Parisi-Sourlas Supertranslation and Scale without Conformal symmetry

Yu Nakayama

REVIEW 3 major objections 5 minor 52 references

Parisi-Sourlas Supertranslation and Scale without Conformal symmetry

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Parisi-Sourlas supertranslation fixes the virial current's dimension to d-1, making eight interacting fixed points scale invariant but not conformal.

desk verdict A genuinely new mechanism for protecting the virial current—Parisi-Sourlas supertranslation—with a clean non-perturbative dimension argument, though the anomaly-free assumption needs scrutiny. read the letter →

arxiv 2411.12934 v2 pith:F7TDP54P submitted 2024-11-19 hep-th cond-mat.dis-nncond-mat.stat-mech

Yu Nakayama This is my paper · ORCID

classification hep-thcond-mat.dis-nncond-mat.stat-mech MSC 81T1781T6082B28

keywords scaleinvariancewithoutconformalParisi-Sourlassupersymmetrysupertranslationvirialcurrentnon-renormalizationrenormalizationgroupfixedpointsrandomfieldIsingmodelepsilonexpansion

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 tries to establish that scale invariance without conformal invariance can occur in an interacting field theory when a Parisi-Sourlas supertranslation symmetry is present. Working in a $4-\epsilon$ dimensional expansion, it studies a quartic superfield potential and finds nine nontrivial scale-invariant renormalization group fixed points, eight of which are not conformal. The usual genericity argument says such fixed points require a non-conserved but non-renormalized vector operator, the virial current, whose existence should be non-generic. The paper shows that, here, the virial current is related to the conserved supercurrent by supertranslation, and a Ward-Takahashi identity fixes its scaling dimension to exactly $d-1$. If correct, this is a second protecting mechanism (after shift symmetry) and makes the eight fixed points scale invariant without conformal symmetry.

What carries the argument

The load-bearing object is the Parisi-Sourlas supertranslation symmetry of the action $S=\int d^d x\, d^2\theta\,(-\tfrac12\Phi\partial^2_\mu\Phi + \text{quartic superfield terms})$, generated by the charge $Q=\int d\Sigma^\mu q_\mu$ with supercurrent $q_\mu=\Psi\partial_\mu\omega-\omega\partial_\mu\Psi$. The identity that carries the proof is $[Q,V_\mu]=q_\mu$, linking the non-conserved virial current $V_\mu=\omega\partial_\mu\phi-\phi\partial_\mu\omega$ to the conserved supercurrent. Combined with the scale-charge commutator $[D,Q]=(\Delta_q-d+1)Q$, it forces $\Delta_V=d-1$; the Ward-Takahashi identity (8) gives the same result from correlation functions. The paper abstracts this as: any symmetry $X$ whose conserved current $x_\mu$ satisfies $[X,V_\mu]=x_\mu$ protects the virial current's dimension. Shift symmetry is the previously known instance, and the supertranslation is the new one.

What would settle it

Compute the two-point function of the virial current at two or more loops at one of the stable scale-only fixed points, for example I-5 or I-6. If its scaling dimension is not exactly $d-1$, or if the supertranslation Ward-Takahashi identity (8) fails, the central claim is false.

Watch

Extended reading notes

Core claim

The central claim is that the virial current $V_\mu = \omega \partial_\mu \phi - \phi \partial_\mu \omega$ carries no anomalous dimension: its scaling dimension is exactly $d-1$, even though the theory is not conformal. The argument is non-perturbative in structure. Let $Q=\int d\Sigma^\mu q_\mu$ be the conserved supertranslation charge and $D$ the scale charge, with $\Delta_q$ the dimension of the supercurrent. From $[D,Q]=(\Delta_q-d+1)Q$ and $[Q,V_\mu]=q_\mu$, acting with $D$ gives $\Delta_Q+\Delta_V=\Delta_q$, hence $\Delta_V=d-1$ regardless of $\Delta_q$. The same conclusion follows by equating scaling dimensions in the supertranslation Ward-Takahashi identity $\langle \partial^\mu q_\mu(x_1)V_\nu(x_2)O(x_3)\rangle=\delta^d(x_1-x_2)\langle q_\nu(x_2)O(x_3)\rangle$. Within the one-loop $\beta$ functions, demanding scale invariance rather than full vanishing of the $\beta$ functions, with the extra parameter $\gamma$ from the virial-current field redefinition, yields one conformal and eight non-conformal fixed points.

Load-bearing premise

The whole argument rests on assuming that the supertranslation charge is actually conserved at the interacting fixed point and that the relation $[Q,V_\mu]=q_\mu$ is exact; if either fails, the formula $\Delta_V=d-1$ collapses.

