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Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions
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Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions Rong Xu, Dong-Hai Zhang∗ Institute of Modern Physics, Shanxi Normal University, Linfen 041004, China
a r t i c l e
i n f o
Article history: Received 21 February 2016 Revised 1 June 2016 Accepted 3 July 2016 Available online xxx PACS: 25.75.-q 25.70.Mn 25.70.Pq 29.40.Rg Keywords: Relativistic heavy ions collision target fragmentation correlation nuclear emulsion
a b s t r a c t Two and three particle short-range correlations among the target evaporated fragments as well as the target recoil protons produced in 12 A GeV 4 He-, 3.7 A GeV 16 O-, 60 A GeV 16 O-, 1.7 A GeV 84 Kr- and 10.7 A GeV 197 Au-AgBr interactions are investigated in emission angle space and azimuthal angle space, respectively. The experimental data exhibit two and three particle correlations among the target fragments in emission angle space, which indicate the occurrence of so-called side splash phenomena and disfavor the evaporation model. In azimuthal angle space the data exhibit strong two particle correlations for both the target evaporated fragments and target recoil protons, and three particle correlations only for target recoil protons. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
1. Introduction In high energy nucleus–nucleus collisions, investigation of correlation effect in terms of two and three particle correlation function is one of the most popular trends because the study of correlation can encapsulate rich information about the space time structure and the dynamics of the emitting source in the late stage of the collision where the nuclear matter is highly excited and diffused [1–3]. The correlation which prevails at the early stage of interaction cannot be expected to survive in the final stage due to the rigors of the initial violent dense stage. The effect of two and three produced particle short range correlations has been investigated in different type of interactions [4–17]. It was observed that in all types of nuclear interactions stated above [4–17], the produced particles shown a tendency to be emitted in a correlated fashion. The reason behind this may be the formation of exotic nuclear matter, hot multi-nuclear fire ball or formation of heavier intermediate states, clusterization or the so called side splash phenomenon [18–23] etc. So up to now it is still an open question whether the particle production processes are basically weakly correlated phenomena or whether strong correlations are present. The matter is further complicated by the fact that conservation laws impose certain kinematical correlation which can not always be separated from correlations of a more dynamical nature. It has been shown by Gulamov et al. [24] from a comparison of correlation functions calculated in the random stars generated according to the cylindrical phase space model and the independent emission model, that the conservation laws lead to the increase of long-range correlations and to the decrease ∗
Corresponding author. Fax: +86 3572051347. E-mail address: [email protected] (D.-H. Zhang).
http://dx.doi.org/10.1016/j.cjph.2016.07.001 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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of short-range correlations, especially in the beam fragmentation region. Therefore, a detailed correlation study is essential to search for the exact reason of correlated emission of particles speculated by different theorists. It should be mentioned that during the previous investigations, emphasis has been given on pions because the pions are most frequently produced particles in relativistic hadron–hadron, hadron–nucleus and nucleus–nucleus collisions and the knowledge of pion production mechanism is essential for understanding the main features of high energy multi-particle production. On the contrary, very little attention was paid on the target fragments. The target fragments include target recoil protons and target evaporated fragments. The target recoil protons are formed due to fast target protons of energy ranging up to 400 MeV, and they are supposed to carry relevant information about the hadronization mechanism because the time scale of emission of these particles is of the same order as that of the produced particles. The target evaporated particles are of low-energy (<30 MeV) singly or multiply charged fragments, and they are emitted at the late stage of nuclear reactions and are expected to remember the parts of the history of interactions. So the study of target evaporated fragments and target recoil protons is of great importance and a potential source of information. In emulsion terminology [25], the target recoil particles are referred to as “grey track particles” and the target evaporated particles are referred to as “black track particles”. According to the cascade evaporation model [26], the shower particle and the grey track particle are emitted from the nucleus very soon after the instant of impact, leaving the hot residual nucleus in a highly excited state. Indeed, the excitation energy may sometimes be comparable with the total binding energy of the nucleus. Emission of black track particles from this state, however, now takes place relatively slowly. In order to escape from the residual nucleus, a particle must await a favorable statistical fluctuation arising out of random collisions among the nucleons. Emission occurs if, due to a fluctuation, the particle is both close to the nuclear boundary, traveling in an outward direction, and if its kinetic energy is greater than its binding energy. After the evaporation of this particle, a further relatively long period on a nuclear time scale, say 10−17 s, will commonly elapse before a second particle is also placed in conditions favorable for escape, and so on. The process continues until the excitation energy of the residual nucleus is so small that transition to the ground state is likely to be effected by the emission of gamma rays. In the rest system of the target nucleus, the emission of evaporated particles is assumed to be isotropic over the whole phase space. The evaporation model is based on the assumption that statistical equilibrium has been established in the decaying system and the lifetime is much longer than the time taken to distribute the energy among nucleons within the nucleus. In our recent studies [27,28], it has been shown that the angular distribution of black track particles from relativistic nucleus–nucleus interactions are not isotropic, and the averaged multiplicities in the forward hemisphere (emission angle θ ≤ 90°) are greater than in the backward hemisphere. These results could not be explained satisfactorily by the evaporation model. The non-isotropic emission means that the target evaporated fragments are not emitted independently. One of the methods to measure these dependent emission is the study of two and three particle correlation. According to our investigation, two and three particle correlations of target fragments were studied by two groups [29–35] in emission angle space and azimuthal space. A strong two and three particle correlation among the target evaporated fragments as well as the target recoil protons was observed in these studies. In this paper, two and three particle correlations of target evaporated fragments as well as target recoil protons emitted from 12 A GeV 4 He-AgBr, 3.7 A GeV 16 O-AgBr, 60 A GeV 16 O-AgBr, 1.7 A GeV 84 Kr-AgBr and 10.7 A GeV 197 Au-AgBr interactions are investigated in emission angle space and azimuthal angle space respectively. 