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arxiv: 2606.28193 · v1 · pith:RTGT3S4Gnew · submitted 2026-06-26 · ✦ hep-th

Quantum anomalies from three-point on-shell bootstrap

Hiren Kakkad , R\'emy Larue , Alexander Ochirov This is my paper

Pith reviewed 2026-06-29 03:11 UTC · model grok-4.3

classification ✦ hep-th
keywords quantum anomalieson-shell bootstrapthree-point amplitudesWeyl anomalychiral anomalygauge anomaly cancellationhelicity spinorsdiffeomorphism anomaly
0
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The pith

Three-point on-shell bootstrap fixes Weyl, chiral, diffeomorphism and Lorentz anomalies up to real constants.

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

The paper demonstrates that quantum anomalies can be bootstrapped from three-point on-shell data alone. It requires only the classical symmetry that may be anomalous, the symmetries preserved by the effective action, and on-shell versions of C, P and T transformations, without needing the explicit form of the action. Massless helicity spinors then determine the possible anomaly structures for the Weyl, chiral (global and local), diffeomorphism and Lorentz cases. The method recovers the standard gauge anomaly cancellation conditions and shows that the Weyl anomaly cannot involve Pontryagin densities.

Core claim

Quantum anomalies are determined by three-point on-shell bootstrap using the input classical symmetry, effective-action symmetries, and discrete C, P, T transformations; this fixes the Weyl, chiral, diffeomorphism and Lorentz anomalies up to real constant prefactors, recovers gauge-anomaly cancellation, and excludes Pontryagin densities from the Weyl anomaly.

What carries the argument

Three-point on-shell bootstrap with massless helicity spinors and on-shell C, P, T transformations.

If this is right

  • The anomalies are fixed up to overall real constants by the input symmetries alone.
  • Gauge anomaly cancellation conditions emerge directly from the three-point data.
  • The Weyl anomaly cannot include Pontryagin densities.
  • Both global and local chiral anomalies are captured by the same procedure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method could be applied to four-point or higher amplitudes to constrain additional structures.
  • It implies that anomaly coefficients are largely fixed by on-shell kinematics and discrete symmetries rather than detailed loop integrals.
  • Similar bootstrap logic might constrain mixed anomalies or anomalies in curved backgrounds without explicit computations.

Load-bearing premise

The classical symmetry that the anomaly may break and the symmetries of the effective action are known, even though the explicit form of the action is not.

What would settle it

An explicit one-loop calculation that produces a Weyl anomaly containing a Pontryagin density while satisfying the same input symmetries would falsify the bootstrap result.

read the original abstract

We bootstrap quantum anomalies using on-shell techniques in the simplest setting: at three points. The necessary field-theoretic input includes the local or global classical symmetry, which the anomaly may break, and the symmetries of the effective action -- though not its explicit form. We use massless helicity spinors together with an on-shell representation of the discrete C, P, and T transformations. Our approach determines the Weyl, chiral (global and local), diffeomorphism, and Lorentz anomalies up to real constant prefactors. In particular, we recover gauge-anomaly cancelation conditions, as well as find that the Weyl anomaly should not involve Pontryagin densities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a three-point on-shell bootstrap to determine quantum anomalies (Weyl, chiral global/local, diffeomorphism, Lorentz) from classical symmetries, effective-action symmetries (without explicit form), massless helicity spinors, and on-shell C/P/T transformations. Anomalies are fixed only up to real constant prefactors; the method recovers gauge-anomaly cancellation conditions and concludes that the Weyl anomaly excludes Pontryagin densities.

Significance. If valid, the work supplies a streamlined on-shell route to anomaly structures that avoids constructing the full effective action, with explicit credit for recovering known cancellation conditions from three-point data alone and for using discrete symmetries to constrain the form. This could simplify anomaly analysis in gauge-gravity theories where higher-point correlators are cumbersome.

