Angular Dependence of φ Meson Production for Different Photon Beam Energies

More than 20 years ago, strangeness enhancement was predicted to be a unique signature of the quark-gluon plasma (QGP) formation in high energy collisions [1]. So far, many experimental results from the Super Proton Synchrotron (SPS) andRelativisticHeavy IonCollider (RHIC) have shown the enhancement of strange particles [2]. At low photon energies, strangeness production starting from threshold is also helpful to understand underlying photoproduction mechanisms because it is far below the region of perturbative QuantumChromodynamics (QCD) and is an energy domain well above the region controlled by low energy theorems [3– 5]. As a particle with hidden strangeness, φ meson can be produced without an associated hyperon. Recently, in the reaction γp → pφ(KSKL), a photoproduction cross-section of the φ meson in its neutral decay mode was measured by the CLAS Collaboration. The experiment is conducted by a tagged photon beam of energy 1.65 ≤ Eγ ≤ 3.6GeV on a liquid hydrogen target at the Thomas Jefferson National Accelerator Facility (TJNAF) [6]. In order to explain the abundant experimental data, some phenomenological models of initial-coherent multiple interactions and particle transport were proposed and developed in recent years [7–12]. The dynamics of the system evolution has been studied by the azimuthal anisotropy of final-state particles produced in high energy collisions. In our previous work [13], the multisource thermal model has been used to describe the elliptic flows of final-state hadrons produced in nucleus-nucleus collisions at RHIC energies. It involves the anisotropic expansion of the participant area in the transverse momentum space. The model is successful in the description of (pseudo)rapidity andmultiplicity distributions of produced particles [14]. In the present work, we focus our attention on the dependence of φ meson photoproduction on the polar angle. A purpose of this paper is to check whether themodel can fit the angular dependence ofφmeson photoproduction. The paper is organized as follows: in Section 2, the multisource thermal model is introduced; in Section 3, we present our results, which are compared with the experimental data; at the end, we provide discussions and conclusions in Section 4.


Introduction
More than 20 years ago, strangeness enhancement was predicted to be a unique signature of the quark-gluon plasma (QGP) formation in high energy collisions [1].So far, many experimental results from the Super Proton Synchrotron (SPS) and Relativistic Heavy Ion Collider (RHIC) have shown the enhancement of strange particles [2].At low photon energies, strangeness production starting from threshold is also helpful to understand underlying photoproduction mechanisms because it is far below the region of perturbative Quantum Chromodynamics (QCD) and is an energy domain well above the region controlled by low energy theorems [3][4][5].As a particle with hidden strangeness,  meson can be produced without an associated hyperon.Recently, in the reaction  → (    ), a photoproduction cross-section of the  meson in its neutral decay mode was measured by the CLAS Collaboration.The experiment is conducted by a tagged photon beam of energy 1.65 ≤   ≤ 3.6 GeV on a liquid hydrogen target at the Thomas Jefferson National Accelerator Facility (TJNAF) [6].
In order to explain the abundant experimental data, some phenomenological models of initial-coherent multiple interactions and particle transport were proposed and developed in recent years [7][8][9][10][11][12].The dynamics of the system evolution has been studied by the azimuthal anisotropy of final-state particles produced in high energy collisions.In our previous work [13], the multisource thermal model has been used to describe the elliptic flows of final-state hadrons produced in nucleus-nucleus collisions at RHIC energies.It involves the anisotropic expansion of the participant area in the transverse momentum space.The model is successful in the description of (pseudo)rapidity and multiplicity distributions of produced particles [14].In the present work, we focus our attention on the dependence of  meson photoproduction on the polar angle.A purpose of this paper is to check whether the model can fit the angular dependence of  meson photoproduction.
The paper is organized as follows: in Section 2, the multisource thermal model is introduced; in Section 3, we present our results, which are compared with the experimental data; at the end, we provide discussions and conclusions in Section 4.

