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arxiv: 2604.24595 · v1 · submitted 2026-04-27 · ✦ hep-ph

Recognition: unknown

Mass spectra of charged mesons and the quenching of vector meson condensation via exact phase-space diagonalization

Jingyi Chao, Kun Xu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:51 UTC · model grok-4.3

classification ✦ hep-ph
keywords Nambu-Jona-Lasinio modelmagnetic catalysisvector meson condensationBethe-Salpeter equationphase-space diagonalizationchiral condensatefinite temperaturetachyonic instability
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The pith

Magnetic catalysis in the NJL model quenches the tachyonic instability of the charged rho meson by raising the continuum threshold above the Zeeman attraction.

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

The paper applies an exact non-commutative phase-space method to the two-flavor NJL model to solve the Bethe-Salpeter equations for charged pions and rhos in a background magnetic field at finite temperature. For the pseudoscalar channel the calculation recovers an exact identity that keeps the generalized Goldstone theorem intact, so the pi+ mass follows the Landau-level zero-point drift. In the vector channel the same framework shows that the magnetic enhancement of the chiral condensate lifts the two-quark continuum threshold faster than the Zeeman term can lower the spin-aligned rho+ mass, eliminating the tachyonic mode. At finite temperature the masses fall monotonically from Pauli blocking yet stay bound until chiral restoration.

Core claim

In the two-flavor NJL model the exact algebraic diagonalization of the Bethe-Salpeter equations reveals that the tachyonic instability of the spin-aligned rho+ state is quenched. The magnetic catalysis of the chiral condensate drives the continuum threshold (2M) upwards, overtaking the Zeeman attraction and preventing vector meson condensation within this mean-field framework.

What carries the argument

The non-commutative phase-space framework obtained via the Wigner-Weyl transform and Moyal star product, which permits algebraic diagonalization of the Bethe-Salpeter kernel for mesons carrying asymmetric fractional charges.

If this is right

  • The generalized Goldstone theorem holds exactly for the charged pion, whose mass tracks the kinematic zero-point energy at order eB.
  • The Zeeman spin-splitting in the vector channel emerges dynamically from microscopic threshold truncations set by the chiral Dirac algebra.
  • All meson modes remain bound at finite temperature and undergo only monotonic thermal suppression without Mott dissociation before chiral restoration.
  • Vector meson condensation is absent in this mean-field description of QCD matter in strong magnetic fields.

Where Pith is reading between the lines

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

  • The result implies that any observation of rho condensation would require either going beyond mean-field or altering the regularization scheme.
  • The phase-space diagonalization technique can be applied to other composite operators or to multi-flavor extensions.
  • The quenching mechanism may constrain the location of possible magnetic phases in the QCD phase diagram relevant to heavy-ion collisions or magnetars.

Load-bearing premise

The two-flavor NJL model in the mean-field approximation with the chosen regularization and truncation of the Bethe-Salpeter kernel remains valid for the vector channel at finite magnetic field and temperature.

What would settle it

A lattice QCD calculation or a beyond-mean-field NJL study that finds the spin-aligned rho+ pole mass becoming imaginary below the two-quark threshold in a sufficiently strong magnetic field would falsify the quenching result.

Figures

Figures reproduced from arXiv: 2604.24595 by Jingyi Chao, Kun Xu.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical evaluation of the generalized triple Laguerre overlap integrals in the internal constituent phase space ( view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The constituent quark mass view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The temperature-induced mass shift ∆ view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The phenomenological vacuum calibration for the vector channel. The constituent quark mass view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The pole masses of the charged view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The temperature-induced mass shifts ∆ view at source ↗
read the original abstract

We investigate the dynamics and mass spectra of charged pseudoscalar ($\pi^+$) and vector ($\rho^+$) mesons in a background magnetic field at finite temperature using the two-flavor Nambu-Jona--Lasinio (NJL) model. By employing a quark propagator that isolates the Schwinger phase from its Landau level expansion, we formulate an exact non-commutative phase-space framework utilizing the Wigner-Weyl transform and the Moyal star product. This approach enables the algebraic diagonalization of the Bethe-Salpeter equations for composite states with asymmetric fractional constituent charges. For the pseudoscalar channel, we analytically verify the exact cancellation between the dynamical random phase approximation spatial sum rules and the vacuum gap equation. This identity preserves the generalized Goldstone theorem, causing the $\pi^+$ pole mass to strictly track the kinematic zero-point energy drift at order of $eB$. In the vector channel, our full phase-space evaluation reveals that the Zeeman spin-splitting emerges dynamically from microscopic threshold truncations governed by the chiral Dirac algebra. Notably, we find that the tachyonic instability of the spin-aligned $\rho^+$ state is quenched. The magnetic catalysis of the chiral condensate drives the continuum threshold ($2M$) upwards, overtaking the Zeeman attraction and preventing vector meson condensation within this mean-field framework. Furthermore, finite-temperature evaluations show a monotonic thermal suppression of the meson masses driven by Pauli blocking, yet all modes remain bound without undergoing Mott dissociation prior to chiral symmetry restoration.

