A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory

Abstract

The goal of this article is to provide a practical method to calculate, in a scalar theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting beta reach a critical value beta_c (with up to 16 significant digits in our actual calculations). We show that the round-off errors on the magnetic susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the systematic errors (finite truncations and volume) can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when beta -> beta_c, this quantity tends to a fixed value which can be determined accurately when D=3 (hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.

Uses revtex with psfig, 31 pages including 15 figures

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Cited by in corpus (15)

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• Nonlinear Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model
• High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model
• Towards Functional Flows for Hierarchical Models
• Non-Universal Quantities from Dual RG Transformations
• Hyperscaling in the Broken Symmetry Phase of Dyson's Hierarchical Model
• Small Numerators Canceling Small Denominators: Is Dyson's Hierarchical Model Solvable?
• Non-Gaussian numerical errors versus mass hierarchy
• Universality in nontrivial continuum limits: a model calculation
• Small numerators canceling small denominators of the high-temperature scaling variables: a systematic explanation in arbitrary dimensions
• Complex singularities of the critical potential in the large-N limit
• Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff
• Universality, Scaling and Triviality in a Hierarchical Field Theory
• Testing perturbative results with non-perturbative methods for the Hierarchical model
• Symmetry Breaking, Duality and Fine-Tuning in Hierarchical Spin Models