Using the effective weak mixing angle as an input parameter in SMEFT

We implement electroweak renormalisation schemes involving the effective weak mixing angle to NLO in Standard Model Effective Field Theory (SMEFT). After developing the necessary theoretical machinery, we analyse a select set of electroweak precision observables in such input schemes. An attractive feature is that large corrections from top-quark loops appearing in other schemes are absorbed into the definition of the effective weak mixing angle. On the other hand, the renormalisation condition which achieves this involves a large number of flavour-specific SMEFT couplings between the $Z$ boson and charged leptons, motivating simple flavour assumptions such as minimal flavour violation for practical applications. The results of this paper provide a valuable new component for estimating systematic uncertainties in SMEFT fits by performing analyses in multiple input schemes.


Introduction
In the absence of direct new physics discoveries, Standard Model Effective Field Theory (SMEFT) [1][2][3] enables to describe in a model-independent fashion small deviations from Standard Model (SM) predictions.SMEFT calculations can be systematically improved by including quantum corrections as well as higher-order terms in the expansion in the new physics scale Λ.The investigation of next-to-leading order (NLO) corrections in dimension-six SMEFT has been a focus of recent studies.While NLO QCD corrections have been fully automated [4], NLO electroweak (EW) corrections and, in a few instances next-to-next-to-leading order (NNLO) QCD corrections, have been calculated on a caseby-case basis for specific processes .
An essential ingredient common to all NLO calculations is the choice of the EW input scheme.In the recent work [56], we systematically examined at NLO in SMEFT the commonly employed α, α µ , and LEP schemes, which are defined in table 1 and use combinations of the Fermi constant G F , the masses of the W and Z bosons, M W and M Z , as well as the electromagnetic coupling α as the three independent EW input parameters.This involved cataloguing the set of Wilson coefficients entering finite parts of observables after the cancellation of UV divergences in different schemes, identifying dominant sets of universal corrections associated with contributions from top-quark loops, and providing a methodology for including these scheme-dependent universal NLO corrections in the LO results, thus extending previous discussions of EW input schemes in SMEFT [57,58].
In the SM, several studies have proposed EW input schemes which use the effective leptonic weak mixing angle sin θ ℓ eff as an input parameter [59][60][61][62][63][64].The effective leptonic weak mixing angle has been measured with per-mille level precision at LEP [65], the Tevatron [66] and the LHC [67][68][69][70].Its numerical precision is (more than an order of magnitude) below that of other commonly used input values such as the mass of the W boson. Future experiments, such as the P2 experiment at MESA [71], as well as the Møller [72] and SoLID [73,74] experiments at Jefferson Laboratory, will test this quantity with similar precision at lower energy scales.
In spite of the past and recent interest in EW input schemes involving sin θ ℓ eff , a discussion in the context of SMEFT has not yet appeared in the literature.The aim of this paper is to fill this gap by incorporating the {G F , sin θ ℓ eff , M Z } and {α, sin θ ℓ eff , M Z } input schemes into the methodology of [56].We find that an attractive feature of these schemes is that large corrections from top-quark loops appearing in other schemes are absorbed into the definition of the effective weak mixing angle.On the other hand, the renormalisation condition which achieves this involves a large number of flavour-specific SMEFT couplings between the Z boson and charged leptons, motivating simple flavour assumptions such as minimal flavour violation for practical applications.
We structure the discussion as follows.In section 2 we cover the construction of UV counterterms in these schemes, assembling and calculating the ingredients needed to implement them into NLO calculations in dimension-6 SMEFT.Then, in section 3 we present a short study of numerical results for a select set of electroweak precision observables, including comparisons between all five EW input schemes listed in table 1 at NLO in the SM and SMEFT, before concluding in section 4. In addition, some explicit results for SMEFT expansion coefficients for derived quantities such as the W -boson mass are given in appendix A, and a short description of minimal flavour violation is provided in appendix B.
