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arxiv: 2404.16026 · v2 · pith:Z4BJVD75new · submitted 2024-04-24 · ✦ hep-ph · hep-ex· hep-lat· hep-th

CP conservation in the strong interactions

Wen-Yuan Ai , Bjorn Garbrecht , Carlos Tamarit This is my paper

Pith reviewed 2026-05-25 08:17 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-lathep-th
keywords CP conservationstrong interactionsQCDtopological sectorsinstantonstheta parameterpath integralchiral effective theory
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The pith

Infinite spacetime volume requires taking the volume limit before summing over topological sectors to conserve CP in the strong interactions.

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

The paper establishes that topological quantization of the strong interaction arises only in the limit of infinite spacetime volume. This ordering of limits—volume to infinity first, before summing over topological sectors—ensures that CP is conserved. The reasoning aligns with building the path integral using steepest-descent contours. Replies are given to objections involving the theta parameter in three-form theories, the dilute instanton gas approximation, and the volume dependence of the partition function. The resulting chiral effective field theory matches analyses using partially conserved axial currents.

Core claim

Topological quantization in the strong interaction is a consequence of an infinite spacetime volume. Because of the ensuing order of limits, i.e. infinite volume prior to summing over topological sectors, CP is conserved. This is consistent with the construction of the path integral from steepest-descent contours.

What carries the argument

The order of limits in the path integral construction, where the infinite-volume limit is taken prior to summing over topological sectors, using steepest-descent contours.

If this is right

  • The chiral effective field theory derived from taking the volume to infinity first is in no contradiction with analyses based on partially conserved axial currents.
  • Objections based on the role of the CP-odd theta-parameter in three-form effective theories are addressed and do not support CP violation.
  • The correct sampling of all configurations in the dilute instanton gas approximation aligns with CP conservation under the proper ordering of limits.
  • The volume dependence of the partition function is consistent with CP conservation when the infinite-volume limit precedes the sum over topological sectors.

Where Pith is reading between the lines

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

  • This ordering of limits may resolve apparent tensions between different effective descriptions of the strong interactions.
  • Finite-volume effects in topological calculations require careful handling to avoid spurious CP violation.
  • Similar considerations of limit ordering could apply to other gauge theories that include a theta term.

Load-bearing premise

The physically correct procedure is to take the infinite-volume limit before summing over topological sectors, and that this ordering is consistent with the construction of the path integral from steepest-descent contours.

What would settle it

An explicit computation of the path integral showing that summing over sectors before the infinite volume limit produces CP-violating effects, or experimental detection of strong CP violation.

Figures

Figures reproduced from arXiv: 2404.16026 by Bjorn Garbrecht, Carlos Tamarit, Wen-Yuan Ai.

Figure 1
Figure 1. Figure 1: Upper panel: The integration contour given by the steepest-descent trajec [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the gauge potential components [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The field strength component F 3 21 for the BPST instanton solution as a function of x1 and x2 with x3 = x4 = 0, ρ = 1. 0 1 2 3 4 5 -100 -80 -60 -40 -20 0 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: trFµνF µν = trFµνFeµν as a function of |x| with ρ = 1 for the BPST instanton. Aμ ... ... ... −S −∞ [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Some projection of steepest-descent contours (thimbles) in the semiclassical [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic representation of the different effective field theories (EFTs) dis [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Leading order contribution to the neutron EDM from the strong interactions [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
read the original abstract

We discuss matters related to the point that topological quantization in the strong interaction is a consequence of an infinite spacetime volume. Because of the ensuing order of limits, i.e. infinite volume prior to summing over topological sectors, CP is conserved. Here, we show that this reasoning is consistent with the construction of the path integral from steepest-descent contours. We reply to some objections that aim to support the case for CP violation in the strong interactions that are based on the role of the CP-odd theta-parameter in three-form effective theories, the correct sampling of all configurations in the dilute instanton gas approximation and the volume dependence of the partition function. We also show that the chiral effective field theory derived from taking the volume to infinity first is in no contradiction with analyses based on partially conserved axial currents.

