α3Corrections to Hyperfine Structure in Hydrogenic Atoms

The ${\alpha }^3$ term in the ratio of the hyperfine splitting in the $2S$ state of the one-electron atom to the hyperfine splitting in the $1S$ state is recalculated, and a new theoretical value for this ratio is obtained which is in agreement with the experimental value, thereby eliminating a previously reported discrepancy. The calculation consists in the evaluation of the low-momentum parts, of order ${\alpha }^3$ hfs, of the expression for the lowest order radiative level shift in the bound interaction representation with external Coulomb and magnetic dipole fields. By rearranging the terms so as to display the gauge invariance of the matrix elements with respect to the external potentials, considerable simplicity is achieved, and the formulas are easily interpreted as a generalization of the expression for the lowest order Lamb shift. The contribution from soft photon intermediate states is obtained by an extension of the method developed by Schwartz and Tiemann for evaluating the Bethe logarithm, and an appendix contains a tabulation of twelve analogous integrals which were integrated numerically, and which may be of use elsewhere. The calculated value of the ratio is $\frac{1}{8}(1.000\mathrm{}0345\pm 0.0000002)$ which agrees with the experimental values for hydrogen: $\frac{1}{8}(1.000\mathrm{}0346\pm 0.0000003)$, and deuterium: $\frac{1}{8}(1.000\mathrm{}0342\pm 0.0000006)$.