Diffusion-model approach to flavor models: A case study for $S_4^\prime$ modular flavor model

TL;DR

The paper tackles the challenge of extracting flavor-model parameters that reproduce experimental observables by treating parameter search as an inverse problem. It introduces a conditional diffusion-model framework, with forward diffusion of model parameters and a trained reverse process that generates plausible parameter sets conditioned on observables like quark masses, the CKM matrix, and the Jarlskog invariant. Applied to the $S_4^\'$ modular flavor model for the quark sector, the method demonstrates spontaneous CP violation driven by the modulus $\tau$ and identifies several viable parameter regions, with transfer learning substantially improving accuracy. The approach provides a versatile, model-agnostic tool for exploring flavor-model landscapes and could yield new physical predictions by efficiently sampling high-dimensional parameter spaces.

Abstract

We propose a numerical method of searching for parameters with experimental constraints in generic flavor models by utilizing diffusion models, which are classified as a type of generative artificial intelligence (generative AI). As a specific example, we consider the $S_4^\prime$ modular flavor model and construct a neural network that reproduces quark masses, the CKM matrix, and the Jarlskog invariant by treating free parameters in the flavor model as generating targets. By generating new parameters with the trained network, we find various phenomenologically interesting parameter regions where an analytical evaluation of the $S_4^\prime$ model is challenging. Additionally, we confirm that the spontaneous CP violation occurs in the $S_4^\prime$ model. The diffusion model enables an inverse problem approach, allowing the machine to provide a series of plausible model parameters from given experimental data. Moreover, it can serve as a versatile analytical tool for extracting new physical predictions from flavor models.

Figures (2)

• Figure 1: The summary of input/output of a neural network in the diffusion process quoted from Ref. Nishimura:2025rsk. The neural network predicts an added noise based on the noised data and conditional labels.
• Figure 2: The distribution of modulus $\tau$ in the 11 solutions that satisfy $\chi^{2}<200.0$, which reproduce the experimental values with relatively high accuracy. The left, middle, and right figures correspond to the points when the total number of generated data is $3\times10^6$, $6\times10^6$, and $9\times10^6$, respectively.