Optimisation of the exact renormalisation group

Daniel F. Litim

REVIEW

Optimisation of the exact renormalisation group

Daniel F. Litim

High Energy Physics - Theory · Submitted 25 May 2000 · 16 references

Pith's verdict: A short, checkable method paper: the optimisation criterion and its C_opt = 2 bound hold up, and the regulator families it selects became standard; the weak spot is the unproven saddle-point step that ties the criterion to amplitude expansions.

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arXiv IDhep-th/0005245 (version 1)

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CitationDaniel F. Litim. "Optimisation of the exact renormalisation group." arXiv:hep-th/0005245 (2000). https://pith.science/paper/TRSY7MZX

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Worth your time, and short enough to read in one sitting. The core is Eq. (4): maximise, over regulators, the minimum over momenta of the coarse-grained inverse propagator P^2(y). The bound C_opt = 2 falls out in two lines from the normalisation r(1) = 1, and — this is the part that makes it more than a definition — it is attained. He gives explicit optimal parameters for seven regulator families: r_mix at b = 2 for every a, r_power at b = 2, r_exp at b = 1/ln 2, with r_mod and r_mexp solved in closed form. The stationarity condition r'(1) = -2 is correct, and the sharp-cutoff result C = 1 is a direct computation. That last number is the paper's real payoff: it explains, rather than just observes, why amplitude expansions on sharp-cutoff flows converge badly. The paper is also honest about the convention — the 2 is a ceiling set by R_k(k^2) = k^2, and he says so rather than dressing it up as a derived constant. Read it that way and there is no circularity problem here, whatever the report's circularity axis suggests. On the soft spots: the identification of the radius of convergence with min_y P^2 in Eqs. (10)-(11) is a saddle-point argument justified only by 'the kernel K is well-behaved'. For his own kernel (9) it is fine — at the optimum r'(1) = -2, so K(1) is nonzero and Laplace applies — but the broader claim that the radius 'does not depend on the kernel K' is not established for general threshold functions of the form (7), where K could vanish at the minimum. That is a real gap in the physical interpretation, not in the criterion. Two smaller ones: b_opt = lim b_n is asserted with 'can be shown' and no demonstration, and the r_step value 1.65 is quoted without the minimisation. The scheme-dependence remark at the end leans on a reference in preparation. None of this is load-bearing. Who should read it: anyone choosing a regulator for a functional RG calculation, and anyone who has wondered why sharp-cutoff polynomial expansions behave so poorly. Send it out; the referees will ask for the saddle-point conditions and the b_n limit, and the paper will be better for it.

Referee report

Significance

If the criterion holds, it supplies a model-independent, essentially parameter-free prescription for choosing the regulator in truncated functional RG calculations, together with closed-form optimal parameters for the regulator families actually used in practice. That is a genuinely useful and reusable result: the criterion refers only to the propagator at vanishing field, so it applies irrespective of the theory, the dimension, or the order of the derivative expansion, and the tabulated values (b_exp = 1/ln 2, b_power = 2, b_mix = 2, and the closed forms for b_mod and b_mexp) are immediately checkable and reproducible. The sharp-cutoff observation, C = 1 versus C_opt = 2, is a clean and falsifiable by-product that connects a folklore numerical problem to a structural property of the regulator. The paper is short and its central inequality is a two-line consequence of the normalisation, which is a strength rather than a weakness: the claim is transparent enough to be verified by the reader.

Major objections (2)

