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--> --> -->Analysis of multi-pion BECs can provide more information about particle-emitting sources, compared with two-pion interferometry [4, 5, 7-36]. In particular, multi-pion BECs are sensitive to the source coherence [9-12, 26, 27]. In Refs. [10, 11], we investigated three- and four-pion BECs for a spherical evolving source of the pion gas with identical boson condensation. However, particle-emitting sources produced in relativistic heavy-ion collisions are anisotropic in space and may have complex structures. It is of interest to explain the experimental measurements of multi-pion BEC suppressions at the LHC using a more realistic model that can also explain the other observables in these collisions.
Event-wise, the initial systems produced in relativistic heavy-ion collisions strongly fluctuate in space. This initial fluctuation may yield inhomogeneous particle-emitting sources, in which there are hot spots and cold valleys. In Refs. [37-40], a granular source model was proposed and developed by Zhang et al., to explain the experimental results of two-pion interferometry at the Relativistic Heavy Ion Collider (RHIC) and LHC [41-45]. In Refs. [46-48], a granular source model was used to systemically study the pion transverse-momentum spectra, elliptic flows, and two-pion BECs in heavy-ion collisions, at the RHIC and LHC. The granular source model reproduced the experimental data of the pion transverse-momentum spectra, elliptic flows, and two-pion interferometry radii [46-48]. Considering that identical pions are emitted from droplets in the granular source model, and considering that the droplet radii are much smaller than the source size, the pion emission from one droplet is perhaps coherent in the case of high pion event multiplicity, owing to the condensation of identical bosons [10, 11, 49, 50].
In this work, we consider a granular source with coherent pion-emission droplets. The droplets in the granular source move with anisotropic velocities and evolve according to viscous hydrodynamics, as described in Ref. [48]. However, identical pion emissions from one droplet are assumed to be completely or partially coherent. We investigate multi-pion BECs in the granular source model with coherent pion-emission droplets. The normalized three- and four-pion correlation functions of granular sources are examined for completely coherent and momentum-dependent partially coherent pion emissions from one droplet.
The rest of this paper is organized as follows. In Sec. II, we examine the three- and four-pion BEC functions for a static granular source with coherent pion-emission droplets. In Sec. III, we investigate the three- and four-pion BECs in the granular source model, in which droplets evolve according to viscous hydrodynamics. We also investigate the normalized multi-pion correlation functions of evolving granular sources in this section. Finally, we provide a summary and discussion in Sec. IV.
$ C_2({{p}}_1,{{p}}_2) = 1+\frac{1}{n}{\rm e}^{-q_{12}^2 r_d^2}+\bigg(1-\frac{1}{n}\bigg) {\rm e}^{-q_{12}^2 (r_d^2+R_G^2)} \equiv 1+\frac{1}{n} {\cal{R}}^d(1,2)+\bigg(1 -\frac{1}{n}\bigg){\cal{R}}^G(1,2), $ ![]() | (1) |
$ \begin{aligned}[b] C_3({{p}}_1,{{p}}_2,{{p}}_3) =& 1+\frac{1}{n}\bigg[{\cal{R}}^d(1,2)+{\cal{R}}^d(1,3) +{\cal{R}}^d(2,3)\bigg]+\bigg(1-\frac{1}{n}\bigg)\bigg[{\cal{R}}^G(1,2) +{\cal{R}}^G(1,3)+{\cal{R}}^G(2,3)\bigg] \\& +\frac{2}{n^2}\bigg[{\cal{R}}^d(1,2){\cal{R}}^d(1,3){\cal{R}}^d(2,3) \bigg]^{\frac{1}{2}}+\frac{2(n-1)}{n^2}\bigg[\Big({\cal{R}}^d(1,3){\cal{R}}^d(2,3)/ {\cal{R}}^d(1,2)\Big)^{\!\frac{1}{2}}{\cal{R}}^G(1,2)\\ &+\Big({\cal{R}}^d(1,2){\cal{R}}^d(2,3)/{\cal{R}}^d(1,3)\Big)^{\!\frac{1}{2}}{\cal{R}}^G(1,3)+\Big({\cal{R}}^d(1,2){\cal{R}}^d(1,3)/ {\cal{R}}^d(2,3)\Big)^{\!\frac{1}{2}}{\cal{R}}^G(2,3)\bigg]\\ &+\frac{2(n-1)(n-2)}{n^2}\bigg[{\cal{R}}^G(1,2){\cal{R}}^G(1,3) {\cal{R}}^G(2,3)\bigg]^{\frac{1}{2}}, \end{aligned} $ ![]() | (2) |
