  A direct investigation of the self-couplings of gauge bosons, completely
described by the non-Abelian gauge symmetry of the Standard Model, is extremely
valuable in understanding the gauge structure of the SM. Any deviation from the
SM predictions on gauge boson self-coupling is to give a hint at the existence
of a new physics beyond the SM, which is defined with a modification of the
self-interactions using an effective field theory approach. In this paper, we
present a detailed Monte Carlo study searching for anomalous quartic gauge
dimension-8 couplings related to $\gamma\gamma\gamma\gamma$ and
$\gamma\gamma\gamma Z$ vertices at the future hadron-hadron collider (FCC-hh)
via tri-photon production at a 100 TeV center of mass energy with an integrated
luminosity L$_{int}$=30 ab$^{-1}$. Events that have been parton showered and
include detector effects are analyzed with a Toolkit for Multivariate Data
Analysis (TMVA) using a boosted decision tree to help distinguish between
signal and background events to achieve the best sensitivities on anomalous
quartic gauge couplings. Our obtained results reveal that the limits on
anomalous quartic gauge couplings $f_{T8}/\Lambda^{4}$ and $f_{T9}/\Lambda^{4}$
at 95\% C.L. without systematic errors are about three orders of magnitude
stronger compared to the best current experimental limits reported by the ATLAS
collaboration at the LHC. Considering a realistic systematic uncertainty such
as 10\% from possible experimental sources, our obtained limits of anomalous
quartic couplings get worse by about one order of magnitude compared to those
without systematic uncertainty but are still two orders of magnitude better
than those recently reported by ATLAS.
