Electroweak input schemes and universal corrections in SMEFT

The choice of an electroweak (EW) input scheme is an important component of perturbative calculations in Standard Model Effective Field Theory (SMEFT). In this paper we perform a systematic study of three different EW input schemes in SMEFT, in particular those using the parameter sets $\{M_W, \, M_Z, \, G_F\}$, $\{M_W, \, M_Z, \, \alpha\}$, or $\{\alpha, \, M_Z, \, G_F\}$. We discuss general features and calculate decay rates of $Z$ and $W$ bosons to leptons and Higgs decays to bottom quarks in these three schemes up to next-to-leading order (NLO) in dimension-six SMEFT. We explore the sensitivity to Wilson coefficients and perturbative convergence in the different schemes, and show that while the latter point is more involved than in the Standard Model, the dominant scheme-dependent NLO corrections are universal and can be taken into account by a simple set of substitutions on the leading-order results. Residual NLO corrections are then of similar size between the different input schemes, and performing calculations in multiple schemes can give a useful handle on theory uncertainties in SMEFT predictions and fits to data.


Introduction
Standard Model Effective Field Theory (SMEFT) is an important tool for investigating small deviations from Standard Model (SM) predictions. Such indirect descriptions of new physics can be made more robust by including quantum corrections not only in the SM, but also in SMEFT. Indeed, the study of next-to-leading order (NLO) corrections (and in a few instances next-to-next-to-leading order (NNLO) corrections) in dimension-six SMEFT has received much attention in recent years, either calculated on a case-by-case basis for specific processes,  or moving towards full automation as in the case of QCD corrections [44].
An important consideration for SMEFT predictions and fits is the choice of the electroweak (EW) input scheme. Ideally, the input parameters should be measured with very high accuracy such that their effect on SMEFT fits is subdominant or even negligible. However, even beyond that, the choice of the input parameters influences perturbative convergence as well as the pattern of Wilson coefficients appearing in leading-order (LO) and NLO predictions. Typical choices of the input parameters include the Fermi constant G F , the mass of the W and Z bosons, M W and M Z , as well as the electromagnetic coupling constant α. Invariably, the NLO SMEFT calculations described above have been performed in one of three different schemes, which use either {M W , M Z , G F } (α µ scheme), {M W , M Z , α} (α scheme) or {α, M Z , G F } (LEP scheme) as inputs. Some discussions of these input schemes can be found in [45,46]. However, there has been no systematic study which elucidates general features of these EW input schemes beyond LO in SMEFT, much less a numerical exploration of benchmark results at NLO in the different schemes. The aim of this paper is to fill this gap.
We structure the discussion as follows. First, in Section 2, we describe the ingredients needed to construct UV renormalised amplitudes in the three schemes, introducing a notation that makes the connections between them transparent. In Section 3 we identify salient features of the different schemes, including patterns of perturbative convergence and Wilson coefficients associated with finite parts of counterterms for typical weak or electromagnetic vertices. We give a first set of NLO results at the level of derived parameters such as M W in the LEP scheme or G F in the α scheme in Section 4, also laying out our method for estimating perturbative uncertainties from scale variations in the SM and SMEFT. In Section 5 we perform a thorough numerical analysis of heavy boson decays at NLO in SMEFT in the three schemes, covering W and Z decay into leptons, and Higgs decay into bottom quarks. Finally, drawing on the insights from the aforementioned sections, we propose in Section 6 a simple procedure which can be used to deduce a set of universal and numerically dominant input-scheme dependent NLO corrections in SMEFT. Concluding remarks are given in Section 7.
While the main focus of the paper is to elucidate the role of EW input schemes in SMEFT, as a by-product we have produced quite a few NLO results which were not available in the literature so far. These have been obtained using an in-house FeynRules [47] implementation of the dimension-six SMEFT Lagrangian, and cross checked with SMEFTsim [48,49]. Matrix elements were computed using FeynArts and FormCalc [50][51][52], analytic results for Feynman integrals were extracted from PackageX [53], and numerical results were obtained with LoopTools [51]. Phase-space integrals arising from the real emission of photons and gluons were calculated analytically using standard methods. The results have been further cross checked by performing calculations in both unitary and Feynman gauge. We include the most important NLO SMEFT results, namely the heavy boson decay rates, ∆r as defined in Eq. (2.20), and the W -boson mass in the LEP scheme, as Mathematica files in the electronic submission of this work.

Three EW input schemes
The dimension-six SMEFT Lagrangian can be written as L = L (4) + L (6) ; L (6) where L (4) denotes the SM Lagrangian and L (6) is the dimension-six Lagrangian with operators O i and the corresponding Wilson coefficients C i = c i /Λ 2 which are inherently suppressed by the new physics scale Λ. For the dimension-six operators we adopt the Warsaw basis [54] -the 59 independent operators in this basis (which in general carry flavour indices) are listed and grouped into eight classes in Table 11. Throughout this work, the SMEFT expansion of a given quantity is truncated to linear order in the Wilson coefficients and thus treated consistently at dimension six. 1 Predictions in SMEFT depend on a number of input parameters and the renormalisation schemes in which they are defined (see for example [56] for an excellent discussion of renormalisation and input schemes in the SM). A number of these are rather standard and are adopted throughout this work. The Wilson coefficients C i ≡ C i (µ) are renormalised in the MS scheme, and are thus functions of the renormalisation scale µ. Moreover, we use on-shell renormalisation for the top-quark mass m t and the Higgs-boson mass M h and set masses of fermions lighter than the top quark equal to zero, with the exception of h → bb decay where we keep a non-zero m b renormalised in the MS scheme in a five-flavour version of QCD×QED as described in Section 5.2 of [57]. Furthermore, we approximate the CKM matrix as the unit matrix.
The difference in EW input schemes used in the literature is related to how the U(1) and SU(2) gauge couplings, denoted by g 1 and g 2 respectively, as well as the vacuum expectation value v T of the Higgs doublet field H, defined by are eliminated in favour of three physical input parameters. In this paper, we consider the following three schemes: 2 (1) The "α scheme", which uses as inputs {α, M W , M Z }, where M W and M Z are renormalised on-shell and α is the fine-structure constant renormalised in a given scheme. 1 Power corrections appearing at dimension eight and beyond come in two distinct types: those which scale as the vacuum expectation value of the theory and those which scale as some kinematic factor p 2 . A powerful formalism where the distinction between these two becomes important is the so-called Geometric SMEFT [55], where corrections of the former kind can in some cases be computed to all orders in the power series. 2 More input schemes have been proposed in the literature for specific processes. For instance, it has been shown that using the sine of the Weinberg angle, sw, as an input parameter leads to good convergence for the prediction of the forward-backward asymmetry at the Z pole [58].
(2) The "α µ scheme", which uses as inputs {G F , M W , M Z }, where G F is the Fermi constant as measured in muon decay. This scheme is sometimes called the "M W scheme" in the SMEFT literature. 3 (3) The "LEP scheme", which uses as inputs {α, G F , M Z }. In contrast to the first two schemes, M W is not an input. This scheme is sometimes called the "α scheme" in the SMEFT literature.
In Sections 2.1-2.3 we discuss the ingredients needed to implement these three EW input schemes to NLO. In order to treat them in a unified fashion, it is convenient to use as a starting point the tree-level Lagrangian written in terms of v T , M W , and M Z . In practice, this is obtained by transforming to the gauge-boson mass-basis using the field rotations defined in [59] and making the substitutions 3) which are valid up to linear order in the Wilson coefficients. The sine and cosine of the Weinberg angle are defined as The renormalised Lagrangian in a given scheme is then obtained by interpreting the treelevel parameters and fields as bare ones, denoted with a subscript 0, and trading them for renormalised quantities through the addition of counterterms appropriate to that scheme. For instance, all three schemes use counterterms δC i for the Wilson coefficients in the MS scheme, which are defined as (2.6) where is the dimensional regulator in d = 4 − 2 space-time dimensions. Explicit results for the δC i at one loop can be derived from [59][60][61].

