  The paper presents for the first time a novel idea of exploiting asymmetry
between differential cross sections of the $Z\gamma$ pair-production in
proton-proton ($pp$) collisions for the final states of a charged-lepton pair
plus a photon $pp \rightarrow Z\gamma \rightarrow l^{+}l^{-}\gamma$ (leptonic
decay) and of a neutrino pair plus a photon $pp \rightarrow Z \gamma
\rightarrow \nu \bar{\nu}\gamma$ (invisible decay). Asymmetry between the
leptonic and invisible decays of the $Z\gamma$ process is investigated by using
fixed-order predictions through inclusion of next-to-next-to-leading (NNLO)
radiative corrections in quantum chromodynamics (QCD) perturbation theory. NNLO
QCD predictions are presented at various $pp$-collision energies as functions
of several key observables including transverse momenta and azimuthal-angle
separation, regarding the $Z\gamma$ decay products. The predicted distributions
for the $Z\gamma$ asymmetry are provided based on realistic fiducial
phase-space requirements in line with the related hadron-collider measurements.
The predicted distributions are assessed at various $pp$-collision energies and
in different phase-space regions such as with higher lepton-pair invariant mass
$m^{l^{+}l^{-}}$ or higher neutrino-pair transverse momentum $p_{\rm{T}}^{\nu
\bar{\nu}}$ requirements. The $Z\gamma$ asymmetry is shown to be significantly
sensitive in different regions of phase space including high-$m^{l^{+}l^{-}}$
and high-$p_{\rm{T}}^{\nu \bar{\nu}}$ regions. The $Z\gamma$ asymmetry can
therefore be translated into an important quantity for probing deviation from
the Standard Model (SM) predictions. In this regard, the asymmetry probe is
proposed as a sensitive indicator for indirect searches for physics beyond the
SM encompassing high-mass resonances and dark-matter sector.
