For this fit, two paramaters were allowed to vary in order to
phenomenologically (or empirically) describe previous data from A4,
SAMPLE, HAPPEX, and G0. These were typically chosen to be the
"strangeness radius" (or something proportional to it e.g. a possible
assumption would be G_{E}^{s}=rho_{s}*tau or
G_{E}^{s}=rho_{s}*tau*G_{dipole}), and
the "strange magnetic moment" (e.g.
G_{M}^{s}=mu_{s} or
G_{M}^{s}=mu_{s}*G_{dipole}).

The fit is also two-dimensional, where the two dimensions are Q2 and Ebeam. In this way all data on hydrogen targets from all experiments can be included without loss of generality. Only hydrogen experiments are yet included, as the same formula can be used to describe the asymmetry for any hydrogen experiment (i.e. for simplicity).

A chi-squared ellipse was then calculated in Minuit (via PAW)
corresponding to Delta chi-squared = 1. The asymmetry at
E_{beam}=1.165 and Q^{2}=0.03 (i.e. for Qweak
kinematics) was then calculated for each point in the chi-square
ellipse. The range of asymmetries allowed was taken to be the 1-sigma
uncertainty due to form factors for the asymmetry measured by the
Qweak experiment.

The affect on the extraction of the parameter Qweak was then estimated by performing a numerical partial derivative dQweak/dA. Doing this derivative gives the result: dQweak/Qweak = 1.38 * dA/A. An alternate method gives dQweak/Qweak = 1.52 * dA/A. The alternate method uses subtracting off the NFF piece and then dividing out by known kinematical factors. I believe this is slightly incorrect, as the NFF piece also depends weakly on Qweak. Another issue is acceptance averaging of Q4 which also introduces a difference also at this level.

Below are some example plots of results from this type of analysis.

- Upper left: chi-sq ellipse for P1 vs P2 (P1 is strange mag mom., P2 is strange rad)
- Upper right: Calculated physics asymmetry for each experiment and results of fit. Note that for each experiment, the strangeness has been artificially set to zero so that the fit results (the asterisks) line up perfectly with the experimental data.
- Lower left: Qweak physics asymmetry vs P1.
- Lower right: Qweak physics asymmetry vs P2. The value of the asymmetry +/- 1 sigma (as determined from this ellipse) is quoted on the graph. The relative error on the asymmetry should be multiplied by 1.38 to get the relative error on Qweak (as explained above, see also summary table below).

- Results for "lattice" assumption.
- Results for "Galster" assumption.
- Results for "linear" assumption.
- Results for "super simple" assumption. Note from the upper right plot that only data below Q2=0.25 were included for this plot.

In reality all experiments (not just the Qweak experiment) are sensitive to the value of sin2thetaW and hence Qweak in some way.

Therefore another way of doing this analysis is to allow Qweak to be a third free parameter to be determined by doing a fit over all the experiments, including Qweak. Doing so implies that the Qweak experiment itself is also somewhat more sensitive to the form factors. Hence the results for Qweak are always somewhat more pessimistic.

In reality, some combination of the two-parameter fit and the three-parameter fit should perhaps be used. It is interesting nonetheless to compare the results of these fits. Again delta chi-squared = 1 was used to define the error ellipses. However, now the value and uncertainty on Qweak simply comes out as the results for one of the fit parameters.

Some typical results of this style of analysis are portrayed below:

- Upper left: P3 vs P2 (Qweak vs strange rad)
- Upper right: P3 vs P1 (Qweak vs strange mag mom)
- Lower left: P2 vs P1 (strange rad vs strange mag mom)
- Lower right: Physics asymmetry and results of fit for each experiment.

- Results for "Galster" assumption.

The following table summarizes the results for the various assumptions for the fit function, the 2par and 3par fits, and the effect of fixing the strangeness moment by some other information.

For the purposes of the PAC Jeopardy proposal of December, 2004. The models titled "super simple" (justifiable over the Q2<0.25 data) and "linear" (used over all data for all Q2) were used for further statistics.

Plot: A/Q2 for HAPPEX and G0FWD compared with range of fit ("low-energy fit"=my "super simple"; "all data fit"=my "linear"). 1-sigma range of the 2-parameter fit are shown. SAMPLE and PVA4 were included in the fit, but are not displayed due to having too different kinematics to line up nicely on the plot.

However, depending on the empirical model chosen for the form factors, the resultant uncertainties for Qweak are at times disconcertingly high. For example, the model called "Galster" gives uncertainties typically more than double the "super simple" and "linear" models.

Possible fixes:

- Lower the average Q2 of Qweak experiment.
- Q2 binning within Qweak (Dave Mack idea).
- include more data: backward angle experiments, deuterium experiments, Helium experiments (currently only proton experiments are considered) in a unified empirical analysis.

Contact *Jeff Martin*
for more information.