Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

Abstract

We present a new efficient analytical approximation scheme to two-point boundary value problems of ordinary differential equations (ODEs) adapted to the study of the derivative expansion of the exact renormalization group equations. It is based on a compactification of the complex plane of the independent variable using a mapping of an angular sector onto a unit disc. We explicitly treat, for the scalar field, the local potential approximations of the Wegner-Houghton equation in the dimension $d=3$ and of the Wilson-Polchinski equation for some values of $d\in ]2,3]$. We then consider, for $d=3$, the coupled ODEs obtained by Morris at the second order of the derivative expansion. In both cases the fixed points and the eigenvalues attached to them are estimated. Comparisons of the results obtained are made with the shooting method and with the other analytical methods available. The best accuracy is reached with our new method which presents also the advantage of being very fast. Thus, it is well adapted to the study of more complicated systems of equations.

Final version to appear in Nucl. Phys. B. 1 reference added

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Cited by in corpus (26)

• Fundamentals of the Exact Renormalization Group
• Analytic continuation of Taylor series and the two-point boundary value problems of some nonlinear ordinary differential equations
• Global Wilson-Fisher fixed points
• Critical $O(N)$ models in the complex field plane
• Comparison of renormalization group schemes for sine-Gordon type models
• The non-perturbative renormalization group in the ordered phase
• Functional renormalization group with a compactly supported smooth regulator function
• Vanishing Beta Function curves from the Functional Renormalisation Group
• Spontaneous symmetry breaking and optimization of functional renormalization group
• Bosonization and Functional Renormalization Group Approach in the Framework of QED_2
• Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking
• Nonlinear trident in the high-energy limit: Nonlocality, Coulomb field and resummations
• Non-universality in the erosion of tilted landscapes
• Scaling solutions in the derivative expansion
• Revisiting the local potential approximation of the exact renormalization group equation
• Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation
• The Numerically Optimized Regulator and the Functional Renormalization Group
• Critical line of the ${\Phi }^4$ theory on a simple cubic lattice in the local potential approximation
• Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators
• New self-dual solutions of SU(2) Yang-Mills theory in Euclidean Schwarzschild space
• Accurate eigenvalues of bounded oscillators
• Rational approximation to Thomas-Fermi equations
• A basis of the gradient flow exact renormalization group for gauge theory
• Rational approximation to the Thomas--Fermi equation
• New Approach to Bounded Quantum--Mechanical Models
• Upper and lower bounds to the eigenvalues of an anharmonic oscillator