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Field-theory approach to flat polymerized membranes

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Pith's one-line read At four-loop order the anomalous stiffness of flat polymerized membranes is η = 0.8670, an exact perturbative result, agreeing with the independent two-field computation and requiring no resummation.

desk verdict Useful review of the authors' three- and four-loop membrane results; the headline exponent is not new, and the unproved vertex factorization (Eq. 33) needs qualification, but the three-loop details and SCSA diagnosis earn a serious referee. read the letter →

arxiv 2412.18490 v2 pith:EHRDA2I6 submitted 2024-12-24 cond-mat.stat-mech hep-th

Simon Metayer , Sofian Teber This is my paper · ORCID
classification cond-mat.stat-mechhep-th PACS 64.60.Fr
keywords polymerizedmembranesanomalousstiffnesseffectiveflexuraltheoryrenormalizationgroupepsilonexpansionfour-loopflatphaseFeynmandiagrams
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a review of the field-theoretic renormalization-group approach to the flat phase of polymerized (crystalline) membranes, and its central claim is a four-loop value for the anomalous stiffness exponent: η(P4) = 0.8670 for a two-dimensional membrane embedded in three-dimensional space. The claim matters because η controls every other critical property of the flat phase — the roughness exponent, the softening of elastic constants, and the enhancement of bending rigidity — and because the authors argue the perturbative series is unusually well behaved. The successive loop values 0.9600, 0.9139, 0.8872, 0.8670 come from directly evaluating the ε-expansion at ε = 1, with no resummation, and they land inside the range 0.7–0.9 accepted from simulations and non-perturbative approaches. The paper also uses the exact order-by-order results to benchmark two resummation schemes (SCSA and NPRG), explaining their success by showing that vertex corrections are unexpectedly small at the stable fixed point.

What carries the argument

The machinery that carries the argument is the effective flexural theory built on two ingredients. First, the quartic flexuron interaction is rewritten as two bare three-point vertices connected by an effective R-propagator, so the dressed four-point vertex is assumed to factorize, to all orders, as V = Γ⁽⁰⁾ R Γ⁽⁰⁾, with R obeying a Dyson equation whose kernel is the polarization Π; the tensor structure of R is diagonalized by two orthogonal projectors M and N onto the shear and bulk channels. Second, renormalization constants are extracted directly from self-energy and polarization functions in the modified minimal-subtraction scheme, avoiding counterterms, and all multiloop integrals are reduced by integration-by-parts to master integrals of transcendental weight up to ζ₅. The all-orders factorization claimed in Eq. (33) is the load-bearing link: it turns the four-point vertex problem into a propagator renormalization problem.

What would settle it

Compute the five-loop coefficient of η(P4): the direct-ε = 1 claim predicts the coefficient sequence 0.9600, −0.04608, −0.02673, −0.02017 continues to decrease in magnitude, so a five-loop term larger than 0.02017 would falsify the claim that the series can be trusted without resummation. Independently, enumerate all five-loop diagrams contributing to the dressed four-point vertex: if any diagram cannot be factored as two bare three-point vertices joined by the Dyson-resummed R-propagator, the exactness of Eq. (33) is disproved.

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Extended reading notes

Core claim

The paper's central discovery is that the effective flexural model — a single transverse (flexural) field interacting through a non-local quartic vertex — is enough to reproduce the known two-field results for the flat phase and to push them to four loops. At the fully interacting, fully stable fixed point P4 of the renormalization-group flow, the anomalous stiffness exponent is computed to four loops and the numerical series reads η(P4) = 0.9600ε − 0.04608ε² − 0.02673ε³ − 0.02017ε⁴ + O(ε⁵), evaluated in the physical case ε = 1 to give successively η1-loop = 0.9600, η2-loop = 0.9139, η3-loop = 0.8872, and η4-loop = 0.8670. Because the series coefficients decrease steadily up to four loops, the authors evaluate the series directly at ε = 1 with no resummation and obtain η = 0.8670, in perfect agreement with Pikelner's independent four-loop computation in the two-field model. They treat this order-by-order result as exact in the perturbative sense, with no resummation involved.

Load-bearing premise

The entire computation rests on assuming that every loop correction to the membrane's four-point interaction can be folded into a single renormalized interaction line, so that the two interaction points themselves never acquire corrections; the paper states this as exact in Eq. (33) but does not prove it.

