Conservation laws, vertex corrections, and screening in Raman spectroscopy

We present a microscopic theory for the Raman response of a clean multiband superconductor accounting for the effects of vertex corrections and long-range Coulomb interaction. The measured Raman intensity, $R(\mathrm{\Omega })$, is proportional to the imaginary part of the fully renormalized particle-hole correlator with Raman form-factors $\gamma (\vec{k})$. In a BCS superconductor, a bare Raman bubble is non-zero for any $\gamma (\vec{k})$ and diverges at $\Omega = 2\Delta +0$,where$\Delta$isthelargestgapalongtheFermisurface\mathrm{.}However,for$\gamma(\vec k) =$const,thefull$R(\Omega)$ is expected to vanish due to particle number conservation. It was long thought that this vanishing is due to the singular screening by long-range Coulomb interaction. We argue that this vanishing actually holds due to vertex corrections from the same short-range interaction that gives rise to superconductivity. We further argue that long-range Coulomb interaction does not affect the Raman signal for $any$ $\gamma (\vec{k})$. We argue that vertex corrections eliminate the divergence at $2\mathrm{\Delta }$ and replace it with a maximum at a somewhat larger frequency. We also argue that vertex corrections give rise to sharp peaks in $R(\mathrm{\Omega })$ at $\mathrm{\Omega }<2\mathrm{\Delta }$, when $\mathrm{\Omega }$ coincides with the frequency of one of collective modes in a superconductor, e.g, Leggett mode, Bardasis-Schrieffer mode, or an excitonic mode.