Dimensional Expansion for the Ising Limit of Quantum Field Theory

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 $\frac{2n}{n-1}$. 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 $D=\frac{2n}{n-1}$. We presume that for values of $D$ greater than this critical value, ${\gamma }_{2n}$ vanishes identically because the corresponding ${\varphi }^{2n}$ scalar quantum field theory is free for $D>\frac{2n}{n-1}$.

8 pages, 2 figures, plain TEX

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