New formulation of gamma-Z box corrections to the weak charge of the proton

We present a new formulation of one of the major radiative corrections to the weak charge of the proton -- that arising from the axial-vector hadron part of the $gamma Z$ box diagram, $Re{rm e}, Box_{gamma Z}^{rm A}$. This formulation, based on dispersion relations, relates the $gamma Z$ contributions to moments of the $F_3^{gamma Z}$ interference structure function. It has a clear connection to the pioneering work of Marciano and Sirlin, and enables a systematic approach to improved numerical precision. Using currently available data, the total correction from all intermediate states is $Re{rm e}, Box_{gamma Z}^{rm A} = 0.0044(4)$ at zero energy, which shifts the theoretical estimate of the proton weak charge from $0.0713(8)$ to $0.0705(8)$. The energy dependence of this result, which is vital for interpreting the Q$_{rm weak}$ experiment, is also determined.