SMEFT Corrections to $Z$ Boson Decays

We compute the one-loop corrections to $Z$ decay properties from dimension-6 operators in the Standard Model Effective Field Theory (SMEFT) that contribute also to anomalous 3-gauge boson couplings and examine the relative sensitivity of the two processes to the anomalous couplings. The size of the contributions is of order a few percent, of the same size as Standard Model electroweak corrections. This is part of a program of computing electroweak quantities to one-loop in the SMEFT: these calculations are needed for a future global fit to limit the coefficients of the dimension-six Wilson coefficients consistently at one loop.


I. INTRODUCTION
The development of the precision electroweak program at the LHC is a major task for the coming decade. At present, the interactions of the Higgs boson and the electroweak gauge bosons appear to have approximately Standard Model (SM) like interactions and there is no sign of new massive particles. These points together imply that deviations from the SM can be analyzed in an effective field theory framework [1,2].
In the Standard Model Effective Field Theory (SMEFT), deviations from the SM are parameterized in terms of a tower of higher dimension operators, O d k , where the operators, O d k , contain only SM fields and are invariant under SU (3) × SU (2) × U (1). The complete set of dimension-6 operators was first compiled in Refs. [3,4] and the Feynman rules in this basis ("Warsaw Basis") are conveniently given in Ref. [5]. The new physics is completely contained in the coefficient functions, C d k . The scale of the assumed UV complete theory is Λ, and we assume Λ v = 246 GeV. For a weakly coupled theory, the corrections to SM predictions are dominated by the dimension-6 contributions.
Predictions for Higgs production and decay, along with V V (W ± , Z, γ) interactions are well known at tree level in the SMEFT [1,2,6,7]. Including also contributions to the oblique parameters, limits on the allowed sizes of the SMEFT coefficients can be extracted in a global fit to Higgs signal rates and gauge boson pair production [8][9][10][11]. A precision Higgs and electroweak physics program, however, requires SMEFT calculations beyond the leading order if matching between the experimental results and theory is to be eventually done at the few percent level.
The program of calculating SMEFT quantities beyond leading order is in its infancy. One-loop calculations exist for H → γγ [12][13][14], H → bb [15,16] and the unphysical H → ZZ and H → W + W − processes [17,18]. The one-loop Yukawa, y t , and λ = M 2 H 2v 2 contributions to Z decays are also known [19]. In addition to effects in the electroweak sector, one-loop contributions from top-quark operators can significantly affect Higgs production rates at the LHC [20,21].
In this paper, we compute the 1-loop corrections to the partial Z decay widths due to the dimension-6 operators that contribute to pp → W + W − and compare the sensitivity of the two processes. These operators are particularly interesting because for transverse gauge boson production they contribute to different helicity amplitudes [22,23], such that their interference with the SM does not grow with energy unless decays or higher order corrections are considered [24,25]. Along with anomalous 3-gauge boson couplings, we include in our calculation the shifts in the Z decay widths due to anomalous fermion couplings, which have important contributions not only to the Z widths [26], but also to gauge boson pair production [23,27,28]. Low energy data places strong limits on deviations from the SM and information from Z decays is particularly interesting due to the precision of the LEP measurements. Consistent fits to the LEP data require the inclusion of the complete set of SMEFT operators, along with the one-loop predictions. Our calculation is a step in this direction, and is related to previous studies of the loop effects of gauge boson self-couplings on precision electroweak observables [29][30][31][32][33].
In Section II, we review the basics of the one-loop SMEFT calculation and in Section III the calculation of Z → f f in the SMEFT is summarized, with analytic formulae presented in a series of appendices. Numerical results are given in Section III.

