Research Article Double-Differential Production Cross...

Research ArticleDouble-Differential Production Cross Sections ofCharged Pions in Charged Pion Induced Nuclear Reactions atHigh Momentums

Ya-Qin Gao,1 Tian Tian,1 S. Fakhraddin,2,3 Magda A. Rahim,2,3 and Fu-Hu Liu1

1 Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China2 Physics Department, Faculty of Science, Sana’a University, P.O. Box 1247, Sana”a, Yemen3 Physics Department, Science & Arts Faculty (Girls Division), Najran University, P.O. Box 1988, Najiran, Saudi Arabia

Correspondence should be addressed to Fu-Hu Liu; [email protected]

Received 5 December 2013; Revised 16 February 2014; Accepted 24 February 2014; Published 30 March 2014

Academic Editor: Seog H. Oh

Copyright © 2014 Ya-Qin Gao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Thepublication of this article was funded by SCOAP3.

The double-differential 𝜋± production cross sections in interactions of charged pions on targets at high momentums are analyzedby using a multicomponent Erlang distribution which is obtained in the framework of a multisource thermal model.The calculatedresults are compared and found to be in agreement with the experimental data at the incident momentums of 3, 5, 8, and 12GeV/cmeasured by the HARP Collaboration. It is found that the source contributions to the meanmomentum of charged particles and tothe distributionwidth of particlemomentums decrease with increase of the emission angle, and the source number and temperaturedo not show an obvious dependence on the emission angle of the considered particle.

1. Introduction

Charged particles are themain products in hadron and heavyion induced nuclear reactions at high energies. In such col-lisions, the particle distributions can provide information onthe properties of interacting system [1–3].These distributionsincludemultiplicity distribution, rapidity and pseudorapiditydistributions, azimuthal and polar angular distributions,momentum and transverse momentum distributions, anddouble-differential cross sections.

Many models have been introduced to describe particleproductions in high energy collisions. For example, theFRITIOF model [4], the VENUS model [5, 6], the RQMDmodel [7–9], the HIJING model [10–12], a multiphase trans-port model (the AMPT model) [13], the Gribov-Glaubermodel [14], the QGSMmodel [15], the color glass condensate(CGC) model [16], the perturbative QCD plus saturationplus hydrodynamics (EKRT)model [17], the ARTmodel [18],the ZPC model [19, 20], the Hydrodynamics model [21, 22],

the string percolationmodel [23], a running coupling nonlin-ear evolution [24], a consistent quantummechanical multiplescattering approach (EPOS) [25, 26], a combinationmodel ofconstituent quarks and Landau hydrodynamics [27], a two-stage gluonmodel or a gluon dominancemodel [28], theKKTmodel [29], a multisource thermal model [30–32], and othersare used in the data analyzes of high energy collisions.

As a challenging investigation, the multisource thermalmodel has been proposed and developed by us to describethe multiplicity distribution of charged particles [33] andisotopic production cross section of emitted fragments [34].It is known that the multisource thermal model is verysimple in describing some experimental data. Particularly, itcan result in a multicomponent Erlang distribution whichdescribes uniformly some distributions. In most cases, theexperimental data have not been presented by the momen-tum distribution, but the transverse momentum distributionof particles. We are interested in analyzing the momentumdistribution in terms of probability or double-differential

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014, Article ID 892582, 20 pageshttp://dx.doi.org/10.1155/2014/892582

2 Advances in High Energy Physics

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0.14𝜋−-Be, 3GeV/c

𝜃= 0.05–0.10 rad, ×2.2(0.55, 4, 1.359)

5GeV/c, ×1.7(0.85, 3, 1.041)

8GeV/c, ×0.7(1.21, 2, 0.444)

12GeV/c, ×1.0(1.61, 2, 1.476)

𝜃 = 0.10–0.15 rad, ×2.2(1.00, 2, 1.988)

×1.3

(0.85, 2, 0.951)×0.7

(1.00, 2, 1.215)×0.9

(1.20, 2, 1.542)

𝜃 = 0.15–0.20 rad, ×2.6(0.68, 2, 0.235)

×1.5

(0.53, 3, 1.664)×0.8

(0.82, 2, 0.623)×0.8

(0.82, 2, 1.727)

𝜃 = 0.20–0.25 rad, ×1.8(0.45, 2, 1.727)

×1.2

(0.34, 4, 1.076)×0.7

(0.32, 5, 1.329)×0.8

(0.31, 5, 1.239)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 1: Double-differential 𝜋+ production cross sections in 𝜋−-Be interactions at high momentums.The circles represent the experimental

data of the HARP Collaboration [35] and the curves are our calculated results. Different panels are scaled by multiplying different amountsmarked in the panels. The three numbers in (⋅ ⋅ ⋅ ) in each panel represent the values of ⟨𝑝

𝑖1⟩,𝑚1, and 𝜒

2/dof, respectively.

cross section for the further test of the multisource thermalmodel.

