Initial-State Temperature of Light Meson Emission Source From Squared Momentum Transfer Spectra in High-Energy Collisions

Abstract

The squared momentum transfer spectra of light mesons, π⁰, π⁺, η, and ρ⁰, produced in high-energy virtual photon-proton (γ*p) → meson + nucleon process in electron-proton (ep) collisions measured by the CLAS Collaboration are analyzed by the Monte Carlo calculations, where the transfer undergoes from the incident γ* to emitted meson or equivalently from the target proton to emitted nucleon. In the calculations, the Erlang distribution from a multi-source thermal model is used to describe the transverse momentum spectra of emitted particles. Our results show that the average transverse momentum (⟨p_T⟩) and the initial-state temperature (T_i) increase from lower squared photon virtuality (Q²) and Bjorken variable (x_B) to higher one. This renders that the excitation degree of emission source, which is described by ⟨p_T⟩ and T_i, increases with increasing of Q² and x_B.

1 Introduction

In the evolution process of high-energy nucleus-nucleus (heavy-ion) collisions, the reaction system undergoes several main stages which are separately the incoming of nuclei, beginning of collisions, strongly-coupled quark-gluon plasma (sQGP) phase or hot-dense matter phase, mixed phase, and hadron gas. In the stage of the incoming of nuclei, two nuclei move toward each other in vacuum tunnel at nearly the speed of light and change the shape to pancake with the Lorentz contraction. The sQGP phase is extremely hot-dense matter and the system can be regarded as a fireball. Considering the effect of pressure gradient, the system begins to inflate and cool down. Then, the hadron matter appears until the system is hadronic. To understand the mechanism of nuclear reaction and the property of system evolution, it is necessary to investigate the characteristics of each stage of collision process. The excitation and equilibrium degrees of the system are among very important characteristics [1–10].

To describe the excitation degree of the system, various temperatures of the system and the average transverse momentum (⟨p_T⟩) of particles are used [11–18]. The various temperatures include, but are not limited to, 1) the initial-state temperature (T_i) which reflects the temperature in the beginning of collisions of two nuclei, 2) the chemical freeze-out temperature (T_ch) which reflects the temperature at chemical freeze-out when inelastic collisions disappear, 3) the kinetic freeze-out or final-state temperature (T_kin or T₀) which reflects the temperature at kinetic freeze-out when elastic collisions disappear, and 4) the effective temperature (T_eff) which is not a “real” temperature, in which the influence of flow effect is not excluded compared with T_kin or T₀. Different kinds of temperatures can be “measured” by different “thermometers” (methods).

As the earliest temperature in collisions, T_i is used to explore the secret of high-energy collisions [11–15]. As we know, T_i is the temperature of emission source or interacting system when the system undergoes the initial-stage of collisions [19]. It is interesting for us to describe the excitation degree of the system by using T_i. Generally, from the transverse momentum (p_T) spectra or fitting the p_T spectra with different distributions or functions, we may obtain T_i. The Erlang distribution [20–22], Tsallis distribution [23, 24], Hagedorn function [25] are usually used, but in this paper, we only choose the Erlang distribution due to its origin of multiple sources in the multi-source thermal model [20–22]. In the special case, such as absent p_T spectra, the squared momentum transfer spectra are alternatively used. Obviously, T_i can not be obtained from the squared momentum transfer spectra directly unless the p_T spectra are transformed to them. From the fit to p_T spectra, ⟨p_T⟩ can be naturally abstracted.

In the transformation of p_T spectra to squared momentum transfer spectra [26], the Monte Carlo method is used. First of all, concrete p_T, satisfying the Erlang distribution [20–22], are produced. Then, the squared momentum transfers are calculated according to the relation between squared momentum transfers and p_T by using the Monte Carlo method. At last, the distribution of squared momentum transfer spectra are obtained and used to fit the experimental data for extracting ⟨p_T⟩ and T_i.