For a small droplet radius, the pion emission from a droplet is significantly coherent [10, 11, 49, 50]. Assuming the pions emitted from one droplet are completely coherent, the two- and three-pion correlation functions for a granular source become
$ C_2({{p}}_1,{{p}}_2) = 1+\frac{(n-1)}{n}{\cal{R}}^G(1,2), $ ![]() | (3) |
$\begin{aligned}[b] C_3({{p}}_1,{{p}}_2,{{p}}_3)\! =& \!1\!+\!\frac{(n-1)}{n}\bigg[{\cal{R}}^G(1,2) \!+\!{\cal{R}}^G(1,3)\!+\!{\cal{R}}^G(2,3)\bigg] \\ &+ \frac{2(n-1)(n-2)}{n^2} \bigg[{\cal{R}}^G(1,2){\cal{R}}^G(1,3){\cal{R}}^G(2,3)\bigg]^{\!\frac{1}{2}}. \end{aligned}$ ![]() | (4) |
$ \begin{aligned}[b] C_4({{p}}_1,{{p}}_2,{{p}}_3,{{p}}_4)\! =& \!1+\frac{(n-1)}{n}\bigg[{\cal{R}}^G(1,2) \!+\!{\cal{R}}^G(1,3)\!+\!{\cal{R}}^G(1,4)\!+\!{\cal{R}}^G(2,3)\!+\!{\cal{R}}^G (2,4)\!+\!{\cal{R}}^G(3,4)\bigg] \\ &+\frac{2(n-1)(n-2)}{n^2}\bigg[\bigg({\cal{R}}^G(1,2){\cal{R}}^G(1,3) {\cal{R}}^G(2,3)\bigg)^{\!\frac{1}{2}}\!+\!\bigg({\cal{R}}^G(1,2){\cal{R}}^G(1,4) {\cal{R}}^G(2,4)\bigg)^{\!\frac{1}{2}}\\ &+\bigg({\cal{R}}^G(2,3){\cal{R}}^G(2,4){\cal{R}}^G(3,4) \bigg)^{\!\frac{1}{2}}\!+\!\bigg({\cal{R}}^G(1,3){\cal{R}}^G(1,4) {\cal{R}}^G(3,4)\bigg)^{\!\frac{1}{2}}\bigg]\\ &+\frac{(n\!-\!1)(n\!-\!2)(n\!-\!3)}{n^3}\bigg[{\cal{R}}^G(1,2) {\cal{R}}^G(3,4)\!+\!{\cal{R}}^G(1,3){\cal{R}}^G(2,4)\!+\!{\cal{R}}^G(2,3){\cal{R}}^G(1,4) \bigg]\\ &+\frac{(n-1)}{n^3}\bigg[{\cal{R}}^G(1,2){\cal{R}}^G(3,4)\bigg({\rm e}^{-2{{q}}_{12}\cdot{{q}}_{34} R_G^2} +{\rm e}^{-2{{q}}_{12}\cdot{{q}}_{43}R_G^2}\bigg) +{\cal{R}}^G(1,3){\cal{R}}^G(2,4) \\ & \times \bigg({\rm e}^{-2{{q}}_{13}\cdot{{q}}_{24}R_G^2} +{\rm e}^{-2{{q}}_{13}\cdot{{q}}_{42}R_G^2}\bigg) +{\cal{R}}^G(1,4){\cal{R}}^G(2,3)\bigg({\rm e}^{-2{{q}}_{14}\cdot{{q}}_{23}R_G^2} +{\rm e}^{-2{{q}}_{14}\cdot{{q}}_{32}R_G^2}\bigg)\bigg] \\ &+\frac{2(n-1)(n-2)}{n^3}\bigg[{\cal{R}}^G(1,2){\cal{R}}^G(3,4)\bigg( {\rm e}^{-{{q}}_{12}\cdot{{q}}_{34}R_G^2} +{\rm e}^{-{{q}}_{12}\cdot{{q}}_{43}R_G^2}\bigg) +{\cal{R}}^G(1,3){\cal{R}}^G(2,4) \\ & \times \bigg({\rm e}^{-{{q}}_{13}\cdot{{q}}_{24}R_G^2} +{\rm e}^{-{{q}}_{13}\cdot{{q}}_{42}R_G^2}\bigg) +{\cal{R}}^G(1,4){\cal{R}}^G(2,3)\bigg({\rm e}^{-{{q}}_{14}\cdot{{q}}_{23}R_G^2} +{\rm e}^{-{{q}}_{14}\cdot{{q}}_{32}R_G^2}\bigg)\bigg] \\ &+\frac{2(n-1)(n-2)(n-3)}{n^3}\bigg[\bigg({\cal{R}}^G(1,2){\cal{R}}^G(2,3){\cal{R}}^G(3,4){\cal{R}}^G(1,4)\bigg)^{\!\frac{1}{2}}\\ &+\bigg({\cal{R}}^G(1,3){\cal{R}}^G(2,3){\cal{R}}^G(2,4){\cal{R}}^G(1,4)\bigg)^{\!\frac{1}{2}}\!+\!\bigg({\cal{R}}^G(1,2){\cal{R}}^G(2,4) {\cal{R}}^G(3,4){\cal{R}}^G(1,3)\bigg)^{\!\frac{1}{2}}\bigg], \end{aligned} $ ![]() | (5) |
In Figs. 1(a) and 1(b), we plot the two-pion correlation functions of static granular sources with chaotic and completely coherent pion-emission droplets, respectively. In Figs. 1(d) and 1(e), we plot the three-pion correlation functions of static granular sources with chaotic and completely coherent pion-emission droplets, respectively. In Figs. 1(g) and 1(h), we plot the four-pion correlation functions of static granular sources with chaotic and completely coherent pion-emission droplets, respectively. The panels (c), (f), and (i) show the ratios of the correlation functions of granular sources with completely coherent pion-emission droplets to the correlation functions of granular sources with chaotic pion-emission droplets. Here, the radii of the granular sources were set to