Editorial extensions

If this is right

  • The flat-phase anomalous stiffness of a two-dimensional polymerized membrane is fixed at η = 0.8670, which through the paper's relations gives the roughness exponent ζ = (4 − d − η)/2 ≈ 0.5665 and the elasticity-softening exponent ηu = 4 − d − 2η ≈ 0.266.
  • The effective flexural model and the two-field model are confirmed to be equivalent at four-loop order, so the simpler one-field formulation can be used for yet higher orders.
  • The ε-expansion for this problem is numerically trustworthy at ε = 1 up to four loops, so further orders can be compared directly with simulations without resummation.
  • An exponential fit to the loop values suggests the all-order limit η ≈ 0.8347, which lies within the generally accepted range [0.7, 0.9].
  • The near-vanishing of vertex corrections at the stable fixed point explains why the SCSA and NPRG approximations, which neglect or truncate vertex corrections, match the exact loop results so closely.

Reading between the lines

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

  • The all-orders factorization in Eq. (33) is asserted rather than derived; the four-loop agreement with the two-field model is a strong consistency check, but a proof of the factorization, or a demonstration that irreducible vertex corrections cancel order by order, remains an open problem.
  • If the decreasing-coefficient pattern persists, perturbation theory for this model may be closer to convergent than typical ε-expansions; a five-loop computation would discriminate between genuine convergence and a merely delayed asymptotic series.
  • The V-dependent bookkeeping introduced in Section 6.1, which tracks the size of vertex corrections, is a ready-made diagnostic for situations such as disordered or finite-temperature membranes where vertex corrections may not be small and SCSA-type approximations could fail.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reviews the field-theoretic renormalization-group approach to the flat phase of polymerized membranes, formulated in the flexural effective model. It introduces the model, an auxiliary QED-like set of Feynman rules with a three-point vertex and an effective R-propagator, and a renormalization scheme based on Dyson equations for the flexuron self-energy and the R-propagator polarization. The paper presents a complete three-loop computation of the renormalization constants, beta functions, and the anomalous stiffness eta at the four fixed points for general d = 4 - 2epsilon and codimension d_c, with explicit master integrals and diagram enumeration. The four-loop contribution is summarized and quoted from the authors' letter [36] and from Pikelner [37], giving eta(P4) = 0.8670 for the physical case d_c = 1 at epsilon = 1, with an exponential extrapolation to 0.8347. The results are compared with SCSA, NPRG, and large-d_c expansions.

Significance. If the results are correct, the paper provides the most complete perturbative determination of the anomalous stiffness exponent for the flat phase to date and a valuable benchmark for non-perturbative approaches. The three-loop material is detailed and carefully documented: diagram generation, symmetry factors, tensor contractions, IBP reduction to master integrals, and numerical checks with FIESTA are all described, and the ancillary files make the RG functions available in computer-readable form. The agreement with Pikelner's independent two-field four-loop computation is a strong consistency check. However, the central derivation rests on the unproven all-orders factorization of the four-point vertex, and the four-loop expressions are not included in this manuscript; as a result, the paper currently supports the three-loop results much more strongly than the four-loop headline value.

