Thermodynamics of two-colour QCD and the Nambu Jona-Lasinio model

Ratti and Weise take the two-colour NJL model with the quark-antiquark and diquark couplings tied by the Fierz transform of the colour-current interaction (G = H = 3G_c/2, eq. 4) and run it against the Hands et al. lattice data. The genuinely new content is the quantitative overlay: chiral and diquark condensates, baryon density and pion mass compared point-by-point with N_c = 2 lattice results, plus a mass-spectrum section that solves the mixed sigma-diquark-antidiquark matrix above mu_c and identifies the massless U(1)_B Goldstone mode. That last piece is a real addition, not a rehash of Kogut et al. The parameter fixing is done the right way round: f_pi, the quark condensate and GMOR, none of them tuned to the lattice observables being compared. The known artifacts, the diquark condensate turning over near the cutoff and the baryon density falling short at large mu, are stated in the text rather than buried. That counts for something. The soft spots are reporting gaps, not structural errors. First, the 'very good agreement' of abstract:s4 rests on a single model curve at m0 = 4.5 MeV laid over lattice points that the captions themselves say correspond to different quark masses. The paper never says which lattice mass matches 4.5 MeV, and there is no error band anywhere. As printed you cannot separate agreement from a curve passing through the middle of a spread. Second, the bridge from SU(2) adjoint staggered fermions to a two-flavour theory is one sentence in section 4; that is the load-bearing identification for the whole comparison and it deserves a paragraph. Third, the truncation that drops the coloured quark-antiquark modes, and thereby enforces Pauli-Gursey, is justified by a private communication [57]. That is the one place where an unpublished note carries real weight. On the reader's circularity worry: mu_c = m_pi/2 does follow from the imposed G = H plus a model-computed m_pi, so it is a consistency check of the symmetry pattern rather than an independent prediction. The paper says 'confirm', which is fair; I would not call this a flaw, just a framing point. Who should read it: anyone modelling colour superconductivity with NJL, and anyone using N_c = 2 lattice data as a benchmark. It deserves referees, mainly to force the mass-matching and error-band issues into the open.

The central empirical claim is abstract:s4, 'Very good agreement is found with lattice QCD (N_c=2) results,' supported by Figs. 2, 5 and 6, which overlay model curves on data from ref. [27]. Two features of that comparison are load-bearing and unexamined. First, the model is computed at a single current quark mass, m0 = 4.5 MeV (Table 1), while the lattice points in Figs. 2, 5 and 6 are explicitly labelled as corresponding to 'different values for the quark masses' (open circles, squares, diamonds). The paper never states which lattice quark mass matches m0 = 4.5 MeV, nor does it show the model curve at the other masses. If the lattice points at different masses do not collapse onto a single curve, then a single model curve cannot be in 'very good agreement' with all of them; it can at best pass through the middle of a spread. Second, the comparison is to SU(2) adjoint staggered fermions, not fundamental quarks; the bridge is one sentence in §4 ('the positive determinant sector behaves like a two-flavour theory') with no demonstration that the Goldstone spectrum and condensate physics coincide at the quantitative level required. Neither point is acknowledged in the text. A related missing support is the private communication [57], cited in the conclusions to justify dropping the coloured qqbar modes that would otherwise break G = H; that truncation is what enforces the Pauli-Gursey structure, so its justification should not rest on an unpublished note.

The paper is a careful mean-field study with a clearly defined model, parameters fixed against hadronic observables rather than the lattice data, and acknowledged artifacts (the |Delta| turnover near Lambda, the density deficit at large mu). The Fierz coefficients G = H = 3/2 G_c are correctly derived in the appendix, and the mu_c = m_pi/2 result is a genuine consistency check of the Pauli-Gursey structure, honestly discussed in §5. The reader's CONDITIONAL verdict is appropriate. My concern sharpens the reader's first point: the 'very good agreement' claim is not merely unquantified, it is a comparison of a single-mass model curve against multi-mass lattice data, and the paper does not establish the mapping. This is fixable by replotting at matched masses. The reader's second point (mu_c = m_pi/2 partly guaranteed) is correct but the paper acknowledges it in §5, so it is a framing issue rather than a correctness issue. The reader's third point (private communication [57]) is a real but minor concern: the justification for dropping coloured qqbar modes rests on an unpublished communication, but the truncation is a standard NJL approximation and the empirical comparison does not depend on it. I would keep the verdict at CONDITIONAL.

The paper provides a quantitatively successful mean-field description of N_c = 2 QCD thermodynamics, with parameters fixed by hadronic observables rather than tuned to the lattice. The G = H relation from the Fierz transform is a principled constraint that ties the chiral and diquark channels, and the resulting Pauli-Gursey symmetry is a genuine feature of two-colour QCD. The comparison to lattice data and chiral EFT is valuable, and the chiral-limit analysis (§5) is instructive. The paper is a useful benchmark for NJL-type quasiparticle models.

