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REVIEW 4 major objections 5 minor 14 references

Singularities of Magnetic Monopoles for Dirac and 't Hooft-Polyakov Theories by Pre-potential Method

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Using the pre-potential $\boldsymbol{C}$ defined by $\boldsymbol{A}=\nabla\times\boldsymbol{C}$, the paper argues that both the Dirac and the 't Hooft-Polyakov solutions are magnetic singularities with zero total charge, not monopoles.

desk verdict The HP monopole conclusion is an artifact of using the asymptotic hedgehog as the full solution; the Dirac derivation is fine but the HP extension collapses under its own regularization. read the letter →

arxiv 2505.15301 v1 pith:HBJQQ2RO submitted 2025-05-21 hep-th

Masakatsu Kenmoku This is my paper · ORCID
classification hep-th PACS 14.80.Hv
keywords magneticmonopoleDiracstringtHooft-Polyakovpre-potentialmethodsingularityhedgehogchargequantizationSO(3)gaugetheory
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 neither the Dirac monopole nor the 't Hooft-Polyakov monopole is an actual magnetic monopole. Using the pre-potential $\boldsymbol{C}$, defined by $\boldsymbol{A}=\nabla\times\boldsymbol{C}$, it splits each magnetic field into a Coulomb pole term and a singular string or hedgehog term. These two pieces carry opposite charges, $+4\pi g\,\delta^{(3)}(\boldsymbol{r})$ and $-4\pi g\,\delta^{(3)}(\boldsymbol{r})$, so the total magnetic charge is zero. The author concludes that only magnetic singularities exist and that the failure to observe monopoles is consistent with this result. A reader should care because, if the argument is right, the standard topological quantization of magnetic charge would have to be rethought.

What carries the argument

The central object is the pre-potential $\boldsymbol{C}$, defined so that $\boldsymbol{A}=\nabla\times\boldsymbol{C}$. The identity $\nabla\times(\nabla\times\boldsymbol{C})=\nabla(\nabla\cdot\boldsymbol{C})-\Delta\boldsymbol{C}$ decomposes any magnetic field built this way into a pole term and a singular potential term. For the 't Hooft-Polyakov case the paper inserts the asymptotic hedgehog gauge field $A_i^a=-\frac{1}{e}\epsilon_{iab}\frac{r^b}{r^2}$ and uses the charge quantization $eg=-1$ to obtain $B_i=g r_i/r^3+g(\Delta\ln r-\frac{1}{r^2})r_i/r$, with the second term called the 'hedgehog-type singularity'. The regularization $r\to r_\epsilon=\sqrt{r^2+\epsilon^2}$ turns the singular combination into $\Delta\ln r_\epsilon-1/r_\epsilon^2=2\epsilon^2/r_\epsilon^4$, making the cancellation of pole and singular charges explicit.

What would settle it

Take the smooth finite-energy solution quoted in the paper's remark (5.7), form its gauge-invariant magnetic field, and integrate $\nabla\cdot\boldsymbol{B}$ over spheres of shrinking radius; if the smooth solution gives no $\delta$-function singularity at the origin, the total magnetic charge is the usual integer and the zero-charge conclusion fails.

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Extended reading notes

Core claim

For the Dirac monopole, the vector potential is written as the curl of $\boldsymbol{C}_D=-g(0,0,\ln(r+z))$, and the magnetic field separates into the Coulomb term $g\boldsymbol{r}/r^3$ plus a singular string along the negative $z$-axis. For the 't Hooft-Polyakov monopole, the pre-potential $\boldsymbol{C}^{HP}=-g\ln r\,\boldsymbol{\tau}$ reproduces the asymptotic hedgehog gauge field, and the field strength separates into a pole term $g r_i r_a/r^4$ and a singular term built from $\Delta\ln r-1/r^2$. Each singular term integrates to a charge of exactly the opposite sign of the pole term, so both configurations have zero total magnetic charge. The paper concludes that both theories describe magnetic singularities rather than monopoles, and that a gauge transformation between them moves the singularity from the $-z$ direction to the local $-r$ direction while preserving zero charge and infinite energy.