[Amplitude expansions, Eqs. (8)-(11)] The identification C = min_{y>=0} P^2(y) in Eq. (11), which carries the entire physical reading of the criterion as an optimisation of amplitude expansions, is asserted rather than derived. The text argues that P^{-n} suppresses the integrand of Eq. (8) except near the minimum of P^2 and then concludes the result 'taking into account that the kernel K is well-behaved', but no condition on K is ever stated and the interchange of the n -> infinity limit with the integral in Eq. (10) is not justified. The subsequent sentence, 'The radius of convergence does not depend on the kernel K', is precisely the point that needs proof: a kernel vanishing at the minimum y_0 would move the effective saddle and change the limit. For the paper's own kernel of Eq. (9) the argument does go through, since at an optimal regulator r'(1) = -2 and hence K(1) != 0, so Laplace's method applies; the gap is in the generality claimed for threshold functions of the form (7). Please state sufficient conditions on K (non-vanishing at the minimum, integrability, appropriate decay at large y), verify them explicitly for the kernels of Eqs. (6) and (9), and either prove the kernel-independence or restrict the claim to kernels satisfying those conditions. This does not affect the criterion Eq. (4) or the bound C_opt = 2, but it does affect how strongly the physical interpretation can be stated.
[Abstract and Applications, discussion of the sharp cutoff] The abstract states that the sharp cutoff's exclusion from the optimal class 'explains the poor convergence of amplitude expansions based on it', and the Applications section repeats that C_sharp = 1 being half the optimal value 'explains why amplitude expansions of sharp cut-off flows have a notoriously bad convergence behaviour'. The quantity computed is the radius of convergence of the amplitude expansion, Eq. (10); the quantity at issue in the cited truncation studies is the rate at which a fixed-order truncation approaches the exact result. These are related but not the same, and the manuscript nowhere bridges them. A concrete demonstration would settle the point: for the N = 1 and N = 3 Wilson-Fisher fixed point in d = 3, compute a standard observable (for instance the critical exponent nu or the anomalous dimension) at two or three successive orders of the derivative expansion for r_exp at b = 1/ln 2 and for the sharp cutoff, and show that the sequence converges faster for the optimal regulator. Absent such a check, the sentence should be softened from 'explains' to 'is consistent with', and the distinction between radius of convergence and rate of convergence should be stated explicitly.

Minor objections (8)

[Applications, text below Fig. 2] The closed form quoted for the power-law regulator, C = b(1-b)^{1/b-1}, is not real for b > 1 and therefore cannot reproduce the stated maximum C = 2 at b_power = 2. Direct evaluation of P^2 = y + y^{1-b} gives min_y P^2 = b(b-1)^{1/b-1}, which does give C = 2 at b = 2 and is maximised there. Please check the printed expression.
[Optimisation, paragraph after Eq. (4)] The paper is commendably explicit that C_opt = 2 follows from combining r(1) = 1 with Eq. (4), but the abstract and introduction do not carry that caveat, and a casual reader may take 2 for a dynamical constant rather than a ceiling fixed by the normalisation convention R_k(k^2) = k^2. It would help to say once, in the introduction, that the absolute value 2 is convention-dependent and that the convention-independent content is the relative improvement, by a factor of two in C, over the sharp cutoff and other regulators with C = 1.
[Amplitude expansions, paragraph on soft/hard equipartition] The statement that the optimal parameter coincides with b_opt = lim_{n->infinity} b_n, where b_n is defined implicitly by a^>_n = a^<_n, is introduced with 'can be shown' and no argument is given. Since this is offered as an independent characterisation of the optimum, a short derivation or an explicit reference would be appropriate; otherwise it should be flagged as a conjecture.
[Table 2 and the r_step entry] For r_step the table quotes b_opt = 4.40 and C_opt = 1.65, and the text explains that the value is limited by two competing minima that become degenerate at b = b_opt. Please state the numerical accuracy of these two numbers and how they were obtained, and give the value of b at which the two minima are degenerate, since that is the quantity that actually defines the optimum in this case.
[Conclusion] The closing remark that the criterion is closely linked to a minimisation of the spurious scheme dependence is supported only by reference [16], which is listed as 'in preparation'. Either substantiate the link briefly here or soften the sentence, so that the paper's claims do not rest on unpublished work.
[Applications, extension to fermions] The fermionic optimal value C_{F,opt} = 4 is stated without derivation. A one-line argument, analogous to the bosonic case, showing that P^2_F = y[1+r_F(y)]^2 with the normalisation r_F(1) = 1 gives min P^2_F <= 4, would make the extension self-contained.
[Figs. 1 and 3] In Fig. 3 the curves are identified only by a clockwise ordering in the caption, which is hard to use in practice. Direct labels on the curves, or a legend, would make the figure considerably more readable; the same applies to the three class labels Ia, Ib, II in Fig. 1.
[Whole manuscript] A few typographical slips should be corrected: 'desribes' in the second paragraph of the section 'Flows, coarse-grainings and propagators', 'classifaction' in the Applications section, and the corrupted author address line on the title page.

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The paper

The paper claims

A single number, the smallest value of the coarse-grained inverse propagator, decides which smoothing of an exact renormalisation group flow gives the most convergent approximations, and its best possible value is 2.