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 develops an exact non-commutative phase-space framework (Wigner-Weyl transform and Moyal star product) to diagonalize the Bethe-Salpeter equations for charged π+ and ρ+ mesons in the two-flavor NJL model at finite B and T. It analytically verifies cancellation identities that preserve the generalized Goldstone theorem for the pseudoscalar channel, so that the π+ pole mass tracks the kinematic zero-point energy. For the vector channel, a full phase-space evaluation shows that the Zeeman spin-splitting emerges from the chiral Dirac algebra in the threshold truncation; the tachyonic instability of the spin-aligned ρ+ is quenched because magnetic catalysis raises the continuum threshold 2M above the Zeeman attraction. Finite-T results indicate monotonic thermal suppression of masses without Mott dissociation prior to chiral restoration.

Significance. If the central results hold, the work provides a technically innovative exact-diagonalization method for composite states with fractional charges in magnetic backgrounds, together with explicit analytical identities that enforce Goldstone-mode protection. These are genuine strengths. The quenching mechanism, however, remains tied to the mean-field NJL ladder approximation and its regularization, so the broader phenomenological implications for heavy-ion collisions or magnetized QCD matter are limited until robustness is demonstrated.

major comments (2)
  1. [vector-channel phase-space evaluation] The quenching of the spin-aligned ρ+ tachyonic mode is the central claim and rests on the competition between the upward shift of 2M (driven by magnetic catalysis in the gap equation) and the Zeeman attraction extracted from the Dirac algebra in the threshold truncation. Because both effects are controlled by the ultraviolet regulator (cutoff on Landau levels or equivalent) and by the precise truncation of the Bethe-Salpeter kernel, the manuscript must show explicitly that 2M overtakes the Zeeman term for the range of NJL parameters (G, Λ) employed; otherwise the result is an artifact of the chosen regularization.
  2. [formalism and gap equation] The self-consistent mean-field treatment determines the chiral condensate M from the same gap equation that sets the continuum threshold 2M. This introduces a built-in correlation that guarantees the quenching outcome inside the model; the paper should clarify whether the cancellation identities and threshold behavior survive when the ladder approximation is relaxed or when higher-order kernel terms are restored.
minor comments (1)
  1. [phase-space framework] The notation for the Moyal star product and the separation of the Schwinger phase from the Landau-level expansion would benefit from an explicit worked example in the formalism section to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment point by point below, providing clarifications on the framework and indicating where revisions will strengthen the presentation. The analysis remains within the consistent mean-field NJL ladder approximation, as stated in the abstract and throughout the text.

read point-by-point responses

  1. Referee: [vector-channel phase-space evaluation] The quenching of the spin-aligned ρ+ tachyonic mode is the central claim and rests on the competition between the upward shift of 2M (driven by magnetic catalysis in the gap equation) and the Zeeman attraction extracted from the Dirac algebra in the threshold truncation. Because both effects are controlled by the ultraviolet regulator (cutoff on Landau levels or equivalent) and by the precise truncation of the Bethe-Salpeter kernel, the manuscript must show explicitly that 2M overtakes the Zeeman term for the range of NJL parameters (G, Λ) employed; otherwise the result is an artifact of the chosen regularization.

    Authors: We agree that an explicit demonstration is necessary to confirm the result is not regularization-dependent. The phase-space evaluation in the manuscript uses standard NJL parameters (G and Λ) fixed by vacuum phenomenology. In the revised version we will add a dedicated subsection and accompanying figure that numerically compares the continuum threshold 2M (from the self-consistent gap equation) against the Zeeman shift for the employed parameter set across the relevant range of eB. This will explicitly verify that magnetic catalysis drives 2M above the Zeeman attraction, quenching the instability. The addition will also include a brief sensitivity check within the phenomenologically allowed window for G and Λ. revision: yes

  2. Referee: [formalism and gap equation] The self-consistent mean-field treatment determines the chiral condensate M from the same gap equation that sets the continuum threshold 2M. This introduces a built-in correlation that guarantees the quenching outcome inside the model; the paper should clarify whether the cancellation identities and threshold behavior survive when the ladder approximation is relaxed or when higher-order kernel terms are restored.

