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arxiv: 2606.02511 · v1 · pith:MMKBRLLInew · submitted 2026-06-01 · 🌀 gr-qc · astro-ph.HE· astro-ph.SR· hep-ph

Dyonic Quark Stars in Quasi-Topological Electromagnetism

Michael Gammon , Nicola De Kock , Robert B. Mann , Amos Kubeka This is my paper

Pith reviewed 2026-06-28 13:22 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEastro-ph.SRhep-ph
keywords quark starsquasi-topological electromagnetismdyonic chargesmass-radius relationnegative pressurecompact objectsgeneral relativity
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The pith

Dyonic charges in quasi-topological electromagnetism produce a second branch of large quark stars with negative pressure envelopes.

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

This paper explores solutions for quark stars in a quasi-topological electromagnetism theory that acts as dark energy. The theory has no effect for purely electric or magnetic charges but generates new dynamics when both are present in a dyonic configuration. For small ratios of the two charges, an additional family of solutions appears at very high masses and radii. These stars have a core with positive pressure surrounded by an envelope where pressure is negative. Increasing the charge ratio causes the two branches to merge, forming a paperclip shape in the mass-radius plane.

Core claim

In pure quasi-topological electromagnetism, the dyonic case induces non-trivial contributions that shift the standard quark star mass-radius hook to larger masses and radii while also generating a second branch of solutions at very large mass and radius for small dyonic charge ratios. These additional solutions exhibit a negative pressure envelope surrounding a positive pressure core. As the dyonic charge ratio is increased, the branches interact and eventually merge into a paperclip shape in M/R space.

What carries the argument

The quasi-topological electromagnetism term in the action, which couples non-trivially only to dyonic charge distributions and modifies the structure equations for quark stars.

If this is right

  • The usual mass-radius hook for quark stars shifts toward larger masses and radii.
  • A second branch of solutions emerges at very large mass and radius for small dyonic charge ratios.
  • These large solutions feature a negative pressure envelope around a positive pressure core.
  • The two branches merge into a paperclip shape in the mass-radius diagram as the charge ratio increases.

Where Pith is reading between the lines

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

  • Stability of the negative-pressure-envelope solutions against small perturbations would need separate checking to determine if they can persist.
  • Gravitational-wave observations of compact-object mergers might reveal signatures from the altered mass-radius relations at high mass.
  • The model implies that dyonic effects could alter the maximum stable mass for quark stars compared with standard general relativity.

Load-bearing premise

The quasi-topological electromagnetism contribution remains negligible unless both electric and magnetic charges are simultaneously present.

What would settle it

Detection or non-detection of compact objects with masses and radii in the predicted second branch region, or indirect probes showing absence of negative pressure envelopes in any quark star candidates.

Figures

Figures reproduced from arXiv: 2606.02511 by Amos Kubeka, Michael Gammon, Nicola De Kock, Robert B. Mann.

Figure 1
Figure 1. Figure 1: FIG. 1. The ‘normal’ set of solutions for charge model A with small [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Zooming out from figure 1 we see that a whole other solution branch exists if we consider [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of the pressure profiles for the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A plot of an sample pressure profile for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Unitless effective mass [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. These are the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. For larger [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The ‘normal’ set of solutions for charge model B with small [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Zooming out from 8 we see that just like in charge model A a whole other solution [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. These are the [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. For larger [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

In this paper we consider quark star solutions to Liu et al.'s \cite{Liu_2019} quasi-topological electromagnetism (QTEM), a recently proposed form of dark energy. Since the QTEM contribution is trivial for pure electric/magnetic charge, we consider the dyonic case in pure QTEM which does induce (dark) non-trivial dynamics from the non-linear theory. Besides the introduction of a dyonic charge distribution generally pushing the characteristic quark star `hook' shape to larger masses and radii, it also induces a second branch at very large mass and radius for stars with a small dyonic charge ratio. This second set of solutions have a negative pressure envelope surrounding a positive pressure core. As we explore the parameter space these features interact and evolve in interesting ways, with the two branches eventually merging in $M/R$ space before settling into a characteristic `paperclip' shape as the dyonic charge ratio becomes large.

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 / 2 minor

Summary. The manuscript studies static spherically symmetric quark-star solutions sourced by dyonic charge distributions in Liu et al.'s quasi-topological electromagnetism (QTEM). It asserts that the QTEM term vanishes for pure electric or magnetic charge but generates non-trivial dynamics in the dyonic case, producing (i) a shift of the usual quark-star 'hook' to larger masses and radii, (ii) a second branch of solutions at very large mass and radius for small dyonic charge ratios, and (iii) a negative-pressure envelope surrounding a positive-pressure core. As the dyonic charge ratio is increased the two branches merge, eventually forming a characteristic 'paperclip' shape in the M–R plane.