In this paper, we will use the multisource thermalmodel to describe the double-differential 𝜋

± productioncross sections in charged pion induced nuclear reactionsat high momentums. The model is described in Section 2.The comparisons with the experimental data at the incidentmomentums of 3, 5, 8, and 12GeV/c measured by the HARP

Collaboration [35] are shown in Section 3. Finally, we give ourconclusions in Section 4.

2. The Model and Method

The model used in the present work can be found in ourprevious work [33, 34] which present related formulations

Advances in High Energy Physics 3

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0.14𝜋−-Be, 3GeV/c

𝜃 = 0.05–0.10 rad, ×0.6(1.15, 3, 0.756)

5GeV/c, ×0.4(1.67, 3, 1.576)

8GeV/c, ×0.5(1.90, 2, 0.713)

12GeV/c, ×0.7(2.12, 2, 0.438)

𝜃 = 0.10–0.15 rad, ×0.7(1.33, 3, 1.896)

×0.8

(1.87, 2, 0.626)×0.5

(1.21, 2, 0.246)×0.7

(1.34, 2, 0.811)

𝜃 = 0.15–0.20 rad, ×1.2(0.88, 2, 0.320)

×1.2

(0.67, 3, 1.505)×0.6

(0.89, 2, 0.608)×0.8

(0.94, 2, 0.811)

𝜃 = 0.20–0.25 rad, ×1.7(0.63, 2, 1.844)

×1.0

(0.63, 2, 1.752)×0.8

(0.92, 2, 0.325)×1.2

(0.92, 2, 0.558)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 2: The same as Figure 1, but showing the results for 𝜋− production cross sections.

in terms of multiplicity distribution of final-state productsand neutron number distribution in isotopes. To give a wholepresentation of the present work, we introduce briefly themodel [33, 34] on the multicomponent Erlang distribution interms of charged particle momentum 𝑝 and its distributionin the following. In the model, many emission sources ofparticles and fragments are assumed to form in collisions.According to different interaction mechanisms or eventsamples, the sources are divided into 𝑙 groups (subsamples).The source number in the 𝑗th group is assumed to be𝑚

𝑗.

It is assumed that each source contributes to momentumdistribution to be an exponential function, that is, themomentum (𝑝

𝑖𝑗) distribution contributed by the 𝑖th source

in the 𝑗th group is given by

𝑓𝑖𝑗(𝑝𝑖𝑗) =

1

⟨𝑝𝑖𝑗⟩

exp(−

𝑝𝑖𝑗

⟨𝑝𝑖𝑗⟩

) , (1)

where ⟨𝑝𝑖𝑗⟩ denotes the meanmomentum contributed by the

𝑖th source in the 𝑗th group [33, 34]. Generally, the mean


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0.14𝜋+-Be, 3GeV/c

𝜃= 0.05–0.10 rad, ×0.5(0.76, 5, 3.280)

5GeV/c, ×0.3(2.25, 3, 1.059)

8GeV/c, ×0.6(2.88, 2, 0.425)

12GeV/c, ×0.5(1.46, 3, 1.935)

𝜃 = 0.10–0.15 rad, ×0.7(1.22, 3, 0.756)

×0.8

(1.19, 3, 0.556)×0.6

(0.90, 3, 0.852)×0.6

(1.03, 3, 0.453)

𝜃 = 0.15–0.20 rad, ×1.0(1.05, 2, 0.376)

×0.9

(0.90, 2, 1.025)×0.7

(0.90, 2, 0.475)×0.5

(0.90, 2, 0.667)

𝜃 = 0.20–0.25 rad, ×1.6(0.68, 2, 0.696)

×1.1

(0.75, 2, 0.710)×1.1

(0.98, 2, 0.513)×1.2

(1.02, 2, 0.330)

d2

+

𝜎𝜋

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(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 3: The same as Figure 1, but showing the results in 𝜋+-Be interactions.

momentum contributed by different sources in the samegroup is assumed to be the same. The particle momentumdistribution contributed by the 𝑗th group is then given by thefolding of𝑚

𝑗exponential functions [33, 34]:

