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Good flavor search in $SU(5)$: a machine learning approach

Fayez Abu-Ajamieh, Shinsuke Kawai, Nobuchika Okada

TL;DR

The work tackles the fermion-mass problem in $SU(5)$ GUTs by leveraging machine learning to compare two well-known fix-its—the $24$-Higgs and $45$-Higgs models—across both nonsupersymmetric and supersymmetric settings. A loss function measuring proximity to the Georgi–Glashow point is minimised via Adam optimization over flavor parameters, consistently finding the $24$-Higgs approach to be more natural than the $45$-Higgs. A one-parameter generalisation with a continuous variable $y$ reveals optimal regions near $y\approx 0.75$–$0.85$ (depending on SUSY) and secondary peaks around $1.2$–$1.3$, suggesting a continuum of natural models and indicating that the most natural structure does not lie exactly at the canonical $24$-Higgs or $45$-Higgs limits. The study demonstrates ML’s effectiveness in probing high-dimensional, theory-space flavor structures and motivates further theoretical interpretation of the emergent optimal parameters.

Abstract

We revisit the fermion mass problem of the $SU(5)$ grand unified theory using machine learning techniques. The original $SU(5)$ model proposed by Georgi and Glashow is incompatible with the observed fermion mass spectrum. Two remedies are known to resolve this discrepancy, one is through introducing a new interaction via a 45-dimensional field, and the other via a 24-dimensional field. We investigate which modification is more natural, defining naturalness as proximity to the original Georgi-Glashow $SU(5)$ model. Our analysis shows that, in both supersymmetric and non-supersymmetric scenarios, the model incorporating the interaction with the 24-dimensional field is more natural under this criterion. We then generalise these models by introducing a continuous parameter $y$, which takes the value 3 for the 45-dimensional field and 1.5 for the 24-dimensional field. Numerical optimisation reveals that $y \approx 0.8$ yields the closest match to the original $SU(5)$ model, indicating that this value corresponds to the most natural model according to our definition.

Good flavor search in $SU(5)$: a machine learning approach

TL;DR

The work tackles the fermion-mass problem in GUTs by leveraging machine learning to compare two well-known fix-its—the -Higgs and -Higgs models—across both nonsupersymmetric and supersymmetric settings. A loss function measuring proximity to the Georgi–Glashow point is minimised via Adam optimization over flavor parameters, consistently finding the -Higgs approach to be more natural than the -Higgs. A one-parameter generalisation with a continuous variable reveals optimal regions near – (depending on SUSY) and secondary peaks around –, suggesting a continuum of natural models and indicating that the most natural structure does not lie exactly at the canonical -Higgs or -Higgs limits. The study demonstrates ML’s effectiveness in probing high-dimensional, theory-space flavor structures and motivates further theoretical interpretation of the emergent optimal parameters.

Abstract

We revisit the fermion mass problem of the grand unified theory using machine learning techniques. The original model proposed by Georgi and Glashow is incompatible with the observed fermion mass spectrum. Two remedies are known to resolve this discrepancy, one is through introducing a new interaction via a 45-dimensional field, and the other via a 24-dimensional field. We investigate which modification is more natural, defining naturalness as proximity to the original Georgi-Glashow model. Our analysis shows that, in both supersymmetric and non-supersymmetric scenarios, the model incorporating the interaction with the 24-dimensional field is more natural under this criterion. We then generalise these models by introducing a continuous parameter , which takes the value 3 for the 45-dimensional field and 1.5 for the 24-dimensional field. Numerical optimisation reveals that yields the closest match to the original model, indicating that this value corresponds to the most natural model according to our definition.

Paper Structure

This paper contains 20 sections, 55 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fermion masses in $SU(5)$ GUT
  3. GUT mass relation of the minimal $SU(5)$ model
  4. The 45-Higgs model
  5. The 24-Higgs model
  6. Beauty of models and loss function
  7. Parametrisation of the mass matrices
  8. Nonsupersymmetric scenario
  9. $SU(5)$ GUT without supersymmetry
  10. Good parameter values: numerical method
  11. 45-Higgs model vs. 24-Higgs model
  12. Supersymmetric scenario
  13. Supersymmetric $SU(5)$ GUT
  14. Good parameter values: numerical method
  15. 45-Higgs model vs. 24-Higgs model
  16. ...and 5 more sections

Figures (9)

  • Figure 1: Renormalisation group flow of $\alpha_i^{-1}\equiv 4\pi/g_i^2$, for the three gauge coupling constants $g_1$, $g_2$, $g_3$. The solid lines are the solution for the $m_D=4.6\times 10^8$ GeV and $m_Q = 2$ TeV case. The dashed lines represent the gauge couplings of the Standard Model.
  • Figure 2: A sample of the loss function evolution for the 45-Higgs model (blue) and the 24-Higgs model (red), in the nonsupersymmetric scenario of Table \ref{['tab:table1']}. We selected $x_0=0$ and employed the identical set of 10 initial parameters for both the 45-Higgs model and the 24-Higgs model. We utilised the Adam algorithm Kingma:2014vow for optimisation and conducted iteration up to $N_{\rm iter}=10^6$ steps.
  • Figure 3: Evolution of the ten parameters, $x_i$, $i=1,2,…,10$, in the same sample of optimisation processes shown in Fig. \ref{['fig:LossEvo']}, for the 45-Higgs model (left panel) and the 24-Higgs model (right). The darkest blue represents the initial random values of the parameters ($N_{\rm iter}=0$), which are chosen to be identical for the two models. The brightest yellow represents the parameter configuration at $N_{\rm iter}=1,000$, with an interval of 10 steps in between. The red lines represent the configurations after $N_{\rm iter}=10^6$ steps. The parameters of the two models begin with the same initial configurations and are observed to be adjusted to distinct optimised values.
  • Figure 4: Distribution of optimised loss function values for $N_{\rm samp} = 1024$ samples, in the nonsupersymmetric $SU(5)$ GUT scenario. The left, middle, and right panels present the results for $x_0=0$, $2\pi/3$, and $4\pi/3$, respectively. The 45-Higgs model (H45) is depicted in blue, while the 24-Higgs model (H24) is shown in red. An optimised value is determined by averaging over the final 100 steps of the total $N_{\rm iter}=10^6$ iterations, thereby mitigating the influence of fluctuations (refer to Fig. \ref{['fig:LossEvo']}). In all three cases, the loss function of the 24-Higgs model exhibits a distribution of smaller values compared to that of the 45-Higgs model.
  • Figure 5: Optimised configurations of ten parameters $x_1,\ldots,x_{10}$ in the nonsupersymmetric $SU(5)$ GUT scenario. The upper (lower) panels present the results for the 45-Higgs (24-Higgs) model. The cases where $x_0=0$, $2\pi/3$, and $4\pi/3$ are depicted on the left, middle, and right panels, respectively. Each panel displays 100 samples representing the 100 smallest values of the loss function out of the 1024 available samples. Darker lines correspond to the smaller loss function values.
  • ...and 4 more figures