Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

Bervillier, C.; Boisseau, B.; Giacomini, Héctor

doi:10.1016/j.nuclphysb.2007.07.005

Cited by 40 publications

(67 citation statements)

References 49 publications

(121 reference statements)

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“…We may thus give the best estimates ever obtained up to now (see [1] for the definitions of both the eigenvalue equation and the exponents, and compare the following values with the estimates obtained in [21]):…”

Section: =mentioning

confidence: 99%

“…Here φ is the (constant) scalar field. Hence the two boundaries associated to the ODEs under study are (see [1]):…”

Section: The Wegner-houghton Flow Equation In the Lpamentioning

confidence: 99%

“…An important difference with [1] however, is that it is easier to follow the zero of interest as M grows. The estimation we obtain this way for r * is excellent since we get 28 stabilized digits for M = 145:…”

Section: Fixed Point Equationmentioning

confidence: 99%

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

2008

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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 ∈ ]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.Key words: Exact renormalisation group, Derivative expansion, Critical exponents, Two-point boundary value problem PACS: 02.30. Hq, 02.30.Mv, 02.60.Lj, 05.10.Cc, 11.10.Gh, 64.60.Fr In a previous article [1] we presented two analytical approaches for studying the derivative expansion of the exact renormalization group equation (ERGE,

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Section: =mentioning

confidence: 99%

“…Here φ is the (constant) scalar field. Hence the two boundaries associated to the ODEs under study are (see [1]):…”

Section: The Wegner-houghton Flow Equation In the Lpamentioning

confidence: 99%

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

2008

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“…By expanding the beta functional of f (x) in powers of g 2 or x, we also modify the nonlinear structure of the flow equation, most importantly removing movable or fixed singularities. Indeed, these singularities of the flow equation are known to lead to rather selective criteria for the existence and physical eligibility of fixed-point solutions [38,39,43,[78][79][80][81][82][83][84]. Hence, the results obtained so far still have to pass the test of global existence and eligibility, as the expansions used so far might decisively change the behavior of f for large fields or close to the origin.…”

Section: E Global Analysis Of the Quasi-fixed-point Potentialsmentioning

confidence: 99%

Non-Abelian Higgs models: Paving the way for asymptotic freedom

Gies¹,

Zambelli²

2017

Phys. Rev. D

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Asymptotically free renormalization group trajectories can be constructed in nonabelian Higgs models with the aid of generalized boundary conditions imposed on the renormalized action. We detail this construction within the languages of simple low-order perturbation theory, effective field theory, as well as modern functional renormalization group equations. We construct a family of explicit scaling solutions using a controlled weak-coupling expansion in the ultraviolet, and obtain a standard Wilsonian RG relevance classification of perturbations about scaling solutions. We obtain global information about the quasi-fixed function for the scalar potential by means of analytic asymptotic expansions and numerical shooting methods. Further analytical evidence for such asymptotically free theories is provided in the large-N limit. We estimate the long-range properties of these theories, and identify initial/boundary conditions giving rise to a conventional Higgs phase.

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“…A powerful method used to consider the phase structure of a model, and consequently to study the appearance of SSB, is the functional renormalization group (FRG) method [15][16][17][18][19][20][21][22][23]. The O(N ) model has extensively studied using FRG approaches: as relevant for our purposes, we mention it was used to study as a function of dimension critical exponents of O(N ) models [10,24,25] and to investigate truncation effects and the regulator-dependence of the FRG equation [26][27][28][29][30][31][32][33][34][35][36][37], while a FRG study of the critical exponents of the Ising model for d < 2 was presented in [24]. The study of single-particle quantum mechanics can be seen as a "low-dimensional" statistical mechanics model: FRG studies addressed double well potential and quantum tunneling [38,39] and quartic anharmonic oscillators [40].…”

Section: Jhep05(2015)141mentioning

confidence: 99%

Truncation effects in the functional renormalization group study of spontaneous symmetry breaking

Defenu

Mati

Márián

et al. 2015

J. High Energ. Phys.

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Abstract:We study the occurrence of spontaneous symmetry breaking (SSB) for O(N ) models using functional renormalization group techniques. We show that even the local potential approximation (LPA) when treated exactly is sufficient to give qualitatively correct results for systems with continuous symmetry, in agreement with the Mermin-Wagner theorem and its extension to systems with fractional dimensions. For general N (including the Ising model N = 1) we study the solutions of the LPA equations for various truncations around the zero field using a finite number of terms (and different regulators), showing that SSB always occurs even where it should not. The SSB is signalled by Wilson-Fisher fixed points which for any truncation are shown to stay on the line defined by vanishing mass beta functions.

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Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

Cited by 40 publications

References 49 publications

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

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

Non-Abelian Higgs models: Paving the way for asymptotic freedom

Truncation effects in the functional renormalization group study of spontaneous symmetry breaking

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