Editorial extensions

If this is right

The eight non-conformal fixed points of Table I are scale-invariant but not conformal interacting fixed points within the $\epsilon$ expansion, with the virial current protected at dimension exactly $d-1$.
Scale invariance without conformal invariance is therefore not ruled out by the genericity argument in this model; supertranslation provides an explicit non-generic mechanism.
The original Parisi-Sourlas model with the larger superrotation symmetry $OSp(d|2)$ remains conformal, so scale-only behavior requires breaking superrotation down to supertranslation.
In the random-field Ising model without fine-tuning, where supertranslation symmetry is lost in lower dimensions, the genericity argument suggests the critical point should be conformal invariant.
The mechanism is not limited to supersymmetry: the parallel shift-current argument protects the virial current at fixed points where a shift symmetry is present.

Reading between the lines

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

Beyond the paper: if the supertranslation charge is not exactly conserved under a particular regulator, the identity $\Delta_V=d-1$ would break; the sharpest test is a two-loop computation of $\langle V_\mu(x)V_\nu(0)\rangle$ at fixed point I-5 or I-6.
The abstract version of the mechanism suggests a search strategy for scale-without-conformal fixed points in other non-unitary models: look for any conserved charge $X$ such that $[X,V_\mu]=x_\mu$, not necessarily tied to supersymmetry.
For the two mysterious non-supersymmetric fixed points with no obvious protecting symmetry, the paper's own genericity logic implies their virial current should acquire an anomalous dimension at higher loops; checking this would either confirm or challenge the genericity argument.
The $PT$-symmetric bosonic reduction of Appendix A offers a simpler laboratory: the scale-only fixed points with only a $Z_2$ symmetry can be studied numerically to see whether the protected dimension $d-1$ survives beyond perturbation theory.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The main novelty here is real: instead of shift symmetry, the paper uses Parisi-Sourlas supertranslation to protect the virial current. The one-loop beta functions in Eq. (6) are explicit and mutually consistent, and the eight non-conformal scale-invariant fixed points in Table I are a concrete counterexample to the genericity argument. The Section IV argument that [Q,V]=q forces Δ_V=d−1 even when Δ_Q is undetermined is elegant, and it is non-perturbative given its assumptions.

The paper is also refreshingly honest. It states plainly that the model is fine-tuned and non-unitary, and it leaves the mysterious fixed points without supertranslation as an open problem rather than overclaiming. The Appendix B discussion of beta-function ambiguities is a useful methodological aside.

The soft spot is the one the stress-test flags: the exact dimension argument relies on a conserved, non-anomalous supertranslation charge and the operator relation [Q,V]=q at the interacting fixed point. These are classical statements. In this non-unitary, analytically continued model, the paper does not show that the renormalized current stays conserved, nor does it specify a regulator that manifestly preserves supertranslation. If a supertranslation anomaly appeared, Eq. (8) would acquire extra terms and Δ_V=d−1 would not follow. That would not necessarily kill the six fixed points, because the SO(1,1) argument provides a separate route for some of them, but it would undercut the paper's central claim. In my view this is a gap to press in review, not a demonstrated flaw. Nothing in the computation suggests an anomaly is actually present.

The SO(1,1) fallback itself assumes a conserved current has no anomalous dimension, which without unitarity is not automatic, but the triangular structure for the λ1=0 cases makes it plausible.

Who is this for: anyone working on scale versus conformal invariance, emergent supersymmetry in random systems, or non-unitary fixed points. It deserves a serious referee. I would send it to review, asking for an explicit anomaly/regulator discussion and a check that the supertranslation current is conserved in dimensional regularization. Minor: the tables are not re-derived, but the beta functions are simple enough to verify.

Referee Report

3 major / 5 minor

Summary. The paper studies a Euclidean field theory of one superfield with a quartic potential that preserves only Parisi-Sourlas supertranslation symmetry, not superrotation symmetry. In d = 4 − ε, the author computes one-loop beta functions (Eq. (6)) and searches for scale-invariant fixed points, relaxing the conformal condition by allowing the trace of the stress tensor to be a divergence of a non-conserved virial current V_μ = γ(ω∂_μ ϕ − ϕ∂_μ ω). This leads to Eq. (7) and, in Table I, nine nontrivial fixed points, one conformal and eight scale-invariant but non-conformal. The central mechanism is proposed in Section IV: the virial current is related to the supertranslation current by a commutator, and the resulting Ward-Takahashi identity fixes Δ_V = d − 1 exactly, explaining the non-renormalization of the virial current. Section VI extends the beta-function analysis to quartic theories without supertranslation symmetry, finding further conformal and scale-invariant fixed points, including some for which no protecting symmetry is identified.

