Abstract
The momentum spectra of charged pions ( and ) and kaons ( and ), as well as protons (p), produced in the beam proton-induced collisions in a 90 cm long graphite target (proton-carbon (p-C) collisions) at the beam momentum GeV/c are studied in the framework of a multisource thermal model by using Boltzmann distribution and the Monte Carlo method. The theoretical model results are approximately in agreement with the experimental data measured by the NA61/SHINE Collaboration. The related free parameters (effective temperature, rapidity shifts, and fraction of nonleading protons) and derived quantities (average transverse momentum and initial quasitemperature) under given experimental conditions are obtained. The considered free parameters and derived quantities are shown to be strongly dependent on the emission angle over a range from 0 to 380 mrad and weakly dependent on longitudinal position (graphite target thickness) over a range from 0 to 90 cm.
1. Introduction
High-energy (relativistic) nucleus-nucleus (heavy ion) collisions with nearly zero impact parameter (central collisions) are believed to form Quark-Gluon Plasma (QGP) or quark matter [1–3] in the laboratory. High-energy nucleus-nucleus collisions with a large impact parameter are not expected to form QGP due to low particle multiplicity yielding lower energy density and temperature [4]. Small collision systems such as proton-nucleus and proton-proton collisions at high energy produce usually low multiplicity, which are not expected to form QGP but are useful to study the multiparticle production processes. However, a few of proton-nucleus and proton-proton collisions at the LHC energies can produce high multiplicity due to the nearly zero “impact parameter,” which are possibly expected to form QGP, where the concept “impact parameter” or “centrality” used in nuclear collisions are used in proton-proton collisions [5]. Degree of collectivity, long-range correlations, strangeness enhancement, etc., which are considered as QGP-like signatures, are recently observed in these high multiplicity events [6–8].
Assuming nucleus-nucleus collisions as a mere superposition of proton-proton collisions in the absence of any nuclear effects, usually one considers proton-proton collisions as the baseline measurements. On the other hand, proton-nucleus collisions [9–13] serve to study the initial state effects and make a bridge between proton-proton [14–18] to nucleus-nucleus collisions [19–23] while studying the multiparticle production processes, though fewer particles are produced in proton-nucleus collisions than in nucleus-nucleus collisions.
There are different types of models or theories being introduced in the studies of high-energy collisions [24, 25]. Among these models or theories, different versions of thermal and statistical models [26–29] characterize some of the aspects of high-energy nuclear collisions, while there are many other aspects that are studied by other approaches. As a basic concept, temperature is ineluctable to be used in analyses. In fact, not only is it that “temperature is surely one of the central concepts in thermodynamics and statistical mechanics” [30], but it is also very important due to its extremely wide applications in experimental measurements and theoretical studies in subatomic physics, especially in high energy and nuclear physics.
In view of this importance, in this paper, we are interested in the study of proton-nucleus collisions at high energy by using the Boltzmann distribution and the Monte Carlo method in the framework of the multisource thermal model [31]. The theoretical model results are compared with the experimental data of the beam proton-induced collisions in a 90 cm long graphite target (proton-carbon (p-C) collisions) at the beam momentum GeV/c measured by the NA61/SHINE Collaboration [32] at the Super Proton Synchrotron (SPS), the European Organisation for Nuclear Research or the European Laboratory for Particle Physics (CERN).
The remainder of this paper is structured as follows. The formalism and method are shortly described in Section 2. Results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.
2. Formalism and Method
According to the multisource thermal model [31], it is assumed that there are many local emission sources to be formed in high-energy collisions due to different excitation degrees, rapidity shifts, reaction mechanisms, and impact parameters (or centralities). In the transverse plane, the local emission sources with the same excitation degree form a (large) emission source. In the rapidity space, the local emission sources with the same rapidity shift form a (large) emission source. In the rest frame of an emission source with a determined excitation degree, the particles are assumed to be emitted isotropically.
