There is no 0-0 gamma decay because the photon must carry at least 1hbar (its spin) of angular momentum.

My question is:

Why the orbital angular momentum (L) and spin momentum (S) of the photon can not sum together to zero to meet the demand of 0-0 gamma decay?

In beta decay, the L and S of leptons(an electron and an antineutrino)  make the 0+  --  0- (first forbidden)beta decay possible  by L+S=0 and |L|=|S|=1.

Then why in gamma decay, the L+S=0 is not possible?