2. Experimental details Five nuclear emulsion stacks, provided by the EMU01 Collaboration, were used in the present investigation. The stacks were exposed horizontally to 12 A GeV 4 He, 3.7 A GeV 16 O, 60 A GeV 16 O, 1.7 A GeV 84 Kr and 10.7 A GeV 197 Au. BA20 0 0 and XSJ-2 microscopes with a 100 × oil immersion objective and 10 × ocular lenses were used to scan the plates. The tracks were picked up at a distance of 5 mm from the edge of the plates and they were carefully followed until they either interacted with emulsion nuclei or escaped from the plates. Interactions within 30 μm of the top or bottom surface of the emulsion plates were not considered for final analysis. In each interaction all of the secondaries were recorded, including shower particles, target recoil protons, target evaporated fragments and projectile fragments. According to the emulsion terminology [25] the particles emitted after interaction are classified as: (a) Shower track particles correspond to the produced single-charged relativistic particles having velocity β ≥ 0.7. Most of these particles belong to pions contaminated with small proportions of fast protons and K mesons. Ionization power of shower particles is less and equal to 1.4I0 , where I0 being the minimum ionization of a singly charged relativistic particle, which is about 30–32grains/100 μm for 12 A GeV 4 He, 3.7 A GeV 16 O, 60 A GeV 16 O and 10.7 A GeV 197 Auemulsion interactions, and about 12grains/100 μm for 1.7 A GeV 84 Kr-emulsion interactions because the emulsion is not enough developed. (b) Grey track particles, with a range in emulsion L > 3 mm and velocity 0.3 < β < 0.7, are mostly recoil protons in the kinetic energy range 26 ≤ Ek ≤ 375 MeV and a few kaons of kinetic energies 20 ≤ Ek ≤ 198 MeV and pions with kinetic energies 12 ≤ Ek ≤ 56 MeV. Ionization power of grey particles lies between 1.4I0 to 9I0 . (c) Black track particles, with a range in emulsion L ≤ 3 mm and velocity β < 0.3, correspond to protons with kinetic energies Ek ≤ 26 MeV and other target fragments of various elements like carbon, lithium, beryllium and helium etc. Ionization power of black particles is greater or equal to 9I0 . Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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3
The grey track particles and black track particles together are called heavily ionized particles or target fragments. Their multiplicity is denoted by nh . (d) Projectile fragments, a charged particle emitted at an angle θ ≤ f (f is the projectile fragmentation angle), without change in its ionization for at least 2 cm from the interaction point, is considered as a non-interacting projectile fragments. The fragmentation angle is defined as f = p f /pbeam , where pbeam stands for the beam momentum and pf for the Fermi momentum of the nucleon in nucleus. The nuclear emulsion is composed of a homogeneous mixture of H, C, N, O, S, I, Br, and Ag nuclei. According to the value of nh the interactions are divided into the following three groups. Events with nh ≤ 1 are due to interactions with H target and peripheral interactions with CNO and AgBr targets, events with 2 ≤ nh ≤ 7 are due to interactions with CNO targets and peripheral interactions with AgBr targets, and events with nh ≥ 8 definitely belong to interactions with AgBr targets. To ensure the targets in the nuclear emulsion are silver or bromine nuclei, we have chosen only the events with at least eight heavy ionizing track particles. 3. Method of analysis Two particle correlation For the phase-space variable z, the two particle normalized correlation function is defined as [36]
R2 ( z1 , z2 ) =
ρ2 ( z 1 , z 2 ) − ρ1 ( z 1 ) ρ1 ( z 2 ) ρ2 ( z 1 , z 2 ) = − 1, ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 1 ) ρ1 ( z 2 )
(1)
where the quantities ρ 1 (z) and ρ 2 (z1 , z2 ) represent one and two particle densities respectively, which are defined as
1 dσ , σin dz1 1 d2 σ ρ2 ( z 1 , z 2 ) = , σin dz1 dz2
ρ1 ( z ) =
and where σ in ,
dσ dz1
and
d2 σ d z1 d z2
(2)
are the inelastic cross-section, the one and the two particle distribution, respectively, z1 and
z2 are the values of a certain physical quantity corresponding to the two particles. R2 can be represented as
R2 (z1 , z2 ) = NT
N2 (z1 , z2 ) −1 N1 (z1 )N1 (z2 )
(3)
where NT is the total number of inelastic events. N1 (z1 ) is the number of particles at the phase space interval of z1 and z1 + dz1 , and N1 (z2 ) is the number of particles at the phase space interval of z2 and z2 + dz2 . N2 (z1 , z2 ) is the number of pairs of particles having one particle between the phase space z1 and z1 + dz1 and the other particle between z2 and z2 + dz2 in an event. The correlation function measures the dependence of the production of the two particles on their own characteristic parameters. The positive value of R2 (z1 , z2 ) means that the presence of one particle with a parameter z1 compels the existence of the other within z2 . A zero value of R2 (z1 , z2 ) means that their emission is independent. On the other hand, a negative value of the correlation function means that the emission of particle at z1 excludes the emission of the other within z2 . If the phase space variable is taken as z = cos θ , θ is emission angle of a particle, the normalized inclusive correlation function can be written as
R2 (cos θ1 , cos θ2 ) = NT
N2 (cos θ1 , cos θ2 ) −1 N1 (cos θ1 )N1 (cos θ2 )
(4)
where N1 (cos θ 1 ) is the number of interactions having a particle between cos θ 1 and cos θ1 + d cos θ1 , and N2 (cos θ 1 , cos θ 2 ) is the number of interactions having at least two particles, the first with angle between cos θ 1 and cos θ1 + d cos θ1 , and the other of angle between cos θ 2 and cos θ2 + d cos θ2 . If the phase space variable is taken as azimuthal angle φ , the normalized inclusive correlation function can be written as
R2 (φ1 , φ2 ) = NT
N2 (φ1 , φ2 ) −1 N1 (φ1 )N1 (φ2 )
(5)
where N1 (φ 1 ) is the number of interactions having a particle between φ 1 and φ1 + dφ1 , and N2 (φ 1 , φ 2 ) is the number of interactions having at least two particles, the first with azimuthal angle between φ 1 and φ1 + dφ1 , and the other of azimuthal angle between φ 2 and φ2 + dφ2 . Experimentally, the two-particle correlation function in the emission angle phase space and azimuthal angle space are calculated as
R2 (cos θ1 , cos θ2 ) =
n1 (cos θ1 )n2 (cos θ2 ) − 1, for cos θ1 = cos θ2 n1 (cos θ1 )n2 (cos θ2 )
(6)
Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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and
n(cos θ )(n(cos θ ) − 1 ) − 1, for cos θ1 = cos θ2 = cos θ n(cos θ )2
R2 (cos θ1 , cos θ2 ) =
(7)
where n1 (cos θ 1 ) and n2 (cos θ 2 ) are the particle multiplicities in a small interval of δ (cos θ ) around cos θ 1 and cos θ 2 .
R 2 ( φ1 , φ2 ) =
n 1 ( φ1 ) n 2 ( φ2 ) − 1, for φ1 = φ2 n1 (φ1 )n2 (φ2 )
(8)
R 2 ( φ1 , φ2 ) =
n(φ )(n(φ ) − 1 ) − 1, for φ1 = φ2 = φ n ( φ )2
(9)
and
where n1 (φ 1 ) and n2 (φ 2 ) are the particle multiplicities in a small interval of δ (φ ) around φ 1 and φ 2 . The variance of R2 can be calculated by [36]
V [R2 ] =
1 [n21 n22 n1 2 n2 2 − 2n21 n2 n1 n2 n1 n2 2 NT n1 4 n2 4 −2n1 n22 n1 n2 n1 2 n2 + n21 n1 n2 2 n2 2 + n22 n1 n2 2 n1 2 +2n1 n2 3 n1 n2 − n1 n2 2 n1 2 n2 2 ] + O(1/N 2 )
(10)
for the case of cos θ 1 = cos θ 2 or the case of φ 1 = φ 2 , and
V [R2 ] =
1 [n41 n1 2 − 4n31 n21 n1 + 4n21 3 − n21 2 n1 2 NT n1 6 +2n31 n1 2 − 4n21 2 n1 + 2n21 n1 3 + n21 n1 2 − n1 4 ] + O(1/N 2 )
for the case of cos θ1 = cos θ2 or the case of φ1 = φ2 . in comparison with the other terms.
