Comprehensive investigation of nucleon decays into one lepton plus two mesons

This paper systematically investigates baryon number violating nucleon decays into one lepton and two mesons using the low-energy effective field theory framework, deriving significantly improved partial lifetime bounds for 31 decay modes by constraining Wilson coefficients with existing experimental data on two-body decays.

The Gist

Imagine the universe as a giant, incredibly stable Lego castle. For decades, physicists have believed that one of the fundamental rules of this castle is that the total number of "matter bricks" (called baryons, like protons and neutrons) can never change. You can rearrange them, but you can't make them disappear or appear out of nowhere. This is the law of Baryon Number Conservation.

However, this paper asks a big "What if?" What if that law isn't absolute? What if, very rarely, a single Lego brick (a proton or neutron) spontaneously crumbles into a tiny explosion of new pieces? This is called Nucleon Decay, and finding it would be a massive discovery, potentially explaining why the universe is made of matter instead of being a void of equal parts matter and antimatter.

Here is a breakdown of what the authors did, using simple analogies:

1. The Setup: The "Two-Piece" vs. "Three-Piece" Puzzle

For a long time, scientists have been hunting for a specific type of decay: a proton turning into one particle (like an electron) and one meson (a type of particle made of quarks, like a pion). Think of this as a Lego brick breaking into exactly two pieces. Experiments have set very strict rules on how long a proton must last before this happens (trillions of trillions of years).

The authors of this paper say: "Wait a minute. If the laws of physics allow a proton to break into two pieces, they almost certainly allow it to break into three pieces as well."

They are investigating three-body decays: A proton breaking into one lepton (an electron or neutrino) and two mesons (like two pions, or a pion and a kaon).

• The Analogy: If you have a rule that says "A brick can break into a red piece and a blue piece," it's logical to assume it could also break into a red piece, a blue piece, and a green piece. The authors are calculating exactly how likely that "three-piece" break is, based on the rules governing the "two-piece" break.

2. The Toolkit: The "Universal Translator"

To do this, the authors used a sophisticated mathematical framework called Effective Field Theory.

• The Analogy: Imagine trying to understand how a car engine works, but you can only see the outside. You can't see the gears inside. "Effective Field Theory" is like a universal translator that lets you predict what's happening inside the engine based on the sounds and vibrations you hear outside.
• In this paper, they translate the complex, invisible interactions of quarks (the tiny bits inside protons) into the language of the particles we can actually detect (protons, electrons, pions). They used a method called Chiral Perturbation Theory, which is like a specific dialect of that translator, perfect for handling the "heavy lifting" of the strong nuclear force.

3. The Calculation: Building the Blueprint

The authors didn't just guess; they built a complete mathematical blueprint for 31 different ways a proton or neutron could decay into three pieces.

• They calculated the "decay width," which is essentially a measure of how fast this crumbling happens.
• They expressed these speeds in terms of "Wilson Coefficients." Think of these as the dials on a control panel. Each dial represents a different possible way the universe could be breaking its own rules.

4. The Strategy: Using the "Known" to Constrain the "Unknown"

Here is the clever part of their work. We don't know the exact settings of those "dials" (the Wilson Coefficients) yet. However, we do know that the two-piece decays (the ones we've been hunting for years) haven't been seen. This means the dials can't be set too high, or we would have seen the two-piece breaks by now.

The authors used this logic:

1. Step 1: Look at the strict limits we already have on the "two-piece" decays.
2. Step 2: Use those limits to figure out the maximum possible setting for the "dials."
3. Step 3: Apply those maximum settings to their new "three-piece" blueprints.

The Result: They found that even if the universe is breaking its rules as much as it possibly can (without us having seen the two-piece breaks yet), the "three-piece" decays must be incredibly rare.

5. The Findings: New, Stricter Limits

The paper provides two main types of results:

• The "Single-Dial" Approach: They assumed only one specific rule-breaker was active at a time. This allowed them to set incredibly tight limits, saying, "If this specific thing is happening, the three-piece decay must happen at least 1,000 to 100,000 times less often than current experiments have checked."
• The "Global" Approach: They considered all the dials turning at once. This is a more realistic but complex scenario. Even here, they found that the three-piece decays are constrained to be hundreds of times rarer than previous estimates.

6. Why This Matters for Future Experiments

The authors aren't saying, "Go build a machine to find this tomorrow." Instead, they are handing the experimentalists a better map.

• The Analogy: Imagine you are looking for a lost coin in a giant field. Previous maps said, "It might be anywhere in this 10-mile radius." This paper provides a new map that says, "Actually, based on the physics of the ground, it's almost certainly in this tiny 100-foot patch, and here is exactly what the coin should look like when you find it."
• They calculated not just if these decays happen, but how the energy is distributed among the three pieces. This helps future experiments (like the massive Super-Kamiokande detector) know exactly what signal to look for, rather than just guessing.

Summary

In short, this paper is a theoretical "stress test" for the universe. It says: "We know the universe is very stable (protons don't decay easily). But if it does break, it might break into three pieces, not just two. We have calculated exactly how rare that would be, using the strict rules we already know about two-piece breaks. We have now told experimentalists exactly where to look and what to expect, making their search much more efficient."

