  We study probes of neutral triple gauge couplings (nTGCs) via $Z^*\gamma$
production followed by off-shell decays $Z^*\to\nu\bar{\nu}$ at the LHC and
future $pp$ colliders, including both CP-conserving (CPC) and CP-violating
(CPV) couplings. We present the dimension-8 SMEFT operators contributing to
nTGCs and derive the correct form factor formulation for the off-shell vertices
$Z^*\gamma V^*$ ($V=Z,\gamma$) by matching them with the dimension-8 SMEFT
operators. Our analysis includes new contributions enhanced by the large
off-shell momentum of $Z^*$, beyond those of the conventional $Z\gamma V^*$
vertices with on-shell $Z\gamma$. We analyze the sensitivity reaches for
probing the CPC/CPV nTGC form factors and the new physics scales of the
dimension-8 nTGC operators at the LHC and future 100TeV $pp$ colliders. We
compare our new predictions with the existing LHC measurements of CPC nTGCs in
the $\nu\bar\nu\gamma$ channel and demonstrate the importance of our new
method.
