# Strong Coupling from Inclusive Semi-leptonic Decay of Charmed Mesons

Jinfei Wu  Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  China Center of Advanced Science and Technology,
Beijing 100190, China  Yuzhi Che  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Yanping Huang  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Gang Li
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  Xinchou Lou  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  University of Texas at Dallas, Richardson,
Texas 75083, USA  University of Chinese Academy of Sciences, Beijing 100049,
China  Manqi Ruan  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Jingbo Ye  Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Employing the heavy quark expansion with kinetic scheme, we study the strong
interaction coupling $\alpha_{S}$ at an energy corresponding to the charm
quark mass $m_{c}$ using the data on the semi-leptonic decays of charmed
mesons from the BESIII and CLEO-c. The values of the $\alpha_{s}(m_{c}^{2})$
from the $D^{0}$, $D^{+}$, and $D_{s}^{+}$ are extracted and found to be
consistent. Combining the semileptonic decay widths of the $D^{0}$, $D^{+}$,
and $D_{s}^{+}$, we determine $\alpha_{s}(m_{c}^{2})$ to be $0.377\pm 0.008\pm
0.114$, where the first and second uncertainty are the experimental and
theoretical uncertainty separately. This result is in good agreement with the
world average of $\alpha_{S}$ running from Z pole to the $m_{c}$.

††preprint: APS/123-QED

##  I Introduction

In the Standard Model (SM) of particle physics, Quantum Chromo-dynamics (QCD)
is the gauge field theory which describes the strong interaction. In this
theory gluons are the force mediators and the effective strong interaction
coupling constant, $\alpha_{S}$, dictates the strength and many features of
the strong interaction [1, 2, 3, 4]. The asymptotic freedom is one of the
important features of QCD, which behaves as the $\alpha_{S}$ increases with
the momentum transfer decreases. The $\alpha_{S}$ is measured in a vast
momentum transfer region from $m_{\tau}$ to 1 $\mathrm{TeV}$, which is
consistent with the theoretical prediction of running behaviour. The
$\alpha_{S}$ below $m_{\tau}$ is not measured till now, since the perturbative
calculation below $m_{\tau}$ may not be reliable. To reveal the nature of QCD
in low momentum transfer region, we perform a determination of $\alpha_{S}$
from the inclusive semi-leptonic decays of charmed mesons, where the momentum
transfer is typically at $m_{c}$ and smaller than all of the existing
measurements of $\alpha_{S}$ [5].

The considerable progress has been achieved in the theoretical description of
the inclusive semi-leptonic decays of charmed and B mesons using the framework
of heavy quark expansion (HQE) over past decades [6, 7, 8, 9, 10, 11, 11, 12,
13, 14]. The theoretical predictions agree well with the experimental
measurements in the inclusive semi-leptonic decays of charmed and B mesons
[15, 12, 14, 16, 17], including the inclusive semi-leptonic decay widths
($\Gamma_{SL}$) as well as several moments of kinematic observables [12, 13,
14, 16]. The observables in the inclusive semi-leptonic decays of charmed
mesons are expanded in terms of $\alpha_{S}$, quark masses, CKM matrix
elements, and non-perturbative parameters of the HQE in the theoretical
framework. Based on the theoretical predictions of the observables, the b
quark masses and $|V_{cb}|$ were extracted from the experimental measurements
of the inclusive semi-leptonic decays of B mesons [18, 15, 19, 20, 21, 22]. In
these studies, the related quark masses and $|V_{cb}|$ are determined from the
measured and predicted observables, where the $\alpha_{S}$ is fixed to the
value running from $\alpha_{S}(m_{Z}^{2})$, and the corresponding uncertainty
is estimated by varying the $\alpha_{S}$ in a conservative way [18, 15, 19,
17]. In addition, the extra constraints on the HQE parameters are imposed to
represent the priori knowledge on the HQE parameters.

The theoretical description of the $\Gamma_{SL}$ of charmed mesons depends on
the $\alpha_{S}$, corresponding quark masses, $|V_{cs}|$, and several non-
perturbative parameters in HQE. Over decades of studies, the definition of
charm quark mass ($m_{c}$) and determination of $|V_{cs}|$ are more precise
[5, 23, 24], which offers an opportunity to determine $\alpha_{S}$ from the
inclusive semi-leptonic decays of charmed mesons. The $\alpha_{S}$, charm
quark mass, $|V_{cs}|$ can be extracted from a combined fit to the
$\Gamma_{SL}$ simultaneously. In 2012, the inclusive semi-leptonic decays of
charmed mesons were investigated by the CLEO-c through the $e^{+}e^{-}$
collision data at $\sqrt{s}=$ 3.774 and 4.170 $\mathrm{GeV}$ [25]. Recently,
the BESIII experiment also accomplished the measurement of
$D_{s}^{+}\rightarrow Xe\nu$ [26] with better precision. With the precise
measurements of lifetimes of charm mesons[5], the $\Gamma_{SL}$ of charmed
mesons can be obtained from the corresponding branch fractions. The
experimental measurements and theoretical calculations make the determination
of the $\alpha_{S}$ using the inclusive semi-leptonic decays of charmed mesons
possible, of which momentum transfer is lower than all of the existing
determinations. In this study, the determination of $\alpha_{S}$ is performed
by imposing extra constraints on the other parameters, such as quark masses.
Furthermore, the moment measurements can also be used to determine the
$\alpha_{S}$, if the correlation matrices among the moment measurements are
available.

