Elong-13-10-01

Cross Section Calculation

For previous calculations, I was using a simplified version of the cross-section where it was assumed that $F_2=2x\cdot F_1$, such that

$\frac{d^2\sigma^u}{d\Omega dE'} = A_X \left( \frac{d\sigma}{d\Omega} \right) _{\mathrm{Mott}_{\mathrm{p}}} \left[ \frac{2\cdot \left(\frac{F_1^{X}}{A_X} \right)}{m_{p}}\right]\cdot \left[\tan^2\left( \frac{\theta_{e'}}{2} \right) + \frac{Q^2 }{2\nu^2} \right] $.

Since we're in a region that isn't DIS, I thought that the difference may be important so I incorporated $F_2$ from Bosted and removed the assumption:

$\frac{d^2\sigma^u}{d\Omega dE'} = A_X \left( \frac{d\sigma}{d\Omega} \right) _{\mathrm{Mott}_{\mathrm{p}}} \left[ \frac{2\cdot \left(\frac{F_1^{X}}{A_X} \right)}{m_{p}}\tan^2\left( \frac{\theta_{e'}}{2} \right) + \frac{\left( \frac{F_2^X}{A_X}\right) }{\nu} \right]$.

This increased the statistical uncertainty, particularly in the high-x region. It also lowered the rates dramatically, which gives us some room to play around with a lower $Q^2$.

Although this changes the cross sections quite a bit,

$F_1 \mathrm{~Only}$	$F_1\mathrm{~and~}F_2$

The dilution factor is nearly identical.

Cross Section Check for b1

To see if this could cause a problem for the b1 statistics, I did a study of the effects of changing the cross section calculation from the simplified to the full for the b1 kinematics.

SHMS 1 $\theta_{e'}=7.35^{\circ}$ $E'=6.80\mathrm{~GeV}$	SHMS 2 $\theta_{e'}=8.96^{\circ}$ $E'=7.45\mathrm{~GeV}$	SHMS 3 $\theta_{e'}=9.85^{\circ}$ $E'=7.96\mathrm{~GeV}$	HMS $\theta_{e'}=12.50^{\circ}$ $E'=7.31\mathrm{~GeV}$

Cross Section Check for Azz

Same as the section above, looking at the difference between the simplified cross section using only $F_1$ and the full cross section using both $F_1$ and $F_2$, but for the Azz kinematics.

HMS $\theta_{e'}=12.45^{\circ}$ $E'=5.80\mathrm{~GeV}$	SHMS $\theta_{e'}=9.51^{\circ}$ $E'=6.07\mathrm{~GeV}$

Comparison to Data

In order to see how my calculations line up with actual data, I've taken the deuterium information from the quasi-elastic scattering archive data page to compare our cross section calculations for b1 and Azz. There are two measurements (both Shutz:1976) that are similar to our settings:

Similar to $A_{zz}$ $E_{\mathrm{beam}}=6.519\mathrm{~GeV}$ $\theta_{e'}=8.00^{\circ}$	Similar to $b_1$ $E_{\mathrm{beam}}=11.671\mathrm{~GeV}$ $\theta_{e'}=8.00^{\circ}$

The top and bottom plots are the same, just that the top are on a log scale and the bottom on a linear scale.

Inelastic and Quasielastic Structure Functions

Previously, I've been shutting off the inelastic part of the structure functions and turning on the quasi-elastic when x=0.75. This creates a discontinuity and may cause trouble for estimating the dilution factor for Azz. As such, I've re-done the calculations of the cross sections and the dilution factors above using

$\frac{d^2\sigma^u}{d\Omega dE'} = A_X \left( \frac{d\sigma}{d\Omega} \right) _{\mathrm{Mott}_{\mathrm{p}}} \left[ \frac{2\cdot \left(\frac{F_{1IE}^X + F_{1QE}^X}{A_X} \right)}{m_{p}}\tan^2\left( \frac{\theta_{e'}}{2} \right) + \frac{\left( \frac{F_{2IE}^X+F_{2QE}^X}{A_X}\right) }{\nu} \right]$,

where $F_{1IE}^X$ and $F_{2IE}^X$ are the inelastic form factors, and $F_{1QE}^X$ and $F_{2QE}^X$ are the quasi-elastic form factors. Below are plots with the combined form factors (in purple) along with just the contribution from DIS (in red) and from QE (in blue).

SHMS 1 $\theta_{e'}=7.35^{\circ}$ $E'=6.80\mathrm{~GeV}$	SHMS 2 $\theta_{e'}=8.96^{\circ}$ $E'=7.45\mathrm{~GeV}$	SHMS 3 $\theta_{e'}=9.85^{\circ}$ $E'=7.96\mathrm{~GeV}$	HMS $\theta_{e'}=12.50^{\circ}$ $E'=7.31\mathrm{~GeV}$

HMS $\theta_{e'}=12.45^{\circ}$ $E'=5.80\mathrm{~GeV}$	SHMS $\theta_{e'}=9.51^{\circ}$ $E'=6.07\mathrm{~GeV}$

--E. Long 18:57, 2 October 2013 (UTC)