Date: Mon, 9 Oct 2006 13:08:19 -0400 (EDT)
From: "William J. Marciano" <marciano@bnl.gov>

Dear Peter and David,

It was nice speaking with you this weekend.  I thought that I would send you 
this email as confirmation of our conversation.

Andrzej Czarnecki, Alberto Sirlin and I have examined the leading radiative 
corrections to muon capture in Hydrogen.  As you mentioned, I had alerted you to 
a relatively large effect several years ago.  Although you should view the 
result I gave you as still somewhat preliminary and we have not assigned an 
error to it, it is important for you to know our findings as a help in 
interpreting your experimental result.

The main effect is a rather universal correction to all semileptonic weak decay 
rates due to normalization relative to muon decay.  That correction (certainly 
its short-distance part) is very solid and dates back to Alberto's pioneering 
work on superallowed nuclear beta decay. There is some issue with the long 
distance part, but it is unlikely to be very large.  We will subsequently 
assign an error.  Using kinematics specific to muon capture, A. Sirlin estimates 
an overall correction of +2.4% to the capture rate from that universal loop 
effect. There is also a correction (first discussed by Goldman) from vacuum 
polarization corrections to the bound state wavefunction.  An update by A. 
Czarnecki finds a slightly smaller +0.4% effect for Hydrogen.  Both effects 
factorize in the capture formula. That means that there is an overall +2.8% 
increase in the predicted capture rate when
the lowest order is expressed in terms of the Fermi constant (from muon decay)
and the four form factors.  That correction leads to about a 15% increase in he 
extracted value of gP.

There are a few other effects that differ somewhat from earlier studies.
For example, Vud=0.9738 and at 0 momentum, gA=1.272. For muon capture, we use
effective values appropriate to the relevand momentum transfer gV=0.976, 
gA=1.247 and gM=3.583.

On effect that I assume you use to correct your final result is the fact that 
the negative muon bound to the proton has non zero velocity. That effect has to 
be corrected for in ordinary mu minus decay in orbit.  The previous experiment 
made such a correction before extracting gP.  It effectively reduced gP via a 
reduction in the measured mu minus decay rate by about 20sec^-1 (if I recall 
properly). It is quite important, almost as big as our radiative correction (but 
opposite in sign).

Let me ask that if you wish to refer to our results, please call them 
preliminary, since we have not formally agreed to a number, and reference all 
three of us Czarnecki, Marciano and Sirlin for the work and communication.

Best regards,
Bill