Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete Set of Dimension-8 Operators in the Standard Model Effective Field Theory

Hao-Lin Li, Zhe Ren, Jing Shu, Ming-Lei Xiao, Jiang-Hao Yu, Yu-Hui Zheng

TL;DR

This paper delivers a complete, non-redundant basis of dimension-8 operators in the Standard Model EFT by a group-theoretic construction that treats Lorentz structures as SU(N) states and resolves IBP redundancies via invariant subspaces. It introduces a systematic framework to handle flavor, permutation symmetry, and repeated fields through permutation representations and SSYTs, enabling explicit flavor-specified operators. The method combines Lorentz and gauge basis construction with inner-product decomposition in the symmetric group to produce a fully explicit, flavor-aware catalog of dim-8 SMEFT operators. The approach is automated, generalizable to higher dimensions, and provides a practical toolkit for phenomenology where dimension-8 contributions can dominate or be distinctive, such as neutral triple gauge couplings and CP-violating interactions.

Abstract

We present a complete list of the dimension 8 operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group, and identify the Lorentz structures as states in a $SU(N)$ group. In this way, redundancy from equations of motion is absent and that from integration-by-part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the $SU(N)$ group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can be easily applied to higher dimensional standard model effective field theory and other effective field theories, making these studies more approachable.

Complete Set of Dimension-8 Operators in the Standard Model Effective Field Theory

TL;DR

This paper delivers a complete, non-redundant basis of dimension-8 operators in the Standard Model EFT by a group-theoretic construction that treats Lorentz structures as SU(N) states and resolves IBP redundancies via invariant subspaces. It introduces a systematic framework to handle flavor, permutation symmetry, and repeated fields through permutation representations and SSYTs, enabling explicit flavor-specified operators. The method combines Lorentz and gauge basis construction with inner-product decomposition in the symmetric group to produce a fully explicit, flavor-aware catalog of dim-8 SMEFT operators. The approach is automated, generalizable to higher dimensions, and provides a practical toolkit for phenomenology where dimension-8 contributions can dominate or be distinctive, such as neutral triple gauge couplings and CP-violating interactions.

Abstract

We present a complete list of the dimension 8 operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group, and identify the Lorentz structures as states in a group. In this way, redundancy from equations of motion is absent and that from integration-by-part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can be easily applied to higher dimensional standard model effective field theory and other effective field theories, making these studies more approachable.

Paper Structure

This paper contains 33 sections, 211 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Standard Model Effective Field Theory
  3. SM Fields: Building Block, and Notation
  4. Invariants at Different Levels: Class, Type, Term, and Operator
  5. Operator Basis for Lorentz and Gauge Symmetries
  6. Motivation and Mathematical Preparation
  7. Why Permutation Symmetry
  8. Group algebra and left ideal
  9. Lorentz Basis: $SU(N) \times \bar{S}$ Irreps
  10. Lorentz Invariance: Enumerating the Classes
  11. Lorentz Structures as $SU(N)$ States
  12. Permutation: Counting and Listing the Lorentz Basis
  13. Gauge Basis: Littelwood-Richardson Rule
  14. Flavor Basis: Inner Product Decomposition
  15. Lists of Operator Basis
  16. ...and 18 more sections

Figures (1)

  • Figure 1: Flow chart for finding all the independent terms at a given dimension. The content above the first dash-dotted line is model-independent and can be applied to any EFT. The content below the second dash-dotted line is our main contribution to this work. We automatize the whole procedure in a Mathematica code.