major comments (3)
  1. [3.3, Eqs. (33)-(34)] The all-orders statement that the dressed four-point vertex factorizes as V = Gamma^(0) R Gamma^(0), with all corrections in the Dyson-dressed R-propagator, is asserted without proof. In the auxiliary cubic representation this is equivalent to saying that the h-h-R three-point vertex is unrenormalized to all orders. The paper's own V-parameter diagnostic in Sec. 6.1, Eqs. (132)-(133), however, introduces a factor V in front of precisely those diagrams that cannot be reduced to a line by removing two-line bubbles, and reports nonzero coefficients at epsilon = 1, e.g., eta(P4) = 0.8667 + 0.0003 V. These diagrams are said to be included in the full computation (V = 1). The manuscript therefore needs either a proof that such diagrams are nevertheless contained in the polarization Pi of Eq. (34), or a qualification that Eq. (33) is an approximation or a scheme choice, together with an analysis of whether a vertex renormalization constant should appear in the beta functions. As written, the exactness claim is unsupported and is load-bearing for the RG functions in Eqs. (108)-(109).
  2. [4.4 and Eq. (126)] The central numerical claim, eta(P4) = 0.8670 at four loops, is not derived in this manuscript. Section 4.4 explicitly states that no four-loop expressions are shown, and Eq. (126) quotes the result from refs. [36] and [37]. If this paper is intended as an original derivation of the four-loop anomalous stiffness, the four-loop renormalization constants, beta functions, or at least the evaluated expressions should be made available, for example in ancillary files. If the paper is intended as a review, the abstract and Sec. 5.2.4 should state more carefully that the four-loop value is reported from prior work rather than derived here. In the present form the reader cannot verify the headline value from this manuscript alone.
  3. [5.2.4, Eqs. (125)-(127)] The reliability claim that the epsilon-series can be evaluated directly at epsilon = 1 rests on four successively decreasing coefficients. This is a necessary but not sufficient condition for practical convergence of an asymptotic series, and no error estimate is provided. The exponential extrapolation eta_all-order = 0.8347 is a two-parameter fit to four data points and should be presented with a quantitative sensitivity caveat, for example by comparing a range of Pade or Borel-Pade estimates and giving an uncertainty interval. This would also make the distinction between the exact four-loop result (0.8670) and the extrapolated estimate (0.8347) clearer.
minor comments (5)
  1. [Eq. (18)] The relation b(d) = lambda / (2 W nu Y) appears dimensionally inconsistent; for d = 2 and lambda = mu, the left side is 4/3 while the right side evaluates to 3/16. Please correct the formula or clarify the intended identity.
  2. [Sec. 5.2.4] The sentence 'a direct substitution of epsilon = 1 in (120)' should refer to Eq. (125), not Eq. (120), when discussing eta(P4).
  3. [Sec. 5.2.2] The phrase 'the the two and four loop contributions' contains a duplicated definite article; it should read 'the two- and four-loop contributions'.
  4. [Fig. 6] The caption should state explicitly that the exponential fit is empirical; with four data points and two fit parameters, it is not a controlled extrapolation, even though the text does label it as a fit.
  5. [Sec. 4.4] Since the four-loop expressions are omitted, the manuscript should at least list the 39 four-loop master integrals and the 113 topological relations in an appendix or ancillary file, or point to where they are tabulated, so that the quoted four-loop result can be checked.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the four-loop anomalous stiffness is independently confirmed by Pikelner's two-field calculation, and the only structural assumption (Eq. 33) is an unproved ansatz rather than a fitted input.

full rationale

The central derivation is not circular: the anomalous stiffness at the flat fixed point, eta(P4)=0.8670, is obtained by solving the beta functions at the fully stable fixed point, with no fitting to the target exponent. The four-loop contribution is imported from the authors' own letter [36], but it is independently corroborated by Pikelner's four-loop two-field computation [37], which the paper explicitly quotes as the first result ('This four-loop result has recently been confirmed by [36]' after crediting [37]). The all-order estimate 0.8347 is transparently presented as an exponential fit to the one- through four-loop values, not as a new prediction. The main caveat is Eq. (33), which asserts that the dressed four-point vertex factorizes exactly into two bare three-point vertices joined by a dressed R-propagator; this is a substantive assumption, not a definition of eta, and the paper gives no proof of its all-orders validity. The V-parameter analysis in Sec. 6.1 shows that non-bubble ('vertex-correction') diagrams contribute a small but nonzero amount to eta(P4), so the factorization claim is not a tautology. Thus the paper contains no reduction of a prediction to its input by construction; the score of 1 reflects only the modest reliance on the authors' own four-loop letter, which is not load-bearing given the independent confirmation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central computation introduces no fitted fundamental parameters; the two couplings μ and b are renormalized couplings of the theory. The only ad hoc fit is the exponential extrapolation in Fig. 6. The main assumptions are the exactness of the flexural-model reduction, the all-orders Dyson factorization, the validity of dimensional regularization for IR divergences, and the direct ε=1 evaluation.