The α scheme
The α and α µ schemes share as common inputs M W and M Z , renormalised in the on-shell scheme. They differ through the way the bare quantity v T,0 is related to renormalised parameters and counterterms. In the α scheme we use  Table 1: Input parameters employed throughout the paper. Note that v α is a derived parameter.
We have introduced the derived parameter where α = e 2 /(4π) is the QED coupling constant defined in a given renormalisation scheme.
The superscripts i and j in the counterterms ∆v (i,j,α) α label the operator dimension and the number of loops (j = 0 for tree-level and j = 1 for one-loop) respectively, while the superscript α refers to the fact that the expansion coefficients are multiplied by explicit factors of v α . The dependence on the perturbative expansion parameter 1/v 2 α ∼ α is then explicit. 4 The expansion coefficients in Eq. (2.7) are determined by the counterterms for the input parameters M W , M Z , and the electric charge e. These are calculated from two-point functions as in [57]. In the α scheme, we relate the bare and renormalised quantities up to NLO as ∆X (4,1,α) + ∆X (6,1,α) , (2.9) where X ∈ {M W , M Z , e} and X 0 are the corresponding bare parameters. We use the same notation for the expansion coefficients of the derived parameters c w and s w , so that, for instance, (2.10) At tree level the relation between v T and v α is given by [57] 1 There are a handful of exceptions to this in ∆v (6,1,α) α ; all appear in tadpoles, with the exception of the contribution from the Class-1 coefficient CW .
Interpreting this as a relation between the bare parameters, renormalising them as in Eq. (2.9), and matching with Eq. (2.7) we find ∆v (6,0 (2.14) When calculating EW corrections to heavy boson decay rates, it is natural to use a renormalisation scheme for α that avoids sensitivity to light fermion masses in counterterms. In the remainder of the paper, unless otherwise stated, we will use the MS definition of α in a five-flavour version of QED×QCD, where the top quark and heavy electroweak bosons have been integrated out and thus contribute finite parts to the ∆e through decoupling constants, see the discussion in Section 4.2 of [57]. 5 This running coupling, α(µ), is related to the effective on-shell definition at The (fixed-order) solution to the RG equation for the running of α to other scales necessary for this work is discussed in more detail in Section 5.2 of [57] and is given as where γ e (M Z ) = α(M Z ) π × 20 9 . Values for these parameters at other scales considered in this work are given as

The α µ scheme
In contrast to the α scheme, the α µ scheme uses G F rather than α as an input parameter. This can be implemented by modifying the counterterm for v T in the α scheme, Eq. (2.7), to read 1 v 2 (2.18) 5 The running α defined in the five-flavour version of QED×QCD is denoted as α ( ) in that reference.
The superscripts on ∆v µ have the same meaning as in Eq. (2.7), so that in particular the superscript µ means that the expansion coefficients multiply distinct powers of We have introduced the derived EW coupling α µ in the final equality of the above equation.
The expansion coefficients in Eq. (2.18) are obtained by a renormalisation condition relating muon decay in SMEFT with that in Fermi theory. We give the technical details of the calculation, and results for the coefficients ∆v (i,j,µ) µ , in Appendix A. A previous result for these coefficients has been given in [18], using a simplified flavour structure for the SMEFT Wilson coefficients, and omitting tadpoles such that the results are gauge dependent and limited to R ξ gauge. While we have made no flavour assumptions and included tadpole contributions in the FJ tadpole scheme [64], so that the coefficients are gauge invariant, we have checked that our results are consistent with those in [18] when the same calculational set-up is used, thus providing a strong check on both sets of results.
For two-body decays of heavy bosons, the SMEFT expansion coefficients in the α µ or α scheme take the form where X = X(M W ), in a slight abuse of notation we have defined and one is to set M W =M W on the right-hand side of the relations in Eq. (2.36). As a simple example, we can relate the counterterm for M W in the on-shell scheme to that in the LEP scheme. Setting X = M W in Eq. (2.35), and recalling that the on-shell definition of M W has no tree-level dimension-six contributions, we can write The terms on the second line as determined from Eq. (2.36) read We emphasise, however, that the LEP scheme uses {α, G F , M Z } as input parameters, so the result is ultimately a function of these parameters and the associated counterterms {∆e,∆v µ ,∆M Z }, which can be obtained from expansion coefficients in the α or α µ scheme similarly to∆M W .

Salient features of the EW input schemes
We are mainly interested in two features of the EW input schemes: the number of Wilson coefficients they introduce into physical observables through renormalisation, and perturbative convergence. Ideally, one would like a small number of coefficients to appear, so that the finite parts of observables are dominated by process-specific Wilson coefficients rather than those related to the EW renormalisation scheme. Furthermore, one would like to avoid large corrections between orders, so that perturbation theory is well behaved and can safely be truncated at a low order. We discuss these two issues in the following subsections.

