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On finite-density QCD at large Nc

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E. Shuster, D. T. Son

High Energy Physics - Phenomenology · Submitted 21 May 1999 · 11 references

Pith's verdict: A careful, honest finite-Nc extension of the DGR instability: the parametric result and the Nc = 3 exclusion hold up, though the abstract states the threshold more sharply than the body supports.

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arXiv IDhep-ph/9905448 (version 2)
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CitationD. T. Son, E. Shuster. "On finite-density QCD at large Nc." arXiv:hep-ph/9905448 (1999). https://pith.science/paper/KQL6ISAG
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The paper

The paper claims

The Deryagin–Grigoriev–Rubakov chiral-wave instability of the dense-QCD Fermi surface survives at finite color number only inside a bounded window of chemical potentials, and needs more than roughly a thousand colors at ordinary densities.

The reading

Deryagin, Grigoriev and Rubakov showed that when the number of colors is taken to infinity, the Fermi surface of dense quark matter is unstable against forming particle-hole pairs whose total momentum is twice the Fermi momentum — a chiral wave — and that this instability survives even at weak coupling, because the paired fermions move in nearly the same direction where gluon exchange is singular. Shuster and Son ask what becomes of that instability when the number of colors $N_c$ is large but finite, which is the only regime in which anything can be said about real QCD. Their answer is that screening by the quark medium cuts off the singularity, so the instability survives only if $N_c$ exceeds a threshold that grows with density; at any fixed $N_c$ the unstable band is bounded above by $mucrit sim exp(gamma ln^2 N_c),LambdaQCD$ with $gamma approx 0.02173$. Numerically the threshold is enormous — of order $1000 N_f$ colors already at $mu = 3LambdaQCD$ — so the infinite-color result is not a guide to three-color quark matter. If this is right, the ground state at reachable densities is a color superconductor, and the chiral-wave phase belongs only to a world with thousands of colors.

What carries the argument

The engine is a (1+1)-dimensional effective theory obtained by integrating out the momentum components perpendicular to the pairing direction, which turns each gluon rung of the ladder into a point-like current-current interaction with a logarithmically running coupling — a Thirring-like model with coupling $lambda(q_parallel) = (g^2/4pi)ln(Delta/q_parallel)$. The double logs are resummed by an RG equation $partial_s lambda(s,u) = (N_c/pi)lambda^2(s,s)$, solved by writing $lambda = pi/[N_c f(s) + (g^2/4pi)(u-s)]$ with $partial_s f = h^2 + f^2$, so $f = h tanh s$ and the Landau pole sits at $s_L = pi/2h$. Screening modifies only the initial condition: the electric and magnetic couplings split into $lambda_pm = (lambda_0 pm lambda_1)/2$, only $lambda_+$ runs, and beyond $s_m = ln(Delta/m)$ its flow equation becomes $partial_s f = f^2 + h^2/6$. That single change is what produces the constant $c$ and hence the entire numerical bound.

Extended reading notes

Core claim

The paper's claim is that the DGR instability is an artifact of the strict $N_c to infty$ limit rather than a property of dense QCD. At finite $N_c$ the gluon propagator is cut off in the infrared by Thomas-Fermi screening for the electric exchange and by Landau damping for the magnetic one, and once the screening mass $m$ exceeds the scale of the would-be Landau pole, the double-logarithmic flow that produced the instability never reaches it. Maximizing over the patch size $Delta$ gives a sharp condition, $m < mu e^{-c/h}$ with $c = sqrt{6}arctan(1/2) + 2arctan sqrt{2/3} approx 2.5051$, which translates into a lower bound on the number of colors, $N_c gtrsim 2N_f h^2 e^{2c/h}$, where $h^2 = g^2N_c/4pi^2$. Combined with the one-loop running of $h$ with $mu$, this confines the instability to a finite window $LambdaQCD < mu < mucrit$, with $mucrit sim exp(gamma ln^2 N_c + O(ln N_c ln ln N_c))LambdaQCD$ and $gamma = 3/(22c^2) = 0.02173$. The authors conclude that for three colors the DGR channel is closed and the ground state is the color superconductor expected from BCS pairing, and they speculate that at very large $N_c$ the true ground state is a superposition of chiral waves from many disjoint patches of the Fermi surface, gaining $mu^4 e^{-2pi/h}$ rather than the $mu^4 e^{-3pi/h}$ of the single standing wave.

Load-bearing premise

The numerical bound on the number of colors, and with it the exclusion of three colors, rests on an order-one coefficient in the threshold condition that the authors explicitly say they have not computed; only the parametric form of the critical chemical potential is independent of it.

Editorial extensions

If this is right

  • At any fixed number of colors the DGR instability occupies only a finite band of chemical potentials, $LambdaQCD < mu < mucrit(N_c)$, so raising the density eventually returns the system to a stable Fermi surface.
  • Because $mucrit$ grows faster than any power of $N_c$, the infinite-color picture is recovered only asymptotically; there is no intermediate regime in which the large-$N_c$ limit can be borrowed for three colors.
  • Above the critical curve the only surviving Fermi-surface instability is BCS pairing, so the ground state there is a color superconductor, even though the gap is exponentially small at large $N_c$.
  • The original DGR standing chiral wave cannot be the ground state, since it breaks translation symmetry along a single direction only; a superposition of waves from many disjoint patches of the Fermi surface lowers the energy further.
  • The phase diagram in the $(N_c, mu)$ plane has a shaded DGR region and a color-superconducting region separated by the curve $N_c sim 2N_f h^2 e^{2c/h}$, with the DGR region drawn only above $mu = 3LambdaQCD$ where the calculation is under control.

