#  DESY 16-234 ISSN 0418-9833
December 2016

On the $b$-quark running mass in QCD and the SM

A.V. Bednyakov<EMAIL_ADDRESS>B.A. Kniehl<EMAIL_ADDRESS>A.F.
Pikelner111On leave of absence from Joint Institute for Nuclear Research,
141980 Dubna, Russia<EMAIL_ADDRESS>O.L. Veretin
<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980 Dubna,
Russia  Dubna State University, 141982, Dubna, Russia  II Institut für
Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

We consider electroweak corrections to the relation between the running
$\overline{\mathrm{MS}}$ mass $m_{b}$ of the $b$ quark in the five-flavor
QCD$\times$QED effective theory and its counterpart in the Standard Model
(SM). As a bridge between the two parameters, we use the pole mass $M_{b}$ of
the $b$ quark, which can be calculated in both models. The running mass is not
a fundamental parameter of the SM Lagrangian, but the product of the running
Yukawa coupling $y_{b}$ and the Higgs vacuum expectation value. Since there
exist different prescriptions to define the latter, the relations considered
in the paper involve a certain amount of freedom. All the definitions can be
related to each other in perturbation theory. Nevertheless, we argue in favor
of a certain gauge-independent prescription and provide a relation which can
be directly used to deduce the value of the Yukawa coupling of the $b$ quark
at the electroweak scale from its effective QCD running mass. This approach
allows one to resum large logarithms $\ln(m_{b}/M_{t})$ systematically.
Numerical analysis shows that, indeed, the corrections to the proposed
relation are much smaller than those between $y_{b}$ and $M_{b}$.

###### keywords:

SM , QCD , Bottom quark

††journal: Physics Letters B

##  1 Introduction

Since its theoretical prediction in 1973 [1] and experimental discovery in
1977 [2], the bottom quark has been serving as a unique probe in studying
various aspects of modern particle physics. Special B factories with
appropriate detectors, e.g., BaBar and Belle, were built and established CP
violation in mesons involving $b$ quarks (see Ref. [3] for a comprehensive
review). Moreover, a dedicated LHC experiment, LHCb, aimed to further improve
our knowledge of the origin of CP violation, was set up at CERN.

Another important point is New Physics (NP), which can manifest itself in rare
(flavor-violating) decays of B mesons. The latter, being suppressed in the
Standard Model (SM), are very sensitive to contributions from beyond-the-SM
(BSM) theories and, at the moment, play a very important role in constraining
parameter spaces of NP scenarios. Finally, both the SM Higgs boson and the top
quark prefer to decay into $b$ quarks. Due to this, the properties of the
bottom quark deserve to be carefully investigated and analyzed.

Theoretical descriptions of the above-mentioned processes usually involve
several different scales, ranging from the long-distance (QCD) scale
$\Lambda_{\mathrm{QCD}}\sim 10^{-1}$ GeV over the electroweak (EW) scale
$M_{Z}\sim 10^{2}$ GeV up to NP scales $\Lambda_{\mathrm{NP}}\gtrsim 10^{3}$
GeV (see, e.g., Ref. [4]). Since the scales are well separated, one can make
use of effective field theories (EFTs) (see, e.g, Ref. [5, 6]) and “factorize”
physics relevant to strong (long-distance) QCD dynamics, which are difficult
to calculate, yet not very interesting from the fundamental point of view,
from short-distance effects due to EW or NP interactions.

In the SM, the $b$-quark mass is generated via the Higgs mechanism due to the
Yukawa interaction of the $b$ quark with the Higgs field condensate
$|\langle\Phi\rangle|^{2}\equiv v^{2}/2$. However, a well-known fact is that
strong interactions prevent quarks from being observed as “free” particles, so
that the notion of physical mass is ill-defined in this case (see, e.g., Ref.
[7]). In such a situation, one can choose a convenient mass definition
depending on the problem considered. Among these definitions are the pole mass
$M_{b}$ [8], the running mass $m_{b}$ and various “threshold” masses, such as
the 1S mass $m_{b}^{\mathrm{1S}}$ [9], the potential-subtracted (PS) mass
$m_{b}^{\mathrm{PS}}$ [10], the renormalon-subtracted (RS) mass
$m_{b}^{\mathrm{RS}}$ [11], etc. (see Ref. [12] for a review). In principle,
all these mass parameters can be related to each other and to the Yukawa
coupling $y_{b}$. The latter enters the fundamental SM Lagrangian and is
definitely very important for precise theoretical predictions both of
$B$-hadron decay properties and the SM Higgs decay width.

It is worth mentioning that the direct measurement of the $b$-quark Yukawa
coupling is very challenging [13]. The estimated uncertainty for the LHC is
about 20%, and a linear collider is needed to reduce it by an order of
magnitude [13, 14].

In this paper, we address the problem of the extraction of the running222In
what follows, we employ modified minimal subtraction
($\overline{\mathrm{MS}}$) scheme. Yukawa coupling $y_{b}(\mu)$ in the SM from
a $b$-quark mass parameter. Our goal is to improve a well-established relation
between $y_{b}(\mu)$ and $M_{b}$ [15, 16, 17] at the two-loop order333In fact,
QCD corrections for the case of a single heavy quark are known through the
four-loop level [18]. However, only three-loop terms [19, 20] are known for
the case when one additional massive quark is in the spectrum (see discussion
below). of perturbation theory (PT) by trading the ill-defined pole mass for a
short-distance mass parameter $\underline{m}_{b}(\mu)\equiv m_{b}^{(5)}(\mu)$
defined in the $\overline{\mathrm{MS}}$ renormalization scheme. The latter
plays the role of an independent Lagrangian parameter of the QCD EFT with five
active quark flavors, which is valid significantly below the EW scale. This
mass parameter is not sensitive to long-distance physics and can be extracted
from experiment with much higher precision. The value of
$\mu_{b}\equiv\underline{m}_{b}(\underline{m}_{b})=4.18\pm 0.03$ GeV quoted by
the Particle Data Group (PDG) [21] will be used here as an input for the
determination of $y_{b}(\mu)$.

This paper is organized as follows. In section 2, we consider various
definitions of running quark mass, all of which are proportional to the
running Yukawa couping $y_{b}(\mu)$, but differ from each other by the
treatment of the vacuum expectation value (vev) $v$. In section 3, we describe
our procedure, which allows us to obtain the relation between $y_{b}(\mu)$ and
$\underline{m}_{b}(\mu)$ at a certain (matching) scale $\mu$. Section 4 is
devoted to our numerical analysis of different corrections to this relation
and the comparison of the latter with the corresponding contributions to the
$y_{b}$-$M_{b}$ relation. Finally, in section 5, some discussions and
conclusions can be found. In A, three-loop renormalization group equations
(RGEs) for the QCD$\times$QED effective theory, utilized to relate the values
of $\mu_{b}$ and $\underline{m}_{b}(\mu)$, are presented. In B, we also
include the three-loop relation between the running mass $m_{b}^{(6)}(\mu)$
and $M_{b}$ in six-flavor QCD, with the account of the heavy top quark. The
latter can be used to extract the running mass of the $b$ quark in the SM from
its given pole mass.

##  2 Running quark masses in the SM

Fermion masses are not fundamental parameters of the SM Lagrangian, but are
induced in the spontaneously broken phase and are proportional to the
corresponding Yukawa couplings:

$\displaystyle m_{f}$ $\displaystyle=\frac{y_{f}v}{\sqrt{2}},$ (1)

with $f=l,q$ denoting leptons and quarks, respectively. The Higgs field
expectation value $v$ corresponds to a minimum of the full effective potential
$V_{\mathrm{eff}}(\phi)$ [22, 23] for the neutral component of the Higgs
doublet $\phi$,

$\displaystyle V_{\mathrm{eff}}(\phi)=V_{\mathrm{tree}}(\phi)+\Delta
V(\phi),\qquad
V_{\mathrm{tree}}(\phi)=-m_{\Phi}^{2}\frac{\phi^{2}}{2}+\lambda\frac{\phi^{4}}{4}.$
(2)

The vev $v$ is a nonperturbative quantity that should be a function of
fundamental SM Lagrangian parameters, i.e., dimensionless gauge and Yukawa
couplings together with the Higgs self-coupling $\lambda$ and mass $m_{\Phi}$
from the tree-level Higgs potential $V_{\mathrm{tree}}$. The fact that $v$
corresponds to a minimum guarantees the absence of “tadpoles,” which are
nothing but $\partial V_{\mathrm{eff}}/\partial\phi(v)=0$.

This simple picture is spoiled by several “technical” obstacles. First of all,
it is very difficult to calculate $V_{\mathrm{eff}}$ beyond the tree-level and
to find an analytic expression for $v$. Another well-known issue is the gauge
dependencies [23, 24] of $V_{\mathrm{eff}}$ (away from extrema) and the field
value $v$ at its minimum. Due to this, various approximations for $v$ are on
the market, which may lead to different PT series if expressed in terms of
dimensionless running couplings and $m_{\Phi}^{2}$.

From the practical point of view, it is possible to avoid the explicit
calculation of $V_{\mathrm{eff}}$ by adjusting the definitions of the
parameters (equivalently, the counterterms) to cancel tadpoles order by order
(see Ref. [25] for the on-shell formulation).

