Complete Set of Dimension-9 Operators in the Standard Model Effective Field Theory

TL;DR

This paper systematically constructs a complete set of dimension-9 operators in the Standard Model effective field theory, revealing new baryon and lepton number violation patterns at higher dimensions.

Contribution

It introduces an automated algorithm to generate an independent, complete basis of dimension-9 operators with explicit flavor symmetries, improving over traditional methods.

Findings

Generated 90,456 operators for three generations, including baryon and lepton number violating ones.

Discovered new violation patterns such as ΔB=2 and ΔL=3 at dimension 9.

Provided a simplified, flavor-symmetry-aware operator basis for future theoretical and phenomenological studies.

Abstract

We present a complete and independent list of the dimension 9 operator basis in the Standard Model effective field theory by an automatic algorithm based on the amplitude-operator correspondence. A complete basis (y-basis) is first constructed by enumerating Young tableau of an auxiliary $SU(N)$ group and the gauge groups, with the equation-of-motion and integration-by-part redundancies all removed. In the presence of repeated fields, another basis (p-basis) with explicit flavor symmetries among them is derived from the y-basis, which further induces a basis of independent monomial operators through a systematic process called de-symmetrization. Our form of operators have advantages over the traditional way of presenting operators constrained by flavor relations, in the simplicity of both eliminating flavor redundancies and identifying independent flavor-specified operators. We list the 90456 (560) operators for three (one) generations of fermions, all of which violate baryon number or lepton number conservation; among them we find new violation patterns as $\Delta B = 2$ and $\Delta L = 3$, which only appear at the dimensions $d \ge 9$.