T-2, Nuclear and Particle Physics, Astrophysics and Cosmology

Operator content of free, conformal field theories in low dimensions

Brian Henning
Yale University

Two of the most pressing phenomenological questions of our era ask can we get a quantitative handle on (1) QCD and (2) multi-particle scattering amplitudes relevant for background and new physics searches at the LHC and future colliders? In this talk, I will suggest that an improved understanding of spacetime kinematics could be of relevance for both of these pursuits. In particular, I look at how spacetime symmetry—either Poincaré or its enhancement to conformal symmetry—organize the Hilbert space of free field theories. Working in momentum space with spinors in d=3 and 4 dimensions, we explicitly decompose the Hilbert space into conformal representations, essentially computing the operator product expansion. For N-particle states, there is an action of U(N) symmetry group in d=4, with a remarkable pairing with the conformal group: the two groups dually control the decomposition. A geometric picture emerges, where observables—in the form of conformal primaries—are identified with the coset space U(N)/U(N-2). A similar story holds in d=3, replacing U(N) with O(N). These states have practical use to the problems raised earlier. They provide necessary input ingredients for a recently revived technique called Hamiltonian truncation, which may offer a numerical window into strongly coupled systems. Looking towards colliders, scattering states are identified with scalar primaries, and correspond to a non-redundant operator basis for effective field theories. We point out applications to the Standard Model effective field theory as well as the chiral Lagrangian.


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