Based on this document describing the filtering, it now appears that the Pythia production cannot be used for estimating the trigger efficiencies.  Ilya's and Alice's studies show that it is unbiased within the trigger requirement, so can be used for estimating conditional probabilites (conditional on requiring the trigger).  Apart from requesting a new, fully reconstructed production, there are then two main avenues to persue: using minbias events or SimpleMC events.

I. Equations if Using Pythia

The full, probabililty distribution equation (which happens to be a Fredholm equation) for using Pythia for estimating the smearing matrix can be written:

This can be read, the probability of a pi0 with reconstructed p_T = p_T^R being reconstructed in an event with trigger "trig" is equal to the integral overall generated pT values of the product of severel conditional probabilities times the probability of generating a pi0 with generated p_T = p_T^G.  The conditional probabilities are 1) the probability of the reconstructed p_T = p_T^R given that the pi0 is reconstructed, the event meets the trigger, and that the generated p_T is p_T^G, 2) the probability of reconstructing the pi0 given the generated pT and that the event met the trigger, and 3) the probability of the event meeting the trigger, given the generated p_T = p_T^G.  The cross section is proportional to p( p_T^G) and the measured data is an estimate of p_{rec}(p_T^R, rec, trig).  The other conditional probabilities must be estimated in other ways.  One then writes the matrix equation

where the indices run over p_T bins and

• is an estimate of the probability of reconstructing with a given pT and trigger
• is an estimate of , the smearing matrix estimated using events of a given trigger
• is an estimate of , the reconstruction efficiency within a given trigger
• is an estimate of , the trigger efficiency

The filtered Pythia production can be used to estimate and , but some other Monte Carlo production must be used to estimate , as it must be independent of whether the pi0 is reconstructed.  Note, however, the Monte Carlo only needs to throw pi0s, though the full GEANT simulation and BFC is needed.  Note, using the SimpleMC will underestimate the trigger efficiency, as the "jet" enviroment the pi0s occur in would effect the likelihood of the trigger firing.  Thus, one would need Pythia, possibly with a filter of a pi0 exists with pT > some threshold.

Ib. Equations if Using Pythia, slightly modified

[added Aug 22nd]

Using a Pythia production without running through GEANT, I can count how many pass the filter and then estimate

.

The effective trigger efficiency, conditioned on the generated pT, is then

The equation then reads

II. Other Equations Using Pythia

If one were to generate more Pythia with exactly the same settings, but no filters and without running it through GEANT, one could then take the ratio between it and the current Pythia to estimate

and could then modify the Fredholm equation to be

This seems a little tricky to ensure (and verify) that the settings are matched between the Pythia productions.  This method requires a Pythia production without GEANT, while option (I) requires GEANT.  In option (II) one must be very careful about matching the Pythia settings and normalizations.  This was not the case in option (I).

III. Equations if Using Min Bias

Using the trigger emulator with MB events can tell us the probability of triggering, conditional on reconstructing pi0 and the pT of the reconstructed pi0.  This can be written

.

To use this, one must modify the master Fredholm equation to be

 Now, however, the given Pythia production cannot be used to estimate the smearing matrix nor the reconstruction efficiency, as these must be independent of the trigger and the filtering causes Pythia to only be unbiased within the trigger cut.  This would require either a full Pythia production with minimal Pythia level filtering (an arduous task) or some fancy use of the SimpleMC.  Even the SimpleMC with just pi0s (and a few muons to define the vertex) takes 10 seconds per event to process (8 for starsim and 2 for BFC), or about 8.3k events per cpu-day.  So 10k per pi0 pT bin would mean about 11 cpu days, 100k per bin (better) would be 110 cpu-days.  This is a fairly substancial cpu-investment.

One could possibly argue that the smearing matrix is the same among events meeting the trigger or not, and thus that Pythia could still be used.  However, to prove this would involve computing the smearing matrix with no trigger requirement (and less filtering), although with possibly less statistics.   However, making the arguement may take a sizeable portion of the effort it would have taken to do Option (III) without the filtered Pythia.

Conclusion

In all cases, more Monte Carlo data is needed.  Option (II) presents the least computation time, but is tricky regarding matching and verifying the proper settings.  Options (I) and (III) seem to require a new reconstructed Pythia production, but it may be possible to use the SimpleMC in some smart way for option (III). Regardless of the generator, Option (III), unfortunately, does require more statistics than option (I).

[added Aug 22nd]

Option Ib seems to be to persue at present.