Derivative expansion of the renormalization group in O(N) scalar field theory

Morris, Tim R.; Turner, Michael

doi:10.1016/s0550-3213(97)00640-8

Cited by 92 publications

(128 citation statements)

References 42 publications

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“…We think that the second possibility is the right explanation. This is because one has already observed in the results of [29] a bad convergence on ω 1 and also on η * as functions of the number N of components of the field in the O (N)-symmetry whereas another choice of cut-off function, as done in [30], seems to be better adapted (see [31] for example). It is very likely that the fundamental reason why the Morris equation is not efficient must be looked for in the constraint imposed to the O (∂ 2 )-equations with a view to satisfy explicitly the reparameterization invariance.…”

Section: Remarkmentioning

confidence: 99%

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: Remarkmentioning

confidence: 99%

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

2008

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“…[12]. The second example concerns the power-like regulator R power (q 2 ) = q 2 (k 2 /q 2 ) b for b = 2 in d = 3 dimensions [22]. It leads to ℓ power (ω) = 2π/ √ 2 + ω.…”

Section: From Erg To Ptrgmentioning

confidence: 99%

“…The case m = 3 2 corresponds to the sharp cut-off [1], m = 2 to the quartic regulator R = k 4 /q 2 [22], and m = 5 2 to the optimal regulator R opt [10]. For m → ∞ we can rely on Eq.…”

Section: Comparison Of Critical Exponentsmentioning

confidence: 99%

Predictive power of renormalisation group flows: a comparison

Litim

Pawlowski

2001

Physics Letters B

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We study a proper-time renormalisation group, which is based on an operator cut-off regularisation of the one-loop effective action. The predictive power of this approach is constrained because the flow is not an exact one. We compare it to the Exact Renormalisation Group, which is based on a momentum regulator in the Wilsonian sense. In contrast to the former, the latter provides an exact flow. To leading order in a derivative expansion, an explicit map from the exact to the proper-time renormalisation group is established. The opposite map does not exist in general. We discuss various implications of these findings, in particular in view of the predictive power of the proper-time renormalisation group. As an application, we compute critical exponents for O(N )-symmetric scalar theories at the Wilson-Fisher fixed point in 3d from both formalisms.

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“…Consequently, in the limit n → ∞, the integrand in (25) is suppressed the least at the minimum of y(1+r), whence (27). Let us normalise all regulators by the requirement r( 1 2 ) = 1, in order to compare their respective radii of convergence.…”

Section: Convergencementioning

confidence: 99%

“…We refer to regulators with C OPT = 1 as optimised. The extremisation of (27) is closely linked to a minimum sensitivity condition [16].…”

Section: Convergencementioning

confidence: 99%

Derivative expansion and renormalisation group flows

Litim

2001

J. High Energy Phys.

118

137

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We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better physical predictions. This is applied to O(N )-symmetric real scalar field theories in 3d, where critical exponents are computed for all N . In comparison to the sharp cut-off regulator, an optimised flow improves the leading order result up to 10%. An analogous reasoning is employed for a proper time renormalisation group. We compare our results with those obtained by other methods.

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Derivative expansion of the renormalization group in O(N) scalar field theory

Cited by 92 publications

References 42 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

Predictive power of renormalisation group flows: a comparison

Derivative expansion and renormalisation group flows

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