To describe the equilibrium degree of the system, one can use the Tsallis distribution [23, 24] or Hagedorn function [25] to fit p_T spectra directly. In the fitting process, the entropy index q can be extracted. The closer to 1 the entropy index q is, the higher the degree of equilibrium of the source or system is. The relation between the two distributions is that the former one covers the later one in which the mass is neglected. Because the universality, similarity, or common characteristics exist in high-energy collisions [27–36], some distributions used in large collision system can be also used in small collision system. Although the equilibrium degree is also important, it is not discussed in this work due to other topics being concerned. We think that the equilibrium degree is enough to use the concept of temperature.

Meson consists of a quark and anti-quark and belongs to hadron. It takes part in the strong interaction and play an important role. Light meson refers to a kind of meson with low mass. The transverse momentum of light meson changes more sensitively than that of the heavy one. Therefore, the study of transverse momentum spectra of light mesons is very important to explore the reaction mechanism and evolution process of high-energy collisions.

Compared with large systems of high-energy nucleus-nucleus collisions, small systems such as high-energy electron-proton, proton-proton, proton-nucleus collisions also produce abundant results. In particular, in electron-proton collisions, the scattered electron exchanges virtual photon (γ*) with the target proton. Then, one may study high-energy γ* induced proton collisions, that is γ*p collisions, experimentally, theoretically as well as phenomenologically.

In this paper, the squared momentum transfer spectra of light mesons, π⁰, π⁺, η, and ρ⁰, produced in high-energy γ*p collisions measured by the CLAS Collaboration [37–40] are fitted by the results originating from the Erlang p_T distribution with the Monte Carlo method. The CLAS experimental data are measured at different squared photon virtuality Q² and Bjorken variable x_B, where Q² and x_B will be discussed later in the Subsection 2.3.

2 Formalism and Method

2.1 The Erlang Distribution

The Erlang distribution is a direct result of the multi-source thermal model [20–22]. One or two-component Erlang distribution can describe the narrow or wide p_T spectra of particles, where the narrow (wide) p_T spectra refers to range less than a few GeV/c (more than 10 GeV/c) [22]. The multi-source thermal model assumes that multiple sources are formed in high-energy collisions. These sources can be nucleons or partons if we study the formation of nucleon clusters (nuclear fragments) or particles.

In this work, we assume that a few (n_s) partons (partons-like) contribute to p_T of a given particle [22]. The contribution of the jth parton is assumed to be an exponential function with variable p_tj which depends on j, and average value ⟨p_t⟩ which is independent of j. We have the normalized exponential function

Here, ⟨p_t⟩ represents the average contribution of participant partons to ⟨p_T⟩ of the considered particles.

The contribution sum of n_s partons is p_T of a given particle. The result convoluting the contributions of n_s partons is the Erlang distribution. We have the Erlang p_T distribution to be

Here N is the number of particles, and the form of (1/N)dN/dp_T results in the normalization of f(p_T) to 1. In fact, the normalization of the Erlang distribution is naturally 1.

We would like to emphasize here the difference between “n_s”, the number of partons and “N”, the number of particles. In γ*p collisions, if three quarks in the proton contributed to p_T, we have n_s = 3. If another pair also contributed to p_T, we have n_s = 3 + 2 = 5. Even in nucleus-nucleus collisions, the value of n_s is not large due to it being determined by the number of contributor partons in a nucleon-nucleon pair, but not collision system itself. This makes sense, in the Fock’s first two terms of the development of the wave function of the proton, as composed by 3 quarks and then 3 quarks plus a pair [41, 42]. As for N, its value may be small in small collision system or peripheral nucleus-nucleus collisions. The value of N may be very large in central nucleus-nucleus collisions at high energy.

2.2 Average Transverse Momentum and Initial-State Temperature

As we know, both the average transverse momentum ⟨p_T⟩ and initial-state temperature T_i [11–15] describe the excitation degree of the system. In particular, in the Erlang distribution, ⟨p_T⟩ can be easily obtained bywhere f (p_T) is normalized to 1. Similarly, ⟨p_t⟩ reflects the excitation degree of participant partons.

According to Refs. [43–45], with a color string percolation method [46], T_i can be regarded aswheredue to f (p_T) is normalized to 1 and is the root-mean-square of p_T. In Eq. 4, F(ξ) is the color suppression factor [46].

