⚛️ phenomenology

Complete Operator Basis for the modular invariant SMEFT

This paper systematically constructs a complete and finite basis of modular-invariant operators within the Standard Model Effective Field Theory (SMEFT) framework by implementing distinct $A_4$ flavor symmetries for quarks and leptons, utilizing Hilbert-series techniques to enumerate independent operators up to dimension 7 under holomorphic assumptions, and extending the formalism to non-holomorphic cases to avoid infinite proliferation while preserving key physical structures like the Weinberg operator.

Original authors: Luo-Jia Kang, Hao Sun, Jiang-Hao Yu

Published 2026-02-02

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Luo-Jia Kang, Hao Sun, Jiang-Hao Yu

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

The Big Picture: Building a Lego City with Rules

Imagine the Standard Model of particle physics as a massive, complex Lego city. We know how the basic bricks (electrons, quarks, photons) fit together to build the world we see. However, physicists suspect there are hidden, heavier bricks (new physics) that are too big to fit into our current models directly.

To study these hidden bricks without seeing them, scientists use a "rulebook" called SMEFT (Standard Model Effective Field Theory). This rulebook lists every possible way the existing Lego bricks could be rearranged to create new, slightly weird structures. The problem is that there are too many possible rearrangements. It's like trying to list every possible shape you could build with a million Lego bricks—it's an impossible task without a strict system.

This paper introduces a new, stricter set of rules to organize this list. The authors use a mathematical concept called Modular Symmetry to act as a "traffic cop," filtering out impossible combinations and giving us a clean, finite list of what is actually allowed.

The Core Concepts

1. The "Flavor" Problem: Why are there three families?

In our Lego city, there are three copies of almost every character: the electron, the muon, and the tau (like three siblings who look similar but have different weights). We don't know why there are three, or why they have the specific masses they do. This is the "Flavor Problem."

Usually, physicists try to solve this by inventing invisible "ghost" fields (called flavons) that arrange the masses. But this paper suggests a different approach: Modular Symmetry.

2. The Modular Symmetry: The Shape-Shifting Torus

Imagine the universe isn't just a flat sheet, but a donut (a torus) that can stretch and twist. The shape of this donut is defined by a single number, called $\tau$ (tau).

The Analogy: Think of $\tau$ as the "knob" on a radio. Turning the knob changes the station (the physics).
The Rule: The paper assumes that the "rules" of how particles interact (their masses and mixing) aren't random numbers we just plug in. Instead, they are determined by the shape of this donut. If you twist the donut in a specific way, the laws of physics must stay the same. This is Modular Symmetry.

3. The "Spurion" Trick: The Invisible Puppeteer

In the Standard Model, the masses of particles are determined by "Yukawa couplings." In this paper, the authors treat these couplings not as random numbers, but as modular forms.

The Analogy: Imagine you are building a Lego castle. Usually, you just grab bricks. Here, the authors say, "No, you can only grab bricks that match the color of the sky right now."
The "color of the sky" is the value of $\tau$ . The "bricks" are the modular forms.
By treating $\tau$ as a background setting (a "spurion") that doesn't move around, they can systematically build every possible interaction that respects the shape of the donut.

What Did They Actually Do?

The authors used a powerful mathematical tool called the Hilbert Series.

The Analogy: Imagine you want to know how many different ways you can stack 10 Lego bricks. You could try to build them all one by one (and get tired). Or, you could use a computer algorithm that counts the possibilities instantly based on the rules of the bricks.
The Hilbert Series is that algorithm. It counts how many unique, non-redundant structures (operators) exist at different "heights" (dimensions).

The Two Scenarios They Explored

Scenario A: The "Holomorphic" Case (The Perfectly Smooth Donut)

The Assumption: They assumed the modular forms are "holomorphic," meaning they are mathematically smooth and predictable, like a perfect circle.
The Result: They found that if you follow these smooth rules, you can generate all the complex interactions using just one basic building block (a weight-2 triplet of modular forms).
The Output: They produced a complete, finite list of allowed structures:
- Dimension 5: 6 unique structures (including the famous "Weinberg operator" which explains why neutrinos have mass).
- Dimension 6: 2,961 unique structures.
- Dimension 7: 360 unique structures.
Why it matters: Previous attempts to list these were messy and contained duplicates. This paper provides a "clean" list where every item is unique and necessary.

