On global vortices in the higher derivative Lorentz-violating scenario

E. Passos; F. A. Brito; M. A. Anacleto; M. Paganelly

REVIEW 4 major objections 4 minor 39 references

On global vortices in the higher derivative Lorentz-violating scenario

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Charged global vortices reveal the Lorentz-violating background through their electric field, while neutral vortices remain unaffected.

desk verdict A mostly coherent BPS construction where the headline LIV-insensitivity of neutral vortices is engineered by the chosen potential, not a generic property of the model; the charged-sector electric fields are new but the paper needs reframing and a numerical fix. read the letter →

arxiv 2501.11588 v2 pith:FQSTQGLV submitted 2025-01-20 hep-th

M. Paganelly , M. A. Anacleto , F. A. Brito , E. Passos This is my paper · ORCID

classification hep-th PACS 11.30.Cp11.27.+d

keywords globalvorticesLorentzinvarianceviolationhigher-derivativeoperatorsBPSfirst-orderformalismenergyregularizationelectricpermittivityBorn-Infeldelectrodynamicstopologicaldefects

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

The paper constructs a (2+1)-dimensional model in which a complex scalar field couples to a fixed three-vector through a higher-derivative term that violates Lorentz invariance (LIV). The authors show that for neutral global vortices, the LIV term cancels out of both the equation of motion and the energy density once a specially chosen potential $V(h,r)$ is used, so those vortices are indifferent to the LIV background. For charged structures, the same cancellation is achieved by moving the LIV dependence into the electric permittivity $P(|\phi|,r)$, making the radial electric field explicitly depend on the LIV parameter $\bar g$ while the energy density stays LIV-free. A Born-Infeld extension produces a finite electric field at the origin whose strength is controlled by $\bar g$, subject to a constraint on the Born-Infeld parameter. The significance is that LIV effects would be invisible in neutral vortex configurations but could leave observable signatures in the electric sector of charged vortices.

What carries the argument

The machinery has three parts. First, the higher-derivative LIV operator $\bar g(\partial_\mu \bar\phi) u^\mu u^\nu (u^\alpha \partial_\alpha)^2 (\partial_\nu \phi)$ couples the complex scalar to the fixed vector $u^\mu$; with LIV in the angular direction it contributes $\bar g n^2 h^2/r^4$ to the energy density. Second, the auxiliary function $W(|\phi|)$ implements the Bogomol'nyi trick: writing the energy density as a perfect square plus $2W'/r$ gives the bound $E \ge 4\pi|W(\infty)-W(0)|$, saturated when $h'=W_{|\phi|}/r$. Third, the potential $V$ in Eq. (8) and the electric permittivity $P$ in Eq. (19) are chosen rather than derived, and this choice cancels the $\bar g$-term from the neutral equations while transferring LIV sensitivity to the electric sector. In the Born-Infeld version, the same $P$ enters inside the square-root nonlinearity, and the parameter $b$ must satisfy $2b^2 > -eE_r/r$ to avoid the pole in Eq. (37).

What would settle it

Repeat the neutral-vortex calculation with a standard translationally invariant potential, e.g., $V(|\phi|) = \lambda( |\phi|^2 - a^2 )^2/4$, instead of the tuned $V(h,r)$ of Eq. (8). If the equation of motion then retains the $\bar g n^2 h^2/r^4$ term and the integrated energy diverges logarithmically with $r$, the claimed LIV-blindness of neutral vortices is an artefact of the chosen ansatz rather than a property of the higher-derivative operator. For the charged sector, evaluating Eq. (25) at $\bar g = 0$ should reproduce the LIV-free regularized field; any mismatch would refute the claimed control of the electric field by the LIV parameter.

Watch

Extended reading notes

Core claim

The central claim is that the higher-derivative Lorentz-violating term in Eq. (1), built from a fixed three-vector $u^\mu$, can be made to disappear from the dynamics of neutral global vortices by choosing the potential $V(h,r)$ in Eq. (8) to contain the same $\bar g n^2 h^2/r^4$ term that appears in the kinetic sector. The first-order BPS equation $h' = W_{|\phi|}/r$ then yields finite-energy solutions with energy $E = 4\pi|W(h(\infty))-W(h(0))|$. For charged vortices, the electric permittivity $P(|\phi|,r)$ in Eq. (19) is tuned so that LIV re-enters through Gauss's law, giving the electric field of Eq. (25), which becomes more negative near the origin as $\bar g$ grows. The Born-Infeld extension of Sec. III C produces a finite field at the origin, reducing to $E_r/2$ in the large-$b$ limit, with the condition $2b^2 > -eE_r/r$ to avoid singularities. The paper concludes that neutral global structures do not see the LIV background, whereas charged structures do, through the intensity of the electric field.

