Soft-Photon Corrections to $\bar{B}\to D{\tau }^{-}{\bar{\nu }}_{\tau }$ Relative to $\bar{B}\to D{\mu }^{-}{\bar{\nu }}_{\mu }$

Abstract

We evaluate long-distance electromagnetic (QED) contributions to $\bar{B}{}^0\to D^{+}{\tau }^{-}{\bar{\nu }}_{\tau }$ and $B^{-}\to D^0{\tau }^{-}{\bar{\nu }}_{\tau }$ relative to $\bar{B}{}^0\to D^{+}{\mu }^{-}{\bar{\nu }}_{\mu }$ and $B^{-}\to D^0{\mu }^{-}{\bar{\nu }}_{\mu }$, respectively, in the standard model. We point out that the QED corrections to the ratios $R(D^{+})$ and $R(D^0)$ are not negligible, contrary to the expectation that radiative corrections are almost canceled out in the ratio of the two branching fractions. The reason is that long-distance QED corrections depend on the masses and relative velocities of the daughter particles. We find that theoretical predictions for $R(D^{+}{)}^{\tau /\mu }$ and $R(D^0{)}^{\tau /\mu }$ can be amplified by $\sim 4\%$ and $\sim 3\%$, respectively, for the soft-photon energy cut in range $20$-$40$ MeV.

Comments: 5 pages, 3 figures; v3: several minor modifications, references added, published version