    Authors: The cancellation identities and the resulting Goldstone protection are derived analytically from the structure of the ladder Bethe-Salpeter kernel being consistent with the gap equation; this self-consistency is essential for the exact algebraic diagonalization via the Moyal product. We will revise the manuscript to state more explicitly that all reported results, including the quenching mechanism, are obtained within the mean-field NJL ladder approximation. A short paragraph will be added noting that relaxing the ladder kernel would require a different non-perturbative framework (e.g., full Dyson-Schwinger equations) and lies beyond the present scope. revision: partial

standing simulated objections not resolved
  • Whether the cancellation identities and quenching mechanism survive when the ladder approximation is relaxed or higher-order kernel terms are restored.

Circularity Check

0 steps flagged

No significant circularity; derivation is model-internal but self-contained

full rationale

The paper solves the NJL gap equation for the chiral condensate M(B,T) under magnetic catalysis, then diagonalizes the Bethe-Salpeter equation in the Wigner-Weyl phase-space formulation to extract meson poles. The continuum threshold 2M and the Zeeman splitting both arise as outputs of the same Dirac algebra and Landau-level sums; the observation that 2M overtakes the splitting is a computed dynamical result, not a redefinition or fit to the target quantity. The exact cancellation identity for the pseudoscalar channel is verified algebraically against the gap equation rather than assumed. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation chain. The framework dependence is explicitly acknowledged but does not reduce the central claim to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard NJL mean-field gap equation, a chosen regularization scheme, and the validity of the Bethe-Salpeter truncation in the presence of the magnetic field.

free parameters (2)
  • NJL four-fermion coupling G
    Fitted to vacuum meson properties and used to determine the chiral condensate self-consistently.
  • Ultraviolet cutoff Lambda
    Regularization parameter required in the NJL model to render integrals finite.
axioms (2)
  • domain assumption Mean-field approximation for the chiral condensate
    Invoked throughout to close the gap equation and determine M(B,T).
  • domain assumption Truncation of the Bethe-Salpeter kernel to ladder level
    Used to obtain the meson bound-state equations.

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Forward citations

Cited by 1 Pith paper

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

  1. $\rho$ mesons in finite magnetic field and finite temperature

    nucl-th 2026-05 unverdicted novelty 4.0

    In the NJL model, rho meson masses for different charge and spin states increase or decrease with magnetic field at zero temperature, decrease with temperature at fixed B, approach constituent quark masses at high T, ...

Reference graph

Works this paper leans on

101 extracted references · 85 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    Mass spectra of charged mesons and the quenching of vector meson condensation via exact phase-space diagonalization

    indicate that while theρ ± mass initially decreases, it ultimately avoids the tachyonic zero. These advancements emphasize that internal quark structure and interactions with the QCD vacuum, such as magnetic catalysis, are critical for stabilizing the vector meson. To describe these composite dynamics, models such as the Nambu-Jona– Lasinio (NJL) model [4...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    = 4iNc π|qπB| ∞X nu,nd=0 Z dωdpz (2π)2 − 1 2 SL J (1) nu,nd−1,0 +J (1) nu−1,nd,0 +J (2) nu,nd,0 Du(ω+q 0)Dd(ω) .(26) Here, the inverse quark propagators governing the longitudinal energy poles areD u(ω+q 0) = (ω+q 0)2 −p 2 z −M 2 − 2nu|quB|,D d(ω) =ω 2 −p 2 z −M 2 −2n d|qdB|andω=p 0 for convenience. We isolate the vacuum condensate background from the dyn...

  3. [3]

    The single-denominator terms constitute the vacuum condensate background

    = J1 +J m(q2 0). The single-denominator terms constitute the vacuum condensate background. BecauseD d(ω) depends exclusively onn d, the infinite summation overn u acts solely on the spatial overlap factor. Utilizing the completeness relation of the Type I overlap integral derived in Appendix B, we have: ∞X nu=0 J (1) nu,nd,0 = π2 2 |qπB||qdB|.(28) Substit...

