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arxiv: 1907.03990 · v1 · pith:INZ4JG3Dnew · submitted 2019-07-09 · ⚛️ nucl-th · hep-ph

Effect of anomalous magnetic moment of quarks on the phase structure and mesonic properties in the NJL model

Nilanjan Chaudhuri , Snigdha Ghosh , Sourav Sarkar , Pradip Roy This is my paper

Pith reviewed 2026-05-25 00:22 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph
keywords NJL modelanomalous magnetic momentinverse magnetic catalysischiral symmetry restorationquark mattermeson massesmagnetic field effects
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The pith

Accounting for quarks' anomalous magnetic moment in the NJL model turns magnetic catalysis into inverse magnetic catalysis.

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

The paper shows that including the anomalous magnetic moment of quarks in the two-flavour Nambu-Jona-Lasinio model reverses the effect of an external magnetic field on the chiral phase transition. Without the anomalous moment the critical temperature rises with field strength, but with it the temperature falls, realizing inverse magnetic catalysis. The same regularization scheme is used for both cases, and the change appears in both the phase diagram and in the masses of scalar and pseudoscalar mesons. A reader would care because this resolves a known tension between model predictions and lattice results on magnetized quark matter at finite temperature and density.

Core claim

When the anomalous magnetic moment of the quarks is incorporated into the NJL model via a field-dependent three-momentum cutoff, the critical temperature of the chiral symmetry restoration transition decreases with increasing magnetic field strength. This is the opposite of the behavior obtained when the anomalous moment is set to zero, which produces the usual magnetic catalysis. The effect also appears in the phase diagram and produces a field-dependent drop in the Mott transition temperature together with a jump in the Goldstone-mode mass.

What carries the argument

The anomalous magnetic moment term added to the quark propagator inside the two-flavour NJL model, regularized with a field-dependent three-momentum cutoff.

If this is right

  • The phase boundary in the temperature-density plane moves to lower temperatures as the magnetic field increases when the anomalous moment is present.
  • The Mott transition temperature for the mesons decreases substantially with rising magnetic field once the anomalous moment is taken into account.
  • The neutral pseudoscalar meson mass shows a sudden jump at and above the Mott temperature whose size is reduced by the anomalous moment.
  • Thermodynamic observables such as pressure and energy density display the inverse-catalysis behavior instead of the usual magnetic catalysis.

Where Pith is reading between the lines

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

  • If the result survives in full QCD, strong magnetic fields would favor the restoration of chiral symmetry at lower temperatures than models without anomalous moments predict.
  • Calculations of meson spectra in magnetized nuclear matter would need to retain the anomalous moment to reproduce the observed field dependence of the Mott temperature.
  • The same regularization choice could be tested at higher densities to see whether inverse catalysis persists in the region relevant for neutron-star cores.

Load-bearing premise

The field-dependent three-momentum cutoff regularization remains valid and consistent once the anomalous magnetic moment term is added to the quark propagator and the thermodynamic potential.

What would settle it

A lattice QCD simulation that includes the anomalous magnetic moment of quarks and directly measures whether the chiral critical temperature rises or falls with magnetic field strength would confirm or refute the inverse-catalysis result.

Figures

Figures reproduced from arXiv: 1907.03990 by Nilanjan Chaudhuri, Pradip Roy, Snigdha Ghosh, Sourav Sarkar.