Load-bearing premise

The potential $V(h,r)$ and the electric permittivity $P(|\phi|,r)$ are chosen by hand rather than derived from the Lagrangian, so the claimed vanishing of LIV effects in neutral vortices and their reappearance in the electric field hold only for these tuned functions; with any other choice the $\bar g$-term in Eq. (4) does not disappear, and the paper does not discuss whether the resulting negative permittivity is physically acceptable.

Editorial extensions

If this is right

Neutral global vortices in this model have the same profile, energy density $\rho = 2W'/r$, and total energy $E = 8\pi$ (for $W=|\phi|^2-|\phi|^3/3$) as in the absence of LIV, so the $\bar g$-term is completely hidden from the neutral sector.
The radial electric field of a charged vortex in Eq. (25) vanishes at the origin and becomes increasingly negative as $\bar g$ grows, meaning the LIV parameter directly controls the field intensity near the core.
All solutions obey the first-order BPS equation, so the energy is minimized and finite, $E = 4\pi|W(h(\infty))-W(h(0))|$.
In the Born-Infeld extension, the electric field at the origin is finite and maximal, with intensity increasing with $\bar g$, and the constraint $b^2 > 8$ (for $e=1$, $n=1$) avoids the divergence of Eq. (37).
The two charged models share the same LIV-free energy density $\rho = 2h'^2$, so the electric field can be reshaped without changing the total energy.

Reading between the lines

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

The LIV-blindness of neutral vortices follows from the chosen forms of $V$ and $P$; with a generic potential, such as the usual symmetry-breaking one, the $\bar g n^2 h^2/r^4$ term remains and the energy diverges, so the paper's conclusion is a property of the constructed model rather than of the LIV operator itself.
The negative effective permittivity implied by the sign of the electric field near the origin is not discussed in the paper; whether a passive medium can realize this requires additional structure, so the physical viability of the medium remains open.
Because the Born-Infeld parameter $b$ only halves the field in the large-$b$ limit, a different auxiliary function could move the LIV signature to intermediate distances, giving a sharper experimental handle in condensed-matter analogues.
One could test the BPS claim by numerically solving the full second-order equations of motion with the tuned potential and comparing the profile to the analytic solution $h(r)=2r^2/(r^2+1)$; a mismatch would indicate the first-order solution is not the true minimizer.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague —

Quick take: this is a model-construction paper in the LIV topological-defect niche. The genuinely new bits are the explicit LIV-dependent electric field in Eq. (25) and its Born-Infeld variant in Eq. (37), along with the BPS reduction that makes the energy finite. The algebra is mostly sound: the first-order equation h' = W_|phi|/r, the solution h(r) = 2r^2/(r^2+1), and the energy density 32r^2/(r^2+1)^4 all check out. The authors also cite the relevant Bazeia-type literature and are honest inside the text that the potential is a phenomenological construction.

But the central physical claim needs re-scoping. The neutral vortex's insensitivity to LIV is not a discovery; it is put in by hand. The LIV term appears in Eq. (4) and Eq. (5), and it disappears only because the potential V(h,r) in Eq. (8) is chosen to contain exactly the opposite g-bar-dependent combination. The paper does acknowledge this in words, but the abstract and conclusion still state that LIV does not affect neutral global structures, which overstates what the model shows. The same issue applies to the permittivity P in Eqs. (19) and (33): those choices make charged structures LIV-sensitive, but the sensitivity is a property of the ansatz, not of the higher-derivative operator itself.

There is also a concrete numerical error. The text says total energy E = 8π, but integrating Eq. (24) gives 2π ∫ 32 r^3/(r^2+1)^4 dr = 16π/3. That matches the BPS bound 4π(W(2)-W(0)) = 16π/3. This has to be corrected.

A softer concern is that the permittivity P becomes negative over part of the domain. That is a physical red flag—negative permittivity can imply instabilities or require external pumping—and the paper never addresses it. The Born-Infeld model has a constraint on b to avoid divergences, which the authors do discuss, but the negative-permittivity issue in the first model deserves at least a comment.

