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The search for lepton flavor violation (LFV) is a powerful probe to look fornew physics beyond the Standard Model. We explored the possibility of searchesfor LFV $Z$ boson couplings to electron and muon pair at the upcomingelectron-proton colliders, namely the Large Hadron Electron Collider (LHeC) andthe Future Circular lepton-hadron Collider (FCC-eh). We employed the study viaa single muon plus an associated jet channel to search for the LFV signal. Weused a multivariate technique to obtain an improved signal-background analysis.By using the condition on nonobservation of any significant deviation of thesignal over the expected background, we provide an upper limit on the LFV $Z$boson coupling and corresponding branching ratio. We find that an upper limitof $2.0\times 10^{-7}$ and $1.5 \times 10^{-7}$ the can be set on BR($Z\to e\mu$) at 95\% C.L. with one year run of LHeC and FCC-eh, respectively, if theLFV coupling is governed by vector or axial-vector coupling. For tensor oraxial-tensor coupling these limits can be improved to $2.9\times 10^{-8}$ and$1.5\times 10^{-8}$ for LHeC and FCC-eh machines, respectively. The projectednumbers improve significantly over the existing limit of $2.6\times 10^{-7}$set by ATLAS.

Indirect Search for Lepton Flavour Violating Signals at the Future Electron-Proton Colliders