Figure 1
Figure 1. Figure 1: Variation of constituent quark mass (M) with temperature (T) at zero quark chemical potential (µq) for three different values of external magnetic field (eB = 0.0 , 0.05 and 0.10 GeV2 ) for (a) κf = 0 and (b) κf , 0. Variation of ∂M ∂T with temperature (T) at zero quark chemical potential (µq) for three different values of external magnetic field (eB = 0.0 , 0.05 and 0.10 GeV2 ) for (c) κf = 0 and (d) κf ,… view at source ↗
Figure 2
Figure 2. Figure 2: T dependence of M at (a) µq = 300 MeV and (b) at µq = 330 MeV for different values of eB with and without considering AMM of quarks. The inset plot in (b) shows the multi-valued nature of M by focusing on the relevant temperature range. µq, after a certain value the imaginary roots become real. This multivaluedness of M is a signature of first order transition. Comparing the solid-red and dashed-blue curve… view at source ↗
Figure 3
Figure 3. Figure 3: µq dependence of Constituent quark mass (M) at (a) T = 30 MeV and (b) at T = 120 MeV for different values of eB and κ. The inset plot in (a) shows the multi-valued nature of M by focusing on the relevant µq range. Next, in Figs. 3(a) and (b), M is plotted as a function of µq at two different temperature (T = 30 and 120 MeV) for four different cases as discussed in the previous paragraph. Here also we get m… view at source ↗
Figure 4
Figure 4. Figure 4: Variation of constituent quark mass (M) at µq = 0 (a) without considering AMM of quarks and (b) considering non-zero values of κf for temperatures of T = 50, 100 and 150 MeV. 0 2 4 6 8 10 0 50 100 150 200 250 300 µq = 0 s/T 3 T (MeV) eB = 0 eB = 0.10 GeV2 , κ = 0 eB = 0.10 GeV2 , κ ≠ 0 Stefan-Boltzmann limit [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of scaled entropy density as function of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of χmm with temperature at different values of quark chemical potential for (a) eB = 0, κf = 0 (b) eB = 0.10 GeV2 , κf = 0, (c) eB = 0.010 GeV2 , κf , 0 and (d) eB = 0.10 GeV2 , κf , 0 80 120 160 200 50 100 150 200 250 300 350 TC (MeV) (µq )C (MeV) eBC = 0 eBC = 0.10 GeV2 , κ = 0 eBC = 0.01 GeV2 , κ ≠ 0 eBC = 0.10 GeV2 , κ ≠ 0 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: TC-(µq)C phase diagram in NJL model for three different conditions. The solid (dashed) lines denote the first-order (crossover) transition. The red, green and blue square points represent CEPs of temperature (chemical potential of quarks) for vanishing AMM of quarks. The opposite effect is realized when AMM of quarks are taken into account. Interestingly, for high values of magnetic field in the later case… view at source ↗
Figure 8
Figure 8. Figure 8: TC-eBC phase diagram in NJL model at (a) (µq)C = 0 and (b) (µq)C = 150 MeV along with the corresponding chiral limits (m = 0). pointing towards the MC effect discussed earlier. On the contrary, when we include AMM, the transition temperature is reduced as (eB)C increases, which leads to the IMC effect, which is obvious from [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Variation of masses of scalar (σ) and neutral pseudoscalar (π 0 ) mesons with temperature at zero external magnetic field and at three different values of quark chemical potential (µq = 0, 200 and 300 MeV). Now, we switch on the external magnetic field. In [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Variation of masses of scalar (σ) and neutral pseudoscalar (π 0 ) mesons with temperature at two different values of external magnetic field (eB = 0 and 0.10 GeV2 ) including and excluding the AMM of the quarks for (a) µq = 0 and (b) µq = 200 MeV. 0 100 200 300 400 500 600 700 800 900 0 50 100 150 200 µq = 0 , eB = 0.10 GeV2 , κ = 0 (a) Mass (MeV) Temperature (MeV) π M 2M 0 100 200 300 400 500 600 700 800… view at source ↗
Figure 11
Figure 11. Figure 11: Variation of π 0 mass, constituent quark mass and twice of the constituent quark mass with temperature at eB = 0.10 GeV2 for (a) µq = 0 and κ = 0, (b) µq = 200 MeV and κ = 0, (c) µq = 0 and κ , 0 and (d) µq = 200 MeV and κ , 0. contrary, while considering the AMM of quarks, the σ mass decreases significantly with the increase in magnetic field in the low temperature region followed by a shift of the minim… view at source ↗
Figure 12
Figure 12. Figure 12: Variation of Mott temperature with external magn [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
read the original abstract

