In discussions of timing spectra, the point is often raised that the complexities of the online trigger must be taken into account, in that sometimes the taggerOR signal and sometimes the bsdAND determined the common time reference for the TDC's. While this consideration does affect the time spectra for individual counters, it is not relevant to the spectrum of a time difference between hits in two counters. For time differences only the logical condition of the trigger, not its precise timing, is relevant. Henceforth in this study it may be assumed that all time spectra use a single hit in a single tagger channel as a time reference.
The choice begs the question of which tagger time is suitable as a reference for any given event, since the rate in the tagger is sufficiently high that the typical event has 2 or more tagger hits within the region of the coincidence peak. The answer is proposed that each tagger hit is an equal candidate for a time reference for an event and all must be treated on exactly the same footing. While it may seem strange to be including a single event more than once in the analysis, the following conceptual argument serves to justify it.
Consider an analysis where the cross section for a well-defined photon energy is to be measured. In that analysis, one might well ignore the TDC information from all but one of the tagging counters, and look only at the coincidences with the relevant tagging channel. For such a case one need not worry about ignoring what other tagging channels would do with any given event, any more than the Radphi analysis needs to concerned with what was happening in regions of the tagger that were not turned on for the experiment. The single-channel tagging spectrum would be shaped very much like the one shown in Fig. 2, with true coincidences appearing as a peak on top of an accidental continuum. The single-channel tagging case is simple to handle tagger because the electronic double-pulse resolution on a single channel of 20 ns prevents more than one hit within a reasonable coincidence window and, even if that were not true, the rate in an individual counter is low enough to make the problem of double-coincidences irrelevant.
The analysis would proceed as follows. A timing spectrum would be formed in which the recoil time for the event is subtracted from the time for every hit in the tagging channel. Based on this timing spectrum, two windows would be assigned, the first containing the coincidence peak (the coincidences window) and the other outside the peak but nearby and of a similar width (the accidentals window). In the subsequent analysis of the information from the detector, two copies of every histogram would be formed: one which is filled on every event with a tagging hit in the coincidences window, and the other on every event in the accidentals. As a final step in the analysis, every coincidence spectrum would have its corresponding accidentals subtracted away. It is apparent that this procedure leads to an unbiased estimate in all experimental variables for the contribution coming only from beam photons within the range of the selected tagging counter.
This procedure can be generalized to cover the case where an inclusive measurement is desired of a photoproduction cross section averaged over a larger region of the focal plane. In such a case, the above single-channel analysis is repeated N times and then the resulting spectra are summed, leading to an average cross section weighted by the bremsstrahlung spectrum times tagging efficiency. The detailed shape of this weighting function is not of interest in the Radphi analysis; diffractive processes are weakly dependent on beam energy. The primary concern is to settle on a method that extracts the tagged component of the experimental spectra without bias and with known errors.