Number of Wilson coefficients
It is a simple matter to count the number of Wilson coefficients appearing in the finite parts of counterterms for the bare parameters M Z,0 , M W,0 and v T,0 in the different input schemes.
The results at LO and NLO are listed in Table 2. Here and below we exclude Wilson 0  0  2  2  NLO  12  29 29  29   α µ  LO  0  0  3  3  NLO  13  30 12  33   LEP  LO  5  0  3  5  NLO  33 30 12 33 in each of the three counterterms considered here. Note that although all schemes use the on-shell renormalisation scheme for M Z , its dimension-six counterterm still differs between the schemes. To see this explicitly, we note that expansion coefficients in α µ and α schemes can be written in the form where here and throughout the remainder of the paper the choice of σ ∈ {µ, α} selects between the α and α µ schemes. An analogous equation holds for the counterterms for M W . The coefficients ∆M (4,1) Z and ∆M (6,1) Z are the same in the two schemes, but differences in the dimension-six piece arise due to the renormalisation of v T . In the LEP scheme one must use σ = µ and in addition apply Eq. (2.33) to trade M W forM W , which gives an additional scheme-dependent contribution.
The specific Wilson coefficients appearing in the various counterterms in the α scheme are determined by the two-point functions shown in Figure 1. The counterterm for the W -boson mass contains the following coefficients: C H and C HW contribute to the two left-most topologies in Figure 1, while C HW B and C W contribute to topologies three and four, which involve vertices with at least three gauge bosons. C HD appears in all four purely bosonic diagrams. We see that 7 of the 12 coefficients appearing are due to flavour-specific W couplings to fermions, arising from the right-most graph in Figure 1. Since in the SM the W boson couples only to left-handed fermions, the SMEFT operators must also be left-handed unless they contain a top-quark loop (in which case a chirality flip is associated with a power of m t ), which explains the relatively small number appearing. For the Z-boson mass, on the other hand, both left and right-handed couplings are relevant even for massless fermions, and operators containing the field-strength tensor for the hypercharge field B µ , namely C HB and C uB , contribute as well. This leads to a much larger number of coefficients compared to M W . The full set is: The counterterm ∆v requires also the counterterm ∆e, as shown in Eq. (2.14). Only those Wilson coefficients appearing in W , top-quark or Higgs loops contribute to the finite parts of the counterterm for electric charge renormalisation (through decoupling constants, as explained in [57]), which limits the result to the following 6 coefficients: All of these are already contained in ∆M , so the set of coefficients contributing to ∆v (6,1,α) α is the same as in Eq. (3.3). In the α µ scheme, one needs the counterterms ∆v (6,j,µ) µ , which are calculated from muon decay in Appendix A. In SMEFT, two kinds of coefficients appear at NLO -those that involve modified couplings of the external fermions, including four-fermion operators of the kind shown in Figure 2, or those that contribute to the W -boson two-point function at vanishing external momentum. The latter condition eliminates some operators compared to what is seen in ∆M W itself (in the case of massless fermions or certain derivative couplings), while the former increases it mainly due to four-fermion operators. The end result is that the following set appears: The counterterms for M W and M Z are also modified compared to the α scheme, as follows from Eq. (3.1); one finds that the α µ scheme contains the four-fermion coefficient C ll 1221 in addition to the α-scheme coefficients listed in Eqs. (3.2, 3.3).
Finally, in the LEP scheme the counterterm∆M ) is a function of those for e, M Z , and v T (renormalised in the α µ scheme), and thus contains the full set of 33 unique coefficients that also appear in the α µ scheme, while no additional coefficients appear in the counterterms for M Z or v T compared to the α µ scheme.
The conclusion of this counting exercise is that there is a large overlap between the set of operators appearing in the NLO counterterms in the different schemes. The main difference is that a handful of four-fermion operators related to muon decay appear in the LEP and α µ schemes but not in the α scheme. The number of Wilson coefficients contributing to observables in the different schemes is process dependent and is determined by the structure of the LO amplitude. For instance, consider a process involving a γ vertex, where is a charged lepton and γ is a photon. In the α scheme, the square of the bare vector-coupling vertex plus SMEFT counterterms (other than from field strength renormalisation) reads In the α µ scheme, on the other hand, the bare vertex plus associated counterterms read ∆r (4,1) − ∆r (6,1) + 2∆r (6,0) ∆r (4,1) .
The two results are equal to each other if Eq. (2.20) is used to relate v µ to v α , but when the numerical value of v µ is used as an input the terms on the second line of Eq. (3.7) contribute a large number of coefficients compared to what one has in the α scheme. The same set of coefficients contributes to muon decay calculated in the α scheme, or in the LEP scheme when M W appears in a tree-level vertex.

Perturbative convergence
Generally speaking, one uses renormalisation schemes that avoid sensitivity to large logarithms of light fermion masses in fixed-order corrections, and also tadpole contributions to finite parts of observables in cases where some parameters are renormalised in the MS scheme and some in the on-shell scheme [57]. As long as those two issues are dealt with, top-quark loops are the main source of enhanced NLO corrections in the finite parts of counterterms. These can be especially important when associated with the counterterm ∆s w , since they involve inverse powers of s 2 w ∼ 0.25 through the relation where the factor of 2 is chosen to match that in Eq. (2.13).
In the SM, enhanced corrections from top-loop contributions to ∆s w related to the renormalisation scheme are easy to trace. First, by analysing the one-loop Feynman diagrams in the large-m t limit, one can show that in the α µ scheme ∆v (4,1,µ) µ mt→∞ ≡ ∆v The subscript "t" here and below refers to the large-m t limit of the given quantity, i.e. the terms containing positive powers of m t in the limit m t → ∞. Second, using Eqs. (2.13, 2.23), along with the fact that the SM contributions to ∆e are subleading in the large-m t limit, the α-scheme result is and we have defined (3.12) The numerical values above use µ = M W to evaluate the running parameter v α , along with the inputs in Table 1. Finally, using Eqs. (3.9, 3.10), the counterterms for v T in the large-m t limit in the two schemes can be written as where δ ασ is the Kronecker delta, and we have used that ∆M For the heavy boson decays considered in this work, the tree-level decay rates all scale as 1/v 2 T . Therefore, Eq. (3.13) produces a simple pattern for the NLO corrections in the α and α µ schemes. In the α µ scheme, the tadpole and divergent contributions in ∆M (4,1) W,t cancel against other such contributions in physical observables, producing oneloop corrections proportional to ∆ρ (4,1) t ∼ 1% in the large-m t limit. In the α scheme, the ∆M (4,1) W,t term is accompanied by a factor of ∆r (4,1) t , which produces a correction of roughly −3.4% compared to the α µ scheme. One indeed sees this pattern in the NLO SM corrections to W decays, Z decays, and Higgs decays into fermions, shown in Tables 3, 4, and 5. Input-scheme dependent NLO corrections to weak vertices are thus better behaved in the α µ scheme, and the numerical differences between the two schemes are nearly process independent. 7 We now ask whether a simple relation between the dominant NLO corrections in the α and α µ schemes also exists in SMEFT. To this end, we first define where z W is the squared wavefunction renormalisation factor of the W -boson field. After replacing the bare quantities on the left-hand side by their renormalised counterparts, it is straightforward to determine the K W,t , and ∆v This yields K at tree level, and substituting in the explicit results for the counterterms leads to the following one-loop expressions in the α scheme: (3) Hq 33 (3. 16) In the α µ scheme one has instead . (3.18) One sees that the SMEFT expansion of K W is tadpole free, finite, and independent of the renormalisation scale up to NLO. This is not an accident -it gives the flavourindependent part of the large-m t limit of W decay into fermions. Furthermore, rearranging the above expressions yields the following result for the v T counterterms: 8 (3.19) The SM part of Eq. (3.19) is identical to Eq. (3.13). The dimension-six parts are the generalisation to SMEFT. In each case, the counterterm for v T is split into two distinct parts: a physical piece that is a finite, gauge and scale-independent quantity (the K W ), plus remaining terms which contain tadpoles and divergent parts that cancel against other such terms in physical observables. While at one-loop in the SM it was simple to identify the physical factor ∆r (4,1) in the α scheme by studying the counterterm v (4,1,α) α alone, in SMEFT it is helpful to choose an observable process in order to split the counterterm into the two distinct parts. While the choice of W decay is not unique, it leads directly to the SM results obtained from studying v T alone.
We can now use our expressions for the counterterms for v T in Eq. (3.19) to check whether, as in the SM, a simple pattern emerges for input-scheme dependent SMEFT corrections to weak vertices. As an example, consider the following expression, which gives a flavour-independent correction to Z-boson decays into fermions: where z Z is the wavefunction renormalisation factor squared of the Z-boson field. 9 The expression on the right-hand side is finite, tadpole free, and scale-independent up to NLO. Writing the counterterms for v T using Eq. (3.19), one has Here we have split each term in the perturbative expansion further into scheme dependent and independent parts (the latter being denoted without the σ superscript). Both the scheme dependent and independent parts are separately scale independent and tadpole free. The results for the scheme-independent pieces are wherek Inverse powers of s w appear only in the α scheme and are absorbed into the factors K (i,j,α) W , so the scheme-independent coefficients k (i,j) Z have an expansion in ∆ρ (4,1) t . In the SM, 9 Compared to Eq. (3.14) an additional factor of CHD arises for Z-boson decays. This arises from the relations between the W/Z-mass and the Lagrangian parameters in SMEFT and can be seen by considering the flavour independent part of Eq. (5.25) in addition to Eq. (5.27) in [59].
it is evident that the scheme-dependent corrections k (4,1,σ) Z follow the pattern discussed after Eq. (3.13). In SMEFT, scheme-dependent corrections appear in the combination K in the last line of Eq. (3.21). Moreover, the K (6,1,σ) W pieces are explicitly µ-dependent, and one normally chooses the scale in a process-dependent way. For these reasons, the numerical pattern of scheme-dependent NLO corrections to weak vertices in SMEFT in the α and α µ schemes is not nearly as regular as in the SM; this is best seen by comparing results for a range of processes, which we leave to Section 5.
We have focussed the above discussion on the α and α µ schemes. Corrections in the LEP scheme are derived from those in the α µ scheme by using Eq. (2.33) to eliminate M W in favour ofM W . The result simplifies considerably in the large-m t limit. To derive it, we first note that the large-m t limit of the expansion coefficients of ∆r defined in Eq. (2.23) can be written in terms of the K W from Eq. (3.14) according to (3.24) We can convert these into expansion coefficients of∆r t using Eq. (2.31). The only nontrivial SMEFT piece is the NLO coefficient, for which we find Inserting these results into Eq. (2.33) gives the following large-m t corrections to the Wboson mass in the LEP scheme within the SM As an example, let us use this to write the factor of M 2 W in Eq. (3.14) in terms ofM 2 W . Denoting the resulting LEP-scheme expansion coefficients asK (i,j,µ) W , one has the NLO SM resultK while the NLO contribution iŝ For other processes, the numerical factors multiplying the∆ W terms are dictated by the dependence of the bare vertex on M W , and are therefore rather process dependent.