The discussion in this paper is focused on providing the building blocks needed for NLO SMEFT calculations in the input schemes involving sin θ ℓ eff .Of course, an essential part of the actual implementation is calculating the loop diagrams (and real emission corrections) needed in the renormalisation process and its application to specific observables, which even for the modest set of processes considered in this work involves a large number of Feynman diagrams and dozens of dimension-6 operators.We have carried out these calculations using an in-house FeynRules [75] implementation of the SMEFT Lagrangian and cross-checked our results with SMEFTsim [76,77].For the calculation of matrix elements we employed FeynArts and FormCalc [78][79][80] and we extracted analytic results for Feynman integrals with PackageX [81].Numerical results were obtained with LoopTools [79].Analytic results for the observables in the v eff µ and v eff α schemes studied in section 3 are provided as electronic files with the arXiv submission of this work.
2 Using sin θ ℓ eff as an input parameter In this section we discuss renormalisation in two EW input schemes involving sin θ ℓ eff : (1) The "v eff µ scheme", which uses as inputs {G F , sin θ ℓ eff , M Z }, where G F is the Fermi constant as measured in muon decay and M Z is on the on-shell Z-boson mass.
Table 1: Nomenclature for the EW input schemes considered in this work.
In what follows, we shall refer to these two schemes collectively as the "v eff σ schemes", where the choice σ ∈ {α, µ} can be used to select between them.
Our treatment of the v eff σ schemes follows closely the notation and procedures introduced in [56].We write the SMEFT Lagrangian up to dimension six as L = L (4) + L (6) ; L (6) where L (4) is the SM Lagrangian and L (6) denotes the dimension-six Lagrangian consisting of the operators Q i in the Warsaw basis [3] and the corresponding Wilson coefficients C i , which are inherently suppressed by two powers of the new physics scale Λ.We list the 59 independent operators, which generally carry flavour indices, in table 8.We expand all quantities up to linear order in the dimension-six Wilson coefficients throughout this work.
In order to implement the v eff σ schemes in a unified notation, we first write the tree-level Lagrangian in terms of {v T , M W , M Z }, where v T is the vacuum expectation value (vev) of the SU(2) doublet Higgs field H The renormalised Lagrangian is obtained by interpreting the tree-level quantities as bare ones which are replaced by renormalised parameters plus counterterms in a particular scheme.For the inputs common to the two schemes, we relate the bare and renormalised parameters according to where here and throughout the paper we indicate bare parameters with a subscript 0, and c w,0 = M W,0 /M Z,0 .The quantities ∆M Z and ∆s w appearing on the right-hand side of the above equations are counterterms, which are calculated in a SMEFT expansion in loops and operator dimension, including tadpoles in the FJ tadpole scheme [82].It will often be convenient to work with the quantity The relation between M eff W and the bare mass can be derived using eq.(2.3).Writing one finds where the . . .indicates terms not needed to NLO in the dimension-6 SMEFT expansion.
In addition to the counterterms for s w and M Z , we also need those for v T .In the v eff µ scheme, one uses while in the v eff α scheme one has instead where we have defined To streamline the notation needed for discussing the v eff σ schemes, our definitions above suppress the superscript "eff" on all quantities except for the scheme names and M eff W , in order to distinguish it from the on-shell W -boson mass M W .It is important to emphasize, however, that the SMEFT expansion coefficients of ∆v α and ∆v µ are not identical to those in the α and α µ input schemes defined in table 1 and discussed in [56], where M W instead of s w is used as an input parameter.In the following two subsections we discuss renormalisation in the v eff σ schemes to NLO in SMEFT.
2.1 The v eff µ scheme In this section we give results for the SMEFT expansion of the counterterms needed for renormalisation in the v eff µ scheme, structuring the discussion in such a way that most results in the v eff α scheme can be obtained by a simple set of substitutions.We begin with the determination of the counterterm ∆s w .To this end, consider the amplitude for Z → ℓℓ decay, where ℓ ≡ ℓ i ∈ {e, µ, τ }.We can write the bare amplitude to NLO in SMEFT in the form where we have introduced the spinor structures with P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2.The ellipsis in eq.