Optimizing Yukawa couplings to suppress Dimension-five Proton Decay in $SU(5)$ GUT

This paper employs machine learning optimization techniques, specifically the Adam optimizer, to navigate the complex 33-dimensional parameter space of a supersymmetric $SU(5)$ GUT extended with $\mathbf{45}$ and $\overline{\mathbf{45}}$ Higgs representations, successfully identifying Yukawa coupling configurations that suppress dimension-five proton decay to satisfy Super-Kamiokande's experimental bounds.

The Gist

Imagine the universe is built on a set of blueprints called the Standard Model. Physicists love these blueprints, but they feel like a rough draft. They want a "Grand Unified Theory" (GUT)—a single, elegant master plan that explains how all the forces of nature fit together. One of the most popular master plans is called SU(5).

However, there's a huge problem with this specific blueprint. According to the math, it predicts that protons (the stable building blocks inside every atom) should fall apart incredibly fast. If this were true, all matter in the universe would have disintegrated billions of years ago. But we are here, and protons are still holding strong.

This paper is about trying to fix that broken blueprint without throwing the whole thing away.

The Problem: The "Leaky Roof"

Think of the proton as a house. In the standard SU(5) model, there is a "leak" in the roof caused by a specific type of particle exchange (colored Higgsinos). This leak allows the house to collapse (proton decay) way too quickly.

Experimental scientists (like those at the Super-Kamiokande detector in Japan) have built a massive underground water tank to catch any falling protons. They have set a rule: If a proton falls apart, it must take at least 59 trillion trillion years. The current SU(5) blueprint predicts it happens much faster than that.

The Proposed Fix: Adding More Bricks

To stop the leak, the authors suggest adding extra "bricks" to the blueprint. Specifically, they add new types of Higgs fields (mathematical representations called 45 and 45).

But here's the catch: Adding these new bricks introduces a massive amount of freedom. It's like trying to tune a radio with 33 different knobs instead of just one. You can twist the knobs (adjusting numbers called "Yukawa couplings") to try and stop the leak.

The problem is that there are so many knobs (33 dimensions) that trying every single combination by hand is impossible. It's like trying to find a specific needle in a haystack the size of a galaxy. This is what scientists call the "curse of dimensionality."

The Solution: The Machine Learning "Tuner"

Instead of trying every combination manually, the authors used Machine Learning (specifically an algorithm called Adam).

Think of the 33 knobs as a giant, complex maze.

1. The Goal: Find the exact spot in the maze where the "leak" (proton decay) is smallest.
2. The Method: The computer starts with thousands of random positions in the maze. It calculates how "leaky" the house is at each spot.
3. The Optimization: The computer acts like a smart hiker. If a spot is very leaky, it knows to move away from it. If a spot is dry, it moves closer. It does this thousands of times, learning the shape of the terrain, until it finds the "dry valleys" where the proton is safe.

What They Found

The authors ran this "smart hiker" simulation for different settings of a variable called tan β (which you can think of as the "tilt" of the universe's gravity).

• The Good News: The computer successfully found specific combinations of the 33 knobs that made the proton stable enough to survive longer than the experimental limit. It proved that the SU(5) model can work, but only if the knobs are set to very specific, non-random values.
• The Bad News: The "sweet spots" are very hard to find. They aren't scattered randomly; they are clustered in tiny, specific islands.
• The Tilt Problem: The authors discovered that as they increased the "tilt" (tan β), it became much harder to keep the proton safe. At high tilts, the "leak" gets bigger, and the computer had to work much harder to find a setting that stopped it. In fact, for the highest tilt they tested, the proton was much more likely to decay, making the model less likely to be correct.

The Bottom Line

This paper doesn't prove that the SU(5) model is definitely right. Instead, it proves that it is possible to make the model work, but only if the universe's internal settings are tuned with extreme precision.

They used a computer to navigate a 33-dimensional maze and found the exit, but the exit is a very narrow door. If the universe's settings (specifically the "tilt" or tan β) are even slightly off, the door closes, and the model fails.

In short: The authors used a digital "smart tuner" to fix a broken cosmic blueprint, showing that while a solution exists, it requires a very specific and delicate arrangement of the universe's fundamental numbers.

Technical Summary

Technical Summary: Optimizing Yukawa Couplings to Suppress Dimension-five Proton Decay in SU(5) GUT

Problem Statement
The minimal supersymmetric (SUSY) $SU(5)$ Grand Unified Theory (GUT) faces a critical phenomenological obstacle: rapid proton decay mediated by dimension-five operators arising from colored-Higgsino exchange. Experimental bounds, specifically the Super-Kamiokande limit of $\tau(p \to K^+ \bar{\nu}) > 5.9 \times 10^{33}$ years, severely constrain the model's viability. While extending the Higgs sector with higher-dimensional representations (specifically 45 and $\overline{45}$) and incorporating non-renormalizable interactions can theoretically resolve this issue, these extensions introduce a vast, high-dimensional parameter space. This space, spanned by numerous mixing angles, CP phases, and mass scales, renders traditional brute-force numerical scans computationally intractable due to the "curse of dimensionality."

