⚛️ phenomenology

Baryon-number-violating nucleon decays into a dark photon particle

This paper establishes a theoretical framework using an extended low-energy effective field theory and chiral perturbation theory to systematically analyze baryon-number-violating nucleon decays into dark photons, deriving decay expressions and setting stringent constraints on these exotic modes by reinterpreting existing experimental data.

Original authors: Jin-Han Liang, Yi Liao, Xiao-Dong Ma, Xiang Zhao

Published 2026-07-27

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CC BY 4.0

Original authors: Jin-Han Liang, Yi Liao, Xiao-Dong Ma, Xiang Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling city where every building is made of tiny, invisible bricks called protons and neutrons. For decades, physicists have been convinced that these bricks are the ultimate, unbreakable foundation of reality. They believed that if you waited long enough, a proton would never simply fall apart; it would just sit there, forever. This idea is so strong that it's written into the laws of physics we use every day. But what if the bricks aren't unbreakable? What if, deep down, they are slowly crumbling into something we can't even see? This is the thrilling mystery of "baryon number violation." It's the search for a crack in the universe's foundation that could explain why we exist at all, why there is more matter than antimatter, and what the mysterious "dark matter" filling the cosmos actually is.

In this high-stakes detective story, scientists are looking for a very specific kind of decay: a proton or neutron breaking apart and spitting out a normal particle (like an electron) and a brand-new, invisible ghost particle. This ghost is called a "dark photon." Think of a dark photon like a silent, invisible balloon released into a crowded room. You can't see the balloon, and you can't hear it, but if you watch the people in the room carefully, you might see them suddenly jump or move in a weird way because the invisible balloon bumped into them. The challenge is that these invisible balloons are so light and so shy that they might be hiding in plain sight, masquerading as missing energy in experiments that were designed to look for something else entirely.

This paper is a masterclass in how to hunt for these invisible balloons. The authors, a team of theoretical physicists, didn't just guess; they built a complete, step-by-step "toolkit" to describe exactly how a proton could break apart to release a dark photon. They created a new set of mathematical rules (called "operators") that act like a blueprint for these decay events. Then, they translated these blueprints into a language that describes how protons and neutrons actually behave, using a method called "chiral perturbation theory," which is like a translator that converts high-energy math into the messy, real-world physics of atomic nuclei.

The team's biggest discovery is a clever way to use old data to find new secrets. They realized that the experiments looking for protons decaying into a normal particle and a meson (a type of particle like a pion) are actually perfect for finding the invisible dark photon version of the same event. Because the dark photon is invisible, the final result looks almost identical to the standard decay, just with a little bit of energy missing. By re-analyzing data from the massive Super-Kamiokande detector in Japan—a giant tank of water that watches for flashes of light from decaying particles—the authors were able to set new, incredibly strict limits on how often these dark-photon decays can happen.

Their calculations show that if these decays are happening, they are incredibly rare. They found that the "lifetime" of a proton before it decays into a dark photon must be at least in the range of $10^{32}$ to $10^{34}$ years. To put that in perspective, the universe is only about $10^{10}$ years old. This means a proton would have to wait trillions of times longer than the current age of the universe to pull off this trick. The paper also maps out exactly what the "footprints" of these decays would look like, predicting how the energy of the visible particles would be distributed. This gives future experiments a clear target: if they see a proton decay with a specific energy pattern that doesn't match the standard rules, they might have just caught a dark photon in the act. While they haven't found the dark photon yet, they have built the most detailed map we have for finding it, turning a wild guess into a precise scientific search.

Technical Summary: Baryon-number-violating nucleon decays into a dark photon particle

Problem Statement
Baryon-number-violating (BNV) nucleon decays involving a light new particle represent a frontier for testing physics beyond the Standard Model (SM), potentially addressing issues such as dark matter nature, matter-antimatter asymmetry, and the neutron lifetime discrepancy. While effective field theory (EFT) formalisms have been established for nucleon decays into scalar particles ( $\phi$ ), axion-like particles ( $a$ ), and sterile neutrinos ( $N$ ), a comparable systematic framework for decays into a spin-1 dark photon ( $X$ ) has remained largely unexplored. Existing experimental searches, primarily by Super-Kamiokande (Super-K), have focused on conventional two-body modes ( $N \to \ell M$ ) or two-body modes with a massless boson. There is a lack of theoretical tools to systematically interpret these data for three-body modes involving a massive dark photon ( $N \to \ell M X$ ) and to constrain the underlying BNV operators.

