aainstitutetext: University of Vienna, Faculty of Physics
Boltzmanngasse 5, A-1090 Wien, Austriabbinstitutetext: Erwin Schrödinger
International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, Austriaccinstitutetext:
Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

# On the Light Massive Flavor Dependence of the Large Order Asymptotic
Behavior and the Ambiguity
of the Pole Mass

André H. Hoang a Christopher Lepenik a,c Moritz Preisser
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We provide a systematic renormalization group formalism for the mass effects
in the relation of the pole mass $m_{Q}^{\rm pole}$ and short-distance masses
such as the $\overline{\mathrm{MS}}$ mass $\overline{m}_{Q}$ of a heavy quark
$Q$, coming from virtual loop insertions of massive quarks lighter than $Q$.
The formalism reflects the constraints from heavy quark symmetry and entails a
combined matching and evolution procedure that allows to disentangle and
successively integrate out the corrections coming from the lighter massive
quarks and the momentum regions between them and to precisely control the
large order asymptotic behavior. With the formalism we systematically sum
logarithms of ratios of the lighter quark masses and $m_{Q}$, relate the QCD
corrections for different external heavy quarks to each other, predict the
${\cal O}(\alpha_{s}^{4})$ virtual quark mass corrections in the
pole-$\overline{\mathrm{MS}}$ mass relation, calculate the pole mass
differences for the top, bottom and charm quarks with a precision of around
$20$ MeV and analyze the decoupling of the lighter massive quark flavors at
large orders. The summation of logarithms is most relevant for the top quark
pole mass $m_{t}^{\rm pole}$, where the hierarchy to the bottom and charm
quarks is large. We determine the ambiguity of the pole mass for top, bottom
and charm quarks in different scenarios with massive or massless bottom and
charm quarks in a way consistent with heavy quark symmetry, and we find that
it is $250$ MeV. The ambiguity is larger than current projections for the
precision of top quark mass measurements in the high-luminosity phase of the
LHC.

††preprint:  UWThPh-2017-13 MIT-CTP 4915

##  1 Introduction

The masses of the heavy charm, bottom and top quarks belong to the most
important input parameters in precise theoretical predictions of the Standard
Model and models of new physics. Due to the effects of quantum chromodynamics
(QCD) and because quarks are states with color charge, however, the mass of a
heavy quark $Q$ is not a physical observable and should, in general, be better
thought of as a renormalized and scheme-dependent parameter of the theory.
This concept is incorporated most cleanly in the so-called
$\overline{\mathrm{MS}}$ mass $\overline{m}_{Q}(\mu)$, which is defined
through the same renormalization prescription as the $\overline{\mathrm{MS}}$
QCD coupling $\alpha_{s}(\mu)$. It can be measured from experimental data very
precisely, but does not have any kinematic meaning, and it can be thought of
incorporating short-distance information on the mass from scales larger than
$\mu$. On the other hand, the so-called pole mass $m_{Q}^{\rm pole}$ is
defined as the single particle pole in correlation functions involving the
massive quark $Q$ as an external on-shell particle, and it determines the
kinematic mass of the quark $Q$ in the context of perturbation theory. It is
therefore unavoidable that the pole mass scheme appears in one way or another
in higher order QCD calculations involving external massive quarks. For
perturbative predictions involving the production of top quarks at hadron
colliders, the pole mass scheme is therefore the main top quark mass scheme
used in the literature, and switching scheme is cumbersome since these
computations are predominantly numerical where the pole scheme provides the
most efficient approach for the computations. In Refs. Tarrach:1980up ;
Gray:1990yh ; Chetyrkin:1999ys ; Chetyrkin:1999qi ; Melnikov:2000qh ;
Marquard:2007uj ; Marquard:2015qpa ; Marquard:2016dcn  the relation between
the $\overline{\mathrm{MS}}$ and the pole mass has been computed up to ${\cal
O}(\alpha_{s}^{4})$ in the approximation that all quarks lighter than $Q$ are
massless. Assuming the values
$\overline{m}_{t}\equiv\overline{m}_{t}(\overline{m}_{t})=163$ GeV,
$\overline{m}_{b}\equiv\overline{m}_{b}(\overline{m}_{b})=4.2$ GeV and
$\overline{m}_{c}\equiv\overline{m}_{c}(\overline{m}_{c})=1.3$ GeV we
obtain111 We assume $\alpha^{(5)}(M_{Z})=0.1180$ for $M_{Z}=91.187$ GeV for
the $\overline{\mathrm{MS}}$ QCD coupling and account for 5-loop evolution
Baikov:2016tgj  and flavor matching at the scales $\overline{m}_{c,b,t}$
Kniehl:2006bg , which gives $\alpha_{s}^{(6)}(\overline{m}_{t})=0.10847$,
$\alpha_{s}^{(5)}(\overline{m}_{b})=0.22430$,
$\alpha_{s}^{(4)}(\overline{m}_{c})=0.38208$.

$\displaystyle m_{t}^{\rm pole}$
$\displaystyle=163\,+\,7.5040\,+\,1.6005\,+\,0.4941\,+\,(0.1944\pm
0.0004)\,\mbox{GeV}\,,$ (1) $\displaystyle m_{b}^{\rm pole}$
$\displaystyle=4.2\,+\,0.3998\,+\,0.1986\,+\,0.1443\,+\,(0.1349\pm
0.0002)\,\mbox{GeV}\,,$ (2) $\displaystyle m_{c}^{\rm pole}$
$\displaystyle=1.3\,+\,0.2108\,+\,0.1984\,+\,0.2725\,+\,(0.4843\pm
0.0005)\,\mbox{GeV}\,,$ (3)

where the terms show the series in powers of the strong coupling
$\alpha_{s}(\overline{m}_{Q})$ in the scheme that includes $Q$ as a dynamical
flavor. The fourth order coefficient displays the numerical uncertainties from
Marquard:2016dcn , which are, however, much smaller than other types of
uncertainties considered in this paper.

The pole mass renormalization scheme is infrared-safe and gauge-invariant
Tarrach:1980up ; Kronfeld:1998di , but suffers from large corrections in the
QCD perturbation series. This is because the pole mass scheme involves
subtractions of on-shell quark self energy corrections containing virtual
gluon and massless quark fluctuations which are linearly sensitive to small
momenta. The on-shell approximation of the self energy diagrams entails that
this sensitivity increases strongly with the order. The effect this has for
the form of the corrections can be seen in Eqs. (1)–(3), which in the
asymptotic large order limit have the form

$\displaystyle m_{Q}^{\rm pole}-\overline{m}_{Q}(\overline{m}_{Q})$
$\displaystyle\sim\mu\,\sum_{n=0}^{\infty}\frac{16}{3}(2\beta_{0}^{(n_{\ell})})^{n}\,n!\,\bigg{(}\frac{\alpha_{s}^{(n_{\ell})}(\mu)}{4\pi}\bigg{)}^{n+1}\,,$
(4)

in the $\beta_{0}$/LL approximation, which means that the terms in the QCD
$\beta$-function,

$\frac{\mathrm{d}\alpha_{s}^{(n_{\ell})}(\mu)}{\mathrm{d}\log\mu}=\beta^{(n_{\ell})}(\alpha_{s}(\mu))\,=\,-\,2\,\alpha_{s}^{(n_{\ell})}(\mu)\sum_{n=0}^{\infty}\beta_{n}^{(n_{\ell})}\bigg{(}\frac{\alpha_{s}^{(n_{\ell})}(\mu)}{4\pi}\bigg{)}^{n+1}\,,$
(5)

beyond the leading logarithmic level (i.e. $\beta_{n>0}$) are neglected. Here
$n_{\ell}$ is the number of massless quark flavors.

The factorially diverging pattern of the perturbation series and the linear
dependence on the renormalization scale $\mu$ of the strong coupling displayed
in Eq. (4) are called the ${\cal O}(\Lambda_{\rm QCD})$ renormalon of the pole
mass Bigi:1994em ; Beneke:1994sw . The form of the series on the RHS of Eq.
(4) implies that at asymptotic large orders, and up to terms suppressed by
inverse powers of $n$, the series becomes independent of its intrinsic
physical scale $m_{Q}$. This and the $n$-factorial growth is an artifact of
the pole mass scheme itself and not related to any physical effect.
Technically this issue entails that for computing differences of series
containing ${\cal O}(\Lambda_{\rm QCD})$ renormalon ambiguities using fixed-
order perturbation theory one must consistently expand in powers of the strong
coupling at the same renormalization scale such that the renormalon can
properly cancel.

The ${\cal O}(\Lambda_{\rm QCD})$ renormalon problem of the pole mass has
received substantial attention in the literature as it turned out to be not
just an issue of pedagogical interest, but one that is relevant
phenomenologically Beneke:1998ui . This is because for $\mu=\overline{m}_{Q}$
the known coefficients of the series in Eqs. (1)–(3) agree remarkably well
with the corresponding large order asymptotic behavior already beyond the
terms of ${\cal O}(\alpha_{s})$ (so that the terms of the series are known
quite precisely to all orders) and because even for orders where the QCD
corrections still decrease with order they can be very large numerically and
make phenomenological applications difficult. The pole mass scheme has
therefore been abandoned in high precision top, bottom and charm quark mass
analyses in favor of quark mass schemes such as $\overline{\mathrm{MS}}$ or
low-scale short distance masses such as the kinetic mass Czarnecki:1997sz ,
the potential-subtracted (PS) mass Beneke:1998rk , the 1S mass Hoang:1998ng ;
Hoang:1998hm ; Hoang:1999ye , the renormalon-subtracted (RS) mass
Pineda:2001zq , the jet mass Jain:2008gb ; Fleming:2007qr  or the MSR mass
Hoang:2008yj ; Hoang:2017suc . These mass schemes do not have an ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon and are called short-distance masses.
It is commonly agreed from many studies that it is possible to determine
short-distance masses with theoretical uncertainties of a few $10$ MeV
Hoang:2000yr ; Olive:2016xmw , and we therefore neglect any principle
ambiguity in their values in this paper.

Using the theory of asymptotic series one can show that the best possible
approximation to the LHS of Eq. (4) is to truncate the series on the RHS at
the minimal term at order $n_{\rm min}$ which is approximately $n_{\rm
min}\approx 2\pi/(\beta_{0}^{(n_{\ell})}\alpha_{s}^{(n_{\ell})}(\mu))$. The
size of the correction of the minimal term is approximately $\Delta(n_{\rm
min})\approx(4\pi\alpha_{s}^{(n_{\ell})}(\mu)/\beta_{0}^{(n_{\ell})})^{1/2}\Lambda_{\rm
QCD}^{(n_{\ell})}$, and there is a region in the orders $n$ around $n_{\rm
min}$ of width $\Delta
n\approx(2\pi^{2}/(\beta_{0}^{(n_{\ell})}\alpha_{s}^{(n_{\ell})}(\mu)))^{1/2}$
in which all series terms have a size close to the minimal term. At orders
above $n_{\rm min}+\Delta n/2$ the series diverges quickly and the series
terms from these orders are useless even if they are known through an
elaborate loop calculation. The uncertainty with which the pole mass can be
determined in principle given the full information about the perturbative
series is called the pole mass ambiguity. It is universal, independent on the
choice of the renormalization scale $\mu$ and exists in equivalent size in any
context without the possibility to be circumvented. However, the
$\mu$-dependence of $n_{\rm min}$, $\Delta(n_{\rm min})$ and $\Delta n$
indicates that the way how the renormalon problem appears in practical
applications based on perturbative QCD can differ substantially depending on
the physical scale of the quantity under consideration and the corresponding
choice of the renormalization scale $\mu$. Using the method of Borel
resummation the pole mass ambiguity can be estimated to be of order
$\Lambda_{\rm QCD}^{(n_{\ell})}$, where the superscript $(n_{\ell})$ stands
for the dependence of the hadronization scale on the number of massless quark
flavors. The norm of the ambiguity, which we call $N_{1/2}^{(n_{\ell})}$ in
this paper, and the resulting pattern of the large order asymptotic behavior
of the series can be determined very precisely and have been studied in many
analyses (see e.g. the recent work of Refs. Ayala:2014yxa ; Beneke:2016cbu ;
Komijani:2017vep ; Hoang:2017suc ). However, when quoting a concrete numerical
size of the ambiguity, criteria common for converging series cannot be
applied, and it is instrumental to consider more global aspects of the series
and the quantity it describes. An essential aspect of the low-energy quantum
corrections in heavy quark masses is heavy quark symmetry (HQS) Isgur:1989vq
on which we put particular focus in this work.

An issue that has received less attention in the literature so far is how the
masses of the lighter massive quarks affect the large order asymptotic
behavior of the pole-$\overline{\mathrm{MS}}$ mass relation, where we refer to
the effects of quarks with masses that are larger than $\Lambda_{\rm QCD}$.
These corrections come from insertions of virtual quark loops and are known up
to ${\cal O}(\alpha_{s}^{3})$ Gray:1990yh ; Bekavac:2007tk  from explicit loop
calculations. It is known that the masses of lighter massive quarks provide an
infrared cutoff and effectively reduce the number $n_{\ell}$ of massless
flavors governing the large order asymptotic behavior Ball:1995ni . Due to the
$n_{\ell}$-dependence of the QCD $\beta$-function the finite bottom and charm
quark masses lead to an increased infrared sensitivity of the top quark pole
mass and a stronger divergence pattern of the series, as can be seen from Eq.
(4). The ambiguity therefore inflates following the $n_{\ell}$-dependent
increase of $\Lambda_{\rm QCD}$. In Refs. Hoang:1999us ; Hoang:2000fm  it was
pointed out that the ${\cal O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$
virtual quark mass corrections are already dominated by the infrared behavior
related to the ${\cal O}(\Lambda_{\rm QCD})$ renormalon. In Ref. Ayala:2014yxa
it was further observed that the ${\cal O}(\alpha_{s}^{3})$ charm mass
corrections in the bottom pole-$\overline{\mathrm{MS}}$ mass relation can be
rendered small when the series is expressed in terms of
$\alpha_{s}^{(n_{\ell}=3)}$ rather than $\alpha_{s}^{(n_{\ell}=4)}$, i.e. the
charm quark effectively decouples. A systematic and precise understanding of
the intrinsic structure of the lighter massive quark effects from the point of
view of disentangling the different momentum modes and their interplay has,
however, not been provided so far in the literature. The task is complicated
since apart from being a problem in connection with the behavior of
perturbation theory at large orders, it also represents a multi-scale problem
with scales given by the quark masses as well as $\Lambda_{\rm QCD}$ and
where, for the top quark, logarithms of mass ratios can be large.

It is the main purpose of this paper to present a formalism that can do
exactly that. It is based on the concept of the renormalization group (RG) and
allows to successively integrate out momentum modes from the
pole-$\overline{\mathrm{MS}}$ mass relation of a heavy quark $Q$ in order to
disentangle the contributions coming from the lighter massive quarks and to
systematically sum logarithms of the mass ratios. The approach allows to
quantify and formulate precisely the effects the masses of the lighter massive
quarks have on the pole-$\overline{\mathrm{MS}}$ mass relation and therefore
on the pole mass itself and may find interesting applications in other
contexts. As the essential new feature the RG formalism entails linear scaling
with the renormalization scale. The common logarithmic scaling, as known for
the strong coupling, cannot capture the linear momentum dependence of QCD
corrections to the heavy quark mass for scales below $m_{Q}$. The formalism is
in particular useful since it fully accounts for all aspects of HQS. It can be
used to concretely formulate and study in a transparent way two important
properties of the heavy quark pole masses following from HQS: (1) The pole
mass ambiguity is independent of the mass of the heavy quark and (2) the
ambiguities of all heavy quarks are equal up to power corrections of order
$\Lambda_{\rm QCD}^{2}/m_{Q}$.

The essential technical tool to set up the formalism is the MSR mass
$m_{Q}^{\rm MSR}(R)$ Hoang:2008yj ; Hoang:2017suc . Like the perturbative
series for the pole-$\overline{\mathrm{MS}}$ mass relation, the pole-MSR mass
relation is calculated from on-shell heavy quark self energy diagrams, but has
also linear dependence on $R$. It is the basis of the RG formalism we propose,
allows to precisely capture the QCD corrections from the different quark mass
scales and, in particular, to encode and study issue (1) coming from HQS. The
renormalization group evolution in the scale $R$ is described by R-evolution
Hoang:2008yj ; Hoang:2017suc , which is free of the ${\cal O}(\Lambda_{\rm
QCD})$ renormalon, and allows to sum large logarithms of ratios of the quark
masses in the evolution between the quark mass scales. Using the concepts of
the MSR mass and the R-evolution it is then possible to relate the
pole-$\overline{\mathrm{MS}}$ masses of the top, bottom and charm quarks to
each other. This allows to systematically encode and study issue (2) coming
from HQS, and to interpret the small effects of HQS breaking as matching
corrections in a renormalization group flow that connects the QCD correction
of the top, bottom and charm quarks. The resulting formula can be used to
specify the heavy quark pole mass ambiguity in the context of lighter massive
quarks and to derive a generalized expression for the large order asymptotic
behavior accounting accurately for the light massive flavor dependence.
Concerning the accuracy of our description of the virtual quark loop mass
effects in the large order asymptotic behavior we reach a precision of a few
MeV, which applies equally for top, bottom and charm quarks.

The second main purpose of this paper is to use the RG formalism to specify
concretely the ambiguity of the top quark pole mass and also the pole mass of
the bottom and charm quarks assuming that their $\overline{\mathrm{MS}}$
masses are given. We in particular address the question how the outcome
depends on different scenarios for treating the bottom and charm quarks as
massive or massless, and we explicitly take into account the consistency
requirements of HQS. The aim is to provide a concrete numerical specification
of the ambiguity of the top quark pole mass beyond the qualitative statement
that the ambiguity is “of order $\Lambda_{\rm QCD}^{(n_{\ell})}$” and to make
a concrete statement up to which principle precision the top quark pole mass
may still be used as a meaningful phenomenological parameter. We stress that
in this context we adopt the view that the pole masses have well-defined and
unique meaning, so that the pole mass ambiguity acquires the meaning of an
intrinsic numerical uncertainty. This differs from the view sometimes used in
high-precision analyses, where the pole mass is employed as an intrinsic
order-dependent parameter to effectively parameterize the use of a short-
distance mass scheme.