II. SMEFT AT ONE-LOOP
In this work we consider modifications of the Zf f and W + W − V (V = Z, γ) vertices. We consider only operators that contribute to both qq → W + W − [23,28] and to Z → f f .
The fermion vertices can be parameterized as, where g Z ≡ e/(c W s W ) = g/c W and f (f ) denotes up-type (down-type) quarks. The SM fermion couplings are: where T f 3 = ± 1 2 and Q f are the weak isospin and electric charge of the fermions, respectively.
Assuming CP conservation, the most general Lorentz invariant 3−gauge boson couplings can be written as [34,35] where g W W γ = e and g W W Z = gc W . For the 3−gauge boson couplings we define Because of gauge invariance we always have δg γ 1 = 0. We assume SU (2) invariance, which implies the coefficients are related by, leaving three independent effective couplings.
We work in the Warsaw basis [3,4] and the dimension-6 operators contributing to the 3-gauge boson vertices are, and Φ is the Higgs doublet field with a vacuum expectation value Φ = (0, v/ √ 2) T . In the mapping from EFT operators to anomalous couplings we have to take into account the EFT shift s 2 W → s 2 W + δs 2 W in the definition of the model input parameters for the gauge couplings, as well as for s W , so that we get back to canonically normalized gauge fields. We take as our input parameters M W , M Z and G µ . The only EFT shift involving C HW B is s 2 where at this order we can use the tree level relations, c W = M W M Z . We find the following mappings between the SMEFT coefficient, C HW B , and the effective couplings, The shifts including all SMEFT operators can be found in Refs. [27,36]. In particular, the SM fermion vertex couplings are For b L , the coefficient in front of C W is 2.7 · 10 −4 rather than 4.5 · 10 −4 because of top mass effects. The tree level contributions of C HW B are contained in the δg Zf L,R contributions as given in Eq. 8.
These effective couplings are bounded by LEP measurements at the Z pole. We proceed to take the limits of [47] on the Z-fermion couplings to constrain the SMEFT operators. We minimize a χ 2 function constructed using the LEP measurements of the quantities (g ν L , g e L , g e R , g u L , g u R , g b L , g b R ) and their correlations. While Z pole measurements constrain all of the operators in Eq. 18, we focus on the implications of our calculation for the operators O W , O HB and O HW , which do not contribute to Z decay at tree level. In Fig. 1, we show the resulting 90% CL limits in 2-dimensional planes of the coefficients of these operators along with that of O HW B , which affects electroweak couplings at tree level. The coefficients of all other operators are set to zero. For comparison, we show bounds on each of the operators from processes to which they contribute at tree level. For O W , we use the limits of [28] obtained by using 8 TeV LHC gauge boson pair production in leptonic final states [48][49][50][51]. For O HB and O HW , we use limits [18] from the calculation of H → γγ [12,14,18] in the SMEFT, as compared to measurements of H → γγ at Run 1 and 2 of the LHC [52][53][54]. O HW B corresponds to the oblique parameter S [55,56], whose limit we take from the Gfitter collaboration [57]. The existing bounds in Fig. 1 are stronger than those that we obtain directly from Z pole measurements. Nevertheless, they provide complementary information, and in particular in the left panel the interplay between the limits on C W and C HW B demonstrates the power of electroweak precision measurements to constrain couplings that only contribute at loop level. In the case of the operators C HB and C HW which directly affect H → γγ, Higgs precision is already significantly more effective than Z pole measurements in setting limits, due to the loop suppression of these operators' contributions to Z decay. We have included further details of our fit procedure as well as the numerical values of the existing experimental bounds in Appendix D.

IV. CONCLUSIONS
Precision measurements of electroweak physics will eventually necessitate higher order calculations of BSM contributions. The SMEFT framework takes a general approach to potential new UV physics by parametrizing its effects in terms of higher dimension operators involving the SM fields. In this work, we have furthered the applicability of the SMEFT to probe new physics by considering the one loop corrections to Z decay from operators which contribute to gauge boson production.
While the contributions of the operators O W , O HB and O HW are small relative to those of the operators that modify the Z coupling to fermions at tree level, the relative size of all of the SMEFT operators is fixed by the new physics. In particular, integrating out a heavy SM singlet scalar could naturally give these operators without changing the leading Z couplings to the fermions [58]. In such a scenario, it would be essential to have the higher order contributions of the BSM physics to all possible processes. In this regard our calculation provides a useful prediction, relating the effects of new physics in Z decay to those in other electroweak processes provided the states responsible for deviations from the SM are heavy enough to be integrated out.
A full calculation of Z decay at one loop in the SMEFT would provide even more complete information about the influence of higher dimensional operators on Z physics. With this as well as other higher order calculations of electroweak processes, in the future a global fit at NLO in the SMEFT could be performed to bound the sizes of all possible dimension-6 SMEFT operators.
The cancellation of UV poles follows from the individual contributions: Numerically with Λ = 1 TeV, the pieces are as follows.
The sum vanishes for any given fermion.

Appendix B: 2-point functions
In this appendix, we show the two-point functions in R ξ gauge due to the SMEFT operators that also contribute to gauge boson pair production. Previous results for the gauge boson two-point functions in other operator bases appear in [31,59].
In D = 4 − 2 dimensions, the two-point function for a massless fermion with weak isospin T f 3 and charge Q f is We have regulated IR divergences with a photon of mass M Z β, and use standard FeynCalc notation [40] for the Passarino-Veltman functions.
This leads to the wave function renormalization For the b L , there are corrections proportional to the top mass, leading to an additional wave function renormalization which in Feynman gauge is The transverse W two-point function is which yields the mass shift The transverse Z two-point function is which yields the mass shift and the wave function renormalization The γ − Z two-point function is which yields the on-shell mixing where the vertex function is   In this appendix, we show the measurements used in Section III to produce Figure 1.
LEP and SLD [47] measured the effective fermion couplings and correlations in Table I. We seek to minimize the quantity (χ 2 ) LEP = ( g SMEFT − g exp ) T V −1 ( g SMEFT − g exp ) (D1) where g = g ν L , g L , g b L , g c L , g R , g b R , g c R and V is the covariance matrix constructed from the errors and correlations above. We use Eq. 18 together with the SM predictions of Table I to calculate g SMEFT . Since we set light fermion masses to zero in our SMEFT analysis, the effective couplings for the down (up) quark apply equally to the b (c) quark, with the exception of the b L for which top quark corrections apply as specified below Eq. 18.
We compare our results to processes in which the SMEFT operators contribute at tree level. The limit of [28], set using LHC Run I data [48][49][50][51], is converted in our notation to −0. 17