(2.10) refers to spinor structures appearing beyond LO in the SMEFT expansion, which do not interfere with those above in the limit of vanishing lepton masses, and the overall factor N 0 is defined by where δ QED refers to QED corrections.We can write the SMEFT expansion of the bare amplitudes as where the superscript (i, j) labels the operator dimension i contribution to the j-loop diagram, and we have pulled out explicit factors of v T,0 such that the coefficients A (i,j) do not depend on v T,0 .1 The notation makes clear that the dimension-6 amplitudes depend on the lepton species ℓ while those in the SM do not.Suppressing the subscript 0, the tree-level SM amplitudes read while in SMEFT where the explicit results for decay into lepton species ℓ i are (2.16) Consider now the definition of the effective weak mixing angle where the G ℓ L,R are experimentally measured form factors at q 2 = M 2 Z [65][66][67][68][69][70].The counterterm ∆s w in eq.(2.3) is determined to all orders in the SMEFT expansion through the renormalisation condition sin 2 θ ℓ eff = s 2 w . (2.18) To implement this renormalisation condition order by order in SMEFT, we first substitute the G ℓ L (G ℓ R ) in eq.(2.17) with the coefficients of S L (S R ) in eq.(2.10), and replace the bare quantities with renormalised ones plus associated counterterms.We write the SMEFT expansions of ∆s w and ∆v µ in the v eff µ scheme as w + ∆s (6,1,µ)   w , µ + ∆v (6,1)   µ . (2.19) The superscripts (6, 0) and (4, 1) have the same meaning as in eq.(2.13), while the coefficient ∆s contains an extra superscript µ to indicate that v T has been renormalised as in eq.(2.7).Isolating the dependence on ∆v µ allows us to write ∆s (6,1,µ)   w = ∆s (6,1)   w − ∆s (4,1)  w ∆v (6,0)   µ + ∆s (6,0)  w ∆v (4,1)   µ , where the coefficient ∆s (6,1) w does not depend on the renormalisation scheme for v T .The construction of renormalised Z → ℓℓ decay amplitudes also requires the on-shell wavefunction renormalisation factors of the external Z-boson and lepton fields.For the lepton fields, we can write the SMEFT expansion of the wavefunction renormalisation factors as In the first line all terms are expressed in terms of the bare parameters v T,0 , s w,0 , while in the second renormalised parameters are used.The notation is such that ∆Z ℓ(4,1) where the derivative in the SMEFT counterterm arises from replacing the implicit dependence on s w,0 in the SM counterterm ∆Z ℓ(4,1) L/R,0 with the right-hand side of eq.(2.19) and performing a SMEFT expansion.We emphasise that the ∆Z ℓ L/R do not include QED corrections, which are instead contained in the factor δ QED in eq.(2.12).Wavefunction renormalisation graphs related to the Z-boson two-point function can be absorbed into the factor N 0 in eq.(2.10).Since N 0 drops out of the ratio in eq. ( 2.17) we do not discuss these two terms further.On the other hand, contributions from the Zγ two-point function, where γ denotes the photon, are included in the definition of A ℓ L/R,0 .Performing a tree-level SMEFT expansion on eq.(2.17) using the above equations yields ∆s (6,0) while the one-loop result in the SM is The part of the one-loop SMEFT result which is independent of the renormalisation scheme for v T is ∆s (6,1) (2.25) A couple of comments are in order concerning the form of this counterterm.First, the quantity ∆ ṡ(6,0) w is obtained from ∆s (6,0) w through the replacement C i → Ċi , where Ċi ≡ dC i /d ln µ; the term involving this quantity arises from MS renormalisation of the Wilson coefficients in d = 4 − 2ϵ dimensions, and the Ċi were calculated at one loop in [83][84][85]. 2econd, the final two lines are related to using eq.(2.19) in the lower-order amplitudes and then performing the SMEFT expansion.
To evaluate the full NLO SMEFT result ∆s (6,1,µ) w in eq. ( 2.20) requires also the counterterm ∆v µ .This counterterm, including tadpoles and without flavour assumptions, was determined at NLO in SMEFT in the α µ scheme in [56], thus generalising the earlier result from [25].Calling the expansion coefficients eq.(2.19) in that scheme ∆v αµ(i,j) µ , the relation with the present work is ∆v (6,0) It will be useful later on to have an expression for the large-m t limit of the loop corrections to the counterterm ∆s w .Here and below, the large-m t limit of a given quantity means the approximation where only terms proportional to positive powers of the top-quark mass m t in the limit m t → ∞ are kept.In the SM, top-quark loops contribute to ∆s L/R,0 .It is easy to show, however, that this two-point function vanishes in the large-m t limit, so ∆s (4,1) where here and below the subscript t refers to the large-m t limit of a given quantity.In SMEFT, there are three contributions in the large-m t limit, which arise from Zγ mixing, top-loop corrections from four-fermion operators, and a scheme-dependent correction proportional to ∆v where as above we omitted divergent and tadpole contributions, and quoted the results in units of ∆ρ (2.30)

