Radiative neutrino mass generation linked to neutrino mixing and $0\nu \beta \beta$-decay predictions

We discuss the connection between the origin of neutrino masses and their mixings, which arises in a class of scenarios with radiatively induced neutrino masses. In these scenarios, the neutrino mass matrix acquires textures with two entries close to zero in the basis where the charged-lepton mass matrix is diagonal. This results in specific constraints on the neutrino mixing parameters, which leads to the prediction of (i) a normal ordering of neutrino masses with the lightest neutrino mass in the $\sim \mathrm{meV}$ range and (ii) testable correlations among the various mixing angles, including a nonzero ${\theta }_{13}$ angle with its exact value correlated with the values of the atmospheric angle ${\theta }_{23}$ and the $CP$ phase $\delta$. We quantify the impact of deviations from exact zeroes in the mass matrix texture and connect it to the amount of hierarchy among Yukawa couplings. These scenarios of radiative neutrino mass generation also give rise to new short-range contributions to neutrinoless double beta decay, which dominate over the usual light-neutrino exchange contribution. As a result, this class of models can have a sizable neutrinoless double beta decay rate, in the range of upcoming experiments despite the normal mass ordering of neutrinos.

Figure 1

Allowed regions for ${\theta }_{13}$ as a function of the lightest neutrino mass $m$ for a neutrino mass matrix with texture $m_{ee}^{\nu }=0$. The regions are for normal (NO/red) and inverted (IO/green) mass ordering, with best-fit measured values of ${\theta }_{12}$, $\mathrm{\Delta }{m}_{13}^2$, and $\mathrm{\Delta }{m}_{21}^2$ (solid region) as given in Ref. [54]. The allowed region with ${\theta }_{12}$, $\mathrm{\Delta }{m}_{13}^2$, and $\mathrm{\Delta }{m}_{21}^2$ within their $3\sigma$ experimental ranges (dotted line) is also shown. The dashed-black horizontal lines enclose the $3\sigma$ allowed experimental range for ${\theta }_{13}$. The grey region is disfavored by cosmological data [57].

Figure 2

Allowed regions for the neutrino mass texture $m_{ee}^{\nu }=m_{e\mu }^{\nu }=0$ in the ($s_{13}^2$, $s_{23}^2$) plane (solid-red), for $\mathrm{\Delta }{m}_{31}^2$ fixed at its $-3\sigma$ value (left), best-fit value (middle), and $+3\sigma$ value (right) and best-fit values of $\mathrm{\Delta }{m}_{21}^2$ and $s_{12}^2$. In each case, the $1\sigma$, $2\sigma$, and $3\sigma$ allowed experimental contours from the global analysis of Ref. [54] are shown in green. The horizontal dashed line indicates maximal mixing ${\theta }_{23}=\pi /4$.

Figure 3

Allowed values of the $CP$ phase $\delta$ (for the neutrino mass texture $m_{ee}^{\nu }=m_{e\mu }^{\nu }=0$) as a function of $s_{23}^2$ along the $1\sigma$ (red), $2\sigma$ (blue), and $3\sigma$ (green) experimental contours from Fig. 2. The values of $\mathrm{\Delta }{m}_{31}^2$, $\mathrm{\Delta }{m}_{21}^2$, and $s_{12}^2$ are set as in Fig. 2. The vertical dashed line indicates maximal mixing ${\theta }_{23}=\pi /4$.

Figure 4

The size of the entries in the neutrino mass matrix $|m_{e\tau }^{\nu }|$, $|m_{\mu \mu }^{\nu }|$, $|m_{\mu \tau }^{\nu }|$, and $|m_{\tau \tau }^{\nu }|$ (in eV) for mass textures with $m_{ee}^{\nu }=m_{e\mu }^{\nu }=0$. The red regions correspond to the allowed values for neutrino oscillation data within $1\sigma$ confidence level, while the green regions correspond to the allowed values for neutrino oscillation data within $3\sigma$ confidence level.

Figure 5

The two-loop diagram generating neutrino masses from ${\mathcal{O}}^9$ in Eq. (4.1). The crosses $\times$ in the fermion propagators represent chirality flips, which (upon electroweak symmetry breaking) make this Feynman amplitude proportional to the masses $m_{l_a}$ and $m_{l_b}$.