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 paper claims that topological quantization in the strong interaction is a consequence of infinite spacetime volume. Due to the order of limits (infinite volume prior to summing over topological sectors), CP is conserved. It shows this reasoning is consistent with constructing the path integral from steepest-descent contours and replies to objections based on the role of the CP-odd theta-parameter in three-form effective theories, the dilute instanton gas approximation, the volume dependence of the partition function, and demonstrates that the resulting chiral EFT is consistent with PCAC analyses.

Significance. If the central claim holds, the work would resolve the strong CP problem in QCD without new fields or mechanisms by reinterpreting the path integral in infinite volume. The explicit engagement with standard objections and the claim of consistency with chiral EFT are strengths that provide concrete points for evaluation.

major comments (2)
  1. [path integral from steepest-descent contours] The section on the path integral from steepest-descent contours asserts consistency with the V→∞ first ordering but does not derive why the contour construction uniquely enforces this ordering (rather than permitting the conventional fixed-θ then V→∞ limit). This choice directly produces the CP-conserving outcome and is load-bearing for the central claim.
  2. [replies to objections] In the replies to the three-form effective theory and dilute-gas objections, the manuscript does not address whether finite-volume lattice regularizations with nonzero θ remain admissible; if they are, the requirement to take V→∞ first would not be enforced by the regularization and the CP-conservation conclusion would not follow.
minor comments (1)
  1. The notation for the order of limits could be made more explicit with mathematical symbols (e.g., lim V→∞ then ∑_n) to reduce ambiguity in the discussion of the path integral.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful comments and constructive feedback. We address each major comment below, providing clarifications and committing to revisions where the presentation can be strengthened.

read point-by-point responses

  1. Referee: [path integral from steepest-descent contours] The section on the path integral from steepest-descent contours asserts consistency with the V→∞ first ordering but does not derive why the contour construction uniquely enforces this ordering (rather than permitting the conventional fixed-θ then V→∞ limit). This choice directly produces the CP-conserving outcome and is load-bearing for the central claim.

    Authors: We acknowledge that an explicit derivation of why the steepest-descent contour construction enforces the V→∞ limit prior to summing over sectors would strengthen the argument. The contours are formulated within the infinite-volume theory, where infinite action barriers separate topological sectors and preclude inter-sector mixing before the volume limit is taken. We will revise the section to include a step-by-step explanation showing that the conventional fixed-θ ordering is incompatible with this contour definition. revision: yes

  2. Referee: [replies to objections] In the replies to the three-form effective theory and dilute-gas objections, the manuscript does not address whether finite-volume lattice regularizations with nonzero θ remain admissible; if they are, the requirement to take V→∞ first would not be enforced by the regularization and the CP-conservation conclusion would not follow.

    Authors: We agree that explicitly addressing the status of finite-volume lattice regularizations with fixed nonzero θ would improve the replies to objections. Such regularizations are admissible as finite-volume approximations but correspond to a different ordering of limits; the physical theory requires the continuum infinite-volume limit to be taken first. We will add a clarifying paragraph explaining that these lattice setups do not contradict the central claim once the proper order of limits is enforced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on path-integral construction rather than self-reference.

full rationale

The paper asserts that infinite volume prior to summing topological sectors implies CP conservation and states that this ordering is consistent with steepest-descent contour construction of the path integral. It also replies to specific objections (three-form theories, dilute gas, partition function). No quoted step reduces the central claim to a fitted input, self-definition, or load-bearing self-citation by construction. The consistency argument is presented as independent verification against an external method (contours), not as a renaming or ansatz smuggled from prior work. The derivation chain therefore remains self-contained against the benchmarks given in the abstract.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument depends on standard QFT path-integral construction and the physical priority of the infinite-volume limit; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Infinite spacetime volume must be taken before summing over topological sectors
    This ordering is presented as the key to CP conservation and consistency with steepest-descent contours.

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Forward citations

Cited by 3 Pith papers

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

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  2. A particle on a ring or: how I learned to stop worrying and love $\theta$-vacua

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    The ACGT order-of-limits prescription fails to reproduce the correct θ-dependent energy spectrum in the quantum rotor and quantum pendulum.

  3. On the $\uptheta$-vacua and CP violation

    hep-th 2026-04 unverdicted novelty 4.0

    The θ-vacuum structure in QCD produces observable CP violation when the theory is consistently quantized with boundary degrees of freedom accounted for in the infinite-volume limit.

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