Magnetic field effect on neutral color superconductivity as a mechanism to boost the inner magnetic field of magnetars

Bo Feng

REVIEW

Magnetic field effect on neutral color superconductivity as a mechanism to boost the inner magnetic field of magnetars

Bo Feng

High Energy Physics - Phenomenology · Submitted 23 Oct 2026 · Published version · 112 references

Pith's verdict: A competent review of the field-boosting mechanism from the author's own group; the central prediction is interesting but rests on a stability threshold that is not derived from first principles.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 26, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2610.32028 (version 1)

Pith IDSZCWJ2XR, Pith's permanent record of this paper and its review

CitationBo Feng. "Magnetic field effect on neutral color superconductivity as a mechanism to boost the inner magnetic field of magnetars." arXiv:2610.32028 (2026). https://pith.science/paper/SZCWJ2XR

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This paper is a review of the magnetic-field amplification mechanism in neutral 2SC quark matter that the author and collaborators published in 2024 [77]. It does a clear job of collecting the formalism, showing the gap equations and Meissner masses, and walking through the vortex-condensate resolution of the chromomagnetic instability. The numerical results are consistent with the prior literature, and the connection to magnetar core fields (~10^17 G) is well motivated.

The main thing to know is that there is no new conceptual or computational advance here. The paper presents itself as a review and draws the central numerical results from [77]. That is fine for a review, but it means the novelty for an expert reader is low.

The soft spot that concerns me most is the stability condition (55): the claim that the lowest charged-gluon mode must not exceed 2Δ to avoid pair-production destabilization. This threshold is presented without derivation from the effective action or any known instability criterion. If the true threshold were Δ or 3Δ, or if no such threshold exists, the equality (56) changes and the predicted amplified field shifts by factors of order unity. The paper lists many limitations (anisotropic polarization, strange quarks, model dependence of GD) but does not flag this ad hoc condition as a separate hidden sensitivity.

That said, the paper acknowledges its own scope honestly—the authors state the limitations and call for future work. The mechanism itself is built on solid ideas (zero-mode problem, paramagnetic response of gluon vortices) that have been explored in other contexts. For a reader working on dense QCD or magnetars, this is a useful consolidation. I would not cite it as a primary source, but I could assign it in a reading group to summarize the mechanism.

Recommendation: Send it to peer review. The topic is of astrophysical interest, the mathematics is internally consistent, and the paper is well written. But the referee should ask for a discussion of the physical basis of condition (55) and a sensitivity estimate of the predicted field to that threshold.

Referee report

Significance

If the mechanism is correct, the paper offers a microphysical and falsifiable scenario for magnetar-strength core fields: the predicted amplification depends on specific model inputs such as the diquark coupling ratio, the gap, and the Meissner mass, and it would not operate if the neutral 2SC phase is not the relevant core state. The manuscript is reasonably complete as a pedagogical synthesis, states its model parameters, and is honest about several model uncertainties, including the diquark coupling convention and the zero-field approximation for the gluon polarization tensor. Its incremental contribution is, however, mostly organizational, since the main quantitative result is taken from Ref. [77]. The central numerical claim also rests on a stability threshold whose derivation and status are not made clear in this paper, which limits the confidence one can place in the quoted 10^17 G value until that point is clarified.

Major objections (3)

[Sec. IV, Eqs. (55)-(56)] Equation (55) is the load-bearing step of the entire numerical claim: the induced field is obtained by solving the equality (56), and Eqs. (58)-(59) show that the predicted B is fixed by the combination 4Delta^2 - m_M^2 once this threshold is imposed. The text introduces (55) with the statement that the lowest energy mode 'must satisfy' the bound 2Delta, but no derivation from the effective action (42)-(44) or from an explicit instability criterion is provided, and no citation is given at the point of use. This threshold is not a harmless convention: replacing 2Delta by Delta or 3Delta changes the right-hand side of Eq. (56) and hence the induced field by order-one factors. The inequality also does not, by itself, exclude a tachyonic lowest mode; the paper should explain why the relevant solution is the saturation of the upper bound rather than, for example, the lower endpoint E^2=0. Since the paper is a review of Ref. [77], it should at minimum attribute the threshold to that work, state explicitly whether it is an assumption, and provide a sensitivity estimate.
[Sec. III.A, Eqs. (44)-(46)] Minimizing the Gibbs free energy (44) with respect to B gives B = H + 4 e |G|^2, whereas Eq. (46) states B = H + 2 e |G|^2. The factor-of-two mismatch is not explained. Equation (46) is subsequently used to eliminate |G|^2 in deriving the dispersion relation (54) and hence Eq. (56), so the inconsistency is not purely cosmetic. If the prefactor in Eq. (44) is a typo, it should be corrected; if the factor reflects the two-component normalization of the vortex field, that normalization should be stated explicitly.
[Sec. IV and Fig. 5] The central numerical claim, namely that an applied field around 10^15 G is amplified by up to two orders of magnitude, is supported only by a verbal report of results from Ref. [77] and by Fig. 5, which plots the induced field versus the chemical potential and not versus the applied field. The text states for eta=0.85 and mu=400 MeV that the boosting is most pronounced near H=10^15 G, and later states that applied fields as low as 10^8-10^9 G produce essentially the same induced field. The review should include a B(H) curve or a table of values for at least one representative mu and eta so that the reader can verify the claimed amplification.

Minor objections (7)

[Eq. (30)] In the electron contribution, the term 2 pi^2 T^2 mu^2 should presumably be 2 pi^2 T^2 mu_e^2; please check the expression.
[Eq. (48)] The definitions of C and D both use g<A^(1)_3>; one of them should presumably refer to A^(2)_3.
[Sec. III.B and Sec. II.C] The notation m_g^2 is used with two different definitions: Sec. II.C defines m_g^2 = 4 alpha_s mu_bar^2/(3 pi), while Sec. III.B writes m_g^2 = 4 alpha_s^2 mu_bar^2/(3 pi). These should be reconciled.
[Sec. IV] The statement that the boosting effect is 'most pronounced' at applied fields around 10^15 G is in tension with the later claim that applied fields of 10^8-10^9 G yield essentially the same induced field, since the latter case corresponds to a much larger amplification factor.
[Fig. 5] The caption of Fig. 5 does not state the value of the applied field for which the induced field is plotted; please add this information.
[Abstract and Introduction] The abstract and parts of the text use active-voice claims such as 'we show' and 'we solve', although the paper is explicitly a review of previously published results; the wording should make the distinction between new and reviewed content clearer.
[Throughout] There are several typographical errors, including 'Mastubara', 'sufficiently' spelled 'suffi ciently', 'relevant' spelled 'relavant', and 'beyond' spelled 'beyong'; an editorial pass is needed.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Gluon vortices in neutral quark matter can amplify seed magnetic fields to magnetar strengths of ~10^17 G.

The reading

This review establishes that in the neutral two-flavor color superconducting (2SC) phase of dense quark matter, a sufficiently strong rotated magnetic field triggers a chromomagnetic instability through the 'zero-mode problem'. The instability is resolved by the formation of a gluon vortex condensate, which in turn induces a paramagnetic response that amplifies the in-medium magnetic field. Seed fields around $10^{15}$ G can be boosted by up to two orders of magnitude, reaching ~$10^{17}$ G, precisely the range inferred for the cores of magnetars. The mechanism works even for much weaker seed fields (down to $10^{8}$–$10^{9}$ G), as the induced field depends primarily on the diquark gap and Meissner mass rather than on the applied field. This offers a plausible microscopic origin for the extreme inner magnetic fields of magnetars, complementary to macroscopic dynamo processes.