Methodology
To address this computational challenge, the authors employ machine learning optimization techniques, specifically the Adam optimizer, to navigate the parameter space of a modified SUSY $SU(5)$ model. The study focuses on a model containing three generations of matter fields ($5 + 10$), electroweak-breaking Higgs fields ($5 + \overline{5} + 45 + \overline{45}$), and a GUT-breaking field ($24$).

The methodology involves the following steps:

1. Model Formulation: The authors derive the GUT-scale Yukawa coupling matrices ($Y_5, Y_{\overline{5}}, Y_{45}, Y_{\overline{45}}$) by matching them to the observed MSSM fermion masses and mixing angles (CKM matrix) at the GUT scale ($\sim 2 \times 10^{16}$ GeV). This involves fine-tuning the mass matrices of the Higgs doublets and triplets to ensure a light $\mu$-term while maintaining heavy colored Higgs triplets.
2. Decay Rate Calculation: The partial decay width for $p \to K^+ \bar{\nu}$ is calculated by integrating out the colored Higgs fields to obtain dimension-five operators. These operators are then evolved via Renormalization Group (RG) equations from the GUT scale down to the hadronic scale, incorporating SUSY particle masses (fixed at 3 TeV for the benchmark spectrum) and hadronic matrix elements.
3. Optimization Setup: The search for viable parameters is framed as an optimization problem. A loss function is defined as the mean log-scaled partial width of the $p \to K^+ \bar{\nu}$ decay.
4. Parameter Space: The optimization varies 33 parameters:
   • 27 angles and phases parameterizing three undetermined unitary matrices ($U_u, U_e, V_e$) that rotate the flavor basis.
   • 6 potential sector parameters ($M, \lambda, \kappa, \kappa', \lambda_a, \lambda_b$) governing the Higgs mass terms and couplings.
   • The parameter $m$ is constrained by the condition $\det(M_{\text{doublet}}) = 0$.
5. Execution: Using TensorFlow, the authors initialize $N=4096$ random parameter configurations and iteratively update them over 100,000 epochs to minimize the loss function. This process is repeated for four distinct values of $\tan \beta$ (3, 10, 30, and 50).

Key Results

• Optimization Efficacy: The Adam optimizer successfully navigated the 33-dimensional space, rapidly reducing the loss function. The optimized parameter sets consistently shifted the proton lifetime distribution toward longer values compared to the initial random configurations.
• Parameter Localization: The analysis revealed that the optimized parameters do not distribute uniformly. Instead, specific mixing angles (particularly those in the unitary matrices $U_u$ and $U_e$) and potential couplings (such as $M$ and $\kappa$) exhibit localized distributions. This suggests that suppressing proton decay requires a highly specific, non-trivial alignment between the Yukawa matrices and the GUT-scale potential sector.
• Dependence on $\tan \beta$: A critical finding is the strong sensitivity of the achievable proton lifetime to $\tan \beta$.
  • For lower values ($\tan \beta = 3, 10$), the optimizer successfully identified regions where the proton lifetime exceeds the Super-Kamiokande bound.
  • As $\tan \beta$ increases (30 and 50), the enhancement of down-type quark and lepton sector contributions significantly amplifies the dimension-five decay operators. Consequently, the optimized proton lifetime decreases, making it increasingly difficult to satisfy the experimental lower bound. In the high $\tan \beta$ regime, the required cancellations within the parameter space become more stringent and less likely to be achieved.

Significance and Claims
The paper claims to demonstrate that machine learning optimization is an effective tool for tackling phenomenological problems in Beyond-the-Standard-Model (BSM) physics that suffer from the curse of dimensionality. By systematically exploring the parameter space of an extended SUSY $SU(5)$ model, the authors show that viable regions exist where dimension-five proton decay is suppressed to acceptable levels.

However, the authors remain modest regarding the universality of these solutions. They emphasize that while the algorithm can identify specific, non-trivial alignments that suppress decay, the viability of the model is highly contingent on the value of $\tan \beta$. The study concludes that while the framework is flexible and powerful, the suppression of triplet-Higgsino-mediated decay in high $\tan \beta$ regimes places severe constraints on the model's capacity to meet experimental bounds. The work serves as a proof-of-concept for applying optimization algorithms to complex GUT parameter spaces, with potential applicability to other GUT groups like $SO(10)$.