Principle of minimal singularity for Green's functions

Analytic continuations of integer-valued parameters can lead to profound insights, such as angular momentum in Regge theory, the number of replicas in spin glasses, the number of internal degrees of freedom, the spacetime dimension in dimensional regularization and Wilson’s renormalization group. In this work, we consider a new kind of analytic continuation of correlation functions, inspired by two recent approaches to underdetermined Dyson-Schwinger equations in $D$-dimensional spacetime. If the Green’s functions $G_n=⟨{\phi }^n⟩$ admit analytic continuation to complex values of $n$, the two different approaches are unified by a novel principle for self-consistent problems: Singularities in the complex plane should be minimal. This principle manifests as the merging of different branches of Green’s functions in the quartic theories. For $D=0$, we obtain the closed-form solutions of the general $g{\phi }^m$ theories, including the cases with complex coupling constant $g$ or non-integer power $m$. For $D=1$, we derive rapidly convergent results for the Hermitian quartic and non-Hermitian cubic theories by minimizing the complexity of the singularity at $n=\mathrm{\infty }$.