1em captionlabel caption

# Two-loop Electroweak Corrections to the Top-Quark Contribution to
$\epsilon_{K}$

Joachim Brod<EMAIL_ADDRESS>Department of Physics, University of
Cincinnati, Cincinnati, OH 45221, USA Sandra Kvedaraitė<EMAIL_ADDRESS>Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
Zachary Polonsky<EMAIL_ADDRESS>Department of Physics, University of
Cincinnati, Cincinnati, OH 45221, USA

###### Abstract

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

The parameter $\epsilon_{K}$ describes CP violation in the neutral kaon system
and is one of the most sensitive probes of new physics. It can be defined as
[1]

$\epsilon_{K}\equiv
e^{i\phi_{\epsilon}}\sin\phi_{\epsilon}\frac{1}{2}\arg\bigg{(}\frac{-M_{12}}{\Gamma_{12}}\bigg{)}\,.$
(1)

Here, $\phi_{\epsilon}=\arctan(2\Delta M_{K}/\Delta\Gamma_{K})$ where $\Delta
M_{K}$ and $\Delta\Gamma_{K}$ are the mass and lifetime differences of the
weak eigenstates $K_{L}$ and $K_{S}$. $M_{12}$ and $\Gamma_{12}$ are the
Hermitian and anti-Hermitian parts of the Hamiltonian that determines the time
evolution of the neutral kaon system. The so-called short-distance
contributions to $\epsilon_{K}$ enter the matrix element $M_{12}=-\langle
K^{0}|\mathcal{L}^{\Delta S=2}_{f=3}|\bar{K}^{0}\rangle/(2\Delta M_{K})$, up
to higher powers in the operator-product expansion.

Experimentally, $\epsilon_{K}$ is well-known, with absolute value
$|\epsilon_{K}|=(2.228\pm 0.011)$ and an uncertainty at the permil level [2].
The standard model (SM) contributions to neutral kaon mixing are conveniently
described by the effective $|\Delta S|=2$ Lagrangian with three active quark
flavors,

$\begin{split}\mathcal{L}^{|\Delta
S|=2}_{f=3}=-\frac{G_{F}^{2}M_{W}^{2}}{4\pi^{2}}\big{[}&\lambda_{u}^{2}C_{S2}^{\prime\prime
uu}(\mu)+\lambda_{t}^{2}C_{S2}^{\prime\prime
tt}(\mu)+\lambda_{u}\lambda_{t}C_{S2}^{\prime\prime
ut}(\mu)\big{]}Q_{S2}^{\prime\prime}+\textrm{h.c.}+\dots\,,\end{split}$ (2)

valid at scales around $\mu=2\,$GeV. Here,

$Q_{S2}^{\prime\prime}=\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu}d_{L}^{\alpha}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu}d_{L}^{\beta}\big{)}\,$
(3)

is the local $|\Delta S|=2$ operator, where $\alpha$ and $\beta$ are color
indices, and the ellipsis denotes contributions of higher dimension local
operators and non-local contributions of $|\Delta S|=1$ operators. The reason
for the appearance of the double primes will become clear later. The elements
of the Cabibbo-Kobayashi-Maskawa (CKM) matrix are combined into the parameters
$\lambda_{i}\equiv V_{is}^{*}V_{id}$. The long-distance SM contributions are
comprised by the hadronic matrix element of the local $|\Delta S|=2$ 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 $\kappa_{\epsilon}=0.94(2)$ [4].

The short-distance contributions are contained in the Wilson coefficients
$C_{S2}^{\prime\prime ij}$. In the SM, the leading-order (LO) contributions to
the Wilson coefficients 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 different representation of the $|\Delta
S|=2$ effective Lagrangian [6], in terms of $\lambda_{c}$ and $\lambda_{t}$.
However, it was shown recently [7] that the parameterization (2) in terms of
$\lambda_{u}$ and $\lambda_{t}$ 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 coefficients are known at next-to-leading order
(NLO) for $C_{S2}^{\prime\prime tt}$ [8] with a remaining uncertainty at the
percent level, and at next-to-next-to-leading order (NNLO) for
$C_{S2}^{\prime\prime ut}$ with a remaining uncertainty below one percent [9,
10, 7]. The coefficient $C_{S2}^{\prime\prime uu}$ is also known at NNLO in
QCD [11, 7], but has no effect on $\epsilon_{K}$ (it does contribute to the
neutral kaon mass difference).

In anticipation of the NNLO QCD calculation of $C_{S2}^{\prime\prime tt}$ [12]
it is worthwhile to consider also the electroweak corrections to the effective
Lagrangian (2). Without an explicit calculation, the electroweak
renormalization scheme of the input parameters is left unspecified and amounts
to an uncertainty of the order of a few percent – which can no longer be
neglected at the current level of precision.

The electroweak corrections to $C_{S2}^{\prime\prime tt}$ 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 $\epsilon_{K}$ 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 first independent check of the
results in Ref. [13].

In this work, we calculate the coefficient $C_{S2}^{\prime\prime tt}$,
proportional to $\lambda_{t}^{2}$, to NLO in the electroweak interactions.
This fixes the renormalization scheme of the electroweak input parameters
contained in the prefactor $G_{F}^{2}M_{W}^{2}$. In fact, when only
considering QCD effects, there are several equivalent ways of rewriting the
prefactor, using the tree-level relation

$G_{F}=\frac{\pi\alpha}{\sqrt{2}M_{W}^{2}s_{w}^{2}}\,,$ (4)

where $\alpha=e^{2}/(4\pi)$ the electromagnetic coupling111Throughout this
paper, the electromagnetic coupling is understood to be fixed in the five-
flavor scheme at scale $\mu=M_{Z}$, unless otherwise noted. and
$s_{w}^{2}=\sin^{2}\theta_{w}$ with the weak mixing angle $\theta_{w}$.
Essentially, this choice specifies which experimental data are used as
parametric input for our prediction. The numerical difference between the
different schemes is expected to be large at LO, as exemplified by the 5%
difference between the on-shell and $\overline{\text{MS}}$ values of
$s_{w}^{2}$.

For our analysis, it is useful to write the effective Lagrangian in the
following form:

$\mathcal{L}^{|\Delta
S|=2}_{f=3}=\lambda_{t}^{2}c_{S2}^{tt}(\mu)Q_{S2}^{(\prime,\prime\prime)}+\textrm{h.c.}+\dots\,,$
(5)

where the ellipsis now also includes contributions not proportional to
$\lambda_{t}^{2}$. Using the tree-level relation (4), we express $c_{S2}^{tt}$
in three different ways:

$c_{S2}^{tt}(\mu)=-\frac{2\pi^{2}}{M_{W}^{2}s_{w}^{4}}C_{S2}^{tt}(\mu)\,,\qquad
c_{S2}^{tt}(\mu)=-\frac{G_{F}}{\sqrt{2}s_{w}^{2}}C_{S2}^{\prime tt}\,,\qquad
c_{S2}^{tt}(\mu)=-\frac{G_{F}^{2}M_{W}^{2}}{4\pi^{2}}C_{S2}^{\prime\prime
tt}\,.$ (6)

As explained in Sec. 2.2, this effectively absorbs different parts of the
radiative corrections into the measured value of the muon decay rate. In the
first and second relation in Eq. (6) we have factored out the powers
$(\alpha/(4\pi))^{2}$ and $\alpha/(4\pi)$, respectively, and absorbed them
into rescaled operators, defined as

$Q_{S2}=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu}d_{L}^{\alpha}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu}d_{L}^{\beta}\big{)}\,,$
(7)

and

$Q_{S2}^{\prime}=\frac{\alpha}{4\pi}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu}d_{L}^{\alpha}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu}d_{L}^{\beta}\big{)}\,,$
(8)

while $Q_{S2}^{\prime\prime}$ has been defined 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

$C_{S2}^{tt}(\mu)=C_{S2}^{\prime tt}(\mu)=C_{S2}^{\prime\prime
tt}(\mu)=S(x_{t})$ (9)

coincide. They are given by the modified [7] Inami-Lim box function [5]

$\mathscr{S}(x_{t})=\frac{4x_{t}-11x_{t}^{2}+x_{t}^{3}}{4(x_{t}-1)^{2}}+\frac{3x_{t}^{3}}{2(x_{t}-1)^{3}}\log
x_{t}\,,$ (10)

where $x_{t}\equiv m_{t}^{2}/M_{W}^{2}$, and we neglect a tiny correction of
${\cal O}(m_{c}^{2}/M_{W}^{2})\sim 10^{-4}$ [7]. We will refer to the first
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. These arbitrary dependences will be
mitigated to a large degree by the NLO electroweak corrections. The aim of
this paper is to show this explicitly, and to provide an updated numerical
prediction for $\epsilon_{K}$ including the electroweak corrections.

This paper is organized as follows. In Sec. 2 we define the effective
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
effects 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. The necessary counterterms in the
effective theory are collected in App. B. We give a formal proof of the scheme
independence of the top-quark contribution to the $|\Delta S|=2$ amplitude
including QED effects in App. C. Finally, the full analytic two-loop result is
presented in App. D.

##  2 Calculation of the two-loop electroweak matching corrections

Our strategy is to perform the matching calculation using the “A2”
normalization in the $\overline{\text{MS}}$ scheme. We can then easily change
to different normalization conventions and renormalization schemes for the
input parameters; this will be discussed in detail in this section. The
variation of the NLO results between the different prescriptions is used as
one way to estimate the remaining theory uncertainty.

###  2.1 Matching calculation

We write the physical five-flavor effective 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 form222The RG evolution to the three-flavor
effective Lagrangian will be considered in Sec. 4.

$\mathcal{L}^{|\Delta
S|=2}_{f=5}=-\frac{2\pi^{2}}{M_{W}^{2}s_{w}^{4}}\lambda_{t}^{2}C_{S2}^{tt}(\mu)Q_{S2}+\textrm{h.c.}+\dots\,.$
(11)

The single physical operator contributing to our results has been defined in
Eq. (7). Our definition of the evanescent operators is given in App. B, thus
fixing the renormalization scheme of our results.

The initial condition for the Wilson coefficient $C_{S2}^{tt}$ at the
electroweak scale is calculated from the difference of the $|\Delta S|=2$
amplitudes in the full SM and the five-flavor EFT. We expand the Wilson
coefficients in the five-flavor EFT as

$C_{S2}^{tt}=C_{S2}^{tt,(0)}+\frac{\alpha^{(5)}}{4\pi}C_{S2}^{tt,(se)}+\ldots\,,$
(12)

where the superscript “(5)” emphasizes that the electroweak coupling is
defined in the five-flavor scheme. We expand the amplitude in the full SM as

${\mathcal{A}}^{tt}=-\frac{2\pi^{2}}{M_{W}^{2}s_{w}^{4}}\lambda_{t}^{2}\sum_{i}\bigg{(}{\mathcal{A}}_{i}^{tt,(0)}+\frac{\alpha^{(6)}}{4\pi}{\mathcal{A}}_{i}^{tt,(se)}+\ldots\bigg{)}\langle
Q_{i}\rangle^{(0)}\,,$ (13)

in terms of the six-flavor coupling. This implies in the matching we have to
include the finite threshold correction [14]

$\alpha^{(6)}=\alpha^{(5)}\bigg{[}1-\frac{\alpha^{(5)}}{4\pi}\bigg{(}\frac{2}{3}+14\log\frac{\mu}{M_{W}}-\frac{32}{9}\log\frac{\mu}{m_{t}}\bigg{)}\bigg{]}\,,$
(14)

in order to express both results in terms of $\alpha^{(5)}$. The sum in Eq.
(13) runs over all operators in the basis.

The LO matching conditions are simply
${\mathcal{A}}_{i}^{tt,(0)}=C_{i}^{tt,(0)}$, and we find the following non-
zero contributions (the terms of order $\epsilon$ in the physical Wilson
coefficient are relevant for the two-loop calculation)

$\displaystyle\begin{split}C_{S2}^{tt,(0)}&=\frac{4x_{t}-11x_{t}^{2}+x_{t}^{3}}{4(x_{t}-1)^{2}}+\frac{3x_{t}^{3}}{2(x_{t}-1)^{3}}\log
x_{t}\\\
&+\epsilon\bigg{[}\frac{4x_{t}-11x_{t}^{2}+x_{t}^{3}}{4(x_{t}-1)^{2}}+\frac{3x_{t}^{3}}{2(x_{t}-1)^{3}}\log
x_{t}\bigg{]}\log\Big(\frac{\mu^{2}}{M_{W}^{2}}\Big{missing})\\\
&+\epsilon\bigg{[}\frac{3x_{t}^{3}-41x_{t}^{2}-12x_{t}}{8(x_{t}-1)^{2}}-\frac{x_{t}^{4}-14x_{t}^{3}-8x_{t}-4x_{t}}{4(x_{t}-1)^{3}}\log
x_{t}-\frac{3x_{t}^{3}}{4(x_{t}-1)^{3}}\log^{2}x_{t}\bigg{]}\\\ &-4\epsilon
C_{E_{S2}}^{tt,(0)}\,,\end{split}$ (15) $\displaystyle C_{E_{S2}}^{tt,(0)}$
$\displaystyle=-\frac{x_{t}+x_{t}^{2}}{4(x_{t}-1)^{2}}+\frac{x_{t}^{2}}{2(x_{t}-1)^{3}}\log
x_{t}\,.$ (16)

We obtained this result by equating the amplitudes in the full SM and the
five-flavor EFT defined above. We set all external momenta and fermion masses
apart from the top quark to zero and employ dimensional regularization in
$d=4-2\epsilon$ dimensions for both ultraviolet (UV) and infrared (IR)
divergences.

