EFT Pathways to $|\Delta B| =2$: Chiral Constructions and Phenomenology

This paper establishes a systematic effective field theory framework using chiral symmetry to connect ultraviolet $|\Delta B|=2$ interactions to low-energy hadronic observables, demonstrating that while baryon-antibaryon oscillations probe a limited subset of operators, dinucleon decay offers a broader sensitivity to previously unexplored channels and parameter space.

The Gist

Imagine the universe is built out of tiny, fundamental Lego bricks called quarks. Usually, these bricks stick together to form larger structures called protons and neutrons (collectively known as baryons). A fundamental rule of our current understanding of physics is that you can't just make these bricks disappear or appear out of nowhere; the total number of "baryon bricks" must stay the same. This is called Baryon Number Conservation.

However, this paper explores a wild possibility: What if that rule isn't actually a law, but just a habit? What if, very rarely, two neutrons could suddenly turn into two anti-neutrons, or two protons could vanish and turn into a burst of particles? This is called Baryon Number Violation.

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

1. The Problem: Too Many Languages

The scientists in this paper are trying to translate a story that is told in three different languages, which are currently hard to understand together:

• The "High-Altitude" Language (UV/Quarks): This is the language of the very small, high-energy world where the story begins. It talks about six quarks interacting in complex ways.
• The "Middle" Language (Chiral Symmetry): This is a set of rules about how these quarks behave when they start to group together. It's like a grammar rule that says, "If you arrange the bricks this way, they must behave that way."
• The "Ground-Level" Language (Hadrons): This is the language of the heavy particles we can actually see in experiments, like protons, neutrons, and pions (mesons).

The problem is that physicists have been trying to connect the "High-Altitude" story directly to the "Ground-Level" story, but they keep getting lost in the translation. They were missing a dictionary.

2. The Solution: Building a Complete Dictionary

The authors built a systematic dictionary (an Effective Field Theory framework) that connects all three languages perfectly.

• The Chiral Map: They used a mathematical tool called "Chiral Symmetry" to sort every possible way six quarks could interact. They made sure they didn't list the same thing twice (non-redundant) and didn't miss anything (complete). Think of it as organizing a massive library where every book is filed under the exact right category so you never lose a page.
• The Translation: They then showed exactly how the "High-Altitude" rules (from the Standard Model) translate into this new "Chiral" language, and finally, how that language translates into the "Ground-Level" particles we can measure in a lab.

3. The Two Experiments: The Oscillator and the Exploder

To test if this "Baryon Number Violation" is real, the paper looks at two different types of experiments, which act like two different detectors for the same invisible signal.

• Experiment A: The Oscillator (Neutron-Antineutron Oscillation)
  Imagine a neutron is a ball bouncing back and forth. Sometimes, it might magically turn into an anti-neutron (a ball made of anti-stuff) and bounce back. This experiment looks for that specific "flip."

  • The Paper's Finding: This experiment is very sensitive, but it only sees a narrow slice of the possible ways the bricks could rearrange. It's like trying to identify a song by only listening to the bass line; you might miss the melody.

• Experiment B: The Exploder (Dinucleon Decay)
  Imagine two protons or neutrons stuck inside a nucleus (the core of an atom). Instead of just flipping, they might suddenly annihilate each other and explode into a shower of new particles (like pions or kaons).

  • The Paper's Finding: This is the "super-detector." Because the two particles are interacting so closely, this experiment can see many more types of rearrangements than the oscillator can. It catches the "melody" that the oscillator misses.

4. The Big Surprise: The "Hidden" Channels

The most exciting part of the paper is that they discovered new channels for these explosions.

• Some types of particle explosions (like two neutrons turning into a Kaon and an anti-Kaon) depend on specific "hidden" rules that the Oscillator experiment can never see.
• The authors calculated that looking for these specific explosions could give us much stronger limits (or even discover the phenomenon) compared to just watching neutrons oscillate. For example, they found that searching for certain decays could be 12 orders of magnitude (a trillion times) more sensitive for some types of interactions than looking for oscillations.

