Two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality

Cheng-Ming Li; Di-Sheng Fan; Li-Kang Yang; Yong-Liang Ma

REVIEW 3 major objections 5 minor 1 cited by

Two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A Fierz-complete NJL analysis of charge-neutral two-flavor quark matter finds a gapless 2SC phase between the normal and gapped superconducting phases.

desk verdict Solid NJL extension with new dominant channels, but the gapless phase is flagged without stability check, so the phase sequence is not established. read the letter →

arxiv 2507.06676 v1 pith:URSGE46X submitted 2025-07-09 hep-ph nucl-th

Li-Kang Yang , Di-Sheng Fan , Cheng-Ming Li , Yong-Liang Ma This is my paper · ORCID

classification hep-phnucl-th

keywords two-flavorcolorsuperconductivitygapless2SCphaseNambu-Jona-LasiniomodelFierztransformationchargeneutralitydensequarkmatterdiagram

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks which interaction channels actually matter for the two-flavor color-superconducting (2SC) phase of dense, charge-neutral quark matter. Working with a general Nambu-Jona-Lasinio model whose many four-fermion couplings are related by Fierz transformations, it solves the gap equations under color and charge neutrality. It claims that, besides the familiar diquark and vector channels, the scalar-isovector, vector-isovector, and vector-isovector-color-octet channels are important, while the rest can be dropped. It also claims that a gapless 2SC phase sits between the normal quark phase and the gapped 2SC phase, with boundaries at $H/G_s^{(0)}\simeq 0.67$ and $0.81$, and a critical endpoint near $H/G_s^{(0)}=1.1$ and $\mu=312$ MeV. If right, neutron-star matter in this density window has a more complex phase sequence and particle composition than standard 2SC studies assume.

What carries the argument

The central machinery is the Fierz-complete form of the NJL Lagrangian, in which all four-fermion interaction channels are generated from a minimal set of couplings by Fierz transformation, reducing the parameter count and relating quark-antiquark condensates to diquark condensates. The argument then runs through a Hartree-Fock-Bogoliubov mean-field treatment of the thermodynamic potential, with constituent quark masses, effective chemical potentials, and 2SC gaps determined by the gap equations, and with the electron and color chemical potentials fixed by the neutrality conditions. The gapless criterion is a zero-energy quasiparticle condition, Eq. (17), which reduces to $\Delta \le |\delta\mu|$ in the limit where the other condensates vanish.

What would settle it

Compute the gluon Meissner mass matrix in the self-consistent gapless solutions of Eq. (14) inside the window $0.67 \lesssim H/G_s^{(0)} \lesssim 0.81$; one negative mass-squared eigenvalue would show the claimed gapless phase is not the stable ground state, and the claimed phase sequence would not hold.

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Extended reading notes

Core claim

The paper works with the most general Lorentz-invariant NJL Lagrangian consistent with the relevant symmetries, then uses Fierz transformations to show that its eight four-fermion couplings are determined by four independent ones, plus the determinant interaction. Solving the Hartree-Fock-Bogoliubov gap equations under color and charge neutrality in the one-gluon-exchange limit, it finds that the surviving condensates are the diquark gap $\Delta$, the vector density $n$, the scalar-isovector condensate $\phi_I$, and the vector-isovector densities $n_I$ and $n_{I8}$; the other channels contribute negligibly. In this solution the system passes, as the chemical potential rises, from a chirally broken normal phase through a gapless 2SC (g2SC) phase into a gapped 2SC phase, with the g2SC-to-2SC transition near $\mu\simeq 480$ MeV. In the $H/G_s^{(0)}$--$\mu$ plane the gapless window is $0.67 \lesssim H/G_s^{(0)} \lesssim 0.81$, and the chiral transition becomes a crossover above $H/G_s^{(0)}=1.1$, with a critical endpoint at $(H/G_s^{(0)},\mu)=(1.1,312\ \mathrm{MeV})$ and $\Delta=138$ MeV. The paper also reports flavor-color density fractions in the 2SC phase, approximately $n_{db}:n_{ur}:n_{dr}:n_{ub}:n_e \simeq 8:4.5:4.5:1:1$, and an up-down constituent mass splitting of up to 140 MeV driven by the scalar-isovector channel.

