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 $(m_c/m_b)^1$

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 $ΔΓ_{s}$ in the $B_s-\bar{B}_s$ system. The considered diagrams contain a closed fermion loop in a gluon propagator and constitute the order $α_s^2 N_f$, where $N_f$ is the number of light quarks. Our results entail a considerable correction in that order, if $ΔΓ_{s}$ 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.

We corrected a typographical mistake in Eq. (4.18), made larger axis labels in Fig.2. Version accepted by JHEP