Figure 1: Sample two-loop Feynman diagrams contributing to the initial
condition of $C_{S2}^{\prime\prime tt}$.

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:

$\begin{split}{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se)}={\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{two-
loop}}+{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{countt}}+{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{one-
loop Z}}+{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{tree-level
Z}}\,.\end{split}$ (17)

Here, ${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{two-loop}}$
denotes the contribution of the ${\mathcal{O}}(30\,000)$ genuine two-loop
diagrams (see Fig. 1 for sample Feynman diagrams). The counterterm
contributions arising from the renormalization of the parameters in the LO
result are

$\begin{split}{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{countt}}=\bigg{[}\bigg{(}4\frac{\delta
e}{e}-2\frac{\delta s_{w}^{2}}{s_{w}^{2}}-\frac{\delta
M_{W}^{2}}{M_{W}^{2}}\bigg{)}{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(0)}+\delta
m_{t}\frac{\partial{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(0)}}{\partial
m_{t}}+\delta
M_{W}^{2}\frac{\partial{\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(0)}}{\partial
M_{W}^{2}}\bigg{]}\,.\end{split}$ (18)

The 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.
The contributions of external field renormalization are comprised by
${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{tree-level Z}}$ and
${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{one-loop Z}}$ (see Fig.
2 for sample diagrams, and App. A for the explicit form of the field
renormalization constants). We include finite matching contributions in the
renormalization of the external fields, such that the fields are normalized in
the same way in the full and effective 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 field and mass renormalizations as well as Goldstone-$W$ mixing. After a
careful application of the Slavnov-Taylor Identities [13], we analytically
verified that the two methods of calculating
${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{countt}}$ are identical.

Figure 2: Sample Feynman diagrams contributing to the off-diagonal field
renormalization: ${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{tree-
level Z}}$ (left panel), and
${\mathcal{A}}_{Q_{S2}^{\prime\prime}}^{tt,(se),\text{one-loop Z}}$ (right
panel). The circled cross denotes a counterterm insertion.

Parameterizing the matrix elements in the EFT as

$\langle Q_{i}\rangle=\sum_{j}r_{ij}\langle Q_{j}\rangle^{(0)}\,,$ (19)

where $\langle Q_{j}\rangle^{(0)}$ denotes the tree-level matrix element, and
writing

$r_{ij}=\delta_{ij}+\frac{\alpha^{(5)}}{4\pi}r_{ij}^{(se)}+\ldots\,,$ (20)

the NLO matching condition is

$C_{i}^{tt,(se)}={\mathcal{A}}_{i}^{tt,(se)}-\sum_{j}C_{j}^{tt,(0)}\Big{(}r_{ji}^{(se)}+Z_{ji}^{(se)}\Big{)}\,,$
(21)

where $Z_{ji}^{(se)}$ denotes the renormalization constants for the Wilson
coefficients; their explicit form is given in App. B. The final result for
$C_{i}^{tt,(se)}$ is given in App. D.

All diagrams have been calculated using self-written FORM [16] routines,
implementing the two-loop recursion presented in Refs. [17, 18]. The
amplitudes were generated using qgraf [19]. As a check on our calculation, we
verified that both IR and UV divergences cancel in the matching, yielding a
finite result. We checked analytically that the full amplitude is independent
of the matching scale to the considered order in the electroweak interaction.

###  2.2 Normalization to muon decay

When considering the prediction of low-energy observables at higher orders in
the electroweak interactions, it is often advantageous to employ the muon
lifetime as one of the experiments fixing 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 coefficient of muon decay in the Fermi effective theory [21]. We write
the effective Lagrangian as

${\mathcal{L}}_{F}=-\frac{4}{\sqrt{2}}\frac{\pi\alpha}{\sqrt{2}M_{W}^{2}s_{w}^{2}}G_{\mu}Q_{\mu}+\text{H.c.}\,,$
(22)

where

$Q_{\mu}=(\bar{\nu}_{\mu,L}\gamma^{\mu}\mu_{L})(\bar{\nu}_{e,L}\gamma_{\mu}e_{L})\,,$
(23)

and we expand

$G_{\mu}=G_{\mu}^{(0)}+\frac{\alpha}{4\pi}G_{\mu}^{(se)}+\ldots\,.$ (24)

The LO matching then just yields $G_{\mu}^{(0)}=1$, while a one-loop matching
calculation gives

$\begin{split}G^{(se)}_{\mu}=&\frac{144x^{2}z^{2}-18xyz^{2}-21y^{2}z^{2}+15yz^{2}+14yz+16y-12z^{2}-24}{24y(z-1)}\\\
&-\frac{22y^{2}z^{3}-53y^{2}z^{2}+28y^{2}z-7yz^{2}+38yz-28y-12z+12}{4y(z-1)^{2}(yz-1)}\log(z)\\\
&+\frac{-108x^{2}z^{2}+27xyz^{2}-32yz+32y}{18y(z-1)}\log(x)+\frac{3y^{2}z^{3}-6yz^{2}}{4(z-1)(yz-1)}\log(y)\\\
&+\frac{216x^{2}z^{2}-54xyz^{2}-27y^{2}z^{2}+27yz^{2}-134yz+188y-54z^{2}-108}{36y(z-1)}\log\Big(\frac{\mu^{2}}{M_{Z}^{2}}\Big{missing})\,,\end{split}$
(25)

with $x\equiv m_{t}^{2}/M_{Z}^{2}$, $y\equiv M_{h}^{2}/M_{Z}^{2}$, and
$z=M_{Z}^{2}/M_{W}^{2}$. (Note that, again, a finite threshold correction has
been included in order to express the result in terms of the five-flavor
$\alpha$.) This result corresponds to the following definition of the relevant
evanescent operator:

$E_{\mu}=(\bar{\nu}_{\mu,L}\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}\mu_{L})(\bar{\nu}_{e,L}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}e_{L})-(16-4\epsilon)Q_{\mu}\,.$
(26)

This form of the evanescent operator ensures the validity at
${\mathcal{O}}(\alpha)$ 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 different normalizations introduced in Sec. 1,
including the NLO corrections. For “GF” normalization we have

$c_{S2}^{tt}=-\frac{1}{\sqrt{2}s_{w}^{2}}G_{F}C_{S2}^{\prime
tt}=-\frac{2\pi^{2}}{M_{W}^{2}s_{w}^{4}}C_{S2}^{tt}\,.$ (27)

Inserting the all-order relation
$G_{F}=\pi\alpha/(\sqrt{2}M_{W}^{2}s_{w}^{2})\times G_{\mu}$ and expanding
$G_{\mu}$, we find

$\frac{\alpha}{4\pi}C_{S2}^{\prime
tt}=C_{S2}^{tt,(0)}+\frac{\alpha}{4\pi}\bigg{[}C_{S2}^{tt,(se)}-C_{S2}^{tt,(0)}G_{\mu}^{(se)}\bigg{]}+\ldots\,.$
(28)

Similarly, for “GF2” normalization we have

$c_{S2}^{tt}=-\frac{M_{W}^{2}}{4\pi^{2}}G_{F}^{2}C_{S2}^{\prime\prime
tt}=-\frac{2\pi^{2}}{M_{W}^{2}s_{w}^{4}}C_{S2}^{tt}\,,$ (29)

or

$\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}C_{S2}^{\prime\prime
tt}=C_{S2}^{tt,(0)}+\frac{\alpha}{4\pi}\bigg{[}C_{S2}^{tt,(se)}-2C_{S2}^{tt,(0)}G_{\mu}^{(se)}\bigg{]}+\ldots\,.$
(30)

(Recall that we have absorbed factors of $\alpha/(4\pi)$ into the definitions
of the operators.) These relations allow us to obtain $C_{S2}^{\prime tt}$ and
$C_{S2}^{\prime\prime tt}$ from our explicit calculations of $C_{S2}^{tt}$ and
$G_{\mu}$.

##  3 Discussion of electroweak renormalization schemes

The purpose of this section is to identify the residual theory uncertainty
with regard to the higher-order electroweak corrections. We will estimate the
uncertainty by studying different renormalization schemes for the input
parameters, and the residual matching scale dependence in the
$\overline{\text{MS}}$ scheme. The immediate problem facing us is the dominant
residual scale dependence with regard to QCD. In order to isolate the
electroweak effects, we will therefore completely ignore QCD in this section.
We define a formally scale- and scheme-independent quantity by multiplying the
Wilson coefficient with the matrix element of the $Q_{S2}$ operator. This
serves to cancel the part of the scale dependence related to the anomalous
dimension of the Wilson coefficient, and to cancel its residual scheme
dependence (see the detailed discussion in Sec. 4). We do not include any
other finite terms in the matrix element. In principle, this matrix element
could be calculated, including QCD effects, using lattice methods. The
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 coefficients. These will be considered,
including QCD, in the next section. To be specific, we multiply the Wilson
coefficient by the partonic matrix element

$\langle
Q_{S2}\rangle\equiv\bigg{[}1+\frac{\alpha}{4\pi}r^{(se)}+\ldots\bigg{]}\langle
Q_{S2}\rangle^{(0)}\,,$ (31)

where $\langle Q_{S2}\rangle^{(0)}$ denotes the tree-level matrix element, and
the LO QED correction is given by

$r^{(se)}=\frac{1}{9}a_{11}-\frac{4}{3}\log\frac{\mu_{t}}{\mu_{\text{had}}}\,,$
(32)

with $\mu_{\text{had}}=2\,$GeV.

$M_{Z}=91.1876(21)\,$GeV |  $M_{t}^{\text{pole}}=172.5(7)\,$GeV
---|---
$M_{W}=80.379(12)\,$GeV | $\sin^{2}\theta_{w}^{\overline{\text{MS}}}=0.23141(4)$
$\alpha_{s}(M_{Z})=0.1179(10)$ | $\alpha(M_{Z})^{-1}=127.952(9)$
$G_{F}=1.1663787(6)\times 10^{-5}\,$GeV-2 |  $M_{h}=125.25(17)\,$GeV
Table 1: Primary input values used in our numerics; all numbers are taken from
the PDG [2].

###  3.1 $\overline{\text{MS}}$ scheme

Our primary input parameters in the $\overline{\text{MS}}$ scheme are

$\alpha_{s}(M_{Z})\,,\quad G_{F}\,,\quad\alpha(M_{Z})\,,\quad M_{Z}\,,\quad
M_{t}\,,\quad M_{h}\,.$ (33)

Their values are collected in Tab. 1. In the $\overline{\text{MS}}$ scheme it
is most convenient to express all physical quantities in terms of the running
parameters

$g_{1}\,,\quad g_{2}\,,\quad g_{s}\,,\quad y_{t}\,,\quad\lambda\,,\quad v\,.$
(34)

To determine the initial conditions of these couplings at $\mu=M_{Z}$, we
first use RunDec [22] to convert the top-quark pole mass to
QCD-$\overline{\text{MS}}$ using three-loop accuracy. We find
$mt(mt)=162.9(7)\,$GeV. We then convert the (electroweak) on-shell masses
$M_{Z}$, $m_{t}(m_{t})$, $M_{h}$ to $\overline{\text{MS}}$ 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 fit 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-$\overline{\text{MS}}$.)

