Rapidity and Energy Dependences of Temperatures and Volume Extracted from Identified Charged Hadron Spectra in Proton-Proton Collisions at a Super Proton Synchrotron (SPS)

The standard (Bose-Einstein/Fermi-Dirac or Maxwell-Boltzmann) distribution from the relativistic ideal gas model is used to study the transverse momentum ($p_T$) spectra of identified charged hadrons (${\pi }^{-}$, ${\pi }^{+}$, $K^{-}$, $K^{+}$, $\bar{p}$, and $p$) with different rapidities produced in inelastic proton-proton ($pp$) collisions at the Super Proton Synchrotron (SPS). The experimental data measured by the NA61/SHINE Collaboration at the center-of-mass (c.m.) energies $\sqrt{s}=6.3$, 7.7, 8.8, 12.3, and 17.3 GeV are fitted well by the distribution. It is shown that the effective temperature ($T_{eff}$ or $T$), kinetic freeze-out temperature ($T_0$), and initial temperature ($T_i$) decrease with the increase in rapidity and increase with the increase in c.m. energy. The kinetic freeze-out volume ($V$) extracted from the ${\pi }^{-}$, ${\pi }^{+}$, $K^{-}$, $K^{+}$, and $\bar{p}$ spectra decreases with the rapidity and increase with the c.m. energy. The opposite tendency of $V$, extracted from the $p$ spectra, is observed to be increasing with the rapidity and decreasing with the c.m. energy due to the effect of leading protons.