Abstract
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation).We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries.In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes.We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most na¨ıve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.Although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.
Highlights
The basic tenets of S-matrix theory and effective field theory are equivalent: the starting point is to take assumed particle content and parameterize all possible scattering experiments
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the effective field theory (EFT)
In the context of effective field theory (EFT), this parameterization is embodied in the operator basis K of the EFT, which is defined to be the set of all operators that lead to physically distinct phenomena
Summary
The basic tenets of S-matrix theory and effective field theory are equivalent: the starting point is to take assumed particle content and parameterize all possible scattering experiments. This is dictated by kinematics and symmetry principles. In the context of effective field theory (EFT), this parameterization is embodied in the operator basis K of the EFT, which is defined to be the set of all operators that lead to physically distinct phenomena. By considering the set K in its own right, we are aiming to get as much mileage from kinematics and selection rules as possible before addressing specific dynamics. At a more fundamental level, we are accounting for Poincare covariance of single particle states together with Poincare invariance of the S-matrix
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