  The electroweak diboson production cross-sections are known to receive large
radiative corrections beyond leading-order (LO), approaching up to 60% at
next-to-leading order (NLO), compared to the scale uncertainties which are in
the range 1-5% at LO. If the scale uncertainties are to be taken seriously, the
NLO predictions are as much as 30 sigma away from their LO counterpart
suggesting a very poor convergence of the perturbation theory. In this paper,
we show that there is a second source of scale uncertainty which has not been
considered in the literature, namely the complex phase of the scales, which can
lead to large perturbative corrections. Using the formalism of soft-collinear
effective theory, we resum these large contributions from the complex phase,
finding that the scale uncertainties in fixed-order calculations can be grossly
underestimated compared to the resummed predictions, which have uncertainties
as large as 13-16% at LO. Even at NLO, we find that the scale uncertainties are
marginally higher than previously estimated, depending on the choice of scale.
Using our method of scale variation, the compatibility of LO and NLO results
within the scale uncertainties is vastly improved so that the perturbation
theory can be relied upon. This method of scale variation can be easily
extended to beyond NLO calculations as well as other LHC processes.
