First order electroweak radiative corrections
to the decay of the polarised $W$ boson
Maria Naeem and Stefan Groote
Füüsika Instituut, Tartu Ülikool, W. Ostwaldi 1, EE-50411 Tartu, Estonia

###### Abstract

Analytic results for the next-to-leading order electroweak radiative
corrections to the decay of the polarised $W$ boson into a pair of heavy
quarks are presented. Finite fermion masses are taken into account throughout
this calculation, and in performing the collinear limit for the final quark
states and all light fermions taking part in this process, the importance of
mass effects is demonstrated. This gives hints for the deviation of the total
decay rate and the branching ratios into massive fermions between experimental
measurements and theoretical predictions.

##  1 Introduction

The $W$ boson as one of the two massive gauge bosons of the electroweak theory
has been subject to intense investigations recently. Besides the mass
discrepancy with the Standard Model (SM) value, challenged by a new
interpretation of old data from 2012 by the CDF collaboration that indicate a
deviation of approximately four standard deviations in 2022 [1], there are
also less profound deviations reported in Ref. [2], for instance in the
measured decay width of the $W$ decay of $\Gamma_{W}=2.137\pm 0.032{\rm\,GeV}$
[3, 4], contrasted with the SM prediction of $\Gamma_{W}=2.0892\pm
0.0008{\rm\,GeV}$ [5, 6] that is $1.5\sigma$ smaller than the measured result.
While measurements of the leptonic branching ratios from ATLAS and CMS are in
good agreement with lepton universality [7, 8], older, less precise data from
LEP 2 indicate a branching ratio fo $W\to\tau+\bar{\nu}_{\tau}$ that is
$2.6\sigma$ larger than the electron–muon average [9]. This and the
experimentally measured larger decay rate might indicate that the difference
is due to mass effects and radiative corrections to the leading order (LO)
Born term result not taken properly into account. In this work we deal with
these issues.

Pairs of $W$ bosons can be produced at lepton or hadron colliders (cf. e.g.
Refs. [10, 11]). However, as the main SM channel for the single $W$ boson is
the decay process $t\to W^{+}+X_{b}$, in Refs. [12, 13, 14, 15, 16] first
order quantum chromodynamics (QCD) and electroweak (EW) radiative corrections
have been calculated in order to obtain next-to-leading order (NLO) results
for the decay processes $t\rightarrow b+W^{+}(\uparrow)$ and the similar
process $b\rightarrow c+W^{-}(\uparrow)$, flipped to the process at hand. Note
that due to the left-handed $V-A$ coupling of the electroweak interaction, the
$W$ boson is polarised [17], and the polarisation can be analysed by looking
at the angular distribution of the decay [18, 19] and can be considered in
$WW$ scattering at the LHC [20, 21]. NLO EW radiative corrections to the
process $W\to\ell^{+}+\nu_{\ell}$ have been calculated in Ref. [22] without
taking into account the polarisation of the decaying particle. This gap was
closed by a master thesis defended at the University of Tartu in 2009 [23].
Taking quark masses into account and using similarities to the process
$Z(\uparrow)\rightarrow q+\bar{q}$ [24, 25, 26], NLO QCD radiative corrections
for the process $W^{+}(\uparrow)\rightarrow Q+\bar{q}$ were calculated in a
master thesis at the University of Tartu in 2010 [27, 28, 29, 30]. In order to
discern from $W$ boson physics beyond the SM [31, 32, 33] and in rare decays
[34, 35] it is inevitable to improve the precision of the SM prediction by
including NLO EW radiative corrections [36] and by taking care of effects from
the masses of the decay products. This is the aim of the present publication.

The central object of interest is the angular dependence of the decay rate for
the polarised $W$ boson, as it can be inferred by looking at Ref [28, 29, 30].
It contains the spin-density matrix $\rho$ that describes the polarisation of
the $W$ boson, the decay functions $H_{mm^{\prime}}$, also known as the
helicity amplitudes and calculated for the decay channel of the $W$ boson, and
the polar angle dependence on $\theta$. The azimuthal angle dependence is
found in the References [12, 14], the decay rate function [30] is

$\displaystyle W(\theta)$ $\displaystyle=$ $\displaystyle\sum_{m=0,\pm
1}\rho_{mm}H_{mm}(\theta)\ =\ \sum_{m,m^{\prime}=0,\pm
1}\rho_{mm}\,d^{1}_{mm^{\prime}}(\theta)\,d^{1}_{mm^{\prime}}(\theta)\,\,H_{m^{\prime}m^{\prime}}\
=$ (1) $\displaystyle=$
$\displaystyle\frac{3}{8}(1+\cos^{2}\theta)\,\Big{(}(\rho_{++}+\rho_{--})\,(H_{++}+H_{--})+2\rho_{00}H_{00}\Big{)}+$
$\displaystyle+\frac{3}{4}\cos\theta\,\Big{(}(\rho_{++}-\rho_{--})\,(H_{++}-H_{--})\Big{)}+$
$\displaystyle+\frac{3}{4}\sin^{2}\theta\,\Big{(}(\rho_{++}+\rho_{--})H_{00}+\rho_{00}(H_{++}+H_{--}-H_{00})\Big{)},$

using the normalized spin density matrix elements $\rho_{mm^{\prime}}$ with
$\rho_{++}+\rho_{00}+\rho_{--}=1$ and the helicity amplitudes
$H_{mm^{\prime}}$ normalised in such a way that $H_{++}+H_{00}+H_{--}=1$ at
the Born term level for massless fermions. While the spin density matrix
elements $\rho_{00}=0.687(5)$, $\rho_{++}=0.0017(1)$ and $\rho_{--}=0.311(5)$
[37] obtained from the production of the $W$ boson in top quark decay are
inputs for our calculation, we have to calculate the helicity amplitudes.

The work is divided up as follows: In Sections 2 and 3 we give the Born term
results and the tree and loop contributions to the EW radiative corrections.
These are incorporated in the helicity amplitudes in Section 4. In Section 5
we give analytical results for the helicity amplitudes, taking also into
account detailed counter terms for the cancellation of IR singularities. A
discussion of the result in terms of the angular dependence of the decay rate
is given in Section 6. Our conclusions are found in Section 7, while detailed
calculations are given in three appendices.

##  2 Tree contributions

For the Born term, the squared absolute value of the matrix element is given
by

$\displaystyle|{\cal M}_{0}|^{2}$ $\displaystyle=$
$\displaystyle\sum_{s_{1},s_{2}}{\cal M}_{0}^{*}{\cal M}_{0}\ =$ (2)
$\displaystyle=$ $\displaystyle\frac{e^{2}}{8s_{W}^{2}}|V_{cb}|^{2}{\rm
tr}\left((p\kern-5.0pt/_{2}-m_{2})\gamma^{\mu}(1-\gamma_{5})(p\kern-5.0pt/_{1}+m_{1})\gamma^{\nu}(1-\gamma_{5})\right)\varepsilon_{\mu}^{*}(q,\lambda)\varepsilon_{\nu}(q,\lambda)\
=$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{s_{W}^{2}}|V_{cb}|^{2}\left[p_{2}^{\mu}p_{1}^{\nu}+p_{2}^{\nu}p_{1}^{\mu}-g^{\mu\nu}(p_{2}p_{1})-p_{2\kappa}p_{1\lambda}i\epsilon^{\kappa\lambda\mu\nu}\right]\varepsilon_{\mu}^{*}(q,\lambda)\varepsilon_{\nu}(q,\lambda),$

with $s_{W}=\sin\theta_{W}$ the sine of the Weinberg angle. For an unpolarised
$W$ boson we sum over the polarisations $\lambda$, and choosing the unitary
gauge $\xi_{W}=\infty$, we obtain [38]

$\sum_{\lambda}\varepsilon_{\mu}^{*}(q,\lambda)\varepsilon_{\nu}(q,\lambda)=-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{W}^{2}}.$
(3)

On mass shell we can use $q^{2}=m_{W}^{2}$. As the $W$ boson is the incoming
particle, instead of summing over the (three) polarisations $+$, $-$ and $0$
we have to calculate the mean value over these. This gives an additional
factor $1/3$ to the decay rate, in compliance with Refs. [5, 30]. Using
$q=p_{1}+p_{2}$, by rearranging accordingly and squaring one obtains

$p_{1}p_{2}=\frac{q^{2}}{2}(1-\mu_{1}-\mu_{2}),\qquad
p_{1}q=\frac{q^{2}}{2}(1+\mu_{1}-\mu_{2}),\qquad
p_{2}q=\frac{q^{2}}{2}(1-\mu_{1}+\mu_{2}),$ (4)

where $m_{1}^{2}=p_{1}^{2}=\mu_{1}q^{2}$ and
$m_{2}^{2}=p_{2}^{2}=\mu_{2}q^{2}$ introduces two dimensionless mass
parameters $\mu_{1}$ and $\mu_{2}$. Therefore, summing over the polarisations
of the final states one has

$\displaystyle\overline{|{\cal M}_{0}|}^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}|V_{cb}|^{2}}{3s_{W}^{2}}\left(p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}-p_{1}p_{2}g^{\mu\nu}+ip_{1\kappa}p_{2\lambda}\epsilon^{\kappa\lambda\mu\nu}\right)\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{W}^{2}}\right)\
=$ (5) $\displaystyle=$
$\displaystyle\frac{e^{2}|V_{cb}|^{2}m_{W}^{2}}{6s_{W}^{2}}\left(2-\mu_{1}-\mu_{2}-(\mu_{1}-\mu_{2})^{2}\right).$

Figure 1: Born-term contribution

###  2.1 The two body decay

In order to calculate the phase space, we have to make explicit the kinematics
of the decay in the rest frame of the $W$ boson. In the two-body decay of the
$W$ boson into up-type quark $Q(p_{1})$ and down-type antiquark
$\bar{q}(p_{2})$, the two quarks are produced back to back. In the quark frame
we take the positive $z$ axis to be the direction of flight of the charm
quark. One obtains

$p_{1}=(E_{1};0,0,|\vec{p}\,|),\qquad p_{2}=(E_{2};0,0,-|\vec{p}\,|),$ (6)

where $p_{1}+p_{2}=q=(\sqrt{q^{2}};0,0,0)$ and

$E_{1}=\frac{1}{2}(1+\mu_{1}-\mu_{2})\sqrt{q^{2}},\quad
E_{2}=\frac{1}{2}(1-\mu_{1}+\mu_{2})\sqrt{q^{2}},\quad|\vec{p}\,|=\frac{1}{2}\sqrt{\lambda(1,\mu_{1},\mu_{2})}\sqrt{q^{2}},$
(7)

with the Källén function given by
$\lambda(a,b,c):=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$. Taking into account that the
axis of flight of the $W$ boson produced in e.g. the dominant process $t\to
W^{+}+b$, this defines another axis. Considering the polarisation of the $W$
boson as important parameter for our analysis, the polar angle $\theta$
between the quark frame and the axis of flight of the $W$ is of importance
here. The rest frame of the $W$ boson with the direction of flight pointing in
positive $z$ direction is called the $W$ frame in the following. The two-
particle phase space depends on the polar angle and is given by

$dPS_{2}=\frac{1}{16\pi q^{2}}\sqrt{\lambda(q^{2},m_{1}^{2},m_{2}^{2})}\
d(\cos\theta)=\frac{1}{16\pi}\sqrt{\lambda(1,\mu_{1},\mu_{2})}\
d(\cos\theta).$ (8)

According to Fermi’s golden rule, for the calculation of the decay rate we
have to combine the squared absolute value of the matrix element and the phase
space factor to obtain

$d\Gamma=\frac{1}{2m_{W}}|{\cal M}|^{2}dPS.$ (9)

After integration over the polar angle, the full unpolarised Born term decay
rate for the process shown in Fig. 1 with an on-shell $W$ boson
($q^{2}=m_{W}^{2}$) is given by

$\Gamma_{0}=\frac{e^{2}|V_{cb}|^{2}m_{W}}{96\pi
s_{W}^{2}}\sqrt{\lambda(1,\mu_{1},\mu_{2})}\left(2-\mu_{1}-\mu_{2}-(\mu_{1}-\mu_{2})^{2}\right).$
(10)

(1)(2)(3)

Figure 2: First order electroweak tree corrections

###  2.2 The three body decay

In this work we are dealing with NLO electroweak corrections. Taking into
account the tree corrections with a real photon emitted, we end up with the
three body decays shown in Fig. 2, with the kinematics shown in Fig. 3. For
the phase space integration it would be appropriate to have the denominator
factors in the easiest shape. For this we define dimensionless quantities
$y_{1}$ and $y_{2}$ by

$\displaystyle y_{1}q^{2}$ $\displaystyle:=$
$\displaystyle(p_{1}+p_{3})^{2}-m_{1}^{2}\ =\
p_{1}^{2}+2p_{1}p_{3}+p_{3}^{2}-m_{1}^{2}\ =\ 2p_{1}p_{3}+p_{3}^{2},$
$\displaystyle y_{2}q^{2}$ $\displaystyle:=$
$\displaystyle(p_{2}+p_{3})^{2}-m_{2}^{2}\ =\
p_{2}^{2}+2p_{2}p_{3}+p_{3}^{2}-m_{2}^{2}\ =\ 2p_{2}p_{3}+p_{3}^{2}$ (11)

with
$q^{2}=(p_{1}+p_{2}+p_{3})^{2}=p_{1}^{2}+2p_{1}p_{2}+2p_{1}p_{3}+p_{2}^{2}+2p_{2}p_{3}+p_{3}^{2}$.
As the integration over the phase space will result in infrared (IR)
divergencies, we use mass regularisation by giving a small mass
$m_{A}^{2}=p_{3}^{2}=\Lambda q^{2}$ to the emitted photon with momentum
$p_{3}$. The scalar products read

$p_{1}p_{2}=\frac{1}{2}\left(1-(\mu_{1}+y_{1})-(\mu_{2}+y_{2})+\Lambda\right)q^{2},\quad
p_{1}p_{3}=\frac{1}{2}(y_{1}-\Lambda)q^{2},\quad
p_{2}p_{3}=\frac{1}{2}(y_{2}-\Lambda)q^{2}.$ (12)

The general ansatz for the kinematics in the quark frame is given by
$p_{1}=(E_{1};0,0,|\vec{p}_{1}|)$,

$p_{2}=(E_{2};|\vec{p}_{2}|\sin\theta_{12},0,|\vec{p}_{2}|\cos\theta_{12}),\qquad
p_{3}=(E_{3};|\vec{p}_{3}|\sin\theta_{13},0,|\vec{p}_{3}|\cos\theta_{13})$
(13)

with

$\displaystyle
E_{1}=\frac{1}{2}\left(1+\mu_{1}-(\mu_{2}+y_{2})\right)\sqrt{q^{2}}$
$\displaystyle|\vec{p}_{1}|=\frac{1}{2}\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\sqrt{q^{2}}$
$\displaystyle
E_{2}=\frac{1}{2}\left(1-(\mu_{1}+y_{1})+\mu_{2}\right)\sqrt{q^{2}}$
$\displaystyle|\vec{p}_{2}|=\frac{1}{2}\sqrt{\lambda(1,\mu_{1}+y_{1},\mu_{2})}\sqrt{q^{2}}$
$\displaystyle E_{3}=\frac{1}{2}(y_{1}+y_{2})\sqrt{q^{2}}$
$\displaystyle|\vec{p}_{3}|=\frac{1}{2}\sqrt{(y_{1}+y_{2})^{2}-4\Lambda}\sqrt{q^{2}}$
(14)

and the cosines and sines of the relative angles given by

$\displaystyle\cos\theta_{12}$ $\displaystyle=$
$\displaystyle\frac{y_{1}y_{2}+(1-\mu_{1}+\mu_{2})y_{1}+(1+\mu_{1}-\mu_{2})y_{2}-\lambda-2\Lambda}{\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\sqrt{\lambda(1,\mu_{1}+y_{1},\mu_{2})}},$
$\displaystyle\sin\theta_{12}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{N(y_{1},y_{2})}}{\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\sqrt{\lambda(1,\mu_{1}+y_{1},\mu_{2})}},$
$\displaystyle\cos\theta_{13}$ $\displaystyle=$
$\displaystyle\frac{-y_{1}(1-\mu_{1}+\mu_{2}+y_{2})+(1+\mu_{1}-\mu_{2}-y_{2})y_{2}+2\Lambda}{\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\sqrt{(y_{1}+y_{2})^{2}-4\Lambda}},$
$\displaystyle\sin\theta_{13}$ $\displaystyle=$
$\displaystyle\frac{-2\sqrt{N(y_{1},y_{2})}}{\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\sqrt{(y_{1}+y_{2})^{2}-4\Lambda}},$
(15)

with

$\displaystyle N(y_{1},y_{2})$ $\displaystyle=$
$\displaystyle(1-\mu_{1}-\mu_{2})y_{1}y_{2}-y_{1}^{2}(\mu_{2}+y_{2})-(\mu_{1}+y_{1})y_{2}^{2}+$
(16)
$\displaystyle+\Lambda\left((1-\mu_{1}+\mu_{2})y_{1}+(1+\mu_{1}-\mu_{2})y_{2}+y_{1}y_{2}-\lambda\right)-\Lambda^{2}.$

The three-body phase space is given by

$dPS_{3}=(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}+p_{3}-q)\prod_{i=1}^{3}\frac{d^{4}p_{i}}{(2\pi)^{4}}(2\pi)\delta(p_{i}^{2}-m_{i}^{2})\theta(p_{i}^{0}).$
(17)

which in the quark frame simplifies to

$dPS_{3}=dPS_{2}\times\frac{q^{2}}{(4\pi)^{2}\sqrt{\lambda(1,\mu_{1},\mu_{2})}}dy_{1}dy_{2},$
(18)

The phase limits in terms of $y_{1}$ and $y_{2}$ read $y_{1-}(y_{2})\leq
y_{1}\leq y_{1+}(y_{2})$ and

$y_{20}:=\Lambda+2\sqrt{\Lambda\mu_{2}}\leq
y_{2}\leq(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}=:y_{2-}.$ (19)

with

$\displaystyle y_{1\pm}(y_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2(\mu_{2}+y_{2})}\Big{(}y_{2}\left(1-\mu_{1}-(\mu_{2}+y_{2})\right)+\Lambda\left(1-\mu_{1}+(\mu_{2}+y_{2})\right)+$
(20)
$\displaystyle\qquad\pm\sqrt{(y_{2}-\Lambda)^{2}-4\Lambda\mu_{2}}\sqrt{\lambda(1,\mu_{1},\mu_{2}+y_{2})}\Big{)}.$

The IR singularity resides in the phase space at $y_{1}=y_{2}=0$ and is
regularised by the parameter $\Lambda$. Details about the calculation of the
phase space integrals and the basic terms contained in the tree contributions
are found in Appendix A.

Figure 3: Kinematics of the three (and two) particle decay

Figure 4: Loop corrections

##  3 Loop contributions

The tree contributions are not the only first order corrections. In addition,
we have to take into account a large number of loop corrections to the vertex
displayed in Fig. 4. The results we obtain from the loop corrections are both
infrared and ultraviolet singular. For the latter, we have to apply the
renormalisation procedure. The result for the vertex correction can in
principle be split up into six contributions according to

$\displaystyle V^{\mu}\ =\
V^{0}_{-}\bar{u}(p_{1})\gamma^{\mu}\Lambda_{-}v(p_{2})+V^{0}_{+}\bar{u}(p_{1})\gamma^{\mu}\Lambda_{+}v(p_{2})+$
$\displaystyle+V^{1}_{-}\bar{u}(p_{1})p_{1}^{\mu}\Lambda_{-}v(p_{2})+V^{1}_{+}\bar{u}(p_{1})p_{1}^{\mu}\Lambda_{+}v(p_{2})+V^{2}_{-}\bar{u}(p_{1})p_{2}^{\mu}\Lambda_{-}v(p_{2})+V^{2}_{+}\bar{u}(p_{1})p_{2}^{\mu}\Lambda_{+}v(p_{2})$

with $\Lambda_{\pm}=(1\pm\gamma_{5})/2$. However, considering only on-shell
$W$ boson decays, the loop corrections are divided up into four form factors
$V_{-}$, $V_{+}$, $V_{1}$ and $V_{2}$ only, defined by

$\displaystyle\frac{\alpha}{4\pi}V_{-}$ $\displaystyle=$
$\displaystyle\mathop{\rm Re}\nolimits(V^{0}_{-}+\delta Z_{\rm CKM}+\delta
Z_{e}-\frac{\delta s_{W}}{s_{W}}+\delta Z_{WW}+\delta Z^{L}_{cc}+\delta
Z^{L}_{bb}),$ $\displaystyle\frac{\alpha}{4\pi}\mu_{1}\mu_{2}V_{+}q^{2}$
$\displaystyle=$ $\displaystyle m_{1}m_{2}\mathop{\rm Re}\nolimits V^{0}_{+},$
$\displaystyle\frac{\alpha}{4\pi}\mu_{1}V_{1}$ $\displaystyle=$ $\displaystyle
m_{1}\mathop{\rm Re}\nolimits(V^{1}_{-}-V^{2}_{-}),$
$\displaystyle\frac{\alpha}{4\pi}\mu_{2}V_{2}$ $\displaystyle=$ $\displaystyle
m_{2}\mathop{\rm Re}\nolimits(V^{1}_{+}-V^{2}_{+}).$ (22)

As will be discussed in Sec. 5.1, IR singularities will be subtracted from
$V_{-}$ to obtain $V_{-}^{*}$. The form factors $V_{-}^{*}$, $V_{+}$, $V_{1}$
and $V_{2}$ are given in Appendix B.