Derived parameters
The simplest observables are "derived parameters", where an input parameter in one scheme is calculated as a SMEFT expansion in another. For the schemes considered here there are three such quantities: α in the α µ scheme, G F in the α scheme, or M W in the LEP scheme. All of these are functions of the expansion coefficients (and their derivatives, in the case of the LEP scheme) of ∆r defined in Eq. (2.20). In this section we briefly examine the latter two cases, and also define the procedure for estimating higher-order corrections in the SMEFT expansion through scale variations used throughout the remainder of the paper. The SMEFT expansion for G F in the α scheme is obtained from Eq. (2.20) and yields The tree-level result (LO) evaluates to and the sum of tree-level and one-loop corrections (NLO) is where in both cases we have used µ = M Z , so that v α = v α (M Z ) and C i = C i (M Z ) in the above equations. In the SM, the LO prediction for G F differs by 3.4% from the measured value while at NLO the difference is −0.8%. Evidently, the large-m t limit contribution in Eq. (3.11) accounts for the bulk of the NLO correction. The LO SMEFT result contains 5 Wilson coefficients which alter the result, while the NLO one contains the full set of 33 Wilson coefficients identified in Table 2.
SMEFT expansions of physical quantities such as G F,α contain a residual dependence on the renormalisation scale µ due to the truncation of the full series at a fixed order in perturbation theory. In the SM this is due to the running of α, while in SMEFT the Wilson coefficients C i also run. It is often useful to use the stability of the results under variations of the scale µ about a default value as an estimate of uncalculated, higher-order corrections in the perturbative expansion. The Wilson coefficients are unknown numerical quantities that we wish to extract from data, so in order to implement their running we must calculate their value at arbitrary scales µ given their value at a default scale choice µ def . For our purposes, it is sufficient to use the fixed-order solution to the RG equation in this calculation, which reads whereĊ i was defined in Eq. (2.6). For the running of α we can also used the fixed-order solution to the RG equation given in Eq. (2.17). Throughout the paper, we estimate uncertainties from scale variations by using the afore mentioned equations to evaluate observables for the three scale choices µ ∈ {µ def , 2µ def , µ def /2}. Central values are given for µ = µ def , and upper and lower uncertainties are determined by values of the observables at the other two choices. 10 Let us apply this method to the calculation of M W in the LEP scheme, which is obtained by evaluating Eq. (2.33). Compared to G F,α , the W -mass is sensitive to a different combination of ∆r as well as its derivatives with respect to M W . The LO result with µ = M Z as the default value and scale uncertainties estimated as described above yields where the . . . indicate contributions where the difference between the upper and lower values obtained from scale variation is less that 1% ofM W when the numerical choice We can understand the qualitative features of these results by studying them in the large-m t limit. Using Eqs. (3.26, 3.27) for the NLO corrections in this limit, the numerical result at the scale µ = M Z is This is clearly a good approximation to Eq. (4.6), where as in that equation we have not included contributions of less than 1%. The SM result is scale invariant in this limit, because the top quark is decoupled from the QED coupling α(µ).
In the above results and throughout the paper we used the MS definition of α in a five-flavour version of QED×QCD. In the literature, one often uses the effective on-shell coupling α O.S. (M Z ) which is related to α(M Z ) using Eq. (2.15). When instead this choice is made, we find the following SM results for G F in the α scheme At LO these two quantities differ by 2 − 3% compared to Eqs. (4.2, 4.5), while at NLO the differences in the two schemes for α are well below the percent level; we have checked that this also holds true in SMEFT. The NLO result for M W in SMEFT generalises the previous result [29] to include the full flavour structure, and resums logarithms of light fermion masses related to the running of α; a more detailed comparison is given Appendix D. The current state-of-the-art in the SM [67] includes complete two-loop corrections as well as a partial set of even higher-order corrections. Adjusted to our numerical inputs, the result derived from the parametrisation in Eq. (6) of that paper, which we refer to as "NNLO", reads which is outside the uncertainties in the NLO result Eq. (4.6). In order to gain insight into the structure of higher-order corrections, we have studied the split of the NNLO result into pure EW, and mixed EW-QCD components, which was given in [67] for the unphysical value M h = 100 GeV. When adjusting our own inputs to that unphysical value, we find that the pure NNLO EW contributions are within our NLO uncertainty estimate, so that the discrepancy is due to mixed EW-QCD effects first appearing at NNLO and unrelated to the running of α. The large-m t limit of these EW-QCD corrections can be obtained by making the following replacement in Eq. (3.26) [68]: Including this correction changes the central value in Eq. (4.6) to 80.41 GeV, which agrees with the NNLO result to better than the per-mille level. Further improvements can be made through resummations of the type discussed in Section 6.

Heavy boson decays at NLO
While the previous sections elucidated some general features of the different input schemes, the aim of this section is to study in detail three benchmark observables to complete NLO in the SMEFT expansion in each scheme: W decay into leptons, Z decay into charged leptons, and Higgs decay into bottom quarks. For the numerical analysis we focus on W → τ ν and Z → τ τ , while in the analytic results submitted in the electronic version we keep the lepton species arbitrary. We write the expansion coefficients of the decay rates to NLO in SMEFT for boson where the superscript s(i, j) refers to dimension-i, j-loop contributions in input scheme s ∈ {α, α µ , LEP}. To study convergence, it is convenient to work instead with expansion coefficients of the decay rate normalised to the LO SM result, namely ∆ s(i,j) Throughout the section numerical values for the decay rates are evaluated using the default value µ def. = m decay , where m decay is the mass of the decaying particle, and scale uncertainties are obtained by varying the scale up and down by a factor of 2 about the default value, as in Section 4. Obviously, results for three decays in three renormalisation schemes and involving a large number of SMEFT Wilson coefficients contain a plethora of information. We have organised it as follows: • Figures 4, 5 and 6 show Eq. (5.2) for the NLO SM corrections as well as corrections appearing at LO and NLO in SMEFT when the choice C i = 1 TeV −2 is made. They also show the large-m t limits of the NLO corrections in cases where top-loops contribute, and group the coefficients such that those appearing solely due to the choice of renormalisation scheme appear on the far right. Figure 3: Representative virtual corrections for W decay into leptons at NLO.
• In Tables 3, 4 and 5 we show the size of the NLO corrections to the SM and SMEFT coefficients which appear at tree-level in the different schemes, for the default scale choices.
• In Appendix B, we give results for the numerically most important contributions to the decay rates at LO and NLO in the SMEFT expansion, including uncertainties as estimated from scale variations.
The following subsections serve to explain and highlight the most noteworthy patterns emerging from these results.

