TTP06-24 NNLO QCD corrections to the ${\bar B}\to X_s\gamma$ matrix elements using interpolation in $m_c$

One of the most troublesome contributions to the NNLO QCD corrections to ${\bar B}\to X_s\gamma$ originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient $\beta_0$ was found in 2003 as an expansion in $m_c/m_b$. In the present paper, we evaluate the asymptotic behaviour of the complete contribution for $m_c \gg m_b/2$. The asymptotic form of the $\beta_0$-part matches the small-$m_c$ expansion very well at the threshold $m_c = m_b/2$. For the remaining part, we perform an interpolation down to the measured value of $m_c$, assuming that the $\beta_0$-part is a good approximation at $m_c=0$. Combining our results with other contributions to the NNLO QCD corrections, we find ${\cal B}({\bar B}\to X_s\gamma) = (3.15 \pm 0.23) \times 10^{-4}$ for $E_{\gamma} > 1.6\;$GeV in the ${\bar B}$-meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: non-perturbative (5\%), parametric (3\%), higher-order %(${\cal O}(\alpha_s^3)$) perturbative (3\%), and the interpolation ambiguity (3\%).