UV cut-off of the Standard Model and proton decays

This paper proposes a composite Higgs scenario with partial fermion compositeness at a scale of approximately $10^{11}$ GeV, which naturally explains small neutrino masses and predicts a proton lifetime consistent with a potential $p \to \pi^0 \mu^+$ decay signal observed at Super-Kamiokande, suggesting that Hyper-Kamiokande will soon detect numerous such events.

The Gist

Imagine the Standard Model of particle physics as a massive, incredibly detailed instruction manual for how the universe's tiniest building blocks behave. For decades, this manual has worked perfectly, but it leaves a few mysteries unsolved: Why do neutrinos have such tiny masses? And why hasn't the proton (a core part of every atom) ever been seen falling apart?

This paper proposes a new way to read the manual, suggesting that there is a "hidden chapter" that kicks in at a very high energy level. Here is the story of that chapter, explained through everyday analogies.

The High-Energy "Ceiling"

Think of the Standard Model as a house we live in. We know the furniture (particles) and the rules of the house (forces). But the authors suggest there is a ceiling to this house, a point called $\Lambda$ (Lambda). Above this ceiling, the rules change. The familiar particles we know might stop being simple points and become something more complex, like a composite object made of smaller, stranger parts.

The paper suggests this ceiling is very high—about $10^{11}$ GeV. To put that in perspective, if the energy of a proton were a single dollar, this ceiling would be a trillion dollars. We can't reach it with our current particle colliders, but we can look for its footprints.

The "Flavor" Recipe and the $\epsilon$ (Epsilon) Scheme

One of the biggest puzzles in physics is why some particles are heavy (like the top quark) and others are light (like the electron). It's like a bakery where some cakes are massive and others are tiny, but the recipe doesn't seem to explain why.

The authors use a concept called "Partial Compositeness." Imagine every particle has a "mixing score" (called $\epsilon$) that tells us how much it is made of the "new stuff" above the ceiling versus the "old stuff" we know.

• Heavy particles (like the top quark) are almost entirely made of the new stuff (a mixing score close to 1).
• Light particles (like the electron) are mostly the old stuff, with just a tiny pinch of the new stuff (a mixing score close to 0).

This "pinch" explains why the masses are so different. It also explains why the particles mix in specific ways, much like how a chef might only use a specific spice blend for certain dishes. The paper shows that if you use this "pinch" recipe, you can perfectly explain the masses of all known particles and the tiny masses of neutrinos.

The Proton: The Unbreakable Brick?

For a long time, physicists thought protons were indestructible. But if there is a new physics ceiling, protons might eventually decay (fall apart) into lighter particles. The big question is: How long does it take?

If the new physics is too close to our energy level, protons would have decayed long ago, and we wouldn't be here. If it's too far away, they would never decay, and we'd never see it.

The authors calculated the "expiration date" of the proton based on their new recipe.

• The Result: They found that if the ceiling is at that specific high energy ($10^{11}$ GeV), the proton's lifespan is just on the edge of what we can detect.
• The Prediction: They predict the proton will likely decay into a pion (a type of particle) and a muon (a heavier cousin of the electron).

The "Ghost" in the Machine

Here is the most exciting part of the paper. The Super-Kamiokande experiment in Japan (a giant tank of water deep underground that watches for particle decays) recently reported seeing one single event that looked like a proton decaying into a pion and a muon.

Usually, scientists are skeptical of single events; it could just be a random glitch or background noise. However, the authors say: "Hey, our theory predicts exactly this kind of event, and it predicts it happens at a rate that matches this one event!"

They aren't claiming this is definitely a discovery yet. Instead, they are saying: "If this one event is real, our theory is a perfect fit."

What Comes Next?

The paper concludes with a call to action for the next generation of detectors, specifically Hyper-Kamiokande.

• If the theory is right, the new detector shouldn't just see one event; it should see many of them soon.
• Crucially, the theory predicts that protons should not decay into electrons (the lighter cousin of the muon) very often. If the new detector sees lots of muons but no electrons, it would be a huge "smoking gun" for this specific theory.

Summary

In simple terms, this paper suggests that the universe has a hidden layer of complexity at extremely high energies. By assuming particles are "partially made" of this new stuff, the authors created a recipe that explains why particles have the masses they do. This same recipe predicts that protons are slowly falling apart into muons and pions. The fact that we might have already seen a single hint of this in the data makes the theory very intriguing, and the next big experiment will tell us if we were just lucky or if we've finally found the key to the universe's hidden rules.

Technical Summary

Technical Summary: UV Cut-off of the Standard Model and Proton Decays

Problem Statement
The Standard Model (SM) is a renormalizable theory that does not predict the scale of new physics ($\Lambda$) without invoking naturalness arguments. While the observation of neutrino oscillations suggests the existence of a dimension-five operator, $\ell\ell HH/\Lambda$, implying a new scale $\Lambda$, the nature of the theory beyond this scale remains open. The authors consider a minimalist scenario where $\Lambda$ represents a UV cut-off (potentially associated with strong dynamics or quantum gravity) where global symmetries of the SM are maximally broken. In this framework, the SM is an effective field theory (EFT) valid below $\Lambda$, and the flavor structure of fermion masses and mixings arises from the "partial compositeness" of fermions. A critical consistency check for this scenario is whether the resulting higher-dimensional operators, specifically dimension-six baryon-number-violating (BNV) operators, induce proton decays at rates compatible with current experimental limits (Super-Kamiokande) while potentially being observable in future experiments (Hyper-Kamiokande).