Employing a field dependent three-momentum cut-off regularization technique, we study the phase structure and mesonic masses using the $2$-flavour Nambu-Jona Lasinio model at finite temperature and density in presence of arbitrary external magnetic field. This approach is then applied to incorporate the effects of the anomalous magnetic moment(AMM) of quarks on constituent quark mass and thermodynamic observables as a function of temperature/baryonic density. The critical temperature for transition from chiral symmetry broken to the restored phase is observed to decrease with the external magnetic field, which can be classified as inverse magnetic catalysis, while an opposite behaviour is realized in the case of a vanishing magnetic moment, implying magnetic catalysis. These essential features are also reflected in the phase diagram. Furthermore, the properties of the low lying scalar and neutral pseudoscalar mesons are also studied in presence of a hot and dense magnetized medium including AMM of the quarks using random phase approximation. For non-zero values of magnetic field, we notice a sudden jump in the mass of the Goldstone mode at and above the Mott transition temperature which is found to decrease substantially with the increase in magnetic field when the AMM of the quarks are taken into consideration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the two-flavor NJL model at finite T and density in an external magnetic field, employing a field-dependent three-momentum cutoff. It incorporates the anomalous magnetic moment (AMM) of quarks and reports that the chiral restoration critical temperature decreases with B when AMM is included (inverse magnetic catalysis), while the opposite (magnetic catalysis) occurs for vanishing AMM. The phase diagram and scalar/pseudoscalar meson masses (via RPA) are also examined, with a noted jump in the Goldstone-mode mass at the Mott transition whose location decreases with B when AMM is present.

Significance. If robust, the result supplies a concrete mechanism within the NJL framework for realizing inverse magnetic catalysis via quark AMM, a phenomenon of direct relevance to the QCD phase structure in strong magnetic fields. The calculation uses a single regularization choice fixed once in vacuum and then applied to finite B without further tuning, which is a clear methodological strength. The lack of cutoff-sensitivity quantification, however, limits the strength of the quantitative claims.

major comments (2)
  1. [Regularization procedure] Regularization section (field-dependent cutoff procedure): The three-momentum cutoff is calibrated to vacuum meson properties in the absence of AMM. Insertion of the AMM term modifies the Landau-level dispersion inside the quark propagator, yet no demonstration is given that the same B-dependent cutoff continues to subtract the same UV divergences, preserves thermodynamic consistency, or respects the relevant Ward identities. Because the sign change in dT_c/dB is the central result, this consistency must be verified explicitly.
  2. [Phase structure and meson properties] Results for critical temperature and Mott transition (phase structure and meson properties sections): The reported decrease of T_c with B (and the substantial decrease of the Mott temperature) is presented without error bands or sensitivity plots under variation of the cutoff parameter. The jump in the Goldstone-mode mass is stated without quantitative uncertainty, making it impossible to assess whether the inverse-catalysis signal survives reasonable regulator variations.
minor comments (1)
  1. [Abstract] Abstract: the phrasing 'a sudden jump in the mass of the Goldstone mode at and above the Mott transition temperature which is found to decrease substantially' is ambiguous; it should be clarified whether the magnitude of the jump or the transition temperature itself decreases with B.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight important aspects of the regularization and the robustness of our numerical results. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: Regularization section (field-dependent cutoff procedure): The three-momentum cutoff is calibrated to vacuum meson properties in the absence of AMM. Insertion of the AMM term modifies the Landau-level dispersion inside the quark propagator, yet no demonstration is given that the same B-dependent cutoff continues to subtract the same UV divergences, preserves thermodynamic consistency, or respects the relevant Ward identities. Because the sign change in dT_c/dB is the central result, this consistency must be verified explicitly.

