# An Analysis of Transverse Momentum Spectra of Various Jets Produced in High Energy Collisions

Authors: Yang-Ming Tai; Pei-Pin Yang; Fu-Hu Liu
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

With the framework of the multisource thermal model, we analyze the experimental transverse momentum spectra of various jets produced in different collisions at high energies. Two energy sources, a projectile participant quark and a target participant quark, are considered. Each energy source (each participant quark) is assumed to contribute to the transverse momentum distribution to be the TP-like function, i.e., a revised Tsallis–Pareto-type function. The contribution of the two participant quarks to the transverse momentum distribution is then the convolution of two TP-like functions. The model distribution can be used to fit the experimental spectra measured by different collaborations. The related parameters such as the entropy index-related, effective temperature, and revised index are then obtained. The trends of these parameters are useful to understand the characteristic of high energy collisions.


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## Body

## 1. Introduction

In central heavy ion (nucleus-nucleus, A-A) collisions at high energy, quark-gluon plasma (QGP) is believed to create possibly [1–3], because the environment of high temperature and density is formed. After the formation, QGP experiences the process of hadronization and then produces lots of final-state particles. Meanwhile, at the early stage of collisions, some products such as various jets are produced and interact subsequently with QGP. Because of the interactions between jets and QGP, jets lost their energies when they go through QGP region. Not only lots of identified particles but also various jets can be measured in experiments at high energies. Indeed, in the abundant data on high energy collisions, the data on various jets are one of the most important constituents. We are interested in analyzing the experimental transverse momentum (pT) spectra of various jets, because they can reflect some information of early collisions of participant quarks or partons.Generally, thepT spectra of various jets are wider than those of identified particles. In fact, both the pT spectra of various jets and identified particles cover a wide pT range. Even if for the later, one may divide the pT range into low- and high-pT regions. It is expected that the spectra in low-pT region are contributed by the soft excitation process, while the spectra in high-pT region are contributed by the hard scattering process. In some cases, the spectra in low- and high-pT regions are still complex. One may divide further the low- and high-pT regions into very low- and low-pT regions as well as high- and very high-pT regions, respectively. It is expected that the spectra in different pT regions can be analyzed by different functions. This means that one needs two-component or even four-component function to fit the wide pT spectra.It is known that perturbative Quantum Chromodynamics (pQCD) successfully describes the processes which involve large momentum transfers. In particular for proton-proton (p-p) collisions, multijet production at highpT is well described if initial- and final-state radiations are considered (see, e.g., ref. [4]). In addition, multiparton interactions may play a role at low pT [5]. However, pQCD is very complex, which limits its wider applications in high energy proton-nucleus (p-A) and A-A collisions. We hope to use an alternative and thermal like or statistical method to describe uniformly the spectra of various jets and identified particles in both low- and high-pT regions in p-p, p-A, and A-A collisions at high energies. As the first step and as an example, we consider the two-component function.There are two methods to superpose the two components in a function [6–8]. The first method uses a weighted sum for the two components, and there are correlations between the parameters of the two components, though the point of linkage is smooth. The second method uses a step function to link the two components [8], and there is a nonsmooth linkage between the two components, though the parameters are uncorrelative. It is imaginable that more issues will appear if we consider four components in a function. Although the two-component function is widely used in literature, it is not an ideal treatment method, not to mention the four-component function. We hope to use a method to treat the two or four components uniformly. Even a single component function is used to fit the spectra in wide pT range.Fortunately, to search for the single component function for the spectra in widepT range is possible, because the similarity, universality, or common law is existent in high energy collisions [9–15]. To search for the single component function, we have tested many potential functions. Finally, we have found that the convolution of two or more revised Tsallis–Pareto-type functions [16, 17] is a suitable choice. For the purpose of doing a convenient description, we call the revised Tsallis–Pareto-type function the TP-like function in our recent work [17] and this paper. In fact, the Tsallis–Pareto-type function is more proper to restrict only to a Tsallis distribution, once one essentially uses the nonextensive statistical mechanics [18]. Because the Tsallis distribution has more than one form of expression, the term of Tsallis–Pareto-type function is used in our work to mention the concrete form.The application of the convolution of two or more functions is a general treatment method with the framework of the multisource thermal model [7], where the considered distributions are assumed from the contributions of two or more energy sources. The considered distributions include at least the multiplicity, transverse energy, and transverse momentum (transverse mass) distributions. At least two energy sources are considered in the collisions. Three or more energy sources are not excluded if the two energy sources are not enough to fit the spectra. The concrete number of energy sources is determined by the quality of fits and the scenario of physics. This fit methodology is suitable and unified for the analyses of spectra in different rapidity intervals, centrality classes, and collision systems.In this paper, in the framework of the multisource thermal model, we assume that a projectile participant quark and a target participant quark take part in the production of various jets, and they contribute to thepT distribution to be the TP-like function [16, 17]. Then, we may use the convolution of two TP-like functions to fit the experimental pT spectra of various jets. The related data quoted in this paper are from proton(anti)proton (p-pp¯), deuteron-gold (d-Au), gold-gold (Au-Au), proton-lead (p-Pb), and lead-lead (Pb-Pb) collisions, with different selection conditions, over a center-of-mass energy (sNN or simplified as s for p-pp¯ collisions) range from 0.2 to 13 TeV.The remainder of this paper is structured in the following. The formalism and method are described in Section2. The results and discussion are given in Section 3. Finally, we give the summary and conclusions in Section 4.


