# Excitation Function of Initial Temperature of Heavy Flavor Quarkonium Emission Source in High Energy Collisions

Authors: Qi Wang; Fu-Hu Liu
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The transverse momentum spectra ofJ/ψ, ψ2S, and YnS,n=1,2,3 produced in proton-proton p+p, proton-antiproton p+p¯, proton-lead p+Pb, gold-gold Au+Au, and lead-lead (Pb+Pb) collisions over a wide energy range are analyzed by the (two-component) Erlang distribution, the Hagedorn function (the inverse power-law), and the Tsallis-Levy function. The initial temperature is obtained from the color string percolation model from the fit by the (two-component) Erlang distribution in the framework of a multisource thermal model. The excitation functions of several parameters such as the mean transverse momentum and initial temperature increase from 39 GeV to 13 TeV, which is considered in this work. The mean transverse momentum and initial temperature decrease (increase slightly or do not change significantly) with the increase of rapidity (centrality). Meanwhile, the mean transverse momentum of YnS,n=1,2,3 is larger than that of J/ψ and ψ2S, and the initial temperature for YnS,n=1,2,3 emission is higher than that for J/ψ and ψ2S emission, which shows a mass-dependent behavior.


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## Body

## 1. Introduction

The excitation functions of some physical quantities are significative to help us to understand the nuclear reaction mechanism and the system evolution characteristic. For instance, the higher the mean transverse momentumpT is, the higher excitation state the emission source stays at. Meanwhile, the higher the initial temperature (Ti) [1–5] is, the more violent the collisions are. By the analysis of the excitation functions of pT and Ti, we can learn more about the process in high energy collisions in which the excitation functions of several parameters such as pT and Ti can be obtained from the pT spectra of produced particles.In a data-driven reanalysis, to obtainpT and Ti, at the first place, we need the pT spectra of particles in experiments. At the second place, we should choose appropriate functions such as the Erlang distribution [6–8], the Hagedorn function or the inverse power-law [9, 10], and the Tsallis-Levy function [11, 12]. At the last place, we use the chosen functions to fit the experiential data on particle spectra. By describing the pT spectra, the parameters from the selected functions can be extracted. By comparing the parameters obtained from the experiential data at different energies, centralities, and rapidities, we can find out the dependences of parameters on these quantities. These dependences are related to excitation and expansion degrees of emission source, which is beneficial for us to understand the mechanism and characteristic of nuclear reactions and system evolution.Besides the two derived parameterspT and Ti, we can obtain other related parameters by using the method which is similar to extract pT and Ti. For example, using the Hagedorn function or the inverse power-law [9, 10] and the Tsallis-Levy function [11, 12] to fit pT spectra, some free parameters such as p0, n0, T, and n in the mentioned functions which will be discussed in Section 2 can be extracted. These free parameters are also useful to understand particle productions and system evolution. Not only the excitation functions of derived parameters pT and Ti but also the trends of free parameters p0, n0, T, and n can be studied from the fit to pT spectra.In this work, the (two-component) Erlang distribution [6–8], Hagedorn function (the inverse power-law) [9, 10], and Tsallis-Levy function [11, 12] are introduced firstly in Section 2. Then, in Section 3, the three distributions or functions are used to preliminarily fit the pT spectra of heavy flavor quarkonia (charmonia and bottomonia) produced in high energy collisions. The function results are compared with the spectra of J/ψ, ψ2S, and YnS,n=1,2,3 measured by the STAR [13, 14], CDF [15–17], ALICE [18], LHCb [19–27], ATLAS [28–30], and CMS Collaborations [31–34] over a wide energy range. Finally, in Section 4, we give our summary and conclusions.


## 2. Formalism and Method

### 2.1. The (Two-Component) Erlang Distribution

According to the multisource thermal model [6–8], a given particle is produced in the collision process where a few partons or quarks have taken part in. Each (the i-th) parton is assumed to contribute to an exponential function fipt of transverse momentum (pt) distribution. Let pt denotes the mean transverse momentum contributed by the i-th parton, we have the probability density function of pt to be
(1)fipt=1ptexp−ptpt,which is normalized to 1. The probability density function of pT contributed by all N partons which have taken part in the collision process is the convolution of N exponential functions [6–8]. We have the pT distribution fEpT (the probability density function of pT) of final state particles to be the Erlang distribution
(2)fEpT=pTN−1N−1!ptNexp−pTpt,which is naturally normalized to 1. The meanpT is pT=Npt.In the two-component Erlang distribution, we have(3)f1pT=kEpTN1−1N1−1!pt1N1exp−pTpt1+1−kEpTN2−1N2−1!pt2N2exp−pTpt2,wherekE denotes the contribution fraction of the first component, N1 (N2) denotes the number of partons in the first (second) component, and pt1pt2 denotes the mean transverse momentum contributed by each parton in the first (second) component. The mean pT is pT=kEN1pt1+1−kEN2pt2, where N1=1–3 in this work and N2=2 if 1−kE≠0.


### 2.2. The (Two-Component) Hagedorn Function

The Hagedorn function is an inverse power-law which is suitable to describe widepT spectra of particles produced in the hard scattering process. In refs. [9, 10], the Hagedorn function or the inverse power-law shows the probability density function of pT to be
(4)fHpT=ApT1+pTp0−n0,wherep0 and n0 are the free parameters and A is the normalization constant which is related to p0 and n0 and results in ∫0∞fHpTdpT=1. Equation (6) is an empirical formula inspired by quantum chromodynamics (QCD). We call Eq. (4) the Hagedorn function or the inverse power-law [9, 10].In the case of using two-component Hagedorn function, we have(5)f2pT=kHA1pT1+pTp01−n01+1−kHA2pT1+pTp02−n02,where kH denotes the contribution fraction of the first component, A1 (A2) is the normalization constant which results in the first (second) component to be normalized to 1, and p01 (p02) and n01 (n02) are free parameters related to the first (second) component. To combine the free parameters of the two components, we have p0=kHp01+1−kHp02 and n0=kHn01+1−kHn02.Generally, Eq. (4) is possible to describe the spectra in both the low- and high-pT regions. In fact, the spectra in the low- and high-pT regions represent similar trend in some cases. This is caused due to the similarity [35–45] which is widely existent in high energy collisions, where the similarity means the common or universality laws existed in different processes or collisions. In addition, one can revise Eq. (4) if needed in different ways [46–52] which suppress in the spectrum itself in low- or high-pT region according to the experimental spectra. To discuss various revisions of the Hagedorn function or the inverse power-law [9, 10] is beyond the focus of this paper. We shall not discuss anymore on this issue. For a very wide pT spectrum, Eq. (5) is possibly needed.


### 2.3. The (Two-Component) Tsallis-Levy Function

The Tsallis statistics [11] has wide applications in high energy collisions. There are various forms of the Tsallis distribution or function. In this work, we use the Tsallis-Levy function [12].
(6)fLpT=CpT1+pT2+m02−m0nT−n,where T and n are free parameters, pT2+m02≡mT is the transverse mass, m0 is the rest mass of the considered particle, and C is the normalized constant which is related to T, n, and m0 and results in ∫0∞fLpTdpT=1.We notice thatfLpT is related to particle mass m0, which is not the case of fEpT and fHpT presented in Eqs. (2) and (4), respectively. Although fLpT is related to m0, this relation is not strong due to m0 appearing only in pT2+m02−m0. The fact that the Tsallis distribution depends on m0 shows that this takes simple kinematics into account, as it is well known that mT or mT−m0 (something like transverse kinetic energy) is a better “scaling variable” for the spectra than pT.In the case of using two-component Tsallis-Levy function, we have(7)f3pT=kLC1pT1+pT2+m02−m0n1T1−n1+1−kLC2pT1+pT2+m02−m0n2T2−n2,where kL denotes the contribution fraction of the first component, C1 (C2) is the normalization constant which results in the first (second) component to be normalized to 1, and T1 (T2) and n1 (n2) are free parameters. To combine the free parameters of the two components, we have T=kLT1+1−kLT2 and n=kLn1+1−kLn2.The temperature parameter in the Tsallis-Levy function is an effective temperature at the final state (the stage of kinetic freeze-out). This effective temperature is not a “real” temperature because it includes not only the contribution of random thermal motion but also the contribution of flow effect. In the case of the first (second) component havingT1 (T2) with the fraction of kL (1−kL), the common effective temperature T of the two components is extracted from the assumed common equilibrium state of the two components. That is T=kLT1+1−kLT2 which has the same form as the parameter n.


