Complete UV Resonances of SMEFT Dim-9 Operators for Short-range Neutrinoless Double Beta Decay

This paper presents a systematic classification of tree-level ultraviolet completions for dimension-nine SMEFT operators relevant to short-range neutrinoless double beta decay, identifying 440 minimal UV realizations and providing the first comprehensive compilation of 324 minimal completions involving vector resonances.

The Gist

The Big Picture: Hunting for Ghosts in the Machine

Imagine the universe is a giant, complex machine. For decades, we've been able to predict how this machine works using a rulebook called the Standard Model. But there's a mystery the rulebook can't solve: Why do neutrinos have mass? And even stranger, why do they seem to be their own antiparticles (like a shadow that is also a mirror image)?

One way to find the answer is to look for a rare event called Neutrinoless Double Beta Decay ($0\nu\beta\beta$).

• Normal Double Beta Decay: Two neutrons in an atom turn into two protons, spitting out two electrons and two invisible neutrinos. It's like a factory making two products and shipping two packages.
• Neutrinoless Decay: The factory makes two protons and two electrons, but no neutrinos are shipped out. The neutrinos were never created, or they ate each other instantly.

If we see this happen, it proves neutrinos are their own antiparticles and that a fundamental law of physics (conservation of lepton number) is broken. This would be a massive discovery, like finding a new continent.

The Problem: The "Black Box" Mystery

Physicists know that if this decay happens, it's caused by some heavy, unknown particle (a "mediator") that acts as a bridge between the neutrons and protons. But we can't see this particle directly yet because it's too heavy for our current particle colliders (like the Large Hadron Collider).

So, we use Effective Field Theory (SMEFT). Think of this as looking at the machine through a foggy window. We can't see the gears inside, but we can see the result of the gears turning. We write down mathematical "recipes" (operators) that describe what happens at low energies.

The problem is: There are too many recipes.
There are hundreds of different ways these heavy particles could be arranged to create the decay. It's like trying to figure out how a cake was baked just by tasting it. Was it a chocolate cake? A vanilla cake? Did they use a mixer or a whisk? Did they use a secret ingredient?

The Solution: The "J-Basis" Blueprint

This paper is a massive organizational project. The authors, Hao-Lin Li and his team, decided to stop guessing and start systematically cataloging every possible way this decay could happen.

They used a new tool called the J-basis framework.

• The Analogy: Imagine you have a pile of 505 different LEGO sets. Some sets look similar, some are just bigger versions of smaller sets, and some use standard LEGO bricks (Standard Model particles) that you already have in your box.
• The Goal: They wanted to find the minimal sets. Which is the smallest, simplest set of new LEGO pieces you need to build a specific structure?

What They Did (The "Cooking" Process)

1. The Ingredients (Operators): They started with the 6 "ingredients" (quarks and electrons) involved in the decay.
2. The Recipe Book (J-basis): They organized these ingredients into every possible legal combination allowed by the laws of physics (symmetry, color, spin).
3. Filtering the Noise:
   • The "SM-like" Filter: Sometimes, a recipe looks like it needs a new heavy ingredient, but it turns out you can just use a standard ingredient (like a Higgs boson) that we already know about. The authors removed these "fake" new particles. They only care about genuinely new heavy particles.
   • The "Minimal" Filter: If a recipe needs three new particles, but you can actually build the same thing with just two, they threw away the three-particle recipe. They only kept the most efficient, "minimal" recipes.

The Big Discoveries

After doing this massive sorting job, they found some surprising things:

1. 505 Recipes, 440 Minimal: They found 505 different ways to build these decay machines. But after removing the duplicates and the ones that could be built with fewer parts, 440 of them are unique, minimal designs.
2. The "Vector" Surprise: For a long time, physicists mostly looked for heavy scalars (like balls) or fermions (like spinning tops) as the culprits. This paper is the first to comprehensively map out the role of vectors (heavy force-carriers, like giant versions of the W or Z bosons).
   • The Result: They found 324 minimal designs that rely on these heavy vectors. This is huge! It means that if we find this decay, there's a very high chance the culprit is a heavy "force-carrier" particle, not just a heavy ball or top.
3. The "Two-Piece" Miracle: Out of all the complex designs, only 12 of them could be built with just two distinct new heavy particles. The rest need three. This gives experimentalists a very short "shopping list" to look for first.

Why This Matters

Think of this paper as a comprehensive map for treasure hunters.

• Before: Hunters were looking for gold in a vast ocean, guessing where the treasure might be buried.
• Now: The authors have drawn a map showing exactly where the 440 most likely treasure spots are. They've labeled the spots: "Here is a spot with a Vector," "Here is a spot with a Scalar," "Here is a spot with only two pieces."

This helps experimentalists at the Large Hadron Collider (LHC) and neutrino detectors know exactly what to look for. Instead of searching blindly, they can say, "If we see this specific signal, it likely means we found a heavy Vector particle with these specific properties."

Summary in One Sentence

This paper is a master catalog that sorts through hundreds of confusing possibilities to tell us exactly which new, heavy particles could be causing a mysterious nuclear decay, revealing that heavy "force-carrier" particles are a much more likely culprit than we previously thought.

Technical Summary

1. Problem Statement

Neutrinoless double beta decay ($0\nu\beta\beta$) is a primary probe for lepton-number violation (LNV) and the Majorana nature of neutrinos. While long-range mechanisms mediated by light neutrinos are well-studied, short-range mechanisms involving heavy mediators at the TeV scale are encoded in dimension-9 ($d=9$) six-fermion operators within the Standard Model Effective Field Theory (SMEFT).