What carries the argument

The central mechanism is the anomalous magnetic moment term $i\tilde{e}\tilde{f}_{\mu\nu} G^{+\mu} G^{-\nu}$ in the effective action for charged gluons in a rotated magnetic field. This term introduces a negative contribution to the squared mass of the gluon modes, causing the zero-mode instability when $\tilde{e}\tilde{B}$ exceeds the Meissner mass squared. The instability is resolved by a gluon vortex condensate $\langle G^\pm_\mu \rangle$, which modifies the ground state and, through the equation $2\tilde{e}|G|^2 - \tilde{B} + \tilde{H}=0$, yields a paramagnetic response that drives the in-medium field $\tilde{B}$ higher than the applied field $\tilde{H}$.

Extended reading notes

Core claim

The core claim is that a gluon vortex condensate forming in neutral 2SC quark matter under a magnetic field not only cures the chromomagnetic instability (negative Meissner mass squared) but also creates a paramagnetic amplification of the field. Solving the coupled equations for the condensate and the induced field shows that the final field strength is set by the diquark gap and the tachyonic Meissner mass, leading to induced fields of order $10^{17}$ G over a wide range of moderate diquark couplings and chemical potentials relevant to neutron star interiors. This amplification is independent of the precise seed field value once the vortex state develops.

Load-bearing premise

The threshold condition (Eq. 55) that the lowest Landau level energy of charged gluons must not exceed twice the quasiparticle gap is motivated by stability against pair production but is not rigorously derived from first principles; if a different threshold is used, the predicted induced field could change significantly.

Editorial extensions

If this is right

If the mechanism operates in magnetar cores, the inner field can reach ~10^17 G without requiring an extremely strong initial seed or a dynamo that exceeds the rotational energy budget.
The amplification is robust for a range of diquark couplings ($\eta = 0.75-0.95$) and chemical potentials ($\mu \sim 320-500$ MeV), suggesting it may be a generic feature of the neutral 2SC phase if it exists in compact stars.
The induced field remains near 10^17 G even for very weak applied fields (down to 10^8 G), so the mechanism can act on any seed present in the stellar interior, independent of the details of the seed's origin.
The formation of a gluon vortex condensate also breaks rotational symmetry, making the medium anisotropic; this could affect transport properties and the equation of state in magnetar cores.

Reading between the lines

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

The same paramagnetic amplification may operate in other color-superconducting phases (e.g., the color-flavor-locked phase) if a similar zero-mode instability exists, potentially broadening the range of stars in which it could occur.
If the gluon-vortex mechanism is active, it predicts that magnetars with quark matter cores should have interior fields clustered near ~10^17 G regardless of their age, unlike the secular decay expected for dynamo-generated fields.
Future observations of gravitational waves from binary neutron star mergers might constrain the equation of state and the presence of color-superconducting cores, indirectly testing whether this amplification channel is viable.
A key test would be to compute the full gluon polarization tensor in the regime $\tilde{e}\tilde{B} \sim \mu^2$; if the induced field deviates significantly from the present estimate, the robustness of the mechanism would be challenged.

Original abstract

In this paper, we review the impact of a strong magnetic field on the neutral two-flavor color superconducting (2SC) phase of dense quark matter. Using a Nambu-Jona-Lasinio (NJL) model approach in the mean-field approximation, we first solve the gap equations for the diquark condensate $\Delta$ and the electric chemical potential $\mu_e$ under a uniform external magnetic field, and discuss the coupled evolution of the chiral and diquark order parameters. We then evaluate the field-dependent Meissner masses of the rotationally charged gluons for a wide range of densities and moderate diquark coupling strengths. Combining the "zero-mode problem" mechanism with the neutrality condition, we show that the gluon masses become tachyonic even at moderate magnetic-field values, signaling a chromomagnetic instability in density regions where the system was stable at zero field. This instability can be resolved through the restructuring of the ground state with the formation of a gluon vortex condensate. A remarkable consequence of the vortex configuration is the paramagnetic amplification of the in-medium magnetic field: seed fields of the order of $10^{15}$ G can be enhanced by up to two orders of magnitude, reaching $\sim 10^{17}$ G, precisely the range predicted for magnetar cores. We compare this microscopic mechanism with the conventional dynamo explanation and discuss how it offers a plausible explanation for the extremely high magnetic fields expected to be realized in magnetar interiors.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-26: 1 request, 5,911 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.26 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper, we review the impact of a strong magnetic field on the neutral two-flavor color superconducting (2SC) phase of dense quark matter. Using a Nambu-Jona-Lasinio (NJL) model approach in the mean-field approximation, we first solve the gap equations for the diquark condensate $\Delta$ and the electric chemical potential $\mu_e$ under a uniform external magnetic field, and discuss the coupled evolution of the chiral and diquark order parameters. We then evaluate the field-dependent Meissner masses of the rotationally charged gluons for a wide range of densities and moderate diquark coupling strengths. Combining the "zero-mode problem" mechanism with the neutrality condition, we show that the gluon masses become tachyonic even at moderate magnetic-field values, signaling a chromomagnetic instability in density regions where the system was stable at zero field. This instability can be resolved through the restructuring of the ground state with the formation of a gluon vortex condensate. A remarkable consequence of the vortex configuration is the paramagnetic amplification of the in-medium magnetic field: seed fields of the order of $10^{15}$ G can be enhanced by up to two orders of magnitude, reaching $\sim 10^{17}$ G, precisely the range predicted for magnetar cores. We compare this microscopic mechanism with the conventional dynamo explanation and discuss how it offers a plausible explanation for the extremely high magnetic fields expected to be realized in magnetar interiors.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we review the impact of a strong magnetic field on the neutral two-flavor color superconducting (2SC) phase of dense quark matter.	contribution 0.84	0.47	0.04	unclear 0.64
s1	Using a Nambu-Jona-Lasinio (NJL) model approach in the mean-field approximation, we first solve the gap equations for the diquark condensate $\Delta$ and the electric chemical potential $\mu_e$ under a uniform external magnetic field, and discuss the coupled evolution of the chiral and diquark order parameters.	method 0.99	0.94	0.03	asserted 0.87
s2	We then evaluate the field-dependent Meissner masses of the rotationally charged gluons for a wide range of densities and moderate diquark coupling strengths.	method 0.62	0.94	0.02	asserted 0.90
s3	Combining the "zero-mode problem" mechanism with the neutrality condition, we show that the gluon masses become tachyonic even at moderate magnetic-field values, signaling a chromomagnetic instability in density regions where the system was stable at zero field.	result 0.99	0.92	0.18	asserted 1.00
s4	This instability can be resolved through the restructuring of the ground state with the formation of a gluon vortex condensate.	result 0.82	0.77	0.07	asserted 0.90
s5	A remarkable consequence of the vortex configuration is the paramagnetic amplification of the in-medium magnetic field: seed fields of the order of $10^{15}$ G can be enhanced by up to two orders of magnitude, reaching $\sim 10^{17}$ G, precisely the range predicted for magnetar cores.	result 0.94	0.86	0.72	asserted 0.98
s6	We compare this microscopic mechanism with the conventional dynamo explanation and discuss how it offers a plausible explanation for the extremely high magnetic fields expected to be realized in magnetar interiors.	interpretation 0.96	0.64	0.23	hedged possibility 0.96

What kind of paper the classifier reports this is: theoretical result 0.73 · evidence from proof or derivation 0.51.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 3.** FIG. 3: The ratio of the squared Meissner mass of charged gluons to the magnetic mass,

**Figure 4.** FIG. 4: The critical field

Figures reproduced from arXiv:2610.32028 by the authors.

