# Hadron transverse momentum distributions of the Tsallis normalized and unnormalized statistics

## TL;DR

This paper derives exact analytical formulas for transverse momentum distributions in Tsallis statistics, compares normalized and unnormalized forms, and explores their implications for high-energy physics phenomena like hadron production.

## Contribution

It provides the first exact series expansion formulas for Tsallis distributions of particles with mass, clarifies the relation between phenomenological and theoretical Tsallis distributions, and introduces a regularization method for divergent cases.

## Key findings

- Exact formulas expressed as series expansions for Tsallis distributions.
- Phenomenological classical Tsallis distribution matches zeroth order of unnormalized Tsallis.
- Tsallis statistics enhances high-$p_T$ hadron production compared to standard distributions.

## Abstract

The exact analytical formulas for the transverse momentum distributions of the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles with nonzero mass in the framework of the Tsallis normalized and Tsallis unnormalized (also known as Tsallis-1 and Tsallis-2) statistics have been consistently derived. The final exact results were expressed in terms of the series expansions in the integral representation. The zeroth term approximation to both quantum and classical statistics of particles has been introduced. We have revealed that the phenomenological classical Tsallis distribution (widely used in high energy physics) is equal to the distribution of the Tsallis unnormalized statistics in the zeroth term approximation, but the phenomenological quantum Tsallis distributions (introduced by definition on the basis of the generalized entropy of the ideal gas) do not correspond to the distributions of the Tsallis statistics. We have found that in the ranges of the entropic parameter relevant to the processes of high-energy physics ($q<1$ for Tsallis-1 and $q>1$ for Tsallis-2) the Tsallis statistics is divergent. Therefore, to obtain physical results, we have regularized the Tsallis statistics by introducing an upper cut-off in the series expansion. The exact numerical results for the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles in the Tsallis normalized and unnormalized statistics have been obtained. We observed that the exact results of the Tsallis statistics strongly enhanced the production of high-$p_{T}$ hadrons in comparison with the usual phenomenological Tsallis distribution function at the same values of $q$. The $q$-duality of the Tsallis normalized and unnormalized statistics for the massive particles was studied.