Apart from specifying the ambiguity of the pole masses we are also interested
in studying the dependence of their value on the different scenarios for
treating the bottom and charm quarks as massive or massless. The issue is of
particular interest for the top quark pole mass which is still widely used for
theoretical predictions and phenomenological studies in top quark physics. The
top quark pole mass is, due to its linear sensitivity to small momenta, also
linearly sensitive to the masses of the lighter massive quarks. Since many
short-distance observables used for top quark pole mass determinations are at
most quadratically sensitive to small momenta, the dominant effects of the
bottom and charm masses may well come from the top quark pole definition
itself. A large dependence of the top quark pole mass value on whether the
bottom and charm quarks are treated as massive or massless would therefore
affect the ambiguity estimate if one considers the top quark pole mass as a
globally defined mass scheme (valid for any scenario for the bottom and charm
quark masses). We can address this question precisely because the RG-formalism
we use allows for very accurate numerical calculations of the lighter quark
mass effects. Within the size of the ambiguity, we do not find any such
dependence. The outcome of our analysis is that the top quark pole mass
ambiguity, and the ambiguity of the bottom and charm quark pole masses, is
around $250$ MeV.

Prior to this work the best estimate and the ambiguity of the top quark pole
mass were studied in Ref. Beneke:2016cbu . They analyzed the top quark
pole-$\overline{\mathrm{MS}}$ mass series of Eq. (1) for
$\mu=\overline{m}_{t}$ and massless bottom and charm quarks and in an extended
analysis also for massive bottom and charm quarks. They argued that the
ambiguity of the top, bottom and charm quark pole masses amounts to $110$ MeV.
We believe that their ambiguity estimate of $110$ MeV is too optimistic, and
we explain this in detail from the requirements of HQS. They also quantified
the bottom and charm mass effects coming from beyond the known corrections at
${\cal O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$ by using a heuristic
prescription based on an order-dependent reduction of the flavor number. This
does not represent a systematic calculation, but we find it to be an adequate
approximation for the task of estimating the top quark pole mass renormalon
ambiguity.

The paper is organized as follows: In Sec. 2 we review the explicitly
calculated corrections up to ${\cal O}(\alpha_{s}^{4})$ for the
pole-$\overline{\mathrm{MS}}$ and the pole-MSR mass relations for the case
that all quarks lighter than quark $Q$ are massless and we explain our
notation for parameterizing the virtual quark mass corrections due to the
light massive quarks. This notation is essential for our setup of the flavor
number dependent RG evolution of the MSR mass, which we also review to the
extend needed for our studies in the subsequent sections. We also review known
basic issues about the large order asymptotic behavior and the renormalon
ambiguity of the pole-$\overline{\mathrm{MS}}$ and the pole-MSR mass
relations, including their dependence on the number of massless quarks. In
Sec. 3 we explain details about the matching procedures that allow to
integrate out the virtual corrections coming from the heavy quark $Q$ and the
lighter massive quarks, and to relate the pole-MSR mass relation of quark $Q$
to the pole-$\overline{\mathrm{MS}}$ mass relation of the next lighter massive
quark, which is based on heavy quark symmetry. These considerations and the
numerical analysis of the latter matching corrections allow us to derive a
prediction for the yet uncalculated ${\cal O}(\alpha_{s}^{4})$ virtual quark
mass corrections and to discuss the large order asymptotic form of the virtual
quark mass corrections. As an application of the RG formalism devised in our
work we compute the difference of the pole masses of the top, bottom and charm
quarks. Since their differences are short-distance quantities we can compute
them with a precision of around $20$ MeV. We also analyze the validity of the
effective flavor decoupling at large orders in the context of the top quark
pole mass. In Sec. 4 we finally discuss in detail the best possible estimate
of the top quark pole mass and in particular its ambiguity in the context of
three different scenarios for the bottom and charm quark masses. We discuss
these three scenarios separately because the pole mass concept, strictly
speaking, depends on the setup for the lighter quark masses, and we also
discuss our results in the context of adopting the view that the top quark
pole mass is a general concept. Finally, in Sec. 5 we conclude. In App. A we
provide explicit results for the virtual quark mass corrections at ${\cal
O}(\alpha_{s}^{3})$ in our notation, using the results from Ref.
Bekavac:2007tk , and we complete them concerning the corrections coming from
the insertion of two quark loops involving quarks with two arbitrary masses.

##  2 Preliminaries and Notation

###  2.1 $\overline{\rm MS}$ Mass

The perturbative series of the difference between the $\overline{\mathrm{MS}}$
mass $\overline{m}_{Q}(\mu)$ at the scale
$\mu=\overline{m}_{Q}(\overline{m}_{Q})$ and the pole mass
$m^{\mathrm{pole}}_{Q}$ of a heavy quark $Q$ is the basic relation from which
we start our analysis of the renormalon ambiguity of the pole mass. To be more
specific we consider

$\displaystyle\overline{m}_{Q}\equiv\overline{m}_{Q}^{(n_{Q}+1)}(\overline{m}_{Q}^{(n_{Q}+1)})\,,$
(6)

which is the $\overline{\mathrm{MS}}$ mass defined for $(n_{Q}+1)$ active
dynamical flavors, where

$\displaystyle n_{Q}\,\equiv\,\mbox{number of flavors lighter than quark
$Q$}\,.$ (7)

In this work we use these two definitions for all massive quarks, and
depending on the context we also use the lower case letter $q$ for massive
quarks. We also define

$\displaystyle n_{\ell}\,\equiv\,\mbox{number of flavors lighter than
$\Lambda_{\rm QCD}$}\,,$ (8)

which we strictly treat in the massless approximation.

Assuming that $q_{1},\dots,q_{n}$ are the massive quarks lighter than $Q$ in
the order of decreasing mass (i.e.
$m_{Q}>m_{q_{1}}>\ldots>m_{q_{n}}>\Lambda_{\mathrm{QCD}}$ with $n<n_{Q}$ and
$n_{\ell}=n_{Q}-n$), the pole-$\overline{\mathrm{MS}}$ mass relation for the
heavy quark $Q$ can be written in the form

$\displaystyle m^{\mathrm{pole}}_{Q}$
$\displaystyle=\overline{m}_{Q}+\overline{m}_{Q}\,\sum_{n=1}^{\infty}\,a_{n}(n_{Q}+1,0)\,\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{n}$
(9)
$\displaystyle+\overline{m}_{Q}\left[\overline{\delta}_{Q}^{(Q,q_{1},\dots,q_{n})}(1,r_{q_{1}Q},\dots,r_{q_{n}Q})+\overline{\delta}_{Q}^{(q_{1},\dots,q_{n})}(r_{q_{1}Q},\dots,r_{q_{n}Q})+\dots+\overline{\delta}_{Q}^{(q_{n})}(r_{q_{n}Q})\right]\,,$

with

$\displaystyle a_{1}(n_{Q},n_{h})$ $\displaystyle={\textstyle\frac{16}{3}}\,,$
(10) $\displaystyle a_{2}(n_{Q},n_{h})$
$\displaystyle=213.437+1.65707\,n_{h}-16.6619\,n_{Q}\,,$ $\displaystyle
a_{3}(n_{Q},n_{h})$
$\displaystyle=12075.+118.986\,n_{h}+4.10115\,n_{h}^{2}-1707.35\,n_{Q}+1.42358\,n_{h}\,n_{Q}+41.7722\,n_{Q}^{2}\,,$
$\displaystyle a_{4}(n_{Q},n_{h})$ $\displaystyle=(911588.\pm
417.)+(1781.61\pm 30.72)\,n_{h}-(60.1637\pm 0.6912)\,n_{h}^{2}$
$\displaystyle\quad-(231.201\pm 0.102)\,n_{h}\,n_{Q}-(190683.\pm
10.)\,n_{Q}+9.25995\,n_{h}^{2}\,n_{Q}$
$\displaystyle\quad+6.35819\,n_{h}^{3}+4.40363\,n_{h}\,n_{Q}^{2}+11105.\,n_{Q}^{2}-173.604\,n_{Q}^{3}\,,$

where $\alpha_{s}^{(n_{Q}+1)}$ is the strong coupling that evolves with
$(n_{Q}+1)$ active dynamical flavors, see Eq. (5).

The coefficients $a_{n}(n_{Q},n_{h})$ encode the QCD corrections to
$m^{\mathrm{pole}}_{Q}-\overline{m}_{Q}$ for the case that the $n_{Q}$ quarks
lighter than $Q$ are assumed to be massless, and $n_{h}=1$ is just an
identifier for the corrections coming from virtual loops of the quark $Q$. The
coefficients $a_{1,2,3}$ are known analytically from Refs. Tarrach:1980up ;
Gray:1990yh ; Chetyrkin:1999ys ; Chetyrkin:1999qi ; Melnikov:2000qh ;
Marquard:2007uj , and $a_{4}$ was determined numerically in Refs.
Marquard:2015qpa ; Marquard:2016dcn , where the quoted numerical uncertainties
have been taken from Ref. Marquard:2016dcn . In Ref. Kataev:2015gvt  an
approach was suggested to further reduce the uncertainties of the
$n_{Q}$-dependent terms. The numerical uncertainties of the coefficient
$a_{4}$ are, however, tiny and irrelevant for the analysis carried out in this
work. We quote them just for completeness throughout this work.

The terms
$\overline{\delta}_{Q}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$
contain the mass corrections coming from the quark $Q$ on-shell self-energy
Feynman diagrams with insertions of virtual massive quark loops. We remind the
reader that the quarks with mass below the hadronization scale are taken as
massless and do not contribute. The superscript $(q,q^{\prime},\dots)$
indicates that each diagram contains at least one insertion of the massive
quark $q$ and in addition all possible insertions of the (lighter) massive
quarks $q^{\prime},\dots$ as well as of massless quark and gluonic loops. From
each diagram the corresponding diagram with all the quark loops in the
massless limit is subtracted in the scheme compatible with the flavor number
scheme for the strong coupling $\alpha_{s}$. The fraction

$\displaystyle
r_{qq^{\prime}}\,\equiv\,\frac{\overline{m}_{q}}{\overline{m}_{q^{\prime}}}\,,$
(11)

stands for the ratio of $\overline{\mathrm{MS}}$ masses for massive quarks $q$
and $q^{\prime}$ as defined in Eq. (6). In the pole-$\overline{\mathrm{MS}}$
mass relation for the heavy quark $Q$ only mass ratios with respect to the
heavy quark mass $\overline{m}_{Q}$ arise. By construction, the sum of all
virtual quark mass corrections contained in the functions
$\overline{\delta}_{Q}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$
are RG-invariant and do not contain effects from quarks heavier than the
external quark $Q$. The effects on the mass of the quark $Q$ related to quarks
heavier than $Q$ are accounted for in the renormalization group evolution of
the $\overline{\mathrm{MS}}$ mass $\overline{m}_{Q}(\mu)$ for scales
$\mu>m_{Q}$ and are not considered here. The virtual quark mass corrections
satisfy the following two relations to all orders of perturbation theory

$\displaystyle\overline{\delta}_{Q}^{(q_{1},q_{2},\dots,q_{n})}(0,0,\dots,0)$
$\displaystyle=0\,,$ (12)
$\displaystyle\overline{\delta}_{Q}^{(Q,q_{1},\dots,q_{n})}(1,0,\dots,0)$
$\displaystyle=\sum_{n=2}^{\infty}\,\left[\,a_{n}(n_{Q},1)-a_{n}(n_{Q}+1,0)\,\right]\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{n}\,.$
(13)

Due to Eq. (13) the pole-$\overline{\mathrm{MS}}$ mass relation of Eq. (9) can
be rewritten in the alternative form

$\displaystyle m^{\mathrm{pole}}_{Q}$
$\displaystyle=\overline{m}_{Q}+\overline{m}_{Q}\,\sum_{n=1}^{\infty}\,a_{n}(n_{Q},1)\,\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{n}$
$\displaystyle\qquad+\overline{m}_{Q}\left[\overline{\delta}_{Q}^{(Q,q_{1},\dots,q_{n})}(1,r_{q_{1}Q},\dots,r_{q_{n}Q})-\overline{\delta}_{Q}^{(Q,q_{1},\dots,q_{n})}(1,0,\dots,0)\right.$
(14) $\displaystyle\hskip
71.13188pt\left.+\;\overline{\delta}_{Q}^{(q_{1},\dots,q_{n})}(r_{q_{1}Q},\dots,r_{q_{n}Q})+\dots+\overline{\delta}_{Q}^{(q_{n})}(r_{q_{n}Q})\right]\,.$

In the limit that all quarks lighter than $Q$ are massless, all
$\overline{\delta}$ terms cancel or vanish in Eq. (2.1), and only the first
line involving the $a_{n}$ coefficients remains.

The perturbative expansion of the virtual quark mass corrections in the
pole-$\overline{\mathrm{MS}}$ mass relation of Eq. (9) and (2.1) can be
written in the form

$\displaystyle\overline{\delta}_{Q}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$
$\displaystyle=\delta_{2}(r_{qQ})\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{2}\,$
$\displaystyle\qquad+\,\sum_{n=3}^{\infty}\,\delta_{Q,n}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{n}\,,$
(15)

which together with Eq. (13) implies that

$\displaystyle\delta_{2}(1)=a_{2}(n_{Q},1)-a_{2}(n_{Q}+1,0)=18.3189\,,$
$\displaystyle\delta_{Q,n}^{(Q,q,q^{\prime},\dots)}(1,0,0,\dots)=a_{n}(n_{Q},1)-a_{n}(n_{Q}+1,0)\,.$
(16)

The ${\cal O}(\alpha_{s}^{2})$ correction comes from the on-shell self energy
diagram of quark $Q$ with the insertion of a loop of the massive quark $q$.
The result was determined analytically in Ref. Gray:1990yh . At ${\cal
O}(\alpha_{s}^{3})$, in Ref. Bekavac:2007tk , the virtual quark mass
corrections were determined in a semi-analytic form for arbitrary quark masses
for insertions of loops of the quark $Q$ and one other massive quark $q$. The
expressions for these virtual quark mass corrections are for convenience
collected in App. A after adapting the results of Ref. Bekavac:2007tk  to our
notation. We also provide the ${\cal O}(\alpha_{s}^{3})$ result for insertions
of loops with two arbitrary massive quarks, which were not given in Ref.
Bekavac:2007tk . The ${\cal O}(\alpha_{s}^{4})$ virtual quark mass corrections
have not been determined through an explicit loop calculation.

One can interpret the $\overline{\mathrm{MS}}$ mass
$\overline{m}_{Q}=\overline{m}_{Q}^{(n_{Q}+1)}(\overline{m}_{Q}^{(n_{Q}+1)})$
as the pole mass minus all self-energy corrections coming from scales at and
below $\overline{m}_{Q}$. So $\overline{m}_{Q}$ only contains mass
contributions from momentum fluctuations from above $\overline{m}_{Q}$, which
illustrates that it is a short-distance mass that is strictly insensitive to
issues related to low momentum fluctuations at the hadronization scale
$\Lambda_{\rm QCD}$. See Fig. 1 for illustration.

Figure 1: Graphical illustration of the physical contributions contained in
the pole, MSR and $\overline{\mathrm{MS}}$ mass schemes coming from the
different momentum scales for the case of the top quark. The quark loops stand
for the contributions of the virtual massive quark loops contained in the
masses.

###  2.2 MSR Mass and R-Evolution

In order to integrate out high momentum contributions and formulate the
renormalization group flow of momentum contributions in the heavy quark masses
we use the MSR mass $m^{\rm MSR}_{Q}(R)$ introduced in Ref. Hoang:2017suc
222In Ref. Hoang:2017suc ; Hoang:2008yj  the natural and the practical MSR
masses were introduced. In this paper we employ the natural MSR mass and call
it just the MSR mass for convenience., extending its definition to account for
the mass effects of the lighter massive quarks.

The MSR mass for the heavy quark $Q$ is derived from on-shell self-energy
diagrams just like the pole-$\overline{\mathrm{MS}}$ mass relation of Eq. (9),
but it does not include any diagrams involving virtual loops of the heavy
quark $Q$, i.e. the contributions from heavy quark $Q$ virtual loops are
integrated out. Like the $\overline{\mathrm{MS}}$ mass, the MSR mass is a
short-distance mass, and since the corrections from the heavy quark $Q$ are
short-distance effects, its relation to the pole mass fully contains the pole
mass ${\cal O}(\Lambda_{\rm QCD})$ renormalon (just as the
pole-$\overline{\mathrm{MS}}$ mass relation of Eqs. (9) and (2.1)).
Furthermore the MSR mass depends on the arbitrary scale $R\lesssim m_{Q}$ to
describe contributions in the mass from the momenta below the scale $m_{Q}$,
and therefore represents the natural extension of the concept of the
$\overline{\mathrm{MS}}$ mass for scales below $m_{Q}$.

Assuming that $q_{1},\dots,q_{n}$ are the massive quarks lighter than $Q$ in
the order of decreasing mass (i.e.
$m_{Q}>m_{q_{1}}>\ldots>m_{q_{n}}>\Lambda_{\mathrm{QCD}}$ with $n<n_{Q}$ and
$n_{\ell}=n_{Q}-n$), the MSR mass $m^{\rm MSR}_{Q}(R)$ is defined by the
relation

$\displaystyle m^{\mathrm{pole}}_{Q}$ $\displaystyle=m^{\rm
MSR}_{Q}(R)+R\,\sum_{n=1}^{\infty}\,a_{n}(n_{Q},0)\left(\frac{\alpha^{(n_{Q})}_{s}(R)}{4\pi}\right)^{n}$
$\displaystyle\qquad+\overline{m}_{Q}\,\left[\delta_{Q}^{(q_{1},\dots,q_{n})}(r_{q_{1}Q},\dots,r_{q_{n}Q})+\dots+\delta_{Q}^{(q_{n})}(r_{q_{n}Q})\right]\,,$
(17)

where the coefficients $a_{n}$ are given in Eqs. (10) and the perturbative
expansion is in powers of the strong coupling in the $n_{Q}$-flavor scheme
since the quark $Q$ is integrated out. The $R$-dependence of the strong
coupling entails that the scale $R$ has to be chosen sufficiently larger than
$\Lambda_{\rm QCD}$ to stay away from the Landau pole. The definition
generalizes the one already provided in Ref. Hoang:2017suc , which only
considered $n_{Q}$ massless quarks.

