Principle of minimal singularity for Green's functions

Abstract

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 $G_n=\langleφ^n\rangle$ 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 $gφ^m$ 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 $n=\infty$.

Figures (4)

• Figure 1: The purely imaginary solutions for $G_1$ in the $D=0$ non-Hermitian theories $\mathcal{L}=-(i^{m}/m)\, \phi^m$ from the general formulae \ref{['0D-general-sol']}, \ref{['periodicity-general']}, \ref{['non-degeneracy-general']}. Many of them are $\mathcal{PT}$ symmetric. The exact solutions at integer $m$ are denoted by black dots. The colored curves from \ref{['0D-general-m']} interpolate the integer-$m$ solutions with the same $k$, according to $\alpha=e^{2\pi i\frac{k}{m}}$.
• Figure 2: The solutions of $G_n$ for the $D=1$ quartic theory \ref{['1D-quartic']} from \ref{['1D-quartic-recursion']} with $\hbar=\lambda=g=1$. The subfigures are labelled by the input $(E, G_2)$. Around the exact values, the three branches of solutions merge into one decay curve at relatively small $n$. At the same precision, the merging of the excited-state results (orange) extends to larger $n$ than the ground-state case (blue).
• Figure 3: The absolute error in the ground-state energy $E_0$ of the $D=1$ quartic theory \ref{['1D-quartic']} from the matching conditions \ref{['matching']}, where $\lambda=g=1$. Note that $M$ is the matching order. The $1/n$ series of $n^{-2/3}F_n$ is truncated to order $n^{-N}$. The estimates converge rapidly to the exact value as $M, N$ increase.
• Figure 4: The even-$n$ Green's functions for the $g=1$ quartic matrix model with different $G_2$. As the input $G_2$ approaches the one-cut value $G_2^\text{(one-cut,+)} =0.51615...$ from above or below, the two branches of Green's functions merge into one smooth curve. We have divided $G_n$ by $2^{n/2}$. Here $G_{4p}$ and $G_{4p+2}$ with $p=0,1,2,\dots$ are denoted by the blue and orange dots. For $G_2=1$, the dots are joined so that the oscillatory behaviors are more clear.