𝑓𝑗(𝑝) =

𝑝𝑚𝑗−1

(𝑚𝑗− 1)!⟨𝑝

𝑖𝑗⟩

𝑚𝑗exp(−

𝑝

⟨𝑝𝑖𝑗⟩

) . (2)

One can see that an Erlang distribution is obtained. Themomentum distribution contributed by the 𝑙 groups is givenby a weighted sum of 𝑙 Erlang distributions:

𝑓 (𝑝) =

𝑙

∑

𝑗=1

𝑘𝑗𝑓𝑗(𝑝) , (3)

where 𝑘𝑗denotes the weight factor and obeys the normaliza-

tion ∑𝑙

𝑗=1𝑘𝑗= 1. To avoid the factorial calculation in (2) in


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0.14𝜋+-Be, 3GeV/c

𝜃 = 0.05–0.10 rad, ×1.5(0.50, 4, 1.906)

5GeV/c, ×1.2(1.45, 2, 0.654)

8GeV/c, ×0.7(1.45, 2, 0.136)

12GeV/c, ×0.8(1.66, 2, 0.370)

𝜃 = 0.10–0.15 rad, ×1.6(0.67, 2, 0.374)

×1.5

(1.00, 2, 0.590)×0.8

(1.00, 2, 0.664)×0.6

(1.12, 2, 0.242)

𝜃 = 0.15–0.20 rad, ×1.4(0.50, 2, 0.627)

×1.4

(0.40, 4, 1.593)×1.0

(0.58, 3, 0.211)×0.8

(0.90, 2, 1.602)

𝜃 = 0.20–0.25 rad, ×2.0(0.15, 8, 0.267)

×1.2

(0.34, 4, 1.076)×0.9

(0.30, 6, 1.185)×0.8

(0.34, 5, 1.092)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 4: The same as Figure 1, but showing the results for 𝜋− production cross sections in 𝜋+- Be interactions.

the case of𝑚𝑗being a large value, the Monte Carlo method is

used to calculate the momentum distribution [33, 34].In the Monte Carlo calculation, let 𝑅

𝑖𝑗denote random

variable in [0, 1]. We have

𝑝𝑖𝑗= −⟨𝑝

𝑖𝑗⟩ ln𝑅

𝑖𝑗(4)

for the 𝑖th source in the 𝑗th group due to (1) which obeys theexponential function.Themomentum contributed by the 𝑗thgroup is

𝑝 = −

𝑚𝑗

∑

𝑖=1

⟨𝑝𝑖𝑗⟩ ln𝑅

𝑖𝑗(5)


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0.14𝜋−-C, 3GeV/c

𝜃= 0.05–0.10 rad, ×1.5(0.60, 4, 0.647)

5GeV/c, ×1.2(1.48, 2, 0.490)

8GeV/c, ×0.6(1.26, 2, 0.538)

12GeV/c, ×0.5(1.50, 2, 0.816)

𝜃 = 0.10–0.15 rad, ×1.3(0.69, 2, 0.765)

×1.3

(0.62, 3, 0.401)×0.7

(1.00, 2, 1.295)×0.5

(0.76, 3, 1.420)

𝜃 = 0.15–0.20 rad, ×1.6(0.65, 2, 0.481)

×1.4

(0.51, 3, 0.479)×1.0

(0.63, 3, 0.989)×0.6

(0.66, 3, 1.908)

𝜃 = 0.20–0.25 rad, ×1.7(0.17, 8, 0.272)

×1.2

(0.37, 4, 1.442)×0.7

(0.27, 6, 1.033)×0.5

(0.32, 5, 1.146)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 5: The same as Figure 1, but showing the results for 𝜋+ production cross sections in 𝜋−-C interactions.

due to the fact that (2) is the folding of 𝑚𝑗exponential

functions. The mean momentum contributed by the 𝑙 groupsis

⟨𝑝⟩ =

𝑙

∑

𝑗=1

𝑘𝑗⟨𝑝𝑖𝑗⟩𝑚𝑗 (6)

due to the fact that (3) is a weighted sum.

To obtain the parameter values, for the purpose ofconvenience, we can use the idea of the least-square method.The values of ⟨𝑝

𝑖𝑗⟩ and𝑚

𝑗are changed from low to high step

by step and the values of 𝜒2 can be obtained. The formerparameter can change continuously and the latter one is aseries of integers.Thebest parameter values correspond to theminimum 𝜒

2 value, and the acceptable 𝜒2 values determinatethe parameter errors.


