Updated study of the $η$-$η'$ mixing and the thermal properties of light pseudoscalar mesons at low temperatures
Within the framework of the $U(3)$ chiral perturbation theory, we revisit the masses, decay constants and the mixing parameters of the light pseudoscalar mesons $π, K, η$ and $η'$. The low energy constants up to next-to-next-to-leading order are determined by including the light-quark mass dependences of the various quantities from different lattice QCD simulations and relevant phenomenological inputs. Then we study the finite-temperature behaviors of the masses of the light pseudoscalar mesons. The thermal behaviors of the $η$-$η'$ mixing angles in singlet-octet and quark-flavor bases are also explored. While the masses of the $π, K$, and $η$ are increased when increasing the temperatures, the mass of the $η'$ turns out to be slightly decreased in the low-temperature region.