$ Q_m = \sqrt{\sum\limits_{i<j\leqslant m}-(p_i-p_j)^\mu (p_i-p_j)_\mu}, \; \; \; \; (m\geqslant 2). $ ![]() | (6) |
The normalized three-pion correlation function
$ r_3(Q_3) = \frac{[c_3(Q_3)-1][n/(n-1)]^{3/2}} {\sqrt{{\cal{R}}^G(1,2)\!(Q_3){\cal{R}}^G(2,3)\!(Q_3){\cal{R}}^G(1,3)\!(Q_3)}}, $ ![]() | (7) |
$ \begin{aligned}[b] c_3(Q_3) =& 1+\frac{2(n-1)(n-2)}{n^2}\\& \times \bigg[{\cal{R}}^G(1,2){\cal{R}}^G(1,3){\cal{R}}^G(2,3) \bigg]^{\!\frac{1}{2}}\!\!\!(Q_3). \end{aligned} $ ![]() | (8) |
$ r_4(Q_4) \!=\! \frac{[c_4(Q_4)-1][n/(n-1)]^2}{\sqrt{{\cal{R}}^G(\!1,2\!)\!(Q_4){\cal{R}}^G(\!2,3\!)\!(Q_4) {\cal{R}}^G(\!3,4\!)\!(Q_4){\cal{R}}^G(\!1,4\!)\!(Q_4)}}, $ ![]() | (9) |
$ \begin{aligned}[b] c_4(Q_4) =& 1+\frac{2(n-1)(n-2)(n-3)}{n^3}\\&\times\bigg[\bigg({\cal{R}}^G(1,2){\cal{R}}^G(2,3) {\cal{R}}^G(3,4){\cal{R}}^G(1,4)\bigg)^{\!\frac{1}{2}}(Q_4)\\&+\bigg({\cal{R}}^G(1,3){\cal{R}}^G(2,3){\cal{R}}^G(2,4){\cal{R}}^G(1,4)\bigg)^{\!\frac{1}{2}}(Q_4)\\&+\bigg({\cal{R}}^G(1,2){\cal{R}}^G(2,4){\cal{R}}^G(3,4) {\cal{R}}^G(1,3)\bigg)^{\!\frac{1}{2}}(Q_4)\bigg]. \end{aligned} $ ![]() | (10) |


Owing to the lack of expansion, the three- and four-pion correlation functions of static granular sources fall rapidly with multi-pion relative momenta
2
A.Evolving granular source model
The model we consider is based on a viscous granular source model developed in Ref. [48], but presently, pions that are emitted from one droplet are assumed to be completely or partially coherent. In this subsection, we briefly present the components of the granular source model used in the work. For a detailed explanation of granular source models, the reader is referred to Refs. [38-40, 46-48].The granular source model was proposed by W. N. Zhang et al. [37, 38], to explain the RHIC HBT puzzle,
The granular source model assumes that the initial spatial inhomogeneity and violent expansion during the early stages of the system produced in ultrarelativistic heavy-ion collisions may break up the system into many hot and dense droplets, leading to the formation of a granular particle-emitting source. During the formation of the initial granular source, the droplet centers are distributed within a cylinder along the collision axis, and the initial energy distribution in a droplet satisfies the Woods-Saxon distribution, as shown in Refs. [47, 48]. The average droplet number,
The evolution of a granular source includes the droplet evolution according to viscous hydrodynamics and the droplet expansion in its entirety, with anisotropic droplet velocities
2
B.Multi-pion correlation functions
In Figs. 4(a) and 4(b), we plot the three-pion correlation functions
In Fig. 4(c) and 4(d), we plot the four-pion correlation functions
Considering that pions with high momenta are more likely to be emitted chaotically from excited states [10, 11, 49, 50], we further studied multi-pion BECs for granular sources with partially coherent pion-emission droplets. We assumed that pions that are emitted from one droplet and have momenta below a fixed value
In Figs. 5(a) and 5(b), we compare the three-pion correlation functions for evolving granular sources with completely coherent pion-emission droplets (corresponding to

In Figs. 5(c) and 5(d), we compare the four-pion correlation functions for evolving granular sources with completely coherent pion-emission droplets (corresponding to
2
C.Normalized multi-pion BEC functions
The normalized multi-pion correlation functionsWe show in Figs. 6(a) and 6(b) the normalized three-pion correlation functions for evolving granular sources with completely coherent pion-emission droplets and different average droplet numbers

