Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations

H. Osborn; J. O'Dwyer

REVIEW

Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations

J. O'Dwyer, H. Osborn

High Energy Physics - Theory · Submitted 20 Aug 2007 · 37 references

Pith's verdict: A careful, honest benchmark paper on epsilon expansions of the Polchinski ERG at multicritical points; the scaling-field truncation result is the real contribution.

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CitationH. Osborn, J. O'Dwyer. "Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations." arXiv:0708.2697 (2007). https://pith.science/paper/RQDGK4NM

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Editor's note

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This one is worth your time if you work on ERG truncations. The core contribution is a systematic epsilon expansion of the Polchinski equation at each multicritical point, done in three schemes side by side: LPA, derivative expansion, and a scaling-field truncation. The scaling-field truncation is the interesting part — it reproduces the perturbative O(epsilon) exponents and, more importantly, gets eta right to O(epsilon^2), which the ordinary derivative expansion does not. The explicit demonstration that the derivative expansion fails for the second class of operators, with the O(epsilon) result (7.23b) depending on the cutoff constants A and B, is a clean negative result that practitioners should know about. Appendix D's exact marginal operator, with the Gaussian check in (D.19), is a nice piece of bookkeeping that ties the whole story together. The paper is honest about overlap with Nicoll-Chang-Stanley and Haagensen et al; the O(epsilon^2) LPA exponents at general n are new, and the authors say so without overselling it. The derivations include real consistency checks: exact eigenfunctions at k=0,1,2n-1, the vanishing sum in Appendix B, the numerical comparisons in Tables 1-3. The stress-test's worry about the regularity class of the cutoff in Appendix C is a fair question but smaller than it looks: (C.9) is anchored to the dimensional-regularization poles in (2.14), so the universality of (8.24) and (8.30) is matched to the standard calculation rather than merely asserted. The genuinely soft spot is the r<=2 truncation in (8.17), which the reader flagged and which the paper treats as an approximation rather than proving. That is an acknowledged limitation, not a hidden one. Section 9's conjecture that the modified expansion has a 'greater chance of predictive success' outside the epsilon expansion is speculative and untested; treat it as a remark, not a result. The paper is long and Appendix A's perturbative results are somewhat orthogonal to the main thrust, but the main line is solid. This deserves a serious referee. It is a benchmark paper for anyone choosing a truncation scheme, and the negative result on the derivative expansion is the kind of thing that should be cited rather than rediscovered.

Referee report

Significance

The paper is a careful and well-organised study of the relation between ERG truncations and the ε-expansion at multicritical points. Its main value is the explicit demonstration that the scaling-field truncation reproduces the perturbative exponents to O(ε) and η to O(ε²), with the cutoff dependence isolated in integrals whose universal values are identified with the log-divergent parts of two-vertex Feynman diagrams. This is a genuine improvement over the standard derivative expansion, and the paper is candid about where the LPA and the derivative expansion fail. The consistency checks are a real strength: the exact eigenfunctions (5.20)-(5.24), the vanishing sum of Appendix B, the Gaussian check of the marginal operator (D.19), and the comparison with the numerical results of [23] in Tables 1-3. The paper does not overclaim, and the conclusion acknowledges that extending the construction to three-vertex graphs is non-trivial and that the status of the derivative expansion remains unclear.

Major objections (1)

[Section 8, Eq. (8.17) and the results (8.23), (8.29)] The truncation (8.17) to operators with r = 0, 2 is the step that makes the scaling-field approach tractable, and the agreement of (8.29) with (2.27) and (2.44), and of (8.23) with (2.19), is the paper's central quantitative claim. The manuscript should state explicitly why the discarded operators cannot feed back at the orders computed. From (8.18a,b) the b_k are generated at O(ε²) and enter ȧ_k only multiplied by some a_l, so their contribution to the a_k flow is O(ε³); likewise the O_{k4} operators generated by the ellipsis in (8.15) are O(ε²) and feed back only at O(ε³). Making this power counting explicit, and connecting it to the statement in Appendix C that the retained integrals are precisely the log-divergent parts of two-vertex graphs, would turn the agreement from a verification of the truncated system into a derivation from the full ERG. As written, a reader cannot tell whether the agreement is explained by the truncation or is a coincidence of it.

