Unified Description of Multiplicity Distributions and Bose-Einstein Correlations at the LHC Based on the Three-Negative Binomial Distribution

TL;DR

This paper introduces a unified approach using the three-negative binomial distribution to describe multiplicity distributions and Bose-Einstein correlations at the LHC, providing new insights into particle interactions and scaling behaviors.

Contribution

The study applies the T-NBD framework to analyze multiplicity and BEC data at LHC energies, revealing consistent interaction ranges and explaining KNO scaling violations.

Findings

T-NBD effectively models multiplicity distributions and BEC at LHC energies.

Estimated interaction ranges R1 and R2 are nearly identical and consistent with pp collisions.

The enlarged KNO scaling function accounts for observed violations at high energies.

Abstract

Using the Monte Carlo data at 7 TeV collected by the ATLAS collaboration (PYTHIA 6), we examine the necessity of applying the three-negative binomial distribution (T-NBD). By making use of the T-NBD formulation, we analyze the multiplicity distribution (MD) and the Bose-Einstein correlation (BEC) at the Large Hadron Collider (LHC). In the T-NBD framework, the BEC is expressed by two degrees of coherence $\lambda_1$ and $\lambda_2$ and two kinds of exchange functions $E_1^2$ and $E_2^2$ that act over interaction ranges $R_1$ and $R_2$, respectively. Using the calculated $\lambda_1$ and $\lambda_2$ based on the T-NBD, along with free $\lambda_1$ and $\lambda_2$, we analyze the BEC data at 0.9 and 7 TeV. The estimated parameters $R_1$ and $R_2$ are almost coincident and seem to be consistent with $pp$ collisions. We also present an enlarged Koba-Nielsen-Olesen (KNO) scaling function based on the T-NBD, and apply it to KNO scaling at LHC energies. The enlarged scaling function describes the observed violation of the KNO scaling.

Tables

Table 1. Table 1: Stochastic properties of the three ensembles computed by PYTHIA 8 in the ATLAS experiment at 7 TeV (see Figs.  5 and 6 in Appendix  A ).

QCD	acceptance and	stochastic property	stochastic description
PYTHIA 6	correction by	of individual ensemble	of sum of three ensembles
7 TeV	ATLAS coll. [10]
ND	$P_{\mathrm{ND}}\left(n\right)$	(sum of)
	$f_{\mathrm{ND}}=\sum P_{\mathrm{ND}}\left(n\right)=0.787$	NBD1 and NBD2	T-NBD Eq. (3)
SD	$P_{\mathrm{SD}}\left(n\right)$		$P_{\left(T-N\right)}(n,⟨n⟩,{\alpha }_i,k_i)=$
	$f_{\mathrm{SD}}=0.121$	NBD2 and NBD3	${\sum }_{i=1}^3{\alpha }_i{P}_{{\mathrm{NBD}}_i}(n,⟨n_i⟩,k_i)$
DD	$P_{\mathrm{DD}}\left(n\right)$
	$f_{\mathrm{DD}}=0.092$	NBD3 and NBD2
	sum of fractions:	NBDi’s are specified	Eq. (3) is a
	$f_{\mathrm{ND}}+f_{\mathrm{SD}}+f_{\mathrm{DD}}=1.0$	by $(⟨n⟩,k)$	possible candidate

Table 2. Table 2: Analysis (by Eq. ( 3 )) of the MD ( P ​ ( n ) 𝑃 𝑛 P(n) ) data at 0.9 and 7.0 TeV collected by the ATLAS and CMS collaborations.

	$i$	${\alpha }_i$	$⟨n_i⟩$	$a_i$	$k_i$
ATLAS	1	0.640$\pm$0.199	13.493$\pm$2.546	116.519$\pm$56.975	1.78$\pm$0.20
0.9 TeV	2	0.250$\pm$0.164	28.488$\pm$3.588	202.892$\pm$142.572	5.01$\pm$1.37
${\chi }^2=5.317$	3	0.111$\pm$0.047	10.998$\pm$0.237	13.426$\pm$5.714	28.1$\pm$24.4
ATLAS	1	0.737$\pm$0.053	21.934$\pm$2.392	354.571$\pm$81.430	1.50$\pm$0.08
7.0 TeV	2	0.183$\pm$0.061	57.214$\pm$2.597	599.040$\pm$206.953	5.67$\pm$0.76
${\chi }^2=5.964$	3	0.080$\pm$0.010	11.164$\pm$0.169	9.971$\pm$1.282	23.4$\pm$8.4
ATLAS	1	0.754$\pm$0.063	22.625$\pm$2.549	385.966$\pm$92.755	1.48$\pm$0.08
7.0 TeV	2	0.164$\pm$0.063	57.936$\pm$2.618	550.479$\pm$217.238	5.94$\pm$0.98
${\chi }^2=6.160$	3	0.082$\pm$0.010	11.177$\pm$0.178	10.244$\pm$1.291	23.4$\pm$8.7
CMS	1	0.743$\pm$0.179	15.852$\pm$2.454	186.705$\pm$73.245	2.08$\pm$0.20
0.9 TeV	2	0.189$\pm$0.170	32.160$\pm$4.567	195.476$\pm$184.382	6.56$\pm$2.85
${\chi }^2=4.289$	3	0.068$\pm$0.032	11.624$\pm$0.814	9.188$\pm$4.511	896$\pm$817
CMS	1	0.739$\pm$0.200	15.830$\pm$2.841	185.092$\pm$83.177	2.10$\pm$0.21
0.9 TeV	2	0.193$\pm$0.179	32.093$\pm$5.002	199.236$\pm$194.669	6.49$\pm$3.03
${\chi }^2=4.941$	3	0.068$\pm$0.031	11.618$\pm$0.810	9.177$\pm$4.384	$\infty$
CMS	1	0.826$\pm$0.091	28.613$\pm$4.126	676.108$\pm$208.928	1.66$\pm$0.12
7.0 TeV	2	0.103$\pm$0.098	67.206$\pm$6.727	465.270$\pm$452.384	6.67$\pm$2.92
${\chi }^2=2.275$	3	0.071$\pm$0.028	13.018$\pm$0.870	12.036$\pm$5.012	38.1$\pm$73.6

Table 3. Table 3: Second and third moments of MD at 0.9 TeV and 7 TeV, measured (data) and calculated by Eqs. ( 2 ) (D-NBD) and ( 3 ) (T-NBD).