In the process of using color string method to obtain T_i in this work, only one string is used, i.e., F(ξ) = 1, in the formation of particle. Although there are probability to have any other strings, they do not affect noticeably T_i. If we consider other strings, according to Ref. [46], one has the minimum F(ξ) ≈ 0.6. This will cause the maximum increase of 29.1% in T_i. Considering the fraction of one string is very large, that of two strings is relative small, and that of multiple strings is very small, the increase in T_i will be much smaller than 29.1%.

2.3 The Squared Momentum Transfer

In the center-of-mass reference frame, in two-body process 2 + 1 → 4 + 3 or two-body-like process of high-energy collisions, there are three Mandelstam variables defined based on the four-momenta of these particles. They have the forms to bewhere P₁, P₂, P₃, and P₄ are four-momenta of particles 1 (target proton), 2 (incident γ*), 3 (emitted nucleon), and 4 (emitted meson), respectively. Here, we assume that particle 1 is incident along the Oz direction and particle 2 is incident along the opposite direction. After collisions, particle 3 is emitted with angle θ relative to the Oz direction and particle 4 is emitted along the opposite direction.

The three Mandelstam variables have different physical meaning. For instance, refers to the center-of-mass energy, and −u is defined as the squared momentum transfer between particles 1 and 4. Here, selected variable −t (the squared momentum transfer between particles 1 and 3) is calculated to fit the experimental data. For convenience, we havewhere E₁ and E₃, and , as well as m₁ and m₃ are the energy, momentum, and rest mass of particles 1 and 3, respectively. In addition, p_3T referred to be perpendicular to the Oz direction component of the transverse momentum of particle 3, which obeys the Erlang distribution, that is Eq. 2 in which p_T = p_3T.

In this paper, the squared momentum transfer spectra of light meson at different squared photon virtuality Q² and Bjorken variable x_B are fitted by calculated results with the Monte Carlo method. Here, Q² is a reflection of hard scale of reaction [47–54]. The harder the reaction is, the higher the excitation degree is. In fact, Q² is the absolute value of the squared mass of γ* (particle 2) that is exchanged between the scattered electron and the target proton (particle 1), and it effectively represents the transverse size of the probe [38]. In addition, −Q² is also the squared momentum transfer to the target proton (particle 1) by the scattered electron [37].

As for the Bjorken variable x_B, it represents contrarily the momentum of particle 1. The lower the x_B is, the higher the momentum of particle 1 is. Generally, [37]. In a symmetric frame, importing ξ′ as skewness, it is half of the longitudinal momentum fraction transferred to the struck parton. The skewness ξ′ can be used to express x_B approximately. That is x_B ≈ 2ξ′/(1 + ξ′) [37].

2.4 The Process of Monte Carlo Calculations

In the calculations of squared momentum transfer, the analytical expression of p_T distribution is difficult to be transformed to that of squared momentum transfer distribution directly by using Eq. 9. Alternatively, we may use the Monte Carlo method to transform p_T to squared momentum transfer. Let R_1,2 and be random numbers distributed evenly in [0,1]. Then, many concrete transverse momentum p_3T satisfied with Eq. 2 and θ are produced. Other quantities such as E₁, m₁, and m₃ in the equation are fixed, though E₁ is treated as a parameter in the present work.

Generally, we may solve the equationwhere δp_T is a small shift relative to p_T. Conveniently, there is a simpler expression due to Eqs 1, 2. In fact, solving the equationwe have

The simpler expression is

The distribution of θ satisfies with the half-sine functionwhich is obtained under the assumption of isotropic emission in the source’s rest frame. Solving the equationwe havewhich is needed in the calculations.

We have check the consistency and correctness of the above expressions in the Monte Carlo method in terms of illustration which is not presented here. After obtaining concrete values of p_3T and θ, and using E₁, m₁, and m₃, the value of |t| can be obtained from Eq. 9. Through repeating the calculations many times, the distribution of |t| is obtained statistically. Based on the method of least squares, the parameter ⟨p_t⟩ and n_s are extracted naturally. Meanwhile, T_i can be obtained from Eq. 4 and can be obtained from Eq. 3 [(5)] or from the statistics. The errors of parameters are obtained by the general method of statistical analysis.

