Two loop virtual corrections to $b\to (d,s){\ell }^{+}{\ell }^{-}$ and $c\to u{\ell }^{+}{\ell }^{-}$ for arbitrary momentum transfer

Abstract

Non-factorizable two loop corrections to heavy to light flavor changing neutral current transitions due to matrix elements of current-current operators are calculated analytically for arbitrary momentum transfer. This extends previous works on $b\to (d,s){\ell }^{+}{\ell }^{-}$ transitions. New results for $c\to u{\ell }^{+}{\ell }^{-}$ transitions are presented. Recent works on polylogarithms are used for the master integrals. For $b\to s{\ell }^{+}{\ell }^{-}$ transitions, the corrections are most significant for the imaginary parts of the effective Wilson coefficients in the large hadronic recoil range. Analytical results and ready-to-use fitted results for a specific set of parameters are provided.

Comments: 14 pages plus references and appendix. The results are supplemented to the source files of this paper; v2: typos corrected and clarifications and references added, supplemented files (+ analytical results in terms of GPLs) and description thereof improved, results unchanged; v3: files and text (fitted results for negative momentum transfer and factorizable results) added, results unchanged