## 2. The Formalism and Method

According to ref. [16], the Tsallis–Pareto-type function which describes empirically the pT spectra of particles with rest mass m0 can be given by
(1)fpTpT=CpT1+pT2+m02−m0nT−n,which is a probability density function and C is the normalization constant because ∫0∞fpTpTdpT=1. In Eq. (1), as an entropy index-related parameter, n is related to the entropy index q because n=1/q−1. Generally, q=1 or n=∞ means an equilibrium state. If q is close to 1 or n is large enough, the system is close to an equilibrium state. The free parameter T in Eq. (1) is an effective temperature that describes the excitation and expansion degree of the emission source for particles. We call T the effective temperature because both the contributions of random thermal motion and flow effect are included.Equation (1) is not flexible enough in the description of pT spectra of particles, in particular for the spectra in low-pT region. Empirically, Eq. (1) can be revised artificially by adding a revised index a0 that is nondimensional as the power index of pT. Then, we have the TP-like function to be [17]
(2)fpTpT=CpTa01+pT2+m02−m0nT−n,where C is the normalization constant which is different from that in Eq. (1). For the purpose of convenience, two normalization constants in Eqs. (1) and (2) are represented by the same symbol C, though they may be different. Although one more parameter is introduced, Eq. (2) is more accurate than Eq. (1). In particular, we can obtain Eq. (1) from Eq. (2) if we use a0=1.With the framework of the multisource thermal model [7], we assume that many quarks or partons take part in the collisions. Each quark or parton is regarded as an energy source. For a given particle or jet, two quarks, i.e., a projectile participant (the first) quark and a target participant (the second) quark, play the main role in the production process. Other quarks that take part in the interactions with weak contributions can be neglected. For the two main quarks, the contribution amount or portion (pti) of each quark to pT is assumed to obey the TP-like function, where i=1 and 2 are for the first and second quarks, respectively. The TP-like function obeyed by pti is [17]
(3)fipti=Ciptia01+pti2+m0i2−m0inT−n,where m0i is empirically the constituent mass of the i-th participant quark.The total amount contributed by the two quarks is the convolution of two TP-like functions. That is [17].
(4)fpTpT=∫0pTf1pt1f2pT−pt1dpt1,or
(5)fpTpT=∫0pTf2pt2f1pT−pt2dpt2,where the functions f1 and f2 are given by Eq. (3) for various jets which are produced by different collisions which are listed in the table in the next section. In most cases, the convolution of two TP-like functions is suitable for the spectra of various jets. Correspondingly, two heavy flavor quarks such as c+c¯, b+b¯, or t+t¯ should be considered due to more effective energy being needed. For two light flavor quarks such as u+u¯, d+d¯, or s+s¯, we do not need to consider them due to too less effective energy for the production of various jets.The method of the convolution of three TP-like functions is similar to that of two TP-like functions. Firstly, we may obtain the convolutionf12pt12 of the first two TP-like functions f1pt1 and f2pt2. Secondly, we may obtain the convolution fpTpT of f12pt12 and f3pt3. Alternatively, we may obtain firstly the convolution f23pt23 of the last two TP-like functions f2pt2 and f3pt3, and then, we may obtain the convolution fpTpT of f1pt1 and f23pt23. The same idea can be used for the convolution of more than three TP-like functions. At present, the convolution of a projectile participant quark and a target participant quark is enough to fit the spectra of pT of various jets. Temporarily, we do not need to consider the convolution of three or more participant quarks.Because of the introduction ofa0, Eq. (2) is more accurate and flexible than Eq. (1). By using a0, the spectra in very low-pT region can be described reasonable. With the framework of the multisource thermal model, the method of the convolution of two or more probability density functions is applicable for not only the spectra of pT but also the spectra of multiplicity and transverse energy. In our analysis, the free parameters are n, T, and a0. The normalization constant is a parameter but not a free parameter. The convolution does not introduce new free parameters but the source number from the collision picture. In the method, to search for the probability density function contributed by a single participant or contributor or source is a key issue. This participant or contributor or source can be quark if we study the spectra of particles or nucleon if we study the spectra of nuclear fragments.