In this letter we will present the determination of $\alpha_{S}$ from a
combined fit to the $\Gamma_{SL}$ of charmed mesons, $D^{+}$, $D^{0}$, and
$D_{s}^{+}$. The Sec.II introduces the theoretical framework of the HQE to
extact $\alpha_{S}$, the Sec.III is the determination of
$\alpha_{S}(m_{c}^{2})$, and the Sec.IV concludes the study.

##  II Heavy quark expansion in the kinetic scheme

The theoretical framework of the $\Gamma_{SL}$ of charmed mesons in [12] is
employed to perform this analysis. In [12], the authors considered the
$\mathcal{O}(\alpha_{S})$ and $\mathcal{O}(\beta_{0}\alpha_{S}^{2})$
corrections from [27, 28] in the case of the inclusive semi-leptonic decays of
charmed mesons, as well as the $\mathcal{O}(1/m_{c}^{3})$ contributions from
[29]. The convergence of the perturbative series of $\Gamma_{SL}$ is strongly
affected by the mass definition of the charm quark [30, 31, 32, 33]. As
pointed out in [23], the pole mass and $\overline{\mathrm{MS}}$ scheme exhibit
bad convergence behaviour in the QCD corrections to $\Gamma_{SL}$. But the
heavy quark masses have a finite infrared limit to any order in the
perturbation theory. The kinetic scheme [30, 32, 34] is introduced to avoid
the divergence, which is used to calculate the $\Gamma_{SL}$ in this study.

In [12], the $\Gamma_{SL}$ is expressed in terms of the
$\alpha_{S}(m_{c}^{2})$, $m_{c}$, $m_{s}$, and non-perturbative corrections,
that is shown in Eq. 1. In the Eq. 1, the $G_{F}$ is the Fermi coupling
constant, the $m_{c}$ is the mass of charm quark, the $r$ is the ratio between
strange and charm quark mass square ($m_{s}^{2}/m_{c}^{2}$), the $|V_{cs}|$ is
the CKM matrix element, $\alpha_{S}\equiv\alpha_{S}(m_{c}^{2})$, the
$\mu^{2}_{\pi,G}$ are the kinetic and chromomagnetic dimension-five operators
[35, 36, 37], and the $\rho_{D,LS}^{3}$ are the Darwin and spin-orbital (LS)
dimension six operators [37] in the HQE. As well as the $B_{WA}$ is the weak
annihilation (WA) contribution, which depends on the spectator quark in
charmed mesons.

$\Gamma_{SL}=\frac{G_{F}^{2}m_{c}^{5}}{192\pi^{3}}|V_{cs}|^{2}[f_{0}(r)+\frac{\alpha_{S}}{\pi}f_{1}(r)+\frac{\alpha_{S}^{2}}{\pi^{2}}f_{2}(r)+\frac{\mu_{\pi}^{2}}{m_{c}^{2}}f_{\pi}(r)+\frac{\mu_{G}^{2}}{m_{c}^{2}}f_{G}(r)+\frac{\rho_{LS}^{3}}{m_{c}^{3}}f_{LS}(r)+\frac{\rho_{D}^{3}}{m_{c}^{3}}f_{D}(r)+\frac{32\pi^{2}}{m_{c}^{3}}B_{WA}]$
(1)

The coefficients of perturbative and non-perturbative corrections,
$f_{0,1,2}(r)$ and $f_{\pi,G,LS,D}(r)$, are calculated in terms of $r$ [12],
as shown in Eq. 2, where the $\beta_{0}$ is the QCD beta function
$\beta_{0}=11-2n_{f}/3$ and $n_{f}$ is the number of light flavors.

