NLL QCD contribution of the electromagnetic dipole operator to B -> X_s gamma gamma with a massive strange quark

Abstract

We calculate the O(alpha_s) corrections to the double differential decay width dGamma_{77}/(ds_1 ds_2) for the process B -> X_s gamma gamma originating from diagrams involving the electromagnetic dipole operator O_7. The kinematical variables s_1 and s_2 are defined as s_i=(p_b - q_i)^2/m_b^2, where p_b, q_1, q_2 are the momenta of b-quark and two photons. We introduce a nonzero mass m_s for the strange quark to regulate configurations where the gluon or one of the photons become collinear with the strange quark and retain terms which are logarithmic in m_s, while discarding terms which go to zero in the limit m_s -> 0. When combining virtual- and bremsstrahlung corrections, the infrared and collinear singularities induced by soft and/or collinear gluons drop out. By our cuts the photons do not become soft, but one of them can become collinear with the strange quark. This implies that in the final result a single logarithms of m_s survives. In principle the configurations with collinear photon emission could be treated using fragmentation functions. In a related work we found that similar results can be obtained when simply interpreting m_s appearing in the final result as a constituent mass. We do so in the present paper and vary m_s between 400 MeV and 600 MeV in the numerics. This work extends a previous paper of us, where only the leading power terms w.r.t. the (normalized) hadronic mass s_3=(p_b-q_1-q_2)^2/m_b^2 were taken into account in the underlying triple differential decay width dGamma_{77}/(ds_1 ds_2 ds_3).