pith. sign in

arxiv: 2312.01355 · v3 · pith:TUJXDEDOnew · submitted 2023-12-03 · ✦ hep-ph

The electroweak magnetic monopole in the presence of KSVZ axion

Tong Li , Rui-Jia Zhang This is my paper

Pith reviewed 2026-05-24 05:29 UTC · model grok-4.3

classification ✦ hep-ph
keywords electroweak monopoleKSVZ axionCho-Maison monopoleWitten effectaxion-photon couplingtopological solutionsmonopole mass
0
0 comments X

The pith

The KSVZ axion modifies the mass, electromagnetic charges, and axion potential of the electroweak magnetic monopole.

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

The paper examines how the KSVZ axion affects the Cho-Maison electroweak monopole through its photon coupling in the Weinberg-Salam theory. Using a spherically symmetric ansatz for the dyon and axion fields, the authors construct an effective Lagrangian, derive the equations of motion, and obtain numerical topological solutions. These solutions yield concrete shifts in monopole mass, electromagnetic charges, and an additional axion potential energy term. A reader would care because the base monopole already sits at a TeV mass scale that motivates dedicated collider searches, so any axion-induced changes would directly alter expected signals or stability.

Core claim

We derive the consequent equations of motion in the presence of the axion-photon coupling and show the numerical results of the topological solutions. We then calculate the changed characteristics of the electroweak monopole such as the monopole mass and the electromagnetic charges, as well as the axion potential energy.

What carries the argument

Effective Lagrangian formed from the electroweak monopole sector plus axion kinetic term and axion-photon interaction, solved via spherically symmetric ansatz for the dyon and axion fields to produce modified equations of motion.

If this is right

  • The monopole mass receives a correction from the axion coupling.
  • The electromagnetic charges of the monopole are shifted.
  • A nonzero axion potential energy appears for the monopole configuration.
  • The Witten-effect dynamics between axion and monopole now operate inside the electroweak theory.
  • Numerical topological solutions continue to exist once the axion-photon term is included.

Where Pith is reading between the lines

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

  • Collider searches for TeV-scale monopoles would need to account for the shifted mass and charge values when setting limits.
  • In a cosmological setting the axion-monopole system could source additional contributions to the axion potential or relic density.
  • Relaxing spherical symmetry might reveal whether the reported solutions remain stable against small perturbations.

Load-bearing premise

The spherically symmetric ansatz for both the electroweak dyon fields and the axion field yields stable topological solutions whose properties can be reliably extracted from the effective Lagrangian.

What would settle it

If the numerical integration of the coupled equations produces monopole mass and charge values that remain identical to the pure Cho-Maison case even after the axion-photon term is turned on, or if the axion potential energy vanishes, the claimed modifications are absent.

Figures

Figures reproduced from arXiv: 2312.01355 by Rui-Jia Zhang, Tong Li.

Figure 1
Figure 1. Figure 1: FIG. 1. The finite-energy dyon solutions in the presence of the KSVZ axion (solid lines) with those in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Behavior of the dyon solutions’ divergence as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The solutions for the SM dyon (left panel) and the dyon in presence of KSVZ axion (right panel) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The total energy (left) and electric charge [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The normalized axion field [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

The Witten effect implies the dynamics of axion and magnetic monopole. The Cho-Maison monopole is a realistic electroweak monopole arisen in the Weinberg-Salam theory. This monopole of TeV scale mass motivates the dedicated search for electroweak monopole at colliders. In this work we investigate the implication of KSVZ axion for the electroweak magnetic monopole. We use the spherically symmetric ansatz for the electroweak dyon and introduce the spherically symmetric function for the axion field. The effective Lagrangian is then shown in terms of the electroweak monopole part, the axion kinetic energy as well as the axion interaction term. We derive the consequent equations of motion in the presence of the axion-photon coupling and show the numerical results of the topological solutions. We then calculate the changed characteristics of the electroweak monopole such as the monopole mass and the electromagnetic charges, as well as the axion potential energy.

