Line-shape effects in the measurement of the positronium hyperfine interval

Abstract

Recently Rich has pointed out that annihilation terms in the effective 4 \ifmmode\times\else\texttimes\fi{} 4 Hamiltonian $H$ for $n=1$ positronium cause the real parts of the Zeeman eigenvalues to be shifted by terms of order ${(\frac{{\ensuremath{\lambda}}_{s}}{4\ensuremath{\pi}}\ensuremath{\Delta}v)}^{2}\ensuremath{\approx}{10}^{\ensuremath{-}5}$ relative to the Breit-Rabi eigenvalues. Here ${\ensuremath{\lambda}}_{s}$ is the annihilation rate of the singlet state and $\ensuremath{\Delta}v$ is the hyperfine interval. Rich observes that the $\ensuremath{\Delta}v$ measurements have not correctly dealt with decay. The Zeeman-resonance line shape is calculated here assuming that the non-Hermitian $H$ describes the motion of the four $n=1$ levels via Schr\"odinger's equation. The deviations of this line shape from a Lorentzian are exhibited. The asymmetry of the line causes a shift in the line center relative to what one would obtain from a Breit-Rabi plus Lorentzian fit to a measured Zeeman-resonance curve. To take this into account, the measurement of $\ensuremath{\Delta}v$ by A. P. Mills, Jr. and G. H. Beaman [Phys. Rev. Lett. 34, 246 (1975)] should be increased by 2.5 ppm to $\ensuremath{\Delta}v$ (Mills and Beaman)=203.3875(16) GHz. When the Egan et al. measurement [P. O. Egan, V. W. Hughes, and M. H. Yam, Phys. Rev. A 15, 251 (1971)], which used a different line shape, is reinterpreted in terms of the line shape calculated here, the Egan et al. $\ensuremath{\Delta}v$ value increases by 21 ppm to $\ensuremath{\Delta}v(\mathrm{Egan} \mathrm{et} \mathrm{al}.)=203.3890(12)$ GHz. The weighted mean of the two corrected measurements is $\ensuremath{\Delta}v=203.3885(10)$ GHz.