## 3. Results and Discussion

### 3.1. Comparison with Data

Figure1 shows the transverse momentum pT spectra of different jets produced in (a) p-p, (b) d-Au, and (c) Au-Au collisions with midpseudorapidity (mid-η, ∣η∣<0.5 for Figures 1(a) and 1(c) and ∣η∣<0.55 for Figure 1(b)), as well as in (d) p-p collisions with non-mid-η (0.8<η<1.8) at sNN=0.2TeV, where Nand Nevt denote the numbers of jets and events, respectively. The symbols are cited from the experimental data measured by the STAR Collaboration [19–21]. In Figures 1(a) and 1(c), the high tower (HT) trigger jets were selected. In Figures 1(a)–1(c), the jet events were selected using a cone radius (R=0.4) and anti-kT algorithm [22], where kT denotes the transverse momentum. In Figures 1(b) and 1(c), the data of d-Au and Au-Au collisions were measured in 0–20% centrality class. In the figure, the curves are our fitted results with Eq. (4). In the fit process, two participant top quarks with constituent mass of 174 GeV/c2 for each one are considered. The values of free parameters (n, T, and a0), normalization constant (N0), χ2, and number of degree of freedom (ndof) are listed in Table 1 in which the parameter trend will be analyzed and discussed in Section 3.2. One can see that the pT spectra of different jets are shown to obey approximately the convolution of two TP-like functions. The values of mean excitation and expansion degree (defined by the effective temperature parameter T) seem to not be related to the size of collision system in the error range.Figure 1
Transverse momentum spectra of different jets produced in (a) p-p, (b) d-Au, and (c) Au-Au collisions with mid-η (∣η∣<0.5 or 0.55), as well as in (d) p-p collisions with non-mid-η (0.8<η<1.8) at sNN=0.2TeV. The symbols are cited from the experimental data measured by the STAR Collaboration [19–21], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)Table 1
Values ofn, T, a0, N0σ0, χ2, and ndof corresponding to the curves in Figures 1–9, where σ0 is only for Figure 2(a).
Figure and typeCollisionnT (GeV)a0N0 [σ0 (mb)]χ2/ndofFigure1(a), p-p, |η∣<0.5t+t¯2.80±0.120.032±0.0080.00±0.013.499±0.002225/19Figure1(b), ∣η∣<0.55t+t¯3.20±0.020.036±0.0080.00±0.010.007±0.0015/1Figure1(c), ∣η∣<0.5t+t¯3.20±0.050.040±0.0090.00±0.052.999±0.002249/19Figure1(d), p-p, 0.8<η<1.8t+t¯6.00±0.050.045±0.0011.50±0.012.499±0.002195/18Figure2(a), p-Pb, 0 − 20%t+t¯3.00±0.010.130±0.0021.00±0.020.999±0.00118/4Figure2(a), p-Pb, 20−40%t+t¯3.00±0.020.130±0.0011.00±0.020.699±0.00211/3Figure2(a), p-Pb, 40−60%t+t¯3.00±0.010.130±0.0031.00±0.020.499±0.00112/2Figure2(a), p-Pb, 60−80%t+t¯3.00±0.030.130±0.0011.00±0.020.299±0.00218/2Figure2(a), p-Pb, 80−100%t+t¯3.00±0.050.130±0.0011.00±0.020.149±0.0026/1Figure2(b), Pb-Pb, 0 − 10%t+t¯3.00±0.110.130±0.0501.00±0.011.799±0.001×10−58/5Figure2(b), Pb-Pb, 10−30%t+t¯3.00±0.050.140±0.0511.00±0.012.299±0.001×10−53/5Figure2(b), Pb-Pb, 30−50%t+t¯3.00±0.210.150±0.0521.00±0.012.999±0.001×10−54/4Figure2(b), Pb-Pb, 50−80%t+t¯3.00±0.310.150±0.0511.00±0.013.999±0.002×10−57/3Figure3(a), p-p¯, Z ⟶ μμt+t¯2.10±0.021.150±0.091−0.45±0.01259969.508±2.200106/31Figure3(b), p-p¯, Z ⟶ eet+t¯2.05±0.051.200±0.092−0.45±0.01239964.205±2.522104/31Figure3(c), p-p, lepton+jett+t¯5.00±0.5513.000±0.853−0.48±0.01199762.511±11.050257/18Figure3(c), p-p, dilepton+jett+t¯5.00±0.8012.000±0.852−0.48±0.0151954.412±2.502201/19Figure3(d), p-p, lepton+b-jetb+b¯10.00±1.5012.000±0.8931.00±0.0119997.136±0.520133/29Figure3(d), p-p, dilepton+b-jetb+b¯10.00±1.5012.000±0.9551.00±0.012599.627±0.22357/25Figure4(a), 7 TeV p-p, e+b-jetb+b¯16.00±0.0315.000±1.5121.00±0.0313999.407±0.620345/12Figure4(b), 7 TeV p-p, μ+b-jetb+b¯16.00±0.0516.000±1.5021.00±0.0114998.827±0.953230/12Figure4(c), 7 TeV p-p, e+jett+t¯3.40±0.153.500±0.5501.60±0.1327400.066±0.965859/20Figure4(d), 7 TeV p-p, μ+jett+t¯3.50±0.123.500±0.5211.50±0.1034924.095±0.856155/20Figure4(e), 8 TeV p-p, e+jett+t¯3.50±0.423.000 ± 0.5521.50±0.01119830.499±2.505509/16Figure4(f), 8 TeV p-p, μ+jett+t¯3.20±0.353.000±0.5121.40±0.05159633.552±3.254832/15Figure5, Leading jett+t¯3.00±0.023.600±0.1951.10±0.121097.905±0.25219/4Figure5, 2nd Jett+t¯3.00±0.021.000±0.1901.70±0.1541991.806±3.25520/3Figure5, 3rd Jett+t¯2.50±0.010.600±0.1851.00±0.1442998.320±3.2356/2Figure5, 4th Jett+t¯2.50±0.010.300±0.0951.00±0.1629999.717±2.2538/1Figure5, 5th Jett+t¯2.50±0.010.150±0.0901.00±0.1224999.942±1.0257/0Figure6(a), Leading b-jetb+b¯9.00±0.0613.000±1.5061.00±0.051097.905±0.25298/20Figure6(b), Subleading b-jetb+b¯6.00±0.104.800±0.5021.00±0.031498.243±0.58611/10Figure6(c), Leading jett+t¯2.01±0.157.000±0.520−0.48±0.014998.740±1.56294/14Figure6(d), Subleading jett+t¯2.10±0.222.050±0.510−0.48±0.016965.727±0.85237/14Figure7(a), Leading light jetc+c¯5.00±0.1811.000±1.5051.00±0.025629.301±0.560147/36Figure7(b), Subleading light jetc+c¯5.50±0.124.500±0.1501.00±0.0111848.861±0.80589/36Figure7(c), Leading jett+t¯2.50±0.155.000±0.5020.00±0.03399824.415±5.60016/32Figure7(d), Subleading jett+t¯2.50±0.172.000±0.1050.00±0.01499979.349±2.35428/32Figure8(a), Small-R e+jett+t¯70.00±10.1070.000±15.100−0.50±0.0239936.022±0.50023/16Figure8(b), Small-R μ+jett+t¯70.00±10.0590.000±15.520−0.50±0.0534825.665±0.850103/16Figure8(c), Large-R e+jett+t¯7.00±0.2125.000±1.5101.00±0.03174997.093±1.2203/15Figure8(d), Large-R μ+jett+t¯7.00±0.2024.000±1.5211.00±0.03174997.733±1.0543/15Figure9(a), Leading, Zjjt+t¯3.00±0.303.200±0.2011.00±0.0589955.257±1.523230/11Figure9(b), Subleading, Zjjt+t¯3.00±0.301.500±0.1001.00±0.0599992.944±1.55440/11Figure9(c), Leading, prefitt+t¯3.00±0.114.800±0.0501.00±0.0293136953.879±15.400191/9Figure9(d), Fourth, prefitt+t¯3.50±0.100.300±0.0101.00±0.0211991257.185±15.60014/6Figure2(a) presents the pT spectra of fast jets produced in p-Pb collisions with different centralities at sNN=5.02TeV, where σ on the vertical axis denotes the cross-section. The pT spectra of charged jets produced in Pb-Pb collisions with different centralities at sNN=2.76TeV are presented in Figure 2(b), where Ncoll on the vertical axis denotes the number of binary nucleon-nucleon collisions. The symbols are cited from the experimental data measured by the ALICE Collaboration [23, 24]. The jet events were selected with a cone radius (R=0.2) and mid-η (∣η∣<0.5). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions describes approximately the experimental data of the mentioned jets. The effective temperature parameter T is the same with changing the centrality percentage in p-Pb collisions. And in Pb-Pb collisions, T increases slightly with the increase of centrality percentage, i.e., T decreases slightly with the increase of centrality itself.Figure 2
Transverse momentum spectra of (a) fast jets in p-Pb collisions atsNN=5.02TeV and (b) charged jets in Pb-Pb collisions at sNN=2.76TeV. The different symbols are cited from the experimental data with different centrality classes measured by the ALICE Collaboration [23, 24], and the curves are our fitted results with Eq. (4).
