Revisiting the three-kaon interaction and its relation with $K(1460)$

TL;DR

The study investigates whether $K(1460)$ arises as a three-body KK\bar{K}$-like molecular state by employing a dynamical coupled-channel framework that enforces three-body unitarity via a spectator-isobar formalism. Two-body subsystems are modeled with LO chiral dynamics to generate light scalar isobars, and seven coupled channels are coupled to form the full three-body amplitude, analyzed through analytic continuation to identify poles and thresholds. In the degenerate-isobar limit, a bound $Kf_0$ and $Ka_0$ configuration emerges, but when realistic isobars are included, the pole is shielded by the complex $Kf_0$ threshold and the spectrum develops strong kinematic structures, notably a triangle singularity at $\sqrt{s}\approx 1.48$ GeV that yields cusp-like enhancements in multiple final states. The findings highlight that three-body dynamics and threshold effects can mimic resonant behavior, complicating the interpretation of $K(1460)$ as a genuine molecular state and providing a framework for disentangling genuine resonances from kinematic effects in kaon spectroscopy.

Abstract

We test the hypothesis of $K(1460)$ being a hadronic $J^P=0^-$ molecule through nonperturbative $S$-wave $K K\bar K$, $Kππ$, $Kπη$ coupled-channel dynamics in a three-body unitary isobar approach. The scalar two-body coupled-channel resonances $f_0(500)$, $f_0(980)$, $a_0(980)$ and $K^*_0(700)$ are generated in the subsytems in amplitudes that match phase shifts from experiment. In a first step, previous results in the limit of mass-degenerate, stable $f_0$ and $a_0$ isobars are reproduced. Once the full seven-coupled channel model is switched on, other $S$-matrix effects obscure and modify the resonance signal, including complex thresholds, a two-body cusp, and a triangle singularity at threshold.

Figures (14)

• Figure 1: Diagrammatic representation of some of the contributions considered in the $T^1$-partition. Particles are represented by horizontal lines. The blobs correspond to two-body interactions, described by two-body $t$-matrices, $t^i$ (with the superscript indicating the spectator particle). When writing the contribution associated with the diagrams, the latter are read from right to left, with the initial (final) state being represented by the ket $|i\rangle$ ($|j\rangle$). In this way, we get, in this case, the series $\langle j|(t^1+t^1 g^{12} t^2+t^1 g^{13} t^3+\cdots)|i\rangle$.
• Figure 2: Top panel, left: Diagram related to the contribution $t^1g^{12}t^2$ in the Faddeev approach. Top panel, right: The same diagram in terms of isobars (double line) to describe the interaction between particles 1 and 3 in the initial state, and 2 and 3 in the final state. Bottom row: Similar to the upper diagrams but for a case with one more interaction between the particles. The elements not included in the LSE, Eq. (\ref{['Ttilde']}), namely the external isobars, are shown in gray.
• Figure 3: A typical production process of $Kf_0^{\pi\pi}$ including the "disconnected" piece and the rescattering part. The $Kf_0$ state can be populated through different intermediate channels. The first diagram indicates that all three particles are in relative $S$-wave. The last diagram shows a near on-shell kaon exchange (highlighted in green) followed by a channel transition $K\bar{K}\to\pi\pi$.
• Figure 4: Left: Thresholds and their cuts (blue dots and lines) and the $K^*$ bound state (red) for mass-degenerate, stable $f_0$ and $a_0$ isobars, corresponding to the situation quoted in Table \ref{['tab:zerolim']}. Right: Same, but for realistic $f_0$, $a_0$ isobars.
• Figure 5: Isobar amplitudes $|\tilde{\tau}_{ji}|^2$ from small (dark) to large (white) values for $\sqrt{s}=(1510-70\,i)$ MeV. Poles are indicated with red dots, cuts with green lines, and the SMCs $C_1$ ($C_2$) from Fig. \ref{['fig:demo']}, mapped to the $\sigma$ planes using Eq. \ref{['eq:sigma']}, with the yellow squares (turquoise circles). For the $K\pi$ isobar there are two different spectator masses ($\pi$ and $\eta$), i.e., two different mappings. The standard choice for the two-body cuts is shown: they run from their respective threshold in the negative Im $\sigma$ direction, i.e., at an angle of $\theta=-\pi/2$ measured from the real axis.