Angular Distribution in the Multisource Thermal Model
According to the multisource model [14], some emission sources of  mesons are assumed to be formed in the reaction process.Each source is considered to emit particles isotropically in a source rest frame.Let the incident beam direction be -axis and let the reaction plane be the  plane, which is scanned by the vector of the beam direction and impact parameter.In the source rest frame, the momentum components    ,    , and    of a considered particle have the same Gaussian distributions with a width .Then, a transverse momentum    = √ 2  +  2  obeys a Rayleigh distribution.
Due to the interactions among the emission sources, the sources will depart from isotropic emissions at the direction of .The corresponding momentum component   of the particle is where   and   indicate a deformation and movement of the emitted source, respectively.As a standard treatment in the Monte Carlo calculation, the random variable   and   are where  1 ,  2 , and  3 are random variables distributed in [0, 1].So the polar angle is given by Generally speaking,   > 1 denotes the expansion of the emission source along the -axis and   > 0 or   < 0 denotes the movement of the source along the positive or negative axes.

Comparison with the CLAS Results
The cos  dependences of  meson differential cross-section / cos  over a photon beam energy range from 1.6 GeV to 2.8 GeV is presented in Figure 1, which is plotted for 0.1 GeV photon energy bins.The symbols denote the experimental data of the photoproduction cross-section of the  meson produced in its neutral decay mode in the reaction  → (    ).The / cos  increases with cos  cm for the photon energy range 1.65 ≤   ≤ 1.95GeV.The solid lines denote the results calculated by the multisource model.One can see that the results are in agreement with the experimental data in the whole observed cos  region for the four energy bins.The values of   and   are obtained by fitting the data.The   increases linearly with the incident photon energies.In order to see the structure of the sources corresponding to different parameter values, the expansions and deformations of the modeling source are shown in Figure 6(a).
In Figure 2, we show the  meson differential crosssection / cos  as a function of cos  cm .The symbols indicate the experimental data for the photon beam energy bins 2.05 ≤   ≤ 2.35 GeV.The solid lines indicate the results calculated by the multisource thermal model.In the calculation, we take the different expansion and Advances in High Energy Physics deformation sources as marked in Figure 6(b).Figures 3,  4, and 5 are detailed comparisons between the results and the experimental data for 2.45 ≤   ≤ 2.75 GeV, 2.85 ≤   ≤ 3.15 GeV, and 3.25 ≤   ≤ 3.55 GeV, respectively.The results are also in agreement with the data from the CLAS Collaboration.The corresponding anisotropic sources are given in Figures 6(c)-6(e).
The parameters   and   obtained in the above comparisons are given in Figure 7.The value of   is around 1.5 and has no obvious regularity.Except for a saturation region, the values of   exhibit a linear dependence on the photon beam energies, which are best fitted by the function   = (3.02± 0.04)  − (4.5 ± 0.08).The movement of the sources along the positive  direction increases approximately with increasing photon beam energy.

Discussions and Conclusions
In the framework of the multisource thermal model, we have investigated the differential cross-section versus the polar angle of  meson photoproduction on hydrogen in the neutral decay mode.The incident photon beam energy range is 1.65-3.55GeV.The results calculated in the model approximately agree with the data of the CLAS Collaboration.In the above discussions, the deformation and movement of the sources are obtained by fitting the experimental data.The local sources of the final-state particles are formed in the reaction plane.Their interactions, which are related to the matter in the sources, result in the anisotropic expansion and the movement along the direction of the beam [12].For the longitudinal structure of the sources,   > 1 presents a longitudinal expansion of the source along -axis.When   ̸ = 0, the source has a movement along -axis.The model involves the anisotropic expansion of the participant area in the transverse momentum space.Therefore, the multisource thermal model can be used to study the elliptic flow of charged hadrons produced in high energy collisions at RHIC energies.The analysis of the elliptic flow shows that the expansion factor is characterized by the impact parameter, which is related to participating nucleons.Moreover, the system expansion can be quantified in the momentum space.In the description of (pseudo)rapidity and multiplicity distributions for the final-state particles [14], this model is also successful.
All final-state particles are emitted from the sources formed in the reaction.The polar angle distributions can be obtained by the statistical method.The movement factor   used in the calculation exhibits a linear dependence on the photon beam energy   < 3.0 GeV.Our previous work mostly described particle production in intermediate energy and high energy collisions.In the description of  meson photoproduction in the reaction  → (    ), this work is a good attempt.

2 AdvancesFigure 1 :
Figure 1: The cos  cm dependence of  meson differential cross-section for different photon beam energy 1.65 ≤   ≤ 1.95 GeV.The scattered symbols represent the experimental data.The solid lines are the calculated results.