An Energy Independent Scaling of Transverse Momentum Spectra of Direct (Prompt) Photons from Two-Body Processes in High-Energy Proton-Proton Collisions

Transverse momentum spectra of direct (prompt) photons from two-body processes in high-energy proton-proton (p+p) collisions are analyzed in this paper. The experimental invariant cross-sections are collected at mid-rapidity in p+p collisions measured by the UA6, CCOR, R806, R110, PHENIX, NA24, CMS, ALICE, and ATLAS Collaborations over a center-of-mass energy range from 24.3 GeV to 13 TeV. In fitting the data, different kinds of functions are used which include the revised Tsallis-Pareto-type function (the TP-like function) at the particle level, the convolution of two TP-like functions at the quark level, and the root-sum-of-squares of two revised Tsallis-like functions in which the quark chemical potentials mu(i) = mu(B)/3 or mu(i) = 0, where mu(B) is the baryon chemical potential. The values of three main free parameters have been extracted: the effective temperature T, power index n(0) (or entropy index q), and correction index a(0). After analyzing the changing trends of the parameters, it is found that T, q, and a(0) increase and n(0) decreases with the increase of collision energy. Based on the analyses of transverse momentum spectra, an energy independent scaling, that is, the x(T) scaling, is obtained.