[b1_ana] Updated b1 method

[O. A. Rondon]

Hi,

Since it looks like the method of extracting b1 described in sec. 6 of
Hoodbhoy, Jaffe and Manohar is sound, I have updated my draft TN on its
practical implementation, as follows:

- I have changed the terms and factors involving \delta p_f to be
explicitly written as relative quantities. This ought to avoid
ambiguities, which might involve dimensional issues for factors like charge.

- I have derived the expression for b1 including differences in charge,
packing fraction and acceptance between the two cups. But I have only
kept differences to first order, which reasonably simplify the expansion
of terms like those in eq. (6) of Dustin's write-up of 4/3/13, "Some
thoughts".

- I did not include an explicit factor for the cup lengths, z, which can
be machined to the same dimensions to better than 0.5%

- I fixed some typos on eqs. (4) and (9).

The draft is here
http://twist.phys.virginia.edu/~or/b1/b1_method-v2.pdf

If we initially neglect the terms that involve the relative differences,
the errors on the normalization factors Q, A, P_zz and s are, of course,
purely relative, so even if we take them to be about 5% for Q, A, and s,
and 10% for P_zz, their root quadratic sum is only 13% relative to b1,
whatever b1 may be. Of course, the same applies to the statistical
error, which only depends on the number of counts.

If we next consider the errors on f and \sigma_He, they are suppressed
by the relative differences, but because f involves the large 14N cross
section, it dominates the systematics. Assuming that the DIS unpolarized
cross sections are known to 2.5% relative, I have estimated that to get
the same relative error on a b1 of the same size as measured by HERMES
at x =0.45, we'd need to control the differences to better than 0.8%.

However, this estimate is based on a conservative 20% P_zz and ND3
target. With a LiD target, for the same 20% P_zz, the same precision on
b1 would require a 1.1% combined differences, because LiD is about 1.88
polarized deuterons.

And I haven't explored the details of the EMC method of combining data
from the different cell configurations, to cancel factors like pf
(=target mass), etc. I urge the collaborators to study the EMC procedure
and see how it would be applied to our proposal. The relevant section of
 R. Piegaia's thesis is on my b1 page on the wiki.

Of course, my derivation also needs thorough independent checking.

Cheers,

Oscar