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

TL;DR

This paper presents a unified analysis of multiplicity distributions and Bose-Einstein correlations at the LHC using double-stochastic models, revealing their effectiveness and parameter relationships.

Contribution

It introduces a combined approach employing D-NBD and D-GGL models for analyzing multiplicity and BEC data at the LHC, demonstrating their comparable performance.

Findings

D-GGL performs as effectively as D-NBD in data analysis.

Parameters from MD analysis relate to those in BEC formulas.

Unified models provide consistent descriptions of multiplicity and BEC data.

Abstract

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.

Tables

Table 1. Table 1: Analyses results of analyses of ATLAS data on the MD ( p t > 100 subscript 𝑝 𝑡 100 p_{t}>100 MeV, | η | < 2.5 𝜂 2.5 |\eta|<2.5 , n ch ≥ 2 subscript 𝑛 ch 2 n_{\rm ch}\geq 2 ) using Eqs. ( 2 ) and ( 9 ).

$\sqrt{s}$	Eq. (2) D-NBD
$[$TeV]	${\alpha }_1$	$k_1$	$⟨n_1⟩$	$k_2$	$⟨n_2⟩$	${\chi }^2/$ndf
0.9	0.23$\pm$0.03	7.52$\pm$0.77	32.9$\pm$1.3	2.70$\pm$0.12	12.6$\pm$0.4	70.2/46
7.0	0.60$\pm$0.02	2.61$\pm$0.10	39.4$\pm$0.6	2.72$\pm$0.09	11.0$\pm$0.1	129/80
$\sqrt{s}$	Eq. (9) D-GGL
$[$TeV]	${\alpha }_1$	$p_1$	$⟨n_1⟩$	$p_2$	$⟨n_2⟩$	${\chi }^2/$ndf
0.9	0.41$\pm$0.05	0.25$\pm$0.04	27.0$\pm$1.4	0.39$\pm$0.02	10.6$\pm$0.4	60.3/46
7.0	0.67$\pm$0.01	0.68$\pm$0.03	36.7$\pm$0.3	0.42$\pm$0.02	10.4$\pm$0.1	114/80

Table 2. Table 2: Analyses results of ATLAS data on BEC ( p t > 100 subscript 𝑝 𝑡 100 p_{t}>100 MeV, | η | < 2.5 𝜂 2.5 |\eta|<2.5 , n ch ≥ 2 subscript 𝑛 ch 2 n_{\rm ch}\geq 2 ) [ 20 ] using Eqs. ( 11 ), ( 12 ), and ( 15 ). The BEC formulas contain the normalization factor and the long range correlation as ( 1 + ε ​ Q ) 1 𝜀 𝑄 (1+\varepsilon Q) . The conventional formula (CF I ) is given as CF = I 1.0 + λ E BE 2 {}_{\rm I}=1.0+\lambda E_{\rm BE}^{2} .

$\sqrt{s}$	Eq. (11) D-NBD			Eq. (11) D-GGL			CFI
[TeV]	$R_1$ [fm]	$R_2$ [fm]	${\chi }^2/$ndf	$R_1$ [fm]	$R_2$ [fm]	${\chi }^2$	$R$ [fm]	$\lambda$	${\chi }^2$
0.9	1.7$\pm$0.4 (E)	1.7$\pm$0.1 (E)	98.6/75	2.8$\pm$0.2 (E)	2.7$\pm$0.1 (E)	148	1.8$\pm$0.1(E)	0.62$\pm$0.01	86.0
7.0	1.8$\pm$0.0 (E)	3.1$\pm$0.1 (E)	743/75	3.6$\pm$0.1 (E)	3.2$\pm$0.1 (E)	629	2.1$\pm$0.0(E)	0.62$\pm$0.01	919
$\sqrt{s}$	Eq. (12) D-NBD			Eq. (12) D-GGL
[TeV]	$R_1$ [fm]	$R_2$ [fm]	${\chi }^2/$ndf	$R_1$ [fm]	$R_2$ [fm]	${\chi }^2$
0.9	0.77$\pm$0.02(G)	1.60$\pm$0.06(G)	105/75	1.69$\pm$0.04(G)	1.44$\pm$0.18(E)	120
7.0	2.28$\pm$0.01(E)	0.68$\pm$0.03(E)	692/75	3.63$\pm$0.01(E)	1.33$\pm$0.11(E)	557

