Scotogenic dark matter from gauged $B-L$

Technical Summary: Scotogenic Dark Matter from Gauged $B-L$

Problem Statement
The Standard Model (SM) fails to address the origin of neutrino masses and lacks a viable, stable Dark Matter (DM) candidate. While the scotogenic model offers a unified explanation by generating neutrino masses at the one-loop level mediated by a DM candidate, the symmetry stabilizing the DM is often an ad hoc global $Z_2$. This work addresses the need for a more fundamental origin of this stability by embedding the scotogenic mechanism within a gauged $U(1)_{B-L}$ extension of the SM. Specifically, the authors investigate a minimal anomaly-free fermion content that differs from the standard three right-handed neutrinos ($B-L = -1$), utilizing a specific charge assignment to generate a residual matter parity that naturally stabilizes the DM candidate.

Methodology
The authors propose an extension of the SM gauge group to $SU(3)_c \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_{B-L}$. The model introduces:

• Fermions: Three electrically neutral right-handed fermions ($N_1, N_2, X$) with $B-L$ charges $(-4, -4, 5)$ to cancel gauge anomalies, and a vector-like fermion $F$ to generate masses for $X$.
• Scalars: Three active scalar singlets ($\phi_2, \phi_6, \phi_8$), a dark inert doublet ($\eta$), and a dark singlet ($\sigma$).

The $U(1)_{B-L}$ symmetry is spontaneously broken by two units via the vacuum expectation values (VEVs) of the singlet scalars. This breaking pattern leaves a residual local $Z_2$ symmetry, identified as matter parity ($MP = (-1)^{3(B-L)+2s}$), which stabilizes the lightest $MP$-odd particle.

The analysis proceeds through the following steps:

1. Boson Spectrum: The scalar potential is constructed, and the minimization conditions are derived. The mass matrices for charged, CP-odd, and CP-even scalars are diagonalized to identify physical states, including the SM-like Higgs, heavy scalars, and Goldstone bosons eaten by the $Z$ and $Z'$ gauge bosons.
2. Neutrino Mass Generation: A scotogenic mechanism is implemented where light active neutrino masses are generated at the one-loop level. The loop involves the neutral fermions $N_1, N_2$ and the dark scalars ($\eta, \sigma$). The Yukawa Lagrangian is analyzed to derive the neutrino mass matrix.
3. Phenomenological Constraints: The model is tested against neutrinoless double beta decay ($0\nu\beta\beta$) limits and Dark Matter relic abundance constraints. The DM candidate is identified as the lightest neutral scalar ($\phi_2$) in the dark sector. The relic density and direct detection cross-sections are calculated using the micrOMEGAS package.

Key Contributions and Results

• Anomaly-Free Charge Assignment: The model successfully utilizes a minimal set of three neutral right-handed fermions with charges $(-4, -4, 5)$ to cancel all gauge anomalies, providing a specific alternative to the standard $B-L=-1$ assignment.
• Rank-2 Neutrino Mass Matrix: Due to the presence of only two dark fermion mediators ($N_1, N_2$) in the scotogenic loop, the resulting light active neutrino mass matrix has a rank of 2. This predicts one massless neutrino state.
  • For Normal Ordering (NO), $m_1 = 0$.
  • For Inverted Ordering (IO), $m_3 = 0$.
• Neutrinoless Double Beta Decay ($0\nu\beta\beta$): The prediction of a massless neutrino prevents the destructive interference that could otherwise lead to a vanishing effective Majorana mass ($\langle m_{\beta\beta} \rangle$). Consequently, the model sets a strict lower bound on $\langle m_{\beta\beta} \rangle$. The authors find that for Inverted Ordering, the predicted effective mass lies within the sensitivity range of next-generation experiments (e.g., nEXO, LEGEND).
• Dark Matter Viability: The lightest neutral scalar ($\phi_2$) is identified as a Weakly Interacting Massive Particle (WIMP) candidate.
  • The model reproduces the correct relic abundance ($\Omega h \approx 0.12$) primarily through Higgs portal interactions (coupling $\lambda_8$).
  • The parameter space is significantly widened by re-scattering effects when the masses of the physical scalars $\phi_1$ and $\phi_2$ are nearly degenerate ($m_{\phi_1} \approx m_{\phi_2}$).
  • The authors identify viable regions below current direct detection limits (LUX-ZEPLIN, PandaX-4T, XENON1T), particularly near the Higgs resonance ($m_{DM} \approx m_h/2$) and for heavy masses ($m_{DM} > 1$ TeV).

Significance
The paper claims to present a minimal, anomaly-free gauge extension where the stability of Dark Matter is not an ad hoc imposition but emerges naturally from the spontaneous breaking of a gauged $B-L$ symmetry. By linking the DM candidate to the mechanism generating neutrino masses, the model offers a unified explanation for two major SM shortcomings. The specific prediction of a massless neutrino state and the resulting non-vanishing lower bound for $0\nu\beta\beta$ decay provide a distinct, testable signature for this class of scotogenic models. The authors note that while a similar model appeared in literature during their revision, their specific field content and charge assignments highlight the existence of a broader class of scotogenic models utilizing fermions with $B-L = -4$ as mediators.