⚛️ phenomenology

Baryon-number-violating nucleon decays in SMEFT extended with a light scalar

This paper systematically investigates baryon-number-violating nucleon decays involving a light scalar within the Standard Model Effective Field Theory framework, deriving stringent constraints from experimental data, predicting distinctive three-body decay signatures and dinucleon transitions, and proposing ultraviolet-complete models to explain the underlying physics.

Original authors: Xiao-Dong Ma, Michael A. Schmidt, Weihang Zhang

Published 2026-02-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiao-Dong Ma, Michael A. Schmidt, Weihang Zhang

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, incredibly complex Lego set. For decades, scientists have been building models of how the universe works using a specific set of rules called the Standard Model. In this model, there are two very important "conservation laws" (like rules that say you can't just destroy a Lego brick without a trace):

Baryon Number: You can't just make a proton (a building block of matter) disappear.
Lepton Number: You can't just make an electron or neutrino appear out of nowhere.

For a long time, we thought these rules were unbreakable. But recently, scientists have started wondering: What if there are new, tiny particles we haven't found yet? And what if these particles allow protons to decay (break apart) in ways we've never seen before?

This paper is a "detective's guide" for the next generation of giant particle detectors (like Super-Kamiokande, JUNO, and Hyper-K) to hunt for these exotic decay events.

Here is the breakdown of the paper's story, using simple analogies:

1. The New Suspect: The "Ghost" Scalar

The authors are investigating a hypothetical new particle called a light scalar (let's call it "The Ghost").

The Analogy: Imagine a proton is a sturdy, locked safe. Usually, it never opens. But if "The Ghost" exists, it might be able to sneak in, unlock the safe, and cause the contents to spill out.
The Twist: When the safe opens, it doesn't just spill out normal debris. It spills out a normal particle (like a positron or a neutrino) plus "The Ghost."
The Problem: "The Ghost" is invisible. It doesn't leave a trace in the detector. It's like a magician pulling a rabbit out of a hat, but the rabbit vanishes the moment it leaves the hat. All the detector sees is the empty hat and the sound of the rabbit's feet hitting the floor, but no rabbit.

2. The Detective's Toolkit: EFT (Effective Field Theory)

Since we don't know exactly what "The Ghost" is or how heavy it is, the authors use a mathematical framework called Effective Field Theory (EFT).

The Analogy: Think of EFT as a "black box" map. Instead of trying to understand the intricate gears inside the machine (the deep, unknown physics), the map just tells us: "If you push this button (a proton), and this specific thing happens (a positron flies out), then there is a hidden mechanism working here."
The authors created a catalog of all the possible ways a proton could break apart with this "Ghost" particle, writing down the mathematical "rules" (operators) for each scenario.

3. The Hunt: Two Types of Clues

The paper looks at two main ways this decay could happen, which act like different types of clues for the detectives:

A. The Two-Body Decay (The "One-Two Punch")

What happens: A proton turns into a positron (or muon) and the invisible Ghost.
The Clue: In a normal decay, the positron would have a very specific speed. But because the invisible Ghost steals some energy, the positron will have a slightly different speed depending on how heavy the Ghost is.
The Strategy: The authors re-analyzed old data from the Super-Kamiokande detector (a giant tank of water in a mine). They looked for positrons with these "suspicious" speeds. They found no Ghosts yet, but they set a strict "bounty" on how heavy the Ghost can be. If the Ghost is too heavy, the decay is impossible.

B. The Three-Body Decay (The "Three-Way Split")

What happens: A neutron turns into a neutrino, a pion (a type of particle), and the invisible Ghost.
The Clue: This is like a three-way tug-of-war. The energy is split three ways. The authors calculated exactly how the pion and neutrino should move around depending on the Ghost's mass.
The Strategy: They predicted what the "fingerprint" (momentum distribution) of these particles would look like. Future detectors can use this fingerprint to spot the Ghost if it ever shows up.

4. The "Double Trouble" (Dinucleon Decay)

The paper also looks at a rarer event where two protons or neutrons decay at the same time.

The Analogy: Imagine two safes next to each other. Usually, they are independent. But if "The Ghost" is heavy, it might act like a bridge connecting them. One safe opens, sends a message to the other via the Ghost, and both explode simultaneously.
Why it matters: If the Ghost is too heavy to be created by a single proton, it might still be created by two protons working together. This gives scientists a second way to catch the Ghost if the first method fails.

5. The "Blueprints" (UV-Complete Models)

Finally, the authors didn't just guess; they built three different "blueprints" (theoretical models) that could naturally produce this "Ghost" particle.