One can distinguish two, yet related, options (for a comprehensive discussion,
see also Refs. [26, 27, 28]). The first option is to write
$v_{\mathrm{eff}}=v_{\mathrm{tree}}+\Delta v$ and move terms due to $\Delta v$
to the interaction part of the SM Lagrangian in the broken phase, so that all
tree-level masses (c.f. Eq.(1)) are proportional to $v_{\mathrm{tree}}$, which
satisfies the tree-level minimization condition, i.e., $\partial
V_{\mathrm{tree}}(v_{\mathrm{tree}})/\partial\phi=0$. The shift $\Delta v$ can
be determined order by order in PT from the requirement that the tree-level
tadpole for the neutral Higgs field $\phi$, which can be written as

$\delta{\mathcal{L}}_{\rm tad}=-\phi t_{\rm tree}=-\phi m_{h}^{2}\Delta
v\left[1+\frac{3}{2}\frac{\Delta
v}{v_{\mathrm{tree}}}+\frac{1}{2}\left(\frac{\Delta
v}{v_{\mathrm{tree}}}\right)^{2}\right],\qquad m_{H}^{2}\equiv 2\lambda
v_{\mathrm{tree}}^{2},$ (3)

cancels the loop-generated ones. In spite of the fact that the one-point
function for the Higgs field is zero, in accordance with minimization
condition, tadpoles in this scheme manifest themselves in every vertex in
which we make the shift $v_{\mathrm{eff}}\to v_{\mathrm{tree}}+\Delta v$. It
turns out that such a kind of contributions can be conveniently taken into
account by considering one-particle-reducible diagrams in which a neutral-
Higgs propagator is allowed to be terminated by a tadpole. This approach was
advocated by Fleischer and Jegerlehner (FJ) in Ref. [29] (see also the recent
discussion in Ref. [30].)

Another prescription, the “tadpole-free” scheme, allows one to avoid the
explicit introduction of $\Delta v$ and implicitly assumes that $v$
corresponds to a gauge-dependent field value at the minimum of
$V_{\mathrm{eff}}$ (see, e.g., Refs. [31, 32, 33, 34, 35, 36]). In this case,
the tree-level tadpole again precisely cancels the loop-generated ones, but
Eq. (3) is rewritten as

$\delta{\mathcal{L}}_{\rm tad}=-\phi t_{\rm tree}=-\phi
v_{\mathrm{eff}}\left(\lambda v_{\mathrm{eff}}-m^{2}_{\Phi}\right),$ (4)

and there is no explicit contribution due to $\Delta v$. Due to this, the
tree-level masses of the would-be Goldstone boson $\chi$ and the Higgs boson
$H$ are given by

$m^{2}_{\chi}=\lambda v_{\mathrm{eff}}^{2}-m_{\Phi}^{2}=0+\frac{t_{\rm
tree}}{v_{\mathrm{eff}}}\,,\qquad m^{2}_{H}=3\lambda
v_{\mathrm{eff}}^{2}-m_{\Phi}^{2}=2\lambda v_{\mathrm{eff}}^{2}+\frac{t_{\rm
tree}}{v_{\mathrm{eff}}}\,,$ (5)

in the broken phase, with $v_{\mathrm{eff}}^{2}\neq m^{2}_{\Phi}/\lambda$.
Since it is a common choice to assume that all tree-level particle masses in
the SM are proportional to a vev, the terms due to $t_{\rm tree}$ in Eqs. (5)
are moved from from the quadratic part of SM Lagrangian to the interaction
part and are traded for loop-generated tadpoles.

It is worth pointing out here that $L$-loop contributions to the Higgs one-
point functions considered in the two above-mentioned approaches, although
being formally of the same loop level, are different, due to the fact that
they are expressed in terms of different "tree-level" running masses. In
addition, in the FJ scheme, Higgs tadpole insertions are allowed, while, in
the “tadpole-free” scheme, only scalar masses are shifted due to tadpoles.

The advantage of the first option is an explicit control of the gauge
dependence of the result, while, in the latter case, the Landau gauge,
$\xi=0$, is usually chosen for the calculation of $V_{\mathrm{eff}}$. In what
follows, we routinely use the FJ tadpole scheme.

For the time being, we say nothing about the utilized regularization and our
choice of renormalization scheme. This is done intentionally, since the
reasoning is equally applicable if Eqs. (1)–(5) are written in terms of either
bare or minimally renormalized parameters. For the FJ prescription, one
defines a running vev $v_{\mathrm{tree}}(\mu)^{2}\equiv
m^{2}_{\Phi}(\mu)/\lambda(\mu)$, the RGE for which can be simply obtained from
those of the unbroken theory (see, e.g., Ref. [37] and references cited
therein). In Refs. [38, 39, 40, 41, 42], a related quantity,
$G_{F}^{\overline{\mathrm{MS}}}(\mu)\equiv 1/(\sqrt{2}v^{2}(\mu)$), is
introduced (see below), and the RGEs are provided in terms of running masses
in the FJ tadpole scheme. All running particle masses are gauge independent in
this case and are proportional to $v_{\mathrm{tree}}(\mu)$ [40]. In addition,
the running Higgs mass is directly related to $m_{\Phi}^{2}$ of the unbroken
Lagrangian, i.e., $m_{H}^{2}=2\lambda v_{\mathrm{tree}}=2m_{\Phi}^{2}$.

One can also define a (gauge-dependent) running vev $v_{\mathrm{eff}}(\mu)$
obtained by minimization of the effective potential of the Higgs field,
renormalized in the $\overline{\mathrm{MS}}$ scheme at scale $\mu$, so that
$v_{\mathrm{eff}}(\mu)=v_{\mathrm{tree}}(\mu)+\Delta v(\mu)$. The scale
dependence of $v_{\mathrm{eff}}(\mu)$ is more involved than that of
$v_{\mathrm{tree}}$, but, in the Landau gauge, it is given by the Higgs field
anomalous dimension (see the discussion in Refs. [43, 44]). The latter can
also be calculated in the unbroken theory.

To summarize, we have discussed the following options to define a running
quark mass in the SM renormalized in the $\overline{\mathrm{MS}}$ scheme:

  * 1.

Gauge-independent running mass $m_{b}(\mu)$:

$\displaystyle m_{b}(\mu)=\frac{y_{b}(\mu)v_{\mathrm{tree}}(\mu)}{\sqrt{2}},$
$\displaystyle\qquad
v_{\mathrm{tree}}(\mu)^{2}\equiv\frac{m^{2}_{\Phi}(\mu)}{\lambda(\mu)}.$ (6)

  * 2.

Gauge-dependent running mass $\tilde{m}_{b}(\mu)$:

$\displaystyle\tilde{m}_{b}(\mu)=\frac{y_{b}(\mu)v_{\mathrm{eff}}(\mu)}{\sqrt{2}},$
$\displaystyle\qquad v_{\mathrm{eff}}(\mu):\left.\frac{\partial
V_{\mathrm{eff}}(\phi,\mu)}{\partial\phi}\right|_{\phi=v_{\mathrm{eff}}}=0.$
(7)

The anomalous dimensions $\gamma_{m}^{b}$ for both quantities, defined as

$\displaystyle\frac{d}{d\ln\mu}m=\gamma_{m}m,\qquad
m\in\\{m_{b},\tilde{m}_{b}\\},$ (8)

can be expressed as sums of the beta function $\beta_{b}$ for $y_{b}(\mu)$ and
the anomalous dimensions $\gamma_{b}$ of the corresponding vevs $v$:

$\displaystyle\frac{d}{d\ln\mu}y_{b}=\beta_{b}y_{b},\qquad\frac{d}{d\ln\mu}v=\gamma_{v}v,\qquad
v\in\\{v_{\mathrm{tree}},v_{\mathrm{eff}}\\}.$ (9)

In the FJ case, we have [41]

$\displaystyle\gamma_{v_{\mathrm{tree}}}=\frac{1}{2}\left(\gamma_{m_{\Phi}^{2}}-\frac{\beta_{\lambda}}{\lambda}\right),$
(10)

with $\gamma_{m_{\Phi}^{2}}\equiv d\ln m_{\Phi}^{2}/\ln\mu$ and
$\beta_{\lambda}\equiv d\lambda/d\ln\mu$. In the “tadpole-free” scheme, there
is no such simple relation between the corresponding anomalous dimension and
RG functions in a general $R_{\xi}$ gauge, but, in Landau gauge, we have

$\displaystyle\gamma_{v_{\mathrm{eff}}}=\gamma_{\Phi},\qquad\gamma_{\Phi}=-\frac{1}{2}\frac{d\ln
Z_{\Phi}}{d\ln\mu},\qquad\mathrm{(Landau~{}gauge!),}$ (11)

with $\gamma_{\Phi}$ being the anomalous dimension of the Higgs doublet
$\Phi$, computed from the Higgs field propagator renormalization constant
$Z_{\Phi}$. The difference in running between $v_{\mathrm{eff}}(\mu)$ and
$v_{\mathrm{tree}}(\mu)$ within the SM was studied numerically in Ref. [37].

It is worth mentioning that, contrary to “tadpole-free” scheme, all the gauge-
fixing parameter dependences of calculable quantities are explicit in the FJ
approach. However, the corresponding expressions in the FJ scheme involve
tadpole contributions, which typically scale like powers of
$[M_{t}^{4}/(M_{W}^{2}M_{h}^{2})\sim 9]$ with $M_{t}$, $M_{W}$, and $M_{H}$
denoting the masses of the top quark, the $W$ boson, and the Higgs boson,
respectively. In the “tadpole-free” scheme, (most of) these dangerous terms
are effectively absorbed in $v_{\mathrm{eff}}(\mu)$.

