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Abstract We consider the impact on W W production of the unique dimension-six operator coupling gluons to the Higgs field. In order to study this process, we have to appropriately model the effect of a veto on additional jets. This requires the resummation of large logarithms of the ratio of the maximum jet transverse momentum and the invariant mass of the W boson pair. We have performed such resummation at the appropriate accuracy for the Standard Model (SM) background and for a signal beyond the SM (BSM), and devised a simple method to interface jet-veto resummations with fixed-order event generators. This resulted in the fast numerical code MCFM-RE, the Resummation Edition of the fixed-order code MCFM. We compared our resummed predictions with parton-shower event generators and assessed the size of effects, such as limited detector acceptances, hadronisation and the underlying event, that were not included in our resummation. We have then used the code to compare the sensitivity of W W and Z Z production at the HL-LHC to the considered higher-dimension operator. We have found that W W can provide complementary sensitivity with respect to Z Z, provided one is able to control theory uncertainties at the percent-level. Our method is general and can be applied to the production of any colour singlet, both within and beyond the SM.