In the rest frame of a given emission source, let denote the temperature parameter. The particles with rest mass produced in the rest frame of the emission source are assumed to have the simplest Boltzmann distribution of momenta [33]. That is, where is the normalization constant which is related to . As a probability density function, Equation (1) is naturally normalized to 1.
If we need to consider multiple sources, we can use a superposition of different equations with different temperatures and fractions. We have where , , and are the fraction, normalization constant, and temperature for the -th source or component, respectively. The average temperature obtained from Equation (2) is due to . The derived parameter is the weighted average over various components, but not the simple weighted sum.
It should be noted that or is not the “real” temperature of the emission source, but the effective temperature due to the fact that the flow effect is not excluded in the momentum spectrum. The “real” temperature is generally smaller than the effective temperature which contains the contribution of the collective radial flow effect. To disengage the thermal motion and collective flow effect, one may use different methods such as the blast-wave model [34, 35] or any alternative method [36, 37]. As an example, we shall discuss shortly the results of the blast-wave model in Section 3.
The contribution of spin being small is not included in Equation (1). The effect of chemical potential () is not included in Equation (1) as well, due to the fact that affects only the normalization, but not the trend, of the spectrum if the spin effect is neglected. Our previous work [38] shows that the spin effect together with or is so small (<1%) that we do not need to consider it in studying momentum or transverse momentum spectra in high-energy collisions. Only the combination of spin and causes an obvious effect, which is not the case in this paper.
In the Monte Carlo method [39, 40], let denote random numbers distributed evenly in [0, 1]. To obtain a concrete value of which satisfies Equation (1) or one of the components in Equation (2), we can perform the solution of where denotes a small shift relative to .
Under the assumption of isotropic emission in the rest frame of emission source, the emission angle of the considered particle has the probability density function: which is a half sine distribution in [0, π], and the azimuth obeys the probability density function which is an even distribution in [0, 2π] [41]. In the Monte Carlo method, satisfies which is the solution of .
Considering and obtained from Equations (3) and (5), we have the transverse momentum to be the longitudinal momentum to be the energy to be and the rapidity to be
In the center-of-mass reference frame or the laboratory reference frame, the rapidity of the considered emission source is assumed to be in the rapidity space. Then, the rapidity of the considered particle in the center-of-mass or laboratory reference frame is due to the additivity of rapidity. Multiple emission sources are assumed to distribute evenly in the rapidity range , where and are the minimum and maximum rapidity shifts of the multiple sources, respectively. In the Monte Carlo method,
In particular, comparing with small mass particles, protons exhibit large effect of leading particles which are assumed to distribute evenly in the rapidity range , where and are the minimum and maximum rapidity shifts of the leading protons, respectively. We have
The fraction of the nonleading (leading) protons in total protons is assumed to be . The effects of leading pions and kaons are small and can be neglected in this paper.
In the center-of-mass or laboratory reference frame, the transverse momentum is the longitudinal momentum is the momentum is and the emission angle is
The whole calculation is performed by the Monte Carlo method, though only random numbers are used for the numerical calculation. To compare the theoretical model results with the experimental momentum spectra in a given range, we analyze the momentum distribution of particles which are in the given range. It should be noted that another experimental selection, i.e., the longitudinal position [32], is not regarded as the selected condition in the theoretical model work due to the fact that is only a reflection of target thickness in a 90 cm long graphite target. From to cm, the beam momentum slightly decreases, which is neglected in this paper. In the calculation using random numbers, the energy-momentum conservation was demanded at each step. The results violating the energy-momentum conservation are not considered for our discussions.
It should be noticed that the Boltzmann distribution, Equation (1), can be used to describe low momentum spectra in the source’s rest frame or low transverse momentum spectra after analytic derivation [41] or via the Monte Carlo method, Equations (3), (5), and (6). In the case of considering high momentum spectra in the source’s rest frame or high transverse momentum spectra, one may use possibly the multicomponent Boltzmann distribution, Equation (2). This paper treats multiple sources moving directly in a rapidity range, or , which results in high momentum in the laboratory reference frame. However, in the rest frame of each source, the total momentum and transverse momentum are small. As a consequence, Equation (1) is valid in all momentum ranges, after the transformation from the source’s rest frame to the laboratory reference frame.