As was stated earlier, we routinely use the FJ scheme to define running
masses. Nevertheless, there is a way to improve the corresponding PT series in
a gauge-invariant way by trading $v_{\mathrm{tree}}(\mu)$ for an “on-shell”
vev,

$\displaystyle
v_{F}\equiv\left(\sqrt{2}G_{F}\right)^{-1/2}=246.21965(6)~{}\mathrm{GeV},$
(12)

which, by definition, is related to a measured quantity, the Fermi constant
extracted from muon decay, $G_{F}\equiv G_{\mu}=1.1663787(6)\times
10^{-5}~{}\mathrm{GeV}^{-2}$ [21]. In what follows, we treat $G_{F}$ as a non-
renormalizable four-fermion coupling of the effective low-energy Fermi theory
valid at scales much less than the EW one (for a discussion of different
definitions of the Fermi constant in the SM, see, e.g., Ref. [41]). The
relation (12) is motivated by the tree-level matching of the SM to the Fermi
theory, i.e., a $W$-boson exchange at low momentum transfer leads to

$\displaystyle\frac{G_{F}}{\sqrt{2}}=\frac{g}{2\sqrt{2}}\times\frac{1}{M_{W}^{2}}\times\frac{g}{2\sqrt{2}}=\frac{1}{2v^{2}},\qquad
M_{W}=\frac{gv}{2},$ (13)

where the $W$-boson mass is proportional to the SU(2) gauge coupling $g$.
Going beyond the tree-level approximation, one needs to perform a more
sophisticated matching by comparing the QED-corrected and, at higher orders,
also QCD-corrected muon lifetime in the EFT with the corresponding expression
in the SM [45]. The corrections to the tree-level matching in Eq. (13) are
usually accumulated in the quantity $\Delta r$ [46],

$\displaystyle\frac{G_{F}}{\sqrt{2}}=\frac{1}{2v^{2}}\left(1+\Delta r\right).$
(14)

The Fermi constant can be treated as a Wilson coefficient of an effective non-
renormalizable operator, which, in general, can be scale dependent. We recall
that Wilson coefficients are indeed scale dependent if the corresponding
operator involves four external quarks [47]. However, muon decay is described
by an effective operator with external leptons,
$O_{F}=\left[\bar{\nu}_{\mu}\gamma^{\alpha}(1-\gamma_{5})\mu\right]\left[\bar{e}\gamma^{\alpha}(1-\gamma_{5})\nu_{e}\right]$,
and the corresponding Wilson coefficient turns out to be scale independent due
to QED Ward–Takahashi identities (see the discussion in Ref. [45]), i.e.,
$G_{F}$ can be treated as a scale-independent “observable” in the SM.

If the right-hand side of Eq. (14) is expressed in terms of
$\overline{\mathrm{MS}}$ parameters, one can invert it in PT to express
$v(\mu)$ in terms of $G_{F}$ and other (dimensionless) parameters. Again, $v$
can be either $v_{\mathrm{tree}}(\mu)$ or $v_{\mathrm{eff}}(\mu)$. It is easy
to convince oneself that both the FJ and “tadpole-free” schemes should lead to
the same PT series, if $v$ is traded for $G_{F}$ in a consistent way.444One
can also use a mixed renormalization scheme, for which the running masses in
$\Delta r$ are rewritten in terms of pole ones. Due to this, in our analysis,
we make use of yet another definition of the running $b$-quark mass,

$\displaystyle
m_{b,Y}(\mu)\equiv\frac{y_{b}(\mu)v_{F}}{\sqrt{2}}\,,\qquad\gamma_{m_{b,Y}}=\beta_{b},$
(15)

discussed in Refs. [48, 41, 17]. Since $v_{F}$ from Eq. (12) is scale
independent, the anomalous dimension of $m_{b,Y}$ coincides with the Yukawa
coupling beta function $\beta_{b}$.

##  3 Details of the matching procedure

Let us now consider the relation between $m_{b,Y}(\mu)$ (or, equivalently, the
$b$-quark Yukawa coupling) and the pole mass $M_{b}$ at the two-loop order,
where we concentrate on EW corrections:

$\displaystyle m_{b,Y}(\mu)$ $\displaystyle\equiv$
$\displaystyle\frac{y_{b}(\mu)v_{F}}{\sqrt{2}}=M_{b}[1+\delta_{b}(\mu)],$ (16)
$\displaystyle\delta_{b}(\mu)$ $\displaystyle=$
$\displaystyle\sum\limits_{i+j=1}^{2}a^{i}(\mu)a_{s}^{j}(\mu)\delta^{(b)}_{ij}(M_{b},M,\mu)$
(17) $\displaystyle=$
$\displaystyle\sum\limits_{i+j=1}^{2}a_{F}^{i}a_{s}^{j}(\mu)\underline{\delta}^{(b)}_{ij}(M_{b},M,\mu),$
(18)

where $M\in\\{M_{W},M_{Z},M_{t},M_{h}\\}$ collectively denotes the “hard”
scales of the problem and $a_{i}(\mu)\equiv\alpha_{i}(\mu)/(4\pi)$. In Eq.
(18), instead of the running coupling $\alpha(\mu)$, a scale-independent
coupling,
$\alpha_{F}\equiv\sqrt{2}G_{F}M_{W}^{2}\sin^{2}\theta_{w}/\pi=1/132.233$ [21],
where $\sin^{2}\theta_{w}=1-M_{W}^{2}/M_{Z}^{2}$, is used (see, e.g., Ref.
[49] for a relation between $\alpha(\mu)$ and $\alpha_{F}$).

We also need an (implicit) relation between the quark pole mass $M_{b}$ in
$n_{f}=5$ QCD$\times$QED555We also consider three charged leptons in the
spectrum. and the running parameters $\underline{m}_{b}(\mu)\equiv
m_{b}^{(5)}(\mu)$, ${a^{\prime}_{s}}(\mu)\equiv a_{s}^{(5)}(\mu)$, and
${a^{\prime}}(\mu)\equiv a^{(5)}(\mu)$:

$\displaystyle M_{b}$ $\displaystyle=$
$\displaystyle\underline{m}_{b}(\mu)\Big{\\{}1+{a^{\prime}_{s}}C_{F}\left(4+3L_{b}\right)+{a^{\prime}}Q_{d}^{2}\left(4+3L_{b}\right)$
(19) $\displaystyle+$ $\displaystyle
2{a^{\prime}}{a^{\prime}_{s}}C_{F}Q_{d}^{2}\left[\frac{121}{8}+30\zeta_{2}+8I_{3}+\frac{27}{2}L_{b}+\frac{9}{2}L_{b}^{2}\right]$
$\displaystyle+$
$\displaystyle{a^{\prime}}^{2}Q_{d}^{2}(Q_{e}^{2}+2Q_{u}^{2})\left(-\frac{71}{2}-24\zeta_{2}-26L_{b}-6L_{b}^{2}\right)$
$\displaystyle+$
$\displaystyle{a^{\prime}}^{2}Q_{d}^{4}\left(-\frac{1019}{8}+30\zeta_{2}+8I_{3}-\frac{129}{2}L_{b}-\frac{27}{2}L_{b}^{2}\right)$
$\displaystyle+$
$\displaystyle{{a^{\prime}_{s}}}^{2}C_{F}\Big{[}C_{F}\left(\frac{121}{8}+30\zeta_{2}+8I_{3}\right)+C_{A}\left(\frac{1111}{24}-8\zeta_{2}-4I_{3}\right)$
$\displaystyle\phantom{\frac{{\alpha^{\prime}_{s}}^{2}}{(4\pi)^{2}}}-T_{f}\left(\left[\frac{71}{6}+8\zeta_{2}\right]n_{f}+12n_{h}(1-2\zeta_{2})\right)$
$\displaystyle\phantom{\frac{{\alpha^{\prime}_{s}}^{2}}{(4\pi)^{2}}}+L_{b}\left(\frac{27}{2}C_{F}+\frac{185}{6}C_{A}-\frac{26}{3}n_{f}T_{f}\right)$
$\displaystyle\phantom{\frac{{\alpha^{\prime}_{s}}^{2}}{(4\pi)^{2}}}+L_{b}^{2}\left(\frac{9}{2}C_{F}+\frac{11}{2}C_{A}-2n_{f}T_{f}\right)\Big{]}\Big{\\}},$

where $Q_{d}=-1/3$, $Q_{u}=2/3$, and $Q_{e}=-1$ are the electric charges of
the SM fermions, $L_{b}=\ln(\mu^{2}/M_{b}^{2})$, and
$I_{3}=3/2\zeta_{3}-6\zeta_{2}\ln 2$. The QCD part for $n_{l}$ light flavors
and $n_{h}$ heavy ones with a common mass, so that $n_{f}=n_{l}+n_{h}$, can be
found, e.g., in Refs. [50, 51, 52].

The pure-QED part can be obtained by the substitutions $C_{A}\to 0$,
$C_{F}^{2}\to Q_{d}^{4}$, $T_{f}n_{f}\to
N_{c}(N_{d}Q_{d}^{2}+N_{u}Q_{u}^{2})+N_{l}Q_{e}^{2}$, and $T_{f}n_{h}\to
Q_{d}^{4}N_{c}n_{h}$, where $N_{u}=2$, $N_{d}=3$, $n_{h}=1$, and $N_{c}=3$.

