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arxiv: 2604.17764 · v3 · pith:DCS3N5RXnew · submitted 2026-04-20 · ✦ hep-ph · nucl-th

Soft mode dynamics associated with QCD critical point and color superconductivity -- pseudogap, anomalous dilepton production and electric conductivity

Masakiyo Kitazawa , Teiji Kunihiro This is my paper

Pith reviewed 2026-05-22 11:21 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords QCD critical pointcolor superconductivitysoft modespseudogapdilepton productionelectric conductivityheavy-ion collisionsNambu-Jona-Lasinio model
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The pith

Soft modes near the QCD critical point and color superconductivity produce a pseudogap in quark spectra and enhance electric conductivity and dilepton rates.

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

The paper shows that collective excitations tied to order parameter fluctuations act as soft modes at the QCD critical point and the two-flavor color superconducting transition within the Nambu-Jona-Lasinio model. These modes develop strong low-energy spectral weight that softens and vanishes at the transition. The diquark soft mode creates a pseudogap by reducing the density of quark states around the Fermi surface just above the critical temperature. Concepts borrowed from condensed matter para-conductivity then imply that the same soft modes produce anomalous rises in electric conductivity and dilepton production, effects that could appear in heavy-ion collision data.

Core claim

Collective excitations coupled to fluctuations of the respective order parameters are the soft modes associated with the phase transitions. They acquire prominent spectral strength in the low-energy and low-momentum region near the transitions, with peak energy that softens and vanishes at the critical point. The diquark soft mode of the 2SC produces a pseudogap, a depression in the density of states of the quark spectra around the Fermi surface above but near the critical temperature. Exploiting para-conductivity ideas from condensed matter physics, these soft modes cause an anomalous enhancement of electric conductivity and the dilepton production rate.

What carries the argument

The diquark soft mode of two-flavor color superconductivity, the collective excitation coupled to diquark condensate fluctuations that gains low-energy spectral weight and depresses the quark density of states near the Fermi surface.

If this is right

  • Electric conductivity receives an anomalous enhancement from the soft modes above the critical temperature.
  • The dilepton production rate increases anomalously in the same temperature window.
  • A pseudogap appears in the quark spectral density near the 2SC transition.
  • These electromagnetic signals become relevant for interpreting data from relativistic heavy-ion collisions.

Where Pith is reading between the lines

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

  • The same soft-mode mechanism may affect other transport coefficients such as shear viscosity in the quark-gluon plasma.
  • Dedicated scans of the phase diagram at future facilities could isolate the predicted conductivity and dilepton enhancements.
  • The mapping of condensed-matter para-conductivity ideas onto QCD suggests similar soft-mode signatures could appear in other high-density phases.

Load-bearing premise

The two-flavor Nambu-Jona-Lasinio model and the direct transfer of para-conductivity concepts from condensed matter physics accurately describe the soft mode dynamics and electromagnetic effects in real QCD near the critical point and color superconductivity transition.

What would settle it

Heavy-ion collision data showing no anomalous rise in dilepton yield or electric conductivity in the temperature and density region expected for the QCD critical point or the 2SC transition.

Figures

Figures reproduced from arXiv: 2604.17764 by Masakiyo Kitazawa, Teiji Kunihiro.