The reading

Exact renormalisation group equations depend on a choice of how short-distance modes are smoothed away, and any practical solution is a truncation, so results drift when that choice changes. This paper proposes a one-line criterion for choosing the smoothing: look only at the coarse-grained propagator at vanishing field, written in dimensionless form as $P^2(y) = y[1+r(y)]$ with $y = q^2/k^2$, and pick the regulator that makes its minimum over momenta as large as possible. Because the normalisation forces $P^2(1) = 2$, the minimum can never exceed 2, and regulators that touch 2 are those whose minimum sits exactly at $y=1$ with $r'(1) = -2$. The same quantity is argued to control two physical things: the radius of convergence of expansions in field-dependent amplitudes, and the distance of the poles of threshold functions from the physical region. Optimal parameters are then written down for a list of standard regulators, and the sharp cutoff comes out at half the attainable radius, which is offered as the explanation for its known slow convergence.

What carries the argument

The load-bearing object is the dimensionless inverse coarse-grained propagator $P^2(y) = y[1 + r(y)]$, where $R_k(q^2) = q^2 r(q^2/k^2)$ and the scale is normalised by $r(1) = 1$. Every regulator must keep $P^2$ strictly positive and growing at large $y$, so $P^2$ has a minimum; the optimisation criterion is the requirement that this minimum be as large as possible, $C_{opt} = max_{RS} min_{y>=0} P^2(y)$. The bound $C_{opt} <= 2$ is immediate from $P^2(1) = 2$, and the extremality condition $dP^2/dy = 1 + r + y r' = 0$ at $y = 1$ gives $r'(1) = -2$, which is what fixes each regulator's optimal parameter. The same minimum is claimed to equal the radius of convergence of amplitude expansions, via a saddle-point reading of $lim_{n->infty} a_n/a_{n+2}$, and to locate the threshold-function pole at $omega_{pole} = -min_y P^2(y)$.

Extended reading notes

Core claim

The paper's claim is that the freedom to choose a regulator can be turned into a well-posed optimisation problem with a universal answer. Define $C = -k^2$ times the inverse coarse-grained propagator at vanishing field, minimised over momenta, and maximise it over regulators: $C_{opt} = 2$, attained by regulators whose $P^2$ bottoms out at $y=1$ with slope $r'(1) = -2$. Two readings are given for why this is the right target. In an expansion of threshold functions in powers of the amplitudes $a_n = -d/2$, the coefficients $a_n = -d/2$ fall off with a radius set by $C = -d/2$, so maximising the minimum of $P^2$ maximises the radius of convergence of amplitude expansions; equivalently, the pole of any threshold function sits at $a_n = -d/2$, and the criterion pushes that pole as far as possible into the unphysical region. Applied to standard regulators, this yields explicit optimal parameters, for example $b = 1/ln 2$ for the exponential regulator, $b = 2$ for the power-law regulator and for the mixed exponential family for every $a$, and numerical values near 3.92 and 1.92 for the mass-like and modified exponential regulators. The sharp cutoff reaches only $C = 1$, half the maximum, and the step regulator is capped at about 1.65 because its $P^2$ has two competing minima.

Load-bearing premise

The step that identifies the radius of convergence of amplitude expansions with the minimum of $P^2$ is a saddle-point argument that assumes the kernel is well-behaved enough for the large-$n$ limit to be taken inside the momentum integral, and no explicit decay or uniformity condition on that kernel is stated.

Editorial extensions

If this is right

Any exact-flow calculation truncated in field-dependent amplitudes has a preferred regulator, selected by $r'(1) = -2$, rather than an arbitrary one, so scheme-dependence error bars can be centred on a principled choice.
The sharp cutoff's radius of convergence is exactly half the attainable value, which supplies a quantitative reason for the slow convergence long observed in amplitude expansions built on it.
Because the criterion refers only to the propagator at vanishing field, it applies unchanged to $N$-component scalars, to fermions (with $P^2_F = y[1+r_F]^2$ and $C_{F,opt} = 4$), to theories with wave-function renormalisation, and to finite-temperature flows.
Optimal parameters are tabulated for exponential, algebraic, mass-like, mixed and step regulators, giving a concrete menu for numerical implementations.
The criterion is compatible with the modified Ward or BRST identities needed for gauge theories, so gauge-invariant flows can be optimised the same way.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the criterion is right, it predicts a monotone relationship that can be checked cheaply: for two regulators with different $C$, the truncation order needed to reach a fixed accuracy at a fixed order parameter should scale inversely with $C$, which is testable on existing flow codes without new formalism.
The two-minima competition that caps the step regulator at about 1.65 suggests a general diagnostic: any regulator whose $P^2$ has a second local minimum away from $y = 1$ will fall short of 2, so scanning a regulator family for extra minima is a quick way to rule out candidates before computing anything.
The argument that the optimal expansion point is always $omega = 0$, while the optimal field expansion point depends on $N$, hints that field-amplitude truncations could be reorganised around the amplitude variable itself rather than around $rho$, which would be a change of truncation scheme rather than of regulator.
The link drawn between the pole location and the smoothing of the transition to a convex effective potential suggests that numerical stability near the infrared, not just convergence order, should improve under the optimal choice, an effect the paper asserts but does not demonstrate numerically.