    Authors: The field-dependent three-momentum cutoff is fixed once by vacuum meson properties (without AMM) and then applied uniformly; the AMM enters only as a finite correction to the dispersion relation inside the already-regularized integrals. In the NJL model the leading UV divergences are independent of the magnetic field and of the AMM term, so the same subtraction continues to remove them. Thermodynamic consistency follows from the fact that the cutoff is applied to the three-momentum magnitude before the Landau-level sum, preserving the standard relation between the gap equation and the thermodynamic potential. Ward identities in the NJL model are satisfied at the level of the chiral Ward-Takahashi identity for the pion; the AMM does not violate this because it is a magnetic-moment insertion that commutes with the chiral rotation. Nevertheless, to make the argument fully explicit we will add a short appendix in the revised manuscript that (i) recomputes the vacuum subtraction with and without AMM at B=0 and (ii) verifies that the thermodynamic potential and its derivatives remain consistent under the same cutoff for several values of B. revision: partial

  2. Referee: Results for critical temperature and Mott transition (phase structure and meson properties sections): The reported decrease of T_c with B (and the substantial decrease of the Mott temperature) is presented without error bands or sensitivity plots under variation of the cutoff parameter. The jump in the Goldstone-mode mass is stated without quantitative uncertainty, making it impossible to assess whether the inverse-catalysis signal survives reasonable regulator variations.

    Authors: We agree that a quantitative assessment of regulator dependence strengthens the claim. In the revised manuscript we will add two supplementary figures: one showing T_c(B) and the Mott temperature for cutoff values varied by ±5 % around the vacuum-fitted value, and a second showing the size of the Goldstone-mode mass jump at the Mott point together with the variation obtained from the same cutoff scan. These plots confirm that the sign of dT_c/dB remains negative and that the downward shift of the Mott temperature with B persists within the explored range. Because the model is effective, we do not claim a precise numerical uncertainty; the figures will nevertheless demonstrate that the qualitative inverse-catalysis signal is robust against reasonable regulator variations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained after vacuum fixing

full rationale

Parameters and the three-momentum cutoff are fixed once in vacuum (B=0) to reproduce meson properties. The thermodynamic potential, including the AMM term in the propagator and the magnetic-field-dependent cutoff, is then minimized at finite T, μ, B to obtain Tc(B). This computation does not reduce to the vacuum inputs by construction; the sign change in dTc/dB when AMM is switched on is a dynamical output of the loop integral rather than a re-statement of the fit. No load-bearing self-citation, self-definitional step, or fitted quantity renamed as prediction is present.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The calculation rests on the standard NJL four-fermion interaction, the mean-field approximation, and a regularization scheme whose magnetic-field dependence is introduced by hand to tame the Landau-level sums. No new particles or forces are postulated.

free parameters (3)
  • vacuum constituent quark mass
    Fitted to reproduce the pion decay constant and constituent mass in vacuum before magnetic field is turned on.
  • scalar coupling G
    Adjusted together with the cutoff to match vacuum observables.
  • field-dependent cutoff parameter
    Chosen so that the thermodynamic potential remains finite at each magnetic-field strength.
axioms (2)
  • domain assumption Mean-field (Hartree) approximation is sufficient to capture the chiral transition.
    Invoked when the gap equation is written and when the meson polarization functions are evaluated in RPA.
  • ad hoc to paper The three-momentum cutoff can be made explicitly dependent on the magnetic field without violating gauge invariance or thermodynamic consistency.
    This is the central technical choice that allows the model to remain finite while incorporating AMM.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

100 extracted references · 100 canonical work pages · cited by 2 Pith papers · 74 internal anchors

  1. [1]

    in the following manner: /vecp2 ⊥ = (√ /barex /barex e f B /barex /barex ( 2n + 1 − sξf )+ M2 − sκf e f B ) 2 − M2 = /barex /barex e f B /barex /barex ( 2n + 1 − sξf )+ (κf e f B)2 − 2sMnf κf e f B. (13) As a cross check, one can see that as κf → 0, we get /vecp2 ⊥ → ( 2n + 1 − sξf ) /barex /barex e f B /barex /barex which is the usual expression of the L...

    work page

  2. [2]

    Note that the first condition will alway s be there for any finite values of eB [ 55] but the second condition is only due to the non-zero values o f AMM of quarks

    will be valid iff Λ2 − /vec p2 ⊥ ≥ 0 and /vecp2 ⊥ ≥ 0 as pz, /vecp⊥ are real quantities. Note that the first condition will alway s be there for any finite values of eB [ 55] but the second condition is only due to the non-zero values o f AMM of quarks. These conditions will constrain the contributing n-vlaues in the sum. From now on we will call these two c...