###  3.1 Renormalization of the vertex

As the CKM matrix element $V_{cb}$ dominating the decay $W^{+}\to c\bar{b}$ is
not close to unit, we have to take into account the mixing of quark states
and, related to this, the renormalisation of the mixing matrix [6, 39, 40, 41,
42, 43, 44, 45, 46]. One has

$\displaystyle{\cal M}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{ie}{\sqrt{2}s_{W}}\Bigg{\\{}\gamma^{\mu}\Lambda_{-}\Bigg{[}V_{cb}\left(1+\delta
Z_{e}-\frac{\delta s_{W}}{s_{W}}+\delta Z_{WW}\right)+$ (23)
$\displaystyle\qquad\qquad\qquad+\delta V_{cb}+\sum_{k}(\delta
Z_{ck}^{L*}V_{kb}+V_{ck}\delta
Z_{kb}^{L})\Bigg{]}+V_{cb}V^{\mu}_{b}\Bigg{\\}},$

where the sum runs over corresponding quark flavours. Using

$\displaystyle\delta V_{cb}=\frac{1}{2}\sum_{k}\left((\delta Z_{ck}^{L}-\delta
Z_{ck}^{L*})V_{kb}-V_{ck}(\delta Z_{kb}^{L}-\delta Z_{kb}^{L*})\right),$ (24)

one obtains

$\displaystyle{\cal M}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{ie}{\sqrt{2}s_{W}}\Bigg{\\{}\gamma^{\mu}\Lambda_{-}\Bigg{[}V_{cb}\left(1+\delta
Z_{e}-\frac{\delta s_{W}}{s_{W}}+\delta Z_{WW}+\delta Z^{L}_{cc}+\delta
Z^{L}_{bb}\right)+$ (25) $\displaystyle\qquad\qquad+\mathop{\rm
Re}\nolimits\delta Z_{cu}^{L}V_{ub}+\mathop{\rm Re}\nolimits\delta
Z_{ct}^{L}V_{tb}+V_{cd}\mathop{\rm Re}\nolimits\delta
Z_{db}^{L}+V_{cs}\mathop{\rm Re}\nolimits\delta
Z_{sb}^{L}\Bigg{]}+V_{cb}V^{\mu}_{b}\Bigg{\\}}.\qquad$

Defining $\delta Z_{\rm CKM}:=\left(\mathop{\rm Re}\nolimits\delta
Z_{cu}^{L}V_{ub}+\mathop{\rm Re}\nolimits\delta
Z_{ct}^{L}V_{tb}+V_{cd}\mathop{\rm Re}\nolimits\delta
Z_{db}^{L}+V_{cs}\mathop{\rm Re}\nolimits\delta Z_{sb}^{L}\right)/V_{cb}$, one
ends up with $V_{-}$ as given in Eq. (3).

The UV singular part within $V^{\mu}_{b}$ is given by
$V^{\mu}_{s}=\gamma^{\mu}\Lambda_{-}V^{0}_{s}$ with

$V^{0}_{s}=\frac{e^{2}}{4m_{W}^{2}s_{W}^{2}}\left[m_{c}^{2}+m_{b}^{2}+12m_{W}^{2}-m_{Z}^{2}\left(1-2(Q_{c}-Q_{b}+2Q_{c}Q_{b})s_{W}^{2}\right)\right],$
(26)

while $\delta Z_{es}+\delta Z_{WWs}-\delta s_{Ws}/s_{W}=-2e^{2}/s_{W}^{2}$ and
[47]

$\displaystyle\delta Z^{Ls}_{bb}$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{8m_{W}^{2}s_{W}^{2}}\left[m_{b}^{2}+\sum_{k}|V_{kb}|^{2}(m_{k}^{2}+2m_{W}^{2})+(1+4Q_{c}Q_{b}s_{W}^{2})m_{Z}^{2}\right],$
$\displaystyle\delta Z^{Ls}_{cc}$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{8m_{W}^{2}s_{W}^{2}}\left[m_{c}^{2}+\sum_{k}|V_{ck}|^{2}(m_{k}^{2}+2m_{W}^{2})+(1+4Q_{c}Q_{b}s_{W}^{2})m_{Z}^{2}\right],$
$\displaystyle\delta Z^{Ls}_{ij}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4m_{W}^{2}s_{W}^{2}}\sum_{k}V_{ik}V_{kj}\frac{m_{i}^{2}m_{j}^{2}-2m_{i}^{2}m_{k}^{2}-m_{j}^{2}m_{k}^{2}+2m_{j}^{2}m_{W}^{2}}{m_{i}^{2}-m_{j}^{2}}.$
(27)

Note that according to Eq. (4.5.27) in Ref. [47], one has $\delta
Z_{ij}^{Ls*}=\delta Z_{ij}^{Ls}|_{m_{i}^{2}\leftrightarrow m_{j}^{2}}$.
Therefore,

$\mathop{\rm Re}\nolimits\delta
Z_{ij}^{Ls}=\frac{-e^{2}}{8m_{W}^{2}s_{W}^{2}}\sum_{k}V_{ik}V_{kj}(m_{k}^{2}+2m_{W}^{2}),\qquad
V_{kj}=V_{jk}^{*}.$ (28)

Without the mixed contributions (and without the general factor $V_{cb}$), one
first obtains

$\displaystyle V^{0}_{s}+\delta Z_{es}-\frac{\delta s_{Ws}}{s_{W}}+\delta
Z_{WWs}+\delta Z^{Ls}_{cc}+\delta Z^{Ls}_{bb}\ =$ (29) $\displaystyle=$
$\displaystyle\frac{e^{2}}{8m_{W}^{2}s_{W}^{2}}\left[m_{c}^{2}+m_{b}^{2}+4m_{W}^{2}-\sum_{k}|V_{ck}|^{2}(m_{k}^{2}+2m_{W}^{2})-\sum_{k}|V_{kb}|^{2}(m_{k}^{2}+2m_{W}^{2})\right].\quad$

On the other hand, the mixed part gives

$\displaystyle\mathop{\rm Re}\nolimits\delta Z_{cu}^{Ls}V_{ub}+\mathop{\rm
Re}\nolimits\delta Z_{ct}^{Ls}V_{tb}+V_{cd}\mathop{\rm Re}\nolimits\delta
Z_{db}^{Ls}+V_{cs}\mathop{\rm Re}\nolimits\delta Z_{sb}^{Ls}\ =$
$\displaystyle=$
$\displaystyle\frac{-e^{2}}{8m_{W}^{2}s_{W}^{2}}\sum_{k}\left(V_{ck}V_{ku}V_{ub}+V_{ck}V_{kt}V_{tb}+V_{cd}V_{dk}V_{kb}+V_{cs}V_{sk}V_{kb}\right)(m_{k}^{2}+2m_{W}^{2})\
=$ $\displaystyle=$
$\displaystyle\frac{-e^{2}V_{cb}}{8m_{W}^{2}s_{W}^{2}}\left[m_{b}^{2}+2m_{W}^{2}-\sum_{k}|V_{ck}|^{2}(m_{k}^{2}+2m_{W}^{2})+m_{c}^{2}+2m_{W}^{2}-\sum_{k}|V_{kb}|^{2}(m_{k}^{2}+2m_{W}^{2})\right].\kern-3.0pt$

This cancels exactly against the previous contribution, gaining an UV finite
form factor.

##  4 Helicity amplitudes

The helicity amplitudes are obtained by contracting with the polarisation
vectors $\varepsilon(q,\lambda)$. First we use the helicity basis in the $W$
frame, i.e., the rest frame of the decaying $W$ boson with the $z$ axis as the
initial direction of flight of the $W$ boson, given by

$\varepsilon(q,\pm)=\frac{1}{\sqrt{2}}(0;\mp
1,-i,0),\quad\varepsilon(q,0)=(0;0,0,1),\quad\varepsilon(q,t)=(1;0,0,0).$ (31)

Note that $\lambda=\pm,0$ describe the different magnetic quantum numbers for
the total angular momentum quantum number $j=1$ while $\lambda=t$ indicates
the magnetic quantum number zero for $j=0$. This time-like component does not
appear for on-shell $W$ bosons and will be skipped in the following. At Born
term level, up to a general factor $e^{2}q^{2}|V_{cb}|^{2}/s_{W}^{2}$ the
helicity amplitudes are given by

$q^{2}H^{\lambda_{1}\lambda_{2}}=\left[p_{1}^{\mu}p_{2}^{\nu}+p_{1}^{\nu}p_{2}^{\mu}-p_{1}p_{2}g^{\mu\nu}+ip_{1\kappa}p_{2\lambda}\epsilon^{\kappa\lambda\mu\nu}\right]\varepsilon_{\mu}(q,\lambda_{1})\varepsilon_{\nu}^{*}(q,\lambda_{2}).$
(32)

This results in the $W$ frame read

$\displaystyle H^{tt}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mu_{1}+\mu_{2}-(\mu_{1}-\mu_{2})^{2}\right),$
$\displaystyle H^{t0}(\theta)$ $\displaystyle=$ $\displaystyle H^{0t}(\theta)\
=\ \frac{1}{2}(\mu_{1}-\mu_{2}){\sqrt{\lambda}}\cos\theta,$ $\displaystyle
H^{t\pm}(\theta)$ $\displaystyle=$ $\displaystyle H^{\pm t}(\theta)\ =\
\mp\frac{1}{2\sqrt{2}}(\mu_{1}-\mu_{2}){\sqrt{\lambda}}\sin\theta,$
$\displaystyle H^{00}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-\mu_{1}-\mu_{2}-\lambda\cos^{2}\theta\right),$
$\displaystyle H^{0\pm}(\theta)$ $\displaystyle=$ $\displaystyle H^{\pm
0}(\theta)\ =\
-\frac{1}{2\sqrt{2}}{\sqrt{\lambda}}(1\mp{\sqrt{\lambda}}\cos\theta)\sin\theta,$
$\displaystyle H^{\pm\pm}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1+\mu_{1}-\mu_{2}\mp{\sqrt{\lambda}}\cos\theta)(1-\mu_{1}+\mu_{2}\mp{\sqrt{\lambda}}\cos\theta),$
$\displaystyle H^{\pm\mp}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\lambda\sin^{2}\theta.$ (33)

Changing to the quark frame as the frame of reference for the angular
distribution (1), one has to take $\theta=0$. Only the the helicity amplitudes
$H^{00}=H^{00}(\theta=0)$, $H^{\pm\pm}=H^{\pm\pm}(\theta=0)$ necessary for the
angular distribution and $H^{\pm\mp}=H^{\pm\mp}(\theta=0)$ (for completeness)
will be considered in the following.

###  4.1 Helicity amplitudes from the tree corrections

Up to a general factor
$e^{2}q^{2}|V_{cb}|^{2}/s_{W}^{2}\times\alpha/(4\pi{\sqrt{\lambda}})$, the NLO
tree contributions read

$\displaystyle H^{00}({\it tree\/})\ =\
(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}\Big{]}{\sqrt{\lambda}}\ell_{\zeta}+$
(34)
$\displaystyle\kern-24.0pt+(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{\zeta}^{\ell}-2t_{z}^{\ell+})+$
$\displaystyle\kern-24.0pt-(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{\zeta
W}^{\ell}+2t_{zW}^{\ell+})+$
$\displaystyle\kern-24.0pt-4\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{\ell}+t_{z}^{-\ell}-t_{z}^{+\ell})+$
$\displaystyle\kern-24.0pt-2{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+\kern-20.0pt$
$\displaystyle\kern-24.0pt+4\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{\ell}+t_{zW}^{-\ell}-t_{zW}^{+\ell})+\kern-9.0pt$
$\displaystyle\kern-24.0pt-{\textstyle\frac{2}{{\sqrt{\mu_{1}}}}}(1+\mu_{1}-\mu_{2})\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{2}}\Big{[}(4-25\mu_{1}-12\mu_{1}^{2}-9\mu_{2}+26\mu_{1}\mu_{2}-2\mu_{1}^{2}\mu_{2}+6\mu_{2}^{2}-3\mu_{1}\mu_{2}^{2}-\mu_{2}^{3})Q_{1}^{2}+$
$\displaystyle-(4+6\mu_{1}+6\mu_{1}^{2}-\mu_{1}^{3}-6\mu_{2}-2\mu_{1}\mu_{2}+9\mu_{1}^{2}\mu_{2}-4\mu_{1}\mu_{2}^{2}+2\mu_{2}^{3})Q_{2}^{2}+$
$\displaystyle-2(4+9\mu_{1}+3\mu_{1}^{2}+\mu_{1}^{3}-7\mu_{2}-7\mu_{1}\mu_{2}-2\mu_{1}^{2}\mu_{2}+2\mu_{2}^{2}-2\mu_{1}\mu_{2}^{2}+\mu_{2}^{3})Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-24.0pt-{\textstyle\frac{1}{2}}\Big{[}(4-23\mu_{1}-16\mu_{1}^{2}+2\mu_{1}^{3}-7\mu_{2}+12\mu_{1}\mu_{2}-6\mu_{1}^{2}\mu_{2}+4\mu_{2}^{2}+6\mu_{1}\mu_{2}^{2}-2\mu_{2}^{3})Q_{1}^{2}+$
$\displaystyle-(4+5\mu_{1}+8\mu_{1}^{2}-2\mu_{1}^{3}-3\mu_{2}-4\mu_{1}\mu_{2}+6\mu_{1}^{2}\mu_{2}-4\mu_{2}^{2}-6\mu_{1}\mu_{2}^{2}+2\mu_{2}^{3})Q_{2}^{2}+$
$\displaystyle-2(4+8\mu_{1}+5\mu_{1}^{2}-12\mu_{2}-12\mu_{1}\mu_{2}+7\mu_{2}^{2})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{4}}\Big{[}(16-67\mu_{1}-15\mu_{1}^{2}-3\mu_{2}+38\mu_{1}\mu_{2}-11\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-(8+11\mu_{1}+11\mu_{1}^{2}-21\mu_{2}-38\mu_{1}\mu_{2}+15\mu_{2}^{2})Q_{2}^{2}+$
$\displaystyle-2(24+\mu_{1}+9\mu_{1}^{2}-31\mu_{2}-14\mu_{1}\mu_{2}+9\mu_{2}^{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}},$
$\displaystyle H^{++}({\it tree\/})\ =\
-\Big{[}Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}\Big{]}{\sqrt{\lambda}}\left((1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}})\ell_{\zeta}+2{\sqrt{\lambda}}\ell_{+}\right)+$
(35)
$\displaystyle\kern-24.0pt-\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta}^{\ell}-2t_{z}^{\ell+})-{\sqrt{\lambda}}(t_{\zeta}^{\ell}-2t_{z}^{\ell-})\right)+$
$\displaystyle\kern-24.0pt+\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta
W}^{\ell}+2t_{zW}^{\ell+})-{\sqrt{\lambda}}(t_{\zeta
W}^{\ell}+2t_{zW}^{\ell-})\right)+$
$\displaystyle\kern-24.0pt+2\Big{[}(1-2\mu_{1}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}t_{z}^{\ell}+$
$\displaystyle\kern-24.0pt+2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{\ell}+t_{z}^{-\ell}-t_{z}^{+\ell})+$
$\displaystyle\kern-24.0pt+{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+\kern-19.0pt$
$\displaystyle\kern-24.0pt-2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(2t_{zW}^{\ell}+t_{zW}^{-\ell}-t_{zW}^{+\ell})+\kern-19.0pt$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{{\sqrt{\mu_{1}}}}}(1+\mu_{1}-\mu_{2})\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-24.0pt+\Big{[}(1-\mu_{1})(5-3\mu_{1}+4\mu_{2})Q_{1}^{2}-(9-10\mu_{1}+\mu_{1}^{2}+6\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}+$
$\displaystyle-2(1-\mu_{1})(5-\mu_{1}+\mu_{2})Q_{W}^{2}\Big{]}\ell_{0}+$
$\displaystyle\kern-24.0pt-{\textstyle\frac{1}{2}}\Big{[}\left(3-8\mu_{1}-4\mu_{1}^{2}-6\mu_{2}+10\mu_{1}\mu_{2}+3\mu_{2}^{2}+(1+5\mu_{1}-\mu_{2}){\sqrt{\lambda}}\right)Q_{1}^{2}+$
$\displaystyle-\mu_{1}(4+7\mu_{1}-4\mu_{2})Q_{2}^{2}-2(1-\mu_{2})(1+5\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{2}}\Big{[}(1-4\mu_{1}-6\mu_{1}^{2}-4\mu_{2}+6\mu_{2}^{2}+8{\sqrt{\lambda}})Q_{1}^{2}-(1+2\mu_{1}+8\mu_{1}^{2}+6\mu_{2}-8\mu_{1}\mu_{2}+8{\sqrt{\lambda}})Q_{2}^{2}+\kern-6.0pt$
$\displaystyle-2(2+3\mu_{1}+\mu_{1}^{2}-5\mu_{2}-6\mu_{1}\mu_{2}+5\mu_{2}^{2}+4{\sqrt{\lambda}})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-24.0pt-{\textstyle\frac{1}{2}}\Big{[}(17+7\mu_{1}-8\mu_{2})Q_{1}^{2}-(13-3\mu_{1}+2\mu_{2})Q_{2}^{2}-6(3+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\lambda_{-}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{12}}\Big{[}9(1+5\mu_{1}+\mu_{2})Q_{1}^{2}+3(15+7\mu_{1}-29\mu_{2})Q_{2}^{2}+$
$\displaystyle+4(29-10\mu_{1}-\mu_{1}^{2}-34\mu_{2}+2\mu_{1}\mu_{2}-\mu_{2}^{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}}+{\textstyle\frac{1}{4}}\Big{[}Q_{1}^{2}-9Q_{2}^{2}+12Q_{W}^{2}\Big{]}\lambda,$
$\displaystyle H^{+-}({\it tree\/})\ =\ H^{-+}({\it tree\/})\ =\
-2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{\ell}+t_{z}^{-\ell}-t_{z}^{+\ell})+$
(36)
$\displaystyle\kern-24.0pt-{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2}\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+\kern-15.0pt$
$\displaystyle\kern-24.0pt+2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{\ell}+t_{zW}^{-\ell}-t_{zW}^{+\ell})+\kern-13.0pt$
$\displaystyle\kern-24.0pt-{\textstyle\frac{1}{{\sqrt{\mu_{1}}}}}(1+\mu_{1}-\mu_{2})\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-24.0pt+\Big{[}(1-6\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+6\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-(1+\mu_{1}+2\mu_{1}^{2}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{2}^{2}+$
$\displaystyle-2(1+\mu_{1}-\mu_{2})^{2}Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-24.0pt-\Big{[}(1-6\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-(1+\mu_{1}+2\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}+$
$\displaystyle-2(1+\mu_{1}-2\mu_{2})(1+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-24.0pt+\Big{[}2(1-5\mu_{1}-\mu_{2})Q_{1}^{2}-(1+3\mu_{1}-\mu_{2})Q_{2}^{2}-2(3+\mu_{1}-3\mu_{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}},$
$\displaystyle H^{--}({\it tree\/})\ =\
-\Big{[}Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}\Big{]}{\sqrt{\lambda}}\left((1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}})\ell_{\zeta}-2{\sqrt{\lambda}}\ell_{+}\right)+$
(37)
$\displaystyle\kern-24.0pt-\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta}^{\ell}-2t_{z}^{\ell+})+{\sqrt{\lambda}}(t_{\zeta}^{\ell}-2t_{z}^{\ell-})\right)+$
$\displaystyle\kern-24.0pt+\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta
W}^{\ell}+2t_{zW}^{\ell+})+{\sqrt{\lambda}}(t_{\zeta
W}^{\ell}+2t_{zW}^{\ell-})\right)+$
$\displaystyle\kern-24.0pt-2\Big{[}(1-2\mu_{1}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}t_{z}^{\ell}+$
$\displaystyle\kern-24.0pt+2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{\ell}+t_{z}^{-\ell}-t_{z}^{+\ell})+$
$\displaystyle\kern-24.0pt+{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+\kern-19.0pt$
$\displaystyle\kern-24.0pt-2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}-t_{zW}^{+\ell})+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{{\sqrt{\mu_{1}}}}}(1+\mu_{1}-\mu_{2})\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-24.0pt-\Big{[}(1-\mu_{1})(5-3\mu_{1}+4\mu_{2})Q_{1}^{2}-(9-10\mu_{1}+\mu_{1}^{2}+6\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}+$
$\displaystyle-2(1-\mu_{1})(5-\mu_{1}+\mu_{2})Q_{W}^{2}\Big{]}\ell_{0}+$
$\displaystyle\kern-24.0pt-{\textstyle\frac{1}{2}}\Big{[}\left(3-8\mu_{1}-4\mu_{1}^{2}-6\mu_{2}+10\mu_{1}\mu_{2}+3\mu_{2}^{2}-(1+5\mu_{1}-\mu_{2}){\sqrt{\lambda}}\right)Q_{1}^{2}+$
$\displaystyle-\mu_{1}(4+7\mu_{1}-4\mu_{2})Q_{2}^{2}-2(1-\mu_{2})(1+5\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{2}}\Big{[}(1-4\mu_{1}-6\mu_{1}^{2}-4\mu_{2}+6\mu_{2}^{2}-8{\sqrt{\lambda}})Q_{1}^{2}-(1+2\mu_{1}+8\mu_{1}^{2}+6\mu_{2}-8\mu_{1}\mu_{2}-8{\sqrt{\lambda}})Q_{2}^{2}+\kern-6.0pt$
$\displaystyle-2(2+3\mu_{1}+\mu_{1}^{2}-5\mu_{2}-6\mu_{1}\mu_{2}+5\mu_{2}^{2}-4{\sqrt{\lambda}})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{2}}\Big{[}(17+7\mu_{1}-8\mu_{2})Q_{1}^{2}-(13-3\mu_{1}+2\mu_{2})Q_{2}^{2}-6(3+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\lambda_{-}+$
$\displaystyle\kern-24.0pt+{\textstyle\frac{1}{12}}\Big{[}9(1+5\mu_{1}+\mu_{2})Q_{1}^{2}+3(15+7\mu_{1}-29\mu_{2})Q_{2}^{2}+$
$\displaystyle+4(29-10\mu_{1}-\mu_{1}^{2}-34\mu_{2}+2\mu_{1}\mu_{2}-\mu_{2}^{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}}-{\textstyle\frac{1}{4}}\Big{[}Q_{1}^{2}-9Q_{2}^{2}+12Q_{W}^{2}\Big{]}\lambda$

with $\lambda_{\pm}:=(1\pm{\sqrt{\mu_{1}}})^{2}-\mu_{2}$ and
$\lambda_{-}\lambda_{+}=\lambda:=\lambda(1,\mu_{1},\mu_{2})$.

