Energy-level displacement of excited n p states of kaonic deuterium in a Faddeev-equation approach

We calculate the energy-level displacement of the excited $np$ state of kaonic deuterium in terms of the $p$-wave scattering length of ${K}^{\ensuremath{-}}d$ scattering. We solve the Faddeev equations for the amplitude of ${K}^{\ensuremath{-}}d$ scattering in the fixed-center approximation and derive the complex $p$-wave scattering length of ${K}^{\ensuremath{-}}d$ scattering in terms of the $s$-wave and $p$-wave scattering lengths of $\overline{K}N$ scattering. The estimated uncertainty of the complex $p$-wave scattering length is of about $15\phantom{\rule{0.16em}{0ex}}%$. For the calculated width ${\ensuremath{\Gamma}}_{2p}=10.203$ meV of the excited $2p$ state of kaonic deuterium we evaluate the yield ${Y}_{{K}^{\ensuremath{-}}d}=0.27%$ of x rays for the ${K}_{\ensuremath{\alpha}}$ emission line of kaonic deuterium. Using the complex $s$-wave and $p$-wave scattering lengths of $\overline{K}N$ scattering, calculated in B. Borasoy, R. Ni\ss{}ler, and W. Weise [Eur. Phys. J. A 25, 79 (2005)] and W. Weise and R. H\"artle [Nucl. Phys. A 804, 173 (2008)], we get the width ${\ensuremath{\Gamma}}_{2p}=2.675$ meV of the excited $2p$ state and the yield ${Y}_{{K}^{\ensuremath{-}}d}=1.90%$ of x rays for the ${K}_{\ensuremath{\alpha}}$ emission line of kaonic deuterium. The results obtained in this paper can be used for planning experiments on the measurements of the energy-level displacement of the ground state of kaonic deuterium, caused by strong low-energy interactions.