Original abstract

A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,023 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given.	contribution 1.00	0.91	0.54	asserted 1.00
s1	It is aimed at improving the convergence of approximate solutions of flow equations.	contribution 0.54	0.45	0.10	asserted 0.84
s2	The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field.	contribution 0.44	0.70	0.15	asserted 1.00
s3	In physical terms, it is understood as an optimisation condition for amplitude expansions.	interpretation 1.00	0.54	0.11	asserted 0.85
s4	Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region.	interpretation 1.00	0.59	0.10	asserted 0.58
s5	The link to expansions in field amplitudes is discussed as well.	interpretation 0.64	0.73	0.04	asserted 0.84
s6	Optimal parameters are given explicitly for a variety of different coarse-grainings.	result 0.87	0.93	0.15	asserted 1.00
s7	As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.	result 0.96	0.94	0.27	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.96 · evidence from proof or derivation 0.88.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The effective inverse propagator for the different regul

**Figure 2.** The dimensionless radii of convergence for amplitud

**Figure 3.** The optimised effective inverse propagators for differe

Figures reproduced from arXiv:hep-th/0005245 by the authors.

References

Works cited (16)

[1]

F.J. Wegner and A. Houghton, Phys. Rev. A8 (1973) 401; K.G. Wilson and I.G. Kogut, Phys. Rep 12 (1974) 75; J. Polchinski, Nucl. Phys. B231 (1984) 269

View on Pith 1973
[2]

C. Wetterich, Nucl. Phys. B352 (1991) 529; Phys. Lett.B301 (1993) 90

View on Pith 1991
[3]

See D.F. Litim and J.M. Pawlowski, On gauge invariant Wilsonian ﬂows , hep-th/9901063, and references therein

View on Pith Pith review arXiv
[4]

N. Tetradis and D.F. Litim, Nucl. Phys. B464 (1996) 492, [hep-th/9512073]

View on Pith Pith review arXiv 1996
[5]

R. D. Ball, P. E. Haagensen, J. I. Latorre and E. Moreno, Ph ys. Lett. B347 (1995) 80 [hep-th/9411122]

View on Pith Pith review arXiv 1995
[6]

D.F. Litim, Phys. Lett. B393 (1997) 103, [hep-ph/9609040]

View on Pith arXiv 1997
[7]

F. Freire and D. F. Litim, Charge cross-over at the U(1)-Higgs phase transition , hep- ph/0002153

View on Pith
[8]

T. R. Morris and J. F. Tighe, JHEP 9908 (1999) 007 [hep-th/9906166]

View on Pith Pith review arXiv 1999
[9]

S. Liao, J. Polonyi and M. Strickland, Nucl. Phys. B567 (2000) 493 [hep-th/9905206]

View on Pith Pith review arXiv 2000
[10]

N. Tetradis and C. Wetterich, Nucl. Phys. B422 (1994) 541 [hep-ph/9308214]

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[11]

T. R. Morris, Phys. Lett. B334 (1994) 355 [hep-th/9405190]

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[12]

B. Bergerhoﬀ, D. Litim, S. Lola and C. Wetterich, Int. J. M od. Phys. A11 (1996) 4273 [cond-mat/9502039]; B. Bergerhoﬀ, F. Freire, D. Litim, S. Lo la and C. Wetterich, Phys. Rev. B53 (1996) 5734 [hep-ph/9503334]

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[13]

N. Tetradis and C. Wetterich, Nucl. Phys. B383 (1992) 197

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K. Aoki, K. Morikawa, W. Souma, J. Sumi and H. Terao, Prog . Theor. Phys. 95 (1996) 409 [hep-ph/9612458]

