-- SaraKnaack - 11 Feb 2009
# Electron Time Spectrum Results

## Consistency with world expectation values.

### Input Assumptions

### Electron fit results.

### Rate Results (Preliminary - subject to some reference checking and documentation)

## El Time Spectrum Scans

## El "nocut"

## El "b120mm"

## Muon Decay Rate Fit Scan

# Work 2/23-3/3

## The Background multiplicity correction decreases the chi-square moderately by ~ .01

## Scan of fits of MC pseudo-data generated from the fit result to the b<120 mm impact-parameter cut data.

## A T_0 scan for the electron time distribution fit.

### A T_0 variable was added to the time distribution function, and fixed in a series of fits to the data while varied between - 100 to 100 ns.

### Conclusions

### T0 Extraction

In order to extract the t0 offset time for the electron time spectrum data I used the high time-resolution histograms in MTA and an error function to apply a fit to the data to obtain a t0 result. In the scan of fits below the data has been broken up into groups of 10 or fewer MuCap data "runs." The transition of the time spectrum at early times is described by an error function, properly parameterized for the amplitude of the early transition in the data.
# 5/6/09 Update - T0 correction and El-Capture time spectrum comparison

- Electron Time Spectrum Results
- Work 2/23-3/3
- 5/6/09 Update - T0 correction and El-Capture time spectrum comparison

- phi = 0.0112 +/- .0001 (known to something like 1 % relative precision)
- C_Ar = 18.5 +/- .5 ppm

- R_ppmu = 1.999 (88) 10^-5 1/ns
- R_pAr_ = 4.468 (17) 10^-5 1/ns
- R_Ar = 1.423 (17) 10^-3 1/ns
- Chi-square currently 1.16, slightly high
- Subject to full consistency and understanding of the electron time spectrum and the assignment of appropriate cuts and systematic errors, mu-Ar-p kinetics and/or apparatus related.

- r_ppmu = R_ppmu/phi = 1.776 (80) 10^-3 1/ns (or )
- Compare to 2.34 (17) 10^-3 1/ns from Bystritsky
- Compare to 1.89 (20) 10^-3 1/ns from Blasser
- This is fairly consistent with a theoretical result ~ 1.8 (1) 10^-3 1/ns. (?)
- Or 2.3 (2) 10^-3 1/ns world average
- r_pAr = 2.156 (61) 10^2 1/ns
- Which is quite higher than the earlier result of 1.63 (9) 10^2 1/ns
- R_Ar = 1.423 (17) 10^-3 1/ns
- Compare to 1.40 (1)10^-3 1/ns (? must double check)

- 120-15000 ns fit window
- "nocut" impact parameter condition

- A improvement in the definition of the pull histogram does not affect the over all chi-square, but better supports the quality of the fit
- Compare to the previous result shown last week

- 120 -15,000 ns
- No background multiplicity correction was taken into account for the fit to the real data
- Question is, does this fit make sense and is it reliable?
- 100 pseudo-data spectra were generated using the fit function resulting from the fit to the data
- In the plots below, the real fit results are in the right most bin of the histograms.
- First I will show the normalized fit chi-square and probability for the 100 samples of pseudo-data
- On the respective axes below the normalized chi-square and probability of the real fit are emphasized with the solid red lines for comparison to the MC fit results

- The important observation here is that the relative to the distribution of MC fit results, the chi-square, (while elevated at 1.12) of the fit to the real data is neither the highest, ~10% being equally or more aberrant from 1 in the >1 direction.
- Like wise the fit to data is also not the least probable in comparison to the pseudo data fit results.

- Just to verify the fits, here are the fit results for the three physical parameters in the fit function, with their "truth values" indicated in read, again these are the values obtained in the fit to the data
- Each parameter shows a random scattering around the truth values, as expected in fits to pseudo-data generated with the same function.

- A T_0 offset can create a time-dependent effect in the fit to the time spectrum, causing a poorer chi-square in the fit.
- Here I have varied a T_0 parameter in the fit function at values from -100 to 100 ns, and fixed this in the fit.
- Again, there was no background multiplicity correction applied to the errors, which would improve the chi-square by another small margine of ~2%, based on the T_0=0 comparison
- First of all the quality of the fit remains at a very similar level, improving slightly in the direction of small negative values at ~ -30 ns.
- I also observe a significant linear response, vs. the value of the T_0 constant, in the fit results of the three physical parameters.
- The shift from T_0=0 ns to T_0=-30 ns is an effect of ~1 sigma for the molecular formation rate and the p-to-argon transfer rate for the muon. Finally the shift is ~3 sigma for the Ar capture rate.
- The chi-square and fit probability indicate that any values for T_0 outside of -100 to 100 ns are increasingly out of agreement with the data
- In fact It is my understanding this brings these results in closer agreement with the capture fit results.

- Finally the full panel of all the scan plots.

- A case is beginning to appear that the somewhat high chi-square may be a result of an improbable distribution which can occur at the ~10% level
- Both the T_0 scan and the background multiplicity correction improve the fit probability, separately, by ~2%.
- There will be more studies necessary to get a handle on the T_0 effect, especially to limit and significant systematics due to this effect, eg. outside information.

The following three plots are fits to the full statistics data with 1.25 ns time bins, in the early -50 - -30 ns , -50 - -38 ns, and -38 - -30 ns fit windows, respectively.

In red are the fit results to the -50 - -30 ns time window, the green points are associated with the -50 - -38 ns time window, and the results in blue are for the -38 - -30 ns fit window condition. The chi-squares of these fits show the same pattern which the three full statistics fit show. The full -50 to -30 ns fit window yields fits which clearly don't fully agree with the data, while the fit to the data in the -50- -38 ns and -38 - -30 ns data are more or less believable, but which yield differing results that are, following the statistics of the fit, fairly inconsistent insofar as the errors assigned don't indicate that the results are overlapping for either t0 or the spread in the transition time.

The first conclusion is that the fit results change little over the groups of data by run, which is also to say that the time t0 offset changes little over the entire data-set. The width of the transition seems to be no wider than + or - 3 ns, as shown in the upper right panel of the plot. Furthermore the fit results for t0 vary little for a given fit time window, but even across time windows the fit results show little spread outside the -38.5 +- 1.5ns. Since the spread in the transition time is greater than the actual variation in the t0 parameter it makes sense to choose a central value in the range of values obtained for t0 (-38), and to assign an uncertainty to that value which covers the expected range of the transition time, ~ 3 ns. In this way we have a reasonable central value, which the data reflects, and an uncertainty which covers the possible range of values in the t0 offset. Following these considerations a -38 +- 3 ns result for t0 seems appropriate for the data, and accounts for the systematic difficulties in obtaining this result.

While the fits in the full -50 to -30 ns fit window show unacceptable chi squares, it's also true that the fit results for the early and late parts of the transition in the data show emphasize that the distortions to this simple transition picture are not of a magnitude that prohibit a reasonable determination of the t0 offset for the analysis. By comparing the fit results for different time windows and noting the stability of these results through-out the data set, it seems to make sense that a -38 +- 3 ns result for t0 will safely account for the uncertainty due to these effects in the time spectrum.

Topic revision: r10 - 2009-05-06 - SaraKnaack

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