(a)(b)Figures3(a) and 3(b) present the pT spectrum of leading jets corresponding to the Z→μμ (Z→ee) channel in p-p¯ collisions at s=1.96TeV. Figures 3(c) and 3(d) present the pT spectra of jets (b-jets) corresponding to the lepton and dilepton channels in p-p collisions at s=7TeV. The symbols are cited from the experimental data measured by the D0 [25, 26], CMS [27], and ATLAS Collaborations [28]. In Figures 3(a) and 3(b), the jet events were selected with a cone radius (R=0.5) and wide η range (∣η∣<2.5). In Figures 3(c) and 3(d), the jet events were selected with ∣η∣<2.4 and ∣η∣<2.5, respectively. The curves are our fitted results with Eq. (4), in which two participant top quarks are considered for Figures 3(a)–3(c), and two participant bottom quarks with constituent mass of 4.19 GeV/c2 for each one are considered for Figure 3(d). The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions provides an approximate description on the data. The effective temperature parameter T obtained from the spectra with the lepton and dilepton channels is almost the same within the error range.Figure 3
Transverse momentum spectra of (a, b) leading jets with (a)Z → μμ and (b) Z → ee in p-p¯ collisions at s=1.96 TeV, as well as (c) jets and (d) b-jets in p-p collisions at sNN=7 TeV with different channels. The symbols are cited from the experimental data measured by the D0 [25, 26], CMS [27], and ATLAS Collaborations [28], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a, b) leading b-jets and (c–f) leading jets corresponding to the (a, c, e) e+jets channel and (b, d, f) μ+jets channel in p-p collisions at (a–d) s=7 and (e, f) 8 TeV are presented in Figure 4. The symbols are cited from the experimental data measured by the ATLAS Collaboration [29–31]. The jet events were selected with ∣η∣<2.5. The curves are our fitted results with Eq. (4), in which two participant bottom quarks are considered for Figures 4(a) and 4(b), and two participant top quarks are considered for Figures 4(c)–4(f). The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions provides an approximate description on the data. The effective temperature parameter T from the e+jets channel and the μ+jets channel is almost the same within the error range.Figure 4
Transverse momentum spectra of (a, b) leadingb-jets and (c–f) leading jets corresponding to the (a, c, e) e+jets channel and (b, d, f) μ+jets channel in p-p collisions at (a–d) s=7 and (e, f) 8 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaborations [29–31], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)(e)(f)The reconstructed jetpT spectra for the leading a 2nd, 3rd, 4th, and 5th order jets in the e+jets channels produced in p-p collisions at s=7 TeV are shown in Figure 5. The symbols are cited from the experimental data measured by the ATLAS Collaboration [32]. The jet events were selected with a cone radius (R=0.4) and wide η range (∣η∣<2.5). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. One can see that the effective temperature parameter T decreases with the growth of jet order O.Figure 5
Transverse momentum spectra of reconstructed jets with different orders in p-p collisions ats=7 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [32], and the curves are our fitted results with Eq. (4).Figure6 presents the pT spectra of (a) leading b-jets, (b) subleading b-jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=7 TeV. The symbols are cited from the experimental data measured by the CMS Collaboration [27, 33]. The jet events were selected with (a, b) ∣η∣<2.1 and (c, d) ∣η∣<2.4. The curves are our fitted results with Eq. (4), in which two participant bottom quarks are considered for Figures 6(a) and 6(b), and two participant top quarks are considered for Figures 6(c) and 6(d). The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. The values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets.Figure 6
Transverse momentum spectra of (a) leadingb-jets, (b) subleading b-jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=7 TeV. The symbols are cited from the experimental data measured by the CMS Collaboration [27, 33], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)Figure7 displays the pT spectra of (a) leading light jets, (b) subleading light jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=8 TeV, where Nobs denotes the number of observation. The symbols are cited from the experimental data measured by the CMS [34] and ATLAS Collaborations [35]. The jet events were selected with ∣η∣<2.4. The curves are our fitted results with Eq. (4), in which two participant charm quarks with constituent mass of 1.27 GeV/c2 for each one are considered for Figures 7(a) and 7(b), and two participant top quarks are considered for Figures 7(c) and 7(d). The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. Once more, the values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets.Figure 7
Transverse momentum spectra of (a) leading light jets, (b) subleading light jets, (c) leading jets, and (d) subleading jets produced in p-p collisions ats=8 TeV. The symbols are cited from the experimental data measured by the CMS [34] and ATLAS Collaborations [35], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a, b) small-R selected and (c, d) large-R jets corresponding to the (a, c) e+jets and (b, d) μ+jets channels produced in p-p collisions at s=13 TeV are shown in Figure 8. The symbols are cited from the experimental data measured by the ATLAS Collaboration [36]. The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. One can see that the values of T from the spectra of e+jets and μ+jets channels are almost the same within the error range.Figure 8
Transverse momentum spectra of (a, b) small-R selected and (c, d) large-R jets corresponding to the (a, c)e+jets and (b, d) μ+jets channels produced in p-p collisions at s=13 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [36], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a) the leading jets and (b) the subleading jets in Zjj baseline region as well as (c) the leading jets and (d) the forth jets with prefit produced in p-p collisions at s=13 TeV are presented in Figure 9. The symbols are cited from the experimental data measured by the ATLAS Collaboration [37, 38]. The jet events were selected with ∣y∣<2.4 for Figures 9(a) and 9(b) and ∣η∣<2.5 for Figures 9(c) and 9(d). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1, in which the normalization is very large due to the fact that it denotes the event number accumulated but not the cross-section or jet number per event. One can see that the values of T from the spectra of the leading jets are much larger than those from the spectra of the subleading and forth jets.Figure 9
Transverse momentum spectra of (a) the leading jets and (b) the subleading jets in Zjj baseline region as well as (c) the leading jets and (d) the forth jets with prefit produced in p-p collisions ats=13 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [37, 38], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)From the above comparison with data, one can see that the data are approximately fitted with the convolution of two TP-like functions. In most cases, the quality of the fits is acceptable due to appropriateχ2/ndof. In a few cases, the result of the fits is worthy of further improvement due to large χ2/ndof. To improve the result, we need more suitable function for the component and/or three or more functions in the convolution. In addition, there are cases in which some parameters do not change, but others suffer significant changes, in that sense is not possible to conclude about the characteristics of high energy collisions. Indeed, as an application of the uniform method, more comparisons with data are needed in the future. Meanwhile, we hope to improve the result of the fits in the future.


