Radiative Dirac neutrino mass, neutrinoless quadruple beta decay, and dark matter in $B-L$ extension of the standard model

We propose that neutrino Dirac masses can be generated through a scotogenic scenario while violating the lepton number by four units by extending the standard model to have an anomaly free $U(1{)}_{B-L}$, which is spontaneously broken to a residual $Z_4$ symmetry. $Z_4$ symmetry ensures the Dirac nature of neutrinos and simultaneously stabilizes dark matter. Although neutrinoless double beta decay is exactly absent, this model predicts a nonzero neutrinoless quadruple beta decay ($0\nu 4\beta$). It is shown that the $0\nu 4\beta$ can be enhanced thanks to the introduction of a new scalar field without affecting tiny neutrino masses and the relic density of dark matter. We perform numerical analysis for a $0\nu 4\beta$, dark matter, and direct detection by imposing cosmological, collider, and theoretical constraints.

Figure 1

Radiative Dirac neutrino mass with $Z_4$ as scotogenic symmetry.

Figure 2

Neutrino mass scale as a function of $v_4$ (black), $m_{\xi }$ (blue) (a), $\frac{Y_D}{Y_M}$ (blue), and $\frac{m_{\xi (s{)}_1}}{m_{\xi (s{)}_2}}$ (black) (b) for $Y_{M,L}\sim O(1)$, $Y_R\sim {10}^{-4}$, ${\theta }_{\xi }=\pi /4$. For the parameters not being scanned in each cases, we fix $v_4\sim {10}^7\mathrm{TeV}$, $m_{\xi }\sim 1\mathrm{TeV}$, $\frac{Y_D}{Y_M}\sim {10}^{-3}$, and $\frac{m_{{\xi }_1}}{m_{{\xi }_2}}-1={10}^{-2}$.

Figure 3

Correlation between $v_4$ and $m_{\xi }$ (a), and $v_4$ and $\frac{Y_D}{Y_M}$ (b) for $\frac{m_{{\xi }_1}}{m_{{\xi }_2}}-1={10}^{-2}$, $Y_{M,L}\sim O(1)$, $Y_R\sim {10}^{-4}$, and ${\theta }_{\xi }=\pi /4$.

Figure 4

Diagram contribution to $0\nu 4\beta$. Blob vertex is explicitly given in Fig. 5.

Figure 5

Quartic effective $\eta$ vertex.

Figure 6

2 two-loop diagrams contributing to $0\nu 4\beta$.

Figure 7

$\frac{Q_{0\nu 4\beta }}{{\mathrm{\Lambda }}^2}$ vs $m_{\xi }$ and $v_4$ for ${\mu }_S={10}^9\mathrm{TeV}$, $\frac{m_{{\xi }_2}}{m_{{\xi }_1}}-1={10}^{-2}$, $\frac{Y_D}{Y_M}={10}^{-3}$, $Y_{M,L}\sim O(1)$, ${\theta }_{\xi }=\pi /4$, and $m_{S_{R(I)}}=0.8(2)\mathrm{TeV}$. A solid curve corresponds to the current half-life constraint on $\frac{Q_{0\nu 4\beta }}{{\mathrm{\Lambda }}^2}$ from NEMO-3 [56] and KURF [58].

Figure 8

Comparison of numerically obtained result from pySecDec (solid circles) with an analytically derived formula in Eq. (17) (curves) for different values of $v_4$, and ${\mu }_S$ for $m_{\xi }=1\mathrm{TeV}$, $\frac{m_{{\xi }_2}}{m_{{\xi }_1}}-1={10}^{-2}$, $\frac{Y_D}{Y_M}={10}^{-3}$, $Y_{M,L}\sim O(1)$, ${\lambda }_2\sim O(1)$, ${\theta }_{\xi }=\pi /4$, and $m_{S_{R(I)}}=0.8(2)\mathrm{TeV}$.

Figure 9

Scatter plot of the $U(1{)}_{B-L}$ gauge coupling $g_{B-L}$ versus the DM mass $|m_N|$ for $\frac{Y_D}{Y_M}\ll 1$ and $10\mathrm{TeV}\lesssim v_4\lesssim 100\mathrm{TeV}$.

Figure 10

Parameter space of $v_4$, $g_{B-L}$, and $m_{\mathrm{DM}}$ constrained by DM relic abundance, $Z^{\prime }$ direct detection, and resonance requirement $m_{Z^{\prime }}=2m_N$.

Figure 11

$S$ decay to neutrino pair diagram.

Figure 12

$S_{R,I}$ annihilation diagrams. $f$ and $h$ denote the SM fermions and the SM-like Higgs, respectively.

Figure 13

$S_{R,I}$ lifetime vs $v_4$, $m_S$, and ${\mu }^x$ for $\frac{m_{{\xi }_1}}{m_{{\xi }_2}}-1={10}^{-2}$ and $Y_{M,L}\sim O(1)$, $Y_R\sim {10}^{-4}$, ${\theta }_{\xi }=\pi /4$, and $Y_D={10}^{-2}$. The solid line indicates the lower limit on ${\tau }_s>{10}^{25}\mathrm{s}$.

Figure 14

${\mathrm{\Omega }}_{\mathrm{DM}}{h}^2$ (a) and $⟨{\sigma }_{\mathrm{eff}}{v}_{\mathrm{rel}}⟩$ (b) vs ${\lambda }_{HS}$ and $m_S$ for $\frac{m_{S_I}}{m_{S_R}}-1={10}^{-2}$, $M_{Z^{\prime }}=4.2\mathrm{TeV}$, and $g_{B-L}=0.127$. The solid lines correspond to ${\mathrm{\Omega }}_S{h}^2=0.120$.

Figure 15

Correlation among ${\lambda }_{HS}$, $m_S$, and $g_{B-L}$ for $\frac{m_{S_I}}{m_{S_R}}-1={10}^{-2}$, $M_{Z^{\prime }}=4.2\mathrm{TeV}$ constrained by DM relic abundance, $0.119<{\mathrm{\Omega }}_S{h}^2<0.121$.

Figure 16

Plot of correlation between $g_{B-L}$ and $v_4$. Unitarity and collider constraints, $g_{B-L}<0.127$ and $M_Z^{\prime }>4.2\mathrm{TeV}$ (Table 4), are imposed. Contours represent $M_Z^{\prime }$ in TeV units.

Figure 17

$N$ annihilation diagrams. (a) $s$-channel, $S_4$ mediator, annihilation to $H$ pair; (b) $s$-channel, $Z^{\prime }$ mediator, annihilation to $f=e,u,d,v,\text{and}\overline{f}$ SM fermion pair; (c) $t$-channel, $\eta$ mediator, annihilation to $L=(v,l)\text{and}L$ pair.