# Multiparticle Production and Initial Quasitemperature from Proton-Induced Carbon Collisions atpLab=31 GeV/c

Authors: Pei-Pin Yang; Mai-Ying Duan; Fu-Hu Liu; Raghunath Sahoo
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The momentum spectra of charged pions (π+ and π−) and kaons (K+ and K−), 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 pLab=31 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.


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## Body

## 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 pLab=31 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 Section2. 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, letT denote the temperature parameter. The particles with rest mass m0 produced in the rest frame of the emission source are assumed to have the simplest Boltzmann distribution of momenta p′ [33]. That is,
(1)fp′p′=Cp′2exp−p′2+m02T,where C is the normalization constant which is related to T. 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(2)fp′p′=∑jkjCjp′2exp−p′2+m02Tj,where kj, Cj, and Tj are the fraction, normalization constant, and temperature for the j-th source or component, respectively. The average temperature obtained from Equation (2) is T=∑jkjTj/∑jkj=∑jkjTj due to ∑jkj=1. The derived parameter T is the weighted average over various components, but not the simple weighted sum.It should be noted thatT or Tj 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 (μ) is not included in Equation (1) as well, due to the fact that μ 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 µ≫m0 or µ≪m0 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 μ≈m0 causes an obvious effect, which is not the case in this paper.In the Monte Carlo method [39, 40], let R1,2,3,4 denote random numbers distributed evenly in [0, 1]. To obtain a concrete value of p′ which satisfies Equation (1) or one of the components in Equation (2), we can perform the solution of
(3)∫0p′fp′p″dp″<R1<∫0p′+δp′fp′p″dp″,where δp′ denotes a small shift relative to p′.Under the assumption of isotropic emission in the rest frame of emission source, the emission angleθ′ of the considered particle has the probability density function:
(4)fθ′θ′=12sinθ′,which is a half sine distribution in [0, π], and the azimuth φ′ obeys the probability density function fφ′φ′=1/2π which is an even distribution in [0, 2π] [41]. In the Monte Carlo method, θ′ satisfies
(5)θ′=2arcsinR2,which is the solution of ∫0θ′1/2sinθ″dθ″=R2.Consideringp′ and θ′ obtained from Equations (3) and (5), we have the transverse momentum pT′ to be
(6)pT′=p′sinθ′,the longitudinal momentum pz′ to be
(7)pz′=p′cosθ′,the energy E′ to be
(8)E′=p′2+m02,and the rapidity y′ to be
(9)y′≡12lnE′+pz′E′−pz′.In the center-of-mass reference frame or the laboratory reference frame, the rapidity of the considered emission source is assumed to beyx in the rapidity space. Then, the rapidity of the considered particle in the center-of-mass or laboratory reference frame is
(10)y=y′+yx,due to the additivity of rapidity. Multiple emission sources are assumed to distribute evenly in the rapidity range ymin,ymax, where ymin and ymax are the minimum and maximum rapidity shifts of the multiple sources, respectively. In the Monte Carlo method,
(11)yx=ymax−yminR3+ymin.In particular, comparing with small mass particles, protons exhibit large effect of leading particles which are assumed to distribute evenly in the rapidity rangeyLmin,yLmax, where yLmin and yLmax are the minimum and maximum rapidity shifts of the leading protons, respectively. We have
(12)yx=yLmax−yLminR4+yLmin.The fraction of the nonleading (leading) protons in total protons is assumed to bek1−k. 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 momentumpT is
(13)pT=pT′,the longitudinal momentum pz is
(14)pz=pT2+m02sinhy,the momentum p is
(15)p=pT2+pz2,and the emission angle θ is
(16)θ=arctanpTpz.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 givenθ range, we analyze the momentum distribution of particles which are in the given θ range. It should be noted that another experimental selection, i.e., the longitudinal position z [32], is not regarded as the selected condition in the theoretical model work due to the fact that z is only a reflection of target thickness in a 90 cm long graphite target. From z=0 to z=90 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, ymin,ymax or yLmin,yLmax, 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

Figures1 and 2 present the momentum spectra, 1/Npotd2n/dpdθ, of charged pions (π+ and π−) produced in p-C collisions at 31 GeV/c in the laboratory reference frame, respectively, where Npot denotes the number of protons on target and n denotes the number of particles. Panels (a–c), (d–f), (g–i), (j–l), (m–o), and (p–q) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. For clarity, spectra in different θ 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 (T, ymax, and ymin), normalization constant (N0), χ2, and number of degree of freedom (ndof) corresponding to the fits for the spectra of π+ and π− are listed in Tables 1 and 2, respectively. In two cases, ndof 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 π+ and π−.Figure 1
Momentum spectra ofπ+ produced in p-C collisions at 31 GeV/c. Panels (a–c), (d–f), (g–i), (j–l), (m–o), and (p–q) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The symbols represent the experimental data [32]. The curves are our results fitted by the multisource thermal model due to Equation (1) and the Monte Carlo method. To show clearly, different spectra are scaled by adding different amounts marked in the panels.Figure 2
Same as Figure1 but showing the spectra of π−.Table 1
Values ofT, ymax, ymin, N0, χ2, and ndof corresponding to the curves in Figure 1 in which different data are measured in different θ and z ranges. In the table, z is in the units of cm and θ is not listed, which appears in Figure 1. In one case, ndof is negative which appears in terms of “–” and the corresponding curve is just for eye guiding purpose.
