Operator bases, $S$-matrices, and their partition functions

TL;DR

The paper develops a unified, algebraic framework to enumerate and construct the operator bases of relativistic EFTs using S-matrix kinematics, via dual position-space (conformal/cohomology) and momentum-space (kinematic-ring) pictures. It introduces the Hilbert series as a counting tool and derives a general matrix-integral formula, applicable in any dimension with arbitrary field content and linearly realized symmetries; it further provides practical algorithms to build operator bases, with explicit n = 5 scalar results. The work also extends to non-linear realizations (CCWZ) and analyzes parity, gauge, and topological terms, clarifying redundancies from EOM, IBP, and Gram determinants. By connecting conformal representation theory, differential forms, and commutative algebra (notably Cohen-Macaulay structures), the approach yields both counting formulas and constructive procedures for EFT operator content, with potential relevance to CFTs and AdS/CFT. Overall, the framework offers a systematic, dimension-agnostic methodology to understand and compute EFT operator spectra and their S-matrix implications, including spinning particles and non-linear symmetries.

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 naïve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.

Figures (3)

• Figure 1: Contact interaction contributions to the $n$-point scattering amplitude from Feynman diagrams associated with the operators $\mathcal{O}^{(n,k)}_\alpha$.
• Figure 2: Growth of the number of independent scalar operators with grading dimension ($[\phi]=1$ and $[\partial]=1$) in the EFT of a real scalar field, up to dimension 80. Red curves are when spacetime rank conditions (Gram conditions) are not enforced, which is equivalent to considering spacetime dimension $d\gg1$; green and blue curves include these rank conditions in $d=2$ dimensions, with and without parity ($O(2)$ and $SO(2)$) imposed on the operator basis, respectively. The solid curves are after imposing both EOM and IBP redundancy; the dashed curves are after only imposing EOM; the dotted curves are without EOM and IBP imposed.
• Figure 3: Relevant Dynkin diagrams: (a) The $D_r$ Dynkin diagram possesses a reflection symmetry about the horizontal axis. This symmetry corresponds to the outer automorphism of the $\mathfrak{so}_{2r}$ Lie algebra under the parity transformation. (b) The Dynkin diagram $B_{r-1}$ (corresponding to Lie algebra $\mathfrak{so}_{2r-1}$) obtained by folding the roots of $\mathfrak{so}_{2r}$ by the outer automorphism. (c) The Dynkin diagram of $C_{r-1}$ (corresponding to Lie algebra $\mathfrak{sp}_{2r-2}$) obtained by folding the co-roots of $\mathfrak{so}_{2r}$ by the outer automorphism.