Large off-shell effects in the $\bar{D}^$ contribution to $B \to \bar{D} \pi \pi$ and $B \to \bar{D} \pi \bar{\ell} \nu_{\ell}$ decays

We stress that, although the D¯* is very narrow (one hundred keV), the difference between the full D¯* contribution to B→D¯ππ and its zero-width limit is surprisingly large: several percent. This phenomenon is a general effect that appears when considering the production of particles that are coupled to an intermediate virtual state, stable or not, and it persists whether the width is large or not. The effects of various cuts and of the inclusion of damping factors at the strong and weak vertices are discussed. It is shown how the zero-width limit, needed to compare with theoretical expectations, can be extracted. One also evaluates the virtual D¯V* contribution, which comes out roughly as found experimentally, but which is however much more dependent on cuts and uncontrollable off-shell effects. We suggest a way to estimate the impact of the damping factors.