Figure
T (GeV)
ymax
ymin
N0 (×0.001)
χ2/ndof
Figure1(a)0<z<18
0.320±0.005
2.30±0.02
1.10±0.02
0.143±0.010
22/2
0.138±0.004
3.25±0.05
1.72±0.06
0.962±0.020
42/12
0.195±0.006
2.47±0.04
1.57±0.03
2.918±0.100
93/12
0.205±0.003
2.36±0.04
1.47±0.03
6.431±0.200
87/9
0.220±0.005
2.09±0.04
1.00±0.03
11.108±0.400
85/9
Figure1(b)0<z<18
0.222±0.004
2.00±0.03
0.80±0.04
14.541±0.300
60/9
0.169±0.003
2.20±0.04
0.78±0.02
16.701±0.340
15/9
0.166±0.003
2.10±0.03
0.45±0.02
17.572±0.260
16/6
0.166±0.002
2.00±0.03
0.65±0.03
19.149±0.300
37/6
0.166±0.002
1.90±0.02
0.65±0.02
18.495±0.340
57/6
Figure1(c)0<z<18
0.166±0.001
1.80±0.02
0.65±0.02
17.839±0.200
70/6
0.136±0.002
2.08±0.06
0.75±0.04
31.163±0.720
35/3
0.160±0.003
1.75±0.03
0.55±0.02
25.050±0.560
28/2
0.115±0.004
2.08±0.08
0.90±0.05
26.483±0.800
12/1
0.115±0.004
2.08±0.04
0.85±0.10
25.278±1.200
2/–
Figure1(d)18<z<36
0.320±0.010
2.50±0.02
1.48±0.01
0.571±0.020
49/3
0.210±0.004
2.73±0.02
1.50±0.04
5.224±0.080
48/12
0.198±0.003
2.56±0.04
1.57±0.03
17.336±0.400
76/12
0.215±0.004
2.30±0.03
1.35±0.02
28.156±1.000
90/9
0.224±0.004
2.09±0.03
1.00±0.02
31.294±0.400
67/9
Figure1(e)18<z<36
0.222±0.005
1.90±0.04
0.70±0.03
31.345±0.800
70/9
0.169±0.004
2.20±0.03
0.80±0.05
30.855±0.400
29/9
0.172±0.005
2.10±0.02
0.45±0.02
28.588±0.600
3/6
0.168±0.002
1.90±0.04
0.55±0.05
25.800±0.400
19/6
0.166±0.001
1.90±0.02
0.65±0.01
24.237±0.200
50/6
Figure1(f)18<z<36
0.167±0.002
1.78±0.03
0.65±0.02
22.902±0.400
64/6
0.137±0.001
1.95±0.02
0.75±0.02
38.121±0.800
44/3
0.160±0.002
1.75±0.03
0.45±0.03
33.929±0.800
22/2
0.155±0.002
1.70±0.03
0.45±0.02
30.541±0.800
6/1
0.182±0.003
0.70±0.02
0.55±0.03
31.012±0.600
7/0
Figure1(g)36<z<54
0.300±0.002
2.50±0.02
1.40±0.02
1.064±0.030
38/3
0.193±0.003
2.85±0.01
1.55±0.02
10.836±0.200
108/12
0.195±0.003
2.60±0.05
1.57±0.03
23.318±0.400
85/12
0.215±0.005
2.30±0.03
0.95±0.03
21.566±0.600
60/9
0.220±0.003
2.09±0.02
0.90±0.02
27.103±0.400
69/9
Figure1(h)36<z<54
0.242±0.002
1.88±0.02
0.25±0.03
26.968±0.600
42/9
0.169±0.003
2.20±0.02
0.70±0.03
26.177±0.400
29/9
0.168±0.003
2.10±0.04
0.40±0.05
22.962±0.400
13/6
0.166±0.005
1.95±0.03
0.30±0.02
21.297±0.600
40/6
0.166±0.002
1.93±0.03
0.35±0.04
19.830±0.500
44/6
Figure1(i)36<z<54
0.166±0.002
1.70±0.07
0.25±0.05
17.778±0.300
97/6
0.138±0.002
1.95±0.03
0.75±0.04
32.081±0.800
37/3
0.160±0.003
1.75±0.05
0.20±0.04
23.055±0.600
5/2
0.155±0.005
1.70±0.05
0.35±0.04
26.196±0.800
6/1
0.186±0.005
0.70±0.07
0.50±0.05
21.033±0.600
22/1
Figure1(j)54 < ·z < 72
0.300±0.004
2.70±0.02
0.80±0.04
1.561±0.040
66/3
0.188±0.001
2.93±0.02
1.60±0.02
11.850±0.200
145/12
0.195±0.003
2.60±0.03
1.50±0.02
18.466±0.400
81/12
0.215±0.005
2.30±0.03
0.80±0.03
21.028±0.500
60/9
0.220±0.004
2.10±0.03
0.80±0.02
21.284±0.400
72/9
Figure1(k)54<z<72
0.242±0.005
1.80±0.02
0.50±0.03
19.705±0.300
76/9
0.169±0.002
2.20±0.02
0.85±0.03
19.159±0.400
36/9
0.168±0.004
2.10±0.04
0.30±0.03
16.753±0.400
13/6
0.165±0.001
1.60±0.01
0.30±0.01
14.510±0.200
87/6
0.166±0.001
1.85±0.03
0.30±0.02
15.317±0.300
35/6
Figure1(l)54<z<72
0.166±0.001
1.70±0.05
0.35±0.03
14.737±0.200
94/6
0.138±0.002
2.20±0.02
0.60±0.02
26.931±0.400
22/3
0.160±0.002
1.75±0.03
0.20±0.03
23.055±0.400
5/2
0.155±0.002
1.50±0.04
0.35±0.03
20.234±0.240
8/1
0.182±0.001
0.85±0.02
0.65±0.01
17.919±0.280
16/1
Figure1(m)54<z<72
0.320±0.005
2.60±0.03
1.00±0.03
1.791±0.020
49/3
0.210±0.003
2.80±0.02
1.60±0.03
10.081±0.300
56/12
0.195±0.002
2.65±0.03
1.30±0.03
13.984±0.400
67/12
0.215±0.004
2.30±0.02
0.70±0.03
15.619±0.300
52/9
0.220±0.003
2.10±0.03
0.50±0.03
15.793±0.400
66/9
Figure1(n)72<z<90
0.242±0.003
1.80±0.02
0.40±0.02
15.019±0.300
64/9
0.169±0.003
2.14±0.02
0.85±0.03
14.177±0.400
73/9
0.168±0.004
2.10±0.03
0.30±0.03
12.801±0.300
9/6
0.210±0.003
1.60±0.03
0.30±0.02
11.752±0.200
14/6
0.175±0.003
1.80±0.05
0.20±0.03
11.763±0.200
36/6
Figure1(o)72<z<90
0.166±0.005
1.70±0.04
0.25±0.05
11.174±0.200
66/6
0.138±0.002
2.00±0.03
0.50±0.03
18.580±0.400
35/3
0.160±0.002
1.75±0.02
0.20±0.02
18.836±0.200
13/2
0.155±0.002
1.50±0.06
0.35±0.02
14.886±0.400
3/1
0.200±0.003
0.85±0.04
0.45±0.03
14.089±0.400
19/1
Figure1(p)z=90
0.320±0.001
2.60±0.01
1.00±0.01
16.231±0.200
46/3
0.200±0.003
2.80±0.02
1.60±0.03
20.333±0.200
97/12
0.195±0.005
2.62±0.03
1.30±0.03
16.733±0.200
88/12
0.210±0.004
2.30±0.02
0.70±0.03
13.641±0.300
50/9
0.210±0.002
2.00±0.01
0.40±0.02
10.621±0.200
104/9
Figure1(q)z=90
0.242±0.003
1.60±0.02
0.10±0.02
14.572±0.400
83/9
0.169±0.003
1.80±0.03
0.90±0.02
8.858±0.400
64/6
0.170±0.002
1.60±0.03
0.60±0.02
7.038±0.120
118/6
0.240±0.002
1.00±0.02
0.20±0.03
4.561±0.240
71/3
0.150±0.003
1.82±0.03
0.10±0.02
3.306±0.200
29/2Table 2
Values ofT, ymax, ymin, N0, χ2, and ndof corresponding to the curves in Figure 2 in which different data are measured in different θ and z ranges. In the table, z is in the units of cm and θ is not listed, which appears in Figure 2. In one case, ndof is negative which appears in terms of “–” and the corresponding curve is just for eye guiding only.
