Probing the couplings of an axion-like particle with leptons via three-lepton final state processes at future $e^{-}p$ colliders

The axion-like particle (ALP) is one of the best motivated particles beyond the Standard Model (SM). We explore the possibility of detecting the couplings of ALP with leptons via three-lepton final state processes $e^- p \to e^- j a~(a \to \ell^+ \ell^-)$ at the LHeC (FCC-eh). For completeness, we investigate the cases where the ALP decays not only into electron and muon pairs but also into tau pairs, and perform a detailed simulation for the decays of taus. We find that the prospective sensitivities of the LHeC (FCC-eh) with the center-of-mass energy $\sqrt{s}=1.3~(3.5)$ TeV and the integrated luminosity $\mathcal{L}=1~ (2)$ ab$^{-1}$ to the ALP-lepton couplings are not only stronger than some existing bounds of the LHC and LEP but also complementary to the expected bounds of the CEPC and FCC-ee.