Handling the time bias

Hi Dick, Glen, Carl

I'm trying to write up the systematic uncertainties
for target depolarisation, but I'm having trouble
with the possible time bias. This is holding me up
from posting my systematics chapter.

For the facts see my recent posting:
 https://twist.phys.ualberta.ca/forum/view.php?site=twist&bn=twist_physics&key=1250713772
Basically we find that applying the asymmetry
analysis to the simulation gives a lambda result
that is 2.6 sigma too small for Ag, but is just
fine for Al.

g474 (nominal gen) is probably throwing off the result
for Ag. If g474 (nominal gen) is excluded, then
the result is lambda = 0.66 +- 0.05 (1.4 sigma
below input).

I complained to Glen that it's awkward to write
this one up and decide how to assign a systematic.
Glen and I brainstormed for a way to investgate
to see where the (potential) bias is emerging, and
whether it really exists.

1) Are the lambda uncertainties from the weighted  
   analysis wrong?
- Carl, I understand you helped Blair implement
  the weighted asymmetry analysis. I will describe
  below what is currently done. Hopefully you can
  advise whether the errors are being done properly
  (I never checked these).
-- The treesum code makes four histograms:
    ai = costh * (x-0.5)/(1.5-x)
    [1] h_gnt_u:      ai   * |ai|^N
    [2] h_gnt_d:      |ai| * |ai|^N
    [3] h_gnt_deriv1: |ai|^{2*N+2}
    [4] h_gnt_deriv2: ai * |ai|^{2*N+1}
-- Then the weighted asymmetry is calculated as
     g = u/d,
   and the error on g as
     sqrt( [1+g^2] * deriv1 - 2*g*deriv2 )/d

2) Is the problem from weighting?
- I would like to see whether the result changes
  for an unweighted asymmetry, but this functionality
  was never ported to the new treesum code (and I'm
  not doing it).
- As an alternative, Glen is going to fit the US
  and DS time distributions separately.
- Glen, here are the two files from 2006
  that give the largest and smallest values
  of lambda:

  /home/e614/analysis_2006_2007/treesum/pass4/
    sum474-1_2_21-1_clk_raw.root
    sum471-1_21-1_clk_raw.root  

  (the results from weighted asymmetry analysis are in
   the posting I ref above)

-- Glen, these treesums do NOT have the
   cut requiring a hit in DC22 / DC23.
   If you really need this, let me know
   and I can ask Anthony to try and make
   the histos by default.

If we can't make any progress, then I need
some advice on how to write this up.
Do I assume the "bias" is the average
of the discrepancy for the 2006 and
2007 sims? Or do I make the lambda
uncertainty large enough to accomodate
the discrepancy? If the latter, how large?

Aside:
In data we find that lambda_Ag = 0.840 +- 0.072.
This propagates into a pmuxi uncertainty
of +- 2.2 x 10^{-4}.
In the worst case scenario, a lambda
uncertainty of +- 0.19 (i.e. 2.6 sigma) would
propagate into a pmuxi uncertainty of
+- 5.6 x 10^{-4}.

Sorry to burden you with this, but I'm
really struggling here.

James