The Analogy: They showed three different ways a factory could be built to manufacture these invisible particles. In these factories, the rules of the universe are slightly tweaked to allow the Ghost to exist without breaking other known laws. This proves that the idea is mathematically possible and not just a wild fantasy.

The Bottom Line

This paper is a roadmap for the future.

For the Detectors: It tells them exactly what to look for (specific speeds and patterns of particles) and how to distinguish a "Ghost" event from background noise.
For the Theory: It sets strict limits on how heavy this new particle can be. If the particle is heavier than the limits set here, it's likely not there (or at least, not interacting in this way).

In short: The universe might be hiding a secret "Ghost" particle that allows matter to vanish in strange ways. This paper gives the next generation of giant particle detectors the exact instructions on how to catch it, ensuring that if it's out there, we won't miss it.

1. Problem Statement

Baryon-number-violating (BNV) nucleon decays are a primary probe for physics beyond the Standard Model (SM). While standard searches focus on two-body decays (e.g., $p \to e^+ \pi^0$ ), recent theoretical interest has shifted toward exotic decay modes involving new light particles (masses $\lesssim$ GeV). These light particles (such as axion-like particles, sterile neutrinos, or dark photons) could explain anomalies like the neutron decay lifetime discrepancy or serve as dark matter candidates.

The specific problem addressed in this work is the systematic investigation of BNV nucleon decays involving a light scalar particle ( $\phi$ ). Existing literature has covered axion-like particles and vector bosons, but a comprehensive Effective Field Theory (EFT) analysis for a light scalar, including momentum distributions, nuclear effects, and UV-complete models, was lacking. The authors aim to:

Construct the relevant EFT operators.
Derive decay rates and momentum distributions for two- and three-body modes.
Reinterpret existing experimental data (Super-Kamiokande, SNO+) to set constraints.
Provide UV-complete models that generate these interactions naturally.

2. Methodology

The authors employ a multi-scale Effective Field Theory approach, bridging the gap between high-energy new physics and low-energy nuclear phenomenology.

A. Theoretical Framework

$\phi$ SMEFT (High Scale): The Standard Model Effective Field Theory is extended by a light singlet scalar $\phi$ $ϕ$ . The authors identify leading-order BNV operators:
- Dimension-7: $\Delta(B-L) = 0$ operators (e.g., $O_{LQdu\phi}$ ).
- Dimension-8: $\Delta(B+L) = 0$ operators (e.g., $O_{LdudH\phi}$ ).
- These are constructed by adding $\phi$ to existing SMEFT dim-6 and dim-7 BNV operators.
$\phi$ LEFT (Electroweak Scale): The theory is matched down to the Low-Energy Effective Field Theory (LEFT) at the electroweak scale ( $\sim 100$ GeV). The heavy SM particles ( $W, Z, t, H$ ) are integrated out, leaving light quarks ( $u, d, s, c, b$ ), leptons, and the scalar $\phi$ . Tree-level matching is performed, accounting for CKM matrix elements and Higgs VEV.
Renormalization Group (RG) Evolution: Wilson coefficients are evolved from the new physics scale ( $\Lambda_{NP}$ ) down to the chiral symmetry breaking scale ( $\Lambda_\chi \approx 1.2$ GeV) using one-loop RG equations, including gluon and top-quark loop corrections.
Baryon Chiral Perturbation Theory (ChPT): To calculate nucleon decay matrix elements, the authors use Baryon ChPT. They map the quark-level operators to hadronic operators involving octet baryons ( $N, \Lambda, \Sigma, \Xi$ ) and pseudoscalar mesons ( $\pi, K, \eta$ ). This allows for the calculation of decay widths for processes like $N \to \ell \phi$ and $N \to \ell M \phi$ .

B. Experimental Reinterpretation

The authors reanalyze data from Super-Kamiokande (Super-K) and SNO+:

Two-body decays ( $p \to \ell^+ \phi$ ): They reinterpret Super-K's search for massless bosons to set limits on massive scalars ( $m_\phi$ ). This involves simulating lepton momentum distributions, including nuclear binding effects (quasi-invariant mass), Fermi motion, and detector resolution.
Three-body decays ( $n \to \bar{\nu} \pi^0 \phi$ ): They recast Super-K's search for $n \to \bar{\nu} \pi^0$ to constrain the three-body channel.
Inclusive/Other modes: Constraints are derived from inclusive searches ( $N \to \ell^+ + \text{anything}$ ) and specific modes like $p \to \bar{\nu} K^+ \phi$ .

