QED factorization of two-body non-leptonic and semi-leptonic $B$ to\n charm decays

Abstract

The QCD$\\times$QED factorization is studied for two-body non-leptonic and\nsemi-leptonic $B$ decays with heavy-light final states. These non-leptonic\ndecays, like $\\bar{B}^0_{(s)}\\to D^+_{(s)} \\pi^-$ and $\\bar{B}_d^0 \\to D^+\nK^-$, are among the theoretically cleanest non-leptonic decays as penguin loops\ndo not contribute and colour-suppressed tree amplitudes are suppressed in the\nheavy-quark limit or even completely absent. Advancing the theoretical\ncalculations of such decays requires therefore also a careful analysis of QED\neffects. Including QED effects does not alter the general structure of\nfactorization which is analogous for both semi-leptonic and non-leptonic\ndecays. For the latter, we express our result as a correction of the tree\namplitude coefficient $a_1$. At the amplitude level, we find QED effects at the\nsub-percent level, which is of the same order as the QCD uncertainty. We\ndiscuss the phenomenological implications of adding QED effects in light of\ndiscrepancies observed between theory and experimental data, for ratios of\nnon-leptonic over semi-leptonic decay rates. At the level of the rate,\nultrasoft photon effects can produce a correction up to a few percent,\nrequiring a careful treatment of such effects in the experimental analyses.\n