O(1/N2 )
(11)
is a polynomial which is negligible when calculating the errors,
Three particle correlation The three particle inclusive correlation function for phase space variable z can be defined as in Ref. [37],
R3 ( z1 , z2 , z3 ) =
ρ3 ( z 1 , z 2 , z 3 ) + 2 ρ 1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 ) ρ2 ( z 1 , z 2 ) ρ1 ( z 3 ) + ρ 2 ( z 2 , z 3 ) ρ1 ( z 1 ) + ρ2 ( z 3 , z 1 ) ρ1 ( z 2 ) − , ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 ) ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 )
(12)
where the quantity
ρ3 ( z 1 , z 2 , z 3 ) =
1
σin
d3 σ dz1 dz2 dz3
(13)
represents the three particle density. ρ 1 and ρ 2 represent the same quantities as before in the case of the two particle correlation. In terms of the number of particles, R3 (z1 , z2 , z3 ) can be written as
R3 (z1 , z2 , z3 ) = NT2
N3 (z1 , z2 , z3 ) N2 (z1 , z2 ) N2 (z2 , z3 ) N2 (z3 , z1 ) − NT − NT − NT + 2, N1 (z1 )N1 (z2 )N1 (z3 ) N1 (z1 )N1 (z2 ) N1 (z2 )N1 (z3 ) N1 (z3 )N1 (z1 )
(14)
where N3 (z1 , z2 , z3 ) is the number of triplets of particles having one particle in the interval z1 to z1 + dz1 , the other particle between z2 to z2 + dz2 and the third particle between z3 to z3 + dz3 in an event. For the phase space variable is taken as cos θ , the three particle correlation function is represented as
N3 (cos θ1 , cos θ2 , cos θ3 ) N2 (cos θ1 , cos θ2 ) + 2 − NT N1 (cos θ1 )N1 (cos θ2 )N1 (cos θ3 ) N1 (cos θ1 )N1 (cos θ2 ) N2 (cos θ2 , cos θ3 ) N2 (cos θ3 , cos θ1 ) − NT − NT . N1 (cos θ2 )N1 (cos θ3 ) N1 (cos θ3 )N1 (cos θ1 )
R3 (cos θ1 , cos θ2 , cos θ3 ) = NT2
(15)
The three particle emission angle correlation function is experimentally obtained as
R3 (cos θ1 , cos θ2 , cos θ3 ) =
n(n − 1 )(n − 2 ) 3n(n − 1 ) − + 2, for cos θ1 = cos θ2 = cos θ3 n 3 n 2
(16)
For the phase space variable is taken as azimuthal angle φ , the three particle correlation function is represented as
R3 (φ1 , φ2 , φ3 ) = NT2
N3 (φ1 , φ2 , φ3 ) N2 (φ1 , φ2 ) N2 (φ2 , φ3 ) N2 (φ3 , φ1 ) − NT − NT − NT + 2. N1 (φ1 )N1 (φ2 )N1 (φ3 ) N1 (φ1 )N1 (φ2 ) N1 (φ2 )N1 (φ3 ) N1 (φ3 )N1 (φ1 )
(17)
Similarly, the three particle azimuthal angle correlation function is experimentally obtained as
R 3 ( φ1 , φ2 , φ3 ) =
n(n − 1 )(n − 2 ) 3n(n − 1 ) − + 2, for φ1 = φ2 = φ3 n 3 n 2
(18)
The variance of this quantity is calculated term by term and instead of giving the long algebraic expression of the net variance. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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1 4
12 A GeV He
0.8
3.7 A GeV 60 A GeV
0.6
16
O
16
1.7 A GeV
O
84
10.7 A GeV
Kr
197
Au
R2
0.4 0.2 0 −0.2 −0.4
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
cosΘ Fig. 1. (Color online) The normalized two particle correlation function in emission angle space for target evaporated fragments.
4. Results and discussion There are 2975 12 A GeV 4 He-AgBr, 927 3.7 A GeV 16 O-AgBr, 521 60 A GeV 16 O-AgBr, 229 1.7 A GeV 84 Kr-AgBr and 619 10.7 A GeV 197 Au-AgBr interacting events are used in present investigation. The two and three particle normalized shortrange correlation function R2 (cos θ 1 , cos θ 2 ) and R3 (cos θ 1 , cos θ 2 , cos θ 3 ) for target evaporated fragments are calculated with the help of relation (7) and (16) respectively. Fig. 1 shows the plot of the two particle normalized correlation function R2 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ ) of the correlation matrices, characterizing the magnitude of the short-range correlation at different phase space intervals. The errors are calculated using Eq. (11). It is evident from Fig. 1 that the short-range correlation values are greater than zero (R2 > 0) in any value of cos θ space intervals for the most of the interactions which implies that the short-range two particle correlation exists over the entire space of emission angle. From the graph it is further seen that the correlation is prominent at cos θ space interval of −1.0 ≤ cos θ < −0.6 or at 0.6 ≤ cos θ < 1.0(values of the correlation functions in these intervals are greater than that in other intervals), it means that the correlation is more likely in the most forward or the most backward direction. Those observations are the same as the result observed in 14.5 A GeV 28 Si-AgBr interactions [30]. An exception is observed in the Fig. 1 that the two particle short-range correlation function in some cos θ space intervals are less than zero and the correlation is prominent at cos θ space intervals of −0.4 ≤ cos θ < 0.0 and 0.6 ≤ cos θ < 0.8 for the case of 60 A GeV 16 O-AgBr interactions. Fig. 2 shows the normalized three particle short-range correlation function R3 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ3 = cos θ ) of the correlation matrices for target evaporated fragments. The variance of R3 is calculated term by term in relation (16). The prominent three particle short-range correlations are in the cos θ space interval of −1.0 ≤ cos θ < −0.8 for most of the interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. The correlation can give direct information about the late stage of the reaction when nuclear matter is highly excited and diffused. According to the model proposed by Stocker et al. [23] using three-dimensional nuclear fluid dynamics, the emission of target fragments in the backward hemisphere can be explained with the help of the side splash phenomenon. In a nucleus–nucleus collision, a head shock zone may be developed during the dividing phase of the projectile nucleus within the target. A strongly compressed and highly excited projectile-like object continues to interpenetrate the target with supersonic velocity and may push the matter sideways. This results in the generation of shock waves that give rise to the correlation in the sideward directions. The two and three particle normalized short-range correlation function R2 and R3 for target recoil protons are calculated using the relation (7) and (16) respectively. Fig. 3 shows the plot of the two particle normalized short-range correlation function R2 against cos θ for the case of cos θ1 = cos θ2 = cos θ . It can be clearly seen from Fig. 3 that the short-range correlation values are greater than zero (R2 > 0) in most value of cos θ space intervals for all of the interactions which implies that there are strong short-range two particle correlation over the entire space of emission angle. The prominent two particle short-range correlations are in the cos θ space intervals of −1.0 ≤ cos θ < −0.8 for 1.7 A GeV 84 Kr- and 3.7 A GeV 16 O-AgBr Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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12 A GeV He
2.5
3.7 A GeV
2
60 A GeV
R3
O
16
1.7 A GeV
1.5
16
O
84
Kr
10.7 A GeV
197
Au
1 0.5 0 −0.5 −1
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
cosΘ Fig. 2. (Color online) The normalized three particle correlation function in emission angle space for target evaporated fragments.
2
4
12 A GeV He 3.7 A GeV
1.5
60 A GeV
16
O
16
1.7 A GeV
O
84
10.7 A GeV
Kr
197
Au
R2
1
0.5
0
−0.5
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
cosΘ Fig. 3. (Color online) The normalized two particle correlation function in emission angle space for target recoil protons.
interactions, but in this region the production of target recoil protons is not too much and the error of correlation function is greater. Fig. 4 shows the normalized three particle short-range correlation function R3 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ3 = cos θ ) of the correlation matrices for target recoil protons. The variance of R3 is calculated term by term in relation (16). The prominent three particle short-range correlations are in the cos θ space intervals of 0.8 ≤ cos θ < 1.0 and intervals of −0.4 ≤ cos θ < 0.0 for most of the interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. Generally speaking, the regions for the prominent three particle short-range correlations should be consistent with that for the prominent two particle short-range correlations. It is true from the comparison of Figs. 1–4 for most of interactions. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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5 4
12 A GeV He 3.7 A GeV
4
60 A GeV
1.7 A GeV
3
16
O
16
O
84
10.7 A GeV
Kr
197
Au
R3
2 1 0 −1 −2
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
cosΘ Fig. 4. (Color online) The normalized three particle correlation function in emission angle space for target recoil protons.