Figure 1
Figure 1. Figure 1: Phase diagram calculated by the mean-field approximation in the 2-flavor NJL model (2.1) [35]. The solid line shows the first-order phase transition calculated with GD = 0.70GS. The dashed, dash-dotted, and dotted lines are the second-order 2SC-PT for GD/GS = 0.70, 0.65, and 0.60, respectively. The QCD-CP is indicated by the circle marker located at (TCP, µCP) ≃ (46.712, 329.34) MeV. at the 2SC-PT. Equatio… view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of Eq. (3.12). The single lines denote the quark propagator. 3.1. Linear response theory The linear-response theory [51] is a useful tool to explore dynamical properties of collective excitations. A key idea of this theory is to disturb the system with an infinitesimal external field represented by the Hamiltonian Hext = R d 3xdteiωt−ik·x f(x, t)O(x, t), where O(x, t) is a boson… view at source ↗
Figure 3
Figure 3. Figure 3: Feynman diagrams representing the quark Green function in the non-self-consistent T-matrix approxi￾mation. The thin lines represent the free propagator G0, while the bold ones represent the full propagator G. points. In this and the next sections, we investigate some such observables in the dense quark matter near the 2SC-PT and QCD-CP. In this section, we focus on the modification of the excitation proper… view at source ↗
Figure 4
Figure 4. Figure 4: The spectral function ρ0 at µ = 400MeV and ε = 0.01 and 0.2. The upper figure is an enlargement of that near the Fermi surface [53]. The peaks at ω = k − µ and ω = −k − µ correspond to the quark and anti-quark quasiparticles, respectively. Notice that there is a depression around ω = 0, which is responsible for the pseudogap formation. 0.4 0.6 0.8 1 1.2 -100 -50 0 50 100 N( ω)/N( ω)free ω µ = 350 MeV ε=0.2… view at source ↗
Figure 5
Figure 5. Figure 5: Density of state at µ = 400MeV and various ε ≡ (T − Tc)/Tc [53]. The Dotted line shows that of the free quarks. A clear pseudogap structure is seen, which survives up to ε ≈ 0.05. remarkable as ϵ decreases. This behavior is in contrast to that of the conventional Fermi liquid, in which the lifetime of the quasiparticles becomes longer as ω approaches the Fermi energy. Substituting this spectral function in… view at source ↗
Figure 6
Figure 6. Figure 6: Contribution of the diquark soft mode to the thermodynamic potential. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Diagrammatic representations of the Aslamazov-Larkin (a), Maki-Thompson (b) and density of states (c, d) terms with the 2SC soft modes with the wavy lines being the photon ones. 5. Electric conductivity and dilepton production rates In this section, we explore the effects of the soft modes on the electric conductivity and dilepton production rates (DPR) near the 2SC-PT and QCD-CP. These quantities are deri… view at source ↗
Figure 8
Figure 8. Figure 8: Contribution of the soft mode of the QCD-CP to the thermodynamic potential. (a) (c) (e) (g) (i) (b) (d) (f) (h) (j) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The diagrammatic representations of the Aslamazov-Larkin (a)–(d), Maki-Thompson (e, f) and density of states (g)–(j) terms with the soft modes of the QCD-CP. The single, double, and wavy lines are quarks, soft modes, and photon, respectively. i.e., the MT and DOS terms cancel out exactly in ImΠRij(k, ω) [59]. Since the electric conductivity and the DPR depend only on ImΠRij(k, ω) as in Eqs. (5.44) and (5.4… view at source ↗
Figure 10
Figure 10. Figure 10: The upper panels: Electric conductivity σ near the 2SC-PT for several values of µ and GD. The thick-red and thin-blue lines are the results of the LE and TDGL approximations, respectively. In the left panels, the lines are plotted at µ = 350, 400, and 500 MeV with fixed GD/GS = 0.7, while the right panels show the results at GD/GS = 0.70, 0.65, and 0.60 for µ = 350 MeV. The dotted lines indicate the criti… view at source ↗
Figure 11
Figure 11. Figure 11: Contour maps of σ/T on the T–µ plane around the CP with GD/GS = 0.70, 0.65 and 0.60. The solid and dashed lines are the first-order and second-order phase transitions, respectively. their difference grows as ϵ becomes larger. The figure confirms that σ/T is insensitive to µ and GD, in accordance with the analytical results in Eq. (5.68). In the lower panel of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Dilepton production rates per unit energy ω and momentum k above Tc of the 2SC at µ = 350 MeV (left) [26] and of the QCD-CP at µ = µCP (right) [27] with GD = 0.7GS. The thick (thin) lines are the contribution of the soft modes (the massless free quark gases). 0 100 200 300 400 M [MeV] 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 d ¡ = d M 2 [G e V ¡ 2 fm ¡ 4 ] GC = 0:7GS; ¹ = 350 [MeV] fluc (T = 1:01Tc) fl… view at source ↗
Figure 13
Figure 13. Figure 13: Dilepton production rates per unit energy ω and momentum k above Tc of the 2SC at µ = 350 MeV (left) [26] and of the QCD-CP at µ = µCP (right) [27] with GD = 0.7GS. The thick (thin) lines are the contribution of the soft modes (the massless free quark gases) [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

We give a systematic account of the soft mode dynamics of QCD critical point and the two-flavor color-superconductivity based on the 2-flavor Nambu--Jona-Lasinio model, and investigate their effects on electromagnetic observables in relativistic heavy-ion collisions (HIC). We first demonstrate that the collective excitations coupled to the fluctuations of the respective order parameters are the soft modes associated with the phase transitions, in the sense that they acquire a prominent spectral strength in the low-energy and low-momentum region near the phase transitions, and the peak energy goes down, i.e., gets softened, and eventually vanishes at the critical point. It is shown that the diquark soft mode of the 2SC gives rise to the pseudogap, i.e., a depression in the density of states of the quark spectra around the Fermi surface above but in the vicinity of the critical temperature. Then, exploiting the ideas that were developed in condensed matter physics for describing the `para-conductivity' in the normal phase of metal superconductors, we show that the soft modes cause an anomalous enhancement of electric conductivity and the dilepton production rate, and discuss their relevance to HIC.

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

1 major / 1 minor

Summary. The manuscript provides a systematic analysis within the two-flavor Nambu-Jona-Lasinio model of soft modes associated with the QCD critical point and the two-flavor color-superconducting (2SC) transition. It demonstrates that collective excitations coupled to order-parameter fluctuations become soft near the transitions, with the diquark soft mode inducing a pseudogap (depression in the density of states around the Fermi surface) above but near Tc. The authors then invoke para-conductivity concepts from condensed-matter physics to argue that these soft modes produce anomalous enhancements in electric conductivity and dilepton production rates, with potential implications for electromagnetic observables in heavy-ion collisions.

