Two-loop Electroweak Corrections to the Top-Quark Contribution to $\epsilon_K$

The parameter $\epsilon_K$ measures $CP$ violation in the neutral kaon system. It is a sensitive probe of new physics and plays a prominent role in the global fit of the Cabibbo-Kobabyashi-Maskawa matrix. The perturbative theory uncertainty is currently dominated by the top-quark contribution. Here, we present the calculation of the full two-loop electroweak corrections to the top-quark contribution to $\epsilon_K$, including the resummation of QED-QCD logarithms. We discuss different renormalization prescriptions for the electroweak input parameters. In the traditional normalization of the weak Hamiltonian with two powers of the Fermi constant $G_F$, the top-quark contribution is shifted by $-1\%$.

1 Introduction e parameter K describes CP violation in the neutral kaon system and is one of the most sensitive probes of new physics. It can be de ned as [1] K ≡ e iφ sin φ Here, φ = arctan(2∆M K /∆Γ K ) where ∆M K and ∆Γ K are the mass and lifetime di erences of the weak eigenstates K L and K S . M 12 and Γ 12 are the Hermitian and anti-Hermitian parts of the Hamiltonian that determines the time evolution of the neutral kaon system. e so-called shortdistance contributions to K enter the matrix element M 12 = − K 0 |L ∆S=2 f =3 |K 0 /(2∆M K ), up to higher powers in the operator-product expansion.
Experimentally, K is well-known, with absolute value | K | = (2.228 ± 0.011) and an uncertainty at the permil level [2]. e standard model (SM) contributions to neutral kaon mixing are conveniently described by the e ective |∆S| = 2 Lagrangian with three active quark avors, valid at scales around µ = 2 GeV. Here, is the local |∆S| = 2 operator, where α and β are color indices, and the ellipsis denotes contributions of higher dimension local operators and non-local contributions of |∆S| = 1 operators. e reason for the appearance of the double primes will become clear later. e elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix are combined into the parameters λ i ≡ V * is V id . e long-distance SM contributions are comprised by the hadronic matrix element of the local |∆S| = 2 operator, and are given in terms of the kaon bag parameter B K = 0.7625(97) [3]. Long-distance contributions that are not included in B K are parameterized by the correction factor κ = 0.94 (2) [4]. e short-distance contributions are contained in the Wilson coe cients C ij S2 . In the SM, the leading-order (LO) contributions to the Wilson coe cients are given by one-loop box diagrams [5]. Higher-order QCD corrections have been calculated in renormalization-group (RG) improved perturbation theory, using a slightly di erent representation of the |∆S| = 2 e ective Lagrangian [6], in terms of λ c and λ t . However, it was shown recently [7] that the parameterization (2) in terms of λ u and λ t leads to a perturbative QCD uncertainty smaller by an order of magnitude. We will adopt this parameterization in this paper.
QCD corrections to the Wilson coe cients are known at next-to-leading order (NLO) for C tt S2 [8] with a remaining uncertainty at the percent level, and at next-to-next-to-leading order (NNLO) for C ut S2 with a remaining uncertainty below one percent [7,9,10]. e coe cient C uu S2 is also known at NNLO in QCD [7,11], but has no e ect on K (it does contribute to the neutral kaon mass di erence).