    work page

  3. [3]

    ( 32) and perform the dk 0d2k⊥ integrals

    into Eq. ( 32) and perform the dk 0d2k⊥ integrals. Some relevant steps for this calculation are provided in Appendix D and we get from Eq. ( D8) ReΠa( q)= /summationdisplay.1 f /summationdisplay.1 sk,sp ∞/summationdisplay.1 l=0 /uniB.dsp √ Λ2− /veck2 ⊥ l 0 dk z π Θ ( /veck2 ⊥ l ) Θ ( /vecp2 ⊥ l ) Θ ( Λ2 − /veck2 ⊥ l ) Θ ( Λ2 − /vec p2 ⊥ l ) P     ...

    work page

  4. [4]

    In all the c ases, M almost remains constant up to T ≈ 100 MeV and the transition from chiral symmetry broken (with M /nequal0) to the restored phase (i.e

    10 GeV2 respectively without considering AMM of quarks. In all the c ases, M almost remains constant up to T ≈ 100 MeV and the transition from chiral symmetry broken (with M /nequal0) to the restored phase (i.e. M ≈ m ≈ 0), is a smooth crossover. Since we have considered non vanishing current quark mass, m = 5. 6 MeV , the chiral symmetry is never restore...

    work page

  5. [5]

    M and using Eq

    w.r.t. M and using Eq. ( A1) we can write ∂n± ∂M = −β(n± )2 exp [ β(Enf s ∓ µ)] ∂Enf s ∂M = −βn± (1 − n± ) M Enf s ( 1 − sκf e f B Mnf ) . (A2) In a similar manner, the following expressions can be derive d: ∂n± ∂µ= ± βn± (1 − n± ) [ 1 ∓ M E f ( 1 − sκf e f B Mnf ) ∂M ∂µ ] (A3) ∂n± ∂T = βn± (1 − n± ) [ Enf s ∓ µ T − M E f ( 1 − sκf e f B Mnf ) ∂M ∂T ] . (...

    work page

  6. [6]

    D. E. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U. Y ee , Lect. Notes Phys. 871, 1 (2013) , arXiv:1211.6245 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  7. [7]

    The Chiral Magnetic Effect

    K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D78, 074033 (2008) , arXiv:0808.3382 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  8. [8]

    D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A803, 227 (2008) , arXiv:0711.0950 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  9. [9]

    D. E. Kharzeev and H. J. Warringa, Phys. Rev. D80, 034028 (2009) , arXiv:0907.5007 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  10. [10]

    G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, JHEP 02, 044 (2012) , arXiv:1111.4956 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    I. A. Shovkovy, Lect. Notes Phys. 871, 13 (2013) , arXiv:1207.5081 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    V . P. Gusynin, V . A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499 (1994) , [Erratum: Phys. Rev. Lett.76,1005(1996)], arXiv:hep-ph/9405262 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  13. [13]

    V . P. Gusynin, V . A. Miransky, and I. A. Shovkovy, Nucl. Phys. B462, 249 (1996) , arXiv:hep-ph/9509320 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  14. [14]

    V . P. Gusynin, V . A. Miransky, and I. A. Shovkovy, Nucl. Phys. B563, 361 (1999) , arXiv:hep-ph/9908320 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  15. [15]

    Inverse magnetic catalysis in dense holographic matter

    F. Preis, A. Rebhan, and A. Schmitt, JHEP 03, 033 (2011) , arXiv:1012.4785 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  16. [16]

    Inverse magnetic catalysis in field theory and gauge-gravity duality

    F. Preis, A. Rebhan, and A. Schmitt, Lect. Notes Phys. 871, 51 (2013) , arXiv:1208.0536 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  17. [17]

    Strongly first order electroweak phase transition induced by primordial hypermagnetic fields

    P. Elmfors, K. Enqvist, and K. Kainulainen, Phys. Lett. B440, 269 (1998) , arXiv:hep-ph/9806403 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  18. [18]

    Ring diagrams and electroweak phase transition in a magnetic field

    V . Skalozub and M. Bordag, Int. J. Mod. Phys. A15, 349 (2000) , arXiv:hep-ph/9904333 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  19. [19]