The paper asks whether the thermodynamics of QCD with two colours can be captured by a Nambu–Jona-Lasinio model: a quark model in which gluons are integrated out and replaced by a local four-fermion contact interaction. The distinguishing move is that the strength of the quark–antiquark channel and the strength of the diquark channel are not tuned independently; a Fierz rearrangement of a single colour-current interaction fixes them equal, $G = H = 3G_c/2$, which is exactly the Pauli-Gürsey symmetry relating pions and scalar diquarks in the two-colour theory. Minimizing the mean-field thermodynamic potential then yields a second-order transition at $μ_c = m_π/2$, above which scalar diquarks Bose-condense and the diquark condensate grows while the chiral condensate shrinks. The authors overlay their curves on lattice data for the two condensates, the baryon density and the pion mass, and report good agreement below about twice the critical chemical potential. The payoff, if the claim holds, is that the low-energy quasiparticle physics of a dense gauge theory is set by symmetry and a single coupling rather than by the details of gluon exchange.

What carries the argument. The load-bearing object is the identification $G = H = 3G_c/2$ obtained by Fierz-transforming the colour-current interaction $-G_c Σ_a J^a_μ J^{μa}$ into colour-singlet quark–antiquark and scalar-diquark channels. This single relation is what makes the model predictive rather than fitted: it forces the pion and the scalar diquark to be related by the Pauli-Gürsey symmetry, so the diquark condensate appears at exactly the chemical potential where a half pion mass can be supplied, $μ_c = m_π/2$. The machinery that turns this into numbers is the Nambu–Gorkov propagator for the bispinor of quarks and charge-conjugated quarks, the mean-field thermodynamic potential of eq. (22) with a sharp three-momentum cutoff $Λ = 0.78$ GeV, and a second-order expansion of the effective action whose mass matrix in the $(π, σ, Δ, Δ^*)$ basis yields the mixed eigenmodes once $|Δ| ≠ 0$.

Core claim. The central claim is that a two-flavour, two-colour NJL model whose quark–antiquark and diquark couplings are tied together by the Fierz transform of the colour-current interaction reproduces the thermodynamics that lattice simulations of $N_c = 2$ QCD observe. Concretely: a second-order phase transition at $μ_c = m_π/2$; a nonzero scalar-diquark condensate for $μ > μ_c$ that tracks the fall of the chiral condensate; a baryon charge density in agreement with lattice data for $μ < 2μ_c$; and a pion mass that rises linearly as $m_π = 2μ$ in the dense phase, matching both the lattice points and the leading-order chiral effective Lagrangian prediction. The authors also work out the bosonic spectrum: below $μ_c$ the diquark and antidiquark masses split as $m_π ∓ 2μ$ according to their baryon number, and above $μ_c$ the $σ$, diquark and antidiquark fields mix, leaving one exactly massless mode — the Goldstone boson of spontaneously broken baryon number — alongside a massive mode following $2μ√(1 + 3(m_π/2μ)^4)$. In the chiral limit the model exhibits the Pauli-Gürsey degeneracy along $σ^2 + |Δ|^2 = $ const, so that the ordering of the two limits $m_0 → 0$ and $μ → 0$ matters.

The premise it rests on. The argument rests on replacing the local colour gauge dynamics by a global-symmetry contact interaction with one coupling and a sharp momentum cutoff, and on treating the auxiliary fields in mean field; if that replacement misses the relevant physics at the densities probed, the agreement with lattice data is a coincidence of the fit rather than a derivation.

The critical chemical potential for the onset of dense matter in two-colour QCD is fixed by the pion mass alone, $μ_c = m_π/2$, with no independent diquark coupling to fit.

In the dense phase the pion mass grows as $m_π = 2μ$, so a measurement of the pion mass at finite density is a direct probe of the condensate's BEC-type nature.

Baryon-number superconductivity in this theory has a definite signature: one scalar mode becomes exactly massless above $μ_c$ while its partner is pushed up to $2μ√(1 + 3(m_π/2μ)^4)$.

The chiral and diquark condensates are not independent order parameters in the chiral limit; they rotate into one another along a fixed radius, so a small quark mass or a small chemical potential is enough to pick the diquark direction.

The model's agreement with lattice data up to $μ ≈ 2μ_c$ and its failure beyond that range locates where mean-field quasiparticles stop being an adequate description of dense two-colour matter.

If the single-coupling relation is what buys the agreement, then the same Fierz bookkeeping should predict the ratio of diquark to chiral observables in any theory where the colour-current interaction dominates — a check that does not require a lattice, only a well-measured meson and diquark spectrum.

The authors note that $|Δ|$ turns over near $μ ∼ Λ$ and call that an artifact of the cutoff; a natural test is to repeat the calculation with a smooth or larger cutoff and see whether the turnover moves with $Λ$ or with a physical scale.

The underestimate of the baryon density beyond $μ ≈ 2μ_c$ is attributed to missing quasiparticle correlations; a beyond-mean-field treatment should therefore improve the density first and the condensates later, which is an ordering that a future calculation could confirm or refute.

Because the transition is second order and driven by a conserved charge, the same structure — a Goldstone mode of broken baryon number appearing at half the meson mass — plausibly recurs in any theory where diquarks are the lightest charged excitations, which makes it a template rather than a two-colour curiosity.

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