Short-Range Hard-Sphere Potential and Coulomb Interaction: Deser-Trueman Formula for Rydberg States of Exotic Atomic Systems

Abstract: In exotic atomic systems with hadronic constituent particles, it is notoriously difficult to estimate the strong-interaction correction to energy levels. It is well known that, due to the strength of the nuclear interaction, the problem cannot be treated on the basis of Wigner-Brioullin perturbation theory. Recently, high-angular-momentum Rydberg states of exotic atomic systems with hadronic constituents have been identified as promising candidates for the search of New Physics in the low-energy sector of the Standard Model. In order to render this endeavor feasible, it is necessary to estimate the strong-interaction correction to the atomic energy levels. We thus derive a generalized Deser-Trueman formula for the induced energy shift for a general hydrogenic bound state with principal quantum number $n$ and orbital angular momentum quantum number~$\ell$, and find that the energy shift is given by the formula delta E = 2 alpha_{n, L} beta_L (a_h/a_0)^{2 L + 1} E_h/n^3, where alpha_{n,0} = 1, alpha_{n,L} is the product from s=1 to s=L of the expression (s^-2 - n^-2), beta_L = (2 L + 1)/[(2 L + 1)!!]^2, where E_h is the Hartree energy, a_h is the hadronic radius and a_0 is the generalized Bohr radius. The square of the double factorial, [(2\ell + 1)!!]^2, in the denominator implies a somewhat fortunate, drastic suppression of the effect for higher angular momenta. The results are verified by numerical calculations on an exponential lattice with 20,000 points.