$\mathcal{O}(\alpha_s^3)$ corrections to semileptonic $b \to c$ decays in the heavy daughter approximation

TL;DR

This paper calculates third-order quantum corrections to semileptonic $b o c$ decays and muon decays, improving theoretical precision for fundamental parameters like $|V_{cb}|$ and $G_F$.

Contribution

It provides the first third-order correction calculations for $b o c$ and muon decays using an expansion around $m_c o m_b$, with improved convergence and scheme-dependent correction reduction.

Findings

Large perturbative corrections in on-shell scheme

Significant reduction of corrections in kinetic scheme

Enhanced precision for $|V_{cb}|$ and $G_F$ determinations

Abstract

We present our recent calculation of the third order corrections to the semileptonic $b \to c$ and the muon decays. The calculation has been performed in an expansion around the limit $m_c \sim m_b$, but shows decent convergence even for $m_c=0$ from which the contribution to the muon decay can be extracted. For the semileptonic $b \to c$ decay we find large perturbative corrections in the on-shell scheme which can be significantly reduced by changing to the kinetic scheme for the bottom quark mass. These results are important input for the inclusive determination of $|V_{cb}|$ and the Fermi coupling constant $G_F$.