It should be noted that the above Monte Carlo calculation is only performed in the transformation from transverse momentum to |t|, in which the physics process such as the radiative corrections for reactions induced by electrons has been taken into account naturally. In fact, the effects of the mentioned process and all other processes are included in the Erlang distribution which is a result of multi-factor interactions. In other words, the Monte Carlo calculation used here is not a simulation for the system evolution from initial to final stages, but the numerical transformation in the final stage.

3 Results and Discussion

3.1 Comparison With Data

Figure 1 shows the differential cross-section, dσ/d|t|, in squared momentum transfer |t| of γ*p → π⁰p process produced in 5.75 GeV electron beam induced collisions in a 2.5 cm long liquid-hydrogen target (ep collisions at beam energy of 5.75 GeV) in different ranges of squared photon virtuality, 1.0 < Q² < 1.5, 1.5 < Q² < 2.0, 2.0 < Q² < 2.5, 2.5 < Q² < 3.0, 3.0 < Q² < 3.5, 3.5 < Q² < 4.0, and 4.0 < Q² < 4.6 GeV², from bottom to up sub-panels, as well as in different ranges of Bjorken variable, 0.10 < x_B < 0.15, 0.15 < x_B < 0.20, 0.20 < x_B < 0.25, 0.25 < x_B < 0.30, 0.30 < x_B < 0.38, 0.38 < x_B < 0.48, and 0.48 0.58, from left to right sub-panels. The sample at the top-left sub-panel shows repeatedly the result in the range of squared photon virtuality, 1.5 < Q² < 2.0 GeV², and the range of Bjorken variable, 0.20 < x_B < 0.25, as an example. The symbols represent the experimental data measured by the CLAS Collaboration [37] and the curves are the statistical results of squared momentum transfer |t|.

FIGURE 1

In Eq. 9, p_3T satisfies the Erlang distribution (Eq. 2) and we obtain it by the Monte Carlo method (Eq. 13). Then, the squared momentum transfer |t| is obtained statistically. In the fitting process, two main parameters, i.e., the average transverse momentum ⟨p_t⟩ contributed by each participant parton and the number n_s of participant partons are extracted naturally. To obtain a better fit result, E₁ is extracted as an insensitive parameter. In addition, a non-free parameter is the normalization constant σ₀. The values of parameters with selection condition (Q² and x_B), χ², and the number of degree of freedom (ndof) are listed in Table 1, where the number of parameters is always 4 which includes E₁, ⟨p_t⟩, n_s, and σ₀. In the case of ndof being less than or equal to the number of parameters, we obtain the curve from a “prediction” or extrapolation based on other reasonable fits in which the tendency of parameters is available. Meanwhile, in these cases, the number of points (nop) is given in a bracket to replace ndof in the table. One can see that the values of χ² are small in most cases, though the (necessary) dense log scale is not easy to judge. The model results are in agreement with the experimental data. From the values of parameters, the average transverse momentum ⟨p_T⟩ and initial temperature T_i are obtained naturally.

Figure 2 presents the differential cross-section, dσ/d|t|, in |t| of γ^∗p → π⁺n process produced in ep collisions at beam energy of 6 GeV at different squared photon virtuality, Q² = 1.75, 2.05, 2.35, 2.65, 2.95, 3.35, 3.85, and 4.35 GeV², as well as at different Bjorken variable, x_B = 0.25, 0.31, 0.37, 0.43, 0.49, and 0.55. The symbols represent the experimental data measured by the CLAS Collaboration [38]. As those in Figure 1, the curves in Figure 2 are also the statistical results of |t| in which p_3T satisfies the Erlang distribution and is obtained by the Monte Carlo method. The values of parameters with selection condition (Q² and x_B), χ², and ndof are listed in Table 1. One can see that the model results are in agreement with the experimental data.