In anticipation of the NNLO QCD calculation of C tt S2 [12] it is worthwhile to consider also the electroweak corrections to the e ective Lagrangian (2). Without an explicit calculation, the electroweak renormalization scheme of the input parameters is le unspeci ed and amounts to an uncertainty of the order of a few percent -which can no longer be neglected at the current level of precision. e electroweak corrections to C tt S2 could, in principle, be adapted from the calculation of the corresponding corrections for B-meson mixing presented in Ref. [13]. However, in our opinion, a reconsideration of the old calculation is worthwhile for a number of reasons. First, the application to K involves a lower energy scale than the one relevant in B-meson mixing; accordingly, the QED-QCD resummation of the appearing logarithms might be necessary. Second, we will discuss the scheme dependence of the corrections in detail, since the usual proofs of scheme independence [6] fail in our case; this topic also has not been addressed before. Last, our calculation presents the rst independent check of the results in Ref. [13].
In this work, we calculate the coe cient C tt S2 , proportional to λ 2 t , to NLO in the electroweak interactions. is xes the renormalization scheme of the electroweak input parameters contained in the prefactor G 2 F M 2 W . In fact, when only considering QCD e ects, there are several equivalent ways of rewriting the prefactor, using the tree-level relation where α = e 2 /(4π) the electromagnetic coupling 1 and s 2 w = sin 2 θ w with the weak mixing angle θ w . Essentially, this choice speci es which experimental data are used as parametric input for our prediction. e numerical di erence between the di erent schemes is expected to be large at LO, as exempli ed by the 5% di erence between the on-shell and MS values of s 2 w . 1 roughout this paper, the electromagnetic coupling is understood to be xed in the ve-avor scheme at scale µ = MZ , unless otherwise noted. For our analysis, it is useful to write the e ective Lagrangian in the following form: where the ellipsis now also includes contributions not proportional to λ 2 t . Using the tree-level relation (4), we express c tt S2 in three di erent ways: As explained in Sec. 2.2, this e ectively absorbs di erent parts of the radiative corrections into the measured value of the muon decay rate. In the rst and second relation in Eq. (6) we have factored out the powers (α/(4π)) 2 and α/(4π), respectively, and absorbed them into rescaled operators, de ned as and while Q S2 has been de ned in Eq. (3). With these conventions, the RG evolution is described by the same anomalous dimension in all three cases, see Sec. 4, and the LO values coincide. ey are given by the modi ed [7] Inami-Lim box function [5] S where x t ≡ m 2 t /M 2 W , and we neglect a tiny correction of O(m 2 c /M 2 W ) ∼ 10 −4 [7]. We will refer to the rst normalization in Eq. (6) as "A2", the second as "GF", and the third as "GF2". While, at LO in the weak interactions, the three parameterizations are equivalent, the numerical prediction depends on the chosen normalization and the scheme of the input parameters. ese arbitrary dependences will be mitigated to a large degree by the NLO electroweak corrections. e aim of this paper is to show this explicitly, and to provide an updated numerical prediction for K including the electroweak corrections.
is paper is organized as follows. In Sec. 2 we de ne the e ective Lagrangian and provide details on our two-loop calculation. Sec. 3 contains the discussion of the various electroweak renormalization schemes and the error estimate on the electroweak corrections. In Sec. 4 we include the e ects of the strong interaction and discuss the scheme dependence of our result. We conclude in Sec. 5. A number of appendices contains more details of our work. In App. A we collect the explicit counterterms that were needed in the renormalization of the SM amplitude. e necessary counterterms in the e ective theory are collected in App. B. We give a formal proof of the scheme independence of the top-quark contribution to the |∆S| = 2 amplitude including QED e ects in App. C. Finally, the full analytic two-loop result is presented in App. D.