$\displaystyle f_{0}(r)$
$\displaystyle=1-8r+8r^{3}-r^{4}-12r^{2}\cdot\mathrm{log}(r),$ (2)
$\displaystyle f_{1}(r)$
$\displaystyle=2.86\sqrt{r}-3.84r\cdot\mathrm{log}(r),$ $\displaystyle
f_{2}(r)$
$\displaystyle=\beta_{0}[8.16\sqrt{r}-1.21r\cdot\mathrm{log}(r)-3.38],$
$\displaystyle f_{\pi}(r)$ $\displaystyle=-f_{0}(r)/2,$ $\displaystyle
f_{G}(r)$ $\displaystyle=\frac{1}{2}f_{0}(r)-2(1-r)^{4},$ $\displaystyle
f_{LS}(r)$ $\displaystyle=-f_{G}(r),$ $\displaystyle f_{D}(r)$
$\displaystyle=\frac{77}{6}+\mathcal{O}(r)+8\mathrm{log}(\frac{\mu_{WA}^{2}}{m_{c}^{2}}),$

Considering the relative low charm mass and QCD scale parameter
($\Lambda_{QCD}$), the infrared cutoff scale in kinetic scheme, $\mu$, is set
to be $0.5\ \mathrm{GeV}$. In the theoretical expression of $f_{D}(r)$, the
$\mu_{WA}$ is the $\overline{\mathrm{MS}}$ renormalization scale associated to
the mixing of the Darwin and WA operators [38, 39], that is chosen to be $0.8\
\mathrm{GeV}$ in the following study as same as values in [12]. In the Eq. 1,
only the process of $c\rightarrow sl\bar{\nu}$ is taken into account, which is
slightly different to the experimental measurements [25, 26] due to missing
the Cabibbo suppressed processes. A corresponding systematical uncertainty in
the determination of $\alpha_{S}(m_{c}^{2})$ is assigned to cover this
difference.

##  III Fit to the $\Gamma_{SL}$ of $D^{0}$, $D^{+}$, and $D_{s}^{+}$

###  III.1 Fit method

A $\chi^{2}$ minimization method is employed to determine the
$\alpha_{S}(m_{c}^{2})$ from a combined fit to the $\Gamma_{SL}$ of $D^{0,+}$
and $D_{s}^{+}$, of which definition is

$\displaystyle\chi^{2}(\alpha_{S},\theta_{j})$
$\displaystyle=\sum_{i}\frac{[\Gamma_{SL,D_{i}}-\hat{\Gamma}_{SL}(\alpha_{S},\theta_{j})]^{2}}{\sigma_{\Gamma_{SL,D_{i}}}^{2}}+\sum_{j}\frac{(\theta_{j}-\theta^{\prime}_{j})^{2}}{\sigma_{\theta^{\prime}_{j}}^{2}},$

The $\Gamma_{SL,D_{i}}$ and $\sigma_{\Gamma_{SL,D_{i}}}$ are the measured
inclusive semi-leptonic decay width and corresponding uncertainty of $D_{i}$
meson. Moreover the HQE parameters are constrained in the $\chi^{2}$ fit with
external inputs. The
$\theta^{\prime}_{j}=\\{m_{c},m_{s},|V_{cs}|,\mu_{G}^{2},\mu_{\pi}^{2},\rho_{D}^{3},\rho_{LS}^{3}\\}$
represents the constrained parameters in the fit, whose values and
uncertainties are the $\theta^{\prime}_{j}$ and
$\sigma_{\theta^{\prime}_{j}}$. The values and uncertainties of
$m_{c},m_{s},|V_{cs}|,\mu_{G}^{2},\mu_{\pi}^{2},\rho_{D}^{3},\mathrm{and}\
\rho_{LS}^{3}$ are discussed in the following sections, as well as the
measured inclusive semi-leptonic decay widths of charmed mesons.

###  III.2 Theoretical inputs

Except the $G_{F}$ and $B_{WA}$, the other parameters in Eq. 1 are floated in
the determination of $\alpha_{S}(m_{c}^{2})$. The $|V_{cs}|$ is set to be
$0.975\pm 0.006$ [5], which is an average of the determinations from leptonic
and semi-leptonic decays. The expected values of $\mu_{G}^{2}$ and
$\rho_{LS}^{3}$ do not run with respect to the kinetic scale in the kinetic
scheme, of which values were determined to be $0.288\pm 0.049\
\mathrm{GeV}^{2}$ and $-0.113\pm 0.090\ \mathrm{GeV}^{3}$ from the $q^{2}$
moments in inclusive semi-leptonic B decays [17]. The $\mu_{\pi}^{2}(0.5\
\mathrm{GeV})$ and $\rho_{D}^{3}(0.5\ \mathrm{GeV})$ are evolved to $\mu=0.5\
\mathrm{GeV}$ using $\mathcal{O}(\alpha_{S}^{2})$ expressions from their
values in $\mu=1\ \mathrm{GeV}$ determined by a global fit of $B\rightarrow
X_{c}l\nu$ [12, 15, 40]. The values of $\mu_{\pi}^{2}(0.5\ \mathrm{GeV})$ and
$\rho_{D}^{3}(0.5\ \mathrm{GeV})$ were determined to be $0.26\pm 0.06\
\mathrm{GeV}^{2}$ and $0.05\pm 0.04\ \mathrm{GeV}^{3}$ separately [12]. The
mass of strange quark is set to be $93.4\pm 8.6\ \mathrm{MeV}$ [5] that has
negligible effect on the $\Gamma_{SL}$, as well as the $G_{F}$ is fixed to
$1.1663788\times 10^{-5}$ [5]. The values of $B_{WA}$ for $D^{+}$, $D^{0}$,
and $D^{+}_{s}$ were extracted as $-0.001$, $-0.001$, and $-0.002\
\mathrm{GeV}^{3}$ in [12].

