1 Introduction↩︎

The strong interaction, one of the fundamental interactions in nature, is described by quantum chromodynamics (QCD). The strong coupling constant, \(\alpha_s(\mu)\), characterizes the strength of this interaction and exhibits a decreasing trend with increasing energy scale \(\mu\). This running behavior is described by the renormalization group equation (RGE) [1], reflecting essential properties of the strong interaction, such as quark confinement at long distances and asymptotic freedom at short distances. Consequently, precise knowledge of \(\alpha_s(\mu)\) across the entire range of energy scale is crucial for a comprehensive understanding and testing of QCD. \(\alpha_s\) at low energy scale has been studied through various methodologies, including hadron production in electron-positron annihilation [2], semileptonic charmed meson [3] and \(\tau\) decays [2], [4]–[9], and inclusive hadronic decay of heavy quarkonia [10], [11]. However, there are relatively few measurements of \(\alpha_s\) in the energy scale range around \(5\mathrm{\,GeV}\).

We consider measuring \(\alpha_s\) from the inclusive semileptonic \(B\) decay which corresponds to the energy scale of \(B\) meson masses. Figure 1 shows the Feynman diagram for the inclusive semileptonic \(B\) decay (\(B \to X \ell\nu\)) at the tree level in the parton model. This process consists of two components: \(B\to X_{c} \ell\nu\) and \(B\to X_{u} \ell\nu\), where \(X_{c}\) represents the charmed system and \(X_{u}\) the light hadron system. The ratio of \(B\to X_{u} \ell\nu\) is approximately 65 times less than the former due to Cabibbo suppression. Using the Heavy Quark Expansion (HQE) method, the branching ratio and the spectral moments of kinematic observables have been parameterized as functions of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, \(|V_{cb}|\) and \(|V_{ub}|\), the strong coupling constant (\(\alpha_s\)), \(b\)-quark mass (\(m_{b}\)), \(c\)-quark mass (\(m_{c}\)) and non-perturbative HQE parameters [12]–[21]. On the experimental side, these observables have been measured by the BaBar, Belle, and Belle II collaborations over the past two decades [22]–[25].

The \(B\to X_c \ell\nu\) process was used to determine \(|V_{cb}|\), \(m_{b}\) and \(m_{c}\), with \(\alpha_s\) fixed at the value extrapolated from the world average of \(\alpha_s(m_{Z})\) [19], [26]–[28]. Nowadays, more precise determinations of the \(|V_{cb}|\), \(m_{b}\) and \(m_{c}\) are available, for example, \(|V_{cb}|\) from exclusive \(B\) decays [29]–[33] or \(W\) decays [34], [35]; heavy quark masses from lattice QCD [36]–[40], \(b\)- and \(c\)-meson masses [41]–[44], or \(e^+e^- \to \text{hadrons}\) cross-section [45]–[49], etc. Therefore, by fixing the values of \(|V_{cb}|\), \(m_{b}\) and \(m_{c}\) according to those progresses, we could extract \(\alpha_s\) using the semileptonic \(B\) decay width at the scale around the \(B\) meson masses.

2 Theoretical model↩︎

Within the Heavy Quark Expansion (HQE) framework, the inclusive semileptonic \(B\) decay width takes the form shown in 1 . \[\begin{align} \Gamma\left(B \rightarrow X_c \ell \bar{\nu}_{\ell}\right)=\Gamma_0[ & C_0-C_{\mu_\pi} \frac{\mu_\pi^2}{2 m_b^2}+C_{\mu_G} \frac{\mu_G^2}{2 m_b^2}\\ & -C_{\rho_D} \frac{\rho_D^3}{2 m_b^3}-C_{\rho_{L S}} \frac{\rho_{L S}^3}{2 m_b^3}+\cdots], \end{align} \label{eq:gammaB}\tag{1}\]