Equations

$$P_{\mathrm{NBD}}(n,k,⟨n⟩)=\frac{\Gamma (n+k)}{\Gamma (n+1)\Gamma (k)}\frac{(⟨n⟩/k{)}^n}{(1+⟨n⟩/k{)}^{n+k}}.$$

$$P_{\mathrm{D}\text{mathchar}45NBD}(n,⟨n⟩)={\alpha }_1{P}_{\mathrm{NB}{\mathrm{D}}_1}(n,k_1,⟨n_1⟩)+{\alpha }_2{P}_{\mathrm{NB}{\mathrm{D}}_2}(n,k_2,⟨n_2⟩).$$

$$P_{\mathrm{GGL}}(n,k,p,⟨n⟩)=\frac{(p⟨n⟩/k{)}^n}{(1+p⟨n⟩/k{)}^{n+k}}\exp \left[-\frac{\gamma p⟨n⟩}{1+p⟨n⟩/k}\right]L_n^{(k-1)}\left(-\frac{\gamma k}{1+p⟨n⟩/k}\right),$$

$$\displaystyle\mbox{Eq.~{}(\ref{eq3})}\ \left\{\begin{array}[]{l}\smash{\mathop{\hbox to28.45274pt{\rightarrowfill}}\limits^{\quad\ k=1\qquad}}\ \mbox{original GL}\ \smash{\mathop{\hbox to28.45274pt{\rightarrowfill}}\limits^{\quad\ p\to 1\qquad}}\ \mbox{Furry distribution}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \smash{\mathop{\hbox to28.45274pt{\rightarrowfill}}\limits^{\quad\ p\to 1\qquad}}\ \mbox{NBD of Eq.~{}(\ref{eq1})}\\ \hskip 85.35826pt\downarrow k\to\infty\\ \smash{\mathop{\hbox to28.45274pt{\rightarrowfill}}\limits^{\quad\ p\to 0\qquad}}\ \mbox{Poisson distribution}.\end{array}\right.$$

$$P_{\mathrm{D}\text{mathchar}45GGL}(n,⟨n⟩)={\alpha }_1{P}_{\mathrm{GG}{\mathrm{L}}_1}(n,k=2,p_1,⟨n_1⟩)+{\alpha }_2{P}_{\mathrm{GG}{\mathrm{L}}_2}(n,k=2,p_2,⟨n_2⟩),$$

$$⟨n⟩$$

$$⟨n(n-1)⟩$$

$$N^{(2+:2-)}/N_{\mathrm{I}}^{\mathrm{BG}}$$

$$N^{(2+:2-)}/N_{\mathrm{II}}^{\mathrm{BG}}$$

$$f(p,E_{\mathrm{BE}})=2p(1-p)E_{\mathrm{BE}}+p^2{E}_{\mathrm{BE}}^2.$$

$$E_{\mathrm{BE}}^2=\exp (-(RQ{)}^n),$$

$$\mathrm{C}{\mathrm{F}}_{\mathrm{II}}=1.0+{\lambda }_1{E}_{\mathrm{B}{\mathrm{E}}_1}^2+{\lambda }_2{E}_{\mathrm{B}{\mathrm{E}}_2}^2,$$

$$P_{\mathrm{T}\text{mathchar}45NBD}(n,⟨n⟩)=\sum _{i=1}^3{\alpha }_i{P}_{{\mathrm{NBD}}_i}(n,k_i,⟨n_i⟩).$$

Full text

1 Introduction

In Large Hadron Collider (LHC) experiments, various multiplicity distributions (MD) have been measured with a pseudorapidity interval ($|\eta|<\eta_{c}$) in proton-proton collision [1, 2, 3]. Additionally, the negative binomial distribution (NBD) has been used to analyze MD.

[TABLE]

However, we cannot understand the behavior of the parameter: $k=7,\,8$ for lower energy (ISR) and $k=2,\,3$ for higher energy (S$p\bar{p}$S).

The double-negative binomial distribution (D-NBD) [4, 5] has been proposed to explain the KNO scaling [6] violation observed by the UA5 [7, 8, 9], using $\alpha_{1}+\alpha_{2}=1$,

[TABLE]

In 2010, we analyzed the ALICE data [10] by the generalized Glauber–Lachs (GGL) formula, and showed that GGL fits as well as NBD [11]:

[TABLE]

where $p=1/(1+\gamma)$ and $\gamma$ reflects the contamination ($K$, $p/\pi$’s) and the degree of superposition of the phase spaces of the particles. Our analyses suggested that the coherent component is necessary. Moreover, Eq. (3) has the following stochastic property:

[TABLE]

To explain the shape of the MD with higher energy than S$p\bar{p}$S, two sources or two collision mechanisms [12, 13, 14] are required; thus, the two-component model is necessary.

