On Two-Ridge Structure in Two-Particle Azimuthal Correlations in Proton-Lead Collisions at Large Hadron Collider Energy

Two-particle azimuthal correlations are studied in the framework of a multisource thermal model. Each source is assumed to produce many particles. Each particle pair measured in final state is considered to be produced at two emission points (subsources) in a single or two sources. The first emission point corresponds to the production of “trigger” particle and the second one corresponds to that of “associated” particle. There are oscillations and other interactions between the two emission points. In the rest frame of the “associated” particle’s emission point, the oscillations and other interactions cause the momentum of the “trigger” particle to depart from the original value.Themodelling results are in agreement with the experimental data of proton-lead (p-Pb) collisions at√sNN = 5.02 TeV, one of the Large Hadron Collider energies, measured by the ALICE and ATLAS Collaborations.


Introduction
Two-particle correlations are important experimental phenomenons in high energy collisions.These correlations contain azimuthal correlations, (pseudo)rapidity correlations, momentum correlations, and so on.As a type of long-range correlation, two-particle azimuthal correlations were studied in recent years.Particularly, at the eras of the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), particle and nuclear physicists have been obtaining some interesting results on the azimuthal correlations in proton-proton (), proton-nucleus, and nucleus-nucleus collisions.Some features are observed in the three types of collisions.
For example, the first studies of two-particle azimuthal correlation function in highest-multiplicity  collision at the LHC present an enhancement of particle pair production at relative azimuth Δ ≈ 0, which results in a "ridge" structure at the "near-side" [1].However, in peripheral proton-lead (-Pb) collisions in which there are only a few nucleons to take part in the collisions and which are similar to  collision, a ridge structure is observed at the "away-side" (Δ ≈ ) [2].The different results between  and peripheral -Pb collisions are possibly caused by the infection of nuclear effects (spectator nucleons).In central -Pb collisions, a double ridge structure is observed, which is consistent with Pb-Pb collisions [3].
To explain the near-side and away-side ridges, many physics mechanisms are proposed in literature, such as the parton saturation [4,5], gluon saturation and color connections [6][7][8][9][10][11], parton-induced interactions [12][13][14], multiparton interactions [15], collective expansion of the final state [16], and collective effects arising in a high-density system [17][18][19][20][21][22][23].If we regard a parton as an emission point and multiple emission points are assumed to form a source, the high energy collisions can be treated by a multisource interacting system.Thus, a multisource thermal model [24][25][26] can be considered in the studies of near-side and awayside ridges in the two-particle azimuthal correlations.The effects listed here are not completely included in this study, but the collective effect is included.
In this paper, in the framework of the multisource thermal model [24][25][26], we consider two emission points (subsources) in a single or two sources.The two emission points are assumed to emit respectively the "trigger" particle and the "associated" particle in a particle pair.After considering oscillations and other interactions between the two emission points, the azimuthal correlations are studied.