Bottom line: this is a competently executed construction in a narrow subfield. It does not resolve a long-standing problem or make contact with data. Its value is as an analytic toy model: if you work on BPS vortex regularization or LIV defect engineering, Eqs. (25) and (37) are citable. But the paper needs a revision before publication: fix the energy value, discuss or justify the negative permittivity, and reframe the LIV-insensitivity claim as a consequence of the chosen potential.

I would send it to peer review, but with the expectation of significant revision. For a reading group, it is a maybe—useful as an example of how engineered ansatze can quietly determine the physics, even when the algebra is correct.

Referee Report

4 major / 4 minor

Summary. The paper studies (2,1)-dimensional global vortices in a theory with a Lorentz-violating higher-derivative coupling between a complex scalar and a fixed vector. The authors add a translation-noninvariant potential V(|φ|, x^2) so that the neutral-sector energy density and equation of motion reduce to a BPS form; they then construct charged generalizations by introducing a scalar-dependent electric permittivity P, in both Maxwell and Born-Infeld versions. The paper reports analytical vortex profiles, first-order equations, finite-energy configurations, and electric fields whose intensity depends on the Lorentz-violating parameter.

Significance. The BPS-completion procedure is transparent and the analytical solutions in the neutral sector are correct; if the charged-sector equations were corrected, the model would be a useful example of how a scalar-controlled permittivity can absorb Lorentz-violating effects in a defect construction. The neutral-sector claim, however, is weaker than stated because it follows from the tuned form of V rather than from the higher-derivative operator alone. The paper also makes a small novel step by combining Born-Infeld regularization with permittivity controlled by the scalar field, although that part currently contains algebraic inconsistencies.

major comments (4)

[II, Eq. (8), Abstract, and Section IV] The claim that neutral global vortices are insensitive to Lorentz-symmetry violation is an artifact of the ansatz. The LIV term \bar g n^2/r^2 is present in Eq. (4) and Eq. (5), and it is eliminated only because Eq. (8) defines V to contain exactly the negative of that term. With any other potential, such as a conventional V(|φ|), the term survives. Section II acknowledges the phenomenological character of V, but the abstract and conclusion do not carry this qualification; the result should be re-scoped to the class of potentials (8) or presented explicitly as an engineered cancellation.
[III A, Eq. (24) and surrounding text] The total energy stated after Eq. (24), E = 8π, is incorrect. Integrating Eq. (24) gives 2π∫_0^∞ 32 r^3/(r^2+1)^4 dr = 16π/3, which coincides with the BPS bound 4π(W(2)−W(0)) = 16π/3 computed from Eq. (21). The value 8π is therefore also inconsistent with the paper's own bound in Eq. (10).
[III A, Eq. (19) and Eq. (25)] The electric permittivity P in Eq. (19) is negative on part of the domain. For n=1 and \bar g>0, the bracket in Eq. (25) is negative both near the origin and asymptotically, so P<0. Because E = e/(Pr), this negative P is responsible for the negative-charge-like field shown in Fig. 2. However, the gauge kinetic term in Eq. (12) has the wrong sign where P<0, and the paper does not discuss the resulting instabilities or justify why such a medium is physically acceptable. The positivity of the final BPS energy density does not remove this concern.
[III C, Eqs. (33)-(37)] I could not reproduce Eq. (37) from the preceding definitions. Substituting Eq. (33) into Eq. (31) and using Eq. (35) gives Θ = 1 + e E_r/(2 r b^2), not the expression in Eq. (36), and then \tilde E_r = E_r (1 + e E_r/(4 r b^2))/(1 + e E_r/(2 r b^2)), which tends to E_r as b→∞ rather than to E_r/2 as claimed. If Eq. (36) is instead used exactly as printed, the magnitude does not reduce to Eq. (37) either. Eq. (37), the associated large-b statement, and Figs. 5-6 therefore need to be re-derived.

minor comments (4)

[Title] The title contains a typo: 'Lorentz-violat ing' should read 'Lorentz-violating'.
[III A, after Eq. (24)] The sentence 'which remains unaﬀected by the constant' is unclear; the constant should be identified explicitly, presumably \bar g.
[III C, Fig. 4 caption] The caption refers to 'g = 0.5' rather than '\bar g = 0.5'; the barred notation should be used consistently.
[III A, Eq. (12)] The charged-sector Lagrangian in Eq. (12) omits the scalar potential V(|φ|, x^2) that appears in Eq. (1); if this omission is intentional, a sentence should state so.

