Electroweak and QCD Corrections to $Z$ and $W$ pole observables in the SMEFT

We compute the next-to-leading order QCD and electroweak corrections to $Z$ and $W$ pole observables using the dimension-6 Standard Model effective field theory and present numerical results that can easily be included in global fitting programs. Limits on SMEFT coefficient functions are presented at leading order and at next-to-leading order under several assumptions.


I. INTRODUCTION
The LHC experiments provide strong evidence that the SU (3) × SU (2) × U (1) Standard Model (SM) gauge theory describes physics at the electroweak scale [1]. To date, there is no evidence of new interactions or high mass particles. Taken together, these features suggest that the weak scale can be described by an effective field theory (SMEFT) having the SM as its low energy limit. The SMEFT is defined by an infinite tower of on-shell and higher operators, involving only the SM particles and assumes that the Higgs boson is part of an SU (2) doublet [2]. The effects of the higher dimension operators are suppressed by powers of a high scale, Λ, and we assume that the most numerically relevant operators are those of dimension-6. All possible new physics phenomena are contained in the coefficient functions.
In this work, we take a major step by computing the next-to-leading order (NLO) EW and QCD corrections in the SMEFT to Z and W pole observables. We assume flavor universality and use the Warsaw basis [30]. We are particularly interested in the numerical effects of the NLO corrections on the global fits. In Section II, we review the basics of the SMEFT theory and in Section III, we describe our NLO calculations. Our results are given in Section IV and Appendix A, which contains numerical expressions for the Z and W pole observables, as well as limits on the SMEFT coefficients at LO and NLO. Section V contains some conclusions and a discussion of the implications of our results for global fits.

II. SMEFT BASICS
The SMEFT parameterizes new physics through an expansion in higher dimensional operators, SM fields and all of the effects of the beyond the SM (BSM) physics reside in the coefficient functions, C k i . We assume all coefficients are real and do not consider the effects of CP violation. We use the Warsaw basis [30] and at tree level (neglecting flavor) there are 10 dimension-6 operators contributing to the Z and W pole observables of our study. These operators are listed in Table I, where φ is the SU (2) L doublet, τ a are the Pauli matrices, At NLO, there are 22 additional operators that contribute: Definitions for these operators can be found in Refs. [30? ]. We use the Feynman rules in R ξ gauge from Ref. [31].
The SMEFT interactions cause the gauge field kinetic energies to have non-canonical normalizations and following Ref. [31], we define "barred" fields and couplings, such that W µ g 2 = W µ g 2 and B µ g 1 = B µ g 1 .
Dimension-6 4-fermion operators give contributions to the decay of the µ, changing the relation between the vev, v, and the Fermi constant G µ , The tree level SMEFT couplings of fermions to the Z and W are given in terms of our input parameters (α, M Z , G µ ), We assume all couplings are flavor independent and we neglect CKM mixing. The weak coupling in Eq. 6 is evaluated using the LO SM relation and Eq 7 serves as the definition of s 2 W , Warsaw Basis Since we are working to O v 2 Λ 2 , we omit dipole type operators that do not interfere with the SM contributions to Z and W pole observables. Similarly, the contributions from righthanded W couplings and the right-handed Zνν interaction do not contribute to our study.
The tree level couplings are, The SMEFT contributions to the effective couplings are listed in Table II [33].