	$⟨n\left(n-1\right)⟩$ ($\times {10}^3$)			$⟨n\left(n-1\right)\left(n-2\right)⟩$ ($\times {10}^4$)
	data	D-NBD	T-NBD	data	D-NBD	T-NBD
ATLAS 0.9 TeV	0.454$\pm$0.026	0.450	0.439	1.55$\pm$0.13	1.54	1.53
ATLAS 7.0 TeV	1.35$\pm$0.08	1.35	1.30	8.75$\pm$0.79	9.09	8.43
CMS 0.9 TeV	0.488$\pm$0.050	0.525	0.511	1.73$\pm$0.22	1.86	1.82
CMS 7.0 TeV	1.57$\pm$0.13	1.67	1.63	10.94$\pm$1.10	11.89	11.52

Table 4. Table 4: Values of λ i ( T − N ) = ( α i ​ ⟨ n i ⟩ 2 / s ) ​ ( 2 / k i ) superscript subscript 𝜆 𝑖 T N subscript 𝛼 𝑖 superscript delimited-⟨⟩ subscript 𝑛 𝑖 2 𝑠 2 subscript 𝑘 𝑖 \lambda_{i}^{\rm(T\mathchar 45\relax N)}=(\alpha_{i}\langle n_{i}\rangle^{2}/s)(2/k_{i}) .

	${\lambda }_1^{\left(T-N\right)}$	${\lambda }_2^{\left(T-N\right)}$	${\lambda }_3^{\left(T-N\right)}$
ATLAS 0.9 TeV, ${\chi }^2=5.317$	0.393$\pm$0.214	0.244$\pm$0.103
ATLAS 7.0 TeV, ${\chi }^2=5.964$	0.490$\pm$0.130	0.219$\pm$0.045
ATLAS 7.0 TeV, ${\chi }^2=6.160$	0.552$\pm$0.152	0.196$\pm$0.050	$𝒪\left({10}^{-3}\sim {10}^{-4}\right)$
CMS 0.9 TeV, ${\chi }^2=4.289$	0.460$\pm$0.241	0.152$\pm$0.102
CMS 0.9 TeV, ${\chi }^2=4.941$	0.448$\pm$0.250	0.156$\pm$0.110
CMS 7.0 TeV, ${\chi }^2=2.275$	0.706$\pm$0.296	0.121$\pm$0.091

Table 5. Table 5: Analysis results of the BEC data from ATLAS  [ 22 ] using Eqs. ( 12 ), ( 13 ), and ( 14 ), where the BEC formulas are normalized with a consistent factor and the long-range correlation is assumed as ( 1 + ε ​ Q ) 1 𝜀 𝑄 (1+\varepsilon Q) (the labels ( ∗ 1 ∗ absent 1 \ast 1 and ∗ 2 ∗ absent 2 \ast 2 ) indicate equivalence between the tabulated results by Eq. ( 12 ) and Eq. ( 14 ), respectively).

ATLAS 0.9 TeV
	$R$ [fm]	$\lambda$ (free)	—	—	${\chi }^2/$ndf
CFI	1.84$\pm$0.07 (E)	0.74$\pm$0.03	—	—	86.0/75
	1.00$\pm$0.03 (G)	0.34$\pm$0.01	—	—	148/75
	$R_1^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_1^{\left(\mathrm{II}\right)}$ (free)	$R_2^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_2^{\left(\mathrm{II}\right)}$ (free)	${\chi }^2$
CFII	4.52$\pm$1.02 (E)	0.98$\pm$0.21	0.81$\pm$0.05 (G)	0.21$\pm$0.04	78.2
	2.82$\pm$0.28(G)	0.47$\pm$0.07	0.87$\pm$0.03 (G)	0.26$\pm$0.02	79.8 ($\ast 1$)
BEC(T-N)	$R_1^{\left(T-N\right)}$ [fm]	${\lambda }_1^{\left(T-N\right)}$ (calcu.)	$R_2$ [fm]	${\lambda }_2^{\left(T-N\right)}$ (calcu.)	${\chi }^2$
MD ${\chi }^2=5.32$	2.55$\pm$0.10 (G)	0.39	0.85$\pm$0.02 (G)	0.25	81.1 ($\ast 1$)
	3.37$\pm$0.22 (E)	0.39	0.89$\pm$0.02 (G)	0.24	101
ATLAS 7.0 TeV
	$R$ [fm]	$\lambda$ (free)	—	—	${\chi }^2/$ndf
CFI	2.06$\pm$0.01 (E)	0.72$\pm$0.01	—	—	919/75
	1.13$\pm$0.01 (G)	0.33$\pm$0.00	—	—	4578/75
	$R_1^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_1^{\left(\mathrm{II}\right)}$ (free)	$R_2^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_2^{\left(\mathrm{II}\right)}$ (free)	${\chi }^2$
CFII	6.54$\pm$0.40 (E)	0.73$\pm$0.05	1.80$\pm$0.02 (E)	0.54$\pm$0.02	465
	1.85$\pm$0.02 (E)	0.59$\pm$0.01	3.51$\pm$0.12 (G)	0.28$\pm$0.01	466 ($\ast 2$)
BEC(T-N)	$R_1^{\left(T-N\right)}$ [fm]	${\lambda }_1^{\left(T-N\right)}$ (calcu.)	$R_2^{\left(T-N\right)}$ [fm]	${\lambda }_2^{\left(T-N\right)}$ (calcu.)	${\chi }^2$
MD ${\chi }^2=5.96$	1.70$\pm$0.01 (E)	0.49	2.52$\pm$0.03 (G)	0.22	609
cf.	2.40$\pm$0.02 (E)	0.49	1.52$\pm$0.02 (E)	0.22	836
MD ${\chi }^2=6.16$	1.80$\pm$0.01 (E)	0.55	2.85$\pm$0.04 (G)	0.20	531 ($\ast 2$)