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0.14𝜋−-C, 3GeV/c

𝜃 = 0.05–0.10 rad, ×0.6(0.88, 3, 1.951)

5GeV/c, ×0.4(2.00, 3, 1.119)

8GeV/c, ×0.5(2.12, 2, 0.359)

12GeV/c, ×0.4(2.10, 2, 0.484)

𝜃 = 0.10–0.15 rad, ×0.6(1.95, 2, 3.136)

×0.7

(1.82, 2, 0.374)×0.5

(1.20, 2, 0.564)×0.4

(0.86, 3, 0.827)

𝜃 = 0.15–0.20 rad, ×0.8(0.66, 3, 0.664)

×0.8

(0.82, 2, 0.759)×0.5

(0.61, 3, 1.004)×0.4

(0.61, 3, 1.297)

𝜃 = 0.20–0.25 rad, ×1.4(0.70, 2, 0.636)

×1.2

(0.72, 2, 1.739)×0.8

(0.85, 2, 0.519)×0.6

(0.83, 2, 0.853)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 6: The same as Figure 1, but showing the results for 𝜋− production cross sections in 𝜋−-C interactions.

In the model, the value of 𝑙 denotes the number of typesof emission sources. Generally, 𝑙 = 1 − 3, which renders asmall number of types of emission sources.The total numberof emission sources can be small or large. A small numberof emission sources means the hadronic sources, and a largenumber of emission sources means the parton degree offreedom. In the charged pion induced nuclear reactions at 3–12GeV/c which are considered in the present work, we expect

a small number of emission sources due to the saturationeffect of target nuclei and not too high incident momentums.

The imbalances in mechanics and geometry render theinteracting system to have kinetic, hydrodynamical, andthermaldynamical evolutions. The system or subsystemsare assumed to stay in an equilibrium state or in localequilibrium states at the stage of chemical freeze-out. Inthe model, the inverse slope (mean transverse momentum)


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0.14𝜋+-C, 3GeV/c

𝜃= 0.05–0.10 rad, ×0.4(1.10, 4, 4.262)

5GeV/c, ×0.3(1.60, 4, 1.949)

8GeV/c, ×0.5(2.64, 2, 0.472)

12GeV/c, ×0.3(2.10, 2, 0.466)

𝜃 = 0.10–0.15 rad, ×0.5(1.25, 3, 1.284)

×0.7

(2.22, 2, 0.395)×0.7

(0.88, 3, 0.635)×0.4

(1.33, 2, 0.812)

𝜃 = 0.15–0.20 rad, ×0.9(1.64, 2, 0.141)

×0.8

(0.95, 2, 0.684)×0.6

(0.97, 2, 0.202)×0.6

(0.81, 3, 0.254)

𝜃 = 0.20–0.25 rad, ×1.9(0.50, 3, 0.133)

×1.0

(0.78, 2, 1.223)×0.6

(0.88, 2, 0.453)×0.7

(0.83, 2, 0.874)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 7: The same as Figure 1, but showing the results for 𝜋+ production cross sections in 𝜋+-C interactions.

in the exponential distribution of transverse momentumsis approximately regarded as the temperature parameter.According to the relation between the transverse momentum𝑝𝑇and the momentum 𝑝, we have the temperature 𝑇 ≈

∑𝑙

𝑗=1𝑘𝑗⟨𝑝𝑖𝑗⟩ sin 𝜃 for a given polar angle 𝜃. Although there is

a 𝜃 in the presentation of 𝑇, we do not expect that there is adependence of 𝑇 on 𝜃.

3. Comparisons with Experimental Data

The double-differential 𝜋+ and 𝜋− production cross sections

measured in different emission angle (𝜃) ranges in 𝜋−-Be

interactions at 3, 5, 8, and 12GeV/c are shown in Figures 1and 2, respectively. Different panels are scaled by multiplyingdifferent amounts to give all of them in a whole figure.The circles represent the experimental data of the HARP


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14𝜋+-C, 3GeV/c

𝜃 = 0.05–0.10 rad, ×1.4(1.05, 4, 1.943)

5GeV/c, ×0.4(1.52, 2, 0.472)

8GeV/c, ×0.8(1.50, 2, 0.464)

12GeV/c, ×0.4(1.20, 3, 1.171)

𝜃 = 0.10–0.15 rad, ×1.4(0.78, 2, 1.101)

×0.7

(0.73, 2, 1.732)×0.6

(0.88, 2, 0.713)×0.4

(1.13, 2, 0.655)

𝜃 = 0.15–0.20 rad, ×1.4(0.56, 2, 0.612)

×1.0

(0.49, 3, 1.841)×0.6

(0.75, 2, 0.748)×0.4

(0.71, 2, 0.296)

𝜃 = 0.20–0.25 rad, ×1.5(0.37, 2, 1.602)

×1.0

(0.49, 2, 1.308)×0.7

(0.38, 4, 0.939)×0.7

(0.88, 2, 0.658)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 8: The same as Figure 1, but showing the results for 𝜋− production cross sections in 𝜋+-C interactions.