1.2 1 0.8
R2
0.6 0.4 0.2 0 −0.2
0
50
100
150
200
250
300
350
Φ Fig. 5. (Color online) The normalized two particle correlation function in azimuthal angle space for target evaporated fragments.
For the cases the prominent three particle short-range correlations with great statistical errors, the two particle short-range correlations are not prominent. The two and three particle normalized short-range correlation function R2 (φ 1 , φ 2 ) and R3 (φ 1 , φ 2 , φ 3 ) for target evaporated fragments are calculated with the help of relation (9) and (18) respectively. Fig. 5 shows the plot of the two particle normalized correlation function R2 against φ for the values of the diagonal elements (φ1 = φ2 = φ ) of the correlation matrices, characterizing the magnitude of the short-range correlation at different phase space intervals. The errors are calculated using Eq. (11). It is evident from Fig. 5 that the short-range correlation values are greater than zero (R2 > 0) in most of φ space intervals which implies that the short-range two particle correlation exists over the entire space of azimuthal angle except for 60 A GeV 16 O-AgBr interaction at azimuthal angle region of 0° ≤ φ < 36°. From the graph it is further seen that the correlation is prominent at φ space interval of 180° ≤ φ < 216° for 10.7 A GeV 197 Au-AgBr, 12 A GeV 4 He-AgBr and Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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3
R3
2
1
0
−1
0
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100
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Φ Fig. 6. (Color online) The normalized three particle correlation function in azimuthal angle space for target evaporated fragments.
1.4 4
12 A GeV He
1.2
3.7 A GeV 60 A GeV
1
16
16
1.7 A GeV
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10.7 A GeV
O
O Kr
197
Au
R2
0.8 0.6 0.4 0.2 0 −0.2
0
50
100
150
200
250
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350
Φ Fig. 7. (Color online) The normalized two particle correlation function in azimuthal angle space for target recoil protons.
3.7 A GeV 16 O-AgBr interaction, of 324° ≤ φ < 360° for 1.7 A GeV 84 Kr-AgBr interaction, and of 108° ≤ φ < 144° for 60 A GeV 16 O-AgBr interaction. Fig. 6 shows the normalized three particle short-range correlation function R3 against φ for the values of the diagonal elements (φ1 = φ2 = φ3 = φ ) of the correlation matrices for target evaporated fragments. The variance of R3 is calculated term by term in relation (18). The prominent three particle short-range correlations is in the φ space interval of 180° ≤ φ < 216° for 12 A GeV 4 He-AgBr, 3.7 A GeV 16 O-AgBr and 10.7 A GeV 197 Au-AgBr interactions, and of 324° ≤ φ < 360° for 1.7 A GeV 84 Kr-AgBr interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. The two and three particle normalized short-range correlation function R2 and R3 for target recoil protons in azimuthal angle space are calculated using the relation (9) and (18) respectively. Fig. 7 shows the plot of the two particle normalized Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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9
6 4
12 A GeV He
5
3.7 A GeV 60 A GeV
4
16
16
1.7 A GeV
O
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10.7 A GeV
O
Kr
197
Au
R3
3 2 1 0 −1
0
50
100
150
200
250
300
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Φ Fig. 8. (Color online) The normalized three particle correlation function in azimuthal angle space for target recoil protons.
short-range correlation function R2 against φ for the case of φ1 = φ2 = φ . It can be clearly seen from Fig. 7 that the shortrange correlation values are greater than zero (R2 > 0) in all of the φ space intervals for all of the interactions which implies that there are strong short-range two particle correlation over the entire space of azimuthal angle. The prominent two particle short-range correlations is in the φ space interval of 252° ≤ φ < 288° for 10.7 A GeV 197 Au-AgBr, 60 A GeV 16 O-AgBr and 3.7 A GeV 16 O-AgBr collision, of 180° ≤ φ < 216° for 1.7 A GeV 84 Kr-AgBr collision, and of 72° ≤ φ < 108° for 12 A GeV 4 He-AgBr collision. Fig. 8 shows the normalized three particle short-range correlation function R3 against φ for the values of the diagonal elements (φ1 = φ2 = φ3 = φ ) of the correlation matrices for target recoil protons. The variance of R3 is calculated term by term in relation (18). It can be seen from Fig. 8 that the short-range correlation values are greater than zero (R3 > 0) in most of the φ space intervals for all of the interactions which implies that there are strong short-range three particle correlation in most of the azimuthal angle space. The prominent three particle short-range correlations is in the φ space interval of 252° ≤ φ < 288° for 10.7 A GeV 197 Au-AgBr and 60 A GeV 16 O-AgBr collisions, of 36° ≤ φ < 72° for 1.7 A GeV 84 Kr-AgBr collision, of 72° ≤ φ < 108° for 12 A GeV 4 He-AgBr collision, and of 144° ≤ φ < 180° for 3.7 A GeV 16 O-AgBr collision. For the cases of the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. In the study of correlations among the target fragments by means of correlation functions, pseudo-correlations may arise from the broad multiplicity distribution and the dependence of the one-particle spectrum on multiplicity, as well as the trivial correlations due to kinematical constraints in individual events. It has been emphasized by Ghosh et al. [38] that the precise model-independent search for dynamical correlations seems to be impossible in hadron–nucleus and nucleus– nucleus collisions. To search for the non-trivial dynamical correlation among the target fragments, the experimental data have to be compared with those calculated from Monte Carlo simulation based on the framework of independent emission hypothesis. The emission of particles from a collectively excited state of the nucleus can be understood by Fermi liquid drop model in which the emission of particles can occur in two ways. First, due to the relaxation processes where the collective energy is transferred to the intrinsic degrees of freedom with subsequent evaporation of particles. Second, a direct nonstatistical emission (splashing) of nucleons is also possible via the dynamical distortion of the Fermi surface accompanying the collective motion. In general, the relative contributions of these mechanisms depend upon the magnitude of the nuclear friction coefficient. Particles emitted from excited nucleus due to both the evaporation and the splashing are due to the collective motion of the nuclear Fermi liquid and are accompanied by direct non-statistical emission of nucleons via the dynamical distortion of the Fermi surface, so they are responsible for the presence of the short range correlation. 5. Conclusions The two and three particle short-range correlation in emission angle space for both target evaporated fragments and target recoil protons produced in 12 A GeV 4 He-, 3.7 A GeV 16 O-, 60 A GeV 16 O-, 1.7 A GeV 84 Kr- and 10.7 A GeV 197 Au-AgBr interactions have been investigated in emission angle space and azimuthal angle space respectively. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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In emission angle space it is found that the evidence of strong two particle short-range correlation exist, and the prominent two particle short-range correlations are in the most forward or backward emission angle space, for both target evaporated fragments and target recoil protons. A weak three particle short-range correlation is observed in the cos θ space intervals of −1.0 ≤ cos θ < −0.8 for target evaporated fragments, but the error is greater. The observation of two and three particle short-range correlation in emission angle space can not explained by evaporation model and favor of the side splash model. In azimuthal angle space it is found that the evidence of both two and three target recoil proton short-range correlation exist, for target evaporated fragment the strong two particle correlation and a weak three particle short-range correlation is observed. The observation of two and three particle short-range correlation can not explained by evaporation model. Acknowledgment This work has been supported by the Chinese National Science Foundation under Grant nos: 11075100 and 11565001, the Natural Foundation of Shanxi Province under Grant 2011011001-2, the Shanxi Provincial Foundation for Returned Overseas Chinese Scholars, China (Grant no. 2011–058). We are grateful to EMU-01 collaboration for supplying emulsion stacks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions Rong Xu, Dong-Hai Zhang∗ Institute of Modern Physics, Shanxi Normal University, Linfen 041004, China
