Absence of CP Violation in the Strong Interaction: Vacuum thwarts Axion

Gerrit Schierholz

REVIEW 4 major objections 4 minor 41 references

Absence of CP Violation in the Strong Interaction: Vacuum thwarts Axion

T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper claims that strong CP violation is absent in QCD because the neutron electric dipole moment vanishes as $\sqrt{\chi_t/V}\,|\theta|$ in the infinite-volume limit, making the axion unnecessary.

desk verdict An important question, but the central EDM scaling is not derived: the paper confuses local zero-mode density with the integrated theta-response, and the axion argument is circular. read the letter →

arxiv 2502.04092 v1 pith:ADH6H5PA submitted 2025-02-06 hep-lat hep-phhep-thnucl-th

Gerrit Schierholz This is my paper · ORCID

classification hep-lathep-phhep-thnucl-th PACS 12.38.Gc11.30.Er14.80.Va

keywords strongCPproblemneutronelectricdipolemomenttopologicalsusceptibilityDiraczeromodesaxionlatticeQCDthetavacuumconfinement

topics The Strong CP Problem

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 argues that the strong CP problem is resolved by QCD itself, without an axion. Its central result is that the neutron electric dipole moment, the observable that would expose CP violation in the strong interaction, vanishes in the infinite-volume limit, scaling as $|d_n| \propto \sqrt{\chi_t/V}\,|\theta|$ rather than approaching a $\theta$-independent constant. The paper further claims that QCD confines only at exactly $\theta=0$: for any nonzero vacuum angle the gluon condensate melts and the theory enters a deconfining phase. Finally, it argues that adding the axion forces the topological charge to zero, $\chi_t=0$, which invalidates the Witten-Veneziano relation and the standard axion-mass and axion-dark-matter predictions.

What carries the argument

The load-bearing object is the topological charge $Q$ written through the zero modes of the overlap Dirac operator via the Atiyah-Singer index theorem, $Q = n_- - n_+$, together with the vanishing theorem that only one chirality of zero modes occurs for a given $Q$. The argument combines the observed locality of these modes with cluster decomposition: zero modes far from the nucleon are invisible to it, so only the probability of a zero mode inside the nucleon's interaction volume matters. That probability is governed by the Gaussian distribution of $Q$ and the density $\langle Q^2\rangle/V = \chi_t$, yielding the $\sqrt{\chi_t/V}$ scaling. For the axion, the constant mode integral over the axion field collapses to a Kronecker delta enforcing $Q=0$ in Eq. (18), which is what makes $\chi_t=0$.

What would settle it

Run a direct lattice calculation of the neutron EDM at fixed $\theta$ on a sequence of volumes from say $(5\,{\rm fm})^4$ to $(20\,{\rm fm})^4$ at physical quark masses; if $|d_n|$ saturates at a nonzero volume-independent value instead of decreasing like $V^{-1/2}$, the paper's central claim fails. A second check is to compute the correlation between the topological charge density and a distant nucleon: observing long-range correlations would contradict the locality premise underlying Eq. (11).

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that $\theta$-dependence in hadronic correlators is carried entirely by localized zero modes of the Dirac operator. Because the density of these modes falls like $1/\sqrt{V}$, the chance of finding one inside a nucleon's interaction volume tends to zero as the volume grows, so the CP-odd part of the nucleon correlator vanishes and the neutron electric dipole moment goes as $\sqrt{\chi_t/V}\,|\theta| \to 0$. Strong CP is therefore conserved by QCD dynamics alone, and the experimental upper bound on $|d_n|$ does not force $\theta$ to be small. In the axion extension, integrating out the constant mode of the axion field restricts the path integral to topological charge $Q=0$, giving $\chi_t=0$ and removing the anomaly and the mass mechanism for the $\eta'$ and for the axion itself.

Load-bearing premise

The whole argument depends on the assumption that only localized zero-energy quark modes carry the $\theta$-dependence felt by a nucleon, and that all distant modes decouple cleanly; if long-range correlations exist, the dipole moment would not have to vanish as the volume grows.