We show in Figs. 6(c) and 6(d) the normalized four-pion correlation functions for evolving granular sources with completely coherent pion-emission droplets and different average droplet numbers
In Figs. 6(e) – 6(h), we compare the normalized three- and four-pion correlation functions for evolving granular sources with completely coherent (
Table 1 presents the results for
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Table1.Results for
By comparing the three- and four-pion correlation functions of evolving granular sources with the experimental data for Pb-Pb collisions at
The normalized multi-pion correlation functions, defined as the ratios of the multi-pion cumulant correlators to the two-pion correlator, can reduce the influence of the resonance decay on themselves. Our investigations indicate that the normalized four-pion correlation function is improved in the
Recently, D. Gangadharan proposed a technique for constructing three- and four-pion correlation functions for partially coherent sources and for estimating the source coherence [9]. Using this technique, the ALICE Collaboration analyzed the three- and four-pion correlation functions for Pb-Pb collisions at
Finally, it should be mentioned that the granular source model used in this paper had the same model parameters as in Ref. [46], but with the assumption of coherent pion emission from one droplet. We noted that this assumption may increase the two-pion interferometry radii by
$ \begin{aligned}[b]C_4({{p}}_1,{{p}}_2,{{p}}_3,{{p}}_4) =& 1+\frac{1}{n}\bigg[{\rm e}^{-{{q}}_{12}^2 r_d^2} +{\rm e}^{-{{q}}_{13}^2 r_d^2}+{\rm e}^{-{{q}}_{14}^2 r_d^2}+{\rm e}^{-{{q}}_{23}^2 r_d^2}+{\rm e}^{-{{q}}_{24}^2 r_d^2} +{\rm e}^{-{{q}}_{34}^2 r_d^2}\bigg] +\frac{(n\!-\!1)}{n}\bigg[{\rm e}^{-{{q}}_{12}^2 (r_d^2+R_G^2)}\\&+\,{\rm e}^{-{{q}}_{13}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{14}^2 (r_d^2+R_G^2)}+{\rm e}^{-{{q}}_{23}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{24}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{34}^2 (r_d^2+R_G^2)} \bigg]\\&+\,\frac{2}{n^2}\bigg[{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)r_d^2/2} \!+\!{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)r_d^2/2}\!+\!{\rm e}^{\!-\!({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)r_d^2/2}\!+\!{\rm e}^{\!-\!({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)r_d^2/2} \bigg] \\ &+\frac{2(n-1)(n-2)}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)(r_d^2+R_G^2)/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)(r_d^2+R_G^2)/2} \\&+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{14}^2+{{q}}_{34}^2)(r_d^2+R_G^2)/2} +{\rm e}^{-({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)(r_d^2+R_G^2)/2} \bigg] \\&+\frac{2(n-1)}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)r_d^2/2} \Big( {\rm e}^{\!-\!{{q}}_{12}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{13}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{23}^2R_G^2} \Big) +{\rm e}^{-({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)r_d^2/2}\\&\times\,\Big({\rm e}^{\!-\!{{q}}_{12}^2 R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{14}^2R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{24}^2 R_G^2}\Big) \!+\!{\rm e}^{\!-\!({{q}}_{13}^2+{{q}}_{14}^2 +{{q}}_{34}^2)r_d^2/2}\Big({\rm e}^{\!-\!{{q}}_{13}^2 R_G^2}+{\rm e}^{\!-\!