W → ν decays
The tree-level decay rate for W → τ ν decays, written in terms of v T , takes the form Renormalisation-scheme dependence thus enters the result through the counterterms for M W and v T . The NLO decay rate is calculated by evaluating virtual corrections such as those shown in Figure 3, and then adding together with UV counterterms and real emission diagrams with an extra photon in the final state to get a finite result. The size of NLO SM corrections in the different schemes is easily understood using the large-m t analysis in Section 3.2. In that limit, the NLO corrections in the α µ scheme vanish, while those in α scheme are roughly −3.4%, a pattern which agrees well with the full results in Table 3. The SM LEP scheme corrections in the large-m t limit arê The first ratio agrees quite well with the estimate G NLO F,α /G F using Eq. (4.3), while the second is consistent with the estimate (M NLO W ) 3 /M 3 W using Eq. (4.6). Once the NLO corrections are included the results between the schemes show (better than) percent-level agreement.
In Figure 4 the corrections in SMEFT are shown. The absolute size of the SMEFT corrections is determined by the choice C i = TeV −2 . For that choice, SMEFT contributions are suppressed by v 2 σ × TeV −2 ≈ 6%, and are anywhere between 10% to below per-mille level of the SM tree-level result depending on the coefficient. The NLO SMEFT results contain a large number of Wilson coefficients. We have organised the coefficients in Figure 4 such that those appearing only due to the renormalisation of v T or M W up to NLO are separated out onto the right part of the figure, while those appearing also in the bare matrix elements or wavefunction renormalisation factors and thus common to all schemes are on the left. In the α µ scheme the coefficients ∆v As in the SM, the numerically dominant NLO SMEFT corrections are related to topquark loops. In the α and α µ schemes, the scheme-dependent corrections in the large-m t limit are nearly all contained in the factors K W given in Eqs. (3.15, 3.17). For the default input choices, the SMEFT contributions evaluate to For coefficients appearing at LO, the NLO corrections are the second term in the parentheses, facilitating a comparison with Table 3. Results also for coefficients first appearing at NLO can be found in Eq. (B.3) and Eq. (B.5). We see the large-m t limit corrections are a good approximation to the full ones. Interestingly, for the coefficients appearing at LO, there is no large hierarchy between the size of NLO corrections in the α scheme compared to the α µ scheme, even though the analytic result for K (6,1,α) W contains 4 (3) inverse powers of s w in the case of C HD (C HW B ). In fact, the largest corrections are from C (1) Hq 33 and C Hu

33
, which appear only due to the scale-dependent logarithmic terms from Eq. (3.16). This illustrates the important point that, unlike the SM, the NLO corrections are strongly scale dependent in SMEFT.
The SMEFT corrections in the LEP scheme can be derived from results in the α µ scheme using Eq. (2.33) to write M W in terms ofM W . The expansion coefficients arising after converting the factor of M 2 W in the large-m t limit,K (6,j,µ) W , were given in Eqs. (3.29, 3.30). In order to calculate the decay rate one must also write the factor of M W arising from 2-body phase space in terms ofM W . We have checked that after doing so the large-m t limit corrections to the coefficients appearing inK W are a good numerical approximation to the full ones.
In addition to the corrections related to the flavour-independent corrections, there are also contributions from the coefficient C , which specifically modifies the τ νW coupling.
The large-m t limit correction to ∆ LEP(6,1) W,t due to this coefficient is given by −2∆ρ The corresponding results in the α and α µ schemes are obtained from the above by settinĝ ∆ are about 4% in the LEP scheme, and 2% in the α and α µ schemes, in rough agreement with Table 3. Compared to the other schemes, the NLO corrections to the coefficients appearing at treelevel in the LEP scheme show a rather irregular pattern due to the complicated dependence on the Weinberg angle. While the size of the NLO corrections studied above is rather scale dependent, the sum of the LO and NLO contributions is independent of the scale (up to uncalculated NNLO terms in the SMEFT expansion) and is thus much less sensitive. To study this effect in detail, in Appendix B.1 we give numerical results in the three schemes including scale variations at LO and NLO. It is seen that in SMEFT, the dominant NLO corrections are typically within the uncertainties of the LO calculation as estimated through scale variations, and that the scale uncertainties in the NLO results are substantially smaller than in the LO ones.

h → bb decays
The tree-level decay rate for h → bb decay is given by The decay h → bb has two important differences with respect to the decays W → ν and Z → (to be discussed in Section 5.3). First, we retain the b-quark mass and, second, the   strong coupling α s (µ) plays a role in the results already at NLO. The Higgs mass M h is evaluated on-shell, but the NLO corrections do not involve its counterterm since it appears through phase space rather than through the amplitude. Therefore, the input-scheme dependence to NLO arises mainly through the counterterm for v T . 11 The decay h → bb receives both QCD and EW corrections at NLO. The two effects are additive and to study the EW input scheme dependence of the results it is useful to quote the QCD and EW corrections separately, as in Table 4. To this order, the QCD corrections are scheme independent. In the α scheme the EW corrections are rather large and depend heavily on the Wilson coefficient considered, ranging from -11% to 4% and thus inducing significant shifts to QCD alone, while in the α µ and LEP schemes the corrections are smaller are more uniform.
We can understand the qualitative features of the NLO EW corrections using the 11 Results in the αµ and LEP scheme differ because one must eliminate MW in favour ofMW in the NLO SM correction, but this is a small effect numerically.
large-m t limit. To this end, we use Eq. (3.19) to write the NLO decay rate in this limit as where K (6,0) h is the SMEFT contribution in Eq. (5.8), and the scheme-dependent part of the NLO SMEFT correction is (5.10) Large-m t limit results in the α scheme have been given previously in [57], while those in the α µ scheme can be extracted from [69]. We make use of those results in what follows, thus employing the "vanishing gauge coupling limit", which in this case amounts to taking the limit M W M h in addition to m t → ∞. The LEP and α µ scheme results are identical in this limit.
In the SM, the scheme-independent NLO correction in the large-m t limit is given by It follows from the discussion in Section 3.2 that the large-m t limit corrections in the α µ scheme are tiny, while those in the α scheme are well approximated by K (4,1,α) W ≈ −3.4%. Clearly, this mimics the features of the exact NLO EW corrections given in Table 4.
In SMEFT, the scheme-independent 12 NLO correction in the large-m t limit is given by where L t = ln(µ 2 /m 2 t ) and we have set N c = 3. The . . . refer to Wilson coefficients which contain no overlap with those appearing in the scheme-dependent pieces in Eq. (5.10). In the α scheme the numerical value of the NLO corrections at µ = M h is , and the order of the numbers inside the parentheses multiplying the Wilson coefficients on the right-hand side of the 12 In fact there is mild dependence on the scheme through the numerical value for vσ. above equation matches the order of the two terms on the left-hand side. In most cases the scheme-dependent parts contained in ∆K (6,1,α) W dominate over the scheme-independent ones. For coefficients not appearing already at NLO, one can verify that the results above are close to the exact NLO results in Eq. (B.12). Combined with the LO result in Eq. (B.11), one infers NLO EW corrections of −9% for C HD , −5% for C HW B in the α scheme. In the α µ scheme, one has (5.14) Contributions from C HW B are completely absent in the α µ scheme, while the NLO EW correction to C HD from the above result and Eq. (B.13) is 3% in the large-m t limit. This explains the pattern of results seen for these coefficients in Table 4. It makes clear that in this case factors of K (i,j,α) W work much the same in SMEFT as in the SM, producing sizeable NLO EW corrections compared to the α µ scheme.
The full set of NLO corrections in the different schemes is shown in Figure 5. In the numerical results in Appendix B.2 we follow [57] and leave in symbolic form enhancement factors of m b /v σ which disappear when Minimal Flavour Violation is assumed. We have not done this in the figure, which explains, for instance, the very large contribution from C dH 33 .
In contrast to the case of W decay, in some cases there are large differences between the large-m t limit and full corrections; this occurs when a Wilson coefficient receives both EW and QCD corrections, the latter invariably being the larger effect. From the perspective of EW input-scheme dependent corrections, the most important feature of the figure is the number of Wilson coefficients appearing. In particular, there are far more in the α scheme, 42 in total, than in the α µ or LEP schemes, both of which receive contributions from the same 29 Wilson coefficients. The main reason is that the renormalisation of v T in the α scheme involves the large set of flavour-specific couplings to fermions identified given in Eq. (3.3), while in the α µ and LEP schemes M Z does not enter the tree-level amplitude and many of these coefficients are therefore absent.

