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  • Open Access

Identifying CP basis invariants in the SMEFT

Neda Darvishi1,*, Yining Wang2,3,†, and Jiang-Hao Yu2,3,4,5,6,‡

  • *Contact author: neda.darvishi@rhul.ac.uk
  • †Contact author: wangyining@itp.ac.cn
  • ‡Contact author: jhyu@itp.ac.cn
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Phys. Rev. D 112, 095009 – Published 7 November, 2025

DOI: https://doi.org/10.1103/c5ff-5h72

Abstract

Building on our automated framework that uses ring diagrams for classifying CP basis invariants Neda Darvishi et al., [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 (νSMEFT) 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.

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Phys. Rev. D 112, 095009– Published 7 November, 2025
  • Received 15 April 2024
  • Accepted 21 August 2025
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