Previous classifications of these operators (e.g., Ref. [1]) were often performed in the broken electroweak phase (after symmetry breaking). This approach suffers from several limitations:

• Redundancy: It merges distinct ultraviolet (UV) completions that appear different in the unbroken phase but project to the same low-energy structure.
• Ambiguity: It does not cleanly separate "genuine" heavy mediators from Standard Model (SM) fields that can be removed via field redefinitions.
• Incomplete Vector Sector: Previous analyses largely neglected massive vector resonances or treated them inconsistently regarding power counting.

The paper aims to provide a systematic, complete, and minimal classification of tree-level UV completions for short-range $0\nu\beta\beta$ in the unbroken electroweak phase, identifying the minimal set of heavy degrees of freedom required to generate each operator.

2. Methodology: The J-basis Framework

The authors utilize the J-basis framework, a group-theoretic and on-shell amplitude-based approach, to organize the SMEFT operator space.

• Operator-Amplitude Correspondence: The method maps local SMEFT operators to on-shell scattering amplitudes. By analyzing the channels (partitions of external legs) in tree-level diagrams, the problem is reduced to diagonalizing Casimir operators.
• Channel Quantum Numbers: For each tree topology (Topology I: boson-fermion-boson; Topology II: three-boson), the authors define partitions of external fermions. They construct commuting sets of partial kinematic generators (Lorentz spin $W^2$) and gauge Casimirs ($SU(3)_C$, $SU(2)_L$, $U(1)_Y$).
• Diagonalization: Diagonalizing these operators yields a "dictionary" where the eigenvalues correspond directly to the quantum numbers (spin, color, weak isospin, hypercharge) of the intermediate UV mediators.
• Filtering Process: To ensure physical relevance to $0\nu\beta\beta$, three filters are applied to the algebraic output:
  1. Leading Power Counting: Only channels contributing at order $\Lambda^{-5}$ via renormalizable vertices are kept. This excludes massive vectors coupled via dimension-5 tensor currents (which would be $d \ge 10$).
  2. Weak Projection: Only UV completions that project onto the specific broken-phase component $u u \bar{d} \bar{d} e e$ are retained.
  3. First-Generation Projection: Constraints arising from the antisymmetry of identical first-generation fermions (Fermi statistics) are applied to eliminate vanishing amplitudes.

3. Key Contributions

A. Minimal UV Realizations

The paper introduces a rigorous definition of minimal UV realizations: the smallest set of genuinely distinct heavy multiplets required to generate an operator.

• SM-like Field Removal: Mediators with quantum numbers identical to SM fields (e.g., Higgs-like scalars) are removed if they can be generated via field redefinitions of lower-dimensional operators.
• Subset Elimination: If a set of mediators $\{A, B, C\}$ generates an operator, but a subset $\{A, B\}$ is sufficient, the larger set is deemed non-minimal and excluded.

B. Comprehensive Vector Sector

This work provides the first comprehensive compilation of UV completions involving massive vector resonances for short-range $0\nu\beta\beta$. Previous studies focused primarily on scalar and fermion mediators. The authors identify 324 minimal UV configurations featuring vector resonances.

C. Automated Tool

The authors provide a Mathematica function (GetUVsForType) built on the ABC4EFT package. This tool automates the J-basis construction, partitioning, and UV dictionary generation for arbitrary SMEFT operators (up to 8 fields), making the methodology reproducible and extensible.

4. Key Results

• Total Combinations: The analysis identifies 505 unique mediator combinations across the six independent SMEFT operator types relevant to short-range $0\nu\beta\beta$.
• Minimal Set: After applying minimality criteria (removing SM-like fields and non-minimal supersets), 440 minimal UV realizations remain.
  • 12 cases require only 2 distinct heavy species ($N_{med}=2$).
  • 428 cases require 3 distinct heavy species ($N_{med}=3$).
  • No cases require only 1 heavy species for a genuine six-fermion tree-level completion.
• Vector Resonances: A significant finding is that 324 of the 440 minimal realizations involve vector resonances. This suggests that models incorporating heavy vectors (e.g., Left-Right symmetric models, vector leptoquarks) constitute a major class of theories capable of generating short-range $0\nu\beta\beta$.
• Comparison with Broken-Phase: The results reproduce the scalar and scalar-fermion-scalar results of Ref. [1] but extend the classification by:
  • Identifying 4 additional scalar-fermion-scalar completions missed in the broken-phase analysis.
  • Systematically cataloging the vector sector, which was previously unexplored in this context.

5. Significance and Impact

• Model Building Guide: The paper provides a "UV dictionary" that allows theorists to immediately identify which heavy particle spectra (multiplets and their quantum numbers) are necessary to explain a potential $0\nu\beta\beta$ signal, bypassing the need for manual diagram enumeration.
• Collider Phenomenology: By working in the unbroken phase and identifying specific $SU(2)_L$ and $SU(3)_C$ representations, the results directly inform searches at the LHC and future colliders. For instance, the identification of color-octet fermions or specific vector leptoquarks provides concrete targets for experimentalists.
• Theoretical Rigor: The use of the J-basis ensures that the classification is complete and free from redundancies caused by gauge symmetry breaking or field redefinitions. It establishes a clear distinction between "genuine" new physics and artifacts of the effective field theory expansion.
• Vector Dominance: The discovery that vector-mediated completions are numerically dominant (324 out of 440) shifts the focus of theoretical model building, suggesting that vector-heavy BSM scenarios are as, if not more, likely candidates for short-range $0\nu\beta\beta$ than previously thought.

In summary, this paper establishes a rigorous, automated, and complete framework for connecting short-range neutrinoless double beta decay to its ultraviolet origins, significantly expanding the known landscape of possible new physics models, particularly those involving heavy vector bosons.