Superconductivity by long-range color magnetic interaction in high-density quark matter

D.T. Son

REVIEW

Superconductivity by long-range color magnetic interaction in high-density quark matter

D.T. Son

High Energy Physics - Phenomenology · Submitted 8 Dec 1998 · 15 references

Pith's verdict: The exponent is right and the mechanism behind it is right; the g^{-5} prefactor is a leading-log artifact rather than a controlled result, and the stress-test objection to it is an arithmetic slip.

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arXiv IDhep-ph/9812287 (version 1)

Pith IDH3UKHL32, Pith's permanent record of this paper and its review

CitationD.T. Son. "Superconductivity by long-range color magnetic interaction in high-density quark matter." arXiv:hep-ph/9812287 (1998). https://pith.science/paper/H3UKHL32

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The thing to take away: Son gets the weak-coupling scaling of the color-superconducting gap right, Δ ~ exp(-3π²/(√2 g)), and he gets it from a mechanism that is physically correct. The magnetic gluon stays unscreened at high density, Landau damping rather than the Meissner effect sets the IR cutoff, and the resulting flow, Eq. (11), has a constant term that feeds the instantaneous channel at every RG step. The exponent is reproduced independently by the Eliashberg equation in Appendix B, and that cross-check is real: the kernel coefficient g²/(18π²) in (B2) is exactly √(AB) from (11), so the two routes agree for a reason rather than by construction. The Sec. III discussion stands on its own — the three standard arguments for a magnetic mass in hot QCD are each checked against finite density and each fails, and the power-counting remark that q0 ~ q³ makes the soft integrals IR finite is the cleanest statement of this I have seen. Credit is due there too: the paper is explicit that Pisarski and Rischke reached a similar conclusion independently [11]. On the prefactor, I checked the stress-test objection and it does not hold. With t = 3π²/(√2g) - 6 ln(1/g), one has e^{-t} = g^{-6} exp(...), so m_D e^{-t} ~ μ g^{-5}. Eq. (13) is internally consistent and the claimed g^{+7} is a sign slip. That said, the g^{-5} is a leading-log output of the matching at δ ~ m_D and the initial condition (12), and Appendix B explicitly does not reproduce it. So it is a plausible prefactor, not a controlled one, and the abstract states it more firmly than the derivation warrants. That is the honest soft spot, and it is a prefactor — it does not touch the headline. Appendix A is an estimate, as the paper says; it matters less than it looks, since an O(g²) correction to A shifts the pole by O(g), a power of g rather than the exponent. The classifier's 'no limitations found' is wrong: the paper says plainly that the RG gives the breakdown scale and not the gap, and that the l-dependence of the channels is unresolved. Who should read it: anyone working on dense QCD, and anyone interested in RG with long-range interactions. Send it out.

Referee report

Significance

If correct, this is an important result: it changes the parametric form of the color-superconductivity gap at asymptotically high density and makes the gap parametrically larger than all earlier estimates, with obvious consequences for the phenomenology of intermediate-density quark matter and for heavy-ion searches. The derivation is parameter-free — no quantity is fitted to the target answer — and the exponent is obtained by two independent routes, the RG flow of Eq. (11) and the Eliashberg equation of Appendix B, which is a genuine cross-check rather than a restatement. The paper also makes a sharp, falsifiable structural claim (the exponent is 3π²/(√2 g), not c/g²) and draws a useful and specific contrast with the finite-temperature magnetic-mass lore and with condensed-matter systems with long-range magnetic interactions.

