Additional ppmu correction concerns

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

Dear all,

I recently sent out an e-mail describing my worries about the different
ppmu corrections generated by first moment vs. best-fit methods.  I am
still trying to understand this issue--in particular, why I observe a
significant difference between the two methods, while Peter K. claims not
to have observed any such difference in his simulations.  In the meantime,
a *second* ppmu-related concern has arisen regarding the two components of
the molecular correction, as I will explain below.

Peter K. yesterday described to me how he came by the 19 Hz "ppm
correction" and the 5 Hz "op correction" which were used at the unblinding
meeting to extract Lambda_S=730 Hz from our unblinded disappearance rate.
Basically, the 19 Hz (which, incidentally, Peter tells me should have
actually been 22 Hz) and 5 Hz numbers come from full-kinetics fits to the
data performed by Steve, and they are presented on slide 9 in Peter's
unblinding talk "Corrections and Uncertainties from Kinetics":

  http://www.npl.uiuc.edu/exp/mucapture/coll/unblind/kinetics.pdf

Steve performed full kinetics fit scans over a range of different
lambda_of (~ lambda_ppmu) and different lambda_op values.  The
corresponding variation in fitted Lambda_S gives us the magnitude of the
molecular correction, and Steve's results appear to basically confirm
Peter's MC results in slides 7 & 8 that the ppmu correction=45 ppm (this
number actually needs some additional manipulations: multiplication by gas
density adjustment 1.16, as well as multiplication by .455 to convert from
ppm to Hz, to get 23.8 Hz).

What worries me is the separation of the molecular correction into a
"ppmu" component and an "op" component, since Steve's study of the
lambda_of (mislabeled as "lambda_pf" in the presentation) variation
already *includes* a nonzero lambda_op component.  It seems to me that the
19/22 Hz correction therefore includes the 5 Hz effect which is tacked on
as a separate "op" correction.

I ran some MC simulations to investigate the situation, and they appear to
confirm my interpretation.  For example:

  Simulation scenario 1:
    lambda_of = 0
    lambda_op = 0
    Mean(best-fit - lambda_0 - Lambda_S) = - 0.5 Hz

  Simulation scenario 2:
    lambda_of = 2.3 E+6 Hz
    lambda_op = 0
    Mean(best-fit - lambda_0 - Lambda_S) = -14.8 Hz

  Simulation scenario 3:
    lambda_of = 2.3 E+6 Hz
    lambda_op = 8 E+4 Hz
    Mean(best-fit - lambda_0 - Lambda_S) = -20.1 Hz

You can see here that the inclusion of lambda_op increases the correction
from 15 Hz to 20 Hz, the latter of which is roughly the magnitude of
Peter's proposed "lambda_ppmu" correction.

Perhaps I have simply misunderstood some aspect of Peter's and Steve's
work, in which case hopefully someone can set me straight.  I want to
emphasize again that I realize that we need not resolve this issue before
Peter's DNP presentation, but I want to raise it now so that we have ample
time for discussion before publication.

Regards,
Tom