Dimensional Expansion for the Ising Limit of Quantum Field Theory

TL;DR

This paper applies the dimensional expansion technique to the Ising limit of a scalar quantum field theory, computing Green's functions and analyzing convergence, revealing insights into the behavior of these functions across dimensions.

Contribution

It introduces the application of dimensional expansion to the Ising limit, providing explicit calculations and convergence analysis for Green's functions at various dimensions.

Findings

Dimensional expansion predicts Green's functions with less than 5% error at D=1.

The radius of convergence for the expansion is approximately 2n/(n-1).

Green's functions increase monotonically with dimension D and diverge at the critical dimension.

Abstract

A recently-proposed technique, called the dimensional expansion, uses the space-time dimension $D$ as an expansion parameter to extract nonperturbative results in quantum field theory. Here we apply dimensional-expansion methods to examine the Ising limit of a self-interacting scalar field theory. We compute the first few coefficients in the dimensional expansion for $\gamma_{2n}$, the renormalized $2n$-point Green's function at zero momentum, for $n\!=\!2$, 3, 4, and 5. Because the exact results for $\gamma_{2n}$ are known at $D\!=\!1$ we can compare the predictions of the dimensional expansion at this value of $D$. We find typical errors of less than $5\%$. The radius of convergence of the dimensional expansion for $\gamma_{2n}$ appears to be ${{2n}\over {n-1}}$. As a function of the space-time dimension $D$, $\gamma_{2n}$ appears to rise monotonically with increasing $D$ and we conjecture that it becomes infinite at $D\!=\!{{2n}\over {n-1}}$. We presume that for values of $D$ greater than this critical value, $\gamma_{2n}$ vanishes identically because the corresponding $\phi^{2n}$ scalar quantum field theory is free for $D\!>\!{{2n}\over{n-1}}$.