  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.
