$Z^\prime$ Portal Dark Matter with Observable $\Delta N_{\rm eff}$

Technical Summary: Z′ Portal Dark Matter with Observable ∆Neff

Problem Statement
The Standard Model (SM) lacks a viable dark matter (DM) candidate and predicts massless neutrinos, both of which contradict cosmological observations and neutrino oscillation data. While the gauged $U(1)_{B-L}$ extension of the SM is a popular framework that introduces right-handed neutrinos ($\nu_R$) to generate neutrino masses via the seesaw mechanism, the nature of these neutrinos remains ambiguous. If $\nu_R$ are Majorana particles, they do not contribute significantly to the effective number of relativistic species ($N_{\text{eff}}$). However, if $\nu_R$ are Dirac particles, they contribute to $\Delta N_{\text{eff}} = N_{\text{eff}} - N_{\text{eff}}^{\text{SM}}$. Current and upcoming cosmological experiments (DESI, P-ACT, CMB-S4, CMB-HD) are tightening constraints on $\Delta N_{\text{eff}}$, potentially reaching sensitivities as low as $\sim 0.027$. The authors investigate how these stringent $\Delta N_{\text{eff}}$ constraints impact the viability of $Z'$ portal dark matter models where $\nu_R$ are Dirac fermions, specifically examining both Weakly Interacting Massive Particle (WIMP) and Feebly Interacting Massive Particle (FIMP) scenarios.

Methodology
The authors construct a minimal $U(1)_{B-L}$ model containing a vector-like Dirac dark matter candidate $\chi$ and Dirac right-handed neutrinos $\nu_R$ with charge $Q_{\nu_R} = -1$. The model is mediated by a $Z'$ boson generated via the Stueckelberg mechanism. The study systematically explores the parameter space defined by the dark matter mass ($m_\chi$), $Z'$ mass ($m_{Z'}$), gauge coupling ($g'$), and the $B-L$ charge of dark matter ($Q_\chi$).

The analysis proceeds in two main regimes based on the production mechanism of $\chi$:

1. WIMP Scenario: Dark matter is produced via thermal freeze-out. The authors consider two distinct cases:
   • Resonance: $m_{Z'} \approx 2m_\chi$, where annihilation occurs through an on-shell $Z'$.
   • Secluded: $m_{Z'} < m_\chi$, where annihilation occurs via $\chi\bar{\chi} \to Z'Z'$.
     The authors calculate the relic density using Boltzmann equations and determine the decoupling temperature of $\nu_R$ ($T_{\nu_R}^{\text{dec}}$). They quantify $\Delta N_{\text{eff}}$ contributions from both thermal processes ($\nu_R\bar{\nu}_R \to f\bar{f}$ and $\nu_R\bar{\nu}_R \to \chi\bar{\chi}$) and non-thermal processes.
2. FIMP Scenario: Dark matter is produced via the freeze-in mechanism, where $\chi$ and $\nu_R$ never reach thermal equilibrium. The authors solve Boltzmann equations for the out-of-equilibrium production of $\chi$ (dominantly via $Z' \to \chi\bar{\chi}$ decay) and $\nu_R$.

The viable parameter space is constrained by:

• Cosmology: $\Delta N_{\text{eff}}$ limits from DESI ($\lesssim 0.4$), P-ACT ($\lesssim 0.17$), and projected sensitivities of CMB-S4 ($\lesssim 0.06$) and CMB-HD ($\lesssim 0.027$).
• Direct Detection: Spin-independent DM-nucleon scattering cross-sections ($\sigma_{\text{SI}}$) constrained by experiments like XENONnT and LZ.
• Indirect Detection: DM annihilation cross-sections ($\langle \sigma v \rangle$) constrained by X-ray, CMB, and gamma-ray observations.
• Collider Searches: Limits on $Z'$ bosons from BaBar, LHCb, LEP, CMS, and ATLAS, considering both visible and invisible decay modes.
• Theoretical Consistency: Perturbative unitarity limits ($g_\chi < \sqrt{4\pi}$) and Big Bang Nucleosynthesis (BBN) constraints on $Z'$ lifetime.

