Production of $K^+K^-$ Pairs Through Decay of $\phi$ Meson

We develop a theoretical framework for the production of $K^+K^-$ pairs through the decay of $\phi$ mesons produced from a thermal background, based on the Nambu-Jona-Lasinio (NJL) model. The differential production rate of $K^+K^-$ is related to the self-energy of the $\phi$ meson, incorporating the contributions of the quark loop at leading order and the kaon loop at next-to-leading order in the $1/N_c$ expansion. We numerically evaluate the invariant mass spectrum of the $K^+K^-$ pair both in vacuum and at finite temperature. The inclusion of the kaon loop results in a finite width of the spectrum, improving agreement with experimental data. We also investigate the spin alignment of the $\phi$ meson induced by its motion relative to the thermal background. In the NJL model with only chiral condensates, we find that deviations from the unpolarized limit of 1/3 are negligible.