Significance. If the results hold, the work could offer a useful theoretical link between soft-mode dynamics near QCD phase transitions and electromagnetic probes in heavy-ion collisions. The NJL-based demonstration of soft modes and the resulting pseudogap constitutes a clear strength, as does the attempt to connect these to HIC-relevant quantities. However, the reliance on direct transfer of condensed-matter formulas without an explicit model calculation of the relevant correlators limits the robustness and immediate applicability of the electromagnetic claims.

major comments (1)
  1. [Section on electromagnetic observables (following the pseudogap discussion)] The central claim of anomalous enhancement in electric conductivity and dilepton production rate rests on invoking para-conductivity ideas from condensed matter without an explicit NJL-model computation of the retarded current-current correlator (Kubo formula) or photon self-energy that includes the soft diquark mode. In the NJL framework the electromagnetic current is carried by quarks whose propagators receive self-energy insertions from the diquark channel; the low-energy enhancement therefore depends on the relativistic dispersion relation, the structure of the four-fermion vertex, and possible vertex corrections. Absent a one-loop or ladder resummation that reproduces or modifies the condensed-matter result, the claimed enhancements for HIC observables are not internally secured by the model calculation itself. This issue is load-bearing for the paper's main phenomenological claims.
minor comments (1)
  1. [Abstract] The abstract states that the authors 'exploit the ideas' from condensed matter but does not clarify that this is an analogy rather than a direct derivation within the NJL model; a brief qualifying phrase would improve clarity for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major concern point by point below.

read point-by-point responses

  1. Referee: The central claim of anomalous enhancement in electric conductivity and dilepton production rate rests on invoking para-conductivity ideas from condensed matter without an explicit NJL-model computation of the retarded current-current correlator (Kubo formula) or photon self-energy that includes the soft diquark mode. In the NJL framework the electromagnetic current is carried by quarks whose propagators receive self-energy insertions from the diquark channel; the low-energy enhancement therefore depends on the relativistic dispersion relation, the structure of the four-fermion vertex, and possible vertex corrections. Absent a one-loop or ladder resummation that reproduces or modifies the condensed-matter result, the claimed enhancements for HIC observables are not internally secured by the model calculation itself. This issue is load-bearing for the paper's main phenomenological claims.

    Authors: We agree that an explicit computation of the current-current correlator (via the Kubo formula) or photon self-energy within the NJL model, including self-energy insertions from the soft diquark mode and possible vertex corrections, would provide stronger internal support for the phenomenological claims. Our manuscript identifies the soft modes and pseudogap directly from the NJL model but then invokes the para-conductivity framework from condensed-matter literature as an established way to connect soft-mode fluctuations to transport enhancements. While the underlying soft-mode physics is analogous, we acknowledge that differences arising from the relativistic quark dispersion, the four-fermion interaction structure, and the need for consistent resummation mean the condensed-matter formulas cannot be transferred without further justification. We will revise the manuscript to (i) clarify these limitations in the relevant section, (ii) outline how a one-loop or ladder calculation could be set up in the NJL model, and (iii) emphasize that the reported enhancements are indicative rather than quantitatively definitive. This constitutes a partial revision that addresses the referee's concern without performing the full new calculation in the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity: soft-mode and pseudogap results derived internally in NJL; EM enhancements via external condensed-matter analogy

full rationale

The paper first demonstrates soft modes and the diquark-induced pseudogap directly within the 2-flavor NJL model via collective excitations coupled to order-parameter fluctuations. The subsequent claim of anomalous enhancement in conductivity and dilepton rate is obtained by exploiting established para-conductivity ideas from condensed-matter physics rather than by any self-referential reduction, fitted-input renaming, or load-bearing self-citation chain. NJL vacuum-parameter fitting is a standard external input and does not force the finite-temperature/density predictions for HIC observables. No quoted step reduces Eq. X to Eq. Y by construction or imports uniqueness from the authors' own prior work.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the NJL model whose parameters are fitted to phenomenology and on the assumption that soft-mode analysis transfers directly from condensed matter; no new entities are postulated.

free parameters (1)
  • NJL coupling strength and cutoff
    Standard NJL parameters adjusted to reproduce vacuum quantities such as pion mass and decay constant; these enter all spectral calculations.
axioms (2)
  • domain assumption The Nambu-Jona-Lasinio model with four-fermion interactions sufficiently approximates low-energy QCD dynamics near the critical point and 2SC phase.
    Invoked as the calculational framework for order-parameter fluctuations and soft modes.
  • domain assumption Para-conductivity concepts developed for metal superconductors apply quantitatively to quark matter soft modes.
    Used to connect soft-mode spectral functions to enhanced conductivity and dilepton rates.