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 studies the effect of the KSVZ axion on the Cho-Maison electroweak monopole (dyon) by adopting spherically symmetric ansatzes for both the electroweak gauge/Higgs fields and the axion field. It constructs the effective Lagrangian including the axion kinetic term and the axion-photon coupling, derives the coupled equations of motion, presents numerical solutions for the resulting topological configurations, and computes the resulting shifts in monopole mass, electromagnetic charges, and axion potential energy.

Significance. If the spherical ansatz remains consistent and the numerical solutions are robust, the results would provide a concrete estimate of how the axion modifies the mass and charges of a TeV-scale electroweak monopole, which is directly relevant to ongoing collider searches. The work performs a direct numerical integration of the coupled system starting from the standard-model-plus-KSVZ Lagrangian without introducing fitted parameters.

major comments (2)
  1. [Ansatz and EOM derivation (abstract and subsequent section on effective Lagrangian)] The manuscript introduces the spherically symmetric ansatz for the axion field (stated in the abstract and used to derive the EOM) without showing that the axion-photon coupling term, which sources the axion from the monopole's topological density, preserves this symmetry or decouples non-spherical modes. This assumption is load-bearing for all reported numerical solutions and extracted quantities.
  2. [Numerical results section] No convergence tests, grid-resolution studies, or error estimates are reported for the numerical integration that yields the modified monopole mass, charges, and axion potential energy. Without these, the quantitative shifts cannot be assessed for reliability.
minor comments (1)
  1. [Abstract] The abstract refers to both 'monopole' and 'dyon'; clarify whether the solutions are purely magnetic or carry electric charge, and ensure consistent terminology throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Ansatz and EOM derivation (abstract and subsequent section on effective Lagrangian)] The manuscript introduces the spherically symmetric ansatz for the axion field (stated in the abstract and used to derive the EOM) without showing that the axion-photon coupling term, which sources the axion from the monopole's topological density, preserves this symmetry or decouples non-spherical modes. This assumption is load-bearing for all reported numerical solutions and extracted quantities.

    Authors: We agree that an explicit justification is needed. The axion-photon interaction term is proportional to a * F ~F. Under the standard spherically symmetric ansatz for the Cho-Maison dyon, the topological density F ~F is itself spherically symmetric. Consequently the source term in the axion equation of motion is spherically symmetric, so a spherically symmetric axion profile is a consistent solution. Linear perturbations around this background would not be sourced by the monopole at leading order. In the revised manuscript we will add a short paragraph immediately after the ansatz is introduced, deriving the symmetry of the source term and confirming consistency of the spherical ansatz. revision: yes

  2. Referee: [Numerical results section] No convergence tests, grid-resolution studies, or error estimates are reported for the numerical integration that yields the modified monopole mass, charges, and axion potential energy. Without these, the quantitative shifts cannot be assessed for reliability.

    Authors: We acknowledge the absence of explicit numerical validation. In the revised version we will add a dedicated subsection describing the numerical integration scheme (shooting method with boundary conditions), the radial grid spacing and cutoff used, and the results of convergence tests obtained by successively halving the step size. We will report the relative change in the monopole mass and electromagnetic charges under these refinements, together with an estimated numerical uncertainty on the quoted shifts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained numerical solution under stated ansatz

full rationale

The paper begins from the Weinberg-Salam Lagrangian augmented by the KSVZ axion-photon coupling term, imposes a spherically symmetric ansatz by assumption for the dyon and axion fields, derives the resulting coupled ODEs, and integrates them numerically to extract mass, charges, and potential. No equation reduces to a prior result by definition, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain whose content is unverified. The output quantities are genuine solutions of the stated boundary-value problem rather than tautological re-expressions of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard electroweak Lagrangian plus the KSVZ axion term and the assumption of spherical symmetry; no explicit free parameters beyond standard model inputs and the axion decay constant are mentioned, and no new entities are postulated.