Calculation of the two-loop electroweak matching corrections
Our strategy is to perform the matching calculation using the "A2" normalization in the MS scheme. We can then easily change to di erent normalization conventions and renormalization schemes for the input parameters; this will be discussed in detail in this section. e variation of the NLO results between the di erent prescriptions is used as one way to estimate the remaining theory uncertainty.

Matching calculation
We write the physical ve-avor e ective Lagrangian, obtained by integrating out the top quark together with the weak gauge bosons W and Z as well as the Higgs boson, in the form 2 e single physical operator contributing to our results has been de ned in Eq. (7). Our de nition of the evanescent operators is given in App. B, thus xing the renormalization scheme of our results. e initial condition for the Wilson coe cient C tt S2 at the electroweak scale is calculated from the di erence of the |∆S| = 2 amplitudes in the full SM and the ve-avor EFT. We expand the Wilson coe cients in the ve-avor EFT as where the superscript "(5)" emphasizes that the electroweak coupling is de ned in the ve-avor scheme. We expand the amplitude in the full SM as in terms of the six-avor coupling. is implies in the matching we have to include the nite threshold correction [14] α (6) in order to express both results in terms of α (5) . e sum in Eq. (13) runs over all operators in the basis. e LO matching conditions are simply A tt,(0) i = C tt,(0) i , and we nd the following non-zero contributions (the terms of order in the physical Wilson coe cient are relevant for the two-loop calculation)  C tt,(0) We obtained this result by equating the amplitudes in the full SM and the ve-avor EFT de ned above. We set all external momenta and fermion masses apart from the top quark to zero and employ dimensional regularization in d = 4 − 2 dimensions for both ultraviolet (UV) and infrared (IR) divergences.
In the remainder of this section, we discuss the matching at NLO in the electroweak interactions. We split the NLO SM amplitude into several contributions as follows: Here, A e explicit forms of the renormalization constants are given in App. A. We have included all tadpole graphs in both the full two-loop amplitude and the counterterms, such that the renormalization constants are gauge independent. e contributions of external eld renormalization are comprised by A tt,(se),tree-level Z Q S2 and A tt,(se),one-loop Z Q S2 (see Fig. 2 for sample diagrams, and App. A for the explicit form of the eld renormalization constants). We include nite matching contributions in the renormalization of the external elds, such that the elds are normalized in the same way in the full and e ective theories (cf. Ref. [15]).
As an additional check, we calculated all one-loop diagrams with single counterterm insertions, including those corresponding to unphysical Goldstone boson eld and mass renormalizations as well as Goldstone-W mixing. A er a careful application of the Slavnov-Taylor Identities [13], we analytically veri ed that the two methods of calculating A tt,(se),coun Q S2 are identical. Parameterizing the matrix elements in the EFT as where Q j (0) denotes the tree-level matrix element, and writing (right panel). e circled cross denotes a counterterm insertion.
the NLO matching condition is where Z (se) ji denotes the renormalization constants for the Wilson coe cients; their explicit form is given in App. B. e nal result for C tt,(se) i is given in App. D. All diagrams have been calculated using self-wri en FORM [16] routines, implementing the twoloop recursion presented in Refs. [17,18]. e amplitudes were generated using qgraf [19]. As a check on our calculation, we veri ed that both IR and UV divergences cancel in the matching, yielding a nite result. We checked analytically that the full amplitude is independent of the matching scale to the considered order in the electroweak interaction.

Normalization to muon decay
When considering the prediction of low-energy observables at higher orders in the electroweak interactions, it is o en advantageous to employ the muon lifetime as one of the experiments xing the values of the input parameters. By a suitable normalization of the results, this allows for the absorption of parts of the radiative corrections into the muon lifetime measurement [20].
In the context of our calculation, we regard the Fermi constant G F as the Wilson coe cient of muon decay in the Fermi e ective theory [21]. We write the e ective Lagrangian as where and we expand e LO matching then just yields G (0) µ = 1, while a one-loop matching calculation gives (Note that, again, a nite threshold correction has been included in order to express the result in terms of the ve-avor α.) is result corresponds to the following de nition of the relevant evanescent operator: is form of the evanescent operator ensures the validity at O(α) of the Fierz relations that have been used to calculate the QED corrections to the muon decay matrix element in the Fermi theory [21].
We can now discuss the di erent normalizations introduced in Sec. 1, including the NLO corrections. For "GF" normalization we have Inserting the all-order relation Similarly, for "GF2" normalization we have (Recall that we have absorbed factors of α/(4π) into the de nitions of the operators.) ese relations allow us to obtain C tt S2 and C tt S2 from our explicit calculations of C tt S2 and G µ .
3 Discussion of electroweak renormalization schemes e purpose of this section is to identify the residual theory uncertainty with regard to the higherorder electroweak corrections. We will estimate the uncertainty by studying di erent renormalization schemes for the input parameters, and the residual matching scale dependence in the MS scheme. e immediate problem facing us is the dominant residual scale dependence with regard to QCD. In order to isolate the electroweak e ects, we will therefore completely ignore QCD in this section. We de ne a formally scale-and scheme-independent quantity by multiplying the Wilson coe cient with the matrix element of the Q S2 operator.
is serves to cancel the part of the scale dependence related to the anomalous dimension of the Wilson coe cient, and to cancel its residual scheme dependence (see the detailed discussion in Sec. 4). We do not include any other nite terms in the matrix element. In principle, this matrix element could be calculated, including QCD e ects, using la ice methods. e absolute value of the matrix element plays no role in our numerics, as in this section we are only interested in the scale and scheme variation, not the absolute values for the Wilson coe cients. ese will be considered, including QCD, in the next section. To be speci c, we multiply the Wilson coe cient by the partonic matrix element Table 1: Primary input values used in our numerics; all numbers are taken from the PDG [2].
where Q S2 (0) denotes the tree-level matrix element, and the LO QED correction is given by with µ had = 2 GeV.