Figure
T (GeV)
ymax
ymin
N0 (×0.001)
χ2/ndof
Figure2(a)0<z<18
0.320±0.005
2.30±0.02
1.10±0.02
0.077±0.002
33/2
0.138±0.004
3.00±0.03
1.72±0.03
0.702±0.020
116/12
0.195±0.004
2.40±0.02
1.00±0.03
1.987±0.060
71/12
0.208±0.003
2.36±0.04
0.70±0.03
4.859±0.100
88/9
0.230±0.003
2.10±0.02
0.40±0.04
8.318±0.120
18/9
Figure2(b)0<z<18
0.260±0.004
1.80±0.02
0.40±0.02
11.530±0.200
33/9
0.169±0.002
2.14±0.05
0.65±0.03
13.579±0.200
18/9
0.168±0.004
2.05±0.03
0.30±0.03
14.905±0.300
27/6
0.200±0.003
1.60±0.02
0.50±0.03
14.783±0.300
50/6
0.175±0.003
1.77±0.02
0.35±0.04
15.571±0.300
37/6
Figure2(c)0<z<18
0.210±0.030
1.35±0.03
0.25±0.02
15.253±0.200
39/6
0.138±0.002
2.00±0.04
0.50±0.03
27.457±0.400
28/3
0.138±0.002
1.75±0.04
0.60±0.03
23.691±0.400
38/2
0.155±0.003
1.05±0.02
0.65±0.02
21.444±0.400
12/1
0.160±0.003
0.85±0.02
0.75±0.03
30.222±0.400
2/–
Figure2(d)18<z<36
0.280±0.004
2.40±0.03
0.06±0.03
0.313±0.020
57/3
0.138±0.002
3.00±0.02
1.52±0.03
3.618±0.140
73/12
0.225±0.003
2.30±0.02
1.00±0.02
11.850±0.200
84/12
0.208±0.003
2.36±0.03
0.70±0.03
19.958±0.300
69/9
0.228±0.002
2.10±0.01
0.50±0.02
24.746±0.400
9/9
Figure2(e)18<z<36
0.262±0.002
1.70±0.02
0.20±0.02
25.438±0.300
57/9
0.182±0.004
2.00±0.01
0.65±0.03
25.160±0.200
64/9
0.169±0.003
2.05±0.03
0.30±0.05
24.000±0.300
21/6
0.200±0.005
1.60±0.03
0.40±0.02
22.373±0.300
30/6
0.175±0.002
1.77±0.01
0.35±0.05
21.727±0.400
55/6
Figure2(f)18<z<36
0.270±0.004
1.00±0.02
0.35±0.03
19.006±0.200
84/6
0.139±0.003
2.00±0.04
0.50±0.03
35.492±0.600
28/3
0.138±0.002
1.75±0.05
0.60±0.03
31.840±0.600
52/2
0.155±0.003
1.00±0.02
0.65±0.02
28.264±0.600
7/1
0.080±0.005
1.40±0.02
1.16±0.03
23.464±0.600
176/0
Figure2(g)36<z<54
0.280±0.004
2.40±0.03
0.06±0.03
0.690±0.020
36/3
0.149±0.003
3.00±0.03
1.52±0.04
7.345±0.159
95/12
0.225±0.003
2.45±0.03
0.60±0.02
16.629±0.360
79/12
0.208±0.002
2.36±0.03
0.60±0.03
19.895±0.300
58/9
0.218±0.002
2.10±0.02
0.30±0.03
20.966±0.400
33/9
Figure2(h)36<z<54
0.260±0.003
1.70±0.02
0.20±0.02
22.186±0.400
48/9
0.182±0.002
2.00±0.03
0.55±0.03
20.936±0.360
63/9
0.170±0.002
2.05±0.03
0.30±0.03
20.232±0.300
25/6
0.210±0.004
1.60±0.04
0.40±0.03
17.743±0.200
9/6
0.175±0.002
1.77±0.01
0.35±0.02
18.830±0.300
41/6
Figure2(i)36<z<54
0.270±0.002
1.00±0.01
0.15±0.01
16.232±0.200
74/6
0.140±0.002
2.00±0.01
0.50±0.02
31.730±0.640
30/3
0.138±0.002
1.75±0.02
0.40±0.02
24.610±0.400
47/2
0.155±0.002
1.05±0.02
0.70±0.03
23.472±0.600
9/1
0.090±0.003
1.40±0.02
0.16±0.01
19.198±0.400
8/0
Figure2(j)54<z<72
0.280±0.004
2.50±0.03
0.06±0.03
0.943±0.020
41/3
0.149±0.003
3.05±0.02
1.52±0.03
7.362±0.200
65/12
0.225±0.004
2.40±0.02
0.60±0.02
13.744±0.200
61/12
0.208±0.003
2.36±0.01
0.60±0.03
15.998±0.200
60/9
0.220±0.003
2.08±0.02
0.30±0.03
16.243±0.240
13/9
Figure2(k)54<z<72
0.260±0.004
1.70±0.02
0.30±0.02
16.868±0.240
50/9
0.184±0.002
2.00±0.02
0.55±0.03
15.986±0.200
70/9
0.170±0.003
2.05±0.03
0.20±0.02
15.533±0.200
28/6
0.210±0.004
1.55±0.02
0.20±0.03
13.369±0.200
48/6
0.175±0.003
1.77±0.03
0.35±0.02
14.123±0.200
25/6
Figure2(l)54<z<72
0.280±0.004
1.00±0.03
0.15±0.03
12.808±0.100
66/6
0.140±0.002
2.00±0.02
0.40±0.02
23.940±0.400
31/3
0.138±0.002
1.75±0.03
0.40±0.04
19.141±0.400
44/2
0.155±0.002
1.05±0.03
0.70±0.02
15.698±0.400
16/1
0.135±0.003
2.70±0.02
0.66±0.02
15.298±0.400
26/1
Figure2(m)72<z<90
0.280±0.002
2.50±0.03
0.06±0.02
1.019±0.020
43/3
0.149±0.003
3.05±0.03
1.52±0.03
6.509±0.240
73/12
0.225±0.002
2.37±0.02
0.60±0.03
9.724±0.200
88/12
0.208±0.002
2.30±0.03
0.40±0.03
11.638±0.200
86/9
0.220±0.002
2.02±0.02
0.25±0.02
12.192±0.200
23/9
Figure2(n)72<z<90
0.260±0.003
1.60±0.02
0.30±0.02
12.207±0.200
51/9
0.188±0.002
1.90±0.03
0.55±0.03
11.864±0.200
33/9
0.170±0.003
2.05±0.02
0.20±0.03
11.571±0.200
7/6
0.210±0.003
1.55±0.02
0.20±0.02
11.074±0.200
45/6
0.175±0.003
1.77±0.02
0.35±0.03
11.769±0.200
41/6
