Research Article | Open Access
Qi Wang, Fu-Hu Liu, "Excitation Function of Initial Temperature of Heavy Flavor Quarkonium Emission Source in High Energy Collisions", Advances in High Energy Physics, vol. 2020, Article ID 5031494, 31 pages, 2020. https://doi.org/10.1155/2020/5031494
Excitation Function of Initial Temperature of Heavy Flavor Quarkonium Emission Source in High Energy Collisions
The transverse momentum spectra of , , and produced in proton-proton , proton-antiproton , proton-lead , gold-gold , and lead-lead () 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 is larger than that of and , and the initial temperature for emission is higher than that for and emission, which shows a mass-dependent behavior.
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 momentum is, the higher excitation state the emission source stays at. Meanwhile, the higher the initial temperature () [1–5] is, the more violent the collisions are. By the analysis of the excitation functions of and , we can learn more about the process in high energy collisions in which the excitation functions of several parameters such as and can be obtained from the spectra of produced particles.
In a data-driven reanalysis, to obtain and , at the first place, we need the 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 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 parameters and , we can obtain other related parameters by using the method which is similar to extract and . For example, using the Hagedorn function or the inverse power-law [9, 10] and the Tsallis-Levy function [11, 12] to fit spectra, some free parameters such as , , , and 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 and but also the trends of free parameters , , , and can be studied from the fit to 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 spectra of heavy flavor quarkonia (charmonia and bottomonia) produced in high energy collisions. The function results are compared with the spectra of , , and measured by the STAR [13, 14], CDF [15–17], ALICE , 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 -th) parton is assumed to contribute to an exponential function of transverse momentum () distribution. Let denotes the mean transverse momentum contributed by the -th parton, we have the probability density function of to be which is normalized to 1. The probability density function of contributed by all partons which have taken part in the collision process is the convolution of exponential functions [6–8]. We have the distribution (the probability density function of ) of final state particles to be the Erlang distribution
which is naturally normalized to 1. The mean is .
In the two-component Erlang distribution, we have
where denotes the contribution fraction of the first component, () denotes the number of partons in the first (second) component, and denotes the mean transverse momentum contributed by each parton in the first (second) component. The mean is , where –3 in this work and if .
2.2. The (Two-Component) Hagedorn Function
The Hagedorn function is an inverse power-law which is suitable to describe wide 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 to be
where and are the free parameters and is the normalization constant which is related to and and results in . 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 where denotes the contribution fraction of the first component, () is the normalization constant which results in the first (second) component to be normalized to 1, and () and () are free parameters related to the first (second) component. To combine the free parameters of the two components, we have and .
Generally, Eq. (4) is possible to describe the spectra in both the low- and high- regions. In fact, the spectra in the low- and high- 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- 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 spectrum, Eq. (5) is possibly needed.
2.3. The (Two-Component) Tsallis-Levy Function
The Tsallis statistics  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 . where and are free parameters, is the transverse mass, is the rest mass of the considered particle, and is the normalized constant which is related to , , and and results in .
We notice that is related to particle mass , which is not the case of and presented in Eqs. (2) and (4), respectively. Although is related to , this relation is not strong due to appearing only in . The fact that the Tsallis distribution depends on shows that this takes simple kinematics into account, as it is well known that or (something like transverse kinetic energy) is a better “scaling variable” for the spectra than .
In the case of using two-component Tsallis-Levy function, we have where denotes the contribution fraction of the first component, () is the normalization constant which results in the first (second) component to be normalized to 1, and () and () are free parameters. To combine the free parameters of the two components, we have and .
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 having () with the fraction of (), the common effective temperature of the two components is extracted from the assumed common equilibrium state of the two components. That is which has the same form as the parameter .
2.4. The Initial Temperature
According to the color string percolation model [53–55], the initial temperature of the emission source is determined by where is the square of the root-mean-square of due to . If the -component () and -component () of the transverse momentum are considered, we have
In the source rest-frame and under the assumption of isotropic emission, if the -component of momentum is , we also have
Although the source rest-frame is the lab-frame for symmetric collisions, we have mentioned the source rest-frame because asymmetric proton-lead (+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 factor to be 1 in the color string percolation model . If we consider more than one strings taking part in the given particle production, the minimum will be nearly 0.6 . Thus, we shall obtain a higher by multiplying a revised factor 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 extract and , though the Hagedorn and Tsallis-Levy functions are physically relevant. In fact, in the two functions, if we let , , , the two functions are the same. Here, is an entropy index that characterizes the excitation degree of the collision system [11, 12]. Generally, or is a sizeable quantity, which results in 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 low and . The second component has a less fraction and high and . 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, (or ) obtained from different fits is approximately the same within a small systematic uncertainty, if different functions fit the data good enough in the region of data available. For example, if simple Maxwell-Boltzmann or Bose-Einstein statistics can fit the data, we may obtain similar (or ) with other functions. In the case of multicomponent Maxwell-Boltzmann or Bose-Einstein statistics being needed, we may also obtain similar (or ).
Indeed, the data itself decides (or ), and (or ) 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 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 , or power-law tails at large ). To reduce the effect, the data should be measured in a sufficiently large 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-) or mid-pseudorapidity (mid-), or the values of mid- or mid- can be regarded as 0 directly. In fact, for the experimental data with non-zero mid- or mid-, we may directly regard them as those with mid- or mid-. 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 pair or if only one pair) for different panels, Figure 1 shows the spectra, (a–c) , (d, e) , (f) , and (g) , of (a–d) , (e) , (f) prompt , and (g) inclusive produced in (a–c) gold-gold (), (d, e) proton-proton (+), (f) proton-antiproton (), and (g) lead-lead () collisions at mid-rapidity (a–d) , (e) , (f) , and forward rapidity (g) at or (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 denotes the number of particles, denotes the cross section, and 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 , , , and are listed in Table 1 with and number of degrees of freedom (ndof). The values of free parameters , , , and are listed in Table 2 with and ndof. One can see that the (two-component) Erlang distribution, the Hagedorn function, and the Tsallis-Levy function fit approximately the experimental spectra of via different decay or production modes in high energy , , , and collisions.