Kaonic Hydrogen and Deuterium in Hamiltonian Effective Field Theory

The anti-kaon nucleon scattering lengths resulting from a Hamiltonian effective field theory analysis of experimental data and lattice QCD studies are presented. The same Hamiltonian is then used to compute the scattering length for the $K^- d$ system, taking careful account of the effects of recoil on the energy at which the $\bar{K}N$ T-matrices are evaluated. These results are then used to estimate the shift and width of the $1S$ levels of anti-kaonic hydrogen and deuterium. The $K^- p$ result is in excellent agreement with the SIDDHARTA measurement. In the $K^- d$ case the imaginary part of the scattering length and consequently the width of the $1S$ state are considerably larger than found in earlier work. This is a consequence of the effect of recoil on the energy of the $\bar{K}N$ energy, which enhances the role of the $\Lambda(1405)$ resonance.