The 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
$\beta_{v}$ 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 $\beta_{v}=\beta_{v}^{(1)}+\beta_{v}^{(2)}+\ldots\,$. We find

$\displaystyle\beta_{v}^{(1)}$
$\displaystyle=\frac{1}{16\pi^{2}}\bigg{(}-\frac{3}{2}\lambda+\frac{3}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}-\frac{3g_{1}^{4}}{4\lambda}-\frac{3g_{1}^{2}g_{2}^{2}}{2\lambda}-\frac{9g_{2}^{4}}{4\lambda}-3y_{t}^{2}+12\frac{y_{t}^{4}}{\lambda}\bigg{)}\,,$
(35)

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 $\beta_{v}^{(1)}$ is
identical to the coefficient of the $\log(\mu^{2})$ term in Eq. (25), after
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. This yields

$\begin{split}\beta_{v}^{(2)}&=\frac{1}{(16\pi^{2})^{2}}\bigg{(}\frac{1}{32}g_{2}^{4}-\frac{63}{16}g_{1}^{2}g_{2}^{2}-\frac{701}{96}g_{1}^{4}+\frac{379}{24}\frac{g_{1}^{6}}{\lambda}+\frac{559}{24}\frac{g_{1}^{4}g_{2}^{2}}{\lambda}\\\
&\hskip
55.00008pt+\frac{289}{24}\frac{g_{1}^{2}g_{2}^{4}}{\lambda}-\frac{305}{8}\frac{g_{2}^{6}}{\lambda}-\frac{3}{2}g_{1}^{2}\lambda-\frac{9}{2}g_{2}^{2}\lambda+\frac{63}{8}\lambda^{2}\\\
&\hskip
55.00008pt-\frac{85}{24}g_{1}^{2}y_{t}^{2}-\frac{45}{8}g_{2}^{2}y_{t}^{2}-\frac{21}{4}y_{t}^{4}+\frac{19}{2}\frac{g_{1}^{4}y_{t}^{2}}{\lambda}\\\
&\hskip
55.00008pt-21\frac{g_{1}^{2}g_{2}^{2}y_{t}^{2}}{\lambda}+\frac{9}{2}\frac{g_{2}^{4}y_{t}^{2}}{\lambda}+\frac{16}{3}\frac{g_{1}^{2}y_{t}^{4}}{\lambda}-60\frac{y_{t}^{6}}{\lambda}+9y_{t}^{2}\lambda-12g_{2}^{2}g_{s}^{2}\bigg{)}\,.\end{split}$
(36)

Part of the matching scale dependence of the Wilson coefficient cancels the
scale dependence of the running parameters in the prefactor of the Lagrangian.
Hence, we define a formally scale and scheme independent Wilson coefficient in
the $\overline{\text{MS}}$ scheme (where we always use “A2” normalization)

$\hat{C}_{S2}^{tt}=\frac{\alpha^{2}(\mu)}{8m_{W}^{2}(\mu)\big{(}s_{w}^{\overline{\text{MS}}}(\mu)\big{)}^{4}}C_{S2}^{tt}(\mu)\langle
Q_{S2}\rangle(\mu)$ (37)

where $\alpha$, $m_{W}$, and $s_{w}^{\overline{\text{MS}}}$ are defined in the
$\overline{\text{MS}}$ scheme, and the matrix element is defined in Eq. (31).
Varying the matching scale $\mu_{t}$ between $60\,$GeV and $320\,$GeV, we see
that the matching scale dependence is reduced from $\pm 12\%$ at LO to $\pm
0.4\%$ at NLO (see Fig. 3, left panel).

Figure 3: Residual electroweak matching scale dependence in the
$\overline{\text{MS}}$ and hybrid schemes. The dashed line shows the LO
result, while the solid line show the NLO result. See text for details.

###  3.2 On-shell scheme

Here, we define all parameters in the on-shell scheme (OS) regarding the
electroweak interaction. (We always treat the top-quark mass as an
$\overline{\text{MS}}$ mass regarding the strong interaction.) The weak mixing
angle is defined, in the on-shell scheme, by
$\sin^{2}\theta_{w}^{\text{OS}}=1-M_{W}^{2}/M_{Z}^{2}$. The $W$-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 define a
formally scale- and scheme-independent Wilson coefficient. Its explicit form
depends on the chosen normalization and is given by

$\hat{C}_{S2}^{tt}=\frac{\alpha^{2}(\mu)}{8M_{W}^{2}\big{(}s_{w}^{\text{OS}}\big{)}^{4}}C_{S2}^{tt}(\mu)\langle
Q_{S2}\rangle(\mu)$ (38)

for “A2” normalization, by

$\hat{C}_{S2}^{tt}=\frac{\alpha(\mu)G_{F}}{4\sqrt{2}\big{(}s_{w}^{\text{OS}}\big{)}^{2}}C_{S2}^{\prime
tt}(\mu)\langle Q_{S2}\rangle(\mu)$ (39)

for “GF” normalization, and by

$\hat{C}_{S2}^{tt}=\frac{G_{F}^{2}M_{W}^{2}}{4\pi^{2}}C_{S2}^{\prime\prime
tt}(\mu)\langle Q_{S2}\rangle(\mu)$ (40)

for “GF2” normalization. Here, $s_{w}^{\text{OS}}$ and $M_{W}$ are defined in
the on-shell scheme, while $\alpha$ is (always) defined as a
$\overline{\text{MS}}$ coupling (note that the dependence on
$s_{w}^{\text{OS}}$ drops out in GF2 normalization).

Figure 4: Residual electroweak matching scale dependence in the full OS
scheme. The dashed line shows the LO result, while the solid line show the NLO
result. See text for details.

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 attributed to the large top-mass
dependence of the on-shell counterterm for the weak mixing angle. Hence, the
on-shell scheme is not suitable for low-energy observables like
$\epsilon_{K}$, and we will not use it in our error estimate.

###  3.3 Hybrid scheme

In this scheme, we define all masses in the on-shell scheme regarding the
electroweak interaction, while the weak mixing angle that appears in the
prefactor is defined in the $\overline{\text{MS}}$ scheme:
$\sin^{2}\theta_{w}^{\overline{\text{MS}}}=g_{1}^{2}/(g_{1}^{2}+g_{2}^{2})$.
Accordingly, we choose a different set of primary input parameters for the
numerics in this scheme, namely,

$\alpha_{s}(M_{Z})\,,\quad\sin^{2}\theta_{w}\,,\quad\alpha(M_{Z})\,,\quad
M_{W}\,,\quad M_{t}\,,\quad M_{h}\,.$ (41)

Again, we fit the initial conditions of the $\overline{\text{MS}}$ parameters
(34) as described in Sec. 3.1, neglecting the strong interaction. We define
formally scale- and scheme-independent Wilson coefficients in this scheme as
follows. For “A2” normalization,

$\hat{C}_{S2}^{tt}=\frac{\alpha^{2}(\mu)}{8M_{W}^{2}\big{(}s_{w}^{\overline{\text{MS}}}(\mu)\big{)}^{4}}C_{S2}^{tt}(\mu)\langle
Q_{S2}\rangle(\mu)\,,$ (42)

while for “GF” normalization,

$\hat{C}_{S2}^{tt}=\frac{\alpha(\mu)G_{F}}{4\sqrt{2}\big{(}s_{w}^{\overline{\text{MS}}}(\mu)\big{)}^{2}}C_{S2}^{\prime
tt}(\mu)\langle Q_{S2}\rangle(\mu)\,.$ (43)

Here, $s_{w}^{\overline{\text{MS}}}(\mu)$ is defined in the
$\overline{\text{MS}}$ scheme, while $M_{W}$ is still defined 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 difference in the $M_{W}$ mass.

The Wilson coefficients for the three different 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 flat. As an additional estimate of the unknown higher-order
electroweak effects, we take half of the absolute value of the difference
between the largest and the smallest NLO value. Again, this translates into a
$\pm 0.4\%$ uncertainty.

##  4 Inclusion of QCD

As is well-known, the dominant corrections to the LO SM prediction of the
$|\Delta S|=2$ weak Lagrangian arise from QCD and have to be taken into
account in the final numerics. In this section, we will summarize the status
of the QCD corrections and combine them with our electroweak corrections.

The large separation between the electroweak scale, $\mu_{t}\sim m_{t},M_{W}$
and the scale $\mu_{\text{had}}\sim 2\,$GeV where the hadronic matrix elements
are evaluated mandate the use of RG-improved perturbation theory, summing
powers of $\alpha_{s}\log(\mu_{t}/\mu_{\text{had}})$ to all orders. Here, we
include also the (small) QED corrections. The formalism is well-known [6, 28].
We now expand the Wilson coefficient, including the QCD terms, as

$C_{S2}^{tt}(\mu)=C_{S2}^{tt,(0)}(\mu)+\frac{\alpha_{s}(\mu)}{4\pi}C_{S2}^{tt,(1)}(\mu)+\frac{\alpha}{\alpha_{s}(\mu)}C_{S2}^{tt,(e)}(\mu)+\frac{\alpha}{4\pi}C_{S2}^{tt,(se)}(\mu)+\ldots\,.$
(44)

The initial conditions $C_{S2}^{tt,(0)}(\mu_{t})$ and
$C_{S2}^{tt,(se)}(\mu_{t})$ have been given in the previous section. The
coefficient $C_{S2}^{tt,(e)}(\mu_{t})$ does not receive a matching
contribution; it is purely generated by the RG flow and corresponds to LL QED
resummed logarithms of the form
$\alpha\alpha_{s}^{n}\log^{n+1}(\mu_{t}/\mu_{\text{had}})$. The last term
contains the summed NLL QED logarithms of the form
$\alpha\alpha_{s}^{n}\log^{n}(\mu_{t}/\mu_{\text{had}})$. The coefficient
$C_{S2}^{tt,(1)}(\mu_{t})$ receives matching contributions from two-loop box
diagrams with gluon exchange. As a check of our setup we calculated this term
and find

$\begin{split}C_{S2}^{tt,(1)}(\mu)&=\frac{8\pi^{2}}{9x_{t}}+\frac{10x_{t}^{5}+55x_{t}^{4}+149x_{t}^{3}+4x_{t}^{2}}{6(x_{t}-1)^{4}}\log^{2}(x_{t})\\\
&\quad+\frac{-15x_{t}^{5}-2x_{t}^{4}-543x_{t}^{3}+388x_{t}^{2}-148x_{t}+32}{6(x_{t}-1)^{4}}\log(x_{t})\\\
&\quad+\frac{25x_{t}^{4}+248x_{t}^{3}+75x_{t}^{2}-92x_{t}+32}{6(x_{t}-1)^{3}}\\\
&\quad+\text{Li}_{2}(1-x_{t})\frac{10x_{t}^{5}+74x_{t}^{4}-2x_{t}^{3}+40x_{t}^{2}-48x_{t}+16}{3x_{t}(x_{t}-1)^{3}}\\\
&\quad+\frac{5x_{t}^{4}+87x_{t}^{2}-20x_{t}}{2(x_{t}-1)^{3}}\log(x_{\mu})+\frac{3x_{t}^{4}-39x_{t}^{3}}{(x_{t}-1)^{4}}\log(x_{t})\log(x_{\mu})\,,\end{split}$
(45)

where $x_{\mu}\equiv\mu^{2}/M_{W}^{2}$, and
$\text{Li}_{2}(x)=-\int_{0}^{x}du\,\ln(1-u)/u$ is the usual dilogarithm. This
expression is in full agreement the well-known result presented in Ref. [8].

The RG evolution is most conveniently written in terms of an evolution matrix,
such that $C_{S2}^{tt}(\mu)=C_{S2}^{tt}(\mu_{0})U(\mu_{0},\mu,\alpha)$. We
expand

$\begin{split}U(\mu_{0},\mu,\alpha)=U^{(0)}(\mu_{0},\mu)+\frac{\alpha_{s}(\mu)}{4\pi}U^{(1)}(\mu_{0},\mu)+\frac{\alpha}{\alpha_{s}(\mu)}U^{(e)}(\mu_{0},\mu)+\frac{\alpha}{4\pi}U^{(se)}(\mu_{0},\mu)\end{split}$
(46)

in terms of the couplings defined at the low scale $\mu$, and find the
following contributions to the Wilson coefficient at the low scale:

$\displaystyle C_{S2}^{tt,(0)}(\mu)$
$\displaystyle=C_{S2}^{tt,(0)}(\mu_{0})U^{(0)}(\mu_{0},\mu)\,,$ (47)
$\displaystyle C_{S2}^{tt,(1)}(\mu)$ $\displaystyle=\eta
C_{S2}^{tt,(1)}(\mu_{0})U^{(0)}(\mu_{0},\mu)+C_{S2}^{tt,(0)}(\mu_{0})U^{(1)}(\mu_{0},\mu)\,,$
(48) $\displaystyle C_{S2}^{tt,(e)}(\mu)$
$\displaystyle=C_{S2}^{tt,(0)}(\mu_{0})U^{(e)}(\mu_{0},\mu)+\eta^{-1}C_{S2}^{tt,(e)}(\mu_{0})U^{(0)}(\mu_{0},\mu)\,,$
(49) $\displaystyle\begin{split}C_{S2}^{tt,(se)}(\mu)&=\eta
C_{S2}^{tt,(1)}(\mu_{0})U^{(e)}(\mu_{0},\mu)+\eta^{-1}C_{S2}^{tt,(e)}(\mu_{0})U^{(1)}(\mu_{0},\mu)\\\
&\quad+C_{S2}^{tt,(se)}(\mu_{0})U^{(0)}(\mu_{0},\mu)+C_{S2}^{tt,(0)}(\mu_{0})U^{(se)}(\mu_{0},\mu)\,,\end{split}$
(50)

where we have introduced the ratio $\eta=\alpha_{s}(\mu_{0})/\alpha_{s}(\mu)$.
The explicit expression for the evolution matrix can be found in Ref. [28] and
involves the anomalous dimension of the Wilson coefficients. It is given by

$\gamma_{S2}^{(0)}=\frac{\alpha_{s}}{\pi}+\bigg{(}\frac{4N_{f}}{9}-7\bigg{)}\frac{\alpha_{s}^{2}}{16\pi^{2}}+\frac{\alpha}{3\pi}-\frac{148}{9}\frac{\alpha\alpha_{s}}{16\pi^{2}}\,.$
(51)

The first two terms are well-known [6], while the QED corrections are new.
They 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].

