TPC T0 vs. TOF corrections

This post is just to document some thoughts...

I think part of the recent back-and-forth about resolving the TPC T0 for fixed target comes from not distinctly separating the following:
  1. the time of the collision event with respect to the trigger, and
  2. the time-of-flight (TOF) of the particles into the TPC.
We have always corrected for these two things separately:
  1. T0 correction when determining the z of TPC hits, and
  2. TOF correction in the tracking code. 
In the past, we have treated T0 as a constant for a given year or colliding system. But now we are finding this to deviate observably from a constant event-by-event because of the longer bunches (more distributed ion arrival times at STAR) in RHIC.
Ideally, one would want to have both of these corrections in place before tracking even begins. However, due to not knowing exactly where in z tracks originate, the TOF correction is held off until we have a better idea of the origin of the tracks, i.e. the z at which they point along the beamline, so that more accurate speed (from momentum under the assumption of a pion mass currently) and distance-of-travel can be calculated for use in determining TOF. Regardless of collision or fixed target or from where in the RHIC bunches the collision comes, this z projection to the beamline is already reasonably accurate in the current code; the time of flight has been corrected since 2007 when Yuri introduced the correction into Sti (see version 2.105 of StiKalmanTrackNode.cxx and subsequent commits). Nothing needs to change for any year or colliding system for the TOF correction code. It is solid. [NB: Lasers tracks are appropriately corrected, by cosmic rays are not.]
The fact that the time of the collisions with respect to the trigger can be considered no longer a constant does not mean the correction for that time should be delayed until after track-finding. In fact, track-finding does not tell us anything about the time of the collision with respect to the trigger. Irakli has found that other detectors with timing information can in fact be used to help determine the collision time with respect to the trigger. For fixed target, this likely means timing information from a single detector, like the east-side EPD. For collision data, this would need to be determined from multiple detectors, like the average of time from the east-side and west-side EPDs, to avoid correcting in the T0 for the position of the collision (which is already handled in the TOF correction).

One last note... The T0 will be wrong for some TPC hits in two cases, which I believe are acceptable to ignore:
  1. hits on tracks that originate from a pile-up collision within the same bunch crossing (this will have a small time displacement from the time of the triggered collision), and
  2. hits on tracks that originate from a pile-up collision from a different bunch crossing (this will have a large time displacement).