The Analytic Wavefunction

TL;DR

This paper investigates the analytic structure of wavefunction coefficients in Minkowski spacetime, revealing singularities and deriving new sum rules that connect low-energy coefficients to UV data, with implications for positivity bounds.

Contribution

It introduces an off-shell wavefunction framework, analyzes its singularities, and derives novel UV/IR sum rules applicable to both Lorentz invariant and boost-breaking theories.

Findings

Wavefunction coefficients are analytic except for negative real axis singularities.

Derived UV/IR sum rules relate low-energy coefficients to UV integral discontinuities.

Verified sum rules at one-loop order in simple scalar models.

Abstract

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes.