  We study a simplified model of two colourless heavy vector resonances in the
singlet representation of $SU(2)_{L}$, with zero and unit hypercharge. We
discuss mixing with the Standard Model gauge bosons due to electroweak symmetry
breaking, semi-analytic formulae for production at proton colliders,
requirements to obey the narrow width approximation and selected low energy
constraints. We show current LHC constraints and sensitivity projections for
the HL-LHC, HE-LHC, SPPC and FCC-hh on the charged and neutral heavy vectors.
The utility of the simplified model Lagrangian is demonstrated by matching
these results onto three explicit models: a weakly coupled abelian extension of
the Standard Model gauge group, a weakly coupled non-abelian extension and a
strongly coupled minimal composite Higgs model. All our results are presented
in terms of physical resonance masses, which are accurate even at vector masses
near the electroweak scale due to a parameter inversion we derive. We discuss
the importance of this inversion and point out that its effect, and the effects
of electroweak symmetry breaking, can remain important up to resonance masses
of several TeV. Finally, we clarify the relation between this simplified model
and the Heavy Vector Triplet (HVT) model, a simplified model for heavy
$SU(2)_{L}$ triplets with zero hypercharge, and provide exact and approximate
matching relations.