### 2.4. The Initial Temperature

According to the color string percolation model [53–55], the initial temperature of the emission source is determined by
(8)Ti≡pT22,where
(9)pT2=∫0∞pT2f1,2,3pTdpT,is the square of the root-mean-square of pT due to ∫0∞f1,2,3pTdpT=1. If the x-component (px) and y-component (py) of the transverse momentum pT are considered, we have
(10)Ti=px2=py2.In the source rest-frame and under the assumption of isotropic emission, if thez-component of momentum is pz′, we also have
(11)Ti=pz′2.Although the source rest-frame is the lab-frame for symmetric collisions, we have mentioned the source rest-frame because asymmetric proton-lead (p+Pb) collisions are also considered in this work.It should be noted that we have used a single string in the cluster for a given particle production because only a projectile participant quark and a target participant quark are mainly considered in our treatment. The assumption of the single string results in the color suppression factorFξ to be 1 in the color string percolation model [54]. If we consider more than one strings taking part in the given particle production, the minimum Fξ will be nearly 0.6 [54]. Thus, we shall obtain a higher Ti by multiplying a revised factor 1∕Fξ in Eqs. (8), (10), and (11). In our opinion, although more than one strings have influences on the given particle production, the main role is the single string.


### 2.5. Discussion on the Functions

We would like to point out that the three types of functions are mainly just used here as parametrizations to achieve a good fit to the data, to be able to extractpT and Ti, though the Hagedorn and Tsallis-Levy functions are physically relevant. In fact, in the two functions, if we let m0=0, p0=nT, n0=n=1/q−1, the two functions are the same. Here, q is an entropy index that characterizes the excitation degree of the collision system [11, 12]. Generally, n0 or n is a sizeable quantity, which results in q to close to 1 and the collision system to close to an equilibrium state.We have used the two-component functions in some cases. The reason for using two-component source, i.e., basically two temperatures is not just used to achieve a better fit to the data. Physically, the first component corresponds to the non-head-on collisions between projectile and target participant quarks. The second component corresponds to the head-on collisions between the two quarks. Generally, the first component has a large fraction and lowpT and Ti. The second component has a less fraction and high pT and Ti. Because the head-on collisions between the two quarks are infrequent, single component function is usually applicable.In principal, no matter what functions are used to fit the experimental data,pT (or Ti) obtained from different fits is approximately the same within a small systematic uncertainty, if different functions fit the data good enough in the pT region of data available. For example, if simple Maxwell-Boltzmann or Bose-Einstein statistics can fit the data, we may obtain similar pT (or Ti) with other functions. In the case of multicomponent Maxwell-Boltzmann or Bose-Einstein statistics being needed, we may also obtain similar pT (or Ti).Indeed, the data itself decidespT (or Ti), and pT (or Ti) can be directly obtained from the data itself. The reason why we use functions is to see the tendency where the data is not available. However, the extrapolation on the tendency should be careful because it is not fully true, as it to the low- and high pT regions (where there is no data) could in principle have a major effect on the tendency (for example in case of very step exponentials near pT=0, or power-law tails at large pT). To reduce the effect, the data should be measured in a sufficiently large pT interval so that the extrapolation does not spoil as far as possible.For different components and functions, we do not need to consider the values of mid-rapidity (mid-y) or mid-pseudorapidity (mid-η), or the values of mid-y or mid-η can be regarded as 0 directly. In fact, for the experimental data with non-zero mid-y or mid-η, we may directly regard them as those with mid-y=0 or mid-η=0. This treatment is performed to subtract the contribution of kinetic energy of directed motion to the temperature.


## 2.1. The (Two-Component) Erlang Distribution

According to the multisource thermal model [6–8], a given particle is produced in the collision process where a few partons or quarks have taken part in. Each (the i-th) parton is assumed to contribute to an exponential function fipt of transverse momentum (pt) distribution. Let pt denotes the mean transverse momentum contributed by the i-th parton, we have the probability density function of pt to be
(1)fipt=1ptexp−ptpt,which is normalized to 1. The probability density function of pT contributed by all N partons which have taken part in the collision process is the convolution of N exponential functions [6–8]. We have the pT distribution fEpT (the probability density function of pT) of final state particles to be the Erlang distribution
(2)fEpT=pTN−1N−1!ptNexp−pTpt,which is naturally normalized to 1. The meanpT is pT=Npt.In the two-component Erlang distribution, we have(3)f1pT=kEpTN1−1N1−1!pt1N1exp−pTpt1+1−kEpTN2−1N2−1!pt2N2exp−pTpt2,wherekE denotes the contribution fraction of the first component, N1 (N2) denotes the number of partons in the first (second) component, and pt1pt2 denotes the mean transverse momentum contributed by each parton in the first (second) component. The mean pT is pT=kEN1pt1+1−kEN2pt2, where N1=1–3 in this work and N2=2 if 1−kE≠0.


## 2.2. The (Two-Component) Hagedorn Function

The Hagedorn function is an inverse power-law which is suitable to describe widepT spectra of particles produced in the hard scattering process. In refs. [9, 10], the Hagedorn function or the inverse power-law shows the probability density function of pT to be
(4)fHpT=ApT1+pTp0−n0,wherep0 and n0 are the free parameters and A is the normalization constant which is related to p0 and n0 and results in ∫0∞fHpTdpT=1. Equation (6) is an empirical formula inspired by quantum chromodynamics (QCD). We call Eq. (4) the Hagedorn function or the inverse power-law [9, 10].In the case of using two-component Hagedorn function, we have(5)f2pT=kHA1pT1+pTp01−n01+1−kHA2pT1+pTp02−n02,where kH denotes the contribution fraction of the first component, A1 (A2) is the normalization constant which results in the first (second) component to be normalized to 1, and p01 (p02) and n01 (n02) are free parameters related to the first (second) component. To combine the free parameters of the two components, we have p0=kHp01+1−kHp02 and n0=kHn01+1−kHn02.Generally, Eq. (4) is possible to describe the spectra in both the low- and high-pT regions. In fact, the spectra in the low- and high-pT regions represent similar trend in some cases. This is caused due to the similarity [35–45] which is widely existent in high energy collisions, where the similarity means the common or universality laws existed in different processes or collisions. In addition, one can revise Eq. (4) if needed in different ways [46–52] which suppress in the spectrum itself in low- or high-pT region according to the experimental spectra. To discuss various revisions of the Hagedorn function or the inverse power-law [9, 10] is beyond the focus of this paper. We shall not discuss anymore on this issue. For a very wide pT spectrum, Eq. (5) is possibly needed.


## 2.3. The (Two-Component) Tsallis-Levy Function

The Tsallis statistics [11] has wide applications in high energy collisions. There are various forms of the Tsallis distribution or function. In this work, we use the Tsallis-Levy function [12].
(6)fLpT=CpT1+pT2+m02−m0nT−n,where T and n are free parameters, pT2+m02≡mT is the transverse mass, m0 is the rest mass of the considered particle, and C is the normalized constant which is related to T, n, and m0 and results in ∫0∞fLpTdpT=1.We notice thatfLpT is related to particle mass m0, which is not the case of fEpT and fHpT presented in Eqs. (2) and (4), respectively. Although fLpT is related to m0, this relation is not strong due to m0 appearing only in pT2+m02−m0. The fact that the Tsallis distribution depends on m0 shows that this takes simple kinematics into account, as it is well known that mT or mT−m0 (something like transverse kinetic energy) is a better “scaling variable” for the spectra than pT.In the case of using two-component Tsallis-Levy function, we have(7)f3pT=kLC1pT1+pT2+m02−m0n1T1−n1+1−kLC2pT1+pT2+m02−m0n2T2−n2,where kL denotes the contribution fraction of the first component, C1 (C2) is the normalization constant which results in the first (second) component to be normalized to 1, and T1 (T2) and n1 (n2) are free parameters. To combine the free parameters of the two components, we have T=kLT1+1−kLT2 and n=kLn1+1−kLn2.The temperature parameter in the Tsallis-Levy function is an effective temperature at the final state (the stage of kinetic freeze-out). This effective temperature is not a “real” temperature because it includes not only the contribution of random thermal motion but also the contribution of flow effect. In the case of the first (second) component havingT1 (T2) with the fraction of kL (1−kL), the common effective temperature T of the two components is extracted from the assumed common equilibrium state of the two components. That is T=kLT1+1−kLT2 which has the same form as the parameter n.


## 2.4. The Initial Temperature

According to the color string percolation model [53–55], the initial temperature of the emission source is determined by
(8)Ti≡pT22,where
(9)pT2=∫0∞pT2f1,2,3pTdpT,is the square of the root-mean-square of pT due to ∫0∞f1,2,3pTdpT=1. If the x-component (px) and y-component (py) of the transverse momentum pT are considered, we have
(10)Ti=px2=py2.In the source rest-frame and under the assumption of isotropic emission, if thez-component of momentum is pz′, we also have
(11)Ti=pz′2.Although the source rest-frame is the lab-frame for symmetric collisions, we have mentioned the source rest-frame because asymmetric proton-lead (p+Pb) collisions are also considered in this work.It should be noted that we have used a single string in the cluster for a given particle production because only a projectile participant quark and a target participant quark are mainly considered in our treatment. The assumption of the single string results in the color suppression factorFξ to be 1 in the color string percolation model [54]. If we consider more than one strings taking part in the given particle production, the minimum Fξ will be nearly 0.6 [54]. Thus, we shall obtain a higher Ti by multiplying a revised factor 1∕Fξ in Eqs. (8), (10), and (11). In our opinion, although more than one strings have influences on the given particle production, the main role is the single string.


