From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy $pp$ and $p\overline{p}$ collisions

Transverse spectra of both jets and hadrons obtained in high-energy $pp$ and $p\overline{p}$ collisions at central rapidity exhibit power-law behavior of $1/p_T^n$ at high $p_T$. The power index $n$ is 4–5 for jet production and is 6–10 for hadron production. Furthermore, the hadron spectra spanning over 14 orders of magnitude down to the lowest $p_T$ region in $pp$ collisions at the LHC can be adequately described by a single nonextensive statistical mechanical distribution that is widely used in other branches of science. This suggests indirectly the possible dominance of the hard-scattering process over essentially the whole $p_T$ region at central rapidity in high-energy $pp$ and $p\overline{p}$ collisions. We show here direct evidences of such a dominance of the hard-scattering process by investigating the power indices of UA1 and ATLAS jet spectra over an extended $p_T$ region and the two-particle correlation data of the STAR and PHENIX collaborations in high-energy $pp$ and $p\overline{p}$ collisions at central rapidity. We then study how the showering of the hard-scattering product partons alters the power index of the hadron spectra and leads to a hadron distribution that may be cast into a single-particle nonextensive statistical mechanical distribution. Because of such a connection, the nonextensive statistical mechanical distribution may be considered as a lowest-order approximation of the hard-scattering of partons followed by the subsequent process of parton showering that turns the jets into hadrons, in high-energy $pp$ and $p\overline{p}$ collisions.

Figure 1

Comparison of the relativistic hard-scattering model results for jet production, Eq. (16) (solid curves), with experimental $d\sigma /d\eta E_{T}d{E}_T{|}_{\eta \sim 0}$ data from the D0 Collaboration [29], for hadron jet production within $|\eta |<0.5$, in $\overline{p}p$ collisions at (a) $\sqrt{s}=1.80\mathrm{TeV}$, and (b) $\sqrt{s}=0.63\mathrm{TeV}$.

Figure 2

Comparison of the relativistic hard-scattering model results for jet production, Eq. (16) (solid curves), with experimental $d\sigma /d\eta E_{T}d{E}_T{|}_{\eta \sim 0}$ data from the ALICE Collaboration [30], for jet production within $|\eta |<0.5$, in $pp$ collisions at 2.76 TeV for $R=0.4$, and $R=0.2$.

Figure 3

Comparison of the relativistic hard-scattering model results for jet production, Eq. (16) (solid curves), with experimental $d\sigma /d\eta E_{T}d{E}_T{|}_{\eta \sim 0}$ data from the CMS Collaboration [31], for jet production within $|\eta |<0.5$, in $pp$ collisions at 7 TeV.

Figure 4

Comparison of the relativistic hard-scattering model results for jet production, Eq. (22) (solid curves), with experimental $d\sigma /d\eta p_{T}d{p}_T$ data from the UA1 Collaboration [32], for jet production within $|\eta |<1.5$, in $\overline{p}p$ collisions at (a) $\sqrt{s}=0.546\mathrm{TeV}$, and (b) $\sqrt{s}=0.63\mathrm{TeV}$.

Figure 5

Comparison of the relativistic hard-scattering model results for jet production, Eq. (22) (solid curves), with experimental $d\sigma /d\eta p_{T}d{p}_T{|}_{\eta \sim 0}$ data from the ATLAS Collaboration [33], for jet production within $|\eta |<0.5$, in $pp$ collisions at $\sqrt{s}=7\mathrm{TeV}$, with (a) $R=0.4$ and (b) $R=0.6$.

Figure 6

PHENIX azimuthal angular distribution of associated particles per trigger in different $p_t^{\text{trig}}\otimes p_t^{\text{assoc}}$ combinations. The open circles are the associated particle yields per trigger, $d{N}_{\text{ch}}/N_{\text{trig}}d\mathrm{\Delta }\varphi$, in $pp$ collisions [37]. The solid curves are the theoretical associated particle yields per trigger calculated with Eq. (23).

Figure 7

Minimum-$p_T$-biased two-particle angular correlation, without a $p_T$ trigger selection, for charged hadrons produced in $pp$ collisions at $\sqrt{s}=200\mathrm{GeV}$ (from Fig. 1 of Ref. [39] of the STAR Collaboration). Here, ${\varphi }_{\mathrm{\Delta }}$ is $\mathrm{\Delta }\varphi$, the difference of the azimuthal angles of two detected charged hadrons, and ${\eta }_{\mathrm{\Delta }}$ is $\mathrm{\Delta }\eta$, the difference of their pseudorapidities.

Figure 8

Comparison of Eq. (57) with the experimental transverse momentum distribution of hadrons in $pp$ collisions at central rapidity $y$. The corresponding Boltzmann-Gibbs (purely exponential) distribution is illustrated as the dashed curve. For a better visualization, both the data and the analytical curves have been divided by a constant factor as indicated. The ratios of data/fit are shown at the bottom, where a roughly log-periodic behavior is observed on top of the $q$-exponential one. Data are taken from Refs. [53, 54, 55, 56].