Load-bearing premise

The load-bearing premise is that the asymptotic hedgehog form of the gauge field, valid only at $r\to\infty$, is treated as the whole solution all the way into the origin, where a smooth finite-energy solution quoted by the paper has the scalar field vanish and no singularity.

Editorial extensions

If this is right

  • If the argument is right, the total magnetic charge of both configurations is exactly zero, so neither emits net radial magnetic flux.
  • The Dirac string and the hedgehog singularity of the 't Hooft-Polyakov field would be the only magnetic structures present; the name 'monopole' would be a misnomer.
  • The homotopy argument $\pi_2(SU(2)/U(1))=\mathbb{Z}$ would no longer imply a nonzero charge inside the winding configuration, because the singular core cancels the contribution from infinity.
  • The observed absence of magnetic monopoles would follow directly from the field equations, without appeal to cosmological dilution or experimental limits.
  • Gauge transformations between the Dirac and 't Hooft-Polyakov forms would change where the singularity sits but not the zero charge or the infinite energy.

Reading between the lines

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

  • The paper only applies the pre-potential decomposition to the asymptotic hedgehog configuration; applying it to the full smooth finite-energy solution with $\phi$ vanishing at the origin is a direct check that would show whether the singularity is real or an artifact of the $r\to\infty$ truncation.
  • If the cancellation is genuine, the standard surface-integral definition of magnetic charge must be extended to include singular contributions; a finite core with a modified charge definition could still yield the topological integer, which would preserve quantization while changing its physical meaning.
  • The same pole-minus-singularity cancellation could be tested on other hedgehog-like configurations, including the electroweak monopole mentioned in the paper's introduction, to see whether it is a general feature of the pre-potential method.
  • The Dirac string carries a localized electric current in the paper's calculation; checking whether the 't Hooft-Polyakov hedgehog singularity has an analogous electric current would be a concrete extension of the bar-magnet picture.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript introduces a 'pre-potential' C satisfying A = ∇×C in order to analyze singularities in the Dirac and 't Hooft-Polyakov monopole theories. For the Dirac monopole it takes C_D = -g(0,0,ln(r+z)), and for the 't Hooft-Polyakov monopole it takes C_HP = -g ln r τ, based on the asymptotic hedgehog (2.9). The paper computes B = ∇×(∇×C) plus, in the non-Abelian case, a commutator term, and obtains magnetic fields with a pole term and a singular term whose divergences cancel, leading to zero total magnetic charge. It concludes that magnetic monopoles do not exist in either theory, only magnetic singularities, and asserts that regularization confirms this. The final section compares the result with the BPS solution and the 't Hooft tensor but does not alter the conclusion.

Significance. If correct, the paper would overturn the standard topological understanding of 't Hooft-Polyakov monopoles, including the nonzero topological charge 4π/e. The pre-potential method is a simple calculational device, and the paper is explicit in its regularized expressions. However, the central claim is not supported: the method is applied to the asymptotic configuration rather than the full BPS solution, the key distributional identity in Eq. (2.23) is false, and the regularization section concedes that no physical scalar field satisfies the equations of motion. The manuscript therefore does not reach the standard for overturning a well-established result.