###  4.2 Helicity amplitudes from the loop corrections

Again up to a general factor
$e^{2}q^{2}|V_{cb}|^{2}/s_{W}^{2}\times\alpha/(4\pi{\sqrt{\lambda}})$, the NLO
loop contributions read

$\displaystyle H^{00}({\it loop\/})$ $\displaystyle=$
$\displaystyle-(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2}){\sqrt{\lambda}}V_{-}+$
$\displaystyle+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+}-\frac{1}{2}\lambda{\sqrt{\lambda}}(\mu_{1}V_{1}+\mu_{2}V_{2}),$
$\displaystyle H^{++}({\it loop\/})$ $\displaystyle=$
$\displaystyle(1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}){\sqrt{\lambda}}V_{-}+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+},$
$\displaystyle H^{+-}({\it loop\/})$ $\displaystyle=$ $\displaystyle
H^{-+}({\it loop\/})\ =\ 0,$ $\displaystyle H^{--}({\it loop\/})$
$\displaystyle=$
$\displaystyle(1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}){\sqrt{\lambda}}V_{-}+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+}.$
(38)

##  5 Full results for the helicity amplitudes

Having used Feynman gauge up to now, it is worth mentioning here that we have
performed the calculation of the helicity amplitude for the tree corrections
also in unitary gauge. Unitary gauge used for the $W$ propagator in Fig. 2.3
allows to drop contributions from the charged Goldstone boson which is not
indicated but implicitly assumed in this diagram. We found that the results
for the helicity amplitudes $H^{00}({\it tree\/})$, $H^{\pm\pm}({\it tree\/})$
and $H^{\pm\mp}({\it tree\/})$ turn out to be exactly the same. Both tree and
loop contributions, however, contain IR singularities which have to cancel
according to the Lee–Nauenberg theorem.

###  5.1 Counter terms for the IR singularities

In order to deal with the IR singularities in a consistent way, we need a
convenient method to extract them. For the first order tree contributions we
have seen that the IR singular parts are contained in $\ell_{\zeta}$,
$t_{\zeta}^{\ell}$ and $t_{\zeta W}^{\ell}$. While $\ell_{\zeta}$ itself can
be used as counter term, for the other two expressions in the first order tree
contributions we use

$t_{\zeta}^{\ell}=t_{\zeta}^{\ell*}+\ell_{1}\ell_{\zeta},\qquad t_{\zeta
W}^{\ell}=t_{\zeta W}^{\ell*}+\ell_{1W}\ell_{\zeta},$ (39)

where the starred quantities are found in Appendix A. On the other hand, the
form factors contain IR singularities only in the three-point functions with
the photon on one of the internal lines. The scalar three-point functions are
given by

$C(p_{1}^{2},p_{2}^{2},p_{3}^{2};m_{1},m_{2},m_{3})=\frac{i}{(4\pi)^{2}}C_{f}(p_{1}^{2},p_{2}^{2},p_{3}^{2};m_{1},m_{2},m_{3})$
(40)

with [6]

$C_{f}(p_{1}^{2},p_{2}^{2},p_{3}^{2};m_{1},m_{2},m_{3})=\frac{-1}{\sqrt{\lambda(p_{1}^{2},p_{2}^{2},p_{3}^{2})}}\sum_{i=1}^{3}\sum_{\pm}\left[{\rm
Li}_{2}\left(\frac{1-y_{i0}}{y_{i\pm}\pm i\epsilon-y_{i0}}\right)-{\rm
Li}_{2}\left(\frac{-y_{i0}}{y_{i\pm}\pm i\epsilon-
y_{i0}}\right)\right],\kern-25.0pt$ (41)

where

$\displaystyle y_{10}$ $\displaystyle=$
$\displaystyle\frac{(p_{2}^{2}+m_{2}^{2}-m_{3}^{2})\left({\sqrt{\lambda^{\prime}}}+p_{1}^{2}-p_{2}^{2}-p_{3}^{2}\right)+2p_{2}^{2}(p_{3}^{2}-m_{1}^{2}+m_{2}^{2})}{2p_{2}^{2}{\sqrt{\lambda^{\prime}}}},$
$\displaystyle y_{20}$ $\displaystyle=$
$\displaystyle\frac{(p_{3}^{2}+m_{1}^{2}-m_{2}^{2})\left({\sqrt{\lambda^{\prime}}}-p_{1}^{2}+p_{2}^{2}-p_{3}^{2}\right)+2p_{3}^{2}(p_{1}^{2}-m_{3}^{2}+m_{1}^{2})}{2p_{3}^{2}{\sqrt{\lambda^{\prime}}}},$
$\displaystyle y_{30}$ $\displaystyle=$
$\displaystyle\frac{(p_{1}^{2}+m_{3}^{2}-m_{1}^{2})\left({\sqrt{\lambda^{\prime}}}-p_{1}^{2}-p_{2}^{2}+p_{3}^{2}\right)+2p_{1}^{2}(p_{2}^{2}-m_{2}^{2}+m_{3}^{2})}{2p_{1}^{2}{\sqrt{\lambda^{\prime}}}}$
(42)

and

$\displaystyle y_{1\pm}$ $\displaystyle=$
$\displaystyle\frac{p_{2}^{2}+m_{2}^{2}-m_{3}^{2}\pm\sqrt{\lambda(p_{2}^{2},m_{2}^{2},m_{3}^{2})}}{2p_{2}^{2}},$
$\displaystyle y_{2\pm}$ $\displaystyle=$
$\displaystyle\frac{p_{3}^{2}+m_{1}^{2}-m_{2}^{2}\pm\sqrt{\lambda(p_{3}^{2},m_{1}^{2},m_{2}^{2})}}{2p_{3}^{2}},$
$\displaystyle y_{3\pm}$ $\displaystyle=$
$\displaystyle\frac{p_{1}^{2}+m_{3}^{2}-m_{1}^{2}\pm\sqrt{\lambda(p_{1}^{2},m_{3}^{2},m_{1}^{2})}}{2p_{1}^{2}}$
(43)

Therefore, we again define starred quantities by

$\displaystyle C_{f}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{A},m_{W},m_{1})$
$\displaystyle=$ $\displaystyle
C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{A},m_{W},m_{1})+\frac{\ell_{1}-\ell_{1W}}{2m_{W}^{2}{\sqrt{\lambda}}}\ell_{\zeta},$
$\displaystyle C_{f}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{W},m_{A},m_{2})$
$\displaystyle=$ $\displaystyle
C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{W},m_{A},m_{2})+\frac{\ell_{1W}}{2m_{W}^{2}{\sqrt{\lambda}}}\ell_{\zeta},$
$\displaystyle C_{f}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{1},m_{2},m_{A})$
$\displaystyle=$ $\displaystyle
C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{1},m_{2},m_{A})-\frac{\ell_{1}+2\pi
i}{2m_{W}^{2}{\sqrt{\lambda}}}\ell_{\zeta}\qquad$ (44)

where

$\displaystyle C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{A},m_{W},m_{1})\ =$
$\displaystyle=$
$\displaystyle-\frac{1}{m_{W}^{2}{\sqrt{\lambda}}}\Bigg{[}{\rm
Li}_{2}\left(\frac{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle-\frac{1}{4}\ell_{1}^{2}+\frac{1}{4}\ell_{1W}^{2}-\frac{1}{2}\ln\left(\frac{\mu_{1}}{\lambda}\right)(\ell_{1}-\ell_{1W})\Bigg{]},$
$\displaystyle C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{W},m_{A},m_{2})\ =$
$\displaystyle=$
$\displaystyle-\frac{1}{m_{W}^{2}{\sqrt{\lambda}}}\Bigg{[}{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle-\frac{1}{4}\ell_{1}^{2}+\frac{1}{4}(\ell_{1}-\ell_{1W})^{2}-\frac{1}{2}\ln\left(\frac{\mu_{2}}{\lambda}\right)\ell_{1W}\Bigg{]},$
$\displaystyle C_{f}^{*}(m_{1}^{2},m_{2}^{2},m_{W}^{2};m_{1},m_{2},m_{A})\ =$
(45) $\displaystyle=$
$\displaystyle-\frac{1}{m_{W}^{2}{\sqrt{\lambda}}}\Bigg{[}{\rm
Li}_{2}\left(\frac{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+\frac{2\pi^{2}}{3}+$
$\displaystyle+\frac{1}{4}(\ell_{1}-\ell_{1W})^{2}+\frac{1}{4}\ell_{1W}^{2}+\frac{1}{2}\ln\left(\frac{\mu_{1}\mu_{2}}{\lambda}\right)(\ell_{1}+2\pi
i)\Bigg{]},$

and inserted into the form factor $V_{-}$, one has

$\displaystyle{\sqrt{\lambda}}V_{-}$ $\displaystyle=$
$\displaystyle{\sqrt{\lambda}}V_{-}^{*}+(Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}){\sqrt{\lambda}}\ell_{\zeta}+(1+\mu_{1}-\mu_{2})Q_{1}Q_{W}(\ell_{1}-\ell_{1W})\ell_{\zeta}+$
(46)
$\displaystyle-(1-\mu_{1}+\mu_{2})Q_{2}Q_{W}\ell_{1W}\ell_{\zeta}+(1-\mu_{1}-\mu_{2})Q_{1}Q_{2}(\ell_{1}+2\pi
i)\ell_{\zeta},$

where the starred form factor $V_{-}^{*}$ contains the starred three-point
functions while the other form factors remain unchanged. The term
$(Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}){\sqrt{\lambda}}\ell_{\zeta}$ originates from
the counter terms of the renormalisation. Using the starred instead of the
usual form factors, one has

$\displaystyle H^{00}({\it loop\/})$ $\displaystyle=$ $\displaystyle
H^{00*}({\it
loop\/})-(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}(Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2}){\sqrt{\lambda}}+$
(47)
$\displaystyle\kern-36.0pt+\left((1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\right)\ell_{1}-\left(Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\right)\ell_{1W}\Big{]}\ell_{\zeta}.\quad$

Therefore, adding first order tree and loop corrections for all five helicity
amplitudes under review, the IR singularities cancel and we obtain

$\displaystyle H^{00}(\alpha)\ =\
-(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2}){\sqrt{\lambda}}V_{-}^{*}+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+}-\frac{1}{2}\lambda{\sqrt{\lambda}}(\mu_{1}V_{1}+\mu_{2}V_{2})+$
$\displaystyle\kern-12.0pt+(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{\zeta}^{\ell*}-2t_{z}^{\ell+})+$
$\displaystyle\kern-12.0pt-(-\mu_{1}+\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{\zeta
W}^{\ell*}+2t_{zW}^{\ell+})+$
$\displaystyle\kern-12.0pt-4\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}-t_{z}^{+\ell}+t_{z}^{\ell})+$
$\displaystyle\kern-12.0pt-2{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+$
$\displaystyle\kern-12.0pt+4\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})+$
$\displaystyle-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}-t_{zW}^{+\ell}+t_{zW}^{\ell})+$
$\displaystyle\kern-12.0pt-\frac{2(1+\mu_{1}-\mu_{2})}{{\sqrt{\mu_{1}}}}\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{2}}\Big{[}\left((4-\mu_{2})\lambda-\mu_{1}(17+16\mu_{1}-32\mu_{2}+\mu_{1}\mu_{2}+5\mu_{2}^{2})\right)Q_{1}^{2}+$
$\displaystyle-\left((4-\mu_{1}+2\mu_{2})\lambda+\mu_{1}(15+8\mu_{2}+5\mu_{1}\mu_{2}+\mu_{2}^{2})\right)Q_{2}^{2}+$
$\displaystyle-2\left((4+\mu_{1}+\mu_{2})\lambda+\mu_{1}(16+\mu_{1}+5\mu_{2}-\mu_{1}\mu_{2}-\mu_{2}^{2})\right)Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-12.0pt-{\textstyle\frac{1}{2}}\Big{[}\left(2(2+\mu_{1}-\mu_{2})\lambda-17\mu_{1}-16\mu_{1}^{2}+3\mu_{2}+20\mu_{1}\mu_{2}-4\mu_{2}^{2}\right)Q_{1}^{2}+$
$\displaystyle-\left(2(2-\mu_{1}+\mu_{2})\lambda+15\mu_{1}+3\mu_{2}+4\mu_{1}\mu_{2}-4\mu_{2}^{2}\right)Q_{2}^{2}+$
$\displaystyle-2(4+8\mu_{1}+5\mu_{1}^{2}-12\mu_{2}-12\mu_{1}\mu_{2}+7\mu_{2}^{2})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-12.0pt+\frac{1}{4}\Big{[}(16-67\mu_{1}-15\mu_{1}^{2}-3\mu_{2}+38\mu_{1}\mu_{2}-11\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-(8+11\mu_{1}+11\mu_{1}^{2}-21\mu_{2}-38\mu_{1}\mu_{2}+15\mu_{2}^{2})Q_{2}^{2}+$
$\displaystyle-2(24+\mu_{1}+9\mu_{1}^{2}-31\mu_{2}-14\mu_{1}\mu_{2}+9\mu_{2}^{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}},$
$\displaystyle H^{++}(\alpha)\ =\
(1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}){\sqrt{\lambda}}V_{-}^{*}+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+}-2(Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2})\lambda\ell_{+}+$
$\displaystyle\kern-12.0pt-\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta}^{\ell*}-2t_{z}^{\ell+})-{\sqrt{\lambda}}(t_{\zeta}^{\ell*}-2t_{z}^{\ell-})\right)+$
$\displaystyle\kern-12.0pt+\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta
W}^{\ell*}+2t_{zW}^{\ell+})-{\sqrt{\lambda}}(t_{\zeta
W}^{\ell*}+2t_{zW}^{\ell-})\right)+$
$\displaystyle\kern-12.0pt+2\Big{[}(1-2\mu_{1}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}t_{z}^{\ell}+$
$\displaystyle\kern-12.0pt+2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}-t_{z}^{+\ell}+t_{z}^{\ell})+$
$\displaystyle\kern-12.0pt+{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+$
$\displaystyle\kern-12.0pt-2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})+$
$\displaystyle-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}-t_{zW}^{+\ell}+2t_{zW}^{\ell})+$
$\displaystyle\kern-12.0pt+\frac{1+\mu_{1}-\mu_{2}}{{\sqrt{\mu_{1}}}}\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-12.0pt+\Big{[}(1-\mu_{1})(5-3\mu_{1}+4\mu_{2})Q_{1}^{2}-(9-10\mu_{1}+\mu_{1}^{2}+6\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}+$
$\displaystyle-2(1-\mu_{1})(5-\mu_{1}+\mu_{2})Q_{W}^{2}\Big{]}\ell_{0}+$
$\displaystyle\kern-12.0pt-{\textstyle\frac{1}{2}}\Big{[}\left(3-8\mu_{1}-4\mu_{1}^{2}-6\mu_{2}+10\mu_{1}\mu_{2}+3\mu_{2}^{2}+(1+5\mu_{1}-\mu_{2}){\sqrt{\lambda}}\right)Q_{1}^{2}+$
$\displaystyle-\mu_{1}(4+7\mu_{1}-4\mu_{2})Q_{2}^{2}-2(1+5\mu_{1}-\mu_{2})(1-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{2}}\Big{[}(1-4\mu_{1}-6\mu_{1}^{2}-4\mu_{2}+6\mu_{2}^{2}+8{\sqrt{\lambda}})Q_{1}^{2}+$
$\displaystyle-(1+2\mu_{1}+8\mu_{1}^{2}+6\mu_{2}-8\mu_{1}\mu_{2}+8{\sqrt{\lambda}})Q_{2}^{2}+$
$\displaystyle-2(2+3\mu_{1}+\mu_{1}^{2}-5\mu_{2}-6\mu_{1}\mu_{2}+5\mu_{2}^{2}+4{\sqrt{\lambda}})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-12.0pt-{\textstyle\frac{1}{2}}\Big{[}(17+7\mu_{1}-8\mu_{2})Q_{1}^{2}-(13-3\mu_{1}+2\mu_{2})Q_{2}^{2}-6(3+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\lambda_{-}+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{12}}\Big{[}3(3+15\mu_{1}+3\mu_{2}+{\sqrt{\lambda}})Q_{1}^{2}+3(15+7\mu_{1}-29\mu_{2}-9{\sqrt{\lambda}})Q_{2}^{2}+$
$\displaystyle+4(29-10\mu_{1}-\mu_{1}^{2}-34\mu_{2}+2\mu_{1}\mu_{2}-\mu_{2}^{2}-9{\sqrt{\lambda}})Q_{W}^{2}\Big{]}{\sqrt{\lambda}},$
$\displaystyle H^{+-}(\alpha)\ =\ H^{-+}(\alpha)\ =\
-2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}-t_{z}^{+\ell}+t_{z}^{\ell})+$
$\displaystyle\kern-12.0pt-{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+$
$\displaystyle\kern-12.0pt+2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})+$
$\displaystyle-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}-t_{zW}^{+\ell}+t_{zW}^{\ell})+$
$\displaystyle\kern-12.0pt-\frac{1+\mu_{1}-\mu_{2}}{{\sqrt{\mu_{1}}}}\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-12.0pt+\Big{[}(1-6\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+6\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-(1+\mu_{1}+2\mu_{1}^{2}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{2}^{2}-2(1+\mu_{1}-\mu_{2})^{2}Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-12.0pt-\Big{[}(1-6\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-(1+\mu_{1}+2\mu_{1}^{2}-\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}-2(1+\mu_{1}-2\mu_{2})(1+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-12.0pt+\Big{[}2(1-5\mu_{1}-\mu_{2})Q_{1}^{2}-(1+3\mu_{1}-\mu_{2})Q_{2}^{2}-2(3+\mu_{1}-3\mu_{2})Q_{W}^{2}\Big{]}{\sqrt{\lambda}},$
$\displaystyle H^{--}(\alpha)\ =\
(1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}){\sqrt{\lambda}}V_{-}^{*}+2\mu_{1}\mu_{2}{\sqrt{\lambda}}V_{+}+2(Q_{1}^{2}+Q_{2}^{2}+Q_{W}^{2})\lambda\ell_{+}+$
(48)
$\displaystyle\kern-12.0pt-\Big{[}(1-\mu_{2})Q_{1}^{2}-\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta}^{\ell*}-2t_{z}^{\ell+})+{\sqrt{\lambda}}(t_{\zeta}^{\ell*}-2t_{z}^{\ell-})\right)+$
$\displaystyle\kern-12.0pt+\Big{[}Q_{1}^{2}-Q_{2}^{2}+(\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\left((1-\mu_{1}-\mu_{2})(t_{\zeta
W}^{\ell*}+2t_{zW}^{\ell+})+{\sqrt{\lambda}}(t_{\zeta
W}^{\ell*}+2t_{zW}^{\ell-})\right)+$
$\displaystyle\kern-12.0pt-2\Big{[}(1-2\mu_{1}-2\mu_{2}-\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}-\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}t_{z}^{\ell}+$
$\displaystyle\kern-12.0pt+2\mu_{1}\Big{[}(1+5\mu_{1}-\mu_{2})Q_{1}^{2}+2\mu_{1}(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}-t_{z}^{+\ell}+t_{z}^{\ell})+$
$\displaystyle\kern-12.0pt+{\sqrt{\mu_{1}}}\Big{[}(1-10\mu_{1}-3\mu_{1}^{2}-2\mu_{2}+2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{1}^{2}+$
$\displaystyle-2\mu_{1}(1+\mu_{1}-\mu_{2})(Q_{2}^{2}-Q_{W}^{2})\Big{]}(t_{z}^{-\ell}+t_{z}^{+\ell})+$
$\displaystyle\kern-12.0pt-2\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})+$
$\displaystyle-(3+2\mu_{1}+\mu_{1}^{2}-4\mu_{2}-2\mu_{1}\mu_{2}+\mu_{2}^{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}-t_{zW}^{+\ell})+$
$\displaystyle\kern-12.0pt+\frac{1+\mu_{1}-\mu_{2}}{{\sqrt{\mu_{1}}}}\Big{[}(1+\mu_{1}-\mu_{2})(Q_{1}^{2}-Q_{2}^{2})-2(1+2\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}(t_{zW}^{-\ell}+t_{zW}^{+\ell})+$
$\displaystyle\kern-12.0pt-\Big{[}(1-\mu_{1})(5-3\mu_{1}+4\mu_{2})Q_{1}^{2}-(9-10\mu_{1}+\mu_{1}^{2}+6\mu_{2}-2\mu_{1}\mu_{2})Q_{2}^{2}+$
$\displaystyle-2(1-\mu_{1})(5-\mu_{1}+\mu_{2})Q_{W}^{2}\Big{]}\ell_{0}+$
$\displaystyle\kern-12.0pt-{\textstyle\frac{1}{2}}\Big{[}\left(3-8\mu_{1}-4\mu_{1}^{2}-6\mu_{2}+10\mu_{1}\mu_{2}+3\mu_{2}^{2}-(1+5\mu_{1}-\mu_{2}){\sqrt{\lambda}}\right)Q_{1}^{2}+$
$\displaystyle-\mu_{1}(4+7\mu_{1}-4\mu_{2})Q_{2}^{2}-2(1+5\mu_{1}-\mu_{2})(1-\mu_{2})Q_{W}^{2}\Big{]}\ell_{1}+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{2}}\Big{[}(1-4\mu_{1}-6\mu_{1}^{2}-4\mu_{2}+6\mu_{2}^{2}-8{\sqrt{\lambda}})Q_{1}^{2}+$
$\displaystyle-(1+2\mu_{1}+8\mu_{1}^{2}+6\mu_{2}-8\mu_{1}\mu_{2}-8{\sqrt{\lambda}})Q_{2}^{2}+$
$\displaystyle-2(2+3\mu_{1}+\mu_{1}^{2}-5\mu_{2}-6\mu_{1}\mu_{2}+5\mu_{2}^{2}-4{\sqrt{\lambda}})Q_{W}^{2}\Big{]}\ell_{1W}+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{2}}\Big{[}(17+7\mu_{1}-8\mu_{2})Q_{1}^{2}-(13-3\mu_{1}+2\mu_{2})Q_{2}^{2}-6(3+\mu_{1}-\mu_{2})Q_{W}^{2}\Big{]}\lambda_{-}+$
$\displaystyle\kern-12.0pt+{\textstyle\frac{1}{12}}\Big{[}3(3+15\mu_{1}+3\mu_{2}-{\sqrt{\lambda}})Q_{1}^{2}+3(15+7\mu_{1}-29\mu_{2}+9{\sqrt{\lambda}})Q_{2}^{2}+$
$\displaystyle+4(29-10\mu_{1}-\mu_{1}^{2}-34\mu_{2}+2\mu_{1}\mu_{2}-\mu_{2}^{2}+9{\sqrt{\lambda}})Q_{W}^{2}\Big{]}{\sqrt{\lambda}}.$

