Unified Analyses of Multiplicity Distributions and Bose-Einstein Correlations at the LHC using Double-Stochastic Distributions

We analyze data on multiplicity distributions (MD) at Large Hadron Collider (LHC) energies using a double-negative binomial distribution (D-NBD) and double-generalized Glauber-Lachs formula (D-GGL). Moreover, we investigate the Bose-Einstein correlation (BEC) formulas based on these distributions and analyze the BEC data using the parameters obtained by analysis of MDs. From these analyses it can be inferred that the D-GGL formula performs as effectively as the D-NBD. Moreover, our results show that the parameters estimated in MD are related to those contained in the BEC formula.

In theoretical analysis of experimental data, MD and the Bose-Einstein correlation (BEC) have been handled as independent observables. In contrast, we have proposed a method for analyzing MD and BEC data using common parameters [16,17,15]. See a review book [18].
To understand the KNO scaling violation, the weight factors (α 1 and α 2 ) are displayed in Fig. 3. The fluctuation of (α 1 and α 2 ) are reflecting the KNO scaling violation.
For BEC, we need the ratio N (2+: 2−) /N BG , where N stands for the number of pairs. The numerator N (2+: 2−) is the number of the pairs of the same charged particles. N BG is that of different charged particles. There are and N BG 2 = n 2 2 . This procedure needs the complete separation of two ensembles, for this aim. Conversely, the second N BG II is computed from two ensembles, such as N BG II = (α 1 n 1 2 + α 2 n 2 2 ). The denominator is the sum of numerators of pairs come from the first and second ensembles. Thus, we have two formulas for BEC, which depend on N BG I and N BG II .
where R is the magnitude of the interaction range and Q = −(p 1 − p 2 ) 2 is the momentum transfer. For BEC based on D-NBD, we obtain the formula by adapting p = 1.0 in Eq. (9). Our results are based on the parameters presented in Table 1 and Eqs. (7)∼(10). The results are displayed in Table 2 and Fig. 4.

Concluding remarks
1) From the analysis of MD with |η| <2.5-3.0, we obtained the energy dependence of the weight factors (α 1 , α 2 ) (Fig. 3). This behavior denotes the degree of the KNO scaling violation.
2) The formula based on N BG II , i.e., Eq. (8), explains the BEC data more reasonably than the formula based on N BG I , i.e., Eq. (7) ( Table 2). At present, the second denominator seems to be available for the study on the BEC. Detailed calculations will be provided elsewhere [21].
3) In 1986, the violation of KNO scaling was discovered by UA5 Collaboration. To explain this phenomenon, the UA5 collaboration proposed the D-NBD, i.e., Eq. (2). Moreover, in experiments at LHC energies, a remarkable violation of the KNO scaling has been found. In such situation, we proposed the unified analyses of MD and BEC at the LHC by making use of D-GGL as well as D-NBD. For BEC, we have derived Eqs. (7) and (8). They are unified as follows:   where λ 2 is the second degree of coherence. Eq. (11) can be named a second conventional formula as CF II . Our result by Eq. (11) is also given in Table 2.

4)
Analyses of ATLAS BEC by the triple-NBD (T-NBD) [21] are added. The T-NBD is the extensive formula of Eq. (2): For T-NBD, see also [22]. The similarities with results by CF II seem to be better those by D-NBD and D-GGL: Results by CF II and those by Eq. (12) with the extensive formula of Eq. (8) are almost the same, except for λ's and χ 2 s.