## 2.5. Discussion on the Functions

We would like to point out that the three types of functions are mainly just used here as parametrizations to achieve a good fit to the data, to be able to extractpT and Ti, though the Hagedorn and Tsallis-Levy functions are physically relevant. In fact, in the two functions, if we let m0=0, p0=nT, n0=n=1/q−1, the two functions are the same. Here, q is an entropy index that characterizes the excitation degree of the collision system [11, 12]. Generally, n0 or n is a sizeable quantity, which results in q to close to 1 and the collision system to close to an equilibrium state.We have used the two-component functions in some cases. The reason for using two-component source, i.e., basically two temperatures is not just used to achieve a better fit to the data. Physically, the first component corresponds to the non-head-on collisions between projectile and target participant quarks. The second component corresponds to the head-on collisions between the two quarks. Generally, the first component has a large fraction and lowpT and Ti. The second component has a less fraction and high pT and Ti. Because the head-on collisions between the two quarks are infrequent, single component function is usually applicable.In principal, no matter what functions are used to fit the experimental data,pT (or Ti) obtained from different fits is approximately the same within a small systematic uncertainty, if different functions fit the data good enough in the pT region of data available. For example, if simple Maxwell-Boltzmann or Bose-Einstein statistics can fit the data, we may obtain similar pT (or Ti) with other functions. In the case of multicomponent Maxwell-Boltzmann or Bose-Einstein statistics being needed, we may also obtain similar pT (or Ti).Indeed, the data itself decidespT (or Ti), and pT (or Ti) can be directly obtained from the data itself. The reason why we use functions is to see the tendency where the data is not available. However, the extrapolation on the tendency should be careful because it is not fully true, as it to the low- and high pT regions (where there is no data) could in principle have a major effect on the tendency (for example in case of very step exponentials near pT=0, or power-law tails at large pT). To reduce the effect, the data should be measured in a sufficiently large pT interval so that the extrapolation does not spoil as far as possible.For different components and functions, we do not need to consider the values of mid-rapidity (mid-y) or mid-pseudorapidity (mid-η), or the values of mid-y or mid-η can be regarded as 0 directly. In fact, for the experimental data with non-zero mid-y or mid-η, we may directly regard them as those with mid-y=0 or mid-η=0. This treatment is performed to subtract the contribution of kinetic energy of directed motion to the temperature.