The task is to relate $\underline{m}_{b}(\mu)$ to $m_{b,Y}(\mu)$ at a certain
scale $\mu$ by introducing the so-called decoupling constants $\zeta(\mu)$:

$\displaystyle\underline{m}_{b}(\mu)$ $\displaystyle=$ $\displaystyle
m_{b,Y}(\mu)\zeta_{m_{b,Y}}(\mu),\quad\zeta_{m_{b,Y}}(\mu)=1+\delta\zeta_{m_{b,Y}}(\mu),$
(20) $\displaystyle\delta\zeta_{m_{b,Y}}(\mu)$ $\displaystyle=$
$\displaystyle\sum\limits_{i+j=1}^{2}a^{i}a_{s}^{j}\delta\zeta^{(b)}_{ij}(M,\mu)$
(21) $\displaystyle=$
$\displaystyle\sum\limits_{i+j=1}^{2}a_{F}^{i}a_{s}^{j}\tilde{\delta}\zeta^{(b)}_{ij}(M,\mu).$
(22)

The key feature of $\zeta_{m_{b,Y}}(\mu)$ is the absence of the dependence on
the “soft” scale $M_{b}$ and the absence of tadpole contributions. The latter
feature can be traced to the fact that, at the leading order, we have
$\underline{m}_{b}(\mu)=m_{b,Y}(\mu)$ and not
$\underline{m}_{b}(\mu)=m_{b}(\mu)\equiv
y_{b}(\mu)v_{\mathrm{tree}}(\mu)/\sqrt{2}$ with a running, gauge-independent
vev. Nevertheless, it is worth mentioning that one can also use the latter
definition for the extraction of the running Yukawa coupling $y_{b}(\mu)$.
This choice corresponds to restructuring the PT series in Eq. (20). In spite
of the fact that the decoupling constant
$\underline{m}_{b}(\mu)=m_{b}(\mu)\zeta_{m_{b}}(\mu)$ in this case involve
tadpole contributions, the latter are canceled when $m_{b}(\mu)$ is divided by
the running vev $v(\mu)$ expressed in terms of $v_{F}$ [49]. However, our
choice seems more natural, since QCD “knows” nothing about tadpoles and it is
tempting to absorb them in the effective mass parameter and not to put them
into the decoupling constant.

The perturbative expansion of $\zeta_{m_{b,Y}}(\mu)$ can be found order by
order by substituting the pole mass of Eq. (19) into Eq. (16) and expressing
$\underline{m}_{b}(\mu)$ in terms of $m_{b,Y}(\mu)$ via Eq. (20). Expanding
$\delta_{ij}^{(b)}$ in the small quantity $M_{b}$ and keeping only leading
(logarithmic) terms, one obtains, at the one-loop order,
$\delta\zeta^{(b)}_{01}=0$, since there are no additional pure-QCD one-loop
diagrams in the full SM, and

$\displaystyle\delta\zeta^{(b)}_{10}$
$\displaystyle=-\frac{5}{18}-\frac{1}{3}L_{Z}+\frac{1}{\sin^{2}\theta_{w}}\left[\frac{41}{36}+\frac{3}{4}L_{W}+\frac{1}{6}L_{Z}\right]-\frac{3}{8}\frac{L_{WZ}}{\sin^{4}\theta_{w}}$
$\displaystyle+\frac{1}{\sin^{2}2\theta_{w}}\left[\frac{13}{9}-\frac{1}{4M_{Z}^{2}}(M_{t}^{2}+M_{H}^{2})+\frac{5}{6}L_{Z}-\frac{3M_{t}^{2}}{2M_{Z}^{2}}L_{t}\right]$
$\displaystyle+\frac{3}{8\sin^{2}\theta_{w}}\left[\frac{M_{H}^{2}}{M_{H}^{2}-M_{W}^{2}}L_{WH}-\frac{M_{t}^{2}}{M_{t}^{2}-M_{W}^{2}}-\frac{M_{t}^{4}}{(M_{t}^{2}-M_{W}^{2})^{2}}L_{Wt}\right],$
(23)

where $L_{X}\equiv\ln(\mu^{2}/M_{X}^{2})$ and
$L_{XY}=\ln(M_{X}^{2}/M_{Y}^{2})$, since, in addition to photon exchange, we
also have contributions involving the heavy EW gauge bosons and the Higgs
boson in the full SM. The expression in Eq. (23) can be obtained from the one-
loop contribution to the ratio $m_{b,Y}(\mu)/M_{b}$ given in Eq. (16) of Ref.
[17] by neglecting terms suppressed by powers of $M_{b}$ and subtracting pure
QCD and QED terms.

At the two-loop order, we have to take into account that the couplings of the
$n_{f}=5$ QCD$\times$QCD effective theory should also be expressed in terms of
more fundamental ones. For the current work, it is sufficient to consider only
one-loop decoupling relations (for results concerning $\alpha(\mu)$, see Refs.
[46, 53, 39, 49]):

$\displaystyle{\alpha^{\prime}_{s}}(\mu)$
$\displaystyle=\alpha_{s}(\mu)\zeta_{\alpha_{s}}(\mu),\quad\zeta_{\alpha_{s}}(\mu)=1+a_{s}\frac{4}{3}T_{f}\ln\frac{M_{t}^{2}}{\mu^{2}}+\ldots$
(24) $\displaystyle{\alpha^{\prime}}(\mu)$
$\displaystyle=\alpha(\mu)\zeta_{\alpha}(\mu),\quad\zeta_{\alpha}(\mu)=1+a\underbrace{\left(\frac{2}{3}+\frac{4}{3}N_{c}Q_{u}^{2}\ln\frac{M_{t}^{2}}{\mu^{2}}-7\ln\frac{M_{W}^{2}}{\mu^{2}}\right)}_{\delta\zeta^{(\alpha)}_{10}}+\ldots$
(25)

The decoupling constants given in Eqs. (24) and (25) can be easily obtained by
expanding the required one-loop Green functions with external light particles
in small external momenta and masses. Only contributions with at least one
heavy particle survive, and, in this simple case, no infrared divergences are
generated. For the fine-structure constant, we also have to take into account
the mixing of the photon with the $Z$ boson in the SM, so that we have

$\displaystyle\delta\zeta^{(1)}_{\alpha}(\mu)$ $\displaystyle=$
$\displaystyle-\delta\zeta^{(1)}_{\gamma\gamma}(\mu)-\frac{\sin\theta_{w}}{\cos\theta_{w}}\delta\zeta^{(1)}_{Z\gamma}(\mu).$
(26)

Here, $\delta\zeta^{(1)}_{\gamma\gamma}(\mu)$ is found from the transverse
part $i\Pi_{\gamma\gamma}(k^{2})$ of the photon self-energy via the relation

$\displaystyle\delta\zeta^{(1)}_{\gamma\gamma}(\mu)$ $\displaystyle=$
$\displaystyle\tilde{\Pi}^{(1)^{\prime}}_{\gamma\gamma}(0),$ (27)

in which the tilde is to indicate that one should only consider contribution
from diagrams with at least one heavy line. For the mixing term in Eq. (26),
we have

$\displaystyle\delta\zeta^{(1)}_{Z\gamma}(\mu)=-\frac{2}{M_{Z}^{2}}\tilde{\Pi}^{(1)}_{Z\gamma}(0).$
(28)

One can notice that Eqs. (26)—(28) resemble expressions corresponding to the
on-shell electric-charge renormalization at one loop (cf. Refs. [25, 38, 39,
45])

The result for the quark mass decoupling constant can be cross-checked by
taking the derivative of Eq. (20) w.r.t. $\mu$, i.e.,

$\displaystyle\gamma^{b}_{m}({\alpha^{\prime}_{s}},{\alpha^{\prime}})$
$\displaystyle=\frac{1}{\zeta_{m_{b,Y}}}\frac{d}{d\ln\mu}\zeta_{m_{b,Y}}(\alpha_{s},\alpha,M,\mu)+\beta_{y_{b}}(\alpha_{s},\alpha,M,\mu).$
(29)

and expressing the effective-theory couplings, which appear on the left-hand
side, in terms of the SM ones, $\alpha(\mu)$ and $\alpha_{s}(\mu)$.

We expanded Eq. (29) in $\alpha_{s}(\mu)$ and $\alpha(\mu)$ through the second
order and proved that the relation indeed holds666 We can also trade the SM
fine-structure constant $\alpha(\mu)$ for $\alpha_{F}$ via Ref. [49] to prove
that the scale dependence of $\tilde{\delta}\zeta^{(b)}_{ij}(\mu)$ is also
reproduced. for
$\delta\zeta^{(b)}_{11}(\mu)=\tilde{\delta}\zeta^{(b)}_{11}(\mu)$,
$\delta\zeta^{(b)}_{20}(\mu)$, and $\tilde{\delta}\zeta^{(b)}_{20}(\mu)$
presented here. The pure-QCD decoupling corrections for the running quark mass
are known through the four-loop level [54, 55]. For convenience, we present
here the $\mathcal{O}(\alpha_{s}^{2})$ term

$\displaystyle\delta\zeta^{(b)}_{02}(\mu)$
$\displaystyle=C_{F}T_{f}\left(\frac{89}{18}-\frac{10}{3}L_{t}+2L_{t}^{2}\right)$
(30)

and refer to Ref. [56] for higher-order corrections.