Significance. If the main claims hold, the paper supplies a new, concrete mechanism for scale invariance without conformal invariance that is not based on shift symmetry, and it gives an explicit non-perturbative argument for the exact scaling dimension of the virial current. The fixed-point classification in a supertranslation-invariant quartic superfield model is also potentially useful for debates about emergent supersymmetry in random systems. The paper is honest about its limitations, including the perturbative nature of the fixed-point search and the existence of unexplained 'mysterious' fixed points in the non-supersymmetric extension. The algebraic commutator argument for Δ_V = d − 1 is elegant and internally consistent; however, it rests on an unverified assumption about the absence of supertranslation anomalies, and one entry in the central fixed-point table does not satisfy the defining equations.

major comments (3)

[Section III, Table I (row I-1)] Row I-1 of Table I does not satisfy the scale-invariance condition (7). Substituting (λ1, λ0, λ−1, λ−2, γ) = (0, 0, ε, −ε, ε/2) into the fourth equation gives β_{λ−2} = −ε(−ε) + 8ε² = 9ε², while the required right-hand side is −4γλ−2 = 2ε². Solving the equations with λ1 = λ0 = 0 and λ−1 = ε gives γ = ε/2 and λ−2 = −8ε, not −ε. This is a concrete error in the central numerical claim that eight scale-invariant non-conformal fixed points were found; the row must be corrected and the remaining rows re-verified.
[Section IV, Eqs. (8) and the commutator argument] The non-perturbative proof that Δ_V = d − 1 presupposes that the supertranslation current q_μ is conserved in the interacting quantum theory, that the operator relation [Q, V_μ] = q_μ holds, and that the Ward-Takahashi identity (8) contains only the displayed contact term. The model is non-unitary and analytically continued, so standard unitarity/reflection-positivity arguments cannot be invoked to exclude a global current anomaly. The paper does not specify a regulator, check scheme independence, or rule out anomalous non-contact terms in (8). If ∂·q develops an anomaly, the dimension counting in (8) no longer yields Δ_V = d − 1, and the eight fixed points would not be established as genuine scale-without-conformal theories. Please either provide a one-loop (or all-orders) check of the supertranslation Ward-Takahashi identity in this model, or state the result as conditional on the absence of such an anomaly.
[Section IV, operator O and fixed points with λ−1 = 0] The Ward-Takahashi derivation in Section IV introduces a Q-invariant operator O whose existence, as stated, requires λ−1 ≠ 0, but several fixed points in Table I (e.g., I-3, I-4, I-5, I-6) have λ−1 = 0. The paper says the condition can be relaxed at the expense of extra terms in (8), but this is only sketched. Since the exact-dimension conclusion is claimed for all scale-invariant fixed points, the argument should be written in a way that covers the λ−1 = 0 cases explicitly, or the scope of the proof should be narrowed and stated.

minor comments (5)

[Section III, Eq. (6) and Section VI, Eq. (11)] The one-loop beta functions are stated without derivation or reference. Given that the entire fixed-point search rests on these expressions, an appendix with the Feynman-diagram computation or an explicit reference to a standard computation would greatly improve verifiability.
[Section IV, notation around q_μ] The current q_μ is Grassmann-odd, so the relation [Q, V_μ] = q_μ should specify whether the bracket is a graded commutator and state the convention used for the supercharge and the superspace measure; this would remove potential ambiguity for readers.
[General presentation] The abstract and introduction speak of 'nine non-trivial scale invariant fixed points' and 'eight scale-invariant but not conformal fixed points'; after the Table I correction, please verify the counts and the wording in the conclusion.
[Section VI, paragraph on mysterious fixed points] The two 'mysterious' fixed points without any identified protecting symmetry are intriguing but also challenge the genericity argument. Consider stating more explicitly that their existence does not invalidate the supertranslation mechanism, but that they point to additional mechanisms or to limitations of the one-loop analysis.
[Typographical and stylistic details] There are minor typographical issues, such as 'Confor mal' in the title line and inconsistent use of 'non-renormalized' / 'not renormalized'; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exact virial dimension follows from the supertranslation algebra, and the fixed-point search solves algebraic equations with gamma as a variable, not as a fitted target.