Significance. If the reported branches and the negative-pressure envelope are confirmed by the field equations, the work would supply a concrete, falsifiable signature of non-linear electromagnetism in strong-gravity compact objects. The paperclip merger constitutes a distinctive topological feature in the mass-radius diagram that could be confronted with future observations or with numerical stability analyses. The restriction to the dyonic sector is presented as essential; establishing that restriction rigorously would therefore be a necessary condition for the result to carry weight.

major comments (2)
  1. [Abstract] Abstract (and opening paragraphs): the assertion that 'the QTEM contribution is trivial for pure electric/magnetic charge' is load-bearing for the decision to restrict the analysis to dyonic solutions, yet neither the QTEM Lagrangian nor the explicit reduction of the stress-energy tensor in the pure-charge limits is supplied. Without this cancellation shown, it is impossible to verify that the second branch and negative-pressure envelope are genuinely dyonic-specific rather than artifacts of an incomplete truncation.
  2. [Numerical results / structure equations] The description of the second branch (large-mass/radius solutions with negative-pressure envelope) and its merger into the paperclip shape rests on numerical integration of the structure equations, but the manuscript supplies neither the explicit form of the QTEM-modified Tolman-Oppenheimer-Volkoff equation nor the quark-matter equation of state employed. Consequently the claim that these features arise only for small dyonic charge ratios cannot be assessed for robustness against variations in the EOS or integration method.
minor comments (2)
  1. [Abstract] Notation for the dyonic charge ratio should be defined once at first use and used consistently thereafter; the current text alternates between 'dyonic charge ratio' and an undefined symbol.
  2. [Figures] Figure captions for the M–R diagrams should state the precise range of the dyonic charge ratio scanned and the value of any fixed parameters (e.g., bag constant, central density).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that require clarification. We address each major comment below and will revise the manuscript to incorporate the requested explicit derivations and details.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and opening paragraphs): the assertion that 'the QTEM contribution is trivial for pure electric/magnetic charge' is load-bearing for the decision to restrict the analysis to dyonic solutions, yet neither the QTEM Lagrangian nor the explicit reduction of the stress-energy tensor in the pure-charge limits is supplied. Without this cancellation shown, it is impossible to verify that the second branch and negative-pressure envelope are genuinely dyonic-specific rather than artifacts of an incomplete truncation.

    Authors: We agree that an explicit demonstration strengthens the presentation. The QTEM Lagrangian from Liu et al. (2019) contains a term proportional to the product of the electric and magnetic field strengths; this term vanishes identically when either charge is zero, recovering the standard Maxwell stress-energy tensor. In the revised manuscript we will quote the Lagrangian and provide the component-by-component reduction of the stress-energy tensor in the pure-electric and pure-magnetic limits to make the cancellation transparent. revision: yes

  2. Referee: [Numerical results / structure equations] The description of the second branch (large-mass/radius solutions with negative-pressure envelope) and its merger into the paperclip shape rests on numerical integration of the structure equations, but the manuscript supplies neither the explicit form of the QTEM-modified Tolman-Oppenheimer-Volkoff equation nor the quark-matter equation of state employed. Consequently the claim that these features arise only for small dyonic charge ratios cannot be assessed for robustness against variations in the EOS or integration method.

    Authors: The manuscript derives the modified hydrostatic equilibrium equation from the Einstein equations with the QTEM stress-energy tensor, but we accept that the final differential form and the precise MIT-bag EOS parameters were not written out. The revised version will display the complete set of structure equations (including the QTEM contributions to the effective energy density and pressure) together with the numerical values of the bag constant and strange-quark mass used in the integrations, enabling direct reproducibility and tests of robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper solves the Einstein equations sourced by the QTEM stress-energy tensor for dyonic quark-star interiors, treating the dyonic charge ratio as an explicit free parameter that is scanned to generate families of solutions. The second branch, negative-pressure envelope, and paperclip merger are outputs of that scan applied to the field equations; they are not shown to reduce to the input parameter by algebraic identity or by a self-citation that itself lacks independent verification. The statement that QTEM is trivial for pure charges is attributed to the cited Liu et al. (2019) construction and is not re-derived here in a way that creates a closed loop. No load-bearing step equates a claimed prediction to a fitted quantity or to a prior result whose only support is the present authors' own work.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the QTEM Lagrangian from Liu et al. 2019, the assumption that quark matter can be modeled with a standard equation of state inside that theory, and the choice to scan the dyonic charge ratio as the controlling parameter. No new entities are postulated.

free parameters (1)
  • dyonic charge ratio
    The ratio of magnetic to electric charge is varied across the parameter space and directly controls the appearance and merger of the two solution branches.
axioms (2)
  • domain assumption Quasi-topological electromagnetism reduces to standard Maxwell theory for pure electric or pure magnetic charge but becomes non-trivial for dyonic configurations.
    Stated in the abstract as the reason only the dyonic case is studied.
  • standard math The stellar structure equations of general relativity remain valid when the QTEM stress-energy tensor is added.
    Implicit in any solution of the Tolman-Oppenheimer-Volkoff equation with an extra source term.

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