22 April 96

A First Study of the JACEE Event


I have made a hand study of the JACEE event, using the "raw data" I 
collected from the JACEE figure during our group meeting in Cleveland. 
The method is simpleminded and would benefit from a bit of statistical 
razzle-dazzle (likelihood analysis, etc.) which I am incapable of providing.
However, what really matters is our detection efficiency for photons, 
because there is a great deal of leverage in that parameter. I will work 
some more on that part of the problem, but it looks like it would be good 
to have some big lead-out runs to play with. The reduced detection 
efficiency may actually be of some advantage if we can still establish 
sensitivity under these circumstances, because we will have much less 
sensitivity to reduced efficiency at high n_gamma because of correlated 
efficiencies, pattern-recognition problems, etc. 
Lead-out uncompressed vs lead-out 
compressed should also be interesting because the amount of 
(scintillator) converter is in the ratio 2:1.

The method: 

We just take our lego acceptance and drop it at random many times onto 
the JACEE event (with center between, say eta = 6.5 and 8.5). Each 
sampling gives us an (N_ch,N_gamma) pair. My experience is that the value 
of N_ch is unremarkable, so that what matters is N_gamma. Quite often it 
is remarkably large. But we do not observe all those gammas. What we 
observe is an absence above a certain value, typically 5-7. So we 
estimate (as done in the next subsection) the probability that we observe 
at least n_gamma photons when N_gamma were produced, and multiply it by 
the probability that N_gamma is found per sampling. This is our net 
efficiency for seeing at least n_gamma photons per JACEE event. From this 
we can estimate from the data the fraction of p-pbar collisions 
which produce a JACEE-like final state. 

Jacee Probabilities:

In Cleveland, I took, after negotiations with Cyrus, a circle of radius 
0.5 as a simple and conservative representation of our lego acceptance, and 
dropped it 16 times onto the lego plot according to a grid I laid 
out. There were no overlaps. (This should be of course automated using 
our real acceptance.) The values of (N_ch,N_gamma) obtained were 
unremarkable in 8 cases, but yielded N_gamma = 7, 13, 9, 10, 8, 7, 6, and 
6 in the other 8 cases. 

Efficiencies:

We need the probability P(n,N) of observing n gammas when N are produced.
In the limit of uncorrelated efficiencies, and assuming the momentum
spectrum of JACEE gammas is similar to what we observe in the data
(dangerous!!), we can get this from our analyses. The formula for P(n,N)
is just a binomial distribution:

    P(n,N) = \frac {\epsilon^{n}(1-\epsilon)^{(N-n)}N!}{n!(N-n)!}

The efficiency $\epsilon$ for finding a produced photon should 
approximately be <n_gamma> / <n_ch> as seen in our data. This number is 
about 0.45. 1 X0 of radiator would give by itself about 0.63, so that 
means our finding efficiency for conversions, relative to the finding 
efficiency for charged tracks, is about 71%. This seems to me to be high, 
but if it is we have other problems beside this analysis. (Experts: 
COMMENTS PLEASE!). In the analysis to follow I have used

     \epsilon = 0.40

With a hand calculator I found it advantageous to calculate the P(n,N) 
iteratively to avoid large numbers appearing in intermediate steps. Maybe 
this matters when this procedure gets automated, so I will describe the 
steps I used here:

1) I declared n less than 4 and N less than 6 as uninteresting and 
calculated directly P(4,6). Using the simple formula for the ratio of 
P(n,N+1)/P(n,N) I then got successively P(4,7)...P(4,13).

2) Using the simple formula for P(n+1,N)/P(n,N), I calculated as needed
P(5,7)...P(7,7), P(5,8)...P(8,8), ... ,P(5,13)...

3) To get the probabilities of finding at least n when N were 
produced I added up the appropriate entries. 

This procedure was so fast and painless that it got done in wee hours of 
this morning as I lay in bed, wide awake with a bad case of jet lag.
The relevant table is as follows:

  N=     6      7      8      9      10     11     12     13

P_4    .179   .289   .403   .517   .621   .703   .772   .823

P_5    .041   .096   .173   .266   .370   .466   .559   .639

P_6    .004   .0188  .050   .0992  .169   .246   .332   .418

P_7           .0016  .0085  .0249  .055   .0985  .1553  .2211

P_8                  .00066 .00380 .0123  .0285  .0585  .0899       

P_9                         .00026 .00168 .0052  .0125  .0243


Here the P_n are the probabilities that at least n photons are observed 
when N were produced into our acceptance (assuming 40% overall detection 
efficiency).

Results:

I took the P_6(N) and multiplied them against the JACEE list of N_gamma as 
given above to get a summed value of .782; dividing by 16 yields the 
chance of our seeing at least 6 gammas in the JACEE final state to be 4.9%.
Doing the same with the P_4 weight gave the chance of our seeing at least 
4 gammas in the JACEE final state to be 3.30/16 = 20.6%. In run 1125
(225K events; lead in) and using Mary's numbers, we see no events with 6 
or more gammas. This gives a 95% confidence limit on JACEE final states 
(3 events allowed when none are seen) per observed ppbar collision of

      \frac{3}{225K x .049} = 2.7 E-4 

For the looser choice, we see 61 events with at least 4 gammas, leading 
to an upper bound on the frequency of JACEE events of

      \frac{61}{225K x .206} = 1.3 E-3

This is still a factor 10 above the 1/70 level of the JACEE experiment, 
although our sensitivity becomes marginal. 

Dependence of photon detection efficiency:

The dependence of P(n,N) on efficiency $\epsilon$ needs a careful study. 
I have not done that yet, but have made scratch estimates of how the 
above results are weakened if the efficiency is reduced. For a reduction 
of 0.4 to 0.3, I only lost a factor 2 in sensitivity. But in reducing 
from 0.4 to 0.2 I lost about a factor 20. Both these cases need to be 
done more carefully, however. If the photon energy spectrum is softened 
by a factor 2 to 3 for the JACEE case, which seems not too unreasonable 
if the phenomenon is associated with low-p_t secondaries, we need to know 
how our efficiencies are affected. This should be straightforward with 
Mary's and Cyrus's GEANT studies. As for the effect of correlation 
effects, losses associated with large n_gamma and/or large n_hits, etc., 
I think the first and easiest line of attack will be to directly look at 
the lead out data to see whether we are sensitive in that case.

Given that the data set used represents less than 10% of the total, I 
think we should have a good shot at making a good statement regarding the 
JACEE stuff.

COMMENTS PLEASE: what else needs to be considered?

                                              bj