where \(\Gamma_0 \equiv \frac{G^2_F |V_{cb}|^2 m_b^5 A_{\mathrm{ew}}}{192 \pi^3}\), with \(G_F = 1.16637886 \times 10^{-5} \mathrm{~GeV^{-2}}\) is the Fermi coupling constant, and \(A_{\mathrm{ew}} = 1.014\) is the electroweak correction factor [51]. The coefficients \(C_{i}\) (\(i=0, \mu_\pi, \mu_G\)) depend on the ratio of the squared \(c\)- and \(b\)-quark masses, \(\rho = \frac{m_c^2}{m_b^2}\), and have perturbative expansions in \(\alpha_s\). The \(\mu_{\pi}^2\), \(\mu_{G}^2\), \(\rho_D^3\) and \(\rho_{L S}^3\) are the HQE parameters corresponding to the kinetic, chromomagnetic, Darwin and spin-orbital terms, respectively. For the purpose of determining \(\alpha_s\), a numerical mapping between \(\Gamma\left(B \rightarrow X_c \ell \bar{\nu}_{\ell}\right)\) and \(\alpha_s\) has been established, considering the leading order power correction up to the fourth order, as well as the leading order contributions from the four terms of high order power corrections. The precision of the HQE parameters, the next-to-leading order contributions of the high order power corrections, and the contributions of order \(\mathcal{O}(1/m_b^{4,5})\) have been included in the error.

The perturbation expansion of \(C_0\) can be expressed as: \(C_0 = \mathbf{c}_0 + \mathbf{c}_1 \frac{\alpha_s}{\pi} + \mathbf{c}_2 \Big(\frac{\alpha_s}{\pi}\Big)^2 + \mathbf{c}_3 \Big(\frac{\alpha_s}{\pi}\Big)^3 + \mathcal{O}(\alpha_s^4)\), where the leading term \(\mathbf{c}_0 = 1 - 8\rho + 8\rho^3 - \rho^4 - 12\rho^2\operatorname{ln} \rho\) is the tree-level phase space factor [13]. The results for the second-order [15]–[17] and third-order [20], [28] perturbative corrections have been provided in the on-shell scheme. For better perturbative convergence, these perturbative QCD results are reformulated in the \(\overline{\text{MS}}\) scheme, at the renormalization scale \(\mu = 5~\mathrm{GeV}\) the mass scale of the decaying \(B\) meson. For the purpose of extracting \(\alpha_s(5\mathrm{\,GeV})\) in 5-flavor scheme, the perturbative correction \(C_0\) is reformulated consistently in terms of the \(\overline{\text{MS}}\)-renormalized quark masses \(\overline{m}_b(\mu)\), \(\overline{m}_c(\mu)\) and \(\alpha_s(\mu)\). The values for the arguments \(\overline{m}_b(5\mathrm{\,GeV})\) and \(\overline{m}_c(5\mathrm{\,GeV})\) (in 5-flavor scheme) are derived by solving the RGE system with the boundary conditions for \(\overline{m}_b(\mu), \overline{m}_c(\mu)\) and \(\alpha_s(\mu)\) set as following: the Particle Data Group (PDG) average values of \(\overline{m}_c(\overline{m}_c) = 1.27 \pm 0.02 \mathrm{\,GeV}\) and \(\overline{m}_b(\overline{m}_b) = 4.18^{+0.03}_{-0.02} \mathrm{\,GeV}\) [50], and the sampled values of \(\alpha_s(5\mathrm{\,GeV})\) in the fit. In this way the perturbative correction to \(\Gamma\left(B \rightarrow X_c \ell \bar{\nu}_{\ell}\right)\) up to \(\mathcal{O}(\alpha_s^3)\) in the leading-power correction is eventually expressed as a numerical function of \(\alpha_s(5\mathrm{\,GeV})\).

The numerical calculations for the coefficients \(C_{\mu_{\pi}}\) and \(C_{\mu_{G}}\) were detailed in [52]. For the \(1/m_b^3\) power corrections, the coefficients related to the Darwin term and spin-orbital were discussed in [53], [54]. The HQE parameters \(\mu_{\pi}^2\), \(\mu_{G}^2\), \(\rho_{D}^3\), and \(\rho_{LS}^3\) have been measured through a simultaneous fit using the spectral moments of semileptonic \(B\) decays [28]. In addition, the \(\mu_{\pi}^2\) and \(\mu_{G}^2\) can also be derived from the mass splitting of \(B\) mesons [55]. The determination of \(\rho_{D}^3\) is expected to be achieved with high precision using \(D\) decay data from BESIII [56]. Using leading-order approximations of \(1/m_b^{2,3}\) power corrections (Eq. (4.1) in [52] and Eq. (23) in [53]) and parameters listed in Table 1, we estimate the high order power corrections to decrease the decay width by \(\sim 7\%\), relative to \(\Gamma_0\mathbf{c}_0\). ¹