FIGURE 2

Figure 3 displays the differential cross-section, dσ/d|t|, in |t| of γ^∗p → ηp process produced in ep collisions at beam energy of 5.75 GeV in different Q² and x_B ranges shown in the panel. As an example, the sample at the top-left sub-panel shows repeatedly the result in 1.5 < Q² < 2.0 GeV² and 0.20 < x_B < 0.25. The symbols represent the experimental data measured by the CLAS Collaboration [39]. The curves are the statistical results of |t| in which p_3T satisfies the Erlang distribution and is obtained by the Monte Carlo method. The values of parameters with selection condition (Q² and x_B), χ², and ndof are listed in Table 2. One can see that the model results are in agreement with the experimental data.

FIGURE 3

Similar to Figures 1–3, Figure 4 presents the differential cross-section, dσ/d|t|, in |t| of γ^∗p → ρ⁰p process produced in ep collisions at beam energy of 5.754 GeV in different Q² and x_B ranges shown in the panel. As an example, the sample at the top-left sub-panel shows repeatedly the result in 2.8 < Q² < 3.1 GeV² and 0.40 < x_B < 0.46 range. The symbols represent the experimental data measured by the CLAS Collaboration [40]. The curves are the statistical results of |t| in which p_3T satisfies the Erlang distribution and is obtained by the Monte Carlo method. The values of parameters with selection condition (Q² and x_B), χ², and ndof are listed in Table 2. One can see that the model results are in agreement with the experimental data.

FIGURE 4

3.2. Parameter Tendency and Discussion

In Figures 1–4, the cross-sections for π⁰p, π⁺n, ηp, and ρ⁰p are fitted to show some differences in concrete values and parameters, and some common features among them in the tendency of curves also appear. This is caused by the fact that different channels have different fraction ratios, and all of them are from the same ep collisions, though the collision energies are slightly different.

The dependences of ⟨p_T⟩ (A, C, E, G) and T_i (B, D, F, H) on Q² in γ*p collisions with different emitted channels [π⁰p (A, B), π⁺n (C, D), ηp (E, F), and ρ⁰p (G, H)] are shown in Figure 5, where ⟨p_T⟩ = n_s⟨p_t⟩ due to Tables 1, 2 and the values of T_i are from Tables 1, 2. Different symbols represent the results for different x_B. One can see that ⟨p_T⟩ and T_i increase generally with an increase in Q². Because Q² represents the hard scale (violent degree) of collisions and a harder scale results in a higher excitation degree, it is natural that larger ⟨p_T⟩ and T_i appear at higher Q².

FIGURE 5

Figure 6 is similar to Figure 5, but it shows the dependences of ⟨p_T⟩ (A, C, E, G) and T_i (B, D, F, H) on x_B in γ*p collisions with emitted channels π⁰p (A, B), π⁺n (C, D), ηp (E, F), and ρ⁰p (G, H). Different symbols represent the results for different Q². One can see that ⟨p_T⟩ and T_i increase generally with an increase in x_B. Because x_B ∝ Q, we may think that x_B also represents the hard scale of collisions and a harder scale results in a higher excitation degree. It is understandable that larger ⟨p_T⟩ and T_i appear at higher x_B.

FIGURE 6

In addition, x_B also represents the longitudinal momentum fraction transferred to the struck parton. In the considered γ*p → meson + nucleon process in ep collisions at given energy, the larger x_B means the larger longitudinal momentum transfer to the struck parton or the system, and hence the more energy deposited to the system. The system naturally stays at higher excitation degree. As a result, larger ⟨p_T⟩ and T_i are observed.

Generally, ⟨p_T⟩ > T_i ≥ T_ch ≥ T₀. If the evolution time of the system is 0, that is if the initial-state, chemical freeze-out, and kinetic freeze-out happen simultaneously, we have T_i = T_ch = T₀. If the evolution time is not negligible, we have T_i > T_ch > T₀. The difference between ⟨p_T⟩ and temperature is explained as the contribution of flow effect. According to Ref. [55], in the final-state, the expected real T₀ ≈ ⟨p_T⟩/3.07. Then, we have the contribution of flow effect to be ⟨p_T⟩ −T₀ ≈ 2.07⟨p_T⟩/3.07. One can see that the flow effect contributes largely to ⟨p_T⟩. It is expected that the contribution of flow effect increases with the increase of evolution time, if ⟨p_T⟩ is fixed from the initial- to final-states.

