Method for high-accuracy multiplicity-correlation measurements

Multiplicity-correlation measurements provide insight into the dynamics of high-energy collisions. Models describing these collisions need these correlation measurements to tune the strengths of the underlying QCD processes which influence all observables. Detectors, however, often possess limited coverage or reduced efficiency that influence correlation measurements in obscure ways. In this paper, the effects of nonuniform detection acceptance and efficiency on the measurement of multiplicity correlations between two distinct detector regions (termed forward-backward correlations) are derived. An analysis method with such effects built in is developed and subsequently verified using different event generators. The resulting method accounts for acceptance and efficiency in a model-independent manner with high accuracy, thereby shedding light on the relative contributions of the underlying processes to particle production.

Figure 1

The plots depict three sets of forward-backward multiplicity pairs with $b=1,0.6,\text{and}0$ for (a)–(c), respectively. The variances (used in the denominator of $b$) are the same for $N_f$ and $N_b$ in all three cases. This demonstrates that the correlation information is essentially contained in the covariance.

Figure 2

(a) The detector is schematically divided into two halves along the line at $\theta =\pi /2$. Each half then consists of solid angles which span a “small” polar angle and the full azimuthal angle. A forward-backward bin consists of two regions where their centers are equidistant from $\theta =\pi /2$. The regions are often termed forward when $\theta <\pi /2$ and backward when $\theta >\pi /2$. Each forward-backward bin is represented by a different color here. (b) The detector regions have been mapped to a two-dimensional figure. The horizontal axis is the pseudorapidity, which is a function of the polar angle, $\theta$. Note that the mapping is not to scale and equal size polar angle bins on the left do not correspond to equal size pseudorapidity bins on the right. Pseudorapidity bin sizes of 1, 0.5, and 0.25 units are used here. The magnitude of the correlation factor depends on this bin width (decreasing as the bin width is reduced).

Figure 3

The diagrams depict sets of correlations between different $\phi$ regions. Note that the solid arrow and dashed arrow can (in general) point to different $\eta$ regions. Correlations between regions shifted by 1, 2, and $3 \phi$ segments for (a)–(c), respectively, are shown. In general this shift can be any value between 0 and the number of $\phi$ segments minus 1. For each shift, the correlation between the regions should be the same (for primary particles) independent of the average $\phi$ angle of the correlated regions. This produces redundant measures of the same correlation. A set of redundant measures is called an “invariant twist.”

Figure 4

The diagrams depict sets of dead regions (colored gray) for a detector that has $5 \phi$ segments. If one assumes the same dead regions apply to both the forward and backward pseudorapidity regions (as an example), one can compute the number of times each invariant twist is measured $\left(N_{\mathrm{meas}}\right)$ for each configuration with a shift of $s$ segments and compute a scale factor that must be applied to give the correct contribution to the variance or covariance. For the same total dead area one can still obtain different variances or covariances, because the configuration of dead regions affects the measurement through different scale factors applied to each invariant twist [shown in (a) and (b)]. (c) shows that if one has less than 50% acceptance, one will lack a measure of an invariant twist, making the measurement impossible to calculate with this method.

Figure 5

Correlation factors (values of $b$) obtained using the particle multiplicities computed at $\eta =\frac{\mathrm{\Delta }\eta }{2}$ and $\eta =-\frac{\mathrm{\Delta }\eta }{2}$ from the pythia6 event generator. Three different tunes were selected: Perugia3, Perugia0, and DW. Three different bin widths, ${\mathrm{\Delta }}_{\mathrm{bin}}$, in $\eta$ were used: (a) ${\mathrm{\Delta }}_{\mathrm{bin}}=1$, (b) ${\mathrm{\Delta }}_{\mathrm{bin}}=0.5$, (c) ${\mathrm{\Delta }}_{\mathrm{bin}}=0.25$.