The notation used for the virtual quark mass corrections involving the
functions $\delta_{Q}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$ is
the same as the one for the $\overline{\mathrm{MS}}$ mass described above, and
their sum is by construction RG-invariant. Their perturbative expansion has
the form

$\displaystyle\delta_{Q}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$
$\displaystyle=\delta_{2}(r_{qQ})\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})}{4\pi}\right)^{2}$
$\displaystyle\qquad+\sum_{n=3}^{\infty}\,\delta_{Q,n}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})}{4\pi}\right)^{n}\,,$
(18)

where the coefficient functions
$\delta_{Q,n}^{(q,q^{\prime},\dots)}(r_{qQ},r_{q^{\prime}Q},\dots)$ are
identical to the ones appearing in Eq. (2.1).

In our definition of the MSR mass, the virtual quark mass corrections are
independent of $R$. This entails that the renormalization group evolution of
the MSR mass in $R$ does not depend on the masses of the $n_{Q}$ lighter
quarks. So $m^{\rm MSR}_{Q}(R)$ is defined in close analogy to the
$\mu$-dependent $\overline{\mathrm{MS}}$ strong coupling and the
$\overline{\mathrm{MS}}$ masses, whose renormalization group evolution only
depends on the number of active dynamical quarks (which is typically the
number of quarks lighter than $\mu$) and where mass effects are implemented by
threshold corrections when $\mu$ crosses a flavor threshold. Moreover, because
the ${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity of the series
proportional to $R$ is independent of $R$ and because the corrections from the
virtual loops of the heavy quark $Q$ are short-distance effects, the series of
the pole-MSR mass relation in Eq. (2.2) suffers from the same ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity as the
pole-$\overline{\mathrm{MS}}$ mass relation of Eqs. (9) and (2.1). It can
therefore also be used to study and quantify the ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon of the pole mass $m_{Q}^{\rm pole}$.

As explained below Eq. (4), in order to expand the difference of MSR masses at
two scales $R$ and $R^{\prime}$ in the fixed-order expansion in powers of
$\alpha_{s}^{(n_{Q})}$ it is necessary to do that at a common renormalization
scale $\mu$ so that the renormalon in the $R$-dependent corrections of Eq.
(2.2) cancels order by order. This unavoidably leads to large logarithms if
the scale separation is large, similarly to when considering the fixed-order
expansion of the difference of the strong coupling at widely separated scales.
To sum the logarithms in the difference of MSR masses we use its RG-evolution
equation in $R$, which reads

$R\frac{\mathrm{d}}{\mathrm{d}R}m_{Q}^{\mathrm{MSR}}(R)=-\,R\,\gamma^{R,(n_{Q})}(\alpha^{(n_{Q})}_{s}(R))=-\,R\sum_{n=0}^{\infty}\gamma_{n}^{R,(n_{Q})}\bigg{(}\frac{\alpha^{(n_{Q})}_{s}(R)}{4\pi}\bigg{)}^{\\!\\!n+1}\;,$
(19)

where the coefficients are known up to four loops and given by Hoang:2008yj ;
Hoang:2017suc

$\displaystyle\gamma_{0}^{R,(n_{Q})}$
$\displaystyle={\textstyle\frac{16}{3}}\,,$ (20)
$\displaystyle\gamma_{1}^{R,(n_{Q})}$
$\displaystyle=96.1039-9.55076\,n_{Q}\,,$
$\displaystyle\gamma_{2}^{R,(n_{Q})}$
$\displaystyle=1595.75-269.953\,n_{Q}-2.65945\,n_{Q}^{2}\,,$
$\displaystyle\gamma_{3}^{R,(n_{Q})}$ $\displaystyle=(12319.\pm
417.)-(9103.\pm 10.)\,n_{Q}+610.264\,n_{Q}^{2}-6.515\,n_{Q}^{3}\,.$

The difference of MSR masses at two scales $R^{\prime}$ and $R$ can then be
computed from solving the evolution equation

$\Delta m^{(n_{Q})}(R,R^{\prime})=m^{\rm MSR}_{Q}(R^{\prime})-m^{\rm
MSR}_{Q}(R)=\sum_{n=0}^{\infty}\,\gamma_{n}^{R,(n_{Q})}\int_{R^{\prime}}^{R}\mathrm{d}R\,\left(\frac{\alpha^{(n_{Q})}_{s}(R)}{4\pi}\right)^{n+1}\,,$
(21)

which accounts for the RG-evolution in the presence of $n_{Q}$ active
dynamical quark flavors.

The RG-equation of the MSR mass has a linear as well as logarithmic dependence
on $R$ and thus differs from the usual logarithmic RG-equations for
$\alpha_{s}$ and the $\overline{\mathrm{MS}}$ mass. Since its linear
dependence on $R$ allows to systematically probe linear sensitivity to small
momenta it can be used to systematically study the ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon behavior of perturbative series
Hoang:2008yj ; Hoang:2017suc . Since this is impossible for usual logarithmic
RG-evolution equations, Eq. (19) was called the R-evolution equation in Refs.
Hoang:2008yj ; Hoang:2017suc . Continuing on the thoughts made at the end of
Sec. 2.1 we note that one can interpret the MSR mass $m^{\rm MSR}_{Q}(R)$ as
the pole mass minus all self-energy contributions coming from scales below $R$
and all virtual quark mass corrections from quarks lighter than $Q$, see Fig.
1. This also illustrates that the MSR mass $m^{\rm MSR}_{Q}(R)$ is a short-
distance mass. The negative overall sign on the RHS of Eq. (19) expresses that
self-energy contributions are added to the MSR mass when $R$ is evolved to
smaller scales, and that $\Delta m^{(n_{Q})}(R,R^{\prime})$ for $R>R^{\prime}$
is positive and represents the self-energy contributions to the mass in the
presence of $n_{Q}$ active dynamical flavors coming from the scales between
$R^{\prime}$ and $R$. This is illustrated in Fig. 2.

Figure 2: Graphical illustration for pole-$\overline{\mathrm{MS}}$ mass
differences, the MSR-$\overline{\mathrm{MS}}$ mass matching corrections and
MSR mass differences for different $R$ scales. They constitute the major
contributions in the RG analysis of the heavy quark pole masses.

In the context of the analyses in this work the essential property is that the
${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity in the series on the
RHS of Eq. (2.2) is $R$-independent. This entails that the R-evolution
equation is free of the ${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon, and
solving the R-evolution equation in Eq. (21) allows to relate MSR masses at
different scales in a way that is renormalon free and, in addition,
systematically sums logarithms $\ln(R/R^{\prime})$ to all orders in a way free
of the ${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon. So the R-evolution
equation resolves the problem of the large logarithms that arise when
computing MSR mass differences in the fixed-order expansion. The integral of
Eq. (21) can be readily computed numerically, and an analytic solution has
been discussed in detail in Hoang:2017suc . The analytic solution also allows
to derive the large-order asymptotic form of the perturbative coefficients
$a_{n}$. To implement renormalization scale variation in Eq. (21) one expands
$\alpha_{s}^{(n_{Q})}(R)$ as a series in $\alpha_{s}^{(n_{Q})}(\lambda R)$,
and by varying $\lambda$ in some interval around unity. We note that in our
analysis we consider the top, bottom and charm mass scales, and using the
R-evolution equation is instrumental for our discussion of the top quark pole
mass.

In Tab. 1 we show numerical results for various MSR mass differences $\Delta
m^{(n_{Q})}$ relevant in our examinations below for $n_{Q}=3,4,5$. We display
the results obtained from using the R-evolution equation at ${\cal
O}(\alpha_{s}^{n})$ for $n=1,2,3,4$. The uncertainties are from $\lambda$
variations in the interval $[0.5,2]$ for the cases where scales above the
charm mass scale $1.3$ GeV are considered, and in the interval $[0.6,2.5]$ for
cases which involve the charm mass scale. We see an excellent convergence and
stability of the results and a significant reduction of scale variation with
the order, illustrating that the mass differences $\Delta
m^{(n_{Q})}(R,R^{\prime})$ are free of an ${\cal O}(\Lambda_{\rm QCD})$
renormalon ambiguity. For our analyses below we use the most precise ${\cal
O}(\alpha_{s}^{4})$ results shown in the respective lowest lines.

${\cal O}(\alpha_{s}^{n})$ |  $\Delta m^{(5)}(163,20)$ |  $\Delta m^{(5)}(163,4.2)$ |  $\Delta m^{(5)}(163,1.3)$
---|---|---|---
$n=1$ |  $7.358\pm 0.811$ |  $8.536\pm 1.008$ |  $8.864\pm 1.047$
$n=2$ |  $8.007\pm 0.168$ |  $9.336\pm 0.225$ |  $9.728\pm 0.311$
$n=3$ |  $8.031\pm 0.024$ |  $9.368\pm 0.035$ |  $9.764\pm 0.066$
$n=4$ |  $8.006\pm 0.009$ |  $9.331\pm 0.016$ |  $9.716\pm 0.023$
${\cal O}(\alpha_{s}^{n})$ |  $\Delta m^{(4)}(163,4.2)$ |  $\Delta m^{(4)}(20,4.2)$ |  $\Delta m^{(4)}(4.2,1.3)$
$n=1$ |  $8.181\pm 1.026$ |  $1.153\pm 0.211$ |  $0.337\pm 0.098$
$n=2$ |  $9.064\pm 0.270$ |  $1.326\pm 0.073$ |  $0.419\pm 0.063$
$n=3$ |  $9.139\pm 0.054$ |  $1.346\pm 0.018$ |  $0.434\pm 0.026$
$n=4$ |  $9.114\pm 0.014$ |  $1.337\pm 0.007$ |  $0.423\pm 0.017$
${\cal O}(\alpha_{s}^{n})$ |  $\Delta m^{(3)}(163,1.3)$ |  $\Delta m^{(3)}(20,1.3)$ |  $\Delta m^{(3)}(4.2,1.3)$
$n=1$ |  $8.009\pm 1.044$ |  $1.419\pm 0.296$ |  $0.328\pm 0.106$
$n=2$ |  $9.008\pm 0.404$ |  $1.691\pm 0.166$ |  $0.418\pm 0.078$
$n=3$ |  $9.130\pm 0.126$ |  $1.741\pm 0.067$ |  $0.440\pm 0.037$
$n=4$ |  $9.111\pm 0.032$ |  $1.729\pm 0.023$ |  $0.434\pm 0.020$
Table 1:  MSR mass differences $\Delta m^{(n_{Q})}(R,R^{\prime})$ computed
from R-evolution, for $n_{Q}=3,4,5$ active dynamical flavors for scale
differences involving top, bottom and charm masses and the scale $20$ GeV. The
central values are obtained for $\lambda=1$ and the uncertainties are
symmetrized $\lambda$ variations in the interval $[0.5,2]$. For entries
involving the scale $\overline{m}_{c}$ the interval $[0.6,2.5]$ is used for
$\lambda$ variations. The numbers for $\Delta m^{(n_{Q})}(R,R^{\prime})$ are
given in units of GeV.

###  2.3 Asymptotic High Order Behavior and Borel Transform for Massless
Lighter Quarks

In this section we review a number of known results relevant for the analyses
in the subsequent parts of the paper. The results are already known since
Refs. Bigi:1994em ; Beneke:1994sw ; Beneke:1998ui . We adapt them according to
our notation and present updated numerical results accounting for the recent
perturbative calculations of the pole-$\overline{\mathrm{MS}}$ mass relation
and the QCD $\beta$-function.

The Borel transform of an $\alpha_{s}$ power series

$f(\alpha_{s}(R))=R\,\sum_{n=0}^{\infty}\,a_{n+1}\left(\frac{\alpha_{s}(R)}{4\pi}\right)^{n+1}\,,$
(22)

is defined as

$B[f](u)=R\,\sum_{n=0}^{\infty}\,a_{n+1}\,\frac{u^{n}}{n!\,\beta_{0}^{n+1}}\,,$
(23)

where $\beta_{0}$ is the one-loop $\beta$-function coefficient in the flavor
number scheme of $\alpha_{s}$. For the approximation that all quarks lighter
than the heavy quark $Q$ are massless (i.e. $n_{\ell}=n_{Q}$) the Borel
transform of the series for the pole-MSR mass reads

$\displaystyle B$ $\displaystyle\left[m^{\mathrm{pole}}_{Q}-m^{\rm
MSR}_{Q}(R)\right](u)\,=$ $\displaystyle\qquad
N_{1/2}^{(n_{\ell})}\;R\,\frac{4\pi}{\beta_{0}^{(n_{\ell})}}\,\sum_{k=0}^{\infty}\,g_{k}^{(n_{\ell})}\frac{\Gamma(1+\hat{b}_{1}^{(n_{\ell})}-k)}{\Gamma(1+\hat{b}_{1}^{(n_{\ell})})}\,(1-2\,u)^{-1-\hat{b}_{1}^{(n_{\ell})}+k}+\dots\,,$
(24)

where the non-analytic (and singular) terms multiplied by the normalization
factor $N_{1/2}^{(n_{\ell})}$ single out the ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon behavior of the pole-MSR mass series
and the ellipses stand for contributions not affected by an ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon. Their form is unambiguously determined
by the coefficients $\beta_{n}^{(n_{\ell})}$ of the QCD $\beta$-function in
Eq. (5), and the sum over $k$ parametrizes the subleading effects due to the
higher order coefficients of the QCD $\beta$-function. The coefficients
$g^{(n_{\ell})}_{k}$ can be determined from the recursion formulae
Hoang:2017suc

$\displaystyle\hat{b}_{n+1}$
$\displaystyle=2\sum_{i\,=\,0}^{n}\,\frac{\hat{b}_{n-i}\,\beta_{i+1}}{(-2\beta_{0})^{i+2}}\,,$
$\displaystyle g_{n+1}$
$\displaystyle=\frac{1}{1+n}\sum_{i=0}^{n}\,(-1)^{i}\,\hat{b}_{i+2}\,g_{n-i}$
(25)

with $\hat{b}_{0}=g_{0}=1$, where we dropped the superscript $(n_{\ell})$ for
simplicity. Currently, coefficients $g^{(n_{\ell})}_{k}$ are known up to
$k=3$. The factor $N_{1/2}^{(n_{\ell})}$ precisely quantifies the overall
normalization of the ${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon behavior
and can be determined quite precisely from the coefficients
$a_{n}(n_{\ell},0)$ known from explicit computations. Accounting for the
coefficients up to ${\cal O}(\alpha_{s}^{4})$ the normalization was determined
with very small errors for the relevant flavor numbers $n_{\ell}=3,4,5$ in
Refs. Ayala:2014yxa ; Beneke:2016cbu ; Hoang:2017suc , all of which are in
agreement. We use the results from Ref. Hoang:2017suc :

$\displaystyle N_{1/2}^{(n_{\ell}=3)}=0.526\pm 0.012\,,$ $\displaystyle
N_{1/2}^{(n_{\ell}=4)}=0.492\pm 0.016\,,$ (26) $\displaystyle
N_{1/2}^{(n_{\ell}=5)}=0.446\pm 0.024\,.$

The uncertainties are not essential for the outcome of our analysis and quoted
for completeness. Their small size reflects that the large-order asymptotic
behavior of the series is known very precisely.

The inverse Borel transform

$\int_{0}^{\infty}\mathrm{d}u\,B[f](u)\;\mathrm{e}^{-\frac{4\pi
u}{\beta_{0}\alpha_{s}(R)}}\,,$ (27)

has the same $\alpha_{s}$ power series as the original series
$f(\alpha_{s}(R))$ and provides the exact result if it can be calculated
unambiguously from the Borel transform $B[f](u)$. However, for the case of Eq.
(2.3), due to the singularity at $u=1/2$ and the cut along the positive real
axis for $u>1/2$, the integral cannot be computed without further prescription
and an ambiguity remains. Using an $i\epsilon$ prescription
$(1-2u)^{\alpha}\to(1-2u-i\epsilon)^{\alpha}$ to shift the cut to the lower
complex half plane, the resulting imaginary part of the integral is

$\displaystyle\Delta m_{\mathrm{Borel}}^{(n_{\ell})}$
$\displaystyle\equiv\left|\mathrm{Im}\int_{0}^{\infty}\mathrm{d}u\,\exp\left(-\frac{4\pi
u}{\beta_{0}^{(n_{\ell})}\alpha_{s}^{(n_{\ell})}(R)}\right)\right.$
$\displaystyle\hskip
65.44142pt\left.\times\left[N_{1/2}^{(n_{\ell})}\,R\,\frac{4\pi}{\beta_{0}^{(n_{\ell})}}\sum_{k=0}^{\infty}g_{k}^{(n_{\ell})}\frac{\Gamma(1+\hat{b}_{1}^{(n_{\ell})}-k)}{\Gamma(1+\hat{b}_{1}^{(n_{\ell})})}\,(1-2u)^{-1-\hat{b}_{1}^{(n_{\ell})}+k}\right]\right|$
$\displaystyle=N_{1/2}^{(n_{\ell})}\,\frac{2\pi^{2}}{\beta_{0}^{(n_{\ell})}\Gamma(1+\hat{b}_{1}^{(n_{\ell})})}\Lambda_{\mathrm{QCD}}^{(n_{\ell})}\,,$
(28)

and represents a quantification of the ambiguity of the pole mass, where
$\Lambda_{\mathrm{QCD}}^{(n_{\ell})}$ is given by the expression
($t_{R}=-2\pi/\beta_{0}^{(n_{\ell})}\alpha_{s}^{(n_{\ell})}(R)$)

$\Lambda_{\mathrm{QCD}}^{(n_{\ell})}=R\,\exp\left(t_{R}+\hat{b}_{1}^{(n_{\ell})}\log(-t_{R})-\sum_{k=2}^{\infty}\frac{\hat{b}_{k}^{(n_{\ell})}}{(k-1)t_{R}^{k-1}}\right).$
(29)

In this work we use this expression as the definition of
$\Lambda_{\mathrm{QCD}}$ for $n_{\ell}$ massless flavors. The RHS is
$R$-independent, and truncating at $k=4$ provides the results

$\displaystyle\Lambda_{\rm QCD}^{(n_{\ell}=3)}$ $\displaystyle=253\text{
MeV}\,,$ $\displaystyle\Lambda_{\rm QCD}^{(n_{\ell}=4)}$
$\displaystyle=225\text{ MeV}\,,$ (30) $\displaystyle\Lambda_{\rm
QCD}^{(n_{\ell}=5)}$ $\displaystyle=166\text{ MeV}\,,$

with uncertainties below $0.5$ MeV. $\Lambda_{\mathrm{QCD}}^{(n_{\ell})}$
increases for smaller flavor numbers $n_{\ell}$ since the scale-dependence of
$\alpha_{s}$, and thus also the infrared sensitivity of QCD quantities,
increases with $n_{\ell}$. The expressions for $\Delta
m_{\mathrm{Borel}}^{(n_{\ell})}$ for the size of the imaginary part of the
inverse Borel transform in Eq. (2.3) provide a parametric estimate for the
ambiguity of the pole mass. Using Eqs. (2.3) and (30) they give $\Delta
m_{\mathrm{Borel}}^{(3,4,5)}=(329\pm 8,295\pm 10,213\pm 11)$ MeV which are
around a factor $1.3$ larger than the corresponding values for $\Lambda_{\rm
QCD}^{(n_{\ell})}$.