Scenario B: The "Non-Holomorphic" Case (The Rougher Donut)

The Problem: Real-world physics isn't always perfectly smooth. Sometimes you need "rougher" math (non-holomorphic forms) to describe reality.
The Danger: If you allow these rough forms, the number of possible structures explodes to infinity. It's like saying you can build with any shape of clay, not just Lego bricks. The list becomes unmanageable.
The Solution: The authors imposed a "minimal assumption." They said, "Let's pretend the rough forms still follow the same basic pattern as the smooth ones, just with a conjugate partner."
The Result: This restriction keeps the list finite again. They successfully reconstructed the standard "Weinberg operator" and the dimension-6 operators using this new, stricter rule for the rough forms.

The "A4" Flavor Group

The paper focuses on a specific mathematical symmetry group called $A_4$ .

The Analogy: Think of a tetrahedron (a pyramid with a triangular base). It has 12 ways you can rotate it so it looks the same.
The authors assign the three generations of particles (like the three electron siblings) to the corners of this pyramid.
By using the $A_4$ symmetry, they ensure that the interactions between these particles respect the geometry of the pyramid. This naturally explains why we see the specific mixing patterns (like the PMNS matrix for neutrinos) that we observe in experiments.

Summary of Claims

Systematic Counting: They used the Hilbert Series to count every possible interaction in the Standard Model that respects Modular Symmetry, removing all duplicates and redundancies.
Finite Basis: They proved that even with complex modular forms, you can organize the interactions into a finite, complete list (a "basis") if you treat the modular forms as the primary building blocks.
Two Approaches:
- In the holomorphic (smooth) case, they found 2,961 dimension-6 operators.
- In the non-holomorphic (rougher) case, they showed that without extra rules, the list is infinite, but with their "minimal assumption," it becomes finite again.
No New Physics Needed: They did not invent new particles. They simply reorganized the existing Standard Model particles using these new symmetry rules to see what is mathematically allowed.

What This Paper Does Not Claim

It does not claim to have discovered a new particle.
It does not claim to solve the "Flavor Problem" definitively (it provides a framework, not a final answer to why masses are what they are).
It does not predict specific experimental outcomes for the LHC right now; it provides the theoretical toolkit (the list of allowed operators) that experimentalists can use to look for deviations.

In short, this paper is a cataloging project. It takes a chaotic library of possible particle interactions and uses the rules of Modular Symmetry to organize them into a neat, finite, and complete index.

Technical Summary: Complete Operator Basis for the Modular Invariant SMEFT

Problem Statement
The Standard Model (SM) successfully describes particle physics but leaves the origin of flavor hierarchies and mixing patterns unexplained. While Minimal Flavor Violation (MFV) addresses this by treating Yukawa couplings as spurions, it does not predict the specific structure of these couplings. Discrete flavor symmetries (e.g., $A_4$ ) offer a bottom-up approach but often require multiple flavon fields and specific vacuum alignments, leading to a large parameter space and reduced predictivity. Modular flavor symmetry provides an alternative where Yukawa couplings arise from holomorphic modular forms of a modulus $\tau$ , eliminating the need for flavons. However, extending this framework to the Standard Model Effective Field Theory (SMEFT) presents challenges:

Redundancy: Naive construction of higher-dimensional operators often leads to linearly dependent structures due to algebraic relations among modular forms.
Holomorphy vs. Non-Holomorphy: While the superpotential in supersymmetric (SUSY) theories is holomorphic, the general SM Lagrangian (and its effective operators) is not. Non-holomorphic modular forms (e.g., polyharmonic Maaß forms) are not closed under multiplication, potentially leading to an infinite proliferation of independent operators if not constrained.
Completeness: A systematic, finite, and complete basis of modular-invariant SMEFT operators up to dimension 7 has not been fully established using modern counting techniques.

Methodology
The authors employ the Hilbert series technique, a powerful algebraic tool for counting independent operators invariant under specific symmetries, while accounting for redundancies from integration by parts (IBP) and equations of motion (EOM).

