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We explore the physics expected sensitivity at the Large Hadron electronCollider (LHeC) and the Future Circular Collider-hadron electron (FCC-he) tosearch for the anomalous quartic $WW\gamma\gamma$ couplings in single $W$-bosonproduction in association with a photon. We study the process $ep \toe^-\gamma^* p \to eW\gamma q'X$ via the subprocess $\gamma^* q \to W\gamma q'$.The center-of-mass energies and luminosities of the LHeC are assumed to be$\sqrt{s}=1.30, 1.98$ TeV, ${\cal L}=10-100$ ${\rm fb^{-1} }$ and for theFCC-he $\sqrt{s}=3.46, 5.29$ TeV and ${\cal L}=100-1000$ ${\rm fb^{-1} }$. Withthese energies and luminosities, we estimate bounds on the anomalous quartic$WW\gamma\gamma$ couplings at $95\%$ C.L., which can be an order of magnitudestringent than the experimental limits report by ATLAS and CMS Collaborationsat the LHC.

Sensitivity physics expected to the measurement of the quartic $WW\gamma\gamma$ couplings at the LHeC and the FCC-he