Methodology
The authors employ a specific flavor ansatz, the $\epsilon$-scheme, to parameterize the suppression factors of fermion operators based on their degree of compositeness.

1. Flavor Parametrization: The Yukawa couplings and neutrino mass terms are assumed to follow a structure where couplings are products of suppression factors $\epsilon_f$ (for $f = q, u, d, \ell, e$) and $\mathcal{O}(1)$ coefficients. These $\epsilon$ factors are determined by fitting the observed fermion mass hierarchies and mixing matrices (CKM and PMNS). Specifically, $\epsilon_{q_i}$ are related to CKM elements, while $\epsilon_{\ell_i}$ and $\epsilon_{e_i}$ are constrained by neutrino masses and charged lepton masses.
2. Effective Field Theory: The theory below $\Lambda$ is described by the Standard Model Effective Field Theory (SMEFT). The leading contribution to proton decay arises from dimension-six BNV operators with $\Delta B = \Delta L = 1$. The Wilson coefficients of these operators are assumed to scale with the same $\epsilon$ factors as the Yukawa interactions, reflecting the flavor structure of the UV completion.
3. Calculation Pipeline:
   • The Wilson coefficients are evolved from the scale $\Lambda$ down to the electroweak scale ($m_W$) using Renormalization Group (RG) equations within SMEFT.
   • At $m_W$, the SMEFT is matched to the Low Energy Effective Field Theory (LEFT), incorporating threshold corrections.
   • The coefficients are further evolved down to the hadronic scale ($\Lambda_{\text{QCD}} \approx 2$ GeV), dominated by QCD interactions.
   • Nucleon decay rates are calculated using hadronic matrix elements determined by Lattice QCD (parameters $\alpha$ and $\beta$) and Baryon Chiral Perturbation Theory (BChPT) for meson-baryon interactions.
4. Parameter Space: The analysis varies three primary free parameters: the cutoff scale $\Lambda$, the top-quark compositeness factor $\epsilon_{q3}$, and the Yukawa coupling strength $\lambda_{\text{yuk}}$. The scale $\Lambda$ is constrained by the requirement that the Higgs quartic coupling remains positive (avoiding vacuum instability), suggesting an upper bound $\Lambda \lesssim 10^{11}$ GeV, and by the requirement to reproduce the tau mass, setting a lower bound $\Lambda \gtrsim 1.3 \times 10^{11}$ GeV.

Key Contributions and Results

• Flavor Structure and Suppression: The $\epsilon$-scheme naturally suppresses proton decay modes involving first-generation leptons ($e^+$) due to the smallness of the corresponding $\epsilon$ factors. Conversely, it predicts that the leading decay modes involve second-generation leptons ($\mu^+$).
• Proton Lifetime Predictions:
  • For the scale $\Lambda \sim 10^{11}$ GeV, the calculated proton lifetimes for first-generation modes (e.g., $p \to \pi^0 e^+$) are significantly longer than current experimental limits ($> 10^{37}$ years), ensuring consistency with Super-Kamiokande data.
  • For second-generation modes, specifically $p \to \pi^0 \mu^+$ and $p \to K^0 \mu^+$, the predicted lifetimes are shorter, ranging from $\sim 10^{34}$ to $10^{35}$ years depending on the specific parameters ($\epsilon_{q3}$ and $\lambda_{\text{yuk}}$).
• Consistency with Data: The paper highlights that the predicted lifetime for $p \to \pi^0 \mu^+$ can be as short as $10^{34}$ years. This is consistent with the current 90% confidence level limits from Super-Kamiokande and, notably, aligns with the observation of a single candidate event for this decay mode in recent Super-Kamiokande data.
• Future Prospects: The analysis indicates that if the single candidate event is indeed a proton decay signal, the Hyper-Kamiokande experiment should observe a large number of $p \to \pi^0 \mu^+$ events in the near future. The paper notes that the non-observation of $p \to \pi^0 e^+$ alongside a potential signal in the muon channel would strongly support the specific flavor structure proposed (the $\epsilon$-scheme with $\Lambda \sim 10^{11}$ GeV).

Significance
The paper argues that the combination of the neutrino mass scale and the observed flavor hierarchy points to a UV cut-off scale of $\Lambda \sim 10^{11}$ GeV. In this specific scenario, the proton lifetime is not arbitrarily long but is minimized to a value that is testable by next-generation experiments. The significance lies in the prediction that the dominant proton decay mode is $p \to \pi^0 \mu^+$ rather than the traditionally favored $p \to \pi^0 e^+$. This provides a concrete, testable hypothesis for the flavor structure of physics beyond the Standard Model. The authors suggest that the observation of this specific decay mode, coupled with the absence of electron-mode decays, would offer direct insight into the nature of fermion hierarchies and the compositeness of the Higgs sector. Furthermore, the scale $\Lambda$ is linked to potential explanations for other SM mysteries, such as dark matter and baryon asymmetry, possibly via an axion associated with the decay constant of order $\Lambda$.