# Improved analysis of the decay width of $t\to Wb$ up to N3LO QCD corrections

Jiang Yan<EMAIL_ADDRESS>Department of Physics, Chongqing Key Laboratory
for Strongly Coupled Physics, Chongqing University, Chongqing 401331, P.R.
China  Xing-Gang Wu<EMAIL_ADDRESS>Department of Physics, Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, P.R. China  Hua Zhou<EMAIL_ADDRESS>School of Science,
Southwest University of Science and Technology, Mianyang 621010, P.R. China
Hong-Tai Li<EMAIL_ADDRESS>Department of Physics, Chongqing Key Laboratory
for Strongly Coupled Physics, Chongqing University, Chongqing 401331, P.R.
China  Jing-Hao Shan<EMAIL_ADDRESS>Department of Physics, Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, P.R. China

###### Abstract

In this paper, we analyze the top-quark decay $t\to Wb$ up to next-to-next-to-
next-to-leading order (N3LO) QCD corrections. For the purpose, we first adopt
the principle of maximum conformality (PMC) to deal with the initial pQCD
series, and then we adopt the Bayesian analysis approach, which quantifies the
unknown higher-order (UHO) terms’ contributions in terms of a probability
distribution, to estimate the possible magnitude of the N4LO QCD corrections.
In our calculation, an effective strong coupling constant $\alpha_{s}(Q_{*})$
is determined by using all non-conformal $\\{\beta_{i}\\}$ terms associated
with the renormalization group equation. This results in the next-to-leading-
log PMC scale $Q_{*}^{(\rm NLL)}=10.3048$ GeV, which could be regarded as the
correct momentum flow of the process. Consequently, we obtain an improved
scale invariant pQCD prediction for the top-quark decay width $\Gamma_{t}^{\rm
tot}=1.3172\pm 0.0038$ GeV, where the error is the squared average of the
uncertainties from the decay width of $W$-boson $\Delta\Gamma_{W}=\pm 0.042$
GeV, the coupling constant $\Delta\alpha_{s}(m_{Z})=\pm 0.0009$ and the
predicted N4LO-level UHO-terms. The magnitude of the top-quark pole mass
greatly affects the total decay width. By further taking the PDG top-quark
pole mass error from cross-section measurements $\Delta m_{t}=\pm 0.7$ GeV
into consideration, we obtain $\Gamma_{t}^{\rm
tot}=1.3172^{+0.0194}_{-0.0192}$ GeV.

##  I Introduction

The top quark is the heaviest elementary particle in the Standard Model (SM),
and it is remarkable for its decay processes. It differs from the other
quarks, which has a much larger mass and a quite shorter lifetime that
precludes hadronization. Its substantial Yukawa coupling to the Higgs boson
give it considerable influence on precision observables of the SM at lower
scales. Furthermore, it can be an exceptional laboratory for probing
fundamental interactions at the Electroweak (EW) symmetry-breaking scale and
beyond.

Within the SM, the top quark decays almost exclusively into a $W$-boson and a
$b$-quark. Thus the top-quark total width, which is inversely proportional to
its lifetime, can be deduced from the partial decay width $\Gamma(t\to Wb)$
and the branching fraction $\mathcal{B}(t\to Wb)$. In 2012, using 5.4 fb-1 of
integrated luminosity collected by the D0 Collaboration at the Tevatron
$p\bar{p}$ Collider, $\Gamma_{t}=2.00_{-0.43}^{+0.47}$ GeV was extracted [1].
In 2014, the CMS Collaboration provided a better determination of the total
width, $\Gamma_{t}=1.36\pm 0.02(\rm stat.)_{-0.11}^{+0.14}(\rm syst.)$ GeV [2]
under the assumption $\mathcal{B}(t\to Wq)=1$ for $q=b,s,d$, where “stat.” and
“syst.” are short notations for statistical and systematic errors,
respectively. In 2017, the initial direct measurement was conducted by an
ATLAS analysis, involving the direct fitting of reconstructed lepton$+$jets
events in 20.2 fb-1 of data at a center-of-mass energy of $\sqrt{s}=8$ TeV.
This resulted in $\Gamma_{t}=1.76\pm 0.33({\rm stat.})_{-0.68}^{+0.79}({\rm
syst.})$ GeV [3]. In 2019, a measurement by ATLAS collaboration, using 139
fb-1 of data at a center-of-mass energy of $\sqrt{s}=13$ TeV, employed a
template fit to the invariant mass of the lepton-$b$-quark in dilepton final
states. This yielded $\Gamma_{t}=1.94_{-0.49}^{+0.52}$ GeV [4]. The Particle
Data Group (PDG) reported the world average as $\Gamma(t\to
Wq)=1.42_{-0.15}^{+0.19}$ GeV and $\mathcal{B}(t\to Wb)=\Gamma(t\to
Wb)/\Gamma(t\to Wq)=0.957\pm 0.034$ [5].