  4. [4]

    =− 4iNc π|qπB| ∞X nu,nd=0 Z dωdpz (2π)2 Nπ+ Du(ω+q 0)Dd(ω) ,(29) where the scalar dynamic mass scale isR 2 π+ =n u|quB|+n d|qdB| −q 2 0/2, and the consolidated transverse spatial weight is: Nπ+ = 1 2 R2 π+ J (1) nu,nd−1,0 +J (1) nu−1,nd,0 − J (2) nu,nd,0.(30) Applying the Matsubara summation formalism (ω→iω m =i(2m+ 1)πT), the kernel splits into vacuum an...

  5. [5]

    The transverse regularization scale is designated to depend exclusively on the Landau level of asinglereference constituent quark

    =− 4Nc π|qπB| ∞X nu,nd=0 Z ∞ −∞ dpz 2π Nπ+ 2EuEd Eu +E d q2 0 −(E u +E d)2 ,(31) Jmed m (q2 0, T) = 4Nc π|qπB| ∞X nu,nd=0 Z ∞ −∞ dpz 2π Nπ+ 2EuEd " Eu +E d q2 0 −(E u +E d)2 (nF (Eu) +n F (Ed)) − Eu −E d q2 0 −(E u −E d)2 (nF (Eu)−n F (Ed)) # .(32) To manage ultraviolet divergences, we apply a 3D soft cutoff regularization with a smooth damping function f...

  6. [6]

    Applying the Matsubara summation, the dynamic kernels separate into vacuum and medium components: Jvac m,λ(q2

    =J 1+J m,∥(q2 0). Applying the Matsubara summation, the dynamic kernels separate into vacuum and medium components: Jvac m,λ(q2

  7. [7]

    =− 4Nc π|qρB| ∞X nu,nd=0 Z ∞ −∞ dpz 2π Nλ 2EuEd Eu +E d q2 0 −(E u +E d)2 ,(51) Jmed m,λ (q2 0, T) = 4Nc π|qρB| ∞X nu,nd=0 Z ∞ −∞ dpz 2π Nλ 2EuEd " Eu +E d q2 0 −(E u +E d)2 (nF (Eu) +n F (Ed)) − Eu −E d q2 0 −(E u −E d)2 (nF (Eu)−n F (Ed)) # ,(52) 10 where the consolidated transverse spatial weights are: N∥ = 1 2 R2 ∥(q2 0, pz) J (1) nu,nd−1,0 +J (1) nu−...

  8. [8]

    =n u|quB|+n d|qdB| −q 2 0/2 andR 2 ∥(q2 0, pz) = R2 ±(q2 0)−2p 2 z. The RPA equations governing the massesm ρ±(λ) are thus: 1−2G V h J1 +J m,λ(q2 0 =m 2 ρ±,λ) i = 0.(55) A fundamental feature of these equations is that the Zeeman splitting emerges dynamically from microscopic threshold truncationsenforced by the chiral Dirac algebra, rather than phenomeno...

  9. [9]

    Type I (Scalar) Overlap Integrals For the standard Laguerre polynomialsL n(x) present in the longitudinal trace, we utilize the generating function in Eq. (A4). To perform the transverse phase-space integration, we define the composite 2D momentum vector v= (k ⊥,p ⊥)T . Substituting the explicit definitions ofα u,α d, andα π into the generating function p...

  10. [10]

    Summing over all possible internal states of one constituent quark must analytically recover a constant density of states

    Completeness and Probability Conservation The physical consistency of the spatial projection framework is verified by evaluating the completeness limit for the macroscopic ground state (n= 0). Summing over all possible internal states of one constituent quark must analytically recover a constant density of states. Recall the integral definition for the ca...

  11. [11]

    Type II (V ector) Overlaps and Auxiliary Recurrence Type II integralsJ (2) involve the bilinear momentum factor (k⊥ ·p⊥). We express this inner product as a symmetric quadratic form overv: k⊥ ·p ⊥ = 1 2 vT Mv,whereM= 0 1 1 0 .(B14) According to the Gaussian moment generating theorem, inserting this quadratic form evaluates to 1 2Tr(MA−1) multiplied by the...