    Improved ring potential of QED at finite temperature and in the presence of weak and strong magnetic field

    N. Sadooghi and K. S. Anaraki, Phys. Rev. D78, 125019 (2008) , arXiv:0805.0078 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  20. [20]

    Symmetry restoration at finite temperature with weak magnetic fields

    J. Navarro, A. Sanchez, M. E. Tejeda- Y eomans, A. Ayala, and G. Piccinelli, Phys. Rev. D82, 123007 (2010) , arXiv:1007.4208 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [21]

    Color neutral 2SC phase of cold and dense quark matter in the presence of constant magnetic fields

    S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D82, 045010 (2010) , arXiv:1005.5022 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  22. [22]

    Phase diagram of hot magnetized two-flavor color superconducting quark matter

    S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D83, 025026 (2011) , arXiv:1009.6125 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  23. [23]

    Phase diagram in an external magnetic field beyond a mean-field approximation

    V . Skokov,Phys. Rev. D85, 034026 (2012) , arXiv:1112.5137 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  24. [24]

    Magnetic catalysis in hot and dense quark matter and quantum fluctuations

    K. Fukushima and J. M. Pawlowski, Phys. Rev. D86, 076013 (2012) , arXiv:1203.4330 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  25. [25]

    M. N. Chernodub, Phys. Rev. Lett. 106, 142003 (2011) , arXiv:1101.0117 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  26. [26]

    M. N. Chernodub, J. Van Doorsselaere, and H. Verschelde , Phys. Rev. D85, 045002 (2012) , arXiv:1111.4401 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  27. [27]

    Estimate of the magnetic field strength in heavy-ion collisions

    V . Skokov, A. Yu. Illarionov, and V . Toneev,Int. J. Mod. Phys. A24, 5925 (2009) , arXiv:0907.1396 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  28. [28]

    Magnetohydrodynamics, charged currents and directed flow in heavy ion collisions

    U. Gursoy, D. Kharzeev, and K. Rajagopal, Phys. Rev. C89, 054905 (2014) , arXiv:1401.3805 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  29. [29]

    Vachaspati, Phys

    T. Vachaspati, Phys. Lett. B265, 258 (1991)

    work page 1991

  30. [30]

    Origin of Cosmic Magnetic Fields

    L. Campanelli, Phys. Rev. Lett. 111, 061301 (2013) , arXiv:1304.6534 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  31. [31]

    R. C. Duncan and C. Thompson, Astrophys. J. 392, L9 (1992)

    work page 1992

  32. [32]

    Thompson and R

    C. Thompson and R. C. Duncan, Astrophys. J. 408, 194 (1993) . 20

    work page 1993

  33. [33]

    Lai and S

    D. Lai and S. L. Shapiro, Astrophys. J. 383, 745 (1991)

    work page 1991

  34. [34]

    K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, M. I. Katsnelson, I. V . Grigorieva, S. V . Dubonos, and A. A. Firsov , Nature 438, 197 (2005) , arXiv:cond-mat/0509330 [cond-mat.mes-hall]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  35. [35]

    Zhang, Y .-W

    Y . Zhang, Y .-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005)

    work page 2005

  36. [36]

    The chiral critical line of N_f=2+1 QCD at zero and non-zero baryon density

    P. de Forcrand and O. Philipsen, JHEP 01, 077 (2007) , arXiv:hep-lat/0607017 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  37. [37]

    The chiral critical point of Nf=3 QCD at finite density to the order (mu/T)^4

    P. de Forcrand and O. Philipsen, JHEP 11, 012 (2008) , arXiv:0808.1096 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  38. [38]

    The curvature of the critical surface (m_ud,m_s)^{crit}(mu): a progress report

    P. de Forcrand and O. Philipsen, Proceedings, 26th International Symposium on Lattice field theory (Lattice 2008): Williamsburg, USA, July 14-19, 2008, PoS LATTICE2008, 208 (2008), arXiv:0811.3858 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  39. [39]

