  We generalize forward real Compton amplitude to the case of the interference
of the electromagnetic and weak neutral current, formulate a low-energy
theorem, relate the new amplitudes to the interference structure functions and
obtain a new set of sum rules. We address a possible new sum rule that relates
the product of the axial charge and magnetic moment of the nucleon to the 0th
moment of the structure function $g_5(\nu,0)$. We apply the GDH and the finite
energy sum rule for constraining the dispersive $\gamma Z$-box correction to
the proton's weak charge.