Table 6. Table 6: Results of the BEC data collected by the CMS  [ 23 ] and analyzed by Eqs. ( 12 ), ( 13 ), and ( 14 ), where the BEC formulas are normalized with a constant factor and the long-range correlation is assumed as ( 1 + ε ​ Q ) 1 𝜀 𝑄 (1+\varepsilon Q) (the labels ( ∗ 1 ∗ absent 1 \ast 1 and ∗ 2 ∗ absent 2 \ast 2 ) indicate equivalence between the tabulated results by Eq. ( 12 ) and Eq. ( 14 ), respcetively).

CMS 0.9 TeV
	$R$ [fm]	$\lambda$ (free)	—	—	${\chi }^2/$ndf
CFI	1.56$\pm$0.02 (E)	0.62$\pm$0.01	—	—	487/194
	0.87$\pm$0.01 (G)	0.30$\pm$0.00	—	—	1157/194
	$R_1^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_1^{\left(\mathrm{II}\right)}$ (free)	$R_2^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_2^{\left(\mathrm{II}\right)}$ (free)	${\chi }^2$
CFII	3.37$\pm$0.19 (E)	0.62$\pm$0.01	0.80$\pm$0.04 (G)	0.14$\pm$0.01	356
	2.06$\pm$0.07 (G)	0.38$\pm$0.02	0.65$\pm$0.01 (G)	0.17$\pm$0.01	384 ($\ast 1$)
BEC(T-N)	$R_1^{\left(T-N\right)}$ [fm]	${\lambda }_1^{\left(T-N\right)}$ (calcu.)	$R_2^{\left(T-N\right)}$ [fm]	${\lambda }_2^{\left(T-N\right)}$ (calcu.)	${\chi }^2$
MD ${\chi }^2=4.29$	2.02$\pm$0.02 (G)	0.46	0.61$\pm$0.01 (G)	0.15	429 ($\ast 1$)
MD ${\chi }^2=4.94$	2.06$\pm$0.02 (G)	0.45	0.62$\pm$0.01 (G)	0.16	422 ($\ast 1$)
cf.	1.29$\pm$0.01 (E)	0.45	2.04$\pm$0.01 (G)	0.15	454
CMS 7.0 TeV
	$R$ [fm]	$\lambda$ (free)	—	—	${\chi }^2/$ndf
CFI	1.89$\pm$0.02 (E)	0.62$\pm$0.01	—	—	738/194
	1.03$\pm$0.01 (G)	0.29$\pm$0.00	—	—	1776/194
	$R_1^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_1^{\left(\mathrm{II}\right)}$ (free)	$R_2^{\left(\mathrm{II}\right)}$ [fm]	${\lambda }_2^{\left(\mathrm{II}\right)}$ (free)	${\chi }^2$
CFII	3.88$\pm$0.18 (E)	0.84$\pm$0.03	0.71$\pm$0.01 (G)	0.12$\pm$0.01	540 ($\ast 2$)
	2.39$\pm$0.07 (G)	0.40$\pm$0.01	0.76$\pm$0.01 (G)	0.16$\pm$0.00	600
BEC(T-N)	$R_1^{\left(T-N\right)}$ [fm]	${\lambda }_1^{\left(T-N\right)}$ (calcu.)	$R_2^{\left(T-N\right)}$ [fm]	${\lambda }_2^{\left(T-N\right)}$ (calcu.)	${\chi }^2$
MD ${\chi }^2=2.27$	3.41$\pm$0.03 (E)	0.71	0.70$\pm$ 0.01 (G)	0.12	559 ($\ast 2$)
	2.07$\pm$0.01 (E)	0.71	12.70$\pm$ 2.35 (E)	0.11	817

Table 7. Table 7: Comparison of R 1 subscript 𝑅 1 R_{1} and R 2 subscript 𝑅 2 R_{2} values marked with ( ∗ 1 ∗ absent 1 \ast 1 and ∗ 2 ∗ absent 2 \ast 2 ) in Tables  5 and 6 , and χ 2 superscript 𝜒 2 \chi^{2} of the comparison.

$\sqrt{s}$ [TeV]	formula	$R_1$ [fm]	$R_2$ [fm]	${\chi }^2$
ATLAS	CFII	2.82$\pm$0.28(G)	0.87$\pm$0.03 (G)	79.8
0.9	BEC(T-N)	2.55$\pm$0.10 (G)	0.85$\pm$0.02 (G)	81.1
7.0	CFII	1.85$\pm$0.02 (E)	3.51$\pm$0.12 (G)	466
	BEC(T-N)	1.80$\pm$0.01 (E)	2.85$\pm$0.04 (G)	531
CMS	CFII	2.06$\pm$0.07 (G)	0.65$\pm$0.01 (G)	384
0.9	BEC(T-N)	2.06$\pm$0.02 (G)	0.62$\pm$0.01 (G)	422
7.0	CFII	3.88$\pm$0.18 (E)	0.71$\pm$0.01 (G)	540
	BEC(T-N)	3.41$\pm$0.03 (E)	0.70$\pm$ 0.01 (G)	559

Table 8. Table 8: Comparisons of the two degrees of coherence for the results marked with ( ∗ 1 ∗ absent 1 \ast 1 and ∗ 2 ∗ absent 2 \ast 2 ) in Tables  5 and 6 .

