Basis for non-derivative baryon-number-violating operators

This paper presents a minimal basis for non-derivative baryon-number-violating operators in the Standard Model Effective Field Theory up to mass dimension 11, as well as specific dimension-12 operators, offering a set of terms that generally features fewer components and simpler contractions than existing results while acknowledging cases where minimality conflicts with structural simplicity.

The Gist

Imagine the Standard Model of particle physics as a massive, incredibly complex Lego set. For decades, physicists have known how to build the standard structures (atoms, protons, electrons) using specific rules. But there's a secret rule in the game: Baryon Number. In our current understanding of the universe, this rule says you can never take a proton (a baryon) and make it disappear or turn into something else without a trace. It's like saying a Lego brick can never vanish.

However, many physicists suspect that this rule might be broken deep down in the universe's code. If it is broken, protons could eventually decay, and the universe would look very different. To find out if this happens, scientists use a "dictionary" of possible ways this rule could be broken. This dictionary is called an Effective Field Theory.

This paper is essentially a massive renovation of that dictionary.

The Problem: A Messy Library

Imagine you are trying to write a catalog of every possible way a Lego brick could vanish.

• The Old Way: Previous scientists wrote down lists of these possibilities. But their lists were messy. They included the same idea written in three different ways (like writing "The cat sat on the mat," "The mat had a cat on it," and "On the mat sat the cat"). They also used complicated, hard-to-read instructions for how to snap the pieces together.
• The Goal: The authors of this paper wanted to create a minimal, clean catalog. They wanted to find the absolute smallest number of unique "sentences" needed to describe every possible way a proton could vanish, without any redundancy, and using the simplest possible instructions.

The Challenge: The "Permutation" Puzzle

The hardest part of this job is dealing with repeated pieces.
Imagine you have a sentence with three identical Lego bricks labeled "Q" (like a quark). If you swap the first "Q" with the second "Q," does the sentence mean something new?

• The Old Approach: Some scientists treated every swap as a new, unique sentence. This made the list huge and bloated.
• The New Approach: The authors realized that swapping identical pieces often just creates a mathematical "echo" of the same idea. They developed a clever counting method (using a tool called Sym2Int) to figure out exactly how many truly unique sentences exist.

The Analogy:
Think of it like a song.

• If you have a chorus with three identical notes, playing them in a different order might sound the same to the ear.
• The authors asked: "How many distinct melodies can we make with these notes?"
• They found that for many complex scenarios, previous lists had 74 different "melodies," but the authors proved that only 2 truly unique melodies are needed to cover all possibilities. They achieved this by mixing and matching the old, messy versions into new, compact ones.

The Method: Building the "Minimal Basis"

The authors didn't just guess; they built a systematic process:

1. Count the Space: They calculated the total "volume" of all possible ways the particles could interact.
2. Find the Minimum: They determined the smallest number of "building blocks" (terms) needed to fill that volume.
3. Simplify the Constructions: They tried to build these blocks using simple, standard Lego connectors (mathematical tools called tensors).
   • The Catch: Sometimes, the math says you need only one block to fill the space. But that one block is so weirdly shaped (a "ugly" mathematical contraction) that it's impossible to build with simple Lego pieces. In those rare cases, they had to use two slightly larger, simpler blocks instead of one giant, confusing one. They call this a "non-minimal but nice" basis.

The Results: A Cleaner Catalog

The paper covers "dimensions" of complexity, ranging from simple interactions (Dimension 6) to very complex ones (Dimension 12).

• Dimensions 6 & 7: They confirmed existing lists were correct.
• Dimensions 8 & 9: They found that previous lists were too long. They trimmed them down, removing redundant entries and simplifying the instructions.
• Dimensions 10, 11, & 12: This is the frontier. No one had fully mapped these complex interactions before. The authors provided the first complete, minimal lists for these high-energy scenarios.

Why This Matters (According to the Paper)

The authors emphasize that this work is about organization and clarity.

• Efficiency: If you want to study how protons might decay, you don't want to check 100 different equations if only 2 are actually unique. This paper tells you exactly which 2 to check.
• Simplicity: They avoided using "vector" or "tensor" operators (which are like using a complex, custom-made 3D-printed connector) whenever possible. Instead, they stuck to simple, standard connectors (scalars), making the math easier for other scientists to read and use.
• Completeness: They mapped out the landscape up to Dimension 12, ensuring that no potential "proton decay" scenario is left off the map.

Summary

In short, this paper is a cleanup crew for the theoretical physics of proton decay. They took a library full of duplicate books and confusing instructions, threw out the redundancies, rewrote the complex chapters into simple language, and organized the whole thing into a minimal, easy-to-use catalog. They didn't discover a new particle or prove that protons do decay; they just made sure that if we ever do find evidence of it, we have the perfect, non-redundant list of theories to compare it against.

Technical Summary

1. Problem Statement

The paper addresses the need for a systematic and minimal classification of Baryon Number Violating (BNV) operators within the Standard Model Effective Field Theory (SMEFT).

• Context: While BNV is a sensitive probe for physics beyond the Standard Model (e.g., nucleon decay), comprehensive studies are hindered by the exponential growth of the number of independent operators as the mass dimension ($d$) increases.
• Gap: Existing literature provides minimal bases for $d \le 9$, and partial results for $d=12$. However, many existing bases are either non-minimal (containing redundant terms) or utilize complex contraction structures (involving vector/tensor currents or explicit linear combinations) that obscure the underlying gauge structure.
• Goal: The authors aim to construct a minimal basis of non-derivative BNV operators up to mass dimension $d=11$, plus specific $\Delta B = 2$ operators at $d=12$. The basis must be minimal in the number of terms and composed of "simple" contractions (using only invariant tensors like $\epsilon$ and $\delta$, avoiding vector/tensor currents).

