Cosmological signature and light Dark Matter in Dirac $L_μ-L_τ$ model

We revisit an anomaly-free extension of the Standard Model (SM) $viz.$ gauged ${L_μ-L_τ}$ model in the Dirac framework, where the local $U(1){L_μ-L_τ}$ symmetry breaks and gives rise to a new gauge boson $Z’$ and corresponding gauge coupling $g{μτ}$. Three additional heavy vector-like fermions, three light right-handed neutrinos and two heavy singlet scalars are added to complete the model framework for Dirac neutrinos. Another singlet vector-like fermion is added with a new gauge charge, which serves as a viable DM candidate, and the correct relic abundance is obtained via the resonance effect. The parameter space is considered after satisfying the current bounds on $M_{Z’}$ and the gauge coupling $g_{μτ}$. The influence of dark radiations coming from the additional light degrees of freedoms are studied in connection with the dark matter. After imposing all relevant theoretical and experimental constraints, the allowed parameter space is found to be highly restricted yet still accessible to ongoing and near-future experiments, rendering the scenario strongly predictive. Moreover, clear correlations among the relevant observables emerge throughout this study, making the model testable in current and future experimental searches.

💡 Research Summary

In this work the authors revisit an anomaly‑free extension of the Standard Model based on the gauged Lμ‑Lτ symmetry, but now formulated in a Dirac framework. The U(1){Lμ‑Lτ} gauge group is spontaneously broken by two singlet scalars Φ₁ and Φ₂, giving mass to a new gauge boson Z′ with M{Z′}=g_{μτ}√{(1‑n_X)²v₁²+v₂²}. Three heavy vector‑like fermions (N_e, N_μ, N_τ) and three right‑handed neutrinos (ν_{eR}, ν_{μR}, ν_{τR}) are introduced to generate Dirac neutrino masses via a type‑I Dirac seesaw. The right‑handed neutrinos carry Lμ‑Lτ charges (0, n_X, ‑n_X) respectively; the charge n_X is a free parameter that can be tuned to control their thermal history.

The neutrino mass matrix is built from Dirac Yukawa couplings to the SM Higgs and the heavy fermions, together with the VEVs of Φ₁ and Φ₂. In the limit M_D, M′D ≪ M{LR}, the light neutrino masses are given by m_ν≈−M′D M{LR}^{‑1} M_D, reproducing the observed mixing solely through the structure of M_{LR} generated by Φ₂. This construction respects lepton‑number conservation, ensuring that Majorana mass terms are absent.

A crucial phenomenological motivation is the muon anomalous magnetic moment. The Z′ contributes at one loop, and the authors use the standard integral expression (Eq. 11) to map the (M_{Z′}, g_{μτ}) plane compatible with the measured Δa_μ. By overlaying bounds from CCFR, BABAR, white‑dwarf cooling, COHERENT, and NA64‑μ, they identify a viable benchmark region: M_{Z′} ≲ 100 MeV with g_{μτ}=3.8×10⁻⁴. This region simultaneously satisfies the (g‑2)_μ discrepancy and evades all current laboratory constraints.

Cosmologically, the extra right‑handed neutrinos contribute to the effective number of relativistic species, ΔN_eff. ν_{eR} decouples well above the electroweak scale through its tiny Yukawa interactions, yielding a modest ΔN_eff≈0.047, comfortably within Planck 2018 limits. The μ and τ right‑handed neutrinos, however, can stay in equilibrium via Z′‑mediated scattering (ν_R ν_R ↔ f \bar f). To prevent them from raising ΔN_eff beyond observational bounds, the authors require the Lμ‑Lτ charge n_X to be ≤ 10⁻⁶, ensuring that ν_{μR} and ν_{τR} never reach thermal equilibrium for the chosen (M_{Z′}, g_{μτ}) values. Figure 4 demonstrates that for such tiny charges the interaction rate remains below the Hubble expansion rate throughout the relevant temperature range.

The dark sector is completed by a singlet vector‑like fermion ψ, charged under U(1){Lμ‑Lτ} but otherwise a Standard Model singlet. ψ is stable due to an imposed Z₂ symmetry and serves as the dark matter (DM) candidate. Its annihilation proceeds dominantly through s‑channel Z′ exchange into light SM neutrinos or other dark sector particles. By tuning the DM mass to lie near the Z′ resonance (2 M_ψ ≈ M{Z′}), the thermally averaged cross section is enhanced, allowing ψ to achieve the observed relic density Ω_{DM} h² ≈ 0.12 even with the very small gauge coupling. Direct detection signals are highly suppressed because kinetic mixing between Z′ and the photon is loop‑induced and negligible (ε ≈ g_{μτ}/70). Consequently, the model predicts observable signals only in low‑energy fixed‑target experiments (e.g., NA64‑μ, LDMX) and in future precision measurements of ΔN_eff or (g‑2)_μ.

Putting all constraints together, the allowed parameter space is tightly confined: 1 MeV ≤ M_{Z′} ≤ 200 MeV, g_{μτ}=3.8×10⁻⁴, n_X ≲ 10⁻⁶, and DM mass M_ψ≈M_{Z′}/2. Within this region the model simultaneously explains the muon (g‑2) anomaly, yields a viable sub‑GeV dark matter candidate, and respects cosmological limits on extra radiation. The authors emphasize that upcoming experiments—especially those probing light gauge bosons, low‑mass dark matter, and precision cosmology—will be able to test the remaining viable slice of the model, making it highly predictive and experimentally falsifiable.