Taming Dyson-Schwinger Equations with Null States.

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 nonperturbative studies, ranging from quantum chromodynamics and hadron physics to strongly correlated electron systems. However, they are notoriously formidable to solve. One of the main obstacles is that a finite truncation of the infinite system is underdetermined. Recently, Bender et al. [Phys. Rev. Lett. 130, 101602 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.101602] 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 Letter, 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 gϕ^{n} type, including the complex solutions.