Z → decays
The tree-level decay rate for Z → τ τ decay, written in terms of v T , takes the form Γ (4,0) where The term inside the square brackets in the first line of Eq. (5.15) is independent of the fermion species into which the Z decays and was considered in Eq. (3.20). The function g τ depends on the charge and weak isospin of the τ lepton, and the terms on second line are specific to Zτ τ couplings in SMEFT. The LO decay rate depends on the full set of parameters M W , M Z , v T , and so scheme-dependent corrections involve the full set of coefficients identified in Section 3.1. The NLO decay rates in the three schemes are shown in Figure 6. In the α scheme the set of coefficients appearing in the renormalisation of v T is the same as that for renormalising M Z and M W , so it does not introduce any unique coefficients at NLO. In the LEP and α µ schemes, on the other hand, the renormalisation of v T introduces a set of 4-fermion coefficients shown on the right-hand side of the figure that would not otherwise appear in the decay rate. In this case the number of coefficients appearing at NLO is quite large: 63 in the α scheme, and 67 in the α µ and LEP schemes.
In order to understand the dominant corrections we study the large-m t limit. Let us first consider the corrections to the SMEFT coefficients specific to Zτ τ couplings, given in the second line of Eq. (5.15). In order to evaluate them in the three schemes, we can use ∆M (4,1) where the . . . signify tadpole contributions which cancel against those in the bare matrix elements. Along with the LEP scheme result it is then easy to show that in the large-m t limit we can replace the tree-level expressions involving C He where the first result is for the α µ (or α scheme after µ → α) and the second line is for the LEP scheme. The results are a good approximation to the exact ones shown in Table 5. The fairly large difference between the LEP and α µ scheme makes clear that the corrections can be quite sensitive to the exact dependence on e.g. s w in the tree-level results. We have checked that the corrections to the remaining coefficients appearing in the second line of Eq. (5.15) are also well-approximated by the large-m t limit. The NLO corrections related to the first line of Eq. (5.15) are more complicated. To study them, we first note that the large-m t limit corrections to the function g τ can be written in the α and α µ schemes as The scheme-independent function g (4,1) τ is obtained by replacing s w → s w (1 + ∆s w ) and isolating the SM corrections; it thus reads The function g (6,1) τ is obtained in the same way, except for in that case one must also include corrections from Z − γ mixing to get a finite and tadpole-free result. The explicit result is .

(5.23)
We can now obtain the NLO corrections to the first line of Eq. (5.15) in the large-m t limit in the α scheme through the replacement . This accounts for the SM corrections in the α and α µ schemes given in Table 5, which as in Higgs and W decay follows the pattern identified in Section 3.2.
Turning to SMEFT, the LO corrections in the α scheme are contained in where the order of the terms on the second line matches that in the first. In the α µ scheme one replaces α → µ in the above equation; in that case it is clear that the tree-level contributions from C HD and C HW B are quite small, since K  Table 5. Clearly, this pattern is quite complicated and is not driven by the scheme-dependent factors K W as in the SM. On the other hand, the coefficients on the second line only appear through K (6,1,α) W , and as seen from the exact results in Eq. (B.18) we see that this factor indeed absorbs the dominant corrections from them, much like K (4,1,α) W in the SM. The LEP scheme results can be obtained from those in the α µ scheme by employing Eq. (2.36). In the large-m t limit the only non-trivial conversions are on the functions g τ , which contain M W dependence already at tree level. For instance, calling the LEP-scheme functionsĝ τ , we have the LO SMEFT result  Table 5: NLO corrections to prefactors of LO Wilson coefficients in the three schemes.
Negative corrections indicate a reduction in the magnitude of the numerical coefficient of a given Wilson coefficient, while < 0.1% indicates changes below 0.1%, both positive and negative. The flavour index j refers to j ∈ 1, 2. and the LEP-scheme version of Eq. (5.26) becomeŝ Compared to the α µ scheme, the LO result for the coefficient C HD is significantly larger, and those from the operators contained in K (6,0,µ) W are slightly smaller, which roughly explains the pattern for those coefficients seen in LEP scheme results Table 5. The result for C HW B is slightly increased, but remains small and for that reason still receives a substantial NLO correction.
We have derived the complete large-m t limit results and verified that they provide a good approximation to the full one, but the explicit expression for the functionĝ (6,1,µ) τ is somewhat lengthy and we do not reproduce it here. In Section B.3 we show detailed LO and NLO results including uncertainties estimated from scale variations. It is clear that in cases where the NLO corrections are large, namely for certain operators in the α µ and the LEP schemes, the uncertainties are underestimated, while in the α-scheme the uncertainty estimates are more reliable. This example highlights very clearly that the issue of NLO corrections in SMEFT is considerably more scheme and process-dependent than in the SM. The general rule that NLO corrections to weak decays are smaller in the LEP and α µ schemes than in the α scheme familiar from the SM does not transfer over to SMEFT. W τ ν , as defined in Eq. (5.2), for the decay W → τ ν in the three schemes. Note that "NLO" in the legends only refers to the NLO corrections and that we write superscripts in the Wilson coefficient names as C Hq3 ≡ C   Figure 4, but for the decay Z → τ τ .