Collaboration [35] and the curves are our calculated results.In the calculation, we have used 𝑙 = 1. The best valuesof ⟨𝑝𝑖1⟩ and 𝑚

1, as well as the value of 𝜒

2 per degree offreedom (𝜒2/dof), are given in the figure in terms of (⟨𝑝

𝑖1⟩,

𝑚1, 𝜒2/dof), where ⟨𝑝

𝑖1⟩ is in the units of GeV/c. The relative

errors for ⟨𝑝𝑖1⟩ are about 6%, which are only statistical errors,

and the systematic errors are eliminated.The error for𝑚1can

be neglected due to that the smallest alteration𝑚1± 1 results

in different distribution shape with an unexpected large 𝜒2.

Similarly, the double-differential 𝜋+ and 𝜋− production cross

sections measured in different 𝜃 ranges in 𝜋+-Be interactions

at 3, 5, 8, and 12GeV/c are shown in Figures 3 and 4,respectively. From Figures 1–4 one can see that the modelwith 𝑙 = 1 describes uniformly the experimental data in mostcases.

Figures 5–8 are the same as Figures 1–4, respectively,but showing the results for 𝜋

∓-C interactions. Figures 9–12are the same as Figures 1–4, respectively, too, but showing


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14𝜋−-Al, 3GeV/c

𝜃= 0.05–0.10 rad, ×1.0(2.65, 2, 0.754)

5GeV/c, ×0.7(1.40, 2, 0.337)

8GeV/c, ×0.3(1.27, 2, 0.719)

12GeV/c, ×0.3(1.40, 2, 0.403)

𝜃 = 0.10–0.15 rad, ×1.0(0.66, 2, 3.600)

×0.6

(0.77, 2, 1.509)×0.4

(0.98, 2, 0.716)×0.3

(1.17, 2, 1.133)

𝜃 = 0.15–0.20 rad, ×1.2(0.59, 2, 0.136)

×0.7

(0.49, 3, 0.747)×0.4

(0.75, 2, 0.731)×0.6

(0.79, 2, 1.248)

𝜃 = 0.20–0.25 rad, ×1.1(0.36, 3, 0.600)

×0.6

(0.44, 3, 0.957)×0.6

(0.32, 5, 0.970)×0.3

(0.40, 4, 0.927)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 9: The same as Figure 1, but showing the results for 𝜋+ production cross sections in 𝜋−-Al interactions.

the results for 𝜋∓-Al interactions. We see that the model

describes uniformly the experimental data in most cases.In [35], the experimental data on 𝜋

∓-Cu, Sn, Ta, and Pbinteractions at high momentums have been given, too. Asimilar conclusion can be obtained in the case of fitting theexperimental data by using the model.

From Figures 1–12 we see that the distribution widthwhich is represented by ⟨𝑝

𝑖1⟩𝑚1shows a decrease with

increases of 𝜃. This phenomenon renders that the emissionsource has a forward movement along the beam direction.

As a result, comparing with the situation in the source restframe, the pion with small 𝜃 has a large momentum and thatwith large 𝜃 has a small momentumdue to the effect of sourcemomentum. We see also from Figures 1–12 that 𝑚

1is only

in the range from 2 to 12 and the most probability is 2. Thesources are obviously incident pion and target nucleons. Thenumber of participant target nucleons is in the range from1 to 11 and the most probability is 1. Except the participantnucleons, other nucleons in target nucleus are the spectatornucleons.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14𝜋−-Al, 3GeV/c

𝜃 = 0.05–0.10 rad, ×0.4(2.65, 2, 0.726)

5GeV/c, ×0.2(3.40, 2, 0.556)

8GeV/c, ×0.2(2.00, 2, 0.539)

12GeV/c, ×0.2(1.98, 2, 0.47)

𝜃 = 0.10–0.15 rad, ×0.4(1.99, 2, 0.506)

×0.4

(1.8 , 2, 0.596)×0.2

(1.08, 2, 0.378)×0.3

(0.79, 3, 0.805)

𝜃 = 0.15–0.20 rad, ×0.6(0.99, 2, 0.144)

×0.5

(0.85, 2, 1.286)×0.3

(0.82, 2, 0.522)×0.2

(0.89, 2, 0.767)

𝜃 = 0.20–0.25 rad, ×1.1(0.49, 3, 0.552)

×0.6

(0.74, 2, 0.406)×0.4

(0.82, 2, 0.739)×0.4

(0.85, 2, 0.720)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 10: The same as Figure 1, but showing the results for 𝜋− production cross sections in 𝜋−-Al interactions.