Analytical expressions for the form factors $V_{-}^{*}$, $V_{+}$, $V_{1}$ and
$V_{2}$ are found in Appendix B.

##  6 Discussion

In general, electroweak corrections are assumed to be small. However, the size
of the correction depends strongly on the subtraction scheme used. Using the
$\alpha(0)$ scheme as the canonical choice from the perturbation series is not
an option for the decay of a massive $W$ boson, as the scale of the problem is
far from being zero. Instead, one might consider the $\alpha(m_{W}^{2})$
scheme with the renormalisation scale fixed to the mass of the $W$ boson. As
argued in Ref. [46], however, the best choice is the $G_{\mu}$ scheme
(traditionally known as the $G_{F}$ scheme related to the Fermi constant as
the best measured quantity in the process). This scheme and the consequences
of its application is discussed in Appendix C.1. Frankly speaking, using the
latter two schemes, large logarithms from light fermion loops are removed from
the result which then is indeed a small correction to the Born term result.

The effect of the different schemes is displayed in the angular distributions
shown in Fig. 5.

Figure 5: Angular distribution in dependence on the subtraction scheme
employed

As it turns out, in the $G_{\mu}$ scheme the electroweak correction is of the
order of $4.5\%$ while for the other schemes, the corrections are larger,
$9.4\%$ and $18.6\%$ for the $\alpha(m_{W}^{2})$ and $\alpha(0)$ schemes,
respectively.111Note that different from Refs. [28, 29, 30] we have chosen the
positive $z$ axis in the quark frame as the direction of motion of the quark.
A different convention means an interchange $\theta\leftrightarrow-\theta$,
i.e., the diagram for the angular distribution will be mirrored at the
vertical axis. This is in accordance to the statement in Ref. [46] that the
$G_{\mu}$ scheme is appropriate for such kind of decay processes. In the
second column of Table 1 we give the helicity amplitudes for the different
schemes.

Table 1: Helicity amplitudes for the Born term contribution and the electroweak radiative correction, the latter being given up to a general factor $\alpha(0)/4\pi{\sqrt{\lambda}}$. Shown are numerical values for the exact fermion masses, and for the fermion masses except for the top quark mass set to zero, i.e., in the collinear limit. scheme, | amplitude | $m_{f}$ |  $m_{f}=0$, $f\neq t$
---|---|---|---
LO | $H^{00}$ | $0.001474$ | $0$
| $H^{++}$ | $7\times 10^{-7}$ | $0$
| $H^{+-}$ | $0$ | $0$
| $H^{--}$ | $0.997045$ | $1$
$\alpha(0)$ | $H^{00}$ | $-0.435863$ | $+7.10152$
| $H^{++}$ | $+0.087951$ | $-3.22712$
| $H^{+-}$ | $+0.183283$ | $+3.55076$
| $H^{--}$ | $+318.292$ | $+338.950$
$\alpha(m_{W}^{2})$ | $H^{00}$ | $-0.203968$ | $+7.10152$
| $H^{++}$ | $+0.087844$ | $-3.22712$
| $H^{+-}$ | $+0.183283$ | $+3.55076$
| $H^{--}$ | $+161.441$ | $+224.913$
$G_{\mu}$ | $H^{00}$ | $-0.078636$ | $+7.10152$
| $H^{++}$ | $+0.087787$ | $-3.22712$
| $H^{+-}$ | $+0.183283$ | $+3.55076$
| $H^{--}$ | $+76.667$ | $+134.212$

Three observables characterising the angular distribution are considered.
These observables are the maximal curvature of the angular distribution,
called the convexity parameter, given by

$c_{f}=\frac{d^{2}W(\theta)}{d(\cos\theta)^{2}}=\frac{3}{4}(\rho_{++}-2\rho_{00}+\rho_{--})(H_{++}-2H_{00}+H_{--}),$
(49)

and the forward–backward asymmetry of the decay distribution defined by

$A_{FB}=\frac{W(F)-W(B)}{W(F)+W(B)}=\frac{3}{4}(\rho_{++}-\rho_{--})(H_{++}-H_{--})$
(50)

with $W(F)=W(0\leq\theta\leq\pi/2)$ and $W(B)=W(\pi/2\leq\theta\leq\pi)$. The
position of the extremum

$\cos\theta\Big{|}_{\rm
extr}=-\frac{A_{FB}}{c_{f}}=-\frac{(\rho_{++}-\rho_{--})}{(\rho_{++}-2\rho_{00}+\rho_{--})}\frac{(H_{++}-H_{--})}{(H_{++}-2H_{00}+H_{--})}$
(51)

as a third possible observable depends on the other two observables. The
values for these observables are found in the first lines in the scheme blocks
of Table 2.

Table 2: Numerical values for the three observables $c_{f}$, $A_{\rm FB}$ and $\cos\theta_{\rm extr}$ to leading order and for three next-to-leading order schemes with exact fermion masses and in the collinear limit scheme, | case | $c_{f}$ | $A_{\rm FB}$ | $\cos\theta_{\rm extr}$
---|---|---|---|---
LO | $m_{f}\neq 0$ | $-0.791277$ | $+0.231289$ | $+0.292299$
| $m_{f}=0$ | $-0.795975$ | $+0.231975$ | $+0.291435$
$\alpha(0)$ | $m_{f}\neq 0$ | $-0.939281$ | $+0.274281$ | $+0.292012$
| $m_{f}=0$ | $-0.945031$ | $+0.278206$ | $+0.294388$
$\alpha(m_{W}^{2})$ | $m_{f}\neq 0$ | $-0.866350$ | $+0.253090$ | $+0.292133$
| $m_{f}=0$ | $-0.892164$ | $+0.262799$ | $+0.294563$
$G_{\mu}$ | $m_{f}\neq 0$ | $-0.826933$ | $+0.241636$ | $+0.292207$
| $m_{f}=0$ | $-0.850115$ | $+0.250544$ | $+0.294718$

On the other hand, we can investigate the dependence on the masses. For this,
the collinear limit $m_{c},m_{b}\to 0$ can be taken, as the collinear
singularities between tree and loop contributions were not only implicitly
cancelled but explicitly truncated on both sides in our approach. The last
column shows the results in the collinear limit where all fermion masses
except for the top quark mass are set to zero. The collinear limit is used
also for the observables in Table 2. Finally, in Figure 6 we have compared the
angular distribution for the LO Born term result and for the NLO result in the
$G_{\mu}$ scheme with the results in the collinear limit in order to study
mass effects.

Figure 6: Mass dependence of the angular distribution in the $G_{\mu}$ scheme

While mass corrections for the Born term results are difficult to discern and
are of the order $-0.2\%$, mass effects in the $G_{\mu}$ scheme amount to
$-3.4\%$, with the mass effects for the $\alpha(m_{W}^{2})$ and $\alpha(0)$
schemes given by $-3.5\%$ and $-1.2\%$, respectively. At the same time, we see
that in the collinear limit the electroweak correction is larger and amounts
to $7.9\%$, $13.1\%$ and $19.8\%$ for the $G_{\mu}$, $\alpha(m_{W}^{2})$ and
$\alpha(0)$ schemes, respectively. This is also echoed in Tab. 2.

Up to a general factor of $e^{2}q^{2}|V_{cb}|^{2}/s_{W}^{2}$, the Born term
results in the collinear limit are given by $H^{00}({\it Born\/})=H^{++}({\it
Born\/})=H^{+-}({\it Born\/})=H^{-+}({\it Born\/})=0$, while the only non-
vanishing amplitude is $H^{--}({\it Born\/})=1$. On the other hand, up to an
additional general factor of $\alpha/(4\pi)$, the analytic expressions for the
NLO EW corrections to the helicity amplitudes in the collinear limit for the
$G_{\mu}$ scheme read

$\displaystyle H^{00}(\alpha)\ =\
4Q_{c}^{2}-2Q_{b}^{2}-12Q_{W}^{2}+\frac{4\pi^{2}}{3}Q_{c}Q_{W}-\frac{8\pi^{2}}{3}Q_{b}Q_{W},$
$\displaystyle H^{++}(\alpha)\ =\
-\frac{15}{2}Q_{c}^{2}+8Q_{b}^{2}+\frac{47}{3}Q_{W}^{2}-\pi^{2}Q_{c}Q_{W}+3\pi^{2}Q_{b}Q_{W},$
$\displaystyle H^{+-}(\alpha)\ =\ H^{-+}(\alpha)\ =\
2Q_{c}^{2}-Q_{b}^{2}-6Q_{W}^{2}+\frac{2\pi^{2}}{3}Q_{c}Q_{W}-\frac{4\pi^{2}}{3}Q_{b}Q_{W},$
$\displaystyle H^{--}(\alpha)\ =\
5Q_{c}^{2}-\frac{9}{2}Q_{b}^{2}+\frac{11}{3}Q_{W}^{2}-\frac{5\pi^{2}}{3}Q_{c}Q_{W}+\pi^{2}Q_{b}Q_{W}+$
(52)
$\displaystyle-4g_{c}^{-}g_{b}^{-}\left(\frac{m_{W}^{2}+m_{Z}^{2}}{m_{W}^{2}}\right)^{2}\left[{\rm
Li}_{2}\left(-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)+\ln\left(1+\frac{m_{W}^{2}}{m_{Z}^{2}}\right)\ln\left(-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)\right]+$
$\displaystyle+4g_{c}^{-}\frac{m_{W}^{2}+2m_{Z}^{2}}{m_{Z}m_{W}s_{W}}\Bigg{[}{\rm
Li}_{2}\left(\frac{2}{1-\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)+{\rm
Li}_{2}\left(\frac{2}{1+\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)+$
$\displaystyle\qquad-{\rm
Li}_{2}\left(\frac{2(1-m_{W}^{2}/m_{Z}^{2})}{1-\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)-{\rm
Li}_{2}\left(\frac{2(1-m_{W}^{2}/m_{Z}^{2})}{1+\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)\Bigg{]}+$
$\displaystyle-4g_{b}^{-}\frac{m_{W}^{2}+2m_{Z}^{2}}{m_{Z}m_{W}s_{W}}\left[{\rm
Li}_{2}\left(\frac{2m_{W}^{2}/m_{Z}^{2}}{1-\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)+{\rm
Li}_{2}\left(\frac{2m_{W}^{2}/m_{Z}^{2}}{1+\sqrt{1-4m_{W}^{2}/m_{Z}^{2}}}\right)\right]+$
$\displaystyle+\frac{m_{Z}^{2}}{m_{W}^{4}}\left(1-2Q_{c}s_{W}^{2}+Q_{b}s_{W}^{2}-4Q_{c}Q_{b}s_{W}^{2}\right)\left[(2m_{W}^{2}+m_{Z}^{2})\left(\ln\left(\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+1\right)+\frac{3\pi
im_{W}^{2}}{2s_{W}^{2}}\right]+\kern-8.0pt$
$\displaystyle-4g_{c}^{-}g_{b}^{-}\frac{\pi
im_{Z}^{2}}{m_{W}^{2}}+\frac{4m_{W}^{2}-m_{Z}^{2}}{m_{W}^{2}s_{W}^{2}}+\frac{1}{2s_{W}^{2}}+(g_{c}^{-2}+g_{b}^{-2})\left(\ln\left(\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+\frac{1}{2}\right)+$
$\displaystyle-\frac{m_{t}^{2}|V_{tb}|^{2}}{8(m_{t}^{2}-m_{W}^{2})^{2}m_{W}^{2}s_{W}^{2}}\left[3(m_{t}^{4}-m_{W}^{4})-2m_{t}^{2}(m_{t}^{2}+2m_{W}^{2})\ln\left(\frac{m_{t}^{2}}{m_{W}^{2}}\right)\right]+$
$\displaystyle+\frac{48m_{W}^{10}+20m_{W}^{8}m_{Z}^{2}-320m_{W}^{6}m_{Z}^{4}+249m_{W}^{4}m_{Z}^{6}-30m_{W}^{2}m_{Z}^{8}-3m_{Z}^{10}}{24m_{Z}^{4}m_{W}^{6}s_{W}^{4}}\ln\left(\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+$
$\displaystyle+\frac{m_{t}^{2}(5m_{t}^{4}-9m_{W}^{4})}{6m_{W}^{6}s_{W}^{2}}\ln\left(\frac{m_{t}^{2}}{m_{W}^{2}}\right)-\frac{(m_{t}^{2}-m_{W}^{2})(5m_{t}^{4}+5m_{t}^{2}m_{W}^{2}-4m_{W}^{4})}{6m_{W}^{6}s_{W}^{2}}\ln\left(\frac{m_{t}^{2}}{m_{W}^{2}}-1\right)+\kern-3.0pt$
$\displaystyle-\frac{3m_{H}^{8}-18m_{H}^{6}m_{W}^{2}+51m_{H}^{4}m_{W}^{4}-64m_{H}^{2}m_{W}^{6}+12m_{W}^{8}}{24(m_{H}^{2}-m_{W}^{2})m_{W}^{6}s_{W}^{2}}\ln\left(\frac{m_{H}^{2}}{m_{W}^{2}}\right)+$
$\displaystyle+\frac{28m_{W}^{6}+52m_{W}^{4}m_{Z}^{2}-13m_{W}^{2}m_{Z}^{4}-m_{Z}^{6}}{2m_{Z}^{2}m_{W}^{6}s_{W}^{2}}\sqrt{4m_{W}^{2}m_{Z}^{2}-m_{Z}^{4}}\arctan\left(\sqrt{\frac{2m_{W}-m_{Z}}{2m_{W}+m_{Z}}}\right)+$
$\displaystyle+\frac{m_{H}^{6}-7m_{H}^{4}m_{W}^{2}+20m_{H}^{2}m_{W}^{4}-28m_{W}^{6}}{2(4m_{W}^{2}-m_{H}^{2})m_{W}^{6}s_{W}^{2}}\sqrt{4m_{H}^{2}m_{W}^{2}-m_{H}^{4}}\arctan\left(\sqrt{\frac{2m_{W}-m_{H}}{2m_{W}+m_{H}}}\right)+$
$\displaystyle+\frac{3m_{H}^{4}-10m_{t}^{4}-10m_{H}^{2}m_{W}^{2}-8m_{t}^{2}m_{W}^{2}-184m_{W}^{4}+38m_{W}^{2}m_{Z}^{2}+3m_{Z}^{4}}{12m_{W}^{4}s_{W}^{2}}+\frac{4\pi
i}{3s_{W}^{2}}+$
$\displaystyle+\frac{3V_{tb}^{2}m_{t}^{2}}{8m_{W}^{2}s_{W}^{2}}\left[\frac{m_{t}^{2}+m_{W}^{2}}{m_{t}^{2}-m_{W}^{2}}-\frac{2m_{t}^{2}(m_{t}^{2}+2m_{W}^{2})}{3(m_{t}^{2}-m_{W}^{2})^{2}}\ln\left(\frac{m_{t}^{2}}{m_{W}^{2}}\right)\right].$

Note that only $H^{--}(\alpha)$ depends on the loop corrections and the
corrections due to the renormalisation factors and, therefore, on the chosen
subtraction scheme.

##  7 Conclusions

In this publication we have given analytic results for the NLO electroweak
corrections to the decay process $W^{+}(\uparrow)\to c\bar{b}$ of a polarised
$W$ boson into quarks, taking into account all fermion and boson masses
contributing to this process. We have estimated the size of the corrections
and shown the dependence on the subtraction scheme of the renormalisation
procedure. Finally, we have performed the collinear limit in order to estimate
the influence of mass effects to this decay process.

### Acknowledgments

SG acknowledges useful and fuitful discussions with Ansgar Denner. The
research was supported by the European Regional Development Fund under Grant
No. TK133, and by the Estonian Research Council under Grant No. PRG356.

