Completing the NLO QCD calculation of B -> Xs gamma

TL;DR

This work completes the NLO QCD calculation for the decay $\bar{B} \to X_s \gamma$ by computing two-loop matrix elements of all four-quark operators with no derivatives, without relying on the $m_c/m_b$ expansion and including the QCD-penguin operators. The authors derive both unrenormalized and MS-bar renormalized amplitudes, update the relevant magic numbers, and provide explicit $z=m_c^2/m_b^2$-dependent results through functions $a(z)$ and $b(z)$. The resulting branching ratio in the SM with $E_\gamma>1.6$ GeV is $BR = (3.57 \pm 0.30) \times 10^{-4}$, with the penguin contributions yielding only about a 1% effect due to small Wilson coefficients. The exact treatment of the charm mass and the inclusion of operators beyond the previously handled set extend the accuracy and applicability of the NLO prediction to SM extensions and new physics analyses.

Abstract

We evaluate two-loop b -> s gamma matrix elements of all the four-quark operators containing no derivatives. Contrary to previous calculations, no expansion in the mass ratio mc/mb is performed, and all the possible Dirac and flavor structures are included. Consequently, we are able to provide the last item in the NLO analysis of B -> Xs gamma that has been missing so far, namely the two-loop matrix elements of the QCD-penguin operators. Due to smallness of the Wilson coefficients of those operators in the Standard Model, their effect on the branching ratio is small: a reduction by roughly 1%. We find BR[ B -> Xs gamma]_{E_gamma > 1.6 GeV} = (3.57 +_ 0.30)*10^{-4}.

Figures (2)

• Figure 1: Two-loop 1PI contributions to the matrix elements of $P_k$.
• Figure 2: Functions a(z) and b(z). Dotted lines represent their expansions at $z=0$ up to ${\cal O}(z^6)$.