full rationale

The central derivation in Section IV is a non-perturbative group-theoretic consequence of operator relations that are derived, not assumed as the conclusion. The paper defines the virial current candidate as V_mu = omega(mu phi) - phi(mu omega) and the supercurrent as q_mu = Psi(mu omega) - omega(mu Psi), then observes that both are components of the same superfield Phi(mu theta-bar theta Phi) - (theta-bar theta Phi)(mu Phi). The relation [Q,V_mu] = q_mu follows from the supertranslation algebra, and the scale-dimension relation [D,Q] = (Delta_q - d + 1)Q is the standard charge-current relation. Combining these with the Jacobi identity gives Delta_V = d - 1 without assuming Delta_V; the conclusion is not an input, so this is not self-definitional. The Ward-Takahashi identity (8) is the standard contact-term identity for a symmetry, and dimensional matching of both sides gives the same result; it does not encode the conclusion. The fixed-point search in Section III solves the one-loop beta-function equations with gamma as an auxiliary variable determined algebraically by the system (7); gamma is not fitted to produce a desired set of fixed points, and the existence of nontrivial solutions is a genuine algebraic result. The beta functions themselves are stated as one-loop computations and the Ba-function identification is justified in Appendix B independently of the cited literature. The self-citations to [8] and [13] provide background on the genericity argument and the shift-symmetry mechanism; the shift-symmetry argument for fixed point I-3 is reproduced in the text and is not needed for the supertranslation mechanism, so the citations are not load-bearing. The paper honestly flags its own gaps: the non-perturbative argument assumes a conserved, non-anomalous supertranslation charge and the relation [Q,V_mu]=q_mu without a regulator/anomaly check, the 'mysterious' fixed points in Section VI are admitted to lack a known protecting mechanism, and Appendix B notes possible higher-order ambiguities. These are limitations or correctness risks, not circular steps. Overall, the derivation chain is self-contained and no prediction reduces by construction to its inputs.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on three main legs: the fine-tuned quartic superfield action, one-loop universality of the beta functions, and conservation or non-anomalousness of supertranslation. The first two are explicit modeling and perturbation assumptions; the third is a standard symmetry assumption but is the most structurally load-bearing for the exact dimension result.

free parameters (1)

gamma (virial parameter / anomalous dimension shift for phi and omega) = 0 at I-0; epsilon/2 at I-1; -epsilon/2 at I-2; epsilon/4 at I-3; -epsilon/2 at I-4; -epsilon/8 at I-5; epsilon/4 at…
Introduced in Eq. (7) to allow scale invariance without beta = 0 by absorbing the SO(1,1) field redefinition direction. It is determined algebraically at each fixed point rather than fitted to data, but it is the extra degree of freedom that enlarges the solution space.

assumptions (3)

ad hoc to paper The fine-tuned quartic superfield action (5) includes all supertranslation-compatible quartic interactions and is the correct starting point for the fixed point search.
The model is deliberately fine-tuned so that all classical dimensions are equal; this is not a generic random-system model but a controlled theoretical laboratory.
domain assumption The one-loop beta functions in Eqs. (6) and (11) are universal and, together with the Ba-function analysis in Appendix B, determine scale invariance at leading order in epsilon.
The scale-invariant fixed points are classified by solving these equations; Appendix B argues that derivative ambiguities cannot alter the one-loop conclusion.
domain assumption The supertranslation symmetry survives quantization as an exact, non-anomalous symmetry at the interacting fixed point, with a conserved supercurrent q_mu and charge Q satisfying [Q,V_mu] = q_mu.
Section IV commutation relations and Ward-Takahashi identity (8) depend on this without an explicit anomaly analysis.

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Cite this review

Pith. "Pith review of Parisi-Sourlas Supertranslation and Scale without Conformal symmetry." pith.science (2026). https://pith.science/paper/F7TDP54P

@misc{pith2026241112934,
  author       = {Pith},
  title        = {Pith review of: Parisi-Sourlas Supertranslation and Scale without Conformal symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7TDP54P}},
  note         = {Machine review of arXiv:2411.12934}
}

abstract

Inspired by the possibility of emergent supersymmetry in critical random systems, we study a field theory model with a quartic potential of one superfield, possessing the Parisi-Sourlas supertranslation symmetry. Within perturbative $\epsilon$ expansion, we find nine non-trivial scale invariant renormalization group fixed points, but only one of them is conformal. We, however, believe scale invariance without conformal invariance cannot occur without a sophisticated mechanism because it predicts the existence of a non-conserved but non-renormalized vector operator called virial current, whose existence must be non-generic. We show that the virial current in this model is related to the supercurrent by supertranslation. The supertranslation Ward-Takahashi identity circumvents the genericity argument, explaining its non-renormalization property.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 12, 2026 · model on record in the stance chip above.

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