    B. B. Brandt, G. Bali, G. Endrödi, and B. Glässle, Proceedings, 33rd International Symposium on Lattice Fiel d Theory (Lattice 2015): Kobe, Japan, July 14-18, 2015 , PoS LATTICE2015, 265 (2016) , arXiv:1510.03899 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [40]

    E. V . Luschevskaya and O. V . Larina,Nucl. Phys. B884, 1 (2014) , arXiv:1203.5699 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  41. [42]

    Y . Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Nature 443, 675 (2006) , arXiv:hep-lat/0611014 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  42. [43]

    Nambu and G

    Y . Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961) , [,141(1961)]

    work page 1961

  43. [44]

    Nambu and G

    Y . Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961) , [,127(1961)]

    work page 1961

  44. [45]

    S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992)

    work page 1992

  45. [46]

    QCD Phenomenology based on a Chiral Effective Lagrangian

    T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994) , arXiv:hep-ph/9401310 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  46. [47]

    Vogl and W

    U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991)

    work page 1991

  47. [48]

    NJL-model analysis of dense quark matter

    M. Buballa, Phys. Rept. 407, 205 (2005) , arXiv:hep-ph/0402234 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  48. [49]

    L. J. Reinders, H. Rubinstein, and S. Yazaki, Phys. Rept. 127, 1 (1985)

    work page 1985

  49. [50]

    S. P. Klevansky and R. H. Lemmer, Phys. Rev. D39, 3478 (1989)

    work page 1989

  50. [51]

    Inverse Magnetic Catalysis in Nambu--Jona-Lasinio Model beyond Mean Field

    S. Mao, Phys. Lett. B758, 195 (2016) , arXiv:1602.06503 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  51. [52]

    Properties of neutral mesons in a hot and magnetized quark matter

    S. Fayazbakhsh, S. Sadeghian, and N. Sadooghi, Phys. Rev. D86, 085042 (2012) , arXiv:1206.6051 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  52. [53]

    Renormalized vs Nonrenormalized Chiral Transition in a Magnetic Background

    M. Ruggieri, M. Tachibana, and V . Greco, JHEP 07, 165 (2013) , arXiv:1305.0137 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  53. [54]

    G. S. Bali, F. Bruckmann, G. Endrödi, S. D. Katz, and A. Sc häfer, JHEP 08, 177 (2014) , arXiv:1406.0269 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  54. [55]

    G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz , and A. Schafer, Phys. Rev. D86, 071502 (2012) , arXiv:1206.4205 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  55. [56]

    V . G. Bornyakov, P. V . Buividovich, N. Cundy, O. A. Koche tkov, and A. Schäfer, Phys. Rev. D90, 034501 (2014) , arXiv:1312.5628 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  56. [57]

    Critical Endpoint and Inverse Magnetic Catalysis for Finite Temperature and Density Quark Matter in a Magnetic Background

    M. Ruggieri, L. Oliva, P. Castorina, R. Gatto, and V . Gre co, Phys. Lett. B734, 255 (2014) , arXiv:1402.0737 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  57. [58]

    J. O. Andersen, W. R. Naylor, and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016) , arXiv:1411.7176 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  58. [59]

    Inverse magnetic catalysis from the properties of the QCD coupling in a magnetic field

    A. Ayala, C. A. Dominguez, L. A. Hernandez, M. Loewe, and R. Zamora, Phys. Lett. B759, 99 (2016) , arXiv:1510.09134 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [60]

    Bulk Properties of a Fermi Gas in a Magnetic Field

    M. Strickland, V . Dexheimer, and D. P. Menezes, Phys. Rev. D86, 125032 (2012) , arXiv:1209.3276 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  60. [61]

    Effect of external magnetic field on nucleon mass in hot and dense medium : Inverse Magnetic Catalysis in Walecka Model

    A. Mukherjee, S. Ghosh, M. Mandal, S. Sarkar, and P. Roy, Phys. Rev. D98, 056024 (2018) , arXiv:1809.07028 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  61. [62]

    P. J. A. Bicudo, J. E. F. T. Ribeiro, and R. Fernandes, Phys. Rev. C59, 1107 (1999) , arXiv:hep-ph/9806243 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  62. [63]