Overall, the numerical mapping from the assumed values of \(\alpha_s(5\mathrm{\,GeV})\) to the decay width \(\Gamma(B\to X_c \ell\nu)\) is shown in Fig. 2. The involved parameters are summarized in Table 1. Within the range of \(\alpha_s(5\mathrm{\,GeV}) \in [0.16,\, 0.26]\), the decay width dependence is parameterized by a fifth-order polynomial: \(\Gamma\left(B \rightarrow X_c \ell \bar{\nu}_{\ell}\right) = (1.01\times 10^{-14}\mathrm{\,GeV})\, \big(-1 + 88.9\, \alpha_s^{\mathrm{fit}} - 904.8\, (\alpha_s^{\mathrm{fit}})^2 + 4946.4\, (\alpha_s^{\mathrm{fit}})^3 - 13467.1\, (\alpha_s^{\mathrm{fit}})^4 + 15502.7\, (\alpha_s^{\mathrm{fit}})^5 \big)\) where \(\alpha_s^{\mathrm{fit}} \equiv \alpha_s(5\mathrm{\,GeV})\).

3 Result and discussion↩︎

We fit the value of \(\alpha_s(5\mathrm{\,GeV})\) to the inclusive semileptonic decay widths of the \(B^{\pm}\) and \(B^{0}\) mesons. The experimental decay width can be obtained from measured values for the lifetime and semileptonic decay branching ratio: \[\Gamma_{sl} = \frac{\hbar}{\tau} \mathcal{B}_{sl} \,. \label{eq:decay95width}\tag{2}\] We quote the world averages for the lifetime [50] and the partial branching ratios (with a cut on the lepton energy, \(E_{l} > 0.4\mathrm{\,GeV}\)) measured by the Belle experiment [24], \[\tau_{B^{\pm}} = 1.638 \pm 0.004 \mathrm{\,ps}, \quad \mathcal{B}(B^{\pm} \to X_{c} \ell\nu) = 10.8 \pm 0.4 \%,\] \[\tau_{B^{0}} = 1.517 \pm 0.004 \mathrm{\,ps}, \quad \mathcal{B}(B^{0} \to X_{c} \ell\nu) = 10.1 \pm 0.4 \%.\] The total branching ratios are obtained by scaling the particle branching ratios by a factor of \(1.015\), according to [26]. As a result, we obtain \(\Gamma(B^{\pm}) = (4.40 \pm 0.16) \times 10^{-14} \mathrm{\,GeV}\) and \(\Gamma(B^{0}) = (4.44 \pm 0.17) \times 10^{-14} \mathrm{\,GeV}\). For the related parameters, \(|V_{cb}|\) is fixed at the PDG world average value [50], which is extracted from the \(\overline{B} \rightarrow D^* \ell \bar{\nu}_{\ell}\) decays (with \(\ell = e, \mu\)) along with the lattice QCD calculation of the form factors, independent of the perturbative \(\alpha_s\). The minimum-\(\chi^2\) fit incorporates the experimental errors in the decay widths, as well as the theoretical uncertainties introduced by parameters and the numerical expression of HQE. These uncertainties are considered to be independent of each other.

The fit yields \(\alpha_s(5\mathrm{\,GeV}) = 0.244\pm 0.013\) from \(\Gamma(B^{\pm}\to X_{c} \ell\nu)\), and \(\alpha_s(5\mathrm{\,GeV}) = 0.246\pm 0.013\) from \(\Gamma(B^{0}\to X_{c} \ell\nu)\). Combining the two fits gives: \[\alpha_s(5\mathrm{\,GeV})= 0.245\pm 0.009. \label{eq:res}\tag{3}\] The combined results are shown and compared with other determinations in Fig. 3.