From the expression for the Borel transform given in Eq. (2.3) one can derive
the large order asymptotic form of the perturbative coefficients $a_{n}$ of
the pole-MSR mass series (which describe the case that all quarks lighter than
$Q$ are massless, i.e. $n_{Q}=n_{\ell}$):

$a_{n}^{\mathrm{asy}}(n_{\ell},n_{h})=a_{n}^{\mathrm{asy}}(n_{\ell},0)=4\pi
N_{1/2}^{(n_{\ell})}(2\beta_{0}^{(n_{\ell})})^{n-1}\sum_{k=0}^{\infty}g_{k}^{(n_{\ell})}\frac{\Gamma(n+\hat{b}_{1}^{(n_{\ell})}-k)}{\Gamma(1+\hat{b}_{1}^{(n_{\ell})})}\,,$
(31)

where the value of $n_{h}$ is insignificant because the virtual effects of
quark $Q$ do not affect the large order asymptotic behavior. The sum in $k$ is
convergent, and truncating at $k=3$ one can use the results for $n>4$ as an
approximation for the yet uncalculated series coefficients. The results up to
$n=12$ for $n_{\ell}=3,4,5$ using the values for the $N_{1/2}^{(n_{\ell})}$
from Eq. (2.3) are displayed in Tab. 2.

With the normalization factors $N_{1/2}^{(n_{\ell})}$, which are known to a
precision of a few percent and which also entails the same precision for
$\Delta m_{\mathrm{Borel}}^{(n_{\ell})}$ and the asymptotic coefficients
$a_{n}^{\rm asy}$, the series for the pole-MSR and also for the
pole-$\overline{\mathrm{MS}}$ mass relation are essentially known to all
orders for the case of $n_{\ell}=n_{Q}$. The task to determine the ambiguity
of the pole mass involves to specify how this precisely known pattern limits
the principle capability to determine the pole mass numerically, see the
discussion in Sec. 4.1. In other words, the ambiguity of the pole mass is
known to be proportional to $\Delta m_{\mathrm{Borel}}^{(n_{\ell})}$ or
$\Lambda_{\rm QCD}^{(n_{\ell})}$, but the factor of proportionality has to be
determined from an additional dedicated analysis.

$n$ |  $a^{\mathrm{asy}}_{n}(n_{\ell}=3,0)$ |  $a^{\mathrm{asy}}_{n}(n_{\ell}=4,0)$ |  $a^{\mathrm{asy}}_{n}(n_{\ell}=5,0)$
---|---|---|---
$5$ |  $(3.394\pm 0.077)\times 10^{7\phantom{0}}$ |  $(2.249\pm 0.075)\times 10^{7\phantom{0}}$ |  $(1.379\pm 0.074)\times 10^{7\phantom{0}}$
$6$ |  $(3.309\pm 0.075)\times 10^{9\phantom{0}}$ |  $(2.019\pm 0.067)\times 10^{9\phantom{0}}$ |  $(1.128\pm 0.060)\times 10^{9\phantom{0}}$
$7$ |  $(3.819\pm 0.087)\times 10^{11}$ |  $(2.147\pm 0.071)\times 10^{11}$ |  $(1.095\pm 0.059)\times 10^{11}$
$8$ |  $(5.093\pm 0.115)\times 10^{13}$ |  $(2.641\pm 0.088)\times 10^{13}$ |  $(1.231\pm 0.066)\times 10^{13}$
$9$ |  $(7.706\pm 0.175)\times 10^{15}$ |  $(3.687\pm 0.123)\times 10^{15}$ |  $(1.572\pm 0.084)\times 10^{15}$
$10$ |  $(1.305\pm 0.030)\times 10^{18}$ |  $(5.762\pm 0.192)\times 10^{17}$ |  $(2.250\pm 0.120)\times 10^{17}$
$11$ |  $(2.443\pm 0.055)\times 10^{20}$ |  $(9.964\pm 0.332)\times 10^{19}$ |  $(3.563\pm 0.191)\times 10^{19}$
$12$ |  $(5.014\pm 0.114)\times 10^{22}$ |  $(1.889\pm 0.063)\times 10^{22}$ |  $(6.190\pm 0.331)\times 10^{21}$
Table 2: Coefficients of the pole-MSR mass series for $a_{n>4}(n_{\ell},0)$
for $n_{\ell}=3,4,5$ estimated from the asymptotic formula of Eq. (31) and
with uncertainties from Eq. (2.3).

##  3 Integrating Out Hard Modes from the Heavy Quark Pole Mass

###  3.1 MSR-$\overline{\mathrm{MS}}$ Mass Matching

Using the MSR mass we can successively separate off, i.e. integrate out, hard
momentum contributions from the pole-$\overline{\mathrm{MS}}$ mass difference,
$m^{\mathrm{pole}}_{Q}-\overline{m}_{Q}$. We start with the matching relation
between the MSR and the $\overline{\mathrm{MS}}$ masses at the common scale
$\mu=R=\overline{m}_{Q}$, which can be obtained by eliminating the pole mass
from Eqs. (2.1) and (2.2). The matching relation accounts for the virtual top
quark loop contributions and can be written in the form

$m^{\rm MSR}_{Q}(\overline{m}_{Q})-\overline{m}_{Q}=\Delta
m_{Q}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})+\delta
m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})\,.$
(32)

The term $\Delta m_{Q}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})$
contains the virtual top quark loop contributions in the approximation that
all $n_{Q}$ quarks lighter than quark $Q$ are massless and has the form
Hoang:2017suc

$\displaystyle\Delta$ $\displaystyle m_{Q}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})=$ (33)
$\displaystyle\qquad\overline{m}_{Q}\left\\{1.65707\,\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{2}+[110.05+1.424\,n_{Q}]\,\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{3}\right.$
$\displaystyle\left.\hskip 45.52458pt+\left[352.\pm 31.-(111.59\pm
0.10)\,n_{Q}+4.40\,n_{Q}^{2}\right]\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{4}+\dots\right\\}\,,$

where we expressed the series in powers of the strong coupling in the
$(n_{Q}+1)$ flavor scheme. The series only contains the hard corrections
coming from the virtual heavy quark $Q$ and therefore does not have any ${\cal
O}(\Lambda_{\rm QCD})$ ambiguity, see Fig. 2 for illustration.

In Tab. 3 the numerical values for $\Delta m_{Q}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})$ are shown at ${\cal O}(\alpha_{s}^{2,3,4})$ for the
top, bottom, and charm quarks for
($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) = ($163,4.2,1.3$) GeV.
Also shown is the variation due to changes in the renormalization scale in the
range $0.5\,\overline{m}_{Q}\leq\mu\leq 2\,\overline{m}_{Q}$, for the top and
bottom quark and $0.65\,\overline{m}_{c}\leq\mu\leq 2.5\,\overline{m}_{c}$ for
the charm quark. The ${\cal O}(\alpha_{s}^{3})$ corrections are quite sizable
compared to the ${\cal O}(\alpha_{s}^{2})$ contributions, but the ${\cal
O}(\alpha_{s}^{4})$ corrections are small indicating that the ${\cal
O}(\alpha_{s}^{4})$ result and the uncertainty estimate based on the scale
variations can be considered reliable. Overall, the matching corrections
amount to $32,4$ and $5$ MeV for the top, bottom and charm quarks,
respectively with an uncertainty at the level of $1$ to $2$ MeV. The numerical
uncertainties of the ${\cal O}(\alpha_{s}^{4})$ coefficients displayed in Eq.
(33) are smaller than $0.1$ MeV for all cases and therefore irrelevant for
practical purposes.

${\cal O}(\alpha_{s}^{n})$ |  $\Delta m_{t}^{(6\rightarrow 5)}(\overline{m}_{t})$ |  $\Delta m_{b}^{(5\rightarrow 4)}(\overline{m}_{b})$ |  $\Delta m_{c}^{(4\rightarrow 3)}(\overline{m}_{c})$
---|---|---|---
$2$ |  $0.021\pm 0.004$ |  $0.003\pm 0.001$ |  $0.002\pm 0.002$
$3$ |  $0.033\pm 0.003$ |  $0.006\pm 0.002$ |  $0.008\pm 0.005$
$4$ |  $0.032\pm 0.001$ |  $0.004\pm 0.001$ |  $0.005\pm 0.002$
Table 3: The MSR-$\overline{\mathrm{MS}}$ mass matching corrections for the
top, bottom and charm quarks for
($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) = ($163,4.2,1.3$) GeV,
given in units of GeV. The uncertainties are obtained from variations of the
renormalization scale in the range $0.5\,\overline{m}_{Q}\leq\mu\leq
2\,\overline{m}_{Q}$ for the top and bottom quark and
$0.65\,\overline{m}_{c}\leq\mu\leq 2.5\,\overline{m}_{c}$ for the charm quark.
The central value is the respective mean of the largest and smallest values
obtained in the scale variation.

The term $\delta m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})$ represents the virtual top quark loop contributions
arising from the finite masses of the lighter massive quarks
$q_{1},\dots,q_{n}$. Since at ${\cal O}(\alpha_{s}^{2})$ only the loop of
quark $Q$ can be inserted, the series for $\delta
m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})$ starts
at ${\cal O}(\alpha_{s}^{3})$, where only self energy diagrams with one
insertion of a loop of quark $Q$ and one insertion of a loop of one of the
lighter massive quarks $q_{1},\dots,q_{n}$ can contribute. At ${\cal
O}(\alpha_{s}^{3})$ $\delta m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})$ has the form

$\displaystyle\delta$ $\displaystyle
m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})=\overline{m}_{Q}\,\Bigg{\\{}\bigg{[}\delta_{Q,3}^{(Q,q_{1},\dots,q_{n})}(1,r_{q_{1}Q},\dots,r_{q_{n}Q})$
$\displaystyle\hskip
156.49014pt-\,\delta_{Q,3}^{(Q,q_{1},\dots,q_{n})}(1,0,\dots,0)\bigg{]}\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{3}+\dots\Bigg{\\}}$
$\displaystyle\qquad=\overline{m}_{Q}\,\left\\{\sum_{i=1}^{n}\bigg{[}14.2222\,r_{q_{i}Q}^{2}-18.7157\,r_{q_{i}Q}^{3}+\bigg{(}7.3689-11.1477\ln(r_{q_{i}Q})\bigg{)}\,r_{q_{i}Q}^{4}\right.$
$\displaystyle\hskip
85.35826pt+\left.\dots\bigg{]}\left(\frac{\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})}{4\pi}\right)^{3}+\dots\right\\}\,,$
(34)

where $r_{qQ}=\overline{m}_{q}/\overline{m}_{Q}$, and for simplicity we
suppress the masses of the quarks $q_{1},\dots,q_{n}$ in the argument of
$\delta m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow n_{Q})}$. Starting at
${\cal O}(\alpha_{s}^{4})$ the finite quark mass corrections in $\delta
m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})$ become
also dependent on the flavor threshold corrections relating
$\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})$ and
$\alpha_{s}^{(n_{Q}+1)}(\overline{m}_{Q})$. In Eq. (3.1) we have also
displayed the first terms of the expansions in the mass ratios $r_{q_{i}Q}$.
They start quadratically in the $r_{q_{i}Q}$ indicating that the corrections
are governed by the scale $\overline{m}_{Q}$ just like the matching term
$\Delta m_{Q}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})$ and do not have
any linear sensitivity to small momenta and the lighter quark masses, in
particular. This feature is realized at any order of perturbation theory.

Because the finite mass corrections $\delta
m_{Q,q_{1},\dots,q_{n}}^{n_{Q}+1\rightarrow n_{Q}}(\overline{m}_{Q})$ start at
${\cal O}(\alpha_{s}^{3})$ and are quadratic in the mass ratios $r_{q_{i}Q}$
they are extremely small and never exceed $0.01$ MeV for the top quark (due to
the finite bottom or charm masses) and the bottom quark (due to the finite
charm mass). We can expect that this is also exhibited at higher orders, so
that $\delta m_{Q,q_{1},\dots,q_{n}}^{(n_{Q}+1\rightarrow
n_{Q})}(\overline{m}_{Q})$ can be neglected for all practical purposes and
will not be considered and discussed any further in this work.

###  3.2 Top-Bottom and Bottom-Charm Mass Matching

Comparing the pole-$\mathrm{MSR}$ mass relation (2.2) for the heavy quark $Q$
to the pole-$\overline{\mathrm{MS}}$ mass relation (9) for the next lighter
massive quark $q$, one immediately notices that for $R=\overline{m}_{q}$ the
corrections are identical in the approximation that in the virtual quark loops
all $n_{Q}$ lighter quarks (i.e. including the quark $q$) are treated as
massless. This identity is a consequence of heavy quark symmetry which states
that the low-energy QCD corrections to the heavy quark masses coming from
massless partons are flavor-independent.

For the top MSR and the bottom $\overline{\mathrm{MS}}$ masses (i.e. for $Q=t$
and $q=b$) the resulting matching relation reads

$\left[m^{\mathrm{pole}}_{t}-m^{\rm
MSR}_{t}(\overline{m}_{b})\right]-\left[m^{\mathrm{pole}}_{b}-\overline{m}_{b}\right]=\delta
m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})\,,$ (35)

where $\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$
encodes the heavy quark symmetry breaking corrections coming from the finite
virtual charm and bottom quark masses. Their form can be extracted directly
from Eqs. (9) and (2.2) and written in the form
($r_{qq^{\prime}}=\overline{m}_{q}/\overline{m}_{q^{\prime}}$)

$\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})=\overline{m}_{t}\left[\delta_{t}^{(b,c)}(r_{bt},r_{ct})+\delta_{t}^{(c)}(r_{ct})\right]-\overline{m}_{b}\left[\overline{\delta}_{b}^{(b,c)}(1,r_{cb})+\bar{\delta}_{b}^{(c)}(r_{cb})\right]\,,$
(36)

where the first term on the RHS (multiplied by $\overline{m}_{t}$) represents
the virtual bottom and charm mass effects from the top quark self energy and
the second term (multiplied by $\overline{m}_{b}$) represents the virtual
bottom and charm mass effects from the bottom quark self energy. Their
explicit form up to ${\cal O}(\alpha_{s}^{3})$ reads

$\displaystyle\overline{m}_{t}$
$\displaystyle\left[\delta_{t}^{(b,c)}(r_{bt},r_{ct})+\,\delta_{t}^{(c)}(r_{ct})\right]=\overline{m}_{t}\left[\,\delta_{2}(r_{bt})+\delta_{2}(r_{ct})\,\right]\left(\frac{\alpha_{s}^{(5)}(\mu)}{4\pi}\right)^{2}$
(37)
$\displaystyle+\overline{m}_{t}\left[\,\delta_{t,3}^{(b,c)}(r_{bt},r_{ct})+\delta_{t,3}^{(c)}(r_{ct})+4\beta_{0}^{(5)}\ln\left(\frac{\mu}{\overline{m}_{t}}\right)\left[\,\delta_{2}(r_{bt})+\delta_{2}(r_{ct})\right]\right]\left(\frac{\alpha_{s}^{(5)}(\mu)}{4\pi}\right)^{3}+\dots\,,$

and

$\displaystyle\overline{m}_{b}$
$\displaystyle\left[\overline{\delta}_{b}^{(b,c)}(1,r_{cb})+\,\overline{\delta}_{b}^{(c)}(r_{cb})\right]=\overline{m}_{b}\left[\,\delta_{2}(1)+\delta_{2}(r_{cb})\,\right]\left(\frac{\alpha_{s}^{(5)}(\mu)}{4\pi}\right)^{2}$
(38)
$\displaystyle+\overline{m}_{b}\left[\,\delta_{b,3}^{(b,c)}(1,r_{cb})+\delta_{b,3}^{(c)}(r_{cb})+4\beta_{0}^{(5)}\ln\left(\frac{\mu}{\overline{m}_{b}}\right)\left[\,\delta_{2}(1)+\delta_{2}(r_{cb})\right]\right]\left(\frac{\alpha_{s}^{(5)}(\mu)}{4\pi}\right)^{3}+\dots\,.$

It is important that the quark mass corrections in (36) are expressed
coherently in powers of $\alpha_{s}$ at the common scale $\mu$ because the
individual $\delta_{n}$ terms carry contributions that modify the infrared
sensitivity and therefore each contain ${\cal O}(\Lambda_{\mathrm{QCD}})$
renormalon ambiguities. In Eq. (35) these renormalon ambiguities mutually
cancel. We also note that $\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})$ also depends on the top quark mass
$\overline{m}_{t}$. We have suppressed $\overline{m}_{t}$ in the argument
since $\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$
encodes symmetry breaking corrections due to the finite bottom and charm quark
masses.