2.2
The v eff α scheme The v eff α scheme differs from the v eff µ scheme through the renormalisation of v T , which is performed as in eq.(2.8).The SMEFT expansion coefficients of that counterterm, as well as those of ∆s w in this scheme, are defined as in eq.(2.19) after the replacement µ → α.
In order to calculate the expansion coefficients of ∆v α , we will need those for M Z and electric charge renormalisation.We define these as ∆e (4,1) + ∆e (6,1) − ∆v (6,0)   α ∆e (4,1) , ( where the coefficients with superscript (6, 1) are calculated as in eq.(2.22).It will also be useful to work with expansion coefficients of the derived counterterm ∆M eff W .We define these coefficients as ∆s (6,1)   w + ∆s (6,0)  w ∆M ∆s (6,0)  w ∆s (4,1)   w . (2.34) With this notation at hand, we can present a compact result for the expansion coefficients of ∆v α .They read ∆v (6,0) + ∆s (4,1)   w − ∆e (4,1) , ∆v + ∆s (6,1)   w − ∆e (6,1) − ∆v (6,0)   α ∆v (4,1)   α In the above, we have denoted the tree-level α-scheme result as which leads to the following tree-level results in the v eff α scheme: The implementation of the v eff α scheme also requires to specify the renormalisation scheme for α.One possible choice is the MS definition α(µ) in five-flavour QED×QCD, where EW scale contributions are included through decoupling constants [35] and perturbative uncertainties can be estimated through scale variations.In the current paper we adopt instead the more conventional on-shell definition α(M Z ) [88].It is simple to convert between these two schemes using the perturbative relation where Q f is the charge of the fermion and N f c = 3 (N f c = 1) for quarks (leptons).