Circularity Check

2 steps flagged · score 6.0 of 10

The central qualitative claims—neutral-vortex LIV insensitivity and charged-vortex LIV sensitivity—are built into the tuned potential V(h,r) and permittivity P(|phi|,r) in Eqs. (8) and (19), rather than being independent consequences of the higher-derivative LIV operator.

self definitional [Sec. II, Eq. (8), and conclusion after Eq. (11)]
"In order to eliminate the dependence of the effective potential on the winding number, we assume that the potential is given by [V(h,r)= W_|phi|^2/r^2 - n^2 h^2/r^2 (1 + gbar n^2/r^2)]. ... we observe that the LIV term disappears from it and the equation of motion, leading to a scenario where the breaking of Lorentz symmetry is not noticed. Consequently, the global structures analyzed here are not influenced by Lorentz-symmetry violation."

Eq. (8) is not derived from the base Lagrangian; it is chosen so that its second term exactly cancels the LIV combination n^2 h^2 (1 + gbar n^2/r^2)/r^2 in the energy density and equation of motion. Therefore the conclusion that neutral global vortices are insensitive to LIV is an algebraic consequence of substituting this ansatz. With any potential not containing this counter-term, the gbar contribution in Eqs. (4) and (5) remains. The paper acknowledges the potential as phenomenological, but the abstract and conclusion state the result unconditionally.
self definitional [Sec. III A, Eqs. (19) and (25), with the text before Eq. (19)]
"Motivated by these works, we make a particular choice of P, writing it in the form [P = e^2/2 ( W_|phi|^2 - n^2 h^2(1 + gbar n^2/r^2) )]. This expression follows the same line of reasoning applied to the potential in Eq. (8). ... Replacing the electric permittivity in Eq. (19) with the above solution in Eq. (23) allows us to rewrite the expression for the electric field in Eq. (15) as follows [E = ... gbar ...]."

The gbar dependence of the electric field is inserted through the permittivity P, which is chosen to contain the same -n^2 h^2(1 + gbar n^2/r^2) combination used in Eq. (8). Eq. (25) is a rearrangement of that choice, so the claim that charged structures are sensitive to LIV is a property of the imposed P rather than an independent consequence of the higher-derivative interaction. The same construction is repeated for the Born-Infeld permittivity P in Eq. (33).

full rationale

The formal derivation is algebraically self-contained: once V(h,r) and P(|phi|,r) are fixed by Eqs. (8), (19), and (33), the first-order BPS equations, the exact profile h(r)=2r^2/(r^2+1), the energy density rho=32r^2/(r^2+1)^4, and the electric-field formulas follow consistently. However, the two headline physical messages are not independent discoveries; they are encoded in those very choices. Eq. (8) is explicitly tuned so that the term (1 + gbar n^2/r^2) cancels from the neutral vortex equation of motion and energy density, and Eq. (19) is explicitly tuned so that the same term reappears in the permittivity and hence in the electric field. The paper's own language, 'we assume', 'we make a particular choice', and 'phenomenological construction', confirms that these are ansatze rather than forced results. The citation to Ref. [28] is motivational and is not used as a load-bearing uniqueness theorem, so no separate self-citation circularity is present. The numerical claim E=8pi is inconsistent with integrating Eq. (24) (the result is 16pi/3), but that is a correctness risk rather than a circularity. Overall, the model is coherent, but the abstract and conclusion should be re-scoped to state that LIV insensitivity in the neutral sector and LIV sensitivity in the charged sector are consequences of the specially constructed V and P, not of the LIV operator alone; hence partial circularity, score 6.