For the bottom MSR and the charm $\overline{\mathrm{MS}}$ masses the
corresponding matching relation reads

$\left[m^{\mathrm{pole}}_{b}-m^{\rm
MSR}_{b}(\overline{m}_{c})\right]-\left[m^{\mathrm{pole}}_{c}-\overline{m}_{c}\right]=\delta
m_{c}^{(b\rightarrow c)}(\overline{m}_{c})\;,$ (39)

with

$\delta m_{c}^{(b\rightarrow
c)}(\overline{m}_{c})=\overline{m}_{b}\delta_{b}^{(c)}(r_{cb})-\overline{m}_{c}\overline{\delta}_{c}^{(c)}(1)\,,$
(40)

where the first term on the RHS (multiplied by $\overline{m}_{b}$) represents
the virtual charm mass effects from the bottom quark self energy and the
second term (multiplied by $\overline{m}_{c}$) represent the virtual charm
mass effects from the charm quark self energy. Their explicit form up to
${\cal O}(\alpha_{s}^{3})$ reads

$\displaystyle\overline{m}_{b}\,\delta_{b}^{(c)}(r_{cb})$
$\displaystyle=\overline{m}_{b}\,\delta_{2}(r_{cb})\left(\frac{\alpha_{s}^{(4)}(\mu)}{4\pi}\right)^{2}$
$\displaystyle\qquad+\overline{m}_{b}\left[\delta_{b,3}^{(c)}(r_{cb})+4\beta_{0}^{(4)}\delta_{2}(r_{cb})\ln\left(\frac{\mu}{\overline{m}_{b}}\right)\right]\left(\frac{\alpha_{s}^{(4)}(\mu)}{4\pi}\right)^{3}+\dots\,,$
(41)

and

$\displaystyle\overline{m}_{c}\,\overline{\delta}_{c}^{(c)}(1)$
$\displaystyle=\overline{m}_{c}\,\delta_{2}(1)\left(\frac{\alpha_{s}^{(4)}(\mu)}{4\pi}\right)^{2}$
$\displaystyle\qquad+\overline{m}_{c}\left[\delta_{c,3}^{(c)}(1)+4\beta_{0}^{(4)}\delta_{2}(1)\ln\left(\frac{\mu}{\overline{m}_{c}}\right)\right]\left(\frac{\alpha_{s}^{(4)}(\mu)}{4\pi}\right)^{3}+\dots\,,$
(42)

where again we expanded both terms consistently for a common renormalization
scale $\mu$ in the strong coupling.

Figure 3:  (a) Top-MSR bottom-$\overline{\mathrm{MS}}$ mass matching correction $\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ at ${\cal O}(\alpha_{s}^{2})$ (red dashed curve) and ${\cal O}(\alpha_{s}^{3})$ (red solid curve) over the renormalization scale $\mu$. The virtual bottom and charm mass effects to the top quark self energy of Eq. (37) (green curves) and the virtual bottom and charm mass effects to the bottom quark self energy of Eq. (38) (blue curves) at ${\cal O}(\alpha_{s}^{2})$ (dashed) and ${\cal O}(\alpha_{s}^{3})$ (solid). For the masses of the top, bottom and charm quarks the values ($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) = ($163,4.2,1.3$) GeV are used. (b) Same quantities as in panel (a) for $\overline{m}_{c}=0$. (c) The bottom-MSR charm-$\overline{\mathrm{MS}}$ mass matching correction $\delta m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$ at ${\cal O}(\alpha_{s}^{2})$ (red dashed curve) and ${\cal O}(\alpha_{s}^{3})$ (red solid curve) over the renormalization scale $\mu$. The virtual charm mass effects to the bottom quark self energy of Eq. (3.2) (green curves) and the virtual charm mass effects to the charm quark self energy of Eq. (3.2) (blue curves) are shown at ${\cal O}(\alpha_{s}^{2})$ (dashed) and ${\cal O}(\alpha_{s}^{3})$(solid).  ${\cal O}(\alpha_{s}^{n})$ |  $\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ |  $\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},0)$ |  $\delta m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$
---|---|---|---
$2$ |  $0.007\pm 0.004$ |  $0.006\pm 0.004$ |  $0.004\pm 0.002$
$3$ |  $0.006\pm 0.001$ |  $0.005\pm 0.001$ |  $0.004\pm 0.001$
Table 4: The top-bottom MSR-$\overline{\mathrm{MS}}$ mass matching
corrections, given in units of GeV, for finite bottom and charm masses (second
column), for finite bottom quark mass and massless charm quark (third column),
and the bottom-charm MSR-$\overline{\mathrm{MS}}$ mass matching correction
(fourth column). For the finite masses of the top, bottom and charm quarks the
values ($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) =
($163,4.2,1.3$) GeV are used. The uncertainties are obtained from variations
of the renormalization scale in the range
$\overline{m}_{b}\leq\mu\leq\overline{m}_{t}$ for $\delta
m_{b,c}^{(t\rightarrow b)}$ and in the range
$\overline{m}_{c}\leq\mu\leq\overline{m}_{b}$ for $\delta m_{c}^{(b\rightarrow
c)}$. The central values are the respective mean of the largest and smallest
values obtained in the scale variation.

In Fig. 3 the top-MSR bottom-$\overline{\mathrm{MS}}$ mass matching correction
$\delta m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ of Eq.
(35) is displayed as a function of the renormalization scale $\mu$ at ${\cal
O}(\alpha_{s}^{2})$ (red dashed line) and ${\cal O}(\alpha_{s}^{3})$ (red
solid line) for ($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) =
($163,4.2,1.3$) GeV. The matching correction at ${\cal O}(\alpha_{s}^{3})$
amounts to $6$ MeV and has a scale variation of only $1$ MeV for
$\overline{m}_{b}\leq\mu\leq\overline{m}_{t}$. Compared to the ${\cal
O}(\alpha_{s}^{2})$ result we see a strong reduction of the scale-dependence
at ${\cal O}(\alpha_{s}^{3})$. The final numerical results at ${\cal
O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$ are shown in the second
column of Tab. 4 where the uncertainties are obtained from variations of the
renormalization scale in the range
$\overline{m}_{b}\leq\mu\leq\overline{m}_{t}$ and the central values are the
respective mean of the largest and smallest values obtained in the scale
variation. The corresponding results for a vanishing charm quark mass are
shown in Fig. 3 and the third column of Tab. 4. We see that the charm mass
effects in the top-MSR bottom-$\overline{\mathrm{MS}}$ mass matching
correction $\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})$ are only around $1$ MeV, and the
stability for $\overline{m}_{c}\rightarrow 0$ shows that the matching
correction is governed by scales of order $\overline{m}_{b}$ and higher, which
reconfirms the range $\overline{m}_{b}\leq\mu\leq\overline{m}_{t}$ for the
variation of the renormalization scale.

In Fig. 3 the bottom-MSR charm-$\overline{\mathrm{MS}}$ mass matching
correction $\delta m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$ of Eq. (39) is
displayed as a function of the renormalization scale $\mu$ for
$\overline{m}_{b}=4.2$ GeV and $\overline{m}_{c}=1.3$ GeV at ${\cal
O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$ using the same color coding
and curve styles as for Figs. 3 and 3. In the fourth column of Tab. 4 the
final numerical results at ${\cal O}(\alpha_{s}^{2})$ and ${\cal
O}(\alpha_{s}^{3})$ are shown using
$\overline{m}_{c}\leq\mu\leq\overline{m}_{b}$ for the renormalization scale
variation. The stability and convergence is again excellent, and at ${\cal
O}(\alpha_{s}^{3})$ the matching correction amounts to $4$ MeV with an
uncertainty of $1$ MeV.

Given that the heavy quark symmetry breaking matching corrections $\delta
m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ and $\delta
m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$ amount to only $4$ to $6$ MeV, we
note that they may be simply neglected in practical applications where they
yield contributions that are much smaller than other sources of uncertainties.
In fact, this also applies to our subsequent studies of the top, bottom and
charm quark pole masses. However, we include them here for completeness. Due
to their small size, we have not explicitly included the heavy quark symmetry
breaking matching corrections in the graphical illustration of Fig. 2.

###  3.3 Light Virtual Quark Mass Corrections at ${\cal O}(\alpha_{s}^{4})$
and Beyond

The excellent perturbative convergence of the top-MSR
bottom-$\overline{\mathrm{MS}}$ mass matching correction $\delta
m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ and of the
bottom-MSR charm-$\overline{\mathrm{MS}}$ mass matching correction $\delta
m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$ discussed in the previous section
illustrates that they both are short-distance quantities and free of an ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity. This is also expected
theoretically due to heavy quark symmetry. However, the facts that the overall
size of the matching corrections only amounts to a few MeV, and that the
${\cal O}(\alpha_{s}^{3})$ corrections are only around $1$ MeV allows us to
draw interesting conceptual implications for the large order asymptotic
behavior of the virtual quark mass corrections in the mass relations of Eqs.
(9), (2.1) and (2.2). We discuss these implications in the following. As a
consequence we can predict the yet uncalculated virtual quark mass corrections
at ${\cal O}(\alpha_{s}^{4})$ to within a few percent without an additional
loop calculation and draw important conclusions on their properties for the
orders beyond.

To be concrete, we consider the matching correction $\delta
m_{q}^{(Q\rightarrow q)}(\overline{m}_{q})$ between the MSR mass of heavy
quark $Q$ and the $\overline{\mathrm{MS}}$ mass of the next lighter massive
quark $q$ assuming the massless approximation for all quarks lighter than
quark $q$ i.e. $n_{Q}=n_{q}+1=n_{\ell}+1$ and $n_{\ell}=n_{q}$ being the
number of massless quarks. This situation applies to the matching relation for
the top-MSR and the bottom $\overline{\mathrm{MS}}$ masses for a massless
charm quark or to the matching relation between the bottom-MSR and the
charm-$\overline{\mathrm{MS}}$ masses.

In Fig. 3 we have displayed separately the virtual bottom and charm mass
effects to the top quark self energy of Eq. (37) (green curves) and the
virtual bottom and charm mass effects to the bottom quark self energy of Eq.
(38) (blue lines) at ${\cal O}(\alpha_{s}^{2})$ (dashed) and ${\cal
O}(\alpha_{s}^{3})$ (solid). In Fig. 3 the charm quark is treated as massless
in the same quantities. In Fig. 3 the virtual charm mass effects to the bottom
quark self energy of Eq. (3.2) and the virtual charm mass effects to the charm
quark self energy of Eq. (3.2) are shown at ${\cal O}(\alpha_{s}^{2})$ and
${\cal O}(\alpha_{s}^{3})$ with the analogous line styles and colors. We see
that both types of contributions each are quite large and furthermore do not
at all converge. The ${\cal O}(\alpha_{s}^{3})$ corrections are even bigger
than the ${\cal O}(\alpha_{s}^{2})$ corrections, which indicates that the
corresponding asymptotic large order behavior already dominates the ${\cal
O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$ corrections.

The origin of this behavior has been already mentioned and is understood: The
mass of the virtual quark $q$ acts as an infrared cutoff and therefore
modifies the infrared sensitivity of the self energy diagrams (of quark $Q$
and of quark $q$) with respect to the case where the virtual loops of quark
$q$ are evaluated in the massless approximation. As a consequence these
corrections individually carry an ${\cal O}(\Lambda_{\mathrm{QCD}})$
renormalon ambiguity. Moreover, at large orders in perturbation theory the
sensitivity of the self energy diagrams to infrared momenta increases due to
high powers of logarithms from gluonic and massless quark loops. As a
consequence, at large orders, the finite mass effects of the virtual loops of
quark $q$ in the self energy diagrams of quark $Q$ and the self energy
diagrams of quark $q$ become equivalent due to heavy quark symmetry. The
strong cancellation in the sum of both types of corrections in $\delta
m_{q}^{(Q\rightarrow q)}(\overline{m}_{q})$ ($\sim 75\%$ at ${\cal
O}(\alpha_{s}^{2})$ and $\gtrsim 90\%$ at ${\cal O}(\alpha_{s}^{3})$ for the
cases displayed in Fig. 3) thus confirms that the known ${\cal
O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$ self energy corrections
coming from virtual quark masses are already dominated by their large order
asymptotic behavior.

From the observations that the series for $\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})$ and $\delta m_{c}^{(b\rightarrow
c)}(\overline{m}_{c})$ converge very well and that their ${\cal
O}(\alpha_{s}^{3})$ corrections amount to only about $1$ MeV, we can therefore
expect that the two types of corrections that enter $\delta
m_{b,c}^{(t\rightarrow b)}(\overline{m}_{b},\overline{m}_{c})$ as well as
$\delta m_{c}^{(b\rightarrow c)}(\overline{m}_{c})$ agree to even better than
$1$ MeV at ${\cal O}(\alpha_{s}^{4})$ and beyond. This allows us to make an
approximate prediction for the yet uncalculated ${\cal O}(\alpha_{s}^{4})$
finite mass corrections from virtual loops of quark $q$ in the
pole-$\overline{\mathrm{MS}}$ mass relations of quark $Q$ of Eqs. (9) and
(2.1) by setting the ${\cal O}(\alpha_{s}^{4})$ correction in $\delta
m_{q}^{(Q\rightarrow q)}(\overline{m}_{q})$ to zero:

$\displaystyle\delta_{Q,4}^{(q)}(r_{qQ})$ (43) $\displaystyle\approx
r_{qQ}\Bigg{[}\,\delta_{q,4}^{(q)}(1)+\left(6\,\beta_{0}^{(n_{Q})}\delta_{q,3}^{(q)}(1)+4\,\beta_{1}^{(n_{Q})}\delta_{2}(1)\right)\ln\left(\frac{\mu}{\overline{m}_{q}}\right)+12\,\delta_{2}(1)\left(\beta_{0}^{(n_{Q})}\ln\left(\frac{\mu}{\overline{m}_{q}}\right)\right)^{2}\Bigg{]}\,$
$\displaystyle\hskip
36.98866pt-\left(6\,\beta_{0}^{(n_{Q})}\delta_{Q,3}^{(q)}(r_{qQ})+4\,\beta_{1}^{(n_{Q})}\delta_{2}(r_{qQ})\right)\ln\left(\frac{\mu}{\overline{m}_{Q}}\right)-12\,\delta_{2}(r_{qQ})\left(\beta_{0}^{(n_{Q})}\ln\left(\frac{\mu}{\overline{m}_{Q}}\right)\right)^{2}\,.$

The prediction has a residual $\mu$-dependence, which would vanish in the
formal limit that the virtual quark $q$ mass corrections are entirely
dominated by their large order asymptotic behavior. Therefore the dependence
on the scale $\mu$ can be used as an uncertainty estimate of our
approximation.

Figure 4:  (a) Prediction for the ${\cal O}(\alpha_{s}^{4})$ virtual quark
mass correction $\delta_{Q,4}^{(q)}(r_{qQ})$ for
$\overline{m}_{q}\leq\mu\leq\overline{m}_{Q}$ (green bands) for
$n_{Q}=n_{\ell}+1=5$ (lower band) and $n_{Q}=n_{\ell}+1=4$ (upper band). The
black dashed lines show the prediction for $\mu=\overline{m}_{Q}$ which gives
the simple approximation formula in Eq. (45). (b) The ${\cal
O}(\alpha_{s}^{3})$ virtual quark mass correction $\delta_{Q,3}^{(q)}(r_{qQ})$
for $n_{Q}=n_{\ell}+1=5$ (red curve). The green band is the prediction for
$\delta_{Q,3}^{(q)}(r_{qQ})$ using the method of panel (a) for
$\overline{m}_{q}\leq\mu\leq\overline{m}_{Q}$ showing excellent agreement to
the exact result within errors.

In Fig. 4 we show the prediction for $\delta_{Q,4}^{(q)}(r_{qQ})$ for
$\overline{m}_{q}\leq\mu\leq\overline{m}_{Q}$ (green bands) for
$n_{Q}=n_{q}+1=n_{\ell}+1=5$ (lower band) and $n_{Q}=n_{q}+1=n_{\ell}+1=4$
(upper band). The prediction satisfies exactly the required boundary condition
$\delta_{Q,4}^{(q)}(0)=0$ and Eq. (16) for $r_{qQ}=1$ and provides an
interpolation for $0<r_{qQ}<1$ with an uncertainty of $\pm 3\%$ (for
$r_{qQ}\lesssim 0.1$) or smaller (for $r_{qQ}>0.1$). To judge the quality of
the prediction we apply the same method at ${\cal O}(\alpha_{s}^{3})$ to
“predict” $\delta_{Q,3}^{(q)}(r_{qQ})$ which gives

$\delta_{Q,3}^{(q)}(r_{qQ})\approx
r_{qQ}\left[\,\delta_{q,3}^{(q)}(1)+4\,\beta_{0}^{(n_{Q})}\delta_{2}(1)\ln\left(\frac{\mu}{\overline{m}_{q}}\right)\right]-4\beta_{0}^{(n_{Q})}\delta_{2}(r_{qQ})\ln\left(\frac{\mu}{\overline{m}_{Q}}\right)\,.$
(44)

The result for the prediction of $\delta_{Q,3}^{(q)}(r_{qQ})$ is shown in Fig.
4 for $n_{Q}=n_{q}+1=n_{\ell}+1=5$. The green band illustrates again the range
of predictions for $\mu$-variations
$\overline{m}_{q}\leq\mu\leq\overline{m}_{Q}$, and represents an uncertainty
of $\pm 10\%$ (for $r_{qQ}\lesssim 0.1$) or smaller (for $r_{qQ}>0.1$).
Compared to the ${\cal O}(\alpha_{s}^{4})$ result, the larger $\mu$ variation
we observe at ${\cal O}(\alpha_{s}^{3})$ is expected because the infrared
sensitivity is weaker and the large order asymptotic behavior is less
dominating at the lower order. The red curve is the exact result for
$\delta_{Q,3}^{(q)}(r_{qQ})$ obtained from the results in Ref. Bekavac:2007tk
, see also Eq. (69). We see that the prediction is fully compatible with the
exact result and that the uncertainty estimate based on the $\mu$-variation is
reliable. The prediction for $\delta_{Q,3}^{(q)}(r_{qQ})$ for
$n_{Q}=n_{q}+1=n_{\ell}+1=4$ has the same good properties but is not displayed
since it is numerically very close to the prediction for
$n_{Q}=n_{q}+1=n_{\ell}+1=5$.

