The mass splitting among the isospin multiplets of light vector mesons

By including the strong isospin symmetry breaking effects and the electromagnetic contributions between the pseudoscalar mesons, we calculate the phase shifts of the $P$-wave $\pi\pi$ and $K\pi$ scattering up to $\mathcal{O}(p^4)$ in the framework of the SU(3) chiral perturbation theory (ChPT) and coupled channel inverse amplitude method. We re-fit the low energy constants with the present meson-meson scattering data and derive the mass differences for the charged and neutral iso-multiplets of $\rho$ and $K^{*}$. Our results show that the mass difference between $\rho^\pm$ and $\rho^0$ is very small while the mass difference between the charged and neutral $K^*$ can reach a relatively large value of $m_{K^{*0}}-m_{K^{*+}}= 2.91^{+1.43}_{-1.41}$ MeV. This full one-loop ChPT calculation would shed some light towards a better understanding of the long-standing puzzle about the $K^*$ mass splitting.