##  Appendix A Analytical expressions for the tree corrections

The integrals necessary for the tree corrections are of the form

$I(n_{1},n_{2};\Lambda)=\int_{y_{20}}^{y_{2-}}dy_{2}\int_{y_{1-}}^{y_{1+}}dy_{1}\frac{y_{1}^{n_{1}}y_{2}^{n_{2}}}{(y_{1}+y_{2})^{2}}.$
(A1)

While the integrals are regular for $n_{1}+n_{2}>0$ and can be calculated
taking $\Lambda=0$, the integrals at the border line $n_{1}+n_{2}=0$ are IR
singular. After performing the integation over $y_{1}$, the integration over
$y_{2}$ cannot be calculated analytically for a general parameter $\Lambda$ in
a closed form. Instead, we split the integral up into a divergent part
$D(n_{1},n_{2};\Lambda)$ and a convergent part $C(n_{1},n_{2})$, where the
divergent part is given by $I(n_{1},n_{2};\Lambda)$ with the integrand
approximated for small values of $y_{2}$, while the convergent part is given
by the limit $\Lambda\to 0$ of the difference
$I(n_{1},n_{2};\Lambda)-D(n_{1},n_{2};\Lambda)$ that is finite by
construction. In technical terms, for the calculation of the divergent part in
terms of a couple of elementary $\zeta$-integrals, we use the substitution
$y_{2}=\Lambda+\sqrt{\Lambda\mu_{2}}(1+\zeta)/\sqrt{\zeta}$ to perform the
integral over $\zeta$ between
$\zeta_{-}=\Lambda\mu_{2}/((1-\sqrt{\mu_{1}})^{2}-\mu_{2})^{2}$ to $1$. After
having performed this substitution, all occurrences of $\Lambda$ in the
(simplified) integrand can be neglected and the integral can be performed. The
simplified integrand is then transformed back to $y_{2}$ and subtracted from
the full integrand and integrated in the limit $\Lambda\to 0$ by using the
substitution $y_{2}=1+\mu_{1}-\mu_{2}-\sqrt{\mu_{1}}(z+1/z)$ to obtain the
convergent part again in terms of a couple of $z$-integrals. The corresponding
limits are given by
$z_{-}=(1+\mu_{1}-\mu_{2}-\sqrt{\lambda(1,\mu_{1},\mu_{2})})/(2\sqrt{\mu_{1}})$
and $1$. A large portion of the $\zeta$\- and $z$-integrals can be expressed
in terms of rational functions and logarithms, containing

$\displaystyle\ell_{\zeta}$ $\displaystyle=$
$\displaystyle\ln\left(\frac{\lambda^{2}}{\Lambda\mu_{1}\mu_{2}}\right),\qquad\ell_{0}\
=\ \ln\left(\frac{1-{\sqrt{\mu_{1}}}}{\sqrt{\mu_{2}}}\right),\qquad\ell_{+}\
=\ \ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}}{{\sqrt{\mu_{1}}}}\right),$
$\displaystyle\ell_{1}$ $\displaystyle=$
$\displaystyle\ln\left(\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right),\qquad\ell_{1W}\
=\
\ln\left(\frac{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right).$
(A2)

Exceptional are integrals containing a logarithm together with $\zeta$ or $z$
to the power of $-1$. These integrals contain dilogarithms and are kept as
closed form terms $t_{\zeta}$ and $t_{z}$, the analytic expressions for these
found in the following. The $\zeta$-terms

$\displaystyle t_{\zeta}^{\ell*}$ $\displaystyle=$ $\displaystyle{\rm
Li}_{2}\left(-\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(-\frac{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle-2{\rm
Li}_{2}\left(\frac{{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+2{\rm
Li}_{2}\left(\frac{-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+2{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-2{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+\ell_{1}\ln\Bigg{(}\frac{4\mu_{1}\left((1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}\right)^{2}}{\lambda^{2}}\Bigg{)},$
$\displaystyle t_{\zeta W}^{\ell*}$ $\displaystyle=$ $\displaystyle{\rm
Li}_{2}\left(-\frac{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(-\frac{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
(A3) $\displaystyle-2{\rm
Li}_{2}\left(\frac{{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)+2{\rm
Li}_{2}\left(\frac{-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+2{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)-2{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+\ell_{1W}\ln\Bigg{(}\frac{4\mu_{1}\left((1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}\right)^{2}}{\lambda^{2}}\Bigg{)}$

are IR subtracted, cf. the discussion in Sec. 5.1. The remaining $z$-terms
read

$\displaystyle t_{z}^{\ell-}\ =\ {\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+$
(A4) $\displaystyle+{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle+\ell_{1}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}\right),$
$\displaystyle t_{z}^{\ell+}\ =\ {\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+2{\rm
Li}_{2}({\sqrt{\mu_{1}}})+$ $\displaystyle+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle-{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle-{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2}\right)-{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}{2}\right)+$
$\displaystyle-\ell_{1}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}}{{\sqrt{\mu_{1}}}}\right)-\ln^{2}\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}\right),$
$\displaystyle t_{z}^{-\ell}\ =\ {\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2(1-{\sqrt{\mu_{1}}})}\right)+$
(A6) $\displaystyle+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2(1-{\sqrt{\mu_{1}}}){\sqrt{\mu_{1}}}}\right),$
$\displaystyle t_{z}^{+\ell}\ =\ -{\rm
Li}_{2}\left(\frac{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}{2(1+{\sqrt{\mu_{1}}})}\right)+{\rm
Li}_{2}\left(-\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}(1+{\sqrt{\mu_{1}}})}\right)+{\rm
Li}_{2}\left(-\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+$
(A7) $\displaystyle-{\rm
Li}_{2}\left(-\frac{1-{\sqrt{\mu_{1}}}}{1+{\sqrt{\mu_{1}}}}\right)+{\rm
Li}_{2}\left(\frac{1-{\sqrt{\mu_{1}}}}{1+{\sqrt{\mu_{1}}}}\right)-{\rm
Li}_{2}(-1)+$
$\displaystyle+\frac{1}{2}\ell_{1}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}}{{\sqrt{\mu_{1}}}(1+{\sqrt{\mu_{1}}})}\right)-\frac{1}{2}\ell_{1W}\ln\left(\frac{1+{\sqrt{\mu_{1}}}}{{\sqrt{\mu_{1}}}}\right)+$
$\displaystyle+\frac{1}{4}\ln\left(\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)\ln\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+\ln({\sqrt{\mu_{1}}})\ln\left(\frac{1-{\sqrt{\mu_{1}}}}{\sqrt{\mu_{2}}}\right),$
$\displaystyle t_{z}^{\ell}\ =\ {\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}{2}\right)+{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2}\right)-2{\rm
Li}_{2}({\sqrt{\mu_{1}}})+$ (A8)
$\displaystyle+\frac{1}{4}\ln^{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right),$
$\displaystyle t_{zW}^{\ell-}\ =\ {\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+$
(A9) $\displaystyle-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle-{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle-\ell_{1W}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}\right),$
$\displaystyle t_{zW}^{\ell+}\ =\ {\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-2{\rm
Li}_{2}({\sqrt{\mu_{1}}})+$ $\displaystyle-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}\right)+{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}+{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1-\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{-2{\sqrt{\lambda}}}{1-\mu_{1}+\mu_{2}-{\sqrt{\lambda}}}\right)-{\rm
Li}_{2}\left(\frac{2{\sqrt{\lambda}}}{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}\right)+$
$\displaystyle+{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2}\right)+{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}{2}\right)+\ell_{1W}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}}{{\sqrt{\mu_{1}}}}\right),$
$\displaystyle t_{zW}^{-\ell}\ =\ {\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+{\rm
Li}_{2}\left(\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2(1-{\sqrt{\mu_{1}}})}\right)+$
(A11) $\displaystyle-{\rm
Li}_{2}\left(-\frac{(1-{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}(1-{\sqrt{\mu_{1}}})}\right),$
$\displaystyle t_{zW}^{+\ell}\ =\ {\rm
Li}_{2}\left(-\frac{(1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right)+{\rm
Li}_{2}\left(\frac{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}{2(1+{\sqrt{\mu_{1}}})}\right)-{\rm
Li}_{2}\left(-\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}(1+{\sqrt{\mu_{1}}})}\right)+$
(A12) $\displaystyle+{\rm
Li}_{2}\left(-\frac{1-{\sqrt{\mu_{1}}}}{1+{\sqrt{\mu_{1}}}}\right)-{\rm
Li}_{2}\left(\frac{1-{\sqrt{\mu_{1}}}}{1+{\sqrt{\mu_{1}}}}\right)-{\rm
Li}_{2}(-1)+$
$\displaystyle+\ln(1+{\sqrt{\mu_{1}}})\ln\left(\frac{1-\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}(1-{\sqrt{\mu_{1}}})}\right)-\ln\left(1+\frac{1}{{\sqrt{\mu_{1}}}}\right)\ln\left(\frac{1-\mu_{1}+\mu_{2}+{\sqrt{\lambda}}}{2(1-{\sqrt{\mu_{1}}})}\right)+$
$\displaystyle-\ell_{1W}\ln\left(\frac{(1+{\sqrt{\mu_{1}}})^{2}-\mu_{2}-{\sqrt{\lambda}}}{2{\sqrt{\mu_{1}}}}\right),$
$\displaystyle t_{zW}^{\ell}\ =\ 2{\rm Li}_{2}({\sqrt{\mu_{1}}})-{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}+{\sqrt{\lambda}}}{2}\right)-{\rm
Li}_{2}\left(\frac{1+\mu_{1}-\mu_{2}-{\sqrt{\lambda}}}{2}\right).$ (A13)