MS scheme
Our primary input parameters in the MS scheme are eir values are collected in Tab. 1. In the MS scheme it is most convenient to express all physical quantities in terms of the running parameters To determine the initial conditions of these couplings at µ = M Z , we rst use RunDec [22] to convert the top-quark pole mass to QCD-MS using three-loop accuracy. We nd mt(mt) = 162.9(7) GeV. We then convert the (electroweak) on-shell masses M Z , m t (m t ), M h to MS masses, using the one-loop expressions from [23,24], and choosing the matching scale to be the respective on-shell masses. We take into account the correction (25) when converting G F to v. We then t the initial conditions of the couplings (34) such that they reproduce the central values of the converted input parameters at their respective scales, employing two-loop running of the couplings. Note that the strong coupling g s is neglected throughout this section (apart from the conversion of the top-quark mass to QCD-MS.) e two-loop beta functions for the SM are taken from Ref. [25]. However, their result for the running of the Higgs vacuum expectation value is not consistent with our treatment of tadpole graphs (in fact, the result for β v given in Ref. [25] is only valid in Landau gauge). A gauge-independent result consistent with our conventions can be extracted from Refs. [23,24]. For convenience, we collect the corresponding beta function here, writing where we counted the contributions of colored fermions in Ref. [24] with multiplicity three. As a check, we obtained the identical result by an explicit calculation of the one-loop renormalization of the Higgs mass, and using the beta function for the Higgs quartic from Ref. [25] (see also Ref. [26]). As a further consistency check, we note that β (1) v is identical to the coe cient of the log µ 2 term in Eq. (25), a er expressing the mass ratios in terms of couplings. Also the two-loop beta function can  be extracted from Refs. [23,24], as a combination of the beta functions for the W mass and the g 2 gauge coupling. is yields Part of the matching scale dependence of the Wilson coe cient cancels the scale dependence of the running parameters in the prefactor of the Lagrangian. Hence, we de ne a formally scale and scheme independent Wilson coe cient in the MS scheme (where we always use "A2" normalization) where α, m W , and s MS w are de ned in the MS scheme, and the matrix element is de ned in Eq. (31). Varying the matching scale µ t between 60 GeV and 320 GeV, we see that the matching scale dependence is reduced from ±12% at LO to ±0.4% at NLO (see Fig. 3, le panel).

On-shell scheme
Here, we de ne all parameters in the on-shell scheme (OS) regarding the electroweak interaction. (We always treat the top-quark mass as an MS mass regarding the strong interaction.) e weak mixing angle is de ned, in the on-shell scheme, by e W -boson mass itself is a function of the primary input parameters; to obtain its numerical value M W = 80.354 GeV, we use the results of Ref. [27]. Again, we de ne a formally scale-and scheme-independent Wilson coe cient. Its explicit form depends on the chosen normalization and is given bŷ  for "A2" normalization, byĈ for "GF" normalization, and byĈ for "GF2" normalization. Here, s OS w and M W are de ned in the on-shell scheme, while α is (always) de ned as a MS coupling (note that the dependence on s OS w drops out in GF2 normalization). As is clearly visible in Fig. 4, the NLO corrections in the on-shell scheme are large, indicating slow convergence of the perturbation series. As discussed in Ref. [14], this can be a ributed to the large topmass dependence of the on-shell counterterm for the weak mixing angle. Hence, the on-shell scheme is not suitable for low-energy observables like K , and we will not use it in our error estimate.

