[Halld-mc] bug fix needed for G4 library to get dirc simulation timing correct, plus a little surprise

[Richard Jones]

Dear all,

After some work, I have finally tracked down the origin of the dirc timing
bug that Justin reported, issue #4 on the HDGeant4 github repo. A pull
request for this fix has been submitted to the Master branch. One of the
dirc_commissioning_2018 branch developers should import this fix from the
master branch into the working branch or branches, whatever they are. The
needed fix is to the core G4 tracking library, to cover a special case that
this dirc geometry introduces which was either never used before, or else
the bug was never noticed. I am submitting the bug report to the G4
developers. Thanks to Justin for pointing this out and providing easy
directions for reproducing it.

I also came across a fun little surprise while I was debugging this, that
the speed of Cerenkov photons in the fused silica is not given by c / n(E)
where n(E) is the refractive index for the given photon energy. I have
worked with RICH/DIRC simulations for the better part of my career and it
never dawned on me that this simple relation v = c / n does not give the
speed of light in a dispersive medium. I have even taught this incorrectly
to students in junior lab! What c/n gives is the phase velocity, not the
propagation (or group) velocity. The correct formula for the speed of
propagation is v = (c / n) * (1 - dn / d log(E)). This correction is
implemented in the core G4 code so it is taken care of automatically if we
provide the appropriate table of n(E) values, which we do. Maybe I am the
only one who until now had overlooked this distinction. The DIRC is one
place where this distinction may actually matter.

-Richard Jones
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