{{q}}_{14}^2 R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{34}^2 R_G^2}\Big) \\&+{\rm e}^{-({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)r_d^2/2}\Big({\rm e}^{-{{q}}_{23}^2 R_G^2} +{\rm e}^{-{{q}}_{24}^2 R_G^2}+{\rm e}^{-{{q}}_{34}^2 R_G^2}\Big)\bigg] +\frac{1}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} \\&+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} +{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} \bigg] +\frac{(n-1)(n-2)(n-3)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)(r_d^2+R_G^2)} \\& +\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)(r_d^2+R_G^2)} +{\rm e}^{-({{q}}_{14}^2 +{{q}}_{23}^2)(r_d^2+R_G^2)} \bigg] +\frac{(n-1)}{n^2}\bigg[{\rm e}^{-{{q}}_{12}^2 r_d^2} {\rm e}^{-{{q}}_{34}^2 (r_d^2+R_G^2)}\\& +\,{\rm e}^{\!-\!{{q}}_{13}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{24}^2 (r_d^2+\!R_G^2)}\!+\!{\rm e}^{\!-\!{{q}}_{14}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{23}^2 (r_d^2+\!R_G^2)} \!+\!{\rm e}^{\!-\!{{q}}_{23}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{14}^2(r_d^2+\!R_G^2)} \!+\!{\rm e}^{\!-\!{{q}}_{24}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{13}^2(r_d^2+\!R_G^2)} \\& +\,{\rm e}^{\!-{{q}}_{34}^2r_d^2}{\rm e}^{\!-{{q}}_{12}^2(r_d^2+R_G^2)} \bigg] +\frac{(n-1)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} {\rm e}^{-({{q}}_{12} +{{q}}_{34})^2 R_G^2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} \\ &\times\,{\rm e}^{-({{q}}_{12} +{{q}}_{43})^2 R_G^2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} {\rm e}^{\!-({{q}}_{13}+{{q}}_{24})^2 R_G^2} +{\rm e}^{\!-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} {\rm e}^{\!-({{q}}_{13}+{{q}}_{42})^2 R_G^2} \\&+\,{\rm e}^{\!-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{\!-({{q}}_{14}+{{q}}_{23})^2 R_G^2} +{\rm e}^{\!-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{\!-({{q}}_{14}+{{q}}_{32})^2 R_G^2}\bigg] \\ &+\,\frac{2(n-1)(n-2)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} {\rm e}^{-({{q}}_{12}^2 +{{q}}_{34}^2) R_G^2/2} \Big({\rm e}^{-({{q}}_{12}+{{q}}_{34})^2 R_G^2/2} +{\rm e}^{-({{q}}_{12}+ {{q}}_{43})^2 R_G^2/2}\Big) \\ &+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2}{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} \Big( {\rm e}^{-({{q}}_{13}+{{q}}_{24})^2R_G^2/2}+{\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} \Big) \\& +\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} \Big( {\rm e}^{-({{q}}_{14}+{{q}}_{23})^2R_G^2/2}+{\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2} \Big) \bigg] \\& +\,\frac{2}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2)r_d^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2}\bigg] \\&+\,\frac{2(n\!-\!1)(n\!-\!2)(n\!-\!3)}{n^3}\bigg[{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{23}^2 +{{q}}_{34}^2+{{q}}_{41}^2)(r_d^2+\!R_G^2)/2} +{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{24}^2 +{{q}}_{43}^2+{{q}}_{31}^2)(r_d^2+\!R_G^2)/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)(r_d^2+\!R_G^2)/2}\bigg] +\frac{2(n\!-\!1)}{n^3}\bigg[{\rm e}^{-({ q}_{12}^2+{ q}_{23}^2+{ q}_{34}^2+{ q}_{41}^2)r_d^2/2}\Big({\rm e}^{-{ q}_{12}^{2}R_{G}^{2}} \\& +\,{\rm e}^{-{{q}}_{23}^2R_G^2}\!+\!e^{-{{q}}_{34}^2R_G^2}+{\rm e}^{-{{q}}_{41}^2R_G^2}\Big) \!+\!{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2\!+\!{{q}}_{31}^2)r_d^2/2}\Big({\rm e}^{-{{q}}_{12}^2 R_G^2}\!+\!{\rm e}^{-{{q}}_{24}^2R_G^2} \\& +\,{\rm e}^{\!-\!{{q}}_{43}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{31}^2R_G^2}\Big)\!+\!{\rm e}^{\!-\!({{q}}_{13}^2 +{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{\!-\!