Combined effect of coherent Z exchange and the hyperfine interaction in the atomic parity-nonconserving interaction

The nuclear spin-dependent parity-nonconserving (PNC) interaction arising from a combination of the hyperfine interaction and the coherent, spin-independent, PNC interaction from Z exchange is evaluated using many-body perturbation theory. For the ${6s}_{1/2}-{7s}_{1/2}$ transition in ${}^{133}\mathrm{Cs},$ we obtain a result that is about 40% smaller than that found previously by Bouchiat and Piketty [Phys. Lett. B 269, 195 (1991)]. Applying this result to ${}^{133}\mathrm{Cs}$ leads to an increase in the experimental value of nuclear anapole moment and exacerbates differences between constraints on PNC meson coupling constants obtained from the Cs anapole moment and those obtained from other nuclear parity violating experiments. Nuclear spin-dependent PNC dipole matrix elements, including contributions from the combined weak-hyperfine interaction, are also given for the ${7s}_{1/2}-{8s}_{1/2}$ transition in ${}^{211}\mathrm{Fr}$ and for transitions between ground-state hyperfine levels in K, Rb, Cs, ${\mathrm{Ba}}^{+},$ Au, Tl, Fr, and ${\mathrm{Ra}}^{+}.$