Figure2(o)72<z<90
0.280±0.003
1.00±0.03
0.15±0.04
10.322±0.200
50/6
0.140±0.003
2.00±0.04
0.30±0.03
18.405±0.400
23/3
0.138±0.004
1.75±0.04
0.40±0.05
15.381±0.400
30/2
0.155±0.004
1.05±0.03
0.70±0.03
12.841±0.600
15/1
0.135±0.002
2.70±0.04
0.06±0.04
11.641±0.400
19/1
Figure2(p)z=90
0.280±0.006
2.60±0.04
0.06±0.01
9.687±0.200
47/3
0.147±0.002
3.10±0.02
1.32±0.02
14.156±0.140
18/12
0.225±0.003
2.32±0.03
0.60±0.03
11.283±0.200
70/12
0.208±0.002
2.30±0.02
0.40±0.03
10.248±0.160
66/9
0.220±0.003
2.00±0.02
0.30±0.02
8.134±0.200
64/9
Figure2(q)z=90
0.260±0.004
1.55±0.01
0.30±0.02
12.255±0.400
91/9
0.188±0.004
1.80±0.02
0.25±0.02
8.593±0.400
30/6
0.165±0.002
1.80±0.03
0.20±0.03
5.816±0.320
37/6
0.210±0.005
1.30±0.01
0.10±0.03
4.394±0.120
13/3
0.175±0.003
1.33±0.03
0.05±0.04
3.019±0.040
34/2Figure3 presents the momentum spectra of (a, b) π+ and (c, d) π− in (a–c) θ=20–40 mrad and (b–d) θ=100–140 mrad in six z 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 T, ymax, ymin, N0, χ2, and ndof corresponding to the fits for the spectra of π+ and π− are listed in Table 3. One can see again that the theoretical model results are approximately in agreement with the experimental data of π+ and π−.Figure 3
Same as Figure1 but showing the spectra of (a–b) π+ and (c–d) π− in (a–c) θ=20–40 mrad and (b–d) θ=100–140 mrad in six z ranges.Table 3
Values ofT, ymax, ymin, N0, χ2, and ndof corresponding to the curves in Figure 3 in which different data are measured in different z and θ ranges. In the table, θ is in the units of mrad and z is not listed, which appears in Figure 3.
Figure
T (GeV)
ymax
ymin
N0 (×0.001)
χ2/ndof
Figure3(a)20<θ<40
0.400±0.004
1.91±0.02
1.50±0.02
1.622±0.040
48/12
0.400±0.004
1.92±0.03
1.37±0.02
6.272±0.100
57/12
0.400±0.004
1.97±0.02
1.37±0.02
10.672±0.200
95/12
0.400±0.003
2.05±0.01
1.30±0.02
11.335±0.240
55/12
0.400±0.005
2.05±0.02
1.30±0.02
8.904±0.240
68/12
0.340±0.001
2.22±0.01
1.20±0.02
19.874±0.400
231/12
Figure3(b)100<θ<140
0.200±0.004
2.01±0.03
0.80±0.02
30.276±0.400
30/9
0.230±0.003
1.90±0.02
0.20±0.02
58.805±1.600
19/9
0.228±0.002
1.89±0.03
0.20±0.02
46.620±1.600
11/9
0.230±0.003
1.86±0.02
0.10±0.02
34.882±0.800
17/9
0.220±0.004
1.85±0.03
0.35±0.02
26.181±0.600
20/9
0.235±0.003
1.75±0.02
0.10±0.02
12.809±0.400
37/9
Figure3(c)20<θ<40
0.440±0.005
1.80±0.02
0.30±0.03
1.034±0.020
57/12
0.400±0.004
1.92±0.03
0.50±0.03
4.136±0.200
16/12
0.400±0.004
1.97±0.03
0.30±0.02
7.232±0.160
52/12
0.400±0.005
2.00±0.03
0.30±0.03
7.329±0.200
23/12
0.444±0.004
1.82±0.02
0.50±0.02
5.935±0.100
26/12
0.345±0.005
2.08±0.02
0.60±0.02
12.610±0.200
40/12
Figure3(d)100<θ<140
0.202±0.002
2.03±0.02
0.30±0.02
24.814±0.480
17/9
0.230±0.005
1.86±0.02
0.10±0.02
48.532±0.800
15/9
0.230±0.004
1.81±0.02
0.10±0.02
40.351±0.600
36/9
0.240±0.004
1.73±0.02
0.23±0.02
30.234±0.800
11/9
0.245±0.002
1.70±0.02
0.22±0.02
23.424±0.520
27/9
0.245±0.002
1.68±0.02
0.10±0.02
10.480±0.520
36/9Similar to Figures1 and 2, Figures 4 and 5 show the momentum spectra of positively and negatively charged kaons (K+ and K−) produced in p-C collisions at 31 GeV/c, respectively. Panels (a–f) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The values of T, ymax, ymin, N0, χ2, and ndof corresponding to the fits for the spectra of K+ and K− 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 K+ and K−.Figure 4
Same as Figure1 but showing the spectra of K+. Panels (a–f) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.Figure 5
Same as Figure1 but showing the spectra of K−. Panels (a–f) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.Table 4
Values ofT, ymax, ymin, N0, χ2, and ndof corresponding to the curves in Figure 4 in which different data are measured in different θ and z ranges. In the table, z is in the units of cm and θ is not listed, which appears in Figure 4.