As the Eq. 1 shows, the $\Gamma_{SL}$ has strong dependence on the $m_{c}$.
Consequently the precision of $m_{c}$ is crucial to the determination of
$\alpha_{S}(m_{c}^{2})$, that induces a significant uncertainty in this study.
The relation between the $\overline{\mathrm{MS}}$ and kinetic mass of charm
quarks had been investigated to three-loop order ($\mathrm{N}^{3}$LO) [23,
24]. For the different choices of scale ($\mu_{s}$) in
$\overline{\mathrm{MS}}$, the $m_{c}(0.5\ \mathrm{GeV})$ in kinetic scheme,
$m_{c}^{kin}(0.5\ \mathrm{GeV})$, is obtained separately using the relation in
[23, 24], as shown in Eq. 3.

$\displaystyle m_{c}^{kin}(0.5\ \mathrm{GeV})$ $\displaystyle=1336\
\mathrm{MeV}\ \mathrm{from}\ \overline{m}_{c}(\mu_{s}=3\ \mathrm{GeV})$ (3)
$\displaystyle m_{c}^{kin}(0.5\ \mathrm{GeV})$ $\displaystyle=1372\
\mathrm{MeV}\ \mathrm{from}\ \overline{m}_{c}(\mu_{s}=2\ \mathrm{GeV})$
$\displaystyle m_{c}^{kin}(0.5\ \mathrm{GeV})$ $\displaystyle=1404\
\mathrm{MeV}\ \mathrm{from}\ \overline{m}_{c}(\mu_{s}=\overline{m}_{c})$

The average of the $m_{c}^{kin}(0.5\ \mathrm{GeV})$ from different choices of
$\mu_{s}$ is treated as the input values of $m_{c}(0.5\ \mathrm{GeV})$ in the
$\chi^{2}$ fit, which is determined to be $1370\ \mathrm{MeV}$. For
conservative consideration, the uncertainty of $m_{c}(0.5\ \mathrm{GeV})$ is
obtained as the largest difference between $m_{c}(0.5\ \mathrm{GeV})$ and
$m_{c}^{kin}(0.5\ \mathrm{GeV})$ in Eq. 3, that is $34\ \mathrm{MeV}$. The
input values of the parameters used in the $\chi^{2}$ fit are summarized in
Tab. 1.

Table 1: Except $\alpha_{S}(m_{c}^{2})$, the input values of the parameters used in the $\chi^{2}$ fit. Parameter | Value
---|---
$G_{F}$ | $1.1663788\times 10^{-5}$
$|V_{cs}|$ | $0.975\pm 0.006$
$m_{c}(0.5\ \mathrm{GeV})$ | $(1.370\pm 0.034)\ \mathrm{GeV}$
$m_{s}(0.5\ \mathrm{GeV})$ | $(93.4\pm 8.6)\ \mathrm{MeV}$
$\mu_{G}^{2}(0.5\ \mathrm{GeV})$ | $(0.288\pm 0.049)\ \mathrm{GeV}^{2}$
$\mu_{\pi}^{2}(0.5\ \mathrm{GeV})$ | $(0.26\pm 0.06)\ \mathrm{GeV}^{2}$
$\rho_{D}^{3}(0.5\ \mathrm{GeV})$ | $(0.05\pm 0.04)\ \mathrm{GeV}^{3}$
$\rho_{LS}^{3}(0.5\ \mathrm{GeV})$ | $(-0.113\pm 0.090)\ \mathrm{GeV}^{3}$
$B_{WA,D^{+,0}}$ | $-0.001\ \mathrm{GeV}^{3}$
$B_{WA,D^{+}_{s}}$ | $-0.002\ \mathrm{GeV}^{3}$

