The width difference in $$ B-\overline{B} $$ beyond mixing at order αs and

Abstract

We complete the calculation of the element $$ {\Gamma}_{12}^q $$ Γ 12 q of the decay matrix in $$ {B}_q-{\overline{B}}_q $$ B q − B ¯ q mixing, q = d, s, to order αs 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 $$ {\alpha}_s^2 $$ α 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 ∆Γs/∆Ms = (5.20 ± 0.69) · 10−3 if $$ {\Gamma}_{12}^s $$ Γ 12 s is expressed in terms of the $$ \overline{\mathrm{MS}} $$ MS ¯ b-quark mass, while we find ∆Γs/∆Ms = (4.70 ± 0.96) · 10−3 instead for the use of the pole mass.

The width difference in the $B-\bar{B}$ system at next-to-next-to-leading order of QCD

“…The state of the art is as follows: QCD corrections to Γ s 12 are only known for the leading term of the Λ QCD /m b expansion ("leading power"). These include NLO QCD corrections to the contributions with current-current and chromomagnetic penguin operators [26][27][28][29], the corresponding NNLO corrections (and NLO corrections involving four-quark penguin operators) enhanced by the number N f of active quark flavours [30][31][32] as well as NLO results with one current-current and one penguin operator [33] or two penguin operators [34]. The latter paper also presents two-loop results with one or two chromomagnetic penguin operators which are part of the NNLO and N 3 LO contributions.…”

“…[30] and Refs. [33,34] have been calculated in an expansion in m c /m b to first and second order, respectively. ∆Γ s /∆M s further involves a well-computed ratio of two hadronic matrix elements [35][36][37].…”

“…For the ∆B = 1 theory one distinguishes current-current and penguin operators. At leading and next-to-leading orders the current-current operators provide about 90% of the total contribution to Γ ab 12 [34]. Thus, in this work we restrict ourselves to the current-current contributions.…”

“…Ref. [34]). It is necessary to compute the corresponding renormalization constants for the operator mixing up to two-loop order.…”

“…The 1/m b -suppression of R 0 beyond tree-level is manifest only if one is able to distinguish between ultraviolet (UV) and infrared (IR) divergences, e.g., by regularizing the latter using a gluon mass m g . Otherwise, R 0 develops an unphysical evanescent piece E R0 that scales as m 0 b [34] and hence must be included into the definition of R 0 to obtain correct matching coefficients. One cannot isolate E R0 from R 0 at the operator level, but one can distinguish evanescent and physical pieces in the matrix elements: We use R 0 from Eq.…”