Taming Dyson-Schwinger equations with null states

Abstract

In quantum field theory, the Dyson-Schwinger equations are an infinite set of coupled equations relating $n$-point Green's functions in a self-consistent manner. They have found important applications in non-perturbative studies, ranging from quantum chromodynamics and hadron physics to strongly correlated electron systems. However, they are notoriously formidable to solve. One of the main problems is that a finite truncation of the infinite system is underdetermined. Recently, Bender et al. [Phys. Rev. Lett. 130, 101602 (2023)] proposed to make use of the large-$n$ asymptotic behaviors and successfully obtained accurate results in $D=0$ spacetime. At higher $D$, it seems more difficult to deduce the large-$n$ behaviors. In this paper, we propose another avenue in light of the null bootstrap. The underdetermined system is solved by imposing the null state condition. This approach can be extended to $D>0$ more readily. As concrete examples, we show that the cases of $D=0$ and $D=1$ indeed converge to the exact results for several Hermitian and non-Hermitian theories of the $g{\varphi }^n$ type, including the complex solutions.

arXiv commentv3: 5 pages, 2 figures, typos corrected, references added, Introduction extended

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• Imaginary-Temperature Zeros for Quantum Phase Transitions
• Analytic trajectory bootstrap for matrix models
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Bootstrapping periodic quantum systems
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations