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We calculate a class of three-loop Feynman diagrams which contribute to thenext-to-next-to-leading logarithmic approximation for the width difference$\Delta\Gamma_{s}$ in the $B_s-\bar{B}_s$ system. The considered diagramscontain 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 entaila considerable correction in that order, if $\Delta\Gamma_{s}$ is expressed interms of the pole mass of the bottom quark. If the $\overline{MS}$ scheme isused instead, the correction is much smaller. As a result, we find a decreaseof the scheme dependence. Our result also indicates that the usually quotedvalue of the NLO renormalization scale dependence underestimates theperturbative error.

the journal of high energy physics/the journal of high energy physics

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 $(m_c/m_b)^1$