Taming Dyson-Schwinger equations with null states

• URL: http://arxiv.org/abs/2303.10978v3
• Date: Fri, 21 Jul 2023 23:33:33 GMT
• Title: Taming Dyson-Schwinger equations with null states
• Authors: Wenliang Li
• Abstract summary: In quantum field theory, the Dyson-Schwinger equations are an infinite set of equations relating $n$-point Green's functions in a self-consistent manner. One of the main problems is that a finite truncation of the infinite system is underdetermined. In this paper, we propose another avenue in light of the null bootstrap.
• Score: 0.913755431537592
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $g\phi^n$ type, including the complex solutions.

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