  12. [12]

    V. M. Kaspi and A. Beloborodov, Magnetars, Ann. Rev. Astron. Astrophys.55, 261 (2017), arXiv:1703.00068 [astro-ph.HE]

    work page arXiv 2017

  13. [13]

    D. E. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U. Yee, ’Strongly interacting matter in magnetic fields’: an overview, Lect. Notes Phys.871, 1 (2013), arXiv:1211.6245 [hep-ph]

    work page arXiv 2013

  14. [14]

    V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rept.576, 1 (2015), arXiv:1503.00732 [hep-ph]

    work page arXiv 2015

  15. [15]

    J. O. Andersen, W. R. Naylor, and A. Tranberg, Phase diagram of QCD in a magnetic field: A review, Rev. Mod. Phys. 88, 025001 (2016), arXiv:1411.7176 [hep-ph]

    work page arXiv 2016

  16. [16]

    Estimate of the magnetic field strength in heavy-ion collisions

    V. Skokov, A. Y. Illarionov, and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A24, 5925 (2009), arXiv:0907.1396 [nucl-th]

    work page Pith review arXiv 2009

  17. [17]

    Event-by-event generation of electromagnetic fields in heavy-ion collisions

    W.-T. Deng and X.-G. Huang, Event-by-event generation of electromagnetic fields in heavy-ion collisions, Phys. Rev. C 85, 044907 (2012), arXiv:1201.5108 [nucl-th]

    work page Pith review arXiv 2012

  18. [18]

    The Chiral Magnetic Effect

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

    work page Pith review arXiv 2008

  19. [19]

    V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field, Nucl. Phys. B462, 249 (1996), arXiv:hep-ph/9509320

    work page arXiv 1996

  20. [20]

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

    work page arXiv 2012

  21. [21]

    Bruckmann, G

    F. Bruckmann, G. Endrodi, and T. G. Kovacs, Inverse magnetic catalysis and the Polyakov loop, JHEP04, 112, arXiv:1303.3972 [hep-lat]

    work page arXiv

  22. [22]

    J. Chao, P. Chu, and M. Huang, Inverse magnetic catalysis induced by sphalerons, Phys. Rev. D88, 054009 (2013), arXiv:1305.1100 [hep-ph]

    work page arXiv 2013

  23. [23]

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

    work page arXiv 2013

  24. [24]

    Ding and D

    H.-T. Ding and D. Zhang, Chiral Properties of (2 + 1)-Flavor QCD in Magnetic Fields at Zero Temperature (2026), arXiv:2601.18354 [hep-lat]

    work page arXiv 2026

  25. [25]

    N. O. Agasian and I. A. Shushpanov, Gell-Mann-Oakes-Renner relation in a magnetic field at finite temperature, JHEP 10, 006, arXiv:hep-ph/0107128

    work page arXiv

  26. [26]

    J. O. Andersen, Thermal pions in a magnetic background, Phys. Rev. D86, 025020 (2012), arXiv:1202.2051 [hep-ph]

    work page arXiv 2012

  27. [27]

    V. D. Orlovsky and Y. A. Simonov, Nambu-Goldstone mesons in strong magnetic field, JHEP09, 136, arXiv:1306.2232 [hep-ph]

    work page arXiv

  28. [28]

    Colucci, E

    G. Colucci, E. S. Fraga, and A. Sedrakian, Chiral pions in a magnetic background, Phys. Lett. B728, 19 (2014), arXiv:1310.3742 [nucl-th]

    work page arXiv 2014

  29. [29]

    S. S. Avancini, R. L. S. Farias, M. Benghi Pinto, W. R. Tavares, and V. S. Tim´ oteo,π 0 pole mass calculation in a strong magnetic field and lattice constraints, Phys. Lett. B767, 247 (2017), arXiv:1606.05754 [hep-ph]

    work page arXiv 2017

  30. [30]

    Brauner and N

    T. Brauner and N. Yamamoto, Chiral Soliton Lattice and Charged Pion Condensation in Strong Magnetic Fields, JHEP 04, 132, arXiv:1609.05213 [hep-ph]

    work page arXiv

  31. [31]

    G. S. Bali, B. B. Brandt, G. Endr˝ odi, and B. Gl¨ aßle, Weak decay of magnetized pions, Phys. Rev. Lett.121, 072001 (2018), arXiv:1805.10971 [hep-lat]

    work page arXiv 2018

  32. [32]

    B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert, J. Schaffner-Bielich, and S. Schmalzbauer, New class of compact stars: Pion stars, Phys. Rev. D98, 094510 (2018), arXiv:1802.06685 [hep-ph]

    work page arXiv 2018

  33. [33]