$\sqrt{s}$ [TeV]	formulas	${\lambda }_1$	${\lambda }_2$
ATLAS	CFII	0.47$\pm$0.07	0.26$\pm$0.02
0.9	BEC(T-N)	0.39$\pm$0.21	0.24$\pm$0.10
7.0	CFII	0.59$\pm$0.01	0.28$\pm$0.01
	BEC(T-N)	0.55$\pm$0.15	0.20$\pm$0.05
CMS	CFII	0.38$\pm$0.02	0.17$\pm$0.01
0.9	BEC(T-N)	0.45$\pm$0.25	0.16$\pm$0.11
7.0	CFII	0.84$\pm$0.03	0.12$\pm$0.01
	BEC(T-N)	0.71$\pm$0.30	0.12$\pm$0.09

Table 9. Table 9: Correspondences among KNO scaling, MD, and BEC  [ 20 , 21 ] (see also  [ 17 , 19 ] ).

KNO scaling	MD	BEC
existence.	Single NBD	CFI
${\psi }_k\left(z\right)=\frac{k{\left(kz\right)}^{\left(k-1\right)}{e}^{-kz}}{\Gamma \left(k\right)},$	Eq. (2)	Eq. (13)
where $z=n/⟨n⟩$.
violation I:	D-NBD	CF${}_{\mathrm{II}}{}^{}=$ Eq. (14)
$\psi \left(z\right)=\sum _{i=1}^2\frac{{\alpha }_i}{r_i}{\psi }_{k_i}\left(z_i\right)$	$P(n,⟨n⟩)$	${\mathrm{BEC}}_{\left(D-N\right)}$
$=\frac{{\alpha }_1}{r_1}{\psi }_{k_1}\left(z_1\right)+\frac{{\alpha }_2}{r_2}{\psi }_{k_2}\left(z_2\right)$,	$=\sum _{i=1}^2{\alpha }_i{P}_{{\mathrm{NBD}}_i}(n,⟨n_i⟩,k_i)$,	$=1.0+{\lambda }_1^{\left(D-N\right)}{E}_{{\mathrm{BEC}}_1}^2$
where $z_i=z/r_i$ [20].	$s=\sum _{i=1}^2{\alpha }_i{⟨n_i⟩}^2=a_1+a_2$.	$+{\lambda }_2^{\left(D-N\right)}{E}_{{\mathrm{BEC}}_2}^2$,
$\sum _{i=1}^2{\alpha }_i=1.0$	$\sum _{i=1}^2{\alpha }_i{r}_i=1.0$	where ${\lambda }_i^{\left(D-N\right)}=\left(a_i/s\right)\left(2/k_i\right)$
		($i=1,\mathrm{\hspace{0.17em}2}$). See Ref. [21]
violation II:	T-NBD	CF${}_{\mathrm{II}}{}^{}=$ Eq. (14)
$\psi \left(z\right)=\sum _{i=1}^3\frac{{\alpha }_i}{r_i}{\psi }_{k_i}\left(z_i\right)$	Eq. (1)	${\mathrm{BEC}}_{\left(T-N\right)}=$ Eq. (12),
$\sum _{i=1}^3{\alpha }_i=1$	$s=\sum _{i=1}^3{\alpha }_i{⟨n_i⟩}^2=\sum _{i=1}^3{a}_i$	where ${\lambda }_i^{\left(T-N\right)}=\left(a_i/s\right)\left(2/k_i\right)$
The third term shows	$\sum _{i=1}^3{\alpha }_i{r}_i=1.0$	($i=1\sim 3$).  Notice  that
the contribution of the		${\lambda }_3^{\left(T-N\right)}=𝒪\left({10}^{-3}\right)$.
Poisson-like distribution.

Table 10. Table 10: Effective ranges calculated by Eq. ( 15 ).

	formulas	$R_E$ [fm]
		0.9 TeV	7 TeV
ATLAS	CFII	1.55$\pm$0.24	2.07$\pm$0.05
	BEC(T-N)	1.21$\pm$0.55	1.56$\pm$0.31
CMS	CFII	0.89$\pm$0.05	3.34$\pm$0.19
	BEC(T-N)	1.03$\pm$0.34	2.51$\pm$1.02

Table 11. Table 11: PYTHIA 6 analysis of MD at 7 TeV by the ATLAS collaboration calculated by Eqs. ( 1 ) and ( 2 ). The magnitude of the error bars is assumed as 10% of the data points. The χ 2 superscript 𝜒 2 \chi^{2} values are markedly improved by D-NBD.

PYTHIA 6	ATLAS	single-NBD		D-NBD
7 TeV	ratio	$⟨n⟩$, $k$	${\chi }^2$	$i$	${\alpha }_i$	$⟨n_i⟩$	$k_i$	${\chi }^2$
ND	0.787	$⟨n⟩=30.17\pm 0.36$	150.4	1	0.814	32.5	2.56	6.84
		$k=2.53\pm 0.04$		2	0.186	13.2	16.85
SD	0.121	$⟨n⟩=7.08\pm 0.08$	257.3	1	0.716	4.63	7.56	11.75
		$k=15.87\pm 0.80$		2	0.284	10.1	206.7
DD	0.092	$⟨n⟩=6.15\pm 0.08$	240.7	1	0.798	4.14	4.34	34.00
		$k=10.46\pm 0.50$		2	0.202	9.88	1000.0

Table 12. Table 12: Estimated parameters of T-NBD in the P tot = P ND + P SD + P DD subscript 𝑃 tot subscript 𝑃 ND subscript 𝑃 SD subscript 𝑃 DD P_{\rm tot}=P_{\rm ND}+P_{\rm SD}+P_{\rm DD} (calculated by Eq. ( 3 )). The magnitude of the error bars is assumed as 10% of the data points.

