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

TL;DR

This paper advances high-temperature lattice calculations of Green's functions in the Ising limit to the eleventh order, providing improved extrapolation methods and new results across various space-time dimensions.

Contribution

It extends previous sixth-order calculations to eleventh order and introduces a better extrapolation technique for all real dimensions between 0 and 4.

Findings

Green's functions calculated at D=2 and D=3 with improved accuracy

Four-point Green's function values: 0.620±0.007 at D=2, 0.98±0.01 at D=3

Six-point Green's function values: 0.96±0.03 at D=2, 1.2±0.2 at D=3

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 $D=3$.