Figure 5: Sample Feynman diagrams contributing to the two-loop anomalous
dimension.

The result (51) is valid for the Wilson coeffients ($C_{S2}^{tt}$,
$C_{S2}^{\prime tt}$, and $C_{S2}^{\prime\prime tt}$) in all three
normalizations conventions, due to our absorbing powers of $\alpha$ into the
definitions of the corresponding operators. Note that the $\alpha_{s}^{2}$
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 verified by
explicit calculation.

This last observation deserves further discussion. Our two-loop result
$C_{S2}^{tt,(se)}$ does depend on the definition of evanescent operators in
Eq. (63). The SM prediction for the observable $\epsilon_{K}$ must, of course,
be independent of such arbitrary choices; in fact, the scheme dependence of
the Wilson coefficient will cancel exactly against the corresponding scheme
dependence of the hadronic matrix element (a proof is given in App. C.) In the
literature on $\epsilon_{K}$, the scheme independent product of Wilson
coefficient and matrix element is usually factorized into two separately
scheme- and scale-independent quantities, namely, the QCD correction factors
$\eta_{tt}$ and $\eta_{ut}$, and the kaon bag factor $\hat{B}_{K}$. This is
achieved by writing the evolution matrix as
$U(\mu_{0},\mu,\alpha)=K^{-1}(\mu_{0},\alpha)U^{(0)}(\mu_{0},\mu,\alpha)K(\mu,\alpha)$,
and combining the $K$ factors, together with the appropriate part of the LO
evolution matrix, with the Wilson coefficients and the matrix elements to
yield scheme-independent quantities (see Refs. [28, 6] for details).

In our case, this strategy fails when including QED corrections, as the
${\mathcal{O}}(\alpha\alpha_{s})$ anomalous dimension is scheme independent by
itself. This 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.
Therefore, the definition of the scheme-invariant correction factor
$\eta_{tt}$ cannot be extended to include QED effects (as a fixed-order
expansion in $\alpha$).

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, the explicit dependence
on the coefficient $a_{11}$, defined in App. B, is

$C_{S2}^{\prime\prime tt}(2\,\text{GeV})=(3.90-0.0003a_{11})\times 10^{-8}\,.$
(52)

This dependence will cancel once a full matrix element is available. However,
we expect the finite shift in the hadronic matrix element to be equally tiny,
of order $\alpha/(4\pi)\sim 10^{-4}$. The bulk of the effect of electroweak
corrections is captured by the matching calculation at the weak scale, not by
the QED effects in the effective theory. We will therefore neglect the scheme
dependence in our numerical discussion.

Figure 6: Residual matching scale dependence with full QCD resummation, in the
$\overline{\text{MS}}$ and hybrid schemes. The dashed line shows the LO
result, the dash-dotted line shows the result including NLO QCD corrections,
while the solid line show the full (QED and QED) NLO result. See text for
details.

To quantify the impact of the electroweak corrections, we examine the
numerical values of the Wilson coefficients in the hybrid renormalization
scheme, as it shows the best convergence properties and least residual scale
dependence (cf. Fig. 6). The values in the three different 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). The
shown uncertainties are half of the difference between maximum and minimum
values when varying the matching scale $\mu_{t}$ between $60\,$GeV and
$320\,$GeV. We see that the shift from NLL QCD to full electroweak is actually
largest, $-1\%$, in the conventionally used “GF2” normalization, while in the
“A2” and “GF” normalizations the shift is $+0.5\%$ and $-0.5\%$,
respectively.333Using the (even) more conservative range
$40\,\text{GeV}\leq\mu_{t}\leq 320\,\text{GeV}$ leads to a $-1\%$ shift in all
three normalizations. The resulting NLO values are in perfect agreement.

| NLL QCD | NLL QCD & NLL QED
---|---|---
$\alpha^{2}/(8M_{W}^{2}s_{w}^{4})C_{S2}^{tt}(2\,\text{GeV})\times 10^{8}$ | $3.96(6)$ | $3.98(6)$
$\alpha G_{F}/(4\sqrt{2}\pi s_{w}^{2})C_{S2}^{\prime tt}(2\,\text{GeV})\times 10^{8}$ | $4.00(4)$ | $3.98(5)$
$G_{F}^{2}M_{W}^{2}/(4\pi^{2})C_{S2}^{\prime\prime tt}(2\,\text{GeV})\times 10^{8}$ | $4.02(5)$ | $3.98(5)$
Table 2: Wilson coefficients. Uncertainty given is scale variation only.

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. The shift in the Wilson
coefficient 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 effects only) is both $\pm 0.4\%$
(see Sec. 3), we suggest the following temporary prescription that is most
easily implemented: Adopt the traditional “GF2” normalization, and multiply
$\eta_{tt}$ by the electroweak correction factor $(1-\Delta_{tt})$, with
$\Delta_{tt}=0.01\pm 0.004$. With this prescription, the SM prediction
presented in Ref. [7] is shifted by $-0.7\%$ to
$|\epsilon_{K}|=2.15(6)(7)(15)\times 10^{-3}$, with the errors corresponding
to short-distance, long-distance, and parametric uncertainties (see Ref. [7]
for details).

##  5 Conclusions

We have presented the complete two-loop electroweak corrections to the top-
quark contribution to the parameter $\epsilon_{K}$. The analogous electroweak
corrections for $B^{0}-\overline{B^{0}}$ 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
different normalizations for the effective Lagrangian and several
renormalization schemes for the electroweak input parameters. While these lead
to different numerical predictions at LO, the final 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 $\pm 0.4\%$ asscociated with unknown higher-
order corrections.

We then considered the full RG evolution in the effective 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 leftover scheme dependence of our result.

Numerically, the inclusion of the electroweak corrections amounts to a $-1\%$
downward shift in the Wilson coefficient for the top-quark contribution
defined at the hadronic scale $\mu_{\text{had}}=2\,$GeV, when using the
conventional normalization of the effective Lagrangian. We defer a more
extensive numerical study of the effect on the SM prediction of $\epsilon_{K}$
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 shift the
usual QCD correction factor $\eta_{tt}$ by a factor $1-0.010(4)$. This shifts
the SM prediction presented in Ref. [7] by $-0.7\%$ to
$|\epsilon_{K}|=2.15(6)(7)(15)\times 10^{-3}$. The errors correspond to short-
distance, long-distance, and parametric uncertainties.

## Acknowledgments

We thank Jens Erler for useful communication regarding the electroweak input
parameters, and Martin Gorbahn and Emmanuel Stamou for many discussions and
comments on the manuscript. JB acknowledges support by DOE grant DE-SC0011784.
The Feynman diagrams have been generated using jaxodraw [31], based on axodraw
[32].

##  Appendix A Mass and field renormalization constants

All mass renormalization constants include the tadpole contributions and are
thus gauge-parameter independent:

$\displaystyle\begin{split}\delta
M_{W}^{2}&=\frac{e^{2}}{16\pi^{2}\epsilon}M_{W}^{2}\bigg{[}\frac{3}{4}-\frac{11}{12s_{w}^{2}}+\frac{3}{4}\frac{s_{w}^{2}}{c_{w}^{2}}-\frac{3}{2s_{w}^{2}}\frac{m_{t}^{2}}{M_{W}^{2}}\\\
&\hskip
64.00003pt-\frac{3}{4s_{w}^{2}}\frac{M_{h}^{2}}{M_{W}^{2}}+\frac{6}{s_{w}^{2}}\frac{m_{t}^{4}}{M_{W}^{2}M_{h}^{2}}-\frac{3}{2s_{w}^{2}c_{w}^{4}}\frac{M_{W}^{2}}{M_{h}^{2}}-\frac{3}{s_{w}^{2}}\frac{M_{W}^{2}}{M_{h}^{2}}\bigg{]}\,,\end{split}$
(53) $\displaystyle\begin{split}\delta
M_{Z}^{2}&=\frac{e^{2}}{16\pi^{2}\epsilon}M_{Z}^{2}\bigg{[}\frac{29}{6c_{w}^{2}}-\frac{11}{12}\frac{1}{s_{w}^{2}c_{w}^{2}}+\frac{11}{3}\frac{s_{w}^{2}}{c_{w}^{2}}-\frac{3}{2s_{w}^{2}c_{w}^{2}}\frac{m_{t}^{2}}{M_{Z}^{2}}\\\
&\hskip
64.00003pt-\frac{3}{4s_{w}^{2}c_{w}^{2}}\frac{M_{h}^{2}}{M_{Z}^{2}}+\frac{6}{s_{w}^{2}c_{w}^{2}}\frac{m_{t}^{4}}{M_{Z}^{2}M_{h}^{2}}-\frac{3}{2s_{w}^{2}c_{w}^{2}}\frac{M_{Z}^{2}}{M_{h}^{2}}-\frac{3}{s_{w}^{2}}\frac{M_{Z}^{2}}{M_{h}^{2}}+3\frac{M_{Z}^{2}}{M_{h}^{2}}\bigg{]}\,,\end{split}$
(54) $\displaystyle\delta M_{h}^{2}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}\epsilon}M_{h}^{2}\bigg{[}-\frac{3}{4s_{w}^{2}c_{w}^{2}}-\frac{3}{2s_{w}^{2}}+\frac{3}{4s_{w}^{2}}\frac{M_{h}^{2}}{M_{W}^{2}}+\frac{3}{2s_{w}^{2}}\frac{m_{t}^{2}}{M_{W}^{2}}\bigg{]}\,,$
(55) $\displaystyle\begin{split}\delta
m_{t}&=\frac{e^{2}}{16\pi^{2}\epsilon}m_{t}\bigg{[}-\frac{3}{4s_{w}^{2}c_{w}^{2}}\frac{M_{Z}^{2}}{M_{h}^{2}}-\frac{3}{2s_{w}^{2}}\frac{M_{W}^{2}}{M_{h}^{2}}-\frac{3}{8s_{w}^{2}}\frac{M_{h}^{2}}{M_{W}^{2}}\\\
&\qquad\qquad\qquad-\frac{1}{3c_{w}^{2}}+\frac{3}{8s_{w}^{2}}\frac{m_{t}^{2}}{M_{W}^{2}}+\frac{3}{s_{w}^{2}}\frac{m_{t}^{4}}{M_{W}^{2}M_{h}^{2}}\bigg{]}\,.\end{split}$
(56)

It follows that

$\frac{\delta s_{w}^{2}}{s_{w}^{2}}=-\frac{c_{w}^{2}}{s_{w}^{2}}\frac{\delta
c_{w}^{2}}{c_{w}^{2}}=\frac{e^{2}}{16\pi^{2}\epsilon}\bigg{[}\frac{11}{3}+\frac{19}{6s_{w}^{2}}\bigg{]}\,.$
(57)

Note that all tadpole contributions to $\delta s_{w}^{2}$ cancel. The charge
renormalization is given by

$\delta
Z_{e}=\frac{e^{2}}{16\pi^{2}\epsilon}\bigg{[}\frac{8N_{u}}{9}+\frac{2N_{d}}{9}+\frac{2N_{\ell}}{3}-\frac{7}{2}\bigg{]}\,.$
(58)

The divergent part of the one-loop off-diagonal field renormalization constant
(in ’t Hooft-Feynman gauge) is

$\displaystyle\delta Z_{sd}^{L}$
$\displaystyle=-\frac{e^{2}}{16\pi^{2}\epsilon}\frac{V_{ts}V_{td}^{*}}{s_{w}^{2}}\frac{x_{t}}{4}\,.$
(59)

The finite part of the one-loop off-diagonal field renormalization constant
(in ’t Hooft-Feynman gauge) is

$\delta
Z_{sd}^{L}=-\frac{e^{2}}{16\pi^{2}}\frac{V_{ts}V_{td}^{*}}{s_{w}^{2}}\bigg{[}\frac{3(x_{t}^{2}+x_{t})}{8(x_{t}-1)}+\frac{x_{t}-4x_{t}^{2}}{4(x_{t}-1)^{2}}\log
x_{t}+\frac{x_{t}}{4}\log\frac{\mu^{2}}{m_{t}^{2}}\bigg{]}\,,$ (60)

where $x_{t}\equiv m_{t}^{2}/M_{W}^{2}$. Both Eqs. (59) and (60) are in
agreement with the results in Ref. [13]. The divergent part of the one-loop
diagonal field renormalization constant for the down-type quarks (in ’t Hooft-
Feynman gauge) is