References

Works cited (112)

[77]

Yuan, S.; Feng, B.; Ferrer, E. J.; Pinero, A. Magnetized neutral 2SC color superconductivity and possible origin of the inner magnetic field of magnetars. Phys. Rev. D 110 (2024) 114038

View on Pith 2024
[1]

Baym, G.; Hatsuda, T.; Kojo, T.; Powell, P. D.; Song, Y. F.; Takatsuka,T. From hadrons to quarks in neutron stars: a review. Rep. Prog. Phys. 81 (5) (2018) 056902

View on Pith 2018
[2]

Dexheimer, V.; et al. Future physics perspectives on the equation of state from heavy ion collisions to neutron stars. J. Phys. G 48 (2021) 7, 073001

View on Pith 2021
[3]

Luo, X. F.; Wang, Q.; Xu, N.; Zhuang, P. F. Properties of QCD matter at high baryon density. Springer (Singapore) 2022

View on Pith 2022
[4]

Weber, F. Strange quark matter and compact stars. Prog. Part Nucl. Phys. 54 (2005) 193

View on Pith 2005
[5]

Sedrakian, A. Astrophysics of dense quark matter in compact stars. Acta Phys. Polon. B 3 (2010) 669

View on Pith 2010
[6]

Annala, E.; Gorda, T.; Kurkela, A.; Nattila, J.; Vuorinen, A. Evidence for quark-matter cores in massive neutron stars. Nat. Phys. 16 (2020) 907

View on Pith 2020
[7]

Fukushimam K. QCD phase diagram and astrophysical implications. arXiv:2501.01907[hep-ph]

View on Pith Pith review arXiv
[8]

Abbott, B. P.; et al. (The LIGO Scientific Collaboration and the Virgo Collaboration). GW170817: Observation of gravitation waves from a binary neutron star inspiral. Phys. Rev. Lett. 119 (2017) 161101

View on Pith 2017
[9]

Abbott, B. P.; et al. (The LIGO Scientific Collaboration and the Virgo Collaboration). GW170817: Measurements of neutron star radii and equation of state. Phys. Rev. Lett. 121 (2018) 161101. 16

View on Pith 2018
[10]

Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108 (1957) 1175

View on Pith 1957
[11]

Barrois, B. C. Superconducting quark matter. Nucl. Phys. B 129 (1977) 390

View on Pith 1977
[12]

Bailin, D.; Love, A. Superfluidity and superconductivity in relativistic fermion systerms. Phys. Rept. 107 (1984) 325

View on Pith 1984
[13]

Iwasaki, M.; Iwado, T. Superconductivity in the quark matter. Phys. Lett. B 350 (1995) 163

View on Pith 1995
[14]

Alford, M. G.; Rajagopal, K.; Wilczek, F. QCD at finite baryon density: Nucleon droplets and color superconductivity. Phys. Lett. B 422 (1998) 247

View on Pith 1998
[15]

Rapp, R.; Schafer, T.; Shuryak, E. V.; Velkovsky, M. Diquark bose condensates in high density matter and instantons. Phys. Rev. Lett. 81 (1998) 53

View on Pith 1998
[16]

Alford, M.; Rajagopal, K.; Wilczek, F. Color-flavor locking and chiral symmetry breaking in high density QCD. Nucl. Phys. B 537 (1999) 443

View on Pith 1999
[17]

Berges, J.; Rajagopal, K. Color superconductivity and chiral symmetry restoration at nonzero baryon density and tem- perature. Nucl. Phys. B 538 (1999) 215

View on Pith 1999
[18]

Rajagopal, K.; Wilczek, F. The condensed matter physics of QCD. arXiv:hep-ph/0011333

View on Pith Pith review arXiv
[19]

Alford, M.; Schmitt. A.; Rajagopal, K.; Schafer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys. 80 (2008) 1455

View on Pith 2008
[20]

Zhang, C.-M.; et al. Five-Hundred-Meter Aperture Spherical Radio Telescope (F AST): A Powerful Explorer of Exotic Pulsars. Universe 11, (2025) 378

View on Pith 2025
[21]

Livingstone, M. A.; Kaspi, V. M. Long-term X-ray monitoring of the young pulsar PSR B1509-58. Astrophys. J 742 (2011) 31

View on Pith 2011
[22]

Wijnands, R.; van der Klis M. A millisecond pulsar in an X-ray binary system. Nature 394 (1998) 344

View on Pith 1998
[23]

Papitto, A.; et al. Swings between rotation and accretion power in a binary millisecond pulsar. Nature 501 (2013) 517

View on Pith 2013
[24]

Tiengo, A.; Esposito, P.; Mereghetti, S.; Turolla, R.; Nobili, L.; Gastaldello, F.; Gotz, D.; Israel, G. L.; Rea, N.; Stella, L.; et al. A variable absorption feature in the X-ray spectrum of a magnetar. Nature 500 (2013) 312

View on Pith 2013
[25]

Thompson, C.; Duncan, R. C. Formation of very strongly magnetized neutron stars: Implications for Gamma-ray bursts. Astrophys. J. 392 (1992) L9

View on Pith 1992
[26]

Turolla, R.; Zane, S.; Watts, A. L. Magnetars: the physics behind observations. A review. Rep. Prog. Phys. 78 (2015) 116901

View on Pith 2015
[27]

Kaspi, V. M.; Beloborodov, A. M. Magnetars. Annu. Rev. Astron. Astrophys. 55 (2017) 261

View on Pith 2017
[28]

Dong, L.; Shapiro, S. L.; Cold equation of state in a strong magnetic field: Effects of inverse beta-decay. Astrophys. J. 383 (1991) 745

View on Pith 1991
[29]

Ferrer, E. J.; de la Incera, V.; Keith, J. P.; Portillo, I.; Springsteen, P. L.; Equation of state of a dense and magnetized fermion system. Phys. Rev. C 82 (2010) 065802

View on Pith 2010
[30]

Uryu, K.; et al. New code for equilibriums and quasiequilibrium initial data of compact objects. IV. Rotating relativistic stars with mixed poloidal and toroidal magnetic fields. Phys. Rev. D 100 (2019) 123019

View on Pith 2019
[31]

Uryu, K.; et al. Equilibriums of extremely magnetized compact stars with force-free magnetotunnels. Phys. Rev. D 107 (2023) 103016

View on Pith 2023
[32]

Sob’yanin, D. N. Ultraslow PSR J0901-4046 with an ultrahigh magnetic field of 3.2 × 1016 G. Phys. Rev. D 107 (2023) L081301

View on Pith 2023
[33]

Hurley-Walker, N.; Rea, N.; McSweeney, S. J.; et al. A long-period radio transient active for three decades. Nature 619 (2023) 487

View on Pith 2023
[34]

Cardall, C. Y.; Prakash, M.; Lattimer, J. M. Effects of strong magnetic fields on neutron star structure. Astrophys. J. 554 (2001) 322

View on Pith 2001
[35]

Feng, B.; Hou, D. f.; Ren, H. C.; Wu, P. P. Single-flavor color superconductivity in a magnetic field. Phys. Rev. Lett. 105 (2010) 042001

View on Pith 2010
[36]

Chatterjee, D.; Elghozi, T.; Novac, J.; Oertel, M. Consistent neutron star models with magnetic-field-dependent equations of state. Mon. Not. Roy. Astron. Soc. 447 (2015) 3785

View on Pith 2015
[37]