From Tables 1, 2, we note that the values of n_s are 3–5 for different channels. As the number of participant partons, n_s is constrained to be integer with uncertainty of 0. For a given channel, n_s is independent of Q² and x_B in most cases. The channel independent n_s renders that the number of participant partons is not too small or big. The number of struck parton(s) is usually regarded as 1 or 2, which is very small. The struck parton(s) and the partons around the struck parton(s) are participant partons. The partons far away from the struck parton(s) are remainder or spectator partons.

Before summary and conclusion, we would like to point out that the discussion about the temperature and flow in this paper is applicable. Although the multiplicity in ep collisions at a few GeV is very limited and the final particles are in a state far from thermal equilibrium, we may use the grand canonical ensemble for lots of events in which the number of total particles is very large and the whole system is in a homogeneous and equilibrium state. Therefore, the temperature used in this paper is comparable to the freeze-out temperatures used in nucleus-nucleus collisions. Of course, we may also regarded the temperature used here as a fitting parameter if necessary.

The initial-temperature T_i is extracted from the root-mean-square of p_T, which is independent of model, though the relation between T_i and is from the color string percolation method [43–46]. As deep inelastic scattering, ep collisions are head-on collisions, and may be harder than nucleus-nucleus collisions at similar energy per nucleon due to the fact that some non-head-on nucleon-nucleon collisions exist in the later. As a hybrid state of head-on and non-head-on nucleon-nucleon collisions, nucleus-nucleus collisions may be weaker than head-on ep collisions. In addition, cold spectator nuclear effect also causes the temperature in nucleus-nucleus collisions to reduce. This renders that T_i obtained in this paper is higher than that in nucleus-nucleus collisions.

It should be emphasized that the parameter T_i reflects the violent degree of collisions. To our knowledge, other groups and other studies where T_i is extracted for hadronic collisions is not available at present, though T_i for nucleus-nucleus collisions is available. In terms of T_i, Erlang distribution, and Monte Carlo calculation, the present work has proposed an alternative method to describe light meson electroproduction data obtained with the JLab-CLAS facility. Typically those data are interpreted in terms of handbag diagram within the formalism of generalized parton distributions, whereas here statistical methods, that were developed for high-energy nucleus-nucleus collisions, are applied. At least, the present work has significance in the application of statistical methods.

4 Summary and Conclusion

In summary, the squared momentum transfer spectra of π⁰, π⁺, η, and ρ⁰ produced in γ*p → meson + nucleon process have been fitted by the calculated results with the Erlang distribution which is obtained from the multi-source thermal model and used to describe the transverse momentum spectra of emitted particles. The squared momentum transfer undergoes from the incident γ* to emitted meson, and also equivalently from the target proton to emitted nucleon. The model results are in agreement with the experimental data measured by the CLAS Collaboration. The values of the related parameters are extracted in the fitting process. The squared photon virtuality Q² and Bjorken variable x_B dependent parameters are obtained.

With increasing of Q², the quantities ⟨p_T⟩ and T_i increase generally. Q² is defined as absolute value of the squared mass of the virtual photon that is exchanged between the electron and the target proton, and it effectively represents the transverse size of the probe. Q² also reflects the hard scale of collisions. A harder scale results in a higher excitation degree of the system, and a larger ⟨p_T⟩ and T_i. At harder scale (larger Q²), the degree of equilibrium decreases because of more disturbance to the equilibrated residual partons in target particle, though the system is at the state of high degree of excitation.

Similar to the tendency of Q², with an increase of x_B, the quantities ⟨p_T⟩ and T_i increase. In the considered γ*p → meson + nucleon process, x_B represents the longitudinal momentum fraction transferred to the struck parton. The larger x_B means the larger longitudinal momentum transfer to the system. It is natural that ⟨p_T⟩ and T_i are larger at larger x_B. In addition, because x_B ∝ Q, one may argue that x_B also represents the hard scale of collisions. Indeed, it is understandable that larger ⟨p_T⟩ and T_i appear at higher x_B.

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