$\displaystyle\delta Z_{dd}^{L}$
$\displaystyle=-\frac{e^{2}}{16\pi^{2}\epsilon}\frac{1}{s_{w}^{2}}\bigg{[}V_{td}V_{td}^{*}\frac{x_{t}}{4}+\frac{13}{18}+\frac{1}{36c_{w}^{2}}\bigg{]}\,.$
(61)

The finite part of the one-loop diagonal field renormalization constant for
the down-type quarks (in ’t Hooft-Feynman gauge) is

$\begin{split}\delta
Z_{dd}^{L}&=-\frac{e^{2}}{16\pi^{2}}\frac{V_{td}V_{td}^{*}}{s_{w}^{2}}\bigg{[}\frac{3(x_{t}^{2}+x_{t})}{8(x_{t}-1)}+\frac{x_{t}-4x_{t}^{2}}{4(x_{t}-1)^{2}}\log
x_{t}+\frac{x_{t}}{4}\log\frac{\mu^{2}}{m_{t}^{2}}\bigg{]}\\\
&\quad+\frac{e^{2}}{16\pi^{2}s_{w}^{2}}\bigg{[}\frac{11}{36}+\frac{1}{72c_{w}^{2}}+\frac{c_{w}^{2}}{18}-\bigg{(}(1+c_{w}^{2})\frac{1}{9}+\frac{1}{36c_{w}^{2}}\bigg{)}\log
x-\frac{1}{2}\log x_{t}\bigg{]}\\\
&\quad-\frac{e^{2}}{16\pi^{2}s_{w}^{2}}\bigg{[}\frac{11}{18}+\frac{1}{36c_{w}^{2}}+\frac{c_{w}^{2}}{9}\bigg{]}\log\frac{\mu^{2}}{m_{t}^{2}}\,.\end{split}$
(62)

The finite part differs from the expression in Ref. [13] due to our using
dimensional regularization for the IR divergences.

##  Appendix B Renormalization constants in the effective theory

The evanescent operators necessary for the calculation of the matching
conditions and anomalous dimensions up to two-loop in QCD and QED are chosen
as [10]

$\displaystyle F_{S2}^{(0)}$
$\displaystyle=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu}d_{L}^{\beta}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu}d_{L}^{\alpha}\big{)}-Q_{S2}\,,$
(63) $\displaystyle E_{S2}^{(1)}$
$\displaystyle=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}d_{L}^{\alpha}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}d_{L}^{\beta}\big{)}-(16-a_{11}\epsilon-4\epsilon^{2})Q_{S2}\,,$
(64) $\displaystyle F_{S2}^{(1)}$
$\displaystyle=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}d_{L}^{\beta}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}d_{L}^{\alpha}\big{)}-(16-b_{11}\epsilon-4\epsilon^{2})\big{(}Q_{S2}+F_{S2}^{(0)}\big{)}\,,$
(65)
$\displaystyle\begin{split}E_{S2}^{(2)}&=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}\gamma_{\mu_{4}}\gamma_{\mu_{5}}d_{L}^{\alpha}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}\gamma^{\mu_{4}}\gamma^{\mu_{5}}d_{L}^{\beta}\big{)}\\\
&\quad-\bigg{(}256-a_{21}\epsilon-\frac{108816}{325}\epsilon^{2}\bigg{)}Q_{S2}\,,\end{split}$
(66)
$\displaystyle\begin{split}F_{S2}^{(2)}&=\bigg{(}\frac{\alpha}{4\pi}\bigg{)}^{2}\big{(}\overline{s}_{L}^{\alpha}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}\gamma_{\mu_{4}}\gamma_{\mu_{5}}d_{L}^{\beta}\big{)}\otimes\big{(}\overline{s}_{L}^{\beta}\gamma^{\mu_{1}}\gamma^{\mu_{2}}\gamma^{\mu_{3}}\gamma^{\mu_{4}}\gamma^{\mu_{5}}d_{L}^{\alpha}\big{)}\\\
&\quad-\bigg{(}256-b_{21}\epsilon-\frac{108816}{325}\epsilon^{2}\bigg{)}\big{(}Q_{S2}+F_{S2}^{(0)}\big{)}\,.\end{split}$
(67)

The terms quadratic in $\epsilon$ do not play a role in our calculation and
are kept only for completeness. Some of the coefficients of the $\epsilon$
terms have been left unspecified as an additional check of our calculation.
The result for the Wilson coefficient at NLO in the electroweak interaction
depends on the coefficient $a_{11}$, 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

$Z=1+\sum_{k=0}^{\infty}\Bigg{[}\sum_{i=1}^{\infty}Z^{(i,k)}\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{i}+Z^{(e,k)}\frac{\alpha}{4\pi}+Z^{(se,k)}\frac{\alpha\alpha_{s}}{(4\pi)^{2}}\Bigg{]}\frac{1}{\epsilon^{k}}\,.$
(68)

The necessary, non-zero $Z$ factors at order $\alpha_{s}$ are

$Z_{Q_{S2},Q_{S2}}^{(1,1)}=2\,,\qquad
Z_{Q_{S2},F_{S2}^{(0)}}^{(1,1)}=3\,,\qquad
Z_{Q_{S2},E_{S2}^{(1)}}^{(1,1)}=-\frac{1}{6}\,,\qquad
Z_{Q_{S2},F_{S2}^{(1)}}^{(1,1)}=\frac{1}{2}\,,$ (69)

and

$\begin{split}Z_{F_{S2}^{(0)},Q_{S2}}^{(1,0)}&=2-\frac{5}{12}a_{11}-\frac{1}{12}b_{11}\,,\qquad
Z_{E_{S2}^{(1)},Q_{S2}}^{(1,0)}=-48-\frac{7}{3}a_{11}+13b_{11}+\frac{1}{6}a_{21}-\frac{1}{2}b_{21}\,,\\\
Z_{F_{S2}^{(1)},Q_{S2}}^{(1,0)}&=48+5a_{11}+\frac{65}{3}b_{11}-\frac{1}{4}a_{21}-\frac{7}{12}b_{21}\,.\end{split}$
(70)

The necessary, non-zero $Z$ factors at order $\alpha$ are

$Z_{Q_{S2},Q_{S2}}^{(e,1)}=\frac{2}{3}\,,\qquad
Z_{Q_{S2},E_{S2}^{(1)}}^{(e,1)}=\frac{1}{9}\,,$ (71)

and

$Z_{F_{S2}^{(0)},Q_{S2}}^{(e,0)}=\frac{1}{9}a_{11}-\frac{1}{9}b_{11}\,,\qquad
Z_{E_{S2}^{(1)},Q_{S2}}^{(e,0)}=\frac{32}{9}a_{11}-\frac{1}{9}a_{21}\,,\qquad
Z_{F_{S2}^{(1)},Q_{S2}}^{(e,0)}=\frac{32}{9}b_{11}-\frac{1}{9}b_{21}\,.$ (72)

At two-loop, we find

$Z_{Q_{S2},Q_{S2}}^{(se,1)}=-\frac{61}{9}-\frac{7}{27}a_{11}+\frac{13}{9}b_{11}+\frac{1}{54}a_{21}-\frac{1}{18}b_{21}\,.$
(73)

Using these renormalization constants, we find that all scheme dependence
cancels in the two-loop anomalous dimension at order $\alpha\alpha_{s}$, as
expected on general grounds [28].

##  Appendix C Scheme independence at order $\alpha$

In this appendix we give a proof of the scheme independence of the prediction
of the top-quark contribution to $\epsilon_{K}$ at NLO in QED. We first recall
the transformation properties of Wilson coefficients and anomalous dimension
at order $\alpha$ by adapting the results in Ref. [10]. To this end, we write
the general transformation among all dimension-six operators as

$\begin{pmatrix}Q^{\prime}\\\ E^{\prime}\end{pmatrix}=\begin{pmatrix}R&0\\\
0&M\end{pmatrix}\begin{pmatrix}1&0\\\ \epsilon
U+\epsilon^{2}V&1\end{pmatrix}\begin{pmatrix}1&W\\\
0&1\end{pmatrix}\begin{pmatrix}Q\\\ E\end{pmatrix}\,.$ (74)

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 $\epsilon$ times physical
operators to the evanescent operators. As explained in detail in Ref. [33],
this transformation implies an additional finite renormalization that is
needed in order to restore the standard $\overline{\text{MS}}$ definition of
the renormalization constants. This finite renormalization, induced by the
change (74), is given in the notation of the previous section by

$\displaystyle Z_{QQ}^{\prime(1,0)}$
$\displaystyle=R\left[WZ_{EQ}^{(1,0)}-\left(Z_{QE}^{(1,1)}+WZ_{EE}^{(1,1)}-\frac{1}{2}\gamma^{(0)}W\right)U\right]R^{-1}\,,$
(75) $\displaystyle Z_{QQ}^{\prime(e,0)}$
$\displaystyle=R\left[WZ_{EQ}^{(e,0)}-\left(Z_{QE}^{(e,1)}+WZ_{EE}^{(e,1)}-\frac{1}{2}\gamma^{(e)}W\right)U\right]R^{-1}\,,$
(76)

by a straightforward generalization of the results in Ref. [10], where also
the higher-order QCD expressions can be found. The corresponding
transformation law for the anomalous dimension matrices is then

$\displaystyle\gamma^{\prime(0)}$ $\displaystyle=R\gamma^{(0)}R^{-1}\,,$ (77)
$\displaystyle\gamma^{\prime(e)}$ $\displaystyle=R\gamma^{(e)}R^{-1}\,,$ (78)
$\displaystyle\gamma^{\prime(se)}$
$\displaystyle=R\gamma^{(se)}R^{-1}-\left[Z_{QQ}^{\prime(e,0)},\gamma^{\prime(0)}\right]-\left[Z_{QQ}^{\prime(1,0)},\gamma^{\prime(e)}\right]\,,$
(79)

in agreement with the results given in Ref. [28]. The Wilson coefficients
change according to

$C^{\prime}=\left[1+\frac{\alpha_{s}}{4\pi}Z^{\prime(1,0)}_{QQ}+\frac{\alpha}{4\pi}Z^{\prime(e,0)}_{QQ}\right]^{T}\\!\\!(R^{-1}\big{)}^{T}C\,.$
(80)

In particular, we find

$\displaystyle C^{\prime(e)}$ $\displaystyle=C^{(e)}\,,$ (81) $\displaystyle
C^{\prime(se)}$
$\displaystyle=C^{(se)}+C^{(0)}Z^{\prime(e,0)}_{QQ}+C^{(e)}Z^{\prime(1,0)}_{QQ}\,.$
(82)

The 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 effects of order $\alpha/\alpha_{s}$ and $\alpha$. (The QCD case has
been discussed extensively in the literature, see e.g. Refs. [34, 35, 33]. The
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 as444Of course,
in reality this matrix element needs to be computed using nonperturbative
methods such as lattice QCD. One can think of Eq. (83) as the perturbative
conversion of the lattice result to the $\overline{\text{MS}}$ scheme.

$\langle
Q\rangle(\mu)=\bigg{(}1+\frac{\alpha_{s}(\mu)}{4\pi}r^{(1)}+\frac{\alpha}{4\pi}r^{(se)}\bigg{)}\langle
Q\rangle^{(0)}\,.$ (83)

Recalling the perturbative expansion of the RG evolution matrix, Eq. (46), and
the expansion in powers of couplings of the low-energy Wilson coefficient, Eq.
(47), the terms at order $\alpha/\alpha_{s}$ and $\alpha$ in the amplitude are
then (we drop obvious function arguments)

$\displaystyle\langle CUQ\rangle^{(e)}$
$\displaystyle=\frac{\alpha(\mu)}{\alpha_{s}(\mu)}\bigg{(}C^{(0)}U^{(e)}+\eta^{-1}C^{(e)}U^{(0)}\bigg{)}\langle
Q\rangle^{(0)}\,,$ (84) $\displaystyle\begin{split}\langle
CUQ\rangle^{(se)}&=\alpha(\mu)\bigg{(}U^{(e)}\big{[}\eta
C^{(1)}+C^{(0)}r^{(1)}\big{]}+\eta^{-1}C^{(e)}\big{[}U^{(1)}+U^{(0)}r^{(1)}\big{]}\\\
&\hskip
40.00006pt+C^{(0)}U^{(se)}+C^{(se)}U^{(0)}+C^{(0)}U^{(0)}r^{(se)}\bigg{)}\langle
Q\rangle^{(0)}\,.\end{split}$ (85)

It is apparent that the order $\alpha/\alpha_{s}$ contribution is scheme
independent, as it involves only scheme-independent quantities. In the order
$\alpha$ contribution, the QCD scheme dependence cancels within the square
brackets in the first line, respectively, as shown in Ref. [34]. In the second
line, note that the first term is scheme independent. The renormalized matrix
element $r^{(se)}$ is given by555We assume that parameter and field
renormalizations have been performed in the usual way.