axioms (2)
  • domain assumption The electroweak theory admits the Cho-Maison monopole solution as a realistic topological defect.
    Invoked in the opening sentences when the authors identify the Cho-Maison monopole as arising in the Weinberg-Salam theory.
  • domain assumption Spherical symmetry is an adequate ansatz for both the dyon and axion fields.
    Stated when the authors 'use the spherically symmetric ansatz for the electroweak dyon and introduce the spherically symmetric function for the axion field'.

pith-pipeline@v0.9.0 · 5690 in / 1567 out tokens · 26564 ms · 2026-05-24T05:29:05.547429+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 7 internal anchors

  1. [1]

    r2 ˙A(r) g2 + ˙B(r) g′2 !#′ +g aγγ I r=∞ d⃗S·a ⃗B =−4πe

    Under appropriate boundary conditions, we can numerically integrate the above differential equations to obtain the electroweak monopole solution with KSVZ axion. Following Ref. [18], we choose the following boundary conditions for UV regularized electroweak monopole ρ(0) = 0, ρ(∞) =v , f(0) = 1, f(∞) = 0, A(0) = 0, B(0) = 0, A(∞) =B(∞) =gv/4 =M W /2,(40) ...

    work page

  2. [2]

    R. D. Peccei and H. R. Quinn, Phys. Rev. Lett.38, 1440 (1977)

    work page 1977

  3. [3]

    R. D. Peccei and H. R. Quinn, Phys. Rev. D16, 1791 (1977)

    work page 1977

  4. [4]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett.40, 223 (1978)

    work page 1978

  5. [5]

    Wilczek, Phys

    F. Wilczek, Phys. Rev. Lett.40, 279 (1978)

    work page 1978

  6. [6]

    Preskill, M

    J. Preskill, M. B. Wise, and F. Wilczek, Phys. Lett. B120, 127 (1983)

    work page 1983

  7. [7]

    Dine and W

    M. Dine and W. Fischler, Phys. Lett. B120, 137 (1983)

    work page 1983

  8. [8]

    The landscape of QCD axion models

    L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, Phys. Rept.870, 1 (2020), arXiv:2003.01100 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  9. [9]

    Sikivie, Reviews of Modern Physics93(2021), 10.1103/revmodphys.93.015004

    P. Sikivie, Reviews of Modern Physics93(2021), 10.1103/revmodphys.93.015004

    work page doi:10.1103/revmodphys.93.015004 2021

  10. [10]

    P. A. M. Dirac, Proc. Roy. Soc. Lond. A133, 60 (1931)

    work page 1931

  11. [11]

    T. T. Wu and C. N. Yang, Phys. Rev. D12, 3845 (1975)

    work page 1975

  12. [12]

    ’t Hooft, Nucl

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

    work page 1974

  13. [13]

    A. M. Polyakov, JETP Lett.20, 194 (1974)

    work page 1974

  14. [14]

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

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  15. [15]

    P. Q. Hung, Nucl. Phys. B962, 115278 (2021), arXiv:2003.02794 [hep-ph]

    work page arXiv 2021

  16. [16]

    Alexandre and N

    J. Alexandre and N. E. Mavromatos, Phys. Rev. D100, 096005 (2019), arXiv:1906.08738 [hep-ph]

    work page arXiv 2019

  17. [17]

    The Price of an Electroweak Monopole

    J. Ellis, N. E. Mavromatos, and T. You, Phys. Lett. B756, 29 (2016), arXiv:1602.01745 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    Lazarides and Q

    G. Lazarides and Q. Shafi, Phys. Rev. D103, 095021 (2021), arXiv:2102.07124 [hep-ph]. 17

    work page arXiv 2021

  19. [19]

    Y . M. Cho, Phil. Trans. Roy. Soc. Lond. A377, 20190038 (2019)

    work page 2019

  20. [20]

    Aad et al

    G. Aad et al. (ATLAS), Phys. Rev. Lett.124, 031802 (2020), arXiv:1905.10130 [hep-ex]

    work page arXiv 2020

  21. [21]

    Magnetic monopole search with the full MoEDAL trapping detector in 13 TeV $pp$ collisions interpreted in photon-fusion and Drell-Yan production