{{q}}_{13}^2R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{32}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{24}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{41}^2R_G^2} \Big)\bigg] \\ &+\,\frac{(n-1)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{12}+{{q}}_{23})^2R_G^2}+{\rm e}^{-({{q}}_{12}+{{q}}_{34})^2R_G^2} +{\rm e}^{-({\bf{q}}_{12}+{\bf{q}}_{41})^2R_G^2} \\&+\,{\rm e}^{-({{q}}_{23}+{{q}}_{34})^2R_G^2}+{\rm e}^{-({{q}}_{23}+{{q}}_{41})^2R_G^2} +{\rm e}^{-({{q}}_{34}+{{q}}_{41})^2R_G^2}\Big) +{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} \\ &\times\,\Big({\rm e}^{\!-\!({{q}}_{12}+{{q}}_{24})^2\!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{12}+{{q}}_{43})^2 \!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{12}+{{q}}_{31})^2\!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{24}+{{q}}_{43})^2 \!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{24}+{{q}}_{31})^2\!R_G^2} \\&+\,{\rm e}^{-({{q}}_{43}+{{q}}_{31})^2R_G^2}\Big) +{\rm e}^{-({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2 +{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{13}+{{q}}_{32})^2R_G^2}+{\rm e}^{-({{q}}_{13}+{{q}}_{24})^2 R_G^2} \\&+{\rm e}^{-({{q}}_{13}+{{q}}_{41})^2R_G^2}+{\rm e}^{-({{q}}_{32}+{{q}}_{24})^2R_G^2} +{\rm e}^{-({{q}}_{32}+{{q}}_{41})^2R_G^2}+{\rm e}^{-({{q}}_{24}+{{q}}_{41})^2R_G^2} \Big) \bigg] \\ &+\,\frac{2(n-1)(n-2)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2) r_d^2/2} \Big({\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)R_G^2/2} \!+\!\,{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} \\& +\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)R_G^2/2} {\rm e}^{-({{q}}_{12}+{{q}}_{43})^2R_G^2/2} \\\end{aligned} $ ![]() |
$ \tag{A1}\begin{aligned}[b]&+\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} {\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2}\Big) \!+\!{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} \Big( {\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{\bf{q}}_{23}^2)R_G^2/2} \\&+\,{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} +{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{24}^2)R_G^2/2} \\&+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} {\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)R_G^2/2} {\rm e}^{-({{q}}_{12}+{{q}}_{43})^2R_G^2/2}\Big) \\& +\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)R_G^2/2} +{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} {\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} +\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} {\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2}\Big)\bigg], \end{aligned} $ ![]() |
The four-pion correlation function of a granular source is complex, including the relative angles of two relative momenta in the double pair correlations and pure quadruplet correlations. For a completely chaotic pion emission droplet,
$ \begin{aligned}[b] C_4({{p}}_1,{{p}}_2,{{p}}_3,{{p}}_4) =& 1+\frac{\lambda}{n}\bigg[{\rm e}^{-{{q}}_{12}^2 r_d^2} +{\rm e}^{-{{q}}_{13}^2r_d^2}+{\rm e}^{-{{q}}_{14}^2 r_d^2}+{\rm e}^{-{{q}}_{23}^2 r_d^2} +{\rm e}^{-{{q}}_{24}^2 r_d^2} +{\rm e}^{-{{q}}_{34}^2r_d^2}\bigg] +\frac{(n\!-\!1)}{n}\bigg[{\rm e}^{-{{q}}_{12}^2 (r_d^2+R_G^2)} \\ &+\,{\rm e}^{-{{q}}_{13}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{14}^2 (r_d^2+R_G^2)}+{\rm e}^{-{{q}}_{23}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{24}^2 (r_d^2+R_G^2)} +{\rm e}^{-{{q}}_{34}^2 (r_d^2+R_G^2)}\bigg] \\ &+\,\frac{2\xi}{n^2}\bigg[{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)r_d^2/2} \!+\!{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)r_d^2/2}\!+\!{\rm e}^{\!-\!({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)r_d^2/2}\!+\!{\rm e}^{\!-\!