Truth, beauty, and goodness in grand unification: a machine learning approach

TL;DR

This work tackles the discrepancy between the minimal SUSY $SU(5)$ GUT’s fermion-mass relations and experimental data by comparing two extensions: a $\overline{\mathbf{45}}$-Higgs and a nonrenormalisable coupling involving the $\mathbf{24}$-Higgs. It defines determinant-based loss functions $L_{45}$ and $L_{24}$ to quantify deviations from minimal $M_5$ and employs machine-learning–style sampling and optimisation to explore the high-dimensional flavour parameter space. Across 1000 random initialisations, the 24-Higgs extension consistently achieves smaller optimised losses, indicating it can reproduce the observed masses with less departure from the minimal model. The results demonstrate a practical, data-driven approach to evaluating GUT flavour structures and point to the 24-Higgs path as a more natural extension under the chosen criteria, with implications for further phenomenology and cosmology.

Abstract

We investigate the flavour sector of the supersymmetric $SU(5)$ Grand Unified Theory (GUT) model using machine learning techniques. The minimal $SU(5)$ model is known to predict fermion masses that disagree with observed values in nature. There are two well-known approaches to address this issue: one involves introducing a 45-representation Higgs field, while the other employs a higher-dimensional operator involving the 24-representation GUT Higgs field. We compare these two approaches by numerically optimising a loss function, defined as the ratio of determinants of mass matrices. Our findings indicate that the 24-Higgs approach achieves the observed fermion masses with smaller modifications to the original minimal $SU(5)$ model.

Figures (4)

• Figure 1: Behaviour of the loss function upon optimisation. A sample of the loss function for the 45-Higgs model (24-Higgs model) is shown in blue (red). Same initial parameters are used, and the optimisation is made up to $N_{\rm iter}=10^6$ iteration steps.
• Figure 2: Distribution of the loss function values after optimisation, for the 45-Higgs model (H45, blue) and the 24-Higgs model (H24, red). Optimisation is made for $N_{\rm iter}=10^6$ iterations, and $N_{\rm samp} = 1000$ samples are collected for each model. The distribution of the loss function for the 24-Higgs model is seen to be peaked at a smaller value than that of the 45-Higgs model.
• Figure 3: Example of the evolution of the eleven parameters $x_i$, $i=0,2,...,10$ in the optimisation process. The darkest blue lines represent the initial random values of the parameters ($N_{\rm iter}=0$), and the brightest yellow lines represent the parameter configuration at $N_{\rm iter}=1,000$, after which the change of the parameter values is found to be small. The lines in-between show parameter configurations at an interval of 10 iteration steps. The left and the right panels show the results for the 45-Higgs and 24-Higgs model, starting with a same initial parameter configuration. The parameters of the two models are seen to evolve differently, toward different optimised configurations.
• Figure 4: Configurations of the eleven parameters $x_i$, $i=0,\cdots 10$ after optimisation. The case of the 45-Higgs model is shown on the left and that of the 24-Higgs model is shown on the right. Each panel displays 100 samples that give rise to the smallest 10% of the optimised loss function values, out of 1000 samples. Darker lines indicate smaller values of the optimised loss function.