Probing anomalous $Z\gamma\gamma\gamma$ couplings at a future muon collider

The sensitivity to anomalous quartic gauge couplings (AQGCs) of the $\gamma\gamma\gamma Z$ interaction is studied in the $\mu^+\mu^- \rightarrow \mu^+\gamma\gamma \mu^-$ scattering at a future muon collider with unpolarized beams. The anomalous $\gamma\gamma\gamma Z$ vertex is described by two couplings, $\zeta_1$ and $\zeta_2$. The differential and total cross sections are calculated for the center-of-mass energies of 3 TeV, 14 TeV, and 100 TeV. For these values of the collision energy the $95\%$ C.L. exclusion regions for AQGCs are obtained depending on the systematic error. In particular, for the 14 TeV muon collider with the integrated luminosity $L = 20$ ab$^{-1}$ the best sensitivities are derived to be $\zeta_1 = 3.1 \times 10^{-5}$ TeV$^{-4}$ and $\zeta_2 = 6.5 \times 10^{-5}$ TeV$^{-4}$. These constraints are three orders of magnitude stronger than the bounds obtained for the 27 TeV HE-LHC with $L = 15$ ab$^{-1}$. At the 100 TeV muon collider with $L = 1000$ ab$^{-1}$ AQGCs can be probed up to $(1.64 \div 3.4) \times 10^{-8}$ TeV$^{-4}$. The partial-wave unitarity constraints on couplings $\zeta_1$, $\zeta_2$ are evaluated. It is shown that the unitarity is not violated in the region of the AQGCs examined in the present paper.