Exploring the flavor structure of leptons via diffusion models

Abstract

We propose a method to explore the flavor structure of leptons using diffusion models, which are known as one of generative artificial intelligence (generative AI). We consider a simple extension of the Standard Model with the type I seesaw mechanism and train a neural network to generate the neutrino mass matrix. By utilizing transfer learning, the diffusion model generates 104 solutions that are consistent with the neutrino mass squared differences and the leptonic mixing angles. The distributions of the CP phases and the sums of neutrino masses, which are not included in the conditional labels but are calculated from the solutions, exhibit non-trivial tendencies. In addition, the effective mass in neutrinoless double beta decay is concentrated near the boundaries of the existing confidence intervals, allowing us to verify the obtained solutions through future experiments. An inverse approach using the diffusion model is expected to facilitate the experimental verification of flavor models from a perspective distinct from conventional analytical methods.

Figures (5)

• Figure 1: The summary of input/output of a neural network in the diffusion process. Based on the noise schedule, noise $\epsilon$ sampled from the standard normal distribution $\mathcal{N} \left(0, 1\right)$ is added to the data $G$. The inputs of the network are the noisy data $x_t$ and the label information $\{L,t\}$, while the output is the predicted noise $\epsilon_\theta$. In particular, $\{L,t\}$ is also input to the intermediate layers for conditional learning. The network is updated to minimize a mean squared error (MSE) defined from the actual added noise $\epsilon$ and predictive noise $\epsilon_\theta$.
• Figure 2: The distribution of absolute values of right-handed neutrino masses. The color bar shows $\chi^2$ values with $P_l=\{ \Delta m_{21}^{2}, \Delta m_{31}^{2}, s_{12}^2, s_{23}^2, s_{13}^2 \}$ in Eq. \ref{['eq:chi_sq']}. In addition, the white region is allowed by the relation $M_{1} \leq M_{2} \leq M_{3}$, but the gray region does not satisfy it.
• Figure 3: The distribution of the Dirac CP phase $\delta_{\mathrm{CP}}$ with respect to sum of neutrino masses in the left panel and the mixing angle $\theta_{23}$ in the right panel. The color bars show $\chi^2$ values with $P_l=\{ \Delta m_{21}^{2}, \Delta m_{31}^{2}, s_{12}^2, s_{23}^2, s_{13}^2 \}$ in Eq. \ref{['eq:chi_sq']}. The solutions are concentrated around $\delta_\mathrm{CP}=106,228\,[\mathrm{deg}]$ and $\Sigma_{i}m_i=60.3\,[\hbox{meV}]$. Note that for the mixing angle in the right panel, all of the points are located within 3$\sigma$ range because we extract the cases that satisfy the experimental constraints from the data generated by the diffusion model.
• Figure 4: The histogram shows the distribution of the Majorana phase $\alpha_{31}$. It turns out that none of the solutions are found near $\alpha_{31}=0\,[\mathrm{deg}]$.
• Figure 5: The distribution of the effective Majorana neutrino mass $m_{\beta\beta}$ with respect to the sum of neutrino masses in the left panel and the mass of electron neutrino in the right panel. The color bars show $\chi^2$ values with $P_l=\{ \Delta m_{21}^{2}, \Delta m_{31}^{2}, s_{12}^2, s_{23}^2, s_{13}^2 \}$ in Eq. \ref{['eq:chi_sq']}. The white area is the 95% CL allowed regions for the normal ordering, and the gray area separated by the red boundary means outside of those regions based on NuFIT 6.0 Esteban:2024eli. All the solutions generated by the diffusion model satisfy the restrictions from experiments along the boundaries.