$r^{(se)}=r^{(se),\text{bare}}+Z^{(e)}\,.$ (86)

According to the discussion at the beginning of this section, under a change
of renormalization scheme with nonzero coefficient $U$ in Eq. (74), this
contribution to the matrix element transforms as $r^{(se)}\to
r^{(se)}-Z^{\prime(e)}$. This finishes the proof of the scheme independence of
the matrix element.

Note that for the cancelation to work, we need to fix the electromagnetic
coupling in the effective theory to its value at $\mu=M_{Z}$, i.e.
$\alpha=\alpha(M_{Z})$. This is consistent with the fixed-order perturbative
expansion in $\alpha$.

##  Appendix D The full two-loop result

$\begin{split}C_{S2}^{tt,(se)}&=\big{[}128y-896yz-384xyz+288xz^{2}+2752yz^{2}+1504xyz^{2}+448x^{2}yz^{2}\\\
&\qquad-1080x^{2}z^{3}-7536xyz^{3}+872x^{2}yz^{3}-256x^{3}yz^{3}-576y^{2}z^{3}+144xz^{4}\\\
&\qquad-864x^{3}z^{4}+608yz^{4}-1240xyz^{4}+1564x^{2}yz^{4}-1664x^{3}yz^{4}+64x^{4}yz^{4}\\\
&\qquad+2016xy^{2}z^{4}+288y^{3}z^{4}-540x^{2}z^{5}+6408x^{4}z^{5}-1632xyz^{5}+4740x^{2}yz^{5}\\\
&\qquad-10694x^{3}yz^{5}+184x^{4}yz^{5}-3438x^{2}y^{2}z^{5}-720xy^{3}z^{5}+432x^{3}z^{6}\\\
&\qquad-5184x^{5}z^{6}+2117x^{2}yz^{6}-1668x^{3}yz^{6}-4050x^{4}yz^{6}+1872x^{3}y^{2}z^{6}\\\
&\qquad+765x^{2}y^{3}z^{6}-36x^{4}z^{7}+432x^{6}z^{7}-373x^{3}yz^{7}+364x^{4}yz^{7}\\\
&\qquad+1080x^{5}yz^{7}-198x^{4}y^{2}z^{7}-225x^{3}y^{3}z^{7}\big{]}/\big{[}288y(z-1)z(xz-1)^{3}\big{]}\\\
&\quad+\pi^{2}(-16x+128xz+16x^{2}z-404xz^{2}-24x^{2}z^{2}+8x^{3}z^{2}+472xz^{3}-156x^{2}z^{3}\\\
&\qquad\quad-8x^{3}z^{3}+32z^{4}-32xz^{4}+148x^{2}z^{4}-134x^{3}z^{4}+48x^{4}z^{4}+8x^{5}z^{4}-16z^{5}\\\
&\qquad\quad+64xz^{5}-24x^{2}z^{5}+44x^{3}z^{5}-24x^{4}z^{5}+8x^{5}z^{5}+2z^{6}-14xz^{6}+2x^{2}z^{6}\\\
&\qquad\quad-96x^{4}z^{6}+2x^{5}z^{6}-18x^{2}y^{2}z^{6}-6x^{2}z^{7}-108x^{5}z^{7}-x^{2}z^{8})/\big{[}216x^{2}(z-1)z^{3}\big{]}\\\
&\quad+\Phi\bigg{(}\frac{1}{4x}\bigg{)}(-256z+2816xz-10624x^{2}z+15232x^{3}z-8192x^{4}z-2048x^{5}z\\\
&\qquad\qquad\qquad+128z^{2}-512xz^{2}-3584x^{2}z^{2}+21888x^{3}z^{2}-27520x^{4}z^{2}+11264x^{5}z^{2}\\\
&\qquad\qquad\qquad+10240x^{6}z^{2}-16z^{3}-32xz^{3}+1304x^{2}z^{3}-4752x^{3}z^{3}+2928x^{4}z^{3}\\\
&\qquad\qquad\qquad-12704x^{5}z^{3}+1024x^{6}z^{3}-14848x^{7}z^{3}+32xz^{4}-1056x^{2}z^{4}+5936x^{3}z^{4}\\\
&\qquad\qquad\qquad-4640x^{4}z^{4}-27952x^{5}z^{4}+72256x^{6}z^{4}-3840x^{7}z^{4}+7168x^{8}z^{4}\\\
&\qquad\qquad\qquad-8x^{2}z^{5}+2304x^{3}z^{5}-19252x^{4}z^{5}+50320x^{5}z^{5}-35164x^{6}z^{5}\\\
&\qquad\qquad\qquad-40352x^{7}z^{5}+19456x^{8}z^{5}-512x^{9}z^{5}-1532x^{4}z^{6}+11756x^{5}z^{6}\\\
&\qquad\qquad\qquad-23860x^{6}z^{6}+4184x^{7}z^{6}+13696x^{8}z^{6}-2816x^{9}z^{6}+41x^{4}z^{7}\\\
&\qquad\qquad\qquad-198x^{5}z^{7}-250x^{6}z^{7}+1612x^{7}z^{7}-252x^{8}z^{7}-1280x^{9}z^{7})\\\
&\qquad\qquad\qquad/(576x^{2}(4x-1)(z-1)(xz-1)^{4})\\\
&\quad+\Phi\bigg{(}\frac{y}{4x}\bigg{)}(-176x^{2}z^{3}+256xyz^{3}-56y^{2}z^{3}+656x^{3}z^{4}-1088x^{2}yz^{4}+304xy^{2}z^{4}\\\
&\qquad\qquad\qquad-16y^{3}z^{4}-1164x^{4}z^{5}+1740x^{3}yz^{5}-564x^{2}y^{2}z^{5}+48xy^{3}z^{5}+728x^{5}z^{6}\\\
&\qquad\qquad\qquad-948x^{4}yz^{6}+324x^{3}y^{2}z^{6}-32x^{2}y^{3}z^{6}-44x^{6}z^{7}+112x^{5}yz^{7}\\\
&\qquad\qquad\qquad-66x^{4}y^{2}z^{7}+14x^{3}y^{3}z^{7}-x^{2}y^{4}z^{7})/(64(z-1)(xz-1)^{4})\\\
&\quad+\Phi\bigg{(}\frac{z}{4}\bigg{)}(256x^{2}z+7872x^{2}z^{2}-512x^{3}z^{2}-23072x^{2}z^{3}-17216x^{3}z^{3}+256x^{4}z^{3}\\\
&\qquad\qquad\qquad+5664x^{2}z^{4}-1680x^{3}z^{4}+2048x^{4}z^{4}+5688x^{2}z^{5}-6240x^{3}z^{5}\\\
&\qquad\qquad\qquad-2768x^{4}z^{5}-1222x^{2}z^{6}-1636x^{3}z^{6}+2032x^{4}z^{6}-73x^{2}z^{7}\\\
&\qquad\qquad\qquad+900x^{3}z^{7}+322x^{4}z^{7}+32x^{3}z^{8}-208x^{4}z^{8}-8x^{4}z^{9})\\\
&\qquad\qquad\qquad/(576(z-1)(xz-1)^{4})\\\ \end{split}$
$\begin{split}&\quad+\Phi\bigg{(}\frac{yz}{4}\bigg{)}(144x^{2}z^{3}+48x^{3}z^{4}-224x^{2}yz^{4}-192x^{4}z^{5}+336x^{3}yz^{5}+132x^{2}y^{2}z^{5}\\\
&\qquad\qquad\qquad-88x^{4}yz^{6}-228x^{3}y^{2}z^{6}-18x^{2}y^{3}z^{6}+98x^{4}y^{2}z^{7}+36x^{3}y^{3}z^{7}\\\
&\qquad\qquad\qquad-x^{2}y^{4}z^{7}-16x^{4}y^{3}z^{8})/(64(z-1)(xz-1)^{4})\\\
&\quad+\Phi\bigg{(}\frac{1}{xz},\frac{1}{x}\bigg{)}(-64+576z+384xz-2128z^{2}-2848xz^{2}-928x^{2}z^{2}+3824z^{3}\\\
&\qquad\qquad\qquad\quad+8704xz^{3}+5664x^{2}z^{3}+1120x^{3}z^{3}-3312z^{4}-12752xz^{4}\\\
&\qquad\qquad\qquad\quad-14056x^{2}z^{4}-5816x^{3}z^{4}-672x^{4}z^{4}+992z^{5}+9472xz^{5}\\\
&\qquad\qquad\qquad\quad+14904x^{2}z^{5}+11672x^{3}z^{5}+3544x^{4}z^{5}+224x^{5}z^{5}+272z^{6}\\\
&\qquad\qquad\qquad\quad-3032xz^{6}-6904x^{2}z^{6}-9866x^{3}z^{6}-6800x^{4}z^{6}-1840x^{5}z^{6}\\\
&\qquad\qquad\qquad\quad-224x^{6}z^{6}-144z^{7}-424xz^{7}+2232x^{2}z^{7}+4678x^{3}z^{7}\\\
&\qquad\qquad\qquad\quad+6914x^{4}z^{7}+4536x^{5}z^{7}+1104x^{6}z^{7}+288x^{7}z^{7}-16z^{8}\\\
&\qquad\qquad\qquad\quad+456xz^{8}-592x^{2}z^{8}-607x^{3}z^{8}+6444x^{4}z^{8}+368x^{5}z^{8}\\\
&\qquad\qquad\qquad\quad-2664x^{6}z^{8}-408x^{7}z^{8}-160x^{8}z^{8}+40xz^{9}-304x^{2}z^{9}\\\
&\qquad\qquad\qquad\quad-1787x^{3}z^{9}+2891x^{4}z^{9}+2268x^{5}z^{9}-2720x^{6}z^{9}\\\
&\qquad\qquad\qquad\quad+1000x^{7}z^{9}-8x^{8}z^{9}+32x^{9}z^{9}-16x^{2}z^{10}+565x^{3}z^{10}\\\
&\qquad\qquad\qquad\quad-1766x^{4}z^{10}+2963x^{5}z^{10}-2524x^{6}z^{10}+1018x^{7}z^{10}\\\
&\qquad\qquad\qquad\quad-272x^{8}z^{10}+32x^{9}z^{10}+41x^{3}z^{11}-403x^{4}z^{11}\\\
&\qquad\qquad\qquad\quad+1093x^{5}z^{11}-1279x^{6}z^{11}+694x^{7}z^{11}-154x^{8}z^{11}+8x^{9}z^{11})\\\
&\qquad\qquad\qquad\quad/(288x^{2}(z-1)z^{4}(xz-1)^{4})\\\
&\quad+\Phi\bigg{(}\frac{1}{xz},\frac{y}{x}\bigg{)}(-16+104xz+64yz-296x^{2}z^{2}-296xyz^{2}-96y^{2}z^{2}\\\
&\qquad\qquad\qquad\quad+476x^{3}z^{3}+544x^{2}yz^{3}+344xy^{2}z^{3}+64y^{3}z^{3}-436x^{4}z^{4}\\\
&\qquad\qquad\qquad\quad-568x^{3}yz^{4}-416x^{2}y^{2}z^{4}-216xy^{3}z^{4}-16y^{4}z^{4}+160x^{5}z^{5}\\\
&\qquad\qquad\qquad\quad+326x^{4}yz^{5}+271x^{3}y^{2}z^{5}+192x^{2}y^{3}z^{5}+64xy^{4}z^{5}+80x^{6}z^{6}\\\
&\qquad\qquad\qquad\quad-106x^{5}yz^{6}-185x^{4}y^{2}z^{6}+19x^{3}y^{3}z^{6}-88x^{2}y^{4}z^{6}-100x^{7}z^{7}\\\
&\qquad\qquad\qquad\quad+114x^{6}yz^{7}-31x^{5}y^{2}z^{7}-24x^{4}y^{3}z^{7}+41x^{3}y^{4}z^{7}+28x^{8}z^{8}\\\
&\qquad\qquad\qquad\quad-78x^{7}yz^{8}+65x^{6}y^{2}z^{8}-7x^{5}y^{3}z^{8}-9x^{4}y^{4}z^{8}+x^{3}y^{5}z^{8})\\\
&\qquad\qquad\qquad\quad/(32x^{2}(z-1)z(xz-1)^{4})\\\
