Comprehensive Effective Field Theory Analysis for Baryon Number Violating Processes

This paper presents a comprehensive effective field theory pipeline that connects baryon number-violating new physics to low-energy hadronic observables by systematically matching Standard Model EFT operators up to dimension nine to Low-Energy EFT and chiral perturbation theory, thereby enabling a model-independent analysis that incorporates higher-dimensional operators and a broader class of ultraviolet completions than previously possible.

The Gist

Imagine the universe as a giant, incredibly complex machine. For decades, physicists have been trying to figure out why there is so much "stuff" (matter) in the universe and almost no "anti-stuff" (antimatter). One of the biggest clues to this mystery is a phenomenon called Baryon Number Violation (BNV).

In simple terms, "Baryon Number" is like a strict accounting rule in the Standard Model of physics: it says that protons and neutrons (the building blocks of atoms) are supposed to be eternal. They shouldn't just disappear or turn into something else. But if they did disappear (a process called proton decay), it would explain why the universe exists as it does today.

This paper is a massive, detailed instruction manual for how to hunt for these disappearing protons, but with a twist: the authors are building a universal translator to connect the "high-energy" world of new physics with the "low-energy" world we can actually measure.

Here is the breakdown of their work using some everyday analogies:

1. The Problem: The Language Barrier

Imagine you are trying to understand a secret message written in a language spoken only by giants (the High-Energy/UV world, where new physics lives). You are a human living in a village (the Low-Energy world).

• The Giants (UV Physics): They speak in complex, high-speed code involving Grand Unified Theories (GUTs) and exotic particles.
• The Humans (Low-Energy Physics): We only see the results: protons turning into pions and positrons.
• The Gap: The giants are too far away to talk to us directly. We need a middleman to translate their complex code into simple instructions we can understand.

2. The Solution: The "Pipeline" Translator

The authors built a three-stage translation pipeline to connect the giants to the humans. Think of it like a relay race where the baton is passed through three different runners:

• Runner 1: SMEFT (The High-Level Translator)
  This is the first layer. It takes the complex theories of the "Giants" (like Grand Unified Theories) and writes them down as a list of potential rules (operators). The authors looked at rules up to Dimension 9.

  • Analogy: Imagine the giants wrote a 1,000-page legal contract. The SMEFT is the lawyer who summarizes that contract into a list of "If this happens, then that might happen" bullet points.

• Runner 2: LEFT (The Middle-Man)
  This is the Low-Energy Effective Field Theory. It takes the summary from Runner 1 and simplifies it further, removing the heavy particles that don't exist at our energy levels.

  • The Big Innovation: Previous studies only looked at the simplest bullet points (Dimension 6). This paper says, "Wait, we need to look at the complex bullet points too (Dimensions 7, 8, and 9)."
  • Why it matters: If you only look at the simple rules, you might miss the subtle clues. By including the complex rules, they found new "shapes" (chiral representations) that the protons could turn into. It's like realizing the giants didn't just write in English, but also in a secret dialect that changes the meaning of the message.

• Runner 3: Chiral Perturbation Theory (The Village Interpreter)
  This is the final step. It translates the particle physics rules into the language of hadrons (protons, neutrons, pions).

  • The Magic Trick: They used a method called the "Spurion Method." Imagine you have a set of Lego bricks (quarks). The "Spurion" is a special tag you put on the bricks to tell them how to snap together to form a specific shape (a proton decaying into a pion).
  • The authors showed that by using the complex rules from Runner 2, they could build a much wider variety of Lego shapes than anyone thought possible before.

3. The "New" Discoveries

The paper highlights three major breakthroughs:

1. Completing the Puzzle: Previous maps of proton decay were like a puzzle with missing corners. By adding the higher-dimensional operators (the complex rules), the authors filled in the missing corners. They found that protons can decay in ways that were previously thought impossible or were simply ignored.
2. The "Universal" Map: They created a complete map that connects any high-energy theory (like a specific type of Grand Unified Theory) to the specific decay rates we can measure in detectors like JUNO, Hyper-K, and DUNE.
3. The "UV" Detective: They didn't just translate; they worked backward. They asked, "If we see a proton decay in a specific way, what kind of giant particle could have caused it?" They listed the specific "suspects" (new particles like scalars, fermions, and vectors) that could be responsible.

