Weak charge of the proton and new physics

We address the physics implications of a precise determination of the weak charge of the proton, $Q_W\mathrm{}\left(p\right),$ from a parity violating elastic electron proton scattering experiment to be performed at the Jefferson Laboratory. We present the standard model (SM) expression for $Q_W\mathrm{}\left(p\right)\mathrm{}$ including one-loop radiative corrections, and discuss in detail the theoretical uncertainties and missing higher order QCD corrections. Owing to a fortuitous cancellation, the value of $Q_W\mathrm{}\left(p\right)\mathrm{}$ is suppressed in the SM, making it a unique place to look for physics beyond the SM. Examples include extra neutral gauge bosons, supersymmetry, and leptoquarks. We argue that a $Q_W\mathrm{}\left(p\right)\mathrm{}$ measurement will provide an important complement to both high energy collider experiments and other low energy electroweak measurements. The anticipated experimental precision requires the knowledge of the $O\left({\alpha }_s\right)$ corrections to the pure electroweak box contributions. We compute these contributions for $Q_W\mathrm{}\left(p\right),$ as well as for the weak charges of heavy elements as determined from atomic parity violation.