free parameters (2)
  • exponential fit amplitude = 0.195
    Figure 6 fits η(L) = η_all-order + 0.195 e^{-0.446 L} to the four computed loop values; the amplitude is chosen by the fit.
  • exponential fit decay rate = 0.446
    Same fit; the decay rate is chosen to reproduce the four loop-order points.
assumptions (5)
  • domain assumption The flexural effective model (12) is exactly equivalent to the two-field phonon-flexuron model (8) after integrating out the phonon field.
    Used throughout as the starting point; the equivalence is supported by prior work [5,10,33] and checked by the μ=0 limit, but it is an assumption about the exactness of the integration.
  • ad hoc to paper The dressed four-point vertex factorizes as Γ^(0) R Γ^(0) to all orders (Eq. 33), so all loop corrections are encoded in the R-propagator polarization satisfying the Dyson equation (34).
    This structural identity is asserted as exact in Sec. 3.3 but not proven; it underpins the two-coupling renormalization scheme.
  • standard math IR divergences from massless tadpoles cancel in dimensional regularization, so only UV poles need renormalization.
    The paper invokes the Smirnov-Chetyrkin theorem [40] for IR-safe propagators; this is a standard result.
  • ad hoc to paper The ε-expansion can be evaluated at ε=1 without resummation because the coefficients decrease up to four loops.
    The paper observes small and decreasing coefficients for P4 and substitutes ε=1 directly; this is an empirical inference from four data points, not a proven convergence property.
  • standard math Canonical power-counting (Eq. 11) establishes renormalizability in d=4-2ε; neglected operators including (Δu)^2 and phonon nonlinearities are irrelevant.
    Power-counting argument standard in the field; stated in Sec. 2.

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Pith. "Pith review of Field-theory approach to flat polymerized membranes." pith.science (2026). https://pith.science/paper/EHRDA2I6

@misc{pith2026241218490,
  author       = {Pith},
  title        = {Pith review of: Field-theory approach to flat polymerized membranes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHRDA2I6}},
  note         = {Machine review of arXiv:2412.18490}
}
abstract

We review the field-theoretic renormalization-group approach to critical properties of flat polymerized membranes. We start with a presentation of the flexural effective model that is entirely expressed in terms of a transverse (flexural) field with non-local interactions. We then provide a detailed account of the full three-loop computations of the renormalization-group functions of the model within the dimensional regularization scheme. The latter allows us to consider the general case of a $d$-dimensional membrane embedded in $D$-dimensional space. Focusing on the critical flat phase of two-dimensional membranes $(d = 2)$ in three-dimensional space $(D = 3)$, we analyse the corresponding flow diagram and present the derivation of the anomalous stiffness. The latter controls all the other critical exponents of the theory such as the roughness exponent and the scaling of the elastic constants. State-of-the-art four-loop results as well as discussions on the structure of the perturbative series and comparison with other approaches are also provided.

Figures

Figures reproduced from arXiv: 2412.18490 by the authors.

Figure 1
Figure 1. One-loop diagrams and their associated symmetry factors (S). [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Two-loop flexuron self-energy diagrams and symmetry factors (S). [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Two-loop vertex self-energy diagrams with their symmetry factors [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Three-loop flexuron self-energy diagrams and their associated [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Three-loop vertex self-energy diagrams and their associated [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Exponential fit of the results found for [PITH_FULL_IMAGE:figures/full_fig_p036_6.png]
Figure 7
Figure 7. Figure 7: RG-flow diagram in the (µr,br) plane. The mechanical stability of the model imposes µr > 0 and br > 0. The corresponding non-physical regions are indicated in red and delimited by the red dashed lines µr = 0, br = 0, on which lie the fixed points P1, P′ 2 and P3 at all…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Critical Ripples and Dirac Fermions in Crystalline Membranes

    cond-mat.other 2026-07 conditional novelty 6.0 of 10

    Flat membrane-Dirac systems asymptotically decouple at charge neutrality; finite-momentum rippling realizes either Wilson-Fisher criticality with spectator fermions or, when the ripple carries Kekulé mass quantum numb...

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Works this paper leans on

104 extracted references · 80 canonical work pages · cited by 1 Pith paper

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    Guitter, F

    E. Guitter, F. David, S. Leibler, and L. Peliti. Thermodynamical behavior of polymerized membranes. J. Phys. France, 50(14):1787–1819, 1989

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    in the equivalent two-field model, recently confirmed by [36] in the effective flexural model. 5.2.4 Flat phase fixed point P 4 The most important fixed point is P 4 which is characterized by P4 : µ∗ 4 = (4π)2 12ε dc + 24+ 1440 (dc + 24)3 − 616 5(dc + 24)2 ε2 + 345600 (dc + 24)5 (122a) + Indeed, a resummation of the series (119) with a simple[1/2] Padé ap...