The next-to-leading order (NLO) quantum chromodynamics (QCD) corrections were
first computed in Refs.[6, 7, 8, 9], while the NLO EW corrections were
provided in Refs.[10, 11]. The next-to-next-to-leading order (NNLO) QCD
corrections using the asymptotic expansion for the $t\to Wb$ decay were given
in Refs.[12, 13, 14, 15, 16], and the complete analytical results can be found
in Ref.[17]. The NNLO polarized decay rates have been calculated in Refs.[18,
19]. Recently, the next-to-next-to-next-to-leading order (N3LO) corrections in
the large-$N_{C}$ limit have been presented in Ref.[20], and the first
complete high-precision numerical results of N3LO QCD corrections have been
computed in Ref.[21]. Their results are perfectly consistent, indicating the
dominance of leading-color contributions and the high reliability of the
approximation of large-$N_{C}$ limit in this process.

Due to large value of the kinematic scale $Q\sim{\cal O}(m_{t})$ and the small
strong coupling constant $\alpha_{s}(m_{t})\sim 0.1$, the N3LO-level QCD
corrections to the $t\to Wb$ decay width exhibit good convergence. It however
still has a sizable renormalization scale dependence due to divergent
renormalon terms [22, 23, 24]. Practically, one usually selects the
renormalization scale as $\mu_{R}=m_{t}$ so as to eliminate the divergent
large logarithmic terms $\ln\left(\mu_{R}^{2}/m_{t}^{2}\right)$, then vary it
within a certain range such as $\mu_{R}\in[m_{t}/\xi,\xi m_{t}]$ with $\xi$
being usually chosen as $2$, $3$, $4$, and etc., to account for its
uncertainty. This treatment is referred to as the conventional scale-setting
approach. It is evident that this approach is arbitrary, and the perturbative
nature of the series heavily relies on the choice of $\xi$, thereby
diminishing the reliability of the final theoretical prediction. As is well-
known, the physical observable, which corresponds to an infinite-order
perturbative series, is scale invariant [25, 26, 27, 28]; and it is important
to known whether such scale invariance can also be achieved for a fixed-order
series. One cannot simply require the fixed-order series to be scale invariant
to achieve the goal, which is not only not true but also explicitly breaks
standard renormalization group invariance (RGI) [29], since it implicitly
assumes that all uncalculated higher-order terms contribute zero. Thus proper
way need to be introduced to improve the perturative series before applying
the scale-setting procedures.

Many scale-setting approaches have been suggested to improve the fixed-order
series so as to achieve a scale invariant prediction. Especially, by using the
$n_{f}$-terms as a guide, the Brodsky–Lepage–Mackenzie (BLM) approach [30]
automatically resums the corresponding gluons as well as the quark vacuum-
polarization contributions and leads to a scheme-and-scale invariant
prediction [31]. This indicates that if good matching of the expansion
coefficients with the corresponding $\alpha_{s}$ can be achieved, exactly
scheme-independent predictions can be achieved. Since the running behavior of
$\alpha_{s}$ is governed by the renormalization group equation (RGE) or the
$\beta$-function [32, 33], to deal with the $\\{\beta_{i}\\}$-terms involved
in the RGE is then more fundamental than to deal with the $n_{f}$-terms.
Lately, the BLM is developed to Principle of Maximum Conformality (PMC) [34,
36, 37, 35, 38], which perfects the idea behind BLM and offers a reliable
extension of the BLM to all orders. The BLM and PMC in the Abelian limit [39]
reduces to the well-known Gell-Mann-Low approach [40] for QED. Subsequently,
the PMC single-scale-setting approach (PMCs), as an effective alternative to
the original PMC multi-scale-setting approach, has also been proposed in
Refs.[41, 42] from two distinct but equivalent perspectives. It has been
demonstrated that the PMC prediction is independent to any choice of
renormalization scheme and scale [43], being consistent with the self-
consistency requirements of the renormalization group [44, 45]. The PMCs
approach also greatly suppresses the residual scale dependence [46] of the PMC
predictions due to unknown even higher-order terms of the pQCD series. In the
present paper, we will adopt the PMCs approach to deal with the $t\to Wb$
decay width up to N3LO QCD corrections.

For any perturbative series, there are some uncertainties caused by the
unknown higher-order terms (UHO-terms). Since the exact pQCD result is
unknown, it would be helpful to quantify the UHO-terms’ contribution in terms
of a probability distribution. Following the idea of Bayesian analysis (BA)
[47, 48, 49, 50], the conditional probability of the unknown perturbative
coefficient is first given by a subjective prior distribution, which is then
updated iteratively according to the Bayes’ theorem as more and more
information has been included. It has been found that to estimate the
contributions of the UHO-terms, the generally more convergent and scheme-and-
scale invariant PMC series provides a more reliable basis for estimating the
contributions from the UHO-terms [51, 52]. In the present paper, we will adopt
the BA approach to estimate the contributions from the unknown N4LO-terms of
the PMC series.

The remaining parts of the paper are organized as follows. Sec.II gives the
formulas for the top-quark decay process, $t\to Wb$, up to N3LO QCD
corrections, and then shows how to apply the PMC scale-setting procedures to
the present process. Sec.III presents numerical results and discussions for
the top-quark decay. Sec.IV is reserved for a summary.