    H. T. Ding, S. T. Li, A. Tomiya, X. D. Wang, and Y. Zhang, Chiral properties of (2+1)-flavor QCD in strong magnetic fields at zero temperature, Phys. Rev. D104, 014505 (2021), arXiv:2008.00493 [hep-lat]

    work page arXiv 2021

  34. [34]

    Z. Xing, J. Chao, L. Chang, and Y.-x. Liu, Exposing the effect of the p-wave component in the pion triplet under a strong magnetic field, Phys. Rev. D105, 114003 (2022), arXiv:2110.01245 [hep-ph]

    work page arXiv 2022

  35. [35]

    N. Wen, X. Cao, J. Chao, and H. Liu, Neutral pion masses within a hot and magnetized medium in a lattice-improved soft-wall AdS/QCD model, Phys. Rev. D109, 086021 (2024), arXiv:2402.06239 [hep-th]

    work page arXiv 2024

  36. [36]

    pion-rho Mixing as a mechanism for non-monotonic charged pion behavior in magnetic fields

    Z. Wang, Residue-Enhanced Pion-Rho Mixing as the Origin of Nonmonotonic Charged Pion Mass in Magnetic Fields (2026), arXiv:2602.15410 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  37. [37]

    Hidaka and A

    Y. Hidaka and A. Yamamoto, Charged vector mesons in a strong magnetic field, Phys. Rev. D87, 094502 (2013), arXiv:1209.0007 [hep-ph]

    work page arXiv 2013

  38. [38]

    Kawaguchi and S

    M. Kawaguchi and S. Matsuzaki, Vector meson masses from a hidden local symmetry in a constant magnetic field, Phys. Rev. D93, 125027 (2016), arXiv:1511.06990 [hep-ph]

    work page arXiv 2016

  39. [39]

    M. A. Andreichikov, B. O. Kerbikov, E. V. Luschevskaya, Y. A. Simonov, and O. E. Solovjeva, The Evolution of Meson Masses in a Strong Magnetic Field, JHEP05, 007, arXiv:1610.06887 [hep-ph]

    work page arXiv

  40. [40]

    Callebaut, D

    N. Callebaut, D. Dudal, and H. Verschelde, Holographic study of rho meson mass in an external magnetic field: Paving the road towards a magnetically induced superconducting QCD vacuum?, PoSF ACESQCD, 046 (2010), arXiv:1102.3103 [hep-ph]

    work page arXiv 2010

  41. [41]

    M. N. Chernodub, Superconductivity of QCD vacuum in strong magnetic field, Phys. Rev. D82, 085011 (2010), 18 arXiv:1008.1055 [hep-ph]

    work page arXiv 2010

  42. [42]

    M. N. Chernodub, Spontaneous electromagnetic superconductivity of vacuum in strong magnetic field: evidence from the Nambu–Jona-Lasinio model, Phys. Rev. Lett.106, 142003 (2011), arXiv:1101.0117 [hep-ph]

    work page arXiv 2011

  43. [43]

    Brauner and X.-G

    T. Brauner and X.-G. Huang, Vector meson condensation in a pion superfluid, Phys. Rev. D94, 094003 (2016), arXiv:1610.00426 [hep-ph]

    work page arXiv 2016

  44. [44]

    D’Elia and F

    M. D’Elia and F. Negro, Chiral Properties of Strong Interactions in a Magnetic Background, Phys. Rev. D83, 114028 (2011), arXiv:1103.2080 [hep-lat]

    work page arXiv 2011

  45. [45]

    E. V. Luschevskaya, O. E. Solovjeva, O. A. Kochetkov, and O. V. Teryaev, Magnetic polarizabilities of light mesons in SU(3) lattice gauge theory, Nucl. Phys. B898, 627 (2015), arXiv:1411.4284 [hep-lat]

    work page arXiv 2015

  46. [46]

    G. S. Bali, B. B. Brandt, G. Endr˝ odi, and B. Gl¨ aßle, Meson masses in electromagnetic fields with Wilson fermions, Phys. Rev. D97, 034505 (2018), arXiv:1707.05600 [hep-lat]

    work page arXiv 2018

  47. [47]

    D’Elia, F

    M. D’Elia, F. Manigrasso, F. Negro, and F. Sanfilippo, QCD phase diagram in a magnetic background for different values of the pion mass, Phys. Rev. D98, 054509 (2018), arXiv:1808.07008 [hep-lat]

    work page arXiv 2018

  48. [48]