3. Results and Discussion
Figures 1 and 2 present the momentum spectra, , of charged pions ( and ) produced in p-C collisions at 31 GeV/c in the laboratory reference frame, respectively, where denotes the number of protons on target and denotes the number of particles. Panels (a–c), (d–f), (g–i), (j–l), (m–o), and (p–q) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. For clarity, spectra in different ranges are scaled by adding different numbers (marked in the panels) represented by different symbols, which are the experimental data measured by the NA61/SHINE Collaboration [32]. The curves are our results fitted by the multisource thermal model using to Equation (1) and the Monte Carlo method. The values of free parameters (, , and ), normalization constant (), , and number of degree of freedom () corresponding to the fits for the spectra of and are listed in Tables 1 and 2, respectively. In two cases, in the fittings are negative which appear in the tables with “–” signs and the corresponding curves are for eye guiding only. One can see that the theoretical model results are approximately in agreement with the NA61/SHINE experimental data of and .
Figure 3 presents the momentum spectra of (a, b) and (c, d) in (a–c) –40 mrad and (b–d) –140 mrad in six ranges with different scaled amounts shown in the panels. The symbols represent the experimental data [32]. The curves are our results fitted by the model. The values of , , , , , and corresponding to the fits for the spectra of and are listed in Table 3. One can see again that the theoretical model results are approximately in agreement with the experimental data of and .
Similar to Figures 1 and 2, Figures 4 and 5 show the momentum spectra of positively and negatively charged kaons ( and ) produced in p-C collisions at 31 GeV/c, respectively. Panels (a–f) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The values of , , , , , and corresponding to the fits for the spectra of and are listed in Tables 4 and 5, respectively. One can see that the theoretical model results are approximately in agreement with the experimental data of and .
Similar to Figure 1, Figure 6 shows the momentum spectra of emitted in p-C collisions at 31 GeV/c. Panels (a, b), (c, d), (e, f), (g, h), (i, j), and (k, l) represent the spectra for –18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The values of , , , , , , , , and corresponding to the fits for the spectra are listed in Table 6. In a few cases, are negative which appear in the table in terms of “–” and the corresponding curves are just for eye guiding only. It should be noted that the contributions of leading protons have to be considered in the spectra. One can see that the theoretical model results are approximately in agreement with the experimental data.
We notice from Tables 1–6 that different for a range of and its dependence with or are observed, but the development of the model in our previous work [42] concludes that is independent of . We would like to explain here that this paper treats as a differential function of or , which is more detailed, while our previous work treats as an integral or mean quantity over . As for which case should be used, it depends on the experimental data.
We now analyze the dependence of free parameters on and . Figures 7 and 8 show, respectively, the dependence of and on (a–e) , which is extracted from the data samples within different ranges for , , , , and , respectively, and on (f) , which are extracted from the data samples within different ranges for and , where we use to denote the difference between and to avoid trivialness in using both and . In particular, in Figure 8(e), the results with are mainly for leading protons and obtained by . One can see that, for and , and decrease slightly with the increase of and do not change obviously with the increase of . The obtained () values for negative and positive pions or kaons seem to be very similar as we expect. The data for the antiproton () are not available in Ref. [32], which forbids in making a comparison for and in this paper. In fact, the situation for is more complex due to the effect of leading protons.
The dependence of and on for the productions of and can be explained by the effect of cascade collisions in the target and by the nuclear stopping of the target. The cascade collisions can cause larger and more energy loss and then lower . The nuclear stopping can cause smaller . Combining with cascade collisions and nuclear stopping, one can obtain low and small at large for the productions of and . Because of the effect of leading particles, the situation for the emissions of is more complex, which shows different trends from those of and . Meanwhile, the flow effect can cause larger , which is related to a more complex mechanism.