Load-bearing premise

The load-bearing premise is that the zero-energy quasiparticle condition defines a real phase, even though the known chromomagnetic instability of gapless 2SC matter may push the system into a crystalline, gluonic, or otherwise different state.

Editorial extensions

If this is right

Neutron-star matter at moderate densities will pass from the normal quark phase through a gapless 2SC phase before entering the gapped 2SC phase, so single-transition models miss a whole density interval.
Any NJL-based study of 2SC matter that keeps only the scalar and diquark channels will misplace phase boundaries: the vector channel shifts them to higher $\mu$, the scalar-isovector channel widens the gapless window, and the isovector channel lowers both boundaries.
The chiral transition in charge-neutral 2SC matter is a crossover rather than a first-order transition once the vector channel is included and $H/G_s^{(0)}>1.1$.
The approximate density fractions $n_{db}:n_{ur}:n_{dr}:n_{ub}:n_e \simeq 8:4.5:4.5:1:1$ in the 2SC phase give concrete input for transport and cooling models of neutron stars.

Reading between the lines

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

Editorial inference: If the chromomagnetic instability survives in the self-consistent gapless solutions, the true ground state in the claimed g2SC window may be a crystalline or gluonic phase, making the NQ-g2SC-2SC sequence a mean-field artifact rather than the physical transition chain.
Editorial inference: The Fierz-complete coupling reduction implies that equations of state computed with only scalar-plus-diquark NJL Lagrangians are not closed under Fierz transformations; comparing tidal deformability predictions with and without isovector channels would test whether these channels matter for neutron-star observables.
Editorial inference: The zero-energy quasiparticle criterion could be applied at finite temperature to map a gapless surface in the $(T,\mu,H/G_s^{(0)})$ space, and the stability of that surface against meson fluctuations is not addressed here.
Editorial inference: The predicted electron fraction $n_e/n \simeq 1/27$ in the 2SC phase is a testable compositional prediction; observations of neutron-star cooling or transport that require a specific charged-lepton content could indirectly discriminate between this 2SC picture and alternatives.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a workmanlike NJL study of neutral two-flavor color superconductivity that expands the interaction channel basis beyond the usual diquark and vector terms. The genuinely new part is the systematic inclusion of the scalar-isovector, vector-isovector, and vector-isovector-color-octet channels under both color and charge neutrality, and the claim that these channels materially shift the 2SC phase boundaries. The lower boundary of the gapless phase, H/G_s ≈ 0.67, agrees with the earlier Huang-Shovkovy result, which is a good sanity check. The paper also gives a useful breakdown of particle fractions.

The main soft spot is the treatment of the gapless 2SC phase. The authors identify it via the zero-energy quasiparticle criterion (Eq. 17) and draw the normal–g2SC–2SC sequence, but they themselves cite the known chromomagnetic instability of g2SC in the introduction and then do not check whether their solutions are affected. That is a load-bearing omission: if the gapless branch is unstable, the phase sequence they describe is not the ground-state sequence, and the quoted boundaries and CEP may describe an unstable branch. This needs to be addressed, either by computing Meissner masses in their model or by explicitly arguing why the instability is irrelevant to their mean-field phase diagram.

A secondary concern: Eq. (14) and the criterion (17) are asserted without derivation. The characteristic equation for the quasiparticle spectra is central to the paper, so a referee should ask for a derivation or at least a reference. Also, the novelty statement in the introduction overstates things a bit: the Fierz framework and the g2SC phase under neutrality are already in the literature, including Ref. [35]. The newness is the specific combination of channels, not the framework itself.