Figure
T (GeV)
ymax
ymin
N0 (×0.001)
χ2/ndof
Figure4(a)0<z<18
0.300±0.003
2.00±0.03
1.20±0.03
0.335±0.012
18/2
0.300±0.003
1.90±0.02
1.10±0.03
3.079±0.120
30/2
0.300±0.003
1.45±0.03
1.10±0.03
4.787±0.180
24/2
0.300±0.004
1.10±0.02
0.60±0.03
6.462±0.300
31/2
Figure4(b)18<z<36
0.400±0.003
2.00±0.03
1.20±0.03
2.569±0.060
8/2
0.300±0.003
2.00±0.03
1.10±0.02
9.552±0.300
26/2
0.400±0.004
1.00±0.02
0.95±0.01
8.080±0.180
27/2
0.320±0.004
1.08±0.02
0.70±0.03
8.773±0.100
25/2
Figure4(c)36<z<54
0.420±0.004
1.75±0.02
1.35±0.02
3.828±0.060
47/4
0.300±0.004
1.85±0.04
1.10±0.03
7.922±0.180
39/2
0.280±0.004
1.45±0.04
1.35±0.03
6.561±0.240
21/2
0.316±0.004
1.15±0.03
0.40±0.04
7.016±0.500
14/2
Figure4(d)54<z<72
0.400±0.005
2.00±0.03
1.30±0.02
4.065±0.180
38/4
0.300±0.003
1.80±0.03
1.05±0.03
5.934±0.300
18/2
0.360±0.005
1.35±0.03
0.60±0.06
4.578±0.180
13/2
0.275±0.002
1.02±0.01
1.00±0.01
5.361±0.200
38/2
Figure4(e)72<z<90
0.400±0.004
2.20±0.03
1.35±0.03
3.806±0.120
14/4
0.300±0.004
1.80±0.03
1.05±0.03
4.641±0.120
32/2
0.280±0.003
1.45±0.03
1.20±0.03
3.491±0.120
17/2
0.300±0.004
1.10±0.03
0.50±0.04
4.315±0.150
22/2
Figure4(f)z=90
0.400±0.003
2.20±0.03
1.00±0.03
6.300±0.120
10/5
0.300±0.006
1.79±0.03
1.00±0.02
3.123±0.060
36/2Table 5
Values ofT, ymax, ymin, N0, χ2, and ndof corresponding to the curves in Figure 5 in which different data are measured in different θ and z ranges. In the table, z is in the units of cm and θ is not listed, which appears in Figure 5.
Figure
T (GeV)
ymax
ymin
N0 (×0.001)
χ2/ndof
Figure5(a)0<z<18
0.280±0.005
1.90±0.03
1.80±0.03
0.191±0.006
27/2
0.300±0.005
1.70±0.03
1.10±0.02
1.173±0.060
18/2
0.300±0.003
1.20±0.01
1.10±0.01
1.997±0.060
67/2
0.290±0.005
1.10±0.04
0.50±0.02
2.685±0.100
19/2
Figure5(b)18<z<36
0.400±0.005
1.73±0.03
1.20±0.03
0.966±0.060
39/2
0.300±0.003
1.50±0.03
1.30±0.03
3.386±0.120
72/2
0.400±0.003
1.00±0.02
0.70±0.01
2.796±0.090
18/2
0.320±0.003
0.91±0.02
0.70±0.02
3.752±0.150
25/2
Figure5(c)36<z<54
0.420±0.005
1.45±0.02
1.40±0.01
1.247±0.030
33/4
0.340±0.003
1.65±0.03
1.10±0.03
3.308±0.150
18/2
0.248±0.003
1.45±0.03
1.30±0.03
2.580±0.090
18/2
0.300±0.003
1.10±0.03
0.60±0.01
2.800±0.100
29/2
Figure5(d)54<z<72
0.440±0.010
1.45±0.07
1.30±0.05
1.211±0.060
67/4
0.300±0.004
1.70±0.03
1.05±0.03
2.225±0.060
21/2
0.330±0.005
1.10±0.03
1.03±0.03
2.058±0.090
17/2
0.288±0.002
1.00±0.02
0.95±0.02
2.131±0.100
55/2
Figure5(e)72<z<90
0.330±0.004
1.85±0.03
1.28±0.02
0.894±0.060
39/4
0.260±0.004
2.00±0.04
0.60±0.04
1.811±0.060
10/2
0.245±0.004
1.45±0.03
1.00±0.02
1.496±0.060
5/2
0.300±0.004
0.95±0.03
0.63±0.02
1.773±0.100
23/2
Figure5(f)z=90
0.350±0.003
2.00±0.02
1.00±0.02
1.640±0.060
36/5
0.300±0.004
1.60±0.04
1.20±0.03
1.283±0.060
39/2Similar to Figure1, Figure 6 shows the momentum spectra of p 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 z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively. The values of T, k, ymax, ymin, yLmax, yLmin, N0, χ2, and ndof corresponding to the fits for the spectra are listed in Table 6. In a few cases, ndof 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.Figure 6
Same as Figure1 but showing the spectra of p. Panels (a, b), (c, d), (e, f), (g, h), (i, j), and (k, l) represent the spectra for z=0–18, 18–36, 36–54, 54–72, 72–90, and 90 cm, respectively.Table 6
Values ofT, k, ymax, ymin, yLmax, yLmax, N0, χ2, and ndof corresponding to the curves in Figure 6 in which different data are measured in different θ and z ranges. In the table, z is in the units of cm and θ is not listed, which appears in Figure 6. In a few cases, ndof are negative which appear in terms of “–” and the corresponding curves are just for eye guiding only.