&\quad+\text{Li}_{2}\bigg{(}1-\frac{1}{z}\bigg{)}(16-112z-48xz+292z^{2}+232xz^{2}+40x^{2}z^{2}-260z^{3}\\\
&\qquad\qquad\qquad\quad-420xz^{3}-128x^{2}z^{3}+16z^{4}+236xz^{4}+114x^{2}z^{4}-40x^{3}z^{4}\\\
&\qquad\qquad\qquad\quad+44z^{5}-64xz^{5}+186x^{2}z^{5}+72x^{3}z^{5}+4z^{6}-42xz^{6}+8x^{2}z^{6}\\\
&\qquad\qquad\qquad\quad-155x^{3}z^{6}-2xz^{7}-54x^{2}z^{7}+31x^{3}z^{7}-4x^{2}z^{8}+36x^{3}z^{8}+2x^{3}z^{9})\\\
&\qquad\qquad\qquad\quad/(36xz^{3}(xz-1)^{2})\\\ \end{split}$
$\begin{split}&\quad+\text{Li}_{2}\big{(}1-yz\big{)}(4-14xz-12yz+20x^{2}z^{2}+24xyz^{2}+12y^{2}z^{2}-13x^{3}z^{3}\\\
&\qquad\qquad\qquad\qquad-12x^{2}yz^{3}-18xy^{2}z^{3}-4y^{3}z^{3}+6x^{3}yz^{4}-6x^{2}y^{2}z^{4}+8xy^{3}z^{4}\\\
&\qquad\qquad\qquad\qquad+9x^{3}y^{2}z^{5}-2x^{2}y^{3}z^{5}-2x^{3}y^{3}z^{6})/(4x(z-1)(xz-1)^{2})\\\
&\quad+\text{Li}_{2}\big{(}1-xz\big{)}(-32+256z+128xz-808z^{2}-816xz^{2}-176x^{2}z^{2}+944z^{3}\\\
&\qquad\qquad\qquad\qquad+2112xz^{3}+896x^{2}z^{3}+80x^{3}z^{3}-2504xz^{4}-1788x^{2}z^{4}\\\
&\qquad\qquad\qquad\qquad-396x^{3}z^{4}+1968x^{2}z^{5}+616x^{3}z^{5}+164x^{4}z^{5}+32x^{5}z^{5}\\\
&\qquad\qquad\qquad\qquad-3x^{3}z^{6}-644x^{4}z^{6}-164x^{5}z^{6}-48x^{6}z^{6}-375x^{4}z^{7}+332x^{5}z^{7}\\\
&\qquad\qquad\qquad\qquad+44x^{6}z^{7}+16x^{7}z^{7}+25x^{5}z^{8}-112x^{6}z^{8}+16x^{7}z^{8}+449x^{6}z^{9}\\\
&\qquad\qquad\qquad\qquad+4x^{7}z^{9}-216x^{7}z^{10})/(72x(z-1)z^{3}(xz-1)^{3})\\\
&\quad+\text{Li}_{2}\bigg{(}1-\frac{x}{y}\bigg{)}(9x^{2}z^{3}-18xyz^{3}+9y^{2}z^{3}-20x^{3}z^{4}+42x^{2}yz^{4}-24xy^{2}z^{4}\\\
&\qquad\qquad\qquad\qquad+2y^{3}z^{4}+8x^{4}z^{5}-18x^{3}yz^{5}+12x^{2}y^{2}z^{5}-2xy^{3}z^{5})\\\
&\qquad\qquad\qquad\qquad/(4(z-1)(xz-1)^{2})\\\
&\quad+\text{Li}_{2}\big{(}1-x\big{)}(-64z+192xz-160x^{2}z+16x^{3}z+16x^{5}z+32z^{2}-32xz^{2}\\\
&\qquad\qquad\qquad\qquad-72x^{2}z^{2}-16x^{3}z^{2}+200x^{4}z^{2}-96x^{5}z^{2}-16x^{6}z^{2}-4z^{3}\\\
&\qquad\qquad\qquad\qquad-2xz^{3}+141x^{2}z^{3}-256x^{3}z^{3}+165x^{4}z^{3}-84x^{5}z^{3}+32x^{6}z^{3}\\\
&\qquad\qquad\qquad\qquad+8x^{7}z^{3}+4xz^{4}-98x^{2}z^{4}+136x^{3}z^{4}-6x^{4}z^{4}-16x^{5}z^{4}\\\
&\qquad\qquad\qquad\qquad-28x^{6}z^{4}+8x^{7}z^{4}+84x^{3}z^{5}-188x^{4}z^{5}+126x^{5}z^{5}\\\
&\qquad\qquad\qquad\qquad-24x^{6}z^{5}+2x^{7}z^{5})/(36x^{2}(z-1)(xz-1)^{2})\\\
&\quad+\log^{2}(x)(256yz-1280xyz+1408x^{2}yz-128x^{3}yz-128x^{5}yz-128yz^{2}\\\
&\qquad\qquad\qquad-512xyz^{2}+4096x^{2}yz^{2}-4352x^{3}yz^{2}-1216x^{4}yz^{2}+768x^{5}yz^{2}\\\
&\qquad\qquad\qquad+512x^{6}yz^{2}+16yz^{3}+256xyz^{3}-376x^{2}yz^{3}-2688x^{3}yz^{3}\\\
&\qquad\qquad\qquad+3776x^{4}yz^{3}+8160x^{5}yz^{3}-2560x^{6}yz^{3}-832x^{7}yz^{3}+1296x^{3}y^{2}z^{3}\\\
&\qquad\qquad\qquad-504x^{2}y^{3}z^{3}-48xyz^{4}+736x^{2}yz^{4}+72x^{3}yz^{4}-3904x^{4}yz^{4}\\\
&\qquad\qquad\qquad-7296x^{5}yz^{4}-12608x^{6}yz^{4}+3008x^{7}yz^{4}+704x^{8}yz^{4}-6480x^{4}y^{2}z^{4}\\\
&\qquad\qquad\qquad+2952x^{3}y^{3}z^{4}-144x^{2}y^{4}z^{4}-5184x^{7}z^{5}+40x^{2}yz^{5}-3120x^{3}yz^{5}\\\
&\qquad\qquad\qquad+6228x^{4}yz^{5}+968x^{5}yz^{5}+13272x^{6}yz^{5}+6000x^{7}yz^{5}-1344x^{8}yz^{5}\\\
&\qquad\qquad\qquad-320x^{9}yz^{5}+13284x^{5}y^{2}z^{5}-6660x^{4}y^{3}z^{5}+576x^{3}y^{4}z^{5}\\\
&\qquad\qquad\qquad+10368x^{8}z^{6}-8x^{3}yz^{6}+3788x^{4}yz^{6}-8232x^{5}yz^{6}+2092x^{6}yz^{6}\\\
&\qquad\qquad\qquad-2760x^{7}yz^{6}+48x^{8}yz^{6}+64x^{9}yz^{6}+64x^{10}yz^{6}-12528x^{6}y^{2}z^{6}\\\
&\qquad\qquad\qquad+6552x^{5}y^{3}z^{6}-720x^{4}y^{4}z^{6}-5184x^{9}z^{7}-41x^{4}yz^{7}-1580x^{5}yz^{7}\\\
&\qquad\qquad\qquad+2772x^{6}yz^{7}+804x^{7}yz^{7}-5584x^{8}yz^{7}-272x^{9}yz^{7}+64x^{10}yz^{7}\\\
&\qquad\qquad\qquad+4824x^{7}y^{2}z^{7}-2736x^{6}y^{3}z^{7}+396x^{5}y^{4}z^{7}-9x^{4}y^{5}z^{7}+41x^{5}yz^{8}\\\
\end{split}$
$\begin{split}&\qquad\qquad\qquad+48x^{6}yz^{8}-208x^{7}yz^{8}-88x^{8}yz^{8}+3696x^{9}yz^{8}+16x^{10}yz^{8}\\\
&\qquad\qquad\qquad-396x^{8}y^{2}z^{8}+396x^{7}y^{3}z^{8}-108x^{6}y^{4}z^{8}+9x^{5}y^{5}z^{8}-864x^{10}yz^{9})\\\
&\qquad\qquad\qquad/(576x^{2}y(z-1)(xz-1)^{5})\\\
&\quad+\log^{2}(y)(9x^{2}z^{3}-18xyz^{3}+7y^{2}z^{3}-20x^{3}z^{4}+42x^{2}yz^{4}-20xy^{2}z^{4}+2y^{3}z^{4}\\\
&\qquad\qquad\qquad+8x^{4}z^{5}-18x^{3}yz^{5}+10x^{2}y^{2}z^{5}-2xy^{3}z^{5})/(8(z-1)(xz-1)^{2})\\\
&\quad+\log(z)(128y-832yz-448xyz+1776yz^{2}+2976xyz^{2}+512x^{2}yz^{2}-192yz^{3}\\\
&\qquad\qquad\qquad-6256xyz^{3}-4784x^{2}yz^{3}-160x^{3}yz^{3}-144y^{2}z^{3}+216xz^{4}\\\
&\qquad\qquad\qquad-648x^{3}z^{4}-16yz^{4}+232xyz^{4}+10336x^{2}yz^{4}+4032x^{3}yz^{4}\\\
&\qquad\qquad\qquad-64x^{4}yz^{4}+684xy^{2}z^{4}-1026x^{2}z^{5}+3888x^{4}z^{5}+40xyz^{5}\\\
&\qquad\qquad\qquad+268x^{2}yz^{5}-5374x^{3}yz^{5}-1576x^{4}yz^{5}+32x^{5}yz^{5}-1134x^{2}y^{2}z^{5}\\\
&\qquad\qquad\qquad+1566x^{3}z^{6}-7128x^{5}z^{6}-491x^{2}yz^{6}-1668x^{3}yz^{6}+9007x^{4}yz^{6}\\\
&\qquad\qquad\qquad+184x^{5}yz^{6}+1377x^{3}y^{2}z^{6}-171x^{2}y^{3}z^{6}-810x^{4}z^{7}+4104x^{6}z^{7}\\\
&\qquad\qquad\qquad+203x^{3}yz^{7}+340x^{4}yz^{7}-1938x^{5}yz^{7}-648x^{4}y^{2}z^{7}+189x^{3}y^{3}z^{7}\\\
&\qquad\qquad\qquad+54x^{5}z^{8}-216x^{7}z^{8}-96x^{4}yz^{8}-78x^{5}yz^{8}+567x^{6}yz^{8}\\\
&\qquad\qquad\qquad+27x^{5}y^{2}z^{8}-72x^{4}y^{3}z^{8})/(144y(z-1)z(xz-1)^{4})\\\
&\quad+\log^{2}(z)(-288y+1872xyz-560x^{3}yz+864y^{2}z-5328x^{2}yz^{2}+1424x^{3}yz^{2}\\\
&\qquad\qquad\qquad+2096x^{4}yz^{2}+128x^{5}yz^{2}-4320xy^{2}z^{2}-864y^{3}z^{2}+2348x^{3}yz^{3}\\\
&\qquad\qquad\qquad-2672x^{4}yz^{3}-3168x^{5}yz^{3}-448x^{6}yz^{3}+8640x^{2}y^{2}z^{3}+3888xy^{3}z^{3}\\\
&\qquad\qquad\qquad+288y^{4}z^{3}+8248x^{3}yz^{4}-5916x^{4}yz^{4}+2400x^{5}yz^{4}+2464x^{6}yz^{4}\\\
&\qquad\qquad\qquad+576x^{7}yz^{4}-9864x^{3}y^{2}z^{4}-6048x^{2}y^{3}z^{4}-1440xy^{4}z^{4}+1296x^{4}z^{5}\\\
&\qquad\qquad\qquad-5184x^{6}z^{5}+2806x^{3}yz^{5}-16040x^{4}yz^{5}+20288x^{5}yz^{5}-2928x^{6}yz^{5}\\\
&\qquad\qquad\qquad-944x^{7}yz^{5}-320x^{8}yz^{5}+7920x^{4}y^{2}z^{5}+3618x^{3}y^{3}z^{5}+2736x^{2}y^{4}z^{5}\\\
&\qquad\qquad\qquad-2592x^{5}z^{6}+10368x^{7}z^{6}-1368x^{3}yz^{6}-2450x^{4}yz^{6}+9708x^{5}yz^{6}\\\
&\qquad\qquad\qquad-11280x^{6}yz^{6}+2240x^{7}yz^{6}+48x^{8}yz^{6}+64x^{9}yz^{6}-4212x^{5}y^{2}z^{6}\\\
&\qquad\qquad\qquad-270x^{4}y^{3}z^{6}-2484x^{3}y^{4}z^{6}+1296x^{6}z^{7}-5184x^{8}z^{7}-73x^{3}yz^{7}\\\
&\qquad\qquad\qquad+2332x^{4}yz^{7}-498x^{5}yz^{7}-1916x^{6}yz^{7}-3060x^{7}yz^{7}-480x^{8}yz^{7}\\\
&\qquad\qquad\qquad+64x^{9}yz^{7}+972x^{6}y^{2}z^{7}-378x^{5}y^{3}z^{7}+1080x^{4}y^{4}z^{7}-9x^{3}y^{5}z^{7}\\\
