Study of the cumulative number distribution of charged particles produced in He12C interactions at 4.2 A GeV/c
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The behavior of the cumulative number distribution along with maximum values of cumulative number of all charged particles (i.e. protons, π±-mesons) produced in He12C-interactions at 4.2 A GeV/c from experimental and simulation data coming from cascade evaporation model has been analyzed in this paper. One could see four different regions in the distributions of cumulative number for protons in which the last region has cumulative number greater than one which correspond to cumulative region. Similarly one could see two different regions for negative and positive pions respectively but there is totally absence of the cumulative area for both positive and negative pions. The simulation data obtained from Cascade code could reproduced agreeably the distributions of the cumulative number for protons as well as for maximum values of cumulative number distribution.
KeywordsCumulative number distribution Nucleus–nucleus interactions protons Mesons Hadron–nucleus interactions
PACS Nos.12.38.Mh 12.40.-y 13.60.Rj 13.75.Cs 13.85.-t
In hot and dense matter [1, 2, 3], the collective behavior for the partons have been observed through experimental results from ultra-relativistic heavy ion collisions. The JINR Cumulative effect  at relativistic hadron–nuclear and nuclear–nuclear interactions and CERN EMC (European Muon Collaboration) effect  in deep inelastic scattering (DIS) experiment could be taken as experimental evidence of nucleon collective phenomenon in the medium.
First signal of a collective behavior for inner nuclear nucleons have been observed through the JINR Cumulative effect . It led to the notion for generation of particles with energies outside the kinematic limit of free nucleon collisions . The effect has been explained in the paper  and some interesting points of these were enumerated in paper  particularly: (1) Energies of pions were observed ~ 8 GeV in D + A reactions at 5 A GeV; (2) In reactions B + A → C + X, the particles C were produced with x > 1, where values of x can be defined, in terms of Mandelstam invariants u and s, as x = u/s ≈ (1/m)(ε − p cosθ), here, m, ε, p and θ are the mass of the C particle, its total energies, 3 momentum and emission angle in the lab frame respectively. The value of x is limited by 1 for free nucleon collisions, whereas energetic particles were emitted with x > 1 in hadron–nuclear interactions at JINR.
Peculiar properties have been observed in the JINR cumulative effect: (1) it has been observed for photon-nuclear; lepton–nuclear; hadron–nuclear and nuclear–nuclear interactions. (2) a strong A-dependences were shown for the invariant inclusive cross sections of the cumulative particles (f (p) = ε dσ/dp).
Theoretical interpretation of the effect suggested that it is a consequence of nucleon collective phenomena and the cumulative particles could be generated from the collected system of nucleons-coherent groups of nucleons. Cumulative particles could be produced as a result of fluctuations of nuclear density , the interaction of the projectile with target nucleons, and nucleon percolation .
Thus, the JINR cumulative effect can be considered as a phenomenon of the nucleons collective behavior and coherent interactions.
CERN EMC (European Muon Collaboration) effect has confirmed the JINR cumulative effect . EMC studied the muon deep inelastic scattering on iron and deuterium. They detected big disagreement between experimental result and theoretical expectations. The experimental result shown that the structure function F2 and quark–gluon distributions of a nucleon bound in a nucleus differ from those of a free nucleon. Popular models could not explain the EMC effect and present a new point of view on the effect as a simple relativistic phenomenon . The effect can also be expressed as a result of nucleon collective phenomena and coherent interactions. Coherent Tube Model (CTM)  could provide clear understanding for production of energetic (cumulative) particle. Here the interaction of a hadron with a target nucleus considers as its simultaneous collision with the tube of nucleons of cross section σ that lie along its path to the target nucleus. For the interaction of projectile with momentum plab the cumulative square of the center-of-mass energy is si ≅ 2implab (i is a number of nucleons, m–a nucleon mass). Unusually strong A dependence (stronger than the commonly assumed A or A2/3) of the cross section has been quantitatively described in paper  for p + A → J/Ψ + X reaction at the incident energies less than 30 GeV, using cumulative effects (via energy rescaling).
The CTM has been discussed in paper  for high energy nucleus–nucleus collisions.
In this case, two tubes have been considered with: i1 and i2 nucleons in incident and target tubes respectively. The c.m. energy squared is approximately given by ~ si1i2 ≅ 2i1i2mplab for this tube–tube collision.
Cumulative particles could provide essential knowledge on collective phenomenon inside the nucleus and on dynamics of coherent particle productions. Though cumulative effect have been studied for many years there are only a few papers  where the effect studied in 4π conditions of measurements. On other hand this condition is necessary to get the information on the correlations in the cumulative events. The main objective of the paper is to study the cumulative effect in conditions of 4π geometry measurements.
2 Experimental detail
Experimental data obtained from 2m propane (C3H8) bubble chamber, of the Laboratory of High Energy (LHE), JINR [15, 16] in Dubna, Russia, has been used for analysis in this paper. Chamber was kept in the magnetic field strength of 1.5 Tesla and then exposed to the beams of helium nuclei accelerated to a momentum of 4.2 A GeV/c in the JINR Synchrophasotron . In 4π geometry, most of all charged particles having energy greater than the threshold energy for the formation of track, were detected. Identification of charged particle  has been carried out based on the track curvature, the momentum of the particle and the magnetic field in the chamber. Each particle of certain mass possessed minimum momentum to shape the particular track. The pion with momentum below 70 MeV/c cannot produce a visible track. All negative particles have been classified as π− mesons except indentified electrons. If the momenta of the charged particle is less than 1.0 GeV/c then positive pions could be identified . The π+ mesons are distinguished from protons by ionization produced in the chamber in the momentum region less than 0.5 GeV/c. Identification momentum range for protons is 0.15–0.5 GeV/c and above this momentum the π+ mesons has contamination with protons [20, 21]. We have created the cumulative number spectrum for the secondary charged particles. For the cumulative number we have used the variable ‘x’ as defined in the introduction. The charged particles comprise of identified protons and π±-mesons. The x spectrum of the protons and positive and negative pions were created separately too. The x-distribution for charged particles with maximum values of the x in an event is also considered. Relevant experimental data have been compared with stimulated data generated by using the Dubna version of the cascade evaporation model  and exponential function y = a (e−bx) was used for fitting, where a and b are free parameters of fitting and e is the base of the natural logarithms.
3 Cascade evaporation model (CEM)
The model is based on Monte-Carlo simulation which has been described in somewhere else where scattering is important [21, 22, 23]. The model has been used for explaining the common prospects of nucleus–nucleus and hadron–nucleus collisions at higher energies. As result of interaction of secondary particles, the newly produced particles may produce supplementary particles . The process continues till either these are absorbed in the target medium or exiting from the target medium. The excited nuclei that are unable to produce new particles after the cascade stage are then described by the statistical theory in the evaporation approximation. The basic assumptions and procedures of the cascade model are presented in papers [23, 24].
4 Results and discussion
The behavior of the cumulative number distribution for the charged particles (proton, π±-mesons) produced in the He12C-interactions at 4.2 A GeV/c along with particles with maximum values of the cumulative number in an event too, has been studied. One has got: (1) four different regions in the cumulative number distributions are observed for all charged particles and protons and the last region is corresponding to cumulative number values greater than one; (2) for both positive and negative pions, there are two regions; (3) Cascade model cannot explain agreeably the distributions of the cumulative protons and cumulative π±-mesons, it gives less number for the mentioned particles. However for particles with maximum values of cumulative number, cascade model can describe the behavior of cumulative number distribution well for protons and for positive pions.
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