Ferrer, E. J.; Hackebill, A. Thermodynamics of neutrons in a magnetic field and its implications for neutron stars. Phys. Rev. C 99 (2019) 065803

View on Pith 2019
[38]

Ferrer, E. J.; de la Incera, V.; Manreza Paret, D.; Perez Martinez, A.; Sanchez, A. Insignificance of the anomalous magnetic moment of charged fermions for the equation of state of a magnetized and dense medium. Phys. Rev. D 91 (2015) 085041

View on Pith 2015
[39]

Ferrer, E. J.; de la Incera, V. Magnetism in Dense Quark Matter. Lect. Notes Phys. 871 (2013) 399

View on Pith 2013
[40]

Gorbar, E. V.; Color superconductivity in an external magnetic field. Phys. Rev. D 62 (2000) 014007

View on Pith 2000
[41]

Casalbuoni, R.; Duan, Z.; Sannino, F. Electroweak physics for color superconductivity. Phys. Rev. D 63 (2001) 114026

View on Pith 2001
[42]

Ferrer, E. J.; de la Incera, V. Magnetic fields boosted by gluon vortices in color superconductivity. Phys. Rev. Lett. 97 (2006) 122301

View on Pith 2006
[43]

Ferrer, E. J.; de la Incera, V. Chromomagnetic instability and induced magnetic field in neutral two-flavor color super- conductivity. Phys. Rev. D 76 (2007) 114012

View on Pith 2007
[44]

Ferrer, E. J.; de la Incera, V. Paramagnetism in color superconductivity and compact stars. J. Phys. A 40 (2007) 6913

View on Pith 2007
[45]

Ferrer, E. J. Abrikosov gluon vortices in color superconductors. AIP Conf. Proc. 1361 (2011) 1, 252

View on Pith 2011
[46]

Rischke, D. H. Debye screening and Meissner effect in a two flavor color superconductor. Phys. Rev. D 62 (2000) 034007

View on Pith 2000
[47]

Rischke, D. H.; Shovkovy, I. A. Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor. Phys. Rev. D 66 (2002) 054019

View on Pith 2002
[48]

Huang, M.; Shovkovy, I. Gapless color superconductivity at zero and at finite temperature. Nucl. Phys. A 729 (2003) 835. 17

View on Pith 2003
[49]

Shovkovy, I. A.; Huang, M. Gapless two flavor color superconductivity. Phys. Lett. B 564 (2003) 205

View on Pith 2003
[50]

Schmitt, A.; Wang, Q.; Rischke, D. H. Mixing and screening of photons and gluons in a color superconductor. Phys. Rev. D 69 (2004) 094017

View on Pith 2004
[51]

Huang, M.; Shovkovy, I. A.; Chromomagnetic instability in dense quark matter. Phys. Rev. D 70 (2004) 051501(R)

View on Pith 2004
[52]

Huang, M.; Shovkovy, I. A.; Screening masses in neutral two-flavor color superconductivity. Phys. Rev. D 70 (2004) 094030

View on Pith 2004
[53]

Alford, M.; Bowers, J.; Rajagopal, K. Crystalline color superconductivity. Phys. Rev. D 63 (2001) 074016

View on Pith 2001
[54]

Reddy, S.; Rupak, G. Phase structure of two-flavor quark matter: Heterogeneous superconductors. Phys. Rev. C 71 (2005) 025201

View on Pith 2005
[55]

Giannakis, I.; Ren, H. C. The Meissner effect in a two flavor LOFF color superconductor. Nucl. Phys. B 723 (2005) 255

View on Pith 2005
[56]

Giannakis, I.; Ren, H. C. Chromomagnetic instability and the LOFF state in a two flavor color superconductivity. Phys. Lett. B 611 (2005) 137

View on Pith 2005
[57]

Giannakis, I.; Hou, D. F.; Ren, H. C. A neutral two-flavor LOFF color superconductor. Phys. Lett. B 631 (2005) 16

View on Pith 2005
[58]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Neutral LOFF state and chromomagnetic instability in two-flavor dense QCD. Phys. Rev. Lett. 96 (2006) 022005

View on Pith 2006
[59]

Fukushima, K.; Characterizing the Larkin-Ovchinnikov-Fulde-Ferrel phase induced by the chromomagnetic instability. Phys. Rev. D 73 (2006) 094016

View on Pith 2006
[60]

Hashimoto, M. Manifestation of instabilities in Nambu-Jona-Lasinio type models. Phys. Lett. B 642 (2006) 93

View on Pith 2006
[61]

Larkin, A. I.; Ovchinnikov, Y. N. Nonuniform state of superconductors. Sov. Phys. JETP 20 (1965) 762

View on Pith 1965
[62]

Fulde, P.; Ferrell, R. A. Superconductivity in a strong spin-exchange field. Phys. Rev. A 135 (1964) 550

View on Pith 1964
[63]

Skalozub, V. V. Sov. J. Nucl. Phys. 23 (1978) 113

View on Pith 1978
[64]

Nielsen, N. K.; Olesen, P. An unstable Yang-Mills field mode. Nucl. Phys. B 144 (1978) 376

View on Pith 1978
[65]

Skalozub, V. V. Sov. J. Nucl. Phys. 43 (1986) 665

View on Pith 1986
[66]

Skalozub, V. V. Sov. J. Nucl. Phys. 45 (1987) 1058

View on Pith 1987
[67]

Ambjorn, J.; Olesen, P. On electroweak magnetism. Nucl. Phys. B 315 (1989) 606

View on Pith 1989
[68]

Ambjorn, J.; Olesen, P. A magnetic condensate solution of the classical electroweak theory. Phys. Lett. B 218 (1989) 67

View on Pith 1989
[69]

Ferrara, S.; Porrati, M. String phase transitions in a strong magnetic field. Mod. Phys. Lett. A 8 (1993) 2497

View on Pith 1993
[70]

Ferrer, E. J.; de la Incera, V. Regge trajectories and critical fields of the charged string in a magnetic background. Int. Jour. of Mod. Phys. A 11 (1996) 3875

View on Pith 1996
[71]

Fayazbakhsh, Sh.; Sadooghi, N. Color neutral 2SC phase of cold and dense quark matter in the presence of constant magnetic fields. Phys. Rev. D 82 (2010) 045010

View on Pith 2010
[72]

Mandal, T.; Jaikumar, P. Neutrality of a magnetized two-flavor quark superconductivity. Phys. Rev. C 87 (2013) 045208

View on Pith 2013
[73]

Mandal, T.; Jaikumar, P. Effect of temperature and magnetic field on two-flavor superconducting quark matter. Phys. Rev. D 94 (2016) 074016

View on Pith 2016
[74]

Mandal, T.; Jaikumar, P. Effect of strong magnetic field on competing order parameters in two-flavor dense quark matter. Adv. High Energy Phys. 2017 (2017) 6472909

View on Pith 2017
[75]

Coppola, M.; Allen, P.; Grunfeld, A. G.; Scoccola, N. N. Magnetized color superconducting quark matter under compact star conditions: Phase structure within the SU (2)f NJL model. Phys. Rev. D 96 (2017) 056013

View on Pith 2017
[76]

Mishra, H.; Colorsuperconductivity in magnetised quark matter: an NJL model approach. Eur. Phys. J. Spec. Top. 231 (2022) 103

View on Pith 2022
[78]

Ritus, V. I. Radiative corrections in quantum electrodynamics with intense field and their analytical properties. Ann. Phys. (N. Y.) 69 (1972) 555

View on Pith 1972
[79]