5. Why This Matters

Before this paper, if you wanted to know if a specific "High-Altitude" theory (a theory about the very beginning of the universe) was true, you had to guess how it would look in a lab. It was like trying to guess what a cloud looks like from the ground without a map.

Now, the authors have provided the map.

• They showed exactly how to trace a signal from the high-energy theory all the way down to the specific particles a detector would see.
• They proved that Dinucleon Decay (the explosion) is not just a backup plan; it is a complementary and often superior way to hunt for these violations, especially for the types of interactions that oscillation experiments can't touch.

In short: The authors built a complete translation guide for a mysterious cosmic rule. They showed that while we've been looking for this rule by watching particles "flip" (oscillate), we might find it much faster by watching particles "explode" (decay), because the explosion reveals secrets the flip hides.

Technical Summary

1. Problem Statement

Baryon number violation (BNV) is a crucial probe for physics beyond the Standard Model (SM). While $|\Delta B| = 1$ processes (like proton decay) are well-studied, $|\Delta B| = 2$ processes (such as neutron-antineutron oscillations and dinucleon decays) offer complementary and qualitatively distinct constraints.
The central challenge addressed in this work is the lack of a systematic Effective Field Theory (EFT) framework that connects high-energy ultraviolet (UV) completions to low-energy hadronic observables for $|\Delta B| = 2$ transitions. Specifically:

• Existing analyses often rely on simplified two-flavor ($SU(2)$) chiral limits, which are insufficient for describing strange baryons (e.g., $\Lambda$) and kaon final states.
• There is no complete, non-redundant classification of dimension-nine six-quark operators under the full three-flavor chiral symmetry $SU(3)_R \times SU(3)_L$.
• The connection between these operators and specific hadronic processes (like dinucleon decay) has not been systematically mapped, limiting the ability to constrain UV models using nuclear decay data.

2. Methodology

The authors develop a rigorous EFT pipeline spanning three energy scales: SMEFT (Standard Model EFT) $\to$ LEFT (Low-Energy EFT) $\to$ B$\chi$PT (Baryon Chiral Perturbation Theory).

A. Chiral Operator Construction

• Symmetry Basis: The framework utilizes the approximate chiral symmetry of QCD, $SU(3)_R \times SU(3)_L$, which is spontaneously broken to $SU(3)_V$.
• Operator Classification: They construct the complete, non-redundant basis of dimension-nine six-quark operators. By decomposing quark bilinears into irreducible representations (irreps) of the chiral group, they identify all allowed structures.
• Color and Flavor: Using color tensors ($T^{SSS}, T^{AAS}$, etc.) and Schouten identities, they eliminate redundant operators. They classify operators by their chirality (RRR, RRL, RLL) and their transformation properties under $SU(3)_R \times SU(3)_L$ (e.g., $27_R \otimes 1_L$, $15'_R \otimes 6_L$, $6_R \otimes 6_L$).
• Counting: They establish that there are 72 independent operators (plus their parity conjugates) for the three-light-flavor sector.

B. Matching to SMEFT

• The authors project the dimension-nine SMEFT operators (which arise from integrating out heavy UV physics) onto the chiral basis.
• They explicitly resolve the $SU(2)_L$ indices of the SMEFT operators to match them onto the Low-Energy EFT (LEFT) structures, and subsequently onto the chiral irreducible representations. This provides a direct link between UV Wilson coefficients and low-energy chiral coefficients.

C. Matching to Baryon Chiral Perturbation Theory (B$\chi$PT)

• Hadronic Realization: The chiral operators are matched onto B$\chi$PT, where degrees of freedom are baryons (octet) and mesons (pseudoscalar octet).
• Construction: They construct composite fields (e.g., $\xi B \xi$, $\xi^\dagger B \xi^\dagger$) that transform correctly under chiral symmetry.
• Low-Energy Constants (LECs): The matching introduces LECs ($\alpha_R$) which encode non-perturbative QCD dynamics. The authors utilize existing lattice QCD results for a subset of these LECs and provide the formalism to compute the rest.
• Parity: Unlike standard B$\chi$PT, the $|\Delta B|=2$ Lagrangian admits both parity-even and parity-odd structures, allowing for interactions with odd numbers of mesons.