To see clearly the changing trends of the parameters,the dependences of ⟨𝑝

𝑖1⟩, 𝑚1, ⟨𝑝𝑖1⟩𝑚1, and 𝑇 on 𝜃 for

𝜋+ produced in 𝜋

−-Be, C, and Al interactions are shownin Figure 13. The different symbols represent the parametervalues obtained at different momentums, and for targets Be,C, and Al, the results are taken from Figures 1, 5, and 9,respectively. Similarly, the corresponding dependences for𝜋− produced in the same interactions are given in Figure 14

which holds the parameter values taking from Figures 2, 6,and 10. The results for 𝜋

+ and 𝜋− produced in 𝜋

+ induced

interactions are presented in Figures 15 and 16 which holdthe parameter values taking from Figures 3, 7, and 11,as well as Figures 4, 8, and 12, respectively. We see that⟨𝑝𝑖1⟩ has a decreasing trend, 𝑚

1and 𝑇 do not show an

obvious change, and ⟨𝑝𝑖1⟩𝑚1has a decreasing trend with

increasing 𝜃.This renders again that the emission source has aforwardmovement along the beam direction, which leads themomentum distribution width of produced pions to be smallat large 𝜃. The target nucleus presents a saturation effect onthe source number and temperature.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14𝜋+-Al, 3GeV/c

𝜃= 0.05–0.10 rad, ×0.3(1.12, 4, 2.911)

5GeV/c, ×0.2(3.00, 3, 1.682)

8GeV/c, ×0.3(2.99, 2, 0.356)

12GeV/c, ×0.2(1.98, 2, 0.237)

𝜃 = 0.10–0.15 rad, ×0.3(1.40, 3, 0.547)

×0.4

(1.80, 2, 0.289)×0.3

(0.87, 3, 0.361)×0.2

(1.30, 2, 0.199)

𝜃 = 0.15–0.20 rad, ×0.6(0.80, 3, 1.084)

×0.4

(0.89, 2, 1.226)×0.2

(0.88, 2, 0.660)×0.2

(0.78, 2, 1.781)

𝜃 = 0.20–0.25 rad, ×1.1(0.62, 2, 0.158)

×0.5

(0.77, 2, 0.656)×0.4

(0.82, 2, 0.671)×0.3

(0.55, 2, 0.497)

d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

+

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 11: The same as Figure 1, but showing the results for 𝜋+ production cross sections in 𝜋+-Al interactions.

From Figures 1–16, we see that the changing trends ofthe parameters on incident momentum and target size arenot obvious. We have not seen an obvious difference ofparameters for 𝜋

+ and 𝜋− produced in reactions induced

by incident 𝜋+ and 𝜋

−. To see clearly the dependences ofparameters on incident momentum (𝑃) and target size, wecombine the results of 𝜋± produced in the four 𝜃 ranges inreactions induced by𝜋± and give themean values of ⟨𝑝

𝑖1⟩,𝑚1,

⟨𝑝𝑖1⟩𝑚1, and 𝑇 (i.e., ⟨𝑝

𝑖1⟩, 𝑚1, ⟨𝑝𝑖1⟩𝑚1, and 𝑇) for different

incident momentums and target nuclei in Figures 17(a)–17(d), respectively, where the error bars for ⟨𝑝

𝑖1⟩, ⟨𝑝𝑖1⟩𝑚1,

and 𝑇 are obtained according to the error transformationformula, and those for 𝑚

1are standard deviations. One

can see that the concerned mean values have no obviousdependence on incident momentum and target size. Wewould like to point out that the nondependence on incident


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)0 2 4 6 8

p (GeV/c)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14𝜋+-Al, 3GeV/c

𝜃= 0.05–0.10 rad, ×0.8(1.70, 3, 0.703)

5GeV/c, ×0.2(1.45, 2, 0.266)

8GeV/c, ×0.2(1.47, 2, 0.399)

12GeV/c, ×0.3(1.00, 3, 0.165)