Leung, C. N.; Ng, Y. J.; Ackley, A. W. Schwinger-Dyson approach to chiral symmetry breaking in an external magnetic field. Phys. Rev. D 54 (1996) 4181

View on Pith 1996
[80]

Lee, D. S.; Leung, C. N.; Ng, Y. J. Chiral symmetry breaking in a uniform external magnetic field. Phys. Rev. D 55 (1997) 6504

View on Pith 1997
[81]

Leung, C. N.; Wang, S. Y. Gauge independent approach to chiral symmetry breaking in a strong magnetic field. Nucl. Phys. B 747 (2006) 266

View on Pith 2006
[82]

Ferrer, E. J.; de la Incera, V. Magnetic catalysis in the presence of scalar fields. Phys. Lett. B 481 (2000) 287

View on Pith 2000
[83]

Elizalde, E.; Ferrer, E. J.; de la Incera, V. Neutrino self-energy and index of refraction in strong magnetic field: A new approach. Ann. Phys. (N. Y.) 295 (2002) 33

View on Pith 2002
[84]

Elizalde, E.; Ferrer, E. J.; de la Incera, V. Neutrino propagation in a strongly magnetized medium. Phys. Rev. D 70 (2004) 043012

View on Pith 2004
[85]

Ferrer, E. J.; de la Incera, V.; Manuel, C. Magnetic color-flavor locking phase in high-density QCD. Phys. Rev. Lett. 95 (2005) 152002

View on Pith 2005
[86]

Ferrer, E. J.; de la Incera, V.; Manuel, C. Color-superconducting gap in the presence of a magnetic field. Nucl. Phys. B 747 (2006) 88

View on Pith 2006
[87]

Ferrer, E. J.; de la Incera, V. Magnetic phases in three-flavor color superconductivity. Phys. Rev. D 76 (2007) 045011

View on Pith 2007
[88]

Feng, B.; Ferrer, E. J.; de la Incera, V. Cooper pairs’ magnetic moment in MCFL color superconductivity. Nucl. Phys. B 853 (2011) 213

View on Pith 2011
[89]

Klevansky, S. P. The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev. Mod. Phys. 64 (1992) 3

View on Pith 1992
[90]

Buballa, M. NJL-model analysis of dense quark matter. Phys. Rept. 407 (2005) 205. 18

View on Pith 2005
[91]

Ruester, S. B.; Werth, V.; Buballa, M.; Shovkovy, I. A.; Rischke, D. H. Phase diagram of neutral quark matter: Self- consistent treatment of quark masses. Phys. Rev. D 72 (2005) 034004

View on Pith 2005
[92]

Shabad, A. E. Polarization of the Vacuum and a Quantum Relativistic Gas in an External Field (Nova Science Publishers, New York, 1991) (Trudy Fizicheskogo Instituta im. P. N. Lebedeva 192, pp 5-152(1988))

View on Pith 1988
[93]

Perez Rojas, H.; Shabad, A. E. Polarization of relativistic electron and positron gas in a strong magnetic field. Propagation of electromagnetic waves. Ann. Phys. (N. Y.)121 (1979) 432

View on Pith 1979
[94]

Feng, B.; Ferrer, E. J.; de la Incera, V. Photon self-energy and electric susceptibility in a magnetized three-flavor color superconductor. Phys. Rev. D 85 (2012) 103529

View on Pith 2012
[95]

Feng, B.; Ferrer, E. J.; Portillo, I. Lack of Debye and Meissner screening in strongly magnetized quark matter at intermediate densities. Phys. Rev. D 101 (2020) 056012

View on Pith 2020
[96]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Gluonic phase in neutral two-flavor dense QCD. Phys. Lett. B 632 (2006) 305

View on Pith 2006
[97]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A.; Shovokovy, I. A. Collective excitations, instabilities, and ground state in dense quark matter. Phys. Rev. D 73 (2006) 111502

View on Pith 2006
[98]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Gluonic phases, vector condensates, and exotic hadrons in dense QCD. Phys. Rev. D 75 (2007) 085012

View on Pith 2007
[99]

Miransky, V. A. Dynamics with vector condensate at finite density in QCD and beyond. Prog. Theor. Phys. Suppl. 168 (2007) 405

View on Pith 2007
[100]

Gorbar, E. V.; Jia, J. J.; Miransky, V. A. Vortices in gauge models at finite density with vector condensates. Phys. Rev. D 73 (2006) 045001

View on Pith 2006
[101]

Thompson, C.; Duncan, R. C. Neutron star dynamos and the origins of pulsar magnetism. Astrophys. J. 408 (1993) 194

View on Pith 1993
[102]

Olausen, S. A.; Kaspi, V. M. The Astrophysical Journal Supplement, 212 (2014) article id. 6, 22 pp. ( McGill Online Magnetar Catalog, http://www.physics.mcgill.ca/ pulsar/magnetar/main.html)

View on Pith 2014
[103]

Raynaud, R.; Guilet, J.; Janka, H.-T.; Gastine, T. Magnetar formation through a convective dynamo in protoneutron stars. Sci. Adv. 6 (2020) eaay2732

View on Pith 2020
[104]

Matsumoto, J.; Takiwaki, T.; Kotake, K.; Asahina, Y.; Takahashi, H. R. Convection and dynamo in newly born neutron stars. Astrophys. J. 924 (2022) 75

View on Pith 2022
[105]

Barrère, P.; Guilet, J.; Raynaud, R.; Reboul-Salze, A. Tayler-Spruit dynamo in stably stratified rotating fluids: Applica- tion to proto-magnetars. Astron. Astrophys. 695 (2025) A183

View on Pith 2025
[106]

Reboul-Salze, A.; Guilet, J.; Raynaud, R.; Bugli, M. A global model of the magnetorotational instability in protoneutron stars. Astron. Astrophys. 645 (2021) A109

View on Pith 2021
[107]

Reboul-Salze, A.; Guilet, J.; Raynaud, R.; Bugli, M. MRI-driven αΩ dynamos in protoneutron stars. Astron. Astrophys. 667 (2022) A94

View on Pith 2022
[108]

Lo, P. M.; Szymanski, M.; Redlich, K.; Sasaki, C. Driving chiral phase transition with ring diagram. Eur. Phys. J. A. 58 (2022) 172

View on Pith 2022
[109]

Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y. Chiral and color superconducting phase transitions with vector interaction in a simple model. Prog. Theor. Phys. 108 (2002) 929; Prog. Theor. Phys. 110 (2003) 185 (erratum)

View on Pith 2002
[110]

Menezes, D. P.; Pinto, M. B.; Castro, L. B.; Costa, P.; Providencia, C. Repulsive vector interaction in three flavor magnetized quark and stellar matter. Phys. Rev. C 89 (2014) 055207

View on Pith 2014
[111]

Farias, R. L. S.; Dallabona, G.; Krein, G.; Battistel, O. A. Extension of the Nambu-Jona-Lasinio model at high densities and temperatures using an implicit regularization scheme. Phys. Rev. C 77 (2008) 065201

View on Pith 2008
[112]

Li, C. F.; Yang, L.; Wen, X. J.; Peng, G. X. Magnetized quark matter with a magnetic-field dependent coupling. Phys. Rev. D 93 (2016) 054005

View on Pith 2016

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [77]

Yuan, S.; Feng, B.; Ferrer, E. J.; Pinero, A. Magnetized neutral 2SC color superconductivity and possible origin of the inner magnetic field of magnetars. Phys. Rev. D 110 (2024) 114038

View on Pith 2024

[2] [1]

