The $\sigma K$ coupling in the chiral unitary approach and the isoscalar $\bar{K}N$, $\bar{K}A$ interaction

TL;DR

This paper investigates the role of sigma exchange in antikaon-nucleon interactions within a chiral unitary framework, revealing it vanishes in certain conditions and questioning its significance in nuclear matter models.

Contribution

It demonstrates that sigma exchange contributions vanish in the forward direction in a chiral approach, challenging assumptions used in relativistic mean field models for antikaon interactions.

Findings

Sigma exchange contribution vanishes in the forward direction.

The elementary sigma exchange potential is smaller than current uncertainties.

Questions the physical meaning of sigma exchange potentials in nuclear models.

Abstract

We evaluate the "$\sigma$" exchange contribution to the $\bar{K}N\to\bar{K}N$ scattering within a chiral unitary approach. We show that the chiral transition potentials for $\pi \pi \to K \bar{K}$ in the $t$-channel lead to a "$\sigma$" contribution that vanishes in the $\bar{K}$ forward direction and, hence, would produce a null "$\sigma$" exchange contribution to the $K^-$ optical potential in nuclear matter in a simple impulse approximation. This is a consequence of the fact that the leading order chiral Lagrangian gives an I=0 $\pi\pi\to K\bar{K}$ amplitude proportional to the squared momentum transfer, $q^2$. This finding poses questions on the meaning or the origin of "$\sigma$" exchange potentials used in relativistic mean field approaches to the $K^-$ nuclear selfenergy. This elementary "$\sigma$" exchange potential in $\bar{K}N\to\bar{K}N$ is compared to the Weinberg-Tomozawa term and is found to be smaller than present theoretical uncertainties but will be relevant in the future when aiming at fitting increasingly more accurate data.