CP-odd effects at NLO in SMEFT $WH$ and $ZH$ production

CP-violation (CPV) is a rare phenomenon in the Standard Model whilst there is compelling indirect evidence for additional CPV sources in the Universe. The search for CPV effects at the LHC is thus one of the best-motivated precision tests of the Standard Model (SM) and an excellent probe of New Physics. NLO QCD corrections can affect the predictions for those measurements substantially. We study the impact of NLO QCD corrections in $WH$ and $ZH$ production in the Standard Model Effective Field Theory with bosonic CP-odd dimension-6 operators. We analyze the angular distributions at LO of those processes that can be used to probe CPV effects. We then show how NLO QCD effects modify those distributions. We encounter that the corrections have a clear angular dependence and differ between the SM, the dimension-6 squared and their interference, emphasising the need for an exact inclusion of NLO QCD in precision computations. We then perform a phenomenological analysis of $WH$ production at the LHC to study the impact of NLO QCD effects on the projected bounds on the CP-odd Wilson Coefficient $c_{\varphi\widetilde{W}}$. NLO QCD effects in the signal improve the bounds by $\sim10\%$ but reduce the significance of the interference.