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

The model is an engineering exercise: the potential, the permittivity, and the auxiliary function are all chosen so that the BPS reduction works and so that LIV effects appear only in the charged electric field. The main free inputs are the LIV coupling, the Born-Infeld constant, and the winding number. No new particles or extra dimensions are introduced.

free parameters (4)

LIV coupling constant gbar = positive, e.g., 0.5, 1.5, 2 in plots
Chosen by hand; controls the electric-field signature in Eq. (25) and defines the model. In the neutral sector it is canceled by the chosen potential.
Born-Infeld parameter b = b squared greater than 8, with b=3 in plots
Chosen to avoid the pole in Eq. (37); controls the saturation scale and the near-origin field profile.
vorticity n = n=1 in plots
Topological integer input inherited from the ansatz in Eq. (3); appears in the engineered potential and permittivity.
Auxiliary function W coefficients = W = |phi|^2 - (1/3)|phi|^3
Hand-picked to solve the first-order equation h' = W_h/r; determines h(r), the energy profile, and the electric field. Many forms are possible, so the results are not generic.

assumptions (4)

domain assumption A fixed background three-vector u_mu exists and takes the angular direction u_theta = (0,0,1) in spherical coordinates.
The LIV operator in Eq. (1) is a model input; the angular choice is used to obtain Eq. (4). Conclusions depend on this choice.
ad hoc to paper The potential V(|phi|,x^2) breaks translation invariance via x^2 dependence to evade Derrick-Hobart and admits the BPS form in Eq. (8).
The potential is not derived; it is engineered so that the LIV term cancels and the first-order formalism closes.
ad hoc to paper The electric permittivity P(|phi|,x^2) is an adjustable function chosen as Eq. (19) and Eq. (33) in the Born-Infeld case.
This choice places the LIV parameter inside the electric field and cancels the divergent energy terms; it is not a consequence of the original Lagrangian.
standard math The BPS energy-splitting ansatz assumes the auxiliary function W exists and that minimum energy is achieved by the first-order equation h' = W_h/r.
The Bogomol'nyi method is standard; the specific choice of W is hand-picked.

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Pith. "Pith review of On global vortices in the higher derivative Lorentz-violating scenario." pith.science (2026). https://pith.science/paper/FQSTQGLV

@misc{pith2026250111588,
  author       = {Pith},
  title        = {Pith review of: On global vortices in the higher derivative Lorentz-violating scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQSTQGLV}},
  note         = {Machine review of arXiv:2501.11588}
}

abstract

We study the influence of Lorentz invariance violation (LIV) background in energy regularization of global structures in $(2, 1)$--dimensions. To this end, we consider a model in which the complex scalar and fixed three-vector couple as a high derivative order term. We show that LIV-background does not affect the energy and the equation of motion of neutral global structures. However, we observe that the charged structures are sensitive to LIV-background by presenting signatures in the electric field whose intensity is controlled by the LIV-parameter. Furthermore, the procedure developed leads to first-order solutions with finite energy and a regularized electric field.

Figures

Figures reproduced from arXiv: 2501.11588 by the authors.

**Figure 2.** FIG. 2: The radial component of the electric field in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: The radial component of the electric field in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: The radial component of the electric field in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: The radial component of electric field in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: The radial component of the electric field in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Works this paper leans on

39 extracted references · 32 canonical work pages

[1]

Colladay and V

D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998)

1998
[2]

V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989)

1989
[3]

A. A. A. Filho, J. R. Nascimento, A. Y. Petrov and P. J. Porf ´ ırio, Phys. Rev. D108, no.8, 085010 (2023)

work page 2023
[4]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, 2000)

2000
[5]

S. M. Carroll and H. Tam, Phys. Rev. D 78, 044047 (2008)

work page 2008
[6]

Horava, Phys

P. Horava, Phys. Rev. D 79, 084008 (2009)

work page 2009
[7]

R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003)

work page 2003
[8]

Casana, M

R. Casana, M. M. Ferreira, Jr, E. da Hora and C. Miller, Phy s. Lett. B 718, 620 (2012)

work page 2012

Show all 39 references

[9]

Belich, T

H. Belich, T. Costa-Soares, M. M. Ferreira, J. A. Helayel -Neto and M. T. D. Orlando, Int. J. Mod. Phys. A 21, 2415 (2006)

2006
[10]

Mavigno, H

E. Mavigno, H. Belich and G. Luchini, Eur. Phys. J. C 84, 1177 (2024)

2024
[11]

N. S. Manton and P. Sutcliﬀe, Topological solitons (Cambridge University Press, 2004)

2004
[12]

T. W. B. Kibble, J. Phys. A 9, 1387 (1976)

1976
[13]

H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973)

1973
[14]

E. B. Bogomolny, Sov. J. Nucl. Phys. 24, 449 (1976)

1976
[15]

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

1975
[16]

Shinjo, T

T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono, S cience 289, 930 (2000)

2000
[17]

