Low-derivative operators of the Standard Model effective field theory via Hilbert series methods

Abstract

In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we conjecture an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the conjectured technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically recreate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Assuming our method is correct, we derive a set of 535 dimension-8 N f = 1 operators.

2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: higher dimension operators in the SM EFT

“…Removing terms proportional to the EOM in this tower means eliminating terms proportional to D 2 φ for scalars, / Dψ for fermions, or D µ X µν L/R for field strengths. This construction is also described by Lehman and Martin [7]. 1 When there is no gauge symmetry, Dµ = ∂µ and the derivatives are automatically symmetric.…”

“…We demonstrate counting up to dimension 15, allowing for an arbitrary number of fermion generations N f , and provide explicit results for the content of the independent operators up to dimension 12. Previous results have appeared in the literature at dimension 5 [2], 6 [3][4][5], 7 [6] and very recently 8 [7]. This particular counting problem has a history of being tricky, due to the somewhat complicated nature of the particle content of the SM, and the intricacies of equations of motion (EOM) and integration by parts (IBP), which yield relations between operators.…”

Multiboson interactions at the LHC

“…The ∆ i = 6 are the most important operators when the EFT is valid unless there is a systematic power counting due to a particular UV interpretation that would suppress the dimensionless Wilson coefficients (Arzt et al, 1995;Liu et al, 2016). At ∆ i = 6 there are already 59 operators in the SM (Buchmuller and Wyler, 1986;Grzadkowski et al, 2010), while for ∆ i = 8 an exhaustive list of 535 operators was finally classified in (Lehman and Martin, 2016). While there are slight differences in number of operators at a fixed dimension in the literature depending on what assumptions are chosen, the operator basis has now been extended through ∆ i = 12 in the SM using more sophisticated mathematical techniques (Henning et al, 2015).…”

The Z decay width in the SMEFT: y t and λ corrections at one loop

“…The number of non redundant operators in L (5) , L (6) , L (7) and L (8) is known [1][2][3][4][5][6][7] and a general algorithm to determine operator bases at higher orders has been established in refs. [6][7][8][9], making the SMEFT, in principle, defined to all orders in the expansion in local operators.…”

“…[6][7][8][9], making the SMEFT, in principle, defined to all orders in the expansion in local operators. In this work, we use a naive power counting in mass dimension so that the operators Q (k) i will be suppressed by k − 4 powers of the cutoff scale Λ, where the C (k) i are the Wilson coefficients.…”