###  III.3 Experimental inputs

The currently available experimental measurements of $\Gamma_{SL}$ are derived
from the inclusive semi-leptonic decay branching fraction ($\mathcal{B}_{SL}$)
[25, 26] and mean lifetime ($\tau$) [5] of $D^{0,+}$ and $D^{+}_{s}$. In the
CLEO-c measurements [25], the inclusive semi-leptonic branching fractions of
$D^{0,+}$ and $D^{+}_{s}$ were measured using $818\ \mathrm{pb}^{-1}$ and
$602\ \mathrm{pb}^{-1}$ open-charm data samples at $E_{CMS}=3.774\
\mathrm{GeV}$ and $4.170\ \mathrm{GeV}$. Due to lack of statistic, the
uncertainty of $\mathcal{B}_{SL,D^{+}_{s}}$ is much higher than the
$\mathcal{B}_{SL,D^{0,+}}$ in the measurements of the CLEO-c. Recently, the
$\mathcal{B}_{SL,D^{+}_{s}}$ was also measured by the BESIII using $3.19\
\mathrm{fb}^{-1}$, $2.08\ \mathrm{fb}^{-1}$, and $1.05\ \mathrm{fb}^{-1}$
$e^{+}e^{-}$ collision data samples at $E_{CMS}=4.178\ \mathrm{GeV}$,
$4.189-4.219\ \mathrm{GeV}$, and $4.225-4.230\ \mathrm{GeV}$ [26]. Using more
data samples, the uncertainty of $\mathcal{B}_{SL,D^{+}_{s}}$ measured by the
BESIII is comparable to the $\mathcal{B}_{SL,D^{0,+}}$ at the CLEO-c. To
improve the precision of the $\alpha_{S}(m_{c}^{2})$, the
$\mathcal{B}_{SL,D^{+}_{s}}$ measured by the BESIII and
$\mathcal{B}_{SL,D^{0,+}}$ measured by the CLEO-c are adopted for the
calculation of $\Gamma_{SL}$. The $\Gamma_{SL}$ is calculated using
$\mathcal{B}_{SL}$ and $\tau$ by Eq. 4.

$\displaystyle\Gamma_{SL,\ D_{i}}$ $\displaystyle=\frac{6.582\times
10^{-25}\cdot\mathcal{B}_{SL}(D_{i}\rightarrow Xe\nu_{e})}{\tau_{D_{i}}}\
\mathrm{GeV},$ (4)

where the $D_{i}$ is $D^{0,+}$ and $D^{+}_{s}$, the $\tau_{D_{i}}$ is the mean
life of $D_{i}$, and $\mathcal{B}_{SL}(D_{i}\rightarrow Xe\nu_{e})$ is the
semi-electronic decay branching fraction of $D_{i}$. The experimental values
of $\mathcal{B}_{SL}(D_{i}\rightarrow Xe\nu_{e})$, $\tau_{D_{i}}$, and
$\Gamma_{SL,D_{i}}$ are summarized in Tab. 2. The $\Gamma_{SL}$ of D mesons
are close to each other, which increases the reliability of the HQE in the
inclusive semi-leptonic decays of charmed mesons.

Table 2:  The experimental values of $\mathcal{B}_{SL}(D_{i}\rightarrow Xe\nu_{e})$, $\tau_{D_{i}}$, and $\Gamma_{SL,D_{i}}$. $D_{i}$ |  $\mathcal{B}_{SL}$ [%] |  $\tau$ [$10^{-13}$s] |  $\Gamma_{SL}$ [$10^{-15}\mathrm{GeV}$]
---|---|---|---
$D^{0}$ | $6.46\pm 0.09\pm 0.11$ | $4.10\pm 0.01$ | $104\pm 2$
$D^{+}$ | $16.13\pm 0.10\pm 0.29$ | $10.33\pm 0.05$ | $103\pm 2$
$D^{+}_{s}$ | $6.30\pm 0.13\pm 0.09\pm 0.04$ | $5.04\pm 0.04$ | $82\pm 2$

Except the $\mathcal{B}_{SL}$, the CLEO-c and BESIII also measured the
distributions of electron’s momentum ($|p_{e^{+}}|$) with $|p_{e^{+}}|>200\
\mathrm{MeV}$ from the inclusive semi-leptonic decays of $D^{0,+}$,
$D_{s}^{+}$, and $\Lambda_{c}^{+}$ [41] at laboratory frame, that are shown in
Fig. 1. Besides, the average $|p_{e^{+}}|$ of the $D^{0}$, $D^{+}$,
$D_{s}^{+}$, and $\Lambda_{c}^{+}$ ($\overline{|p_{e^{+}}|}$) is also plot in
Fig. 1. The distributions of $|p_{e^{+}}|$ agree well among $D^{0,+}$,
$D_{s}^{+}$, and $\Lambda_{c}^{+}$, even in the inclusive semi-leptonic decay
of $\Lambda_{c}^{+}$.

