  We study probes of neutral triple gauge couplings (nTGCs) at the LHC and the
proposed 100TeV $pp$ colliders, and compare their sensitivity reaches with
those of the proposed $e^+ e^-$ colliders. The nTGCs provide a unique window to
the new physics beyond the Standard Model (SM) because they can arise from SM
effective field theory (SMEFT) operators that respect the full electroweak
gauge group $SU(2)_L\otimes U(1)_Y$ of the SM only at the level of dimension-8
or higher. We derive the neutral triple gauge vertices (nTGVs) generated by
these dimension-8 operators in the broken phase and map them onto a newly
generalized form factor formulation, which takes into account only the residual
U(1)$_{\rm{em}}$ gauge symmetry. Using this mapping, we derive new relations
between the form factors that guarantee a truly consistent form factor
formulation of the nTGVs and remove large unphysical energy-dependent terms. We
then analyze the sensitivity reaches of the LHC and future 100TeV hadron
colliders for probing the nTGCs via both the dimension-8 nTGC operators and the
corresponding nTGC form factors in the reactions $ pp(q\bar{q})\to Z\gamma$
with $Z\to\ell^+\ell^-,\nu\bar{\nu}$. We compare their sensitivities with the
existing LHC measurements of nTGCs and with those of the high-energy $e^+e^-$
colliders. In general, we find that the prospective LHC sensitivities are
comparable to those of an $e^+ e^-$ collider with center-of-mass energy $\leq
1$TeV, whereas an $e^+ e^-$ collider with center-of-mass energy $(3 - 5)$TeV
would have greater sensitivities, and a 100TeV $pp$ collider could provide the
most sensitive probes of the nTGCs.