View on Pith Pith review arXiv 1996
[15]

D.F. Litim, Wilsonian ﬂow equations and thermal ﬁeld theory , [hep-th/9811272]

View on Pith arXiv
[16]

D.F. Litim, in preparation. 11

View on Pith

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hep-th 2010-02 conditional novelty 5.0 of 10

Running the gravitational coupling with distance in d-dimensional Schwarzschild black holes yields a minimum black hole mass, a cold remnant with vanishing temperature, and a suppressed but semi-classical collider cro...
Renormalization Group Flow in Scalar-Tensor Theories. I
hep-th 2009-11 conditional novelty 5.0 of 10

In a two-derivative truncation of scalar-tensor gravity, the only continuous fixed point in any dimension d > 2 is one where all scalar self-couplings vanish while Newton's constant and the cosmological constant stay nonzero.
Fixed points of quantum gravity in extra dimensions
hep-th 2006-02 conditional novelty 5.0 of 10

A non-trivial ultraviolet fixed point exists for quantum gravity in 4 to 11 dimensions within the Einstein-Hilbert truncation, with scaling exponents stable across regulators.
Ward-constrained melonic renormalization group flow for the rank-four $\phi^6$ tensorial group field theory
hep-th 2019-08 reject novelty 4.0 of 10

For the rank-4 φ6 tensorial group field theory, the authors derive a Ward-constrained renormalization group flow and report a nontrivial fixed point, but the fixed point's reported values violate their own consistency...
Impact of topology in foliated Quantum Einstein Gravity
hep-th 2017-05 conditional novelty 4.0 of 10

The non-Gaussian fixed point that makes gravity asymptotically safe on a circle-times-sphere background is robust against changes in parametrization and regulator scheme, and closely matches the fixed point found on a...
The phase diagram of nuclear and quark matter at high baryon density
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The review argues that a third regime, quarkyonic matter, sits between confined nuclear matter and deconfined quark matter in large-Nc QCD, with pressure scaling as O(Nc).
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gr-qc 2016-03 accept novelty 2.0 of 10

Tensor-invariant bubble interactions supply group field theories with enough locality to renormalize, yielding a classified table of power-counting renormalizable models and hints of Wilson-Fisher-like fixed points.
Renormalisation group and the Planck scale
hep-th 2011-02 accept novelty 2.0 of 10

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An asymptotically safe guide to quantum gravity and matter
hep-th 2018-10 accept novelty 1.0 of 10

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How later papers used it

163 later papers cite it

Evidence: substantial. 148 of the citing papers share no author with it, and 103 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 238 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

12 later papers by other authors are listed below: 12 used its method or tool.

Uses its method or tool 12 of 112 papers with this label

Chose the optimized cutoff of this paper to perform the integrations in the flow equation trace in closed form.

Renormalization Group Flow in Scalar-Tensor Theories. I (2009) · 0.99 · paraphrased: the licence does not allow quoting
Used the optimised cut-off of this paper to write the non-perturbative beta functions for the dimensionless couplings.

Asymptotically Safe Starobinsky Inflation (2013) · 0.93 · paraphrased: the licence does not allow quoting
Used the Wetterich equation with the optimised regulator of this paper, called the Litim regulator.

Renormalisation Group for Gravity and Dimensional Reduction (2018) · 1.00 · paraphrased: the licence does not allow quoting
Employed bosonic and fermionic Litim-type regulators of this paper for its calculations.

Asymptotic safety in the dark (2018) · 1.00 · paraphrased: the licence does not allow quoting
Performed all subsequent calculations using the Litim regulator of this paper.

Renormalisation Group approach to General Relativity (2025) · 1.00 · paraphrased: the licence does not allow quoting
For the regulator function, we chose a modified version of the Litim optimized regulator [ref] , allowing to make analytic computations: [equation] which obviously satisfy the requirements $1-4$ given after equation [ref].

Revisited functional renormalization group approach for random matrices in the large-$N$ limit (2019) · Local potential approximation · 0.89
Employed regulators of the type given in another work and used a Litim-type shape function of this paper.

