Prediction of the non-SUSY AdS conjecture on the lightest neutrino mass revisited

We study the constraint of the nonsupersymmetric (non-SUSY) anti–de Sitter (AdS) conjecture on the three-dimensional vacua obtained from the compactification of the Standard Model coupled to Einstein gravity on a circle where the three-dimensional components of the four-dimensional metric are general functions of both noncompact and compact coordinates. We find from studying the wave function profile of the three-dimensional metric in the compactified dimension that the radius of the compactified dimension must be quantized. Consequently, the three-dimensional vacua are constrained by not only the non-SUSY AdS conjecture but also the quantization rule of the circle radius, leading to both upper and lower bounds for the mass of the lightest neutrino as $\sqrt{2}\le m_{\nu }/\sqrt{{\mathrm{\Lambda }}_4}<\sqrt{3}$, where ${\mathrm{\Lambda }}_4\simeq 5.06\times {10}^{-84}{\mathrm{GeV}}^2$ is the observed cosmological constant. This means that the lightest neutrino should have a mass around ${10}^{-32}\mathrm{eV}$ or it would be approximately massless. With this prediction, we reconstruct the light neutrino mass matrix that is fixed by the neutrino oscillation data and in terms of three new mixing angles and six new phases for both the normal ordering and inverted ordering. In the situation where the light neutrino mass matrix is Hermitian, we calculate its numerical value in the $3\sigma$ range.

Figure 1

The scaled radion potential $\overline{V}(R)\equiv V(R)/(2\sqrt{\pi }{r}^3{\mathrm{\Lambda }}_4^6)$ versus the scaled $S^1$ radius $R\sqrt{{\mathrm{\Lambda }}_4}$ for various values of $m_{\nu }/\sqrt{{\mathrm{\Lambda }}_4}$ with $M_{\mathrm{Pl}}^2{\mathrm{\Lambda }}_3/{\mathrm{\Lambda }}_4^2=0.1$. The red and orange curves correspond to the AdS vacuum and runaway dS vacuum, respectively, whereas the blue and purple curves lead to the dS vacuum.

Figure 2

The lightest neutrino mass versus the three-dimensional cosmological constant under the quantization rule (20) and the constraint of the non-SUSY AdS conjecture.