##  Appendix B Form factors for the vertex correction

$\displaystyle V_{-}^{*}\ =\
\frac{m_{c}^{2}+m_{b}^{2}+2m_{W}^{2}-2(2Q_{c}-1)(2Q_{b}+1)m_{Z}^{2}s_{W}^{2}}{4m_{W}^{2}s_{W}^{2}}+$
$\displaystyle\kern-12.0pt+\delta_{\rm CKM}+\delta Z_{e}-\frac{\delta
s_{W}}{s_{W}}+\delta Z_{WW}+\delta Z^{L}_{cc}+\delta Z^{L}_{bb}+$
$\displaystyle\kern-12.0pt+2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})Q_{c}C_{f}^{*}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{A},m_{W},m_{c})+$
$\displaystyle\kern-12.0pt+2(m_{c}^{2}-m_{b}^{2}-m_{W}^{2})Q_{b}C_{f}^{*}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{A},m_{b})+$
$\displaystyle\kern-12.0pt+2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})Q_{c}Q_{b}C_{f}^{*}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{A})+$
$\displaystyle\kern-12.0pt-\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{Z},m_{W},m_{c})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{c}^{2}\left(m_{b}^{2}(3m_{Z}^{2}-4m_{W}^{2})m_{Z}^{2}+(4m_{W}^{2}+m_{Z}^{2})\lambda^{\prime}\right)+$
$\displaystyle+2\Big{(}m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+3m_{W}^{2})m_{Z}^{4}+2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})m_{W}^{2}\lambda^{\prime}+$
$\displaystyle\qquad+2\left(m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})+2\lambda^{\prime}\right)m_{W}^{2}m_{Z}^{2}\Big{)}(2Q_{c}s_{W}^{2}-1)\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{Z},m_{b})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{b}^{2}\left(m_{c}^{2}(3m_{Z}^{2}-4m_{W}^{2})m_{Z}^{2}+(4m_{W}^{2}+m_{Z}^{2})\lambda^{\prime}\right)+$
$\displaystyle-2\Big{(}m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+3m_{W}^{2})m_{Z}^{4}+2(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})m_{W}^{2}\lambda^{\prime}+$
$\displaystyle\qquad+2\left(m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})+2\lambda^{\prime}\right)m_{W}^{2}m_{Z}^{2}\Big{)}(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{Z})}{2\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\times$
$\displaystyle\times(m_{c}^{2}+m_{b}^{2}-m_{W}^{2}-m_{Z}^{2})\Big{(}m_{W}^{2}m_{Z}^{2}+\lambda^{\prime}\Big{)}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle\kern-12.0pt+\frac{m_{c}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{H},m_{W},m_{c})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}(m_{H}^{2}-4m_{W}^{2})\Big{(}m_{b}^{2}m_{H}^{2}+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{b}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{H},m_{b})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}(m_{H}^{2}-4m_{W}^{2})\Big{(}m_{c}^{2}m_{H}^{2}+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{H})}{\lambda^{\prime}m_{W}^{2}s_{W}^{2}}m_{c}^{2}m_{b}^{2}\Big{(}m_{W}^{2}m_{H}^{2}+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{2}{\lambda}^{\prime}B_{f}(m_{c}^{2};m_{c},m_{A})Q_{c}(Q_{b}+1)\Big{(}m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{2}{\lambda}^{\prime}B_{f}(m_{b}^{2};m_{b},m_{A})Q_{b}(Q_{c}-1)\Big{(}m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{2m_{W}^{2}}{\lambda^{\prime}}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})(Q_{c}-Q_{b})B_{f}(m_{W}^{2};m_{W},m_{A})+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{c},m_{Z})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{c}^{2}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})(4m_{W}^{2}-3m_{Z}^{2})+$
$\displaystyle+4\left(2m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})m_{W}^{2}+2m_{W}^{2}\lambda^{\prime}-(m_{c}^{2}-m_{W}^{2})(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})m_{Z}^{2}\right)(2Q_{c}s_{W}^{2}-1)+\kern-33.0pt$
$\displaystyle-2\left(2m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})+(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})m_{Z}^{2}+2\lambda^{\prime}\right)m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{)}+\kern-21.0pt$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{b},m_{Z})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{b}^{2}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})(4m_{W}^{2}-3m_{Z}^{2})+$
$\displaystyle-4\Big{(}2m_{W}^{2}m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})+2m_{W}^{2}\lambda^{\prime}-(m_{b}^{2}-m_{W}^{2})(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})m_{Z}^{2}\Big{)}(2Q_{b}s_{W}^{2}+1)+\kern-29.0pt$
$\displaystyle-2\left(2m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})+(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})m_{Z}^{2}+2\lambda^{\prime}\right)m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{)}+\kern-21.0pt$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{W},m_{Z})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}\left((m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}+\lambda^{\prime}\right)(4m_{W}^{2}-3m_{Z}^{2})+$
$\displaystyle-4\left(2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{4}+2(m_{c}^{2}-m_{W}^{2})m_{W}^{2}m_{Z}^{2}+m_{Z}^{2}\lambda^{\prime}\right)(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+4\left(2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{4}+2(m_{b}^{2}-m_{W}^{2})m_{W}^{2}m_{Z}^{2}+m_{Z}^{2}\lambda^{\prime}\right)(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{m_{c}^{2}B_{f}(m_{c}^{2};m_{c},m_{H})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{H}^{2}+4m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})+4\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{m_{b}^{2}B_{f}(m_{b}^{2};m_{b},m_{H})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{H}^{2}+4m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})+4\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{W},m_{H})}{8\lambda^{\prime}m_{W}^{2}s_{W}^{2}}(m_{H}^{2}-4m_{W}^{2})\Big{(}(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}+\lambda^{\prime}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{c}^{2};m_{b},m_{W})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{c}^{2}m_{b}^{2}(m_{H}^{2}-3m_{Z}^{2})+(m_{b}^{2}+6m_{W}^{2})\lambda^{\prime}+$
$\displaystyle+2m_{c}^{2}(3m_{b}^{2}-3m_{c}^{2}+5m_{W}^{2})m_{W}^{2}+2m_{c}^{2}(m_{b}^{2}-m_{c}^{2}+3m_{W}^{2})m_{Z}^{2}(2Q_{b}+1)s_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{b}^{2};m_{W},m_{c})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}m_{c}^{2}m_{b}^{2}(m_{H}^{2}-3m_{Z}^{2})+(m_{c}^{2}+6m_{W}^{2})\lambda^{\prime}+$
$\displaystyle+2m_{b}^{2}(3m_{c}^{2}-3m_{b}^{2}+5m_{W}^{2})m_{W}^{2}-2m_{b}^{2}(m_{c}^{2}-m_{b}^{2}+3m_{W}^{2})m_{Z}^{2}(2Q_{c}-1)s_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{c},m_{b})}{4\lambda^{\prime}m_{W}^{2}s_{W}^{2}}\Big{(}\left(\lambda^{\prime}-4m_{c}^{2}m_{b}^{2}-2(m_{c}^{2}+m_{b}^{2}-2m_{W}^{2})m_{W}^{2}\right)m_{W}^{2}+$
$\displaystyle+4m_{W}^{4}m_{Z}^{2}(1+Q_{b}-Q_{c})s_{W}^{2}-\left(\lambda^{\prime}-2(m_{c}^{2}+m_{b}^{2}-m_{Z}^{2})m_{W}^{2}\right)m_{Z}^{2}(2Q_{c}-1)(2Q_{b}+1)s_{W}^{2}\Big{)},\kern-9.0pt$
$\displaystyle V_{+}\ =\
-\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{Z},m_{W},m_{c})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}(4m_{W}^{2}+m_{Z}^{2})\lambda^{\prime}-2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}m_{Z}^{2}+$
(B2)
$\displaystyle+(2m_{c}^{2}+m_{b}^{2}-2m_{W}^{2})m_{Z}^{4}-2(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})(2m_{W}^{2}+m_{Z}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{Z},m_{b})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}(4m_{W}^{2}+m_{Z}^{2})\lambda^{\prime}-2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}m_{Z}^{2}+$
$\displaystyle+(2m_{b}^{2}+m_{c}^{2}-2m_{W}^{2})m_{Z}^{4}+2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})(2m_{W}^{2}+m_{Z}^{2})m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{Z})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{W}^{2}m_{Z}^{2}+2\lambda^{\prime}+2\lambda^{\prime}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle-2\lambda^{\prime}(2Q_{b}s_{W}^{2}+1)+4m_{W}^{2}m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{H},m_{W},m_{c})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{b}^{2}m_{H}^{2}+\lambda^{\prime}+2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{H},m_{b})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{c}^{2}m_{H}^{2}+\lambda^{\prime}+2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{W}^{2}m_{H}^{2}+2\lambda^{\prime}+2(m_{c}^{2}+m_{b}^{2}-m_{W}^{2})m_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{2m_{W}^{2}}{\lambda^{\prime}}(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})Q_{c}(Q_{b}+1)B_{f}(m_{c}^{2};m_{c},m_{A})+$
$\displaystyle\kern-12.0pt-\frac{2m_{W}^{2}}{\lambda^{\prime}}(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})Q_{b}(Q_{c}-1)B_{f}(m_{b}^{2};m_{b},m_{A})+\frac{4m_{W}^{4}}{\lambda^{\prime}}(Q_{c}-Q_{b})B_{f}(m_{W}^{2};m_{W},m_{A})+\kern-15.0pt$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{c},m_{Z})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{c}^{2}(4m_{W}^{2}-3m_{Z}^{2})-2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})(2m_{W}^{2}+m_{Z}^{2})(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{b},m_{Z})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{b}^{2}(4m_{W}^{2}-3m_{Z}^{2})+2(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})(2m_{W}^{2}+m_{Z}^{2})(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle+2(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{W}^{2}B_{f}(m_{W}^{2};m_{W},m_{Z})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}4m_{W}^{2}-5m_{Z}^{2}+$
$\displaystyle-4(2m_{W}^{2}+m_{Z}^{2})(2Q_{c}s_{W}^{2}-1)+4(2m_{W}^{2}+m_{Z}^{2})(2Q_{b}s_{W}^{2}+1)\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{c}^{2}B_{f}(m_{c}^{2};m_{c},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{H}^{2}+2(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{b}^{2}B_{f}(m_{b}^{2};m_{b},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{H}^{2}+2(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})\Big{)}+$
$\displaystyle\kern-12.0pt+\frac{m_{W}^{2}B_{f}(m_{W}^{2};m_{W},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{H}^{2}-4m_{W}^{2}\Big{)}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{b},m_{W})}{4\lambda^{\prime}s_{W}^{2}}\left(m_{c}^{2}(m_{H}^{2}+3m_{Z}^{2})+4(m_{c}^{2}-m_{b}^{2}+m_{W}^{2})(Q_{b}m_{Z}^{2}s_{W}^{2}+m_{W}^{2})\right)+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{W},m_{c})}{4\lambda^{\prime}s_{W}^{2}}\left(m_{b}^{2}(m_{H}^{2}+3m_{Z}^{2})-4(m_{b}^{2}-m_{c}^{2}+m_{W}^{2})(Q_{c}m_{Z}^{2}s_{W}^{2}-m_{W}^{2})\right)+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{W}^{2};m_{c},m_{b})}{4\lambda^{\prime}s_{W}^{2}}\Big{(}m_{W}^{2}(m_{H}^{2}-m_{Z}^{2})+2\lambda^{\prime}+2(m_{c}^{2}+m_{b}^{2}+m_{W}^{2})m_{W}^{2}+$
$\displaystyle-4m_{Z}^{2}m_{W}^{2}s_{W}^{2}(2Q_{c}-1)(2Q_{b}+1)\Big{)},$
$\displaystyle V_{1}\ =\
\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{Z},m_{W},m_{c})}{2\lambda^{\prime
2}s_{W}^{2}}\times$ (B3)
$\displaystyle\Big{\\{}\lambda^{\prime}(\lambda^{\prime}+2m_{b}^{2}m_{Z}^{2})(4m_{W}^{2}+m_{Z}^{2})+(7\lambda^{\prime}+3m_{b}^{2}m_{Z}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}m_{Z}^{2}+$
$\displaystyle+2\Big{[}\lambda^{\prime}\left(\lambda^{\prime}-2m_{b}^{2}(2m_{W}^{2}-m_{Z}^{2})\right)-2(\lambda^{\prime}+3m_{b}^{2}m_{Z}^{2})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\Big{]}m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)\Big{\\}}+\kern-28.0pt$
$\displaystyle\kern-12.0pt+\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{Z},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\Big{\\{}m_{b}^{2}\left(\lambda^{\prime}(2m_{W}^{2}+m_{Z}^{2})+6m_{c}^{2}m_{W}^{2}m_{Z}^{2}\right)+2\Big{[}\lambda^{\prime
2}-4\lambda^{\prime}(m_{c}^{2}-m_{W}^{2})m_{W}^{2}+$
$\displaystyle\qquad\qquad+2\left(\lambda^{\prime}(m_{c}^{2}+m_{W}^{2})+3m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{b}s_{W}^{2}+1)\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2},m_{c},m_{b},m_{Z})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\Big{\\{}m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})(\lambda^{\prime}+3m_{W}^{2}m_{Z}^{2})+2\lambda^{\prime}(\lambda^{\prime}+2m_{W}^{2}m_{Z}^{2})(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle-2(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})(2\lambda^{\prime}+3m_{W}^{2}m_{Z}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2},m_{H},m_{W},m_{c})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\times\Big{\\{}\lambda^{\prime}\left(\lambda^{\prime}+4m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\right)+$
$\displaystyle-
m_{b}^{2}\left(\lambda^{\prime}-3(m_{c}^{2}-m_{b}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{b}^{2}m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{H},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}2\lambda^{\prime}(m_{W}^{2}-2m_{b}^{2}+2m_{c}^{2})-\left(\lambda^{\prime}-6m_{c}^{2}(m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}\Big{\\}}+\kern-40.0pt$
$\displaystyle\kern-12.0pt-\frac{3m_{b}^{2}m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{H})}{2\lambda^{\prime
2}s_{W}^{2}}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})\left(\lambda^{\prime}+m_{W}^{2}m_{H}^{2}\right)+$
$\displaystyle\kern-12.0pt+\frac{4m_{W}^{2}}{\lambda^{\prime}}Q_{c}(Q_{b}+1)(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})B_{f}(m_{c}^{2};m_{c},m_{A})+\frac{8m_{W}^{2}}{\lambda^{\prime}}(Q_{c}-1)Q_{b}m_{b}^{2}B_{f}(m_{b}^{2};m_{b},m_{A})+\kern-26.0pt$
$\displaystyle\kern-12.0pt-\frac{4m_{W}^{2}}{\lambda^{\prime}}(Q_{c}-Q_{b})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})B_{f}(m_{W}^{2};m_{W},m_{A})+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{c}^{2};m_{c},m_{Z})}{4\lambda^{\prime
2}m_{c}^{2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{c}^{2}\Big{[}m_{c}^{2}(8m_{W}^{2}+m_{Z}^{2})+8(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}+3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{Z}^{2}\Big{]}+\kern-37.0pt$
$\displaystyle-4\lambda^{\prime}m_{c}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)-4\Big{[}\lambda^{\prime}(2m_{c}^{2}-m_{Z}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}+$
$\displaystyle\qquad+\left(\lambda^{\prime}m_{c}^{2}(m_{c}^{2}+m_{b}^{2})+6m_{c}^{2}m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+2\Big{[}2\lambda^{\prime}m_{c}^{2}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})+$
$\displaystyle\qquad-\left(\lambda^{\prime}(m_{W}^{2}-m_{b}^{2}+2m_{c}^{2})-6m_{c}^{2}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{\\}}+\kern-48.0pt$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{b},m_{Z})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{b}^{2}m_{Z}^{2}+$
$\displaystyle-4\Big{[}4\lambda^{\prime}m_{b}^{2}m_{W}^{2}-\left(\lambda^{\prime}(2m_{b}^{2}-m_{W}^{2})-6m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle-2\Big{[}4\lambda^{\prime}m_{b}^{2}+(\lambda^{\prime}+12m_{b}^{2}m_{W}^{2})m_{Z}^{2}\Big{]}m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{W}^{2};m_{W},m_{Z})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{W}^{2}(16m_{W}^{2}+m_{Z}^{2})+4\lambda^{\prime}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}+$
$\displaystyle+3\left(\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})+4m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{Z}^{2}+$
$\displaystyle-4\Big{[}2\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}+\left(\lambda^{\prime}(m_{c}^{2}-m_{b}^{2})+12m_{b}^{2}m_{W}^{4}\right)m_{Z}^{2}\Big{]}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+4\Big{[}2\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}+\left(\lambda^{\prime}(m_{c}^{2}-m_{b}^{2})+6(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{4}\right)m_{Z}^{2}\Big{]}(2Q_{b}s_{W}^{2}+1)\Big{\\}}+\kern-40.0pt$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{c}^{2};m_{c},m_{H})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{c}^{2}m_{H}^{2}+$
$\displaystyle-\lambda^{\prime}(4m_{c}^{2}+m_{H}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})+12m_{c}^{2}m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{b}^{2}B_{f}(m_{b}^{2};m_{b},m_{H})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}8\lambda^{\prime}m_{b}^{2}-3\left(\lambda^{\prime}+4m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})\right)m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{W},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{\\{}4(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}-(m_{c}^{2}-m_{b}^{2})m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{b},m_{W})}{2\lambda^{\prime
2}m_{c}^{2}s_{W}^{2}}\Big{\\{}\lambda^{\prime 2}m_{W}^{2}+(\lambda^{\prime
2}-12m_{c}^{4}m_{W}^{4})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})+$
$\displaystyle+\lambda^{\prime}m_{c}^{2}\left(2m_{b}^{2}m_{W}^{2}-(5m_{W}^{2}-m_{c}^{2})(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})\right)+$
$\displaystyle-
m_{c}^{2}m_{b}^{2}(\lambda^{\prime}+6m_{c}^{2}m_{W}^{2})m_{Z}^{2}-m_{c}^{2}m_{b}^{2}\left(\lambda^{\prime}-6m_{c}^{2}(m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}+$
$\displaystyle-4m_{c}^{2}\Big{[}\lambda^{\prime}(m_{c}^{2}+m_{W}^{2})+3m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\Big{]}m_{Z}^{2}(2Q_{b}+1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{W},m_{c})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime
2}-\lambda^{\prime}(3m_{W}^{2}+m_{b}^{2})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-12m_{b}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{4}+\kern-23.0pt$
$\displaystyle-
m_{b}^{2}\left(2\lambda^{\prime}+3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}+m_{b}^{2}\left(\lambda^{\prime}-3(m_{c}^{2}-m_{b}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}+\kern-23.0pt$
$\displaystyle-4m_{b}^{2}\Big{[}\lambda^{\prime}-3(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\Big{]}m_{Z}^{2}(2Q_{c}-1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{c},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}(\lambda^{\prime}+4m_{W}^{4})+\left(\lambda^{\prime}(m_{c}^{2}+m_{b}^{2})+6m_{W}^{6}\right)(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})+$
$\displaystyle+3m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}(m_{Z}^{2}-m_{H}^{2})-6(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{4}m_{Z}^{2}(2Q_{c}-1)s_{W}^{2}+$
$\displaystyle+2\left(2\lambda^{\prime}+3(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\right)m_{W}^{2}m_{Z}^{2}(2Q_{b}+1)s_{W}^{2}+$
$\displaystyle-(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})(\lambda^{\prime}+6m_{W}^{2}m_{Z}^{2}s_{W}^{2})m_{Z}^{2}(2Q_{c}-1)(2Q_{b}+1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{\ln(m_{c}/\bar{\mu})-1}{\lambda^{\prime}s_{W}^{2}}\Big{\\{}m_{c}^{4}-m_{c}^{2}m_{b}^{2}+4m_{c}^{2}m_{W}^{2}-m_{b}^{2}m_{W}^{2}+m_{W}^{4}+$
$\displaystyle+(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{Z}^{2}(2Q_{c}-1)(2Q_{b}+1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{b}^{2}(\ln(m_{b}/\bar{\mu})-1)}{\lambda^{\prime}m_{c}^{2}s_{W}^{2}}\Big{\\{}m_{b}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-2m_{W}^{4}-2m_{c}^{2}m_{Z}^{2}(2Q_{c}-1)(2Q_{b}+1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{W}^{2}(\ln(m_{W}/\bar{\mu})-1)}{\lambda^{\prime}m_{c}^{2}s_{W}^{2}}\Big{\\{}2m_{c}^{4}-m_{c}^{2}m_{b}^{2}+2m_{c}^{2}m_{W}^{2}-m_{b}^{4}-m_{b}^{2}m_{W}^{2}+2m_{W}^{4}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{Z}^{2}(2\ln(m_{Z}/\bar{\mu})-1)}{4\lambda^{\prime}m_{c}^{2}s_{W}^{2}}\Big{\\{}m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-4(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+8m_{c}^{2}m_{W}^{2}(2Q_{b}s_{W}^{2}+1)+2(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)(2Q_{b}s_{W}^{2}+1)\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{H}^{2}(2\ln(m_{H}/\bar{\mu})-1)}{4\lambda^{\prime}s_{W}^{2}}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-\frac{m_{b}^{2}+2m_{W}^{2}}{2m_{c}^{2}s_{W}^{2}},$
$\displaystyle V_{2}\ =\
\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2},m_{Z},m_{W},m_{c})}{2\lambda^{\prime
2}s_{W}^{2}}\times$ (B4)
$\displaystyle\Big{\\{}m_{c}^{2}\left(\lambda^{\prime}(2m_{W}^{2}+m_{Z}^{2})+6m_{b}^{2}m_{W}^{2}m_{Z}^{2}\right)-2\Big{[}\lambda^{\prime
2}+4\lambda^{\prime}(m_{W}^{2}-m_{b}^{2})m_{W}^{2}+$
$\displaystyle\qquad+2\left(\lambda^{\prime}(m_{W}^{2}+m_{b}^{2})+3m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{c}s_{W}^{2}-1)\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2},m_{W},m_{Z},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime
2}(4m_{W}^{2}+m_{Z}^{2})+8\lambda^{\prime}m_{c}^{2}m_{W}^{2}m_{Z}^{2}+$
$\displaystyle+7\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}m_{Z}^{2}+m_{c}^{2}\left(2\lambda^{\prime}+3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\right)m_{Z}^{4}+$
$\displaystyle-2\Big{[}\lambda^{\prime
2}-2\lambda^{\prime}m_{c}^{2}(4m_{W}^{2}-m_{Z}^{2})-2\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}+$
$\displaystyle\qquad-6m_{c}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}m_{Z}^{2}\Big{]}m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{Z}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{Z})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\Big{\\{}m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})(\lambda^{\prime}+3m_{W}^{2}m_{Z}^{2})-2\lambda^{\prime}(\lambda^{\prime}+2m_{W}^{2}m_{Z}^{2})(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle-2(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})(2\lambda^{\prime}+3m_{W}^{2}m_{Z}^{2})m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)(2Q_{c}s_{W}^{2}-1)\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{c}^{2}m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{H},m_{W},m_{c})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\Big{\\{}2\lambda^{\prime}(m_{W}^{2}-2m_{c}^{2}+2m_{b}^{2})-\left(\lambda^{\prime}+6m_{b}^{2}(m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{W},m_{H},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\times$
$\displaystyle\Big{\\{}\lambda^{\prime}\Big{[}\lambda^{\prime}+4m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})-3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\Big{]}+$
$\displaystyle-
m_{c}^{2}\left(\lambda^{\prime}+3(m_{c}^{2}-m_{b}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{3m_{c}^{2}m_{H}^{2}C_{f}(m_{c}^{2},m_{b}^{2},m_{W}^{2};m_{c},m_{b},m_{H})}{2\lambda^{\prime
2}s_{W}^{2}}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})(\lambda^{\prime}+m_{W}^{2}m_{H}^{2})+$
$\displaystyle\kern-12.0pt+\frac{8m_{W}^{2}}{\lambda^{\prime}}Q_{c}(Q_{b}+1)m_{c}^{2}B_{f}(m_{c}^{2};m_{c},m_{A})+\frac{4m_{W}^{2}}{\lambda^{\prime}}(Q_{c}-1)Q_{b}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})B_{f}(m_{b}^{2};m_{b},m_{A})+\kern-26.0pt$
$\displaystyle\kern-12.0pt-\frac{4m_{W}^{2}}{\lambda^{\prime}}(Q_{c}-Q_{b})(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})B_{f}(m_{W}^{2};m_{W},m_{A})+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{c},m_{Z})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{c}^{2}m_{Z}^{2}+$
$\displaystyle+4\Big{[}4\lambda^{\prime}m_{c}^{2}m_{W}^{2}+\left(\lambda^{\prime}(m_{W}^{2}-2m_{c}^{2})+6m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle-2\left(4\lambda^{\prime}m_{c}^{2}+(\lambda^{\prime}+12m_{c}^{2}m_{W}^{2})m_{Z}^{2}\right)m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)(2Q_{c}s_{W}^{2}-1)\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{b}^{2};m_{b},m_{Z})}{4\lambda^{\prime
2}m_{b}^{2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{b}^{2}\Big{[}m_{b}^{2}(8m_{W}^{2}+m_{Z}^{2})+8(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}+3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{Z}^{2}\Big{]}+\kern-39.0pt$
$\displaystyle-4\Big{[}\lambda^{\prime}(m_{Z}^{2}-2m_{b}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}+$
$\displaystyle\qquad-\left(\lambda^{\prime}m_{b}^{2}(m_{c}^{2}+m_{b}^{2})+6m_{c}^{2}m_{b}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}\Big{]}(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle+4\lambda^{\prime}m_{b}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{Z}^{2}(2Q_{c}s_{W}^{2}-1)+2\Big{[}2m_{b}^{2}(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})(\lambda^{\prime}+3m_{W}^{2}m_{Z}^{2})+$
$\displaystyle\qquad-\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}+2m_{b}^{2})m_{Z}^{2}\Big{]}m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)(2Q_{c}s_{W}^{2}-1)\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{W}^{2};m_{W},m_{Z})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}m_{W}^{2}(16m_{W}^{2}+m_{Z}^{2})+4\lambda^{\prime}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}+$
$\displaystyle\qquad+3\left(\lambda^{\prime}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})+4m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}\right)m_{Z}^{2}+$
$\displaystyle+4\Big{[}2\lambda^{\prime}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}-\left(\lambda^{\prime}(m_{c}^{2}-m_{b}^{2})-12m_{c}^{2}m_{W}^{4}\right)m_{Z}^{2}\Big{]}(2Q_{b}s_{W}^{2}+1)+$
$\displaystyle-4\Big{[}2\lambda^{\prime}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}-\left(\lambda^{\prime}(m_{c}^{2}-m_{b}^{2})-6(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{4}\right)m_{Z}^{2}\Big{]}(2Q_{c}s_{W}^{2}-1)\Big{\\}}+\kern-39.0pt$
$\displaystyle\kern-12.0pt-\frac{m_{c}^{2}B_{f}(m_{c}^{2};m_{c},m_{H})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}(8m_{c}^{2}-3m_{H}^{2})-12m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{b},m_{H})}{4\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}(4m_{b}^{2}+m_{H}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})-m_{b}^{2}\left(\lambda^{\prime}+12m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})\right)m_{H}^{2}\Big{\\}}+\kern-39.0pt$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{W},m_{H})}{4\lambda^{\prime}s_{W}^{2}}\Big{\\{}4(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}+(m_{c}^{2}-m_{b}^{2})m_{H}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{c}^{2};m_{b},m_{W})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime
2}-\lambda^{\prime}(3m_{W}^{2}+m_{c}^{2})(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})-12m_{c}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{4}+\kern-10.0pt$
$\displaystyle-
m_{c}^{2}\Big{[}2\lambda^{\prime}+3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\Big{]}m_{Z}^{2}+m_{c}^{2}\left(\lambda^{\prime}+3(m_{c}^{2}-m_{b}^{2})(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}+\kern-11.0pt$
$\displaystyle+4m_{c}^{2}\left(\lambda^{\prime}-3(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{Z}^{2}(2Q_{b}+1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{B_{f}(m_{b}^{2};m_{W},m_{c})}{2\lambda^{\prime
2}m_{b}^{2}s_{W}^{2}}\Big{\\{}\lambda^{\prime 2}m_{W}^{2}+(\lambda^{\prime
2}-12m_{b}^{4}m_{W}^{4})(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})+$
$\displaystyle+\lambda^{\prime}m_{b}^{2}\left(2m_{c}^{2}m_{W}^{2}-(5m_{W}^{2}-m_{b}^{2})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})\right)+$
$\displaystyle-
m_{c}^{2}m_{b}^{2}(\lambda^{\prime}+6m_{b}^{2}m_{W}^{2})m_{Z}^{2}-m_{c}^{2}m_{b}^{2}\left(\lambda^{\prime}+6m_{b}^{2}(m_{c}^{2}-m_{b}^{2})\right)m_{H}^{2}+$
$\displaystyle+4m_{b}^{2}\Big{[}\lambda^{\prime}(m_{W}^{2}+m_{b}^{2})+3m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\Big{]}m_{Z}^{2}(2Q_{c}-1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{B_{f}(m_{W}^{2};m_{c},m_{b})}{2\lambda^{\prime
2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}(\lambda^{\prime}+4m_{W}^{4})+\left(\lambda^{\prime}(m_{c}^{2}+m_{b}^{2})+6m_{W}^{6}\right)(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})+$
$\displaystyle+3m_{c}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{W}^{2}(m_{Z}^{2}-m_{H}^{2})+6(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{4}m_{Z}^{2}(2Q_{b}+1)s_{W}^{2}+$
$\displaystyle-2\left(2\lambda^{\prime}+3(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})m_{W}^{2}\right)m_{W}^{2}m_{Z}^{2}(2Q_{c}-1)s_{W}^{2}+$
$\displaystyle-(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})(\lambda^{\prime}+6m_{W}^{2}m_{Z}^{2}s_{W}^{2})m_{Z}^{2}(2Q_{b}+1)(2Q_{c}-1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{c}^{2}(\ln(m_{c}/\bar{\mu})-1)}{\lambda^{\prime}m_{b}^{2}s_{W}^{2}}\Big{\\{}2m_{W}^{4}-m_{c}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})+2m_{b}^{2}m_{Z}^{2}(2Q_{b}+1)(2Q_{c}-1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{\ln(m_{b}/\bar{\mu})-1}{\lambda^{\prime}s_{W}^{2}}\Big{\\{}2m_{b}^{2}m_{W}^{2}+$
$\displaystyle+(m_{W}^{2}+m_{b}^{2})(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})+(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{Z}^{2}(2Q_{b}+1)(2Q_{c}-1)s_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt+\frac{m_{W}^{2}(\ln(m_{W}/\bar{\mu})-1)}{\lambda^{\prime}m_{b}^{2}s_{W}^{2}}\Big{\\{}\lambda^{\prime}-3m_{b}^{2}(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})-3(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{Z}^{2}(2\ln(m_{Z}/\bar{\mu})-1)}{4\lambda^{\prime}m_{b}^{2}s_{W}^{2}}\Big{\\{}m_{b}^{2}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})+$
$\displaystyle+4(m_{W}^{2}-m_{c}^{2}-m_{b}^{2})m_{W}^{2}(2Q_{b}s_{W}^{2}+1)-8m_{b}^{2}m_{W}^{2}(2Q_{c}s_{W}^{2}-1)+$
$\displaystyle+2(m_{W}^{2}-m_{c}^{2}+m_{b}^{2})m_{Z}^{2}(2Q_{b}s_{W}^{2}+1)(2Q_{c}s_{W}^{2}-1)\Big{\\}}+$
$\displaystyle\kern-12.0pt-\frac{m_{H}^{2}(2\ln(m_{H}/\bar{\mu})-1)}{4\lambda^{\prime}s_{W}^{2}}(m_{W}^{2}-m_{b}^{2}+m_{c}^{2})-\frac{2m_{W}^{2}+m_{c}^{2}}{2m_{b}^{2}s_{W}^{2}},$

where $\lambda^{\prime}=\lambda(m_{W}^{2},m_{c}^{2},m_{b}^{2})$. $\bar{\mu}$
is the $\overline{\rm MS}$ renormalisation scale which for the decay process
considered here is set to be the mass of the $W$ boson. The IR subtracted main
form factor $V_{-}^{*}$ contains also the UV finite parts of the counter terms
which are listed in Appendix C.