Editorial extensions

If this is right

The experimental upper bound on the neutron electric dipole moment no longer constrains $\theta$; values of order one would be consistent with the measured $|d_n|$.
Strong CP is conserved by QCD itself, so the Peccei-Quinn mechanism is not needed to explain the absence of observed CP violation.
QCD is confining only at $\theta = 0$; at nonzero $\theta$ the gluon condensate vanishes and a deconfining phase sets in with screening length $\lambda_D = 0.31/|\theta|$ fm, so nucleons could disintegrate for $|\theta| \gtrsim 0.4$.
Other CP-violating hadronic quantities, such as the pion-nucleon coupling $\bar g_{\pi NN}$, also scale to zero with $\sqrt{\chi_t/V}$.
In the axion extension, $\chi_t = 0$ invalidates the Witten-Veneziano relation and the axion mass formula $m_a = \sqrt{\chi_t}/f_a$, including the 50(4) μeV axion dark-matter mass estimate.

Reading between the lines

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

Editorial extension: if the $\sqrt{\chi_t/V}$ scaling is universal, current finite-volume lattice computations of the neutron EDM (at volumes near $(10\,{\rm fm})^4$) should show a marked volume dependence, and the published nonzero estimates may drop substantially once extrapolated to $V\to\infty$.
Editorial extension: the same zero-mode counting argument could be applied to other CP-odd hadronic matrix elements and to $\theta$-dependent spectroscopy; a generic $V^{-1/2}$ falloff would be a clean signature to look for in existing ensembles.
Editorial extension: because the axion restriction to $Q=0$ suppresses topology entirely, the paper implies that axion phenomenology built on high-temperature $\chi_t$ would need to be rebuilt from the $Q=0$ sector, where confinement itself is in question.

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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 paper is a proceedings talk that packages the author's previous claims about QCD theta-dependence into a single narrative. The genuinely new material is limited: two plots of the gluon condensate as a function of theta (Fig. 3) and of the ratio of condensates in the Q=0 sector (Fig. 4). Both figures lack error bars and a description of the scale setting, which makes them hard to assess.

The talk is clearly written and the questions it asks are worth asking. The zero-mode localization pictures are nice, and the author is right that the standard naturalness argument for the strong CP problem is worth interrogating. But the central argument does not hold up.

The load-bearing step is Eq. (10) to (11). The author replaces the integrated topological charge Q by a sum over localized zero modes and then says the neutron EDM is proportional to the probability of finding a zero mode in a reference volume. That confuses a local density with a collective response. The first-order Hellmann-Feynman expression is a spacetime integral of a connected correlator; cluster decomposition says the integrand decays exponentially, so the integral converges to a finite constant. It does not produce a 1/sqrt(V) factor. The cited Leutwyler-Smilga paper does not support the exclusion of long-range correlations in this context. If the same logic were applied to the pion mass or chiral condensate, all theta-dependence would vanish in the thermodynamic limit, contradicting the standard theta-dependence that the paper itself relies on in Eq. (20). This is not a minor gap; it is the core.

The axion invalidation is circular. Integrating out the constant axion mode yields a delta function in Q, which presupposes that the axion has no QCD-generated potential. Then chi_t = 0 is a consequence of the assumption, not a result.

The deconfinement claim for theta > 0 rests on a single observable (the gluon condensate) from a 24^4 lattice with no errors, and the screening length is taken from a self-cited prior work. There is also no engagement with published lattice results on theta-dependence that disagree with these claims.

I would not cite this paper in my own work, and I wouldn't bring it to a reading group as a source of facts. But I would send it to a referee if it were submitted to a journal, purely because the claims are significant and the flaws are worth documenting in a report. A desk rejection would be defensible, but referee time is cheap relative to the importance of the question.

Recommendation: send to review, but expect rejection.

Referee Report

4 major / 4 minor

Summary. The paper claims that strong CP is conserved by QCD dynamics alone: the neutron electric dipole moment vanishes in the thermodynamic limit as |d_n| proportional to sqrt(chi_t/V) |theta| (Eq. 11), QCD deconfines for theta > 0 (Section 4), and the axion extension ends up with chi_t = 0 (Eq. 19), invalidating the Witten-Veneziano relation and axion-mass predictions. The argument is built on the distribution of overlap-Dirac zero modes, a Gaussian topological-charge distribution, and lattice data for the gluon condensate. The central derivation, however, relies on an unproved locality assumption in the passage from Eq. (10) to Eq. (11), and the axion conclusion depends on a self-cited treatment in which integrating out the constant axion mode projects onto Q = 0. The manuscript does not provide a self-contained derivation of either step, and the conclusions contradict standard chiral perturbation theory and existing lattice results.