    Ding, S.-T

    H.-T. Ding, S.-T. Li, S. Mukherjee, A. Tomiya, and X.-D. Wang, Meson masses in external magnetic fields with HISQ fermions, PoSLA TTICE2019, 250 (2020), arXiv:2001.05322 [hep-lat]

    work page arXiv 2020

  49. [49]

    Endrodi, QCD with background electromagnetic fields on the lattice: A review, Prog

    G. Endrodi, QCD with background electromagnetic fields on the lattice: A review, Prog. Part. Nucl. Phys.141, 104153 (2025), arXiv:2406.19780 [hep-lat]

    work page arXiv 2025

  50. [50]

    Wang and P

    Z. Wang and P. Zhuang, Meson properties in magnetized quark matter, Phys. Rev. D97, 034026 (2018), arXiv:1712.00554 [hep-ph]

    work page arXiv 2018

  51. [51]

    S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys.64, 649 (1992)

    1992

  52. [52]

    H. Liu, X. Wang, L. Yu, and M. Huang, Neutral and charged scalar mesons, pseudoscalar mesons, and diquarks in magnetic fields, Phys. Rev. D97, 076008 (2018), arXiv:1801.02174 [hep-ph]

    work page arXiv 2018

  53. [53]

    Chaudhuri, S

    N. Chaudhuri, S. Ghosh, S. Sarkar, and P. Roy, Effect of the anomalous magnetic moment of quarks on the phase structure and mesonic properties in the NJL model, Phys. Rev. D99, 116025 (2019), arXiv:1907.03990 [nucl-th]

    work page arXiv 2019

  54. [54]

    S. S. Avancini, M. Coppola, N. N. Scoccola, and J. C. Sodr´ e, Light pseudoscalar meson masses under strong magnetic fields within the SU(3) Nambu–Jona-Lasinio model, Phys. Rev. D104, 094040 (2021), arXiv:2109.01911 [hep-ph]

    work page arXiv 2021

  55. [55]

    Cao, Recent progresses on QCD phases in a strong magnetic field: views from Nambu–Jona-Lasinio model, Eur

    G. Cao, Recent progresses on QCD phases in a strong magnetic field: views from Nambu–Jona-Lasinio model, Eur. Phys. J. A57, 264 (2021), arXiv:2103.00456 [hep-ph]

    work page arXiv 2021

  56. [56]

    Sheng, Y

    B. Sheng, Y. Wang, X. Wang, and L. Yu, Pole and screening masses of neutral pions in a hot and magnetized medium: A comprehensive study in the Nambu–Jona-Lasinio model, Phys. Rev. D103, 094001 (2021), arXiv:2010.05716 [hep-ph]

    work page arXiv 2021

  57. [57]

    K. Xu, J. Chao, and M. Huang, Effect of the anomalous magnetic moment of quarks on magnetized QCD matter and meson spectra, Phys. Rev. D103, 076015 (2021), arXiv:2007.13122 [hep-ph]

    work page arXiv 2021

  58. [58]

    Chao and Y.-X

    J. Chao and Y.-X. Liu, Dimensional reduction and the generalized pion in a magnetic field within the NJL model, Phys. Rev. D107, 074038 (2023), arXiv:2202.05090 [hep-ph]

    work page arXiv 2023

  59. [59]

    W.-L. Yuan, J. Chao, and A. Li, Diquark and chiral condensates in a self-consistent NJL-type model, Phys. Rev. D108, 043008 (2023), arXiv:2304.12050 [hep-ph]

    work page arXiv 2023

  60. [60]

    Chang and C

    L. Chang and C. D. Roberts, Sketching the Bethe-Salpeter kernel, Phys. Rev. Lett.103, 081601 (2009), arXiv:0903.5461 [nucl-th]

    work page arXiv 2009

  61. [61]

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

    1972

  62. [62]

    Zhang, W.-j

    R. Zhang, W.-j. Fu, and Y.-x. Liu, Properties of Mesons in a Strong Magnetic Field, Eur. Phys. J. C76, 307 (2016), arXiv:1604.08888 [hep-ph]

    work page arXiv 2016

  63. [63]

    Liu and S

    Z. Liu and S. Mao, Vector meson in magnetic field at finite temperature (2026), in preparation. While finalizing this work, we noted a concurrent investigation into the equivalence between the Schwinger proper time and Ritus formalisms. Our non-commutative Wigner-Weyl approach and their findings represent independent yet mutually reinforcing perspectives ...

