Correlation Between Proton Decay Channels and the Axion Mass in an Extended SU(5) GUT

This paper investigates a renormalizable SU(5) grand unified theory with a 45-dimensional Higgs field and a DFSZ axion sector, demonstrating how Georgi--Jarlskog flavor constraints and one-loop gauge coupling unification establish a direct correlation between viable proton decay channels (specifically $p \to e^+ \pi^0$) and the QCD axion mass.

The Gist

Imagine the universe as a giant, intricate machine built from tiny Lego bricks. For decades, physicists have been trying to figure out how these bricks fit together to make the whole machine work. This paper is like a team of engineers proposing a specific, upgraded blueprint for that machine.

Here is the story of their blueprint, explained in simple terms:

1. The Problem: A Broken Blueprint

The current "Standard Model" of physics is a great blueprint, but it has two major cracks:

• The Unification Gap: It treats three of the fundamental forces (like electricity and magnetism) as separate things, but the authors believe they are actually just three different faces of a single, giant force that existed at the very beginning of the universe.
• The "Strong CP" Glitch: There is a weird glitch in how matter behaves that the current blueprint can't explain. It's like a car engine that runs perfectly but makes a strange noise that shouldn't be there.

2. The Solution: Adding a New Room and a Secret Door

To fix these cracks, the authors propose an "Extended SU(5) Grand Unified Theory." Think of this as adding a new wing to a house and a secret door that connects everything.

• The New Wing (The 45-Dimensional Higgs): In the old blueprint, the "Higgs field" (which gives particles their weight) was a simple, small room. The authors say, "Let's build a massive, 45-dimensional mansion instead." This new room helps fix the math so that the weights of different particles (like electrons and quarks) make sense.
• The Secret Door (The Axion): To fix the "glitch" in the engine, they introduce a new particle called the Axion. Think of the Axion as a "pressure valve" or a "damping spring" that automatically adjusts the engine to stop the weird noise. This Axion is also a leading candidate for Dark Matter, the invisible stuff that holds galaxies together.

3. The Tightrope Walk: Balancing the Forces

The authors had to walk a very fine line. They needed to make sure that when they added this new "45-dimensional mansion," the three fundamental forces still managed to merge into one at the right moment (a process called Gauge Coupling Unification).

• The Thresholds: Imagine the new mansion has several floors. Some floors are very heavy, and some are light. The authors found that if the "lighter floors" (specifically certain particle states) are at just the right height, the forces merge perfectly. If they are too high or too low, the whole blueprint collapses.

4. The Safety Test: Will the Proton Explode?

In these big unified theories, there is a risk that protons (the stable building blocks of atoms) might suddenly decay or "explode."

• The Stress Test: The authors ran a simulation to see if their new blueprint would cause protons to fall apart too quickly. They checked against real-world data from giant detectors (like Super-Kamiokande) that watch for protons decaying.
• The Flavor Filter: They used a specific rule called the Georgi-Jarlskog structure to organize how particles talk to each other. This rule acts like a strict bouncer at a club, ensuring that only the right particles interact. This helped them prove that their blueprint is safe: the protons won't explode anytime soon, but they might decay in a very specific, predictable way that future experiments could catch.

5. The Big Connection: Linking the Invisible to the Massive

This is the paper's "magic trick." Usually, the Axion (the tiny, invisible particle) and the Grand Unified Theory (the massive, high-energy blueprint) are studied separately.

• The Link: The authors showed that in their specific blueprint, the mass of the Axion is directly tied to the size of the Grand Unified Theory.
• The Result: Because they know the rules for the big theory (from the proton safety test), they can now predict exactly how heavy the Axion should be and how it should interact with light.

6. The Treasure Map for Future Hunters

The paper concludes by drawing a "treasure map" for scientists:

• Where to look for the Axion: They predict the Axion's mass will be in a very specific, narrow range (around a few nano-electron-volts).
• How to find it: They tell experimentalists exactly what kind of signals to look for.
  • ABRACADABRA: A detector looking for the Axion talking to light (photons).
  • CASPEr-Electric: A detector looking for the Axion making neutrons wobble (electric dipole moment).

In Summary:
This paper builds a new, more complex house for the universe. It fixes the math of particle weights, adds a secret "pressure valve" (the Axion) to fix a glitch, and proves that this house is safe from collapsing protons. Most importantly, it creates a direct link between the massive scale of the universe's beginning and the tiny, invisible Axion, giving scientists a precise target to hunt for this mysterious particle in the near future.

Technical Summary

Technical Summary: Correlation Between Proton Decay Channels and the Axion Mass in an Extended SU(5) GUT

Problem Statement
The Standard Model (SM) successfully describes electroweak interactions but fails to explain gauge coupling unification and the strong CP problem. While Grand Unified Theories (GUTs) like SU(5) offer a framework for unification, the minimal Georgi–Glashow model predicts unrealistic fermion mass relations (specifically $Y_d = Y_e^T$ at the unification scale). Furthermore, the strong CP problem requires a dynamical solution, typically provided by the Peccei–Quinn (PQ) mechanism and the QCD axion. In generic axion models, the PQ scale is often independent of the GUT scale, limiting predictive power. This paper addresses the need for a unified framework that simultaneously resolves fermion mass hierarchies, achieves gauge coupling unification (GCU), satisfies proton decay constraints, and provides correlated predictions for axion phenomenology.