The results for the purely EW and mixed two-loop corrections can be cast into
the following expressions777The same expansion holds for
$\tilde{\delta}\zeta^{(b)}_{ij}(\mu)$, but with coefficients denoted by
$\tilde{X}^{(k)}_{ij}$. with an auxiliary scale $\mu_{0}$:

$\displaystyle\delta\zeta^{(b)}_{ij}(\mu)=\left(\frac{M_{t}^{2}}{M_{W}^{2}s_{W}^{2}}\right)^{i}\left(X^{(0)}_{ij}+X^{(1)}_{ij}\ln\frac{\mu^{2}}{\mu_{0}^{2}}+X^{(2)}_{ij}\ln^{2}\frac{\mu^{2}}{\mu_{0}^{2}}\right),\quad
i+j=2,$ (31)

where the large ratio $M_{t}^{2}/(M_{W}^{2}\sin^{2}\theta_{w})\simeq 20.8(2)$
was factored out. We refrain from writing down a lengthy analytical result for
the coefficients $X^{(k)}_{ij}$, but evaluate them at $\mu_{0}=175$ GeV and
provide the following numerical formulas:

$\displaystyle X^{(2)}_{11}$ $\displaystyle=\frac{3}{2},$ (32a) $\displaystyle
X^{(1)}_{11}$ $\displaystyle=1.9647-0.0192\times\Delta
M_{t}-0.0002\times\Delta M_{W},$ (32b) $\displaystyle X^{(0)}_{11}$
$\displaystyle=-5.7665-0.0123\times\Delta M_{t}+0.0015\times\Delta
M_{W}-0.0002\times\Delta M_{Z},$ (32c) $\displaystyle X^{(2)}_{02}$
$\displaystyle=-0.365+0.001\times\Delta M_{t},$ (32d) $\displaystyle
X^{(1)}_{02}$ $\displaystyle=-0.329+0.016\times\Delta M_{t},$ (32e)
$\displaystyle X^{(0)}_{02}$ $\displaystyle=-0.971+0.020\times\Delta
M_{t}-0.003\times\Delta M_{W}+0.002\times\Delta M_{H},$ (32f)
$\displaystyle\tilde{X}^{(2)}_{02}$ $\displaystyle=-0.389+0.001\times\Delta
M_{t},$ (32g) $\displaystyle\tilde{X}^{(1)}_{02}$
$\displaystyle=-0.669+0.008\times\Delta M_{t},$ (32h)
$\displaystyle\tilde{X}^{(0)}_{02}$ $\displaystyle=+0.569+0.012\times\Delta
M_{t},$ (32i)

where $\Delta M_{i}\equiv(M_{i}-M_{i}^{\mathrm{PDG}})/\delta
M^{\mathrm{PDG}}_{i}$ with $M_{i}^{\mathrm{PDG}}$ and $\delta
M_{i}^{\mathrm{PDG}}$ corresponding to the central value and experimental
error for $M_{i}$ quoted by the PDG [21]. We have checked that Eq. (32)
reproduces the full analytic results888Available upon request from the
authors. within the 3$\sigma$ region around the central values of the input
parameters.

The value of $m_{b,Y}(\mu)$ or, equivalently, the running Yukawa coupling in
the SM can be found from $\underline{m}_{b}(\mu)$ by inverting the relation in
Eq. (20) and expressing $\alpha_{s}(\mu)$ and $\alpha(\mu)$ in terms of the
effective-theory couplings ${\alpha^{\prime}_{s}}(\mu)$ and
${\alpha^{\prime}}(\mu)$:

$\displaystyle m_{b,Y}(\mu)$
$\displaystyle=\zeta^{-1}_{m_{b,Y}}(\mu)\underline{m}_{b}(\mu)=\underline{m}_{b}(\mu)\sum\limits_{i+j=1}^{2}\left({a^{\prime}}\right)^{i}\left({a^{\prime}_{s}}\right)^{j}\underline{\delta}\zeta^{(b)}_{ij}(\mu),$
(33) $\displaystyle\underline{\delta}\zeta^{(b)}_{10}$
$\displaystyle=-\delta\zeta^{(b)}_{10},\quad\underline{\delta}\zeta^{(b)}_{11}=-\delta\zeta^{(b)}_{11},\quad\underline{\delta}\zeta^{(b)}_{02}=-\delta\zeta^{(b)}_{02},$
(34) $\displaystyle\underline{\delta}\zeta^{(b)}_{20}$
$\displaystyle=-\delta\zeta^{(b)}_{20}+\left(\delta\zeta^{(\alpha)}_{10}+\delta\zeta^{(b)}_{10}\right)\delta\zeta^{(b)}_{10}.$
(35)

Another option is to use the scale-independent coupling $\alpha_{F}$ in place
of ${\alpha^{\prime}}(\mu)$. In this case, one needs to replace
$\delta\zeta^{(b)}_{20}(\mu)$ by $\tilde{\delta}\zeta^{(b)}_{20}(\mu)$ in Eq.
(35) and exclude the contributions from $\delta\zeta^{(\alpha)}_{10}(\mu)$,
which originate in Eq. (33) due to the conversion
$\alpha(\mu)\to{\alpha^{\prime}}(\mu)$.

For illustration, let us present numerical values of the different corrections
at some fixed scale, e.g., $\mu=M_{Z}$. In the case of the $m_{b,Y}-M_{b}$
relation, one obtains

$\displaystyle m_{b,Y}(M_{Z})=M_{b}\big{(}1$
$\displaystyle-\underbrace{0.2682}_{\alpha_{s}}-\underbrace{0.0776}_{\alpha_{s}^{2}}-\underbrace{0.0330}_{\alpha_{s}^{3}}$
$\displaystyle-\underbrace{0.0101}_{\alpha}+\underbrace{0.0032}_{\alpha\alpha_{s}}+\underbrace{0.0003}_{\alpha^{2}}+\ldots\big{)},$
(36)

while the $m_{b,Y}-\underline{m}_{b}$ relation yields

$\displaystyle m_{b,Y}(M_{Z})=\underline{m}_{b}(M_{Z})\big{(}1$
$\displaystyle-\underbrace{0.00074}_{\alpha_{s}^{2}}-\underbrace{0.00023}_{\alpha_{s}^{3}}+\underbrace{0.00002}_{\alpha_{s}^{4}}+\Delta\zeta_{b}\big{)},$
(37) $\displaystyle\Delta\zeta_{b}=$
$\displaystyle-\underbrace{0.00838}_{\alpha_{F}}+\underbrace{0.00068}_{\alpha_{F}\alpha_{s}}-\underbrace{0.00005}_{\alpha_{F}^{2}}+\ldots$
(38) $\displaystyle=$
$\displaystyle-\underbrace{0.00865}_{\alpha}+\underbrace{0.00070}_{\alpha\alpha_{s}}+\underbrace{0.00029}_{\alpha^{2}}+\ldots$
(39)

From the comparison of Eqs. (36) and (37), one can see that the decoupling
corrections in Eq. (33) are much smaller than the pole-mass corrections in Eq.
(16), since the latter involve large logarithms, which are resummed in
$\underline{m}_{b}(\mu)$ in the former case. As for the contribution due to EW
interactions, the leading one-loop term dominates in Eq. (37), while the
subleading mixed corrections of order $\mathcal{O}(\alpha_{F}\alpha_{s})$ tend
to cancel the two-loop $\mathcal{O}(\alpha_{s}^{2})$ contribution.

Equation (33) is written for some fixed scale and is typically applied for
$\mu\sim\mu_{0}$ close to the EW scale. The value of
$\underline{m}_{b}(\mu_{0})$ can be found from the known value
$\mu_{b}\equiv\underline{m}_{b}(\underline{m}_{b})=4.18\pm 0.03$ GeV999We use
here the PDG value for conservative estimates, instead of the more precise
value $\underline{m}_{b}(\underline{m}_{b})=4.136\pm 0.016$ GeV [57]. by
solving the coupled RGEs of the QCD$\times$QED effective theory,

$\displaystyle\frac{\underline{m}_{b}(\mu)}{\underline{m}_{b}(\mu_{b})}$
$\displaystyle=\exp\left[~{}\int\limits_{\mu_{b}}^{\mu}\gamma_{m}^{b}\left[{\alpha^{\prime}}(\mu^{\prime}),{\alpha^{\prime}_{s}}(\mu^{\prime})\right]d\ln\mu^{\prime}~{}\right]\equiv
C_{\scriptstyle\mathrm{QCD}\times\mathrm{QED}}(\mu,\mu_{b}).$ (40)

In pure QCD, the integration over $\ln\mu$ in Eq. (40) can be traded for the
integration over ${\alpha^{\prime}_{s}}$. Due to this, the analogous factor
$C_{\scriptstyle\mathrm{QCD}}(\mu,\mu_{b})$ can be cast into the form

$\displaystyle
C_{\scriptstyle\mathrm{QCD}}(\mu,\mu_{b})=\frac{c({\alpha^{\prime}_{s}}(\mu)/\pi)}{c({\alpha^{\prime}_{s}}(\mu_{b})/\pi)},$
(41)

with $c(x)$ given, e.g., in the recent Ref. [58].

Collecting all the factors, the final formula for $m_{b,Y}(\mu)$ reads:

$\displaystyle
m_{b,Y}(\mu)=\mu_{b}C_{\mathrm{QCD}\times\mathrm{QED}}(\mu,\mu_{b})\zeta^{-1}_{m_{b,Y}}(\mu).$
(42)

##  4 Numerical analysis of matching relations

To begin with, we study the impact of additional QED correction to the running
of ${\alpha^{\prime}_{s}}(\mu)$ and $\underline{m}_{b}(\mu)$. This running
corresponds to the resummation of logarithmically enhanced terms due to EW
interactions. In Fig. 1, we present the scale dependence of these quantities
computed by means of the five-loop QCD RGE [58, 59] accompanied by the three-
loop QED corrections given in A. It turns out that the difference between
three-loop and five-loop results are negligible when compared to experimental
uncertainties in the boundary values. Nevertheless, for illustrative purpose,
we provide the scale dependencies of the relative contributions to the five-
loop strong-coupling beta function and the quark mass anomalous dimension. For
the strong coupling, the two-loop QED contribution to $\beta_{\alpha_{s}}$ is
comparable with the four-loop QCD terms, while the three-loop electromagnetic
effects compete with the five-loop pure-QCD contribution. As for the $b$-quark
mass, the situation is similar, and the leading one-loop QED corrections is of
the same order as the four-loop pure-QCD terms. It is interesting to note that
the corresponding two-loop contributions are slightly less than the three-loop
QED terms. This is due to a cancellation of $\mathcal{O}(\alpha^{2})$ and
$\mathcal{O}(\alpha\alpha_{s})$ corrections to $\gamma_{\underline{m}_{b}}$.