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Works this paper leans on

63 extracted references · 63 canonical work pages · 26 internal anchors

  1. [1]

    Introduction One of the central problems in the modern physics is to reveal the properties of hot and dense matter as realized in the cores of compact stars or in the early universe. Since the extremely hot and dense matter should be described in terms of quarks and gluons governed by quantum chromody- namics(QCD), such an effort can be tantamount to deve...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    [35], L = ¯ψi(/∂ − m)ψ + GS[( ¯ψψ)2 + ( ¯ψiγ5⃗τψ)2] + GD( ¯ψiγ5τ2λAψC)( ¯ψCiγ5τ2λAψ), (2.1) where ψ(x) is the quark field and ψC(x) = iγ2γ0 ¯ψT(x) denotes its charge conjugation

    Model Lagrangian and phase diagram To explore the effects of critical fluctuations in dense quark matter, we employ a simple 2-flavor NJL model with a current quark mass, as was done in Ref. [35], L = ¯ψi(/∂ − m)ψ + GS[( ¯ψψ)2 + ( ¯ψiγ5⃗τψ)2] + GD( ¯ψiγ5τ2λAψC)( ¯ψCiγ5τ2λAψ), (2.1) where ψ(x) is the quark field and ψC(x) = iγ2γ0 ¯ψT(x) denotes its charge ...

    work page

  3. [3]

    We show that these fields exhibit collective excitations with prominent peaks of the strength function near these phase transitions, respectively [16,17,24,25,35]

    Collective diquark/particle-hole excitations as the soft modes of the phase transitions In this section, we discuss the dynamical properties of fluctuations of ∆ and M near the 2SC-PT and QCD-CP , respectively, based on the linear response theory. We show that these fields exhibit collective excitations with prominent peaks of the strength function near t...

    work page

  4. [4]

    The emergence of the soft modes can in turn naturally modify properties of various physical observables near the transition 10 of 22 T T

    Emergence of pseudogap in quark excitation spectra In the previous sections, we have seen that the fluctuations of ∆ and M are enhanced near the 2SC-PT and QCD-CP , respectively, and well-developed collective modes, the soft modes, are formed, which become massless at the critical points of the second-order nature. The emergence of the soft modes can in t...

    work page

  5. [5]

    These quantities are derived from the retarded photon self-energy ΠRµν (k, ω) = Z d4xeiωt−ik·x⟨[jµ (x, t), jν(0, 0)]⟩θ(t), (5.43) with the electric current operator jµ (x, t)

    Electric conductivity and dilepton production rates In this section, we explore the effects of the soft modes on the electric conductivity and dilepton production rates (DPR) near the 2SC-PT and QCD-CP . These quantities are derived from the retarded photon self-energy ΠRµν (k, ω) = Z d4xeiωt−ik·x⟨[jµ (x, t), jν(0, 0)]⟩θ(t), (5.43) with the electric curre...

    work page

  6. [6]

    Brief summary and concluding remark In this article, we have made a unified account of the soft modes of QCD-CP and 2SC-CP and their relevance to HIC, based on the 2-flavor Nambu-Jona-Lasinio (NJL) model. We started by discussing not only static but also dynamical fluctuations of physical quantities coupled to the order parameters of both the second-order...

    work page

  7. [7]

    The phase diagram of dense QCD

    Fukushima, K.; Hatsuda, T. The phase diagram of dense QCD. Rept. Prog. Phys. 2011, 74, 014001, [arXiv:hep-ph/1005.4814]. https://doi.org/10.1088/0034-4885/74/1/014001

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0034-4885/74/1/014001 2011

  8. [8]

    Color superconductivity in dense quark matter

    Alford, M.G.; Schmitt, A.; Rajagopal, K.; Schäfer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys. 2008, 80, 1455–1515, [arXiv:hep-ph/0709.4635]. https://doi.org/10.1103/RevModPhys.80.1455

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/revmodphys.80.1455 2008

  9. [9]

    Chiral Restoration at Finite Density and Temperature

    Asakawa, M.; Yazaki, K. Chiral Restoration at Finite Density and Temperature. Nucl. Phys. A 1989, 504, 668–684. https://doi.org/10.1016/0375-9474(89)90002-X

    work page doi:10.1016/0375-9474(89)90002-x 1989

  10. [10]

    Chiral Symmetry Breaking in QCD at Finite Temperature and Density

    Barducci, A.; Casalbuoni, R.; De Curtis, S.; Gatto, R.; Pettini, G. Chiral Symmetry Breaking in QCD at Finite Temperature and Density. Phys. Lett. B 1989, 231, 463–470. https://doi.org/10.1016/0370-2693(89)90695-3

    work page doi:10.1016/0370-2693(89)90695-3 1989

  11. [12]

    Quark number susceptibility and fluctuations in the vector channel at high temperatures

    Kunihiro, T. Quark number susceptibility and fluctuations in the vector channel at high temperatures. Phys. Lett. B 1991, 271, 395–402. https://doi.org/10.1016/0370-2693(91)90107-2

    work page doi:10.1016/0370-2693(91)90107-2 1991

  12. [13]