## 3. Results and Discussion

Ordered by center-of-mass energy per nucleon pairsNN or s if only one pair) for different panels, Figure 1 shows the pT spectra, (a–c) 1/2πpTd2N∕dpTdy, (d, e) 1/2πpTd2σ/dpTdy, (f) d2σ/dpTdy, and (g) d2Y/dpTdy, of (a–d) J/ψ→e+e−, (e) J/ψ→μ+μ−, (f) prompt J/ψ, and (g) inclusive J/ψ produced in (a–c) gold-gold (Au+Au), (d, e) proton-proton (p+p), (f) proton-antiproton (p+p¯), and (g) lead-lead (Pb+Pb) collisions at mid-rapidity (a–d) ∣y∣<1, (e) ∣y∣<0.4, (f) ∣y∣<0.6, and forward rapidity (g) 2.5<y<4 at sNN or s= (a) 39, (b) 62.4, (c) 200, (d) 500, and (e) 510 GeV, as well as (f) 1.8 and 1.96, and (g) 2.76 TeV, where N denotes the number of particles, σ denotes the cross section, and Y denotes the yield. The symbols represent the experimental data [13–16, 18] and the curves are our fitted results. In the calculations, the method of least square is used to obtain the best free parameters. The values of free parameters pt1, pt2, N1, and kE are listed in Table 1 with χ2 and number of degrees of freedom (ndof). The values of free parameters p0, n0, T, and n are listed in Table 2 with χ2 and ndof. One can see that the (two-component) Erlang distribution, the Hagedorn function, and the Tsallis-Levy function fit approximately the experimental pT spectra of J/ψ via different decay or production modes in high energy p+p, p+p¯, Au+Au, and Pb+Pb collisions.Figure 1
Transverse momentum spectra, (a–c)1/2πpTd2N/dpTdy, (d, e) 1/2πpTd2σ/dpTdy, (f) d2σ/dpTdy, and (g) d2Y/dpTdy of J/ψ in different decay or production modes in (a–c) Au+Au, (d, e) p+p, (f) p+p¯, and (g) Pb+Pb collisions at energies (a) 39, (b) 62.4, (c) 200, (d) 500, and (e) 510 GeV, as well as (f) 1.8 and 1.96, and (g) 2.76 TeV. The experimental data represented by the symbols are measured by the (a–e) STAR [13, 14], (f) CDF [15, 16], and (g) ALICE Collaborations [18] with different centrality classes such as (a–c) 0-20%, 20-40%, 40-60%, and 0-60%, and (g) 0-20%, 20-40%, and 40-90%, as well as with different cross sections, e.g., (d, e) full and fiducial cross sections. Some data points are scaled by different amounts marked in the panels. The data points are fitted by the (two-component) Erlang distribution (Eq. (3), the solid curve), the Hagedorn function (Eq. (4), the dashed curve), and the Tsallis-Levy function (Eq. (6), the dotted curve), respectively.
(a)(b)(c)(d)(e)(f)(g)Table 1(a)
Values ofpt1, pt2, N1, kE, pT, Ti, and χ2/ndof corresponding to the solid curves in Figures 1 and 2. In all cases, N2=2, which is not listed in the table. In the case of ndof ≤0, we use “−” to mention
FigureMain selectionpt1 (GeV/c)pt2 (GeV/c)N1kEpT (GeV/c)Ti (GeV)χ2/ndof1(a)0–20%0.88±0.05—2±011.760±0.1001.520±0.0701.87/-20–40%0.86±0.05—2±011.720±0.1001.490±0.0701.67/-40–60%0.85±0.10—2±011.700±0.2001.472±0.1400.38/-0–60%0.86±0.05—2±011.720±0.1001.490±0.0701.34/-1(b)0–20%0.91±0.05—2±011.820±0.1001.576±0.0702.10/-20–40%0.89±0.05—2±011.780±0.1001.542±0.0701.05/-40–60%0.88±0.05—2±011.760±0.1001.524±0.0702.64/-0–60%0.89±0.01—2±011.780±0.2001.542±0.1402.81/-1(c)0–20%0.92±0.05—2±011.840±0.1001.593±0.0700.66/220–40%0.91±0.05—2±011.820±0.1001.576±0.0702.47/240–60%0.90±0.05—2±011.800±0.1001.559±0.0702.65/20–60%0.91±0.05—2±011.820±0.1001.576±0.0701.99/21(d)Full cross-section0.91±0.022.20±0.052±00.99 ± 0.011.846±0.0341.663±0.03713.40/14Fiducial cross-section0.95±0.012.50±0.052±00.99 ± 0.011.931±0.0301.709±0.04054.91/141(e)Full cross-section0.98±0.02—2±011.960±0.0401.697±0.0305.50/2Fiducial cross-section1.02±0.02—2±012.040±0.0401.767±0.03014.50/21(f)1.8 TeV0.98±0.052.30±0.102±00.98 ± 0.012.013±0.0561.743±0.0406.54/61.96 TeV1.02±0.052.41±0.102±00.98 ± 0.012.096±0.0571.815±0.04117.34/191(g)0–20%0.72±0.02—3±012.160±0.0601.764±0.04025.90/1020–40%0.71±0.02—3±012.130±0.0601.739±0.04020.90/1040–90%0.70±0.02—3±012.100±0.0601.715±0.04030.75/102(a)1.5<y<2.01.44±0.05—2±012.880±0.1002.494±0.0354.00/52.0<y<2.51.39±0.05—2±012.780±0.1002.408±0.0354.40/52.5<y<3.01.37±0.05—2±012.740±0.1002.373±0.0353.95/53.0<y<3.51.32±0.05—2±012.640±0.1002.286±0.0353.96/53.5<y<4.01.25±0.05—2±012.500±0.1002.165±0.0354.84/52(b)1.5<y<2.01.65±0.05—2±013.300±0.1002.858±0.0353.01/52.0<y<2.51.60±0.05—2±013.200±0.1002.771±0.0355.99/52.5<y<3.01.55±0.05—2±013.100±0.1002.685±0.0358.89/53.0<y<3.51.50±0.05—2±013.000±0.1002.598±0.03511.77/53.5<y<4.01.40±0.05—2±012.800±0.1002.425±0.03513.44/52(c)2.0<y<2.51.40±0.02—2±012.800±0.0402.425±0.01410.37/112.5<y<3.01.38±0.02—2±012.760±0.0402.390±0.01410.43/113.0<y<3.51.36±0.02—2±012.720±0.0402.356±0.01410.12/113.5<y<4.01.26±0.02—2±012.520±0.0402.182±0.0147.96/104.0<y<4.51.15±0.02—2±012.300±0.0401.992±0.0145.97/82(d)2.0<y<2.51.62±0.02—2±013.240±0.0402.806±0.0147.49/112.5<y<3.01.60±0.02—2±013.200±0.0402.771±0.0147.18/113.0<y<3.51.56±0.02—2±013.120±0.0402.702±0.0148.82/113.5<y<4.01.42±0.02—2±012.840±0.0402.460±0.0145.13/104.0<y<4.51.34±0.02—2±012.680±0.0402.321±0.0146.36/82(e)2.0<y<2.51.42±0.02—2±012.840±0.0402.460±0.0149.57/112.5<y<3.01.39±0.02—2±012.780±0.0402.408±0.0147.10/113.0<y<3.51.37±0.02—2±012.740±0.0402.373±0.0149.77/113.5<y<4.01.32±0.02—2±012.640±0.0402.286±0.0149.62/114.0<y<4.51.20±0.02—2±012.400±0.0402.078±0.0147.60/112(f)2.0<y<2.51.72±0.02—2±013.440±0.0402.979±0.0148.11/112.5<y<3.01.70±0.02—2±013.400±0.0402.944±0.0148.99/113.0<y<3.51.60±0.02—2±013.200±0.0402.771±0.0149.57/113.5<y<4.01.55±0.02—2±013.100±0.0402.685±0.0149.89/114.0<y<4.51.40±0.02—2±012.800±0.0402.425±0.0147.99/112(g)2.0<y<2.51.46±0.02—2±012.920±0.0402.529±0.0148.79/112.5<y<3.01.43±0.02—2±012.860±0.0402.477±0.0149.08/113.0<y<3.51.40±0.02—2±012.800±0.0402.425±0.0149.44/113.5<y<4.01.35±0.02—2±012.700±0.0402.338±0.0149.93/114.0<y<4.51.32±0.02—2±012.640±0.0402.286±0.01414.04/112(h)2.0<y<2.51.80±0.02—2±013.600±0.0403.118±0.01410.58/112.5<y<3.01.77±0.02—2±013.540±0.0403.066±0.0149.08/113.0<y<3.51.73±0.02—2±013.460±0.0402.996±0.0149.17/113.5<y<4.01.60±0.02—2±013.200±0.0402.771±0.0149.47/114.0<y<4.51.50±0.02—2±013.000±0.0402.598±0.01412.09/11(b)
Values ofpt1, pt2, N1, kE, pT, Ti, and χ2/ndof corresponding to the solid curves in Figures 3 and 4. In all cases, N2=2, which is not listed in the table. In the case of ndof ≤0, we use “−” to mention
FigureMain selectionpt1 (GeV/c)pt2 (GeV/c)N1kEpT (GeV/c)Ti (GeV)χ2/ndof3(a)1.8 TeV1.00±0.052.70±0.202±00.98±0.012.068±0.0601.791±0.0453.80/-1.96 TeV1.30±0.053.30±0.102±00.99±0.012.640±0.0642.286±0.05511.16/203(b)Inclusiveψ2S1.58±0.10—2±013.160±0.2002.737±0.07113.01/2Promptψ2S1.61±0.05—2±013.220±0.1002.789±0.0355.49/2ψ2S from b1.65±0.10—2±013.300±0.2002.858±0.0718.51/23(c)0.00<∣y∣<0.751.18±0.033.75±0.203±00.98±0.013.619±0.0692.962±0.0742.39/-0.75<∣y∣<1.501.14±0.023.65±0.053±00.98±0.013.498±0.0562.863±0.0703.50/-1.50<∣y∣<2.001.12±0.013.60±0.053±00.98±0.013.437±0.0462.813±0.0694.25/-3(d)0.00<∣y∣<0.751.20±0.033.80±0.103±00.97±0.013.720±0.0683.049±0.0743.23/-0.75<∣y∣<1.501.16±0.033.70±0.103±00.97±0.013.598±0.0672.948±0.0727.64/-1.50<∣y∣<2.001.14±0.033.65±0.103±00.97±0.013.536±0.0492.898±0.0712.33/-3(e)0<∣y∣<0.93.50±0.10—1±013.500±0.1003.031±0.0351.45/-0.9<∣y∣<1.53.30±0.10—1±013.300±0.1002.858±0.0351.81/-1.5<∣y∣<1.933.10±0.10—1±013.100±0.1002.685±0.0352.75/-1.93<∣y∣<2.43.00±0.10—1±013.000±0.1002.598±0.0353.50/-3(f)∣y∣<1.22.10±0.20—2±014.200±0.0403.637±0.1414.51/61.2<∣y∣<1.61.90±0.20—2±013.800±0.0403.291±0.1416.30/41.6<∣y∣<2.41.70±0.20—2±013.400±0.0402.944±0.1415.71/43(g)∣y∣<1.22.60±0.20—2±015.200±0.0404.503±0.1414.19/61.2<∣y∣<1.62.10±0.20—2±014.200±0.0403.637±0.1416.61/41.6<∣y∣<2.41.90±0.20—2±013.800±0.0403.291±0.1415.58/44(a)∣y∣<0.754.90±0.20—1±014.900±0.2004.900±0.14116.68/70.75<∣y∣<1.54.85±0.20—1±014.850±0.2004.850±0.14112.64/71.5<∣y∣<2.04.60±0.20—1±014.600±0.2004.600±0.14114.92/74(b)∣y∣<0.755.70±0.20—1±015.700±0.2005.700±0.1419.50/70.75<∣y∣<1.55.60±0.20—1±015.600±0.2005.600±0.14112.60/71.5<∣y∣<2.04.90±0.20—1±014.900±0.2004.900±0.14123.39/74(c)Promptψ2S4.50±0.50—1±014.500±0.5004.500±0.35431.00/2Non-promptψ2S5.00±0.50—1±015.000±0.5005.000±0.35424.00/24(d)0.0<∣y∣<0.36.00±0.20—1±0¡¡16.000±0.2006.000±0.14178.14/60.3<∣y∣<0.65.90±0.20—1±015.900±0.2005.900±0.141149.64/60.6<∣y∣<0.95.80±0.20—1±015.800±0.2005.800±0.141191.75/60.9<∣y∣<1.25.70±0.20—1±015.700±0.2005.700±0.141444.14/6(c)
Values ofpt1, pt2, N1, kE, pT, Ti, and χ2/ndof corresponding to the solid curves in Figures 5 and 6. In all cases, N2=2, which is not listed in the table. In the case of ndof ≤0, we use “−” to mention