Numerical Results
In this section we present a brief numerical analysis of select electroweak precision observables in the v eff σ schemes, covering derived input parameters in section 3.1 before turning to heavy EW boson decays in section 3.2.We study perturbative convergence and the number of Wilson coefficients associated with these schemes, and make qualitative and quantitative comparisons with the widely used α, α µ , and LEP schemes.
In all cases we use the numerical input parameters given in table 2. Furthermore, we approximate the CKM matrix by the unit matrix.Experimentally, sin 2 θ ℓ eff is typically averaged over measurements involving electrons and muons.In SMEFT, using an average leads to some difficulties because it would require a combination of first and second-generation Wilson coefficients entering the counterterms, depending on the ratio of electron and muon data entering the combination.To avoid this issue, we use the most precise available measurement of sin θ ℓ eff from the couplings to electrons only, namely the ATLAS measurement with one central and one forward electron [68].
Results in SMEFT also depend on the renormalisation scale µ appearing in the Wilson coefficients C i ≡ C i (µ).When variations of this renormalisation scale are used as a measure of perturbative uncertainties, we use the following fixed-order expression for the RG running, where µ def is the default scale for the particular analysis, and the one-loop expressions for Ċi are taken from [83][84][85].

Derived parameters
In each of the five input schemes in table 1, two parameters in the set {M W , α, G F , sin θ ℓ eff } are derived parameters which can be calculated in a SMEFT expansion.For instance, in the v eff σ schemes, the on-shell W -boson mass M W is given by where ∆ W is a finite shift.Similarly, the v σ themselves are related according to ∆r eff(4,1) + ∆r eff (6,1) , where in the second equality we made explicit that the expansion coefficients are the same whether v µ or v α is used.The form above allows to determine G F in the v eff α scheme after setting σ = α, whereas α in the v eff µ scheme is easily obtained after setting σ = µ.We derive the SMEFT expansions for ∆ W and ∆r eff , including explicit results in the large-m t limit, in appendix A. Results for sin θ ℓ eff in the α, α µ and LEP schemes are obtained by evaluating eq.(2.17), while results for all other derived parameters have been given in [56]. 3he derived parameters are useful for two reasons.First, from a practical perspective, they are the simplest examples of EW precision observables and therefore play an important role in global analyses of data.Second, they are the key ingredients for converting results and understanding differences between EW input schemes.For example, if one calculates a quantity in the α µ scheme, one can convert it to the v eff µ scheme by substituting M W with eq.(3.2) with σ = µ and performing a SMEFT expansion.In the SM, if the derived value of M W in the v eff µ agrees well with the measured value at a given order, then results for other observables in the α µ and v eff µ scheme will show similar level of numerical agreement.This should also be true in SMEFT, but in that case the derived value of M W depend on the Wilson coefficients, whose values are not precisely known and are left symbolic.Differences in observables between schemes show up in non-trivial patterns of Wilson coefficients and the level of agreement between schemes is less obvious.In the remainder of this section we examine derived parameters in the SM across all schemes, and use the prediction of M W in the v eff σ schemes to illustrate some of their important features.