### 3.2. Parameter Trend and Discussion

Although the comparison with data is not perfect, one can also see some trends in most cases. To show the trends of main parameters, Figure10(a) presents the relation of the effective temperature T and the centrality percentage C in Pb-Pb collisions at sNN=2.76 TeV. The symbols represent the values of T obtained from Figure 2 and are listed in Table 1. The curve is our fit by an exponential function
(6)T=−0.03±0.01exp−C17.00±2.00+−0.15±0.01,Figure 10
The relations of (a) the effective temperature parameterT and the centrality percentage C in Pb-Pb collisions at s=2.76TeV, as well as (b) T and the jet order O in p-p collisions at s=7 TeV. The symbols represent the values of T obtained between Figures 2 and 5 and are listed in Table 1. The curves are our fitted results with Eqs. (6) and (7), respectively.
(a)(b)in whichT and C are in GeV and %, respectively. One can see that T increases slightly with the increase of C, or T is almost the same within the error range when C varies. The relation between T and C renders that QGP formed in central Pb-Pb collisions has less influence on the jet transport. Or in the transport process of jets in QGP in central Pb-Pb collisions, jets lost less energy.Figure10(b) presents the relation of the effective temperature T and the jet order O in p-p collisions at s=7 TeV. The symbols represent the values of T obtained from Figure 5 and are listed in Table 1. The curve is our fit by an exponential function
(7)T=11.00±0.10exp−O0.80±0.01+0.20±0.01,in which T is in GeV. One can see that T decreases with the growth of O. This trend is natural due to the fact that the jet with high order corresponds to the source with less excitation degree.Figure11 shows the relations of the effective temperature T and (a) the size of interacting system, (b) ℓ and di-ℓ channels, (c) μ (μμ) and e (ee) channels, and (d) leading and subleading jets. The symbols represent the values of T obtained from the above figures and are listed in Table 1. One can see that T seems to not be related to the system size in the error range. This is in agreement with the conclusion from Figure 10(a) in which central collisions correspond to large system, and peripheral collisions correspond to small system. In the error range, different lepton channels show nearly the same effective temperature, which renders nearly the same excitation degree of source. At the same time, the values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets, which is the same as the conclusion from Figure 10(b).Figure 11
The relations of the effective temperature parameterT and (a) the size of interacting system, (b) the ℓ and di-ℓ channels, and (c) the e (ee) and μ (μμ) channels, as well as (d) the leading and subleading jets. The symbols represent the values of T obtained from various figures and are listed in Table 1.
(a)(b)(c)(d)As a parameter determining the curvature in middle-pT region and the extended range in high-pT region, n is related to the entropy index q because n=1/q−1. In most cases, q≥1.2 which is not close to 1 because n≤5 which is not large. This implies that the source of jets does not stay at the equilibrium state. In a few cases, n is large, and q is close to 1. This happens coincidentally but not implies that the source of jets stay at the equilibrium state. This situation is different from the source of identified particles. Generally, the source of identified particles stays approximately at the equilibrium or local equilibrium state. These nonequilibrium and (local) equilibrium states do not mean a contradiction between the productions of various jets and identified particles. In fact, the two types of products are produced at different stages of system evolution. Generally, various jets are produced at the initial stage, and the identified particles are produced at the kinetic freeze-out stage. From the initial to kinetic freeze-out stages, the collision system evolves quickly from nonequilibrium to (local) equilibrium states.As a parameter determining the slope of the curve in low-pT region, a0 is elastic from negative to positive values. A negative a0 results in a cocked up distribution and a positive a0 results in a falling distribution. In many cases, a0≠1 which means that it is necessary introducing a0 in the Tsallis–Pareto-type function. Due to the introduction of a0, the revised Tsallis–Paretotype function, i.e., the TP-like function, becomes more flexible. The convolution of two or more TP-like functions is expected to fit more pT spectra in high energy collisions. The effective temperature T from the spectra of leading jets is much larger than that from the spectra of subleading jets because the leading jets undergone more violent scattering. However, a0 parameter does not change usually with the order of jet because a0 determines only the spectra at low pT. To determine T, the spectra at medium and high pT play the main role. Generally, the parameters related to various jets are not associated with the flow because various jets are produced at the initial stage where the flow is not formed. The flow is associated with identified particles which are finally produced at the kinetic freeze-out stage where the flow is formed.It is accepted that regarding central A-A collisions at high energies, jets lose considerable energy as they propagate through the hot and dense medium [39]. Due to the loss of energy, the pT spectra are reduced, and we obtain seemingly a relative small T and/or large n (small q) comparing with peripheral A-A collisions. However, due to large error, T in A-A collisions also shows a nearly invariant trend. In central p-A collisions, jets do not lose considerable energy, and the pT spectra are not reduced due to similar small participant volume comparing with peripheral p-A collisions. This renders that the main parameters in p-A collisions are independent of centrality, as what appears in A-A collisions. Mostly, the spectra of jets cited from p-A and A-A collisions are not under the same condition; it is hard to compare the parameters directly.In A-A collisions, the medium effect on the probe of hard process is commonly referred to as jet quenching. The effect has been observed via deviations from a well-calibrated baseline established in the absence of a medium (e.g., in minimum-bias p-p collisions). The generic strategy is illustrated with the nuclear modification factor (RAA), which evaluates the deviation of a single particle inclusive spectra away from the baseline.RAA is expected to be one in the absence of medium effect. Jet measurements in central A-A collisions at high energies have shown that RAA is pT dependent, and it is smaller than one. For peripheral A-A collisions, RAA approaches to unity, and it is nearly flat (see, e.g., [40]). An analogous analysis in p-A collisions gives a RpA which is consistent to one (see, e.g., [23]).It seems that these features of data onRAA are not captured by the parameters presented in Table 1. For instance, the extracted parameters which characterize the spectral shape (a0, T, and n) are the same for central A-A collisions and 0-100% p-A collisions. Probably, this is not a surprise because based on the reduced χ2, the proposed function seems not to describe all the data which are analyzed in the paper. This seems to illustrate that the paper also lacks consistent results. In fact, these opinions cannot be obtained because the parameters between A-A and p-A collisions cannot be compared directly. As we know, in Figure 2, the spectra in A-A and p-A collisions are from different kinds of jets. The same parameters for the two collisions are coincidental due to the similar trends of data quoted.Before Summary and Conclusions, we would like to emphasize the functions of parameters. As what we discussed in our recent work [17], the power index a0 describes flexibly the shapes of spectra at low pT. From negative to positive a0, the spectra bend from up to down over a pT range from 0 to 1 GeV/c. Correspondingly, the spectra at medium and high pT change higher due to the result of normalization. With the increase of T and by fixing a0 and n, the spectra become wider. Meanwhile, with the increase of n and by fixing a0 and T, the spectra become narrower. However, the changes with the increases of T and n cannot be offset. Anyhow, the introduction of a0 makes the TP-like function more flexible.We would like to point out that the effective temperature is one of the reported parameters; however, the kinetic freeze-out temperature (T0) should be more important to disentangle the T0 and flow. We have been worked on it for identified particles in our previous work [41]. In fact, that paper reports an increase of T0 with the collision energy, as well as the mass dependence of the T0. The present work and our previous work are not contradictory. However, we cannot compare them directly due to different stages of considerations. The identified particles are studied at the stage of kinetic freeze-out, while various jets are produced at the stage of initial collisions. In addition, different fit functions which correspond to different “thermometers” are used, which also increases the degree of difficulty for direct comparisons.