    Quark-gluon plasma: From big bang to little bang ; Vol

    Yagi, K.; Hatsuda, T.; Miake, Y. Quark-gluon plasma: From big bang to little bang ; Vol. 23, 2005

    work page 2005

  13. [14]

    Future facilities for high µ B physics

    Galatyuk, T. Future facilities for high µ B physics. Nucl. Phys. A 2019, 982, 163–169. https://doi.org/10.101 6/j.nuclphysa.2018.11.025

    work page 2019

  14. [15]

    Mapping the Phases of Quantum Chromodynamics with Beam Energy Scan

    Bzdak, A.; Esumi, S.; Koch, V .; Liao, J.; Stephanov, M.; Xu, N. Mapping the Phases of Quantum Chromodynamics with Beam Energy Scan. Phys. Rept. 2020, 853, 1–87, [arXiv:nucl-th/1906.00936]. https://doi.org/10.1016/j.physrep.2020.01.005

    work page doi:10.1016/j.physrep.2020.01.005 2020

  15. [16]

    Abdallah, M.; et al. Cumulants and correlation functions of net-proton, proton, and antiproton multiplicity distributions in Au+Au collisions at energies available at the BNL Relativistic Heavy Ion Collider. Phys. Rev. C 2021, 104, 024902, [arXiv:nucl-ex/2101.12413]. https://doi.org/10.1103/PhysRevC.104.024902

    work page doi:10.1103/physrevc.104.024902 2021

  16. [17]

    Signatures of the Tricritical Point in QCD

    Stephanov, M.A.; Rajagopal, K.; Shuryak, E.V . Signatures of the tricritical point in QCD.Phys. Rev. Lett. 1998, 81, 4816–4819, [hep-ph/9806219]. https://doi.org/10.1103/PhysRevLett.81.4816

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.81.4816 1998

  17. [18]

    A Theory of highly condensed matter

    Walecka, J.D. A Theory of highly condensed matter. Annals Phys. 1974, 83, 491–529. https://doi.org/10.101 6/0003-4916(74)90208-5

    work page 1974

  18. [19]

    Fermi Liquid Properties of Nuclear Matter in a Relativistic Mean - Field Theory

    Matsui, T. Fermi Liquid Properties of Nuclear Matter in a Relativistic Mean - Field Theory. Nucl. Phys. A 1981, 370, 365–388. https://doi.org/10.1016/0375-9474(81)90103-2

    work page doi:10.1016/0375-9474(81)90103-2 1981

  19. [20]

    Fluctuations of conserved charges in relativistic heavy ion collisions: An introduction

    Asakawa, M.; Kitazawa, M. Fluctuations of conserved charges in relativistic heavy ion collisions: An introduction. Prog. Part. Nucl. Phys. 2016, 90, 299–342, [arXiv:nucl-th/1512.05038]. https://doi.org/10.1016/ j.ppnp.2016.04.002

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  20. [21]

    Dynamics of critical fluctuations: Theory – phenomenology – heavy-ion collisions

    Bluhm, M.; et al. Dynamics of critical fluctuations: Theory – phenomenology – heavy-ion collisions. Nucl. Phys. A 2020, 1003, 122016, [arXiv:nucl-th/2001.08831]. https://doi.org/10.1016/j.nuclphysa.2020.122016

    work page doi:10.1016/j.nuclphysa.2020.122016 2020

  21. [22]

    Scalar density fluctuation at critical end point in NJL model

    Fujii, H. Scalar density fluctuation at critical end point in NJL model. Phys. Rev. D 2003, 67, 094018, [hep-ph/0302167]. https://doi.org/10.1103/PhysRevD.67.094018

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.67.094018 2003

  22. [23]

    Sigma and hydrodynamic modes along the critical line

    Fujii, H.; Ohtani, M. Sigma and hydrodynamic modes along the critical line. Phys. Rev. D 2004, 70, 014016, [hep-ph/0402263]. https://doi.org/10.1103/PhysRevD.70.014016

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.70.014016 2004

  23. [24]

    Functional renormalization group analysis of the soft mode at the QCD critical point

    Yokota, T.; Kunihiro, T.; Morita, K. Functional renormalization group analysis of the soft mode at the QCD critical point. PTEP 2016, 2016, 073D01, [arXiv:hep-ph/1603.02147]. https://doi.org/10.1093/ptep/ptw062

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptw062 2016

  24. [25]

    Tachyonic instability of the scalar mode prior to QCD critical point based on Functional renormalization-group method

    Yokota, T.; Kunihiro, T.; Morita, K. Tachyonic instability of the scalar mode prior to the QCD critical point based on the functional renormalization-group method in the two-flavor case. Phys. Rev. D 2017, 96, 074028, [arXiv:hep-ph/1707.05520]. https://doi.org/10.1103/PhysRevD.96.074028

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.96.074028 2017

  25. [26]

