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In the post-LHC era, the muon collider represents a frontier project capableof providing high-energy and high-luminosity leptonic collisions among futurelepton-lepton particle accelerators. In addition, it provides significantlycleaner final states than those produced in hadron collisions. With thisexpectation in mind, in this article, we research the sensitivity of theanomalous neutral gauge boson couplings $Z\gamma\gamma\gamma$, $ZZ\gamma\gamma$, and $ZZZ\gamma$ defined by dimension-8 operators, through the$\mu^+\mu^- \to \mu^+\mu^-Z\gamma$ signal, with the $Z$-boson decaying toneutrino pair. The projections of new physics at the future muon collider withthe center-of-mass energy of $\sqrt{s}=10$ TeV, integrated luminosity of ${\calL}=10$ $\rm ab^{-1}$, and systematic uncertainties of $\delta_{sys}=0\%, 3\%,5\%$, for extraction of expected sensitivity on the anomalous $f_{T,j}/\Lambda^4$ couplings at $95\%$ confidence level, are of the order of${\cal O}(10^{-4}- 10^{-3})$. Compared with the research of the ATLAS and CMSCollaborations on the anomalous quartic gauge couplings, we find that thehigh-luminosity future muon collider could have better sensitivity.

The future muon collider for the research of the anomalous neutral quartic $Z\gamma\gamma\gamma$, $ZZ\gamma\gamma$, and $ZZZ\gamma$ couplings