Reading between the lines

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

  • The competition at work here — a singular forward-scattering channel versus screening by the medium — is generic to dense gauge matter, so the window formula is a template for judging any proposed particle-hole instability rather than a statement about one channel.
  • The parametric result is robust but the numerical exclusion of three colors is not: fixing the order-one coefficient in the threshold condition is a finite, well-posed calculation that would turn a scaling statement into a sharp statement about which color numbers are ruled out.
  • If a chiral wave ever formed, its signature would be a periodic modulation of $langle barpsi psi rangle$ with spatial period $pi/mu$, a distinctive target should a controllable large-$N_c$ simulation ever become available.
  • The closing suggestion of a state resembling a tomographic Luttinger liquid — chiral symmetry unbroken but the chiral response singular at wave number $2mu$ — is a concrete conjecture about a non-BCS, non-Fermi-liquid ground state that a large-$N_c$ model study could in principle probe.
Original abstract

Deryagin, Grigoriev, and Rubakov (DGR) have shown that in finite-density QCD at infinite Nc the Fermi surface is unstable with respect to the formation of chiral waves with wavenumber twice the Fermi momentum, while the BCS instability is suppressed. We show here that at large, but finite Nc, the DGR instability only occurs in a finite window of chemical potentials from above Lambda_QCD to mu_critical = exp(gamma ln^2 Nc + O(ln Nc ln ln Nc))Lambda_QCD, where gamma = 0.02173. Our analysis shows that, at least in the perturbative regime, the instability occurs only at extremely large Nc, Nc > 1000 Nf, where Nf is the number of flavors. We conclude that the DGR instability is not likely to occur in QCD with three colors, where the ground state is expected to be a color superconductor. We speculate on possible structure of the ground state of finite-density QCD with very large Nc.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,466 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.86 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.91 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Deryagin, Grigoriev, and Rubakov (DGR) have shown that in finite-density QCD at infinite Nc the Fermi surface is unstable with respect to the formation of chiral waves with wavenumber twice the Fermi momentum, while the BCS instability is suppressed. We show here that at large, but finite Nc, the DGR instability only occurs in a finite window of chemical potentials from above Lambda_QCD to mu_critical = exp(gamma ln^2 Nc + O(ln Nc ln ln Nc))Lambda_QCD, where gamma = 0.02173. Our analysis shows that, at least in the perturbative regime, the instability occurs only at extremely large Nc, Nc > 1000 Nf, where Nf is the number of flavors. We conclude that the DGR instability is not likely to occur in QCD with three colors, where the ground state is expected to be a color superconductor. We speculate on possible structure of the ground state of finite-density QCD with very large Nc.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Deryagin, Grigoriev, and Rubakov (DGR) have shown that in finite-density QCD at infinite Nc the Fermi surface is unstable with respect to the formation of chiral waves with wavenumber twice the Fermi momentum, while the BCS instability is suppressed. background 1.00 0.03 0.05 asserted 1.00
s1 We show here that at large, but finite Nc, the DGR instability only occurs in a finite window of chemical potentials from above Lambda_QCD to mu_critical = exp(gamma ln^2 Nc + O(ln Nc ln ln Nc))Lambda_QCD, where gamma = 0.02173. result 0.99 0.94 0.25 asserted 1.00
s2 Our analysis shows that, at least in the perturbative regime, the instability occurs only at extremely large Nc, Nc > 1000 Nf, where Nf is the number of flavors. result 0.96 0.93 0.13 asserted 0.97
s3 We conclude that the DGR instability is not likely to occur in QCD with three colors, where the ground state is expected to be a color superconductor. interpretation 0.95 0.71 0.82 hedged possibility 0.91
s4 We speculate on possible structure of the ground state of finite-density QCD with very large Nc. future work 0.93 0.26 0.03 speculative 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. The particle-hole pair
Figure 2. FIG. 2. Ladder approximation
Figure 3. FIG. 3. Reducing one-gluon exchange to a point-like interaction in th
Figure 4. FIG. 4. The zero-sound diagram
Figure 5. FIG. 5. Region of DGR instability in the (

Figures reproduced from arXiv:hep-ph/9905448 by the authors.

References
How later papers used it

58 later papers cite it

Evidence: substantial. 58 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 90 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Compares with it 1 of 4 papers with this label
How the literature uses it

51 arXiv papers cite this in 81 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 56 sentences without a confident label

Uses its method or tool 1 paper
  • The $L= 0$ amplitude is [equation] The idea [ref] (see also [ [ref] , [ref] , [ref] ] for earlier developments towards this approach) is to treat the integral in Eq. [ref] in a Wilsonian way, by decimating only fast bosonic modes.

    Enhanced Pairing of Quantum Critical Metals Near d=3+1 · Log squared divergences and their interpretation · 0.92

Compares with it 3 papers
Cites it as support 1 paper
Cites it as background 20 papers
  • In contrast, at large $N_c$, at high density ( $1≪μ/Λ_QCD ∼N_c^0$ ) there is another instability, first noted by Deryagin, Grigoriev, and Rubakov (DGR), toward the formation of a chiral density wave, i.e., a $⟨q̅(x) q(y)⟩$ condensate [ref] .

    Color Superconductivity at Large N: A New Hope · Introduction · 0.84

  • This phase, originally introduced in [ref] and studied in several follow-up papers, e.g. [ref] , is characterized by quark-hole pairs having a finite total momentum, and leading to standing waves.

    The magnetic dual chiral density wave phase in a rotating cold quark matter · Introduction · 0.99

  • Theoretical explorations have proposed diverse modulated structures such as the chiral density wave [ref] , crystalline color superconductivity [ref] , chiral magnetic spiral [ref] , and chiral magnetic wave [ref] .

    Is holographic quark-gluon plasma homogeneous? · Introduction · 0.98

Discussion

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