a r t i c l e
i n f o
Article history: Received 21 February 2016 Revised 1 June 2016 Accepted 3 July 2016 Available online xxx PACS: 25.75.-q 25.70.Mn 25.70.Pq 29.40.Rg Keywords: Relativistic heavy ions collision target fragmentation correlation nuclear emulsion
a b s t r a c t Two and three particle short-range correlations among the target evaporated fragments as well as the target recoil protons produced in 12 A GeV 4 He-, 3.7 A GeV 16 O-, 60 A GeV 16 O-, 1.7 A GeV 84 Kr- and 10.7 A GeV 197 Au-AgBr interactions are investigated in emission angle space and azimuthal angle space, respectively. The experimental data exhibit two and three particle correlations among the target fragments in emission angle space, which indicate the occurrence of so-called side splash phenomena and disfavor the evaporation model. In azimuthal angle space the data exhibit strong two particle correlations for both the target evaporated fragments and target recoil protons, and three particle correlations only for target recoil protons. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
1. Introduction In high energy nucleus–nucleus collisions, investigation of correlation effect in terms of two and three particle correlation function is one of the most popular trends because the study of correlation can encapsulate rich information about the space time structure and the dynamics of the emitting source in the late stage of the collision where the nuclear matter is highly excited and diffused [1–3]. The correlation which prevails at the early stage of interaction cannot be expected to survive in the final stage due to the rigors of the initial violent dense stage. The effect of two and three produced particle short range correlations has been investigated in different type of interactions [4–17]. It was observed that in all types of nuclear interactions stated above [4–17], the produced particles shown a tendency to be emitted in a correlated fashion. The reason behind this may be the formation of exotic nuclear matter, hot multi-nuclear fire ball or formation of heavier intermediate states, clusterization or the so called side splash phenomenon [18–23] etc. So up to now it is still an open question whether the particle production processes are basically weakly correlated phenomena or whether strong correlations are present. The matter is further complicated by the fact that conservation laws impose certain kinematical correlation which can not always be separated from correlations of a more dynamical nature. It has been shown by Gulamov et al. [24] from a comparison of correlation functions calculated in the random stars generated according to the cylindrical phase space model and the independent emission model, that the conservation laws lead to the increase of long-range correlations and to the decrease ∗
Corresponding author. Fax: +86 3572051347. E-mail address: [email protected] (D.-H. Zhang).
http://dx.doi.org/10.1016/j.cjph.2016.07.001 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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of short-range correlations, especially in the beam fragmentation region. Therefore, a detailed correlation study is essential to search for the exact reason of correlated emission of particles speculated by different theorists. It should be mentioned that during the previous investigations, emphasis has been given on pions because the pions are most frequently produced particles in relativistic hadron–hadron, hadron–nucleus and nucleus–nucleus collisions and the knowledge of pion production mechanism is essential for understanding the main features of high energy multi-particle production. On the contrary, very little attention was paid on the target fragments. The target fragments include target recoil protons and target evaporated fragments. The target recoil protons are formed due to fast target protons of energy ranging up to 400 MeV, and they are supposed to carry relevant information about the hadronization mechanism because the time scale of emission of these particles is of the same order as that of the produced particles. The target evaporated particles are of low-energy (<30 MeV) singly or multiply charged fragments, and they are emitted at the late stage of nuclear reactions and are expected to remember the parts of the history of interactions. So the study of target evaporated fragments and target recoil protons is of great importance and a potential source of information. In emulsion terminology [25], the target recoil particles are referred to as “grey track particles” and the target evaporated particles are referred to as “black track particles”. According to the cascade evaporation model [26], the shower particle and the grey track particle are emitted from the nucleus very soon after the instant of impact, leaving the hot residual nucleus in a highly excited state. Indeed, the excitation energy may sometimes be comparable with the total binding energy of the nucleus. Emission of black track particles from this state, however, now takes place relatively slowly. In order to escape from the residual nucleus, a particle must await a favorable statistical fluctuation arising out of random collisions among the nucleons. Emission occurs if, due to a fluctuation, the particle is both close to the nuclear boundary, traveling in an outward direction, and if its kinetic energy is greater than its binding energy. After the evaporation of this particle, a further relatively long period on a nuclear time scale, say 10−17 s, will commonly elapse before a second particle is also placed in conditions favorable for escape, and so on. The process continues until the excitation energy of the residual nucleus is so small that transition to the ground state is likely to be effected by the emission of gamma rays. In the rest system of the target nucleus, the emission of evaporated particles is assumed to be isotropic over the whole phase space. The evaporation model is based on the assumption that statistical equilibrium has been established in the decaying system and the lifetime is much longer than the time taken to distribute the energy among nucleons within the nucleus. In our recent studies [27,28], it has been shown that the angular distribution of black track particles from relativistic nucleus–nucleus interactions are not isotropic, and the averaged multiplicities in the forward hemisphere (emission angle θ ≤ 90°) are greater than in the backward hemisphere. These results could not be explained satisfactorily by the evaporation model. The non-isotropic emission means that the target evaporated fragments are not emitted independently. One of the methods to measure these dependent emission is the study of two and three particle correlation. According to our investigation, two and three particle correlations of target fragments were studied by two groups [29–35] in emission angle space and azimuthal space. A strong two and three particle correlation among the target evaporated fragments as well as the target recoil protons was observed in these studies. In this paper, two and three particle correlations of target evaporated fragments as well as target recoil protons emitted from 12 A GeV 4 He-AgBr, 3.7 A GeV 16 O-AgBr, 60 A GeV 16 O-AgBr, 1.7 A GeV 84 Kr-AgBr and 10.7 A GeV 197 Au-AgBr interactions are investigated in emission angle space and azimuthal angle space respectively. 