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arxiv: 2604.17764 · v2 · submitted 2026-04-20 · ✦ hep-ph · nucl-th

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

Masakiyo Kitazawa ad Teiji Kunihiro
Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:06 UTC · model grok-4.3

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

Soft modes near the QCD critical point and color superconductivity produce a pseudogap and anomalously enhance electric conductivity and dilepton production in heavy-ion collisions.

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

This paper employs the two-flavor Nambu-Jona-Lasinio model to analyze collective excitations tied to the chiral and diquark order parameters at the QCD critical point and the two-flavor color-superconducting transition. It shows that these excitations qualify as soft modes because they develop strong low-momentum spectral weight above the critical temperatures and their peak positions move downward until they vanish at the critical points. The diquark soft mode creates a pseudogap by depressing the quark density of states near the Fermi surface. Adapting the para-conductivity concept from ordinary superconductors, the authors demonstrate that these soft modes drive measurable increases in electric conductivity and the rate of dilepton production, effects they link directly to observables in relativistic heavy-ion collisions.

Core claim

Collective excitations coupled to fluctuations of the respective order parameters qualify as soft modes for both the QCD critical point and the two-flavor color-superconducting transition: they acquire prominent low-energy, low-momentum spectral strength above the critical temperatures, with their peak energies softening and eventually vanishing at the critical points. The diquark soft mode induces a pseudogap, a depression in the density of states of the quark spectra around the Fermi surface above but near the critical temperature. Applying ideas developed for para-conductivity in the normal phase of metal superconductors, the soft modes produce an anomalous enhancement of electric conduct

What carries the argument

The soft modes, i.e., collective excitations that gain prominent spectral strength in the low-energy and low-momentum region above the critical temperatures and whose dispersion softens to zero at the critical points.

If this is right

  • The diquark soft mode above the critical temperature for two-flavor color superconductivity creates a pseudogap by depressing the quark density of states around the Fermi surface.
  • Soft modes produce an anomalous increase in electric conductivity in the normal phase above both the QCD critical point and the color-superconducting transition.
  • The same soft modes generate an anomalous enhancement of the dilepton production rate above the critical temperatures.
  • These electromagnetic enhancements constitute observable signals that could be used to locate the phase transitions in relativistic heavy-ion collisions.

Where Pith is reading between the lines

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

  • Confirmation would suggest that dilepton spectra at low invariant mass offer an independent experimental handle on the location of the QCD critical point.
  • The direct mapping of condensed-matter conductivity formulas onto QCD matter implies that similar soft-mode effects may appear in other transport coefficients near the critical endpoint.
  • If the enhancements survive more realistic hydrodynamic evolution, they could alter the interpretation of existing dilepton data from RHIC and LHC runs at intermediate beam energies.
  • The work opens the possibility that conductivity measurements or related observables in neutron-star mergers might also carry imprints of color-superconducting soft modes.

Load-bearing premise

The para-conductivity framework developed for metal superconductors applies directly to the quark matter case in the NJL model, and the resulting enhancements are both sizable and detectable in realistic heavy-ion collision environments.

What would settle it

Absence of the predicted enhancement in the low-mass dilepton spectrum or in proxies for electric conductivity when heavy-ion collisions are tuned to the beam energies and centralities expected to pass near the QCD critical point.

Figures

Figures reproduced from arXiv: 2604.17764 by Masakiyo Kitazawa ad 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(QCD-CP) and the two-flavor color-superconductivity(2SC-CP) 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 to the respective phase transitions, in the sense that they acquire a prominent spectral strength in the low-energy and low-momentum region above the respective critical temperatures, and the peak energy of the respective spectral functions goes down, i.e., gets softened, and eventually vanishes at the 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

2 major / 3 minor

Summary. The paper uses the two-flavor Nambu-Jona-Lasinio model to analyze soft mode dynamics near the QCD critical point and the 2SC phase transition. It identifies collective excitations as soft modes that acquire low-energy spectral strength and soften toward the critical points, demonstrates a pseudogap in the quark density of states induced by diquark fluctuations above Tc, and adapts para-conductivity ideas from condensed-matter superconductors to argue for anomalous enhancements in electric conductivity and dilepton production rates, with discussion of relevance to heavy-ion collisions.