##  II calculation technology

By neglecting the corrections caused by the finite $b$-quark mass and the off-
shell $W$-boson effect, the total decay width of the top-quark decay $t\to Wb$
up to N3LO QCD corrections can be expressed as

$\displaystyle\Gamma_{t\to Wb}^{\rm QCD}=\frac{G_{F}m_{t}^{3}|V_{\rm
tb}|^{2}}{8\sqrt{2}\pi}\sum_{i=0}^{3}f_{i}(\omega;\mu_{R})\alpha_{s}^{i}(\mu_{R}),$
(1)

where $\mu_{R}$ is the renormalization scale, $G_{F}$ is the Fermi constant,
$m_{t}$ is the top-quark pole mass, $V_{\rm tb}$ is the Cabibbo-Kobayashi-
Maskawa (CKM) matrix element, and $\omega=m_{W}^{2}/m_{t}^{2}$. The LO decay
width that is caused by weak interaction provides dominant contribution to the
total decay width. Because the $W$-boson will decay promptly into leptons or
quarks, we rewrite the top-quark decay width as

$\displaystyle\Gamma_{t\to W^{*}b}^{\rm QCD}=\Gamma_{\rm
LO}\left[1+R_{t}(\mu_{R})\right],$ (2)

where $W^{*}$ indicates the $W$-boson may be off-shell and the QCD corrections

$\displaystyle
R_{t}(\mu_{R})=\sum_{i=1}^{3}r_{i}(\mu_{R})\alpha_{s}^{i}(\mu_{R})+{\cal
O}(\alpha_{s}^{4}),$ (3)

where for the coefficients $r_{i}$ ($i\in[1,3]$), we have [6, 17]

$\displaystyle r_{i}(\mu_{R})=\frac{1}{\Gamma_{\rm
LO}}\frac{G_{F}m_{t}^{3}|V_{\rm
tb}|^{2}}{8\sqrt{2}\pi}\frac{\omega\gamma}{\pi}\int_{0}^{\bar{\varepsilon}}\frac{f_{i}(\varepsilon,x;\mu_{R})}{(x-\omega)^{2}+\omega^{2}\gamma^{2}}\,{\rm
d}x.$ (4)

Here $\varepsilon=m_{b}^{2}/m_{t}^{2}$,
$\bar{\varepsilon}=(1-m_{b}/m_{t})^{2}$ and $\gamma=\Gamma_{W}/m_{W}$ with
$\Gamma_{W}$ being the $W$-boson total decay width. The LO decay width is
scale invariant and equals to

$\displaystyle\Gamma_{\rm LO}=\frac{G_{F}m_{t}^{3}|V_{\rm
tb}|^{2}}{8\sqrt{2}\pi}\frac{\omega\gamma}{\pi}\int_{0}^{\bar{\varepsilon}}\frac{f_{0}(\varepsilon,x)}{(x-\omega)^{2}+\omega^{2}\gamma^{2}}\,{\rm
d}x.$ (5)

To get the analytic expression, the integration involving the polylogarithm
functions can be accomplished with the assistance of the PolyLogTools [53] and
GiNaC [54] packages. In Eqs.(4, 5), the finite $b$-quark mass contributions
have been included, e.g. the coefficients $f_{i}(\omega;\mu_{R})$ in Eq.(2)
have been replaced by $f_{i}(\varepsilon,\omega;\mu_{R})$ and the upper limit
of the integral in Eqs.(4, 5) becomes $\bar{\varepsilon}$ instead of $1$.
Currently, the finite $b$-quark mass effect is known only up to NLO accuracy
[6]. For convenience, we list the analytic coefficients $f_{0}$ and $f_{1}$ at
the scale $\mu_{R}=m_{t}$, including and excluding $b$-quark mass effects, in
the Appendix. The analytic form of the coefficients $f_{2}$ and $f_{3}$ can be
found in Refs.[17, 20], where $f_{3}$ contains only the most dominant leading
color contribution that give the most dominant contributions.

Following the standard PMC single-scale setting procedures [41, 42], by using
the QCD degeneracy relations among different orders [55], one can distribute
the original $n_{f}$-series into conformal and non-conformal terms, and then
use the RGE-involved non-conformal $\\{\beta_{i}\\}$-terms to determine an
overall effective running coupling $\alpha_{s}(Q_{*})$ for $t\to Wb$ decay,
where $Q_{*}$ is referred to as the PMC scale, which can be viewed as the
effective momentum flow for the process. More explicitly, the coefficients
$r_{i}$ in Eq.(2) can be parameterized as follows:

$\displaystyle r_{1}$ $\displaystyle=r_{1,0},$ (6) $\displaystyle r_{2}$
$\displaystyle=r_{2,0}+\beta_{0}r_{2,1},$ (7) $\displaystyle r_{3}$
$\displaystyle=r_{3,0}+\beta_{1}r_{2,1}+2\beta_{0}r_{3,1}+\beta_{0}^{2}r_{3,2}.$
(8)

Here, $r_{i,0}$ are scale-invariant conformal coefficients and $r_{i,j(\neq
0)}$ are non-conformal coefficients. At present, the
$\\{\beta_{i}\\}$-functions have been computed up to the five-loop level in
the modified minimal-subtraction scheme ($\overline{\rm MS}$-scheme) [56, 57],
e.g. $\beta_{0}=\left(11-2n_{f}/3\right)/(4\pi)$ and
$\beta_{1}=\left(102-38n_{f}/3\right)/(4\pi)^{2}$ for $n_{f}$ active flavors.