$i$	${\alpha }_i$	$⟨n_i⟩$	$k_i$	${\chi }^2$
1	0.711$\pm$0.131	15.37$\pm$3.62	1.56$\pm$0.31
2	0.255$\pm$0.129	47.86$\pm$5.69	7.40$\pm$2.88	0.168
3	0.035$\pm$0.023	12.99$\pm$2.05	1000.0$\pm$551.0

Table 13. Table 13: Analysis (by Eq. ( 17 )) of KNO data at 0.9 and 7.0 TeV collected by the ATLAS and CMS collaboration.

	$i$	${\alpha }_i$	$r_i$	$k_i$
ATLAS	1	0.761$\pm$0.134	0.84$\pm$0.10	1.79$\pm$0.08
0.9 TeV	2	0.166$\pm$0.006	1.87$\pm$0.03	5.75$\pm$1.62
${\chi }^2=25.6$	3	0.074$\pm$0.031	0.67$\pm$0.01	11.0$\pm$3.2
ATLAS	1	0.664$\pm$0.067	0.68$\pm$0.09	1.54$\pm$0.10
7.0 TeV	2	0.275$\pm$0.011	1.91$\pm$0.02	4.37$\pm$0.54
${\chi }^2=27.4$	3	0.061$\pm$0.011	0.41$\pm$0.01	10.0$\pm$1.8
ATLAS	1	0.692$\pm$0.069	0.73$\pm$0.11	1.50$\pm$0.10
8.0 TeV	2	0.235$\pm$0.003	2.00$\pm$0.01	4.61$\pm$0.61
${\chi }^2=39.28$	3	0.073$\pm$0.015	0.38$\pm$0.01	7.77$\pm$1.34
ATLAS	1	0.751$\pm$0.037	0.81$\pm$0.06	1.27$\pm$0.04
13 TeV	2	0.168$\pm$0.050	2.18$\pm$0.09	4.88$\pm$0.49
${\chi }^2=43.26$	3	0.081$\pm$0.007	0.35$\pm$0.01	7.33$\pm$0.66
CMS	1	0.575$\pm$0.079	0.75$\pm$0.09	1.51$\pm$0.17
0.9 TeV	2	0.286$\pm$0.050	1.66$\pm$0.04	5.0 (lower limit)
${\chi }^2=18.80$	3	0.139$\pm$0.043	0.67$\pm$0.04	7.0 (lower limit)
CMS	1	0.809$\pm$0.019	0.83$\pm$0.02	1.42$\pm$0.05
7.0 TeV	2	0.153$\pm$0.009	2.06$\pm$0.05	5.18$\pm$0.33
${\chi }^2=7.83$	2	0.038$\pm$0.020	0.45$\pm$0.04	15.2$\pm$10.9

Equations

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

$$P_{(\mathrm{D}\text{mathchar}45N)}(n,⟨n⟩,{\alpha }_i,k_i)=\sum _{i=1}^2{\alpha }_i{P}_{{\mathrm{NBD}}_i}(n,⟨n_i⟩,k_i),$$

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

$$N^{(2+:2-)}/N^{\mathrm{BG}}=1+(a_1/s)E_1^2+(a_2/s)E_2^2,$$

$$⟨n⟩$$

$$⟨n(n-1)⟩$$

$$⟨n(n-1)(n-2⟩$$

$$⟨n^a(n^a-1)⟩=\sum _{i=1}^3{\alpha }_i⟨n_i^a{⟩}^2\left(1.0+\frac{2.0}{k_i}\right),$$

$$Eq.(\text{ref}eq5):$$

$$\displaystyle\frac{N^{\rm(2+:\,2-)}}{N^{\rm BG}}=\dfrac{\sum_{i=1}^{3}\alpha_{i}(\langle n_{i}^{+}\rangle^{2}+\langle n_{i}^{-}\rangle^{2})\left(1.0+\frac{2.0}{k_{i}}\right)}{\sum_{i=0}^{3}2\times(\mbox{The number of pairs of different charged particles $(+-)$ in MD with $\alpha_{i}$})}.$$

$$N^{\mathrm{BG}}$$

$$\frac{N^{(2+:2-)}}{N^{\mathrm{BG}}}=\sum _{i=1}^3\left(\frac{a_i}{s}\right)\left(1.0+\frac{2}{k_i}{E}_{{\mathrm{BEC}}_i}^2\right)$$

$$\displaystyle E_{\rm BEC}^{2}=\ \left\{\begin{array}[]{l}\exp(-RQ)\mbox{ (Exponential function) (E)},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \exp(-(RQ)^{2})\mbox{ (Gaussian distribution) (G)},\end{array}\right.$$

$${\mathrm{BEC}}_{(\mathrm{T}\text{mathchar}45N)}$$

$$\mathrm{C}{\mathrm{F}}_{\mathrm{I}}=1.0+\lambda E_{\mathrm{BEC}}^2,$$

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

$$R_{\mathrm{E}}=R_1\times {\lambda }_1+R_2\times {\lambda }_2.$$

$$\displaystyle\left\{\begin{array}[]{l}\mbox{1) The first NBD weighted with }\alpha_{1}\leftrightarrow\mbox{ the main part of }\sigma_{\rm ND}.\\ \mbox{2) The second NBD weighted with }\alpha_{2}\leftrightarrow\mbox{ the main part of }\sigma_{\rm SD}\mbox{, and parts of }\sigma_{\rm ND}\mbox{ and }\sigma_{\rm DD}.\\ \mbox{3) The third NBD weighted with }\alpha_{3}\leftrightarrow\mbox{ the main part of }\sigma_{\rm DD}\mbox{, and a part of }\sigma_{\rm SD}.\\ \end{array}\right.$$

$${\psi }_{(\mathrm{T}\text{mathchar}45N)}(z=n/⟨n⟩,{\alpha }_i,k_i,r_i;i=1\sim 3)=\sum _{i=1}^3{\alpha }_i\frac{1}{r_i}\frac{k_i^{k_i}(z/r_i{)}^{k_i-1}}{\Gamma (k_i)}{e}^{-k_{i}z/r_i}.$$

$${\int }_0^{\infty }dz{\psi }_{(\mathrm{T}\text{mathchar}45N)}(z,{\alpha }_i,k_i,r_i;i=1\sim 3)=\sum _{i=1}^3{\alpha }_i=\mathrm{1.0.}$$