Hybrid scheme
In this scheme, we de ne all masses in the on-shell scheme regarding the electroweak interaction, while the weak mixing angle that appears in the prefactor is de ned in the MS scheme: sin 2 θ MS w = g 2 1 /(g 2 1 + g 2 2 ). Accordingly, we choose a di erent set of primary input parameters for the numerics in this scheme, namely, Again, we t the initial conditions of the MS parameters (34) as described in Sec. 3.1, neglecting the strong interaction. We de ne formally scale-and scheme-independent Wilson coe cients in this scheme as follows. For "A2" normalization, while for "GF" normalization,Ĉ Here, s MS w (µ) is de ned in the MS scheme, while M W is still de ned in the on-shell scheme. Note that the hybrid scheme coincides with the on-shell scheme for the "GF2" normalization, up to the tiny numerical di erence in the M W mass.
e Wilson coe cients for the three di erent normalizations are shown in Fig. 3 (right three panels). While the LO results in "A2" and "GF" normalization still show sizeable scale dependence, all NLO curves are essentially at. As an additional estimate of the unknown higher-order electroweak e ects, we take half of the absolute value of the di erence between the largest and the smallest NLO value. Again, this translates into a ±0.4% uncertainty.

Inclusion of QCD
As is well-known, the dominant corrections to the LO SM prediction of the |∆S| = 2 weak Lagrangian arise from QCD and have to be taken into account in the nal numerics. In this section, we will summarize the status of the QCD corrections and combine them with our electroweak corrections. e large separation between the electroweak scale, µ t ∼ m t , M W and the scale µ had ∼ 2 GeV where the hadronic matrix elements are evaluated mandate the use of RG-improved perturbation theory, summing powers of α s log(µ t /µ had ) to all orders. Here, we include also the (small) QED corrections. e formalism is well-known [6,28]. We now expand the Wilson coe cient, including the QCD terms, as e initial conditions C tt,(0) S2 (µ t ) and C tt,(se) S2 (µ t ) have been given in the previous section. e coe cient C tt,(e) S2 (µ t ) does not receive a matching contribution; it is purely generated by the RG ow and corresponds to LL QED resummed logarithms of the form αα n s log n+1 (µ t /µ had ). e last term contains the summed NLL QED logarithms of the form αα n s log n (µ t /µ had ). e coe cient C tt,(1) S2 (µ t ) receives matching contributions from two-loop box diagrams with gluon exchange. As a check of our setup we calculated this term and nd C tt,(1) where is expression is in full agreement the well-known result presented in Ref. [8].
e RG evolution is most conveniently wri en in terms of an evolution matrix, such that C tt S2 (µ) = C tt S2 (µ 0 )U (µ 0 , µ, α). We expand in terms of the couplings de ned at the low scale µ, and nd the following contributions to the Wilson coe cient at the low scale: where we have introduced the ratio η = α s (µ 0 )/α s (µ). e explicit expression for the evolution matrix can be found in Ref. [28] and involves the anomalous dimension of the Wilson coe cients. It is given by e rst two terms are well-known [6], while the QED corrections are new. ey have been calculated by extracting the UV poles of the relevant one-and two-loop diagrams (see Fig. 5 for examples) using the infrared rearrangement described in Ref. [29]. e result (51) is valid for the Wilson coe ents (C tt S2 , C tt S2 , and C tt S2 ) in all three normalizations conventions, due to our absorbing powers of α into the de nitions of the corresponding operators. Note that the α 2 s term depends on the explicit form of the evanescent operators, given in App. B, while the other three terms are scheme independent, as we have veri ed by explicit calculation.
is last observation deserves further discussion. Our two-loop result C tt,(se) S2 does depend on the de nition of evanescent operators in Eq. (63). e SM prediction for the observable K must, of course, be independent of such arbitrary choices; in fact, the scheme dependence of the Wilson coe cient will cancel exactly against the corresponding scheme dependence of the hadronic matrix element (a proof is given in App. C.) In the literature on K , the scheme independent product of Wilson coe cient and matrix element is usually factorized into two separately scheme-and scale-independent quantities, namely, the QCD correction factors η tt and η ut , and the kaon bag factorB K .
In our case, this strategy fails when including QED corrections, as the O(αα s ) anomalous dimension is scheme independent by itself. is is consistent with the general expression for the scheme dependence of anomalous dimensions given in Ref. [28] and App. C, because here the anomalous dimension is a one-dimensional matrix, i.e. just a number. erefore, the de nition of the schemeinvariant correction factor η tt cannot be extended to include QED e ects (as a xed-order expansion in α).
It follows that, in the absence of a determination of the hadronic matrix element including QED corrections, our result will be scheme dependent. However, the scheme dependence is tiny. Numerically,  (5) 3.98(5) the explicit dependence on the coe cient a 11 , de ned in App. B, is