2. Experimental details Five nuclear emulsion stacks, provided by the EMU01 Collaboration, were used in the present investigation. The stacks were exposed horizontally to 12 A GeV 4 He, 3.7 A GeV 16 O, 60 A GeV 16 O, 1.7 A GeV 84 Kr and 10.7 A GeV 197 Au. BA20 0 0 and XSJ-2 microscopes with a 100 × oil immersion objective and 10 × ocular lenses were used to scan the plates. The tracks were picked up at a distance of 5 mm from the edge of the plates and they were carefully followed until they either interacted with emulsion nuclei or escaped from the plates. Interactions within 30 μm of the top or bottom surface of the emulsion plates were not considered for final analysis. In each interaction all of the secondaries were recorded, including shower particles, target recoil protons, target evaporated fragments and projectile fragments. According to the emulsion terminology [25] the particles emitted after interaction are classified as: (a) Shower track particles correspond to the produced single-charged relativistic particles having velocity β ≥ 0.7. Most of these particles belong to pions contaminated with small proportions of fast protons and K mesons. Ionization power of shower particles is less and equal to 1.4I0 , where I0 being the minimum ionization of a singly charged relativistic particle, which is about 30–32grains/100 μm for 12 A GeV 4 He, 3.7 A GeV 16 O, 60 A GeV 16 O and 10.7 A GeV 197 Auemulsion interactions, and about 12grains/100 μm for 1.7 A GeV 84 Kr-emulsion interactions because the emulsion is not enough developed. (b) Grey track particles, with a range in emulsion L > 3 mm and velocity 0.3 < β < 0.7, are mostly recoil protons in the kinetic energy range 26 ≤ Ek ≤ 375 MeV and a few kaons of kinetic energies 20 ≤ Ek ≤ 198 MeV and pions with kinetic energies 12 ≤ Ek ≤ 56 MeV. Ionization power of grey particles lies between 1.4I0 to 9I0 . (c) Black track particles, with a range in emulsion L ≤ 3 mm and velocity β < 0.3, correspond to protons with kinetic energies Ek ≤ 26 MeV and other target fragments of various elements like carbon, lithium, beryllium and helium etc. Ionization power of black particles is greater or equal to 9I0 . Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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3
The grey track particles and black track particles together are called heavily ionized particles or target fragments. Their multiplicity is denoted by nh . (d) Projectile fragments, a charged particle emitted at an angle θ ≤ f (f is the projectile fragmentation angle), without change in its ionization for at least 2 cm from the interaction point, is considered as a non-interacting projectile fragments. The fragmentation angle is defined as f = p f /pbeam , where pbeam stands for the beam momentum and pf for the Fermi momentum of the nucleon in nucleus. The nuclear emulsion is composed of a homogeneous mixture of H, C, N, O, S, I, Br, and Ag nuclei. According to the value of nh the interactions are divided into the following three groups. Events with nh ≤ 1 are due to interactions with H target and peripheral interactions with CNO and AgBr targets, events with 2 ≤ nh ≤ 7 are due to interactions with CNO targets and peripheral interactions with AgBr targets, and events with nh ≥ 8 definitely belong to interactions with AgBr targets. To ensure the targets in the nuclear emulsion are silver or bromine nuclei, we have chosen only the events with at least eight heavy ionizing track particles. 3. Method of analysis Two particle correlation For the phase-space variable z, the two particle normalized correlation function is defined as [36]
R2 ( z1 , z2 ) =
ρ2 ( z 1 , z 2 ) − ρ1 ( z 1 ) ρ1 ( z 2 ) ρ2 ( z 1 , z 2 ) = − 1, ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 1 ) ρ1 ( z 2 )
(1)
where the quantities ρ 1 (z) and ρ 2 (z1 , z2 ) represent one and two particle densities respectively, which are defined as
1 dσ , σin dz1 1 d2 σ ρ2 ( z 1 , z 2 ) = , σin dz1 dz2
ρ1 ( z ) =
and where σ in ,
dσ dz1
and
d2 σ d z1 d z2
(2)
are the inelastic cross-section, the one and the two particle distribution, respectively, z1 and
z2 are the values of a certain physical quantity corresponding to the two particles. R2 can be represented as
R2 (z1 , z2 ) = NT
N2 (z1 , z2 ) −1 N1 (z1 )N1 (z2 )
(3)
where NT is the total number of inelastic events. N1 (z1 ) is the number of particles at the phase space interval of z1 and z1 + dz1 , and N1 (z2 ) is the number of particles at the phase space interval of z2 and z2 + dz2 . N2 (z1 , z2 ) is the number of pairs of particles having one particle between the phase space z1 and z1 + dz1 and the other particle between z2 and z2 + dz2 in an event. The correlation function measures the dependence of the production of the two particles on their own characteristic parameters. The positive value of R2 (z1 , z2 ) means that the presence of one particle with a parameter z1 compels the existence of the other within z2 . A zero value of R2 (z1 , z2 ) means that their emission is independent. On the other hand, a negative value of the correlation function means that the emission of particle at z1 excludes the emission of the other within z2 . If the phase space variable is taken as z = cos θ , θ is emission angle of a particle, the normalized inclusive correlation function can be written as
R2 (cos θ1 , cos θ2 ) = NT
N2 (cos θ1 , cos θ2 ) −1 N1 (cos θ1 )N1 (cos θ2 )
(4)
where N1 (cos θ 1 ) is the number of interactions having a particle between cos θ 1 and cos θ1 + d cos θ1 , and N2 (cos θ 1 , cos θ 2 ) is the number of interactions having at least two particles, the first with angle between cos θ 1 and cos θ1 + d cos θ1 , and the other of angle between cos θ 2 and cos θ2 + d cos θ2 . If the phase space variable is taken as azimuthal angle φ , the normalized inclusive correlation function can be written as
R2 (φ1 , φ2 ) = NT
N2 (φ1 , φ2 ) −1 N1 (φ1 )N1 (φ2 )
(5)
where N1 (φ 1 ) is the number of interactions having a particle between φ 1 and φ1 + dφ1 , and N2 (φ 1 , φ 2 ) is the number of interactions having at least two particles, the first with azimuthal angle between φ 1 and φ1 + dφ1 , and the other of azimuthal angle between φ 2 and φ2 + dφ2 . Experimentally, the two-particle correlation function in the emission angle phase space and azimuthal angle space are calculated as
R2 (cos θ1 , cos θ2 ) =
n1 (cos θ1 )n2 (cos θ2 ) − 1, for cos θ1 = cos θ2 n1 (cos θ1 )n2 (cos θ2 )
(6)
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and
n(cos θ )(n(cos θ ) − 1 ) − 1, for cos θ1 = cos θ2 = cos θ n(cos θ )2
R2 (cos θ1 , cos θ2 ) =
(7)
where n1 (cos θ 1 ) and n2 (cos θ 2 ) are the particle multiplicities in a small interval of δ (cos θ ) around cos θ 1 and cos θ 2 .
R 2 ( φ1 , φ2 ) =
n 1 ( φ1 ) n 2 ( φ2 ) − 1, for φ1 = φ2 n1 (φ1 )n2 (φ2 )
(8)
R 2 ( φ1 , φ2 ) =
n(φ )(n(φ ) − 1 ) − 1, for φ1 = φ2 = φ n ( φ )2
(9)
and
where n1 (φ 1 ) and n2 (φ 2 ) are the particle multiplicities in a small interval of δ (φ ) around φ 1 and φ 2 . The variance of R2 can be calculated by [36]
V [R2 ] =
1 [n21 n22 n1 2 n2 2 − 2n21 n2 n1 n2 n1 n2 2 NT n1 4 n2 4 −2n1 n22 n1 n2 n1 2 n2 + n21 n1 n2 2 n2 2 + n22 n1 n2 2 n1 2 +2n1 n2 3 n1 n2 − n1 n2 2 n1 2 n2 2 ] + O(1/N 2 )
(10)
for the case of cos θ 1 = cos θ 2 or the case of φ 1 = φ 2 , and
V [R2 ] =
1 [n41 n1 2 − 4n31 n21 n1 + 4n21 3 − n21 2 n1 2 NT n1 6 +2n31 n1 2 − 4n21 2 n1 + 2n21 n1 3 + n21 n1 2 − n1 4 ] + O(1/N 2 )
for the case of cos θ1 = cos θ2 or the case of φ1 = φ2 . in comparison with the other terms.