Full text

1 Introduction

1.1 Negative Binomial Distribution (NBD)

Approximately three decades ago, the UA5 collaboration [1] discovered a violation of Koba-Nielsen-Olesen (KNO) scaling [2] at CERN S$\bar{p}p$S collider. To explain those data, the UA5 collaboration assumed a double-negative binomial distribution (D-NBD) of the data. The NBD is given as

[TABLE]

where $\langle n\rangle$ and $k$ are the averaged multiplicity and the intrinsic parameter of the NBD, respectively. The D-NBD is expressed as

[TABLE]

where $\alpha_{1}+\alpha_{2}=1$. Using Eqs. (1) and (2), several authors [3, 4, 5, 6] have analyzed the multiplicity distribution (MD) of data at high energies, such as the high-energy data of the Large Hadron Collider (LHC) [7, 8]. Among those analyses, the three-NBD (T-NBD) was applied by Zborovsky [9]:

[TABLE]

where $\alpha_{1}+\alpha_{2}+\alpha_{3}=1.0$ (see also [6]).

Here, we address the question “Why must the MD of LHC data be analyzed in the T-NBD framework?” To find a plausible answer, we must consider the constraints adopted by the ATLAS and CMS collaborations (for example, $|\eta|<2.5$ and $|\eta|<2.4$). Even in restricted $\eta$ regions, Monte Carlo (MC) estimations have revealed that three processes, i.e., the non-diffractive dissociation (ND), the single-diffractive dissociation (SD), and the double-diffractive dissociation (DD) contribute to the MD at LHC energies [10, 11, 12].

For the reader’s convenience, our analysis results of MC data at 7 TeV collected by the ATLAS collaboration are presented in Tables 11 and 12, and Figs. 5 and 6 of Appendix A. The individual data ensembles of ND, SD, and DD, defined by the partial probability distributions $P_{\rm ND}(n)$, $P_{\rm SD}(n)$, and $P_{\rm DD}(n)$ respectively, are reasonably described by the D-NBD (see Table 1). Accordingly, the total probability $P_{\rm tot}=P_{\rm ND}+P_{\rm SD}+P_{\rm DD}$ is also described by the T-NBD. This behavior implies an important role for the T-NBD in MD analyses at LHC energies.

Very recently, Zborovsky [13] revealed the stochastic structure of the T-NBD studying the oscillations in combinations of T-NBDs. As pointed out by Wilk and Wlodarczyk [14], Zborovsky’s work supports the theoretical plausibility of the T-NBD. See also recent study on this subject [15]. Regarding recent T-NBD investigations, we approach the T-NBD from a different perspective, namely, the identical particle effect [16, 17, 18] observed at the LHC (see Refs. [19, 20, 21] and [22, 23, 24] for related theoretical and empirical studies, respectively)111In Ref. [20], for $N^{\rm BG}$, an identical separation between two ensembles with $\alpha_{1}$and $\alpha_{2}$ is assumed. For no-separation between them, the following formula is obtained:

$N^{\rm(2+:\,2-)}/N^{\rm BG}=1+(a_{1}/s)E_{1}^{2}+(a_{2}/s)E_{2}^{2},$

where $s=a_{1}+a_{2}=\alpha_{1}\langle n_{1}\rangle^{2}+\alpha_{2}\langle n_{2}\rangle^{2}$ (see succeeding Ref. [21]). .

1.2 Bose-Einstein correlation at the LHC

The moments of a charged-particle distributions are calculated as

[TABLE]

When the particles are identical, we obtain the following relation (where the charge sign $a$ is $+$ or $-$):

[TABLE]

Eq. (5) can be interpreted as

[TABLE]

Meanwhile, the authors of [20] recently studied the interrelation between the MD and the Bose-Einstein correlation (BEC) under the D-NBD assumption. To extend the framework of T-NBD, we compute $N^{\rm BG}$ as a function of $\{\alpha_{i},\,\langle n_{i}\rangle:\,i=1\sim 3\}$ (see Ref. [21]). The BEC in the extended framework is given by

[TABLE]

To simplify the calculations, we denote $\langle n_{i}^{+}\rangle=\langle n_{i}^{-}\rangle=\langle n_{i}\rangle/2$. In the denominator $N^{\rm BG}$ of Eq. (6), the three coefficients $\alpha_{i}\langle n_{i}\rangle^{2}$ are given by

[TABLE]

In the framework of the T-NBD assumption, we have

[TABLE]

To describe the BEC in the $0\leq Q\leq 2$ GeV region, we assume the following exchange function $E_{\rm BEC}^{2}$

[TABLE]

where $R$ and $Q$ are the interaction range and the momentum-transfer squared function, respectively. The latter is calculated as $Q=\sqrt{-(p_{1}-p_{2})^{2}}$, where $p_{1}$ and $p_{2}$ are momenta of identical particles.