After applying the PMC, the divergent renormalon terms, which grow as various
powers of $\beta_{0}$ under the approximation
$\beta_{i}\simeq\beta_{0}^{i+1}$, have been removed, we then obtain a more
convergent pQCD series as

$\displaystyle R_{t}\big{|}_{\rm
PMC}=\sum_{i=1}^{3}r_{i,0}\alpha_{s}^{i}(Q_{*})+{\cal O}(\alpha_{s}^{4}),$ (9)

where $Q_{*}$ is fixed by requiring all non-conformal terms to vanish. Using
the present known pQCD series up to N3LO level, the PMC scale $Q_{*}$ can be
fixed up to next-to-leading-log (NLL) accuracy, i.e.,

$\displaystyle\ln\frac{Q_{*}^{2}}{Q^{2}}=S_{0}+S_{1}\alpha_{s}(Q_{*})+{\cal
O}\left(\alpha_{s}^{2}\right),$ (10)

where the coefficients $S_{0,1}$ are

$\displaystyle S_{0}$ $\displaystyle=-\frac{\hat{r}_{2,1}}{\hat{r}_{1,0}},$
(11) $\displaystyle S_{1}$
$\displaystyle=\frac{2(\hat{r}_{2,0}\hat{r}_{2,1}-\hat{r}_{1,0}\hat{r}_{3,1})}{\hat{r}_{1,0}^{2}}+\frac{\hat{r}_{2,1}^{2}-\hat{r}_{1,0}\hat{r}_{3,2}}{\hat{r}_{1,0}^{2}}\beta_{0},$
(12)

where $\hat{r}_{i,j}\equiv r_{i,j}|_{\mu_{R}=Q}$. Since all the non-conformal
$\\{\beta_{i}\\}$-terms of the series (3) which are associated with the RGE
have been absorbed into $\alpha_{s}(Q_{*})$, the resulting pQCD series (9)
becomes conformal series with the magnitude of $\alpha_{s}$ being well matched
with its coefficients, yielding naturally scheme-independent theoretical
predictions at any fixed order [58, 29, 43, 45, 59].

##  III Numerical results and discussions

For numerical calculations, the following values are taken as input parameters
[5]:

$\displaystyle m_{b}$ $\displaystyle=4.78\ {\rm GeV},$ $\displaystyle m_{t}$
$\displaystyle=172.5\pm 0.7\ {\rm GeV},$ $\displaystyle m_{W}$
$\displaystyle=80.377\ {\rm GeV},$ $\displaystyle\Gamma_{W}$
$\displaystyle=2.085\pm 0.042\ {\rm GeV},$ $\displaystyle m_{Z}$
$\displaystyle=91.1876\ {\rm GeV},$ $\displaystyle G_{F}$
$\displaystyle=1.1663788\times 10^{-5}\ {\rm GeV}^{2},$

and $\alpha_{s}(m_{Z})=0.1179\pm 0.0009$. The NLO EW correction $\Delta_{\rm
EW}$ can be calculated using the formulas given in Refs.[10, 11, 60] together
with the recently issued Mathematica program TopWidth [17, 20], and given the
above inputs, we obtain $\Delta_{\rm EW}=0.0249$ GeV.

###  III.1 Basic properties

Using Eqs.(10,11,12), we can obtain the wanted LL-accuracy and NLL-accuracy
PMC scales that correspond to NNLO and N3LO pQCD series, respectively, e.g.,

$\displaystyle Q_{*}^{(\rm LL,NLL)}=\\{15.2143,10.3048\\}\ {\rm GeV}.$ (13)

Using the NLL-accuracy $Q_{*}$, we then obtain the N3LO-level pQCD approximant
$R_{t}$ under the PMC scale-setting approach

$R_{t}\big{|}_{\rm PMC}=-0.1146.$ (14)

We observe that the improved pQCD series after applying the PMC becomes
independent to any choice of renormalization scale. On the other hand, we
observe that the scale dependence of the original pQCD series does become
smaller when more loop terms have been included, being consistent with
conventional wisdom. As for the present N3LO-level series, the net error under
conventional scale-setting approach is small which is about $2.5\%$ for
$\mu_{R}\in[m_{t}/2,2m_{t}]$, which will be extended to $\simeq 4.8\%$ for a
broader choice of scale range $\mu_{R}\in[m_{t}/4,4m_{t}]$. More explicitly,
we give the N3LO-level conventional prediction in the following,

$R_{t}\big{|}_{\rm Conv.}=-0.1122_{-0.0019-0.0013}^{+0.0009+0.0014},$ (15)

whose central value corresponds to $\mu_{R}=m_{t}$, the first errors are for
$\mu_{R}\in[m_{t}/2,2m_{t}]$ and the second ones are additional errors by
using a broader range $\mu_{R}\in[m_{t}/4,4m_{t}]$. The net N3LO $R_{t}$ for
conventional and PMC series are consistent with each other under proper choice
of scale range for conventional scale-setting approach. Thus we need to be
careful of discussing the scale uncertainties under conventional scale-setting
approach. For convenience, if not specially stated, we will adopt the usual
choice of $\mu_{R}\in[m_{t}/2,2m_{t}]$ to do our discussions. It is found that
the perturbative behavior of conventional series still depends heavily on the
choice of scale. To show this point more clearly, we define a $\kappa$-factor
for the series (3) or (9), i.e.,

