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 GEs=rhos*tau or GEs=rhos*tau*Gdipole), and the "strange magnetic moment" (e.g. GMs=mus or GMs=mus*Gdipole).
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 Ebeam=1.165 and Q2=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.
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:
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.
Summary for Mark
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.
Contact Jeff Martin
for more information.