The uncertainty on \(\alpha_s\) arises from three sources: the experimental measurements of the branching ratio and lifetime, the theoretical uncertainty induced by uncertainties on the used parameters and perturbation expansion for the coefficients of HQE. These terms of the \(\alpha_s\) uncertainty are listed in Table 2, taking the fit with \(\Gamma(B^{\pm} \to X_{c} \ell\nu)\) as an example. The uncertainty in \(|V_{cb}|\) propagates to \(\Gamma_{sl}\) via \(\Gamma_{0}\) as follows: \[\left.\frac{\sigma (\Gamma)}{\Gamma}\right|_{|V_{cb}|} = 2 \frac{\sigma(|V_{cb}|)}{|V_{cb}|}\] contributing a relative uncertainty of \(3.0\%\) on \(\Gamma_{sl}\). The errors induced by the uncertainties on the input values of \(\overline{m}_{c}(\overline{m}_{c})\) and \(\overline{m}_{b}(\overline{m}_{b})\) are estimated by taking the largest deviations from varying their values within the respective error bounds. The results show that uncertainties in \(\overline{m}_{c}(\overline{m}_{c})\) and \(\overline{m}_{b}(\overline{m}_{b})\) contribute about \(2\%\) and \(3\%\) relative uncertainty to the \(\Gamma_{sl}\) prediction, respectively. The uncertainty due to the remnant renormalization scale dependence (R-scale uncertainty) of the leading-order power correction is estimated by varying \(\mu\) from \(2.5\mathrm{\,GeV}\) to \(10\mathrm{\,GeV}\). It leads to about \(-2\%\) to \(-4.4\%\) variations in the perturbative corrections relative to the result at \(\mu = 5\mathrm{\,GeV}\). The larger variation is taken as the estimation of the uncertainty. The uncertainty of the high order power corrections are estimated by summing up two contributions quadratically. First, the errors on the \(\mu_{\pi}^2\), \(\mu_{G}^2\), \(\rho_{D}^3\), \(\rho_{LS}^3\), and \(m_{b}^{\text{kin}}\) are considered, leading an uncertainty of about \(0.9\%\) on the theoretical prediction for \(\Gamma\left(B \rightarrow X_c \ell \bar{\nu}_{\ell}\right)\). The second component is the truncation error. The next-to-leading order contributions of \(C_{\mu_\pi} \frac{\mu_\pi^2}{2 m_b^2}\), \(C_{\mu_G} \frac{\mu_G^2}{2 m_b^2}\), \(C_{\rho_D} \frac{\rho_D^3}{2 m_b^3}\) are estimated according to [52] and [57]. By summing the absolute values of these next-leading-order corrections, the truncation error for \(1/m_b^{2,3}\) power corrections are estimated conservatively. Meanwhile, the \(\mathrm{O}(1/m_b^{4,5})\) order power corrections are evaluated to cause a reverse influence by a factor of \(1.3\%\) in [57], which is also added into the total truncation error. As a result, we assign a truncation error of \(2.3\%\) on the high order power corrections. ² The experimental uncertainty of the branching ratio and life-time also contributes to \(\alpha_s(5\mathrm{\, GeV})\) uncertainty.

