  We propose an approach for investigating the many body effects on phonon
spectrum in an electron-phonon coupled system by taking into account the
electron self-energy and vertex corrections which respect the Ward identity
representing the conservation of electronic charge. Our approach provides a
systematic diagrammatic expansion of the phonon self-energy
$\Pi(\mathbf{q},\omega)$ in powers of the electron-phonon scattering strength
$(g^2)$.
  In this approach the many body corrections to phonon spectrum vanish
identically in the regime of long-wavelength excitations ($\mathbf{q}
\rightarrow 0$) due to an exact cancellation between the contributions arising
from electron self-energy and vertex corrections. Our results demonstrate that
the contributions of electron self-energy and vertex corrections are not only
comparable but they also tend to cancel each oher so that the phonon spectrum
remains nearly unaffected due to many body effects in the regime of
long-wavelength excitations ($\mathbf{q} \rightarrow 0$)