Major objections (3)

[Abstract and Eq. (13)] The exponent is robust, but the g^{-5} prefactor is not under control and the abstract presents it as a derived result. The arithmetic of Eq. (13) is internally consistent: with m_D ~ gμ and t = 3π²/(√2 g) - 6 ln(1/g), one has e^{-t} = g^{-6} exp(-3π²/(√2 g)), so m_D e^{-t} ~ μ g^{-5} exp(...). I note this because a natural misreading of the sign of the -6 ln(1/g) term gives g^{+7}; that reading is incorrect and I do not hold it against the paper. The real issue is different: the power of g is fixed by the coefficient 6, which comes entirely from the leading-log matching condition f_0(0) = -(2g²/(3μ²)) ln(1/g) at δ ~ m_D in Eq. (12). The O(1) constant inside that logarithm (the precise lower matching scale q_min = c·m_D and the upper scale at which the logarithm is cut off) is never computed, and a shift of that constant shifts t_pole by an additive constant, i.e., changes the power of g while leaving the exponent untouched. Appendix B says as much explicitly, reproducing 'only the leading exponential behavior ... but not the power (g^{-5}) part.' The abstract should therefore present g^{-5} as indicative of the expected prefactor rather than as a result of the calculation, and Eq. (13) should not read as though the power follows from the preceding line without further input.
[Sec. IV, Eqs. (9) and (12)] The factor of 2 between Eq. (9) and Eq. (12) is exactly the factor that controls the prefactor power and should be stated explicitly. Eq. (9) gives f_0 ≈ -(g²/3) ln(μ/q_min), which is the magnetic contribution alone; Eq. (12) quotes -(2g²/(3μ²)) ln(1/g), i.e., twice as large, presumably because the Debye-screened electric term in Eq. (8) contributes an equal logarithm cut off at m_D. Since the coefficient c in f_0(0) = -c(g²/μ²) ln(1/g) propagates as t_pole → t_pole + 9c ln(1/g) and hence to a prefactor g^{1-9c}, the difference between c = 1/3 (magnetic only, giving g^{-2}) and c = 2/3 (electric plus magnetic, giving g^{-5}) is precisely the quoted power. The text should say in one sentence where the factor 2 comes from and that the electric contribution is being retained despite being subleading in the exponent, so that a reader can see that the g^{-5} rests on this matching rather than on the RG flow itself.
[Sec. IV opening and Appendix A] The non-renormalization of the non-instantaneous sector is the assumption the whole construction rests on, and the paper is right to flag Appendix A as an estimate rather than a proof. It would strengthen the paper to state explicitly that this assumption is bounded in its consequences: the one-loop correction estimated in Eq. (A1) is down by O(g²) relative to the tree amplitude, so it can shift the constant A = g²/(9μ²) in Eq. (11) only by a relative O(g²), which shifts √(AB) and hence t_pole by O(g) — a prefactor power, not the exponent. Making that bound explicit would convert the paper's most vulnerable premise into a demonstration that the headline exponent 3π²/(√2 g) is insensitive to it, and would also make clear that the same O(g) ambiguity feeds the prefactor discussed above.

Minor objections (5)

[Sec. V, 'The possibility of breaking rotational symmetry'] The statement that Δf_l ≈ (g²/9μ²)P_l(1) holds 'provided that l is not parametric on g' is not the right condition. The window in θ is of width ~ δ/μ, which shrinks as the RG proceeds down to δ ~ Δ ~ μ e^{-c/g}; the approximation P_l(cos θ) ≈ 1 across the window requires l δ/μ ≪ 1, i.e., l ≪ e^{c/g}, not merely l not parametric in g. The conclusion that all l evolve identically to leading order is presumably unaffected, but the stated condition should be corrected.
[Sec. III, Eq. (7)] The hard-dense-loop magnetic propagator of Eq. (7), with its Landau-damping term (π/2)m_D²|q_0|/q, is quoted without a reference. Since the entire argument about the absence of magnetic screening at finite density turns on the q_0 ~ q³ scaling implied by this form, a citation to the resummation literature would help the reader verify the starting point independently of the paper's own reasoning.
[Sec. III, paragraph on infrared finiteness] The claim that perturbation theory at finite μ is infrared finite for momenta of order g^n μ, based on the modified q_0 ~ q³ scaling, is stated in a single sentence and attributed in a footnote. A short explicit power-counting display for the representative integral ∫d⁴q D²(q) would make this argument, which is doing real work in Sec. III, checkable at a glance.
[Appendix B, Eq. (B2)] The replacement of (π/2)m_D² by μ² in the logarithm is justified on the grounds that only exponential scales matter, which is fine for the exponent. Given the discussion above about the prefactor, however, it would be worth stating explicitly that this replacement is one of the places where the power of g is lost, so that the reader does not expect Appendix B to confirm the g^{-5}.
[Sec. V, 'The very soft magnetic gluons'] The estimate B² ~ m_D²Δ² compared with a critical field of order μΔ is presented as a crude argument, and it is appropriately hedged. It would be clearer to state the resulting ratio of magnetic energy density to condensation energy density explicitly, so that the reader can see how much margin the weak-coupling limit provides.

