Relative positions of the2Δpeaks in Raman and tunneling spectra ofd-wave superconductors

We study $B_{1g}$ Raman intensity $R\left(\Omega \right)$ and the density of states $N\left(\omega \right)$ in isotropic two-dimensional d-wave superconductors. For an ideal gas, $R\left(\Omega \right)$ and $N\left(\omega \right)$ have sharp peaks at $\Omega =2\mathrm{}\Delta$ and $\omega =\Delta ,$ respectively, where $\Delta$ is the maximum value of the gap. We study how the peak positions are affected by the fermionic damping due to impurity scattering. We show that while the damping generally shifts the peak positions to larger frequencies, the peak in $R\left(\Omega \right)$ still occurs at almost twice the peak position in $N\left(\omega \right)$ and therefore cannot account for the experimentally observed downturn shift of the peak frequency in $R\left(\Omega \right)$ in underdoped cuprates compared to twice that in $N\left(\omega \right).\mathrm{}$ We also discuss how the fermionic damping affects the dynamical spin susceptibility.