Low-energy $Nφ$ scattering from a pole-enhanced triangle diagram

TL;DR

This work identifies a pole-enhanced triangle diagram, featuring two-kaon exchange promoted by a near-threshold $\Lambda(1405)$ in the $N\bar{K}$ subsystem, as a leading mechanism for low-energy $N\phi$ scattering. Using an unphysical kaon mass (to mimic lattice input) within a non-relativistic effective-range framework, the authors show the convergent part of the triangle diagram yields an attractive $N\phi$ interaction with a real scattering length $a$ roughly in the range $-1.5$ to $-0.5$ fm, in agreement with HAL QCD and ALICE results. A key result is the threshold behavior of $a$ with respect to $\tilde{\delta}$, the $K\bar{K}$-threshold–to–$\phi$ mass difference, where $a \propto \tilde{\delta}^n$ with $n$ between 0 and 1/2 depending on the regulator, evidencing genuine three-body dynamics. The paper demonstrates that threshold-driven three-body effects can dominate near-threshold two-body observables and provides a unified perspective that can be tested in lattice simulations with varying light-quark masses and in future experiments.

Abstract

We investigate low-energy $Nφ$ scattering driven by a pole-enhanced triangle diagram, in which the two-kaon-exchange contribution is promoted by the near-threshold $Λ(1405)$ pole in the $N\bar K$ subsystem. Using an unphysical kaon mass motivated by lattice simulations, we evaluate the $Nφ$ scattering length and find that this mechanism generates an attractive interaction with a magnitude of $-1.5$ to $-0.5,\rm{fm}$. Spin-dependent effects are not treated explicitly and are expected to provide subleading corrections in the near-threshold region. We further analyze the low-energy behavior of the triangle-diagram amplitude and show that the scattering length performs a characteristic power-law dependence on the parameter $\tildeδ$, defined as the mass difference between the $K\bar K$ threshold and the $φ$ meson. This threshold-driven behavior differs from that associated with van der Waals-type forces or the long-range tail of two-pion exchange, highlighting the role of three-body dynamics encoded in the pole-enhanced triangle diagram in shaping the near-threshold $Nφ$ interaction.

Figures (9)

• Figure 1: Schematic illustration of the two-kaon-exchange contribution to $N\phi$ scattering through a triangle diagram. The dashed, solid, and bold lines represent the kaon, $\phi$ meson, and nucleon, respectively. The corresponding particle mass and momentum are labeled in the figure.
• Figure 2: The $N\phi$ scattering driven by two-loop diagrams with Kaon scattering off nucleon. The dashed, solid, and bold lines represent Kaon, $\phi$ meson, and $N$, respectively, where the bubble represents $\mathcal{V}+\mathcal{V}\mathcal{G}\mathcal{V}\dots$ with the Weinberg-Tomozawa interaction $\mathcal{V}$ and $N\bar{K}$ loop $\mathcal{G}$.
• Figure 3: The $N\phi$ scattering length from two-Kaon-Exchange with $\tilde{\delta}=\left(1.0\pm 0.2 \right)\, \rm{MeV}$, where the solid, dashed, and dotted lines correspond to the central value, lower and upper limits of $\delta$, respectively. The magenta and green bands represent the real part of the scattering lengths from HAL QCD Lyu:2022imf and ALICE collaboration ALICE:2021cpv, respectively.
• Figure 4: The $N\phi$ scattering length from two-Kaon-Exchange with uncertainty in $R_{pole}$ deduced by physical couplings.
• Figure 5: The $N\phi$ scattering length from two-Kaon-Exchange with uncertainty in $g_i$.