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 of keV), the difference between the full $D^*$ contribution to $B \to \bar{D} \pi \pi$ and its zero width limit, which stems from the resonance tail, is surprisingly large~: several percents. This phenomenon is a general effect which 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 expected from theory, 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.