolve all light quark couplings to Z independently, leading to an approximate flat direction that is only lifted by less precise measurements. Second, some of the Wilson coefficients affecting Higgs observables only are strongly constrained when they compete with the SM loop-induced processes (C gg , C γγ ), or with small SM Yukawa couplings The best fit point has ∆χ 2 = χ 2 SM − χ 2 min ≈ 36.7. This translates to a p-value of approximately 20% for the SM hypothesis. Thus there is no significant hint of physics beyond the SM in the fit, even though some of the individual couplings in Table II display pulls (defined simply as the number of Gaussian standard deviations away from 0) of order 3σ.
The correlation matrix is shown in Fig. 1. The change of variables in Eq. (4) and Eq. (6) disentangles most of the large correlations present if the orignal Warsaw basis variables. Some O(1) correlations remain, however. Notably, there are large correlations between the combinations c zz and c z . This is due to the fact that the current Higgs data poorly disentangle different possible Lorentz structures of the Higgs couplings to electroweak gauge bosons. This situation can be somewhat improved by including in the analysis, in addition to the signal strength, transversemomentum distributions in the gluon fusion or invariant mass distributions in the associated Higgs production [53,[59][60][61]. We note that for strongly correlated Wilson coefficients the best fit values and the magnitude of the errors may be sensitive to including in the observables quadratic (O(Λ −4 )) effects in Wilson coefficients. Indeed, for c zz and c z we find that the error change by ∼ 50% upon including the quadratic corrections, with smaller or negligible effects for other Wilson coefficients.
In closing we remark that it is possible to further relax the assumptions of this global fit without losing a stable    Table II. minimum. Namely, it is possible to leave also the Wilson coefficient [c uϕ ] 22 as a free parameter in the fit. This is thanks to the direct measurement of the h → cc signal strength at the LHC, which constrains the possible magnitude of charm Yukawa coupling modifications due to [c uϕ ] 22 . In the relaxed 32-parameter fit the results are mostly unchanged with respect to those displayed in Table II, except for a threefold increase of the error on the parameter C z . That increase happens because introducing [c uϕ ] 22 opens an approximately flat direction corresponding to simultaneously increasing both the total Higgs width and the hV µ V µ couplings. The flat direction is lifted precisely thanks to the direct h → cc measurement [44].

A. Oblique parameters
Our first example application is the fit to oblique parameters starting from our general likelihood. To this end we assume that, at the energy scale µ ∼ m Z , only two Wilson coefficients c ϕW B and c ϕD are non-negligible. We set all the remaining parameters to zero, and minimize the resulting two-dimensional likelihood. This procedure is directly related to the classic assumption that new physics enters via the so-called oblique S and T parameters [62,63], with the identification  7). We show the 1σ preferred region separately for Higgs (blue) and electroweak (orange) data. The red contours mark the 1σ, 2σ, and 3σ preferred regions using the combined likelihood.
Obviously, this simple example is not using the full flavourful power of our approach. Nevertheless it is useful to present here in order to connect and compare to previous works. For the Wilson coefficients at the scale m Z we find with the correlation coefficient ρ = −0.74. This translates to S = 0.039 ± 0.038, T = 0.071 ± 0.036. The best fit ellipses are shown in Fig. 2, for the combined likelihood, and for the electroweak and Higgs likelihoods separately. It can be seen that the LHC Higgs data contribute to constraining the S parameter, mostly via measurements of the h → γγ rate [14].