({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)r_d^2/2} \bigg] \\& +\frac{2(n-1)(n-2)}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)(r_d^2+R_G^2)/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)(r_d^2+R_G^2)/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{14}^2+{{q}}_{34}^2)(r_d^2+R_G^2)/2} +{\rm e}^{-({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)(r_d^2+R_G^2)/2} \bigg] \\ &+\frac{2(n-1)\lambda}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)r_d^2/2} \Big( {\rm e}^{\!-\!{{q}}_{12}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{13}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{23}^2R_G^2} \Big) +{\rm e}^{-({{q}}_{12}^2+{{q}}_{14}^2+{{q}}_{24}^2)r_d^2/2} \\&\times\,\Big({\rm e}^{\!-\!{{q}}_{12}^2 R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{14}^2R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{24}^2 R_G^2}\Big) \!+\!{\rm e}^{\!-\!({{q}}_{13}^2+{{q}}_{14}^2 +{{q}}_{34}^2)r_d^2/2}\Big({\rm e}^{\!-\!{{q}}_{13}^2 R_G^2}+{\rm e}^{\!-\!{{q}}_{14}^2 R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{34}^2 R_G^2}\Big)\end{aligned} $ ![]() |
$ \tag{A2}\begin{aligned}[b] & +{\rm e}^{-({{q}}_{23}^2+{{q}}_{24}^2+{{q}}_{34}^2)r_d^2/2}\Big({\rm e}^{-{{q}}_{23}^2 R_G^2} +{\rm e}^{-{{q}}_{24}^2 R_G^2}+{\rm e}^{-{{q}}_{34}^2 R_G^2}\Big)\bigg]+\frac{\lambda^2}{n^2}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} \\ & +\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} +{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} \bigg] +\frac{(n-1)(n-2)(n-3)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)(r_d^2+R_G^2)} \\&+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)(r_d^2+R_G^2)} +{\rm e}^{-({{q}}_{14}^2 +{{q}}_{23}^2)(r_d^2+R_G^2)} \bigg] +\frac{(n-1)\lambda}{n^2}\bigg[{\rm e}^{-{{q}}_{12}^2 r_d^2} e^{-{{q}}_{34}^2 (r_d^2+R_G^2)} \\ &+\,{\rm e}^{\!-\!{{q}}_{13}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{24}^2 (r_d^2+\!R_G^2)}\!+\!{\rm e}^{\!-\!{{q}}_{14}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{23}^2 (r_d^2+\!R_G^2)} \!+\!{\rm e}^{\!-\!{{q}}_{23}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{14}^2(r_d^2+\!R_G^2)} \!+\!{\rm e}^{\!-\!{{q}}_{24}^2 r_d^2} {\rm e}^{\!-\!{{q}}_{13}^2(r_d^2+\!R_G^2)} \\ & +\,{\rm e}^{\!-{{q}}_{34}^2r_d^2}{\rm e}^{\!-{{q}}_{12}^2(r_d^2+R_G^2)} \bigg] +\frac{(n-1)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} {\rm e}^{-({{q}}_{12} +{{q}}_{34})^2 R_G^2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} \\ & \times\,{\rm e}^{-({{q}}_{12}+{{q}}_{43})^2 R_G^2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} {\rm e}^{\!-({{q}}_{13}+{{q}}_{24})^2 R_G^2} +{\rm e}^{\!-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2} {\rm e}^{\!-({{q}}_{13}+{{q}}_{42})^2 R_G^2} \\ &+\,{\rm e}^{\!-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{\!-({{q}}_{14}+{{q}}_{23})^2 R_G^2} +{\rm e}^{\!-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{\!-({{q}}_{14}+{{q}}_{32})^2 R_G^2}\bigg] \\ & +\,\frac{2(n-1)(n-2)}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)r_d^2} {\rm e}^{-({{q}}_{12}^2 +{{q}}_{34}^2) R_G^2/2} \Big({\rm e}^{-({{q}}_{12}+{{q}}_{34})^2 R_G^2/2} +{\rm e}^{-({{q}}_{12}+ {{q}}_{43})^2 R_G^2/2}\Big) \\ & +\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)r_d^2}{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} \Big( {\rm e}^{-({{q}}_{13}+{{q}}_{24})^2R_G^2/2}+{\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} \Big) \\& +\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)r_d^2} {\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} \Big( {\rm e}^{-({{q}}_{14}+{{q}}_{23})^2R_G^2/2}+{\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2} \Big) \bigg] \\ & +\,\frac{2\eta}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2)r_d^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2}\bigg] \\ &+\,\frac{2(n\!-\!1)(n\!-\!2)(n\!