By making use of those calculations mentioned above, the BEC is then formulated as

[TABLE]

where $\lambda_{i}^{\rm(T\mathchar 45\relax N)}=(a_{i}/s)(2/k_{i})$ ($i=1,\,2$). It should be noted that the third component (with coefficient $\alpha_{3}$) exhibits a Poisson property. Because the $k_{3}$ values are large, the third term ($i=3$) does not numerically contribute to BEC(T-N).

In our BEC analysis of LHC data, we note that all three collaborations (ATLAS, CMS, and LHCb [22, 23, 24]) applied the well-known conventional formula

[TABLE]

Regarding Eq. (12) as another conventional formula, we would like to propose that

[TABLE]

where $\lambda_{1}^{\rm(II)}$ and $\lambda_{2}^{\rm(II)}$ are free parameters. By analyzing the BEC data, we can compare $\lambda_{1}^{\rm(II)}$ and $\lambda_{2}^{\rm(II)}$ in Eq. (14) with the terms $\lambda_{i}^{\rm(T\mathchar 45\relax N)}=(a_{i}/s)(2/k_{i})$ ($i=1,\,2$) in Eq. (12), and the terms $R_{i}^{\rm(T\mathchar 45\relax N)}$ and $R_{i}^{\rm(II)}$ in Eqs. (12) and (14).

The second section of this paper analyzes the MD at 0.9 and 7 TeV through Eq. (3). The second and third moments, and $a_{i}$’s and $(a_{i}/s)(2/k_{i})$’s are displayed in this section. Section 3 analyzes the BEC through Eqs. (12), (13) and (14). This paper concludes with remarks and discussions in Section 4. The appendices analyze the MC data at 7 TeV collected by the ATLAS collaboration (Appendix A) and the KNO scaling data by an enlarged KNO scaling function based on the T-NBD assumption (Appendix B).

2 Multiplicity Distribution ($P(n)$) Analysis

We begin by analyzing the MD at 0.9 TeV and 7 TeV obtained by the ATLAS [7] and CMS [8] collaborations under the T-NBD assumption. The MINUIT program is initialized by assigning random variables to the physical quantities. The estimated parameters are displayed in Fig. 1 and Table 2. Note that both collaborations obtained similar minimum $\chi^{2}$ values at 0.9 TeV. To compare our results with those of Zborovsky [9], we adopt the same treatments to the probability distributions. Specifically, we renormalize Eq. (3) without the $P(0)$ and $P(1)$ as the MD obtained by the ATLAS collaboration, and also renormalize the MD obtained by the CMS collaboration after excluding $P(0)$.

Our results in Table 2 almost match those of Zborovsky [9]. Moreover, the empirical values of the second and third moments almost equal those of the D-NBD and T-NBD (Table 3). From the values in Table 2, we obtain $\lambda_{i}^{\rm(T\mathchar 45\relax N)}=(\alpha_{i}\langle n_{i}\rangle^{2}/s)(2/k_{i})$ ($i=1,\,2$). The results are shown in Table 4.

3 Analyses of BEC data by Eqs. (12), (13) and (14)

Our BEC results are displayed in Fig. 2 and Tables 5 and 6. In Tables 5 and 6, the combinations exhibiting high coincidence are indicated by ($\ast 1$ and $\ast 2$). In the BEC(T-N) analysis, we apply the calculated $\lambda_{i}^{\rm(T\mathchar 45\relax N)}$ ($i=1,\,2$) values in Table 4, which were fixed in the MINUIT computations. Contrarily, the four parameters of the CFII, calculations {$R_{i}^{\rm(II)}$, $\lambda_{1}^{\rm(II)}$, $i=1,\,2$} are free; however, the four parameters in Eq. (14) and the set of two parameters ($R_{i}^{\rm(T\mathchar 45\relax N)}$, $i=1,\,2$) and two fixed parameters ($\lambda_{i}^{\rm(T\mathchar 45\relax N)}$, $i=1,\,2$) give very similar results. We emphasize that the geometrical combinations G$+$G at 0.9 TeV and E$+$G at 7 TeV are identical in the CFII and BEC(T-N) formulations. This coincidence is likely attributable to the common stochastic properties of the MD and BEC ensembles (see Tables 5 and 6).

4 Concluding remarks and discussions

Observing the results of Table 1 and Appendix A, the T-NBD appears to adequately describe the MD at LHC energies. The MD data are contributed by three processes, ND, SD, and DD. The total probability distribution $P_{\rm tot}$ is expressed by the T-NBD (see Table 1 and Appendix A).

Figure 3 compares the workflows of the T-NBD and CFII computations. The estimated interaction ranges and geometrical combinations are comparable between the two approaches.

The main achievements of the study are summarized below.

C1)

The values in Table 2 are estimated after initializing the MD($P(n)$) values with random variables in the MINUIT application. Our calculations consider the lack of $P(0)$ and $P(1)$ in the ATLAS collaboration and the exclusion problem on $P(0)$ in the CMS collaboration [9]. Renormalization is the necessary step in the application of Eq. (3).

C2)

Utilizing the $\lambda_{i}^{\rm(T\mathchar 45\relax N)}$ in Table 4, we obtain the BEC data by Eq. (12). The $\chi^{2}$ values at 7 TeV are higher in CFI than those in BEC(T-N) and CFII (see also point 3), probably because the MD($P(n)$) at the LHC is governed by stochastic effects [25, 26].

C3)

Using the CFII values calculated by Eq. (14), we estimate the numerical values of the four-parameter set {$R_{i}^{\rm(II)}$, $\lambda_{1}^{\rm(II)}$, $i=1,\,2$}. The results are presented in Table 7. The $R_{1}$ and $R_{2}$ values estimated by CFII and BEC(T-N) are satisfactorily similar. Table 8 summarizes the two degrees of coherence for the results indicated by ($\ast 1$ and $\ast 2$) in Tables 5 and 6. Despite the large error bars in $\lambda_{i}^{\rm(T\mathchar 45\relax N)}$’s, the $\lambda_{i}^{\rm(II)}$ and $\lambda_{i}^{\rm(T\mathchar 45\relax N)}$ values are reasonably coincident, possibly reflecting the common stochastic properties of the MD and BEC ensembles, which are both described by the T-NBD.

