Searching for Heavy Leptophilic $Z'$: from Lepton Colliders to Gravitational Waves

We study the phenomenology of leptophilic $Z'$ gauge bosons at the future high-energy $e^+e^-$ and $\mu^+\mu^-$ colliders, as well as at the gravitational wave observatories. The leptophilic $Z'$ model, although well-motivated, remains largely unconstrained from current low-energy and collider searches for $Z'$ masses above ${\cal O}(100~{\rm GeV})$, thus providing a unique opportunity for future lepton colliders. Taking $U(1)_{L_\alpha-L_\beta}~(\alpha,\beta=e,\mu,\tau)$ models as concrete examples, we show that future $e^+e^-$ and $\mu^+\mu^-$ colliders with multi-TeV center-of-mass energies provide unprecedented sensitivity to heavy leptophilic $Z'$ bosons. Moreover, if these $U(1)$ models are classically scale-invariant, the phase transition at the $U(1)$ symmetry-breaking scale tends to be strongly first-order with ultra-supercooling, and leads to observable stochastic gravitational wave signatures. We find that the future sensitivity of gravitational wave observatories, such as advanced LIGO-VIRGO and Cosmic Explorer, can be complementary to the collider experiments, probing higher $Z'$ masses up to ${\cal O}(10^4~{\rm TeV})$, while being consistent with naturalness and perturbativity considerations.