Figure 6

Correlation factors (values of $b$) obtained when only 60% of the total acceptance exists in $\phi$ equivalently over all $\eta$, correcting the found correlation factor from Perugia3 (P3), Perugia0 (P0), or DW by the ratio of the primary-to-found correlation factor from one of the other tunes. Three different bin widths, ${\mathrm{\Delta }}_{\mathrm{bin}}$, in $\eta$ are shown: (a) ${\mathrm{\Delta }}_{\mathrm{bin}}=1$, (b) ${\mathrm{\Delta }}_{\mathrm{bin}}=0.5$, (c) ${\mathrm{\Delta }}_{\mathrm{bin}}=0.25$. The bottom part of each plot shows the ratio of the corrected correlation factor to the actual primary correlation factor for the tune that the data was generated from.

Figure 7

Four simple examples of the acceptance maps of the $\eta$ bins. (a)–(d) show the inactive regions with acceptances of 90%, 80%, 70%, and 60%, respectively.

Figure 8

(a) Attenuation of the measured correlation factors as a result of decreased $\phi$ acceptance in the $\eta$ bins (see Fig. 7). (b) The obtained correlation factors computed using Eq. (20), which does not exploit $\phi$ segmentation. In both (a) and (b), the bottom part of each plot shows the ratio of the obtained correlation factors to the primary correlation factors. The simple assumption of uniform detection probability reduces the discrepancy from the primary correlation factor by more than a factor of 3, but still leaves substantial discrepancies between the obtained correlation factors and those from the generator output.

Figure 9

(a) Correlation factors computed using $10 \phi$ segments to account for the azimuthal event shape. (b) Correlation factors obtained when 70% acceptance and a varied number of $\phi$ segments in each $\eta$ bin exists. Note that there is no improvement when increasing from 10 to $20 \phi$ segments, because both segmentations produce the same result analytically for the acceptance map applied here. In both (a) and (b), the bottom part of each plot shows the ratio of the obtained correlation factors to the primary correlation factors. Note that the scale in the bottom part of (a) has a much smaller range than that from the previous plots with no $\phi$ segmentation.

Figure 10

(a) Acceptance map with 20 randomly placed inactive regions. The individual regions span 1 unit in $\eta$ and $\frac{2\pi }{10}$ in azimuth. (b) The correlation factors found after accounting for randomly placed inactive regions using different $\phi$ segmentations. The bottom part shows the ratio of the obtained correlation factors to the primary correlation factors.

Figure 11

(a) The efficiency map from a sine function of the form $\varepsilon \left(\phi \right)=0.6sin(\phi /2)+0.2$. The efficiencies take on values in the range $0.2\le \varepsilon \le 0.8$ with the azimuthal angular range of $0\le \phi \le 2\pi$. Note that the colors indicate the average efficiency within the bins and not the values from the efficiency function itself. (b) Resultant correlation factors obtained using Eq. (32) with efficiency values extracted from the shown efficiency map using different azimuthal segmentations. The bottom part shows the ratio of the obtained correlation factors to the primary correlation factors.

Figure 12

(a) The efficiency map from a sine function of the form $\varepsilon \left(\eta \right)=0.6cos(\eta \pi /8)+0.2$. The efficiencies take on values in the range $0.2\le \varepsilon \le 0.8$ with the pseudorapidity range of $-4\le \eta \le 4$. The colors indicate the average efficiency within the bins and not the values from the efficiency function itself. (b) Resultant correlation factors obtained using Eq. (36) with efficiency values extracted from the shown efficiency map using different pseudorapidity segmentations for each measured point. The bottom part shows the ratio of the obtained correlation factors to the primary correlation factors.

Figure 13

Resultant correlation factors obtained using different tunes of pythia and three different $\eta$ bin widths [${\mathrm{\Delta }}_{\mathrm{bin}}=1,0.5,\text{and}0.25$ for (a)–(c), respectively]. Data from each tune are subjected to the same geometrical acceptance (60%). The bottom part of each plot shows the ratio of the obtained correlation factor to the primary value for that tune (shown in Fig. 5). No significant dependence of the deviations on the tune is observed.