Hyperfine and Zeeman interactions of the $a(1)[^3\Sigma^+_1]$ state of PbO

The role of the interaction with the nearest electronic state $^3\Sigma^+_{0^-}$ on the hyperfine structure and magnetic properties of the $a(1)[^3\Sigma^+_1]$ state of PbO is assessed. The accounting for this contribution leads to difference between $g$-factors of the $J=1$ $\Omega$-doublet levels, $\Delta g = 37\times10^{-4}$, that is in a good agreement with the experimental datum $\Delta g = 30(8)\times10^{-4}$. The contribution of this interaction rapidly grows with $J$. For $J=30$ the difference of $g$-factors of $\Omega$-doublet states reaches 100%; for hyperfine constants it is 18%. These differences also depend on the electric field and for $E=11$ V/cm for $^{207}$PbO the difference in $g$-factors turn to zero. The latter is important for suppressing systematic effects in the electron electric dipole moment search experiment.