Searching for elusive dark Higgs boson in spin-1/2 inelastic dark matter models at Belle II

TL;DR

This work analyzes spin-1/2 inelastic dark matter with a spontaneously broken $U(1)_D$, where the dark Higgs $h_2$ is required to restore unitarity and generate both the dark photon mass and DM mass splitting. It develops a concrete model with a dark scalar $ ext{Φ}$ and fermion $ ext{χ}$, leading to Majorana states $ ext{χ}_1$ and $ ext{χ}_2$, and investigates $h_2$ production and decays at Belle II through dark Higgs-strahlung and rare $B$ decays, including semi-visible and invisible decay modes. The analysis delineates constraints on the parameter space, identifies two complementary search regimes (large $ ext{α}_D$ with small $ ext{sin} heta$ for Higgs-strahlung and small $ ext{α}_D$ with moderate $ ext{sin} heta$ for $B$ decays), and demonstrates how inclusive two-displaced-vertex signatures plus missing energy can indicate the dark Higgs, with L-GAZELLE further extending sensitivity to long-lived states. Additionally, invisible $h_2$ decays can address the Belle II $B o K uar{ u}$ excess, while L-GAZELLE broadens exploration of parameter space beyond Belle II’s capabilities. Overall, the study provides a roadmap for probing elusive dark Higgs physics at Belle II and future LLP detectors.

Abstract

Spin-1/2 inelastic dark matter (DM) models are popular among sub-GeV to GeV thermal DM scenarios due to the dominant role of co-annihilation in determining the DM relic abundance. In these models, the dark Higgs boson plays a crucial role in generating the mass of the new gauge boson, the dark photon ($A'$), and in establishing the mass splitting between the excited ($χ_2$) and ground ($χ_1$) states of DM. In particular, the Compton scattering $χ_1 A' \rightarrow χ_2^* \rightarrow χ_1 A'$ and its $t$-channel crossed process, $χ_1 χ_1 \rightarrow A' A'$, remain unitary for high energy longitudunal dark photon, only if the contribution of the dark Higgs boson is included. However, experimental searches for the dark Higgs boson have received relatively little attention. In particular, when the dark Higgs boson mass exceeds twice that of the DM excited state, its decay signatures become semi-visible or invisible, making detection challenging with current light scalar search strategies. In this work, we explore the prospects for detecting the elusive dark Higgs boson in spin-1/2 inelastic DM models at Belle II via dark Higgs-strahlung and rare $B$ meson decay processes. Our analysis indicates that both the inclusive signature of two displaced dilepton vertices and the additional missing energy from dark Higgs boson decays serve as robust indicators of its presence. Furthermore, we assess the future potential for detecting the dark Higgs boson with the proposed far detector related to Belle II, GAZELLE.

Figures (16)

• Figure 1: The decay branching ratio of $h_2$. Two benchmark points: $m_{h_2} = \frac{5}{6}m_{A^{\prime}} = 2.5 M_{\chi_1}$ (left panel) and $m_{h_2} = 2.5 m_{A^{\prime}} = 7.5 M_{\chi_1}$ (right panel). Here we fix $\Delta_{\chi} = 0.1 M_{\chi_1}$, $\sin\theta = 10^{-3}$ and $\alpha_D = 10^{-4}$.
• Figure 2: The decay branching ratio of $A^{\prime}$. Two benchmark points: $m_{A'}=3 M_{\chi_1}$ (left panel) and $m_{A'}=2.01 M_{\chi_1}+\Delta_{\chi}$ (right panel) are considered. Here we fix $\epsilon = 10^{-3}$, and $\Delta_{\chi} = 0.1 M_{\chi_1}$, but varying $\alpha_D = 10^{-4}$, and $10^{-1}$.
• Figure 3: The constraints on $h_2$ are shown in the $(m_{h_2} \text{ (GeV)},\log_{10}\sin\theta)$ plane. We consider four benchmark points: $(\alpha_D, \Delta_{\chi}/M_{\chi_1}) =$$(10^{-4}, 0.1)$ (upper-left), $(10^{-1}, 0.1)$ (upper-right), $(10^{-4}, 0.2)$ (lower-left), and $(10^{-1}, 0.2)$ (lower-right). In these benchmarks, the mass spectrum is fixed at $m_{h_2} = \frac{5}{6} m_{A'} = 2.5 M_{\chi_1}$.
• Figure 4: The Feynman diagrams for $e^{+}e^{-}\rightarrow A^{\prime} h_2$, $h_2 \rightarrow \chi_2 \chi_2$, $A'\rightarrow \chi_1\chi_2$ and $e^{+}e^{-}\rightarrow \Upsilon(4s)\rightarrow B \Bar{B}$, $B \rightarrow K h_2$, $h_2 \rightarrow \chi_2 \chi_2$ where $\chi_2\rightarrow\chi_1 f\bar{f}$ for both of these two processes.
• Figure 5: The relations between $m_{h_2}$ (GeV) and $\sigma$ (fb) for $e^{+}e^{-}\rightarrow A^{\prime} h_2, h_2 \rightarrow \chi_2 \chi_2$ (red lines) and $e^{+}e^{-}\rightarrow \Upsilon(4s)\rightarrow B \Bar{B}, B \rightarrow K h_2, h_2 \rightarrow \chi_2 \chi_2$ (blue line) and $e^{+}e^{-}\rightarrow \Upsilon(4s)\rightarrow B \Bar{B}, B \rightarrow K^{\ast} h_2, h_2 \rightarrow \chi_2 \chi_2$ (green line). Here, the other free parameters are fixed: $m_{A^{\prime}} = 3 M_{\chi_1}$, $\alpha_D = g^{2}_D/4\pi = 0.1$, $\epsilon = 10^{-3}$, and $\Delta_{\chi}=0.1M_{\chi_1}$ for these signal processes.