    Thermal fluctuations of gauge fields and first order phase transitions in color superconductivity

    Matsuura, T.; Iida, K.; Hatsuda, T.; Baym, G. Thermal fluctuations of gauge fields and first order phase transitions in color superconductivity. Phys. Rev. D 2004, 69, 074012, [hep-ph/0312042]. https://doi.org/10 .1103/PhysRevD.69.074012

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  26. [27]

    Gauge Field Fluctuations and First-Order Phase Transition in Color Superconductivity

    Giannakis, I.; Hou, D.f.; Ren, H.c.; Rischke, D.H. Gauge field fluctuations and first-order phase transition in color superconductivity. Phys. Rev. Lett. 2004, 93, 232301, [hep-ph/0406031]. https://doi.org/10.1103/ PhysRevLett.93.232301

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  27. [28]

    Absence of the London limit for the first-order phase transition to a color superconductor

    Noronha, J.L.; Ren, H.c.; Giannakis, I.; Hou, D.; Rischke, D.H. Absence of the London limit for the first-order phase transition to a color superconductor. Phys. Rev. D 2006, 73, 094009, [hep-ph/0602218]. https://doi.org/10.1103/PhysRevD.73.094009

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.73.094009 2006

  28. [29]

    Functional renormalization group approach to color superconducting phase transition

    Fej˝ os, G.; Yamamoto, N. Functional renormalization group approach to color superconducting phase transition. JHEP 2019, 12, 069, [arXiv:hep-ph/1908.03535]. https://doi.org/10.1007/JHEP12(2019)069

    work page doi:10.1007/jhep12(2019)069 2019

  29. [30]

    Precursor of Color Superconductivity in Hot Quark Matter

    Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y. Precursor of color superconductivity in hot quark matter. Phys. Rev. D 2002, 65, 091504, [nucl-th/0111022]. https://doi.org/10.1103/PhysRevD.65.091504

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.65.091504 2002

  30. [32]

    Anomalous enhancement of dilepton production as a precursor of color superconductivity

    Nishimura, T.; Kitazawa, M.; Kunihiro, T. Anomalous enhancement of dilepton production as a precursor of color superconductivity. PTEP 2022, 2022, 093D02, [arXiv:hep-ph/2201.01963]. https://doi.org/10.1093/ ptep/ptac100

    work page arXiv 2022

  31. [33]

    Enhancement of dilepton production rate and electric conductivity around the QCD critical point

    Nishimura, T.; Kitazawa, M.; Kunihiro, T. Enhancement of dilepton production rate and electric conductivity around the QCD critical point. PTEP 2023, 2023, 053D01, [arXiv:hep-ph/2302.03191]. https://doi.org/10.1 093/ptep/ptad051

    work page arXiv 2023

  32. [34]

    Dynamical Model of Elemen tary Particles Based on an Analogy with Superconductivity. 1.,

    Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Supercon- ductivity. 1. Phys. Rev. 1961, 122, 345–358. https://doi.org/10.1103/PhysRev.122.345. 21 of 22

    work page doi:10.1103/physrev.122.345 1961

  33. [35]

    Dynamical Model Of Elemen tary Particles Based On An Analogy With Superconductivity. Ii,

    Nambu, Y.; Jona-Lasinio, G. Dynamical model of elementary particles based on an analogy with supercon- ductivity. II. Phys. Rev. 1961, 124, 246–254. https://doi.org/10.1103/PhysRev.124.246

    work page doi:10.1103/physrev.124.246 1961

  34. [36]

    The Nambu and Jona Lasinio model: Its implications for hadrons and nuclei.Prog

    Vogl, U.; Weise, W. The Nambu and Jona Lasinio model: Its implications for hadrons and nuclei. Prog. Part. Nucl. Phys. 1991, 27, 195–272. https://doi.org/10.1016/0146-6410(91)90005-9

    work page doi:10.1016/0146-6410(91)90005-9 1991

  35. [37]

    The Nambu-Jona-Lasinio model of quantum chromodynamics.Rev

    Klevansky, S.P . The Nambu-Jona-Lasinio model of quantum chromodynamics. Rev. Mod. Phys. 1992, 64, 649–708. https://doi.org/10.1103/RevModPhys.64.649

    work page doi:10.1103/revmodphys.64.649 1992

  36. [38]

    QCD Phenomenology based on a Chiral Effective Lagrangian

    Hatsuda, T.; Kunihiro, T. QCD phenomenology based on a chiral effective Lagrangian. Phys. Rept. 1994, 247, 221–367, [hep-ph/9401310]. https://doi.org/10.1016/0370-1573(94)90022-1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0370-1573(94)90022-1 1994

  37. [39]

    Effective hadron theory of QCD.Prog

    Ebert, D.; Reinhardt, H.; Volkov, M.K. Effective hadron theory of QCD.Prog. Part. Nucl. Phys. 1994, 33, 1–120. https://doi.org/10.1016/0146-6410(94)90043-4

    work page doi:10.1016/0146-6410(94)90043-4 1994

  38. [40]

