Breit Equation with Form Factors in the Hydrogen Atom

TL;DR

This paper extends the Breit equation to include proton structure effects via form factors, providing analytical and numerical insights into hyperfine splittings in hydrogen and muonic hydrogen.

Contribution

It introduces a finite size corrected potential with form factors into the Breit equation, offering new analytical expressions and numerical results for hyperfine structures.

Findings

Finite size corrections significantly affect hyperfine splittings.

Analytical expressions for hyperfine potentials with form factors are derived.

Numerical results align with experimental data and other methods.

Abstract

The Breit equation with two electromagnetic form-factors is studied to obtain a potential with finite size corrections. This potential with proton structure effects includes apart from the standard Coulomb term, the Darwin term, retarded potentials, spin-spin and spin-orbit interactions corresponding to the fine and hyperfine structures in hydrogen atom. Analytical expressions for the hyperfine potential with form factors and the subsequent energy levels including the proton structure corrections are given using the dipole form of the form factors. Numerical results are presented for the finite size corrections in the 1S and 2S hyperfine splittings in the hydrogen atom, the Sternheim observable $D_{21}$ and the 2S and 2P hyperfine splittings in muonic hydrogen. Finally, a comparison with some other existing methods in literature is presented.