Probing $Z^\prime$ Mediated Charged Lepton Flavor Violation with Taus at the LHeC

While charged lepton flavor violation (cLFV) with taus is often expected to be largest in many extensions of the Standard Model (SM), it is currently much less constrained than cLFV with electrons and muons. We study the sensitivity of the LHeC to $e$-$\tau$ (and $e$-$\mu$) conversion processes $p e^- \to \tau^- + j$ (and $p e^- \to \mu^- + j$) mediated by a $Z'$ with flavor-violating couplings to charged leptons in the $t$-channel. Compared to current tests at the LHC, where cLFV decays of the $Z'$ (produced in the s-channel) are searched for, the LHeC has sensitivity to much higher $Z'$ masses, up to O(10) TeV. For cLFV with taus, we find that the LHeC sensitivity from the process $p e^- \to \tau^- + j$ can exceed the current limits from collider and non-collider experiments in the whole considered $Z'$ mass range (above $500$ GeV) by more than two orders of magnitude. In particular for extensions of the SM with a heavy $Z'$, where direct production at colliders is kinematically suppressed, $e-\tau$ conversion at LHeC provides an exciting new discovery channel for this type of new physics.