SM.
In table 3 we show LO and NLO results for derived parameters in the SM, where NLO is defined as LO plus the NLO correction.In all cases, the NLO and measured values agree to roughly half a percent or better.In the v eff σ schemes, the deviation between the derived parameters and the experimental values is already below the per-mille level at LO, while the α and α µ schemes involve percent-level NLO corrections to sin θ ℓ eff , α or G F .Such corrections originate mainly from large-m t corrections to the counterterm for s w in those schemes; for instance, in the α µ scheme the one-loop SM result is where the . . .refers to terms which are subleading in the large-m t limit, and c αµ w = M W /M Z .The same result holds in the α scheme after the replacement v µ → 2M W s αµ w / √ 4πα.A noticeable feature of the v eff σ schemes is that the NLO corrections to G F or α are extremely small.These corrections are related to ∆r eff (4,1) in eq.(3.3), and an estimate from the top-loop contribution in eq.(A.11) gives −∆ρ where the ordering of the numerical terms on the second line matches those of the analytic expressions above.We note an accidental cancellation between the large-m t limit result and that related to the running of α in the on-shell scheme 4 ; the latter correction can be eliminated by converting to the MS definition as in eq.(2.38).By contrast, the NLO corrections to M W in the v eff σ scheme do not depend on the counterterm for α.The top-loop contribution in eq.(A.5) is a good estimate for the NLO correction, as seen in the result where the order of numerical contributions in the second line matches that on the first and we set v σ = v µ to obtain the numerical value.
SMEFT.Results for derived parameters in SMEFT depend on a number of Wilson coefficients and are thus quite lengthy.For brevity, we focus the discussion on M W in the v eff σ schemes, leaving a comparison of observables across schemes to the heavy-boson decay rates presented in section 3.2.
We show in figure 1 the LO and NLO SMEFT corrections to M W in the v eff σ schemes.The numerical contribution from each Wilson coefficient at the scale µ = M Z is obtained by making the choice C i (M Z ) = 1/v 2 σ , and the results are given in units of M eff W ; in other words, we are quoting results for the expansion coefficients of ∆ W as defined in eq.(3.2).A remarkable feature is the large number of Wilson coefficients contributing to M W for arbitrary flavour structure; the exact number of coefficients at LO (NLO) is 5 (63) in both schemes.Out of these, 34 Wilson coefficients appearing at NLO correspond to different 4 A similar cancellation occurs in the NLO correction to sin 2 θ ℓ eff in the LEP scheme, whose large-mt correction is obtained from eq. (3.4) by the replacement c 2 w /s 2 w → −c2w/c 2 w and is roughly −1.5% numerically.flavour structures of ten four-fermion operators.The number of Wilson coefficients is reduced when we employ additional flavour assumptions.For concreteness, we consider the scenario of minimal flavour violation (MFV) where the top Yukawa [89] is the only source of the breaking of the U (3) 5 symmetry in SMEFT, see appendix B. Under the MFV assumption, 34 Wilson coefficients contribute to ∆ W at NLO, 16 of which correspond to different flavour structures of the ten four-fermion operators.
It is clear from figure 1 that many of the NLO SMEFT corrections to M W are numerically small when all Wilson coefficients are set to a common value.In table 4, we give numerical results at LO and NLO (defined as the sum of LO plus the NLO correction) for those SMEFT operators whose NLO contribution is larger than 1% for the default choice All of these coefficients receive NLO corrections from top loops, and to show their significance we give results where only the large-m t limit of these corrections is used (NLO t in the table).In each case, we include scale uncertainties obtained by evaluating the prediction for the three scale choices µ ∈ [M Z , 2M Z , M Z /2], using eq.(3.1) to express the results in terms of C i (M Z ).In most cases there is a good convergence between

Table 4:
The numerical prefactors of the Wilson coefficients in the v eff µ and v eff α schemes contributing to ∆ W for the LO, NLO and NLO t (large-m t limit) perturbative approximations.The SM tree-level approximation along with v 2 µ has been factored out.The results have been evaluated at µ = M Z and varied up and down by a factor of 2 to give the uncertainties.Only Wilson coefficients whose numerical prefactor is greater than 1% at NLO have been included.
LO and NLO when scale uncertainties are included.The large-m t limit results are generally an improvement for central values, but come with small scale uncertainties which do not always overlap with the complete NLO result.

