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Identifying CP Basis Invariants in SMEFT

Neda Darvishi, Yining Wang, Jiang-Hao Yu

TL;DR

This work extends the automated ring-diagram framework for classifying CP basis invariants and integrates Cayley-Hamilton reduction to construct basic and joint invariants with explicit CP properties. It applies the method to SMEFT with a dim-6 core (including Yukawa and four-fermion operators) and to SMEFT with sterile neutrinos up to dim-7 and Type-I seesaw, comparing results with Hilbert-Poincare series to validate invariant counts and reveal explicit structures. The approach provides CP-odd and joint invariants at low orders and scales to high-rank tensors and complex flavor symmetries, offering a complementary and more constructive alternative to HS/PL. The technique improves understanding of CP violation and invariant syzygies, with potential extensions to multi-Higgs doublet EFTs and broader beyond-SM frameworks.

Abstract

Building on our automated framework that uses ring diagrams for classifying CP basis invariants [Phys. Rev. D 108, 115030 (2023)], this paper broadens the application of the methodology with more extensive examples and a wider scope of theoretical frameworks. Here, we showcase its versatility through detailed analyses of specific operators in the Standard Model effective field theory (SMEFT), such as a four-fermion operator at dimension-6 and a Yukawa operator extended up to dimension-2n terms while maintaining a dimension-6 core, as well as in SMEFT with sterile neutrinos up to dimension-7. By integrating the ring-diagram technique with the Cayley-Hamilton theorem, we have developed a system that not only simplifies the process of identifying basic and joint invariants but also enables the automatic differentiation between CP-even and CP-odd invariants from the lowest orders. Additionally, this work presents a comparison of our results with those derived using the traditional Hilbert-Poincaré series and its Plethystic logarithm. While these conventional approaches primarily yield the numerical count of invariants, our framework provides a complete structure of invariants, thereby surpassing the limitations of these traditional methods.

Identifying CP Basis Invariants in SMEFT

TL;DR

This work extends the automated ring-diagram framework for classifying CP basis invariants and integrates Cayley-Hamilton reduction to construct basic and joint invariants with explicit CP properties. It applies the method to SMEFT with a dim-6 core (including Yukawa and four-fermion operators) and to SMEFT with sterile neutrinos up to dim-7 and Type-I seesaw, comparing results with Hilbert-Poincare series to validate invariant counts and reveal explicit structures. The approach provides CP-odd and joint invariants at low orders and scales to high-rank tensors and complex flavor symmetries, offering a complementary and more constructive alternative to HS/PL. The technique improves understanding of CP violation and invariant syzygies, with potential extensions to multi-Higgs doublet EFTs and broader beyond-SM frameworks.

Abstract

Building on our automated framework that uses ring diagrams for classifying CP basis invariants [Phys. Rev. D 108, 115030 (2023)], this paper broadens the application of the methodology with more extensive examples and a wider scope of theoretical frameworks. Here, we showcase its versatility through detailed analyses of specific operators in the Standard Model effective field theory (SMEFT), such as a four-fermion operator at dimension-6 and a Yukawa operator extended up to dimension-2n terms while maintaining a dimension-6 core, as well as in SMEFT with sterile neutrinos up to dimension-7. By integrating the ring-diagram technique with the Cayley-Hamilton theorem, we have developed a system that not only simplifies the process of identifying basic and joint invariants but also enables the automatic differentiation between CP-even and CP-odd invariants from the lowest orders. Additionally, this work presents a comparison of our results with those derived using the traditional Hilbert-Poincaré series and its Plethystic logarithm. While these conventional approaches primarily yield the numerical count of invariants, our framework provides a complete structure of invariants, thereby surpassing the limitations of these traditional methods.
Paper Structure (2 sections, 35 equations, 5 figures, 1 table)

This paper contains 2 sections, 35 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Introduction of Ring-diagrams

Figures (5)

  • Figure 1: The steps for derivation of invariants are visualised. In the fifth block, the acronyms TP and CHT are used for tensor products and the Cayley-Hamilton Theorem.
  • Figure 1: Ring-diagram for the SM.
  • Figure 2: The steps for distinguishing invariants are visualised.
  • Figure 2: The basic structure of a Ring-diagram.
  • Figure 3: Ring-diagram for three rotations and four blocks A,B,C and D