major comments (2)
  1. [Abstract] Abstract and the three-point bootstrap construction: the claim that three-point on-shell amplitudes plus the listed inputs suffice to fix all listed anomalies (and exclude Pontryagin terms) rests on the unverified assumption that every Wess-Zumino consistency condition is already encoded at three points; standard anomaly polynomials and descent equations involve integrated higher-point or multi-form data that are not directly probed here, and no explicit cross-check against four-point consistency or the full descent is supplied.
  2. [Weyl anomaly section] Weyl-anomaly derivation: the exclusion of Pontryagin densities is obtained from three-point helicity amplitudes; because the known four-dimensional Weyl anomaly polynomial contains a Pontryagin term whose coefficient is fixed by independent methods, an explicit demonstration that this term is ruled out already at three points (rather than only by higher-order conditions) is required to support the claim.
minor comments (1)
  1. [Notation and setup] The on-shell representation of discrete C, P, T transformations is stated but the explicit action on the helicity spinors and little-group factors is not tabulated; adding a short table or equations would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the three-point bootstrap construction: the claim that three-point on-shell amplitudes plus the listed inputs suffice to fix all listed anomalies (and exclude Pontryagin terms) rests on the unverified assumption that every Wess-Zumino consistency condition is already encoded at three points; standard anomaly polynomials and descent equations involve integrated higher-point or multi-form data that are not directly probed here, and no explicit cross-check against four-point consistency or the full descent is supplied.

    Authors: The three-point bootstrap is constructed so that the classical symmetry, effective-action symmetries (without explicit form), massless helicity spinors, and on-shell C/P/T transformations together constrain the anomaly structures. These inputs are chosen precisely to enforce the local consistency requirements that appear in the Wess-Zumino conditions at the lowest non-trivial order; the resulting anomaly expressions automatically satisfy the algebraic relations needed for consistency because any term violating them is excluded by the discrete symmetries or helicity counting. Gauge-anomaly cancellation emerges directly from this three-point data. While we do not supply an independent four-point verification, the method is not intended to replace the full descent but to isolate the anomaly coefficients from three-point information alone. We will add a clarifying sentence in the abstract and introduction stating that the listed inputs encode the relevant consistency conditions at three points. revision: partial

  2. Referee: [Weyl anomaly section] Weyl-anomaly derivation: the exclusion of Pontryagin densities is obtained from three-point helicity amplitudes; because the known four-dimensional Weyl anomaly polynomial contains a Pontryagin term whose coefficient is fixed by independent methods, an explicit demonstration that this term is ruled out already at three points (rather than only by higher-order conditions) is required to support the claim.

    Authors: The Pontryagin density is excluded because its on-shell three-point reduction produces helicity amplitudes that are incompatible with the assumed C, P, T transformation properties of the Weyl anomaly (specifically, the parity-odd structures generated by the epsilon tensor and curvature insertions fail to match the required transformation under the discrete symmetries while preserving the effective-action symmetries). This incompatibility is visible directly in the three-point helicity counting and does not rely on higher-point data. We will expand the relevant paragraph in the Weyl-anomaly section to display the explicit three-point amplitude structures that rule out the Pontryagin contribution. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external classical symmetries as independent input

full rationale

The paper states its inputs as the local/global classical symmetries (which the anomaly may break), symmetries of the effective action (without explicit form), and on-shell C/P/T representations. These are presented as external field-theoretic data, not derived within the paper. The output is the anomaly structures fixed only up to real constants, with no equations shown that reduce the anomaly coefficients to a fit or self-definition of the input symmetries. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The three-point bootstrap is therefore self-contained against the stated external benchmarks; higher-point consistency concerns belong to correctness risk rather than circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that classical symmetries and effective-action symmetries are known inputs, that an on-shell representation of C/P/T exists, and that three-point amplitudes suffice to fix anomaly structures up to constants. No new particles or forces are introduced. The real constant prefactors are free parameters left undetermined by the bootstrap.

free parameters (1)
  • real constant prefactors for each anomaly
    The bootstrap fixes the anomaly structures only up to overall real numbers; these constants are not derived from the three-point data.
axioms (2)
  • domain assumption The local or global classical symmetry and the symmetries of the effective action are known inputs (abstract).
    These symmetries are supplied externally and not derived inside the bootstrap.
  • domain assumption An on-shell representation of discrete C, P, and T transformations exists and is compatible with massless helicity spinors.
    Invoked to encode the discrete symmetries in the three-point amplitudes.

pith-pipeline@v0.9.1-grok · 5629 in / 1613 out tokens · 40725 ms · 2026-06-29T03:11:58.804922+00:00 · methodology

discussion (0)

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    But such a denominator may be absorbed by a judicious choice of the basis vector: Abµ gA(1−2,2 −1,3) =κtrt b r C−⟨1|σµ|2]⟨12⟩3,(30b) Abµ gA(1+2,2 +1,3) =κtrt b r C+[1|¯σµ|2⟩[12]3.(30c) Here the gauge field must clearly be abelian and, byCin- variance in the Dirac-fermion case, axial. Moreover,CPT symmetry impliesC + =−C ∗ − =: iQ′C, whereas parity convert...

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