Model Setup:
- The flavor group is taken as $A_4^{(q)} \times A_4^{(e)}$ , with distinct moduli $\tau_q$ and $\tau_e$ for the quark and lepton sectors.
- Left-handed fermions are assigned to $A_4$ triplets, while right-handed fermions are assigned to inequivalent singlets ( $1, 1', 1''$ ).
- The Higgs doublet carries modular weight 0.
- MFV-like Assumption: Flavor breaking is controlled solely by the vacuum expectation values (VEVs) of the moduli, $\langle \tau \rangle$ . Modular forms act as the fundamental spurions.
- Holomorphic Case: The authors assume the leading-order Yukawa sector is holomorphic. Higher-weight modular forms are generated via symmetric tensor products of the fundamental weight-2 triplet $Y_3^{(2)}$ .
- Non-Holomorphic Case: The authors consider polyharmonic Maaß forms. To avoid an infinite operator basis, they impose a minimal organizing principle: higher-dimensional operators are constructed as $[Y_r^{(k_Y)} Y_{r'}^{(k'_Y)*} \mathcal{O}]_1$ , where the modular weight compensation is supplied entirely by independent modular forms, rather than arbitrary powers of $\text{Im}\tau$ .
Computational Framework:
- The Hilbert series is computed for the continuous group (Lorentz and gauge symmetries) and the discrete group ( $A_4$ ).
- Two equivalent approaches are used to verify the basis:
  1. Direct counting using the Hilbert series with modular forms treated as spurions.
  2. Explicit construction via tensor products of modular forms and field operators, projecting onto the trivial singlet.
- Constraints specific to $A_4$ modular forms (e.g., $Y_2^2 + 2Y_1Y_3 = 0$ for weight 2) are explicitly incorporated to remove redundant structures.

Key Contributions and Results

Holomorphic $A_4$ Modular Invariant SMEFT:
- Dimension-5: The authors identify 6 independent operators (3 complex structures and their conjugates) for the Weinberg operator. They demonstrate that previous constructions (e.g., Ref. [89]) contained redundancies due to linear dependencies among tensor products of modular forms. The Hilbert series correctly isolates the 3 independent singlet structures: $Y_3^{(4)}[(LH)(LH)]_3$ , $Y_1^{(4)}[(LH)(LH)]_1$ , and $Y_{1'}^{(4)}[(LH)(LH)]_{1''}$ .
- Dimension-6: A complete enumeration of 2,961 independent operators is provided. This includes:
  - 15 bosonic operators.
  - 24 $\psi^2 H^3$ operators.
  - 66 $\psi^2 X H$ operators.
  - 67 $\psi^2 H^2 D$ operators.
  - 2,789 four-fermion operators ( $\psi^4$ ).
- The authors explicitly construct all baryon- and lepton-number conserving dimension-6 operators, detailing the flavor contractions for bilinear and quadrilinear structures.
- Dimension-7: The Hilbert series yields 360 independent operators at dimension 7.
Non-Holomorphic Extension (Polyharmonic Maaß Forms):
- The paper addresses the case where modular forms are non-holomorphic. Unlike the holomorphic case, products of non-holomorphic forms do not generally satisfy the Laplacian condition, meaning they cannot be generated from a single seed form.
- To retain a finite basis, the authors adopt the ansatz $[Y_r^{(k_Y)} Y_{r'}^{(k'_Y)*} \mathcal{O}]_1$ .
- Using the benchmark lepton model from Ref. [90], they reconstruct the dimension-5 Weinberg operator and dimension-6 lepton operators. They show that under this minimal assumption, the structure reproduces the benchmark results while avoiding the infinite tower of operators that would arise from treating all non-holomorphic combinations as independent.
Comparison with Existing Literature:
- The work refines the operator counting in Ref. [89]. The authors show that the "stringy factorization" ansatz used in [89] leads to overcounting because it fails to account for the algebraic relations (kernel of the multiplication map) inherent in the ring of modular forms. The Hilbert series method systematically removes these redundancies.

Significance and Claims
The paper claims to provide the first complete and independent operator basis for SMEFT under modular $A_4$ flavor symmetry, utilizing the Hilbert series to rigorously handle flavor redundancies.

Systematic Classification: It establishes that higher-dimensional operators can be formally organized as singlets of the form $[Y_r^{(k_Y)}, Y_{r'}^{(k'_Y)*}, \mathcal{O}]_1$ .
Finite Basis: By treating modular forms as the sole sources of flavor breaking (an MFV-like approach) and neglecting explicit $\text{Im}\tau$ dependence in higher-dimensional operators, the authors ensure the operator basis remains finite and manageable, even in the non-holomorphic case.
Predictivity vs. Complexity: The authors modestly note that while the modular constraints reduce the number of operators compared to a general flavor-violating SMEFT, the reduction is not as dramatic as in standard MFV (where the count drops from ~3000 to ~900). In their setup, the count is ~2,961, suggesting that while modular symmetry provides a structured origin for flavor, it does not yet drastically reduce the parameter space without additional constraints.
UV Connection: The holomorphic construction is motivated by a potential connection to a modular-invariant SUSY UV completion, where the effective operators descend from F-term interactions. The non-holomorphic extension offers a framework for scenarios where D-term contributions or non-SUSY completions are relevant.

In summary, the work provides a rigorous mathematical foundation for constructing modular-invariant effective field theories, correcting previous overcounting issues and offering a complete catalog of operators up to dimension 7 for phenomenological studies.