III. W AND Z POLE OBSERVABLES TO NLO
The observables we consider are: The SM results for these observables are quite precisely known, and as a by product of our study we recover the known NLO QCD and NLO EW results as a check of our calculation [34].
The next-to-leading order contributions to Z and W pole observables require the calculation of one loop virtual diagrams in the SMEFT and in most cases, the contribution also of real photon and gluon emission diagrams. Since the SMEFT theory is renormalizable order by order in the (v 2 /Λ 2 ) expansion, we retain only terms of O(v 2 /Λ 2 ). The one-loop SMEFT calculations contain both tree level and one-loop contributions from the dimension-6 operators, along with the full electroweak and QCD one-loop SM amplitudes. Sample diagrams contributing to the Z decay widths at NLO are shown in Fig. 1 Corrections to this formula are of higher order and we do not include them [35]. Nonresonant contributions, such as photon exchange, box diagrams and 4-fermions interactions [33], are also not included because they do not contribute to the observables on the Z pole to O( 1 Λ 2 ). We employ a modified on-shell (OS) scheme, where the SM parameters are renormalized in the OS scheme. The effective field theory coefficients of the dimension-6 operators are treated as M S parameters and the poles of the one-loop coefficients C i are known from Refs. [32,36,37], where µ is the renormalization scale, γ ij is the one-loop anomalous dimension, The renormalized SM gauge boson masses are, where Π V V (M 2 V ) is the one-loop correction to the 2-point function for Z or W computed on-shell and tree level quantities are denoted with the subscript 0 in this section. The gauge The circles represent potential insertions of dimension-6 SMEFT operators.
boson 2-point functions in the SMEFT can be found analytically in Refs. [38,39]. The one-loop relation between the vacuum expectation value and the Fermi constant is, where v 0 is the unrenormalized minimum of the potential and ∆r is obtained from the oneloop corrections to µ decay. Complete analytic expressions for ∆r in both the SM and the SMEFT at dimension-6 are given in Ref. [24]. Finally, the on-shell renormalization of α is extracted from the renormalization of the llγ vertex.
We obtain the relevant amplitudes for the virtual contributions using FeynArts [40] with a model file generated by FeynRules [41] and the Feynman rules of Ref. [31]. Then we use FeynCalc [42,43] to manipulate and reduce the integrals and LoopTools [44] for the numerical evaluation.
The Z decays to charged fermions receive contributions from one-loop virtual diagrams and from real photon emission that are separately IR divergent and we regulate these divergences with a photon mass. Since we only consider the inclusive quantities of Eq. 10, the photon mass dependence cancels after integration over the photon phase space and there is no need for a photon energy cut. The complicated form of the SMEFT vertices makes direct integration of the phase space difficult, so we use the method of Ref. [45], where the integration over the photon phase space is replaced with a loop integration. This is possible after we use the identity, After making this replacement, we treat the momenta of the outgoing particles as internal loop momenta, the integration over the phase space becomes an integration over the loop momenta and we can use the IBP relations to reduce the loop integrals to known master integrals. In the case of Z → f f γ, the integrals are 2-point 2-loop integrals, for which a generic basis of master integrals is known [46,47] and the reduction can be done using FIRE [48]. This is identical to the technique we applied in the calculation of the real contributions to H → W + W − γ in Ref. [20].
We do not include the effective weak leptonic mixing angle in our fit since it can be directly derived from other observables, but present it here for completeness.
The NLO corrections to sin 2 θ l,ef f change the numerical effects of the coefficients appearing at tree level by O(5 − 10%), and introduce dependencies on other coefficients.
For the W mass and total width, we find the predictions, It is interesting to note that some of the contributions to the W mass and width change by more than 10% when going from LO to NLO in the SMEFT. The NLO SMEFT contributions to the other observables of Eq. 10 given in Appendix IV.
We fit to the experimental data given in Table III, (omitting sin 2 θ ef f ) since it can be directly derived from other observables). The most accurate SM predictions are given in the right-hand column and we use these values in our fits, as opposed to the LO or NLO SM contributions directly calculated. The pole observables we consider are [59][60][61]: where we assume lepton universality and the experimental correlations can be found in Ref. [51]. We include the measurements of A l from LEP and SLD as separate data points.
The χ 2 is computed from, Using the LO SMEFT expressions for the observables and taking Λ = 1 T eV , we find 1 , where φq , C φq , C φl , C and we find χ 2 SM ∼ 13.42. The symmetric matrix M LO is, 1 Our results are consistent with SMEFT fits to purely LEP observables using slightly different sets of inputs[9, 59-61].
Using the NLO SMEFT expressions we find χ 2 N LO , (for Λ = 1 T eV ) , where, where the numerical form of M N LO is given in the supplemental material. At NLO, the   fit to the electroweak observables, resolving these 22 blind directions requires input from other processes and/or assumptions about which operators can be safely neglected.
We chose to perform our fits setting C φe = 0 and C φq = 0, along with setting all of the operators that first appear at NLO to 0. We then marginalize over the remaining operators to study the numerical impacts of the NLO contributions. These results are shown in Tab.
VI. We see that the effects of the NLO corrections can be significant, although the numerical results are sensitive to which operators are set to 0. Our results suggest that including the NLO corrections in the global fits (where the complete set of operators can potentially be bounded) may be important.
As another way of examining the impact of the NLO contributions, we consider the oblique parameters. The tree level SMEFT contributions are, For the NLO oblique parameter fit, we set all coefficients to 0, except C φW B and C φD . The resulting limits are shown in Fig. 2   the coefficients from a 2 parameter fit to our observables). In this example, the effect of the NLO SMEFT corrections is small. At NLO, new coefficients can influence the oblique parameters, and the complete one-loop SMEFT result is given in Ref. [38].
Our calculation includes only the resonant Z and W contributions to the precision elec- troweak observables. In the SMEFT, there are tree level non-resonant contributions due to 4-fermion operators that can complicate the experimental extraction of the widths from the data. The size of these effects has been estimated in Ref. [33]. For example, where C 4f is a generic 4-fermion operator. These effects could potentially be similar in size to the electroweak corrections we have computed. There are also off-shell corrections proportional to q 2 Λ 2 , where q 2 is the momentum running through the Z boson propagator. The experimental cuts [51] were designed to extract predominantly the on-shell Z events, so although the size of these effects is expected to be small a detailed theoretical study would be needed to quantify them.

V. CONCLUSIONS
We have computed the NLO electroweak and QCD corrections to the SMEFT predictions for the precision electroweak observables. Our results are presented in a numerical form that can easily be incorporated in the global fitting programs. We also present numerical results for the LO and NLO χ 2 that can be customized for the reader's use. Our studies suggest that the NLO SMEFT corrections may have a sizable effect on the global fits. Numerical results for the SMEFT NLO expressions for the observables considered here, along with the χ 2 LO , χ 2 N LO and the matrix M N LO , are posted at https://quark.phy.bnl.gov/Digital_ Data_Archive/dawson/ewpo_19.

Appendix A: Observables to LO and NLO in the SMEFT
In this appendix, we report the contributions to the observables of Table II The SMEFT contributions to the total Z width are, The ratios are defined to be The asymmetries are defined as, and the SMEFT contributions are, φl + 0.12C (1) φq + 0.12C Finally, the forward backward asymmetries are defined as where defining θ to be the angle between the incoming l − and the outgoing f i , σ F has θ between (0, π 2 ) and σ B has θ between ( π 2 , π). The SMEFT results are, +0.0006C ed + 0.0017C ee − 0.0114C eu − 0.0028C φd − 0.0191C (1) φq + 0.0159C φq + 0.0263C φu +0.0005C ld + 0.0011C le + 0.0068C (1) lq + 0.0022C