Figure 1:  The distributions of $|p_{e^{+}}|$ with $|p_{e^{+}}|>200\
\mathrm{MeV}$ from the inclusive semi-leptonic decays of $D^{0,+}$,
$D_{s}^{+}$, and $\Lambda_{c}^{+}$ at laboratory frame. The blue dots are
$D_{s}^{+}$ measured by the BESIII, the orange diamonds are $D^{0}$ measured
by the CLEO-c, the greed triangles are $D^{+}$ measured by the CLEO-c, the
purple diamonds are $\Lambda_{c}^{+}$ measured by the BESIII, and the gray
dashed line is the average of $D^{0}$, $D^{+}$, $D_{s}^{+}$, and
$\Lambda_{c}^{+}$.

As well as the moments of $|p_{e^{+}}|$ can be employed to determine
$\alpha_{S}(m_{c}^{2})$. But the theoretical predictions of these moments are
in the rest frame of D meson, which can not be retrieved from available
experimental measurements conveniently. Hence only the $\Gamma_{SL}$ is used
to extract the $\alpha_{S}(m_{c}^{2})$ in this study. The $\Gamma_{SL,D_{i}}$
and $\sigma_{\Gamma_{SL,D_{i}}}$ of $D_{i}$ meson are shown in Tab. 2.

The Kolmogorov-Smirnov (KS) tests on the distributions of $|p_{e^{+}}|$ and
$\overline{|p_{e^{+}}|}$ are performed to further validate the reliability of
HQE in the inclusive semi-leptonic decays of charmed hadrons, which is
implemented by [42]. The results of the KS tests are listed in Tab. 3, where
the null hypothesis is that the two distributions are identical.
Experimentally, the reliability of HQE in the inclusive semi-leptonic decays
of charmed mesons is validated by,

  * •

The $\Gamma_{SL}$ are close among different charmed mesons, which are listed
in Tab. 2,

  * •

The KS tests indicate the distributions of $|p_{e^{+}}|$ among different
charmed hadrons are consistent, which are listed in Tab. 3.

Table 3:  The results of the KS tests are listed, where the null hypothesis is that the two distributions are identical. Test Samples | Test Statistic | P Value
---|---|---
$|p_{e^{+},D^{0}}|$ and $\overline{|p_{e^{+}}|}$ | $0.125$ | $0.999$
$|p_{e^{+},D^{+}}|$ and $\overline{|p_{e^{+}}|}$ | $0.063$ | $1.000$
$|p_{e^{+},D^{+}_{s}}|$ and $\overline{|p_{e^{+}}|}$ | $0.132$ | $0.992$
$|p_{e^{+},\Lambda^{+}_{c}}|$ and $\overline{|p_{e^{+}}|}$ | $0.125$ | $0.999$

###  III.4 Fit results

In the $\chi^{2}$ fit, the high order perturbative corrections need to be
taken into account for the semi-leptonic decays of charmed mesons. In the
kinetic scheme, the perutrbative behavior of $\Gamma_{SL}$ is well under
control, especially in the semileptonic decay of b quark [14]. In [14], the
order $\alpha_{S}^{3}$ correction to the decay $b\rightarrow cl\bar{\nu}$ is
dertermined to be less than 1% in the kinetic scheme. For conservative
consideration, 5 times larger than the order $\alpha_{S}^{3}$ correction in
$b\rightarrow cl\bar{\nu}$ is taken as the high order perturbative corrections
in the semi-leptonic decays of charmed mesons, which is 5% of $\Gamma_{SL}$.
Furthermore, the theoretical prediction of the $\Gamma_{SL}$ of charmed mesons
in Eq. 1 is the contributions of $c\rightarrow sl\bar{\nu}$, which misses the
Cabibbo suppressed processes of $c\rightarrow dl\bar{\nu}$. The
$|V_{cd}|^{2}/(|V_{cd}|^{2}+|V_{cs}|^{2})=5\%$ is treated to be the
uncertainty caused by the Cabibbo suppressed processes. In total, the
$\Gamma_{SL}$ is varied by 10% to estimate the uncertainty caused by un-
perfect theoretical prediction in the $\chi^{2}$ fit. The authors need to
emphasize that the high order perturbative correction plays an important role
in this study, and an advanced theoretical prediction of the high order
perturbative corrections is highly desirable.

The $\alpha_{S}(m^{2}_{c})$ is extracted from different combinations of
$\Gamma_{SL,D^{+}}$, $\Gamma_{SL,D^{0}}$ and $\Gamma_{SL,D^{+}_{S}}$,
including

  * •

$\Gamma_{SL,D^{+}}$, $\Gamma_{SL,D^{0}}$ and $\Gamma_{SL,D^{+}_{S}}$
separately,

  * •

$\Gamma_{SL,D^{+}}$ and $\Gamma_{SL,D^{0}}$,

  * •

$\Gamma_{SL,D^{+}}$, $\Gamma_{SL,D^{0}}$ and $\Gamma_{SL,D^{+}_{S}}$ together.

