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 $\Delta {\Gamma }_s$ 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.

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