Inelastic Majorana Dark Matter and Its Self-Interactions in a Gauged $U(1)_{L_μ- L_τ}$ Model

Abstract

We develop an inelastic Majorana dark matter model gauged by $U(1)_{L_μ-L_τ}$ symmetry, featuring a complex scalar and fermions with a small Dirac mass. The model exhibits strong interactions between the dark fermion-antifermion pair and the scalar, with a large coupling generating a Majorana mass after symmetry breaking. The dark fermion splits into two nearly degenerate Majorana eigenstates, with the $U(1)_{L_μ-L_τ}$ gauge boson $Z^\prime$ interacting via inelastic axial vector currents. The lighter Majorana serves as dark matter (DM), while the heavier may also contribute. The model predicts DM masses ranging from about 10 GeV to hundreds of GeV, with the scalar favored below 100 MeV. Strong coupling causes DM particles to mainly annihilate into two scalars, which determines the relic abundance, while thermal equilibrium with the bath occurs through $Z^\prime$ interactions. Self-interacting DM particles scatter via ladder exchange of strongly coupled light scalars, helping resolve small-scale issues. The $Z^\prime$ interacts with the muon, affecting its magnetic moment and explaining the $(g-2)_μ$ discrepancy. Effects on experiments, $N_{\text{eff}}$, and the Hubble tension from kinetic mixing between $Z^\prime$ and the photon, originating from high-energy scales, are discussed. The model details and parameter constraints used in experiments are also covered.

Figures (15)

• Figure 1: The central value (magenta solid curve) and the parameter regions within 2$\sigma$ (red) are determined by the muon $(g-2)_\mu$ measurements for the present iDM model with the $U(1)_{L_\mu - L_\tau}$ gauge boson $Z^\prime$. Exclusions from various experiments, including BABAR BaBar:2016sci, CCFR CCFR:1991lplAltmannshofer:2014pba, white dwarf (WD) cooling Dreiner:2013tjaBauer:2018onh, and Borexino Bellini:2011rxKamada:2018zxi, are displayed. The hatched regions, which satisfy $3.2 < N_{\rm eff} <3.33$Planck:2018vyg, can alleviate the Hubble tension ($H_0$), with colors in brown, orange, and cyan representing $|\epsilon_A| =g_{\mu\tau}/69, 10^{-7}, 0$, respectively. The region to the left of the Hubble tension boundary line, where $N_{\rm eff} > 3.33$, is disfavored by Planck 2018 data (TT, TE, EE+lowE+lensing+BAO at 95%CL) Planck:2018vyg.
• Figure 2: Left panel: $x_f$ as a function of $m_-$. Right panel: The coupling constant $f$ that can produce the correct DM relic density as a function of $m_-$. The solid curve corresponds to $\delta=1$, while the dashed and dotdashed curves are for $\delta =1.5$ and $0.5$, respectively. The unsmooth part of the curves is due to the QCD phase transition at $T_{\rm QCD}=150$ MeV.
• Figure 3: Thermally averaged annihilation cross sections for processes, $\chi_\mp \chi_\mp \to SS$ (black), $\chi_- \chi_+ \to Z^\prime S$ (red), and $\chi_\mp \chi_\mp \to Z^\prime Z^\prime$ (blue), where the freeze-out temperature parameter $x_f$ corresponding to the value shown in Fig. \ref{['fig:xf-xs-mx']} with $\delta=1$ is adopted.
• Figure 4: The evolution of the ratios $T/T_{\nu_e}$ (red) and $T_{\nu_\mu} / T_{\nu_e}$ (brown) for $|\epsilon_A| =7.6\times 10^{-6}$ (solid line), $10^{-7}$ (dashed line), and $0$ (dotted line), with $m_{Z^\prime}=12$ MeV and $m_S=30$ MeV in the left panel, and $m_{Z^\prime}=8$ MeV and $m_S=20$ MeV in the right panel. The resulting $N_{\rm eff}$ value is also given.
• Figure 5: $N_{\rm eff}$ as a function of $m_{Z^\prime}$ values for several settings of the kinetic mixing parameter, $|\epsilon_A|$, with the Standard Model (SM) value marked by the black line. The shaded gray and brown areas represent the constraints of $N_{\rm eff} = 2.99^{+0.34}_{-0.33}$ from Planck TT, TE, EE + lowE + lensing + BAO at a 95% confidence level. The brown region, where $3.2< N_{\rm eff}<3.33$, hints at a potential reduction in the Hubble tension.