The Model and Method
In the multisource thermal model [24][25][26], we assume that many emission points (subsources) are formed in high energy collisions.These subsources can form a few sources which stay in local equilibrium sates.The Maxwell, Boltzmann, and other distributions can be used to describe spectrums of final-state particles in which productions of particle pairs are included.For a particle pair measured in final state, both the two emission points take part in the production process.The two emission points may be in a single or two sources.The first emission point corresponds to the production of "trigger" particle and the second one corresponds to the production of "associated" particle.There are oscillations between the two emission points, which results in the momentum of produced particle to depart from the original one.In the meantime, the multiple partons interactions [15], collective expansion of the final state [16], and other collective effects [17][18][19][20][21][22][23] can affect the momentum spectrums of final-state particles.Because the total physical extent of the interacting system is about 2 times the nuclear size, the average extent of a source is only a few fm in the case of considering two or three sources (temperatures).
Let the beam direction be the  axis, and let the transverse momentum direction of the "associated" particle be the positive direction of the  axis.A right-hand reference frame is established.In the rest frame of the second emission point, let   and   denote, respectively, the and -components of the "trigger" particle momentum with oscillations and other interactions, and let    and    denote those without oscillations and other interactions.The simplest relations between   and    , as well as   and    are linear: where  , and  , are parameters to characterize the strength of oscillations, and  is a parameter to characterize the distribution width of original momentum components.The default values of  , and  , are 1 and 0, respectively.From the original   , to the final-state  , , a detailed consideration on the relativistic effect needs Lorentz transformation.Although there is no particular consideration on the relativistic effect in the above equations for the purpose of convenience, the present expression reflects approximately the mean result of the relativistic effect [24].Meanwhile, the conservation laws are applicable in the interacting system.We would like to point out that in the above consideration both the "trigger" and "associated" particles are firstly assumed to emit isotropically from two subsources which have no oscillations and other interactions at this assumption state.Then, the momentum components are transformed from original   , to final-state  , due to the oscillations and other interactions between the two subsources.
As the first approximation, the original momentum components   , are assumed to obey Gaussian distribution with the width , which results in the transverse momentum, momentum, and nonrelativistic kinetic energy spectrums to be Rayleigh, Maxwell, and Boltzmann distributions, respectively.In the Monte Carlo calculation, let  1,2,3,4 be random numbers distributed evenly in (0, 1].We have which distributes the azimuth of the particle randomly.The relative azimuth between the "trigger" and "associated" particles is due to   = 1 and   = 0 for the near-side and away-side ridge structures in general.A statistical calculation can give the normalization distribution (Δ) of Δ.
We would like to point out that (3) simply calculates the  of the "trigger" particle using  = arctan(  /  ).Although implied in the description of the coordinate system, it might be specified reiterated that Δ =  trig −  assoc =  trig due to  assoc = 0 (the transverse momentum direction of the "associated" particle being the positive direction of the  axis) can be represented using (3), where  trig and  assoc denote the azimuths of "trigger" and "associated" particles, respectively.Then, (3) uses ( 1) to obtain Δ.In addition, because we focus our attention on the relative magnitudes of   and   (  and   ), we may choose   = 1 (  = 0) or   = 1 (  = 0).Conventionally, we have   = 1 and   = 0.
The physics condition gives that   ≥ 1.Generally,   = 1 and   = 0 describe the state without oscillations and other interactions.  > 1 reflects an expansion of the subsource along  axis in the momentum space.  > 0 and   < 0 present respectively a near-displacement and an awaydisplacement of the first subsource to the second one along the  axis.The near-side and away-side phenomenons are partly determined by   > 0 and   < 0, respectively, and partly determined by   > 1.
Before giving comparisons with experimental data, we need to introduce two representations which are used in the literature [2,3,27].The first representation uses unidentified charged tracks as "trigger" particles and combines them with  and  as "associated" particles (denoted by ℎ −  and ℎ − , resp.)[2,3,27].The correlation (Δ, Δ) is expressed in terms of the "associated" yield per "trigger" particle where both particles are from a given transverse momentum (  ) interval and pseudorapidity () region: where  trig and  assoc are the numbers of "trigger" and "associated" particles, respectively, and (Δ, Δ) and (Δ, Δ) are the signal and background distributions [3,27] and constructed from the same event and "mixed events, " respectively Because the background (Δ) constructed from the isotropic "mixed events" in our model is a constant, we have where  1 is the normalization constant and the signal (Δ) ≡ (Δ).
To perform the calculation, we probe a set of   and   and use (3) to give a lots of Δ by using many random numbers.Then, the normalized distribution (Δ) which is normalized to 1 can be obtained.Introducing (Δ) into (6) (or (8)) which is normalized to the experimental cross section or yield, the parameter  1 (or the parameters  2 and  0 ) can be sounded out.By changing the parameters step by step, many repeating calculations can determine the best parameters and their uncertainties.In the real calculation, we can use the idea of the least-square fitting method.The minimum  2 corresponds to the best parameter values, and the acceptable  2 determines the uncertainties of the parameters.uncertainties can be determined.From the table, one can see that the model describes well the experimental data of the ALICE Collaboration.The subsource has an expansion and an away-displacement along the  axis.

Comparison with Experimental Data and Discussions
Figure 2 shows (Δ)-|Δ| relations in -Pb collisions at √  = 5.02 TeV.The closed and open circles represent, respectively, the central (denoted "" in the panel) and peripheral collisions (denoted "" in the panel) measured by the ATLAS Collaboration [2], and the curves are our results calculated by the model.From Figures 2(a) to 2(f), different transverse momentum intervals for the "trigger" particle (   ) and for the "associated" particle (   ) are shown in the panels.The values of related parameters   ,   ,  2 , and  0 as well as  2 /dof are listed in Table 1.Once more, the model describes well the experimental data of the ATLAS Collaboration.In most cases, the subsource in central collisions has a larger expansion and a smaller away-displacement along the  axis, while the subsource in peripheral collisions is opposite.These differences between the central and peripheral collisions are rendered by the number of participant nucleons.
The difference ( − ) on (Δ)-|Δ| relations between the central and peripheral -Pb collisions at √  = 5.02 TeV is shown in Figure 3.The circles represent the experimental data of the ATLAS Collaboration [2] and the curves are our results calculated by the model.The values of related parameters and  2 /dof are given in Table 1.We see that the model describes the the difference between the central and peripheral collisions.The values of   (=1.04) and   (≈0) obtained from Figure 3 are consistent with those obtained from Figures 1(b) and 1(c).Because there is no direct proportional relation between the parameter values and the correlation magnitude, we cannot obtain simply the parameter values for Figure 3 by the method of central minus peripheral collisions.
From the above comparisons we see that we have used the same method to describe the central, peripheral, and central-peripheral collisions.Different participant nucleon numbers (different spectator nucleon numbers) in central and peripheral collisions reflect different parameter values.As an integrative result, the oscillation and other interactions existing between the two subsources cause   to be greater than 1 and   to be less than 0. Generally,   in central collisions is greater than that in peripheral collisions, and |  | in central collisions is less than that in peripheral collisions, due to the effect of participant nucleon number.By comparison with Figures 1(a) and 2, as the difference between the central and peripheral collisions, Figures 1(b), 1(c), and 3 do not contain new content but the effect of participant nucleon number.
In the comparisons, because our calculation is performed on a purely signal sample, and experimental data are usually presented by a admixture of signal and background samples, we need the normalized constants  1 and  2 .Our results show that  1 > 1 and  2 > 1 for both the central and peripheral collisions, which render that the magnitude of signal is greater than that of background.There are strong correlations between two particles in the production in -Pb collisions at LHC energy.collisions.These phenomenons may be caused by different numbers of participant nucleons in peripheral and central collisions.