𝜃 = 0.10–0.15 rad, ×0.8(0.55, 2, 1.605)

×0.5

(0.80, 2, 0.734)×0.3

(0.95, 2, 0.451)×0.2

(0.74, 3, 0.549)

𝜃 = 0.15–0.20 rad, ×0.6(0.38, 2, 0.844)

×0.6

(0.45, 3, 1.389)×0.3

(0.72, 2, 0.803)×0.2

(0.74, 2, 1.640)

𝜃 = 0.20–0.25 rad, ×0.9(0.11, 12 , 1.624)

×0.5

(0.47, 2, 1.313)×0.4

(0.52, 3, 1.564)×0.3

(0.20, 10 , 0.688)

d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)d2

−

𝜎𝜋

/dpdΩ

(bar

n/G

eV c−

1sr

)

Figure 12: The same as Figure 1, but showing the results for 𝜋− production cross sections in 𝜋+-Al interactions.

momentum may be a result of the too narrow range ofincident momentums, and the nondependence on target sizerenders the saturation effect of target nuclei in charged pioninduced reactions at the considered incident momentums.Particularly, the extracted mean temperatures are in therange from (107 ± 7)MeV to (147 ± 9)MeV which arelower than the critical temperature (156MeV) for phasetransition from hadron matter to quark matter obtained bythe thermaldynamic model [36].

4. Conclusions and Discussions

To conclude, the multisource thermal model is used to givea new and simple description of the double-differential 𝜋±production cross sections measured in different emissionangle ranges in interactions of 𝜋

∓ on different targets atthe considered incident momentums. In most cases, themodel describes well the experimental data of the HARPCollaboration. In the calculation we have used 𝑙 = 1. This


0

0.5

1

1.5

2

2.5

3

5

8

12GeV/c

0

2

4

6

8

0

1

2

3

4

5

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25

𝜃 (rad)

𝜋+

3

5

8

12GeV/c3

5

8

12GeV/c

𝜋−-Be 𝜋

−-C 𝜋−-Al

⟨pi1⟩

(GeV

/c)

m1

⟨pi1⟩m1

(GeV

/c)

T(G

eV)

𝜃 (rad) 𝜃 (rad)

Figure 13: Dependences of ⟨𝑝𝑖1⟩,𝑚1, ⟨𝑝𝑖1⟩𝑚1, and 𝑇 on 𝜃 for 𝜋+ produced in 𝜋

−-Be, C, and Al interactions. The different symbols representthe parameter values obtained at different momentums. The parameter values for targets Be, C, and Al are taken from Figures 1, 5, and 9,respectively.

means that there is only one type of emission sources inthe considered emission angle ranges. The parameter 𝑚

1is

small, which renders that the emission sources are incidentpion and target nucleons.The structures of pion and nucleon

do not play an important role at the considered incidentmomentums.

The parameter𝑚1does not show an obvious change with

increases of 𝜃. For a given interaction, it is a natural result that


0

0.5

1

1.5

2

2.5

3

3.5

0

1

2

3

4

0

1

2

3

4

5

6

7

0

0.05

0.1

0.15

0.2

0.25

0.3

𝜋−-Be 𝜋

−-C 𝜋−-Al

3

5

8

12GeV/c

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25

𝜃 (rad)

3

5

8

12GeV/c3

5

8

12GeV/c


⟨pi1⟩

(GeV

/c)

m1

⟨pi1⟩m1

(GeV

/c)

T(G

eV)

𝜋−

Figure 14: The same as Figure 13, but showing the results for 𝜋−. The parameter values for targets Be, C, and Al are taken from Figures 2, 6,and 10, respectively.

the source number has no relation with the emission angleof produced particle. The parameter ⟨𝑝

𝑖1⟩ and the product

⟨𝑝𝑖1⟩𝑚1decrease obviously with increases of 𝜃. Considering

the noneffect of 𝑚1, ⟨𝑝𝑖1⟩ has the same behavior as ⟨𝑝

𝑖1⟩𝑚1.

As the distribution width of particle momentum, ⟨𝑝𝑖1⟩𝑚1

has a small value at large 𝜃 because the emission source

has a forward movement along the beam direction and theparticle has amomentum transformation from the source restframe to the laboratory reference frame.