Overall, the examination shows that the prediction and the uncertainty
estimate for $\delta_{Q,4}^{(q)}(r_{qQ})$ can be considered reliable. We can
also provide a very simple closed analytic expression by evaluating Eq. (43)
for $\mu=\overline{m}_{Q}$, which gives

$\displaystyle\delta_{Q,4}^{(q)}(r_{qQ})$ (45) $\displaystyle\approx
r_{qQ}\bigg{[}\,\delta_{q,4}^{(q)}(1)-\left(6\,\beta_{0}^{(n_{Q})}\delta_{q,3}^{(q)}(1)+4\,\beta_{1}^{(n_{Q})}\delta_{2}(1)\right)\ln\left(r_{qQ}\right)+12\,\delta_{2}(1)\left(\beta_{0}^{(n_{Q})}\ln(r_{qQ})\right)^{2}\bigg{]}$
$\displaystyle=r_{qQ}\bigg{[}\,(203915.\pm
32.)-22962.\,n_{Q}+525.2\,n_{Q}^{2}+(-130946.+13831.\,n_{Q}-328.5\,n_{Q}^{2})\,\ln(r_{qQ})$
$\displaystyle\hskip
42.67912pt+(26599.1-3224.1\,n_{Q}+97.70\,n_{Q}^{2})\,\ln(r_{qQ})^{2}\,\bigg{]}\,.$

The expression depends via the boundary condition of Eq. (16) entirely on the
coefficients $a_{n}(n_{q},n_{h})$ of Eq. (10), which for this case describe
the corrections to the heavy quark $q$ self energy for the case that all
lighter quarks are massless, and the coefficients of the $\beta$-function. The
expression is shown as the black dashed lines in Fig. 4 for
$n_{Q}=n_{\ell}+1=5$ (lower line) and $n_{Q}=n_{\ell}+1=4$ (upper line). This
approximation for $\delta_{Q,4}^{(q)}(r_{qQ})$ has a simple overall linear
behavior on the mass ratio $r_{qQ}=\overline{m}_{q}/\overline{m}_{Q}$. The
behavior is just a manifestation of $\delta_{Q,4}^{(q)}(r_{qQ})$ being
dominated by the large order asymptotic behavior due to its ${\cal
O}(\Lambda_{\rm QCD})$ renormalon ambiguity which is related to linear
sensitivity to small scales. The overall linear dependence of
$\delta_{Q,4}^{(q)}(r_{qQ})$ on $\overline{m}_{q}$ arises since the mass of
quark $q$ represents an infrared cut and thus represents the characteristic
physical scale that governs $\delta_{Q,4}^{(q)}(r_{qQ})$. This also explains
the origin of the logarithms shown in Eq. (45): They arise because all virtual
quark mass corrections in Eqs. (9), (2.1) and (2.2) are defined in an
expansion in $\alpha_{s}(\overline{m}_{Q})$. We note that for the ${\cal
O}(\alpha_{s}^{3})$ virtual massive quark correction
$\delta_{Q,3}^{(q)}(r_{qQ})$ these aspects were already discussed in Ref.
Hoang:2000fm  and later in Ref. Ayala:2014yxa , where a direct comparison to
the explicit calculations from Ref. Bekavac:2007tk  could be carried out.
These analyses were, however, using generic considerations and were not
carried out within a systematic RG framework.

The expression of Eq. (45) is a special case of the general statement that the
asymptotic large order behavior of the coefficients
$\delta_{Q,n}^{(q)}(r_{qQ})$ can be obtained from the relation

$\displaystyle\delta_{2}(r_{qQ})$
$\displaystyle\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})}{4\pi}\right)^{2}+\sum_{n=3}^{\infty}\,\delta_{Q,n}^{(q)}(r_{qQ})\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})}{4\pi}\right)^{n}$
(46) $\displaystyle\approx
r_{qQ}\,\overline{\delta}_{q}^{(q)}(1)=r_{qQ}\left[\delta_{2}(1)\,\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{q})}{4\pi}\right)^{2}+\sum_{n=3}^{\infty}\delta_{q,n}^{(q)}(1)\left(\frac{\alpha_{s}^{(n_{Q})}(\overline{m}_{q})}{4\pi}\right)^{n}\,\right]\,,$

where on the RHS of the approximate equality
$\alpha_{s}^{(n_{Q})}(\overline{m}_{q})$ has to be expanded in powers of
$\alpha_{s}^{(n_{Q})}(\overline{m}_{Q})$, and we have $\delta_{2}(1)=18.3189$,
$\delta_{q,3}^{(q)}(1)=1870.79-82.1208\,n_{Q}$ and
$\delta_{q,4}^{(q)}(1)=(203915.\pm 32.)-22961.6\,n_{Q}+525.216\,n_{Q}^{2}$.
The terms $\delta_{q,n}^{(q)}(1)$ for $n>4$ can be obtained from using Eqs.
(2.1) and (16) together with the large order asymptotic form of the
coefficients $a_{n}$ shown in Eq. (31), giving

$\displaystyle\delta_{q,n>4}^{(q)}(1)$ $\displaystyle\approx
a_{n}^{\mathrm{asy}}(n_{q})-a_{n}^{\mathrm{asy}}(n_{q}+1)=a_{n}^{\mathrm{asy}}(n_{Q}-1)-a_{n}^{\mathrm{asy}}(n_{Q})\,,$
(47)

where we would like to remind the reader that for the case we consider here we
have $n_{Q}=n_{q}+1=n_{\ell}+1$. Our examination at ${\cal O}(\alpha_{s}^{3})$
and ${\cal O}(\alpha_{s}^{4})$ above showed that this relation provides an
approximation for $\delta_{Q,4}^{(q)}$ within a few percent. For the higher-
order terms $\delta_{Q,n}^{(q)}$ with $n>4$ it should be even more precise,
and we therefore believe that it should be sufficient for essentially all
future applications in the context of studies of the pole mass scheme.

To conclude we note that it is straightforward to extend Eq. (43) from the
case of having only one massive quark $q$ being lighter than heavy quark $Q$,
i.e. $n_{Q}=n_{q}+1=n_{\ell}+1$, to the case of having a larger number of
lighter massive quarks. For example for the case that there are two massive
quarks lighter than quark $Q$ (let’s say $q$ and $q^{\prime}$, in order of
decreasing mass) with $n_{Q}=n_{q}+1=n_{q^{\prime}}+2=n_{\ell}+2$, the
generalization of the approximation formula (43) reads

$\displaystyle\delta_{Q,4}^{(q,q^{\prime})}(r_{qQ},r_{q^{\prime}Q})+\delta_{Q,4}^{(q^{\prime})}(r_{q^{\prime}Q})\approx
r_{qQ}\,\Bigg{\\{}\,\delta_{q,4}^{(q,q^{\prime})}(1,r_{q^{\prime}q})+\delta_{q,4}^{(q^{\prime})}(r_{q^{\prime}q})$
$\displaystyle\hskip
28.45274pt+\left[6\,\beta_{0}^{(n_{Q})}\left(\delta_{q,3}^{(q,q^{\prime})}(1,r_{q^{\prime}q})+\delta_{q,3}^{(q^{\prime})}(r_{q^{\prime}q})\right)+4\,\beta_{1}^{(n_{Q})}\left(\delta_{2}(1)+\delta_{2}(r_{q^{\prime}q})\right)\right]\ln\left(\frac{\mu}{\overline{m}_{q}}\right)$
$\displaystyle\hskip
28.45274pt+12\,\left(\delta_{2}(1)+\delta_{2}(r_{q^{\prime}q})\right)\,\left(\beta_{0}^{(n_{Q})}\ln\left(\frac{\mu}{\overline{m}_{q}}\right)\right)^{2}\Bigg{\\}}$
(48) $\displaystyle\hskip
28.45274pt-\left[6\,\beta_{0}^{(n_{Q})}\left(\delta_{Q,3}^{(q,q^{\prime})}(r_{qQ},r_{q^{\prime}Q})+\delta_{Q,3}^{(q^{\prime})}(r_{q^{\prime}Q})\right)+4\,\beta_{1}^{(n_{Q})}\left(\delta_{2}(r_{qQ})+\delta_{2}(r_{q^{\prime}Q})\right)\right]\ln\left(\frac{\mu}{\overline{m}_{Q}}\right)$
$\displaystyle\hskip
28.45274pt-12\,\left(\delta_{2}(r_{qQ})+\delta_{2}(r_{q^{\prime}Q})\right)\,\left(\beta_{0}^{(n_{Q})}\ln\left(\frac{\mu}{\overline{m}_{Q}}\right)\right)^{2}\,.$

###  3.4 Pole Mass Differences

Using the MSR mass we have set up a conceptual framework to systematically
quantify the contributions to the pole mass of a heavy quark coming from the
different momentum regions contained in the on-shell self energy diagrams. The
pole mass of a heavy quark $Q$ contains the contributions from all momenta,
while the $\overline{\mathrm{MS}}$ mass $\overline{m}_{Q}(\mu)$ and the MSR
mass $m^{\rm MSR}_{Q}(R)$ contain the contributions from above the scales
$\mu$ and $R$, respectively (see Fig. 1). The MSR mass is the natural
extension of the $\overline{\mathrm{MS}}$ mass, which is applied for scales
$\mu>m_{Q}$, to scales $R<m_{Q}$, and obeys a RG-evolution equation that is
linear in $R$, called R-evolution Hoang:2008yj ; Hoang:2017suc . The
R-evolution equation quantifies in a way free of the ${\cal O}(\Lambda_{\rm
QCD})$ renormalon the change in the MSR mass when contributions from lower
momenta are included into the mass when $R$ is decreased, as long as
$R>\Lambda_{\mathrm{QCD}}$.

In Sec. 3.1 we discussed the matching corrections $\Delta
m_{Q}^{(n_{Q}+1\rightarrow n_{Q})}(\overline{m}_{Q})$ that arise when the
virtual loop contributions of quark $Q$ are integrated out by switching from
$\overline{m}_{Q}$ to $m^{\rm MSR}_{Q}(\overline{m}_{Q})$. In Sec. 2.2 we
discussed the MSR mass difference $\Delta m^{(n_{Q})}(R,R^{\prime})=m^{\rm
MSR}_{Q}(R^{\prime})-m^{\rm MSR}_{Q}(R)$, which is determined from solving the
R-evolution equation of the MSR mass and which systematically sums logarithms
of $R/R^{\prime}$. In Sec. 3.2 we examined the matching between the QCD
corrections to the MSR mass of the heavy quark $Q$ and the
$\overline{\mathrm{MS}}$ mass of the next lighter massive quark $q$, $\delta
m_{q,q^{\prime},\dots}^{(Q\rightarrow
q)}(\overline{m}_{q},\overline{m}_{q^{\prime}},\dots)$ accounting for the mass
effects of the quarks $q,q^{\prime},\dots\;$. This matching is based on heavy
quark symmetry and the small numerical size of $\delta
m_{q,q^{\prime},\dots}^{(Q\rightarrow
q)}(\overline{m}_{q},\overline{m}_{q^{\prime}},\dots)$ reflects that the
symmetry breaking effects due to the finite quark masses are quite small.
These two types of matching corrections and the R-evolution of the MSR mass
each are free of ${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguities and
show excellent convergence properties in QCD perturbation theory.

An interesting application is the determination of the difference of the pole
masses of two massive quarks. Due to heavy quark symmetry, the differences of
two heavy quark pole masses are also free of ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguities and can therefore be
determined to high precision. The matching corrections discussed above and the
R-evolution of the MSR mass allow us to systematically sum logarithms of the
mass ratios that would remain unsummed in a fixed-order calculation, and to
achieve more precise perturbative predictions Hoang:2017suc . Taking the
example of the top and bottom mass one can then write the difference of the
top quark pole-$\overline{\mathrm{MS}}$ mass relation and the bottom quark
pole-$\overline{\mathrm{MS}}$ mass relation in the form

$\left[m^{\mathrm{pole}}_{t}-\overline{m}_{t}\right]-\left[m^{\mathrm{pole}}_{b}-\overline{m}_{b}\right]=\Delta
m_{t}^{(6\rightarrow 5)}(\overline{m}_{t})+\Delta
m^{(5)}(\overline{m}_{t},\overline{m}_{b})+\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})\,.$ (49)

The analogous relation for the bottom and charm quarks reads

$\left[m^{\mathrm{pole}}_{b}-\overline{m}_{b}\right]-\left[m^{\mathrm{pole}}_{c}-\overline{m}_{c}\right]=\Delta
m_{b}^{(5\rightarrow 4)}(\overline{m}_{b})+\Delta
m^{(4)}(\overline{m}_{b},\overline{m}_{c})+\delta m_{c}^{(b\rightarrow
c)}(\overline{m}_{c})\,.$ (50)

Each of the mass differences is the sum of universal matching and evolution
building blocks which each can be computed to high precision, as shown in
Tabs. 1, 3, 4.

The resulting relations between the top, bottom and charm quark pole masses
read

$\displaystyle m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{b}$
$\displaystyle=\left[\overline{m}_{t}-\overline{m}_{b}\right]+\Delta
m_{t}^{(6\rightarrow 5)}(\overline{m}_{t})+\Delta
m^{(5)}(\overline{m}_{t},\overline{m}_{b})+\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})\,,$ (51) $\displaystyle
m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$
$\displaystyle=\left[\overline{m}_{b}-\overline{m}_{c}\right]+\Delta
m_{b}^{(5\rightarrow 4)}(\overline{m}_{b})+\Delta
m^{(4)}(\overline{m}_{b},\overline{m}_{c})+\delta m_{c}^{(b\rightarrow
c)}(\overline{m}_{c})\,,$ (52) $\displaystyle
m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{c}$
$\displaystyle=\left[\overline{m}_{t}-\overline{m}_{c}\right]+\Delta
m_{t}^{(6\rightarrow 5)}(\overline{m}_{t})+\Delta
m^{(5)}(\overline{m}_{t},\overline{m}_{b})+\delta m_{b,c}^{(t\rightarrow
b)}(\overline{m}_{b},\overline{m}_{c})$ $\displaystyle\hskip 60.6045pt+\Delta
m_{b}^{(5\rightarrow 4)}(\overline{m}_{b})+\Delta
m^{(4)}(\overline{m}_{b},\overline{m}_{c})+\delta m_{c}^{(b\rightarrow
c)}(\overline{m}_{c})\,,$ (53)

and can be readily evaluated from the highest order results given in Tabs. 1,
3, 4 for the case ($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) =
($163,4.2,1.3$) GeV:

$\displaystyle m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{b}$
$\displaystyle=158.800+(0.032\pm 0.001)+(9.331\pm 0.016)+(0.006\pm
0.001)\;{\rm GeV}$ $\displaystyle=168.169\pm 0.016\;{\rm GeV}\,,$ (54)
$\displaystyle m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$
$\displaystyle=2.9+(0.004\pm 0.001)+(0.423\pm 0.017)+(0.004\pm 0.001)\;{\rm
GeV}$ $\displaystyle=3.331\pm 0.017\;{\rm GeV}\,,$ (55) $\displaystyle
m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{c}$ $\displaystyle=171.500\pm
0.024\;{\rm GeV}\,,$ (56)

where we have added all uncertainties quadratically. We can compare our
results for the bottom-charm pole mass difference
$m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$ to the result obtained in Ref.
Hoang:2005zw  using a fixed-order expansion at ${\cal O}(\alpha_{s}^{3})$ for
the mass difference. Their result was based on a linear approximation for the
virtual charm quark mass effects derived in Ref. Hoang:2000fm  which is
similar to Eq. (44), but used a numerical calculation of the coefficient
linear in $r_{qQ}$ from Ref. Melles:1998dj . In this analysis the pole mass
difference was used to eliminate the charm quark mass as a primary parameter
in the predictions. They determined
$m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}=3.401\pm 0.013$ GeV and obtained
$\overline{m}_{c}=1.22\pm 0.06$ GeV from the fits using
$\overline{m}_{b}=4.16\pm 0.05$ GeV as input. Their result for
$m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$ is consistent with ours, but one
should keep in mind that logarithms of $\overline{m}_{c}/\overline{m}_{b}$
were not systematically summed and that their result also included nontrivial
QCD corrections to semileptonic B-meson decay spectra for $B\to X_{c}\ell\nu$
and $B\to X_{s}\gamma$ which were only known to ${\cal O}(\alpha_{s}^{2})$.
The mutual agreement is reassuring (also for the theoretical approximations
made in the context of the B meson analyses) and in particular shows that the
summation of logarithms of $\overline{m}_{c}/\overline{m}_{b}$ is not
essential for bottom and charm masses, which is expected, and that the ${\cal
O}(\alpha_{s}^{4})$ corrections are tiny, which can also be seen explicitly in
our results. The larger error we obtain in our computation of
$m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$ arises from the renormalization
scale scale variation in $\Delta m^{(4)}(\overline{m}_{b},\overline{m}_{c})$
which includes scales as low as $0.6\,\overline{m}_{c}$ while in their
analysis the lowest renormalization scale was $\overline{m}_{c}$. Similar
determinations of bottom and charm quark masses from B-meson decay spectra
were carried out in Ref. Buchmuller:2005zv ; Gambino:2016jkc , and they are
also consistent with our result for
$m^{\mathrm{pole}}_{b}-m^{\mathrm{pole}}_{c}$.