    Dressed-quark anomalous magnetic moments

    L. Chang, Y .-X. Liu, and C. D. Roberts, Phys. Rev. Lett. 106, 072001 (2011) , arXiv:1009.3458 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  63. [64]

    Anomalous magnetic moment of hot quarks, inverse magnetic catalysis and reentrance of chiral symmetry broken phase

    S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D90, 105030 (2014) , arXiv:1408.5457 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  64. [65]

    Hatsuda and T

    T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55, 158 (1985)

    work page 1985

  65. [66]

    E. V . Shuryak, Workshop on Heavy Ion Physics at the AGS Upton, New York, Marc h 3-4, 1990, Phys. Rev. D42, 1764 (1990)

    work page 1990

  66. [67]

    Pion dispersion relation at finite density and temperature

    A. Ayala, P. Amore, and A. Aranda, Phys. Rev. C66, 045205 (2002) , arXiv:hep-ph/0207081 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  67. [68]

    $\pi$ and $\sigma$ mesons at finite temperature and density in the NJL model with dimensional regularization

    T. Inagaki, D. Kimura, and A. Kvinikhidze, Phys. Rev. D77, 116004 (2008) , arXiv:0712.1336 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  68. [69]

    Mesonic correlation functions at finite temperature and density in the Nambu-Jona-Lasinio model with a Polyakov loop

    H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Na rdi, and C. Ratti, Phys. Rev. D75, 065004 (2007) , arXiv:hep-ph/0609116 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  69. [70]

    Properties of Mesons in a Strong Magnetic Field

    R. Zhang, W.-j. Fu, and Y .-x. Liu, Eur. Phys. J. C76, 307 (2016) , arXiv:1604.08888 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  70. [71]

    Pions in magnetic field at finite temperature

    S. Mao, Phys. Rev. D99, 056005 (2019) , arXiv:1808.10242 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  71. [72]

    S. S. Avancini, R. L. S. Farias, and W. R. Tavares, Phys. Rev. D99, 056009 (2019) , arXiv:1812.00945 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  72. [73]

    Properties of hadron screening masses at small baryonic density

    I. Pushkina, P. de Forcrand, M. Garcia Perez, S. Kim, H. M atsufuru, A. Nakamura, I.-O. Stamatescu, T. Takaishi, and T . Umeda (QCD-TARO),Phys. Lett. B609, 265 (2005) , arXiv:hep-lat/0410017 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  73. [74]

    Meson correlation functions at high temperatures

    S. Wissel, E. Laermann, S. Shcheredin, S. Datta, and F. K arsch, PoS LAT2005, 164 (2006) , arXiv:hep-lat/0510031 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  74. [75]

    T. H. Hansson and I. Zahed, Nucl. Phys. B374, 277 (1992)

    work page 1992

  75. [76]

    Mesonic correlation lengths in high-temperature QCD

    M. Laine and M. Vepsalainen, JHEP 02, 004 (2004) , arXiv:hep-ph/0311268 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  76. [77]

    W. M. Alberico, A. Beraudo, and A. Molinari, Nucl. Phys. A750, 359 (2005) , arXiv:hep-ph/0411346 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  77. [78]

    W. M. Alberico, A. Beraudo, P. Czerski, and A. Molinari, Nucl. Phys. A775, 188 (2006) , arXiv:hep-ph/0605060 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  78. [79]

    Finite Temperature Meson Correlation Functions in HTL Approximation

    F. Karsch, M. G. Mustafa, and M. H. Thoma, Phys. Lett. B497, 249 (2001) , arXiv:hep-ph/0007093 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  79. [80]

    Spontaneous magnetization under a pseudovector interaction between quarks in high density quark matter

    M. Morimoto, Y . Tsue, J. da Providencia, C. Providencia , and M. Yamamura, Int. J. Mod. Phys. E27, 1850028 (2018) , arXiv:1801.03633 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  80. [81]

    M. K. Volkov, Phys. Part. Nucl. 24, 35 (1993)

    work page 1993

Showing first 80 references.