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    Bowick and Alex Travesset

    Mark J. Bowick and Alex Travesset. The Statistical mechanics of membranes. Phys. Rept., 344:255–308, 2001

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    This result displays an interesting structure in the perturbative series with denominators in powers of 1/(4 +dc)

    = 4ε dc + 4+ 16 (dc + 4)3 − 20 3(dc + 4)2 + 2 3(dc + 4) ε2 + 128 (dc + 4)5 (113) ∥ See footnote 2 Field-theory approach to flat polymerized membranes 33 − 16(10368ζ3 + 2029) 3375(dc + 4)4 − 4(5184ζ3 + 58177) 3375(dc + 4)3 + 4(3888ζ3 + 27239) 3375(dc + 4)2 − 37 9(dc + 4) ε3 + O(ε4). This result displays an interesting structure in the perturbative series w...

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    Numerically, this series evaluates to η(P′

    = 4ε 5 − 2ε2 375 + (119232ζ3 − 120079)ε3 2109375 (114) + (−51994931 + 7803552ζ3 + 26827200ζ4 + 13512960ζ5)ε4 316406250 + O(ε5), with each term in the series getting divided by an increasing powers of1/5. Numerically, this series evaluates to η(P′

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    = 0.8000ε − 0.005333ε2 + 0.01102ε3 + 0.001369ε4 + O(ε5), (115) with surprisingly small coefficients as already noticed in [33, 35]. As discussed in these papers, perturbative series are asymptotic in nature but the case of polymerized membranes is quite peculiar in the sense that various factors (such as increasing powers of 1/(dc + 4) in the case of P′

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    The series therefore effectively looks convergent even in the case of interestε = 1

    conspire to numerically reduce the coefficient of the epsilon-series over several orders. The series therefore effectively looks convergent even in the case of interestε = 1. As a matter of fact, note that the the two and four loop contributions in (115) are not only numerically small but also smaller than the one- loop coefficient by two orders of magnit...

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    = 0.8000, η 2-loop(P′

Show all 104 references
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    = 0.7947, η 3-loop(P′

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    = 0.8057, η 4-loop(P′

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    (116) The one-loop result has been first obtained in [6], the two-loop result (32 year later) in [32, 33], the three-loop result in [35] and the four-loop one in [36]

    = 0.8071. (116) The one-loop result has been first obtained in [6], the two-loop result (32 year later) in [32, 33], the three-loop result in [35] and the four-loop one in [36]. 5.2.3 Infinitely compressible fixed point P3 The infinitely compressible fixed point P3 is characte...

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    = 0.7983 or η[1/2](P′

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    = 0.8057 and the four-loop oneη[2/2] = 0.8074, which are all very close if not indistinguishable from the non-resummed results of (116). Field-theory approach to flat polymerized membranes 34 b∗ 3 = 0 + O(ε4) (Infinitely compressible) (117b) with a non-trivial value for the sh...

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    (120) Remarkably, all the coefficients appearing in (120) are small and even decreasing up to three loops

    Indeed, in the physical casedc = 1, including explicitly the four-loop contribution, the coefficients simplify as η(P3) = 20ε 21 − 94ε2 1323 − (312336ζ3 − 9011)ε3 5250987 (119) − (14383003505 + 36705338304ζ3 + 59031504ζ4 − 56435313600ζ5)ε4 661624362 + O(ε5), and numerically ev...

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    = 4ε dc + 4+ 16 (dc + 4)3 − 20 3(dc + 4)2 + 2 3(dc + 4) ε2 (130b) + 128 (dc + 4)5 + 128 3(dc + 4)4 − 400 3(dc + 4)3 + 406 9(dc + 4)2 − 37 9(dc + 4) ε3 + O(ε4), ηSCSA(P3) = 20ε dc + 20+ 2000 (dc + 20)3 + 1180 3(dc + 20)2 − 74 3(dc + 20) ε2 (130c) + 400000 (dc + 20)5 + 584000 3(...

    2000

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    = 0.800ε − 0.00533ε2 + 0.0248ε3 − 0.00339ε4 + O(ε5), (131b) η(P′

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    The first order is exact as expected from such a technique

    = 0.800ε − 0.00533ε2 + 0.0110ε3 + 0.00137ε4 + O(ε5), ηSCSA(P3) = 0.952ε − 0.0667ε2 − 0.0560ε3 − 0.0519ε4 + O(ε5), (131c) η(P3) = 0.952ε − 0.0711ε2 − 0.0698ε3 − 0.0748ε4 + O(ε5), ηSCSA(P4) = 0.960ε − 0.0476ε2 − 0.0280ε3 − 0.0177ε4 + O(ε5), (131d) η(P4) = 0.960ε − 0.0461ε2 − 0.0...