Baym, G.; Hatsuda, T.; Kojo, T.; Powell, P. D.; Song, Y. F.; Takatsuka,T. From hadrons to quarks in neutron stars: a review. Rep. Prog. Phys. 81 (5) (2018) 056902

View on Pith 2018

[3] [2]

Dexheimer, V.; et al. Future physics perspectives on the equation of state from heavy ion collisions to neutron stars. J. Phys. G 48 (2021) 7, 073001

View on Pith 2021

[4] [3]

Luo, X. F.; Wang, Q.; Xu, N.; Zhuang, P. F. Properties of QCD matter at high baryon density. Springer (Singapore) 2022

View on Pith 2022

[5] [4]

Weber, F. Strange quark matter and compact stars. Prog. Part Nucl. Phys. 54 (2005) 193

View on Pith 2005

[6] [5]

Sedrakian, A. Astrophysics of dense quark matter in compact stars. Acta Phys. Polon. B 3 (2010) 669

View on Pith 2010

[7] [6]

Annala, E.; Gorda, T.; Kurkela, A.; Nattila, J.; Vuorinen, A. Evidence for quark-matter cores in massive neutron stars. Nat. Phys. 16 (2020) 907

View on Pith 2020

[8] [7]

Fukushimam K. QCD phase diagram and astrophysical implications. arXiv:2501.01907[hep-ph]

View on Pith Pith review arXiv

[9] [8]

Abbott, B. P.; et al. (The LIGO Scientific Collaboration and the Virgo Collaboration). GW170817: Observation of gravitation waves from a binary neutron star inspiral. Phys. Rev. Lett. 119 (2017) 161101

View on Pith 2017

[10] [9]

Abbott, B. P.; et al. (The LIGO Scientific Collaboration and the Virgo Collaboration). GW170817: Measurements of neutron star radii and equation of state. Phys. Rev. Lett. 121 (2018) 161101. 16

View on Pith 2018

[11] [10]

Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108 (1957) 1175

View on Pith 1957

[12] [11]

Barrois, B. C. Superconducting quark matter. Nucl. Phys. B 129 (1977) 390

View on Pith 1977

[13] [12]

Bailin, D.; Love, A. Superfluidity and superconductivity in relativistic fermion systerms. Phys. Rept. 107 (1984) 325

View on Pith 1984

[14] [13]

Iwasaki, M.; Iwado, T. Superconductivity in the quark matter. Phys. Lett. B 350 (1995) 163

View on Pith 1995

[15] [14]

Alford, M. G.; Rajagopal, K.; Wilczek, F. QCD at finite baryon density: Nucleon droplets and color superconductivity. Phys. Lett. B 422 (1998) 247

View on Pith 1998

[16] [15]

Rapp, R.; Schafer, T.; Shuryak, E. V.; Velkovsky, M. Diquark bose condensates in high density matter and instantons. Phys. Rev. Lett. 81 (1998) 53

View on Pith 1998

[17] [16]

Alford, M.; Rajagopal, K.; Wilczek, F. Color-flavor locking and chiral symmetry breaking in high density QCD. Nucl. Phys. B 537 (1999) 443

View on Pith 1999

[18] [17]

Berges, J.; Rajagopal, K. Color superconductivity and chiral symmetry restoration at nonzero baryon density and tem- perature. Nucl. Phys. B 538 (1999) 215

View on Pith 1999

[19] [18]

Rajagopal, K.; Wilczek, F. The condensed matter physics of QCD. arXiv:hep-ph/0011333

View on Pith Pith review arXiv

[20] [19]

Alford, M.; Schmitt. A.; Rajagopal, K.; Schafer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys. 80 (2008) 1455

View on Pith 2008

[21] [20]

Zhang, C.-M.; et al. Five-Hundred-Meter Aperture Spherical Radio Telescope (F AST): A Powerful Explorer of Exotic Pulsars. Universe 11, (2025) 378

View on Pith 2025

[22] [21]

Livingstone, M. A.; Kaspi, V. M. Long-term X-ray monitoring of the young pulsar PSR B1509-58. Astrophys. J 742 (2011) 31

View on Pith 2011

[23] [22]

Wijnands, R.; van der Klis M. A millisecond pulsar in an X-ray binary system. Nature 394 (1998) 344

View on Pith 1998

[24] [23]

Papitto, A.; et al. Swings between rotation and accretion power in a binary millisecond pulsar. Nature 501 (2013) 517

View on Pith 2013

[25] [24]

Tiengo, A.; Esposito, P.; Mereghetti, S.; Turolla, R.; Nobili, L.; Gastaldello, F.; Gotz, D.; Israel, G. L.; Rea, N.; Stella, L.; et al. A variable absorption feature in the X-ray spectrum of a magnetar. Nature 500 (2013) 312

View on Pith 2013

[26] [25]

Thompson, C.; Duncan, R. C. Formation of very strongly magnetized neutron stars: Implications for Gamma-ray bursts. Astrophys. J. 392 (1992) L9

View on Pith 1992

[27] [26]

Turolla, R.; Zane, S.; Watts, A. L. Magnetars: the physics behind observations. A review. Rep. Prog. Phys. 78 (2015) 116901

View on Pith 2015

[28] [27]

Kaspi, V. M.; Beloborodov, A. M. Magnetars. Annu. Rev. Astron. Astrophys. 55 (2017) 261

View on Pith 2017

[29] [28]

Dong, L.; Shapiro, S. L.; Cold equation of state in a strong magnetic field: Effects of inverse beta-decay. Astrophys. J. 383 (1991) 745

View on Pith 1991

[30] [29]

Ferrer, E. J.; de la Incera, V.; Keith, J. P.; Portillo, I.; Springsteen, P. L.; Equation of state of a dense and magnetized fermion system. Phys. Rev. C 82 (2010) 065802

View on Pith 2010

[31] [30]

Uryu, K.; et al. New code for equilibriums and quasiequilibrium initial data of compact objects. IV. Rotating relativistic stars with mixed poloidal and toroidal magnetic fields. Phys. Rev. D 100 (2019) 123019

View on Pith 2019

[32] [31]

Uryu, K.; et al. Equilibriums of extremely magnetized compact stars with force-free magnetotunnels. Phys. Rev. D 107 (2023) 103016

View on Pith 2023

[33] [32]

Sob’yanin, D. N. Ultraslow PSR J0901-4046 with an ultrahigh magnetic field of 3.2 × 1016 G. Phys. Rev. D 107 (2023) L081301

View on Pith 2023

[34] [33]

Hurley-Walker, N.; Rea, N.; McSweeney, S. J.; et al. A long-period radio transient active for three decades. Nature 619 (2023) 487

View on Pith 2023

[35] [34]

Cardall, C. Y.; Prakash, M.; Lattimer, J. M. Effects of strong magnetic fields on neutron star structure. Astrophys. J. 554 (2001) 322

View on Pith 2001

[36] [35]

Feng, B.; Hou, D. f.; Ren, H. C.; Wu, P. P. Single-flavor color superconductivity in a magnetic field. Phys. Rev. Lett. 105 (2010) 042001

View on Pith 2010

[37] [36]

Chatterjee, D.; Elghozi, T.; Novac, J.; Oertel, M. Consistent neutron star models with magnetic-field-dependent equations of state. Mon. Not. Roy. Astron. Soc. 447 (2015) 3785

View on Pith 2015

[38] [37]

Ferrer, E. J.; Hackebill, A. Thermodynamics of neutrons in a magnetic field and its implications for neutron stars. Phys. Rev. C 99 (2019) 065803

View on Pith 2019

[39] [38]