Significance. If the claims were correct, they would overturn the standard understanding of theta-dependence in QCD, eliminate the need for the axion, and change axion dark-matter predictions. The paper does offer some concrete lattice observations about zero-mode localization and the topological-charge distribution, and it cites explicit numerical results in Figs. 1-4. However, the load-bearing derivations are not established: the volume suppression of the neutron EDM is based on a locality assumption that is not derived, and the axion section concludes chi_t = 0 on the basis of a circular treatment of the constant axion mode. The extraordinary conclusions therefore do not follow from the presented evidence.

major comments (4)

[Section 3, Eq. (10)-(11)] The derivation of the volume suppression is not valid. Equation (10) is a first-order expansion in theta in which the topological charge Q is a full spacetime integral; the connected correlator <O q(x)>_c decays exponentially for large separations by cluster decomposition, so the integral gives a finite, volume-independent contribution in the thermodynamic limit. The text instead replaces Q by a sum over local zero modes, restricts attention to a reference volume V0 = (2.5 fm)^4, and interprets the result as the probability of finding a zero mode in the nucleon's interaction range. This is not equivalent to the collective response to a uniform topological-charge density Q/V. The statement "We can exclude long-range correlations [16]" is not established by the cited Leutwyler-Smilga paper, which derives finite-volume spectral sum rules but does not address the locality of theta-dependence in nucleon correlators. If the same logic were applied to the pion mass or the chiral condensate, all theta-dependence would vanish in the thermodynamic limit, contradicting the theta-dependence assumed in Eq. (20). The paper does not derive Eq. (11) from QCD; it imposes a locality assumption whose content is precisely the claim at issue.
[Section 5, Eq. (19)] The conclusion chi_t = 0 in the axion extension is obtained by citing the author's own treatment [31] in which integrating out the constant axion mode produces a Kronecker delta delta_{Q,0}. This step presupposes that the QCD theta-dependence of the integrand is absent, since a non-trivial theta-dependence would give the constant mode a potential and the integral would not project onto Q = 0. The manuscript does not supply an independent derivation; it merely states that "the limitation to trivial topology is just a question of choice." That is circular: one cannot conclude that the QCD vacuum has no topological susceptibility from an integration over the axion field unless one has already assumed that the axion has no QCD-generated potential. This is load-bearing for the claimed invalidation of the Witten-Veneziano relation and axion mass predictions.
[Section 4, Figs. 3-4] The claim that "the theory does not confine for theta > 0" is based on a single lattice observable, the gluon condensate on a 24^4 lattice at one lattice spacing, plus a screening radius taken from the author's earlier work [30]. No direct evidence is given that the static potential is screened, that the nucleon disintegrates, or that a genuine phase transition occurs for |theta| < pi. The Gaussian width in Fig. 3 is not extrapolated to the continuum or thermodynamic limit, and the scale/scheme dependence of the width is acknowledged but not controlled. As stated, the deconfinement conclusion is not supported by the presented data.
[Section 2, Eq. (14)] The infrared behavior alpha(mu) = Lambda^2/mu^2 is asserted to follow from a proof in the author's prior work [24], but the current manuscript does not reproduce or summarize the argument. Since the scale dependence of the gluon condensate in Fig. 4 and the deconfinement conclusion depend on this relation, the result is not self-contained; a self-citation cannot carry this load.

minor comments (4)

[Throughout] There are numerous typographical and OCR artifacts, for example "asyptotically linear" in Section 4, "ration" for "ratio" in Section 3, and "the set nonperturbative framework" in the Conclusions; these should be corrected in a revised version.
[References] Reference [1] appears garbled: the neutron EDM bound is not Phys. Lett. B 427 (1998) 125; the authors should cite the actual experimental paper (Abel et al., 2020) with the correct journal and year.
[Fig. 2] The caption of Fig. 2 is unclear about the units and normalization of the horizontal axis; the notation "Q (fm4/V)" should be explained, and the relation between Q and the number of zero modes N should be stated explicitly.
[Section 3] The "femto universe" reference volume V0 is introduced heuristically, and the text says its exact size does not matter for the final conclusion; however, the numerical estimates (N = 0.06 on a (10 fm)^4 volume and N = 0.015 on a (20 fm)^4 volume) depend on the chosen V0, so the role of this parameter should be clarified.

