The distribution of low-pT Υ(1<i>S</i>) in hadron gas

The <inline-formula><tex-math id="M17">\begin{document}$ \Upsilon(1S) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20230990_M17.jpg"/><graphic xlink:href="19-20230990_M17.png"/></alternatives></inline-formula> meson serves as a reliable probe in heavy-ion collisions, the regeneration process QGP(Quark-Gluon Plasma) is negligible compared to id="M18">\begin{document}$ J/\psi xlink:href="19-20230990_M18.jpg"/><graphic xlink:href="19-20230990_M18.png"/></alternatives></inline-formula>. Therefore, distribution of id="M19">\begin{document}$ xlink:href="19-20230990_M19.jpg"/><graphic xlink:href="19-20230990_M19.png"/></alternatives></inline-formula> hadron gas provides valuable information about QGP. Consequently, its study holds great significance. influenced by flow, quantum, and strong interaction effects. Previous models have predominantly focused on one or two these effects while neglecting others, resulting inclusion unconsidered fitted parameters. In this paper, we aim comprehensively examine all three simultaneously from novel fractal perspective through physical calculations, rather than relying solely data fitting. Close critical temperature, combined action leads formation two-meson structure comprising id="M20">\begin{document}$ xlink:href="19-20230990_M20.jpg"/><graphic xlink:href="19-20230990_M20.png"/></alternatives></inline-formula> nearest neighboring meson. However, with evolution system, most states undergo disintegration. To describe process, establish two-particle model(TPF model). Our model proposes that, under influence near self-similarity emerges, involving id="M21">\begin{document}$ xlink:href="19-20230990_M21.jpg"/><graphic xlink:href="19-20230990_M21.png"/></alternatives></inline-formula>-<i>π</i> state id="M22">\begin{document}$ xlink:href="19-20230990_M22.jpg"/><graphic xlink:href="19-20230990_M22.png"/></alternatives></inline-formula>-<i>π</i> two-quark state. As system evolves, gradually disintegrates. We introduce an influencing factor, id="M23">\begin{document}$ q_{fqs} xlink:href="19-20230990_M23.jpg"/><graphic xlink:href="19-20230990_M23.png"/></alternatives></inline-formula>, account for effects, well escort id="M24">\begin{document}$ q_2 xlink:href="19-20230990_M24.jpg"/><graphic xlink:href="19-20230990_M24.png"/></alternatives></inline-formula>, represent binding force between <i>b</i> id="M25">\begin{document}$ \bar{b} xlink:href="19-20230990_M25.jpg"/><graphic xlink:href="19-20230990_M25.png"/></alternatives></inline-formula> impact By solving probability entropy equations, derive values id="M26">\begin{document}$ xlink:href="19-20230990_M26.jpg"/><graphic xlink:href="19-20230990_M26.png"/></alternatives></inline-formula> id="M27">\begin{document}$ xlink:href="19-20230990_M27.jpg"/><graphic xlink:href="19-20230990_M27.png"/></alternatives></inline-formula> at various collision energies. Substituting value id="M28">\begin{document}$ xlink:href="19-20230990_M28.jpg"/><graphic xlink:href="19-20230990_M28.png"/></alternatives></inline-formula> into function, successfully obtain transverse momentum spectrum low-<inline-formula><tex-math id="M29">\begin{document}$ p_T xlink:href="19-20230990_M29.jpg"/><graphic xlink:href="19-20230990_M29.png"/></alternatives></inline-formula> id="M30">\begin{document}$ xlink:href="19-20230990_M30.jpg"/><graphic xlink:href="19-20230990_M30.png"/></alternatives></inline-formula>, which demonstrates good agreement experimental data. Additionally, analyze id="M31">\begin{document}$ xlink:href="19-20230990_M31.jpg"/><graphic xlink:href="19-20230990_M31.png"/></alternatives></inline-formula> temperature. Interestingly, observe that id="M32">\begin{document}$ xlink:href="19-20230990_M32.jpg"/><graphic xlink:href="19-20230990_M32.png"/></alternatives></inline-formula> greater 1 decreases temperature decreases. This behavior arises fact reduce number microstates, leading id="M33">\begin{document}$ q_{fqs}>1 xlink:href="19-20230990_M33.jpg"/><graphic xlink:href="19-20230990_M33.png"/></alternatives></inline-formula>. decrease id="M34">\begin{document}$ xlink:href="19-20230990_M34.jpg"/><graphic xlink:href="19-20230990_M34.png"/></alternatives></inline-formula> aligns understanding diminishes expands. future, TPF can be employed investigate other mesons resonance states.