Two-loop QCD Corrections to $b \to c$ Transitions at Zero Recoil: Analytical Results

TL;DR

The paper resolves the $O(\alpha_s^2)$ QCD corrections to zero-recoil $b\to c$ transitions by delivering analytical expressions for the corrections to the axial and vector form factors, $\eta_A$ and $\eta_V$. It develops a robust two-scale two-loop calculation framework, employing partial fractioning, IBP reductions, and a dispersion-based treatment of massive fermion loops to obtain closed-form results involving polylogarithms and the auxiliary functions $R_1$, $R_2$. The results are decomposed by color factors and shown to agree with prior zerorecoil approximations, while providing full dependence on the mass ratio $\omega=m_c/m_b$ and yielding useful expansions near $\omega=1$ and $\omega=0$. The work enhances the precision of $|V_{cb}|$ extractions from exclusive decays and demonstrates a practical methodology for complex multi-loop, multi-mass calculations in heavy-quark physics.

Abstract

We present analytical results for the $O(α_s ^2)$ contributions to the functions $η_A$ and $η_V$ which parameterize QCD corrections to semileptonic $b \to c$ transitions at zero recoil. Previously obtained approximate results are confirmed. The methods of computing the relevant two-loop diagrams with two mass scales are discussed in some detail.

Figures (2)

• Figure 1: Two-loop QCD corrections to the $b\to c$ transitions at zero recoil. Symbols $\otimes$ mark places where a virtual $W$ boson can possibly couple to the quark line.
• Figure 2: Coefficient functions $\eta^F$ (solid), $\eta^{H}$ (dashed), $\eta^{AF}$ (dotted), for the vector (a) and axial case (b), plotted against $\omega = 0..0.8$; none of these functions varies noticably between $\omega = 0.8$ and $\omega = 1$.