$\displaystyle\kappa_{\rm Conv.}^{\rm N^{\it i}LO}$
$\displaystyle=\frac{r_{i}(\mu_{R})\alpha_{s}^{i}(\mu_{R})}{r_{1}\alpha_{s}(\mu_{R})},$
$\displaystyle\kappa_{\rm PMC}^{(i)}$
$\displaystyle=\frac{r_{i,0}\alpha_{s}^{i}(Q_{*})}{r_{1,0}\alpha_{s}(Q_{*})}.$
(16)

Numerically, we have

$\displaystyle\kappa_{\rm Conv.}$
$\displaystyle=\\{1,0.1702,0.0476\\},\quad\mu_{R}=m_{t}/2$ (17)
$\displaystyle\kappa_{\rm Conv.}$
$\displaystyle=\\{1,0.2450,0.0792\\},\quad\mu_{R}=m_{t}$ (18)
$\displaystyle\kappa_{\rm Conv.}$
$\displaystyle=\\{1,0.3096,0.1155\\},\quad\mu_{R}=2m_{t}$ (19)
$\displaystyle\kappa_{\rm PMC}$ $\displaystyle=\\{1,0.1210,0.0548\\}.$ (20)

The scale-independent convergent behavior of the PMC series (9) could be
regarded as the intrinsic perturbative nature of $R_{t}$. According to
Eqs.(14, 15), after including the known NLO EW correction $\Delta_{\rm EW}$,
the total top-quark decay width of $t\to Wb$ are

$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm Conv.}$
$\displaystyle=1.3156_{-0.0027}^{+0.0014}\ {\rm GeV},$ (21)
$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm PMC}$ $\displaystyle=1.3120\
{\rm GeV},$ (22)

where the errors for the series under conventional scale-setting approach is
for $\mu_{R}\in[m_{t}/2,2m_{t}]$.

Figure 1: The total top-quark decay width of $t\to Wb$ ($\Gamma_{t}^{\rm
tot}$) up to N3LO QCD corrections versus the renormalization scale $\mu_{R}$
using conventional (Conv.) (black lines) and PMC (red lines) scale-setting
approaches, respectively.

We present the total top-quark decay width $\Gamma_{t}^{\rm tot}$ up to N3LO
QCD corrections versus the renormalization scale $\mu_{R}$ (divided by
$m_{t}$) before and after applying the PMC scale-setting approach in FIG.1.
After applying the PMC, as shown in FIG.1, the scale dependence of the top-
quark decay width is eliminated at any fixed-order. The difference between
N2LO-level and the N3LO-level PMC predictions are much smaller than that of
the conventional ones, indicating that the convergence of the pQCD series is
significantly improved.

###  III.2 Predictions of the uncalculated N4LO contributions using the
Bayesian analysis approach

It has been noted that the improved series by using the PMC scale-setting
approach not only provides more precise predictions for the fixed-order pQCD
series but also establishes a robust basis for estimating the potential
contributions from the UHO-terms, thus significantly improving the predictive
power of the perturbation theory.

In the following, we adopt the BA to estimate the effects from the
uncalculated N4LO-terms from the known initial N3LO-level series (3) and the
PMC series (9), respectively. The BA quantifies the contributions of the UHO-
terms in terms of a probability distribution. It becomes most effective when
the series has a good convergent behavior. A detailed introduction to the BA
can be found in Refs. [47, 48, 49, 50] and its combination with the PMC can be
found in Refs.[52, 42]. Following the idea of the BA, for a fixed degree-of-
belief (DoB) or equivalently the Bayesian probability, the estimated UHO-
coefficient $r_{p+1}$, given known coefficients
$\\{r_{1},r_{2},\cdots,r_{p}\\}$, will fall within the following specific
credible interval (CI) $r_{p+1}\in\left[-r_{p+1}^{(\rm DoB)},r_{p+1}^{(\rm
DoB)}\right]$, where

$\displaystyle r_{p+1}^{(\rm DoB)}=\left\\{\begin{aligned}
&\bar{r}_{(p)}\frac{p+1}{p}{\rm DoB},&{\rm DoB}&\leq\frac{p}{p+1}\\\
&\bar{r}_{(p)}\left[(p+1)(1-{\rm DoB})\right]^{-1/p},&{\rm
DoB}&\geq\frac{p}{p+1}\end{aligned}\right.$ (23)

with $\bar{r}_{(p)}={\rm max}\\{|r_{1}|,|r_{2}|,\cdots,|r_{p}|\\}$. For
definiteness and without loss of generality, we take ${\rm DoB}\equiv 95.5\%$
to estimate the contributions from the UHO-terms.

Figure 2: Comparison of the calculated central values of the total top-quark
decay width of $t\to Wb$ ($\Gamma_{t}^{\rm tot}$) for the known series
(labeled as “exact value”) with the predicted credible intervals of
$\Gamma_{t}^{\rm tot}$ up to N4LO-level QCD corrections. The blue hollow
diamonds and red hollow quadrates represent the calculated central values of
the fixed-order pQCD predictions using conventional (Conv.) and PMC scale-
setting approaches, respectively. The blue stars and blue solid diamonds and
red solid quadrates with error bars represent the predicted credible intervals
using the Bayesian approach based on the known Conv. series and the PMC
series, respectively. ${\rm DoB}=95.5\%$.

