Identifying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:math> basis invariants in the SMEFT

Building on our automated framework that uses ring diagrams for classifying <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>C</a:mi> <a:mi>P</a:mi> </a:math> basis invariants Neda Darvishi , [], 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- <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mn>2</c:mn> <c:mi>n</c:mi> </c:math> terms while maintaining a dimension-6 core, as well as in SMEFT with sterile neutrinos ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>ν</e:mi> <e:mi>SMEFT</e:mi> </e:math> ) 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>C</g:mi> <g:mi>P</g:mi> </g:math> -even and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>C</i:mi> <i:mi>P</i:mi> </i:math> -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.