&\qquad\qquad\qquad+105x^{4}yz^{8}-1188x^{5}yz^{8}+142x^{6}yz^{8}+3524x^{8}yz^{8}+16x^{9}yz^{8}\\\
&\qquad\qquad\qquad+54x^{6}y^{3}z^{8}-180x^{5}y^{4}z^{8}+9x^{4}y^{5}z^{8}-40x^{5}yz^{9}+224x^{6}yz^{9}\\\
&\qquad\qquad\qquad-864x^{9}yz^{9}+8x^{6}yz^{10})/(576xy(z-1)(xz-1)^{5})\\\
&\quad+\log\bigg(\frac{\mu}{M_{W}}\bigg{missing})((432xz-1408xyz-1620x^{2}z^{2}+1072xyz^{2}+5280x^{2}yz^{2}\\\
&\qquad\qquad\qquad\quad+216xz^{3}+432x^{3}z^{3}-312xyz^{3}-3732x^{2}yz^{3}-4224x^{3}yz^{3}\\\
&\qquad\qquad\qquad\quad+108xy^{2}z^{3}-810x^{2}z^{4}+3132x^{4}z^{4}+1323x^{2}yz^{4}-312x^{3}yz^{4}\\\
&\qquad\qquad\qquad\quad+352x^{4}yz^{4}-405x^{2}y^{2}z^{4}+648x^{3}z^{5}-2592x^{5}z^{5}-171x^{3}yz^{5}\\\
&\qquad\qquad\qquad\quad+137x^{4}yz^{5}+324x^{3}y^{2}z^{5}-54x^{4}z^{6}+216x^{6}z^{6}+78x^{4}yz^{6}\\\
&\qquad\qquad\qquad\quad-135x^{5}yz^{6}-27x^{4}y^{2}z^{6})/(72y(z-1)(xz-1)^{3})\\\
\end{split}$
$\begin{split}&\qquad\qquad\qquad\quad+(108x^{3}z^{3}-352x^{3}yz^{3}-108x^{4}z^{4}+436x^{3}yz^{4}+352x^{4}yz^{4}\\\
&\qquad\qquad\qquad\quad+54x^{3}z^{5}-216x^{5}z^{5}-180x^{3}yz^{5}+113x^{4}yz^{5}+27x^{3}y^{2}z^{5}\\\
&\qquad\qquad\qquad\quad-54x^{4}z^{6}+216x^{6}z^{6}+27x^{4}yz^{6}-54x^{5}yz^{6}-27x^{4}y^{2}z^{6})\\\
&\qquad\qquad\qquad\quad\times\log(xz)/(12y(z-1)(xz-1)^{4})\\\
&\quad+\log(y)((-16yz^{2}+40xyz^{3}+16y^{2}z^{3}-30x^{2}yz^{4}-40xy^{2}z^{4}+7x^{3}yz^{5}\\\
&\qquad\qquad\qquad+43x^{2}y^{2}z^{5}-x^{4}yz^{6}-13x^{3}y^{2}z^{6})/(16(z-1)(xz-1)^{3})\\\
&\qquad\qquad\qquad+(16-88xz-48yz+208x^{2}z^{2}+192xyz^{2}+48y^{2}z^{2}-268x^{3}z^{3}\\\
&\qquad\qquad\qquad-288x^{2}yz^{3}-168xy^{2}z^{3}-16y^{3}z^{3}+168x^{4}z^{4}+284x^{3}yz^{4}\\\
&\qquad\qquad\qquad+168x^{2}y^{2}z^{4}+64xy^{3}z^{4}+8x^{5}z^{5}-210x^{4}yz^{5}-39x^{3}y^{2}z^{5}\\\
&\qquad\qquad\qquad-88x^{2}y^{3}z^{5}-72x^{6}z^{6}+120x^{5}yz^{6}-48x^{4}y^{2}z^{6}+58x^{3}y^{3}z^{6}\\\
&\qquad\qquad\qquad+28x^{7}z^{7}-50x^{6}yz^{7}+21x^{5}y^{2}z^{7}-12x^{4}y^{3}z^{7})\\\
&\qquad\qquad\qquad\times\log(z)/(32x(z-1)(xz-1)^{4}))\\\
&\quad+\log(x)((-64y+256xy+448yz-1600xyz-768x^{2}yz-1248yz^{2}+2528xyz^{2}\\\
&\qquad\qquad\qquad+6688x^{2}yz^{2}+384x^{3}yz^{2}-64yz^{3}+5360xyz^{3}-10944x^{2}yz^{3}\\\
&\qquad\qquad\qquad-13504x^{3}yz^{3}+768x^{4}yz^{3}+288y^{2}z^{3}-1152xy^{2}z^{3}+648x^{3}z^{4}\\\
&\qquad\qquad\qquad-2592x^{4}z^{4}-296yz^{4}+1496xyz^{4}-11792x^{2}yz^{4}+19904x^{3}yz^{4}\\\
&\qquad\qquad\qquad+14048x^{4}yz^{4}-896x^{5}yz^{4}-1188xy^{2}z^{4}+4752x^{2}y^{2}z^{4}-144y^{3}z^{4}\\\
&\qquad\qquad\qquad+576xy^{3}z^{4}-3888x^{4}z^{5}+15552x^{5}z^{5}+1104xyz^{5}-4900x^{2}yz^{5}\\\
&\qquad\qquad\qquad+11174x^{3}yz^{5}-7328x^{4}yz^{5}-6432x^{5}yz^{5}+256x^{6}yz^{5}+1764x^{2}y^{2}z^{5}\\\
&\qquad\qquad\qquad-7056x^{3}y^{2}z^{5}+504xy^{3}z^{5}-2016x^{2}y^{3}z^{5}-108x^{3}z^{6}+432x^{4}z^{6}\\\
&\qquad\qquad\qquad+7128x^{5}z^{6}-28512x^{6}z^{6}-1432x^{2}yz^{6}+7409x^{3}yz^{6}-20447x^{4}yz^{6}\\\
&\qquad\qquad\qquad+33124x^{5}yz^{6}+864x^{6}yz^{6}-1710x^{3}y^{2}z^{6}+6840x^{4}y^{2}z^{6}-576x^{2}y^{3}z^{6}\\\
&\qquad\qquad\qquad+2304x^{3}y^{3}z^{6}+108x^{4}z^{7}-432x^{5}z^{7}-4104x^{6}z^{7}+16416x^{7}z^{7}\\\
&\qquad\qquad\qquad+1099x^{3}yz^{7}-4722x^{4}yz^{7}+5226x^{5}yz^{7}-7528x^{6}yz^{7}+720x^{4}y^{2}z^{7}\\\
&\qquad\qquad\qquad-2880x^{5}y^{2}z^{7}+315x^{3}y^{3}z^{7}-1260x^{4}y^{3}z^{7}+216x^{7}z^{8}-864x^{8}z^{8}\\\
&\qquad\qquad\qquad-115x^{4}yz^{8}+439x^{5}yz^{8}-783x^{6}yz^{8}+2268x^{7}yz^{8}-36x^{5}y^{2}z^{8}\\\
&\qquad\qquad\qquad+144x^{6}y^{2}z^{8}-45x^{4}y^{3}z^{8}+180x^{5}y^{3}z^{8})\\\
&\qquad\qquad\qquad/(144(4x-1)y(z-1)z(xz-1)^{4})\\\
&\qquad\qquad\qquad+(16-88xz-48yz+208x^{2}z^{2}+192xyz^{2}+48y^{2}z^{2}-340x^{3}z^{3}\\\
&\qquad\qquad\qquad-144x^{2}yz^{3}-224xy^{2}z^{3}-16y^{3}z^{3}+472x^{4}z^{4}-340x^{3}yz^{4}\\\
&\qquad\qquad\qquad+440x^{2}y^{2}z^{4}+48xy^{3}z^{4}-448x^{5}z^{5}+750x^{4}yz^{5}-495x^{3}y^{2}z^{5}\\\
&\qquad\qquad\qquad-40x^{2}y^{3}z^{5}+216x^{6}z^{6}-504x^{5}yz^{6}+272x^{4}y^{2}z^{6}+10x^{3}y^{3}z^{6}\\\
&\qquad\qquad\qquad-36x^{7}z^{7}+94x^{6}yz^{7}-59x^{5}y^{2}z^{7}+4x^{4}y^{3}z^{7})\\\
&\qquad\qquad\qquad\times\log(y)/(32x(z-1)(xz-1)^{4})\\\ \end{split}$
$\begin{split}&\qquad\qquad\qquad+((-64y+512yz+384xyz-1616yz^{2}-2656xyz^{2}-928x^{2}yz^{2}\\\
&\qquad\qquad\qquad+2064yz^{3}+7456xyz^{3}+5568x^{2}yz^{3}+1120x^{3}yz^{3}-1104yz^{4}\\\
&\qquad\qquad\qquad-8312xyz^{4}-13640x^{2}yz^{4}-6008x^{3}yz^{4}-672x^{4}yz^{4}+432y^{2}z^{4}\\\
&\qquad\qquad\qquad-112yz^{5}+4512xyz^{5}+11544x^{2}yz^{5}+12608x^{3}yz^{5}+3704x^{4}yz^{5}\\\
&\qquad\qquad\qquad+224x^{5}yz^{5}-2160xy^{2}z^{5}-432y^{3}z^{5}+160yz^{6}+248xyz^{6}\\\
&\qquad\qquad\qquad-6200x^{2}yz^{6}-6630x^{3}yz^{6}-3968x^{4}yz^{6}-1776x^{5}yz^{6}-224x^{6}yz^{6}\\\
&\qquad\qquad\qquad+4320x^{2}y^{2}z^{6}+1944xy^{3}z^{6}+144y^{4}z^{6}+16yz^{7}-640xyz^{7}\\\
&\qquad\qquad\qquad+176x^{2}yz^{7}+5732x^{3}yz^{7}-10218x^{4}yz^{7}-2248x^{5}yz^{7}+1072x^{6}yz^{7}\\\
&\qquad\qquad\qquad+288x^{7}yz^{7}-4716x^{3}y^{2}z^{7}-3024x^{2}y^{3}z^{7}-720xy^{4}z^{7}+648x^{4}z^{8}\\\
&\qquad\qquad\qquad-5184x^{6}z^{8}-56xyz^{8}+760x^{2}yz^{8}+1197x^{3}yz^{8}-6256x^{4}yz^{8}\\\
&\qquad\qquad\qquad+25032x^{5}yz^{8}+520x^{6}yz^{8}-472x^{7}yz^{8}-160x^{8}yz^{8}+3474x^{4}y^{2}z^{8}\\\
&\qquad\qquad\qquad+1755x^{3}y^{3}z^{8}+1368x^{2}y^{4}z^{8}-1296x^{5}z^{9}+10368x^{7}z^{9}+56x^{2}yz^{9}\\\
&\qquad\qquad\qquad-870x^{3}yz^{9}-53x^{4}yz^{9}+2044x^{5}yz^{9}-11776x^{6}yz^{9}+1120x^{7}yz^{9}\\\
&\qquad\qquad\qquad+24x^{8}yz^{9}+32x^{9}yz^{9}-1242x^{5}y^{2}z^{9}-351x^{4}y^{3}z^{9}-1170x^{3}y^{4}z^{9}\\\
&\qquad\qquad\qquad+648x^{6}z^{10}-5184x^{8}z^{10}-57x^{3}yz^{10}+952x^{4}yz^{10}-2187x^{5}yz^{10}\\\
&\qquad\qquad\qquad+1820x^{6}yz^{10}-2962x^{7}yz^{10}-240x^{8}yz^{10}+32x^{9}yz^{10}\\\
&\qquad\qquad\qquad-558x^{6}y^{2}z^{10}+243x^{5}y^{3}z^{10}+450x^{4}y^{4}z^{10}-9x^{3}y^{5}z^{10}\\\
&\qquad\qquad\qquad+41x^{4}yz^{11}-362x^{5}yz^{11}+731x^{6}yz^{11}-548x^{7}yz^{11}\\\
&\qquad\qquad\qquad+3418x^{8}yz^{11}+8x^{9}yz^{11}+450x^{7}y^{2}z^{11}-135x^{6}y^{3}z^{11}\\\
&\qquad\qquad\qquad-72x^{5}y^{4}z^{11}+9x^{4}y^{5}z^{11}-864x^{9}yz^{12})\\\
&\qquad\qquad\qquad\times\log(z))/(288xy(z-1)z^{3}(xz-1)^{5}))\,.\end{split}$
(87)

Here, $x\equiv m_{t}^{2}/M_{Z}^{2}$, $y\equiv M_{h}^{2}/M_{Z}^{2}$, and
$z\equiv M_{Z}^{2}/M_{w}^{2}$. The functions $\Phi$ are defined in Ref. [17].

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