The dependence of and on , which are extracted from the data samples within different ranges for (a) , (b) , (c) , (d) , and (e) , are given in Figures 9 and 10, respectively. In particular, large in Figure 10(e) represent mainly the rapidity shifts of leading protons. In principle, there is no obvious increase or decrease in and with the increase of , but there are some statistical fluctuations in a few cases. This result is natural due to the fact that is not the main factor in a 90 cm long graphite target. It is expected that and will decrease with the increase of in a very long graphite target in which the energy loss of the beam protons has to be considered. The NA61/SHINE experimental data analyzed in this paper are not obtained from a long graphite target, and hence, it is not necessary to consider the energy loss of the beam protons.
Figure 11 displays the dependence of fraction of nonleading protons on (a) and (b) , which are extracted from the data samples within different and ranges, respectively. One can see that there is no obvious change in the dependence of on , but some statistical fluctuations. There is a slight increase in the dependence of on with the increase of , which can be explained by more energy loss of the beam protons at larger . This energy loss is small in a not too large range, which does not affect obviously other free parameters such as and due to their less sensitivity at the energy in the range considered in this paper. It is natural that the larger (fewer) fraction of protons appears as nonleading (leading) particles at lower energy or larger . Indeed, the fraction is mainly determined by the collision energy, and the leading protons are considerable at the SPS. In fact, the leading protons are those that existed in the projectile with high momentum and small emission angle, but not the produced protons. With the increase of collision energy up to dozens of GeV and above at which meson-dominated final states appear [43], will increase due to the increase of accompanied produced protons. With the decrease of collision energy down to several GeV and below at which baryon-dominated final states appear [43], will also increase due to the increase of target stopping which causes the decrease of leading protons.
Figures 12 and 13 show, respectively, the dependence of average and on (a–e) , which are extracted from the data samples within different ranges for , , , , and , respectively, and on (f) , which are extracted from the data samples within different ranges for and , where denotes the initial quasitemperature which is given by the root-mean-square over 2 according to the color string percolation model [44–46]. It should be noted that in Refs. [44–46] is regarded as the initial temperature. In that model, there are free parameters associated with the medium created in a high-energy collision, which is not the case for this paper at low energy. So we call the initial quasitemperature in this paper. The dependence of and on are presented in Figures 14 and 15, respectively, which are extracted from the data samples within different ranges. One can see that, for pions and kaons, there are increases in and when increases. The situation is complex for protons due to the effect of leading protons which have high momenta and result in high and at small . The produced protons which are nonleading should have a similar trend in and as those for pions and kaons. As a combination, the final protons are the sum of leading and produced protons. There is no obvious change in and when increases due to not too large energy loss in a 90 cm long graphite target.
We would like to point out that there are different definitions [47] for leading particles in experiments. There are at least four production mechanisms [48, 49] for leading protons in electron-induced deep-inelastic scattering on protons. Among these mechanisms, at HERA energy, diffractive deep-inelastic scattering [50, 51] in which 72% of leading protons have momentum being larger than occupy about 26% [48] of leading protons, which are not enough to cover all leading protons. In particular, for leading protons with momenta being , a large fraction (77%) comes from nondiffractive deep-inelastic scatterings. In proton-proton and proton-nucleus collisions at the considered energy level of this paper, the fraction of the diffractive process is about 20% [52] in inelastic events, which is only half of the fraction of leading protons. Even in nucleus-nucleus collisions, the effect of leading protons in the forward rapidity region is also obvious [42, 53–55], which also reflects in a high momentum region and is not only from diffractive process.
Naturally, there are other additional arguments to explain the behavior of Figures 12 and 13 for the proton case. In fact, there are multiple or cascade secondary scatterings among produced particles and target nucleons. As low mass particles, the emission angles of pions and kaons increase obviously after multiple scatterings. This results in large and due to large for pions and kaons. Contrary to this, the emission angles of protons increase in smaller amount after multiple scatterings due to higher mass of protons compared to pions and kaons. This results in small and due to small for protons. However, nonnegligible leading protons which have high momenta and smaller angles do not experience much multiple scatterings, which renders large and at small . As a competitive result, protons present a different case from pions and kaons.
