Moment Analysis of New Data bj (4/17/96) In this note we have a try at looking at higher factorial and cumulant moments. The source document from which most of the formulae flow is entitled "Analysis Strategy" (cf my web page), updated 4/15/96 and hereafter called AS. The data are new runs analyzed by Ken and Mary, with hand-calculation of moments by me. Therefore there is the real possibility of arithmetic errors. I have discovered a few just from noticing discrepancies from expectations. A general analysis assuming a mix of DCC and generic production is beyond the scope of this note. For the most part we will assume generic production plus possibly mundane sources of error, such as leakage of fake gammas into the data set or losses of real tracks which are due to correlated effects. As an example of the method I will start with Ken's analysis of run 1125, as reported on the listserve 11 April. Notation is as follows: f_jk = normalized factorial moments as defined in AS, eqn. 25. k_jk = normalized cumulant moments as defined in AS, eqns 48-55. F_jk = non-normalized factorial moments; the numerator in eqn. 25. N_jk = N . F_jk, where N = N_00 = number of events in the sample. The quantity N_jk is very directly calculable from the input table of joint multiplicities. To do this I form two tables from the input table. The first is weighted with n_ch, and the second by n_gamma. Rows and columns of each are summed, then resummed with weight n_ch-1 or n_g-1 as appropriate, then resummed again with weight n-2, etc. In this way all the N_ij with either i or j no larger than unity are obtained. I stopped at the sum of i and j no larger than 4, so this leaves only F_22 to determine. This was done brute force. It is important to tabulate and retain the N_jk, so that different runs can be combined without difficulty; clearly it is the N_jk's that are additive. From the N_jk, division by N gives the F_jk, and then division by the appropriate powers of N_10 and N_01 produces the f_jk. From the latter the k_jk can be obtained. It has become clear during this analysis that the robust discriminants r_j and \rho_j (AS, eqns 30, 56) should be generalized. We define the quantities r_jk and \rho_jk as follows: r_jk = f_jk / f_(j+k)0 \rho_jk = k_jk / k_(j+k)0 Evidently for k=0 these are trivial, and for k=1 they reduce to the quantities previously defined. For larger k, these quantities are in the absence of DCC bounded below by unity, and are unity in the limit of the efficiency parameter \xi vanishing. For nonvanishing \xi, there are small, controllable corrections, as shown in what follows. For pure DCC, it is straightforward to calculate the idealized values for the r_jk (I have not tried to do it for the k_jk; Ken K where are you?). The answer I get is r_jk (DCC) = j! (2k-1)!! / (j + k)! This reduces correctly for the cases k=0 and k=1; as one goes to the photon-weighted side r_jk grows larger than unity. For example, r_04 = 35/8 We now tabulate all these quantities for Ken's run 1125: Indices(jk) N_jk F_jk f_jk k_jk r_jk rho_jk 00 222 628 1.00000 1.00000 0.00000 10 108 062 .48539 1.00000 1.00000 20 73 652 .33083 1.40106 0.40106 30 67 452 .30298 2.64937 0.44619 40 73 944 .33214 5.98354 0.30987 01 49 212 .22105 1.00000 1.00000 11 33 730 .15151 1.41207 0.41207 1.00786 1.02745 21 32 004 .14376 2.76036 0.53516 1.04189 1.19940 31 37 254 .16734 6.61958 0.63273 1.10630 2.04192 02 15 406 .06920 1.41621 0.41621 1.01081 1.03778 12 13 780 .06190 2.60974 0.36939 0.98504 0.82788 22 15 800 .07097 6.16474 -1.58383 1.03028 -5.11127 03 4 706 .02114 1.95704 -0.29159 0.73868 -0.65351 13 6 336 .02846 5.42841 0.61258 0.90722 1.97689 04 2 136 .00960 4.01846 1.16787 0.67159 3.76890 In terms of the normalized parent-pion factorial moments, which we denote F_i, and the efficiency factor \xi, the F_jk have rather simple expressions. They are f_10 = F_1 = 1 f_01 = F_1 = 1 f_20 = F_2 f_11 = F_2 F_30 = F_3 f_21 = F_3 F_40 = F_4 f_31 = F_4 f_02 = F_2 + \xi F_1 f_12 = F_3 + \xi F_2 f_22 = F_4 + \xi F_3 f_03 = F_3 + 3 \xi F_2 f_13 = F_4 + 3 \xi F_3 f_04 = F_4 + 6 \xi F_3 + 3 \xi^2 F_2 It is reasonable to identify the F_j with the f_j0. Then the r_j1 quantities test the predicted values of the f_j1. The predicted values of the f_jk with k > 1 are then bounded below. This basic expectation is borne out only for f_02 and f_22. However in almost all cases the difference is quite small, so that what one has is a rough bound on the magnitude of \xi and some kind of correction which has not been taken into account. Upon constructing the r_jk, one sees that the \xi-dependent terms have coefficients which are an integer factor times the ratio F_i-m / F_i, with m the power to which \xi is raised. These ratios are generically less than unity, both from the data amd from a priori expectations. For example, for the negative-binomial distribution the above ratio equals F_(i-m) / F_i = k^m (k+i-m-1)! / (k+i-1)! which is always less than unity. It appears that in this data set there is little effect of the two-photon contribution proportional to \xi. It also appears that there is remarkable internal consistency in the moments, at least out to order 3. The moments of order 4 are perhaps not even statistically robust, and additional data is necessary to evaluate discrepancies.