FigureMain selectionpt1 (GeV/c)pt2 (GeV/c)N1kEpT (GeV/c)Ti (GeV)χ2/ndof5(a)Y1S2.35±0.10—2±014.700±0.2004.070±0.07114.28/11Y2S2.65±0.10—2±015.300±0.2004.590±0.07111.50/6Y3S2.70±0.10—2±015.400±0.2004.677±0.0717.99/65(b)Y1S2.35±0.10—2±014.700±0.2004.070±0.0717.79/3Y2S2.65±0.10—2±015.300±0.2004.590±0.0714.25/3Y3S2.75±0.10—2±015.500±0.2004.763±0.0712.36/35(c)Y1S2.75±0.10—2±015.500±0.2004.763±0.07113.89/3Y2S3.04±0.10—2±016.080±0.2005.265±0.07111.00/-Y3S3.28±0.10—2±016.540±0.2005.681±0.07112.00/-5(d)2.0<y<2.52.87±0.10—2±015.740±0.2004.971±0.07110.30/212.5<y<3.02.74±0.10—2±015.480±0.2004.746±0.07112.57/213.0<y<3.52.66±0.10—2±015.320±0.2004.607±0.07113.98/213.5<y<4.02.30±0.10—2±014.600±0.2003.984±0.07111.90/184.0<y<4.52.15±0.10—2±014.300±0.2003.724±0.0718.28/125(e)2.0<y<2.53.24±0.10—2±016.480±0.2005.612±0.07114.04/212.5<y<3.03.07±0.10—2±016.140±0.2005.317±0.07111.76/213.0<y<3.52.98±0.10—2±015.960±0.2005.162±0.07110.98/213.5<y<4.02.60±0.10—2±015.200±0.2004.503±0.0719.67/184.0<y<4.52.30±0.10—2±014.600±0.2003.984±0.0719.78/125(f)2.0<y<2.53.58±0.10—2±017.160±0.2006.201±0.07110.56/212.5<y<3.03.36±0.10—2±016.720±0.2005.820±0.0718.31/213.0<y<3.53.15±0.10—2±016.300±0.2005.456±0.07110.95/213.5<y<4.02.80±0.10—2±015.600±0.2004.850±0.07110.59/184.0<y<4.52.50±0.10—2±015.000±0.2004.330±0.0719.28/126(a)2.0<y<2.52.93±0.10—2±015.860±0.2005.075±0.07113.98/212.5<y<3.02.80±0.10—2±015.600±0.2004.850±0.07117.21/213.0<y<3.52.71±0.10—2±015.420±0.2004.694±0.07115.08/213.5<y<4.02.50±0.10—2±015.000±0.2004.330±0.0717.56/184.0<y<4.52.18±0.10—2±014.360±0.2003.776±0.0718.70/126(b)2.0<y<2.53.25±0.10—2±016.500±0.2005.629±0.07113.31/212.5<y<3.03.18±0.10—2±016.360±0.2005.508±0.07115.28/213.0<y<3.53.00±0.10—2±016.000±0.2005.196±0.07112.05/213.5<y<4.02.80±0.10—2±015.600±0.2004.850±0.07110.23/184.0<y<4.52.40±0.10—2±014.800±0.2004.157±0.0719.45/126(c)2.0<y<2.53.65±0.10—2±017.300±0.2006.322±0.07110.23/212.5<y<3.03.47±0.10—2±016.940±0.2006.010±0.07112.87/213.0<y<3.53.30±0.10—2±016.600±0.2005.716±0.0719.25/213.5<y<4.03.00±0.10—2±016.000±0.2005.196±0.0717.25/184.0<y<4.52.60±0.10—2±015.200±0.2004.503±0.0719.06/126(d)1.5<y<2.03.00±0.10—2±016.000±0.2005.196±0.0714.47/42.0<y<2.52.95±0.10—2±015.900±0.2005.110±0.0832.43/42.5<y<3.02.85±0.10—2±015.700±0.2004.936±0.0823.25/43.0<y<3.52.75±0.10—2±015.500±0.2004.763±0.0813.64/43.5<y<4.02.60±0.10—2±015.200±0.2004.503±0.0714.52/36(e)2.0<y<2.53.25±0.10—2±016.500±0.2005.629±0.07118.79/212.5<y<3.03.00±0.10—2±016.000±0.2005.196±0.07117.87/223.0<y<3.52.80±0.10—2±015.600±0.2004.850±0.07118.85/213.5<y<4.02.75±0.10—2±015.500±0.2004.763±0.07116.42/194.0<y<4.52.70±0.10—2±015.400±0.2004.677±0.07111.03/136(f)2.0<y<2.53.35±0.10—2±016.700±0.2005.802±0.07117.71/212.5<y<3.03.30±0.10—2±016.600±0.2005.716±0.07123.75/223.0<y<3.53.20±0.10—2±016.400±0.2005.543±0.07116.94/213.5<y<4.03.10±0.10—2±016.200±0.2005.369±0.07120.21/194.0<y<4.53.00±0.10—2±016.000±0.2005.196±0.07110.58/136(g)2.0<y<2.53.80±0.10—2±017.600±0.2006.582±0.07113.87/212.5<y<3.03.60±0.10—2±017.200±0.2006.235±0.07114.69/223.0<y<3.53.50±0.10—2±017.000±0.2006.062±0.07112.10/213.5<y<4.03.30±0.10—2±016.600±0.2005.716±0.07116.61/194.0<y<4.53.20±0.10—2±016.400±0.2005.543±0.07111.17/13Table 2(a)
Values ofp0, n0, and χ2/ndof corresponding to the dashed curves in Figures 1 and 2, as well as values of T, n, and χ2/ndof corresponding to the dotted curves in Figures 1 and 2
FigureMain selectionp0 (GeV/c)n0χ2/ndofT (GeV)nχ2/ndof1(a)0–20%6.20±0.507.80±0.501.07/-0.39±0.025.70±1.000.80/-20–40%6.10±0.507.90±0.502.11/-0.37±0.026.00±1.001.01/-40–60%5.95±0.508.05±0.500.23/-0.36±0.026.20±1.001.11/-0–60%6.00±0.508.00±0.500.59/-0.37±0.026.00±1.000.44/-1(b)0–20%6.60±0.508.00±0.503.00/-0.41±0.025.90±1.001.40/-20–40%6.50±0.508.10±0.501.73/-0.39±0.026.20±1.001.07/-40–60%6.35±0.508.25±0.502.16/-0.38±0.026.30±1.000.78/-0–60%6.40±0.308.20±0.302.94/-0.39±0.026.20±1.001.37/-1(c)0–20%7.00±0.509.30±0.502.36/20.43±0.026.10±1.000.84/220–40%6.80±0.209.50±0.204.43/20.40±0.026.30±1.001.86/240–60%6.70±0.509.60±0.503.85/20.39±0.026.40±1.001.46/20–60%6.80±0.509.50±0.503.86/20.40±0.026.30±1.000.41/21(d)Full cross-section6.80±0.3011.50±0.3035.57/160.44±0.028.00±0.2024.30/16Fiducial cross-section7.00±0.3011.00±0.3070.35/160.47±0.017.50±0.2054.75/161(e)Full cross-section7.50±0.5011.70±1.0011.89/20.45±0.028.10±1.002.22/2Fiducial cross-section8.00±0.3011.20±0.504.31/20.50±0.037.60±1.0010.25/21(f)1.8 TeV8.10±0.3011.80±0.304.87/80.55±0.028.50±0.204.20/81.96 TeV8.20±0.3012.00±0.5026.09/210.57±0.038.90±0.5021.40/211(g)0–20%12.40±0.5011.50±0.50217.00/100.53±0.0511.50±2.0011.31/1020–40%12.20±0.5011.80±0.5090.75/100.52±0.0511.60±2.0010.92/1040–90%12.00±0.5012.00±0.5041.80/100.52±0.0511.65±2.0015.71/102(a)1.5<y<2.013.00±0.5012.50±0.508.97/50.85±0.0510.00±2.002.16/52.0<y<2.512.90±0.5012.60±0.508.52/50.84±0.0510.50±2.002.66/52.5<y<3.012.60±0.5012.90±0.505.91/50.83±0.0511.00±2.002.46/53.0<y<3.512.40±0.5013.10±0.505.91/50.82±0.0511.50±2.003.10/53.5<y<4.012.10±0.5013.50±0.505.85/50.80±0.0512.50±2.003.61/52(b)1.5<y<2.014.50±0.5012.00±0.503.10/51.00±0.109.50±2.003.65/52.0<y<2.514.40±0.5012.10±0.503.75/50.98±0.059.70±2.003.62/52.5<y<3.014.30±0.5012.20±0.506.39/50.95±0.0510.00±2.004.60/53.0<y<3.514.10±0.5012.40±0.506.75/50.93±0.0510.20±2.006.60/53.5<y<4.013.50±0.5013.00±0.5011.66/50.87±0.0511.00±2.005.21/52(c)2.0<y<2.513.50±0.5014.00±0.508.91/110.85±0.0512.50±2.007.93/112.5<y<3.013.30±0.5014.20±0.508.66/110.84±0.0513.00±2.009.52/113.0<y<3.513.00±0.5014.40±0.508.41/110.82±0.0514.00±2.009.55/113.5<y<4.012.60±0.5014.80±0.508.30/100.80±0.0515.00±2.008.99/104.0<y<4.512.20±0.5015.00±0.507.80/80.74±0.0516.00±2.004.91/82(d)2.0<y<2.515.00±0.5013.00±0.503.19/111.02±0.0511.00±2.005.85/112.5<y<3.014.80±0.5013.20±0.504.93/111.00±0.0511.50±2.006.44/113.0<y<3.514.30±0.5013.50±0.505.32/110.98±0.0512.00±2.005.98/113.5<y<4.014.10±0.5014.00±0.508.86/100.93±0.0513.00±2.006.55/104.0<y<4.514.00±0.5014.20±0.509.91/80.90±0.0513.50±2.005.26/82(e)2.0<y<2.513.70±0.5014.20±0.5010.12/110.88±0.0513.00±2.007.17/112.5<y<3.013.50±0.5014.40±0.505.75/110.87±0.0513.50±2.008.75/113.0<y<3.513.30±0.5014.60±0.505.78/110.86±0.0514.00±2.008.16/113.5<y<4.013.10±0.5014.80±0.508.76/110.84±0.0515.00±2.0010.56/114.0<y<4.512.50±0.5015.20±0.509.92/110.77±0.0516.50±2.007.25/112(f)2.0<y<2.515.60±0.5013.10±0.505.45/111.07±0.0511.50±2.005.17/112.5<y<3.015.40±0.5013.30±0.508.64/111.05±0.0512.00±2.006.79/113.0<y<3.515.00±0.5013.70±0.505.17/111.02±0.0512.50±2.007.16/113.5<y<4.014.70±0.5014.10±0.5011.56/110.98±0.0513.50±2.007.07/114.0<y<4.514.20±0.5014.40±0.509.57/110.93±0.0514.00±2.005.81/112(g)2.0<y<2.515.30±0.5014.80±0.509.38/110.93±0.0514.00±2.006.16/112.5<y<3.015.20±0.5014.90±0.508.95/110.91±0.0514.50±2.008.12/113.0<y<3.515.00±0.5015.00±0.507.32/110.89±0.0514.80±2.009.17/113.5<y<4.014.40±0.5015.30±0.5011.22/110.86±0.0515.30±2.009.72/114.0<y<4.514.20±0.5015.40±0.508.86/110.84±0.0516.80±2.0013.48/112(h)2.0<y<2.518.00±0.5013.80±0.5010.77/111.20±0.1012.00±2.008.03/112.5<y<3.017.80±0.5013.90±0.505.92/111.15±0.1012.50±2.008.62/113.0<y<3.517.60±0.5014.00±0.508.20/111.12±0.1012.80±2.0010.17/113.5<y<4.016.50±0.5014.40±0.509.56/111.05±0.1014.20±2.008.75/114.0<y<4.515.50±0.5014.60±0.5013.83/111.00±0.0514.70±2.0010.58/11(b)
Values ofp0, n0, and χ2/ndof corresponding to the dashed curves in Figures 3 and 4, as well as values of T, n, and χ2/ndof corresponding to the dotted curves in Figures 3 and 4. The parameter values in the first (second) row in each panel for Figure 4(d) is for the first (second) component, where ∗ denotes kH and ∗∗ denotes kL. In other cases, only the single component is used
FigureMain selectionp0 (GeV/c)n0χ2/ndofT (GeV)nχ2/ndof3(a)1.8 TeV9.00±0.5012.10±0.504.00/20.65±0.059.00±1.005.50/21.96 TeV10.30±0.5012.20±0.5017.50/220.75±0.059.30±1.009.99/223(b)Inclusiveψ2S14.10±0.5013.10±0.5019.96/21.07±0.1012.30±2.004.51/2Promptψ2S14.20±0.5013.00±0.5010.43/21.10±0.1012.00±1.002.36/2ψ2S from b14.30±0.5012.80±0.5015.75/21.13±0.1011.70±2.007.59/23(c)0.00<∣y∣<0.7514.50±0.5013.50±0.507.75/21.16±0.1012.80±1.007.00/20.75<∣y∣<1.5014.30±0.5013.60±0.505.14/21.14±0.1013.00±1.003.22/21.50<∣y∣<2.0014.10±0.5013.70±0.503.14/21.12±0.1013.20±1.003.14/23(d)0.00<∣y∣<0.7516.00±0.5013.00±0.506.25/21.32±0.1012.50±1.004.33/20.75<∣y∣<1.5015.90±0.5013.10±0.509.89/21.31±0.1012.60±1.007.39/21.50<∣y∣<2.0015.80±0.5013.20±0.5011.25/21.30±0.1012.70±1.0010.25/23(e)0<∣y∣<0.915.50±0.5011.20±0.502.00/-1.50±0.0512.00±1.002.21/-0.9<∣y∣<1.515.30±0.5011.30±0.501.85/-1.48±0.0512.20±1.002.69/-1.5<∣y∣<1.9315.10±0.5011.40±0.502.75/-1.45±0.0512.50±1.003.50/-1.93<∣y∣<2.414.90±0.5011.50±0.503.50/-1.42±0.0512.70±1.004.84/-3(f)∣y∣<1.216.00±0.5014.50±0.502.39/61.50±0.1014.50±1.003.39/61.2<∣y∣<1.615.50±0.5014.80±0.503.80/41.38±0.1015.00±1.006.67/41.6<∣y∣<2.415.00±0.5015.10±0.502.10/41.28±0.1015.50±1.003.64/43(g)∣y∣<1.218.00±0.5013.00±0.504.17/61.65±0.0514.00±1.003.98/61.2<∣y∣<1.617.00±0.5013.50±0.504.23/41.57±0.0514.50±1.004.86/41.6<∣y∣<2.416.00±0.5014.00±0.503.49/41.50±0.0515.00±1.002.89/44(a)∣y∣<0.7525.00±0.5014.00±0.506.84/71.85±0.0514.00±1.007.88/70.75<∣y∣<1.524.60±0.5014.20±0.506.81/71.82±0.0514.30±1.007.69/71.5<∣y∣<2.024.20±0.5014.40±0.506.64/71.79±0.1014.60±1.006.67/74(b)∣y∣<0.7525.50±0.5012.50±0.507.78/72.20±0.1013.00±1.008.77/70.75<∣y∣<1.525.30±0.5012.60±0.505.70/72.10±0.1013.20±1.008.45/71.5<∣y∣<2.024.50±0.5013.00±0.508.56/72.00±0.1013.40±1.007.03/74(c)Promptψ2S26.00±0.5014.60±0.5016.25/21.90±0.0514.50±1.0017.27/2Non-promptψ2S26.50±0.5013.00±0.509.40/22.30±0.1013.50±1.008.59/24(d)0.0<∣y∣<0.335.00±1.0018.00±1.000.80±0.05∗2.35±0.2018.00±2.000.75 ± 0.05∗∗20.00±1.009.00±0.503.94/32.35±0.108.40±0.504.17/30.3<∣y∣<0.634.50±1.0018.50±1.000.80±0.05∗2.30±0.2018.50±2.000.75 ± 0.05∗∗20.00±1.009.00±0.204.08/32.30±0.108.20±0.504.47/30.6<∣y∣<0.934.00±1.0019.00±1.000.80±0.05∗2.25±0.2019.00±2.000.75 ± 0.05∗∗20.00±1.009.00±0.207.50/32.25±0.108.00±0.506.95/30.9<∣y∣<1.233.50±1.0019.50±1.000.80±0.05∗2.20±0.2019.50±2.000.75 ± 0.05∗∗20.00±1.009.00±0.209.81/32.20±0.107.80±0.306.12/3(c)
