REVIEW
QCD at Finite Baryon Density: Nucleon Droplets and Color Superconductivity
High Energy Physics - Phenomenology · Submitted 19 Nov 1997 · 14 references
Pith's verdict: The color-superconductivity result is the durable one and it holds up; the negative-pressure droplet argument is a statement about this particular Hamiltonian, not about QCD, and the authors half-know it.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
Using a variational model of quark matter, the authors argue that the uniform phase with the usual chiral condensate is mechanically unstable — it has negative pressure and must break into droplets of chirally restored matter — and that at,
The reading
What carries the argument
The engine is a variational wave function: a BCS-style product of particle-antiparticle pairs for the chiral condensate, and of particle-particle and hole-hole pairs for the color superconducting condensate, with the pairing angles fixed by minimizing the energy. Minimization gives a momentum-independent gap parameter $Delta$ through a self-consistency equation, Eq. (3.4) for the chiral case and Eq. (4.4) for the color case; the finite-density version is obtained simply by raising the lower limit of the gap integral from $0$ to the Fermi momentum $p_F$. The instability claim is then carried by a thermodynamic identity rather than by the gap itself: the pressure is $P = n,partial e/partial n - e$, so a state whose energy density rises faster than linearly with density has negative pressure and cannot be uniform. The crucial structural difference the authors exploit is that a chiral gap acts as an effective mass, so the chemical potential is $mu^2 = p_F^2 + Delta^2 F^4(p_F)$, whereas a pairing gap in a quark-quark channel does not, so $p_F = mu$.
Extended reading notes
Core claim
The paper makes two linked claims. First, in the instanton-inspired model, a uniform state of quark matter that keeps the conventional chiral condensate has negative pressure with respect to empty space at all but the very lowest densities, so it is mechanically unstable; the authors read this as a derivation, inside a definite model, of the physical picture behind the MIT bag, with nucleons as zero-pressure droplets of density $n_0$ in which chiral symmetry is restored, surrounded by vacuum. Second, at every density above the point $n_c$ where the chiral gap vanishes, the model finds an attractive channel for quark-quark pairing, and since a pair of quarks cannot be a color singlet, the resulting $<qq>$ Cooper-pair condensate breaks color symmetry. A plausible ordering scheme gives a substantial gap in a Lorentz scalar channel involving two colors — of order $50$ to $150$ MeV at $n_0$ — and a much smaller gap, at most a few keV, in an axial vector channel involving the third color, which would also break rotation invariance. The authors note that the equation of state in the color superconducting phase differs from the free-quark one by only a few percent, so the condensate is hard to see thermodynamically.
Load-bearing premise
The whole picture rests on the instanton-inspired four-quark interaction, softened by factorized form factors, being a faithful enough stand-in for real QCD that its phase structure survives; the authors say so themselves, and report that for $nu = 1$ with $2.2 < Lambda < 3.2$ GeV the zero-pressure phase has higher energy per quark than a dilute massive-quark gas, while for $Lambda > 3.2$ GeV the pressure is positive at every density, so the droplet picture fails outright in those parameter ranges.
Editorial extensions
If this is right
- Nucleons acquire a derived justification as droplets of chirally restored matter at zero pressure, rather than as an assumed bag construction.
- Chiral symmetry restoration at zero temperature and finite density proceeds by percolation as droplets merge, which is a different mechanism from the finite-temperature transition.
- At densities above $n_0$, quark matter is a color superconductor: some gluons acquire mass by the Higgs mechanism, color $SU(3)$ breaks to $SU(2)$, and a residual combination of electromagnetism and color hypercharge stays massless.
- The large scalar gap suppresses neutrino emission from single thermally excited quarks, which is a possible answer to the observed slow cooling of neutron stars.
- The small axial vector gap for the third color gives a preferred spin direction on part of the Fermi surface, which could in principle show up as correlations between the polarizations of $Lambda$ baryons in a single heavy-ion event.
Reading between the lines
- The negative-pressure argument is really a statement about the shape of the energy-density curve $e(n)$; the same logic would apply to any model whose chiral condensate makes $e(n)$ concave, so the droplet conclusion is a testable property of the equation of state rather than of the instanton vertex specifically.
- Because the two condensates compete for the same quarks, a joint variational treatment would likely interpolate between the droplet regime and the color superconducting regime rather than producing a sharp boundary; the authors say they have begun this calculation, so the size of the shift in $n_0$ is an open quantitative question.
- The five-order-of-magnitude split between the scalar and axial vector gaps is a consequence of the model's color structure, not of a symmetry; a different four-fermion interaction, such as one from single-gluon exchange, would change the axial gap far more than the scalar one, which makes the axial condensate the discriminating prediction.
- If the droplet picture is right, the bag constant and the quark density inside a nucleon are not independent parameters but two readings of the same zero-pressure condition, which suggests a cross-check against measured nucleon charge radii.
Original abstract
We use a variational procedure to study finite density QCD in an approximation in which the interaction between quarks is modelled by that induced by instantons. We find that uniform states with conventional chiral symmetry breaking have negative pressure with respect to empty space at all but the lowest densities, and are therefore unstable. This is a precisely defined phenomenon which motivates the basic picture of hadrons assumed in the MIT bag model, with nucleons as droplets of chiral symmetry restored phase. At all densities high enough that the chirally symmetric phase fills space, we find that color symmetry is broken by the formation of a <qq> condensate of quark Cooper pairs. A plausible ordering scheme leads to a substantial gap in a Lorentz scalar channel involving quarks of two colors, and a much smaller gap in an axial vector channel involving quarks of the third color.
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.
