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We calculate the set of O(alpha_s) corrections to the double differentialdecay width dGamma_77/(ds_1*ds_2) for the process B -> X_s gamma gammaoriginating 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. While the(renormalized) virtual corrections are worked exactly for a certain range ofs_1 and s_2, we retain in the gluon bremsstrahlung process only the leadingpower w.r.t. the (normalized) hadronic mass s_3=(p_b-q_1-q_2)^2/m_b^2 in theunderlying triple differential decay width dGamma_77/(ds_1*ds_2*ds_3). Thedouble differential decay width, based on this approximation is free ofinfrared- and collinear singularities when combining virtual- andbremsstrahlung corrections. The corresponding results are obtainedanalytically. When retaining all powers in s_3, the sum of virtual- andbremstrahlung corrections contains uncanceled 1/epsilon singularities (whichare due to collinear photon emission from the s-quark) and other concepts,which go beyond perturbation theory, like parton fragmentation functions of aquark or a gluon into a photon, are needed which is beyond the scope of ourpaper.

NLL QCD contribution of the Electromagnetic Dipole operator to B-> X_s gamma gamma