Figure 6

Left: For NO, the allowed region in the ($m,s_{13}^2$) plane as the upper bound on $|m_{ee}^{\nu }|$ is increased. Right: For $m_{ee}^{\nu }=0$, the allowed region in the ($s_{13}^2,s_{23}^2$) plane as the upper bound on $|m_{e\mu }^{\nu }|$ is increased. In both cases, we consider the best-fit values for the other neutrino oscillation parameters [54].

Figure 7

Left: The minimal amount of Yukawa hierarchy $\mathrm{\Pi }$ [defined in Eq. (4.4)] as a function of the lightest neutrino mass $m$. A clear minimum in the hierarchy appears in a narrow range of $m$, where $\mathrm{\Pi }\simeq 5.5$ and $|m_{ee}^{\nu }|\simeq {10}^{-7}\mathrm{eV}$. Right: Values of the various neutrino mass entries $|m_{ab}^{\nu }|$ as a function of $m$. The parameters ${\theta }_{12}$, ${\theta }_{13}$, ${\theta }_{23}$, $\mathrm{\Delta }{m}_{13}^2$, and $\mathrm{\Delta }{m}_{21}^2$ are fixed to their best-fit measured values [54].

Figure 8

Top: Effective $0\nu \beta \beta$ contribution from ${\mathcal{O}}^9$. Bottom left: $0\nu \beta \beta$ from tree-level BSM completions of ${\mathcal{O}}^9$. Bottom right: $0\nu \beta \beta$ from one-loop BSM completions of ${\mathcal{O}}^9$.

Figure 9

Current limits and future sensitivities on ${\epsilon }_3$ and $|m_{ee}^{\nu }|$ from the ${}^{76}\mathrm{Ge}$ experiments GERDA and MAJORANA (horizontal dashed-black lines), derived from $T_{1/2}^{0\nu \beta \beta }$ limits with $0\nu \beta \beta$ decay dominated by either short-distance physics (${\epsilon }_3$) or light-neutrino exchange ($|m_{ee}^{\nu }|$). Allowed $|m_{ee}^{\nu }|$ regions for NO (red) and IO (green) are shown for neutrino oscillation parameters fixed to best-fit values (filled regions) and if allowed to vary within their $3\sigma$ uncertainty range (solid lines). The rectangle (blue dots) represents a generic region in the ($m,{\epsilon }_3$) plane for scenarios featuring the short-distance ${\mathcal{O}}^9$ operator in Eq. (4.1) (recall that these scenarios have NO with $|m_{ee}^{\nu }|\simeq 0$). Its enclosed horizontal solid-black lines are concrete Yukawa choices $C_{ee}={10}^{-3},{10}^{-4},{10}^{-5}$ (one loop), and $C_{ee}=1,{10}^{-1}$ (tree level) for the tree-level/one-loop renormalizable completion of ${\mathcal{O}}^9$ discussed in the text. The grey regions are currently excluded by cosmological observations and present $0\nu \beta \beta$ decay limits.

Figure 10

Same as Fig. 9, for the ${}^{136}\mathrm{Xe}$ experiments EXO, NEXT, and KamLAND-Zen (top) and the ${}^{150}\mathrm{Nd}$ experiments NEMO3, $\mathrm{SNO}+$, and SuperNEMO (bottom).

Figure 11

Same as Fig. 9, for the ${}^{130}\mathrm{Te}$ experiments CUORICINO and CUORE and the ${}^{82}\mathrm{Se}$ experiments NEMO3 and SuperNEMO. Note that ${}^{130}\mathrm{Te}$ and ${}^{82}\mathrm{Se}$ can be presented in the same plot, as their $|{\mathcal{M}}^{\mathrm{SD}}|/|{\mathcal{M}}^{\nu }|$ ratios are numerically very similar (see the text and Table 1).

Figure 12

For $m_{ee}=m_{e\mu }=0$, the allowed region $|\cos \delta |\le 1$ from Eq. (a4). Left: in the (${\theta }_{12}$, ${\theta }_{23}$) plane. Middle: in the (${\theta }_{13}$, ${\theta }_{23}$) plane. Right: in the ($\mathrm{\Delta }{m}_{21}^2$, $\mathrm{\Delta }{m}_{31}^2$) plane. All slices are taken at the point of the remaining neutrino oscillation parameters fixed to their best-fit values from Ref. [54]. Normal ordering ($|A|<1$) and inverted ordering ($|A|>1$) are shown in the plots as (red) horizontal-hashed and (blue) vertical-hashed regions, respectively. The black star corresponds to the experimentally best-fit value of the neutrino oscillation parameters [54].