NLL QCD NLL QCD & NLL QED
is dependence will cancel once a full matrix element is available. However, we expect the nite shi in the hadronic matrix element to be equally tiny, of order α/(4π) ∼ 10 −4 . e bulk of the e ect of electroweak corrections is captured by the matching calculation at the weak scale, not by the QED e ects in the e ective theory. We will therefore neglect the scheme dependence in our numerical discussion.
To quantify the impact of the electroweak corrections, we examine the numerical values of the Wilson coe cients in the hybrid renormalization scheme, as it shows the best convergence properties and least residual scale dependence (cf. Fig. 6). e values in the three di erent normalization conventions are given in Tab. 2, both for the NLL QCD result only, and including the full NLO electroweak corrections (with NLL QED resummation). e shown uncertainties are half of the di erence between maximum and minimum values when varying the matching scale µ t between 60 GeV and 320 GeV. We see that the shi from NLL QCD to full electroweak is actually largest, −1%, in the conventionally used "GF2" normalization, while in the "A2" and "GF" normalizations the shi is +0.5% and −0.5%, respectively. 3 e resulting NLO values are in perfect agreement.
With the three-loop QCD corrections in the top sector and the electroweak corrections in the charm-top sector still outstanding, we refrain from an extensive numerical discussion of our results. e shi in the Wilson coe cient is small, of order one percent, in the hybrid scheme in all normalizations. Since both the residual scale dependence and the spread between NLO values (considering electroweak e ects only) is both ±0.4% (see Sec. 3), we suggest the following temporary prescription that is most easily implemented: Adopt the traditional "GF2" normalization, and multiply η tt by the electroweak correction factor (1 − ∆ tt ), with ∆ tt = 0.01 ± 0.004. With this prescription, the SM prediction presented in Ref. [7] is shi ed by −0.7% to | K | = 2.15(6)(7)(15) × 10 −3 , with the errors corresponding to short-distance, long-distance, and parametric uncertainties (see Ref. [7] for details).

Conclusions
We have presented the complete two-loop electroweak corrections to the top-quark contribution to the parameter K . e analogous electroweak corrections for B 0 − B 0 mixing have been presented previously in Ref. [13]; as a check, we reproduced their numerical results using the input parameters given in that reference. In our calculation we used three di erent normalizations for the e ective Lagrangian and several renormalization schemes for the electroweak input parameters. While these lead to di erent numerical predictions at LO, the nal results agree well if the electroweak corrections are taken into account. We assess the theoretical uncertainty by studying the residual electroweak matching scale dependence and the residual dependence on the renormalization scheme for the input parameters, leading to a error of ±0.4% asscociated with unknown higher-order corrections.
We then considered the full RG evolution in the e ective theory at NLO in QCD and mixed QED-QCD. In particular, we discuss how the scheme dependence, related to the freedom in choosing the evanescent operators, cancels against the hadronic matrix element if evaluated including QED corrections. While these corrections are currently unknown, they are expected to be tiny, as is the le over scheme dependence of our result.
Numerically, the inclusion of the electroweak corrections amounts to a −1% downward shi in the Wilson coe cient for the top-quark contribution de ned at the hadronic scale µ had = 2 GeV, when using the conventional normalization of the e ective Lagrangian. We defer a more extensive numerical study of the e ect on the SM prediction of K to the future, when the three-loop QCD corrections in the top sector [12], the two-loop electroweak corrections in the charm-top sector [30], and possibly an updated hadronic matrix element will be available. Instead, we just shi the usual QCD correction factor η tt by a factor 1 − 0.010 (4). is shi s the SM prediction presented in Ref. [7] by −0.7% to | K | = 2.15(6)(7)(15) × 10 −3 . e errors correspond to short-distance, long-distance, and parametric uncertainties.