##  Appendix C UV finite parts of the counter terms

$\displaystyle\delta Z^{L}_{ij}\ =\
\frac{-1}{8m_{W}^{2}s_{W}^{2}}\sum_{k}V_{ik}V_{kj}\Bigg{\\{}m_{k}^{2}A_{f}(m_{k})-m_{W}^{2}A_{f}(m_{W})-2m_{W}^{2}+$
$\displaystyle+\frac{B_{f}(m_{i}^{2};m_{k},m_{W})}{m_{i}^{2}-m_{j}^{2}}\left((m_{i}^{2}-m_{k}^{2})(m_{j}^{2}-m_{k}^{2})+(2m_{i}^{2}-m_{j}^{2}+m_{k}^{2}-2m_{W}^{2})m_{W}^{2}\right)+$
$\displaystyle+\frac{B_{f}(m_{j}^{2};m_{k},m_{W})}{m_{j}^{2}-m_{i}^{2}}\left((m_{j}^{2}-m_{k}^{2})(m_{i}^{2}-m_{k}^{2})+(2m_{j}^{2}-m_{i}^{2}+m_{k}^{2}-2m_{W}^{2})m_{W}^{2}\right)\Bigg{\\}},$
$\displaystyle\delta Z_{e}\ =\
-\frac{1}{3}\Bigg{\\{}1-\frac{21}{2}\left(1-A_{f}(m_{W})\right)+2\sum_{f}Q_{f}^{2}\left(1-A_{f}(m_{f})\right)\Bigg{\\}},$
$\displaystyle\frac{\delta s_{W}}{s_{W}}\ =\
\frac{-1}{72m_{W}^{2}m_{Z}^{6}s_{W}^{4}}\Bigg{\\{}4m_{W}^{2}(m_{W}^{2}-m_{Z}^{2})(36m_{W}^{4}+24m_{W}^{2}m_{Z}^{2}+m_{Z}^{4})+$
$\displaystyle+3(4m_{W}^{2}-m_{Z}^{2})(12m_{W}^{4}+20m_{W}^{2}m_{Z}^{2}+m_{Z}^{4})\times$
$\displaystyle\qquad\times\left(m_{W}^{2}B_{f}(m_{Z}^{2};m_{W},m_{W})-m_{W}^{2}A_{f}(m_{W})-m_{Z}^{2}B_{f}(m_{W}^{2};m_{W},m_{Z})+m_{Z}^{2}A_{f}(m_{Z})\right)+\kern-8.0pt$
$\displaystyle+3m_{Z}^{4}\Big{(}(12m_{W}^{4}-4m_{W}^{2}m_{H}^{2}+m_{H}^{4})B_{f}(m_{W}^{2};m_{W},m_{H})-m_{W}^{2}m_{H}^{2}(A_{f}(m_{H})-A_{f}(m_{W}))\Big{)}+\kern-11.0pt$
$\displaystyle-3m_{W}^{2}m_{Z}^{2}\Big{(}(12m_{Z}^{4}-4m_{Z}^{2}m_{H}^{2}+m_{H}^{4})B_{f}(m_{Z}^{2};m_{Z},m_{H})-m_{Z}^{2}m_{H}^{2}(A_{f}(m_{H})-A_{f}(m_{Z}))\Big{)}+\kern-14.0pt$
$\displaystyle+144m_{W}^{4}m_{Z}^{2}(m_{W}^{2}-m_{Z}^{2})B_{f}(m_{W}^{2},m_{W},m_{A})+6m_{W}^{4}m_{Z}^{2}(42m_{W}^{2}-m_{Z}^{2})(A_{f}(m_{W})-A_{f}(m_{Z}))+\kern-33.0pt$
$\displaystyle+6m_{W}^{2}m_{Z}^{2}(m_{W}^{2}-m_{Z}^{2})(18m_{W}^{2}-5m_{Z}^{2})A_{f}(m_{Z})+3(m_{W}^{2}-m_{Z}^{2})m_{Z}^{2}m_{H}^{4}A_{f}(m_{H})+$
$\displaystyle\kern-12.0pt-
m_{Z}^{4}\sum_{i}\Big{[}4m_{W}^{2}(m_{W}^{2}-3m_{\nu_{i}}^{2}-3m_{\ell_{i}}^{2})+$
$\displaystyle+6\left((m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})^{2}+(m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2}-2m_{W}^{2})m_{W}^{2}\right)B_{f}(m_{W}^{2};m_{\nu_{i}},m_{\ell_{i}})+$
$\displaystyle-6(m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})\left(m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})-m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}})\right)+12m_{W}^{2}\left(m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})+m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}})\right)\Big{]}+$
$\displaystyle\kern-12.0pt-
m_{Z}^{4}\sum_{i,j}|V_{ij}|^{2}\Big{[}4m_{W}^{2}(m_{W}^{2}-3m_{i}^{2}-3m_{j}^{2})+$
$\displaystyle+6\left((m_{i}^{2}-m_{j}^{2})^{2}+(m_{i}^{2}+m_{j}^{2}-2m_{W}^{2})m_{W}^{2}\right)B_{f}(m_{W}^{2};m_{i},m_{j})+$
$\displaystyle-6(m_{i}^{2}-m_{j}^{2})\left(m_{i}^{2}A_{f}(m_{i})-m_{j}^{2}A_{f}(m_{j})\right)+12m_{W}^{2}\left(m_{i}^{2}A_{f}(m_{i})+m_{j}^{2}A_{f}(m_{j})\right)\Big{]}+$
$\displaystyle\kern-12.0pt-8m_{W}^{4}(m_{W}^{2}-m_{Z}^{2})\sum_{f}\Big{[}(g_{f}^{-2}+g_{f}^{+2})\left(m_{Z}^{2}-6m_{f}^{2}+6m_{f}^{2}A_{f}(m_{f})\right)+$
$\displaystyle+3\left((g_{f}^{-2}+g_{f}^{+2}-6g_{f}^{-}g_{f}^{+})m_{f}^{2}-(g_{f}^{-2}+g_{f}^{+2})m_{Z}^{2}\right)B_{f}(m_{Z}^{2};m_{f},m_{f})\Big{]}\Bigg{\\}},$
$\displaystyle\delta Z_{WW}\ =\
\frac{1}{72m_{W}^{4}m_{Z}^{2}s_{W}^{2}}\Bigg{\\{}4m_{W}^{4}m_{Z}^{2}+$
$\displaystyle+3(48m_{W}^{6}-16m_{W}^{4}m_{Z}^{2}+6m_{W}^{2}m_{Z}^{4}+m_{Z}^{6})B_{f}(m_{W}^{2};m_{W},m_{Z})+$
$\displaystyle-3m_{Z}^{2}(2m_{W}^{2}-m_{H}^{2})m_{H}^{2}B_{f}(m_{W}^{2};m_{W},m_{H})-144m_{W}^{4}(m_{W}^{2}-m_{Z}^{2})B_{f}(m_{W}^{2};m_{W},m_{A})+$
$\displaystyle+3m_{W}^{2}(4m_{W}^{2}-m_{Z}^{2})(12m_{W}^{4}+20m_{W}^{2}m_{Z}^{2}+m_{Z}^{4})B^{\prime}_{f}(m_{W}^{2};m_{W},m_{Z})+$
$\displaystyle-3m_{W}^{2}m_{Z}^{2}(12m_{W}^{4}-4m_{W}^{2}m_{H}^{2}+m_{H}^{4})B^{\prime}_{f}(m_{W}^{2};m_{W},m_{H})+$
$\displaystyle+144m_{W}^{6}m_{Z}^{2}s_{W}^{2}B^{\prime}_{f}(m_{W}^{2};m_{W},m_{A})-3m_{W}^{2}m_{Z}^{2}(2m_{W}^{2}-m_{Z}^{2}-m_{H}^{2})A_{f}(m_{W})+$
$\displaystyle+3m_{Z}^{2}(m_{W}^{2}-m_{Z}^{2})(8m_{W}^{2}+m_{Z}^{2})A_{f}(m_{Z})+3m_{Z}^{2}(m_{W}^{2}-m_{H}^{2})m_{H}^{2}A_{f}(m_{H})+$
$\displaystyle\kern-12.0pt-
m_{Z}^{2}\sum_{i}\Big{[}3\Big{(}2\left((m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})^{2}+2m_{W}^{4}\right)B_{f}(m_{W}^{2};m_{\nu_{i}},m_{\ell_{i}})+$
$\displaystyle\qquad-2m_{W}^{2}\left((m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})^{2}+(m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2}-2m_{W}^{2})m_{W}^{2}\right)B^{\prime}_{f}(m_{W}^{2};m_{\nu_{i}},m_{\ell_{i}})+$
$\displaystyle\qquad-2(m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})(m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})-m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}}))\Big{)}-4m_{W}^{4}\Big{]}+$
$\displaystyle\kern-12.0pt-
m_{Z}^{2}\sum_{i,j}|V_{ij}|^{2}\Big{[}3\Big{(}2\left((m_{i}^{2}-m_{j}^{2})^{2}+2m_{W}^{4}\right)B_{f}(m_{W}^{2};m_{i},m_{j})+$
$\displaystyle\qquad-2m_{W}^{2}\left((m_{i}^{2}-m_{j}^{2})^{2}+(m_{i}^{2}+m_{j}^{2}-2m_{W}^{2})m_{W}^{2}\right)B^{\prime}_{f}(m_{W}^{2};m_{i},m_{j})+$
$\displaystyle\qquad-2(m_{i}^{2}-m_{j}^{2})(m_{i}^{2}A_{f}(m_{i})-m_{j}^{2}A_{f}(m_{j}))\Big{)}-4m_{W}^{4}\Big{]}\Big{\\}},$
$\displaystyle\delta Z^{L}_{cc}\ =\
\frac{-1}{16m_{c}^{2}m_{W}^{2}s_{W}^{2}}\Bigg{\\{}2m_{c}^{4}+2m_{c}^{2}\left(4(Q_{c}^{2}+g_{c}^{-2})m_{W}^{2}s_{W}^{2}+m_{c}^{2}\right)(A_{f}(m_{c})-1)+$
$\displaystyle+m_{Z}^{2}(8g_{c}^{-2}m_{W}^{2}s_{W}^{2}+m_{c}^{2})\left(B_{f}(m_{c}^{2};m_{c},m_{Z})-A_{f}(m_{Z})\right)+$
$\displaystyle+m_{c}^{2}m_{H}^{2}\left(B_{f}(m_{c}^{2};m_{c},m_{H})-A_{f}(m_{H})\right)+$
$\displaystyle+2m_{c}^{2}\left(8g_{c}^{-2}(2m_{c}^{2}-m_{Z}^{2})m_{W}^{2}s_{W}^{2}-32g_{c}^{-}g_{c}^{+}m_{c}^{2}m_{W}^{2}s_{W}^{2}-m_{c}^{2}m_{Z}^{2}\right)B^{\prime}_{f}(m_{c}^{2};m_{c},m_{Z})+$
$\displaystyle+2m_{c}^{4}(4m_{c}^{2}-m_{H}^{2})B^{\prime}_{f}(m_{c}^{2};m_{c},m_{H})-32m_{c}^{4}m_{W}^{2}s_{W}^{2}Q_{c}^{2}B^{\prime}_{f}(m_{c}^{2};m_{c},m_{A})+$
$\displaystyle\kern-12.0pt+\sum_{k}|V_{ck}|^{2}\Big{[}-4m_{c}^{2}m_{W}^{2}+2(m_{k}^{2}+2m_{W}^{2})\times$
$\displaystyle\qquad\times\left((m_{c}^{2}-m_{k}^{2}+m_{W}^{2})B_{f}(m_{c}^{2};m_{k},m_{W})+m_{k}^{2}A_{f}(m_{k})-m_{W}^{2}A_{f}(m_{W})\right)+$
$\displaystyle-4m_{c}^{2}(m_{c}^{2}m_{k}^{2}-m_{k}^{4}-2m_{c}^{2}m_{W}^{2}-m_{k}^{2}m_{W}^{2}+2m_{W}^{4})B^{\prime}_{f}(m_{c}^{2};m_{k},m_{W})\Big{]}\Bigg{\\}},$
$\displaystyle\delta Z^{L}_{bb}\ =\
\frac{-1}{16m_{b}^{2}m_{W}^{2}s_{W}^{2}}\Bigg{\\{}2m_{b}^{4}+2m_{b}^{2}\left(4(Q_{b}^{2}+g_{b}^{-2})m_{W}^{2}s_{W}^{2}+m_{b}^{2}\right)(A_{f}(m_{b})-1)+$
(C1)
$\displaystyle+m_{Z}^{2}(8g_{b}^{-2}m_{W}^{2}s_{W}^{2}+m_{b}^{2})\left(B_{f}(m_{b}^{2};m_{b},m_{Z})-A_{f}(m_{Z})\right)+$
$\displaystyle+m_{b}^{2}m_{H}^{2}\left(B_{f}(m_{b}^{2};m_{b},m_{H})-A_{f}(m_{H})\right)+$
$\displaystyle+2m_{b}^{2}\left(8g_{b}^{-2}(2m_{b}^{2}-m_{Z}^{2})m_{W}^{2}s_{W}^{2}-32g_{b}^{-}g_{b}^{+}m_{b}^{2}m_{W}^{2}s_{W}^{2}-m_{b}^{2}m_{Z}^{2}\right)B^{\prime}_{f}(m_{b}^{2};m_{b},m_{Z})+$
$\displaystyle+2m_{b}^{4}(4m_{b}^{2}-m_{H}^{2})B^{\prime}_{f}(m_{b}^{2};m_{b},m_{H})-32m_{b}^{4}m_{W}^{2}s_{W}^{2}Q_{b}^{2}B^{\prime}_{f}(m_{b}^{2};m_{b},m_{A})+$
$\displaystyle\kern-12.0pt+\sum_{k}|V_{kb}|^{2}\Big{[}-4m_{b}^{2}m_{W}^{2}+2(m_{k}^{2}+2m_{W}^{2})\times$
$\displaystyle\qquad\times\left((m_{b}^{2}-m_{k}^{2}+m_{W}^{2})B_{f}(m_{b}^{2};m_{k},m_{W})+m_{k}^{2}A_{f}(m_{k})-m_{W}^{2}A_{f}(m_{W})\right)+$
$\displaystyle-4m_{b}^{2}(m_{b}^{2}m_{k}^{2}-m_{k}^{4}-2m_{b}^{2}m_{W}^{2}-m_{k}^{2}m_{W}^{2}+2m_{W}^{4})B^{\prime}_{f}(m_{b}^{2};m_{k},m_{W})\Big{]}\Bigg{\\}}.$

In order to save the symmetry, the expressions presented here contain also
vanishing contributions from e.g. the neutrinos. However, in order to handle
two-point functions containing the neutrino masses appropriately, a couple of
limiting cases has to be calculated. Calculating
$B_{f}(m_{W}^{2};m_{\nu},m_{\ell})$ and the derivative
$B^{\prime}_{f}(m_{W}^{2};m_{\nu},m_{\ell})$, we start with the expansion of
the square root of Källén function for this particular mass configuration,

$\sqrt{\lambda(m_{W}^{2},m_{\nu}^{2},m_{\ell}^{2})}=m_{W}^{2}-m_{\ell}^{2}-\frac{p^{2}+m_{\ell}^{2}}{p^{2}-m_{\ell}^{2}}m_{\nu}^{2}+O(m_{\nu}^{4}).$
(C2)

The finite part of the two-point function
$B(p^{2};m_{1},m_{2})=i\bar{\mu}^{-2\varepsilon}/(4\pi)^{2}(1/\varepsilon+B_{f}(p^{2};m_{1},m_{2}))$
(for simplicity, $\bar{\mu}$ is taken to be the $\overline{\rm MS}$ scale) is
written as (cf. (B.1) in Ref. [48])

$\displaystyle B_{f}(p^{2};m_{1},m_{2})\ =\
2-\frac{1}{2}\left(\ln\left(\frac{m_{1}^{2}}{\bar{\mu}^{2}}\right)+\ln\left(\frac{m_{2}^{2}}{\bar{\mu}^{2}}\right)\right)-\frac{m_{1}^{2}-m_{2}^{2}}{2p^{2}}\ln\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)+\frac{1}{p^{2}}\times$
(C3)
$\displaystyle\kern-24.0pt\cases{\displaystyle-\sqrt{(m_{1}+m_{2})^{2}-p^{2}}\sqrt{(m_{1}-m_{2})^{2}-p^{2}}\ln\left(\frac{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}-\sqrt{(m_{1}-m_{2})^{2}-p^{2}}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}+\sqrt{(m_{1}-m_{2})^{2}-p^{2}}}\right)\cr\displaystyle-2\sqrt{(m_{1}+m_{2})^{2}-p^{2}}\sqrt{p^{2}-(m_{1}-m_{2})^{2}}\arctan\left(\frac{\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}}\right)\cr\displaystyle\sqrt{p^{2}-(m_{1}+m_{2})^{2}}\sqrt{p^{2}-(m_{1}-m_{2})^{2}}\left(\ln\left(\frac{\sqrt{p^{2}-(m_{1}+m_{2})^{2}}-\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}{\sqrt{p^{2}-(m_{1}+m_{2})^{2}}+\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}\right)+i\pi\right)\cr}\kern-17.0pt$

in the cases $p^{2}<(m_{1}-m_{2})^{2}$,
$(m_{1}-m_{2})^{2}<p^{2}<(m_{1}+m_{2})^{2}$, and $p^{2}>(m_{1}+m_{2})^{2}$,
respectively. The derivative of this finite part with respect to $p^{2}$
results in

$\displaystyle B^{\prime}_{f}(p^{2};m_{1},m_{2})\ =\
\frac{m_{1}^{2}-m_{2}^{2}}{2p^{4}}\ln\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)-\frac{1}{p^{2}}+\frac{1}{p^{4}}\times$
(C4)
$\displaystyle\kern-24.0pt\cases{\displaystyle\frac{(m_{1}^{2}-m_{2}^{2})^{2}-(m_{1}^{2}+m_{2}^{2})p^{2}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}\sqrt{(m_{1}-m_{2})^{2}-p^{2}}}\ln\left(\frac{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}-\sqrt{(m_{1}-m_{2})^{2}-p^{2}}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}+\sqrt{(m_{1}-m_{2})^{2}-p^{2}}}\right)\cr\displaystyle-2\frac{(m_{1}^{2}-m_{2}^{2})^{2}-(m_{1}^{2}+m_{2}^{2})p^{2}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}\arctan\left(\frac{\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}{\sqrt{(m_{1}+m_{2})^{2}-p^{2}}}\right)\cr\displaystyle-\frac{(m_{1}^{2}-m_{2}^{2})^{2}-(m_{1}^{2}+m_{2}^{2})p^{2}}{\sqrt{p^{2}-(m_{1}+m_{2})^{2}}\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}\left(\ln\left(\frac{\sqrt{p^{2}-(m_{1}+m_{2})^{2}}-\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}{\sqrt{p^{2}-(m_{1}+m_{2})^{2}}+\sqrt{p^{2}-(m_{1}-m_{2})^{2}}}\right)+i\pi\right)\cr}\kern-28.0pt$

in the same cases. The power series expansions are given by

$\displaystyle B_{f}(m_{W}^{2};m_{\nu},m_{\ell})$ $\displaystyle=$
$\displaystyle
2-\frac{m_{\ell}^{2}}{m_{W}^{2}}\ln\left(\frac{m_{\ell}^{2}}{\bar{\mu}^{2}}\right)-\frac{m_{W}^{2}-m_{\ell}^{2}}{m_{W}^{2}}\left(\ln\left(\frac{m_{W}^{2}-m_{\ell}^{2}}{\bar{\mu}^{2}}\right)-i\pi\right)+O(m_{\nu}^{2}),$
$\displaystyle B^{\prime}_{f}(m_{W}^{2};m_{\nu},m_{\ell})$ $\displaystyle=$
$\displaystyle-\frac{1}{m_{W}^{2}}+\frac{m_{\ell}^{2}}{m_{W}^{4}}\ln\left(\frac{m_{\ell}^{2}}{\bar{\mu}^{2}}\right)-\frac{m_{\ell}^{2}}{m_{W}^{4}}\left(\ln\left(\frac{m_{W}^{2}-m_{\ell}^{2}}{\bar{\mu}^{2}}\right)-i\pi\right)+O(m_{\nu}^{2}).$

