The width difference in \bbm\ at order $\alpha_s$ and beyond

TL;DR

This paper completes the calculation of the decay matrix element 12^q in B_q-12_q mixing at order lpha_s, including next-to-next-to-leading-order corrections, refining Standard Model predictions for elta_s/elta_s.

Contribution

It provides the first comprehensive calculation of 12^q at order lpha_s and beyond, including two-loop penguin and chromomagnetic operator contributions.

Findings

Standard-Model prediction elta_s/elta_s = (5.20 b1 0.69) d7 10^{-3} with ar MS b-quark mass.

Alternative prediction elta_s/elta_s = (4.70 b1 0.96) d7 10^{-3} using pole mass.

Includes two-loop penguin and chromomagnetic operator corrections at NNLO.

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 $\overline{\rm MS}$ 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.