Ferrer, E. J.; de la Incera, V.; Manreza Paret, D.; Perez Martinez, A.; Sanchez, A. Insignificance of the anomalous magnetic moment of charged fermions for the equation of state of a magnetized and dense medium. Phys. Rev. D 91 (2015) 085041

View on Pith 2015

[40] [39]

Ferrer, E. J.; de la Incera, V. Magnetism in Dense Quark Matter. Lect. Notes Phys. 871 (2013) 399

View on Pith 2013

[41] [40]

Gorbar, E. V.; Color superconductivity in an external magnetic field. Phys. Rev. D 62 (2000) 014007

View on Pith 2000

[42] [41]

Casalbuoni, R.; Duan, Z.; Sannino, F. Electroweak physics for color superconductivity. Phys. Rev. D 63 (2001) 114026

View on Pith 2001

[43] [42]

Ferrer, E. J.; de la Incera, V. Magnetic fields boosted by gluon vortices in color superconductivity. Phys. Rev. Lett. 97 (2006) 122301

View on Pith 2006

[44] [43]

Ferrer, E. J.; de la Incera, V. Chromomagnetic instability and induced magnetic field in neutral two-flavor color super- conductivity. Phys. Rev. D 76 (2007) 114012

View on Pith 2007

[45] [44]

Ferrer, E. J.; de la Incera, V. Paramagnetism in color superconductivity and compact stars. J. Phys. A 40 (2007) 6913

View on Pith 2007

[46] [45]

Ferrer, E. J. Abrikosov gluon vortices in color superconductors. AIP Conf. Proc. 1361 (2011) 1, 252

View on Pith 2011

[47] [46]

Rischke, D. H. Debye screening and Meissner effect in a two flavor color superconductor. Phys. Rev. D 62 (2000) 034007

View on Pith 2000

[48] [47]

Rischke, D. H.; Shovkovy, I. A. Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor. Phys. Rev. D 66 (2002) 054019

View on Pith 2002

[49] [48]

Huang, M.; Shovkovy, I. Gapless color superconductivity at zero and at finite temperature. Nucl. Phys. A 729 (2003) 835. 17

View on Pith 2003

[50] [49]

Shovkovy, I. A.; Huang, M. Gapless two flavor color superconductivity. Phys. Lett. B 564 (2003) 205

View on Pith 2003

[51] [50]

Schmitt, A.; Wang, Q.; Rischke, D. H. Mixing and screening of photons and gluons in a color superconductor. Phys. Rev. D 69 (2004) 094017

View on Pith 2004

[52] [51]

Huang, M.; Shovkovy, I. A.; Chromomagnetic instability in dense quark matter. Phys. Rev. D 70 (2004) 051501(R)

View on Pith 2004

[53] [52]

Huang, M.; Shovkovy, I. A.; Screening masses in neutral two-flavor color superconductivity. Phys. Rev. D 70 (2004) 094030

View on Pith 2004

[54] [53]

Alford, M.; Bowers, J.; Rajagopal, K. Crystalline color superconductivity. Phys. Rev. D 63 (2001) 074016

View on Pith 2001

[55] [54]

Reddy, S.; Rupak, G. Phase structure of two-flavor quark matter: Heterogeneous superconductors. Phys. Rev. C 71 (2005) 025201

View on Pith 2005

[56] [55]

Giannakis, I.; Ren, H. C. The Meissner effect in a two flavor LOFF color superconductor. Nucl. Phys. B 723 (2005) 255

View on Pith 2005

[57] [56]

Giannakis, I.; Ren, H. C. Chromomagnetic instability and the LOFF state in a two flavor color superconductivity. Phys. Lett. B 611 (2005) 137

View on Pith 2005

[58] [57]

Giannakis, I.; Hou, D. F.; Ren, H. C. A neutral two-flavor LOFF color superconductor. Phys. Lett. B 631 (2005) 16

View on Pith 2005

[59] [58]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Neutral LOFF state and chromomagnetic instability in two-flavor dense QCD. Phys. Rev. Lett. 96 (2006) 022005

View on Pith 2006

[60] [59]

Fukushima, K.; Characterizing the Larkin-Ovchinnikov-Fulde-Ferrel phase induced by the chromomagnetic instability. Phys. Rev. D 73 (2006) 094016

View on Pith 2006

[61] [60]

Hashimoto, M. Manifestation of instabilities in Nambu-Jona-Lasinio type models. Phys. Lett. B 642 (2006) 93

View on Pith 2006

[62] [61]

Larkin, A. I.; Ovchinnikov, Y. N. Nonuniform state of superconductors. Sov. Phys. JETP 20 (1965) 762

View on Pith 1965

[63] [62]

Fulde, P.; Ferrell, R. A. Superconductivity in a strong spin-exchange field. Phys. Rev. A 135 (1964) 550

View on Pith 1964

[64] [63]

Skalozub, V. V. Sov. J. Nucl. Phys. 23 (1978) 113

View on Pith 1978

[65] [64]

Nielsen, N. K.; Olesen, P. An unstable Yang-Mills field mode. Nucl. Phys. B 144 (1978) 376

View on Pith 1978

[66] [65]

Skalozub, V. V. Sov. J. Nucl. Phys. 43 (1986) 665

View on Pith 1986

[67] [66]

Skalozub, V. V. Sov. J. Nucl. Phys. 45 (1987) 1058

View on Pith 1987

[68] [67]

Ambjorn, J.; Olesen, P. On electroweak magnetism. Nucl. Phys. B 315 (1989) 606

View on Pith 1989

[69] [68]

Ambjorn, J.; Olesen, P. A magnetic condensate solution of the classical electroweak theory. Phys. Lett. B 218 (1989) 67

View on Pith 1989

[70] [69]

Ferrara, S.; Porrati, M. String phase transitions in a strong magnetic field. Mod. Phys. Lett. A 8 (1993) 2497

View on Pith 1993

[71] [70]

Ferrer, E. J.; de la Incera, V. Regge trajectories and critical fields of the charged string in a magnetic background. Int. Jour. of Mod. Phys. A 11 (1996) 3875

View on Pith 1996

[72] [71]

Fayazbakhsh, Sh.; Sadooghi, N. Color neutral 2SC phase of cold and dense quark matter in the presence of constant magnetic fields. Phys. Rev. D 82 (2010) 045010

View on Pith 2010

[73] [72]

Mandal, T.; Jaikumar, P. Neutrality of a magnetized two-flavor quark superconductivity. Phys. Rev. C 87 (2013) 045208

View on Pith 2013

[74] [73]

Mandal, T.; Jaikumar, P. Effect of temperature and magnetic field on two-flavor superconducting quark matter. Phys. Rev. D 94 (2016) 074016

View on Pith 2016

[75] [74]

Mandal, T.; Jaikumar, P. Effect of strong magnetic field on competing order parameters in two-flavor dense quark matter. Adv. High Energy Phys. 2017 (2017) 6472909

View on Pith 2017

[76] [75]

Coppola, M.; Allen, P.; Grunfeld, A. G.; Scoccola, N. N. Magnetized color superconducting quark matter under compact star conditions: Phase structure within the SU (2)f NJL model. Phys. Rev. D 96 (2017) 056013

View on Pith 2017

[77] [76]

Mishra, H.; Colorsuperconductivity in magnetised quark matter: an NJL model approach. Eur. Phys. J. Spec. Top. 231 (2022) 103

View on Pith 2022

[78] [78]

Ritus, V. I. Radiative corrections in quantum electrodynamics with intense field and their analytical properties. Ann. Phys. (N. Y.) 69 (1972) 555

View on Pith 1972

[79] [79]