4. Why Should You Care?

Think of the universe as a locked safe. We know the combination exists, but we don't know the numbers.

• Proton Decay is the sound of the safe clicking open.
• This Paper is the new, high-tech lock-picking tool.

Before this paper, if you heard a click, you might guess the combination based on a rough sketch. Now, thanks to this "pipeline analysis," scientists have a high-definition blueprint. When the next generation of detectors (JUNO, Hyper-K, DUNE) starts listening for protons decaying, they won't just be guessing. They will have a precise dictionary to translate "clicks" into "new physics."

If they find a proton decaying, this paper tells them exactly which "Giant" theory is correct. If they don't find it, this paper tells them exactly which theories are now ruled out.

In a nutshell: The authors built a super-accurate translation dictionary that connects the wild, high-energy theories of the universe's origin to the quiet, slow decay of protons we can measure today, ensuring we don't miss any clues in the search for why we exist.

Technical Summary

Here is a detailed technical summary of the paper "Comprehensive Effective Field Theory Analysis for Baryon Number Violating Processes" by Chuan-Qiang Song and Jiang-Hao Yu.

1. Problem Statement

Baryon number (B) violation is a generic prediction of many Beyond the Standard Model (BSM) theories, particularly Grand Unified Theories (GUTs), and is essential for explaining the matter-antimatter asymmetry of the universe. Despite decades of searching (e.g., Super-Kamiokande), no proton decay has been observed. With next-generation experiments like JUNO, Hyper-K, and DUNE poised to improve sensitivity by an order of magnitude, there is an urgent need for a model-independent, precise theoretical framework to interpret potential signals.

Current theoretical analyses often rely on specific GUT models or are limited to dimension-6 operators in the Effective Field Theory (EFT) framework. This limitation is problematic because:

• Higher-dimensional operators (dimension 7, 8, and 9) can arise from various UV completions and may dominate specific decay channels.
• Conventional dimension-6 analyses miss crucial chiral representations (specifically the $(6,3)$ and $(10,1)$ representations) required to describe the full set of three-quark operators with derivatives.
• There is a lack of a systematic pipeline connecting high-energy UV physics to low-energy hadronic observables for these higher-dimensional operators.

2. Methodology

The authors construct a comprehensive "pipeline" analysis connecting three energy scales:

1. UV Scale ($\Lambda_{GUT}$): Standard Model Effective Field Theory (SMEFT).
2. Intermediate Scale ($\Lambda_{EW} \sim 100$ GeV): Low-Energy Effective Field Theory (LEFT).
3. Low-Energy Scale ($\Lambda_{\chi} \sim 1$ GeV): Chiral Perturbation Theory (ChPT).

Key Methodological Steps:

• SMEFT to LEFT Matching: The authors consider SMEFT operators up to dimension 9. They perform tree-level matching to derive the complete set of LEFT operators up to dimension 8. This includes operators involving covariant derivatives and field-strength tensors.
• Renormalization Group Evolution (RGE): They incorporate QCD renormalization group equations to evolve Wilson coefficients from the electroweak scale down to the chiral scale.
• Systematic Spurion Method: To match LEFT operators to ChPT, the authors employ a systematic spurion method. They introduce spurion fields ($T_L, T_R$) transforming under $SU(3)_L \times SU(3)_R$ to classify the flavor structures of the quark operators.
• Chiral Lagrangian Construction: Using Young tensor techniques, they construct the full chiral Lagrangian for BNV processes. Crucially, they identify that higher-dimensional LEFT operators generate complete chiral representations (including $(3,3)$, $(8,1)$, $(6,3)$, and $(10,1)$), whereas dimension-6 operators only yield $(3,3)$ and $(8,1)$.
• UV Model Reconstruction: They derive the complete set of tree-level UV resonances (scalars, fermions, vectors, and tensors) that generate dimension-6 and dimension-7 SMEFT operators, and representative models for higher dimensions. They also map these to specific GUT breaking chains (Flipped $SU(5)$ and Pati-Salam within $SO(10)$).