Asymptotic safety of gravity-matter systems (2015) · 1.00 · paraphrased: the licence does not allow quoting
The fermionic regulator $R^f_v⃗(k,p)$ can be derived from the well-known Litim regulator [ref] as [equation] where $p_∥=v⃗·p⃗$, $k=Λe^-t$, and $r_f(k,p_∥)$ is given as [equation] and the integration measure contained in the supertrace in Eq. ([ref]) is understood as [equation] where we used $δ^(2)(p⃗_)=(1/μ_q^2)δ^(2) (v⃗)$ and $d^2p_ = μ_q^2 d v⃗$.

Functional Renormalization Group analysis of the quark-condensation pattern on the Fermi surface: A simple effective-model approach (2024) · The exact RG flow equation in the mean-field approximation for the bosonic fields · 0.98
Chose the Litim regulator, which is optimized in the sense of this paper.

Field theoretical approach for signal detection in nearly continuous positive spectra I: Matricial data (2020) · 0.92 · paraphrased: the licence does not allow quoting
For the regulator $R_k$, we adopt the Litim's cut-off [ref] , in which we set $e^k→k$: [equation] and computing the first derivative with respect to $k$, we find: [equation] Hence, we are now in position to extract the flow equations for each couplings, which is the subject of the next section.

Functional renormalization group for the $U(1)$-$T_5^6$ tensorial group field theory with closure constraint (2016) · Truncation and Regularization · 1.00
Adopted a slightly modified, spectrally adapted Litim-type cutoff of this paper as its regulator.

Functional Renormalization Group for a Rank-4 Renormalizable Tensorial Group Field Theory with Derivative Necklace Couplings (2026) · 0.99 · paraphrased: the licence does not allow quoting
Used a Litim-shaped regulator of this paper to keep its equations completely analytic.

Towards apparent convergence in asymptotically safe quantum gravity (2016) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

118 arXiv papers cite this in 180 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 71 sentences without a confident label

Contests it 1 paper

The beta functions written in [ref] contain a number of transcription errors, which however do not affect the subsequent results.] Here the integrals have already been performed, using an optimized cutoff [ref] , so the beta functions are in closed form and completely explicit.

Renormalization Group Flow in Scalar-Tensor Theories. I · Explicit beta functions · contest 0.83

Uses its method or tool 83 papers

With the “optimised" cut–off of Ref. [ref] and in the Landau type gauge of [ref] , the full, non-perturbative beta functions for the dimensionless couplings $G ≡G/k^2$ and $b$ take the form

Asymptotically Safe Starobinsky Inflation · Explicit beta functions and the UVFP2 · 0.93
The RG equation is of the form [ref] : [equation] where $R$ is an appropriately defined momentum cutoff at the scale $p$ and is usually determined by the so-called optimized cutoff process [ref] .

Black holes in an asymptotically safe gravity theory with higher derivatives · Asymptotically safe high-derivative gravity · 0.84
We exploit this freedom to obtain explicit analytical RG flows for all couplings [ref] adopting the ideas of [ref] .

Further evidence for asymptotic safety of quantum gravity · Gravitational renormalisation group · 1.00

Compares with it 3 papers

This is in contrast to the vacuum calculations in [ref] , which were performed with a smoothed version of the flat regulator [ref] .

Non-perturbative finite-temperature Yang-Mills theory · Regulator and truncation dependence · 0.95
This is evident within a comparison to LPA and LPA$^'$ results with the flat or Litim regulator, [ref] , which is the optimised one for LPA but not beyond [ref] .

Critical scaling for spectral functions · Benchmarks and extensions · 0.82
For instance, in comparison to the notation used in [ref] , we have the following correspondence [equation] Let us finally list the explicit forms of the threshold functions as they are needed for the present work in $d=4$, employing the linear regulator [ref] for $r_B$ and $r_F$:

Asymptotically Safe QED · appendix · 0.90

Cites it as background 6 papers

A first detailed and general discussion of optimization of functional RG flows has been given by Litim in a series of ground-breaking articles [ref] .

Optimization and Stabilization of Functional Renormalization Group Flows · Introduction · 0.92
Feasible truncations in QCD applications are often more limited than in simpler model studies such that one relies on more general optimization criteria [ref] .

Regulator scheme dependence of the chiral phase transition at high densities · Summary and Conclusions · 0.88
Various incarnations of the (exact) renormalisation group are available [ref] , which, combined with systematic approximation schemes [ref] , give access to the relevant physics without being tied to weak coupling.

Critical $O(N)$ models in the complex field plane · Introduction · 0.84

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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