we specialize to the 2p to 1s decay,

perform the radial integration,

and perform the angular integration.

Lets assume the initial state is unpolarized, so we will sum over and divide by 3, the number of different allowed.

Our result is independent of photon polarization since we assumed the initial state was unpolarized, but,
we must still sum over photon polarization.
Lets assume that we are not interested in measuring the photon's polarization.
The polarization vector is constrained to be perpendicular to the photons direction

so there are two linearly independent polarizations to sum over. This just introduces a factor of two as we sum over final polarization states.

The integral over photon direction clearly just gives a factor of
since there is no
direction dependence left in the integrand (due to our assumption of an unpolarized initial state).

Jim Branson 2013-04-22