# Conservation laws, vertex corrections, and screening in Raman spectroscopy

## TL;DR

This paper develops a microscopic theory for the Raman response in multiband superconductors, emphasizing the roles of vertex corrections and Coulomb interactions, and predicts how these factors influence spectral features and collective modes.

## Contribution

It demonstrates that vertex corrections, not long-range Coulomb interactions, are responsible for the vanishing of the Raman response for constant form-factors and predicts new spectral peaks at collective mode frequencies.

## Key findings

- Vertex corrections eliminate divergence at 2Δ, replacing it with a maximum at higher frequency.
- Long-range Coulomb interaction does not affect Raman response for any form-factor.
- Sharp peaks in Raman intensity occur at collective mode frequencies below 2Δ.

## Abstract

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, $R(\Omega)$, 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 $\Delta$ is the largest gap along the Fermi surface. However, for $\gamma(\vec k) =$ const, the full $R(\Omega)$ 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 $2\Delta$ and replace it with a maximum at a somewhat larger frequency. We also argue that vertex corrections give rise to sharp peaks in $R(\Omega)$ at $\Omega < 2\Delta$, when $\Omega$ coincides with the frequency of one of collective modes in a superconductor, e.g, Leggett mode, Bardasis-Schrieffer mode, or an excitonic mode.