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arxiv: 2606.14641 · v2 · pith:3BD7LHWHnew · submitted 2026-06-12 · ✦ hep-th · math-ph· math.MP· nlin.SI

On the gauge-invariant dynamical charges and densities of the 1-instanton solution

C. A. da Silva , L. A. Ferreira This is my paper

Pith reviewed 2026-06-27 04:28 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPnlin.SI
keywords instantonYang-Mills theorygauge-invariant chargesnon-Abelian Gauss lawmagnetic fluxelectric fluxSU(2)theta vacuum
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The pith

The 1-instanton solution carries non-zero observable gauge-invariant magnetic and electric fluxes at its center.

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

The paper applies gauge-invariant charges previously built from integral equations on generalized loop spaces to the explicit 1-instanton solution of SU(2) Yang-Mills theory in four-dimensional Euclidean space. It evaluates the integral non-Abelian Gauss law to extract the flux of the magnetic and electric fields through spheres centered at the instanton. From these fluxes it defines corresponding charge densities inside thin spherical shells whose radius is scaled by the instanton size parameter. The resulting fluxes are shown to be non-zero and reparameterization-invariant for both the instanton and anti-instanton when the scaled radius equals one and Euclidean time is zero. This supplies a concrete picture of the internal charge distribution inside the instanton and links it to properties of the Yang-Mills theta vacuum.

Core claim

The magnetic and electric fluxes of the 1-instanton and anti-instanton, evaluated via the integral non-Abelian Gauss law on spherical surfaces centered at the origin, are non-zero and gauge-invariant at r=1 and x^4=0, where r is the scaled radial coordinate; these fluxes therefore define observable gauge-invariant charge densities that describe the internal structure of the instanton.

What carries the argument

Integral non-Abelian Gauss law applied to spherical surfaces centered on the instanton, which converts the conserved charges into gauge-invariant magnetic and electric fluxes.

If this is right

  • The magnetic and electric fluxes remain non-zero and observable for both instanton and anti-instanton at the stated surface.
  • Charge densities obtained from the flux through infinitesimal shells are reparameterization-invariant.
  • The internal structure of the instanton is characterized by a non-trivial distribution of these gauge-invariant charges.
  • The same construction may constrain the role of instantons in the Yang-Mills theta vacuum.

Where Pith is reading between the lines

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

  • Observable fluxes could serve as a diagnostic for single-instanton configurations in lattice calculations.
  • Non-vanishing central fluxes suggest that instanton-induced tunneling carries an intrinsic charge signature that might modify vacuum-to-vacuum amplitudes.
  • The method could be applied to multi-instanton configurations to extract interaction-induced charge redistributions.

Load-bearing premise

The gauge-invariant charges built from loop-space integral equations remain dynamically conserved when inserted into the explicit 1-instanton field configuration.

What would settle it

Numerical evaluation of the integral non-Abelian Gauss law on the standard BPST 1-instanton solution that returns exactly zero flux at scaled radius r=1 and x^4=0.

Figures

Figures reproduced from arXiv: 2606.14641 by C. A. da Silva, L. A. Ferreira.

Figure 1
Figure 1. Figure 1: Intermediary surface scanning Ω, as constructed in [14]. The point xR is at (−∞, 0, 0). With this intermediary surface, one then simply inflate it to scan Ω, until the sphere reaches ζ = ζf . In fact, this expansion must start from the infinitesimal closed surface at the reference point xR . In other words, before inflating the surface of the sphere, one must first inflate the thin cylinder until it reache… view at source ↗
Figure 2
Figure 2. Figure 2: Gauge-invariant magnetic flux (enclosed magnetic charge) for the instanton at x 4 = 0. From this graph we observe that the flux goes to zero as r increases, indicating that ΦB = 0 as r → ∞. This result can be shown analytically, for any Euclidean time x 4 , and it means that the instanton and anti-instanton solutions, when considering all space, have no net magnetic and electric charges coming from the non… view at source ↗
Figure 3
Figure 3. Figure 3: Euclidean time symmetry of the gauge-invariant magnetic flux for the instanton. Now, for the Euclidean time evolution of the flux (4.13) for the instanton and anti￾instanton solutions, we can plot the following graphs, going from x 4 = 0 to x 4 = 1 [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Euclidean time evolution of the gauge-invariant magnetic flux for the instanton. Here we see that the magnetic (or electric) flux goes to zero as the Euclidean time increases. In fact, from the symmetry indicated in [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Gauge-invariant magnetic charge density for the instanton at x 4 = 0. This graph must be compared with the one at [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Euclidean time evolution of the gauge-invariant magnetic charge density for the ins￾tanton. As we see, the charge density will eventually vanish for x 4 → ∞ (and also for x 4 → −∞, from the Euclidean time symmetry of the solutions), which is in accordance with the evolution of the flux ΦB plotted in [PITH_FULL_IMAGE:figures/full_fig_p036_6.png] view at source ↗
read the original abstract

We study the gauge-invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group $SU(2)$. Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-Abelian fields through spherical surfaces centered at the origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r\equiv\sqrt{x_i \, x^i}/\lambda$, with $\lambda$ being the parameter of the instanton solution, defining its size, and $x_i \, x^i = (x^1)^2 + (x^{2})^2 + (x^{3})^2$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at $r=1$ and $x^4 = 0$, $x^4$ being the Euclidean time, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $\theta$-vacuum.

