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

TL;DR

This paper derives a generalized Deser-Trueman formula to estimate strong-interaction energy shifts in high-angular-momentum Rydberg states of exotic atoms with hadronic constituents, aiding searches for new physics.

Contribution

It introduces a new formula for calculating energy corrections in exotic atomic systems, extending previous models to include arbitrary angular momentum states.

Findings

Derived a generalized Deser-Trueman formula for energy shifts.

Showed the suppression of strong-interaction effects at higher angular momenta.

Provided explicit expressions for the energy correction depending on quantum numbers.

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 drastic suppression of the effect for higher angular momenta.