The fitted and measured $\Gamma_{SL}$ in the combination of $D^{0,+}$ and
$D^{+}_{s}$ are shown in Fig. 2. Since the conservative theoretical
uncertainty (10%) is taken into account, the uncertainties of fitted
$\Gamma_{SL}$ are much larger than measured $\Gamma_{SL}$.

Figure 2: The fitted and measured $\Gamma_{SL}$ in the fit to combination of
$D^{0,+}$ and $D^{+}_{s}$. The open dot is the fitted $\Gamma_{SL}$, and the
solid dot is the measured $\Gamma_{SL}$. The error bar represents the
uncertainty.

The fitted $\alpha_{S}(m^{2}_{c})$ of each combination is extracted at
$m_{c}^{fit}$ and run to $m_{Z}$ using $\mathbf{RunDec}$ [43], which is shown
in Fig. 3 and Tab. 4. As the Tab. 4 shows, the $m_{c}$ is fitted to be
$1.392\pm 0.033\ \mathrm{GeV}$ when the combination of $D^{0,+}$ and
$D_{s}^{+}$ is used in the fit, that is slight different with the other
combinations. The difference of $m_{c}^{fit}$ among the combinations
demonstrates that there is also a slight constraint power on $m_{c}$ in this
study. The fitted values of $\alpha_{S}(m_{c}^{2})$ for different combination
are comparable to the value running from $\alpha_{S}(m^{2}_{Z})$ [5] within
1$\sigma$, which confirms the asymptotic freedom of QCD at scale of $1.4\
\textrm{GeV}$ and represents a remarkable success of the QCD calculations.

Figure 3: The $\alpha_{S}(m_{c}^{2})$ of each combination is extracted and run to $m_{Z}$, where the $m_{c}$ and $|V_{cs}|$ are floated in the fit. The grey shade is the world average of $\alpha_{S}$, that is $0.1180\pm 0.0009$ at $m_{Z}^{2}$ [5]. The inner error bar is the experimental uncertainty and outer error bar is the total uncertainty. Table 4:  The $m_{c}$ and $|V_{cs}|$ are floated in the fit. The $m_{c}$ and $\alpha_{S}(m_{c}^{2})$ of each combination are listed. The first uncertainty of $\alpha_{S}$ is the experimental uncertainty and second is the theoretical uncertainty. Combination | $m_{c}^{\mathrm{float}}[\mathrm{GeV}]$ |  $\alpha_{S}(m_{c}^{2})$ [$10^{-3}$] |  $\alpha_{S}(m_{Z}^{2})$ [$10^{-4}$]
---|---|---|---
$D^{0}$ | $1.370\pm 0.034$ | $448\pm 13\pm 114$ | $1177^{+7+50}_{-7-83}$
$D^{+}$ | $1.370\pm 0.034$ | $444\pm 12\pm 115$ | $1175^{+6+51}_{-7-85}$
$D^{+}_{s}$ | $1.370\pm 0.034$ | $400\pm 14\pm 113$ | $1148^{+9+60}_{-10-102}$
$D^{0,+}$ | $1.370\pm 0.034$ | $445\pm 9\pm 114$ | $1176^{+5+51}_{-5-84}$
$D^{0,+}$, $D^{+}_{s}$ | $1.392\pm 0.033$ | $377\pm 8\pm 114$ | $1134^{+6+68}_{-6-117}$

The $\chi^{2}/dof$ of the combination of $D^{0,+}$ and $D^{+}_{s}$ is $7.7/6$,
which demonstrate the combined fit of $D^{0,+}$ and $D_{s}^{+}$ is reasonable.
The profile contours of different combinations are shown in Fig. 4, that
indicate the consistence within 1$\sigma$ among different combinations. The
consistent $\alpha_{S}(m_{c}^{2})$ determinations of different combination
also increase the confidence of this study. The correlation matrix in the fit
of the combination of $\Gamma_{SL,D^{+}}$, $\Gamma_{SL,D^{0}}$ and
$\Gamma_{SL,D^{+}_{S}}$ is shown Fig. 5. The large negative correlation
between $m_{c}$ and $\alpha_{S}$ confirms the strong $m_{c}$ dependence in Eq.
1.

Figure 4: The profile contours of different combinations at 68% confidence
level. The solid blue curve and blue star are the contour and best fit value
for the combination of $D^{0,+}$ and $D_{s}^{+}$, the orange dashed curve and
cross are for the combination of $D^{0}$ and $D^{+}$, the green, red, and
purple dashed curve and cross are for the $D^{0}$, $D^{+}$, and $D^{+}_{s}$
separately. Figure 5: The correlation matrix among different parameters in the
fit of combination of $\Gamma_{SL,D^{+}}$, $\Gamma_{SL,D^{0}}$ and
$\Gamma_{SL,D^{+}_{S}}$.