This is sufficient for the parts coming from the $W$ boson self energy.
However, for the parts from the $Z$ boson self energy in $\delta s_{W}$ we
have to expand $B_{f}(m_{Z}^{2};m_{\nu},m_{\nu})$ and
$B^{\prime}_{f}(m_{Z}^{2};m_{\nu},m_{\nu})$ in $m_{\nu}$. With

$\sqrt{\lambda(m_{Z}^{2},m_{\nu}^{2},m_{\nu}^{2})}=\sqrt{m_{Z}^{2}(m_{Z}^{2}-4m_{\nu}^{2})}=m_{Z}^{2}-2m_{\nu}^{2}-\frac{2m_{\nu}^{4}}{m_{Z}^{2}}+O(m_{\nu}^{6})$
(C6)

one obtains

$\displaystyle B_{f}(m_{Z}^{2};m_{\nu},m_{\nu})$ $\displaystyle=$
$\displaystyle
2-\ln\left(\frac{m_{\nu}^{2}}{\bar{\mu}^{2}}\right)+\sqrt{1-\frac{4m_{\nu}^{2}}{m_{Z}^{2}}}\left(\ln\left(\frac{1-\sqrt{1-4m_{\nu}^{2}/m_{Z}^{2}}}{1+\sqrt{1-4m_{\nu}^{2}/m_{Z}^{2}}}\right)+i\pi\right),$
(C7) $\displaystyle B^{\prime}_{f}(m_{Z}^{2};m_{\nu},m_{\nu})$
$\displaystyle=$
$\displaystyle-\frac{1}{m_{Z}^{2}}+\frac{2m_{\nu}^{2}}{m_{Z}^{4}\sqrt{1-4m_{\nu}^{2}/m_{Z}^{2}}}\left(\ln\left(\frac{1-\sqrt{1-4m_{\nu}^{2}/m_{Z}^{2}}}{1+\sqrt{1-4m_{\nu}^{2}/m_{Z}^{2}}}\right)+i\pi\right).\qquad$
(C8)

For the $G_{\mu}$ scheme dealt with later in this Appendix, we have to
calculate $B_{f}(0;m_{1},m_{2})$, As $p^{2}=0$, we have to use the first case.
Without loss of generality, we assume that $m_{1}^{2}>m_{2}^{2}$ (the opposite
inequality will lead to the same result). We have

$\sqrt{\lambda(p^{2},m_{1}^{2},m_{2}^{2})}=\sqrt{(m_{1}^{2}-m_{2}^{2})^{2}-2p^{2}(m_{1}^{2}+m_{2}^{2})+p^{4}}=m_{1}^{2}-m_{2}^{2}-\frac{m_{1}^{2}+m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}p^{2}+O(p^{4}).$
(C9)

In this case $p^{2}<(m_{1}-m_{2})$. Therefore

$\displaystyle B_{f}(p^{2};m_{1},m_{2})$ $\displaystyle=$ $\displaystyle
2-\frac{1}{2}\left(\ln\left(\frac{m_{1}^{2}}{\bar{\mu}^{2}}\right)+\ln\left(\frac{m_{2}^{2}}{\bar{\mu}^{2}}\right)\right)-\frac{m_{1}^{2}-m_{2}^{2}}{2p^{2}}\ln\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)+$
(C10)
$\displaystyle-\frac{1}{2p^{2}}\left(m_{1}^{2}-m_{2}^{2}-p^{2}\frac{m_{1}^{2}+m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\right)\left(\ln\left(\frac{m_{2}^{2}}{m_{1}^{2}}\right)+\frac{2p^{2}}{m_{1}^{2}-m_{2}^{2}}\right)\
=$ $\displaystyle=$ $\displaystyle
1-\frac{1}{m_{1}^{2}-m_{2}^{2}}\left(m_{1}^{2}\ln\left(\frac{m_{1}^{2}}{\bar{\mu}^{2}}\right)-m_{2}^{2}\ln\left(\frac{m_{2}^{2}}{\bar{\mu}^{2}}\right)\right)+O(p^{2})$

and

$B_{f}(0;m_{1},m_{2})=\frac{m_{1}^{2}A_{f}(m_{1})-m_{2}^{2}A_{f}(m_{2})}{m_{1}^{2}-m_{2}^{2}},$
(C11)

were the finite part $A_{f}(m)$ of the one-point function is defined via

$A(m)=\frac{im^{2}\bar{\mu}^{-2\varepsilon}}{(4\pi)^{2}}\left(\frac{1}{\varepsilon}+A_{f}(m)\right),\qquad
A_{f}(m)=1-\ln\left(\frac{m^{2}}{\bar{\mu}^{2}}\right).$ (C12)

The same approximation we need for equal masses. Starting with

$\sqrt{\lambda(p^{2},m^{2},m^{2})}=\sqrt{(4m^{2}-p^{2})p^{2}}=\sqrt{4m^{2}p^{2}}\sqrt{1-\frac{p^{2}}{4m^{2}}}=p^{2}\sqrt{\frac{4m^{2}}{p^{2}}}\left(1-\frac{p^{2}}{8m^{2}}\right)+O(p^{2}),$
(C13)

with a short Taylor series expansion one obtains

$B_{f}(p^{2};m,m)=2-\ln\left(\frac{m^{2}}{\bar{\mu}^{2}}\right)-2\sqrt{\frac{4m^{2}}{p^{2}}}\arctan\left(\sqrt{\frac{p^{2}}{4m^{2}}}\right)+O(p^{2})=-\ln\left(\frac{m^{2}}{\bar{\mu}^{2}}\right)+O(p^{2})$
(C14)

and, therefore, $B_{f}(0;m,m)=A_{f}(m)-1$. With a longer expansion one has

$\displaystyle B^{\prime}_{f}(m_{A}^{2};m,m)\ =\
-\frac{1}{p^{2}}+\frac{4m^{2}}{p^{2}\sqrt{(4m^{2}-p^{2})p^{2}}}\arctan\left(\sqrt{\frac{p^{2}}{4m^{2}-p^{2}}}\right)\
=$ (C15) $\displaystyle=$
$\displaystyle-\frac{1}{p^{2}}+\frac{1}{p^{2}}\sqrt{\frac{4m^{2}}{p^{2}}}\left(1+\frac{p^{2}}{8m^{2}}\right)\sqrt{\frac{p^{2}}{4m^{2}}}\left(1+\frac{p^{2}}{24m^{2}}\right)+O(p^{2})\
=\ \frac{1}{6m^{2}}+O(p^{2})\qquad$

and $B^{\prime}_{f}(0;m,m)=1/(6m^{2})$.

###  C.1 Applying the $G_{\mu}$ scheme

In the electroweak theory with its multiple parameters depending on each
other, it is essential to decide which of the parameters are independent. In
the so-called $\alpha$ scheme, the coupling and the masses of the particles
are used as such. However, depending on energy of the process, more and more
fermion loops can be resummed to the coupling, so that $\alpha(0)$ is changed
to e.g. $\alpha(m_{W}^{2})$. This is shown in Sec. 8.2.1 of Ref. [6]. In the
quite detailed review article in Ref. [46] is is argued that for processes
involving the electroweak vector bosons, the $G_{F}$ (or $G_{\mu}$) scheme
based on the Fermi constant (in muon decay) and discussed in Ref. [49] is
preferable. Therefore, for our work we will deal with this scheme. From Eqs.
(426) and (423) of Ref. [46] we deduce that

$\displaystyle\delta Z_{e}\Big{|}_{G_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Pi_{AA}(m_{A}^{2})-\frac{s_{W}}{c_{W}}\frac{\Sigma_{AZ}^{T}(m_{A}^{2})}{M_{Z}^{2}}-\frac{1}{2}\Delta
r\ =$ (C16) $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}\frac{c_{W}^{2}}{s_{W}^{2}}\left(\frac{\Sigma_{ZZ}^{T}(m_{Z}^{2})}{m_{Z}^{2}}-\frac{\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}\right)-\frac{\Sigma_{WW}^{T}(0)-\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}+$
$\displaystyle-\frac{2\Sigma_{AZ}^{T}(0)}{s_{W}c_{W}m_{Z}^{2}}-\frac{\alpha(0)}{4\pi
s_{W}^{2}}\left(6+\frac{7-4s_{W}^{2}}{2s_{W}^{2}}(A_{f}(m_{Z})-A_{f}(m_{W})\right)\Bigg{)}\qquad$

(cf. also Eq. (4.6.11) in Ref. [47]). For the self energy of the $W$ boson at
$p^{2}=0$ one obtains

$\displaystyle\Pi^{WW}_{T}(m_{A}^{2})\ =\
\frac{\alpha}{4\pi}\Bigg{[}\frac{-1}{2m_{W}^{2}s_{W}^{2}\varepsilon}\left(4m_{W}^{2}-2m_{Z}^{2}+\sum_{i}(m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2})+N_{c}\sum_{i,j}|V_{ij}|^{2}(m_{i}^{2}+m_{j}^{2})\right)+$
(C17)
$\displaystyle-\frac{1}{6m_{W}^{2}s_{W}^{2}}\left(18m_{W}^{2}+m_{Z}^{2}+m_{H}^{2}-2\sum_{i}(m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2})-2N_{c}\sum_{i,j}|V_{ij}|^{2}(m_{i}^{2}+m_{j}^{2})\right)+$
$\displaystyle-\frac{(14m_{W}^{2}+m_{Z}^{2})m_{W}^{2}-(16m_{W}^{2}-m_{Z}^{2})m_{H}^{2}}{3(m_{H}^{2}-m_{W}^{2})m_{Z}^{2}s_{W}^{4}}A_{f}(m_{W})+$
$\displaystyle-\frac{(4m_{W}^{2}-m_{Z}^{2})(2m_{W}^{2}+3m_{Z}^{2})}{3m_{W}^{2}m_{Z}^{2}s_{W}^{4}}A_{f}(m_{Z})+\frac{2m_{H}^{2}A_{f}(m_{H})}{3(m_{H}^{2}-m_{W}^{2})s_{W}^{2}}+$
$\displaystyle-\frac{1}{3m_{W}^{2}s_{W}^{2}}\sum_{i}\left(m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})+m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}})+\frac{m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2}}{2(m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})}\left(m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})-m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}})\right)\right)+\kern-28.0pt$
$\displaystyle-\frac{N_{c}}{3m_{W}^{2}s_{W}^{2}}\sum_{ij}|V_{ij}|^{2}\left(m_{i}^{2}A_{f}(m_{i})+m_{j}^{2}A_{f}(m_{j})+\frac{m_{i}^{2}+m_{j}^{2}}{2(m_{i}^{2}-m_{j}^{2})}\left(m_{i}^{2}A_{f}(m_{i})-m_{j}^{2}A_{f}(m_{j})\right)\right)\Bigg{]}+\kern-32.0pt$
$\displaystyle\kern-24.0pt+\frac{\alpha}{4\pi}\Bigg{[}\frac{-1}{4m_{H}^{2}m_{W}^{2}s_{W}^{2}\varepsilon}\left(6(2m_{W}^{4}+m_{Z}^{4})+(2m_{W}^{2}+m_{Z}^{2})m_{H}^{2}+3m_{H}^{4}-8\sum_{f}m_{f}^{4}\right)+$
$\displaystyle+\frac{1}{4m_{H}^{2}m_{W}^{2}s_{W}^{2}}\Bigg{(}4(2m_{W}^{4}+m_{Z}^{4})-2m_{W}^{2}(m_{H}^{2}+6m_{W}^{2})A_{f}(m_{W})+$
$\displaystyle\qquad\qquad\qquad-
m_{Z}^{2}(m_{H}^{2}+6m_{Z}^{2})A_{f}(m_{Z})-3m_{H}^{4}A_{f}(m_{H})+8\sum_{f}m_{f}^{4}A_{f}(m_{f})\Bigg{)}\Bigg{]}.$

It can be shown that

$\displaystyle\Delta r$ $\displaystyle=$
$\displaystyle\Pi_{AA}^{T}(m_{A}^{2})+\frac{2c_{W}}{s_{W}}\frac{\Sigma_{AZ}^{T}(m_{A}^{2})}{m_{Z}^{2}}+\frac{\Sigma_{WW}^{T}(0)-\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}+$
(C18)
$\displaystyle-\frac{c_{W}^{2}}{s_{W}^{2}}\left(\frac{\Sigma_{ZZ}^{T}(m_{Z}^{2})}{m_{Z}^{2}}-\frac{\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}\right)+\frac{\alpha}{4\pi
s_{W}^{2}}\left(6+\frac{7-4s_{W}^{2}}{2s_{W}^{2}}\ln c_{W}^{2}\right)$

is UV-finite. This has been done by using $\sum_{f}(1-4I_{f}^{3}Q_{f})=0$,
which in detail results in $Q_{1}-Q_{2}=Q_{W}$ for all quark and lepton
generations. In replacing the $\alpha(0)$-scheme result $\delta Z_{e}=-\delta
Z_{AA}-\delta Z_{ZA}s_{W}/c_{W}$ by Eq. (C16) means that the contribution
$-\delta Z_{AA}$ producing the large mass logarithms no longer appears. We can
use Eq. (C16) to see that in this new scheme $\delta Z_{e}$ together with the
counter term for the sine of the Weinberg angle simplifies to

$\displaystyle\left[-\frac{\delta s_{W}}{s_{W}}+\delta Z_{e}\right]_{G_{\mu}}\
=\
\frac{c_{W}^{2}}{2s_{W}^{2}}\left(\frac{\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}-\frac{\Sigma_{ZZ}^{T}(m_{Z}^{2})}{m_{Z}^{2}}\right)+$
$\displaystyle+\frac{1}{2}\Bigg{[}\frac{c_{W}^{2}}{s_{W}^{2}}\left(\frac{\Sigma_{ZZ}^{T}(m_{Z}^{2})}{m_{Z}^{2}}-\frac{\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}\right)-\frac{\Sigma_{WW}^{T}(0)-\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}+$
$\displaystyle-\frac{2\Sigma_{AZ}^{T}(0)}{s_{W}c_{W}m_{Z}^{2}}-\frac{\alpha(0)}{4\pi
s_{W}^{2}}\left(6+\frac{7-4s_{W}^{2}}{2s_{W}^{2}}(A_{f}(m_{Z})-A_{f}(m_{W})\right)\Bigg{]}\
=$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Bigg{[}\frac{2\Sigma_{AZ}^{T}(0)}{s_{W}c_{W}m_{Z}^{2}}+\frac{\Sigma_{WW}^{T}(0)-\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}+\frac{\alpha(0)}{4\pi
s_{W}^{2}}\left(6+\frac{7-4s_{W}^{2}}{2s_{W}^{2}}(A_{f}(m_{Z})-A_{f}(m_{W})\right)\Bigg{]},\kern-14.0pt$

which means that in this scheme, for practical reasons, we can set

$\displaystyle\frac{\delta s_{W}}{s_{W}}\Big{|}_{G_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\frac{\Sigma_{WW}^{T}(0)-\Sigma_{WW}^{T}(m_{W}^{2})}{m_{W}^{2}}+\frac{\alpha(0)}{4\pi
s_{W}^{2}}\left(6+\frac{7-4s_{W}^{2}}{2s_{W}^{2}}(A_{f}(m_{Z})-A_{f}(m_{W})\right)\Bigg{]},$
$\displaystyle\delta Z_{e}|_{G_{\mu}}$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{AZ}^{T}(0)}{s_{W}c_{W}m_{Z}^{2}}$ (C20)

with UV singular contributions

$\displaystyle\frac{\delta
s_{Ws}}{s_{W}}\Big{|}_{G_{\mu}}=\frac{e^{2}}{12s_{W}^{2}}\left(19-\sum_{f}1\right),\qquad\delta
Z_{es}\Big{|}_{G_{\mu}}=-\frac{2e^{2}}{s_{W}^{2}}$ (C21)

and UV regular contributions

$\displaystyle\frac{\delta
s_{Wr}}{s_{W}}\Big{|}_{G_{\mu}}=e^{2}\Bigg{[}\frac{-72m_{W}^{2}+182m_{Z}^{2}+\sum_{f}m_{Z}^{2}}{36m_{Z}^{2}s_{W}^{2}}+$
(C22)
$\displaystyle-\frac{1}{24(m_{H}^{2}-m_{W}^{2})m_{W}^{2}m_{Z}^{4}s_{W}^{4}}\Big{(}m_{W}^{2}(48m_{W}^{6}-46m_{W}^{4}m_{Z}^{2}+17m_{W}^{2}m_{Z}^{4}-m_{Z}^{6})+$
$\displaystyle\qquad\qquad\qquad-
m_{H}^{2}(48m_{W}^{6}-39m_{W}^{4}m_{Z}^{2}+10m_{W}^{2}m_{Z}^{4}-m_{Z}^{6})+m_{H}^{4}m_{Z}^{4}s_{W}^{2}\Big{)}A_{f}(m_{W})+$
$\displaystyle+\frac{16m_{W}^{6}-51m_{W}^{4}m_{Z}^{2}+16m_{W}^{2}m_{Z}^{4}+m_{Z}^{6}}{24m_{W}^{4}m_{Z}^{2}s_{W}^{4}}A_{f}(m_{Z})+$
$\displaystyle+(4m_{W}^{2}-m_{Z}^{2})\frac{12m_{W}^{4}+20m_{W}^{2}m_{Z}^{2}+m_{Z}^{4}}{24m_{W}^{4}m_{Z}^{2}s_{W}^{2}}B_{f}(m_{W}^{2};m_{W},m_{Z})+$
$\displaystyle+m_{H}^{2}\frac{11m_{W}^{4}-4m_{W}^{2}m_{H}^{2}+m_{H}^{4}}{24(m_{H}^{2}-m_{W}^{2})m_{W}^{4}s_{W}^{2}}A_{f}(m_{H})-\frac{12m_{W}^{4}-4m_{W}^{2}m_{H}^{2}+m_{H}^{4}}{24m_{W}^{4}s_{W}^{2}}B_{f}(m_{W}^{2};m_{W},m_{H})+\kern-8.0pt$
$\displaystyle+\frac{1}{12m_{W}^{4}s_{W}^{2}}\sum_{i}\Bigg{(}\left((m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})^{2}+(m_{\mu}^{2}+m_{\ell_{i}}^{2})m_{W}^{2}-2m_{W}^{4}\right)B_{f}(m_{W}^{2};m_{\nu_{i}},m_{\ell_{i}})+$
$\displaystyle\qquad\qquad\qquad\qquad-\left((m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2})^{2}+(m_{\nu_{i}}^{2}+m_{\ell_{i}}^{2})m_{W}^{2}\right)\frac{m_{\nu_{i}}^{2}A_{f}(m_{\nu_{i}})-m_{\ell_{i}}^{2}A_{f}(m_{\ell_{i}})}{m_{\nu_{i}}^{2}-m_{\ell_{i}}^{2}}\Bigg{)}+$
$\displaystyle+\frac{1}{12m_{W}^{4}s_{W}^{2}}\sum_{i,j}|V_{ij}|^{2}\Bigg{(}\left((m_{i}^{2}-m_{j}^{2})^{2}+(m_{i}^{2}+m_{j}^{2})m_{W}^{2}-2m_{W}^{4}\right)B_{f}(m_{W}^{2};m_{i},m_{j})+$
$\displaystyle\qquad\qquad\qquad\qquad-\left((m_{i}^{2}-m_{j}^{2})^{2}+(m_{i}^{2}+m_{j}^{2})m_{W}^{2}\right)\frac{m_{i}^{2}A_{f}(m_{i})-m_{j}^{2}A_{f}(m_{j})}{m_{i}^{2}-m_{j}^{2}}\Bigg{)}\Bigg{]},$
$\displaystyle\delta Z_{er}\Big{|}_{G_{\mu}}\ =\
\frac{2e^{2}}{s_{W}^{2}}\left(1-A_{f}(m_{W})\right).$ (C23)

Together with

$\delta Z_{WWs}=\frac{e^{2}}{12s_{W}^{2}}\left(19-\sum_{f}1\right)$ (C24)

from above, one has $\delta Z_{WWs}-\delta s_{Ws}/s_{W}+\delta
Z_{es}|_{G_{\mu}}=-2e^{2}/s_{W}^{2}$ as before, only that the first two UV
singularities cancel each other and the only singularity that is left is the
one from $\delta Z_{e}|_{G_{\mu}}$.

## References

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