## 3.1. Comparison with Data

Figure1 shows the transverse momentum pT spectra of different jets produced in (a) p-p, (b) d-Au, and (c) Au-Au collisions with midpseudorapidity (mid-η, ∣η∣<0.5 for Figures 1(a) and 1(c) and ∣η∣<0.55 for Figure 1(b)), as well as in (d) p-p collisions with non-mid-η (0.8<η<1.8) at sNN=0.2TeV, where Nand Nevt denote the numbers of jets and events, respectively. The symbols are cited from the experimental data measured by the STAR Collaboration [19–21]. In Figures 1(a) and 1(c), the high tower (HT) trigger jets were selected. In Figures 1(a)–1(c), the jet events were selected using a cone radius (R=0.4) and anti-kT algorithm [22], where kT denotes the transverse momentum. In Figures 1(b) and 1(c), the data of d-Au and Au-Au collisions were measured in 0–20% centrality class. In the figure, the curves are our fitted results with Eq. (4). In the fit process, two participant top quarks with constituent mass of 174 GeV/c2 for each one are considered. The values of free parameters (n, T, and a0), normalization constant (N0), χ2, and number of degree of freedom (ndof) are listed in Table 1 in which the parameter trend will be analyzed and discussed in Section 3.2. One can see that the pT spectra of different jets are shown to obey approximately the convolution of two TP-like functions. The values of mean excitation and expansion degree (defined by the effective temperature parameter T) seem to not be related to the size of collision system in the error range.Figure 1
Transverse momentum spectra of different jets produced in (a) p-p, (b) d-Au, and (c) Au-Au collisions with mid-η (∣η∣<0.5 or 0.55), as well as in (d) p-p collisions with non-mid-η (0.8<η<1.8) at sNN=0.2TeV. The symbols are cited from the experimental data measured by the STAR Collaboration [19–21], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)Table 1
Values ofn, T, a0, N0σ0, χ2, and ndof corresponding to the curves in Figures 1–9, where σ0 is only for Figure 2(a).
Figure and typeCollisionnT (GeV)a0N0 [σ0 (mb)]χ2/ndofFigure1(a), p-p, |η∣<0.5t+t¯2.80±0.120.032±0.0080.00±0.013.499±0.002225/19Figure1(b), ∣η∣<0.55t+t¯3.20±0.020.036±0.0080.00±0.010.007±0.0015/1Figure1(c), ∣η∣<0.5t+t¯3.20±0.050.040±0.0090.00±0.052.999±0.002249/19Figure1(d), p-p, 0.8<η<1.8t+t¯6.00±0.050.045±0.0011.50±0.012.499±0.002195/18Figure2(a), p-Pb, 0 − 20%t+t¯3.00±0.010.130±0.0021.00±0.020.999±0.00118/4Figure2(a), p-Pb, 20−40%t+t¯3.00±0.020.130±0.0011.00±0.020.699±0.00211/3Figure2(a), p-Pb, 40−60%t+t¯3.00±0.010.130±0.0031.00±0.020.499±0.00112/2Figure2(a), p-Pb, 60−80%t+t¯3.00±0.030.130±0.0011.00±0.020.299±0.00218/2Figure2(a), p-Pb, 80−100%t+t¯3.00±0.050.130±0.0011.00±0.020.149±0.0026/1Figure2(b), Pb-Pb, 0 − 10%t+t¯3.00±0.110.130±0.0501.00±0.011.799±0.001×10−58/5Figure2(b), Pb-Pb, 10−30%t+t¯3.00±0.050.140±0.0511.00±0.012.299±0.001×10−53/5Figure2(b), Pb-Pb, 30−50%t+t¯3.00±0.210.150±0.0521.00±0.012.999±0.001×10−54/4Figure2(b), Pb-Pb, 50−80%t+t¯3.00±0.310.150±0.0511.00±0.013.999±0.002×10−57/3Figure3(a), p-p¯, Z ⟶ μμt+t¯2.10±0.021.150±0.091−0.45±0.01259969.508±2.200106/31Figure3(b), p-p¯, Z ⟶ eet+t¯2.05±0.051.200±0.092−0.45±0.01239964.205±2.522104/31Figure3(c), p-p, lepton+jett+t¯5.00±0.5513.000±0.853−0.48±0.01199762.511±11.050257/18Figure3(c), p-p, dilepton+jett+t¯5.00±0.8012.000±0.852−0.48±0.0151954.412±2.502201/19Figure3(d), p-p, lepton+b-jetb+b¯10.00±1.5012.000±0.8931.00±0.0119997.136±0.520133/29Figure3(d), p-p, dilepton+b-jetb+b¯10.00±1.5012.000±0.9551.00±0.012599.627±0.22357/25Figure4(a), 7 TeV p-p, e+b-jetb+b¯16.00±0.0315.000±1.5121.00±0.0313999.407±0.620345/12Figure4(b), 7 TeV p-p, μ+b-jetb+b¯16.00±0.0516.000±1.5021.00±0.0114998.827±0.953230/12Figure4(c), 7 TeV p-p, e+jett+t¯3.40±0.153.500±0.5501.60±0.1327400.066±0.965859/20Figure4(d), 7 TeV p-p, μ+jett+t¯3.50±0.123.500±0.5211.50±0.1034924.095±0.856155/20Figure4(e), 8 TeV p-p, e+jett+t¯3.50±0.423.000 ± 0.5521.50±0.01119830.499±2.505509/16Figure4(f), 8 TeV p-p, μ+jett+t¯3.20±0.353.000±0.5121.40±0.05159633.552±3.254832/15Figure5, Leading jett+t¯3.00±0.023.600±0.1951.10±0.121097.905±0.25219/4Figure5, 2nd Jett+t¯3.00±0.021.000±0.1901.70±0.1541991.806±3.25520/3Figure5, 3rd Jett+t¯2.50±0.010.600±0.1851.00±0.1442998.320±3.2356/2Figure5, 4th Jett+t¯2.50±0.010.300±0.0951.00±0.1629999.717±2.2538/1Figure5, 5th Jett+t¯2.50±0.010.150±0.0901.00±0.1224999.942±1.0257/0Figure6(a), Leading b-jetb+b¯9.00±0.0613.000±1.5061.00±0.051097.905±0.25298/20Figure6(b), Subleading b-jetb+b¯6.00±0.104.800±0.5021.00±0.031498.243±0.58611/10Figure6(c), Leading jett+t¯2.01±0.157.000±0.520−0.48±0.014998.740±1.56294/14Figure6(d), Subleading jett+t¯2.10±0.222.050±0.510−0.48±0.016965.727±0.85237/14Figure7(a), Leading light jetc+c¯5.00±0.1811.000±1.5051.00±0.025629.301±0.560147/36Figure7(b), Subleading light jetc+c¯5.50±0.124.500±0.1501.00±0.0111848.861±0.80589/36Figure7(c), Leading jett+t¯2.50±0.155.000±0.5020.00±0.03399824.415±5.60016/32Figure7(d), Subleading jett+t¯2.50±0.172.000±0.1050.00±0.01499979.349±2.35428/32Figure8(a), Small-R e+jett+t¯70.00±10.1070.000±15.100−0.50±0.0239936.022±0.50023/16Figure8(b), Small-R μ+jett+t¯70.00±10.0590.000±15.520−0.50±0.0534825.665±0.850103/16Figure8(c), Large-R e+jett+t¯7.00±0.2125.000±1.5101.00±0.03174997.093±1.2203/15Figure8(d), Large-R μ+jett+t¯7.00±0.2024.000±1.5211.00±0.03174997.733±1.0543/15Figure9(a), Leading, Zjjt+t¯3.00±0.303.200±0.2011.00±0.0589955.257±1.523230/11Figure9(b), Subleading, Zjjt+t¯3.00±0.301.500±0.1001.00±0.0599992.944±1.55440/11Figure9(c), Leading, prefitt+t¯3.00±0.114.800±0.0501.00±0.0293136953.879±15.400191/9Figure9(d), Fourth, prefitt+t¯3.50±0.100.300±0.0101.00±0.0211991257.185±15.60014/6Figure2(a) presents the pT spectra of fast jets produced in p-Pb collisions with different centralities at sNN=5.02TeV, where σ on the vertical axis denotes the cross-section. The pT spectra of charged jets produced in Pb-Pb collisions with different centralities at sNN=2.76TeV are presented in Figure 2(b), where Ncoll on the vertical axis denotes the number of binary nucleon-nucleon collisions. The symbols are cited from the experimental data measured by the ALICE Collaboration [23, 24]. The jet events were selected with a cone radius (R=0.2) and mid-η (∣η∣<0.5). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions describes approximately the experimental data of the mentioned jets. The effective temperature parameter T is the same with changing the centrality percentage in p-Pb collisions. And in Pb-Pb collisions, T increases slightly with the increase of centrality percentage, i.e., T decreases slightly with the increase of centrality itself.Figure 2
Transverse momentum spectra of (a) fast jets in p-Pb collisions atsNN=5.02TeV and (b) charged jets in Pb-Pb collisions at sNN=2.76TeV. The different symbols are cited from the experimental data with different centrality classes measured by the ALICE Collaboration [23, 24], and the curves are our fitted results with Eq. (4).
(a)(b)Figures3(a) and 3(b) present the pT spectrum of leading jets corresponding to the Z→μμ (Z→ee) channel in p-p¯ collisions at s=1.96TeV. Figures 3(c) and 3(d) present the pT spectra of jets (b-jets) corresponding to the lepton and dilepton channels in p-p collisions at s=7TeV. The symbols are cited from the experimental data measured by the D0 [25, 26], CMS [27], and ATLAS Collaborations [28]. In Figures 3(a) and 3(b), the jet events were selected with a cone radius (R=0.5) and wide η range (∣η∣<2.5). In Figures 3(c) and 3(d), the jet events were selected with ∣η∣<2.4 and ∣η∣<2.5, respectively. The curves are our fitted results with Eq. (4), in which two participant top quarks are considered for Figures 3(a)–3(c), and two participant bottom quarks with constituent mass of 4.19 GeV/c2 for each one are considered for Figure 3(d). The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions provides an approximate description on the data. The effective temperature parameter T obtained from the spectra with the lepton and dilepton channels is almost the same within the error range.Figure 3
