On Truncations of the Exact Renormalization Group

Tim R. Morris

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On Truncations of the Exact Renormalization Group

Tim R. Morris

High Energy Physics - Theory · Submitted 30 May 1994

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim No completely reliable method exists to reject the many spurious solutions that are also found.

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We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices.	contribution 0.58	0.79	0.03	asserted 0.90
s1	In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian.	contribution 0.66	0.89	0.09	asserted 0.97
s2	Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point.	result 0.96	0.94	0.55	asserted 1.00
s3	No completely reliable method exists to reject the many spurious solutions that are also found.	limitation 0.90	0.58	0.32	asserted 1.00
s4	These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.	interpretation 0.92	0.88	0.11	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from proof or derivation 0.63.

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Original abstract

We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.

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Cited by 13 Pith papers

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How the literature uses it

26 arXiv papers cite this in 48 sentences

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Uses its method or tool 6 papers

Since we define the effective model with non-perturbative four-quark interactions at intermediate densities, the effective average action of the system will be investigated in the Functional Renormalization Group (FRG) framework [ref] , instead of a renormalization-group (RG) analysis of the couplings at a given finite order of the perturbative expansion.

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We want to search for fixed point solutions of equation ([ref]) and ([ref]), i.e. search for scale invariant solutions such that $∂_t _k = 0$, requiring them to be globally analytic [ref] .

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Here, we use direct numerical integration techniques to find the fixed point solution of fp, without primarily relying on a polynomial approximation [ref] .

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Whether our fixed points $A$ and $B$ have anything to do with [ref] is unclear.] One way to test this idea would be by extending the truncation of $u_t$ to higher orders and check stability of $A$ and $B$, taking carefully into account a number of subtleties associated with the polynomial truncation method [ref] .

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In single component scalar field theory in the LPA it has been shown that unphysical fixed point solutions persist to high degrees in a polynomial truncation of the potential [ref] , while the correct results are reproduced when a general potential is retained in the effective action.

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It satisfies an exact RG equation known as the Wetterich equation [ref] .

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