  We show that it is essential to include renormalisation group (RG) effects
for determining the SMEFT parameter space consistent with the CDF W-mass
anomaly at the matching scale. This is because operators that are only weakly
constrained--for instance those probed by diboson and Higgs data--can have a
contribution to the W-boson mass via one-loop RG effects, that is comparable to
the tree-level contribution of the much more strongly constrained operators
related to electroweak precision observables. We RG evolve the low energy SMEFT
parameter space consistent with the anomaly to the matching scale and find a
much larger allowed region. In particular we find that it is possible to have a
vanishing or even negative T -parameter at the matching scale. This will
hopefully lead to a larger set of UV completions that can explain the anomaly.
For the one-loop contributions to be important a relatively low new physics
scale around 800 GeV is required. This enhances the possibility of probing this
new physics in direct and indirect searches in the recent future.