major comments (4)
  1. [Sec. 2.2, Eqs. (2.9) and (2.13)] The pre-potential C_HP = -g ln r τ in (2.13) is built from the asymptotic hedgehog (2.9), which is valid only at r → ∞. The paper then differentiates this expression at all r, including the origin, where the singularity is claimed to sit. The BPS solution cited by the paper in (5.7) is regular at the origin (in the standard solution φ^a(0) = 0 and A_i^a(0) = 0), but the paper never substitutes (5.7) into the gauge-invariant 't Hooft tensor (5.9) or the projected field (2.21). Without an analysis of the core region, the claimed r = 0 singularity and zero total charge are unsupported; the asymptotic form cannot be extrapolated inward.
  2. [Sec. 2.2, Eq. (2.23)] The identity ∂_i[(Δln r - 1/r²) r_i/r] = g ∂_r(Δln r - 1/r²) = g Δ(1/r) = -4πgδ³(r) is incorrect. In three dimensions Δln r = 1/r² as a distribution, so Δln r - 1/r² = 0 identically, and its derivative vanishes. Moreover, the paper's own regularization (3.13) gives Δln r_ε - 1/r_ε² = 2ε²/r_ε⁴, which integrates to zero as ε → 0 (∫ d³x 2ε²/r_ε⁴ = 2π²ε), not to -4πδ³(r). Since the cancellation in (2.22)-(2.23) relies on this identity, the zero-total-charge conclusion is not established.
  3. [Sec. 3.2, Eq. (3.15)] The paper explicitly states that for the regularized gauge potential (3.8), no scalar field satisfies the equation of motion: D_i φ_ε^a ≠ 0. Consequently the regularized magnetic field B_{i,ε}^{HP} = B_{ia,ε}^{HP} φ^a/|φ| is not gauge-invariant and the colored charge density (3.14) is not a physical magnetic charge. The regularization thus confirms only an algebraic identity for the pointlike hedgehog, not the existence of a singularity in the actual 't Hooft-Polyakov solution.
  4. [Sec. 5, remarks (1) and (3)] The paper states that its conclusion is unchanged by the BPS solution and dismisses the standard 't Hooft tensor analysis as taking account of only the pole term. However, the standard magnetic charge is defined by a surface integral at spatial infinity of (5.9); for the BPS hedgehog this integral gives a nonzero topological charge. The paper never performs that surface integral, nor does it identify a step in the standard derivation that fails. A concrete test of the paper's claim would be to compute the surface integral of (5.9) using (5.7) and to show that the result vanishes; without such a computation, the central conclusion contradicts a well-established result without providing a specific error in it.
minor comments (5)
  1. [Eq. (2.24)] The expression δ(x̄)δ(ȳθ(−z̄))er is not well formed; presumably δ(x̄)δ(ȳ)θ(−z̄)er is intended. Please correct.
  2. [Eq. (2.15)] The antisymmetric field strength contains a repeated term ∂_j A_k^{HP} - ∂_j A_k^{HP}; this must be ∂_j A_k^{HP} - ∂_k A_j^{HP}.
  3. [Eq. (3.11)] The covariant derivative in the colored charge density should act on a gauge-covariant object; B_{ia,ε}^{HP} is not gauge-covariant, so the definition needs justification (or should be labeled as a formal expression).
  4. [General notation] The notation (2,1), (3,10) etc. is nonstandard; use (2.1), (3.10) for consistency with the text references.
  5. [References] Reference [10] (PDG 2012) is outdated; a current edition should be cited if the observational statement is meant to be up to date.

Circularity Check

3 steps flagged · score 7.0 of 10

HP 'singularity' is an artifact of substituting the asymptotic hedgehog (2.9) into the pre-potential ansatz (2.12) and differentiating it at r=0; the paper's own BPS citation (5.7) and admitted regularization failure (3.15) do not rescue the no-monopole conclusion.

  1. self definitional [Sec. 2.2, eqs. (2.12)-(2.14)]
    "The pre-potential for the ’t Hooft-Polyakov monopole is defined to the asymptotic solution as C^{a,HP}_i = −g ln r δ^a_i, (2,12) ... The gauge potential is derived by the curl operation to the pre-potential: A^{HP} = ∇ × C^{HP} = − g/r^2 (yτ_z − zτ_y, zτ_x − xτ_z, xτ_y − yτ_x), (2,14), which is coincident with the HP solution of eq.(2,9)."

    C^HP is defined solely to reproduce the r→∞ hedgehog (2.9); differentiating it at all r makes the r=0 singularity an input of the representation. The ln r factor is singular at the origin, and eqs. (2.22)-(2.23) recover that singularity as δ-functions. No independent input from the full BPS solution (5.7) is used; the paper later asserts the conclusion is unchanged without evaluating the core. Thus the asserted HP singularity is equivalent to the chosen ansatz, not derived from the theory.