The result of \(\alpha_s\) fit in 3 is extrapolated to the scale of \(m_Z\), \(\alpha_s(m_{Z}) = 0.1266\pm 0.0023\). As shown in the second plot of Fig. 3, the equivalent \(\alpha_s(m_{Z})\) from this study exhibits accuracy comparable to the PDG pre-averages from other \(\alpha_s\) determination fields. The primary source of the uncertainty arises from the RG-scale uncertainty, which will be refined by future perturbative calculations. Based on the observed reduction in the conventional perturbative QCD scale uncertainty from \(\mathcal{O}(\alpha_s^2)\) to \(\mathcal{O}(\alpha_s^3)\) for \(b \rightarrow c \ell\nu\) and \(b \rightarrow u \ell\nu\), it is plausible to anticipate that the knowledge of the next order result may halve this perturbative uncertainty. The recent lattice QCD results have determined the quark mass with uncertainties around the \(10\mathrm{\,MeV}\) level and further improvements are anticipated. The \(|V_{cb}|\) measurements from \(W\) boson decays are expected to achieve the accuracy of \(\sim 0.7\%\) on the future electron-positron collider [34], [35]. The current measurements of the semileptonic \(B\) decay branching ratios are derived from the \(140\mathrm{\,fb^{-1}}\) of data collected by Belle [24], with statistical and systematic uncertainties being comparable. Among these, the statistical term will decrease by a factor of approximately 20 when the data set increases to \(50\mathrm{\,ab^{-1}}\) on the Belle II [58]. Assuming the systematic uncertainty remains at the same level, the experimental uncertainty of \(B\) decay width will be \(\sim 0.3\%\). All these improvements are scaled to the perspective uncertainties on the \(\Gamma_{sl}\) and \(\alpha_s(5\mathrm{\,GeV})\), and marked in parenthesis in Table 2. Taking into account those advancements, the \(\alpha_s(m_{Z})\) determination could eventually reach \(\pm 0.0018\), halving the precision conducted by this research. This precision is comparable to the current precision of \(\alpha_s\) measurement from \(\tau\) decays, which is considered one of the most precise approaches.

In addition, this fit uses the exclusive determination of \(|V_{cb}|\) as an external input. However, the world average value of the exclusive \(|V_{cb}|\) shows \(3\sigma\) tension with the inclusive one [67]. Consequently, the \(\alpha_s\) determination is entangled with this \(|V_{cb}|\) puzzle. Figure 3 illustrates the impact of the \(|V_{cb}|\) value on the resulting \(\alpha_s(5\mathrm{\,GeV})\). More recently, the exclusive \(|V_{cb}|\) has been determined to be \(|V_{cb}| = (41.0\pm 0.7)\times 10^{-3}\) using the Belle data [33], which is also compared in Fig. 3.

4 Summary↩︎

This manuscript discusses the feasibility to determine \(\alpha_s(5\mathrm{\,GeV})\) from the inclusive semileptonic \(B\) decay width. The theory model is based on the framework of HQE and includes the \(|V_{cb}|\), \(\overline{m}_b(\overline{m}_b)\), \(\overline{m}_c(\overline{m}_c)\), and four HQE parameters. By constrain the other parameters at external determinations listed in Table 1, it is possible to achieve an \(\alpha_s(5\mathrm{\,GeV})\) determination of \(\alpha_s= 0.245\pm 0.009\), corresponding to \(\alpha_s(m_Z) = 0.1266\pm 0.0023\).

The uncertainty is estimated to be comparable to the averages of \(\alpha_s(m_{Z})\) from other experimental methods. The main sources of the uncertainty are estimated in Table 2. With further improvements in perturbation calculations and the measurement accuracy of related parameters, the uncertainty of this method could be halved.

It should be noted that as a fundamental parameter, \(\alpha_s\) influences QCD predictions through multiple parameters and calculations. A challenge in \(\alpha_s\) determination is that theoretical models correlate with prior \(\alpha_s\) assumptions. To address this challenge, we use the measurement of \(|V_{cb}|\) from exclusive \(B\) decays, which is, in principle, independent of the perturbative \(\alpha_s\). We also include the \(\alpha_s\) dependency of the scheme transformation and the scale evolution of the quark masses in the \(\alpha_s\) fitting. However, the values of \(\overline{m}_b(\overline{m}_b)\), \(\overline{m}_c(\overline{m}_c)\) used in our present analysis are taken from the PDG averages and the HQE parameters are determined from simultaneous fit using spectral moments of inclusive \(B\to X_{c}\ell\nu\) decays. These parameters depend on the assumptions of the perturbation \(\alpha_s\). To further mitigate this implicit correlation, one potential future avenue is to reformulate the perturbative corrections using the renormalized quark masses defined in the regularization-independent momentum-subtraction schemes [69]–[73], which can be directly extracted from lattice calculations independently of the perturbative \(\alpha_s\). Alternatively, it is also worth considering a future global fitting of \(\alpha_s\), \(m_{b}\), and \(m_{c}\) using a broader range of observables, such as spectral moments of semileptonic \(B\) and \(D\) decays, and the masses of \(B\) mesons.

References↩︎