Circularity Check

2 steps flagged · score 7.0 of 10

Two load-bearing conclusions reduce to the paper's own ansatz/self-citations: Eq. (11)'s 1/sqrt(V) EDM suppression is obtained by replacing the exact spacetime integral of Eq.

ansatz smuggled in via citation [Section 3, between Eq. (10) and Eq. (11), femto universe paragraph]
"Given that the zero mode eigenfunctions extend over a fraction of a fermi only, the dipole moment (10) can be visualized by a fermion-line disconnected diagram of two rather local operators. The result will depend on the probability of finding a zero mode within the interacting range. If the zero modes are out of range, it does not matter whether they have positive or negative chirality, due to the absence of long-range interactions and the cluster decomposition property [19]. ... We can exclude long-range correlations [16]."

Eq. (10) is exact: d_n ~ theta times a connected correlator with Q = sum_i integral psi_i^+ gamma5 psi_i integrated over all spacetime. Cluster decomposition makes the connected correlator decay with separation, so the integral converges to a volume-independent constant. The factor 1/sqrt(V) in Eq. (11) comes instead from counting expected zero modes in the fixed 'femto universe' V0, i.e. from the density (6), not from Eq. (10). The justification for discarding distant modes is the self-citation [19] (same author, 'Absence of strong CP violation') plus [16] (Leutwyler-Smilga), which does not state any such suppression for hadronic correlators. Thus Eq. (11) is the locality/counting ansatz restated as a prediction.
self definitional [Section 5, 'The axion intrusion', Eqs. (16)-(19)]
"The integral over bar-phi can be carried out [31], which results in a Kronecker delta function, 2 pi f_a delta_{Q,0}. ... The limitation to the sector Q=0 has far reaching consequences. First and foremost chi_t = <Q^2>/V = 0 (19)."

The axion action (15) is constructed with exact shift symmetry and no axion potential; integrating out the constant mode is the Fourier projection onto Q=0. chi_t=0 (19) is therefore already fixed by the ansatz, not derived from QCD. The delta-function step is attributed to [31], the author's own prior axion treatment. Moreover, the chi_t used in (20) and (21) is the fixed-theta QCD susceptibility; substituting the full-axion susceptibility (19) and declaring those relations invalid redefines the quantity, making the 'invalidation' definitional.

full rationale

The paper's genuinely new lattice data (Fig. 3, Fig. 4) are not themselves circular, and the zero-mode localization observations [5-7] might be legitimate empirical input. However, the two central conclusions of the talk are not derived from QCD alone. (1) The EDM result Eq. (11) follows from Eq. (10) only after the unproved step of replacing the full spacetime integral over topological-charge density by the probability of finding a zero mode in a fixed subvolume. That step is justified by self-citation [19] (the author's prior 'Absence of strong CP violation') and by a citation to Leutwyler-Smilga that does not establish the claimed 1/sqrt(V) suppression for a hadronic correlator; indeed cluster decomposition, which the paper invokes, would give a volume-independent integral. (2) The claimed invalidation of Witten-Veneziano and the axion mass is a consequence of the shift-symmetric axion ansatz: integrating the constant axion mode yields delta_{Q,0}, so chi_t=0 is built into the starting Lagrangian (15)-(16). The citation [31] for this step is again the author's own treatment. These are not cases of malicious circularity but of load-bearing assumptions being carried by self-citations and by the paper's own construction. Because the central 'prediction' (chi_t=0) is equivalent by construction to the input, and Eq. (11) is an ansatz restated with a citation rather than a derivation, the overall circularity score is 7. The correct independent content (lattice gluon condensate at theta>0, zero-mode localization) is ancillary and does not rescue the two main claims from circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central claims rest on several non-standard assumptions: the locality and rarity of zero modes as the sole carriers of theta-dependence, the Gaussian form of the topological charge distribution, the self-cited proof of infrared slavery, and the self-cited delta-function treatment of the axion zero mode. None of these are derived in this paper or verified against independent results.

free parameters (3)