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    The three-loop order is very close for both P3 and P4 but differs by a factor of two for P′

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    Interestingly, at four loops, the SCSA is very close numerically for both P3 and P4 but gives the wrong sign and misses a factor of 3 for P′ 2 indicating the limits of the SCSA approximation. In order to clarify the approximation involved in the SCSA calculations, we have reco...

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    A comparison with (131) then reveals that the approximation made in the SCSA is to neglect all vertex corrections (V = 0)

    = 0.800ε − 0.0053ε2 + (0.0248 − 0.0138V )ε3 − (0.00339 − 0.00476V )ε4 + O(ε5), η(P3) = 0.952ε − (0.0667 + 0.0043V )ε2 − (0.0560 + 0.0138V )ε3 − (0.0519 + 0.0228V )ε4 + O(ε5), η(P4) = 0.960ε − (0.0476 − 0.0015V )ε2 − (0.0280 − 0.0012V )ε3 − (0.0177 + 0.0025V )ε4 + O(ε5), and ar...

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    The renormalization-group functions generalized for allV are available in computer readable files as ancillary files to the arXiv version of the letter [36]

    = 0.8161 − 0.0090V , (133b) η(P3) = 0.7777 − 0.0409V , (133c) η(P4) = 0.8667 + 0.0003V , (133d) and shows that vertex corrections are surprisingly small overall, e.g., of the order of 1% for P′ 2, 5% for P3 and even 0.03% for the physical stable fixed point P4. The renormaliza...

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    As for the SCSA approach, the NPRG reproduce well the1/(dc + n) structure previously observed with n = 4,20,24

    = 4ε dc + 4+ 8 3(dc + 4)3 − 14 3(dc + 4)2 + 1 dc + 4 ε2 (136b) + 32 9(dc + 4)5 − 76 9(dc + 4)4 + 25 3(dc + 4)3 − 65 18(dc + 4)2 + 1 2(dc + 4) ε3 + O(ε4), ηNPRG(P3) = 20ε dc + 20+ 1000 3(dc + 20)3 + 1330 3(dc + 20)2 − 23 dc + 20 ε2 (136c) + 100000 9(dc + 20)5 + 204500 9(dc + 20...

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    = 0.800ε + 0.0347ε2 + 0.0099ε3 + 0.00305ε4 + O(ε5), η(P′

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    Just as for the SCSA, this might be due to the fact that the perturbative series behaves remarkably well for the present problem

    = 0.800ε − 0.0053ε2 + 0.0110ε3 + 0.00137ε4 + O(ε5), (137b) ηNPRG(P3) = 0.952ε − 0.0540ε2 − 0.0519ε3 − 0.0485ε4 + O(ε5), η(P3) = 0.952ε − 0.0711ε2 − 0.0698ε3 − 0.0748ε4 + O(ε5), (137c) ηNPRG(P4) = 0.960ε − 0.0367ε2 − 0.0266ε3 − 0.0178ε4 + O(ε5), Field-theory approach to flat po...

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    (139d) For our purposes, we may now setd = 4− 2ε in the above equations and expand them up to O(ε4/dc), which yields ηSCSA(P1) = 0 + O(ε5/d2 c), (140a) ηSCSA(P′

    =ηSCSA d, dcd(d − 1) 2 , (139b) ηSCSA(P3) =ηSCSA d, dcd(d − 1) (d − 2)(d + 1) , (139c) ηSCSA(P4) =ηSCSA(d,dc). (139d) For our purposes, we may now setd = 4− 2ε in the above equations and expand them up to O(ε4/dc), which yields ηSCSA(P1) = 0 + O(ε5/d2 c), (140a) ηSCSA(P′

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    = 1 dc 4ε + 2ε2 3 − 37ε3 9 + 4 479 216 − 2ζ3 ε4 + O(ε5) + O 1/d2 c , (140b) ηSCSA(P3) = 1 dc 20ε − 74ε2 3 − 155ε3 9 + 20 769 1080 − 2ζ3 ε4 + O(ε5) + O 1/d2 c , (140c) ηSCSA(P4) = 1 dc 24ε − 24ε2 − 64ε3 3 + 24 26 27 − 2ζ3 ε4 + O(ε5) + O 1/d2 c . (140d) Field-theory approach to ...

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