On the plus side, the formalism is standard and the numerics appear consistent. The paper does not provide code or data, which would have helped, but it is a self-contained effective-model study. The writing has some typos (“gaped,” “the emergency”) but nothing that obscures the logic.

This paper is for people working on NJL models of dense quark matter and neutron-star phenomenology. It is a moderate incremental contribution, not a breakthrough. It deserves a serious referee, but the referee should insist on the stability check and derivations before acceptance.

Referee Report

3 major / 5 minor

Summary. The paper studies two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality. Using Fierz identities, the authors reduce the number of independent couplings and construct a Fierz-complete set of interaction channels. Solving the gap equations in the one-gluon-exchange parametrization, they identify the scalar-isovector, vector-isovector, and vector-isovector-color-octet channels as important in addition to the diquark and vector channels. They predict a gapless 2SC (g2SC) phase between the normal quark phase and the gapped 2SC phase, with boundaries around H/G_s^(0) \approx 0.67 and 0.81, and a critical endpoint at H/G_s^(0)=1.1, mu=312 MeV. They also compute particle fractions in the gapped and gapless phases and discuss how individual channels affect the phase diagram.

Significance. The paper's Fierz-complete treatment is a useful systematic step beyond the usual diquark-plus-vector NJL studies, and its dense-matter results are parameter-free in the sense that couplings are determined by vacuum observables and Fierz relations rather than by fitting the phase diagram. The reproduction of the known lower g2SC boundary from Ref. [35] provides a nontrivial check. If the gapless region is actually the ground state, the predicted phase boundaries and particle fractions would be directly relevant to NJL-model studies of neutron-star matter. However, the central phase label rests on an unverified stability assumption, so the significance of the claimed phase sequence depends on the outcome of a stability analysis.

major comments (3)

[Sec. IV A, Eq. (17), and Figs. 8–13] The g2SC phase boundaries are located by the zero-energy quasiparticle criterion (17), which is an existence condition for a zero in one dispersion branch and is not a thermodynamic or dynamical stability criterion. The paper itself cites Refs. [38,39], which show that g2SC (and 2SC for |delta mu| > Delta/sqrt(2)) suffers from chromomagnetic instability in closely related NJL treatments. Since the central claim is that a g2SC region exists between the normal and gapped 2SC phases, the manuscript should compute the Meissner masses (or the full curvature matrix) for its solutions and compare the free energy of the g2SC branch with candidate competitor phases such as crystalline, solitonic, or gluonic phases. Without this check, the phase sequence in Figs. 8–13 describes a branch of solutions rather than an established phase.
[Eqs. (14) and (17)] The characteristic equation (14) and the g2SC criterion (17) are load-bearing for the definition and location of the g2SC phase, but they are asserted without derivation. The reader cannot verify that inequality (17) is the correct condition for a zero eigenvalue of (14) when delta M_r and Delta_0 are nonzero. Please provide a derivation or a precise reference that contains it.
[Sec. IV (general)] The paper does not state how competing self-consistent branches are compared when the normal, gapped 2SC, and gapless 2SC solutions coexist. Drawing the phase diagrams in Figs. 8–13 requires the global minimum of the thermodynamic potential Omega over all branches; the zero-mode boundaries alone do not locate first-order transitions or select the stable phase. Please clarify whether each point in these figures is chosen by free-energy minimization, and if so, show the relevant free-energy comparisons.

minor comments (5)

[Eq. (5)] In the expression for G3, the term 'C2' appears where 'C'_2' seems required for consistency with Eq. (3); please verify this relation.
[Throughout] The manuscript contains numerous language and typographical errors, including 'obgained' in Sec. II, 'In additiotn' in Sec. III, 'gaped' for 'gapped' in the abstract, and '[24?–26]' in the Introduction; a careful proofread is needed.
[Abstract and Sec. IV] The upper boundary of the g2SC region is quoted as 0.81 in the abstract and Sec. IV A, but as 0.8 in Sec. IV B ('0.67−0.8 to 0.67−0.85'); please use a consistent value or clarify that the boundary is chemical-potential dependent.
[Sec. III B] The statement that the effects of n_I8 'can be mimicked by n_I totally' is not self-evident; please show the combination of couplings that enters the gap equations and quantify the error made by neglecting n_I8 in the later analysis.
[Sec. III B, Eq. (25)] The approximate particle fractions in Eq. (26) are leading-order results obtained after dropping terms of order G mu^2 in Eq. (25); the text should state more explicitly that fractions such as n_e/n \approx 1/27 receive corrections and are not exact.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dense-matter phase diagram is computed from Fierz relations and vacuum-fitted parameters, not fitted to the target phase boundaries.