C. Dinucleon Decays

The paper investigates $\Delta B = 2$ dinucleon decays ( $NN \to \ell \ell$ ) mediated by the scalar $\phi$ via $t$ - and $u$ -channel diagrams. This provides complementary constraints, particularly when $m_\phi > m_n$ , where single nucleon decays are kinematically forbidden.

D. UV-Complete Models

Three specific models are constructed to generate the effective operators at tree level without generating lower-dimensional SMEFT operators (ensuring the scalar is necessary for BNV):

Leptoquark Model (Dim-7): Involves a scalar leptoquark ( $R_1$ ) and a scalar diquark ( $S_1$ ) with a $Z_2$ lepton parity symmetry.
Vector-Like Quark Model (Dim-7): Involves a leptoquark and a vector-like quark ( $D$ or $U$ ).
Leptoquark Model (Dim-8): Generates $\Delta(B+L)=0$ dim-8 operators using a different leptoquark ( $R_2$ ) and diquark ( $S_1$ ).

3. Key Contributions

Systematic Operator Construction: The first complete classification of $\phi$ SMEFT and $\phi$ LEFT BNV operators involving a light scalar up to dimension 8.
General Decay Formalism: Derivation of general expressions for two- and three-body nucleon decay widths, including the full dependence on the scalar mass $m_\phi$ and Wilson coefficients.
Momentum Distributions: Calculation of normalized momentum distributions for final-state leptons and mesons. The authors demonstrate that these distributions are sensitive to the scalar mass and can distinguish between different chiral structures of the underlying operators.
Nuclear Effects: A rigorous treatment of nuclear binding energy and Fermi motion in the reinterpretation of Super-K data, showing significant impacts on the derived limits for heavy scalars.
Complementary Probes: Demonstration that dinucleon decays provide unique constraints in the region $m_\phi > m_n$ , where single nucleon decays are kinematically blocked.

4. Results

A. Experimental Constraints

Two-body Proton Decay ( $p \to e^+ \phi, \mu^+ \phi$ ): The reanalysis of Super-K data sets stringent lower bounds on the partial lifetimes. For $m_\phi \lesssim 0.8$ GeV, the limits are $\tau > 10^{32}$ years. The limits weaken significantly as $m_\phi$ approaches the kinematic threshold due to phase space suppression and background dominance.
Three-body Neutron Decay ( $n \to \bar{\nu} \pi^0 \phi$ ): Constraints are set on the lifetime, with limits around $\tau > 10^{32}$ years for low masses.
Wilson Coefficients: The effective scales ( $\Lambda_{eff}$ ) for the $\phi$ LEFT operators are constrained to be $\gtrsim 10^9$ GeV for dim-7 operators and $\gtrsim 10^7$ GeV for dim-8 operators, assuming order-1 couplings.
Dinucleon Decays: For $m_\phi > m_n$ , dinucleon decays ( $pp \to \ell^+ \ell^+$ , $nn \to \bar{\nu}\bar{\nu}$ ) constrain the effective scale to $\gtrsim 1$ TeV, complementing the single nucleon bounds.

B. Predictions

The authors predict lower bounds on the partial lifetimes of various three-body modes (e.g., $p \to e^+ \pi^0 \phi$ , $n \to \bar{\nu} K^0 \phi$ ) based on the constraints from the two-body modes. These predicted lifetimes range from $10^{32}$ to $10^{36}$ years.
Momentum Spectra: The paper provides benchmark momentum distributions for different scalar masses (0, 0.1, 0.3 GeV), showing that the endpoint of the distribution is a direct probe of $m_\phi$ .

C. Model Implications

The UV-complete models successfully generate the required dim-7 and dim-8 operators while suppressing lower-dimensional BNV operators.
In these models, the scalar $\phi$ is stable against decay (due to $Z_2$ parity), making it a viable scalar dark matter candidate.

5. Significance

This work provides a comprehensive theoretical and phenomenological toolkit for searching for BNV nucleon decays involving light scalars.

Experimental Guidance: By providing momentum distributions and reinterpreted limits, the paper guides future experiments (JUNO, Hyper-Kamiokande, DUNE) on how to search for these exotic signatures, specifically looking for deviations in momentum spectra rather than just total rates.
Dark Matter Connection: The link between BNV and light scalar dark matter is strengthened, suggesting that nucleon decay experiments are also direct probes of dark matter production mechanisms.
Completeness: It fills a gap in the literature by covering the full EFT chain from high-scale UV models down to hadronic matrix elements, including the often-overlooked dim-8 operators and dinucleon channels.

In summary, the paper establishes that light scalars are a viable and testable extension to BNV physics, with current experiments already probing new physics scales up to $10^9$ GeV, and next-generation detectors poised to improve sensitivity by orders of magnitude.