Figure
T (GeV)
k
ymax
ymin
yLmax
yLmin
N0 (×0.001)
χ2/ndof
Figure6(a)0<z<18
0.400±0.004
0.40±0.01
1.60±0.05
1.00±0.10
4.00±0.07
3.55±0.08
2.996±0.200
10/–
0.280±0.004
0.40±0.02
1.50±0.10
1.20±0.10
3.10±0.10
2.70±0.10
1.337±0.020
118/9
0.200±0.005
0.40±0.02
1.50±0.08
1.20±0.10
3.45±0.10
2.00±0.08
2.145±0.100
86/7
0.127±0.003
0.40±0.01
1.55±0.07
1.50±0.06
3.45±0.05
2.20±0.05
7.216±0.160
202/6
0.125±0.005
0.45±0.01
1.55±0.05
0.90±0.05
3.45±0.08
2.00±0.05
10.156±0.400
128/6
Figure6(b)0<z<18
0.115±0.003
0.40±0.01
0.98±0.02
0.80±0.02
3.90±0.08
1.30±0.05
11.629±0.400
50/3
0.150±0.004
0.54±0.02
0.80±0.03
0.70±0.03
2.80±0.07
0.60±0.10
13.653±0.400
57/3
0.164±0.004
0.53±0.02
0.80±0.03
0.70±0.03
3.50±0.10
0.60±0.20
13.653±0.400
57/–
0.177±0.005
0.53±0.02
0.80±0.05
0.70±0.05
3.00±0.12
1.50±0.10
12.291±0.800
28/–
0.140±0.010
0.35±0.02
0.40±0.05
0.30±0.05
3.50±0.10
1.00±0.05
27.176±0.800
5/–
Figure6(c)18<z<36
0.450±0.010
0.53±0.02
1.60±0.10
1.00±0.10
4.00±0.10
3.55±0.05
8.988±0.400
9/–
0.280±0.006
0.43±0.02
1.50±0.20
1.20±0.20
3.10±0.03
2.70±0.03
7.496±0.300
117/9
0.200±0.003
0.49±0.01
1.50±0.05
1.20±0.06
3.45±0.05
2.10±0.03
14.518±0.400
42/7
0.130±0.006
0.45±0.02
1.60±0.02
1.30±0.03
3.45±0.03
2.20±0.01
28.178±0.800
171/6
0.125±0.005
0.45±0.02
1.55±0.03
0.85±0.03
3.45±0.05
2.00±0.05
22.773±0.480
118/7
Figure6(d)18<z<36
0.115±0.003
0.40±0.01
1.00±0.03
0.80±0.03
3.90±0.04
1.40±0.04
20.745±0.600
40/3
0.120±0.003
0.54±0.01
0.65±0.03
0.60±0.03
2.80±0.05
0.90±0.05
20.636±0.600
50/3
0.158±0.003
0.57±0.03
0.60±0.04
0.50±0.04
3.50±0.10
0.80±0.05
19.404±0.400
18/–
0.167±0.002
0.55±0.02
0.65±0.02
0.55±0.02
3.00±0.30
1.50±0.08
19.730±0.400
18/–
0.180±0.006
0.34±0.01
0.40±0.03
0.36±0.02
3.50±0.06
1.50±0.02
47.883±0.800
6/–
Figure6(e)36<z<54
0.455±0.005
0.53±0.01
1.60±0.05
1.00±0.06
4.06±0.06
3.55±0.05
18.555±0.400
11/–
0.290±0.004
0.53±0.01
1.50±0.15
1.20±0.15
3.10±0.02
2.70±0.03
15.862±0.300
94/9
0.200±0.005
0.50±0.01
1.50±0.08
1.20±0.10
3.45±0.05
2.10±0.03
22.391±0.600
35/7
0.130±0.004
0.45±0.01
1.60±0.01
1.30±0.03
3.45±0.03
2.15±0.01
28.281±0.400
173/6
0.125±0.003
0.45±0.02
1.55±0.02
0.88±0.02
3.45±0.02
2.00±0.02
20.981±0.400
118/6
Figure6(f)36<z<54
0.115±0.002
0.35±0.01
1.00±0.02
0.80±0.02
4.00±0.03
1.50±0.02
19.587±0.400
42/3
0.120±0.004
0.58±0.02
0.70±0.02
0.50±0.02
2.80±0.02
0.90±0.03
18.925±0.400
48/3
0.164±0.001
0.60±0.02
0.60±0.03
0.50±0.03
3.50±0.20
0.80±0.08
17.285±0.400
23/–
0.155±0.005
0.55±0.01
0.65±0.04
0.55±0.04
3.00±0.20
1.50±0.10
16.519±0.400
12/–
0.178±0.004
0.43±0.01
0.53±0.03
0.36±0.03
3.50±0.15
1.50±0.02
37.825±0.960
7/–
Figure6(g)54<z<72
0.450±0.004
0.53±0.01
1.60±0.20
1.00±0.20
4.00±0.05
3.55±0.05
24.718±0.500
13/–
0.290±0.003
0.53±0.01
1.50±0.03
1.20±0.15
3.10±0.02
2.70±0.03
19.202±0.240
32/9
0.200±0.005
0.50±0.01
1.50±0.10
1.20±0.10
3.30±0.06
2.10±0.05
18.359±0.200
31/7
0.130±0.002
0.45±0.01
1.60±0.01
1.30±0.01
3.45±0.03
2.15±0.01
23.629±0.400
174/6
0.130±0.003
0.45±0.01
1.55±0.02
0.88±0.02
3.45±0.03
2.00±0.03
17.502±0.400
129/6
Figure6(h)54<z<72
0.145±0.003
0.45±0.02
0.90±0.02
0.70±0.03
4.00±0.10
0.95±0.03
15.343±0.400
32/3
0.115±0.005
0.55±0.01
0.65±0.03
0.50±0.03
3.00±0.05
1.00±0.05
14.933±0.400
66/3
0.160±0.003
0.60±0.02
0.55±0.03
0.50±0.03
3.50±0.20
0.80±0.02
14.798±0.400
9/–
0.150±0.004
0.55±0.02
0.65±0.05
0.55±0.05
3.00±0.10
1.50±0.05
13.657±0.400
16/–
0.170±0.003
0.43±0.01
0.53±0.03
0.36±0.03
3.50±0.07
1.50±0.03
29.279±0.960
37/–
Figure6(i)72<z<90
0.450±0.004
0.53±0.01
1.60±0.03
1.00±0.05
4.00±0.10
3.55±0.10
26.965±0.600
6/–
0.290±0.003
0.53±0.01
1.50±0.10
1.20±0.15
3.10±0.03
2.70±0.03
17.031±0.280
33/9
0.160±0.004
0.50±0.01
1.50±0.03
1.20±0.02
3.45±0.03
2.10±0.03
13.722±0.240
19/7
0.140±0.004
0.60±0.01
1.60±0.03
0.50±0.03
3.60±0.02
1.80±0.03
18.999±0.480
99/6
0.130±0.004
0.50±0.01
1.55±0.03
0.88±0.02
3.45±0.03
0.90±0.03
13.342±0.520
133/6
Figure6(j)72<z<90
0.145±0.005
0.53±0.01
0.90±0.04
0.40±0.05
4.00±0.04
0.95±0.05
12.922±0.520
8/3
0.120±0.001
0.55±0.01
0.65±0.02
0.50±0.01
3.00±0.02
1.00±0.03
11.877±0.480
51/3
0.160±0.002
0.63±0.02
0.55±0.01
0.50±0.01
3.50±0.15
0.80±0.05
11.686±0.320
10/–
0.155±0.006
0.60±0.03
0.65±0.03
0.55±0.03
3.00±0.15
1.20±0.06
10.773±0.400
25/–
0.170±0.003
0.50±0.01
0.53±0.02
0.36±0.03
3.50±0.20
1.50±0.05
20.968±0.400
56/–
Figure6(k)z=90
0.450±0.010
0.53±0.02
1.60±0.10
1.00±0.04
4.15±0.15
3.55±0.15
193.715±0.460
44/1
0.290±0.005
0.53±0.01
1.50±0.10
1.20±0.10
3.10±0.01
2.70±0.03
36.734±0.460
186/9
0.160±0.002
0.55±0.01
1.50±0.03
1.20±0.02
3.45±0.04
2.10±0.03
15.771±0.500
192/9
0.140±0.004
0.60±0.01
1.60±0.03
0.50±0.05
3.60±0.07
1.80±0.06
14.012±0.400
104/6
Figure6(l)z=90
0.130±0.005
0.48±0.01
1.55±0.03
0.88±0.02
3.45±0.12
1.90±0.03
6.526±0.200
146/6
0.160±0.004
0.60±0.01
0.65±0.05
0.30±0.03
2.30±0.02
0.60±0.04
4.851±0.200
30/1
0.120±0.001
0.55±0.01
0.65±0.03
0.50±0.02
3.00±0.02
1.00±0.04
3.167±0.200
51/1We notice from Tables1–6 that different T for a range of z and its dependence with θ or y are observed, but the development of the model in our previous work [42] concludes that T is independent of y. We would like to explain here that this paper treats T as a differential function of θ or y, which is more detailed, while our previous work treats T as an integral or mean quantity over y. As for which case should be used, it depends on the experimental data.We now analyze the dependence of free parameters onθ and z. Figures 7 and 8 show, respectively, the dependence of T and Δy=ymax−ymin on (a–e) θ, which is extracted from the data samples within different z ranges for π+, π−, K+, K−, and p, respectively, and on (f) z, which are extracted from the data samples within different θ ranges for π+ and π−, where we use Δy to denote the difference between ymax and ymin to avoid trivialness in using both ymax and ymin. In particular, in Figure 8(e), the results with Δy>1 are mainly for leading protons and obtained by yLmax−yLmin. One can see that, for π± and K±, T and Δy decrease slightly with the increase of θ and do not change obviously with the increase of z. The obtained T (Δy) values for negative and positive pions or kaons seem to be very similar as we expect. The data for the antiproton (p¯) are not available in Ref. [32], which forbids in making a comparison for p and p¯ in this paper. In fact, the situation for p is more complex due to the effect of leading protons.Figure 7