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The paper

The paper claims

The color magnetic field in dense quark matter is never screened, and that long-range force makes the superconducting gap fall off only as $e^{-c/g}$ rather than $e^{-c/g^2}$.

The reading

The paper asks what happens to the pairing gap of quark matter at asymptotically high density, where the strong coupling $g$ is small and perturbation theory should work. The obstacle is that the color magnetic part of one-gluon exchange is not screened in a dense medium, so the interaction between quarks on opposite sides of the Fermi surface is long-ranged and the standard BCS treatment, which assumes a short-range force, breaks down. The author argues that the magnetic field stays unscreened at any finite order in $g$ (the hot-medium argument for a magnetic mass of order $g^2 T$ does not carry over to finite density), and then redoes the renormalization-group analysis near the Fermi surface so that softer and softer gluon exchanges keep feeding the pairing channel as the cutoff is lowered. The result is a gap $Delta sim mu g^{-5} exp(-3pi^2/(sqrt{2} g))$, exponentially larger at weak coupling than every earlier estimate of the form $e^{-c/g^2}$, which matters because it raises the expected pairing scale and hence the chance of seeing a color superconductor at accessible densities.

What carries the argument

The load-bearing object is the modified renormalization-group equation for the $s$-wave scattering amplitude, $d f_0/dt = -g^2/(9 mu^2) - (mu^2/(2pi^2)) f_0^2$, where $t = -ln delta$ runs toward the Fermi surface. The second term is the familiar BCS term of the Fermi-surface RG; the first is new and records that gluons with momentum in the window $(e^{-1/3} q_delta, q_delta)$ graduate from the retarded to the instantaneous sector during each step, so the long-range magnetic force acts as an infinite reservoir that keeps replenishing the pairing interaction. With the initial condition $f_0(0) = -(2g^2/3mu^2) ln(1/g)$ at $delta sim m_D$, the solution is a tangent that hits the Landau pole at $t = 3pi^2/(sqrt{2} g) - 6 ln(1/g)$, giving the gap scale. Because the increment $Delta f(theta)$ is concentrated at small angles, its partial-wave coefficients are essentially $l$-independent, so the same flow governs every angular momentum channel.

Extended reading notes

Core claim

The central claim is that the unscreened color magnetic interaction changes the asymptotic form of the color-superconducting gap qualitatively, from $exp(-c/g^2)$ to $exp(-c/g)$ with $c = 3pi^2/sqrt{2}$. The author first argues that the magnetic sector is not screened at the scale $g^2 mu$: there is no dimensional reduction at finite chemical potential, the long-range magnetic perturbation theory is infrared finite because Landau damping makes $q_0$ scale like $q^3$, and the Bose-enhancement argument that produces a magnetic mass at high temperature has no finite-density analogue. He then splits one-gluon exchange into an instantaneous part (gluons with momentum above $q_delta = m_D^{2/3} delta^{1/3}$, where Landau damping is negligible) and a retarded part below it. Lowering the cutoff transfers a calculable slice of the retarded part into the instantaneous part at every step, which adds a constant negative term to the flow of the partial-wave amplitudes and drives them to the Landau pole much sooner. The resulting gap, $Delta sim mu g^{-5} exp(-3pi^2/(sqrt{2} g))$, is confirmed in its leading exponential by an Eliashberg gap equation in Appendix B.

