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Cross-talk correction for the multiplicity

It is clear that the cross-talk between neighbouring channels not only influences the correlation/fluctuation content of the observed events, but, being effectively positive, makes the measured multiplicity systematically larger.

Figure 6.9: A distribution of the covariance matrix elements, that represent correlations between adjacent inner channels of sectors. Matrix elements involving dead channels are not shown. Run 3192.

In this section I convert the knowledge of the 9% cross-talk effect (subsection 6.5.4) into a down-scaling correction factor for the total multiplicity. The simplest way of doing it is to say that a fitted multiplicity, based on weights $w_0$, $w_1$, $w_2$ ... (see formula 3.2), can be approximated by taking the total energy deposited in a pad over the run and dividing it by mean energy $\langle dE_{MIP}\rangle$. This is equivalent to saying that multiplicity is, effectively, proportional to the amplitude. Then, the extra multiplicity induced by the cross-talk from a typical adjacent channel is, on the average, a 9% addition, and this number needs to be multiplied by 2 for a typical channel which has two neighbours. More accurately, 14 out of 16 channels have two neighbours and 2 have 1 neighbour, therefore on the average the factor is $(14\times2+2\times1)/16 = 15/8$ rather than 2. The best is to take the subsample of the channel pairs from Fig. 6.7 such that both channels are inner channels, and use Eq. 6.33 to get the effective coupling $c$. Then the multiplicity correction factor becomes $1/(1+15c/8)$. The distribution of the covariance matrix elements for such pairs is shown in Fig. 6.9. With its input, the correction factor becomes 0.86. A necessary caveat is that the 9% cross-talk is merely an ``effective'', approximate and integral way of looking at the problem. The level of approximateness implied by using the single correction for all channels is therefore not a reduction of accuracy, as compared to what has been done in Subsection 6.5.4.