    2026

  64. [64]

    J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc.45, 99 (1949)

    1949

  65. [65]

    Curtright, D

    T. Curtright, D. Fairlie, and C. K. Zachos, Features of time independent Wigner functions, Phys. Rev. D58, 025002 (1998), arXiv:hep-th/9711183

    work page arXiv 1998

  66. [66]

    C. K. Zachos and T. Curtright, Phase space quantization of field theory, Prog. Theor. Phys. Suppl.135, 244 (1999), arXiv:hep-th/9903254

    work page arXiv 1999

  67. [67]

    J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev.82, 664 (1951)

    1951

  68. [68]

    J. Chao, L. Yu, and M. Huang, Zeta function regularization of the photon polarization tensor for a magnetized vacuum, Phys. Rev. D90, 045033 (2014), [Erratum: Phys.Rev.D 91, 029903 (2015)], arXiv:1403.0442 [hep-th]

    work page arXiv 2014

  69. [69]

    Sheng, D

    X.-l. Sheng, D. H. Rischke, D. Vasak, and Q. Wang, Wigner functions for fermions in strong magnetic fields, Eur. Phys. J. A54, 21 (2018), arXiv:1707.01388 [hep-ph]

    work page arXiv 2018

  70. [70]

    S. S. Avancini, W. R. Tavares, and M. B. Pinto, Properties of magnetized neutral mesons within a full RPA evaluation, Phys. Rev. D93, 014010 (2016), arXiv:1511.06261 [hep-ph]

    work page arXiv 2016

  71. [71]

    Mao, Pions in magnetic field at finite temperature, Phys

    S. Mao, Pions in magnetic field at finite temperature, Phys. Rev. D99, 056005 (2019), arXiv:1808.10242 [nucl-th]

    work page arXiv 2019

  72. [72]

    Berra-Montiel, H

    J. Berra-Montiel, H. Garcia-Compean, and A. Molgado, Star exponentials from propagators and path integrals, Annals Phys.468, 169744 (2024), arXiv:2404.08815 [math-ph]

    work page arXiv 2024

  73. [73]

    Coppola, D

    M. Coppola, D. G´ omez Dumm, and N. N. Scoccola, Charged pion masses under strong magnetic fields in the NJL model, Phys. Lett. B782, 155 (2018), arXiv:1802.08041 [hep-ph]

    work page arXiv 2018

  74. [74]

    Coppola, D

    M. Coppola, D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Neutral and charged pion properties under strong magnetic 19 fields in the NJL model, Phys. Rev. D100, 054014 (2019), arXiv:1907.05840 [hep-ph]

    work page arXiv 2019

  75. [75]

    J. Li, G. Cao, and L. He, Gauge independence of pion masses in a magnetic field within the Nambu–Jona-Lasinio model, Phys. Rev. D104, 074026 (2021), arXiv:2009.04697 [nucl-th]

    work page arXiv 2021

  76. [76]

    Li and S

    L. Li and S. Mao, Inverse magnetic catalysis effect and current quark mass effect on mass spectra and Mott transitions of pions under external magnetic field, Phys. Rev. D108, 054001 (2023), arXiv:2308.12491 [hep-ph]

    work page arXiv 2023

  77. [77]

    H. Liu, L. Yu, and M. Huang, Charged and neutral vectorρmesons in a magnetic field, Phys. Rev. D91, 014017 (2015), arXiv:1408.1318 [hep-ph]

    work page arXiv 2015

  78. [78]

    Ghosh, A

    S. Ghosh, A. Mukherjee, M. Mandal, S. Sarkar, and P. Roy, Spectral properties ofρmeson in a magnetic field, Phys. Rev. D94, 094043 (2016), arXiv:1612.02966 [nucl-th]

    work page arXiv 2016

  79. [79]

    J. P. Carlomagno, D. Gomez Dumm, M. F. I. Villafa˜ ne, S. Noguera, and N. N. Scoccola, Charged pseudoscalar and vector meson masses in strong magnetic fields in an extended NJL model, Phys. Rev. D106, 094035 (2022), arXiv:2209.10679 [hep-ph]

    work page arXiv 2022

  80. [80]

    Gomez Dumm, S

    D. Gomez Dumm, S. Noguera, and N. N. Scoccola, Charged meson masses under strong magnetic fields: Gauge invariance and schwinger phases, Phys. Rev. D108, 016012 (2023), arXiv:2306.04128 [hep-ph]

    work page arXiv 2023

Showing first 80 references.