Comparison of the calculated central values using known series (simply labeled
as “exact value”) of the total top-quark decay width $\Gamma_{t}^{\rm tot}$
with the predicted credible intervals of $\Gamma_{t}^{\rm tot}$ up to N4LO-
level QCD corrections are given in FIG.2. Due to the scale-dependence of the
coefficients $r_{i>1}$ in the conventional pQCD series, the BA can only be
employed after specifying the renormalization scale, which introduces
additional uncertainties to the total decay width. And in FIG.2, we give two
predictions for $\mu_{R}\in[m_{t}/2,2m_{t}]$ and $\mu_{R}\in[m_{t}/4,4m_{t}]$,
respectively. In contrast, the conformal coefficients $r_{i,0}$ in the PMC
series are scale-independent, providing a more reliable foundation for
constraining estimations from UHO contributions. FIG.2 shows that the
probability distribution become more accurate, simultaneously the resultant
credible intervals become smaller for the same DoB, when more loop terms have
been included. Practically, we can treat the magnitude of unknown N4LO-terms
as one of the errors of the given N3LO-level $\Gamma_{t}^{\rm tot}$. If taking
the scale range $\mu_{R}\in[m_{t}/2,2m_{t}]$ to estimate the UHO-
contributions, such errors by using the BA are

$\displaystyle\Delta\Gamma_{t}^{\rm tot}\big{|}_{\rm Conv.}^{\rm UHO}$
$\displaystyle=\left({}_{-0.0041}^{+0.0037}\right)\ {\rm GeV},$ (24)
$\displaystyle\Delta\Gamma_{t}^{\rm tot}\big{|}_{\rm PMC}^{\rm UHO}$
$\displaystyle=\pm 0.0035\ {\rm GeV}.$ (25)

The predicted error range for the perturbative series under conventional
scale-setting approach will be extended to $\Delta\Gamma_{t}^{\rm
tot}\big{|}_{\rm Conv.}^{\rm UHO}=\left({}_{-0.0054}^{+0.0060}\right)\ {\rm
GeV}$ for $\mu_{R}\in[m_{t}/4,4m_{t}]$. The above two equations show that
after applying the PMC, the errors $\Delta\Gamma_{t}^{\rm tot}$ from the UHO-
terms are only slightly smaller than those of the pQCD series under
conventional scale-setting approach. Both of which are small, e.g.
$\kappa^{\rm N^{4}LO}\sim 0.03$ for all cases, due to the fact that the N3LO-
level QCD corrections already show good convergent behavior as indicated by
Eqs.(17-20).

###  III.3 Uncertainties for the total decay width $\Gamma_{t}^{\rm tot}$

|  $\Delta\Gamma_{t}^{\rm tot}\big{|}^{\Delta\alpha_{s}}$ |  $\Delta\Gamma_{t}^{\rm tot}\big{|}^{\Delta\Gamma_{W}}$ |  $\Delta\Gamma_{t}^{\rm tot}\big{|}^{\Delta m_{t}}$
---|---|---|---
Conv. | $\pm 0.0015$ | $\pm 0.0004$ | $\left({}_{-0.0189}^{+0.0190}\right)$
PMC | $\pm 0.0015$ | $\pm 0.0004$ | $\left({}_{-0.0188}^{+0.0190}\right)$
Table 1: Additional uncertainties (in unit: GeV) arising from
$\Delta\alpha_{s}(m_{Z})=\pm 0.0009$, $\Delta\Gamma_{W}=\pm 0.042$ GeV, and
$\Delta m_{t}=\pm 0.7$ GeV under conventional and PMC scale-setting
approaches, respectively.

In addition to the above mentioned scale uncertainties and the uncertainties
caused by the UHO-terms, there are also other error sources for the
determination of $t\to Wb$ decay width. When discussing the uncertainty for
one of the error sources, the other error sources are set to be their central
values. We put the uncertainties arising from $\Delta\alpha_{s}(m_{Z})=\pm
0.0009$, $\Delta\Gamma_{W}=\pm 0.042$ GeV, and $\Delta m_{t}=\pm 0.7$ GeV
under conventional and PMC scale-setting approaches in TABLE 1. It shows that
the errors are dominated by $\Delta m_{t}$, whose effect to the total decay
width is about five to ten times larger than those of other error sources.
Thus a more precise $m_{t}$ will greatly improve the precision of the
theoretical predictions. The squared average of the uncertainties to the total
top-quark decay width arising from $\mu_{R}\in[m_{t}/2,2m_{t}]$,
$\Delta\alpha_{s}(m_{Z})$, $\Delta\Gamma_{W}$, and the predicted N4LO-terms
are

$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm Conv.}$
$\displaystyle=1.3207_{-0.0051}^{+0.0042}\ {\rm GeV},$ (26)
$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm PMC}$ $\displaystyle=1.3172\pm
0.0038\ {\rm GeV}.$ (27)

