Complete set of dimension-nine operators in the standard model effective field theory

We present a complete and independent list of the dimension-nine 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 the 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 desymmetrization. Our form of operators has 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 = 9.