-\!3)}{n^3}\bigg[{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{23}^2 +{{q}}_{34}^2+{{q}}_{41}^2)(r_d^2+\!R_G^2)/2} +{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{24}^2 +{{q}}_{43}^2+{{q}}_{31}^2)(r_d^2+\!R_G^2)/2} \\ &+\,{\rm e}^{\!-\!({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)(r_d^2+\!R_G^2)/2}\bigg] +\frac{2(n\!-\!1)\xi}{n^3}\bigg[{\rm e}^{\!-\!({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2) r_d^2/2}\Big({\rm e}^{-{{q}}_{12}^2R_G^2} \\ & +\,{\rm e}^{-{{q}}_{23}^2R_G^2}\!+\!{\rm e}^{-{{q}}_{34}^2R_G^2}+{\rm e}^{-{{q}}_{41}^2R_G^2}\Big) \!+\!{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2\!+\!{{q}}_{31}^2)r_d^2/2}\Big({\rm e}^{-{{q}}_{12}^2 R_G^2}\!+\!{\rm e}^{-{{q}}_{24}^2R_G^2} \\ &+\,{\rm e}^{\!-\!{{q}}_{43}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{31}^2R_G^2}\Big)\!+\!{\rm e}^{\!-\!({{q}}_{13}^2 +{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{\!-\!{{q}}_{13}^2R_G^2} \!+\!{\rm e}^{\!-\!{{q}}_{32}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{24}^2R_G^2}\!+\!{\rm e}^{\!-\!{{q}}_{41}^2R_G^2} \Big)\bigg] \\ &+\,\frac{(n-1)\lambda^2}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2+{{q}}_{41}^2) r_d^2/2} \Big({\rm e}^{-({{q}}_{12}+{{q}}_{23})^2R_G^2}+{\rm e}^{-({{q}}_{12}+{{q}}_{34})^2R_G^2} +{\rm e}^{-({{q}}_{12}+{{q}}_{41})^2R_G^2} \\& +\,{\rm e}^{-({{q}}_{23}+{{q}}_{34})^2R_G^2}+{\rm e}^{-({{q}}_{23}+{{q}}_{41})^2R_G^2} +{\rm e}^{-({{q}}_{34}+{{q}}_{41})^2R_G^2}\Big) +{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} \\ & \times\,\Big({\rm e}^{\!-\!({{q}}_{12}+{{q}}_{24})^2\!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{12}+{{q}}_{43})^2 \!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{12}+{{q}}_{31})^2\!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{24}+{{q}}_{43})^2 \!R_G^2}\!+\!{\rm e}^{\!-\!({{q}}_{24}+{{q}}_{31})^2\!R_G^2} \\ & +\,{\rm e}^{-({{q}}_{43}+{{q}}_{31})^2R_G^2}\Big) +{\rm e}^{-({{q}}_{13}^2+{{q}}_{32}^2+{{q}}_{24}^2 +{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{13}+{{q}}_{32})^2R_G^2}+{\rm e}^{-({{q}}_{13}+{{q}}_{24})^2 R_G^2} \\& +{\rm e}^{-({{q}}_{13}+{{q}}_{41})^2R_G^2}+{\rm e}^{-({{q}}_{32}+{{q}}_{24})^2R_G^2} +{\rm e}^{-({{q}}_{32}+{{q}}_{41})^2R_G^2}+{\rm e}^{-({{q}}_{24}+{{q}}_{41})^2R_G^2} \Big) \bigg] \\ &+\,\frac{2(n-1)(n-2)\lambda}{n^3}\bigg[{\rm e}^{-({{q}}_{12}^2+{{q}}_{23}^2+{{q}}_{34}^2 +{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)R_G^2/2} \!+\!\,{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)R_G^2/2} {\rm e}^{-({{q}}_{12}+{{q}}_{43})^2R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} {\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2}\Big) \!+\!{\rm e}^{-({{q}}_{12}^2+{{q}}_{24}^2+{{q}}_{43}^2+{{q}}_{31}^2)r_d^2/2} \Big( {\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} +{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{24}^2)R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} {\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} +{\rm e}^{-({{q}}_{12}^2+{{q}}_{34}^2)R_G^2/2} {\rm e}^{-({{q}}_{12}+{{q}}_{43})^2R_G^2/2}\Big) \\ &+\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{32}^2+{{q}}_{24}^2+{{q}}_{41}^2)r_d^2/2} \Big({\rm e}^{-({{q}}_{12}^2+{{q}}_{13}^2+{{q}}_{23}^2)R_G^2/2} +{\rm e}^{-({{q}}_{12}^2 +{{q}}_{14}^2+{{q}}_{24}^2)R_G^2/2} \\ &+\,{\rm e}^{-({{q}}_{13}^2 +{{q}}_{14}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{23}^2 +{{q}}_{24}^2+{{q}}_{34}^2)R_G^2/2} +{\rm e}^{-({{q}}_{13}^2+{{q}}_{24}^2)R_G^2/2} {\rm e}^{-({{q}}_{13}+{{q}}_{42})^2R_G^2/2} \\ & +\,{\rm e}^{-({{q}}_{14}^2+{{q}}_{23}^2)R_G^2/2} {\rm e}^{-({{q}}_{14}+{{q}}_{32})^2R_G^2/2}\Big)\bigg], \end{aligned} $ ![]() |