full rationale

The derivation chain is self-contained and non-circular. The Fierz transformations in Sec. II are exact mathematical identities that relate the many four-fermion couplings; they are not fitted to the phase diagram. The model parameters (cutoff Λ = 631 MeV, current mass m = 5.5 MeV, and scalar coupling G_s^(0) = 2.188/Λ^2) are fixed by vacuum observables (⟨¯uu⟩, f_π, m_π) and cited to Ref. [63], not to any dense-matter output. The one-gluon-exchange choice L_int = -g(ψbar γ^μ λ^a ψ)^2 fixes the relative channel couplings, including H/G_s^(0) = 3/4, as an input assumption, but the resulting condensates, the location of the g2SC region, the particle fractions, and the critical endpoint at H/G_s^(0) = 1.1, µ = 312 MeV are all self-consistent solutions of the gap equations (18)-(20) with the neutrality conditions (20), and none of these outputs is used to adjust the input constants. The g2SC phase is defined by the zero-energy quasiparticle criterion Eq. (17), but the existence and boundaries of the region where this criterion holds are computed results rather than inserted by hand. The paper's acknowledgement that g2SC may suffer from chromomagnetic instability (Refs. [38,39]) is a physical stability caveat, not a circularity: it undermines the interpretation of which phase is the ground state, but it does not mean the derivation assumes its conclusion. The only self-citation (Ref. [54], by one of the authors) appears in a list of previous NJL studies and plays no load-bearing role in the derivation. Therefore no circular step reduces the paper's claims to its inputs by construction.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

All numerical inputs are standard NJL vacuum parameters plus the OGE coupling choice; the phase-diagram scans use hand-picked coupling ratios. The model relies on Fierz completeness, HFB mean field, hard cutoff regularization, neutrality constraints, and neglect of color-sextet diquarks. No new particles, forces, dimensions, or conserved quantities are introduced.

free parameters (8)

Momentum cutoff Lambda = 631 MeV
Hard cutoff in the thermodynamic potential (13); fitted together with m and G_s^(0) to vacuum condensate, f_pi and m_pi (Ref [63]).
Current quark mass m = 5.5 MeV
Input current mass in Eq. (18), fitted to vacuum observables.
Scalar coupling G_s^(0) = 2.188 / Lambda^2
Coupling of the scalar channel fixed by chiral symmetry breaking and pion properties; cited from Ref [63].
OGE coupling g = approximately 2.46 / Lambda^2
Single coupling in Eq. (6); chosen so the Fierz-derived G_s^(0) equals 2.188 / Lambda^2, which fixes all other channel couplings through Eq. (11).
Diquark coupling ratio H/G_s^(0) = 0.75 in OGE; scanned in Fig. 8
Ratio obtained from the OGE/Fierz dictionary; the phase diagram is computed as this ratio is varied by hand.
Scalar-isovector coupling ratio G_sIv^(0)/G_s^(0) = 1.0 (chosen)
Hand-picked value in Sec. IV.B used to illustrate the phi_I channel effect.
Vector coupling ratio G_v^(0)/G_s^(0) = -0.3 (chosen)
Hand-picked value in Sec. IV.B; the negative sign is motivated by the vector-meson mass relation cited from Ref [67].
Vector-isovector coupling ratio G_vIv^(0)/G_s^(0) = -1.0 (chosen)
Hand-picked value in Sec. IV.B used to illustrate the n_I channel effect.