Values ofp0, n0, and χ2/ndof corresponding to the dashed curves in Figures 5 and 6, as well as values of T, n, and χ2/ndof corresponding to the dotted curves in Figures 5 and 6
FigureMain selectionp0 (GeV/c)n0χ2/ndofT (GeV)nχ2/ndof5(a)Y1S24.00±1.0012.50±1.0056.89/111.60±0.109.50±2.0021.67/11Y2S26.00±2.0012.70±1.009.75/61.75±0.109.70±2.007.50/6Y3S28.00±2.0012.90±1.0014.97/61.85±0.1010.50±2.007.43/65(b)Y1S27.00±2.0012.70±1.0039.17/31.65±0.1011.00±2.009.54/3Y2S28.00±2.0012.80±1.004.76/31.90±0.1011.50±2.004.53/3Y3S29.00±2.0012.90±1.002.73/32.00±0.1012.00±2.002.90/35(c)Y1S29.00±2.0012.90±1.006.60/32.20±0.2016.00±2.005.39/3Y2S33.00±2.0013.00±1.006.64/-2.50±0.2016.30±2.006.69/-Y3S34.00±2.0013.20±1.0011.25/-2.60±0.2016.50±2.0011.50/-5(d)2.0<y<2.531.00±2.0015.00±1.0011.65/212.30±0.2019.00±2.007.17/212.5<y<3.030.50±2.0015.50±1.009.45/212.20±0.2020.00±2.009.48/213.0<y<3.530.00±2.0016.00±1.007.89/212.10±0.2021.00±2.006.39/213.5<y<4.029.50±2.0016.50±1.0010.06/181.95±0.2023.00±2.008.07/184.0<y<4.528.50±2.0017.50±1.009.64/121.80±0.2025.00±2.007.98/125(e)2.0<y<2.536.00±2.0015.20±1.0015.98/212.60±0.2020.00±2.0013.23/212.5<y<3.035.50±2.0015.70±1.008.34/212.50±0.2021.00±2.0010.26/213.0<y<3.535.20±2.0016.20±1.0012.84/212.40±0.2022.00±2.009.34/213.5<y<4.034.30±1.0017.00±0.5010.64/182.10±0.2024.00±2.008.31/184.0<y<4.531.00±1.0017.50±0.5014.06/121.90±0.2026.00±2.0011.34/125(f)2.0<y<2.540.00±2.0015.50±1.0014.73/212.70±0.2021.00±2.0015.45/212.5<y<3.039.50±2.0016.00±1.0010.85/212.60±0.2022.00±2.007.95/213.0<y<3.539.00±2.0016.50±1.0012.37/212.50±0.2023.00±2.006.78/213.5<y<4.038.50±2.0017.80±1.0014.56/182.30±0.2025.00±2.008.31/184.0<y<4.537.00±2.0018.30±1.0011.37/122.10±0.2026.50±2.007.53/126(a)2.0<y<2.532.00±2.0015.50±1.0014.23/212.35±0.2022.00±2.009.12/212.5<y<3.031.50±2.0016.00±1.007.53/212.25±0.2023.00±2.0014.76/213.0<y<3.531.00±2.0016.50±1.0012.33/212.15±0.1024.00±2.007.44/213.5<y<4.030.50±2.0017.00±1.008.72/182.05±0.1025.00±2.009.75/184.0<y<4.529.00±2.0017.50±0.5013.73/121.85±0.1027.00±2.0012.03/126(b)2.0<y<2.538.00±2.0016.00±1.0016.54/212.70±0.2024.00±2.0010.98/212.5<y<3.037.50±2.0016.50±1.007.14/212.60±0.1025.00±2.0012.54/213.0<y<3.537.00±2.0017.00±1.0011.25/212.50±0.1026.00±2.008.34/213.5<y<4.036.50±2.0017.20±0.5010.31/182.30±0.1027.50±2.009.51/184.0<y<4.535.00±2.0017.60±0.5014.24/122.00±0.1028.50±2.005.67/126(c)2.0<y<2.544.00±3.0016.50±1.0015.89/212.90±0.1025.00±2.009.48/212.5<y<3.043.00±3.0017.00±1.007.53/212.80±0.2025.50±2.0010.68/213.0<y<3.542.00±3.0017.50±1.0013.59/212.70±0.1026.50±2.007.53/213.5<y<4.041.00±3.0018.00±1.0012.42/182.50±0.1028.00±2.007.34/184.0<y<4.538.00±3.0018.50±1.0013.59/122.23±0.1029.00±2.008.31/126(d)1.5<y<2.034.00±2.0015.20±1.005.35/42.45±0.1022.00±2.003.85/42.0<y<2.533.00±2.0015.70±1.004.79/42.35±0.1023.00±2.002.20/42.5<y<3.032.50±2.0016.00±1.004.25/42.25±0.1024.00±2.002.69/43.0<y<3.532.00±2.0016.50±1.006.72/42.15±0.1025.00±2.003.70/43.5<y<4.031.50±2.0017.00±1.007.40/32.05±0.1026.00±2.004.40/36(e)2.0<y<2.537.00±3.0016.00±1.0016.58/212.60±0.1026.00±2.0019.66/212.5<y<3.036.00±3.0016.50±1.0010.31/222.50±0.1027.00±2.0014.83/223.0<y<3.535.00±2.0017.00±0.5010.28/212.40±0.1028.00±2.0013.91/213.5<y<4.034.50±2.0017.50±0.5013.42/192.30±0.1029.00±2.0012.85/194.0<y<4.534.30±1.0017.70±0.509.64/132.25±0.1029.50±2.008.53/136(f)2.0<y<2.541.00±3.0016.50±1.0020.71/212.80±0.1029.00±2.0012.23/212.5<y<3.040.00±2.0017.00±0.5018.20/222.70±0.1030.00±2.0024.00/223.0<y<3.539.50±2.0017.20±0.507.92/212.65±0.1030.50±2.008.31/213.5<y<4.039.20±1.0017.40±0.5016.12/192.60±0.1031.00±2.0014.11/194.0<y<4.539.00±1.0017.80±0.5014.17/132.50±0.1032.00±2.0010.81/136(g)2.0<y<2.547.00±2.0017.00±1.0016.98/213.20±0.1030.00±2.0012.63/212.5<y<3.046.00±2.0017.50±1.0019.68/223.10±0.1031.00±2.0010.16/223.0<y<3.545.50±2.0017.70±0.5014.69/213.00±0.1032.00±2.0010.40/213.5<y<4.045.30±2.0018.50±1.0015.60/192.90±0.1033.00±2.0011.71/194.0<y<4.545.00±2.0018.80±0.5014.66/132.80±0.1034.00±2.008.86/13ThepT spectra, d2σ/dpTdy, of (a, c, e, and g) prompt J/ψ and (b, d, f, and h) J/ψ from b produced in (a, b) p+Pb and (c–h) p+p collisions at sNN or s= (a, b) 5, (c, d) 7, (e, f) 8, and (g, h) 13 TeV are presented in Figure 2. The symbols represent the experimental data [19–22], and the curves are our fitted results. The method of least square is used to obtain the best parameter values which are listed in Tables 1 and 2 with χ2 and ndof. One can see that the experimental pT spectra of J/ψ via different production modes in different rapidity intervals in p+p and p+Pb collisions at high energies are approximately fitted by the Erlang distribution, the Hagedorn function, and the Tsallis-Levy function.Figure 2
Transverse momentum spectra,d2σ/dpTdy, of J/ψ in different production modes in (a, b) p+Pb and (c–h) p+p collisions at (a, b) 5, (c, d) 7, (e, f) 8, and (g, h) 13 TeV. The different symbols represent the experimental data measured by the LHCb Collaboration [19–22] in the rapidity intervals of (a, b) 1.5<y<2.0, 2.0<y<2.5, 2.5<y<3.0, 3.0<y<3.5, and 3.5<y<4.0 and (c–h) 2.0<y<2.5, 2.5<y<3.0, 3.0<y<3.5, 3.5<y<4.0, and 4.0<y<4.5, and scaled by different amounts marked in the panels. The solid, dashed, and dotted curves represent our results fitted by the Erlang distribution (Eq. (3)), the Hagedorn function (Eq. (4)), and the Tsallis-Levy function (Eq. (6)), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure3 shows the pT spectra, (a, c–g) d2σ/dpTdy and (b) dσ/dpT, of ψ2S via different production modes. The symbols represent the experimental data [15, 16, 23, 28, 31, 32], and the curves are our fitted results. The values of free parameters are listed in Tables 1 and 2 with χ2 and ndof. One can see that the experimental pT spectra of ψ2S via different production modes in different rapidity intervals in p+p, p+p¯, and p+Pb collisions at high energies are approximately fitted by the (two-component) Erlang distribution, the Hagedorn function, and the Tsallis-Levy function.Figure 3
Transverse momentum spectra, (a, c–g)d2σ/dpTdy and (b) dσ/dpT, of ψ2S from different production modes in (a) p+p¯, (b) p+Pb, and (c–g) p+p collisions at energies (a) 1.8 and 1.96, (b) 5, (c–e) 5.02, and (f–g) 7 TeV. The different symbols represent the experimental data of the (a, c, e, and f) prompt ψ2S, (b) inclusive ψ2S, prompt ψ2S, and ψ2S from b, and (d, g) non-prompt ψ2S measured by the (a) CDF [15, 16], (b) LHCb [23], (c, d) ATLAS [28], and (e–g) CMS Collaborations [31, 32] in (a) p+p¯, (b) p+Pb, (c–g) p+p collisions at sNN or s= (a) 1.8 and 1.96, (b) 5, (c–e) 5.02, and (f, g) 7 TeV with (a) ∣y∣<0.6, (b) 1.5<y<4.0, (c, d) 0.00<∣y∣<0.75, 0.75<∣y∣<1.50, and 1.50<∣y∣<2.00, (e) 0<∣y∣<0.9, 0.9<∣y∣<1.5, 1.5<∣y∣<1.93, and 1.93<∣y∣<2.4, and (f, g) ∣y∣<1.2, 1.2<∣y∣<1.6, and 1.6<∣y∣<2.4, where different collaborations have used different precisions for the rapidity intervals. Some data points are scaled by different amounts marked in the panels. The solid, dashed, and dotted curves represent our results fitted by the (two-component) Erlang distribution (Eq. (3)), the Hagedorn function (Eq. (4)), and the Tsallis-Levy function (Eq. (6)), respectively.
(a)(b)(c)(d)(e)(f)(g)Figure4 shows the pT spectra, d2σ/dpTdy, of ψ2S via different production modes in p+p collisions at s= (a, b) 7, (c) 8, and (d) 13 TeV. The symbols represent the experimental data [29, 30, 33], and the curves are our fitted results. The values of free parameters are listed in Tables 1 and 2 with χ2 and ndof. For Figure 4(d), the two-component Eqs. (5) and (7) are used, where the free parameters for the first (second) component are listed in the first (second) row. One can see that the experimental pT spectra of ψ2S via different production modes in different rapidity intervals in p+p collisions at high energies are also approximately fitted by the Erlang distribution, the (two-component) Hagedorn function, and the (two-component) Tsallis-Levy function.Figure 4