Reference volume V0 (femto universe) = ~ (2.5 fm)^4
Chosen as the scale over which strong interactions are confined to estimate the probability of encountering a zero mode. The paper claims the exact size does not matter, but the prefactor in the EDM estimate depends on it.
Gaussian width of the gluon condensate vs theta = not reported
Fig. 3 is 'well approximated by a Gaussian' whose width is said to decrease with the renormalization scale mu; no fit parameters or uncertainties are given.
Screening length prefactor = 0.31 fm per unit |theta|
Taken from the author's prior work [30] and used to argue that the nucleon disintegrates for |theta| >~ 0.4.

assumptions (6)

standard math On a compact manifold with topological charge Q, the Dirac operator has exactly |Q| zero modes of one chirality only (vanishing theorem).
Invoked in Section 2 to justify that Q = n_- - n_+ with only one chirality present; proven in [9-11].
domain assumption The probability distribution of the topological charge is Gaussian and the topological susceptibility is volume-independent.
Assumed in Eq. (5) and used to derive the zero-mode density Eq. (6). Common in large-volume QCD but not rigorously established to the precision required here.
domain assumption Zero modes are localized objects and zero modes far from the nucleon decouple via cluster decomposition, so only zero modes within the nucleon's interaction range contribute to the electric dipole moment.
This is the load-bearing assumption of Section 3; the paper states 'We can exclude long-range correlations [16]' and uses it to conclude d_n ∝ sqrt(chi_t/V). This assumption is not derived and is contested by chiral perturbation theory.
domain assumption For small theta the vacuum is unchanged to first order, so first-order perturbation theory in theta applies to the nucleon correlator.
Stated in Section 3 and used to write Eq. (10); the paper later claims the vacuum changes drastically for any theta>0, which casts doubt on the consistency of this assumption.
ad hoc to paper The infrared running coupling obeys alpha(mu) = Lambda^2 / mu^2 (infrared slavery), with a proof in the author's prior work [24].
Invoked in Section 4 to connect the gluon condensate to confinement; the cited proof is by the same author and not independently established.
ad hoc to paper Integrating out the constant axion mode yields a Kronecker delta delta(Q), restricting the theory to the topologically trivial sector.
This step, taken from the author's prior paper [31], is central to the axion invalidation but is treated without the axion potential generated by QCD; it leads to chi_t=0 by construction.

invented entities (1)

Femto universe
purpose: A notional reference volume of about (2.5 fm)^4 used to argue that the probability of finding a zero mode within the nucleon's interaction range is small and vanishes with the total volume.
Introduced as a heuristic device in Section 3; it is not a physical object and has no independent experimental handle.

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Cite this review

Pith. "Pith review of Absence of CP Violation in the Strong Interaction: Vacuum thwarts Axion." pith.science (2026). https://pith.science/paper/ADH6H5PA

@misc{pith2026250204092,
  author       = {Pith},
  title        = {Pith review of: Absence of CP Violation in the Strong Interaction: Vacuum thwarts Axion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADH6H5PA}},
  note         = {Machine review of arXiv:2502.04092}
}

abstract

QCD admits a contribution to the action, the $\theta$ term, which potentially gives rise to nontrivial phases and violates CP. This is essentially a question of how the vacuum reacts to the $\theta$ term. In this talk I will address the problem using new developments on the lattice. The overall solution is contrasted with the axion `solution'.

Figures

Figures reproduced from arXiv: 2502.04092 by the authors.

**Figure 1.** The density of a typical zero mode in a given time slice on the 12 3 × 24 lattice at lattice spacing = 0.125 fm. Shown is the isosurface | †| = 0.0005 as given in [5]. Results are from a cluster analysis [7]. modes, but never zero modes of both chiralities. In mathematical language dim ker/ = |+ − − |. This so-called ‘vanishing theorem’ has been proven rigorously in [9–11]. On the lattice the absence of zero modes o… view at source ↗

**Figure 2.** The fraction of zero modes in a subvolume of one fermi cubed as a function of = ± for total volumes of (5 fm) 4 , (10 fm) 4 and (20 fm) 4 . Scale setting is described in the text. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The gluon condensate in lattice units as a function of on the 244 lattice at lattice spacing = 0.082 fm for flow time / 2 = 60. The gluon condensate in the vacuum, (), is obtained from (8). The determination follows the same pattern as the calculation of the running coupling constant [25]. In [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The ratio of gluon condensates =0/ as a function of the scale parameter on the 324 lattice at lattice spacing = 0.082 fm. There are indications that the Peccei-Quinn theory does not even confine. In [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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