For the case ($\overline{m}_{t},\overline{m}_{b},\overline{m}_{c}$) =
($163,4.2,0$) GeV, the difference between the top and bottom pole masses reads

$\displaystyle m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{b}$
$\displaystyle=158.800+(0.032\pm 0.001)+(9.331\pm 0.016)+(0.005\pm
0.001)\;{\rm GeV}$ $\displaystyle=168.168\pm 0.016\;{\rm GeV}\,.$ (57)

This result differs from Eq. (54) by only $1$ MeV showing that the effects of
the finite charm quark mass are tiny in the difference of the top and bottom
pole masses. The uncertainties in the pole mass differences are between $16$
and $24$ MeV and should be considered as conservative estimates of the
theoretical uncertainties due to missing higher order corrections.

###  3.5 Lighter Massive Flavor Decoupling

Another very instructive application of the RG framework to quantify and
separate the contributions to the pole mass of a heavy quark coming from the
different physical momentum regions is to examine the effective massive flavor
decoupling at large orders. It was observed in Ref. Ayala:2014yxa  that the
sum of the known ${\cal O}(\alpha_{s}^{2})$ and ${\cal O}(\alpha_{s}^{3})$
charm quark mass effects in the bottom quark pole-$\overline{\mathrm{MS}}$
mass series expressed in four flavor coupling
$\alpha_{s}^{(4)}(\overline{m}_{b})$ (where they amount to about $35$ MeV) are
essentially fully captured simply by expressing the series in the three flavor
coupling $\alpha_{s}^{(3)}(\overline{m}_{b})$ (where they amount to only $-2$
MeV). This observation entails that one can simply neglect the charm quark
mass corrections by computing the bottom quark pole-$\overline{\mathrm{MS}}$
mass relation right from start in the three flavor theory without any charm
quark (which corresponds to an infinitely heavy charm quark). This effective
decoupling of lighter massive quarks is obvious and truly happening at
asymptotic large orders. The importance of the observation made in Ref.
Ayala:2014yxa  was that the finite charm quark mass corrections in the
decoupled calculation at ${\cal O}(\alpha_{s}^{2})$ and ${\cal
O}(\alpha_{s}^{3})$ were so tiny that there was no need to compute them
explicitly in the first place. If this decoupling property would be true in
general (i.e. the remaining light quark mass correction become negligible) it
would represent a great simplification because it may make an explicit
calculation of the lighter massive quark corrections and also the summation of
the associated logarithms irrelevant.

Using the RG framework for the lighter massive flavor dependence of the pole
mass we can examine systematically in which way this effective lighter massive
quark decoupling property is realized. In the following we analyze this issue
for $(\overline{m}_{t},\overline{m}_{b},\overline{m}_{c})=(163,4.2,1.3)$ GeV.
We start with the effects of the charm quark mass in the bottom
pole-$\overline{\mathrm{MS}}$ mass relation examined in Ref. Ayala:2014yxa .
Applying the same considerations as for the pole mass differences in Sec. 3.4
for this case we can write down the relation

$\displaystyle m^{\mathrm{pole}}_{b}-$
$\displaystyle\bigg{[}\,\overline{m}_{b}+\overline{m}_{b}\sum_{n=1}^{\infty}a_{n}(n_{\ell}=3,0)\left(\frac{\alpha_{s}^{(3)}(\overline{m}_{b})}{4\pi}\right)^{n}\,\bigg{]}$
(58) $\displaystyle=$ $\displaystyle\,\,\Delta m_{b}^{(5\to
4)}(\overline{m}_{b})+\Delta m^{(4)}(\overline{m}_{b},\overline{m}_{c})+\delta
m_{c}^{(b\to c)}(\overline{m}_{c})+\Delta m_{c}^{(4\to 3)}(\overline{m}_{c})$
$\displaystyle-\Delta m^{(3)}(\overline{m}_{b},\overline{m}_{c})\,$
$\displaystyle=$ $\displaystyle\,\,(0.004\pm 0.001)+(0.423\pm 0.017)+(0.004\pm
0.001)+(0.005\pm 0.002)$ $\displaystyle\,\,-(0.434\pm 0.020)\;{\rm GeV}$
$\displaystyle=$ $\displaystyle\,\,0.002\pm 0.026\;{\rm GeV}\,.$ (59)

The RHS represents a computation of the charm quark mass corrections that
remain within a calculation where the charm mass effects are approximated by
making the charm infinitely heavy (i.e. $n_{\ell}=3$). The individual
numerical results have been taken from the highest order results in Tabs. 1, 3
and 4, and for the final numerical result we have conservatively added all
uncertainties quadratically. We see that these remaining corrections are
essentially zero, fully confirming the observation of Ref. Ayala:2014yxa .
This is not surprising since the bottom and charm quark masses are similar in
size and the ratio $\overline{m}_{c}/\overline{m}_{b}$ does not lead to large
logarithms. So the summation of these logarithms which is contained in our
computation does not make an improvement, and the agreement with Ref.
Ayala:2014yxa  simply represents a computational cross check of both
calculations. The scale uncertainty is larger than the one shown in Ref.
Ayala:2014yxa  because we considered variations of the renormalization scale
down to $\mu=0.6\,\overline{m}_{c}$, which were not considered by them, and
because we do not attempt to eliminate the strong correlation in scale-
dependence between $\Delta m^{(4)}(\overline{m}_{b},\overline{m}_{c})$ and
$\Delta m^{(3)}(\overline{m}_{b},\overline{m}_{c})$ from these low scales
here.

Let us now investigate the case of the bottom quark mass corrections in the
top quark pole-$\overline{\mathrm{MS}}$ mass relation assuming a massless
charm quark. We can simply adapt Eq. (58) through trivial modifications and
obtain the relation

$\displaystyle m^{\mathrm{pole}}_{t}-$
$\displaystyle\bigg{[}\,\overline{m}_{t}+\overline{m}_{t}\sum_{n=1}^{\infty}a_{n}(n_{\ell}=4,0)\left(\frac{\alpha_{s}^{(4)}(\overline{m}_{t})}{4\pi}\right)^{n}\,\bigg{]}$
(60) $\displaystyle=$ $\displaystyle\,\,\Delta m_{t}^{(6\to
5)}(\overline{m}_{t})+\Delta m^{(5)}(\overline{m}_{t},\overline{m}_{b})+\delta
m_{b,c}^{(t\to b)}(\overline{m}_{b},0)+\Delta m_{b}^{(5\to
4)}(\overline{m}_{b})$ $\displaystyle-\Delta
m^{(4)}(\overline{m}_{t},\overline{m}_{b})$ $\displaystyle=$
$\displaystyle\,\,(0.032\pm 0.001)+(9.331\pm 0.016)+(0.005\pm 0.001)+(0.004\pm
0.001)$ $\displaystyle\,\,-(9.114\pm 0.014)\;{\rm GeV}$ $\displaystyle=$
$\displaystyle\,\,0.258\pm 0.021\;{\rm GeV}\,.$

We see that using the approximation of an infinitely heavy bottom quark for a
calculation of the bottom mass effects in the top quark
pole-$\overline{\mathrm{MS}}$ mass relation gives a result that is about $260$
MeV too small.

We can now go one step further and also consider the case where the masses of
both the bottom and charm quark are accounted for. Generalizing the previous
two calculations to this case is straightforward and we obtain

$\displaystyle m^{\mathrm{pole}}_{t}-$
$\displaystyle\bigg{[}\,\overline{m}_{t}+\overline{m}_{t}\sum_{n=1}^{\infty}a_{n}(n_{\ell}=3,0)\left(\frac{\alpha_{s}^{(3)}(\overline{m}_{t})}{4\pi}\right)^{n}\,\bigg{]}$
(61) $\displaystyle=$ $\displaystyle\,\,\Delta m_{t}^{(6\to
5)}(\overline{m}_{t})+\Delta m^{(5)}(\overline{m}_{t},\overline{m}_{b})+\delta
m_{b,c}^{(t\to b)}(\overline{m}_{b},\overline{m}_{c})$ $\displaystyle+\Delta
m_{b}^{(5\to 4)}(\overline{m}_{b})+\Delta
m^{(4)}(\overline{m}_{b},\overline{m}_{c})+\delta m_{c}^{(b\to
c)}(\overline{m}_{c})+\Delta m_{c}^{(4\to 3)}(\overline{m}_{c})$
$\displaystyle-\Delta m^{(3)}(\overline{m}_{t},\overline{m}_{c})\,.$
$\displaystyle=$ $\displaystyle\,\,(0.032\pm 0.001)+(9.331\pm 0.016)+(0.006\pm
0.001)$ $\displaystyle\,\,+(0.004\pm 0.001)+(0.423\pm 0.017)+(0.004\pm
0.001)+(0.005\pm 0.002)$ $\displaystyle\,\,-(9.111\pm 0.032)\;{\rm GeV}$
$\displaystyle=$ $\displaystyle\,\,0.694\pm 0.040\;{\rm GeV}\,.$

In this case using the approximation of infinitely heavy bottom and charm
quarks for a calculation of the bottom and charm mass effects in the top quark
pole-$\overline{\mathrm{MS}}$ mass relation gives a result that is almost
$700$ MeV too small.

Our results show that the approximation of computing the lighter heavy flavor
mass corrections in a theory where these heavy flavors are decoupled is an
excellent approximation for the charm mass corrections in the bottom quark
pole mass, but it is considerably worse for the top quark, where the
discrepancy even reaches the $1$ GeV level. The reason is that the decoupling
limit can in general not capture the true size of the lighter quark mass
effects if the hierarchy of scales is large. One should therefore not use this
approximation to determine bottom or charm quark mass effects for the top
quark.

##  4 The Top Quark Pole Mass Ambiguity

Figure 5:  Top quark pole mass as a function of order obtained from the MSR
mass $m_{t}^{\rm MSR}(\overline{m}_{t})$ (black) and
$\overline{m}_{t}=\overline{m}_{t}(\overline{m}_{t})=163$ GeV (gray) for
massless bottom and charm quarks. The central dots refer to the
renormalization scale $\mu=\overline{m}_{t}$ for the strong coupling. The
error bars arise from renormalization scale variation
$\overline{m}_{t}/2\leq\mu\leq 2\,\overline{m}_{t}$. The gray horizontal band
represents the region $m_{t}^{\rm pole}=173.10\pm 0.07$, which indicates the
top quark pole mass and its scale uncertainty obtained from $m_{t}^{\rm
MSR}(\overline{m}_{t})$ at the 8th order.

###  4.1 General Comments and Estimation Method

In this section we address the question of the best possible approximation and
the ambiguity of the top quark pole mass $m_{t}^{\rm pole}$ using the RG
formalism for the top mass described in the earlier sections. As a reminder
and for illustration we show in Fig. 5 $m^{\mathrm{pole}}_{t}$ as a function
of the order obtained from the series for $m^{\mathrm{pole}}_{t}-m^{\rm
MSR}_{t}(\overline{m}_{t})$ in powers of $\alpha_{s}^{(5)}$ given in Eq. (2.2)
for massless bottom and charm quarks, where the central dots are obtained for
the default choice of renormalization scale $\mu=\overline{m}_{t}$ in the
strong coupling and the error bars represent the scale variation
$\overline{m}_{t}/2\leq\mu\leq 2\,\overline{m}_{t}$. The corresponding results
from the series for $m^{\mathrm{pole}}_{t}-\overline{m}_{t}$ given in Eq. (9)
in powers of $\alpha_{s}^{(6)}$, also for massless bottom and charm quarks,
are shown in gray. We have used the asymptotic form of the perturbative
coefficients shown in Tab. 2 for the series coefficients beyond
$\mathcal{O}(\alpha_{s}^{4})$333The uncertainties of the normalization factors
$N_{1/2}^{(n_{Q})}$ are about an order of magnitude smaller than the
renormalization scale variation of the series beyond ${\cal
O}(\alpha_{s}^{4})$ and therefore not significant for our analysis.. We note
that focusing on the approximation of massless bottom and charm quarks by
itself is phenomenologically valuable because it is employed for most current
predictions in the context of top quark physics, and since the analytic
expressions are most transparent for this case.

The graphics illustrates visually the problematic features associated to the
top quark ${\cal O}(\Lambda_{\mathrm{QCD}})$ pole mass renormalon, and in
particular the specific properties of the series for $\mu\sim m_{t}$ already
mentioned in Sec. 1: The minimal term of the series is obtained at order
$n_{\rm min}=8$, which according to the theory of asymptotic series is the
order that provides the best possible approximation for the top quark pole
mass. Furthermore, the corrections are numerically close to the eighth order
correction for the orders in the range 6 to 10, i.e. $\Delta n\approx 5$, for
which the partially summed series increases linearly with the order. According
to the theory of asymptotic series it is this region of orders that is
relevant for the size of the principle uncertainty of this best approximation.
We also see two very important practical issues appearing already at lower
orders which can make dealing with the pole mass in mass determinations
difficult: First, the higher order corrections are much larger than indicated
by usual renormalization scale variations of the lower order prediction and,
second, the common renormalization scale variation at any given truncation
order is not an appropriate tool to estimate the perturbative uncertainty. In
this context it is easy to understand that specifying a concrete numerical
value for the principle uncertainty of the top quark pole mass is non-trivial
even if the series is known precisely to all orders. So to obtain a top quark
pole mass determination with uncertainties close to the principle uncertainty
within a phenomenological analysis based on a usual truncated finite order
calculation may be quite difficult. As a comparison let us recall the much
better perturbative behavior of a series that is free of an ${\cal
O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity such as the MSR mass
differences $\Delta m^{(n_{Q})}(R,R^{\prime})$ of Eq. (21) with numerical
evaluations given in Tab. 1.

Prior to this work the issue of the best possible estimate and the ambiguity
of the top quark pole mass were already studied in Ref. Beneke:2016cbu . They
examined the pole-$\overline{\mathrm{MS}}$ mass relation of Eq. (9) for
massless bottom and charm quarks (i.e. $n_{Q}=n_{t}=n_{\ell}=5$) and their
analysis addressed the numerical uncertainty of the top quark pole mass
accounting for all series terms displayed in Fig. 5 for
$\mu=\overline{m}_{t}$. They adopted a prescription given in Ref.
Beneke:1998ui , which defined the top quark pole mass uncertainty as the
imaginary part of the inverse Borel integral of Eq. (2.3), $\Delta m_{\rm
Borel}^{(n_{\ell}=5)}$, divided by $\pi$, which gives about $65$ MeV. Since
this agrees in size with the minimal series term444 In Ref. Beneke:1998ui  the
order of the minimal series term $n_{\rm min}$ and the size of the minimal
term $\Delta(n_{\rm min})$ were not chosen from the set of the actual series
terms but computed from the minimum of a quadratic fit to the series terms in
the vicinity of the minimum, so that their $n_{\rm min}$ was a non-integer
value and their $\Delta(n_{\rm min})$ value is slightly smaller than the
minimal term in the series. There are neither practical nor conceptual
advantages of this procedure, and the numerical results are unchanged within
their errors if $\Delta(n_{\rm min})$ is taken as the minimal terms in the
series. , which arises at order $\alpha_{s}^{8}$, they argued that $\Delta
m_{\rm Borel}^{(n_{\ell}=5)}/\pi$ (or the size of the minimal term) is a
reliable quantification of the top quark pole mass ambiguity, which they
finally specified as $70$ MeV. Interpreting the specification like a numerical
uncertainty, this gives $m_{t}^{\rm pole}=173.10\pm 0.07$, which is shown in
Fig. 5 as the thin gray horizontal band. The uncertainty band is about the
same size as the renormalization scale variation of the series truncated at
the eighth order.

We believe that quoting $70$ MeV for the top quark pole mass ambiguity for
massless bottom and charm quarks is too optimistic. Given (i) the overall bad
behavior of the series, (ii) that there is a sizable range of orders where the
corrections have very similar size and (iii) that the partially summed series
increases linearly with the order in the range $6$ to $10$ ($\Delta n\approx
5$), we see no compelling reason to truncate precisely at the order $n_{\rm
min}=8$ and to quote a number at the level of the scale variation of the
truncated series or the size of the correction at this order as the principle
uncertainty. Our view is also supported by heavy quark symmetry (HQS)
Isgur:1989vq  which states that the pole mass ambiguity is independent of the
mass of the heavy quark up to power corrections of ${\cal O}(\Lambda_{\rm
QCD}^{2}/m_{Q})$. This is the first aspect following from HQS we discussed in
Sec. 1. HQS requires that the criteria and the outcome of the method used to
determine the top quark pole mass ambiguity are independent of the top mass
value (as long as it is sufficiently bigger than $\Lambda_{\rm QCD}$). So it
is straightforward to carry out a test concerning HQS by changing the value of
$\overline{m}_{t}$ while keeping $\mu/\overline{m}_{t}=1$ and checking whether
the approach to estimate the ambiguity provides stable results.

Concerning Ref. Beneke:2016cbu  this check is best carried out in the five-
flavor scheme for the strong coupling, and we therefore evaluate the size of
the minimal term in the series for $m_{t}^{\rm pole}-m_{t}^{\rm
MSR}(\overline{m}_{t})$. Adopting the values $163$, $20$, $4.2$, $2$ and $1.3$
GeV for $\overline{m}_{t}$ we obtain $62$, $75$, $91$, $113$ and $131$ MeV for
the minimal term $\Delta(n_{\rm min})$. This behavior is roughly described by
the approximate formula $\Delta(n_{\rm
min})\approx(4\pi\alpha_{s}^{(n_{\ell}=5)}(\mu)/\beta_{0}^{(n_{\ell}=5)})^{1/2}\Lambda_{\rm
QCD}^{(n_{\ell}=5)}$, already mentioned in Sec. 1 and shows that the basic
dependence on $\mu$ is logarithmic. We can even render the minimal term
arbitrarily small if we adopt for $\overline{m}_{t}$ values much larger than
$163$ GeV. We see that $\Delta m_{\rm Borel}^{(n_{\ell}=5)}/\pi$, which is
independent of the top mass value and therefore proportional to the ambiguity,
agrees with the size of the minimal term only for $\mu\sim 163$ GeV, but
disagrees for other choices. So the line of reasoning used for the analysis of
the top quark pole mass ambiguity in Ref. Beneke:2016cbu  is not independent
of the top quark mass value, and one has to conclude that the ambiguity must
be larger than $\Delta m_{\rm Borel}^{(n_{\ell}=5)}/\pi$ and certainly larger
than $130$ MeV, which is the size of the minimal term for a very small value
of $\overline{m}_{t}$. Concerning the quoted numbers, we emphasize that we
still discuss the case of massless bottom and charm quarks. From the relation
$\Delta n\times\Delta(n_{\rm min})\propto\pi^{2}\Lambda_{\rm
QCD}^{(n_{\ell})}/\beta_{0}\propto\Delta m_{\rm Borel}$ we see in particular
that a reliable method consistent with HQS has to explicitly account for the
range $n_{\rm min}\pm\Delta n/2$ in orders for which the terms in the series
have values close to $\Delta(n_{\rm min})$. We stress that the latter issue is
not at all new and has been known since the work of Refs. Bigi:1994em ;
Beneke:1994sw . It was also argued in Beneke:2016cbu  that their approach to
estimate the size of the top quark pole mass ambiguity is consistent
concerning that issue. However, their approach did not account for the actual
size of $\Delta n$, which is about $5$ for the case discussed in
Beneke:2016cbu  and also shown in Fig. 5.