The abstract, sentence by sentence
We use a variational procedure to study finite density QCD in an approximation in which the interaction between quarks is modelled by that induced by instantons. We find that uniform states with conventional chiral symmetry breaking have negative pressure with respect to empty space at all but the lowest densities, and are therefore unstable. This is a precisely defined phenomenon which motivates the basic picture of hadrons assumed in the MIT bag model, with nucleons as droplets of chiral symmetry restored phase. At all densities high enough that the chirally symmetric phase fills space, we find that color symmetry is broken by the formation of a <qq> condensate of quark Cooper pairs. A plausible ordering scheme leads to a substantial gap in a Lorentz scalar channel involving quarks of two colors, and a much smaller gap in an axial vector channel involving quarks of the third color.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We use a variational procedure to study finite density QCD in an approximation in which the interaction between quarks is modelled by that induced by instantons. | method 0.94 | 0.93 | 0.03 | asserted 0.97 |
| s1 | We find that uniform states with conventional chiral symmetry breaking have negative pressure with respect to empty space at all but the lowest densities, and are therefore unstable. | result 1.00 | 0.95 | 0.21 | asserted 1.00 |
| s2 | This is a precisely defined phenomenon which motivates the basic picture of hadrons assumed in the MIT bag model, with nucleons as droplets of chiral symmetry restored phase. | interpretation 1.00 | 0.54 | 0.19 | asserted 0.99 |
| s3 | At all densities high enough that the chirally symmetric phase fills space, we find that color symmetry is broken by the formation of a <qq> condensate of quark Cooper pairs. | result 1.00 | 0.95 | 0.31 | asserted 1.00 |
| s4 | A plausible ordering scheme leads to a substantial gap in a Lorentz scalar channel involving quarks of two colors, and a much smaller gap in an axial vector channel involving quarks of the third color. | result 1.00 | 0.85 | 0.18 | hedged possibility 0.71 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-ph/9711395 by the authors.
References
How the literature uses it
144 arXiv papers cite this in 189 sentences
Papers whose main result relates to this one's
States the same result 4 papers
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At all densities high enough that the chirally symmetric phase fills space, color symmetry is broken by the formation of a <qq> condensate of quark Cooper pairs.
QCD at Finite Baryon Density: Chiral Symmetry Restoration and Color Superconductivity · 1.00
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At all densities high enough that the chirally symmetric phase fills space, we find that color symmetry is broken by the formation of a $< qq >$ condensate of quark Cooper pairs.
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It results in chiral symmetry restoration due to the onset of diquark condensation, a `colour superconductor', at finite density.
Chiral Symmetry Breaking and Color Superconductivity in the Instanton Picture · 0.83
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It results in chiral symmetry restoration due to the onset of diquark condensation, a `colour superconductor', at finite density.
Papers citing it · 86 sentences without a confident label
Builds on it 1 paper
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Generalizing models used to study color superconductivity in which quark-quark correlations at the Fermi surface produce Cooper pairs to include three-quark interactions at the Fermi surface could realize our proposal within the framework of Nambu-Jona-Lasino model [ref] .
Quarkyonic Matter and Neutron Stars · Introduction · 0.83
Uses its method or tool 1 paper
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For the two-flavor case, a nonlocal NJL model [ref] is adopted here, which has the Lagrangian density, [equation] where [equation] and $C=iγ_0γ_2$ stands for the Dirac charge conjugation matrix.
Vector interaction, charge neutrality and multiple chiral critical point structures · Model · 0.91
Compares with it 7 papers
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Nevertheless, the results for the gap from Refs. [ref] and [ref] are still in accordance at lower densities.
Zero-temperature thermodynamics of dense asymmetric strong-interaction matter · Parameter constraints from QCD · 0.96
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Note that, towards smaller chemical potentials (associated with densities $n/n_0 ≲5$, where $n_0$ is the nuclear saturation density), the results from the Fierz-complete study in Ref. [ref] are remarkably consistent with those from low-energy models (e.g., Ref. [ref] ), see caption of Fig. [ref] and also Ref. [ref] for a discussion.
From quarks and gluons to color superconductivity at supranuclear densities · Diquark gap · 0.88
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This is in qualitative agreement with the model calculations mentioned above [ref] .
Phase diagram of a generalized Stephanov model for finite-density QCD · Introduction · 0.90
Cites it as support 2 papers
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Although the two are not distinguished by a single-gluon exchange interaction, it has been shown [ref] that the instanton-induced interaction favors the former.
Chiral Lagrangian and spectral sum rules for dense two-color QCD · Symmetry breaking patterns · 0.99
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Note that this is in accordance with an RG analysis of symmetry breaking patterns in dense QCD matter with two quark flavors, where it has been found that a broken $U(1)_A$ symmetry is required to render 2SC pairing correlations to be the most dominant ones at high density [ref] , see also Refs. [ref] for early model studies.]
Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order · Expansion of the QCD effective action · 0.97
Cites it as background 86 papers
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The densest phase of matter according to standard model physics is the color-flavor-locked (CFL) phase [ref] .
Stability of superfluid vortices in dense quark matter · Introduction · 1.00
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(Inclusion of small up and down quark masses is straightforward.) Based on the Bardeen-Cooper-Schrieffer (BCS) theory, the Fermi surface of quarks at high density is unstable against the formation of quark-quark pairing by the attractive one-gluon exchange interaction and instanton-induced interaction in the color antisymmetric channel, leading to the two-flavor color superconductivity (2SC) [ref] .
Quantum Hall liquids in high-density QCD · Review of two-flavor color superconductivity · 0.93
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We consider the RG evolution of effective coupling strengths between gapped and ungapped quarks appearing in the two-flavor color superconducting (2SC) phase [ref] .
Emergent QCD Kondo effect in two-flavor color superconducting phase · Introduction · 0.84
Discussion
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