The QED nonfactorizable correction to the semileptonic charmed three-body $B$ decays

In this paper, the nonfactorizable photonic corrections to the effective four-fermion vertex for the $\bar{B}$ $\to$ $D^{(\ast )}$ $+$ ${\mathrm{\ell }}^{-}$ $+$ ${\bar{\nu }}_{\mathrm{\ell }}$ decays are considered at the quark level within SM, where $\mathrm{\ell }$ $=$ $e$, $\mu$ and $\tau$. It is found that the QED corrections are closely related with the mass of the charged lepton. The QED contributions will enhance branching ratios, which results in the slight reduction of ratios $R(D^{(\ast )})$ and the relationship $R(D^{(\ast )}{)}_e$ $<$ $R(D^{(\ast )}{)}_{\mu }$.