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

We evaluate the $sigma$ exchange contribution to the $bar{K}Ntobar{K}N$ 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 $pipito Kbar{K}$ 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 $bar{K}Ntobar{K}N$ 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.