C4)

Interesting interrelations are found between the results of BEC(T-N) and CFII marked with ($\ast 1$ and $\ast 2$) in Tables 5 and 6. The BEC(T-N) and CFII may both reasonably describe the BEC at the LHC. Moreover, the combination of $E_{\rm BEC}^{2}$’s at 0.9 TeV satisfies the double-Gaussian distribution (G$+$G), whereas those at 7 TeV are combined exponential and Gaussian distribution (E$+$G). This finding implies different production mechanisms at 0.9 TeV and 7 TeV.

C5)

Possible correspondences are found among the KNO scaling, the MD, and the BEC (see Table 9). These correspondences might be attributed to the violation of KNO scaling discovered in 1989 by the UA5 collaboration, that first proposed the D-NBD. The KNO scaling based on the T-NBD at LHC energies is calculated in Appendix B.

C6)

Taking into account the $\lambda_{i}^{\rm(II)}$’s as weight factors, the effective interaction ranges can be estimated as,

[TABLE]

The estimated effective interaction ranges are displayed in Table 10 and Fig. 4. The $R_{\rm E}$ values appear reasonable because they are larger at the higher colliding energy (7.0 TeV) than at the lower energy (0.9 TeV).

D1)

We must also elucidate the physical meanings of the three intrinsic parameters $k_{i}$ and weight factor $\alpha_{i}$. From the MC data in [10, 11, 12] and the results of Table 1, we infer the following correspondences:

[TABLE]

Here, $\sigma_{\rm ND}$, $\sigma_{\rm SD}$, and $\sigma_{\rm DD}$ are the cross sections of the ND, SD, and double-diffractive dissociation (DD), respectively.

D2)

In future work, we are planning the following improvements:

The large error bars of $\lambda_{i}^{\rm(T\mathchar 45\relax N)}\ (i=1,\,2)$ must be reduced in future work. For this purpose we must improve the framework of the MD analysis.

D3)

To properly validate the present theoretical formulation [27], we will analyze the MD and BEC ($2.0<\eta<4.5$) obtained by the LHCb collaboration.

Acknowledgments. One of the authors (M.B.) would like to thank his colleagues at the Department of Physics of Shinshu University for their kindness. T. Mizoguchi would like to acknowledge the funding provided by Pres. Y. Hayashi.

Appendix A Monte Carlo data at 7 TeV collected by the ATLAS collaboration and analyzed by Eqs. (1)–(3)

The MC data collected at 0.9 TeV and 7 TeV are presented in [10], which reported the MD results. The total MD is decomposed into three processes: ND, SD, and double-diffractive dissociation (DD), with probability distributions defined by $P_{\rm ND}$, $P_{\rm SD}$, and $P_{\rm DD}$, respectively. The total probability distribution is expressed as $P_{\rm tot}=P_{\rm ND}+P_{\rm SD}+P_{\rm DD}$. The partial and total probability distributions are plotted in Figs. 5 and 6, respectively.

The NBD and D-NBD are calculated by Eqs. (1) and (2), respectively, and the results are shown in Table 11. Obviously, the single-NBD cannot describe the MD. The D-NBD probably constitutes three ensembles with different $\langle n\rangle$ and $k$ values: the first set with ($\langle n\rangle\cong 33$, $k=$2–4), the second set with ($\langle n\rangle=$4–13, $k=$4–16), and the third set with ($\langle n\rangle\cong 10$ and $k=$200–1000). These ensembles appear to reasonably validate the T-NBD framework in the analysis of MD at the LHC. The T-NBD is computed from the total MC data at 7 TeV by Eq. (3). The analysis results are shown in Table 12.

We also analyze the MC data at 0.9 TeV by PYTHIA 6, and at 7 TeV by PHOJET and PYTHIA 8. The results are similar to those in Tables 11 and 12.

Appendix B An Enlarged KNO scaling function for LHC energies

In this Appendix, we analyze the KNO scaling at LHC energies. Recall that the D-NBD was proposed to explain the KNO scaling violation found at the S$\bar{p}p$S energy ($\sqrt{s}=546$ GeV) [1]. The KNO scaling function in the framework of the T-NBD [20] is given by

[TABLE]

After integratin of the KNO scaling variables $z$, the KNO scaling function becomes

[TABLE]

The violation of the KNO scaling can be understood studying the energy dependences of the parameters $\alpha_{i}$, $k_{i}$, and $r_{i}$.

The results of Eq. (17) are shown in Fig. 7 and Table 13. The ratio $r_{i}=\langle n_{i}\rangle/\langle n\rangle$ should be large, because the average multiplicities $\langle n_{i}\rangle$ are requaired to be large as $\langle n\rangle$ itself; specifically, $r_{i}\geq 0.33$ ($\langle n_{i}\rangle\geq\langle n\rangle/3$). The KNO scaling functions at 0.9 TeV obtained by the ATLAS and CMS collaborations differed from from those at 7, 8, and 13 TeV (Fig 7). The 0.9 TeV data collected by the CMS collaboration must be constrained by $k_{2}>5$ and $k_{3}>7$ because $z_{3}$ is large (800 or infinity) in the MD analysis. Our KNO scaling analysis also includes the $P(0)$ data at 0.9 and 7 TeV obtained by the CMS collaboration. As shown in Table 13, the violation of KNO scaling occurs through the parameters $\alpha_{i}$ and $r_{i}=\langle n_{i}\rangle/\langle n\rangle$ ($i=1\sim 3$).