[b1_ana] Update to the Updated b1 method

[O. A. Rondon]

Hi,

At the risk of boring all to tears, here is an update to the update. Two
items:

1- I took a look at another of the four differences that can be formed
when data are taken for two periods of alternating the configuration of
polarized and unpolarized cells, as I had pointed out in the draft's
section on the errors due to the acceptance.

The original method just considers the upstream unpolarized minus
downstream polarized difference, but with two alternating periods, four
differences can be formed. I have added a discussion of the same-cup
difference, that was also mentioned by Donal and Dustin. It converts the
acceptance and packing fraction into exact normalizations, with the
corresponding favorable error propagation, while the upstream-downstream
differences can control the time dependence.
http://twist.phys.virginia.edu/~or/b1/b1_method-v2.pdf

2- Since the brute force upstream-downstream difference method is most
sensitive to the cross section errors, I took a look at the error on F1d
from the ancient SLAC F2GLOB and R1990 parameterizations. The error on
F2d is nothing, 0.07%, but the error on F1 is 3.2% totally dominated by
the R1990 error. Since R1998 is available, and even more, there is the
JLab F2F109 parameterization of Peter, Eric and Vahe, I'm sure the error
must be no worse than the 2.5% I used in the estimate, most likely
better, including nuclear corrections for the N and He cross sections.

Cheers,

Oscar

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