And if further including the uncertainty caused by $\Delta m_{t}$, we finally
get

$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm Conv.}$
$\displaystyle=1.3207\pm 0.0195\ {\rm GeV},$ (28)
$\displaystyle\Gamma_{t}^{\rm tot}\big{|}_{\rm PMC}$
$\displaystyle=1.3172_{-0.0192}^{+0.0194}\ {\rm GeV}.$ (29)

Figure 3: Total decay width of $t\to Wb$ ($\Gamma_{t}^{\rm tot}$) versus the
top-quark pole mass $m_{t}$, where the light-blue and dark-red bands are
results for conventional (Conv.) and PMC scale-setting approaches,
respectively. The dashed and solid lines are their central values. The shaded
bands are for the uncertainties arising from $\mu_{R}\in[m_{t}/2,2m_{t}]$,
$\Delta\alpha_{s}(m_{Z})$, $\Delta\Gamma_{W}$, and the predicted N4LO-terms.
Figure 4: Comparison between the theoretical predictions with their errors
given in Eqs.(28, 29) and the experimental measurements for the total decay
width $\Gamma_{t}^{\rm tot}$. The errors of theoretical predictions are for
the uncertainties arising from $\mu_{R}\in[m_{t}/2,2m_{t}]$,
$\Delta\alpha_{s}(m_{Z})$, $\Delta\Gamma_{W}$, $\Delta m_{t}$, and the
predicted N4LO-terms.

FIG.3 depicts the relationship between the total decay width $\Gamma_{t}^{\rm
tot}$ and the top-quark pole mass $m_{t}$, which indicates an approximately
proportional relationship between the top-quark’s decay widths and its pole
mass. The total decay width is highly sensitive to the top-quark pole mass.
FIG.4 shows the total decay width $\Gamma_{t}^{\rm tot}$ for the theoretical
predictions given by Eqs.(28, 29). As a comparison, various experimental
measurements [1, 3, 4, 2] and the world average value reported by the PDG [5]
have been included in FIG.4. Clearly, while the top quark’s decay width is
highly sensitive to its pole mass, the large uncertainties in experimental
measurements preclude us from deriving a reliable reference value for the top-
quark pole mass from these data.

In above analysis, we have implicitly taken the CKM matrix element $|V_{\rm
tb}|=1$ to do our numerical calculation. As a final remark, we give an inverse
determination of the CKM matrix element $|V_{\rm tb}|$ by using the PDG
averaged values on the top-quark total decay width and the branching fraction
$\mathcal{B}(t\to Wb)$. That’s by using the PDG’s averaged total decay width
$\Gamma_{t,\rm PDG}=1.42_{-0.15}^{+0.19}$ GeV [5], the branching fraction
$\mathcal{B}(t\to Wb)=0.957\pm 0.034$, along with the theoretical predictions
(28, 29) under conventional and PMC scale-setting approaches, respectively, we
inversely obtain 111The errors are estimated by using the usual error
propagation formulas. That is, the error of a quantity $Z=X/Y$ is calculated
by $\Delta Z=\left(X_{0}/Y_{0}\right)\sqrt{\left(\Delta
X/X_{0}\right)^{2}+\left(\Delta Y/Y_{0}\right)^{2}}$, where $X=X_{0}\pm\Delta
X$, $Y=Y_{0}\pm\Delta Y$ and $Z=\left(X_{0}/Y_{0}\right)\pm\Delta Z$.

$\displaystyle|V_{\rm tb}|_{\rm Conv.}$
$\displaystyle=1.033_{-0.116}^{+0.144}.$ (30) $\displaystyle|V_{\rm tb}|_{\rm
PMC}$ $\displaystyle=1.036_{-0.116}^{+0.144}.$ (31)

Both of them are consistent with the average of Tevatron and LHC values [5],
$|V_{\rm tb}|=1.014\pm 0.029$. By confining the prior within the SM region
$[0,1]$, we then establish a lower limit of $|V_{\rm tb}|>0.917$ and $|V_{\rm
tb}|>0.919$ for conventional and PMC scale-setting approaches, respectively.

##  IV Summary

In the paper, we have presented an improved analysis of the total decay width
of the top-quark decay $t\to Wb$ up to N3LO QCD corrections by applying the
PMC scale-setting approach. In contrast to previous literature [61], our
present treatment is achieved after taking both the off-shell $W$-boson
contributions and the finite $b$-quark mass effects into account. Because the
N3LO-level QCD corrections to the total decay width $\Gamma(t\to Wb)$ already
show good convergent behavior as indicated by Eqs.(17-20), the predictions
under conventional scale-setting approach are close to the PMC predictions.
Especially because the errors are dominated by $\Delta m_{t}$, which dilute
the great improvements on perturbative nature of the series by applying the
PMC. However, the improved pQCD series after applying the PMC is independent
to any choice of renormalization scale, which not only leads to a more precise
prediction, but also provides a better basis for estimating the contributions
from UHO-terms. The errors of the PMC series caused by
$\Delta\alpha_{s}(m_{Z})$ and the predicted N4LO-terms are comparable to each
other, and there are also sizable renormalization scale error for conventional
scale-setting approach. FIG.1 indicates that the difference between N2LO-level
and the N3LO-level PMC predictions are much smaller than the conventional one,
indicating that the convergence of the pQCD series is significantly improved
and the PMC prediction shows a quicker trends of approaching to its
physical/measured value. Thus our present results emphasize the importance of
using proper scale-setting approaches to achieve precise fixed-order pQCD
predictions.