The $m_{c}$ and $|V_{cs}|$ are fixed in the $\chi^{2}$ fit to investigate the
impact from $m_{c}$ and $|V_{cs}|$ and validity of this study. The $m_{c}$ is
fixed to $1.370\pm 0.034\ \mathrm{GeV}$ listed in Tab. 1. Usually, the
$|V_{cs}|$ is obtained from the semi-leptonic $D$ or leptonic $D_{s}$ decays,
which may lead circular argument problem in this study. Hence a value of
$|V_{cs}|$ without involving semi-leptonic $D$ is necessary to validate this
study. With measured $|V_{cd}|$ [5], $|V_{cb}|$ [5], and the unitarity of the
CKM matrix, a $|V_{cs}|$ without involving semi-leptonic $D$ is calculated
using Eq. 5, that is determined to be $0.974\pm 0.001$.

$\displaystyle|V_{cs}|=\sqrt{1-|V_{cd}|^{2}-|V_{cb}|^{2}}$ (5)

The Fig. 6 depicts the corresponding fitted $\alpha_{S}(m_{c}^{2})$ values for
different combinations with fixed $m_{c}$ and $|V_{cs}|$, whose values are
listed in Tab. 5.

Figure 6: The corresponding fitted $\alpha_{S}$ values for different combinations with fixed $m_{c}$ and $|V_{cs}|$. Table 5:  The $m_{c}$ and $|V_{cs}|$ are fixed in the fit. The $m_{c}^{\mathrm{fix}}$ and $\alpha_{S}(m_{c}^{2})$ of each combination are listed. The first uncertainty of $\alpha_{S}$ is the experimental uncertainty and second is the theoretical uncertainty. Combination |  $m_{c}^{\mathrm{fix}}$ [$\mathrm{GeV}$] |  $\alpha_{S}(m_{c}^{2})$ [$10^{-3}$] |  $\alpha_{S}(m_{Z}^{2})$ [$10^{-4}$]
---|---|---|---
$D^{0}$ | $1.370\pm 0.034$ | $450\pm 13\pm 116$ | $1178^{+7+55}_{-7-75}$
$D^{+}$ | $444\pm 12\pm 116$ | $1175^{+6+56}_{-7-77}$
$D^{+}_{s}$ | $402\pm 14\pm 114$ | $1149^{+9+70}_{-10-98}$
$D^{0,+}$ | $447\pm 9\pm 115$ | $1177^{+5+56}_{-5-76}$
$D^{0,+}$, $D^{+}_{s}$ | $433\pm 7\pm 115$ | $1168^{+4+61}_{-4-80}$

The results with fixed and floated $m_{c}$ and $|V_{cs}|$ are consistent
within 1$\sigma$, which further increases the confidence of this study.

##  IV Conclusion

We have determined the $\alpha_{S}(m_{c}^{2})$ for the first time by a
combined fit to the semi-leptonic decay widths of charmed mesons. The
$\alpha_{S}(m_{c}^{2})$ at $m_{c}=1.4\ \textrm{GeV}$ is determined to be
$0.377\pm 0.008_{\textrm{exp.}}\pm 0.114_{\textrm{the.}}$, which is in good
agreement with the value running from $\alpha_{S}(m^{2}_{Z})$. As well as the
values of $\alpha_{S}(m_{c}^{2})$ determined from different charmed mesons are
consistent. The measurement is dominated by the theoretical uncertainty, which
can be improved to achieve more precise $\alpha_{S}(m_{c}^{2})$ in future. In
addition, the leptonic energy moments and $q^{2}$ moments [15, 19, 22] can
also be utilized to extract the $\alpha_{S}(m_{c}^{2})$ when the experimental
measurements and theoretical predictions are available. Furthermore, the
inclusive radiative decay of charmed mesons can also help to measure
$\alpha_{S}(m_{c}^{2})$ using the huge data samples at BESIII. Another
possible improvement from the experimental measurements is the separation of
$c\rightarrow sl\nu$ from the inclusive semi-leptonic decays of charmed
mesons. Hence, the precise measurements of the inclusive semi-leptonic decays
of charmed hadrons are necessary, which can be benefited from the huge data
sets at BESIII and its upgrade programme [44]. All in all, this study propose
a new method to measure $\alpha_{S}$ at the momentum transfer region lower
than all of existing measurements, consequently an approach to investigate the
nature of QCD from non-perturbative to perturbative region.

###### Acknowledgements.

The authors thank H. B. Li and X. T. Huang for fruitful discussions. This work
is supported by the National Natural Science Foundation of China under grants
No.12247119 and No. 12042507.

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