Dependence of Thermodynamic Quantities at Freeze-out on Pseudorapidity and Collision Energy in pp Collisions at LHC Energies

Abstract

Abstract The transverse momentum distributions of charged hadrons produced in proton-proton collisions at center-of-mass energies ($\sqrt{s}$) of 0.9 TeV and 2.36 TeV, as measured by the CMS detector at the Large Hadron Collider (LHC), have been analyzed within various pseudorapidity classes utilizing the thermodynamically consistent Tsallis distribution. The fitting procedure resulted in the key parameters, namely the effective temperature ($T$), non-extensivity parameter ($q$), and kinetic freezeout volume ($V$). Additionally, the mean transverse momentum and the initial temperature of the particle source are determined through the fit function and the string percolation method, respectively. An alternative method is employed to calculate the kinetic freezeout temperature and transverse flow velocity from the $T$. Furthermore, thermodynamic quantities at the freezeout, including energy density ($\varepsilon$), particle density ($n$), entropy density ($s$), pressure ($P$), and squared speed of sound (${C_s}^2$), are computed using the extracted $T$ and $q$.
It is also observed that, with a decrease in pseudorapidity, all thermodynamic quantities, except the $V$ and $q$, increase. This trend is attributed to greater energy transfer along the mid pseudorapidity. The $q$ increases towards higher values of pseudorapidity, indicating that particles close to the beam axis are far from equilibrium. The $V$, however, remains nearly independent of pseudorapidity. The excitation function of these parameters ($q$) shows a direct (inverse) correlation with collision energy. The $\varepsilon$, $n$, $s$, and $P$ show a strong dependence on collision energies at low pseudorapidities. Explicit verification of the thermodynamic inequality $\varepsilon \geq 3 P$ suggests the formation of a highly dense droplet-like Quark-Gluon Plasma (QGP). Additionally, the inequality $T_i > T > T_0$ is explicitly confirmed, aligning with the evolution of the produced fireball.