  The influence of electron-phonon interaction on the transmission phase shift
of an electron passing through a quantum dot is investigated by using the
scattering theory. The transmission phase versus the intra-dot level shows a
serial of phonon-induced dips. These dips are highly sensitive to
electron-phonon interaction strength $\lambda$, and they are much more
pronounced than phonon-assisted sub-peaks appeared in the conductance.
Phonon-induce dephasing is also studied, and the results show that the
dephasing probability $T_d$ monotonically increases with the electron-phonon
interaction strength $\lambda$. The dephasing probability $T_d
\propto\lambda^2$ for small $\lambda$ but $T_d\propto \lambda$ at large
$\lambda$.
