# Final analysis

## Contents

# Total Systematics

Abs Err [ppm] | Rel Err | |

Pol. | 0.007 | 1.1% |

Det. Lin. | 0.007 | 1.1% |

BCM Lin. | 0.007 | 1.1% |

Rescat. | 0.000 | 0.0% |

Trans. Pol. | 0.001 | 0.2% |

Q2 | 0.003 | 0.5% |

Targ Thick | 0.001 | 0.2% |

A_12C | 0.002 | 0.4% |

Inelas. | 0.000 | 0.0% |

Beam | 0.007 | 1.2% |

TOTAL | 0.014 | 2.3% |

# Overall Sign

http://ace.phys.virginia.edu/HAPPEX/2601

Positive Moller asymmetry corresponds to positive spin (i.e. along momentum) when IHWP is OUT and helicity bit is TRUE.

When the Wien is right (solenoid = 90 degrees) the precession is ~360 degrees. Wien right, HWP out gives positive Moller asymmetry and positive reported polarization. From the large asymmetry runs, there has historically been an overall sign difference between the Moller DAQ and and the HAPPEX DAQ (haven't confirmed this for PREX).

So to get a L-R asymmetry,

- Multiply by -1 when HWP is IN
- Multiply by -1 when Wien is LEFT, solenoid = -90 degrees

From this we get a positive R-L sign for lead, which is predicted.

# <math>Q^2</math>

From Kiad

Left average | <math>Q^{2} = 0.009330</math> GeV<math>{}^{2}</math> |

Right average | <math>Q^{2} = 0.008751</math> GeV<math>{}^{2}</math> |

How to average:

To get the effective Q2 when both arms are up, we have to weight by the ADC signals of the two arms. These were continually adjusted by HV changes such that they would be equal. The overall weighting factors for the whole run between the two arms (for both-arm-up data only)

Left weight | 0.5020 |

Right weight | 0.4980 |

Averaging the two weighting <math>N/\sigma^2</math>, over three data sets of the left and right arms are up with the detector signal weighted average, left arm up, and right arm up, I get

Average: <math>Q^{2} = 0.009068</math> GeV<math>{}^{2}</math> (updated from 0.009066 with the latest numbers)

The overall uncertainty is 1%

I propagate <math>Q^2</math> uncertainty to <math>A_{Pb}</math> uncertainty through the derivative of this fit:

http://ace.phys.virginia.edu/HAPPEX/2598

Which gives me 0.5%

# Polarimetry

Compton

88.20 +/- 0.12 (stat) +/- 1.04 (sys) %

Moller

90.32 +/- 0.07 (stat) +/- 1.12 (sys) %

Taking a weighted average between the two

89.18 +/- 1.04%

## Compton

Mindy's HAPLOG post:

http://ace.phys.virginia.edu/HAPPEX/2582

Mindy's result: 87.41 +/- 0.12 (stat)%

chi^2/NDoF = 1.09 for 13 points over the experiment

Compton slug-by-slug:

Kent's post on PMT gain correction: http://ace.phys.virginia.edu/HAPPEX/2599

This increases the polarization by 0.9% +/- 0.9%

Final result:

88.20 +/- 0.12 (stat) +/- 1.04 (sys) %

### Systematic Error

Rel Uncer. (%) | |||
---|---|---|---|

Laser Pol. | 0.7 | http://ace.phys.virginia.edu/HAPPEX/2530 | |

Gain shift | 0.9 | analysis from Kent, see above | |

Collimator Pos. | 0.02 | http://ace.phys.virginia.edu/HAPPEX/2568 | |

Nonlinearity | 0.3 | http://ace.phys.virginia.edu/HAPPEX/2569 | |

TOTAL | 1.18 |

Systematic Error Notes

- From Megan over gain shift (obsolete):

I don't think I'm going to be able to get a number for the gain shift during PREX, although it should scale linearly with signal-background size. Using the 1% gain shift we saw during HAPPEX (signal+bkg to background = 122e6 to 53e6), that would give us a 0.8% gain shift for PREX (signal+bkg to background = 48e6 to 30e6 at 1kHz * 3.3 to convert to 30Hz). This would give us a change in the final polarization number of 1.3%, if it's folded into the background subtraction ( which comes from comparing a background subtraction of (48e6-30e6*(1.008))/(48e6-30e6) ). So I'd say the possible gain shift gives you a 1.3% systematic error.

- From Megan over radiative correction:

I just ran through the PREX data to add in a radiative correction, and I get that this increases the analyzing power by 0.3% (from 0.01828903 to 0.01834393). This is consistent with what I've seen, so I think it's safe to add that in there.

There is an increase in analyzing power of 0.3%, which corresponds to a decrease in beam polarization by 0.3%. Therefore, the final analyzing power should multiply by (1+0.003) due to A_real = A_exp * (1+0.3%)

w/o the radiative correction:

analyzing power = 0.01824526 +- 0.00002583 (sta.)

w/ the radiative correction:

analyzing power = 0.018299996 +- 0.00002583 (sta.)