O(1/N2 )
(11)
is a polynomial which is negligible when calculating the errors,
Three particle correlation The three particle inclusive correlation function for phase space variable z can be defined as in Ref. [37],
R3 ( z1 , z2 , z3 ) =
ρ3 ( z 1 , z 2 , z 3 ) + 2 ρ 1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 ) ρ2 ( z 1 , z 2 ) ρ1 ( z 3 ) + ρ 2 ( z 2 , z 3 ) ρ1 ( z 1 ) + ρ2 ( z 3 , z 1 ) ρ1 ( z 2 ) − , ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 ) ρ1 ( z 1 ) ρ1 ( z 2 ) ρ1 ( z 3 )
(12)
where the quantity
ρ3 ( z 1 , z 2 , z 3 ) =
1
σin
d3 σ dz1 dz2 dz3
(13)
represents the three particle density. ρ 1 and ρ 2 represent the same quantities as before in the case of the two particle correlation. In terms of the number of particles, R3 (z1 , z2 , z3 ) can be written as
R3 (z1 , z2 , z3 ) = NT2
N3 (z1 , z2 , z3 ) N2 (z1 , z2 ) N2 (z2 , z3 ) N2 (z3 , z1 ) − NT − NT − NT + 2, N1 (z1 )N1 (z2 )N1 (z3 ) N1 (z1 )N1 (z2 ) N1 (z2 )N1 (z3 ) N1 (z3 )N1 (z1 )
(14)
where N3 (z1 , z2 , z3 ) is the number of triplets of particles having one particle in the interval z1 to z1 + dz1 , the other particle between z2 to z2 + dz2 and the third particle between z3 to z3 + dz3 in an event. For the phase space variable is taken as cos θ , the three particle correlation function is represented as
N3 (cos θ1 , cos θ2 , cos θ3 ) N2 (cos θ1 , cos θ2 ) + 2 − NT N1 (cos θ1 )N1 (cos θ2 )N1 (cos θ3 ) N1 (cos θ1 )N1 (cos θ2 ) N2 (cos θ2 , cos θ3 ) N2 (cos θ3 , cos θ1 ) − NT − NT . N1 (cos θ2 )N1 (cos θ3 ) N1 (cos θ3 )N1 (cos θ1 )
R3 (cos θ1 , cos θ2 , cos θ3 ) = NT2
(15)
The three particle emission angle correlation function is experimentally obtained as
R3 (cos θ1 , cos θ2 , cos θ3 ) =
n(n − 1 )(n − 2 ) 3n(n − 1 ) − + 2, for cos θ1 = cos θ2 = cos θ3 n 3 n 2
(16)
For the phase space variable is taken as azimuthal angle φ , the three particle correlation function is represented as
R3 (φ1 , φ2 , φ3 ) = NT2
N3 (φ1 , φ2 , φ3 ) N2 (φ1 , φ2 ) N2 (φ2 , φ3 ) N2 (φ3 , φ1 ) − NT − NT − NT + 2. N1 (φ1 )N1 (φ2 )N1 (φ3 ) N1 (φ1 )N1 (φ2 ) N1 (φ2 )N1 (φ3 ) N1 (φ3 )N1 (φ1 )
(17)
Similarly, the three particle azimuthal angle correlation function is experimentally obtained as
R 3 ( φ1 , φ2 , φ3 ) =
n(n − 1 )(n − 2 ) 3n(n − 1 ) − + 2, for φ1 = φ2 = φ3 n 3 n 2
(18)
The variance of this quantity is calculated term by term and instead of giving the long algebraic expression of the net variance. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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cosΘ Fig. 1. (Color online) The normalized two particle correlation function in emission angle space for target evaporated fragments.
4. Results and discussion There are 2975 12 A GeV 4 He-AgBr, 927 3.7 A GeV 16 O-AgBr, 521 60 A GeV 16 O-AgBr, 229 1.7 A GeV 84 Kr-AgBr and 619 10.7 A GeV 197 Au-AgBr interacting events are used in present investigation. The two and three particle normalized shortrange correlation function R2 (cos θ 1 , cos θ 2 ) and R3 (cos θ 1 , cos θ 2 , cos θ 3 ) for target evaporated fragments are calculated with the help of relation (7) and (16) respectively. Fig. 1 shows the plot of the two particle normalized correlation function R2 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ ) of the correlation matrices, characterizing the magnitude of the short-range correlation at different phase space intervals. The errors are calculated using Eq. (11). It is evident from Fig. 1 that the short-range correlation values are greater than zero (R2 > 0) in any value of cos θ space intervals for the most of the interactions which implies that the short-range two particle correlation exists over the entire space of emission angle. From the graph it is further seen that the correlation is prominent at cos θ space interval of −1.0 ≤ cos θ < −0.6 or at 0.6 ≤ cos θ < 1.0(values of the correlation functions in these intervals are greater than that in other intervals), it means that the correlation is more likely in the most forward or the most backward direction. Those observations are the same as the result observed in 14.5 A GeV 28 Si-AgBr interactions [30]. An exception is observed in the Fig. 1 that the two particle short-range correlation function in some cos θ space intervals are less than zero and the correlation is prominent at cos θ space intervals of −0.4 ≤ cos θ < 0.0 and 0.6 ≤ cos θ < 0.8 for the case of 60 A GeV 16 O-AgBr interactions. Fig. 2 shows the normalized three particle short-range correlation function R3 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ3 = cos θ ) of the correlation matrices for target evaporated fragments. The variance of R3 is calculated term by term in relation (16). The prominent three particle short-range correlations are in the cos θ space interval of −1.0 ≤ cos θ < −0.8 for most of the interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. The correlation can give direct information about the late stage of the reaction when nuclear matter is highly excited and diffused. According to the model proposed by Stocker et al. [23] using three-dimensional nuclear fluid dynamics, the emission of target fragments in the backward hemisphere can be explained with the help of the side splash phenomenon. In a nucleus–nucleus collision, a head shock zone may be developed during the dividing phase of the projectile nucleus within the target. A strongly compressed and highly excited projectile-like object continues to interpenetrate the target with supersonic velocity and may push the matter sideways. This results in the generation of shock waves that give rise to the correlation in the sideward directions. The two and three particle normalized short-range correlation function R2 and R3 for target recoil protons are calculated using the relation (7) and (16) respectively. Fig. 3 shows the plot of the two particle normalized short-range correlation function R2 against cos θ for the case of cos θ1 = cos θ2 = cos θ . It can be clearly seen from Fig. 3 that the short-range correlation values are greater than zero (R2 > 0) in most value of cos θ space intervals for all of the interactions which implies that there are strong short-range two particle correlation over the entire space of emission angle. The prominent two particle short-range correlations are in the cos θ space intervals of −1.0 ≤ cos θ < −0.8 for 1.7 A GeV 84 Kr- and 3.7 A GeV 16 O-AgBr Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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cosΘ Fig. 3. (Color online) The normalized two particle correlation function in emission angle space for target recoil protons.
interactions, but in this region the production of target recoil protons is not too much and the error of correlation function is greater. Fig. 4 shows the normalized three particle short-range correlation function R3 against cos θ for the values of the diagonal elements (cos θ1 = cos θ2 = cos θ3 = cos θ ) of the correlation matrices for target recoil protons. The variance of R3 is calculated term by term in relation (16). The prominent three particle short-range correlations are in the cos θ space intervals of 0.8 ≤ cos θ < 1.0 and intervals of −0.4 ≤ cos θ < 0.0 for most of the interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. Generally speaking, the regions for the prominent three particle short-range correlations should be consistent with that for the prominent two particle short-range correlations. It is true from the comparison of Figs. 1–4 for most of interactions. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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1.2 1 0.8
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Φ Fig. 5. (Color online) The normalized two particle correlation function in azimuthal angle space for target evaporated fragments.
