$\Lambda (1405)$ Contribution to Kaon-Nucleon Scattering Lengths In Chiral Perturbation Theory

We examine the role of the $\Lambda(1405)$ in kaon-nucleon scattering lengths using chiral perturbation theory. The leading nonanalytic SU(3) corrections reduce the coupling of the $\Lambda (1405)$ to $KN$ compared to $\Sigma\pi$. S-wave $K^-p$ scattering is the only channel significantly affected by the $\Lambda (1405)$ pole which substantially cancels against the leading term fixed by the vector current. This cancellation and the close proximity of the $\Lambda (1405)$ pole makes this SU(3) correction to $a(K^-p)$ large and at lowest order leaves the sign of $a(K^-p)$ undetermined. We extract two linear combinations of constants appearing at higher order in the chiral expansion from measurements of $KN$ scattering lengths. These constants are important for the offshell behaviour of $KN$ scattering amplitudes that play a central role in kaon condensation. The recent claims about the calculability of these constants are discussed.