[Frost] Dilution factor question/comment

[Michael Dugger]

Hi,

I am trying to figure out the use of dilution factors and am beginning to 
think that -with my current understanding- dilution factors only work for 
a special case.

I have difficulty understanding the use of dilution factors when 
an asymmetry can be expected for the bound nucleons.

The E observable is a special case where there needs to be a double 
polarization of beam and target. Because of this, and the assumption that 
the bound nucleons within the butanol target are expected to have no 
residual polarization, we can assume that there is no E asymmetry coming 
from the bound nucleons within the butanol target. This results in a very 
nice simplification to the numerator of the E observable.

I'm going to write a few simple equations so that I am clear about what I 
am saying. Please bear with me.

First I'll start off with an expression for E.

We can express the E observable through the equation:
P*E = [N_H1 - N_H2]/[N_H1 + N_H2],
where

P = (beam polarization) * (target polarization)
N_H1 = Number of hydrogen events with polarization anti-parallel
N_H2 = Number of hydrogen events with polarization parallel

If we want to express the E observable in terms of butanol events we need 
a few more definitions.

Let
N_B1 = Number of butanol events (anti-parallel)
N_B2 = Number of butanol events (parallel)

and

N_N1 = Number of bound nucleon events (anti-parallel)
N_N2 = Number of bound nucleon events (parallel)

(note: there is no polarization on the bound nucleons, the designation of 
parallel and anti-parallel goes with the hydrogen definition and just acts 
as a way to partition the bound nucleon events)

we can now write the simplification of the numerator:

[N_B1 - N_B2] = [N_H1 - N_H2]

This lets us express the E observable as

P*E*D = [N_B1 - N_B2]/[N_B1 + N_B2],
where
D = dilution factor
and
D = [N_H1 + N_H2]/[N_B1 + N_B2]

This dilution factor can be rewritten in terms of scale factors, carbon 
target events, and butanol target events as Sung does.

But what happens if we could not make the nice simplification that
[N_B1 - N_B2] = [N_H1 - N_H2] ?

For linear beam we do not need the beam*target polarization to obtain the 
Sigma observable, we just need beam polarization. So this question about 
the absence of the numerator simplification is of importance for Sigma.

If we wrote
P*Sigma*D = [N_B1 - N_B2]/[N_B1 + N_B2],

where this time
P = linear beam polarization
and
1 => polarization perpendicular to floor
2 => polarization parallel to the floor

we would get

P*Sigma*D = [N_H1 - N_H2]/[N_B1 - N_B2] + [N_N1 - N_N2]/[N_B1 - N_B2]

In this case, Sigma would not represent the Sigma observable for hydrogen. 
Instead, Sigma would be some strange hybrid of hydrogen and bound 
nucleon beam asymmetry.

Also, this is lack of numerator simplification should be present in the 
Cx and Cz extraction.

If I do not properly understand the use of the dilution factors please let 
me know, and I will take this as a learning experience.

Thanks for your time.

Sincerely,
Michael