    B. Acharya et al. (MoEDAL), Phys. Rev. Lett.123, 021802 (2019), arXiv:1903.08491 [hep-ex]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Witten, Phys

    E. Witten, Phys. Lett. B86, 283 (1979)

    work page 1979

  23. [23]

    Fischler and J

    W. Fischler and J. Preskill, Phys. Lett. B125, 165 (1983)

    work page 1983

  24. [24]

    A. V . Sokolov and A. Ringwald, (2022), arXiv:2205.02605 [hep-ph]

    work page arXiv 2022

  25. [25]

    A. V . Sokolov and A. Ringwald, (2023), 10.1002/andp.202300112, arXiv:2303.10170 [hep-ph]

    work page doi:10.1002/andp.202300112 2023

  26. [26]

    Heidenreich, J

    B. Heidenreich, J. McNamara, and M. Reece, (2023), arXiv:2309.07951 [hep-ph]

    work page arXiv 2023

  27. [27]

    Li, R.-J

    T. Li, R.-J. Zhang, and C.-J. Dai, JHEP03, 088 (2023), arXiv:2211.06847 [hep-ph]

    work page arXiv 2023

  28. [28]

    M. E. Tobar, C. A. Thomson, B. T. McAllister, M. Goryachev, A. Sokolov, and A. Ringwald, Annalen Phys.2023, 2200594 (2022), arXiv:2211.09637 [hep-ph]

    work page arXiv 2023

  29. [29]

    B. T. McAllister, A. Quiskamp, C. O’Hare, P. Altin, E. Ivanov, M. Goryachev, and M. Tobar, Annalen Phys.2023, 2200622 (2022), arXiv:2212.01971 [hep-ph]

    work page arXiv 2023

  30. [30]

    Li, C.-J

    T. Li, C.-J. Dai, and R.-J. Zhang, Phys. Rev. D109, 015026 (2024), arXiv:2304.12525 [hep-ph]

    work page arXiv 2024

  31. [31]

    Li and R.-J

    T. Li and R.-J. Zhang, Chin. Phys. C47, 123104 (2023), arXiv:2305.01344 [hep-ph]

    work page arXiv 2023

  32. [32]

    M. E. Tobar, A. V . Sokolov, A. Ringwald, and M. Goryachev, Phys. Rev. D108, 035024 (2023), arXiv:2306.13320 [hep-ph]

    work page arXiv 2023

  33. [33]

    Patkos, (2023), arXiv:2309.05523 [hep-ph]

    A. Patkos, (2023), arXiv:2309.05523 [hep-ph]

    work page arXiv 2023

  34. [34]

    C.-J. Dai, T. Li, and R.-J. Zhang, EPL148, 54002 (2024), arXiv:2401.14195 [hep-ph]

    work page arXiv 2024

  35. [35]

    Agrawal, K

    P. Agrawal, K. V . Berghaus, J. Fan, A. Hook, G. Marques-Tavares, and T. Rudelius, inSnowmass 2021 (2022) arXiv:2203.08026 [hep-ph]

    work page arXiv 2021

  36. [36]

    J. E. Kim, Phys. Rev. Lett.43, 103 (1979)

    work page 1979

  37. [37]

    M. A. Shifman, A. I. Vainshtein, and V . I. Zakharov, Nucl. Phys. B166, 493 (1980)

    work page 1980

  38. [38]

    W. S. Bae and Y . M. Cho, J. Korean Phys. Soc.46, 791 (2005), arXiv:hep-th/0210299

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  39. [39]

    Y . M. Cho, K. Kimm, and J. H. Yoon, Mod. Phys. Lett. A31, 1650053 (2016), arXiv:1212.3885 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  40. [40]

    Y . M. Cho, K. Kim, and J. H. Yoon, Eur. Phys. J. C75, 67 (2015), arXiv:1305.1699 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  41. [41]

    Zhang, L.-P

    P. Zhang, L.-P. Zou, and Y . M. Cho, Eur. Phys. J. C80, 280 (2020), arXiv:2001.08866 [hep-th]

    work page arXiv 2020

  42. [42]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, 2007). 18

    work page 2007