In Search of an Invisible $Z^\prime$

We consider extending the Standard Model by an anomaly-free and possibly flavour non-universal $U(1)_X$ gauge symmetry, whose breaking gives a $Z^\prime$ boson that does not affect electroweak precision observables at tree-level or via 1-loop renormalisation group (RG) running. Provided it does not also couple to electrons, such a $Z^\prime$ boson would be largely invisible to an electroweak precision machine like FCC-ee or CEPC (up to small finite 1-loop matching contributions that we quantify). We show that, while this class of $Z^\prime$ models can also evade tests of quark flavour violation, the constraint of anomaly-cancellation implies that valence quarks, muons, and taus are all charged under $U(1)_X$, with the up quark charge being necessarily large. The conclusion holds even if one augments the SM by three right-handed neutrinos to try and absorb anomalies. This means such $Z^\prime$ bosons cannot simultaneously hide below the TeV scale from $pp \to \ell\ell$ Drell--Yan measurements at the LHC and, even if we entertain esoteric models in which the lepton charges are numerically very small, we cannot escape dijet searches at the LHC. For equitable quark and lepton charges, $pp\to\ell\ell$ already excludes such a $Z^\prime$ up to $M/g \gtrsim$ 10 TeV, with a reach of 20 TeV expected by the end of the High-Luminosity LHC. The dijet bounds currently sit around 5 TeV, while sensitivity up to 10 TeV could be achieved at HL-LHC. We thus find an excellent complementarity between FCC-ee and HL-LHC in covering all anomaly-free $Z^\prime$ bosons up to several TeV.