Determination of the Strong Coupling Constant $α_s$ from Inclusive Semi-leptonic $B$ Meson Decays

TL;DR

This work investigates determining the strong coupling constant at a low-to-intermediate scale by analyzing the inclusive semileptonic $B$ decays within the Heavy Quark Expansion. By expressing the decay width $Γ(B\to X_c\ell\nu)$ in the $\overline{MS}$ scheme at $μ=5$ GeV and mapping it to $α_s(5\mathrm{GeV})$, the authors extract $α_s(5\mathrm{GeV})$ from world-average $B$ decay widths while fixing $|V_{cb}|$, $\overline{m}_b(\overline{m}_b)$, and $\overline{m}_c(\overline{m}_c)$ and including HQE parameters. The combined fit yields $α_s(5\mathrm{GeV})=0.245\pm0.009$, corresponding to $α_s(m_Z)=0.1266\pm0.0023$, with the dominant uncertainties arising from the perturbative expansion and $|V_{cb}|$; future higher-order calculations and improved measurements could reduce the total uncertainty to about $Δα_s(m_Z)\sim0.0018$, rivaling the precision from $τ$ decays. This demonstrates a viable, complementary method to determine $α_s$ at around the $5$ GeV scale and informs future collider programs.

Abstract

We demonstrate the feasibility of determining the strong coupling constant, $α_s$, from the inclusive semileptonic decay width of $B$ mesons. We express the semileptonic $B$ decay width as a function of $α_s(5\mathrm{\,GeV})$, the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$, $b$- and $c$-quark masses in the $\overline{\mathrm{MS}}$ scheme. We fit $α_s(5\mathrm{\,GeV})$ to current world averages of the $B^{\pm}$ and $B^{0}$ semileptonic decay widths. This yields $α_s(5\mathrm{\,GeV}) = 0.245 \pm 0.009$, corresponding to a 5-flavor extrapolation of $α_s(m_{Z}) = 0.1266 \pm 0.0023$. The primary uncertainty contributions arise from the uncertainty on the perturbative expansion and the value of $|V_{cb}|$. Future advancements including higher-order perturbative calculations, and precise measurements of $|V_{cb}|$ and $B$ decay widths from upcoming $B$ and $Z$ factories, could enable this method to determine $α_s(m_{Z})$ with a competitive precision of $Δα_s(m_{Z}) \sim 0.0018$. This precision is comparable to the current accuracy of $α_s(m_{Z})$ measurements from $τ$-lepton decays, which is regarded as the most precise experimental approach.

Figures (3)

• Figure 1: The parton level Feynman diagram of semileptonic $B$ decay.
• Figure 2: The numerical function of $\Gamma(B\to X_{c}\ell \nu)$ versus $\alpha_s(5\mathrm{\,GeV})$, compared with the $\Gamma(B^{\pm}\to X_{c} \ell\nu) = (4.34\pm 0.16)\times 10^{-14} \mathrm{\,GeV}$ derived from Eq. \ref{['eq:decay_width']}. The numerical function is parameterized by a polynomial function in the $\alpha_s(5\mathrm{\,GeV})$ range from $0.16$ to $0.26$.
• Figure 3: (Top) The combined $\alpha_s(5\mathrm{\, GeV})$ result (Eq. \ref{['eq:res']}) compared with the $\alpha_s$ measurements at other energy scales Wu:2024jyfBoito:2020xliBaikov:2008jhboito2018StrongaNarison:2018xbjCMS:2016lnaD0:2009wsrD0:2012xifATLAS:2017qirCMS:2014mnaSchieck:2012mpDissertori:2009ikBethke:2009ehn. (Bottom) The comparison of the $\alpha_s(m_{Z})$ pre-averages from six experimental sub-fields in PDG pdg and the extrapolated values from this work. Additionally, the $\alpha_s(m_{Z})$ derived using different values of $|V_{cb}|$ are also compared for reference, including the PDG average pdg for inclusive determination ($|V_{cb}|^{\text{PDG}}_{\text{inc}}$), exclusive determination ($|V_{cb}|^{\text{PDG}}_{\text{exc}}$), their average ($|V_{cb}|^{\text{PDG}}_{\text{ave}}$), exclusive $|V_{cb}|$ from HFLAV group ($|V_{cb}|^{\text{HFLAV}}_{\text{exc}}$) Banerjee:2024znd, and recent exclusive determination from Belle ($|V_{cb}|^{\text{Belle 2024}}_{\text{exc}}$) Belle:2023xgj. The extrapolation of $\alpha_s$ along the energy scale is conducted using the RunDec package chetyrkin2000RunDec.