On pionic hydrogen. Quantum field theoretic, relativistic covariant and model-independent approach

Abstract

We consider pionic hydrogen $A_{\pi p}$ , the bound $\pi^- p$ state. Within the quantum field theoretic and relativistic covariant approach we calculate the shift and width of the energy level of the ground state of pionic hydrogen caused by strong low-energy interactions treated perturbatively. The generalization of the Deser-Goldberger-Baumann-Thirring (DGBT) formulas (S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96, 774 (1954)) is given. The generalized DGBT formulas for the energy level displacement of the ground state of pionic hydrogen contain the non-perturbative and model-independent correction of about $1\%$ , caused by the relativistic covariant smearing of the wave function of the ground state around the origin. This non-perturbative correction is very important for the precise extraction of the S-wave scattering lengths of the $\pi N$ scattering from the experimental data on the energy level displacements in pionic hydrogen by the PSI Collaboration. In addition, the shift of the energy level of the ground state of pionic hydrogen is improved by the second-order correction of strong low-energy interactions which is about 0.1%. This testifies the applicability of the perturbative treatment of strong low-energy interactions to the analysis of pionic hydrogen. We show that the width of the energy level of the ground state of pionic hydrogen is valid to all orders of the perturbation theory in strong low-energy interactions.