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].

2 Double-GGL formula (D-GGL)

Herein, we propose the D-GGL formula[19] for analyzing the multiplicity distribution at LHC:

[TABLE]

where $k=2$ reflects the degree of freedom for the $(+-)$ particle ensembles (Fig. 1). Our results are presented in Fig. 2 and Table 1.

To understand the KNO scaling violation, the weight factors ($\alpha_{1}$ and $\alpha_{2}$) are displayed in Fig. 4. The fluctuation of ($\alpha_{1}$ and $\alpha_{2}$) are reflecting the KNO scaling violation.

3 Bose–Einstein correlations (BEC)

By using Eq. (9), we obtain the moments as

[TABLE]

For BEC, we need the ratio $N^{(2+:\,2-)}/N^{\rm BG}$, where $N$ stands for the number of pairs. The numerator $N^{(2+:\,2-)}$ is the number of the pairs of the same charged particles. $N^{\rm BG}$ is that of different charged particles. There are two possibilities for calculating $N^{\rm BG}$. Whole events can be categorized as two ensembles labeled $\alpha_{1}$ and $\alpha_{2}$. Each $N_{1}^{\rm BG}$ and $N_{2}^{\rm BG}$ are calculated inside those ensembles: $N_{1}^{\rm BG}={}_{\langle n_{1}^{(+)}\rangle}C_{1}\times{}_{\langle n_{1}^{(-)}\rangle}C_{1}=\langle n_{1}^{(+)}\rangle\langle n_{1}^{(-)}\rangle=\langle n_{1}\rangle^{2}$ and $N_{2}^{\rm BG}=\langle n_{2}\rangle^{2}$. This procedure needs the complete separation of two ensembles, for this aim.

Conversely, the second $N_{\rm II}^{\rm BG}$ is computed from two ensembles, such as $N_{\rm II}^{\rm BG}=(\alpha_{1}\langle n_{1}\rangle^{2}+\alpha_{2}\langle n_{2}\rangle^{2}$). 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 $N_{\rm I}^{\rm BG}$ and $N_{\rm II}^{\rm BG}$.

[TABLE]

where, $\Gamma=(\alpha_{2}\langle n_{2}\rangle^{2})/(\alpha_{1}\langle n_{1}\rangle^{2})$ and the normalization $(1+\Gamma)^{-1}+(1+\Gamma^{-1})^{-1}=1$, and

[TABLE]

The exchange function between the same charged particles is expressed by the exponential function (E, $n=1$) or the Gaussian function (G, $n=2$).

[TABLE]

where $R$ is the magnitude of the interaction range and $Q=\sqrt{-(p_{1}-p_{2})^{2}}$ is the momentum transfer. For BEC based on D-NBD, we obtain the formula by adapting $p=1.0$ in Eq. (13).

Our results are based on the parameters presented in Table 1 and Eqs. (11)$\sim$(14). The results are displayed in Table 2 and Fig. 4.

4 Concluding remarks

**1) ** From the analysis of MD with $|\eta|<$2.5–3.0, we obtained the energy dependence of the weight factors $(\alpha_{1},\,\alpha_{2})$ (Fig. 4). This behavior denotes the degree of the KNO scaling violation.

**2) ** The formula based on $N_{\rm II}^{\rm BG}$, i.e., Eq. (12), explains the BEC data more reasonably than the formula based on $N_{\rm I}^{\rm BG}$, i.e., Eq. (11) (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. (11) and (12). They are unified as follows:

[TABLE]

where $\lambda_{2}$ is the second degree of coherence. Eq. (15) can be named a second conventional formula as CFII. Our result by Eq. (15) 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):

[TABLE]

For T-NBD, see also [22]. The similarities with results by CFII seem to be better those by D-NBD and D-GGL: Results by CFII and those by Eq. (16) with the extensive formula of Eq. (12) are almost the same, except for $\lambda$’s and $\chi^{2}$s.

*Acknowledgments. * T.M. would like to appreciate the special budget given by Pres. Y. Hayashi. M.B. would like to thank the colleagues at the Department of Physics in Shinshu University for their kindness.