Same as Figure3, but showing the results at (a, b) 7, (c) 8, and (d) 13 TeV. The different symbols represent the experimental data of (a, d) prompt ψ2S, (b) non-prompt ψ2S, and (c) prompt ψ2S and non-prompt ψ2S measured by the (a–c) ATLAS [29, 30] and (d) CMS Collaborations [33] in (a, b) ∣y∣<0.75, 0.75<∣y∣<1.5, and 1.5<∣y∣<2.0, (c) ∣y∣<0.75, and (d) 0.0<∣y∣<0.3, 0.3<∣y∣<0.6, 0.6<∣y∣<0.9, and 0.9<∣y∣<1.2, where some data points are scaled by different amounts marked in the panels. The data points are fitted by the Erlang distribution (Eq. (3), the solid curve), the Hagedorn function (Eq. (4), the dashed curves), and the Tsallis-Levy function (Eq. (6), the dotted curves). In particular, the two-component Hagedorn function (Eq. (5), the dashed curves) and the two-component Tsallis-Levy function (Eq. (7), the dotted curves) are used in Figure 4(d).
(a)(b)(c)(d)In Figure5, the pT spectra, (a, c–f) d2σ/dpTdy and (b) dσ/dpT, of (a–c) YnS,n=1,2,3⟶μ+μ−, (d) Y1S⟶μ+μ−, (e) Y2S⟶μ+μ−, and (f) Y3S⟶μ+μ− induced in (a) p+p¯ and (b–f) p+p collisions at s= (a) 1.8, (b) 2.76, (c) 5.02, and (d–f) 7 TeV are given. The symbols represent the experimental data [17, 24, 25, 34], and the curves are our fitted results. The parameter values are listed in Tables 1 and 2 with χ2 and ndof. One can see that the experimental pT spectra of YnS,n=1,2,3⟶μ+μ− in different rapidity intervals in p+p and p+p¯ collisions at high energies are approximately fitted by the Erlang distribution, the Hagedorn function, and the Tsallis-Levy function.Figure 5
Transverse momentum spectra, (a, c–f)d2σ/dpTdy and (b) dσ/dpT, of (a–c) YnS,n=1,2,3⟶μ+μ−, (d) Y1S⟶μ+μ−, (e) Y2S⟶μ+μ−, and (f) Y3S⟶μ+μ− in (a) p+p¯ and (b–f) p+p collisions at (a) 1.8, (b) 2.76, (c) 5.02, and (d–f) 7 TeV. The symbols shown in (a–c) represent the experimental data measured by the (a) CDF [17], (b) LHCb [24], and (c) CMS Collaborations [34] in (a) p+p¯ and (b, c) p+p collisions in (a) ∣y∣<0.4, (b) 2.0<y<4.5, and (c) ∣y∣<2.4, respectively. The symbols shown in panels (d–f) represent the experimental data measured by the LHCb Collaboration [25] in p+p collisions in 2.0<y<2.5, 2.5<y<3.0, 3.0<y<3.5, 3.5<y<4.0, and 4.0<y<4.5 and scaled by different amounts shown in the panels. The data points are fitted by the Erlang distribution (Eq. (3)), the Hagedorn function (Eq. (4)), and the Tsallis-Levy function (Eq. (6)) by the solid, dashed, and dotted curves, respectively.
(a)(b)(c)(d)(e)(f)In Figure6, the pT spectra, d2σdpTdy, of (a, d, and e) Y1S⟶μ+μ−, (b, f) Y2S⟶μ+μ−, and (c, g) Y3S⟶μ+μ− induced in (a–c) and (e–g) p+p and (d) p+Pb collisions at sNN or s= (a–c) 8, (d) 8.16, and (e–g) 13 TeV are given. The symbols represent the experimental data [25–27], and the curves are our fitted results. The parameter values are listed in Tables 1 and 2 with χ2 and ndof. Once again, one can see that the experimental pT spectra of YnS,n=1,2,3⟶μ+μ− in different rapidity intervals in p+p and p+Pb collisions at high energies are approximately fitted by the Erlang distribution, the Hagedorn function, and the Tsallis-Levy function.Figure 6
Transverse momentum spectra,d2σ/dpTdy, of (a, d, e) Y1S⟶μ+μ−, (b, f) Y2S⟶μ+μ−, and (c, g) Y3S⟶μ+μ− in (a–c) and (e–g) p+p and (d) p+Pb collisions at (a–c) 8, (d) 8.16, and (e–g) 13 TeV. The symbols represent the experimental data measured by the LHCb Collaboration [25–27]. The rapidity intervals for panels (a–c) and (e–f) are 2.0<y<2.5, 2.5<y<3.0, 3.0<y<3.5, 3.5<y<4.0, and 4.0<y<4.5. The rapidity intervals for (d) are 1.5<y<2.0, 2.0<y<2.5, 2.5<y<3.0, 3.0<y<3.5, and 3.5<y<4.0. Different sets of data points are scaled by different amounts shown in the panels. The data points are fitted by the Erlang distribution (Eq. (3)), the Hagedorn function (Eq. (4)), and the Tsallis-Levy function (Eq. (6)) by the solid, dashed, and dotted curves, respectively.
(a)(b)(c)(d)(e)(f)(g)Before discussing the trends of parameters, we would like to point out the usability of the concept of temperature inp+p (p+p¯) collisions, which are small in size. As in refs. [56–59], in this work, we have treated p+p (p+p¯) collisions as where a medium was formed, or at least there is some degree of thermalization, enough to have a temperature for the emission source. On the other hand, the temperature parameter of the emission source is a reflection of the average kinetic energy of given particles. This means that we may use the concept of temperature. Even if the collision system is not enough large, we may use the temperature parameter to characterize the average kinetic energy of given particles over many events.Figure7 shows the dependences of (a, c, and e) pT and (b, d, and f) Ti on sNN (or s) for (a, b) J/ψ, (c, d) ψ2S, and (e, f) YnS,n=1,2,3. The different symbols represent the parameter values derived from free parameters extracted from Figures 1–6 and listed in Tables 1 and 2, where only the (two-component) Erlang distribution in the pT region of data available is used as an example. It is expected that the results corresponding to the Hagedorn and Tsallis-Levy functions are very close to the plot, because the two functions also describe approximately the data. As what we discussed in Subsection 2.5, no matter what functions are used to fit the experimental data, one should obtain similar pT (or Ti), if different functions fit the data good enough in the pT region of data available. By using the mentioned three functions which fit the data good enough, one can obtain pT (or Ti) within a systematic uncertainty of 8%. One can see from Figure 7 that pT and Ti increase significantly with the increase of collision energy. Meanwhile, pT and Ti increase with the increase of particle mass.Figure 7
Dependences of (a, c, and e)pT and (b, d, and f) Ti on sNN (or s) for (a, b) J/ψ, (c, d) ψ2S, and (e, f) YnS,n=1,2,3. The different symbols represent the parameter values derived from Figures 1–6 and listed in Tables 1 and 2 where only the (two-component) Erlang distribution in the pT region of data available is used as an example. By using the mentioned three functions which fit the data good enough in the pT region of data available, one can obtain similar pT (or Ti) within a systematic uncertainty of 8%.
(a)(b)(c)(d)(e)(f)Figure8 is the same as Figure 7, but showing the dependences of (a, c, e, and g) pT and (b, d, f, and h) Ti on (a, b) centrality C and (c-h) rapidity y for (a–d) J/ψ, (e, f) ψ2S, and (g, h) YnS,n=1,2,3. The different symbols represent the parameter values derived from free parameters extracted from Figures 1–6 and listed in Tables 1 and 2, where only the (two-component) Erlang distribution is used as an example. One can see that pT and Ti increase slightly with the increase of event centrality from peripheral to central collisions and decrease with the increase of rapidity from mid-rapidity to forward rapidity. Meanwhile, pT and Ti increase with the increases of collision energy and particle mass.Figure 8
Same as Figure7, but showing the dependences of (a, c, e, and g) pT and (b, d, f, and h) Ti on (a, b) C and (c–h) y for (a–d) J/ψ, (e, f) ψ2S, and (g, h) YnS,n=1,2,3.
(a)(b)(c)(d)(e)(f)(g)(h)The above parameter tendencies show that the temperature is mass dependent. This is also a reflection of the formation time dependence. According to the hydrodynamic behavior, “massive particles coming out of the system earlier in time with smaller radial flow velocities” [59]. This means that with the increase of mass, the formation time decreases, the temperature increases, and the flow velocity decreases. It should be noted that the fact that the massive particles coming out the system earlier is not caused by the high excitation of the system, but the leaver over due to the inertia of massive particles, in the hydrodynamic evolution.We may explain the tendency of derivedpT and Ti which have similar tendency with pT. With the increase of collision energy, the violent degree of collisions increases significantly due to large energy transfer, which results in the obvious increase of pT. With the increase of centrality, the degree of multiple-scattering increases due to more participant nucleons and produced particles taking part in the scattering process, which results in a slight increase of emission angle and then a slight increase of pT. With the increase of rapidity, the energy transfer decreases due to larger penetrability between participant nucleons. Meanwhile, the degree of multiple-scattering also decreases due to less produced particles taking part in the scattering process. These two factors result in the decrease of pT. It is natural that pT increases with the increase of m0.The free parameters in the Erlang distribution are directly reflected inpT, which will not be discussed anymore. The free parameters in the Hagedorn and Tsallis-Levy functions will be discussed in the Appendix, because no clear physical conclusion can be drawn from them at present, though the tendencies of them can be seen from this work.