Dependence ofT on (a–e) θ, which are extracted from the data samples within different z ranges for π+, π−, K+, K−, and p, respectively, and on (f) z, which are extracted from the data samples within different θ ranges for π+ and π−.Figure 8
Same as Figure7 but showing the dependence of ∆y. Large ∆y=yLmax−yLmin>1 in panel (e) represents mainly the rapidity shifts for leading protons.The dependence ofT and Δy on θ for the productions of π± and K± 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 θ and more energy loss and then lower T. The nuclear stopping can cause smaller Δy. Combining with cascade collisions and nuclear stopping, one can obtain low T and small Δy at large θ for the productions of π± and K±. Because of the effect of leading particles, the situation for the emissions of p is more complex, which shows different trends from those of π± and K±. Meanwhile, the flow effect can cause larger T, which is related to a more complex mechanism.The dependence ofT and Δy on z, which are extracted from the data samples within different θ ranges for (a) π+, (b) π−, (c) K+, (d) K−, and (e) p, are given in Figures 9 and 10, respectively. In particular, large Δy=yLmax−yLmin>1 in Figure 10(e) represent mainly the rapidity shifts of leading protons. In principle, there is no obvious increase or decrease in T and Δy with the increase of z, but there are some statistical fluctuations in a few cases. This result is natural due to the fact that z is not the main factor in a 90 cm long graphite target. It is expected that T and Δy will decrease with the increase of z 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 9
Dependence ofT on z, which are extracted from the data samples within different θ ranges for (a) π+, (b) π−, (c) K+, (d) K−, and (e) p.Figure 10
Same as Figure9 but showing the dependence of ∆y. Large ∆y=yLmax−yLmin>1 in panel (e) represents mainly the rapidity shifts for leading protons.Figure11 displays the dependence of fraction k of nonleading protons on (a) θ and (b) z, which are extracted from the data samples within different z and θ ranges, respectively. One can see that there is no obvious change in the dependence of k on θ, but some statistical fluctuations. There is a slight increase in the dependence of k on z with the increase of z, which can be explained by more energy loss of the beam protons at larger z. This energy loss is small in a not too large z range, which does not affect obviously other free parameters such as T and Δy due to their less sensitivity at the energy in the z range considered in this paper. It is natural that the larger (fewer) fraction k1−k of protons appears as nonleading (leading) particles at lower energy or larger z. 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], k 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], k will also increase due to the increase of target stopping which causes the decrease of leading protons.Figure 11
Dependence ofk on (a) θ and (b) z, which are extracted from the data samples within different z and θ ranges, respectively.Figures12 and 13 show, respectively, the dependence of average pTpT and Ti on (a–e) θ, which are extracted from the data samples within different z ranges for π+, π−, K+, K−, and p, respectively, and on (f) z, which are extracted from the data samples within different θ ranges for π+ and π−, where Ti denotes the initial quasitemperature which is given by the root-mean-square pTpT2 over 2 pT2/2 according to the color string percolation model [44–46]. It should be noted that pT2/2 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 pT2/2 the initial quasitemperature in this paper. The dependence of pT and Ti on z are presented in Figures 14 and 15, respectively, which are extracted from the data samples within different θ ranges. One can see that, for pions and kaons, there are increases in pT and Ti when θ increases. The situation is complex for protons due to the effect of leading protons which have high momenta and result in high pT and Ti at small θ. The produced protons which are nonleading should have a similar trend in pT and Ti 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 pT and Ti when z increases due to not too large energy loss in a 90 cm long graphite target.Figure 12
Dependence ofpT on (a–e) θ, which are extracted from the data samples within different z ranges for π+, π−, K+, K−, and p, respectively, and on (f) z, which are extracted from the data samples within different θ ranges for π+ and π−.Figure 13
Same as Figure12 but showing the dependence of Ti.Figure 14
Dependence ofpT on z, which are extracted from the data samples within different θ ranges for (a) π+, (b) π−, (c) K+, (d) K−, and (e) p.Figure 15