Acknowledgements: This work was supported in part by the Chongqing Graduate
Research and Innovation Foundation under Grant No.CYB23011 and No.ydstd1912,
and by the Natural Science Foundation of China under Grant No.12175025 and
No.12347101.

## Appendix

The first two scale-independent coefficients $f_{0}(\varepsilon,\omega)$ and
$f_{1}(\varepsilon,\omega)$ at the scale $\mu_{R}=m_{t}$ for $t\to Wb$ which
contain finite $b$-quark mass effects are [6]

$\displaystyle f_{0}(\varepsilon,\omega)=$
$\displaystyle\lambda^{1/2}(1,\varepsilon,\omega)\left[(1-\varepsilon)^{2}+\omega(1+\varepsilon)-2\omega^{2}\right],$
(32) $\displaystyle f_{1}(\varepsilon,\omega)=$
$\displaystyle-\frac{C_{F}}{2\pi}\Bigg{\\{}\left[(1-\varepsilon)^{2}+\omega(1+\varepsilon)-2\omega^{2}\right](1+\varepsilon-\omega)\Bigg{[}\pi^{2}+2{\rm
Li}_{2}(u_{\rm W})-2{\rm Li}_{2}(1-u_{\rm W})-4{\rm Li}_{2}(u_{q})-4{\rm
Li}_{2}(u_{q}u_{\rm W})$
$\displaystyle\,+\ln\frac{1-u_{q}}{\omega}\ln(1-u_{q})-\ln^{2}(1-u_{q}u_{\rm
W})+\frac{1}{4}\ln^{2}\frac{\omega}{u_{\rm W}}-\ln u_{\rm
W}\ln\frac{(1-u_{q}u_{\rm W})^{2}}{1-u_{q}}-2\ln
u_{q}\ln\left[(1-u_{q})(1-u_{q}u_{\rm W})\right]\Bigg{]}$
$\displaystyle\,-2f_{0}(\varepsilon,\omega)\left(\ln\omega+\frac{3}{2}\ln\varepsilon-2\ln\lambda(1,\varepsilon,\omega)\right)+2(1-\varepsilon)\left[(1-\varepsilon)^{2}+\omega(1+\varepsilon)-4\omega^{2}\right]\ln
u_{\rm W}$
$\displaystyle\,+\frac{1}{2}\bigg{[}(3-\varepsilon+11\varepsilon^{2}-\varepsilon^{3})+\omega(6-12\varepsilon+2\varepsilon^{2})-\omega^{2}(21+5\varepsilon)+12\omega^{3}\bigg{]}\ln
u_{q}$
$\displaystyle\,+\frac{3}{2}\lambda^{1/2}(1,\varepsilon,\omega)(1-\varepsilon)(1+\varepsilon-\omega)\ln\varepsilon+\frac{1}{2}\lambda^{1/2}(1,\varepsilon,\omega)\left[-5+22\varepsilon-5\varepsilon^{2}-9\omega(1+\varepsilon)+6\omega^{2}\right]\Bigg{\\}},$
(33)

where $\omega=m_{W}^{2}/m_{t}^{2}$, $\varepsilon=m_{b}^{2}/m_{t}^{2}$,
$C_{F}=4/3$ is SUc(3) color factor, the Källen function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+xz+yz)$, the polylogarithm function
${\rm Li}_{n}(z)=\sum_{k=1}^{\infty}z^{k}/k^{n}$. The functions $u_{q}$ and
$u_{\rm W}$ are defined as

$\displaystyle u_{q}$
$\displaystyle=\frac{1+\varepsilon-\omega-\lambda^{1/2}(1,\varepsilon,\omega)}{1+\varepsilon-\omega+\lambda^{1/2}(1,\varepsilon,\omega)},$
$\displaystyle u_{\rm W}$
$\displaystyle=\frac{1-\varepsilon+\omega-\lambda^{1/2}(1,\varepsilon,\omega)}{1-\varepsilon+\omega+\lambda^{1/2}(1,\varepsilon,\omega)}.$
(34)

In the case where the $b$-quark mass effects have been ignored, one has

$\displaystyle f_{0}(\omega)\equiv$
$\displaystyle\,f_{0}(0,\omega)=(1-\omega)^{2}(1+2\omega),$ (35)
$\displaystyle f_{1}(\omega)=$
$\displaystyle\,-\frac{C_{F}}{2\pi}\Bigg{\\{}f_{0}(\omega)\Bigg{[}\pi^{2}+2{\rm
Li}_{2}(\omega)-2{\rm Li}_{2}(1-\omega)\Bigg{]}$
$\displaystyle\,+2\omega(1+\omega)(1-2\omega)\ln\omega+(1-\omega)^{2}(5+4\omega)\ln(1-\omega)$
$\displaystyle\,-\frac{1}{2}(1-\omega)\left(5+9\omega-6\omega^{2}\right)\Bigg{\\}}.$
(36)

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