## 4. Summary and Conclusions

In summary, the transverse momentum spectra ofJ/ψ, ψ2S, and YnS,n=1,2,3 produced in p+p, p+p¯, p+Pb, Au+Au, and Pb+Pb collisions over an energy range from 39 GeV to 13 TeV have been analyzed by the (two-component) Erlang distribution, the Hagedorn function, and the Tsallis-Levy function. The function results are approximately in agreement with the experimental data measured by several international collaborations. The values of related parameters are extracted from the fits, and the excitation functions of these parameters are obtained.The excitation functions of parameterspT and Ti increase from 39 GeV to 13 TeV. Meanwhile, pT and Ti increase (slightly) with event centrality and particle mass and decrease from mid-rapidity to forward rapidity. These tendencies render that these parameters describe the excitation and expansion degrees of the system. At higher energy, larger energy transfer had happened, which results in higher excitation and expansion degrees of the system. In central collisions and at mid-rapidity, larger energy transfer and further multiple-scattering had happened, which also results in higher excitation and expansion degrees of the system.The parametersp0 (T) and n0 (n) increase with the collision energy, which reflects the degree of energy deposition and transfer. In given collisions, there is a negative correlation between p0 (T) and n0 (n). At different energies, there is a positive correlation between p0 (T) and n0 (n). Indeed, there are correlations between p0 (T) and n0 (n) when we determine these parameters. The correlation between p0 (T) and n0 (n) is similar to that between kinetic freeze-out temperature and transverse flow velocity. If p0 (T) is similar to kinetic freeze-out temperature, n0 (n) should be similar to transverse flow velocity.



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*Source: 5031494-2020-08-17.xml*