Towards next-to-next-to-leading-log accuracy for the width difference in the $B_s-\bar{B}_s$ system: fermionic contributions to order $(m_c/m_b)^0$ and $(m_c/m_b)^1$

TL;DR

This paper computes three-loop fermionic contributions to the width difference in the $B_s-ar{B}_s$ system, improving the precision of theoretical predictions at next-to-next-to-leading-logarithmic accuracy.

Contribution

It provides the first calculation of fermionic three-loop diagrams for $ au ext{-} ar{ au}$ width difference, reducing scheme dependence and refining perturbative error estimates.

Findings

Significant correction in pole mass scheme

Smaller correction in $ar{MS}$ scheme

Reduced scheme dependence and better error estimation

Abstract

We calculate a class of three-loop Feynman diagrams which contribute to the next-to-next-to-leading logarithmic approximation for the width difference $\Delta\Gamma_{s}$ in the $B_s-\bar{B}_s$ system. The considered diagrams contain a closed fermion loop in a gluon propagator and constitute the order $\alpha_s^2 N_f$, where $N_f$ is the number of light quarks. Our results entail a considerable correction in that order, if $\Delta\Gamma_{s}$ is expressed in terms of the pole mass of the bottom quark. If the $\overline{MS}$ scheme is used instead, the correction is much smaller. As a result, we find a decrease of the scheme dependence. Our result also indicates that the usually quoted value of the NLO renormalization scale dependence underestimates the perturbative error.