A Mass and field renormalization constants
All mass renormalization constants include the tadpole contributions and are thus gauge-parameter independent: It follows that δs 2 Note that all tadpole contributions to δs 2 w cancel. e charge renormalization is given by e divergent part of the one-loop o -diagonal eld renormalization constant (in 't Hoo -Feynman gauge) is e nite part of the one-loop o -diagonal eld renormalization constant (in 't Hoo -Feynman gauge) is where x t ≡ m 2 t /M 2 W . Both Eqs. (59) and (60) are in agreement with the results in Ref. [13]. e divergent part of the one-loop diagonal eld renormalization constant for the down-type quarks (in 't Hoo -Feynman gauge) is (61) e nite part of the one-loop diagonal eld renormalization constant for the down-type quarks (in 't Hoo -Feynman gauge) is (62) e nite part di ers from the expression in Ref. [13] due to our using dimensional regularization for the IR divergences.
e terms quadratic in do not play a role in our calculation and are kept only for completeness. Some of the coe cients of the terms have been le unspeci ed as an additional check of our calculation. e result for the Wilson coe cient at NLO in the electroweak interaction depends on the coe cient a 11 , as discussed in Sec. 4. Our results given in this paper and all plots use the conventional choice a 11 = 4.
We expand the renormalization constants as e necessary, non-zero Z factors at order α s are and e necessary, non-zero Z factors at order α are Z (e,1) and Z (e,0) At two-loop, we nd Using these renormalization constants, we nd that all scheme dependence cancels in the two-loop anomalous dimension at order αα s , as expected on general grounds [28].

C Scheme independence at order α
In this appendix we give a proof of the scheme independence of the prediction of the top-quark contribution to K at NLO in QED. We rst recall the transformation properties of Wilson coe cients and anomalous dimension at order α by adapting the results in Ref. [10]. To this end, we write the general transformation among all dimension-six operators as Here, the matrices R and M parameterize a linear transformation among the physical and evanescent operators Q and E, respectively, W parameterizes the addition of multiples of evanescent operators to the physical operators, and U parameterizes the addition of multiples of times physical operators to the evanescent operators. As explained in detail in Ref. [33], this transformation implies an additional nite renormalization that is needed in order to restore the standard MS de nition of the renormalization constants. is nite renormalization, induced by the change (74), is given in the notation of the previous section by Z (e,0) by a straightforward generalization of the results in Ref. [10], where also the higher-order QCD expressions can be found. e corresponding transformation law for the anomalous dimension matrices is then in agreement with the results given in Ref. [28]. e Wilson coe cients change according to In particular, we nd e matrix elements change with the inverse transformation, thus ensuring the the scheme dependence cancels in the amplitude [10]. Hence, we only need to show that the RG evolution does not upend this cancelation. We will consider only the e ects of order α/α s and α. ( e QCD case has been discussed extensively in the literature, see e.g. Refs. [33][34][35]. e proof of scheme dependence including QED in Ref. [28] fails in our case as the anomalous dimension is itself scheme independent.) We expand the hadronic matrix element in powers of couplings as 4 Recalling the perturbative expansion of the RG evolution matrix, Eq. (46), and the expansion in powers of couplings of the low-energy Wilson coe cient, Eq. (47), the terms at order α/α s and α in the amplitude are then (we drop obvious function arguments) It is apparent that the order α/α s contribution is scheme independent, as it involves only schemeindependent quantities. In the order α contribution, the QCD scheme dependence cancels within the square brackets in the rst line, respectively, as shown in Ref. [34]. In the second line, note that the rst term is scheme independent. e renormalized matrix element r (se) is given by 5 According to the discussion at the beginning of this section, under a change of renormalization scheme with nonzero coe cient U in Eq. (74), this contribution to the matrix element transforms as r (se) → r (se) − Z (e) . is nishes the proof of the scheme independence of the matrix element.
Note that for the cancelation to work, we need to x the electromagnetic coupling in the e ective theory to its value at µ = M Z , i.e. α = α(M Z ). is is consistent with the xed-order perturbative expansion in α.