Methodology
The authors construct a renormalizable SU(5) GUT without supersymmetry, extended by:

1. A 45-dimensional Higgs representation ($\Phi_{45}$) to implement the Georgi–Jarlskog mechanism, correcting the down-type quark and charged-lepton mass relations.
2. A DFSZ axion sector, realized through an extended Higgs structure involving $\Phi_5$, $\Phi'_5$, and $\Phi_{45}$, coupled with a complex adjoint scalar $\Sigma \sim 24$.

The analysis proceeds through the following steps:

• Symmetry Breaking and Axion Derivation: The model imposes a global $U(1)_{PQ}$ symmetry with specific charge assignments. The authors derive the physical axion mode by analyzing the vacuum expectation values (VEVs) of the neutral Higgs components and the adjoint singlet. They establish a direct relation between the axion decay constant ($f_a$) and the SU(5) breaking scale ($v_\Sigma$), which is tied to the GUT scale ($M_{GUT}$).
• Gauge Coupling Unification (GCU): A one-loop renormalization group equation (RGE) analysis is performed, explicitly including threshold corrections from light scalar multiplets arising from $\Phi_{45}$ (specifically the color-octet $S_8$, color-triplet $S_3$, and color anti-triplet $S_1$). The analysis identifies viable regions in the $(M_{GUT}, M_{S1})$ parameter space that satisfy the unification condition $B_{23}/B_{12} = 0.717$.
• Proton Decay Constraints: The study evaluates proton decay rates mediated by both heavy gauge bosons and the scalar $S_1$.
  • For gauge-mediated decay, the $p \to K^+ \bar{\nu}$ mode provides a robust constraint independent of flavor mixing, while $p \to e^+ \pi^0$ depends on the charged-lepton flavor parameter $V_e$.
  • For scalar-mediated decay, the Georgi–Jarlskog texture is used to estimate the flavor structure of the $S_1$ couplings. The authors impose the constraint $Y^U_{45} = 0$ to suppress dangerous $S_3$-mediated contributions, focusing on $S_1$ exchange.
• Axion Phenomenology: Using the constrained parameter space, the authors calculate the resulting axion mass ($m_a$), axion-photon coupling ($g_{a\gamma\gamma}$), and axion-induced neutron electric dipole moment (EDM) coupling ($g_{aD}$). They assume a pre-inflationary PQ breaking scenario to avoid domain wall issues.

Key Contributions and Results

• Correlated Parameter Space: The analysis identifies a viable region in the $(M_{GUT}, M_{S1})$ plane where GCU, proton decay limits, and LHC bounds on scalar masses ($M_{S8} > 3$ TeV) are simultaneously satisfied.
• Flavor Dependence: The allowed region is sensitive to the charged-lepton flavor parameter $V_e$. For representative values ($V_e = 0.1, 1.0, 2.2$), the lower bound on $M_{GUT}$ shifts from $\sim 1.5 \times 10^{15}$ GeV to $\sim 7.3 \times 10^{15}$ GeV.
• Axion Mass Prediction: The model predicts a narrow window for the axion mass, inversely correlated with $M_{GUT}$. For $V_e = 1.0$, the predicted mass range is $1.94 \text{ neV} \leq m_a \leq 3.84 \text{ neV}$.
• Experimental Reach:
  • Axion-Photon Coupling: With $E/N = 8/3$, the predicted coupling falls within the projected sensitivity of the ABRACADABRA Phase-III experiment.
  • Axion-Induced EDM: The predicted coupling $g_{aD}$ is shown to be within the projected reach of CASPEr–Electric Phase-III.
• Scalar Thresholds: The study demonstrates that the hierarchical mass spectrum of the $\Phi_{45}$ scalars is crucial for achieving successful GCU while satisfying proton decay bounds, particularly the constraint $M_{S1} \geq 3.4 \times 10^{13}$ GeV derived from the $p \to K^+ \bar{\nu}$ mode.

Significance
The paper claims that this extended SU(5) framework provides a "motivated" and "predictive" link between high-scale physics and low-energy observables. By tying the PQ scale to the GUT scale through the specific Higgs structure required for realistic fermion masses, the model reduces the freedom typically found in axion models. The authors emphasize that the model correlates four distinct areas: fermion mass generation, gauge coupling unification, proton decay, and axion searches.

The significance lies in the fact that the allowed parameter region is not arbitrary; it is "GUT-selected." Consequently, future proton decay experiments (like Hyper-Kamiokande) and axion searches (ABRACADABRA, CASPEr) provide complementary probes. If the model is correct, improvements in proton decay limits will directly reshape the predicted axion mass window, and axion experiments will be able to test the specific parameter space selected by the unification and proton decay constraints. The work serves as a concrete example of how ultraviolet physics (GUT structure) can be diagnosed through infrared observables (axion properties and nucleon decay).