Minor objections (7)

[Section 7, Eq. (7.23b) versus Eq. (2.44)] The claim that the derivative expansion 'cannot agree with (2.44) in general' is correct but under-argued. The k-dependence of (7.23b) involves (k-2n)!/(k-3n+1)!, a product of n-1 factors, whereas (2.44) involves (k-2n+1)!/(k-3n+1)!, a product of n factors; no choice of AB can reconcile these for all k. Stating this explicitly would make the negative result about the derivative expansion robust rather than leaving the reader to wonder whether a special AB exists.
[Appendix C, Eqs. (C.7)-(C.12)] The derivation of the universal values (C.9) and (C.12) is sound: the r→0 boundary term in (C.7) vanishes because r^{d+2l}G̃(x)^{m+1} → 0 as r→0 for d+2l > 0, so the concern that a small-r contribution could contaminate the l = 0, 1 coefficients does not arise. It would nonetheless help to state the smoothness assumed of K, since (C.4) and (C.10) use K' and the asymptotic form (C.8) with corrections falling faster, while Section 8 states only K(0) = 1 and 'sufficient rapid fall off'.
[Table 3 caption] The caption reads 'ε-expansion at O(ε)' but the text and Eq. (4.16) give the result to O(ε²); the caption should be corrected.
[Appendix D, Eq. (D.18)] The general-η solution for a(x) is quoted without derivation, and the printed expression is not immediately transparent. Since the η = 0 limit is stated separately as a(x) = 1, b(x) = (K(x)-1)/x, it would help to show explicitly that the general formula reduces to this, or to give the intermediate step from the two first-order equations above (D.18).
[Section 9, Eqs. (9.4a,b)] The equivalence of the truncated system (8.18a,b) with the coupled differential equations (9.4a,b) is asserted rather than shown. A short derivation, or a check of one coefficient against (9.2) and (9.3), would make the passage easier to follow.
[Sections 5 and 7, Eqs. (5.4) and (7.3)] The eigenvalue λ_k is related to λ̂_k by different normalisations in (5.4) and (7.3), as the text notes. A short summary table of the definitions and of the resulting λ_k for the LPA and the derivative expansion would help the reader keep track.
[Section 4, Tables 1-3] The comparison with the numerical results of [23] is only for V''_*(0). A comparison of the LPA ε-expansion exponents (5.18) with the numerical LPA exponents of [23] would test the ε-expansion itself, rather than only the fixed-point potential, and would strengthen the discussion of where the ε series ceases to be useful.

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The paper

The paper claims

The paper asks whether truncated exact renormalisation group equations can reproduce the epsilon expansion at every multicritical fixed point, and shows that a scaling-field truncation does so to order $O(1)$ in the exponents and $O(1)$ in,

The reading

Exact renormalisation group equations are formally exact but in practice must be cut down to a finite set of coupled differential equations, and the standard worry is that the answers then depend on the cutoff and on the truncation rather than on the physics. This paper turns that worry into a controlled test: near each multicritical dimension $d_n = 2n/(n-1)$, where the operator $V(1)$ becomes marginal, the nonlinear fixed-point equation linearises to Hermite's equation, so the epsilon expansion becomes an ordinary perturbative expansion of a differential equation. That gives a common yardstick against which three truncations can be measured at once: the local potential approximation, the derivative expansion, and a truncation built from scaling fields, that is, from exact eigenfunctions of the linearised flow. The result the authors want to establish is that the scaling-field truncation, keeping only operators with zero and two derivatives, reproduces the standard perturbative critical exponents to order epsilon and the field anomalous dimension to order epsilon squared, with coefficients that do not depend on the cutoff. The derivative expansion, by contrast, only manages this if two cutoff-dependent constants are tuned to a particular relation, and fails outright for the second class of operators.

What carries the argument

Two pieces of machinery carry the argument. The first is the linearisation: at $d = d_n$ the linearised fixed-point equation becomes $V(1)$ with $V(2)$ the Hermite operator, whose globally bounded solutions are $H_k(x)$, and the marginality condition $n = 2d/(d-2)$ is exactly the condition that $H_{2n}$ is a zero mode. The nonlinear term $v'^2$ then fixes the coefficient $c_n$ and generates the higher-order terms through Hermite orthogonality and the three-polynomial integral $G_{klm}$. The second is the set of universal integrals in the scaling-field truncation: $V(3)$ and $V(4)$ at $d=d_n$ are shown in Appendix C to be total derivatives of the position-space propagator product, so they reduce to surface terms and take cutoff-independent values, matching the logarithmic divergences of two-vertex Feynman diagrams in dimensional regularisation. Those two numbers, and nothing else, are what make the $O(1)$ exponents and the $O(1)$ anomalous dimension come out right. The paper then rewrites the truncation as a pair of coupled differential equations closely resembling the derivative expansion but with no linear $z$ term in the potential equation and with $V(5)$ fixed to $V(6)$.

