Analytical results for the classical and quantum Tsallis hadron transverse momentum spectra. The zeroth order approximation and beyond

We derive analytical expressions for the first and second order terms in the hadronic transverse momentum spectra obtained from the Tsallis normalized (Tsallis-1) statistics. We revisit the zeroth order quantum Tsallis distributions in this formulation and obtain the corresponding analytical closed form expressions. It is observed that unlike the classical case, the phenomenological distributions used in the literature do not resemble the analytical closed forms of the zeroth order quantum spectra after the q $\to$ q${}^{-1}$ substitution, where q is the Tsallis entropic parameter. Though the factorization approximation increases the extent of similarity, it does not make them exactly the same. Since our results are based on the basic formulations of the statistical mechanics, the zeroth order Tsallis quantum distributions derived in this paper will be a better choice than their phenomenological counterparts.