Wachowiak, J

A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgens tern and R. Wiesendanger, Science 298, 577 (2002)

2002
[18]

G. W. Rayﬁeld and F. Reif, Phys. Rev. Lett. 11, 305 (1963)

1963
[19]

Donadello, S

S. Donadello, S. Seraﬁni, M. Tylutki, L. P. Pitaevskii, F. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev. Lett. 113, 065302 (2014)

2014
[20]

Building on these ﬁndings, we examined the inﬂuence of LIV-eﬀects o n the energy and equation of motion of topological defects in (2 , 1)–dimensions

has focused on global vortices, introducing a method that regula rizes the energy, thus enabling new possibilities. Building on these ﬁndings, we examined the inﬂuence of LIV-eﬀects o n the energy and equation of motion of topological defects in (2 , 1)–dimensions. The main fo...

2025 arXiv
[21]

Bazeia, M

D. Bazeia, M. A. Marques and R. Menezes, EPL 122, 61001 (2018)

2018
[22]

K. E. L. de Farias, M. A. Anacleto, F. A. Brito, E. Passos a nd H. Mota, Phys. Lett. B 852, 138641 (2024)

2024
[23]

Bazeia, J

D. Bazeia, J. Menezes and R. Menezes, Phys. Rev. Lett. 91, 241601 (2003)

2003
[24]

Bazeia, M

D. Bazeia, M. A. Marques and R. Menezes, Eur. Phys. J. Plu s 138, no.8, 735 (2023)

2023
[25]

Jubert, R

P.-O. Jubert, R. Allenspach, and A. Bischof, Phys. Rev. B 69, 220410(R) (2004)

2004
[26]

Bazeia, J

D. Bazeia, J. G. G. S. Ramos and E. I. B. Rodrigues, J. Magn . Magn. Mater. 423, 411 (2017)

2017
[27]

Bazeia, M

D. Bazeia, M. M. Doria and E. I. B. Rodrigues, Phys. Lett. A 380, 1947-1952 (2016)

2016
[28]

Bazeia and E

D. Bazeia and E. I. B. Rodrigues, J. Magn. Magn. Mater. 432, 112 (2017)

2017
[29]

Bazeia, M

D. Bazeia, M. A. Marques and M. Paganelly, Eur. Phys. J. P lus 136, 990 (2021)

2021
[30]

Bazeia, M

D. Bazeia, M. A. Marques and M. Paganelly, Eur. Phys. J. C 82, no.11, 1036 (2022)

2022
[31]

Bazeia, M

D. Bazeia, M. A. Marques and R. Menezes, Phys. Lett. B 780, 485-490 (2018)

2018
[32]

Bazeia, M

D. Bazeia, M. A. Marques and R. Menezes, Phys. Rev. D 97, no.10, 105024 (2018)

2018
[33]

Born and L

M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144, 425 (1934)

1934
[34]

Y. F. Alam and A. Behne, Review of Born-Infeld electrodynamics . arXiv:2111.08657

arXiv
[35]

S. I. Kruglov, Mod. Phys. Lett. A 32, 1750201 (2017)

2017
[36]

Han, Proc

X. Han, Proc. Roy. Soc. Lond. A 472, 20160012 (2016)

2016
[37]

S. K. Moayedi, M. Shafabakhsh and F. Fathi, Adv. High Ene rgy Phys. 01, 1885 (2015)

2015
[38]

Bazeia, M

D. Bazeia, M. A. Marques and R. Menezes, Phys. Rev. D 104, L121703 (2021)

2021
[39]

J. L. Kim and G. Geshnizjani, JCAP 03, 104 (2021)

2021

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Colladay and V

D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998)

1998

[2] [2]

V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989)

1989

[3] [3]

A. A. A. Filho, J. R. Nascimento, A. Y. Petrov and P. J. Porf ´ ırio, Phys. Rev. D108, no.8, 085010 (2023)

work page 2023

[4] [4]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, 2000)

2000

[5] [5]

S. M. Carroll and H. Tam, Phys. Rev. D 78, 044047 (2008)

work page 2008

[6] [6]

Horava, Phys

P. Horava, Phys. Rev. D 79, 084008 (2009)

work page 2009

[7] [7]

R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003)

work page 2003

[8] [8]

Casana, M

R. Casana, M. M. Ferreira, Jr, E. da Hora and C. Miller, Phy s. Lett. B 718, 620 (2012)

work page 2012