One can see naturally the coincident trend for and in different and ranges. Due to the flow effect not being excluded, the trend of is inconsistent with that of . As an all-around result, the effects of transverse and longitudinal flows are complex. The flow effect can obviously affect , which is model dependent. The flow effect also affects and which are also model dependent. Therefore, we mention here that is not a “real” temperature, but the effective temperature. In our opinion, the temperature and flow velocity should be independent of models, which is usually not the case more often, as some formalism is used to extract the radial flow and the real/thermal temperature, which estimate the real temperature of the system being dependent of models.
The experimental data cannot be clearly distinguished into two parts: One part is the contribution of thermal motion, which reflects the “real” temperature at the kinetic freeze-out. The other part is the contribution of the collective flow. The current blast-wave model [34, 35] treats the thermal motion and flow effect by using the kinetic freeze-out temperature and transverse flow velocity, respectively. After fitting the spectra with and using coverage as widely as possible (–3 GeV/c), our study using the blast-wave model with flow profile parameter being 2 can obtain similar fit results as the curves in Figures 1–6. To protrude the fit results of the thermal model, the fit results of the blast-wave model are not displayed in these figures. The relation between and for different cases from the spectra of positive particles is plotted in Figure 16, where the circles, squares, and triangles represent the results from , , and spectra, respectively. One can see considerable flow-like effect in p-C collisions at 31 GeV/c, which shows a positive correlation between and . The kinetic freeze-out temperature is about from 0.080 to 0.135 GeV. The corresponding transverse flow velocity is about from 0.21 to . Massive particles such as correspond to larger and smaller comparing to at the same or similar , which is in agreement with hydrodynamic-type behavior. The flow-like effect observed in this work is slightly less than the flow velocity ( in peripheral and in central gold-gold collisions) obtained from the yield ratio of in a simple afterburner model [56]. The difference is due to the fact that a lower energy small system with minimum-bias sample is studied in this paper. In some cases, the results on kinetic freeze-out temperature or transverse flow velocity obtained from different models are not always harmonious [36, 37].
It should be noted that there is entanglement in determining and . For a given spectrum, and are negatively correlated, which means an increase in should result in a decrease of . But for a set of spectra, after determining and for each spectrum, the correlation between and is possibly positive or negative, which depends on the choices of flow profile function and coverage. If the correlation is negative, one may increase and decrease by changing the flow profile function and coverage and obtain possibly positive correlation. If the correlation is positive, one may decrease and increase by changing the flow profile function and coverage and obtain possibly negative correlation. Unlike experimental papers, where one finds a single and a common by fitting the blast-wave model to the bulk part of the spectra (in a very narrow coverage which is particle dependent and much less than 3 GeV/c) by performing a simultaneous fitting to the identified particle spectra using a changeable (from 0 to 4.3) [57], here we have considered a differential freeze-out scenario and have restricted uniformly the fitting up to 3 GeV/c for different particles and have used always . The value of () in positive correlation is larger (less) than that in negative correlation. Positive correlation means high excitation and quick expansion, while negative correlation means longer lifetime (lower excitation) and quicker expansion. In our opinion, although both positive and negative correlations are available, one needs other methods to check which one is suitable. In fact, the positive correlation in Figure 16 is in agreement with the alternative method used in our previous works [36, 37].