As the source number,𝑚1is not related to the concerned

incident momentum and target size.The source contribution(⟨𝑝𝑖1⟩) to the mean momentum of charged particles and


0

0.5

1

1.5

2

2.5

3

3.5

0

1

2

3

4

5

0

1

2

3

4

5

6

7

0

0.05

0.1

0.15

0.2

0.25

0.3

𝜋+-Be 𝜋

+-C 𝜋+-Al

3

5

8

12GeV/c

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25

𝜃 (rad)

3

5

8

12GeV/c3

5

8

12GeV/c


⟨pi1⟩

(GeV

/c)

m1

⟨pi1⟩m1

(GeV

/c)

T(G

eV)

𝜋+

Figure 15: The same as Figure 13, but showing the results for 𝜋+ produced in 𝜋+-Be, C, and Al interactions. The parameter values for targets

Be, C, and Al are taken from Figures 3, 7, and 11, respectively.

the distribution width (⟨𝑝𝑖1⟩𝑚1) of the particle momentums

are not related to the concerned incident momentum andtarget size, too. The nonrelation to the concerned incidentmomentum is caused by the narrow momentum range, andthe nonrelation to the target size is the result of saturation

effect of target nuclei in the concerned reactions. It is expectedthat the concerned parameters will increase at a high enoughincident momentum.

The extracted temperatures do not show an obviousdependence on the incident momentum, target size, and


0

0.5

1

1.5

2

0

2

4

6

8

10

12

0

1

2

3

4

5

0

0.05

0.1

0.15

0.2

3

5

8

12GeV/c

0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25

𝜃 (rad)

𝜋−

3

5

8

12GeV/c3

5

8

12GeV/c


⟨pi1⟩

(GeV

/c)

m1

⟨pi1⟩m1

(GeV

/c)

T(G

eV)

𝜋+-Be 𝜋

+-C 𝜋+-Al

Figure 16: The same as Figure 13, but showing the results for 𝜋− produced in 𝜋+-Be, C, and Al interactions. The parameter values for targets

Be, C, and Al are taken from Figures 4, 8, and 12, respectively.

polar angle in the considered collisions. The target nucleuspresents a saturation effect on the source temperature. Theextracted mean temperatures over polar angles are in therange from (107 ± 7)MeV to (147 ± 9)MeV which are lowerthan the critical temperature (156MeV) for phase transition

from hadronmatter to quarkmatter obtained by the thermal-dynamic model [36]. It is expected that higher temperatureswill be obtained at higher incident momentums.

Because the transverse momentum 𝑝𝑇

= 𝑝 sin 𝜃, thedistribution of 𝑝

𝑇is similar to that of 𝑝 at a given 𝜃.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12 14

P (GeV/c)

⟨pi1⟩

(GeV

/c)

(a)

0 2 4 6 8 10 12 14

P (GeV/c)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

m1

(b)

BeCAl

0 2 4 6 8 10 12 14

P (GeV/c)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

⟨pi1⟩m1(G

eV/c

)

(c)

0 2 4 6 8 10 12 14

P (GeV/c)

0

0.05

0.1

0.15

0.2

0.25

T(G

eV)

BeCAl

(d)

Figure 17: Dependences of (a) ⟨𝑝𝑖1⟩, (b)𝑚

1, (c) ⟨𝑝

𝑖1⟩𝑚1, and (d) 𝑇 on incident momentum 𝑃 for 𝜋± produced in 𝜋

± induced reactions. Thedifferent symbols represent the corresponding values obtained for different targets.The original values of the concerned parameters are takenfrom Figures 1–12.

The present modelling description can be used for the trans-versemomentumdistribution if we use𝑝

𝑇instead of𝑝 in (1)–

(3). In fact, in our previous work [37], the transverse momen-tum distributions of identified particles (𝜋±, 𝜋0, 𝐾±, 𝑝, 𝑝,

and 𝐽/𝜓) produced in proton-proton, proton- (deuteron-) nucleus, and nucleus-nucleus collisions at the RelativisticHeavy Ion Collider and related energies are studied by usingthe multisource thermal model. This means that the present


model can be used in different interacting systems in a widerenergy region.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was partly finished at the State University of NewYork at Stony Brook, USA. One of the authors Fu-Hu Liuthanks Professor Dr. Roy A. Lacey and the members of theNuclear Chemistry Group of Stony BrookUniversity for theirhospitality. The authors acknowledge the supports of theNational Natural Science Foundation of China (under Grantno. 10975095, no. 11005071, and no. 11247250), the ChinaNational Fundamental Fund of Personnel Training (underGrant no. J1103210), theOpenResearch Subject of theChineseAcademy of Sciences Large-Scale Scientific Facility (underGrant no. 2060205), the Shanxi ScholarshipCouncil of China,and the Overseas Training Project for Teachers at ShanxiUniversity.

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