Description of high-energy $pp$ collisions using Tsallis thermodynamics: Transverse momentum and rapidity distributions

A systematic analysis of transverse momentum and rapidity distributions measured in high-energy proton-proton ($pp$) collisions for energies ranging from 53 GeV to 7 TeV using Tsallis thermodynamics is presented. The reasonable description of all transverse momentum spectra obtained in earlier analyses is confirmed and extended. All energies can be described by a single Tsallis temperature of $68\pm 5\mathrm{MeV}$ at all beam energies and particle types investigated (43 in total). The value of the entropic index, $q$, shows a wider spread but is always close to $q\approx 1.146$. These values are then used to describe the rapidity distributions using a superposition of two Tsallis fireballs along the rapidity axis. It is concluded that the hadronic system created in high-energy $pp$ collisions between 53 GeV and 7 TeV can be seen as obeying Tsallis thermodynamics.

Figure 1

Fits to the $p_T$ distribution given by Eq. (6) to experimental data. (a) ${\pi }^{+}$, $\sqrt{s}=0.2\mathrm{TeV}$ [22], (b) $\mathrm{\Lambda }$, $\sqrt{s}=0.9\mathrm{TeV}$ [5], (c) $K^{-}$, $\sqrt{s}=2.76\mathrm{TeV}$ [23], and (d) $\varphi$, $\sqrt{s}=7.0\mathrm{TeV}$ [25].

Figure 2

Results for (a) the temperature $T$ and for (b) the entropic index, $q$, for all the sets of data analyzed in this work. The numbering on the abscissa corresponds to the entries in Table 1.

Figure 3

Results for (a) the temperature $T$ and for (b) the entropic index, $q$, for 200 GeV $pp$ collisions as a function of the charged hadron multiplicity. Different symbols identify different particles, the cross referring to any particle different from pion, kaon and proton.

Figure 4

Fit using Eq. (6) to experimental data for rapidity distribution from $pp$ collision at 900 GeV from the UA5 Collaboration [31]. Data for $y<0$ are obtained symmetrically from $y>0$. The full line represents the best fit using the $q$-Gaussian function ($q^{\prime }=q$) and the dashed line corresponds to the best fit using the Gaussian function ($q^{\prime }=1$) in Eq. (7). The dotted line shows the results when a delta function is used for the fireball rapidity distribution, i.e., $\nu (y_f)=\delta (y_f)$.

Figure 5

Values for the parameters $\sigma$ (a) and $y_0$ (b) in Eq. (6) as a function of the collision energy.

Figure 6

The particle multiplicity $N$ as a function of $\sqrt{s}$. The solid line is a linear fit to the data for $\sqrt{s}>800\mathrm{GeV}$. The dashed line represents the logarithm function. The dotted line is the multiplicity as a function of energy when the coefficient $K$ is used.

Figure 7

The charged hadrons rapidity distribution predicted for 13 GeV $pp$ collisions. The full line corresponds to the multiplicity calculated through the linear extrapolation and the dashed line corresponds to the multiplicity obtained with the logarithm extrapolation, as given in Fig. 6.

Figure 8

The charged hadrons $p_T$ distribution predicted for 13 GeV $pp$ collisions. The shaded regions show the expected distribution considering the uncertainties in the parameters $T$ and $q$. The upper region with full lines corresponds to the multiplicity calculated through the linear extrapolation and the bottom region with dashed lines corresponds to the multiplicity obtained with the logarithm extrapolation, as given in Fig. 6.

Figure 9

Fits to rapidity distributions measured in high-energy $pp$ collisions (see Table 2) with Eq. (6). The experimental data are from (a) UA5 Collaboration [31], (b) UA5 Collaboration [31], (c) PHOBOS Collaboration [32], (d) PHOBOS Collaboration [32], (e) UA5 Collaboration [31].

Figure 10

Series of fittings of experimental data on rapidity distribution (see Table 2) with Eq. (6). The experimental data are from (a) UA5 Collaboration [31], (b) CMS Collaboration [34], (c) CDF Collaboration [35], (d) CMS Collaboration [34], (e) CMS Collaboration [36].