## Moller

http://www.jlab.org/~moller/e02-006.html

Just taking the last 4 runs, which are relevant for production data taking:

Final result weighted average: 90.32 +/- 0.07 (stat) +/- 1.12 (sys) %

### Systematic Error

Sasha's report at the Jan 2011 collaboration meeting

Uncer. (%) | |
---|---|

Fe Pol. | 0.25 |

Targ Discrep. | 0.5 |

Targ Saturation | 0.3 |

Analyzing power | 0.3 |

Levchuk | 0.5 |

Targ temp | 0.02 |

Dead time | 0.3 |

Background | 0.3 |

Others | 0.5 |

Current diff | 0.3 |

TOTAL | 1.12 |

# Finite Acceptance

The correction for the difference between <math>< A ></math> and <math>A(<Q^2>)</math> is calculated here:

It was found that our measured asymmetry needs to be increased by 1.2%.

The radiative effects/multiple scattering/etc. leads to an effective increase in the measured <math>Q^2</math> of 0.8%

From HAMC, the asymmetry as a function of <math>Q^{2}</math> can be well represented by a third order polynomial:

<math> A = -0.0303 + 134.6Q^{2} - 6142Q^{4} + 29700Q^{6} </math>

The fractional difference between <math>A_{obs}</math> and <math>A_{vertex}</math> is 0.31%.

The total scaling going from <math><A></math> -> <math>A(<Q^2 {}_{obs}>)</math> is an increase of 1.5%

# Background

## Inelastic

See Kiad's presentation

http://ace.phys.virginia.edu/HAPPEX/2583

The acceptance for the first excited state of Pb is <0.1%. It's expected to have an asymmetry ~1.3 of the elastic asymmetry so it is totally negligable.

The first excited state of carbon is outside our acceptance and does not contribute

## Carbon

Carbon was done by Kiad

An interpolation was done between runs using integrated charge. The starting target thicknesses were from:

http://hallaweb.jlab.org/parity/prex/runinfo/PREX_Target_Info.doc

The lead and carbon cross sections vs. <math>Q^2</math> are given by:

http://ace.phys.virginia.edu/HAPPEX/2618

The cross section ratio between carbon and lead was taken from an cross section parameterization and found to be about 0.0186. The carbon qsq was 8.3% higher than lead.

The correction is applied by:

<math> A_{Pb} = \frac{A_{meas}}{P} + D\left(\frac{A_{meas}}{P} - A_{C}\right) </math>

where

<math> D = \frac{N_{C}}{N_{Pb}} = \frac{208}{12} \frac{t_{C}}{t_{Pb}} \frac{ \sigma_C}{\sigma_{Pb}} </math>

and

<math> A_{C} = 4\frac{G_F Q^2 \sin^2\theta_W}{4\pi\alpha\sqrt{2}} </math>

The average value for <math>A_C</math> is 817 ppb (with the higher effective <math>Q^2</math>. The average for <math>D</math> is 0.067.

I assigned a 10% uncertainty to <math>D</math> for the target thickness and a 5% uncertainty to <math>A_C</math>. For a 0.6 ppm measurement, this corresponds to contributions of 0.2% and 0.4% uncertainty, respectively.

## Rescattering

Results show that this correction is < 0.1%.

http://ace.phys.virginia.edu/HAPPEX/2620

Relative asymmetry in the tail and the detector asymmetry vs. dp from HAMC:

http://ace.phys.virginia.edu/HAPPEX/2598

# Nonlinearity

Analysis from Seamus:

This analysis does not include considerations for BCM non-linearity and does not account for the changes that we see in pedestals.

From Kent, the BCM nonlinearity contribution in the past has been about 1%. Because the slopes are approximately 1% (and do tend to cancel over the course of the experiment), the overall detector contribution to the asymmetry is likely less than 1%. So assigning a BCM systematic of 1.5% of AQ (so ~1.2ppb) and 1% detector systematic would be reasonable.

AQ from BCM3 using the weights we get from dithering was 84ppb.

# Transverse Asymmetry

The transverse polarization of the beam was about 1 degree:

http://ace.phys.virginia.edu/HAPPEX/2596

Calculations from Bob: http://ace.phys.virginia.edu/HAPPEX/2594

Comments from Kent: http://ace.phys.virginia.edu/HAPPEX/2606

Pb Transverse asymmetries from Ahmed (needs to go in the HAPLOG): On CUE: /home/riordan/Ahmed_TransLeadCarbon-1.pdf

Pb Transverse asymmetry on Pb from Jon is ~200ppb +/- 200ppb

From Kent's arguments delta = 0.2 ppm * sin(3 deg) * 0.1 = 1.0 ppb (0.2% for 0.5ppm measurement)