Kinetic freeze-out properties from transverse momentum spectra of pions in high energy proton-proton collisions

Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions over a center-of-mass energy, $\sqrt{s}$, range from a few GeV to above 10 TeV are analyzed by the blast-wave fit with Boltzmann (Tsallis) distribution. The blast-wave fit results are well fitting to the experimental data measured by several collaborations. In a particular superposition with Hagedorn function, both the excitation functions of kinetic freeze-out temperature ($T_0$) of emission source and transverse flow velocity ($\beta_T$) of produced particles obtained from a given selection in the blast-wave fit with Boltzmann distribution have a hill at $\sqrt{s}\approx10$ GeV, a drop at dozens of GeV, and then an increase from dozens of GeV to above 10 TeV. However, both the excitation functions of $T_0$ and $\beta_T$ obtained in the blast-wave fit with Tsallis distribution do not show such a complex structure, but a very low hill. In another selection for the parameters or in the superposition with the usual step function, $T_0$ and $\beta_T$ increase generally quickly from a few GeV to about 10 GeV and then slightly at above 10 GeV, there is no such the complex structure, when also studying nucleus-nucleus collisions.
Comment: 24 pages, 8 figures. Invited contribution to the special issue "Statistical Approaches in High Energy Physics" in Physics, accepted