Universal corrections in SMEFT
A recurring theme of the previous sections was that EW corrections are dominated by top loops. While the numerical patterns in EW input-scheme dependent top-loop corrections in the SM are quite regular, those in SMEFT are more process and Wilson-coefficient dependent. The purpose of this section is to show that the dominant scheme-dependent EW corrections in SMEFT can nonetheless be taken into account by a certain set of simple substitutions in the LO results, similarly to the well-studied case of the SM. Let us begin the discussion with the SM, where an important feature is that weak vertices in the α scheme receive corrections proportional to ∆r (4,1) t , related to the renormalisation of v T . It is simple to resum such corrections to all orders in perturbation theory. Using the large-m t limit result in Eq. (3.10), and keeping for the moment only the ∆r (4,1) t terms (i.e. terms enhanced in the limit c 2 w /s 2 w 1, in which case the ∆M W,t piece is subleading), we have This resums the ∆r Expressing the counterterm for v T as an expansion inṽ α rather than v α will obviously lead to a quicker convergence between orders. For example, the SM prediction to NLO for the derived quantity G F in such a "α scheme" is Numerically, including uncertainties from scale variation using the procedure described in Section 4, where G LO F,α refers to the first term in Eq. (6.3). This shows considerably improved convergence compared to the fixed-order α scheme expression in Eq. (4.1), and scale variations in the LO result give a good estimate of the NLO corrections.
To the best of our knowledge, a resummation of the type described above was first derived in [70], at the level of the W -boson mass in the LEP scheme (and also including subleading two-loop terms in the limit s w → 0). In that case, similar reasoning using Eq. (2.20) as a starting point leads to the resummed LO prediction The NLO result within the resummation formalism, modified to avoid double counting, is (4,1) , ∆r (4,1) = ∆r (4,1) − ∆r where in both cases the sequence of three numbers after the first equality are the fixedorder LO, the fixed-order NLO correction, and the beyond NLO corrections, respectively. Clearly, the NLO expansions of the resummed formulas approximate the full results at sub percent-level precision, so a fixed-order implementation suffices for practical applications. Universal NLO corrections to weak vertices implied by resummation can be obtained through a procedure of substitutions on LO results. The remaining, non-universal NLO corrections need to be calculated on a case-by-case basis, but these are typically small compared to the ones already included at LO through the aforementioned substitutions. While such procedures for universal corrections are well known in the SM (see for instance [71]), we give here a first implementation within SMEFT.
Step-by-step, it works as follows (2) Make EW-input scheme dependent replacements on the LO amplitudes. In the α or α µ scheme, these read where as usual σ ∈ {α, µ} and the K W are given in Eqs. (3.15, 3.17).
In the LEP scheme, make the above replacements with σ = µ in the LO amplitude. Subsequently, eliminate M W in favour ofM W using Eq. (2.33), in both the replacements and everywhere else in the LO observable (so that factors of M W related to phase space are also taken into account).
(3) Expand the resulting expressions to NLO in a fixed-order SMEFT expansion before evaluating numerically.
We shall refer to results obtained from the above procedure as "LO K " accurate.
In the SM, the substitutions in Eq. (6.9) are sufficient to capture NLO corrections proportional to ∆r (4,1) t . Beyond that, writing M Z = M W /c w before performing the shifts ensures that the large-m t limits of both W and Z decay are reproduced. In SMEFT, the substitution for v T is motivated by Eq. (3.19), which splits the counterterm for v T into a "physical", µ-independent order-by-order in perturbation theory and tadpole free part, K W , and an "unphysical" part, which is tadpole dependent and divergent. The physical part captures the most singular large-m t corrections as s w → 0 in SMEFT, as well as µdependent logarithms. The substitutions for s w also capture such pieces of its counterterm, including a piece proportional to C which is easily shown to be proportional to the NLO SM result. Finally, in both SMEFT and the SM, the shift for c w is chosen to maintain s 2 w + c 2 w = 1. While Eq. (6.9) is not unique, other reasonable choices would differ only by terms proportional to ∆ρ (4,1) t rather than ∆r (4,1) t and thus agree with the above to roughly the percent level. 13 In Table 6, we compare various perturbative approximations to heavy-boson decay rates in the SM within the α and LEP schemes, in each case normalised to the NLO result in the α µ scheme at the default scale choice. The LO and NLO results refer to fixedorder perturbation theory, NLO t refers to the large-m t limit of NLO, and LO K refers to the sum of LO and NLO corrections obtained through the above procedure. For the case of W and Z decay in the α scheme, the convergence between LO K and NLO is greatly improved compared to pure fixed order, and varying the scale in the LO K results gives a  Table 6: SM results in the α and LEP schemes. For each process, the results are normalised to the SM NLO results in the α µ scheme.
good estimate for the residual corrections contained in the full NLO result. Also in Higgs decay LO K is a marked improvement over LO, although in that case the results in all schemes are subject to a roughly -1% scheme-independent correction which is unrelated to the large-m t limit and not captured through scale variations. We next turn to SMEFT, focusing on cases where LO K results involve corrections proportional to ∆r (4,1) t . In Table 7 we show heavy-boson decay rates in SMEFT in the α scheme, listing the prefactors of Wilson coefficients appearing in K (6,1,α) W . In this case, the NLO t (but not LO K ) results use the large-m t limit of Eq. (4.4) for scale variations of the Wilson coefficients. We see that also in SMEFT, the LO K description improves perturbative convergence compared to pure fixed order, taking into account especially the dominant scheme-dependent corrections. This works best for W decay, where the central values of LO K reproduce the NLO t results by construction, and perturbative uncertainties are reduced compared to LO while still showing a good overlap with the NLO results. In Higgs decay, Wilson coefficients that receive significant scheme-independent corrections as shown Eq. (5.12), such as C uW

33
, display the biggest deviations from the NLO t and NLO results at LO K accuracy, although scale variations generally give a good indication of the size of the missing pieces. The case of Z decay is similar, although in contrast to Higgs and W decay the form of the LO amplitude in Eq. (5.15) implies that the shifts of s w in Eq. (6.9) also play a role. This latter effect is even more important in Z decay in the α µ scheme; as shown in Table 8, LO K accuracy largely takes into account the very large corrections to C HD and C HW B (as well as the more moderate but still significant corrections to C He 33 ) seen in Table 5. The LO K results for Higgs and W decay in the α µ scheme, and for all decays in the LEP scheme, show similar levels of improvement as the cases discussed above -detailed tables can be found in Appendix C.    Table 8: The numerical prefactors of the Z decay SMEFT Wilson coefficients in the α µ scheme appearing leading to dominant corrections at various perturbative approximations. The tree-level decay rate as well as v 2 µ have been factored out and the results have been evaluated at the scale of the process.

