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How to use the Si pad array

Given the quality of the Si amplitude resolution in 1995, the traditional method of multiplicity analysis, based on the channel-by-channel comparison of amplitude with the MIP expectation, can not be reliably used for the whole run period, because the amplitude resolution was affected by radiation damage. Multiplicity analysis can be performed under considerably relaxed requirements to the amplitude resolution, if the probability distribution $P(n)$ of number of tracks $n$, crossing an individual pad of the array, is known. Our eventual goal, sufficient for the task of normalization as it is formulated in the section  4.2.1, is finding a single number, $dN/d\eta$, characterizing the average multiplicity of the whole detector, averaged over all events of a physics run. Therefore it is acceptable to deal with one $P(n)$ for the whole detector, over the whole run (or a set of runs taken with a specific trigger and angular setting). Then, number $H$ of the hit pads ( i.e., pads with signal above the threshold, set during the calibration and subtracted during the DST -production) out of $M$ pads in total, is

$$H = M(1-P(0))$$ (21)

As a measure of Si multiplicity, I use the Si occupancy calculated event by event as

$$N = -M\ln(1-H/M).$$ (22)

Here $N$ is the number of tracks that cross the array of $M$ channels and create $H$ hits. Equation 4.20 uses the Poisson law for $P(n)$. $N$ is linearly correlated with the total amplitude (the sum of the channel amplitudes), and for low occupancy cases I use the amplitude (properly calibrated in the units of $N$) instead because in those cases contribution of noise hits (due to drifting pedestals) or absent hits (again due to drifting pedestals) may be significant.

The multiplicity via occupancy expression (Eq.  4.20) has been justified as an integral, or ``average'', multiplicity over the Si acceptance. Variation of multiplicity from ring to ring is thus neglected. The technical reason for that was the absence of individual ring data in the existing DSTs. If the track density is not uniform over the Si acceptance, the formula results in a systematic error, which increases for steeper $\,dN/\,d\eta$. Let's compare the approximate expression for multiplicity with the exact one.

Approximate (Eq.  4.20 detailed):

$$N = -16 m\ln(1-\frac{\sum_{i=1}^{16} H_i}{16m}),$$ (23)

where $m$ is number of pads in a ring, $H_i$ is number of hits, and $N$ is the number of tracks crossing the detector.

Exact:

$$N = \sum_{i=1}^{16} N_i$$ (24)

where every

$$N_i = -m\ln(1-H_i/m)$$ (25)

- from every ring. Therefore the exact formula is

$$N = -m\ln(\prod_{i=1}^{16}(1-H_i/m))$$ (26)

To summarize, the approximate expression uses the arithmetic average in the logarithm, while the exact one replaces it by the geometrical average. We also see that the approximate and exact expression become identical if

$$H_1=H_2=H_3=...=H_{16}$$ (27)

i.e., if the distribution is uniform over the 16 rings of the detector.

Thus we see that knowledge of the distribution can serve to correct for the systematic error. In principle there are two choices:

- purist: use real Si data to extract the shape of the distribution, and use that information to get the correction.

- pragmatic: we know (and use this knowledge elsewhere in the analysis) that RQMD reproduces the shape of the non-identified charged $\,dN/\,dy$ (and therefore $\,dN/\,d\eta$) fairly well. Compare, e.g., the NA49 data points in Fig 4.2 with the corresponding RQMD histogram plotted as a line.

Figure: $\,dN/\,dy$ distributions for negative hadrons: solid and open points - from NA49 measurements [41]; the histogram - from RQMD events of comparable centrality.

So, we can use RQMD to get the correction.

I followed the second path. The average number of tracks hitting a pad in a specific ring can be extracted from a properly binned RQMD charged track ($+$ and $-$) $\,dN/\,d\eta$ distribution:

$$\frac{N}{M} = \frac{\Delta N}{\Delta \eta} \times \frac{\Delta \e... ... \frac{(3.3-1.5)}{16} \frac{1}{32} = 0.0035\times \frac{\Delta N}{\Delta \eta}$$

because 16 rings (with equal pseudorapidity coverage) of the detector cover pseudorapidity range 1.5 to 3.3, and each ring has 32 pads. Inverting Eq.  4.20, the 16 values of $H_i/M_i$ were obtained - they range between 0.5 and 0.8. Plugging them into the ``approximate'' and ``exact'' expressions above yields the correction:

$$\frac{\,dN/\,d\eta(exact)}{\,dN/\,d\eta(approximate)} = 1.022$$

Conclusion: the multiplier 1.022 needs to be applied to account for deviation of the $\,dN/\,d\eta$ distribution from uniformity over the Si acceptance.