In the following subsections we apply a method to determine the best possible
estimate and the ambiguity of the top quark pole mass which explicitly
accounts for the range $n_{\rm min}\pm\Delta n/2$ in orders where the
$\Delta(n)$ are very close to $\Delta(n_{\rm min})$. It also accounts for the
practical problems in an order-by-order determination of the pole mass from a
series containing the ${\cal O}(\Lambda_{\rm QCD})$ renormalon which we
discussed above in the context of Fig. 5. To describe the method we define,
for a given series to calculate the top quark pole mass,

$\Delta(n)\equiv m^{\mathrm{pole}}_{t}(n)-m^{\mathrm{pole}}_{t}(n-1)\,,$ (62)

where $m^{\mathrm{pole}}_{t}(n)$ is the partial sum at ${\cal
O}(\alpha_{s}^{n})$ of the series for the top quark pole mass that contains
the ${\cal O}(\Lambda_{\mathrm{QCD}})$ pole mass renormalon, and thus
$\Delta(n)$ is the $n$-th order correction. The method we use is as follows:

  1. 1.

We determine the minimal term $\Delta(n_{\rm min})$ and the set of orders
$\\{n\\}_{f}\equiv\\{n:\,\Delta(n)\leq f\,\Delta(n_{\rm min})\\}$ in the
series for a default renormalization scale, where $f$ is a number larger but
close to unity.

  2. 2.

We use half of the range of values covered by $m^{\mathrm{pole}}_{t}(n)$ with
$n\in\\{n\\}_{f}$ evaluated for this setup and include renormalization scale
variation in a given range as an estimate for the ambiguity of the top quark
mass. We use the midpoint of the covered range as the central value.

While $n_{\rm min}$, $\Delta(n_{\rm min})$ and $\Delta n$ each can vary
substantially depending on which setup one uses to determine $m_{t}^{\rm
pole}$, the method provides results that are setup-independent and is
therefore consistent with HQS. Through the RG formalism we developed in the
previous sections we can explicitly implement the other important requirement
of HQS, namely that the ambiguities of the pole masses of all heavy quarks
agree. To do this we apply our method for three different scenarios which
differ on whether the bottom and charm quarks are treated as massive or
massless and we furthermore study the pole-MSR mass difference for different
values of $R$.

###  4.2 Massless Bottom and Charm Quarks

For the case that the bottom and charm quarks are treated as massless we can
calculate the top quark pole mass from the top MSR mass $m^{\rm MSR}_{t}(R)$
at different scales $R\leq\overline{m}_{t}$. Using the
$\overline{\mathrm{MS}}$-MSR mass matching contribution $\Delta
m_{t}^{(6\rightarrow 5)}(\overline{m}_{t})$ of Eq. (33) and R-evolution from
the scale $\overline{m}_{t}$ to $R$ of Eq. (21) with $n_{t}=5$ active
dynamical flavors one can write the top quark pole mass as

$m^{\mathrm{pole}}_{t}=\overline{m}_{t}+\Delta m_{t}^{(6\rightarrow
5)}(\overline{m}_{t})+\Delta
m^{(5)}(\overline{m}_{t},R)+R\,\sum_{n=1}^{\infty}a_{n}(n_{\ell}=5,0)\left(\frac{\alpha_{s}^{(5)}(R)}{4\pi}\right)^{n}\,,$
(63)

where the sum of the second and third term on the RHS is just $m^{\rm
MSR}_{t}(R)-\overline{m}_{t}$. The terms $\Delta m_{t}^{(6\rightarrow
5)}(\overline{m}_{t})$ and $\Delta m^{(5)}(\overline{m}_{t},R)$ are free of an
${\cal O}(\Lambda_{\mathrm{QCD}})$ renormalon ambiguity and can be evaluated
to the highest order given in Tabs. 1 and 3. We can then determine the best
estimate of the top quark pole mass and its ${\cal O}(\Lambda_{\mathrm{QCD}})$
renormalon ambiguity from the $R$-dependent series which is just equal to
$m^{\mathrm{pole}}_{t}-m^{\rm MSR}_{t}(R)$. The outcome of the analysis using
the method described in Sec. 4.1 for $\overline{m}_{t}=163$ GeV and
$R=163,20,4.2$ and $1.3$ GeV and $f=5/4$ is shown in the upper section of Tab.
5.

$\overline{m}_{t}=163\,{\rm
GeV},\quad\overline{m}_{b}=\overline{m}_{c}=0\,{\rm GeV},\quad
n_{\ell}=n_{t}=5$
---
$R$ |  $m^{\rm MSR}_{t}(R)-\overline{m}_{t}$ |  $n_{\rm min}$ |  $\Delta(n_{\rm min})$ |  $\sum_{n=5}^{n_{\rm min}}\Delta(n)$ |  $\\{n\\}_{5/4}$ |  $m^{\mathrm{pole}}_{t}\qquad\quad$ $-\;m^{\rm MSR}_{t}(R)$ |  $m^{\mathrm{pole}}_{t}$
$163$ |  $0.032(1)\hphantom{0}$ |  $8$ |  $0.062(3)$ |  $0.310(17)$ |  $\\{6,7,8,9\\}$ |  $10.054(157)$ |  $173.086(157)$
$20$ |  $8.038(9)\hphantom{0}$ |  $6$ |  $0.075(4)$ |  $0.150(8)\hphantom{0}$ |  $\\{5,6,7\\}$ |  $\hphantom{0}2.140(166)$ |  $173.178(166)$
$4.2$ |  $9.363(16)$ |  $4$ |  $0.091\hphantom{(0)}$ |  $0$ |  $\\{3,4,5\\}$ |  $\hphantom{0}0.832(217)$ |  $173.195(218)$
$1.3$ |  $9.748(23)$ |  $3$ |  $0.098\hphantom{(0)}$ |  $0$ |  $\\{2,3,4\\}$ |  $\hphantom{0}0.394(186)$ |  $173.142(187)$
$\overline{m}_{t}=163\,{\rm GeV},\quad\overline{m}_{b}=4.2\,{\rm
GeV},\quad\overline{m}_{c}=0\,{\rm GeV},\quad n_{\ell}=n_{t}-1=4$
$R$ |  $m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{b}\hskip 19.91684pt$ $+\;m^{\rm MSR}_{b}(R)-\overline{m}_{t}$ |  $n_{\rm min}$ |  $\Delta(n_{\rm min})$ |  $\sum_{n=5}^{n_{\rm min}}\Delta(n)$ |  $\\{n\\}_{5/4}$ |  $m^{\mathrm{pole}}_{b}\qquad\quad$ $-\;m^{\rm MSR}_{b}(R)$ |  $m^{\mathrm{pole}}_{t}$
$163$ |  $0.258(21)$ |  $7$ |  $0.087(3)$ |  $0.324(11)$ |  $\\{6,7,8,9\\}$ |  $9.904(227)$ |  $173.162(228)$
$20$ |  $8.035(17)$ |  $5$ |  $0.104(3)$ |  $0.104(3)$ |  $\\{4,5,6\\}$ |  $2.120(211)$ |  $173.155(212)$
$4.2$ |  $9.372(16)$ |  $4$ |  $0.135$ |  $0$ |  $\\{3,4\\}$ |  $0.855(211)$ |  $173.227(212)$
$1.3$ |  $9.795(23)$ |  $2$ |  $0.124$ |  $0$ |  $\\{1,2,3\\}$ |  $0.331(214)$ |  $173.126(215)$
$\overline{m}_{t}=163\,{\rm GeV},\quad\overline{m}_{b}=4.2\,{\rm
GeV},\quad\overline{m}_{c}=1.3\,{\rm GeV},\quad n_{\ell}=n_{t}-2=3$
$R$ |  $m^{\mathrm{pole}}_{t}-m^{\mathrm{pole}}_{c}\hskip 19.91684pt$ $+\;m^{\rm MSR}_{c}(R)-\overline{m}_{t}$ |  $n_{\rm min}$ |  $\Delta(n_{\rm min})$ |  $\sum_{n=5}^{n_{\rm min}}\Delta(n)$ |  $\\{n\\}_{5/4}$ |  $m^{\mathrm{pole}}_{c}\qquad\quad$ $-\;m^{\rm MSR}_{c}(R)$ |  $m^{\mathrm{pole}}_{t}$
$163$ |  $0.694(40)$ |  $7$ |  $0.098(2)$ |  $0.355(8)$ |  $\\{6,7,8,9\\}$ |  $9.471(260)$ |  $173.165(263)$
$20$ |  $8.076(33)$ |  $5$ |  $0.116(3)$ |  $0.116(3)$ |  $\\{4,5,6\\}$ |  $2.085(243)$ |  $173.161(245)$
$4.2$ |  $9.371(31)$ |  $3$ |  $0.154$ |  $0$ |  $\\{3,4\\}$ |  $0.888(257)$ |  $173.259(259)$
$1.3$ |  $9.805(24)$ |  $2$ |  $0.128$ |  $0$ |  $\\{1,2,3\\}$ |  $0.354(243)$ |  $173.159(244)$
Table 5:  Details of the numerical results of our method to determine
$m_{t}^{\rm pole}$ for the cases of massless bottom and charm quarks (upper
section), massless charm quarks (middle section) and finite bottom and charm
quarks (lower section) and exploring different setups to determine $m_{t}^{\rm
pole}$. The final respective results for $m_{t}^{\rm pole}$ are shown in the
last column. See the text for details. All numbers for masses and mass
differences are in units of GeV. Errors are quoted in parentheses.

The entries are as follows: The second column shows $m^{\rm
MSR}_{t}(R)-\overline{m}_{t}=\Delta m_{t}^{(6\rightarrow
5)}(\overline{m}_{t})+\Delta m^{(5)}(\overline{m}_{t},R)$ at the highest
order. The third and fourth column show the order $n_{\rm min}$ and
$\Delta(n_{\rm min})$ for the default renormalization scale $\mu=R$ for the
cases $R=163,20$ and $4.2$ GeV and $\mu=2\overline{m}_{c}$ for $R=1.3$ GeV.
The values for $\Delta(n_{\rm min})$ for $R=163$ and $20$ GeV have an
uncertainty because for these cases $n_{\rm min}>4$ and the values for
$\Delta(n>4)$ are determined from the asymptotic large order values given in
Tab. 2 which have a numerical uncertainty from the normalization factor
$N_{1/2}^{(5)}$ in Eqs. (2.3). The fifth column shows the sum of the
perturbative corrections beyond the explicitly calculated ${\cal
O}(\alpha_{s}^{4})$ terms up to order $n_{\rm min}$ showing the amount of
extrapolation needed to obtain the best possible top quark mass based on the
asymptotic approximation. The sixth column shows the set of orders
$\\{n\\}_{f=5/4}$ for which $\Delta(n)\leq f\,\Delta(n_{\rm min})$ and which
are used for determining the best estimate and the uncertainty of the top
quark pole mass. The seventh column then contains the best estimate and the
ambiguity of the series for $m^{\mathrm{pole}}_{t}-m^{\rm MSR}_{t}(R)$ using
the method from Sec. 4.1. To obtain the uncertainties we used renormalization
scale variation for $\alpha_{s}^{(5)}(\mu)$ in the range $R/2\leq\mu\leq 2\,R$
for the cases $R=163,20,4.2$ GeV and in the range $1.5\,{\rm GeV}\leq\mu\leq
5$ GeV for $R=1.3$ GeV. For $R=1.3$ GeV we always use renormalization scales
$\mu$ of the strong coupling that are larger than $1.5$ GeV because the
dependence on the renormalization scale grows rapidly for smaller scales. The
last column contains the final result for $m^{\mathrm{pole}}_{t}$ combining
the results for $m^{\rm MSR}_{t}(R)-\overline{m}_{t}$ and
$m^{\mathrm{pole}}_{t}-m^{\rm MSR}_{t}(R)$ where the uncertainties of both are
added quadratically to give the final number for the ambiguity of $m_{t}^{\rm
pole}$. These results are also displayed graphically in Figs. 6-6 as the gray
hatched horizontal bands.

Figure 6:  Top quark pole mass $m_{t}^{\rm pole}$ as a function of order
taking $\overline{m}_{t}=\overline{m}_{t}(\overline{m}_{t})=163$ GeV as input
and using different methods to obtain the best possible estimate and the
ambiguity. The central dots are obtained for the default renormalization
scales for the strong coupling and the error bands represent the scale
variation as explained in the text. The light colored hatched horizontal bands
bounded by equal colored lines show the best possible estimate for the
respective method also given in the last column in Tab. 5. All results
obtained for massless bottom and charm quarks are in black, all results for
$(\overline{m}_{b},\overline{m}_{c})=(4.2,0)$ GeV are in red, all results for
$(\overline{m}_{b},\overline{m}_{c})=(4.2,1.3)$ GeV are in blue. Panel (a)
shows results for $R=163$ GeV, panel (b) for $R=20$ GeV, panel (c) for $R=4.2$
GeV and panel (d) for $R=1.3$ GeV.

In Figs. 6 we have also shown in black the results for
$m^{\mathrm{pole}}_{t}(n)$ over the order $n$ for the different setups where
the dots are the results for the default renormalization scales that are used
to determine $n_{\rm min}$, $\Delta(n_{\rm min})$ and $\\{n\\}_{f}$. The error
bars represent the range of values at each order of the truncated series
coming from the variations of the renormalization scale of the strong
coupling. The black dot at $n=0$ visible in Figs. 6, 6 shows the highest order
result for $m^{\rm MSR}_{t}(R)$.

We see that the results for the top quark pole mass $m_{t}^{\rm pole}$ for the
different $R$ values are fully compatible to each other. In particular, the
ambiguity estimates based on our method agree within $\pm 15\%$ and average to
$182$ MeV. Furthermore, the central values for the best estimates vary by at
most $110$ MeV and average to $173.150$ GeV. It is reassuring that the spread
of the central values is smaller than the size of the ambiguity. We emphasize
that the consistency of our results for the different $R$ values to each other
cannot be interpreted in any way statistically since the analyses for
different $R$ values are not theoretically independent. The agreement just
shows that our method is consistent since the best estimate (and also the
ambiguity) of the top quark pole mass is independent of $R$. Interestingly our
estimate for the ambiguity of the top quark pole mass agrees quite well with
$\Lambda_{\mathrm{QCD}}^{(n_{\ell}=5)}=166$ MeV given in Eq. (29).

As already pointed out in Sec. 4.1, the minimal correction $\Delta(n_{\rm
min})$ increases from around $60$ MeV for $R=163$ GeV to about $100$ MeV555
This number is obtained for the default renormalization scale
$\mu=2\,\overline{m}_{c}=2.6$ GeV. In the short analysis of Sec. 4.1 we quoted
$131$ MeV for the size of the minimal term for $R=1.3$ GeV, which was obtained
for $\mu=1.3$ GeV. for $R=1.3$ GeV. At the same time, the order $n_{\rm min}$
where the minimal correction $\Delta(n_{\rm min})$ arises decreases from
$n_{\rm min}=8$ at $R=\overline{m}_{t}$ down to $n_{\rm min}=4$ and $3$ for
$R=4.2$ and $1.3$ GeV. Moreover, the contribution in the best estimate for
$m_{t}^{\rm pole}$ from orders beyond $n=4$ until order $n_{\rm min}$
decreases from about $310$ MeV at $R=\overline{m}_{t}$ to about $150$ MeV at
$R=20$ GeV. For $R$ scales around the bottom quark mass and below, where
$n_{\rm min}\leq 4$, there is no need any more to extrapolate beyond the
explicitly calculated four orders to get the best value for $m_{t}^{\rm
pole}$. This information is not just of academic importance but it is also
relevant for phenomenology: The MSR mass $m_{t}^{\rm MSR}(R)$ for some low
scale $R$ can serve as a low-scale short-distance mass for a physical
application where the characteristic physical scale is $R$. Typical examples
include the top pair inclusive cross section at the production threshold where
$R\sim m_{t}\alpha_{s}\sim 25$ GeV Hoang:2000yr , or the reconstructed
invariant top quark mass distribution where $R$ is in the range of $5$ to $10$
GeV Fleming:2007qr ; Butenschoen:2016lpz ; Hoang:2017kmk . The behavior of the
series for $m_{t}^{\rm pole}-m_{t}^{\rm MSR}(R)$ thus reflects the typical
behavior of the QCD corrections to the mass for the respective physical
applications. The observations we make for the $R$-dependence of the behavior
of the series show that the best possible determination of the top quark mass
from an observable characterized by a low characteristic physical scale can in
general be achieved at a lower order and also involves smaller perturbative
corrections compared to an observable characterized by high characteristic
physical scales (such as inclusive top pair cross sections at high energies or
virtual top quark effects). This general property is also reflected visually
in the graphical illustrations shown in Fig. 6.

We note that our numerical analysis has a rather weak overall dependence on
the choice of $f$ and that the results change by construction in a non-
continuous way. Using $f=4/3$ only the outcome for $R=20$ GeV is modified to