Heavy boson decays at NLO
In this section we analyse W → τ ν and Z → τ τ decays, focussing on a comparison between input schemes.We define SMEFT expansion coefficients for the decay Moreover, we write LO and NLO results as The LO results for s ∈ {v eff µ , v eff α } are given by where ℓ ≡ ℓ i is the charged lepton species used in the definition of ∆s w and g (4,0) = 1 − 4s 2 w + 8s 4 w . (3.10) To derive the result for W decay we have written W -mass dependence arising both through two-body phase-space and in the matrix element squared in terms of M eff W .Note that in Z decay the flavour-independent coupling G (6,0) given in eq.(2.16) has dropped out of the decay rate due to a cancellation against ∆s (6,0) w .Further simplifications occur only if ℓ = τ or a flavour symmetry such as MFV is imposed, in which case the contribution in the final line vanishes.
The LO and NLO results for W and Z decay in the SM are shown in table 5, where we have normalised the results to the experimentally measured values.The NLO corrections in the SM bring results from all five schemes into close numerical agreement.These corrections are at the 1.5% level for W decay in the v eff σ and LEP schemes, where M W is not an input and hence factors of 3∆ s(4,1) W /v 2 σ arise compared to the α µ scheme.Corrections of around −3% arise in the α scheme, which are mainly due to the top-loop corrections to s w shown in eq.(3.4).As explained in section 3.1, the close agreement between decay rates at NLO different schemes is a consequence of that for the derived parameters in table 3.
The situation in SMEFT is different, because in that case the relations between input parameters in different schemes depend on the Wilson coefficients and there can be non- Regarding the first two points, NLO corrections in the v eff σ schemes tend to be milder than in the α or α µ schemes because in the latter case ∆s w gets scale-independent corrections of the type shown in eq.(3.4).In that case including universal corrections from the large-m t limit using the procedure outlined in [56] can improve convergence between orders. 5The specific pattern of contributions described above is specific to Z decay, but the important point that the size of SMEFT contributions related to a particular Wilson coefficient is highly scheme specific is general.
The discussion so far highlights the non-trivial pattern of perturbative convergence across schemes.It is also interesting to study the number of Wilson coefficients characteristic of each scheme.In table 7, we show the total number of Wilson coefficients contributing to W and Z decay at LO and NLO in the five input schemes for different flavour assumptions.For both decays, significantly more coefficients are involved in the v eff σ schemes than in the others, when no flavour restrictions are made; this is because the counterterm for s w is determined from Z → ee decay amplitudes, and involves a number of flavour specific left and right-handed fermion gauge couplings in addition to four-fermion operators, which would not otherwise appear in Z → τ τ decay.Indeed, if we consider the decay Z → ee instead, the number of Wilson coefficients appearing throughout all five schemes is far more similar.The same statement applies if MFV is used -in fact, the number of Wilson coefficients entering the combination of the two decays is the same.
In a full analysis of electroweak precision observables including gauge-boson decays to quarks and Z decay to neutrinos, the total number of Wilson coefficients appearing is further increased through contributions from four-quark operators.For MFV the total number of operators appearing grows from 34 in the leptonic Z and W decays considered here to 56 in the full set of electroweak precision observables [52].

