Nucleon decays into three leptons: contact contributions

This paper systematically investigates contact contributions to baryon number violating nucleon decays into three leptons by constructing a complete basis of dimension-9 operators in low-energy effective field theory, matching them to chiral perturbation theory to calculate decay widths, and deriving stringent constraints from experimental data while illustrating connections to ultraviolet-complete models.

The Gist

The Big Picture: Hunting for Ghosts in the Atom

Imagine the atom as a tiny, incredibly stable fortress. Inside this fortress, protons and neutrons (collectively called nucleons) are the guards. For over a century, physicists have believed these guards are immortal. They thought a proton could never just "die" or disappear.

However, some theories suggest that these guards can vanish, but only if they break a fundamental rule of the universe called Baryon Number Conservation. It's like a bank vault where the total amount of money (baryon number) must always stay the same. If a proton disappears, something else must appear to balance the books.

Usually, when scientists look for this "proton decay," they expect to see a proton turn into a particle and a light message (a meson), like a guard turning into a messenger and a flash of light. This is the "two-body" decay.

This paper is about a much stranger, rarer possibility: What if a proton doesn't just turn into two things, but explodes into three particles at once, all of which are "leptons" (a family of particles that includes electrons and neutrinos)?

Think of it like this:

• Standard Decay: A guard opens the door, hands you a note, and walks away.
• This Paper's Decay: The guard suddenly vanishes, and three invisible ghosts (leptons) pop out of thin air simultaneously.

The Problem: The "Non-Contact" Dead End

In a previous study, the authors looked at how this might happen through a "long-distance" interaction. Imagine the proton trying to talk to the leptons by throwing a ball (a force carrier) across the room. They found that the rules of the universe make this "ball-throwing" method so incredibly weak that it's practically impossible to detect. It's like trying to hear a whisper from across a stadium; the signal is too faint.

The Solution: The "Contact" Handshake

This paper says, "Let's look at a different way." Instead of throwing a ball, what if the proton and the leptons just shake hands directly? In physics, this is called a "contact interaction."

To study this, the authors built a massive "Lego set" of theoretical tools.

1. The Blueprint (Dimension-9 Operators): They created a complete catalog of every possible way three quarks (inside the proton) could instantly transform into three leptons. They call these "Dimension-9 operators." Think of these as the specific instructions for how the atoms rearrange themselves during the handshake.
2. The Translation (Chiral Perturbation Theory): The instructions were written in "quark language," but we can't see quarks directly; we only see protons and neutrons. The authors used a translation tool called Chiral Perturbation Theory to convert the quark instructions into "proton language." This allowed them to calculate exactly how fast this decay would happen if it were real.

The Results: Setting the Speed Limits

Once they had the math, they compared their predictions against real-world data from giant underwater detectors like Super-Kamiokande (a massive tank of water that watches for particles).

• The Findings: They didn't find any ghosts yet (no proton decay has been observed).
• The Constraint: However, because they know how to look, they can set a "speed limit" on how fast this decay could possibly be happening. They calculated that if this decay happens, the proton must live for at least hundreds of trillions of years (specifically, effective energy scales of 300–700 TeV).
• The Analogy: It's like saying, "We haven't seen a unicorn, but if one exists, it must be so rare that you'd have to search the entire ocean for a million years to find one."

The "Why" and the "How": A New Physics Model

The paper doesn't just list rules; it also builds a story about why this might happen. They introduced a fictional "New Physics" model involving Leptoquarks.

• The Leptoquark: Imagine a magical particle that is half-lepton and half-quark. It acts like a translator that allows quarks and leptons to talk to each other directly.
• The Connection: The authors showed that if these Leptoquarks exist, they would naturally create the "contact handshake" the paper describes, while preventing the boring, common decays we've already ruled out. This makes the theory very elegant.

Why Does This Matter?

You might ask, "Why care about a proton that hasn't died yet?"

1. Future Detectors: The next generation of neutrino detectors (like Hyper-Kamiokande) will be huge—eight times bigger than current ones. They will be able to see much fainter signals. This paper provides the "map" for these detectors to know exactly what to look for.
2. Momentum Patterns: The authors calculated that if a proton does decay into three leptons, the leptons won't fly out randomly. They will fly out in specific patterns (like a specific dance move). If a detector sees that specific dance, scientists will know exactly which "New Physics" theory is correct.
3. Testing the Universe: Finding this decay would prove that the universe is not as stable as we thought and would open a door to understanding why there is more matter than antimatter in the universe.

Summary in One Sentence

This paper builds a complete theoretical map for a rare, exotic way protons might decay into three particles at once, calculates exactly what that would look like, and sets strict limits on how rare it must be, guiding future giant detectors on how to hunt for the first sign of new physics beyond our current understanding.

Technical Summary

1. Problem Statement

Baryon number violation (BNV) is a crucial probe for physics beyond the Standard Model (SM). While conventional nucleon decay searches focus on two-body modes (e.g., $p \to e^+ \pi^0$), three-body decays into three leptons ($N \to 3\ell$) offer a unique window into exotic BNV interactions.

• Context: Previous work by the authors analyzed "noncontact" contributions to these decays arising from dimension-6 (dim-6) operators. These were found to be severely suppressed due to stringent constraints from two-body decay limits.
• Gap: There was a lack of a systematic framework for the "contact" contributions arising from dimension-9 (dim-9) operators in the Low-Energy Effective Field Theory (LEFT). These operators can induce $\Delta F_L = 3$ processes (where $F_L$ is lepton flavor) and may dominate $\Delta F_L = 1$ processes where lower-dimensional effects are suppressed.
• Objective: To construct a complete basis of dim-9 LEFT operators responsible for nucleon triple-lepton decays, match them to hadronic physics, calculate decay widths, and derive constraints on the effective scales of new physics.