Extended reading notes

Core claim

At each multicritical point the authors show that the Polchinski fixed-point equation, written in the rescaled variable $x$ and potential $v(x)$, has $v_*(x) = c_n H_{2n}(x)$ at leading order in epsilon, with $H_{2n}$ a Hermite polynomial and $c_n$ fixed by the nonlinear term through an orthogonality condition. This makes epsilon a genuine bookkeeping parameter and lets the local potential approximation be solved order by order, giving explicit polynomials and values such as $V''_*(0)$ to $O(1)$ for the first three multicritical points. The central claim is then a comparison: the local potential approximation reproduces the perturbative exponent formula in (2.27) at order epsilon but disagrees at order epsilon squared, and it cannot produce the field anomalous dimension at all. The derivative expansion introduces $A$ and $B$, two cutoff-dependent constants, and produces an $O(1)$ anomalous dimension that depends on the combination $AB$; only for one class of operators do the exponents come out right. The alternative truncation in Section 8, expanding the full flow in scaling fields $O_{k0}$ and $O_{k2}$ and keeping only those with at most two derivatives, gives $O(1)$ exponents and an $O(1)$ anomalous dimension that agree exactly with perturbation theory, because the two integrals on which they depend, $V(1)$ and $V(2)$, are cutoff-independent and equal to $n/(2n-1)$ and zero respectively. Finally, an exact marginal operator is constructed for the full flow, generating a line of equivalent fixed points, which is what allows the equations to determine the anomalous dimension at all.

Load-bearing premise

The load-bearing premise is that keeping only operators with zero and two derivatives in the scaling-field expansion is enough, that is, that operators with four or more derivatives do not feed back into the exponents at order epsilon or the anomalous dimension at order epsilon squared.

Editorial extensions

If this is right

The local potential approximation is confirmed as a correct $O(1)$ scheme for critical exponents at every multicritical point, and its $O(1)$ failure is quantified explicitly by the second-order polynomial corrections in (5.18).
The derivative expansion with two derivatives is shown to be insufficient as a universal scheme: its second operator class carries an explicit dependence on the cutoff constants through the combination $AB$, so it cannot be matched to perturbation theory without tuning.
The modified equations (9.6) provide a truncation that is structurally close to the derivative expansion but whose coefficients are fixed by universality rather than chosen, giving a candidate scheme for nonperturbative use.
The exact marginal operator in Appendix D establishes that a line of physically equivalent fixed points exists, which is the structural reason the flow can determine the field anomalous dimension rather than leaving it as an input.
Appendix A shows that at the next order in the coupling, transcendental numbers such as $V(7)$ and $V(8)$ enter, so matching a single truncation to perturbation theory simultaneously at all multicritical points becomes hard.

Reading between the lines

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

If the modified equations (9.6) are solved numerically at $d=3$ without any epsilon expansion, the resulting anomalous dimension is a parameter-free prediction that can be compared with the known three-dimensional value; that comparison is the natural next test and is not attempted here.
The exact marginal operator suggests a selection principle for truncations that the paper only gestures at: a truncation that preserves the marginal direction should automatically fix the anomalous dimension, so any truncation that leaves it undetermined can be diagnosed as having broken the field-rescaling symmetry.
The two universal integrals that make the scaling-field truncation work are exactly the coefficients of one-loop divergences, which hints that the agreement extends only as far as one-loop universal data; matching three-vertex graphs would require the two-loop analogue of Appendix C, which the paper does not supply.

Original abstract

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $\epsilon$ and also for the field anomalous dimension to order $\epsilon^2$. An exact marginal operator for the full RG equations is also constructed.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $\epsilon$ and also for the field anomalous dimension to order $\epsilon^2$. An exact marginal operator for the full RG equations is also constructed.

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	The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion.	contribution 0.53	0.86	0.10	asserted 1.00
s1	The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion.	method 0.52	0.89	0.21	asserted 0.61
s2	The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory.	method 0.43	0.80	0.10	asserted 0.98
s3	The results provide a test for the validity of the local potential approximation and also the derivative expansion.	interpretation 0.92	0.79	0.18	asserted 0.97
s4	An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $\epsilon$ and also for the field anomalous dimension to order $\epsilon^2$.	result 0.68	0.88	0.19	asserted 1.00
s5	An exact marginal operator for the full RG equations is also constructed.	result 0.67	0.92	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.98.