They have essentially upgraded the "Wanted" posters for these missing particles, giving the police (the scientists) a much sharper description of the suspect.

Technical Summary

Technical Summary: Comprehensive Investigation of Nucleon Decays into One Lepton Plus Two Mesons

Problem Statement
Baryon number violation (BNV) is a fundamental requirement for explaining the matter-antimatter asymmetry of the Universe and is predicted by various Beyond the Standard Model (BSM) scenarios. While experimental searches have established stringent limits on conventional two-body nucleon decays ($N \to \ell M$, where $M$ is a pseudoscalar meson), the corresponding three-body decay modes ($N \to \ell M_1 M_2$) have remained largely unexplored. Theoretically, these three-body processes are inevitable consequences of the same BNV interactions driving two-body decays. However, existing theoretical estimates are sparse, and experimental bounds on these modes are generally one to two orders of magnitude weaker than those for two-body modes. This work addresses the gap in the systematic theoretical understanding of these three-body decays and aims to derive improved constraints on them by leveraging the tight experimental limits on their two-body counterparts.

Methodology
The authors employ a model-independent framework based on Low-Energy Effective Field Theory (LEFT) and Chiral Perturbation Theory (ChPT).

1. Effective Field Theory (LEFT): The study focuses on leading-order dimension-6 (dim-6) BNV operators involving light $u, d, s$ quarks. These operators are classified into five categories based on quark field configurations: $\ell uud$, $\ell uus$, $\bar{\ell} dds$, $\nu udd$, and $\nu uds$. The analysis considers operators that conserve either $\Delta(B-L)=0$ or $\Delta(B+L)=0$.
2. Chiral Perturbation Theory (ChPT): To handle non-perturbative QCD effects, the quark-level operators are matched onto hadronic degrees of freedom (octet baryons and pseudoscalar mesons) using ChPT. The authors derive the relevant BNV interaction vertices up to quadratic order in meson fields, incorporating both baryon-lepton mixing terms and contact interactions.
3. Decay Width Calculation: The calculation includes four leading-order tree-level Feynman diagrams involving intermediate baryon states and standard strong interaction vertices. The authors derive general analytical expressions for the squared matrix elements and the resulting decay widths for 31 kinematically allowed three-body modes. These expressions are parameterized in terms of the Wilson coefficients (WCs) of the LEFT operators.
4. Constraint Analysis: Two distinct approaches are used to derive improved bounds on the three-body partial lifetimes:
   • Single-Operator Dominance: Assuming one operator dominates at a time, the authors use the most stringent experimental limits on specific two-body modes to constrain the corresponding WCs. These constraints are then applied to the three-body decay widths.
   • Global Analysis: A more robust, assumption-free approach where all relevant WCs for a given field configuration are varied simultaneously. By using multiple experimentally constrained two-body modes to confine the multidimensional WC parameter space to a closed region, the authors derive conservative lower bounds on the three-body lifetimes.
5. Vector Meson Contributions: The study also estimates contributions mediated by octet vector mesons ($\rho, K^*$) within the resonance ChPT framework, finding they can significantly enhance decay rates for specific modes involving kaons.

Key Contributions and Results

• Systematic Derivation: The paper provides the first comprehensive derivation of general decay width expressions for 31 three-body nucleon decay modes ($N \to \ell \pi\pi, \ell \pi\eta, \ell \pi K$) within the LEFT framework, expressed explicitly in terms of Wilson coefficients.
• Improved Bounds (Single-Operator): Under the single-operator-dominance assumption, the authors derive new partial lifetime bounds that are three to eight orders of magnitude stronger than existing experimental limits (e.g., from IMB-3 and Super-K). For modes involving charged leptons ($e^\pm, \mu^\pm$), the bounds reach $O(10^{34} - 10^{37})$ years.
• Improved Bounds (Global Analysis): The global analysis, which avoids the ad hoc assumption of single-operator dominance, yields conservative bounds that are still more than three orders of magnitude stronger than current experimental limits. For modes involving $\pi\pi$ or $\pi\eta$, the bounds are generally around $O(10^{34} - 10^{36})$ years.
• Neutrino Modes: The study provides the first theoretical bounds for three-body modes involving neutrinos or antineutrinos, which have not been previously constrained in this context.
• Vector Meson Impact: The inclusion of vector-meson contributions is found to affect the decay rates by less than 30% for modes with two non-identical pions but can increase rates by factors of several for modes involving a kaon and an electron or neutrino.

Significance
The authors assert that their framework and derived bounds serve as a critical benchmark for future experimental searches. By establishing the theoretical inevitability and providing robust, improved limits on three-body nucleon decays, this work facilitates the design and interpretation of upcoming large-fiducial-mass neutrino experiments (such as Hyper-Kamiokande, DUNE, and JUNO). The results demonstrate that if BNV interactions exist at the scale probed by current two-body limits, the corresponding three-body modes should be within the reach of next-generation detectors, offering a vital channel to fully determine the interaction structure and uncover the nature of the underlying BNV scenario.