    NJL model analysis of quark matter at large density

    Buballa, M. NJL model analysis of quark matter at large density. Phys. Rept. 2005, 407, 205–376, [hep- ph/0402234]. https://doi.org/10.1016/j.physrep.2004.11.004

    work page doi:10.1016/j.physrep.2004.11.004 2005

  39. [41]

    Electromagnetic response of dense quark matter around color- superconducting phase transition and QCD critical point

    Nishimura, T.; Kitazawa, M.; Kunihiro, T. Electromagnetic response of dense quark matter around color- superconducting phase transition and QCD critical point. Annals Phys. 2024, 469, 169768, [arXiv:hep- ph/2405.09240]. https://doi.org/10.1016/j.aop.2024.169768

    work page doi:10.1016/j.aop.2024.169768 2024

  40. [42]

    Soviet Solid State 10, 875 (1968)

    Aslamazov, L.; Larkin, A. Soviet Solid State 10, 875 (1968). Phys. Lett. A 1968, 26, 238

    work page 1968

  41. [43]

    Critical fluctuation of the order parameter in a superconductor

    Maki, K. Critical fluctuation of the order parameter in a superconductor. I. Progress of Theoretical Physics 1968, 40, 193–200

    work page 1968

  42. [44]

    Microwave, flux flow, and fluctuation resistance of dirty type-II superconductors

    Thompson, R.S. Microwave, flux flow, and fluctuation resistance of dirty type-II superconductors. Physical Review B 1970, 1, 327

    work page 1970

  43. [45]

    Fluctuation phenomena in superconductors ; Springer, 2008; pp

    Larkin, A.; Varlamov, A. Fluctuation phenomena in superconductors ; Springer, 2008; pp. 369–458

    work page 2008

  44. [46]

    Introduction to superconductivity; Courier Corporation, 2004

    Tinkham, M. Introduction to superconductivity; Courier Corporation, 2004

    work page 2004

  45. [47]

    Transport coefficients in high temperature gauge theories: (I) Leading-log results

    Arnold, P .B.; Moore, G.D.; Yaffe, L.G. Transport coefficients in high temperature gauge theories. 1. Leading log results. JHEP 2000, 11, 001, [hep-ph/0010177]. https://doi.org/10.1088/1126-6708/2000/11/001

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2000/11/001 2000

  46. [48]

    Transport coefficients in high temperature gauge theories: (II) Beyond leading log

    Arnold, P .B.; Moore, G.D.; Yaffe, L.G. Transport coefficients in high temperature gauge theories. 2. Beyond leading log. JHEP 2003, 05, 051, [hep-ph/0302165]. https://doi.org/10.1088/1126-6708/2003/05/051

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2003/05/051 2003

  47. [49]

    Finite Temperature Spectral Densities of Momentum and R-Charge Correlators in $\N=4$ Yang Mills Theory

    Teaney, D. Finite temperature spectral densities of momentum and R-charge correlators in N=4 Yang Mills theory. Phys. Rev. D 2006, 74, 045025, [hep-ph/0602044]. https://doi.org/10.1103/PhysRevD.74.045025

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.74.045025 2006

  48. [50]

    Electrical Conductivity of Hot QCD Matter

    Cassing, W.; Linnyk, O.; Steinert, T.; Ozvenchuk, V . Electrical Conductivity of Hot QCD Matter.Phys. Rev. Lett. 2013, 110, 182301, [arXiv:hep-ph/1302.0906]. https://doi.org/10.1103/PhysRevLett.110.182301

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.110.182301 2013

  49. [51]

    Electric conductivity of the quark-gluon plasma investigated using a perturbative QCD based parton cascade

    Greif, M.; Bouras, I.; Greiner, C.; Xu, Z. Electric conductivity of the quark-gluon plasma investigated using a perturbative QCD based parton cascade. Phys. Rev. D 2014, 90, 094014, [arXiv:nucl-th/1408.7049]. https://doi.org/10.1103/PhysRevD.90.094014

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.094014 2014

  50. [52]

    Electrical conductivity of the quark-gluon plasma: perspective from lattice QCD

    Aarts, G.; Nikolaev, A. Electrical conductivity of the quark-gluon plasma: perspective from lattice QCD. Eur. Phys. J. A 2021, 57, 118, [arXiv:hep-lat/2008.12326]. https://doi.org/10.1140/epja/s10050-021-00436-5

    work page doi:10.1140/epja/s10050-021-00436-5 2021

  51. [53]

    Spectral and Transport Properties from Lattice QCD

    Kaczmarek, O.; Shu, H.T. Spectral and Transport Properties from Lattice QCD. Lect. Notes Phys. 2022, 999, 307–345, [arXiv:hep-lat/2206.14676]. https://doi.org/10.1007/978-3-030-95491-8_8

    work page doi:10.1007/978-3-030-95491-8_8 2022

  52. [54]