For the cases the prominent three particle short-range correlations with great statistical errors, the two particle short-range correlations are not prominent. The two and three particle normalized short-range correlation function R2 (φ 1 , φ 2 ) and R3 (φ 1 , φ 2 , φ 3 ) for target evaporated fragments are calculated with the help of relation (9) and (18) respectively. Fig. 5 shows the plot of the two particle normalized correlation function R2 against φ for the values of the diagonal elements (φ1 = φ2 = φ ) of the correlation matrices, characterizing the magnitude of the short-range correlation at different phase space intervals. The errors are calculated using Eq. (11). It is evident from Fig. 5 that the short-range correlation values are greater than zero (R2 > 0) in most of φ space intervals which implies that the short-range two particle correlation exists over the entire space of azimuthal angle except for 60 A GeV 16 O-AgBr interaction at azimuthal angle region of 0° ≤ φ < 36°. From the graph it is further seen that the correlation is prominent at φ space interval of 180° ≤ φ < 216° for 10.7 A GeV 197 Au-AgBr, 12 A GeV 4 He-AgBr and Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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Φ Fig. 7. (Color online) The normalized two particle correlation function in azimuthal angle space for target recoil protons.
3.7 A GeV 16 O-AgBr interaction, of 324° ≤ φ < 360° for 1.7 A GeV 84 Kr-AgBr interaction, and of 108° ≤ φ < 144° for 60 A GeV 16 O-AgBr interaction. Fig. 6 shows the normalized three particle short-range correlation function R3 against φ for the values of the diagonal elements (φ1 = φ2 = φ3 = φ ) of the correlation matrices for target evaporated fragments. The variance of R3 is calculated term by term in relation (18). The prominent three particle short-range correlations is in the φ space interval of 180° ≤ φ < 216° for 12 A GeV 4 He-AgBr, 3.7 A GeV 16 O-AgBr and 10.7 A GeV 197 Au-AgBr interactions, and of 324° ≤ φ < 360° for 1.7 A GeV 84 Kr-AgBr interactions. For the other cases the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. The two and three particle normalized short-range correlation function R2 and R3 for target recoil protons in azimuthal angle space are calculated using the relation (9) and (18) respectively. Fig. 7 shows the plot of the two particle normalized Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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Φ Fig. 8. (Color online) The normalized three particle correlation function in azimuthal angle space for target recoil protons.
short-range correlation function R2 against φ for the case of φ1 = φ2 = φ . It can be clearly seen from Fig. 7 that the shortrange correlation values are greater than zero (R2 > 0) in all of the φ space intervals for all of the interactions which implies that there are strong short-range two particle correlation over the entire space of azimuthal angle. The prominent two particle short-range correlations is in the φ space interval of 252° ≤ φ < 288° for 10.7 A GeV 197 Au-AgBr, 60 A GeV 16 O-AgBr and 3.7 A GeV 16 O-AgBr collision, of 180° ≤ φ < 216° for 1.7 A GeV 84 Kr-AgBr collision, and of 72° ≤ φ < 108° for 12 A GeV 4 He-AgBr collision. Fig. 8 shows the normalized three particle short-range correlation function R3 against φ for the values of the diagonal elements (φ1 = φ2 = φ3 = φ ) of the correlation matrices for target recoil protons. The variance of R3 is calculated term by term in relation (18). It can be seen from Fig. 8 that the short-range correlation values are greater than zero (R3 > 0) in most of the φ space intervals for all of the interactions which implies that there are strong short-range three particle correlation in most of the azimuthal angle space. The prominent three particle short-range correlations is in the φ space interval of 252° ≤ φ < 288° for 10.7 A GeV 197 Au-AgBr and 60 A GeV 16 O-AgBr collisions, of 36° ≤ φ < 72° for 1.7 A GeV 84 Kr-AgBr collision, of 72° ≤ φ < 108° for 12 A GeV 4 He-AgBr collision, and of 144° ≤ φ < 180° for 3.7 A GeV 16 O-AgBr collision. For the cases of the results of three particle short-range correlation function R3 are close to zero within experimental error, it means that there is not a three particle short-range correlation. In the study of correlations among the target fragments by means of correlation functions, pseudo-correlations may arise from the broad multiplicity distribution and the dependence of the one-particle spectrum on multiplicity, as well as the trivial correlations due to kinematical constraints in individual events. It has been emphasized by Ghosh et al. [38] that the precise model-independent search for dynamical correlations seems to be impossible in hadron–nucleus and nucleus– nucleus collisions. To search for the non-trivial dynamical correlation among the target fragments, the experimental data have to be compared with those calculated from Monte Carlo simulation based on the framework of independent emission hypothesis. The emission of particles from a collectively excited state of the nucleus can be understood by Fermi liquid drop model in which the emission of particles can occur in two ways. First, due to the relaxation processes where the collective energy is transferred to the intrinsic degrees of freedom with subsequent evaporation of particles. Second, a direct nonstatistical emission (splashing) of nucleons is also possible via the dynamical distortion of the Fermi surface accompanying the collective motion. In general, the relative contributions of these mechanisms depend upon the magnitude of the nuclear friction coefficient. Particles emitted from excited nucleus due to both the evaporation and the splashing are due to the collective motion of the nuclear Fermi liquid and are accompanied by direct non-statistical emission of nucleons via the dynamical distortion of the Fermi surface, so they are responsible for the presence of the short range correlation. 5. Conclusions The two and three particle short-range correlation in emission angle space for both target evaporated fragments and target recoil protons produced in 12 A GeV 4 He-, 3.7 A GeV 16 O-, 60 A GeV 16 O-, 1.7 A GeV 84 Kr- and 10.7 A GeV 197 Au-AgBr interactions have been investigated in emission angle space and azimuthal angle space respectively. Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001
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In emission angle space it is found that the evidence of strong two particle short-range correlation exist, and the prominent two particle short-range correlations are in the most forward or backward emission angle space, for both target evaporated fragments and target recoil protons. A weak three particle short-range correlation is observed in the cos θ space intervals of −1.0 ≤ cos θ < −0.8 for target evaporated fragments, but the error is greater. The observation of two and three particle short-range correlation in emission angle space can not explained by evaporation model and favor of the side splash model. In azimuthal angle space it is found that the evidence of both two and three target recoil proton short-range correlation exist, for target evaporated fragment the strong two particle correlation and a weak three particle short-range correlation is observed. The observation of two and three particle short-range correlation can not explained by evaporation model. Acknowledgment This work has been supported by the Chinese National Science Foundation under Grant nos: 11075100 and 11565001, the Natural Foundation of Shanxi Province under Grant 2011011001-2, the Shanxi Provincial Foundation for Returned Overseas Chinese Scholars, China (Grant no. 2011–058). We are grateful to EMU-01 collaboration for supplying emulsion stacks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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Please cite this article as: R. Xu, D.-H. Zhang, Two and three particle correlations in target fragmentation at relativistic nucleus–nucleus collisions, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.07.001