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

1 major / 2 minor

Summary. The paper applies gauge-invariant dynamical charges and densities, previously constructed via integral equations on generalized loop spaces in references [1,2], to the explicit BPST 1-instanton (and anti-instanton) solution of SU(2) Yang-Mills theory in four-dimensional Euclidean space. It uses the integral non-Abelian Gauss law to compute the magnetic and electric fluxes through spherical surfaces centered at the instanton origin, defines corresponding charge densities within infinitesimal spherical shells of radius r = √(x_i x^i)/λ at x^4=0, discusses reparameterization invariance, and reports that the fluxes at r=1 are non-zero and thus observable, yielding an internal structure for the instanton with possible implications for the Yang-Mills θ-vacuum.

Significance. If the central applicability claim holds, the work provides an explicit gauge-invariant assignment of dynamical charges to the 1-instanton that is independent of the usual topological winding number, which could be relevant for non-perturbative aspects of the θ-vacuum. The explicit evaluation on the BPST solution and the reparameterization-invariance discussion are concrete strengths that go beyond the abstract constructions in [1,2].

major comments (1)
  1. [Application to the 1-instanton solution and results on fluxes] The constructions of the integral equations and conserved charges in references [1,2] are formulated for real-time (Minkowski) Yang-Mills theory. The manuscript applies these directly to the Euclidean 1-instanton without deriving or verifying that the integral equations and dynamical conservation survive the Wick rotation or remain valid under the self-duality condition F=*F. This verification is load-bearing for the claim that the computed fluxes at r=1, x^4=0 are dynamically conserved and observable (see the paragraph beginning 'We use the integral non-Abelian Gauss law...' and the results paragraph on non-zero fluxes).
minor comments (2)
  1. [Definition of charge densities] The definition of the radial coordinate r ≡ √(x_i x^i)/λ is introduced without an explicit statement of how the scaling with the instanton size λ affects the normalization of the reported fluxes; a brief clarification would improve readability.
  2. [Results paragraph] The abstract and results paragraph state that the fluxes 'are non-zero and observable' but do not include a short table or explicit numerical values for the instanton versus anti-instanton cases; adding these would make the central claim easier to assess.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the significance of our results. We address the single major comment below.

read point-by-point responses

  1. Referee: [Application to the 1-instanton solution and results on fluxes] The constructions of the integral equations and conserved charges in references [1,2] are formulated for real-time (Minkowski) Yang-Mills theory. The manuscript applies these directly to the Euclidean 1-instanton without deriving or verifying that the integral equations and dynamical conservation survive the Wick rotation or remain valid under the self-duality condition F=*F. This verification is load-bearing for the claim that the computed fluxes at r=1, x^4=0 are dynamically conserved and observable (see the paragraph beginning 'We use the integral non-Abelian Gauss law...' and the results paragraph on non-zero fluxes).

    Authors: We agree that an explicit verification is required to support the dynamical conservation claims. The integral non-Abelian Gauss law follows directly from the Yang-Mills equations of motion, which retain the same differential form after Wick rotation to Euclidean signature. Self-dual (and anti-self-dual) configurations satisfy these equations, so the integral form and the associated conservation should carry over. To address the referee's concern rigorously, the revised manuscript will include a new subsection that (i) performs the Wick rotation on the integral equations of [1,2], (ii) confirms that the resulting Euclidean integral Gauss law holds for any solution of the Euclidean Yang-Mills equations, and (iii) notes that self-duality is sufficient but not necessary for the conservation statement. This addition will make the applicability to the BPST solution fully explicit. revision: yes

Circularity Check

1 steps flagged

Central gauge-invariant charges and conservation laws imported from authors' prior works [1,2] without independent Euclidean verification

specific steps
  1. self citation load bearing [Abstract]
    "Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-Abelian fields through spherical surfaces centered at the origin of the instanton solution. ... show that the magnetic and electric fluxes for the instanton and anti-instanton, at r=1 and x^4 = 0 ... are non-zero and observable."

    The dynamical conservation property that allows the Gauss law to define observable fluxes is justified solely by citation to [1,2] (prior works with author overlap). The manuscript applies the construction directly to the Euclidean 1-instanton without deriving or confirming survival under Wick rotation/self-duality, so the non-zero flux result reduces to the imported premise rather than an independent check.

full rationale

The paper's core construction of dynamically conserved charges and the use of the integral non-Abelian Gauss law to obtain observable fluxes on the 1-instanton rests entirely on the applicability of the integral equations and conservation properties defined in references [1,2]. These are cited as the source of the charges, with no derivation or explicit check in the present manuscript that the Minkowski-derived equations survive the Wick rotation or self-duality condition F=*F when evaluated on the BPST solution. This matches self_citation_load_bearing: the dynamical status and observability of the reported non-zero magnetic/electric fluxes at r=1, x^4=0 are not independently established here but inherit from the self-cited framework. The reparameterization-invariance discussion does not substitute for that verification. No other circular patterns (self-definition, fitted predictions, etc.) are exhibited by the quoted text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the prior construction of charges in [1,2] and the applicability of the integral non-Abelian Gauss law to the instanton; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The integral non-Abelian Gauss law can be used to evaluate gauge-invariant fluxes for the 1-instanton solution
    Invoked to define the magnetic and electric fluxes through spherical surfaces.

pith-pipeline@v0.9.1-grok · 5798 in / 1240 out tokens · 40419 ms · 2026-06-27T04:28:29.663934+00:00 · methodology

discussion (0)

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Reference graph

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