    Estimation of electric conductivity of the quark gluon plasma via asymmetric heavy-ion collisions

    Hirono, Y.; Hongo, M.; Hirano, T. Estimation of electric conductivity of the quark gluon plasma via asymmetric heavy-ion collisions. Phys. Rev. C 2014, 90, 021903, [arXiv:nucl-th/1211.1114]. https://doi.org/ 10.1103/PhysRevC.90.021903

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevc.90.021903 2014

  53. [55]

    Relativistic resistive magneto-hydrodynamics code for high-energy heavy-ion collisions

    Nakamura, K.; Miyoshi, T.; Nonaka, C.; Takahashi, H.R. Relativistic resistive magneto-hydrodynamics code for high-energy heavy-ion collisions. Eur. Phys. J. C 2023, 83, 229, [arXiv:nucl-th/2211.02310]. https://doi.org/10.1140/epjc/s10052-023-11343-y

    work page doi:10.1140/epjc/s10052-023-11343-y 2023

  54. [56]

    BHAC-QGP: three-dimensional MHD simula- tions of relativistic heavy-ion collisions, II

    Mayer, M.; Rezzolla, L.; Elfner, H.; Inghirami, G.; Rischke, D.H. BHAC-QGP: three-dimensional MHD simula- tions of relativistic heavy-ion collisions, II. Application to Au-Au collisions 2024. [arXiv:hep-ph/2403.08669]

    work page arXiv 2024

  55. [57]

    Quantum theory of many-particle systems ; Courier Corporation, 2012

    Fetter, A.L.; Walecka, J.D. Quantum theory of many-particle systems ; Courier Corporation, 2012

    work page 2012

  56. [58]

    Perturbation theory in statistical mechanics and the theory of superconductivity

    Thouless, D.J. Perturbation theory in statistical mechanics and the theory of superconductivity. Annals of Physics 1960, 10, 553–588

    work page 1960

  57. [59]

    Pseudogap of Color Superconductivity in Heated Quark Matter

    Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y. Pseudogap of color superconductivity in heated quark matter. Phys. Rev. D 2004, 70, 056003, [hep-ph/0309026]. https://doi.org/10.1103/PhysRevD.70.056003

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.70.056003 2004

  58. [60]

    Emergence of soft quark excitations by the coupling with a soft mode of the QCD critical point

    Kitazawa, M.; Kunihiro, T.; Nemoto, Y. Emergence of soft quark excitations by the coupling with a soft mode of the QCD critical point. Phys. Rev. D 2014, 90, 116008, [arXiv:hep-ph/1409.3733]. https: //doi.org/10.1103/PhysRevD.90.116008. 22 of 22

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.116008 2014

  59. [61]

    Spectral properties of massless and massive quarks coupled with massive boson at finite temperature

    Kitazawa, M.; Kunihiro, T.; Mitsutani, K.; Nemoto, Y. Spectral properties of massless and massive quarks coupled with massive boson at finite temperature. Phys. Rev. D 2008, 77, 045034, [arXiv:hep-ph/0710.5809]. https://doi.org/10.1103/PhysRevD.77.045034

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.77.045034 2008

  60. [62]

    Novel Collective Excitations and the Quasi-Particle Picture of Quarks Coupled with a Massive Boson at Finite Temperature

    Kitazawa, M.; Kunihiro, T.; Nemoto, Y. Novel Collective Excitations and Quasi-particle Picture of Quarks Coupled with a Massive Boson at Finite Temperature.Prog. Theor. Phys. 2007, 117, 103–138, [hep-ph/0609164]. https://doi.org/10.1143/PTP .117.103

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1143/ptp 2007

  61. [63]

    Quark Spectrum above but near Critical Temperature of Chiral Transition

    Kitazawa, M.; Kunihiro, T.; Nemoto, Y. Quark spectrum above but near critical temperature of chiral transition. Phys. Lett. B 2006, 633, 269–274, [hep-ph/0510167]. https://doi.org/10.1016/j.physletb.2005.11. 076

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2005.11 2006

  62. [64]

    Possible generation of anomalously soft quark excitations at nonzero temperature: Nonhyperbolic dispersion of parapion and van Hove singularity

    Kitazawa, M.; Kunihiro, T.; Nemoto, Y. Possible generation of anomalously soft quark excitations at nonzero temperature: Nonhyperbolic dispersion of the parapion and van Hove singularity. Phys. Rev. D 2014, 89, 056002, [arXiv:hep-ph/1312.3022]. https://doi.org/10.1103/PhysRevD.89.056002

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.89.056002 2014

  63. [65]

    Enhanced Dilepton production near the color superconducting phase and the QCD critical point 2023

    Nishimura, T.; Nara, Y.; Steinheimer, J. Enhanced Dilepton production near the color superconducting phase and the QCD critical point 2023. [arXiv:hep-ph/2311.14135]. Disclaimer/Publisher’s Note:The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(...

    work page arXiv 2023