Virtual- and bremsstrahlung corrections to b -> d l+ l- in the standard model

We present the calculation of the virtual- and bremsstrahlung corrections of O(alpha_s) to the matrix elements <d l+ l-|O_i|b>. This is the missing piece in the NNLL results for various observables associated with the process B-> X_d l+ l-, like the branching ratio, the CP-rate asymmetry and the forward-backward asymmetry. This paper is an extension of analogous calculations done by some of us for the process B-> X_s l+ l-. As the contributions of the diagrams induced by the operators O_1^u and O_2^u with a u-quark running in the quark loop are strongly CKM suppressed, they were omitted in the analysis of B->X_s l+ l-. This is no longer possible for B-> X_d l+ l-, as the corresponding contributions are not suppressed. The main new work therefore consists of calculating the O(alpha_s) corrections to <d l+ l-|O_{1,2}^u|b>. In this paper we restrict ourselves to the range 0.05 < s/m_b^2 < 0.25 (s is the invariant mass of the lepton pair), which lies above the rho- and omega-resonances and below the J/psi-resonance. We present the analytic results for the mentioned observables related to the process B-> X_d l+ l- as expansions in the small parameters s/m_b^2, z = m_c^2/m_b^2 and s/(4 m_c^2). In the phenomenological analysis at the end of the paper we discuss the impact of the NNLL corrections on the observables mentioned above.