Operators For Generic Effective Field Theory at any Dimension: On-shell Amplitude Basis Construction

Abstract

We describe a general procedure to construct the independent and complete operator bases for generic Lorentz invariant effective field theories, given any kind of gauge symmetry and field content, up to any mass dimension. By considering the operator as contact on-shell amplitude, the so-called amplitude operator correspondence, we provide a unified construction of the Lorentz and gauge and flavor structures by Young Tableau tensor. Several bases are constructed to emphasize different aspects: independence (y-basis and m-basis), repeated fields with flavors (p-basis), and conserved quantum numbers (j-basis). We also provide new algorithms for finding the m-basis by defining inner products for group factors and the p-basis by constructing the matrix representations of the Young symmetrizers from group generators. The on-shell amplitude basis gives us a systematic way to convert any operator into such basis, so that the conversions between any other operator bases can be easily done by linear algebra. All of these are implemented in a Mathematica package: ABC4EFT (Amplitude Basis Construction for Effective Field Theories).

Theoretical developments in the SMEFT at dimension-8 and beyond

“…A new development is the systematic approach of Refs. [13,14,16], which can generate bases of operators to any mass dimension. This new, systematic approach based on Young Tensors is reviewed in 3.2.…”

“…Ref. [28] constructed LEFT operators bases through dimension-9 using the Young Tensor approach of [13,14,16], and Ref. [29] also used this systematic approach to construct bases of operators through dimension-9 in the SMEFT extended with sterile neutrinos.…”

“…[13,32,16,14], a systematic approach based on the applications of Young tableaux of symmetric groups and gauge groups is demonstrated for obtaining the explicit forms of a complete operator basis including the detailed contraction patterns for gauge and Lorentz indices, while the subtlety of the repeated fields is carefully tackled. The full results therein for dimension 8 and 9 SMEFT are generated by an automated Mathematica package ABC4EFT [16], which could also generate higher dimensional SMEFT operators basis up to any mass dimension. Due to the advantage of the Young tensor approach, it is possible to systematically convert any operator into this basis via the ABC4EFT code.…”

“…With the operator-amplitude correspondence introduced in Ref. [13,32,16], one can easily relate the T a 1 ,...,a N and (˜ ) to the group factor and brackets of the local on-shell amplitudes generated by these operators in the spinor-helicity formalism. Therefore the powerful Young tableaux technique developed for the Lorentz and gauge structures of amplitudes can be used for constructing operator basis, which is the reason why we name it as the y-basis.…”

“…Recently the complete dimension-8 SMEFT basis was constructed via brute force in [12] and was systematically generated in [13] and the complete dimension-9 SMEFT basis was constructed as well [14,15]. More recently a general procedure, implemented in a Mathematica package ABC4EFT [16], has been proposed to construct the independent and complete SMEFT operator bases up to any mass dimension.…”