  2. fitted input called prediction [Sec. 5, remark (1), eqs. (2.9) and (5.7)]
    "Bogomol’nyi, Prasad and Sommerfeld found the analytic solution to approach the ’t Hooft-Polyakov asymptotic solution (2.9), which is called as BPS monopoles [11][12]: A^a_i = ε_{aij} r^a/(e r^2)(1 − ρ/sinhρ), φ^a = r^a/(e r^2)(cothρ − 1/ρ), with ρ := er|φ|. The magnetic charge is obtained by the surface integral of the magnetic field, which is determined by the asymptotic behavior of the gauge potential and the scalar field. And our analysis in sec. 2.2 and sec. 3.2 is satisfied and the conclusion is unchanged."

    This is the only place the full solution is mentioned. Because (2.9) is only the asymptotic form, using it in (2.12) and (2.21) amounts to fitting the boundary data and then 'predicting' a core singularity. The BPS solution is smooth at r=0 (φ^a → O(r) for the standard ρ = e v r), so the zero-charge conclusion follows only if one refuses to evaluate the core. The asserted 'conclusion is unchanged' is an assertion, not a calculation from (5.7); the singularity is not present in the cited full solution.

1 more flagged steps
  1. other [Sec. 3.2, eq. (3,15)]
    "We note that in the regularized scalar field solution φ^a_ε cannot be obtained because the scalar field equation does not satisfied: D_i φ^a_ε = ∂φ^a_ε + e ε^{abc} A^b_{iε} φ^c_ε , 0, (3,15) for a given regularized gauge potential A^b_{iε} in eq.(3.8). Therefore the regularized magnetic field B^{HP}_{i,ε} = B^{HP}_{ia,ε} φ^a/|φ| cannot be gauge invariant and we derive the colored magnetic charge density without using the gauge invariant regularized form."

    The regularization section is supposed to confirm the singularity, but it explicitly fails to produce a physical configuration: no scalar field satisfies the equation of motion, so the projected magnetic field and its divergence are not the gauge-invariant 't Hooft magnetic field (5.9). The zero total charge is therefore a property of a gauge-variant colored charge density constructed for the calculation, i.e., the conclusion is baked into the quantity being regulated rather than following from the HP theory.

full rationale

The Dirac-monopole part of the paper is a standard consistency check of a chosen singular pre-potential and is not itself circular. The circularity is concentrated in the 't Hooft-Polyakov analysis. The pre-potential C^HP = -g ln r τ is introduced in (2.12)-(2.13) precisely to reproduce the asymptotic hedgehog (2.9), and every subsequent singularity and cancellation follows from differentiating this ln r ansatz at r=0. The paper never substitutes the BPS solution (5.7), which it itself cites, into the 't Hooft tensor (5.9) or examines the core; it simply asserts that the boundary-determined charge calculation is enough and 'the conclusion is unchanged.' The regularization section then admits in (3.15) that no regularized scalar field satisfies the equations of motion, so the regularized 'confirmation' is computed from a gauge-variant colored charge density rather than from a physical HP configuration. Because the central no-monopole conclusion is forced by the boundary-fitted ansatz and by a regularization that explicitly lacks a physical scalar field, the derivation reduces to its own input representation. The paper does not rely on problematic self-citation; the difficulty is definitional and structural. Score 7 reflects that the central claim is largely a restatement of the chosen pre-potential, while the Dirac analysis and the cited BPS solution provide some independent physical context.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

There are no data-fitted numbers. The central inputs are the functional choice of the pre-potentials, which already contain the singularities, and the assumption that the asymptotic hedgehog describes the full solution. The distributional identities used to convert the singular terms into delta functions are standard, but one identity in eq. (2,23) is asserted without justification and conflicts with the paper's own regularized computation.

assumptions (4)
  • standard math Distributional identities Δ(1/r) = -4πδ³(r) and Δln r = 1/r² for r ≠ 0
    Used in eqs. (2,5), (2,6), (2,22), (2,23) to convert singular terms into delta functions; standard but requires care at the origin.
  • domain assumption The asymptotic HP solution (2,9) is used as the full solution throughout space, including the origin
    The hedgehog configuration is only the r→∞ limit of the 't Hooft-Polyakov solution; the core region is not described. This is the load-bearing modeling choice.
  • ad hoc to paper ∂_r(Δln r - 1/r²) = Δ(1/r) as distributions
    Eq. (2,23) asserts this identity to obtain -4πgδ³(r); for r>0 both sides are zero, and the regularized computation in eq. (3,13)-(3,14) gives an expression that integrates to zero, so the identity is not justified.
  • ad hoc to paper The regularized field configuration continues to describe the HP solution for ε→0
    Sec. 3.2 admits in eq. (3,15) that the regularized scalar field does not satisfy its equation of motion for ε ≠ 0, so the regularization limit is not tied to the actual monopole solution.

