An explicit construction of the dimension-9 operator basis in the standard model effective field theory

Abstract

We investigate systematically dimension-9 operators in the standard model effective field theory which contains only standard model fields and respects its gauge symmetry. With the help of the Hilbert series approach to classifying operators according to their lepton and baryon numbers and their field contents, we construct the basis of operators explicitly. We remove redundant operators by employing various kinematic and algebraic relations including integration by parts, equations of motion, Schouten identities, Dirac matrix and Fierz identities, and Bianchi identities. We confirm counting of independent operators by analyzing their flavor symmetry relations. All operators violate lepton or baryon number or both, and are thus non-Hermitian. Including Hermitian conjugated operators there are $$ {\left.384\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.10\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.4\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.236\right|}_{\Delta B=\pm 1}^{\Delta L=\mp 1} $$ 384 Δ B = 0 Δ L = ± 2 + 10 Δ B = ± 2 Δ L = 0 + 4 Δ B = ± 1 Δ L = ± 3 + 236 Δ B = ± 1 Δ L = ∓ 1 operators without referring to fermion generations, and $$ {\left.44874\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.2862\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.486\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.42234\right|}_{\Delta B=\mp 1}^{\Delta L=\pm 1} $$ 44874 Δ B = 0 Δ L = ± 2 + 2862 Δ B = ± 2 Δ L = 0 + 486 Δ B = ± 1 Δ L = ± 3 + 42234 Δ B = ∓ 1 Δ L = ± 1 operators when three generations of fermions are referred to, where ∆L, ∆B denote the net lepton and baryon numbers of the operators. Our result provides a starting point for consistent phenomenological studies associated with dimension-9 operators.

Top, Higgs, diboson and electroweak fit to the Standard Model effective field theory

“…This SM Effective Field Theory (SMEFT) is a powerful tool for analyzing the consistency of the SM and searching indirectly for possible BSM physics. 1 There have been intensive theoretical efforts in recent years to formulate consistently the SMEFT and to classify the sets of independent operators appearing at each dimension [73,74,[81][82][83][84][85][86][87][88][89][90]. The first non-redundant basis of dimension-6 operators, known as the Warsaw basis, was laid out in ref.…”

Two-loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes

“…There have been many studies of constructing operator basis in standard model EFT (see e.g. [77][78][79][80][81][82][83][84][85]) and consider their renormalization using both conventional and on-shell formalism (see e.g. [86][87][88][89][90][91][92][93][94]).…”

Effective operator bases for beyond Standard Model scenarios: an EFT compendium for discoveries

“…The second one, i.e., the bottom-up approach is concerned with the construction of complete and independent operator sets at various mass dimensions based on group-theoretic ideas [29][30][31][32][33]. Complete and independent sets of SMEFT operators have been constructed for mass dimensions 6 [34], 7 [35], 8 [36,37], and 9 [38,39]. Several ingeniously built modern tools such as GrIP [40], BasisGen [41], Sym2Int [42], ECO [43], and DEFT [44] have made the construction of higher dimensional operators straightforward and convenient.…”