Radiative-recoil contributions to the Lamb shift

• Published in: Annals of physics Vol. 178; no. 1; pp. 1 - 47
• Main Authors: Bhatt, G.C., Grotch, H.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.08.1987
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/S0003-4916(87)80011-8

We consider the proton mass corrections to the radiative Lamb shift in hydrogen. A procedure is developed to incorporate these contributions in the standard approach for obtaining the nonrecoil shifts. An equation given by Grotch and Yennie is used as a starting point to describe the hydrogen atom. This equation includes the Coulomb and the convection potentials as well as the proton kinetic energy. The radiative corrections then produce the desired radiative-recoil contributions. The complete lowest order ( α( Zα) 4 m 2/ M) contributions as well as part of the higher order ( α( Zα) 5 m 2/ M) contributions are derived analytically in the so-called external field approximation. In order to obtain residual α( Zα) 5 m 2/ M contributions, we then consider perturbation kernels involving the two-photon exchange diagrams which are not already included in the external field approximation calculation. This is done utilizing the Fried-Yennie gauge for the radiative photon and the Coulomb gauge for the exchanged photons. The behavior of the perturbation kernels in the low-momentum region of the exchanged photon is analyzed in considerable detail. For the evaluation of the energy shift, the integrals are carried out numerically. The α( Zα) 5 m 2/ M contribution beyond the usual reduced mass correction is ΔE n = [ α( Zα) 5/ n 3] m 2/ M [-1.574 + (-0.415 ± 0.004)] δ l0 . For the 2 S 1/2 - 2 P 1/2 Lamb shift in hydrogen, this amounts to a contribution of (-2.527 ± 0.005) kHz. We conclude by reviewing the known contributions to the Lamb shift and by emphasizing the importance of calculating some of the unknown contributions.