2. Methodology

The authors employed a systematic Effective Field Theory (EFT) approach combined with Chiral Perturbation Theory (ChPT):

1. Operator Basis Construction:

   • They classified all possible nucleon triple-lepton decay modes based on lepton number change ($\Delta L$) and flavor change ($\Delta F_L$).
   • Using the Basisgen package and Fierz transformations, they constructed a complete, independent basis of dim-9 LEFT operators.
   • The operators consist of three quarks ($u, d$) and three leptons. They were organized into irreducible representations (irreps) of the QCD chiral group $SU(3)_L \otimes SU(3)_R$.
   • The basis includes various Lorentz structures (scalar, vector, tensor) and chiralities ($L/R$), categorized by their chiral irreps: $\{8_L \otimes 1_R, \bar{3}_L \otimes 3_R, 6_L \otimes 3_R, 10_L \otimes 1_R\}$ and their chiral partners.

2. Chiral Matching:

   • To calculate physical decay rates, the quark-level operators were matched to Chiral Perturbation Theory (ChPT) using the spurion field method.
   • This translated the quark operators into four-fermion interactions involving a nucleon and three leptons.
   • The matching relied on low-energy constants (LECs) $\alpha, \beta, c_3$ derived from recent Lattice QCD calculations and naive dimensional analysis.

3. Calculation of Observables:

   • Decay Widths: Analytical expressions for the partial decay widths were derived for all allowed channels (e.g., $p \to e^+e^+e^-$, $n \to \mu^-e^+\bar{\nu}$, etc.) as functions of the Wilson Coefficients (WCs).
   • Momentum Distributions: The authors calculated normalized double-differential momentum distributions ($d^2\Gamma/d|p_1|d|p_2|$) to demonstrate how different operator structures (vector vs. scalar vs. tensor) yield distinct kinematic signatures.

4. Constraints and UV Connection:

   • Experimental lower bounds on partial lifetimes (from Super-Kamiokande, IMB-3, etc.) were applied to derive limits on the effective scales ($\Lambda_{eff}$) of the operators.
   • A specific UV-complete model involving leptoquarks and a discrete $Z_2$ symmetry was analyzed to demonstrate how high-scale physics generates these dim-9 operators while forbidding lower-dimensional dim-6/7 operators.

3. Key Contributions

• Complete Operator Basis: The paper provides the first complete classification of dim-9 LEFT operators for nucleon triple-lepton decays, organized by chiral symmetry properties. This includes operators for $\Delta L = \pm 1, \pm 3$ and $\Delta F_L = 1, 3$.
• Systematic ChPT Matching: The authors successfully matched these complex dim-9 operators to the hadronic level, providing explicit Lagrangian terms and decay amplitude formulas.
• Kinematic Signatures: A novel contribution is the analysis of momentum distributions. The authors show that different operator classes (e.g., vector-like vs. scalar) produce distinct shapes in the momentum spectra of the final-state leptons, offering a potential tool for experimental discrimination.
• UV Model Demonstration: The paper explicitly connects a specific leptoquark model to the LEFT framework, showing how to map model parameters (Yukawa couplings, masses) to the effective Wilson coefficients.

4. Key Results

• Constraints on Effective Scales:
  • Using current experimental data, the effective scales ($\Lambda_{eff}$) for operators involving three charged leptons are constrained to be $\gtrsim 300 - 700$ TeV.
  • Constraints for modes involving neutrinos are weaker (by roughly two orders of magnitude) due to the difficulty of detecting neutrino final states, but future experiments like Hyper-Kamiokande and JUNO are expected to improve these limits significantly.
• Operator Hierarchy:
  • Operators in the chiral irreps $6_L \otimes 3_R$ generally face slightly weaker constraints compared to those in $\bar{3}_L \otimes 3_R$ or $8_L \otimes 1_R$, partly due to the uncertainty in the LEC $c_3$.
• Kinematic Features:
  • Vector-like operators ($O_{VRVLS}$) concentrate momentum in specific regions ($0.1 \lesssim |p| \lesssim 0.4$ GeV).
  • Scalar operators show a mild dependence on one lepton's momentum but a strong dependence on the other.
  • Tensor operators peak at the kinematic boundaries.
• UV Model Implications:
  • In the leptoquark model studied, the new physics scale ($\Lambda_{NP} = (M_1 M_2 M_3)^{1/3}$) can be probed up to several tens of TeV with $\mathcal{O}(0.1)$ couplings, making these scenarios testable at future colliders.

5. Significance

• Theoretical Framework: This work establishes a robust, model-independent framework for studying exotic nucleon decays, filling a critical gap between SMEFT (high scale) and experimental observables.
• Experimental Guidance: By providing detailed momentum distributions and specific constraints for various operator types, the paper guides future neutrino experiments (Hyper-K, DUNE, THEIA, JUNO) on how to optimize their searches and analyze data to distinguish between different BNV mechanisms.
• New Physics Reach: The derived constraints push the scale of new physics to the multi-hundred TeV range, suggesting that while direct discovery at current colliders is challenging, precision nucleon decay experiments remain a powerful probe for high-scale physics, particularly for models that suppress lower-dimensional operators.
• Future Prospects: The authors highlight that upcoming detectors with larger fiducial masses will significantly improve sensitivity, potentially reaching scales where these dim-9 contact contributions could be observable or tightly constrained.