B. Custodial vector resonance model
Another example we consider here consists in an SU (2) triplet V I µ of massive vector resonances coupled to the SM Higgs, lepton l and quark q doublets as where i is the generation index, and we allow the couplings to be flavour-non-universal. This kind of resonances and interactions arises e.g. in composite Higgs or warped extra-dimensional scenarios. The parameter space of our simplified model is characterized by 7 couplings g k and the resonance mass M . Assuming U (3) q × U (3) l flavour symmetry would reduce the number of independent couplings to three: g H , g l and g q . Integrating out the massive resonance leads to the SMEFT with the following Wilson coefficients of the operators in Table I: where is the Yukawa coupling of the fermion f i , f = u, d, l, i = 1 . . . 3. As usual, only the ratios coupling/mass are available to a low-energy observer. Thus the SMEFT parameter space describing our simplified model is 7-dimensional in the generic case, and 3-dimensional in the U (3) q × U (3) l limit. We ignore the effects of the operator O ϕ , which only affects double Higgs production and is very weakly constrained at present. Four-fermion g l /M g l 1 /M g l 2 /M g l 3 /M gq/M gq 1 /M gq 2 /M gq 3 /M Higgs 0 ± 0.07 0 ± 0.06 0 ± 0.06 0 ± 0.06 0 ± 0.06 0 ± 0.06 0 ± 0.06 0 ± 0.06 Electroweak 0.04 ± 0.06 −0.03 ± 0.06 −0.06 +0.04 −0.08 0 ± 0.07 0 ± 0.04 0 ± 0.06 0 ± 0.06 0 ± 0.06 Combined −0.04 ± 0.06 −0.02 ± 0.05 −0.06 +0.04 −0.08 0 ± 0.05 0 ± 0.03 0 ± 0.04 0 ± 0.05 0 ± 0.05  (9). We show separately the limits (in units of 1/TeV) obtained using the Higgs, the electroweak, and the combined data. The 1σ limits in each column are obtained by leaving a single resonance-fermion coupling free, and setting the remaining fermion couplings to zero. In the 2nd and 5th column we assume generation-independent couplings to leptons and quarks, respectively, while in the remaining columns we assume the resonance couples to a single fermionic generation. In all cases the resonance-Higgs couplings gH is assumed ten times the corresponding fermionic coupling.
operators, also generated in this model, are neglected in this analysis, which is justified assuming |g H | |g fi |. Such couplings hierarchies can arise naturally in the composite/extra-dimensional scenario, e.g. via fermion localization in an extra dimension. We note that, in the limit |g H | |g fi |, the constraints from the Higgs data can be comparable or superior to those from the electroweak data, which further motivates combining the two. Table III illustrates the difference between fits with and without assuming generation-independent couplings. To this end we study simple one-parameter sections of the full parameter space. The difference between the U (3) q ×U (3) l and the generic cases amounts to up to a factor of two difference in the size of the confidence intervals.
More pronounced differences appear as soon as we study the large parameter space of our simplified model. In the U (3) q × U (3) l case we find the 1σ confidence intervals g H = 0.00 ± 0.63 TeV −1 , g l g H = 0.022 ± 0.017, g q g H = 0.007 ± 0.032.
We see for the resonance coupling to the Higgs field the limits are fairly independent on how the former couples to fermions. This is because the limit is set mostly by the Higgs data. However, the limits on resonance coupling to the SM fermions are sensitive to whether or not U (3) q × U (3) l is assumed. The difference is most dramatic for the couplings to the first two generation of quarks, which are allowed to be an order of magnitude larger in the generic case than in the U (3) q × U (3) l case.

VI. SUMMARY
In this paper we presented an updated fit of the dimension-6 SMEFT operators to combined Higgs and electroweak precision data. We include the most recent ATLAS and CMS combinations of run-2 Higgs data, and also complemented the LEO electroweak data with a couple of recent measurements in hadronic machines. The analysis is based on an open source Python conde, which allows the readers to easily to scrutinize, reproduce, or update our analysis. The code and conventions follow the WCxf format, so that our results can easily be imported by other analysis programs.
At the physics level, the main improvement compared to earlier works is that we allow for a completely general flavour structure of dimension-6 operators. Thanks to that, the likelihood we provide can be used to constrain a broad class of BSM models beyond the U (3) 5 or minimal flavour violation paradigm.