(a)
$\beta_{\alpha_{s}}\equiv\sum\limits_{i+j=1}^{3}\beta^{\alpha_{s}}_{ij}a^{i}a_{s}^{j}+\beta^{\alpha_{s}}_{04}a_{s}^{4}+\beta^{\alpha_{s}}_{05}a_{s}^{5}$.

(b)
$\gamma_{m}^{b}\equiv\sum\limits_{i+j=1}^{3}\gamma^{b}_{ij}a^{i}a_{s}^{j}+\gamma^{b}_{04}a_{s}^{4}+\gamma^{b}_{05}a_{s}^{5}$.

Fig. 1:  The running of $\alpha_{s}$ and $m_{b}$ in QCD$\times$QED with five
active quark flavors obtained from the given input by means of five-loop RGEs.
QED corrections are only included through the three-loop order. The effect of
QED is negligible as compared to the uncertainty in the input parameters.
Nevertheless, in the case of $\alpha_{s}^{(5)}\equiv{\alpha^{\prime}_{s}}$,
the two-loop QED contribution to $\beta_{\alpha_{s}}$ is comparable with the
four-loop QCD terms, while the three-loop corrections due to QED are of the
same order as the five-loop QCD result [59]. For the $b$-quark mass parameter
$m_{b}^{(5)}\equiv\underline{m}_{b}$, the one-loop QED correction to
$\gamma^{b}_{m}$ has the same order as the four-loop QCD term, while the five-
loop contribution [58] due to $\gamma^{b}_{05}$ is much larger than the two-
and three-loop QED corrections for $\mu\ll M_{t}$. For $\mu>M_{t}$, they
become comparable. It is also worth mentioning that, because of cancellations
between terms due to $\gamma_{20}$ and $\gamma_{11}$, the two-loop QED
corrections are even less than the three-loop ones.

Let us now perform a numerical analysis of the corrections to our matching
formulas. In Fig. 2(a), the scale dependencies of the different contributions
to the relation in Eq. (16) computed by means of the program package mr [60]
are presented. Note that the analytic expressions for the two- and three-loop
QCD corrections including finite top-quark mass effects were taken from Ref.
[19]. The three-loop master integrals [20] were reevaluated numerically and by
means of asymptotic expansion for the case of additional heavy quarks. Good
agreement was found between numerical Mellin-Barnes integration and the
lowest-order expansion. The corresponding expressions in the form of
asymptotic series in the small parameter $z=M_{b}/M_{t}$ are given in B.

(a) Relation between $m_{b,Y}$ and $M_{b}$ (16).

(b) Relation between $m_{b,Y}$ and $\underline{m}_{b}$ (33).

Fig. 2: The scale dependencies of different corrections to the relation
between $m_{b,Y}$ and (a) the pole mass $M_{b}$ and (b) the running mass
$\underline{m}_{b}$. The reference scale is $M_{t}=173.21$ GeV.

From Fig. 2(a) and Eq. (36), one can see that pure-QCD contributions dominate
the $m_{b,Y}-M_{b}$ relation. If one formally takes the value of the (total)
three-loop term as an estimate of the theoretical uncertainty, the precision
of the $m_{b,Y}-M_{b}$ matching in Eq. (16) is currently limited to be a few
percent due to the $\mathcal{O}(\alpha_{s}^{3})$ contribution. On the
contrary, the PT series for $m_{b,Y}-\underline{m}_{b}$ in Eq. (33) behaves
much better. Pure-QCD corrections involving $\ln(M_{b}/\mu)$ are resummed
together with the QED ones, so that the relation is saturated by (one-loop) EW
corrections, which are about 1–2%. Two-loop EW terms are approximately of the
same order as three- and four-loop pure-QCD contributions. If compared to the
uncertainty of the input value of $\mu_{b}$, only the one-loop EW corrections
turn out to be important in the $m_{b,Y}-\underline{m}_{b}$ relation for the
considered matching scales, while dominant QCD corrections in the
$m_{b,Y}-M_{b}$ relation can be resummed by means of the three-loop pure-QCD
RGE.

Let us make one more comment about power-suppressed corrections of
$\mathcal{O}(m_{b}/M)$ to the relation between $m_{b,Y}$ and $M_{b}$. In our
approach, we consistently neglect them. This also corresponds to dropping
terms of the order of $\alpha/(4\pi)m_{b}^{2}/M_{W}^{2}\simeq
y_{b}^{2}/(16\pi^{2})\sim 10^{-6}$ in Eq. (33). The estimated contribution is
an order of magnitude less than the typical size of the threshold corrections
considered in this paper (c.f. Eq. (38)). Due to this, the inclusion of power-
suppressed contributions is not necessary at the moment.

Finally, we consider the dependence on the matching scale $\mu_{\rm th}$ of
the running $b$-quark mass parameters in $n_{f}=6$ QCD, $m_{b}(M_{t})$, and
the full SM, $m_{b,Y}(M_{t})$, at a fixed scale $\mu=M_{t}$. The running from
$\mu_{b}$ to $\mu_{\rm th}$ is governed by the $n_{f}=5$ effective-theory
RGEs, while the RG evolution from $\mu_{\rm th}$ to $\mu=M_{t}$ is described
by either QCD with active top quark or the full SM (see Refs. [61, 62, 63] for
three-loop RGEs). In Fig. 3, one can see how the dependence is reduced due to
new higher-order terms both in the RGEs and the matching. The $L$-loop RGEs
are supplemented by $(L-1)$-loop threshold corrections in the pure-QCD case
(see Fig. 3(a)). In Fig. 3(a), we also indicated our conservative estimates
for the corresponding values of $m_{b}(M_{t})$ together with their theoretical
uncertainty due to matching scale variation $0.1\leq\mu_{\rm th}/M_{t}\leq
10$.

In the SM, we lack three- and four-loop EW contributions to the
$m_{b,Y}-\underline{m}_{b}$ relation. Moreover, four-loop EW corrections to
the SM RGE are only partially known in the literature [64, 65]. Due to this,
we restrict ourselves in Fig. 3(b) to the four-loop order. The reduction of
the matching-scale dependence is clearly visible when one goes from two to
three loops in a self-consistent procedure, while the partial addition of
four-loop (RG) terms does not improve the situation. From Fig. 3(b), it is
clear that, if we neglect the EW contribution in the matching relation as
indicated by the dashed lines with the label “no EW,” the dependence becomes
more pronounced, thus, signifying the role of EW corrections in a consistent
analysis.

(a) QCD RGEs and threshold corrections.

(b) SM RGEs and threshold corrections.

Fig. 3: The dependence on the matching scale $\mu_{\rm th}$ of the $b$-quark
running mass parameter (a) in pure $n_{f}=6$ QCD, $m_{b}(M_{t})$, and (b) in
the SM, $m_{b,Y}(M_{t})$, at $L=2,3,4,5$ loops. Pure-QCD threshold corrections
are included through the $(L-1)$-loop level, and the corresponding values of
$m_{b}(M_{t})$ are indicated together with their theoretical uncertainties due
to the $\mu_{\rm th}$ variation by a factor of ten. In the case of the SM, EW
and mixed contributions (collectively labeled EW) are taken into account only
through two loops. Four-loop contributions to the SM RGEs include pure-QCD
corrections to the beta functions of the strong and quark Yukawa couplings
together with recent results from Refs. [64, 65]. The necessity of EW
threshold corrections in the SM can be deduced from the $\mu_{\rm th}$ scale
dependence of the dashed curves, which lack the latter. In addition, the five-
loop pure-QCD curve from Fig. 3(a) is indicated.

In Fig. 3(b), we also add the line corresponding to the four-loop pure-QCD
result from Fig. 3(a). Clearly, if one treats the QCD result as $m_{b,Y}(\mu)$
with neglected EW corrections, this overestimates $m_{b,Y}(\mu)$, and the
shift is comparable with the experimental uncertainty in the input value of
$\mu_{b}$, which is about 0.7%. Our final estimates for $m_{b,Y}(M_{t})$ and
the corresponding theoretical uncertainties are given by

$\displaystyle m_{b,Y}(M_{t})$ $\displaystyle=2.710\pm 0.012_{\rm
th}~{}\mathrm{GeV}\qquad\mbox{(2 loops)},$ $\displaystyle m_{b,Y}(M_{t})$
$\displaystyle=2.681\pm 0.003_{\rm th}~{}\mathrm{GeV}\qquad\mbox{(3 loops)},$
(43)

from which the three-loop value of the corresponding Yukawa coupling can be
easily obtained as

$y_{b}(M_{t})=0.01539\pm 0.00002_{\rm th}.$ (44)

On can see that, thanks to resummation of $\ln M_{b}/M_{t}$, the theoretical
uncertainty is significantly reduced as compared to our previous analysis
based on the $y_{b}-M_{b}$ relation [66].

##  5 Conclusions

The $b$ quark plays a significant role in modern particle physics, and the
precise knowledge of the corresponding mass parameters is necessary for
accurate theoretical predictions.