Conclusions
We have implemented to NLO in SMEFT two EW input schemes involving sin θ ℓ eff as an input parameter.These "v eff σ schemes" share as common inputs sin θ ℓ eff and M Z , but differ through the use of the fine structure constant α (v eff α scheme) or the Fermi constant G F (v eff µ scheme) as the third independent input parameter.Details of the renormalisation procedure in these schemes were given in section 2, and numerical results for a select set of electroweak precision observables, including comparisons with the other commonly used EW input schemes listed in table 1, were given in section 3. Analytic results in the v eff σ schemes which form that basis of that numerical analysis are given in electronic form in the arXiv submission of this paper.
An attractive feature of the v eff σ schemes in SMEFT is that sizeable corrections to the sine of the Weinberg angle related to top-quark loops appearing in other schemes are absorbed into the definition of the parameter sin θ ℓ eff .On the other hand, the renormalisation conditions for sin θ ℓ eff are implemented at the level of form factors for two-body Z → ℓℓ decay, and are thus subject to a large number of flavour-specific Z-fermion SMEFT couplings, including four-fermion operators.For instance, the SMEFT expansion for M W in these schemes receives contributions from five Wilson coefficients at LO, but 63 already at NLO, and as shown in table 7 even a simple process such as Z → τ τ is subject to roughly 90 coefficients at NLO. Flavour assumptions on the SMEFT Wilson coefficients imposed by symmetries such as MFV may therefore be an essential ingredient to practical implementations of this scheme in global fits to data.
Regarding such fits, observables in each input scheme are subject to a different pattern of higher-order corrections in both loops and operator dimensions.Therefore, performing fits in multiple EW input schemes can provide an important estimate on the significance of such missing corrections, and is a valuable consistency check on the results in any one scheme.The results of this paper provide an important new component for such analyses.
the assumption of minimal flavour violation (MFV).In this appendix, we give details on this flavour scenario.
In the SM, the U (3) 5 symmetry for the SM fermions is broken only by the Yukawa couplings [90,91].The MFV scenario extends this requirement to SMEFT [89].Since we consider all fermions except the top quark to be massless, we only thus only allow the breaking of the U (3) 5 symmetry by the top Yukawa coupling Y t .
In the MFV case, we thus distinguish Wilson coefficients involving the top quark from those involving first and second-generation up-type quarks.
We change from the flavour-general scenario to MFV by making a set of replacements on the Wilson coefficients, see e.g.[52].For operators with two flavour indices involving leptons or down-type quarks, we can suppress the flavour indices Hl , C Hl , C Hd . (B.2) For the Wilson coefficients with two flavour indices involving up-type quark fields, we explicitly distinguish top-quark couplings → C HQ , C → C HQ . (B.3) For C uB and C uW only Wilson coefficients with third-family indices contribute in the first place so no replacement is necessary.For four-fermion operators with two different fermion bilinears as well as C ee , which is simplified by a Fierz identity, there is a single coefficient contributing under the MFV assumption when no up-type quarks are involved For C ll , which involves two fermion currents of the same species and chirality, there are two U (3) 5 symmetric combinations, which we distinguish with a prime (q p γ µ q r )(q s γ µ q t ) Q (3) qq (q p γ µ σ I q r )(q s γ µ σ I q t ) Q (1) lq ( lp γ µ l r )(q s γ µ q t ) Q (3) lq ( lp γ µ σ I l r )(q s γ µ σ I q t ) 8 : ( RR)( RR) Q qe (q p γ µ q r )(ē s γ µ e t ) Q (1) qu (q p γ µ q r )(ū s γ µ u t ) Q (8) qu (q p γ µ T A q r )(ū s γ µ T A u t ) Q (1) qd (q p γ µ q r )( ds γ µ d t ) Q (8) qd (q p γ µ T A q r )( ds γ µ T A d t ) 8 : ( LR)( RL) + h.c.

Q
(1) quqd (q j p u r )ϵ jk (q k s d t ) ( lj p e r )ϵ jk (q k s u t ) Q (3) lequ ( lj p σ µν e r )ϵ jk (q k s σ µν u t ) Table 8: The 59 independent baryon number conserving dimension-six operators built from Standard Model fields, in the notation of [83].The subscripts p, r, s, t are flavour indices, and σ I are Pauli matrices.

Figure 1 :
Figure 1: SMEFT corrections to the W boson mass in the v eff µ (top) and v eff α (bottom) schemes, with ∆s w determined from Z → ee decay, so that ℓ = 1.The Wilson coefficients are evaluated at C i = 1/v 2 σ .The flavour indices i and j run over values j ∈ 1, 2, and i ∈ 1, 2, 3.

→→
C x for C x ∈ C ee , C le , C ld , C ed .(B.4)In Wilson coefficients involving up-type quark fields we distinguish the third generationC x iijj → C x for j ∈ 1, 2, C C qe for j ∈ 1, 2, C qe 33ii → C Qe .(B.5)

Table 2 :
Input parameters employed throughout the paper.Note that v α is a derived parameter.

Table 3 :
SM results for derived parameters in scheme s relative to the experimental values in table 2.

Table 5 :
Deviations of the SM predictions for Z → ℓℓ and W → ℓν decay rates in scheme s from the experimental measurements of Γ exp Zℓℓ = 83.98MeV and Γ exp W ℓν = 226.4MeV [88].

Table 7 :
Number of Wilson coefficients appearing in heavy boson decay rates under general flavour assumptions and MFV.Note that assuming MFV the number of operators appearing in Γ Zτ τ and Γ Zee is the same and hence only one value is given.