The width difference in \bbm\ at order ${\alpha }_s$ and beyond

Abstract

We complete the calculation of the element ${\Gamma }_{12}^q$ of the decay matrix in $B_q-{\bar{B}}_q$ mixing, $q=d,s$, to order ${\alpha }_s$ in the leading power of the Heavy Quark Expansion. To this end we compute one- and two-loop contributions involving two four-quark penguin operators. Furthermore, we present two-loop QCD corrections involving a chromomagnetic operator and either a current-current or four-quark penguin operator. Such contributions are of order ${\alpha }_s^2$, i.e. next-to-next-to-leading-order. We also present one-loop and two-loop results involving two chromomagnetic operators which are formally of next-to-next-to-leading and next-to-next-to-next-to-leading-order, respectively. With our new corrections we obtain the Standard-Model prediction $\Delta {\Gamma }_s/\Delta M_s=(5.20\pm 0.69)\cdot 10^{-3}$ if ${\Gamma }_{12}^s$ is expressed in terms of the $\overline{\mathrm{MS}}$ b-quark mass, while we find $\Delta {\Gamma }_s/\Delta M_s=(4.70\pm 0.96)\cdot 10^{-3}$ instead for the use of the pole mass.

Comments: 36 pages