The two-photon invariant mass is given (in the lab frame) by

M = Sqrt(2E₁E₂(1 - Cos(theta)))

where E₁ and E₂ are the energies of the two photons and theta is the angle between those photons.  For every real photon we should measure ~135 MeV, the rest mass of the pi0.  Of course, the detectors have finite resolution and there is some uncertainty in our measurement of each of the three quantities above, so we should end up measuring some spread around 135 MeV.  

But it is not that simple.  We do not see a simple spread around the true pion mass.  Instead, we see some pt dependence in the mean reconstructed mass.  

The above left plot shows the two-photon invariant mass distribution separated into 1 GeV bins.  The pion peak region (between ~.1 and .2 GeV) has been fit with a gaussian.  The mean of each of those gaussians has been plotted in the above right as a function of Pt.  Obviously the mean mass is increasing with Pt.  This effect is not particularly well understood.  It's possible that the higher the Pt of the pion, the closer together the two photons will be in the detector and the more likely it is that some of the energy from one photon will get shifted to the other photon in reconstruction.  This artificially increases the opening angle and thus artificially increases the invariant mass.  Which is essentially to say that this is a detector effect and should be reproducible in simulation.  Indeed...

The above plot overlays (in red) the exact same measurement made with full-pythia monte carlo.  The same behavior is exhibited in the simulation.  Linear fits to the data and MC yield very similar results...

-- M = 0.1134 + 0.0034*Pt  (data)

-- M = 0.1159 + 0.0032*Pt (simulation)

If we repeat this study using single-particle simulations, however, we find some thing slightly different.

-- M = 0.1045 + 0.0033*Pt

So even in single-particle simulation we still see the characteristic rise in mean reconstructed mass.  (This is consistent with the detector-effect explanation, which would be present in single-particle simulation.)  However, the offset (intercept) of the linear fit is different.  These reconstructed pions are 'missing' ~11 MeV.  This is probably the effect of jet background, where other 'stuff' in a jet get mixed in with the two decay photons and slightly boost their energies, leading to an overall increase in measured mass.

The upshot of this study is that we need to correct any single-particle simulations by adding a slight amount of extra energy to each photon.