Transverse momentum spectra of (a, b) leading jets with (a)Z → μμ and (b) Z → ee in p-p¯ collisions at s=1.96 TeV, as well as (c) jets and (d) b-jets in p-p collisions at sNN=7 TeV with different channels. The symbols are cited from the experimental data measured by the D0 [25, 26], CMS [27], and ATLAS Collaborations [28], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a, b) leading b-jets and (c–f) leading jets corresponding to the (a, c, e) e+jets channel and (b, d, f) μ+jets channel in p-p collisions at (a–d) s=7 and (e, f) 8 TeV are presented in Figure 4. The symbols are cited from the experimental data measured by the ATLAS Collaboration [29–31]. The jet events were selected with ∣η∣<2.5. The curves are our fitted results with Eq. (4), in which two participant bottom quarks are considered for Figures 4(a) and 4(b), and two participant top quarks are considered for Figures 4(c)–4(f). The values of n, T, a0, N0, χ2, and ndof are listed in Table 1. One can see that the convolution of two TP-like functions provides an approximate description on the data. The effective temperature parameter T from the e+jets channel and the μ+jets channel is almost the same within the error range.Figure 4
Transverse momentum spectra of (a, b) leadingb-jets and (c–f) leading jets corresponding to the (a, c, e) e+jets channel and (b, d, f) μ+jets channel in p-p collisions at (a–d) s=7 and (e, f) 8 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaborations [29–31], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)(e)(f)The reconstructed jetpT spectra for the leading a 2nd, 3rd, 4th, and 5th order jets in the e+jets channels produced in p-p collisions at s=7 TeV are shown in Figure 5. The symbols are cited from the experimental data measured by the ATLAS Collaboration [32]. The jet events were selected with a cone radius (R=0.4) and wide η range (∣η∣<2.5). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. One can see that the effective temperature parameter T decreases with the growth of jet order O.Figure 5
Transverse momentum spectra of reconstructed jets with different orders in p-p collisions ats=7 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [32], and the curves are our fitted results with Eq. (4).Figure6 presents the pT spectra of (a) leading b-jets, (b) subleading b-jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=7 TeV. The symbols are cited from the experimental data measured by the CMS Collaboration [27, 33]. The jet events were selected with (a, b) ∣η∣<2.1 and (c, d) ∣η∣<2.4. The curves are our fitted results with Eq. (4), in which two participant bottom quarks are considered for Figures 6(a) and 6(b), and two participant top quarks are considered for Figures 6(c) and 6(d). The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. The values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets.Figure 6
Transverse momentum spectra of (a) leadingb-jets, (b) subleading b-jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=7 TeV. The symbols are cited from the experimental data measured by the CMS Collaboration [27, 33], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)Figure7 displays the pT spectra of (a) leading light jets, (b) subleading light jets, (c) leading jets, and (d) subleading jets produced in p-p collisions at s=8 TeV, where Nobs denotes the number of observation. The symbols are cited from the experimental data measured by the CMS [34] and ATLAS Collaborations [35]. The jet events were selected with ∣η∣<2.4. The curves are our fitted results with Eq. (4), in which two participant charm quarks with constituent mass of 1.27 GeV/c2 for each one are considered for Figures 7(a) and 7(b), and two participant top quarks are considered for Figures 7(c) and 7(d). The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. Once more, the values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets.Figure 7
Transverse momentum spectra of (a) leading light jets, (b) subleading light jets, (c) leading jets, and (d) subleading jets produced in p-p collisions ats=8 TeV. The symbols are cited from the experimental data measured by the CMS [34] and ATLAS Collaborations [35], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a, b) small-R selected and (c, d) large-R jets corresponding to the (a, c) e+jets and (b, d) μ+jets channels produced in p-p collisions at s=13 TeV are shown in Figure 8. The symbols are cited from the experimental data measured by the ATLAS Collaboration [36]. The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1. One can see that the values of T from the spectra of e+jets and μ+jets channels are almost the same within the error range.Figure 8
Transverse momentum spectra of (a, b) small-R selected and (c, d) large-R jets corresponding to the (a, c)e+jets and (b, d) μ+jets channels produced in p-p collisions at s=13 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [36], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)ThepT spectra of (a) the leading jets and (b) the subleading jets in Zjj baseline region as well as (c) the leading jets and (d) the forth jets with prefit produced in p-p collisions at s=13 TeV are presented in Figure 9. The symbols are cited from the experimental data measured by the ATLAS Collaboration [37, 38]. The jet events were selected with ∣y∣<2.4 for Figures 9(a) and 9(b) and ∣η∣<2.5 for Figures 9(c) and 9(d). The curves are our fitted results with Eq. (4), in which two participant top quarks are considered. The experimental data are approximately fitted with the convolution of two TP-like functions, and the values of related parameters are given in Table 1, in which the normalization is very large due to the fact that it denotes the event number accumulated but not the cross-section or jet number per event. One can see that the values of T from the spectra of the leading jets are much larger than those from the spectra of the subleading and forth jets.Figure 9
Transverse momentum spectra of (a) the leading jets and (b) the subleading jets in Zjj baseline region as well as (c) the leading jets and (d) the forth jets with prefit produced in p-p collisions ats=13 TeV. The symbols are cited from the experimental data measured by the ATLAS Collaboration [37, 38], and the curves are our fitted results with Eq. (4).
(a)(b)(c)(d)From the above comparison with data, one can see that the data are approximately fitted with the convolution of two TP-like functions. In most cases, the quality of the fits is acceptable due to appropriateχ2/ndof. In a few cases, the result of the fits is worthy of further improvement due to large χ2/ndof. To improve the result, we need more suitable function for the component and/or three or more functions in the convolution. In addition, there are cases in which some parameters do not change, but others suffer significant changes, in that sense is not possible to conclude about the characteristics of high energy collisions. Indeed, as an application of the uniform method, more comparisons with data are needed in the future. Meanwhile, we hope to improve the result of the fits in the future.


