We present a microscopic theory for the Raman response of a clean multiband superconductor, with emphasis on the effects of vertex corrections and long-range Coulomb interaction. The measured Raman intensity, R(Ω), is proportional to the imaginary part of the fully renormalized particle-hole correlator with Raman form factors γ(k - ). In a BCS superconductor, a bare Raman bubble is nonzero for any γ(k - ) and diverges at Ω=2Δmax, where Δmax is the largest gap along the Fermi surface. However, for γ(k - ) = constant, the full R(Ω) is expected to vanish due to particle number conservation. It was sometimes stated that this vanishing is due to the singular screening by long-range Coulomb interaction. In our general approach, we show diagrammatically 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 γ(k - ). We argue that vertex corrections eliminate the divergence at 2Δmax. We also argue that vertex corrections give rise to sharp peaks in R(Ω) at Ω<2Δmin (the minimum gap along the Fermi surface), when Ω coincides with the frequency of one of the collective modes in a superconductor, e.g., Leggett and Bardasis-Schrieffer modes in the particle-particle channel, and an excitonic mode in the particle-hole channel.
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