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We propose an approach for investigating the many body effects on phononspectrum in an electron-phonon coupled system by taking into account theelectron self-energy and vertex corrections which respect the Ward identityrepresenting the conservation of electronic charge. Our approach provides asystematic 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 vanishidentically in the regime of long-wavelength excitations ($\mathbf{q}\rightarrow 0$) due to an exact cancellation between the contributions arisingfrom electron self-energy and vertex corrections. Our results demonstrate thatthe contributions of electron self-energy and vertex corrections are not onlycomparable but they also tend to cancel each oher so that the phonon spectrumremains nearly unaffected due to many body effects in the regime oflong-wavelength excitations ($\mathbf{q} \rightarrow 0$)

A Ward identity preserving approach for investigation of phonon spectrum with self-energy and vertex corrections