## 3.2. Parameter Trend and Discussion

Although the comparison with data is not perfect, one can also see some trends in most cases. To show the trends of main parameters, Figure10(a) presents the relation of the effective temperature T and the centrality percentage C in Pb-Pb collisions at sNN=2.76 TeV. The symbols represent the values of T obtained from Figure 2 and are listed in Table 1. The curve is our fit by an exponential function
(6)T=−0.03±0.01exp−C17.00±2.00+−0.15±0.01,Figure 10
The relations of (a) the effective temperature parameterT and the centrality percentage C in Pb-Pb collisions at s=2.76TeV, as well as (b) T and the jet order O in p-p collisions at s=7 TeV. The symbols represent the values of T obtained between Figures 2 and 5 and are listed in Table 1. The curves are our fitted results with Eqs. (6) and (7), respectively.
(a)(b)in whichT and C are in GeV and %, respectively. One can see that T increases slightly with the increase of C, or T is almost the same within the error range when C varies. The relation between T and C renders that QGP formed in central Pb-Pb collisions has less influence on the jet transport. Or in the transport process of jets in QGP in central Pb-Pb collisions, jets lost less energy.Figure10(b) presents the relation of the effective temperature T and the jet order O in p-p collisions at s=7 TeV. The symbols represent the values of T obtained from Figure 5 and are listed in Table 1. The curve is our fit by an exponential function
(7)T=11.00±0.10exp−O0.80±0.01+0.20±0.01,in which T is in GeV. One can see that T decreases with the growth of O. This trend is natural due to the fact that the jet with high order corresponds to the source with less excitation degree.Figure11 shows the relations of the effective temperature T and (a) the size of interacting system, (b) ℓ and di-ℓ channels, (c) μ (μμ) and e (ee) channels, and (d) leading and subleading jets. The symbols represent the values of T obtained from the above figures and are listed in Table 1. One can see that T seems to not be related to the system size in the error range. This is in agreement with the conclusion from Figure 10(a) in which central collisions correspond to large system, and peripheral collisions correspond to small system. In the error range, different lepton channels show nearly the same effective temperature, which renders nearly the same excitation degree of source. At the same time, the values of T from the spectra of leading jets are much larger than those from the spectra of subleading jets, which is the same as the conclusion from Figure 10(b).Figure 11
The relations of the effective temperature parameterT and (a) the size of interacting system, (b) the ℓ and di-ℓ channels, and (c) the e (ee) and μ (μμ) channels, as well as (d) the leading and subleading jets. The symbols represent the values of T obtained from various figures and are listed in Table 1.
(a)(b)(c)(d)As a parameter determining the curvature in middle-pT region and the extended range in high-pT region, n is related to the entropy index q because n=1/q−1. In most cases, q≥1.2 which is not close to 1 because n≤5 which is not large. This implies that the source of jets does not stay at the equilibrium state. In a few cases, n is large, and q is close to 1. This happens coincidentally but not implies that the source of jets stay at the equilibrium state. This situation is different from the source of identified particles. Generally, the source of identified particles stays approximately at the equilibrium or local equilibrium state. These nonequilibrium and (local) equilibrium states do not mean a contradiction between the productions of various jets and identified particles. In fact, the two types of products are produced at different stages of system evolution. Generally, various jets are produced at the initial stage, and the identified particles are produced at the kinetic freeze-out stage. From the initial to kinetic freeze-out stages, the collision system evolves quickly from nonequilibrium to (local) equilibrium states.As a parameter determining the slope of the curve in low-pT region, a0 is elastic from negative to positive values. A negative a0 results in a cocked up distribution and a positive a0 results in a falling distribution. In many cases, a0≠1 which means that it is necessary introducing a0 in the Tsallis–Pareto-type function. Due to the introduction of a0, the revised Tsallis–Paretotype function, i.e., the TP-like function, becomes more flexible. The convolution of two or more TP-like functions is expected to fit more pT spectra in high energy collisions. The effective temperature T from the spectra of leading jets is much larger than that from the spectra of subleading jets because the leading jets undergone more violent scattering. However, a0 parameter does not change usually with the order of jet because a0 determines only the spectra at low pT. To determine T, the spectra at medium and high pT play the main role. Generally, the parameters related to various jets are not associated with the flow because various jets are produced at the initial stage where the flow is not formed. The flow is associated with identified particles which are finally produced at the kinetic freeze-out stage where the flow is formed.It is accepted that regarding central A-A collisions at high energies, jets lose considerable energy as they propagate through the hot and dense medium [39]. Due to the loss of energy, the pT spectra are reduced, and we obtain seemingly a relative small T and/or large n (small q) comparing with peripheral A-A collisions. However, due to large error, T in A-A collisions also shows a nearly invariant trend. In central p-A collisions, jets do not lose considerable energy, and the pT spectra are not reduced due to similar small participant volume comparing with peripheral p-A collisions. This renders that the main parameters in p-A collisions are independent of centrality, as what appears in A-A collisions. Mostly, the spectra of jets cited from p-A and A-A collisions are not under the same condition; it is hard to compare the parameters directly.In A-A collisions, the medium effect on the probe of hard process is commonly referred to as jet quenching. The effect has been observed via deviations from a well-calibrated baseline established in the absence of a medium (e.g., in minimum-bias p-p collisions). The generic strategy is illustrated with the nuclear modification factor (RAA), which evaluates the deviation of a single particle inclusive spectra away from the baseline.RAA is expected to be one in the absence of medium effect. Jet measurements in central A-A collisions at high energies have shown that RAA is pT dependent, and it is smaller than one. For peripheral A-A collisions, RAA approaches to unity, and it is nearly flat (see, e.g., [40]). An analogous analysis in p-A collisions gives a RpA which is consistent to one (see, e.g., [23]).It seems that these features of data onRAA are not captured by the parameters presented in Table 1. For instance, the extracted parameters which characterize the spectral shape (a0, T, and n) are the same for central A-A collisions and 0-100% p-A collisions. Probably, this is not a surprise because based on the reduced χ2, the proposed function seems not to describe all the data which are analyzed in the paper. This seems to illustrate that the paper also lacks consistent results. In fact, these opinions cannot be obtained because the parameters between A-A and p-A collisions cannot be compared directly. As we know, in Figure 2, the spectra in A-A and p-A collisions are from different kinds of jets. The same parameters for the two collisions are coincidental due to the similar trends of data quoted.Before Summary and Conclusions, we would like to emphasize the functions of parameters. As what we discussed in our recent work [17], the power index a0 describes flexibly the shapes of spectra at low pT. From negative to positive a0, the spectra bend from up to down over a pT range from 0 to 1 GeV/c. Correspondingly, the spectra at medium and high pT change higher due to the result of normalization. With the increase of T and by fixing a0 and n, the spectra become wider. Meanwhile, with the increase of n and by fixing a0 and T, the spectra become narrower. However, the changes with the increases of T and n cannot be offset. Anyhow, the introduction of a0 makes the TP-like function more flexible.We would like to point out that the effective temperature is one of the reported parameters; however, the kinetic freeze-out temperature (T0) should be more important to disentangle the T0 and flow. We have been worked on it for identified particles in our previous work [41]. In fact, that paper reports an increase of T0 with the collision energy, as well as the mass dependence of the T0. The present work and our previous work are not contradictory. However, we cannot compare them directly due to different stages of considerations. The identified particles are studied at the stage of kinetic freeze-out, while various jets are produced at the stage of initial collisions. In addition, different fit functions which correspond to different “thermometers” are used, which also increases the degree of difficulty for direct comparisons.


## 4. Summary and Conclusions

We summarize here our main observations and conclusions.(a)
The transverse momentumpT spectra of various jets selected in different conditions and produced in different collisions over an energy range from 0.2 to 13 TeV are fitted by the convolution of two TP-like functions, where the TP-like function is a revised Tsallis–Pareto-type function. The experimental data recorded by various collaborations are approximately fitted by the mentioned convolution(b)
From the fit on thepT spectra of charged jets produced in Pb-Pb collisions at s=2.76 TeV with different centrality intervals, we know that the effective temperature T increases slightly with increasing the centrality percentage, or T is almost the same in the error range when the centrality changes. Meanwhile, T from the spectra of jets in p-p, d-Au, and Au-Au collisions at 0.2 TeV does not show the size dependence. This is consistent to the nearly independence of T on centrality(c)
The values ofT from the spectra of leading jets are much larger than those from the spectra of subleading jets due to the leading jets undergone more violent scattering. As expected, T extracted from the reconstructed jets produced in p-p collisions at s=7 TeV decreases with the growth of the jet order. In addition, T from the lepton and dilepton channels is almost the same, which means that these jets have common property(d)
The parametern determines the curvature in middle-pT region and the extended range in high-pT region. Meanwhile, n is related to the entropy index q because n=1/q−1. Generally, n is not too large. This means that q is not close to 1, and the source of jets does not stay at the equilibrium state. This is different from the source of identified particles which stays approximately at the equilibrium or local equilibrium state(e)
The parametera0 determines the slope of the curve in low-pT region. A negative a0 results in a cocked up distribution, and a positive a0 results in a falling distribution. Due to the introduction of a0 in the Tsallis–Pareto-type function, the revised function, i.e., the TP-like function, becomes more flexible. The convolution of two or more TP-like functions is expected to have more applications



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