Same as Figure14 but showing the dependence of Ti.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 0.9pLab occupy about 26% [48] of leading protons, which are not enough to cover all leading protons. In particular, for leading protons with momenta being 0.5–0.98pLab, 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 Figures12 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 pT and Ti due to large θ 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 pT and Ti due to small θ for protons. However, nonnegligible leading protons which have high momenta and smaller angles do not experience much multiple scatterings, which renders large pT and Ti at small θ. As a competitive result, protons present a different case from pions and kaons.One can see naturally the coincident trend forpT and Ti in different θ and z ranges. Due to the flow effect not being excluded, the trend of T is inconsistent with that of Ti. As an all-around result, the effects of transverse and longitudinal flows are complex. The flow effect can obviously affect T, which is model dependent. The flow effect also affects pT and Ti which are also model dependent. Therefore, we mention here that T 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 ndof>1 and using pT coverage as widely as possible (pT=0–3 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 T0 and βT for different cases from the spectra of positive particles is plotted in Figure 16, where the circles, squares, and triangles represent the results from π+, K+, and p spectra, respectively. One can see considerable flow-like effect in p-C collisions at 31 GeV/c, which shows a positive correlation between T0 and βT. The kinetic freeze-out temperature T0 is about from 0.080 to 0.135 GeV. The corresponding transverse flow velocity βT is about from 0.21 to 0.42c. Massive particles such as p correspond to larger T0 and smaller βT comparing to π+ at the same or similar θ, which is in agreement with hydrodynamic-type behavior. The flow-like effect observed in this work is slightly less than the flow velocity (0.3c in peripheral and 0.5c in central gold-gold collisions) obtained from the yield ratio of p/π 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].Figure 16
Relation betweenT0 and βT from the blast-wave model. The symbols represent the results from the spectra of positive particles.It should be noted that there is entanglement in determiningT0 and βT. For a given pT spectrum, T0 and βT are negatively correlated, which means an increase in T should result in a decrease of βT. But for a set of pT spectra, after determining T0 and βT for each pT spectrum, the correlation between T0 and βT is possibly positive or negative, which depends on the choices of flow profile function and pT coverage. If the correlation is negative, one may increase T0 and decrease βT by changing the flow profile function and pT coverage and obtain possibly positive correlation. If the correlation is positive, one may decrease T0 and increase βT by changing the flow profile function and pT coverage and obtain possibly negative correlation. Unlike experimental papers, where one finds a single T0 and a common βT by fitting the blast-wave model to the bulk part of the pT 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 n0 (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 n0=2. The value of T0 (βT) 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 usepT 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 pT/2 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 k0. Then, the kinetic freeze-out temperature is k0pT/2, and the transverse flow velocity is 1−k0pT/2m0γ¯, where γ¯ is the mean Lorentz factor of the considered particles in the rest frame of emission source. If we take k0≈0.3 and at large θ, 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 (0.2c for pion emission and 0.1c for proton emission) are qualitatively in agreement with those from the blast-wave model [34, 35] and the afterburner model [50]. The treatment of pT/2 is also model dependent and in agreement with hydrodynamic-type behavior. In addition, larger pT/2 results in larger T0 and βT, which shows positive correlation between T0 and βT. The positive correlation in Figure 16 is also in agreement with the treatment of pT/2.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 χ2/ndof, 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 ofπ+, π−, K+, K−, and p 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, θ, and ranges and longitudinal positions, z, measured by the NA61/SHINE Collaboration at the SPS
(b)
The effective temperatureT and rapidity shifts Δy from the spectra under given experimental conditions which limit various θ and z ranges are obtained. For π± and K±, T and Δy decrease slightly with the increase of θ and do not change obviously with the increase of z. The situation for p is more complex due to the effect of leading protons. There is no obvious change in T and Δy when z increases due to a not too large energy loss in a not too long graphite target. Both T and Δy depend on models. In particular, T contains the contribution of the flow effect, which is not ideal to describe the excitation degree of the emission source
(c)
The fractionk1−k of nonleading (leading) protons in total protons from the spectra in various θ and z ranges are obtained. There is no obvious change in the dependence of k1−k on θ, but some statistical fluctuations. There is a slight increase (decrease) in the dependence of k1−k on z with the increase of z due to more energy loss of the beam protons in the target at larger z. The effect of leading protons cannot be neglected at the SPS energies. It is expected that k1−k will be larger (smaller) at both lower (≤ several GeV) and higher energies (≥ dozens of GeV)
(d)
The average transverse momentumpT and initial quasitemperature Ti from the spectra in various θ and z ranges are obtained. For π± and K±, there are increases in pT and Ti when θ increases. The situation for p is complex due to the effect of leading protons. There is no obvious change in pT and Ti when z increases due to a not too large energy loss in a not too long graphite target. Both pT and Ti 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



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*Source: 9542196-2020-01-17.xml*