3. Key Contributions

• Extension to Dimension 8 in LEFT: The paper provides the first complete classification of dimension-8 LEFT operators relevant for BNV nucleon decays. This is a significant expansion over previous works limited to dimension 6.
• Complete Chiral Representations: The authors demonstrate that dimension-8 operators are necessary to embed the full set of chiral representations for three-quark operators with derivatives. Specifically, they derive chiral Lagrangian terms for the $(6,3)$ and $(10,1)$ representations, which were previously inaccessible or incomplete in dimension-6 analyses.
• Systematic Matching Pipeline: They establish a rigorous, automated-compatible framework for matching:
  • SMEFT (Dim 6–9) $\to$ LEFT (Dim 6–8).
  • LEFT $\to$ Chiral Lagrangian (including NLO terms for derivatives).
• Decay Rate Formulas: They derive explicit analytical formulas for two-body and multi-body nucleon decay rates ($N \to M \ell$), incorporating both direct contributions and indirect contributions via baryon poles.
• UV Completeness: They list all possible tree-level UV completions for dimension-6 and dimension-7 SMEFT operators and link them to specific GUT breaking chains in $SO(10)$.

4. Key Results

• Operator Basis:
  • Dimension 6: 12 operators (standard basis).
  • Dimension 7: 12 operators (involving derivatives).
  • Dimension 8: 48 operators (including those with covariant derivatives and photon field strengths $F_{\mu\nu}$).
• Chiral Lagrangian:
  • The dimension-6 operators map to leading-order (LO) chiral operators with Low-Energy Constants (LECs) $\alpha$ and $\beta$ (calculated via Lattice QCD).
  • Dimension-7 operators require Next-to-Leading Order (NLO) chiral operators involving derivatives ($u_\mu$).
  • Dimension-8 operators introduce new LECs ($C_1, C_2, D_1, D_2, E_1, \dots, F_{15}$), which are estimated using Naive Dimensional Analysis (NDA) due to the lack of direct lattice data.
• Decay Channels:
  • The analysis covers both $\Delta(B-L)=0$ and $|\Delta(B-L)|=2$ processes.
  • It includes channels with photons and multi-body final states (e.g., $N \to 3$ particles), which become relevant for higher-dimensional operators.
  • Explicit decay rate formulas are provided for processes like $p \to \pi^0 e^+$ and $n \to \pi^- e^+$, showing dependence on the specific Wilson coefficients and final-state masses.
• UV Models:
  • Identified specific scalar ($S$), fermion ($F$), vector ($V$), and tensor ($T$) particles that generate these operators.
  • Mapped these particles to representations in $SO(10)$ GUTs, showing how specific breaking chains (e.g., via Flipped $SU(5)$ or Pati-Salam) lead to specific BNV operators.

5. Significance

• Theoretical Robustness: This work fills a critical gap in the literature by providing a complete, model-independent framework for BNV processes up to dimension 8 in the LEFT and dimension 9 in the SMEFT.
• Experimental Readiness: As next-generation detectors (DUNE, Hyper-K) approach their design sensitivities, they may probe decay channels suppressed in dimension-6 but enhanced in higher dimensions. This paper provides the necessary theoretical tools to interpret such signals without bias toward specific GUT models.
• Phenomenological Impact: By including the full set of chiral representations, the authors ensure that no potential decay channel is overlooked. The inclusion of dimension-8 operators allows for the study of processes involving photons and more complex final states, significantly broadening the search space for new physics.
• Future Directions: The framework is designed to be extensible to other BNV processes (e.g., neutron-antineutron oscillations) and other EFTs (e.g., sterile neutrinos). The authors also highlight the need for future Lattice QCD calculations to determine the unknown LECs for the higher-dimensional operators, which will further refine these predictions.

In summary, this paper establishes a rigorous, systematic, and comprehensive bridge between high-scale BSM physics and low-energy hadronic observables, significantly enhancing the theoretical precision required for the next era of nucleon decay searches.