Weak effects in proton beam asymmetries at polarised RHIC and beyond

We report on a calculation of the full one-loop weak corrections through the order $\alpha_{\mathrm{S}}^2\alpha_{\mathrm{W}}$ to parton-parton scattering in all possible channels at the Relativistic Heavy Ion Collider (RHIC) running with polarised $pp$ beams (RHIC-Spin). This study extends the analysis previously carried out for the case of $2\to2$ subprocesses with two external gluons, by including all possible four-quark modes with and without an external gluon. The additional contributions due to the new four-quarks processes are extremely large, of order 50 to 100% (of either sign), not only in the case of parity-violating beam asymmetries but also for the parity-conserving ones and (although to a more limited extent) the total cross section. Such ${\cal O}(\alpha_{\mathrm{S}}^2\alpha_{\mathrm{W}})$ effects on the CP-violating observables would be an astounding 5 times larger for the case of the LHC with polarised beams -- which has been discussed as one of the possible upgrades of the CERN machine -- whereas they would be much reduced for the case of the CP-conserving ones as well as the cross section.