Eleventh-Order Calculation of Green's Functions in the Ising Limit for Arbitrary Space-Time Dimension $D$

Abstract

This paper extends an earlier high-temperature lattice calculation of the renormalized Green's functions of a $D$-dimensional Euclidean scalar quantum field theory in the Ising limit. The previous calculation included all graphs through sixth order. Here, we present the results of an eleventh-order calculation. The extrapolation to the continuum limit in the previous calculation was rather clumsy and did not appear to converge when $D>2$. Here, we present an improved extrapolation which gives uniformly good results for all real values of the dimension between $D=0$ and $D=4$. We find that the four-point Green's function has the value $0.620 \pm 0.007$ when $D=2$ and $0.98 \pm 0.01$ when $D=3$ and that the six-point Green's function has the value $0.96 \pm 0.03$ when $D=2$ and $1.2 \pm 0.2$ when $D=3$.