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Cite this review

Pith. "Pith review of Singularities of Magnetic Monopoles for Dirac and 't Hooft-Polyakov Theories by Pre-potential Method." pith.science (2026). https://pith.science/paper/HBJQQ2RO

@misc{pith2026250515301,
  author       = {Pith},
  title        = {Pith review of: Singularities of Magnetic Monopoles for Dirac and 't Hooft-Polyakov Theories by Pre-potential Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBJQQ2RO}},
  note         = {Machine review of arXiv:2505.15301}
}
abstract

The magnetic monopole is one of the important problems in the early stage of universe as well as observations and experiments on Earth. We study the existence or non-existence of the Dirac and the 't Hooft-Polyakov magnetic monopole theories using the pre-potential $\boldsymbol{C}$, which is defined to derive the vector potential by the curl operation as $\boldsymbol{A}=\nabla \times \boldsymbol{C}$ . We assert that the magnetic singularity exists for the 't Hooft-Polyakov monopole in SO(3) gauge theory, as well as for the Dirac monopole in U(1) gauge theory. The regularization method confirms our assertion.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

  1. [1]

    Dirac, Proceeding of the Royal Society,A133, 60 (1931)

    P.A.M. Dirac, Proceeding of the Royal Society,A133, 60 (1931)

    work page 1931

  2. [2]

    H. B. Nielsen and P. Olsen, Nucl. Phys.B61, 45 (1973)

    work page 1973

  3. [3]

    ’t Hooft, Nucl

    G. ’t Hooft, Nucl. Phys. ,B79, 276 (1974)

    work page 1974

  4. [4]

    Polyakov, JETP Letters,30, 194 (1974); Soviet Physics JETP,41, 988 (1976)

    A.M. Polyakov, JETP Letters,30, 194 (1974); Soviet Physics JETP,41, 988 (1976)

    work page 1974

  5. [5]

    Cho and D

    Y .M. Cho and D. Maison, Phys. Lett.B391, 360 (1997), hep-th/9601028

    arXiv 1997

  6. [6]

    Masakatsu Kenmoku, Singularities of Magnetic Monopoles in the Universe -Analysis by the Pre-potential Method-, Talk given at ICGAC-15, July 3-7 2023, Gyeongju Korea. 7

    work page 2023

  7. [7]

    Boulware, L.S

    D.G. Boulware, L.S. Brown, R.N. Cahn, S.D. Ellis and C. Lee, Phys. Rev.D14, 2708 (1976)

    work page 1976

  8. [8]

    Ryder, Quantum Field Theory, section 10 Topological objects in field theory, Cambridge University Press (2012)

    See for example, L.H. Ryder, Quantum Field Theory, section 10 Topological objects in field theory, Cambridge University Press (2012)

    work page 2012

Show all 14 references
  1. [9]

    Weinberg, The Quantum Theory of Field, vol

    S. Weinberg, The Quantum Theory of Field, vol. II Modern Applications, Cambridge University Press (1996)

    1996

  2. [10]

    Beringer et al.., (Particle Data Group), Phys

    J. Beringer et al.., (Particle Data Group), Phys. Rev.D84010001 (2012)

    2012

  3. [11]

    E. B. Bogomol’nyi, Sov. J. Nucl. Phys.24, 449 (1976)

    1976

  4. [12]

    M. K. Prasad and C. M. Sommerfeld, Phys. Rev. Lett.35, 760 (1975)

    1975

  5. [13]

    Lipkin, W.I

    H.J. Lipkin, W.I. Wesberger and M. Peshkin, Ann. of Phys.,53, 203 (1969)

    1969

  6. [14]

    Arafune, P.G.O

    J. Arafune, P.G.O. Frend and C.J. Goebel, Journal of Mathematical Physics,16, 433 (1975). 8

    1975

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