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Abstract
Interference between the photonic dipole operator Q7 and the current-current operators Q1,2 gives one of the most important QCD corrections to the ( $$ \overline{B}\to {X}_s\gamma $$ decay rate. So far, the $$ \mathcal{O}\left({\alpha}_{\mathrm{s}}^2\right) $$ ) part of this correction has been known in the heavy charm quark limit only (m c ≫ m b /2). Here, we evaluate this part at m c = 0, and use both limits in an updated phenomenological study. Our prediction for the CP- and isospin-averaged branching ratio in the Standard Model reads $$ {\mathrm{\mathcal{B}}}_{s\gamma}^{\mathrm{SM}}=\left(3.36\pm 0.23\right)\times 1{0}^{-4} $$ for E γ > 1.6 GeV.
Highlights
The final state strangeness in eq (1.2) (−1 for Xs and +1 for Xs) as well as the neutral B-meson flavours have been specified upon ignoring effects of the B0B0 and K0K 0 mixing
Taking the B0B0 mixing into account amounts to using in Γ0 the time-integrated decay rates of mesons whose flavour is fixed at the production time
The steps are as follows: 1. We begin with performing the calculation precisely as it was described in ref. [34] but only shifting from B(B → Xsγ) to Bsγ, which amounts to CP-averaging the perturbative decay widths
Summary
Typical diagrams that had to be evaluated for the present project are shown in figure 1. A few leading terms in the series for each Ii can be found by calculating products of massive tadpole integrals up to three loops and massless propagator ones up to four loops, as illustrated in figure 2. Nl and nb denote numbers of massless and massive (m = mb) quark flavours, while κ = 1 marks contributions from the diagrams in figure 3b describing interferences involving four-body sqqγ final states and no ccγ couplings. Some of the numbers in eq (2.3) have been given in an exact form even though our calculation of the master integrals at x = 1 is purely numerical. Some of the diagrams with Q4 insertions contain b-quark tadpoles that are the only source of 1/ǫ2 terms in G(417)bare, and 1/ǫ terms in G(407)m Such divergences are necessary to renormalize the 1/ǫ3 poles in eq (2.3). The remaining LO and NLO results are extensions of the known ones by another power of ǫ, as necessary for the current calculation.
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