We would rather like to use directly in the determination of kinetic freeze-out temperature and transverse flow velocity. For example, the contribution of one participant in each binary collision in the Erlang distribution is which is regarded as effective temperature [58] contributed by the thermal motion and flow effect. We could assume the contribution fraction of the thermal motion to be . Then, the kinetic freeze-out temperature is , and the transverse flow velocity is , where is the mean Lorentz factor of the considered particles in the rest frame of emission source. If we take and at large , the obtained kinetic freeze-out temperatures (0.05 GeV for pion emission and 0.10 GeV for proton emission) are in agreement with those from the blast-wave model [34, 35] and transverse flow velocities ( for pion emission and for proton emission) are qualitatively in agreement with those from the blast-wave model [34, 35] and the afterburner model [50]. The treatment of is also model dependent and in agreement with hydrodynamic-type behavior. In addition, larger results in larger and , which shows positive correlation between and . The positive correlation in Figure 16 is also in agreement with the treatment of .
Before the summary and conclusions, we would like to point out that the kinetic freeze-out temperature and transverse flow velocity obtained in this paper are mass dependent, which renders a scenario for multiple kinetic freeze-out (differential freeze-out) [59]. The afterburner model [56] uses a mass-independent flow velocity, which renders a scenario for single kinetic freeze-out. There are arguments on the kinetic freeze-out scenario, which is beyond the focus of this paper, so we shall not discuss it anymore. In addition, it should be noted that in the absence of a required number of experimental data points, the fittings using the current model in few cases yield negative , making the description unphysical, though the corresponding curves could be used as eye guiding only.
4. Summary and Conclusions
We summarize here our main observations and conclusions: (a)The momentum spectra of , , , , and produced in p-C collisions at 31 GeV/c are analyzed in the framework of a multisource thermal model by using the Boltzmann distribution and Monte Carlo method. The results are approximately in agreement with the experimental data in various emission angles, , and ranges and longitudinal positions, , measured by the NA61/SHINE Collaboration at the SPS(b)The effective temperature and rapidity shifts from the spectra under given experimental conditions which limit various and ranges are obtained. For and , and decrease slightly with the increase of and do not change obviously with the increase of . The situation for is more complex due to the effect of leading protons. There is no obvious change in and when increases due to a not too large energy loss in a not too long graphite target. Both and depend on models. In particular, contains the contribution of the flow effect, which is not ideal to describe the excitation degree of the emission source(c)The fraction of nonleading (leading) protons in total protons from the spectra in various and ranges are obtained. There is no obvious change in the dependence of on , but some statistical fluctuations. There is a slight increase (decrease) in the dependence of on with the increase of due to more energy loss of the beam protons in the target at larger . The effect of leading protons cannot be neglected at the SPS energies. It is expected that will be larger (smaller) at both lower (≤ several GeV) and higher energies (≥ dozens of GeV)(d)The average transverse momentum and initial quasitemperature from the spectra in various and ranges are obtained. For and , there are increases in and when increases. The situation for is complex due to the effect of leading protons. There is no obvious change in and when increases due to a not too large energy loss in a not too long graphite target. Both and are model dependent due to the fact that they are obtained from the model which fits the data(e)The behaviors of effective temperature, rapidity shifts, fraction of nonleading (leading) protons, average transverse momentum, and initial quasitemperature obtained from the fits of a multisource thermal model to the NA61/SHINE data can be explained in terms of cascade collisions in the target, stopping power of the target, energy loss of the beam protons in the target, and so on. This paper provides new evidence for the effectiveness of the multisource thermal model, though there is no connection with a possible formation of a Quark-Gluon Plasma as what was being considered was a small system
Data Availability
The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.
Ethical Approval
The authors declare that they are in compliance with ethical standards regarding the content of this paper.
Disclosure
The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Communications from Debasish Das are highly acknowledged. Authors P.P.Y., M.Y.D., and F.H.L. acknowledge the financial supports from the National Natural Science Foundation of China under Grant Nos. 11575103 and 11847311, the Shanxi Provincial Innovative Foundation for Graduate Education Grant No. 2019SY053, the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) under Grant No. 201802017, the Shanxi Provincial Natural Science Foundation under Grant No. 201701D121005, and the Fund for Shanxi “1331 Project” Key Subjects Construction. Author R.S. acknowledges the financial supports from ALICE Project No. SR/MF/PS-01/2014-IITI(G) of the Department of Science and Technology (DST), Government of India.