Three-body unitary determination of the $f_1(1285)$ and $f_1(1420)$ pole positions

Abstract

We study the $I^G(J^{PC})=0^+(1^{++})$ $K\bar K\pi$ system in an infinite-volume three-body unitary framework, focusing on the pole content of the region of the $f_1(1285)$ and $f_1(1420)$ resonances. The coupled $\pi a_0$-$K\bar K^*$ amplitude is constructed in the spectator-isobar representation, where the one-particle-exchange interaction required by three-body unitarity automatically incorporates the triangle-singularity mechanism. The short-range three-body interaction is constrained by fitting the $0^+(1^{++})$ component of the BESIII $K^0_SK^0_S\pi^0$ invariant-mass distribution in the $J/\psi\to\gamma(K^0_SK^0_S\pi^0)$ decay. Analytically continuing the fitted amplitude to the relevant unphysical Riemann sheets, we find two robust poles: \begin{align} \sqrt{s_{f_1(1285)}}&= \left(1277\pm2\pm1\right) -i\left(12\pm1\pm0\right)\text{MeV}\,,\notag\\ \sqrt{s_{f_1(1420)}}&= \left(1435\pm2\pm7\right) -i\left(40\pm2\pm1\right)\text{MeV}\,.\notag \end{align} The pole trajectories indicate that the $f_1(1285)$ originates from dressing a bare state introduced in the potential. In contrast, the $f_1(1420)$ is predominantly dynamically generated, and a single-channel analysis traces it to an $S$-wave $K\bar K^*$ quasi-bound state mixed with the nearby bare state, supporting its hadronic-molecule interpretation. We also find an additional pole deeper in the complex plane in the best-fit amplitude on the same Riemann sheet as the $f_1(1285)$. This additional pole is generated by the $P$-wave $\pi a_0$ contact interaction alone. It has a sizable cutoff and two-body-input dependence, and leaves little visible imprint on the physical lineshape. Finally, we provide a detailed and pedagogical appendix on how three-body cuts affect the solution of the integral equation.