Loops and trees in generic EFTs

Nathaniel Craig,Ying-Ying Li,Dave Sutherland,Minyuan Jiang

doi:10.1007/jhep08(2020)086

Abstract

We consider aspects of tree and one-loop behavior in a generic 4d EFT of massless scalars, fermions, and vectors, with a particular eye to the high-energy limit of the Standard Model EFT at operator dimensions 6 and 8. First, we classify the possible Lorentz structures of operators and the subset of these that can arise at tree-level in a weakly coupled UV completion, extending the tree/loop classification through dimension 8 using functional methods. Second, we investigate how operators contribute to tree and one-loop helicity amplitudes, exploring the impact of non-renormalization theorems through dimension 8. We further observe that many dimension 6 contributions to helicity amplitudes, including rational parts, vanish exactly at one-loop level. This suggests the impact of helicity selection rules extends beyond one loop in non-supersymmetric EFTs.

Highlights

Considering the higher dimension tree- and loop-level contributions to helicity amplitudes results in powerful ‘non-interference’ [15] and ‘non-renormalization’ [16] theorems, with direct relevance to the structure of helicity amplitudes in the Standard Model in the high energy limit. The former illuminates LHC sensitivity to new physics in diboson channels, while the latter explains the surprising pattern of zeroes [17] appearing in the one-loop matrix of anomalous dimensions for dimension 6 operators in the Standard Model EFT [18,19,20,21,22,23], and more broadly illustrates the pattern of possible loop effects within the EFT itself
The utility of this classification relies on perturbativity of the UV completion and must be used with care [32,33,34], it can provide useful guidance in estimating the relative size of some physical effects and often has intriguing overlap with classification schemes based on non-interference and nonrenormalization theorems
We have presented a simple method for enumerating the classes of Lorentz structures of operators that arise in a generic massless EFT of scalars, fermions and vectors

Summary

Classification of higher dimensional operators

In this manuscript we consider a gauge theory of massless scalars (φ), fermions (ψ), and vectors (V ). As an aside, that the absence of derivatives in three field operators which contribute to the S-matrix can be understood more from the structure of the 3-point helicity amplitude: three particle special kinematics imply either all angle bracket or all square bracket products vanish, and the amplitude is proportional to either 12 a 23 b 31 c or [12]a[23]b[31]c, for some integer a, b, c It contains no momentum factors pi ∼ |i [i| that would arise from a partial derivative. We must check that performing this shift does not inadvertently generate new classes of tree-level dimension 8 operators (necessarily containing field strengths) via the last two terms. Nor the last two terms in (3.7) contain no field strengths, and do not generate any new classes of dimension 8 operator at tree-level. With the tree-level maps in hand, we turn to construct the one-loop amplitudes in the general (massless) EFT

One-loop helicity amplitudes

Di3 Di4

Absent rational terms at dimension 6

Phenomenological implications

Discussion

A Simplified models for one loop calculations