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 μ⁺μ⁻ 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 O(100 GeV), thus providing a unique opportunity for future lepton colliders. Taking U1_Lα−Lβ (α, β = e, μ, τ) models as concrete examples, we show that future e⁺e⁻ and μ⁺μ⁻ 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 O(10⁴ TeV), while being consistent with naturalness and perturbativity considerations.