Leung, C. N.; Ng, Y. J.; Ackley, A. W. Schwinger-Dyson approach to chiral symmetry breaking in an external magnetic field. Phys. Rev. D 54 (1996) 4181

View on Pith 1996

[80] [80]

Lee, D. S.; Leung, C. N.; Ng, Y. J. Chiral symmetry breaking in a uniform external magnetic field. Phys. Rev. D 55 (1997) 6504

View on Pith 1997

[81] [81]

Leung, C. N.; Wang, S. Y. Gauge independent approach to chiral symmetry breaking in a strong magnetic field. Nucl. Phys. B 747 (2006) 266

View on Pith 2006

[82] [82]

Ferrer, E. J.; de la Incera, V. Magnetic catalysis in the presence of scalar fields. Phys. Lett. B 481 (2000) 287

View on Pith 2000

[83] [83]

Elizalde, E.; Ferrer, E. J.; de la Incera, V. Neutrino self-energy and index of refraction in strong magnetic field: A new approach. Ann. Phys. (N. Y.) 295 (2002) 33

View on Pith 2002

[84] [84]

Elizalde, E.; Ferrer, E. J.; de la Incera, V. Neutrino propagation in a strongly magnetized medium. Phys. Rev. D 70 (2004) 043012

View on Pith 2004

[85] [85]

Ferrer, E. J.; de la Incera, V.; Manuel, C. Magnetic color-flavor locking phase in high-density QCD. Phys. Rev. Lett. 95 (2005) 152002

View on Pith 2005

[86] [86]

Ferrer, E. J.; de la Incera, V.; Manuel, C. Color-superconducting gap in the presence of a magnetic field. Nucl. Phys. B 747 (2006) 88

View on Pith 2006

[87] [87]

Ferrer, E. J.; de la Incera, V. Magnetic phases in three-flavor color superconductivity. Phys. Rev. D 76 (2007) 045011

View on Pith 2007

[88] [88]

Feng, B.; Ferrer, E. J.; de la Incera, V. Cooper pairs’ magnetic moment in MCFL color superconductivity. Nucl. Phys. B 853 (2011) 213

View on Pith 2011

[89] [89]

Klevansky, S. P. The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev. Mod. Phys. 64 (1992) 3

View on Pith 1992

[90] [90]

Buballa, M. NJL-model analysis of dense quark matter. Phys. Rept. 407 (2005) 205. 18

View on Pith 2005

[91] [91]

Ruester, S. B.; Werth, V.; Buballa, M.; Shovkovy, I. A.; Rischke, D. H. Phase diagram of neutral quark matter: Self- consistent treatment of quark masses. Phys. Rev. D 72 (2005) 034004

View on Pith 2005

[92] [92]

Shabad, A. E. Polarization of the Vacuum and a Quantum Relativistic Gas in an External Field (Nova Science Publishers, New York, 1991) (Trudy Fizicheskogo Instituta im. P. N. Lebedeva 192, pp 5-152(1988))

View on Pith 1988

[93] [93]

Perez Rojas, H.; Shabad, A. E. Polarization of relativistic electron and positron gas in a strong magnetic field. Propagation of electromagnetic waves. Ann. Phys. (N. Y.)121 (1979) 432

View on Pith 1979

[94] [94]

Feng, B.; Ferrer, E. J.; de la Incera, V. Photon self-energy and electric susceptibility in a magnetized three-flavor color superconductor. Phys. Rev. D 85 (2012) 103529

View on Pith 2012

[95] [95]

Feng, B.; Ferrer, E. J.; Portillo, I. Lack of Debye and Meissner screening in strongly magnetized quark matter at intermediate densities. Phys. Rev. D 101 (2020) 056012

View on Pith 2020

[96] [96]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Gluonic phase in neutral two-flavor dense QCD. Phys. Lett. B 632 (2006) 305

View on Pith 2006

[97] [97]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A.; Shovokovy, I. A. Collective excitations, instabilities, and ground state in dense quark matter. Phys. Rev. D 73 (2006) 111502

View on Pith 2006

[98] [98]

Gorbar, E. V.; Hashimoto, M.; Miransky, V. A. Gluonic phases, vector condensates, and exotic hadrons in dense QCD. Phys. Rev. D 75 (2007) 085012

View on Pith 2007

[99] [99]

Miransky, V. A. Dynamics with vector condensate at finite density in QCD and beyond. Prog. Theor. Phys. Suppl. 168 (2007) 405

View on Pith 2007

[100] [100]

Gorbar, E. V.; Jia, J. J.; Miransky, V. A. Vortices in gauge models at finite density with vector condensates. Phys. Rev. D 73 (2006) 045001

View on Pith 2006

[101] [101]

Thompson, C.; Duncan, R. C. Neutron star dynamos and the origins of pulsar magnetism. Astrophys. J. 408 (1993) 194

View on Pith 1993

[102] [102]

Olausen, S. A.; Kaspi, V. M. The Astrophysical Journal Supplement, 212 (2014) article id. 6, 22 pp. ( McGill Online Magnetar Catalog, http://www.physics.mcgill.ca/ pulsar/magnetar/main.html)

View on Pith 2014

[103] [103]

Raynaud, R.; Guilet, J.; Janka, H.-T.; Gastine, T. Magnetar formation through a convective dynamo in protoneutron stars. Sci. Adv. 6 (2020) eaay2732

View on Pith 2020

[104] [104]

Matsumoto, J.; Takiwaki, T.; Kotake, K.; Asahina, Y.; Takahashi, H. R. Convection and dynamo in newly born neutron stars. Astrophys. J. 924 (2022) 75

View on Pith 2022

[105] [105]

Barrère, P.; Guilet, J.; Raynaud, R.; Reboul-Salze, A. Tayler-Spruit dynamo in stably stratified rotating fluids: Applica- tion to proto-magnetars. Astron. Astrophys. 695 (2025) A183

View on Pith 2025

[106] [106]

Reboul-Salze, A.; Guilet, J.; Raynaud, R.; Bugli, M. A global model of the magnetorotational instability in protoneutron stars. Astron. Astrophys. 645 (2021) A109

View on Pith 2021

[107] [107]

Reboul-Salze, A.; Guilet, J.; Raynaud, R.; Bugli, M. MRI-driven αΩ dynamos in protoneutron stars. Astron. Astrophys. 667 (2022) A94

View on Pith 2022

[108] [108]

Lo, P. M.; Szymanski, M.; Redlich, K.; Sasaki, C. Driving chiral phase transition with ring diagram. Eur. Phys. J. A. 58 (2022) 172

View on Pith 2022

[109] [109]

Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y. Chiral and color superconducting phase transitions with vector interaction in a simple model. Prog. Theor. Phys. 108 (2002) 929; Prog. Theor. Phys. 110 (2003) 185 (erratum)

View on Pith 2002

[110] [110]

Menezes, D. P.; Pinto, M. B.; Castro, L. B.; Costa, P.; Providencia, C. Repulsive vector interaction in three flavor magnetized quark and stellar matter. Phys. Rev. C 89 (2014) 055207

View on Pith 2014

[111] [111]

Farias, R. L. S.; Dallabona, G.; Krein, G.; Battistel, O. A. Extension of the Nambu-Jona-Lasinio model at high densities and temperatures using an implicit regularization scheme. Phys. Rev. C 77 (2008) 065201

View on Pith 2008

[112] [112]

Li, C. F.; Yang, L.; Wen, X. J.; Peng, G. X. Magnetized quark matter with a magnetic-field dependent coupling. Phys. Rev. D 93 (2016) 054005

View on Pith 2016