Conclusions
We have performed a systematic study of three commonly used EW input schemes to NLO in dimension-six SMEFT. After introducing a unified notation which makes transparent the connections between the α, α µ , and LEP schemes, thus facilitating both NLO calculations in the schemes directly or conversions between them, we studied the structure of the SMEFT expansion in the different schemes. This was done at the generic level in Section 3, at the level of derived parameters such as the W -boson mass in the LEP scheme or G F in the α scheme in Section 4, and at the level of heavy boson decay rates in Section 5. In all cases these NLO calculations are either original or generalise previous results to include the full flavour structure of SMEFT. They will be useful for benchmarking automated tools for NLO EW corrections in SMEFT, when they become available, and we have therefore included the analytic results as computer files in the electronic submission of this work.
In the SM, the dominant differences between EW input schemes are mainly taken into account by NLO top-loop corrections to the sine of the Weinberg angle, s w . As an example, for decay rates of heavy bosons, these appear in our formalism through the renormalisation of the Higgs vacuum expectation value v T , and given that such decay rates scale as 1/v 2 T a regular pattern of roughly -3.5% corrections in the α scheme compared to the α µ and LEP schemes is observed. In SMEFT, the dominant corrections related to the renormalisation of v T still arise from top loops, but these involve µ-dependent logarithmic corrections related to the running of Wilson coefficients, in addition to more complicated dependence on the Weinberg angle than in the SM, and as a result the numerical results across Wilson coefficients and processes are not nearly as regular. Nonetheless, we identified the analytic structure of the dominant scheme-dependent NLO corrections in SMEFT, and gave in Section 5 a simple procedure for including these universal NLO corrections in the LO results. Once these are taken into account, residual NLO corrections in different schemes are of similar size; these corrections can be approximated by calculating process dependent top-loop corrections, or eliminated altogether through an exact NLO calculation.
We end with a comment on theory uncertainties and the choice of an EW input scheme in fits of SMEFT Wilson coefficients from data. Observables in SMEFT exhibit schemedependent sensitivity to the full set of SMEFT Wilson coefficients because input parameters across schemes are related through SMEFT expansions. However, once a comprehensive set of observables is combined and dominant scheme-dependent corrections have been taken into account, there is no strong argument in favour of one scheme or another, and the consistency of Wilson coefficients obtained from global fits to data in different input schemes provides a valuable check on the robustness of such analyses.
Science for its kind hospitality and support through the SRITP and the Benoziyo Endowment Fund for the Advancement of Science.

A The α µ scheme at NLO
The α µ scheme is defined by the renormalisation condition that the relation in Eq. (2.19), , holds to all orders in perturbation theory. The Fermi constant G F is a Wilson coefficient appearing in the effective Lagrangian The four-fermion operator Q µ mediates tree-level muon decay, and radiative corrections are obtained through Lagrangian insertions of a five-flavour version of QED×QCD, where the top-quark is integrated out. We will work only to NLO in the couplings, so QCD couplings will not appear and we can drop the QCD Lagrangian in what follows. The Fermi constant G F is calculated by matching SMEFT onto the effective Lagrangian above, by integrating out the heavy electroweak bosons and the top quark. In practice, this is done by ensuring that renormalised Green's functions match order by order in perturbation theory, to leading order in the EFT expansion parameter m µ /M W 1. The matching can be performed with any convenient choice of external states. We work with massless fermions, and set all external momenta to zero. In that case the loop corrections to the bare tree-level amplitude in the EFT are scaleless and vanish, so the renormalised amplitude is just given by the tree-level one plus UV counterterms. The main task is thus to evaluate the renormalised NLO matrix element for the muon decay in SMEFT.
To write the matrix element for the process µ → ν µ eν e , we first define the spinor product where P L = (1 − γ 5 )/2 and it is understood that the arguments of the Dirac spinors u and v are evaluated at p i = 0. Furthermore, we define expansion coefficients of the bare one-loop amplitude in terms of the bare parameter v T,0 as The . . . in the above equations refer either to spinor structures with different chirality structure, which we do not interfere with the tree-level SM result and can thus be neglected, or matrix elements of evanescent operators. Evanescent operators, which vanish in four dimensions as a result of their γ-matrix structure, no longer vanish in dimensional regularization where we work in d dimensions. The definition of the evanescent operators depends on the definition of the γ 5 matrix in d dimensions [73]. We choose to define γ 5 in naive dimensional regularization, where it anti-commutes with the other γ matrices, {γ 5 , γ µ } = 0. For the muon decay only one evanescent operator appears in the one-loop diagrams with a four-fermion interaction and a boson connecting the two fermion bilinears. It is defined in the chiral basis as [74] where P R = (1 + γ 5 )/2 and the ⊗ indicates a direct product of γ matrices (as in Eq. (A.3) after removing the external spinors). The scheme choice for the evanescent operators impacts the finite pieces at one-loop when multiplied with 1/ terms. The evanescent operator E LL itself can be removed by an appropriate counterterm. The renormalised amplitude in the α µ scheme to one-loop order then takes the form In where the superscript L has been used to indicate left-handed fermions and the ∆Z f are expanded as usual At one loop, ∆Z f receives contributions from photon graphs, which vanish, and heavyparticle graphs (Z and W exchanges), which give finite contributions that must be taken into account. The explicit results for the one-loop coefficients in Eq. (2.18) are relatively compact, and we list them here for convenience. In the SM, one has where the tadpole contribution in unitary gauge is and In SMEFT we find 16π 2 ∆v (6,1,µ) (3) Hl 11

+ 10C
(3) Hl 22 lq 2233 + 16π 2 ∆v where the tadpole contribution in unitary gauge is Note that the expansion coefficients are only gauge invariant when tadpoles are included -the split that we have given above is unique to unitary gauge.

B Numerical results for the decay rates
Here we present numerical results for the decay rates considered in Section 5 in the three schemes. We use the notation Γ s X,LO ≡ Γ s(4,0) where the quantities appearing on the right-hand side are defined in Eq. To evaluate scale uncertainties for h → bb we also require the running of m b (µ) and α s (µ). As with the running of α(µ), we again use a one-loop fixed-order solution to the RG equations for m b (µ) and α s (µ) which are given by , and n f = 5 .
In the α scheme we find

C Numerical results using universal corrections in SMEFT
For completeness, we present here numerical results for the prefactors of the Wilson coefficients at different perturbative orders for W , h and Z decay which have not been shown in Section 6 yet. Table 9 shows the results for W and h decay in the α scheme. For the LEP scheme, Table 10 shows the results for W , and Z decay, respectively. The h decay results for the LEP scheme have been omitted since they only have very small (numerical) differences with respect to the numbers in the α µ scheme, which are presented in Table 9.  For results in the α scheme and Z decay in the α µ scheme, we refer to Tables 7 and 8 in Section 6.

D Comparison with previous literature
Electroweak precision observables at NLO in SMEFT have been calculated previously in [29,72]. In this section we compare the LEP-scheme results for M W and the Z → decay rate with results given in that work. 14 In order to do so, we must take into account some differences in calculational set-ups.
First, in contrast to the present paper, those works use α O.S. (0) as an input, so that large logarithms of lepton masses and hadronic contributions appear in fixed order. We can convert our results to that renormalisation scheme by eliminating α(M Z ) through use of Eq. Expanding observables to linear order in α O.S. (0) and ∆α then yields SM predictions as given in [29,72]. After making this conversion and using a common set of input parameters also for heavy-particle masses, we can exactly reproduce the SM values for the W mass at LO and NLO: The SMEFT results for the W -boson mass also agree, when the same set of flavour assumptions is made.
14 The decay rate for the W boson has not been compared since a leptonic partial branching fraction is not provided in the previous literature.  for various perturbative approximations. The tree-level decay rate as well as v 2 µ have been factored out and the results have been evaluated at the scale of the process. We show the results for W decay (top) and Z decay (bottom).
For the Z → decay rate, we agree with an analytic result in the α µ scheme provided to us by the authors of [29,72] (after using the flavour assumptions of those papers). This forms the basis for LEP-scheme results. In our case these are obtained by using Eqs. (2.33) and (2.35) to express the result in terms ofM W , while [29,72] re-organise the SM part of the loop expansion in a way that is specified in the recent preprint [43]. Taking into account these differences, as well as the renormalisation of α discussed above, we find numerical agreement with [29,72]. Table 11: The 59 independent baryon number conserving dimension-six operators built from Standard Model fields, in the notation of [60]. The subscripts p, r, s, t are flavour indices, and σ I are Pauli matrices.