Dimensional expansion for the Ising limit of quantum field theory

A recently proposed technique, called 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 of ${\gamma }_{2\mathit{n}}$, the renormalized 2n-point Green’s function at zero momentum, for n=2, 3, 4, and 5. Because the exact results for ${\gamma }_{2\mathit{n}}$ are known at D=1 we can compare the predictions of the dimensional expansion at this value of D. We find typical accuracies of less than 5%. The radius of convergence of the dimensional expansion for ${\gamma }_{2\mathit{n}}$ appears to be 2n/(n-1). As a function of the space-time dimension D, ${\gamma }_{2\mathit{n}}$ appears to rise monotonically with increasing D and we conjecture that it becomes infinite at D=2n/(n-1). We presume that for values of D greater than this critical value ${\gamma }_{2\mathit{n}}$ vanishes identically because the corresponding ${\mathrm{\phi }}^{2\mathit{n}}$ scalar quantum field theory is free for D>2n/(n-1).