assumptions (8)

standard math Fierz completeness: all four-fermion interactions can be generated from an independent Fierz-complete basis, reducing the independent couplings from eight to five.
Used in Sec. II, Eqs. (3)-(5), to connect the coupling channels.
domain assumption The one-gluon-exchange interaction (6) fixes the relative strengths of all NJL channels through Eq. (11).
Sec. III uses OGE to identify the dominant channels; no empirical dense-QCD input is used.
domain assumption The Hartree-Fock-BCS mean-field approximation is valid for the NJL model.
Eq. (9) and the thermodynamic potential (13) are built on HFB; fluctuations are ignored.
domain assumption Color-sextet diquark channels are negligible.
Stated after Eq. (10) with justification from Refs [13,31]; no numerical estimate is given within this model.
domain assumption Red-green color symmetry is preserved in the 2SC phase.
Used before Eq. (16) to reduce the gap equations; color-asymmetric ground states are not considered.
ad hoc to paper The gapless phase boundary is determined by the zero-mode criterion of Eq. (17), and unstable modes do not invalidate the phase labels.
The criterion follows from setting lambda=0 in the spectrum; the paper cites the chromomagnetic instability of g2SC (Refs [38,39]) but does not check the stability of the phase diagram.
domain assumption Hard momentum cutoff regularization with Lambda=631 MeV.
The integrals in Eq. (13) use the cutoff; results may be cutoff-sensitive when mu approaches Lambda.
domain assumption Charge and color neutrality constraints n_Q=0 and n_8=0 apply to neutron star matter.
Eq. (20); appropriate for bulk neutron star matter but not for isolated finite systems.

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Cite this review

Pith. "Pith review of Two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality." pith.science (2026). https://pith.science/paper/URSGE46X

@misc{pith2026250706676,
  author       = {Pith},
  title        = {Pith review of: Two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URSGE46X}},
  note         = {Machine review of arXiv:2507.06676}
}

read the original abstract

By using a general NJL model including as many interaction channels as possible, and taking into account the constraints imposed by color and charge neutrality, we analyze how the different interaction channels contribute to the charge-neutral two-flavor color-superconducting matter. After taking the Fierz transformation, the number of parameters in the NJL model is reduced and thereby quark-antiquark condensates and diquark condensates are related. Through a self-consistent solution of the gap equations, we find that in addition to the diquark and vector channels, the scalar-isovector, vector-isovector, and vector-isovector-color-octet channels are also important, while other channels can be neglected. Moreover, between the normal quark matter phase and two-flavor color-superconducting phase, there is a gapless two-flavor color-superconducting phase. The fractions of quarks with different flavors and colors are calculated in both gaped and gapless two-flavor color-superconducting phases. In addition, We illustrate the phase diagrams in the diquark coupling and chemical potential plane, and discuss in detail how the dominant channels influence these phase diagrams.

Figures

Figures reproduced from arXiv: 2507.06676 by the authors.

**Figure 1.** FIG. 1. The chemical potential dependence of the constituent [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 3.** FIG. 3. The chemical potential dependence of the quark and [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. The particle density fractions of quarks and electrons [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 6.** Figure 6: displays the spectra ϵ i r (p) of the gap equations at µ = 380 MeV. In this diagram, the ϵ 2 r and ϵ 3 r are particle modes, and ϵ 1 r and ϵ 4 r are antiparticle modes. If we ignore ∆0 and δMr they are expressed as ϵ 1 r (p) = p (Er(p) + µ) 2 + ∆2 − |δµ|, ϵ 2 r (p) = p…

**Figure 5.** Figure 5: FIG. 5. The contributions of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: FIG. 7. Chemical potential dependence of gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. The phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. The chemical potential dependence of the con [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 12.** Figure 12: FIG. 12. Effects of [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

**Figure 13.** Figure 13: FIG. 13. Comparison of the g2SC phase boundaries with dif [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

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