Looking at the results in
In heavy ion collisions, the current consensus is that a primary component of the ridge effect is caused by fluctuations in the initial state geometry with a major contribution from "triangular flow, " which generates a significant third Fourier coefficient contribution to the azimuthal correlations.These may be the reasons which cause oscillations and other interactions between the two subsources.In fact, other reasons can cause similar results.
High energy collisions contain abundant contents such as multiplicity, transverse energy, entropy [33], phase transition [34], and flow effects [35].There are strong relations between flow effects and azimuthal distributions and correlations.Particularly, the azimuthal distribution and correlations of produced particles, target fragments, and projectile fragments have important worth of studies.The present work can be referenced in further.

Conclusions
From the above discussions, we obtain following conclusions.
(a) The near-side and away-side ridge structures in two-particle azimuthal correlation produced in high energy collisions can be explained by the multisource thermal model.The modelling results are in agreement with the experimental data of -Pb collisions at √  = 5.02 GeV measured by the ALICE and ATLAS Collaborations.This renders that our modelling assumption is correct.The two correlated particles are initially assumed to produce isotropically in two rest subsources.Then, the momentum components are transformed from original one to final-state one due to the oscillations and other interactions between the two subsources.
(b) Two emission points (subsources) are assumed and used to perform the calculation.One subsource corresponds to the production of "trigger" particle, and the other subsource corresponds to the production of "associated" particle.There are oscillations and other interactions between the two subsources, which results in the momentum of "trigger" particle, in the rest frame of "associated" particle's source, to depart from the original one.
(c) There are two main parameters,   and   , in the modelling calculation on correlations.  = 1 and   = 0 describe the state without oscillations and other interactions.  > 1 reflects an expansion of the subsource along the  axis in the momentum space.  > 0 and   < 0 present, respectively, a near-displacement and an away-displacement of the "trigger" particle's subsource to the "associated" particle's subsource along the  axis.The magnitude of near-side and away-side ridges is partly determined by   > 0 and   < 0, respectively, and partly determined by   > 1.
(d) In central and peripheral -Pb collisions at the LHC energy, our modelling results show that   > 1 and   < 0. In most cases, the subsource in central collisions has a larger expansion and a smaller away-displacement along the  axis, while the subsource in peripheral collisions is opposite.The difference between the central and peripheral collisions shows a very small   (≈0) which means a nearly zero displacement of the subsource.The values of   and   extracted from the central and difference ( − ) data of the ALICE and ATLAS Collaborations are consistent with each other.
(e) The present model describes the central and peripheral -Pb collisions by using a uniform method.Although there are different participant nucleon numbers (or spectator nucleon numbers) between the central and peripheral collisions, the interacting mechanisms which include the oscillations and other interactions between the two subsources are the same except the intensity.The treatment on the difference  −  does not introduce new content but the effect of participant nucleon number.

Figure 1
Figure 1 presents the correlations versus Δ in transverse momentum interval 1.5 <   < 2.0 GeV/c in -Pb collisions at √  = 5.02 TeV, one of the LHC energies.The circles represent the experimental data of the ALICE Collaboration[3,27] and the curves are our results calculated by the multisource thermal model.Figures1(a), 1(b), and 1(c) correspond to ℎ −  in centrality 0-20%, ℎ −  in centrality (0-20%)-(60-100%), and ℎ −  in centrality (0-20%)-(60-100%), respectively.The values of free parameters   and   , normalization constant  1 , and  2 per degree of freedom ( 2 /dof) obtained by fitting the experimental data are listed in Table1.In the Monte Carlo calculation, we have used the idea of least-square fitting method.Many tries on the calculation have been applied to get the minimum and acceptable  2 .Then, the best parameter values and their

Figure 2 :
Figure 2: The relation (Δ)-|Δ| in -Pb collisions at √  = 5.02 TeV.The closed and open circles represent, respectively, the central and peripheral collisions measured by the ATLAS Collaboration[2], and the curves are our results calculated by the multisource thermal model.From Figures2(a) to 2(f), different transverse momentum intervals for the "trigger" particle (   ) and for the "associated" particle (   ) are shown in the panels.

Figure 3 :
Figure 3: The same as Figure 2, but showing the difference between the central and peripheral -Pb collisions at √  = 5.02 TeV.

Table 1 :
Values of parameters and  2 /dof corresponding to the curves in Figures1-3.
[2,28].If we integrate over Δ in the above equation or if Δ is a small value in general, we have

Table 1 ,
the source with oscillations becomes less displaced in peripheral collisions compared to central collisions.The standard deviation of the momentum components also becomes larger in peripheral Advances in High Energy Physics