In this paper, we left aside low-energy problems related to confinement and
considered high-energy, or short-distance, definitions of the $b$-quark mass.
Given the value of the running $\overline{\mathrm{MS}}$ mass in effective
five-flavor QCD, the EFT approach was used to relate it to the quantity of our
interest, the running parameter in the SM, $m_{b,Y}(\mu)$, or, equivalently,
the $b$-quark Yukawa coupling $y_{b}(\mu)$. We concentrated mainly on the two-
loop EW corrections, which, although being suppressed with respect to the QCD
ones, can play an appreciable role in precise analyses.

We demonstrated how effective theories can be used to resum certain types of
logarithmic corrections, e.g., $\ln(M_{b}/M)$, which appear in the relation
between $y_{b}(M)$ and the pole mass $M_{b}$. Our analysis shows that the
effect of QED logarithm resummation can be safely ignored at the moment, while
EW matching plays an important role in the estimation of the running parameter
$y_{b}(\mu)$ at $\mu\geq M_{Z}$.

The obtained results for $y_{b}(\mu)$ mainly affect high-energy processes
involving $b$ quarks, in which its (kinematic) mass can be neglected and only
Yukawa interactions matter. As an example, we refer to the dominant Higgs
decay mode $H\to\bar{b}b$ (see Refs. [67, 68, 69, 70] and recent Ref. [71] for
the EW corrections and Refs. [72, 73, 74, 75, 76, 77, 78, 79] for the
corrections due to QCD).

## Acknowledgments

The authors are grateful to M.Yu. Kalmykov and A.I. Onischenko for fruitful
discussions. Multiple comments on the manuscript by M.Yu. Kalmykov are also
acknowledged. This work was supported in part by the German Federal Ministry
for Education and Research BMBF through Grant No. 05H15GUCC1, by the German
Research Foundation DFG through the Collaborative Research Centre No. SFB 676
Particles, Strings and the Early Universe: the Structure of Matter and Space-
Time, by the Russian Foundation for Basic Research RFBR through Grant No.
14-02-00494, and by the Heisenberg-Landau Programme.

##  Appendix A  RGEs in effective QCD$\times$QED

The RGEs for the effective-theory couplings are given by (see also Refs. [80,
53, 81])

$\displaystyle\mu^{2}\frac{d\alpha_{i}}{d\mu^{2}}$
$\displaystyle=\alpha_{i}\beta^{i},\quad\alpha_{i}\in\left\\{\alpha,\alpha_{s}\right\\},\quad\beta_{i}=\sum\limits_{k,l=1}^{3}\beta^{i}_{kl}\left(a\right)^{k}\left(a_{s}\right)^{l}+\ldots,$
(45) $\displaystyle\beta^{\alpha}_{10}$
$\displaystyle=\frac{4}{3}\left[N_{l}Q_{e}^{2}+N_{c}\left(N_{d}Q_{d}^{2}+N_{u}Q_{u}^{2}\right)\right],\qquad\beta^{\alpha}_{0j}\equiv
0,\quad j=1,...$ (46) $\displaystyle\beta^{\alpha}_{11}$
$\displaystyle=4C_{F}N_{c}\left[N_{d}Q_{d}^{2}+N_{d}Q_{u}^{2}\right],$ (47)
$\displaystyle\beta^{\alpha}_{20}$
$\displaystyle=4\left[N_{l}Q_{e}^{4}+N_{c}\left(N_{d}Q_{d}^{4}+N_{u}Q_{u}^{4}\right)\right],$
(48) $\displaystyle\beta^{\alpha}_{30}$
$\displaystyle=-\frac{44}{9}N_{c}^{2}\left[N_{d}^{2}Q_{d}^{6}+N_{d}^{2}Q_{u}^{6}+N_{u}N_{d}\left(Q_{u}^{2}+Q_{d}^{2}\right)Q_{u}^{2}Q_{d}^{2}\right]$
$\displaystyle-\frac{44}{9}N_{l}Q_{e}^{2}\left[N_{c}\left[Q_{e}^{2}\left(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2}\right)+N_{u}Q_{u}^{4}+N_{d}Q_{d}^{4}\right]+N_{l}Q_{e}^{4}\right]$
$\displaystyle-2\left[N_{c}(N_{u}Q_{u}^{6}+N_{d}Q_{d}^{6})+N_{l}Q_{e}^{6}\right],$
(49) $\displaystyle\beta^{\alpha}_{21}$
$\displaystyle=-4C_{F}N_{c}\left[N_{u}Q_{u}^{4}+N_{d}Q_{d}^{2}\right],$ (50)
$\displaystyle\beta^{\alpha}_{12}$
$\displaystyle=C_{F}N_{c}\left(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2}\right)\left[\frac{133}{9}C_{A}-2C_{F}-\frac{44}{9}T_{f}(N_{u}+N_{d})\right],$
(51)

$\displaystyle\beta^{\alpha_{s}}_{01}$
$\displaystyle=-\frac{11}{3}C_{A}+\frac{4}{3}T_{f}n_{f},\qquad\beta^{\alpha_{s}}_{j0}\equiv
0,\quad j=1,...$ (52) $\displaystyle\beta^{\alpha_{s}}_{11}$
$\displaystyle=4T_{F}\left[N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2}\right],$ (53)
$\displaystyle\beta^{\alpha_{s}}_{02}$
$\displaystyle=-\frac{34}{3}C_{A}^{2}+T_{f}n_{f}\left[4C_{F}+\frac{20}{3}C_{A}\right],$
(54) $\displaystyle\beta^{\alpha_{s}}_{03}$
$\displaystyle=-\frac{2857}{54}C_{A}^{3}+\frac{1415}{27}C_{A}^{2}T_{f}n_{f}+\frac{205}{9}C_{A}C_{F}T_{f}n_{f}-\frac{158}{27}C_{A}T_{f}^{2}n_{f}^{2}$
$\displaystyle-2C_{F}^{2}T_{f}n_{f}-\frac{44}{9}C_{F}T_{f}^{2}n_{f}^{2},$ (55)
$\displaystyle\beta^{\alpha_{s}}_{12}$
$\displaystyle=-\frac{44}{9}T_{f}\left[N_{c}(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2})^{2}+N_{l}Q_{e}^{2}(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2})\right]$
$\displaystyle-2T_{f}(N_{u}Q_{u}^{4}+N_{d}Q_{d}^{4}).$ (56)

The anomalous dimension of the $b$-quark mass in QED$\times$QCD (for the pure-
QCD part, see Refs. [82, 83]) can be cast into the form

$\displaystyle\mu^{2}\frac{dm_{b}}{d\mu^{2}}$
$\displaystyle=\gamma^{b}_{m}m_{b},\qquad\gamma_{m}^{b}=-\sum\limits_{i+j=1}^{3}\gamma^{b}_{ij}\left(a\right)^{i}\left(a_{s}\right)^{j}+\ldots,$
(57) $\displaystyle\gamma^{b}_{01}$
$\displaystyle=3C_{F},\qquad\gamma^{b}_{10}=3Q_{d}^{2},\qquad\gamma^{b}_{11}=3C_{F}Q_{d}^{2},$
(58) $\displaystyle\gamma^{b}_{02}$
$\displaystyle=\frac{3}{2}C_{F}^{2}+\frac{97}{6}C_{F}C_{A}-\frac{10}{3}C_{F}T_{f}n_{f},$
(59) $\displaystyle\gamma^{b}_{20}$
$\displaystyle=\frac{3}{2}Q_{d}^{4}-\frac{10}{3}Q_{d}^{2}\left[N_{c}\left(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2}\right)+N_{l}Q_{e}^{2}\right]$
(60) $\displaystyle\gamma^{b}_{03}$
$\displaystyle=-\frac{129}{4}C_{F}^{2}C_{A}+\frac{11413}{108}C_{F}C_{A}^{2}+C_{F}C_{A}T_{f}n_{f}\left(-\frac{556}{27}-48\zeta_{3}\right)$
$\displaystyle+\frac{129}{2}C_{F}^{3}-\frac{140}{27}C_{F}T^{2}_{f}n^{2}_{f}+C_{F}^{2}T_{f}n_{f}(-45+48\zeta_{3})-C_{F}^{2}T_{f}n_{f},$
(61) $\displaystyle\gamma^{b}_{12}$
$\displaystyle=-\frac{129}{4}C_{F}C_{A}Q_{d}^{2}+3\frac{129}{2}C_{F}^{2}Q_{d}^{2}$
$\displaystyle-
C_{F}T_{f}(N_{u}+N_{d})Q_{d}^{2}+C_{F}T_{f}(-45+48\zeta_{3})(N_{d}Q_{d}^{2}+N_{u}Q_{u}^{2})$
(62) $\displaystyle\gamma^{b}_{21}$
$\displaystyle=3\frac{219}{2}C_{F}Q_{d}^{4}-C_{F}Q_{d}^{2}\left[N_{e}Q_{e}^{2}+N_{c}(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2})\right]$
$\displaystyle+C_{F}Q_{d}^{2}N_{c}(-45+48\zeta_{3})(N_{d}Q_{d}^{2}+N_{u}Q_{u}^{2})$
(63) $\displaystyle\gamma^{b}_{30}$
$\displaystyle=\frac{129}{2}Q_{d}^{6}-\frac{140}{27}Q_{d}^{2}\left[N_{e}Q_{e}^{2}+N_{c}(N_{u}Q_{u}^{2}+N_{d}Q_{d}^{2})\right]^{2}$
$\displaystyle-
Q_{d}^{4}\left[N_{e}Q_{e}^{4}+N_{c}(N_{u}Q_{u}^{4}+N_{d}Q_{d}^{4})\right].$
(64)

##  Appendix B  Three-loop corrections to the pole mass of the $b$ quark in
$n_{f}=6$ QCD

Fig. 4: The scale dependencies of the relative differences $\Delta_{i}\equiv
1-\delta^{(b)}_{0i}(n_{f}=6)/\delta^{(b)}_{0i}(n_{f}=5)$ of the pure-QCD
coefficients of the $i$-loop contributions to the relation between the running
and pole masses of the $b$ quark in $n_{f}=5$ and $n_{f}=6$ QCD (cf. Eq.
(16)).