2. Methodology

The authors employ a rigorous three-step framework combining group theory, permutation symmetry, and explicit linear algebra construction.

A. Definitions and Spaces

• Operator Type ($T$): Defined by field content (e.g., $e H e Q^3 L^2$) without specific gauge/Lorentz contractions.
• Term: A specific gauge- and Lorentz-invariant contraction pattern (flavor-unexpanded).
• Flavor-Permutation Space ($W_T$): A vector space where flavor labels are treated formally. This allows the authors to analyze linear dependencies arising from gauge/Lorentz symmetries before fixing specific flavor generations.
• Minimal Basis: A set of terms such that the union of their flavor-permuted operators spans the full operator space $V_T$ with the fewest possible terms.

B. The Construction Algorithm

1. Dimension Counting: Using tools like GroupMath, the authors calculate the dimension of the flavor-permutation space ($\dim W_T$), representing the number of linearly independent gauge/Lorentz singlets.
2. Term Counting: Using the Sym2Int algorithm (based on permutation group representations), they determine the theoretical minimum number of terms ($N_T$) required for a minimal basis. This is calculated as $N_T = \lceil \max_\lambda (m_\lambda / \dim \lambda) \rceil$, where $m_\lambda$ is the multiplicity of irreducible representations.
3. Constructive Search:
   • The authors iteratively guess candidate terms with minimal permutation symmetry (to maximize the subspace generated by flavor permutations).
   • They explicitly expand these terms into tensor monomials in Mathematica, accounting for Grassmann signs.
   • They solve linear systems to verify that the chosen set of terms spans the full space $\dim W_T$.
   • If a minimal basis cannot be formed using "nice" (simple) contractions, they either present a slightly non-minimal basis or an awkward minimal one (documented in Appendix A).

3. Key Contributions

• Extension of Scope: Provides the first explicit minimal bases for non-derivative BNV operators up to $d=11$ and specific $\Delta B=2$ operators at $d=12$.
• Simplification of Structure: Unlike previous works (e.g., Murphy, He & Ma) that often use vector/tensor currents or complex linear combinations, this paper prioritizes bases built entirely from scalar bilinears and simple invariant tensors ($\epsilon_{\alpha\beta\gamma}, \delta_{ij}$, etc.).
• Resolution of Redundancy: Demonstrates that several previously proposed bases (e.g., for $d=8$ and $d=12$) were not minimal. The authors provide explicit linear relations showing how larger bases reduce to their smaller, minimal counterparts.
• Identification of Obstructions: Highlights cases (Appendix A) where a minimal basis exists mathematically but cannot be realized using simple contraction patterns due to structural conflicts between flavor permutations and gauge/Lorentz symmetries (e.g., the $eHLed^3LH$ operator).

4. Key Results

The paper classifies operators by mass dimension, $(\Delta B, \Delta L)$, and field content. Key statistics include:

• Dimensions 6 & 7: Reproduces known minimal bases from literature (refs [10, 11]).
• Dimension 8: Reduces the number of terms for certain operators (e.g., $eHQ^3LH$) from 3 terms (Murphy) to 2 terms.
• Dimension 9: Verifies the minimality of Liao and Ma's basis but replaces tensor-current operators with scalar bilinears for better usability.
• Dimensions 10 & 11:
  • Systematically lists bases for all non-derivative BNV types.
  • For example, the operator type $eHeQ^3L^2$ ($d=10$) is shown to have a minimal basis of 3 terms (down from 17 in the permutation-symmetry basis).
• Dimension 12:
  • Focuses on $\Delta B = 2$ operators (relevant for dinucleon decay).
  • Compares with He and Ma [30], finding that their bases often contain more terms than necessary (e.g., for $Q^6L^2$, they use 2 terms, but the authors confirm minimality; for $u^2d^2Q^2L^2$, He and Ma use 6 terms, while the minimal count is 4).
  • Identifies that for some $d=12$ operators, a minimal basis of simple terms is impossible (e.g., $u^2d^2Q^2L^2$ requires 6 terms in the authors' "nice" basis, though the mathematical minimum is 4).

Table Summary of Operator Types (Non-derivative, $\Delta B > 0$):

Dimension	Total Types	$\Delta B=1$	$\Delta B=2$
6	4	4	0
7	4	4	0
8	7	7	0
9	26	23	3
10	54	54	0
11	60	53	7
12	178	164	14

5. Significance

• Phenomenological Utility: By providing bases with fewer terms and simpler structures, the paper facilitates more efficient matching to UV completions and clearer interpretation of experimental limits on nucleon decay.
• Systematic Framework: The methodology (combining Hilbert series counting with explicit constructive verification) sets a standard for handling high-dimensional EFT operators where manual counting becomes intractable.
• Clarification of Literature: The paper resolves ambiguities regarding the minimality of existing bases, showing that "minimal" in the literature sometimes referred to permutation-symmetry bases rather than the true minimal number of independent terms.
• Foundation for Future Work: The authors note that derivative operators are significantly more complex and will be treated separately, but this work establishes the essential non-derivative foundation for the BNV SMEFT landscape.

In conclusion, this work provides the most comprehensive, minimal, and user-friendly catalog of non-derivative baryon-number-violating operators to date, essential for precision studies of nucleon decay and baryogenesis.