Re: muon kinetics equations

[Tom Banks <tbanks@socrates.berkeley.edu>]

Dear Peter et al.,

I have a few items to report concerning our latest kinetics discussions:

(1) Yesterday I reported a ~ 6 Hz discrepancy between my best-fits to
lifetimes with and without an accidental background.  That discrepancy was
real, but I realized today that the background value that I had been
using, B=2000, is far too large and does not accurately reflect the actual
Prod-50 background level. (I have no idea how I mistakenly settled on
2000.)  When I employ a more realistic B=100 value (or, equivalently
B/N(first bin) = 100/1e6 = 1e-4), I find that the discrepancy is slightly
reduced and lies in the range 2-3 Hz.  In this respect it now appears that
my results nicely agree with Peter's results as posted on

  http://twiki.npl.uiuc.edu/twiki/bin/view/Main/GeneralPK

Namely, we both find that the ppmu "corrections" for the different cases
are roughly:

    i. first moment = best-fit with N=1.6e9, B=0: 26 Hz
   ii. best-fit with N=1.6e9, B=100:              23-4 Hz
  iii. best-fit with N=1.6e9, B=2000:             20-21 Hz

The 24 Hz correction Peter quoted at the unblinding meeting and DNP
presentation was thus probably correct after all, because the two recent
adjustments--(a) including a nonzero accidental background, and (b)
updating the ppmu capture rates--seem to effectively cancel each other
out.  I think this subject still demands further study in many respects,
but at least the Illinois and Berkeley estimates now seem to be basically
consistent.

(2)
    >> This might partially explain different results from say impact/
    >> no impact cut spectra, as the accidental level changes there.

I investigated the possibility, as suggested today by Peter, that
accidental background effects on the fit could be partially responsible
for the large, not-well-understood rate difference between the
no-impact-cut/120-mm-impact-cut decay rates.  Unfortunately, the
background effect in this case appears to be negligible.  The Run8
no-cut/cut rate difference is about 10 Hz in the Berkeley analysis (14 Hz
in the Illinois analysis), and this is far too large to be explained
simply by variations in the accidental background levels, which are
B(no-cut)=250 and B(cut)=100.  Note, for example, that the difference
between B=100 and B=2000 is only about 2-3 Hz, as I described in (1)
above.

(3) Peter, can you clarify your earlier statement that the accidental
background effect "has nothing to do with a discrepancy between moment and
fit of the *same* time distibution"?  For the record, I should make clear
that background levels only affect my best-fit results, since the first
moment is always calculated analytically and depends only upon the input
kinetics parameters--i.e. the first moment is an ideal calculation that
does not include experimental realities like an accidental background.

I will continue to study kinetics issues--there's a lot left to do--but
it's nice that we have largely cleared up some of the recent confusion.

Regards,
Tom