Let us consider the relation between the pole mass $M_{q}$ and the running
mass $m_{q}(z,\mu)$ of a heavy quark in QCD with $n_{l}$ massless quarks,
$n_{h}$ quarks with pole mass $M_{q}$, and $n_{m}$ quarks with pole mass
$M_{f}$. Defining $z\equiv M_{q}/M_{f}$, one can write the following relation:

$\displaystyle\frac{m_{q}(z,\mu)}{M_{q}}=1$
$\displaystyle+\left(\delta_{\mathrm{QCD}}(\mu)\right)_{n_{m}=0}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}n_{m}X_{2,1}(M_{q},z,\mu)$
$\displaystyle+\left(\frac{\alpha_{s}}{4\pi}\right)^{3}\left[n_{m}X_{3,1}(M_{q},z,\mu)+n_{m}^{2}X_{3,2}(M_{q},z,\mu)\right].$
(65)

In the case of the $b$ quark in $n_{f}=6$ QCD, we have $n_{l}=4$,
$n_{h}=n_{m}=1$, $M_{q}=M_{b}$, and $z=M_{b}/M_{t}$.

The part independent of the heavy-quark masses can be found in Refs. [84, 85,
49]. The contributions from loops of $n_{m}$ heavy quarks are contained in the
coefficient $X_{2,1}$ at two loops and in the coefficients $X_{3,1}$ and
$X_{3,2}$ at three loops. The exact result for $X_{2,1}$ is available from
Ref. [50]. The expansions of $X_{3,1}$ and $X_{3,2}$ in the limit $z\to\infty$
are known from Ref. [19]. Here, we present results for $X_{3,1}$ and $X_{3,2}$
in the opposite limit $z\to 0$. The relation between the masses is obtained
from a general result [20], in which the analytically known integrals were
substituted and the unknown $\mathcal{O}(\epsilon)$ parts of four master
integrals were calculated by means of asymptotic expansion in the large
internal masses.

For convenience, we present here the two-loop result in expanded form,

$\displaystyle X_{2,1}=C_{F}T_{F}$
$\displaystyle\left(-\frac{89}{18}+\frac{26}{3}L_{M}+2L_{M}^{2}+\frac{52}{3}\ln(z)-8\ln^{2}(z)\right.$
$\displaystyle\left.+z^{2}\left(\frac{152}{75}-\frac{32}{15}\ln(z)\right)+\mathcal{O}(z^{4})\right),$
(66)

together with the leading terms of the three-loop results,101010The expansions
up to the $\mathcal{O}(z^{10})$ terms can be found in an attachment to the
arXiv version of this paper.

$\displaystyle X_{3,2}=$ $\displaystyle
C_{F}T_{F}^{2}\left[\frac{3370}{243}-\frac{224}{9}\zeta_{3}+\frac{496}{27}L_{M}-\frac{104}{9}L_{M}^{2}-\frac{16}{9}L_{M}^{3}\right.$
$\displaystyle+$
$\displaystyle\left(\frac{992}{27}-\frac{416}{9}L_{M}\right)\ln(z)-\left(\frac{416}{9}-\frac{64}{3}L_{M}\right)\ln^{2}(z)+\frac{256}{9}\ln^{3}(z)$
$\displaystyle+$
$\displaystyle\left.z^{2}\left(\frac{368}{81}-\frac{1216}{225}L_{M}-\left(\frac{2432}{225}-\frac{256}{45}L_{M}\right)\ln(z)+\frac{512}{45}\ln^{2}(z)\right)+\mathcal{O}(z^{4})\right],$
(67)

$\displaystyle X_{3,1}=$ $\displaystyle
C_{F}^{2}T_{F}\left[\frac{547}{3}+\frac{88}{45}\pi^{4}+\frac{32}{3}\pi^{2}\ln^{2}2-\frac{32}{3}\ln^{4}2-256a_{4}-114\zeta_{3}\right.$
$\displaystyle+$ $\displaystyle
L_{M}\left(\frac{367}{6}+\frac{40}{3}\pi^{2}-\frac{64}{3}\pi^{2}\ln
2-16\zeta_{3}\right)-26L_{M}^{2}-6L_{M}^{3}$ $\displaystyle+$
$\displaystyle\ln(z)\left(\frac{8}{3}+\frac{80}{3}\pi^{2}-\frac{128}{3}\pi^{2}\ln
2-52L_{M}-32\zeta_{3}\right)$ $\displaystyle+$ $\displaystyle
24L_{M}\ln^{2}(z)+z^{2}\left(\frac{1001648}{30375}+\frac{128}{135}\pi^{2}-\frac{308}{9}\zeta_{3}\right.$
$\displaystyle-$
$\displaystyle\left.\left.\frac{152}{25}L_{M}-\left(\frac{26816}{2025}+\frac{32}{5}L_{M}\right)\ln(z)-\frac{128}{45}\ln^{2}(z)\right)\right]$
$\displaystyle+$ $\displaystyle
C_{F}T_{F}^{2}\left[-\frac{5308}{243}(n_{h}+n_{l})+\frac{128}{9}\zeta_{3}(n_{h}+n_{l})-\left(40n_{h}+8n_{l}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\frac{64}{9}n_{h}\pi^{2}+\frac{32}{9}n_{l}\pi^{2}\right)L_{M}-\frac{208}{9}(n_{h}+n_{l})L_{M}^{2}-\frac{32}{9}(n_{h}+n_{l})L_{M}^{3}$
$\displaystyle+$
$\displaystyle\left(-80n_{h}-16n_{l}+\frac{128}{9}n_{h}\pi^{2}-\frac{64}{9}n_{l}\pi^{2}-\frac{416}{9}(n_{h}+n_{l})L_{M}\right)\ln(z)$
$\displaystyle+$
$\displaystyle\frac{64}{3}(n_{h}+n_{l})L_{M}\ln^{2}(z)+\frac{128}{9}(n_{h}+n_{l})\ln^{3}(z)$
$\displaystyle+$ $\displaystyle
z^{2}\left(-\frac{98624}{3375}n_{h}+\frac{33856}{3375}n_{l}+\frac{256}{135}n_{h}\pi^{2}-\frac{128}{135}n_{l}\pi^{2}-\frac{1216}{225}(n_{h}+n_{l})L_{M}\right.$
$\displaystyle+$
$\displaystyle\left.\left.\left(\frac{512}{25}n_{h}-\frac{128}{25}n_{l}+\frac{256}{45}(n_{h}+n_{l})L_{M}\right)\ln(z)+\frac{256}{45}(n_{h}+n_{l})\ln^{2}(z)\right)\right]$
$\displaystyle+$ $\displaystyle
C_{F}C_{A}T_{F}\left[-\frac{20083}{243}-\frac{62}{45}\pi^{4}-\frac{16}{3}\pi^{2}\ln^{2}2+\frac{16}{3}\ln^{4}2+128a_{4}+\frac{629}{9}\zeta_{3}\right.$
$\displaystyle+$ $\displaystyle
L_{M}\left(\frac{1210}{9}-\frac{32}{9}\pi^{2}+\frac{32}{3}\pi^{2}\ln
2+32\zeta_{3}\right)+\frac{746}{9}L_{M}^{2}+\frac{88}{9}L_{M}^{3}$
$\displaystyle+$
$\displaystyle\ln(z)\left(\frac{2612}{9}-\frac{64}{9}\pi^{2}+\frac{64}{3}\pi^{2}\ln
2+\frac{1144}{9}L_{M}+64\zeta_{3}\right)$ $\displaystyle-$
$\displaystyle\left(\frac{232}{3}+\frac{176}{3}L_{M}\right)\ln^{2}(z)-\frac{352}{9}\ln^{3}(z)+z^{2}\left(-\frac{996881}{60750}-\frac{25}{54}\pi^{2}+\frac{124}{9}\zeta_{3}\right.$
$\displaystyle+$
$\displaystyle\left.\left.\frac{3344}{225}L_{M}+\left(\frac{115877}{4050}-\frac{704}{45}L_{M}\right)\ln(z)-\frac{454}{15}\ln^{2}(z)\right)\right]+\mathcal{O}(z^{4}),$
(68)

where the abbreviations $L_{M}=\ln(\mu^{2}/M_{q}^{2})$ and
$a_{4}=\mathrm{Li}_{4}(1/2)$ are utilized.

The effect due to the new terms is illustrated in Fig. 4, in which we present
the relative differences of the two- and three-loop pure-QCD coefficients
$\delta^{(b)}_{0i}$ in Eq. (16) with and without the effect of the top quark
as functions of the renormalization scale. We observe that, for low-mass
scales $\mathcal{O}(M_{b})$, the $n_{f}=5$ result underestimates the full
$n_{f}=6$ corrections by more than 20%, while, for very large scales
$\mathcal{O}(1~{}\mathrm{TeV})$, the effect is opposite both for the two-loop
($\Delta_{2}$) and three-loop ($\Delta_{3}$) terms. One can see that, for
scales of $\mathcal{O}(M_{t})$, the difference is not so pronounced.111111
However, one should keep in mind that here we only compare coefficients of
powers of $\alpha_{s}$.

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