Methodology
The authors develop a comprehensive low-energy EFT framework, termed XLEFT, extended with a dark photon field $X$ . The study proceeds through the following steps:

Operator Construction: The authors construct a complete set of leading-order BNV operators at dimensions seven and eight. Two parametrizations of the dark photon are considered:
- Case A: Parametrized by a four-potential $X_\mu$ (Dimension 7).
- Case B: Parametrized by a field strength tensor $X_{\mu\nu}$ (Dimension 8).
  The operator basis includes a dark photon, a lepton, and three light quark fields ( $u, d, s$ ). Operators are classified according to their transformation properties under the approximate QCD chiral symmetry $G_\chi \equiv SU(3)_L \otimes SU(3)_R$ .
Chiral Matching: To handle nucleon matrix elements systematically, the XLEFT operators are matched onto Chiral Perturbation Theory (ChPT). The quark fields are traded for octet baryons and pseudoscalar mesons. This matching utilizes spurion fields to organize the interactions into irreducible representations (irreps) of the chiral group, deriving interaction vertices involving the dark photon, octet baryons, and mesons.
Decay Width Calculation: Within the chiral framework, the authors derive general expressions for the decay widths of both two-body ( $N \to \ell X$ ) and three-body ( $N \to \ell M X$ ) nucleon decays. The calculations include both contact diagrams (direct BNV vertices) and non-contact diagrams (mediated by intermediate baryon states via SM strong interactions). The results are expressed in terms of XLEFT Wilson coefficients (WCs) and hadronic low-energy constants (LECs).
Kinematic Analysis: The momentum distributions of visible final-state particles (charged leptons and mesons) in three-body decays are analyzed to distinguish between different operator structures and to probe the dark photon mass.
Experimental Reinterpretation: The authors reinterpret existing Super-Kamiokande data for conventional two-body decays ( $p \to \ell K^0$ , $n \to \ell \pi^-$ , $n \to \nu \pi^0$ , $p \to \nu K^+$ ). By accounting for nuclear effects (Fermi motion, binding energy, intranuclear interactions) and detector resolutions, they derive lower bounds on the partial lifetimes of the corresponding three-body modes with an additional dark photon.

Key Contributions and Results

Formalism Development: The paper provides the first systematic EFT formalism for BNV nucleon decays into a dark photon. It establishes a complete basis of dimension-7 and dimension-8 operators and derives the corresponding chiral Lagrangian vertices.
Decay Width Expressions: General analytical expressions for decay widths are provided for 3 two-body and 17 three-body decay modes. These expressions depend on the dark photon mass, the XLEFT Wilson coefficients, and hadronic LECs.
Kinematic Signatures: The analysis of momentum distributions reveals that different operator structures (classified by chiral irreps) yield distinct spectral shapes. Specifically, operators in the $6_L \otimes 3_R$ irrep can produce different distributions compared to those in $8_L \otimes 1_R$ or $\bar{3}_L \otimes 3_R$ , and the endpoints of these distributions are sensitive to the dark photon mass.
Experimental Constraints: By reanalyzing Super-K data, the authors set lower bounds on the partial lifetimes for representative three-body modes across a range of dark photon masses ( $m_X$ $m_{X}$ ).
- For $p \to \ell^+ K^0 X$ , bounds reach $\sim (2-6) \times 10^{32}$ yr.
- For $n \to \ell^- \pi^+ X$ and $n \to \ell^- K^+ X$ , bounds are approximately $6 \times 10^{31}$ yr and $2 \times 10^{32}$ yr, respectively.
- For $n \to \nu \pi^0 X$ , bounds reach $\sim O(10^{32-34})$ yr.
- For $p \to \nu K^+ X$ , a conservative bound of $> 5.9 \times 10^{33}$ yr is derived.
Effective Scale Limits: Translating these lifetime bounds into constraints on the effective scales ( $\Lambda_{\text{eff}}$ ) of the XLEFT operators, the authors find that nucleon decays constrain these scales to be in the range of $O(10^6 - 10^7)$ GeV over a broad mass range.
Predictions for Other Modes: Using the derived constraints on Wilson coefficients, the paper predicts lower lifetime limits for other correlated decay modes not directly constrained by the reanalyzed data, such as $p \to \ell^+ \pi^0 X$ , $n \to \nu \eta X$ , and $n \to \nu K^0 X$ . These predicted limits range from $O(10^{30})$ to $O(10^{40})$ yr depending on the specific mode and operator structure.

Significance
The paper claims to provide a "toolkit" for future experimental and theoretical studies of exotic nucleon decays. Its significance lies in:

Bridging Theory and Experiment: It offers a model-independent framework to interpret current and future neutrino experiment data (e.g., JUNO, Hyper-Kamiokande, DUNE) in the context of dark photon searches via nucleon decay.
Operator Discrimination: The derived momentum distributions provide a method to distinguish between different underlying BNV operator structures in the event of a signal.
Comprehensive Constraints: It establishes the first stringent constraints on a wide class of BNV operators involving a dark photon, covering both two-body and three-body final states.
Guidance for Future Searches: The results help experimentalists identify the most promising channels for discovery and provide theoretical predictions for modes that have not yet been specifically searched for.

The authors emphasize that their formalism allows for efficient calculations from a UV-complete perspective by integrating out heavy particles to derive matching results for Wilson coefficients, which can then be directly incorporated into the provided decay width expressions.

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