D. Phenomenological Application

• Baryon Oscillations: They calculate the oscillation parameters ($\delta m_B$) for $n-\bar{n}$ and $\Lambda-\bar{\Lambda}$, relating them to specific combinations of Wilson coefficients.
• Dinucleon Decay: They perform a systematic calculation of scattering amplitudes for dinucleon decays ($NN \to hh$) involving:
  • Contact interactions: Direct $|\Delta B|=2$ vertices.
  • Exchange diagrams: $t$- and $u$-channel exchanges involving intermediate baryons and mesons.
• They derive analytical expressions for decay rates, including kinematic factors and spin-summed amplitudes, and evaluate them for specific channels (e.g., $pp \to \pi^+\pi^+$, $nn \to K^+K^-$).

3. Key Contributions

1. Complete Operator Basis: The first derivation of a complete, non-redundant set of $|\Delta B|=2$ dimension-nine operators under full $SU(3)_R \times SU(3)_L$ symmetry, extending previous $SU(2)$ analyses.
2. Systematic EFT Pipeline: A unified framework connecting UV SMEFT operators $\to$ Chiral operators $\to$ Hadronic B$\chi$PT Lagrangians. This includes explicit mapping tables (provided in the appendices and a supplementary Mathematica notebook).
3. Dinucleon Decay Formalism: A first-principles calculation of dinucleon decay amplitudes within B$\chi$PT, going beyond previous approximate treatments. This includes the identification of new decay channels sensitive to previously unconstrained Wilson coefficients.
4. Identification of Unconstrained Regions: The discovery that dinucleon decay probes operator structures (specifically those involving $\Sigma$ baryons and contact terms) that are inaccessible to baryon-antibaryon oscillation searches.

4. Key Results

• Complementarity of Probes:
  • Neutron-Antineutron ($n-\bar{n}$): Oscillations provide the strongest bounds on the $C_{nn}$ coefficient (constraining it to $\sim 10^{-33}$ GeV).
  • Lambda-Anti-Lambda ($\Lambda-\bar{\Lambda}$): Oscillation bounds are weak ($\sim 10^{-18}$ GeV). However, dinucleon decay provides indirect constraints on $C_{\Lambda\Lambda}$ that are 12 orders of magnitude stronger ($\sim 10^{-30}$ GeV), effectively closing a large gap in parameter space.
  • New Channels: The channel $nn \to K^+K^-$ is shown to be sensitive to the $C_{\Sigma^+\Sigma^-}$ coefficient, which is completely unconstrained by oscillation searches due to quantum number mismatches.
• UV Scale Sensitivity: Assuming $O(1)$ couplings, the derived constraints imply sensitivity to UV scales of $\Lambda \sim O(10^3)$ TeV for dimension-nine operators.
• Specific Constraints: The paper provides numerical bounds for various dinucleon decay channels (e.g., $pp \to \pi^+\pi^+$, $nn \to \pi^0\pi^0$) in terms of the underlying Wilson coefficients, demonstrating that nuclear decay limits are competitive with or superior to oscillation limits for many operator structures.

5. Significance

This work establishes a robust theoretical foundation for interpreting $|\Delta B|=2$ experimental data.

• For Theory: It allows for the consistent propagation of flavor assumptions from UV models down to hadronic observables, enabling precise comparisons between different BSM scenarios.
• For Experiment: It highlights the critical importance of dinucleon decay searches (e.g., in Hyper-Kamiokande or DUNE) as a primary discovery channel for BNV, potentially surpassing the sensitivity of free neutron oscillation experiments for specific operator classes.
• For Lattice QCD: It identifies specific Low-Energy Constants that need to be computed on the lattice to further reduce theoretical uncertainties in the matching procedure.

In summary, the paper transforms the study of $|\Delta B|=2$ physics from a collection of isolated phenomenological models into a systematic, predictive EFT framework, revealing that nuclear decay processes are a far more powerful probe of baryon number violation than previously appreciated.