Significance. If the adaptation of the para-conductivity construction is shown to be valid in the relativistic NJL setting, the results could provide a concrete mechanism linking soft-mode fluctuations to observable electromagnetic signals in HIC, aiding searches for the QCD critical point and color superconductivity. The systematic treatment of soft modes for both transitions and the explicit link to the pseudogap constitute a strength of the model-based approach.

major comments (2)
  1. [§ on electric conductivity and dilepton rates] § on electric conductivity and dilepton rates (the Kubo-formula application of para-conductivity): the central claim of anomalous enhancement rests on importing the Aslamazov-Larkin-type fluctuating-pair contributions without an explicit derivation confirming that the electromagnetic current vertices for colored quarks in the relativistic NJL model and the soft-mode damping (from quark loops) produce the same structure and magnitude as in non-relativistic BCS theory. This is load-bearing for the quantitative relevance to HIC.
  2. [§ on pseudogap from diquark soft mode] § on pseudogap from diquark soft mode: while the depression in the density of states around the Fermi surface is shown qualitatively, the absence of quantitative measures (e.g., depth relative to Tc or width in momentum) leaves open whether the effect is sizable enough to influence dilepton or conductivity observables in realistic collision environments.
minor comments (3)
  1. [Abstract] The abstract contains a typographical repetition ('at the the critical point'); this should be corrected.
  2. [References] Ensure that the original references for the para-conductivity formalism (e.g., Aslamazov-Larkin) and any prior NJL studies of soft modes are explicitly cited to clarify the incremental contribution.
  3. [Figures] If spectral-function or conductivity plots are present, confirm that axes, units, and the location of the softening peak are clearly labeled for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major points below and indicate the revisions that will be incorporated to strengthen the presentation and quantitative support for our claims.

read point-by-point responses

  1. Referee: § on electric conductivity and dilepton rates (the Kubo-formula application of para-conductivity): the central claim of anomalous enhancement rests on importing the Aslamazov-Larkin-type fluctuating-pair contributions without an explicit derivation confirming that the electromagnetic current vertices for colored quarks in the relativistic NJL model and the soft-mode damping (from quark loops) produce the same structure and magnitude as in non-relativistic BCS theory. This is load-bearing for the quantitative relevance to HIC.

    Authors: We acknowledge that the adaptation of the para-conductivity formalism requires a clearer justification within the relativistic NJL setting to underpin the quantitative relevance to heavy-ion collisions. The manuscript outlines the diagrammatic correspondence between the soft-mode propagators (obtained via RPA resummation) and the electromagnetic current correlator, but does not provide a full step-by-step derivation of the vertex factors for colored quarks. In the revised version we will add an explicit calculation of the leading Aslamazov-Larkin diagram, demonstrating that the current vertices retain the requisite structure and that the soft-mode damping generated by the quark-loop self-energy produces an enhancement of the same functional form as in the non-relativistic case. This addition will be placed in a dedicated subsection or appendix and will directly address the load-bearing nature of the claim. revision: yes

  2. Referee: § on pseudogap from diquark soft mode: while the depression in the density of states around the Fermi surface is shown qualitatively, the absence of quantitative measures (e.g., depth relative to Tc or width in momentum) leaves open whether the effect is sizable enough to influence dilepton or conductivity observables in realistic collision environments.

    Authors: The pseudogap is exhibited through the quark spectral function and the integrated density of states, which display a clear suppression near the Fermi surface for temperatures above but close to Tc. The original presentation emphasized the qualitative mechanism arising from diquark fluctuations. We agree that quantitative characterization is needed to assess phenomenological impact. In the revision we will extract and report specific measures from our existing numerical results, including the relative depth of the DOS depression at the Fermi momentum (DOS(T)/DOS_normal) for several temperatures above Tc and the momentum width of the depleted region. These numbers will be added to the relevant figures and discussed in relation to their possible influence on dilepton and conductivity observables. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard NJL effective model plus external condensed-matter formulas to compute observables.

full rationale

The paper's chain proceeds by (i) solving the NJL model to identify soft modes and pseudogap above Tc, then (ii) inserting the resulting spectral functions into the Kubo formula using the para-conductivity construction imported from condensed-matter literature. Neither step reduces to its own output by definition: the NJL parameters are fixed externally to vacuum phenomenology (standard practice, not a fit-then-predict loop on the target HIC quantities), and the conductivity/dilepton formulas are taken from non-relativistic BCS theory rather than derived from the present authors' prior results. No self-citation is invoked as a uniqueness theorem or to justify the vertex structure; the calculation is therefore self-contained against external benchmarks even if the applicability of the imported formalism remains a separate correctness question.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The analysis rests on the two-flavor NJL model (with its standard parameters) and on the assumption that soft-mode dynamics and para-conductivity concepts transfer from condensed matter to QCD without additional verification.

free parameters (1)
  • NJL model parameters (couplings, cutoff)
    Standard NJL parameters are fitted to vacuum meson masses and decay constants; their specific values control the location of the critical points and the strength of the soft-mode effects.
axioms (2)
  • domain assumption Collective excitations coupled to order-parameter fluctuations become soft modes that soften and vanish at the critical point
    Invoked to identify the diquark and chiral soft modes above Tc.
  • ad hoc to paper Para-conductivity framework from metal superconductors applies to quark matter
    Used to argue for anomalous enhancement of conductivity and dilepton rate.
invented entities (1)
  • diquark soft mode no independent evidence
    purpose: Produces the pseudogap in the quark density of states above Tc
    Introduced within the NJL calculation as the collective excitation tied to the 2SC order parameter.

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Reference graph

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