Key Contributions and Results

• Impact of $\Delta N_{\text{eff}}$ on WIMP Scenarios:

  • Resonance Case: The study finds that for the resonance scenario ($r_{Z'} \approx 2$), the contribution of $\nu_R$ to $\Delta N_{\text{eff}}$ is dominated by thermal production. The minimum achievable $\Delta N_{\text{eff}}$ in this scenario is $\sim 0.14$ (due to three generations of $\nu_R$). Consequently, the P-ACT constraint ($\Delta N_{\text{eff}} \lesssim 0.17$) excludes dark matter masses $m_\chi \lesssim 400$ GeV. The viable parameter space is restricted to $400 \text{ GeV} \lesssim m_\chi \lesssim 1.54 \times 10^5 \text{ GeV}$ and $0.7 \lesssim Q_\chi \lesssim 3400$.
  • Secluded Case: In the secluded scenario ($r_{Z'} = 0.5$), the off-shell $Z'$ suppresses the $\nu_R\bar{\nu}_R \to \chi\bar{\chi}$ process. For large $Q_\chi$ ($\gtrsim 5 \times 10^6$), $\nu_R$ can be produced non-thermally, allowing $\Delta N_{\text{eff}}$ to drop below 0.14. This opens a viable region for $m_\chi \gtrsim 250$ GeV and $Q_\chi \gtrsim 30$ under P-ACT constraints. The secluded scenario is shown to be more promising for detection than the resonance case, particularly if future experiments observe $\Delta N_{\text{eff}} < 0.14$.

• FIMP Scenario Results:

  • In the FIMP regime, $\Delta N_{\text{eff}}$ is generated non-thermally. The authors identify that to produce an observable $\Delta N_{\text{eff}}$, the DM charge must be small ($Q_\chi \lesssim 10^{-2}$).
  • The viable parameter space is constrained to $1.6 \times 10^{-11} \lesssim Q_\chi \lesssim \mathcal{O}(10^{-3})$ and $0.1 \text{ GeV} \lesssim m_\chi \lesssim 10^4 \text{ GeV}$ (for $r_{Z'}=10$).
  • For very small $Q_\chi$, the predicted $\Delta N_{\text{eff}}$ falls below the sensitivity of future CMB experiments, while for larger $Q_\chi$, the constraints are dominated by P-ACT.

• Interplay of Constraints:

  • The study demonstrates that $\Delta N_{\text{eff}}$ constraints are often more restrictive than direct or indirect detection limits for the WIMP resonance scenario, particularly ruling out lower mass ranges.
  • In the secluded and FIMP scenarios, the ability to generate non-thermal $\nu_R$ allows the model to evade the $\Delta N_{\text{eff}} \approx 0.14$ floor, making these scenarios robust against future null results in $\Delta N_{\text{eff}}$ measurements.

Significance
The paper claims that the interplay between Dirac neutrinos and $Z'$ portal dark matter creates a unique signature in the effective number of relativistic species. The authors argue that upcoming precision measurements of $\Delta N_{\text{eff}}$ by CMB-S4 and CMB-HD will be decisive:

1. If $\Delta N_{\text{eff}} < 0.14$ is observed, the minimal $Z'$ portal resonance scenario with Dirac neutrinos would be completely excluded, as it cannot produce values below this thermal floor.
2. Conversely, the secluded and FIMP scenarios remain viable even if $\Delta N_{\text{eff}}$ is not detected above the SM prediction, as they can produce $\Delta N_{\text{eff}}$ values below the current and future experimental sensitivities.
3. The study provides a comprehensive map of the "promising" parameter space for future experimental verification, highlighting that the secluded and FIMP scenarios offer a broader range of detectability compared to the resonance case.