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Figures

**Figure 1.** Graphs for v∗(x) for n = 2 from ε-expansion at O(ε 2 ) and numerical solution with v∗(0) = −V ′′ ∗ (0)/d from

**Figure 2.** n = 3 v∗(x) from ε-expansion at O(ε 2 ) and numerical solution for d = 2.9, 2.8 and 2.6. Finally, we consider the n = 4 case. Following the same programme as above, the coefficients in the expansion are a0 = 432 35 , a2 = 288 5 , a4 = 324 5 , a6 = 36 , a10 = − 27 100 , a12 = − 3 800 , a14 = − 1 39200 , (4.14) and a8 = 23904 875 . (4.15) 14

**Figure 3.** d Numerical ε-expansion at O(ε) 8 3 0 0 2.6 −0.001 −0.001 2.5 −0.008 −0.005 2.4 −0.028 −0.012 2.3 −0.082 −0.022 2.2 −0.226 −0.035

Figures reproduced from arXiv:0708.2697 by the authors.

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Builds on it 4 papers

Now we build on the method of Ref. [ref] by relaxing the requirement that the solutions in eq.

Critical and multicritical Lee-Yang fixed points in the local potential approximation · Odd models · 0.96
The functional form of the RG flow in $d=8/3-ϵ$ can be obtained by promoting the single-field case of Ref. [ref] to arbitrary number of fields, which can be done unambiguously at LO and NLO without further computations [ref] .

RG and logarithmic CFT multicritical properties of randomly diluted Ising models · beta dc=8/3 · 0.95
In fact, by analysing the form of the $N=1$ beta functionals given in [ref] , one realises that there is only one way to enhance them to the multicomponent case.

Platonic Field Theories · Multicomponent beta functionals · 0.88

Uses its method or tool 12 papers

For this purpose, we use the powerful functional perturbative RG methods recently developed [ref] .

New universality class in three dimensions: The critical Blume-Capel model · Introduction · 1.00
We here outline how the perturbative results [ref] and [ref] are obtained following the background field methods, with a background field $$ and dimensional regularisation, used in [ref] .

Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation · Perturbative Calculations · 0.87
In this paper we consider the leading renormalization and $ϵ$-expansion of all the infinitely many multicritical models $ϕ^2n$ (and some other generalizations as well) in curved space using the formalism of functional perturbation theory [ref] .

Renormalization of multicritical scalar models in curved space · Introduction · 0.88

Uses its data or values 2 papers

According to the results obtained in [ref] b_n/a_n^2 = d_n/2(d_n+2) .

Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation · Equations for a Single Component Field · 0.86
However, in [ref] , the anomalous dimension $ϕ^k$ was given on closed form to order $^2$: [equation] where $k≠5$.[ It would be desirable to compute this anomalous dimension to the next order.

The tricritical Ising CFT and conformal bootstrap · Six-loop renormalisation · 0.88

Compares with it 9 papers

Interestingly enough we find conditions which can be simplified to match exactly the fixed point condition of the RG approach in its functional form [ref] which we dubbed functional perturbative RG approach.

Leading order CFT analysis of multi-scalar theories in d>2 · "Fixed point" equation from CFT · 1.00
In respect of these terms [ref] is identical with an analogous equation obtained in [ref] using an expansion in terms of scaling fields which is similar in spirit to the derivative expansion.

Reparameterisation Invariance and RG equations: Extension of the Local Potential Approximation · Equations for a Single Component Field · 1.00
This gives, after using [ref], η= 4(n-1)^2 n!^6/(2n)!^3ϵ^2 , in agreement with [ref] and recent CFT based computations [ref] .

Functional perturbative RG and CFT data in the $\epsilon$-expansion · General $ϕ^2n$ universality class · 0.95

Cites it as background 7 papers

Further results and a discussion on the universal character of the above results can be found in Ref. [ref] .

Critical and multicritical Lee-Yang fixed points in the local potential approximation · Even models · 0.85
Apart from the standard cases $ϕ^3,ϕ^4$ and $ϕ^6$ corresponding, respectively, to integer $d_c = 6,4,3$ and which have been extensively studied [ref] , upper critical dimensions are generally rational and their universal leading order (LO) and next-to-leading order (NLO) contributions appear at loop orders higher than one; for this reason they have attracted attention only recently [ref] .

Platonic Field Theories · Introduction · 0.82
Soon after the early developments of the Wilsonian action, it has been observed that the perturbative RG too can be conveniently formulated at functional level [ref] .

Leading order CFT analysis of multi-scalar theories in d>2 · Introduction · 0.88

Discussion

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