Some analytic results on the Uehling correction to hyperfine splitting in a muonic atom

Abstract

We analyze the Uehling correction to the hyperfine structure of the ground state in hydrogenlike electronic and muonic atoms with a pointlike nucleus. The results are obtained analytically without any expansion over ~k. PACS No.: 31.20D Resume : Nous analysons la correction d’Uehling a la structure hyperfine du fondamental d’atomes hydrogenoides, electroniques ou muoniques, avec un noyau ponctuel, les resultats sont analytiques, sans recours a une expansion en puissances de ~k. [Traduit par la redaction] This work is a direct continuation of a previous paper [1] devoted to the Uehling correction to the Lamb shift (Fig. 1a) in a hydrogenlike atom with a pointlike nucleus. Here we calculate the spin-dependent part of the correction (Figs. 1b and 1c). In ref. 1, we examined the kind of integral for the complete Dirac wave function and the Uehling potential as a perturbation. More details will be found in a later paper1 where the method will be developed for any single-photon potential diagram including the one in Fig. 1b. The hyperfine structure (hfs) is more sensitive to the nuclear structure than the Lamb shift. As is well known, the theoretical uncertainty of the hyperfine splitting value for a hydrogenlike atom comes mainly from estimates of the nuclear magnetization distribution correction (the Bohr–Weisskopf effect). That is correct both for muonic and electronic atoms, but in the first case this correction is significantly larger and investigations of the hyperfine structure of muonic atoms can be used as a way to measure that effect and to apply the result to an electronic atom with the same nucleus. The hyperfine structure in a muonic atom being of the order +] ,p @ps is more important 1 S.G. Karshenboim. Manuscript in preparation. Received June 8, 1998. Accepted July 2, 1998. S.G. Karshenboim. D.I. Mendeleev Institute for Metrology, 198005 St. Petersburg, Russia. Telephone: 7-812-2591051; FAX: 7-812-1130114; e-mail: sgk@onti.vniim.spb.su V.G. Ivanov. Pulkovo Observatory, 196140, St. Petersburg, Russia. e-mail: ivanovv@gao.pnpi.spb.ru V.M. Shabaev. St. Petersburg State University, 198904, St. Petersburg, Russia. e-mail: shabaev@pobox.spbu.ru Can. J. Phys. 76: 503–506 (1998) © 1998 NRC Canada 504 Can. J. Phys. Vol. 76, 1998 Fig. 1. The Uehling corrections to the energy. The orbiting particle with mass 6 is either a muon or an electron and a particle in vacuum polarization loop is an electron.