Neutrino hierarchies from a gauge symmetry

We consider the phenomenology of the gauged Abelian symmetry $B+3(L_e-L_{\mu }-L_{\tau })$. Right-handed neutrinos necessary to cancel triangle anomalies are used in a type-I seesaw scheme to create active neutrino masses. Breaking the $B+3(L_e-L_{\mu }-L_{\tau })$ symmetry spontaneously below the seesaw scale generates low-energy neutrino mass matrices with the approximate symmetries $L_e$ (leading to normal hierarchy) or $L_e-L_{\mu }-L_{\tau }$ (inverted hierarchy). For the latter we need to introduce a ${\mathbb{Z}}_2$ symmetry which decouples one of the right-handed neutrinos. If exact, this ${\mathbb{Z}}_2$ leads to a Majorana dark matter candidate that interacts with the standard model via the $Z^{\prime }$ and a scalar $s$ originating from spontaneous breaking of the new symmetry. The measured relic abundance of the dark matter particle can be obtained around the scalar and $Z^{\prime }$ resonances, while direct detection experiments are mainly sensitive to scalar exchange, which is induced by mass mixing of $s$ with the standard Higgs.

Figure 1

Left panel: scatter plots using the broken $B+3\overline{L}$ low-energy neutrino mass matrix (5) ($\epsilon =0.05$) that leads to NH. Right panel: scatter plots using the neutrino mass matrix (11) ($\epsilon =0.1$) with five right-handed neutrinos and a ${\mathbb{Z}}_2$, which leads to IH. The accepted values of the mixing parameters satisfy the $3\sigma$ bounds from Ref. 18, except for ${\theta }_{23}$, because it can be arbitrarily adjusted by the charged-lepton contribution.

Figure 2

${\chi }^2$ contours (90%, 95% and 99% C.L.) in the $M_2\mathrm{\text{−}}\sin (\theta )$ plane. The horizontal dashed line is the 95% C.L. lower limit from LEP-2 [29].

Figure 3

Left panel: Relic density of $\chi$ for the parameters $m_1=125\mathrm{GeV}$, $m_2=500\mathrm{GeV}$, $v_S=2.3\mathrm{TeV}$, $g^{\prime }=0.25$, $N_1=1.9\mathrm{TeV}$, $N_2=2.5\mathrm{TeV}$, and $\sin \alpha =0.5$ (blue), 0.3 (red) and 0.1 (black). This puts the ${\varphi }_1$, ${\varphi }_2$ and $Z^{\prime }$ resonances at $\sim 60\mathrm{GeV}$, 250 GeV and 1.7 TeV, respectively. The green band shows the $3\sigma$ range measured by WMAP. Right panel: Relative contribution to the relic density by the processes $\chi \chi \to q\overline{q}$ (sum over all quarks), leptons (including neutrinos), $ZZ$, etc., for $\sin \alpha =0.3$.

Figure 4

Spin-independent cross section of $\chi$ with a proton with the same parameters as Fig. 3, i.e. $m_1=125\mathrm{GeV}$, $m_2=500\mathrm{GeV}$, $v_S=2.3\mathrm{TeV}$, $g^{\prime }=0.25$, $N_1=1.9\mathrm{TeV}$, $N_2=2.5\mathrm{TeV}$, and $\sin \alpha =0.5$ (blue line), 0.3 (red, dashed line) and 0.1 (black, dotted line). Also shown is the XENON100 90% C.L. exclusion from Ref. 36.

Figure 5

Decay of the dark matter candidate $\chi$ due to small $Z_2$ breaking $\epsilon$ in the Dirac mass matrix $m_D$.