2.1. Analysis of the Spectra
In high-energy physics, even the smallest hadron–hadron (
) collisions are rather complicated processes. One usually separates two main regimes of hadron production: one is a soft multiparticle production, dominant at low transverse momenta, where the spectra can also be fitted by an exponential behavior [
23], cf. the curve
in
Figure 1. We realize that
describes well this part of the spectra even in
collisions. As
gets higher (
3 GeV), the spectrum displays a power-law tail. They are predicted by perturbative QCD, owing to the hard scattering of current quarks and gluons. In a number of publications [
16,
17,
18,
19,
20], the Tsallis statistical distribution was successfully applied to describe data for
collisions over a wide range of the transverse momenta because of its two limits: the exponential shape at small
and the power-like distribution at large
,
We focus on the fittings of the produced charged particle spectra in elementary collisions with the non-extensive functions in Equation (
3). Data were taken for pions, kaons, and protons in
collisions at
GeV, 200 GeV from the PHENIX Collaboration [
18] and at 900 GeV [
16], 2.76 TeV [
19], 5.02 TeV, and 7 TeV [
17] from the ALICE Collaboration. We restrict our analysis to the midrapidity region
within the
ranges, as shown in
Table 2. Note that in the following,
,
K, and
p mark the spectra of
,
, and
, respectively.
Figure 1 shows that all of the different non-extensive functions we used fit the pion and kaon spectra very well for various kinds of beam energies at midrapidity. The ratios of
of the relevant fits are given in
Table 1. Specifically, the first two distributions (
and
) of
and
of
show close-fitting results. The distribution,
, derived thermodynamically, does not display large differences in the goodness of fit either. Checking the fitting parameters
A,
T, and
, we observe that, as we expected and introduced in the previous section, all these functions share the same Tsallis parameter
n. The two
functions (
and
) lead to fitting values of the temperature
T, which are different from the pure
fit (
). This indicates that the normalization constant does not affect the fitted
T and
q parameters but the integrated yield
. Namely, by normalizing the momentum spectrum
with the
normalization constant and the condition of
, we obtain the integral over
from 0 to its maximal values
:
Moving towards physical interpretation issues, we investigate the temperature,
T, and the non-extensive parameter,
q. Investigations in [
18,
24] showed that both of them express
dependence. In this paper, we found that they are also dependent on the hadron mass,
m. The
dependence, as a result, is studied in order to analyze hadron spectra parameters within the non-extensive approach. Following the phenomenological observations in [
25,
26], a QCD-like evolution can be introduced for both the parameters
T and
q. While analyzing data, we found that the temperature
T had a weak logarithmic
dependence. Thus, here we assume a linear
dependence to analyze the temperature
T, but the non-extensive parameter
q is kept with the stronger logarithmic distribution:
In summary, our work indicates that the BG distribution is not suitable for describing the hadron spectra over a wide range of
. Comparisons of their corresponding fitting errors
show that both
and
functions share the same goodness between
and
, cf. Equation (
3). Together with the thermodynamically derived
, all the non-extensive approaches (
) follow the experimental data accurately. The fitting temperature,
T, is nearly constant when changing the ratio of the collision energy to hadron mass,
. Specifically, distributions of
,
,
, and
are described best with such a connection, as shown in the left panel of
Figure 2. From
Table 3, we also see that the slope parameters in these four cases are almost zero, which means that they are constant around some values. The non-extensive parameter
q, on the other hand, follows a logarithmic dependence, agreeing with a pQCD-based motivation, cf. [
21]. Note that our results on
T and
q are different from the work by Cleymans et al. [
24]. Those authors parameterized this relation as a power-law.
2.2. Analysis of the and Results
In
[
17] collisions at 5.02 TeV and in
[
27,
28,
29,
30] collisions at 2.76 TeV, more kinds of hadron spectra are analyzed within the formulas of Equation (
3). Data are taken from the ALICE Collaboration within wide
ranges, as seen in
Table 4. We observe that all of them present good fittings over the whole range of
for each hadron at various kinds of centrality bins. On the other hand, similar to the
cases, the BG formula can still perform well just in the low
region (
GeV).
In this work, as an example, we analyzed the fitting results of
spectra of pions and kaons produced in all kinds of collisions mentioned above. It is instructive to plot the relationship between the fitting temperature
T and the Tsallis parameter
q for the same hadron spectra for different centralities in the same heavy-ion collisions. The results of pions and kaons in
collisions are also analyzed as comparisons. In
Figure 3, we show the linear correlating appearances for both
and
K in
at 2.76 TeV [
17] and in
at 5.02 TeV [
27,
28] as well as the
results in all kinds of collision energies [
16,
17,
18,
19] in this paper. In fact, whatever kinds of particle we study, all these non-extensive fittings give a similar dependence of
T on the parameter
q:
which agrees with our previous work [
21,
22] and that of others [
31].
Note that the slope parameter
in
Table 5 turns negative and
is nearly zero for the
case, as discussed in [
22]. Results of fittings on pion spectra, typically in
collisions at 5.02 TeV, fail in the obvious linear combinations probably due to the small mass of pions and high multiplicities. It is found that all forms of non-extensive distributions feature a similar relation between the temperature
T and non-extensive parameter
q. This, in turn, hopefully promotes a better understanding of the meaning of the non-extensive parameter
q.