Load-bearing premise

The derivation assumes that the retarded, soft magnetic part of the interaction is not itself renormalized as the cutoff is lowered, so that the only effect of each RG step is to transfer a calculable slice of soft gluon exchange into the instantaneous pairing channel, an assumption supported by the order-of-magnitude one-loop estimate in Appendix A rather than by a controlled calculation.

Editorial extensions

If this is right

At weak coupling the color-superconducting gap is parametrically larger than in any four-fermion or massive-gluon model, so the pairing temperature at a given density is higher than earlier estimates suggested.
The same RG treatment applies to every partial wave, so the leading exponential does not discriminate between $s$-wave and higher-$l$ pairing; a condensate with nonzero angular momentum, breaking rotational symmetry as in the A phase of $^3$He, is not excluded at leading order.
The wave-function renormalization from the magnetic interaction stays near one, $Z = 1 + O(g)$, at the scale where pairing sets in, so no non-Fermi-liquid behavior competes with superconductivity in this regime.
Residual quantum fluctuations of the almost static magnetic field are of order $g mu Delta$, small compared with the critical field of order $mu Delta$, so they should not destroy the condensate.
The leading exponential can be recovered from an Eliashberg gap equation, giving an independent route to the exponent and a practical starting point for numerical estimates at moderate density.

Reading between the lines

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

If the exponent really is $3pi^2/sqrt{2} g$, the gap at moderate couplings is large enough that the prefactor $g^{-5}$ and the order-one constants inside the logarithm become the dominant uncertainty for any phenomenological claim, so the natural next step is a numerical solution of the Eliashberg equation rather than more asymptotic analysis.
The argument that the retarded sector is unrenormalized is what fixes the coefficient of the linear term in the flow; a two-loop treatment that renormalizes the soft gluon exchange would shift the constant and hence the exponent, so the exponent is the sharpest test of the whole scheme.
The $l$-independence of the flow suggests the degeneracy among angular-momentum channels is broken only at relative order $g$, which would put the selection of the ground-state pairing channel in the same category as the subleading corrections to the gap itself.
The same reservoir mechanism should apply to any dense fermion system with an unscreened gauge field, for example a U(1) magnetic interaction in a metal, where the exponent would change from the BCS form even though the sign of the interaction there is repulsive.

Original abstract

We argue that in quark matter at high densities, the color magnetic field remains unscreened and leads to the phenomenon of color superconductivity. Using the renormalization group near the Fermi surface, we find that the long-range nature of the magnetic interaction changes the asymptotic behavior of the gap $\Delta$ at large chemical potential $\mu$ qualitatively. We find $\Delta\sim\mu g^{-5}\exp(-{3\pi^2\over\sqrt{2}}{1\over g})$, where $g$ is the small gauge coupling. We discuss the possibility of breaking rotational symmetry by the formation of a condensate with nonzero angular momentum, as well as interesting parallels to some condensed matter systems with long-range forces.

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, 3,722 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.86 · compares with prior results 0.25 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We argue that in quark matter at high densities, the color magnetic field remains unscreened and leads to the phenomenon of color superconductivity. Using the renormalization group near the Fermi surface, we find that the long-range nature of the magnetic interaction changes the asymptotic behavior of the gap $\Delta$ at large chemical potential $\mu$ qualitatively. We find $\Delta\sim\mu g^{-5}\exp(-{3\pi^2\over\sqrt{2}}{1\over g})$, where $g$ is the small gauge coupling. We discuss the possibility of breaking rotational symmetry by the formation of a condensate with nonzero angular momentum, as well as interesting parallels to some condensed matter systems with long-range forces.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We argue that in quark matter at high densities, the color magnetic field remains unscreened and leads to the phenomenon of color superconductivity.	result 0.51	0.42	0.89	asserted 0.61
s1	Using the renormalization group near the Fermi surface, we find that the long-range nature of the magnetic interaction changes the asymptotic behavior of the gap $\Delta$ at large chemical potential $\mu$ qualitatively.	method 0.58	0.96	0.12	asserted 1.00
s2	We find $\Delta\sim\mu g^{-5}\exp(-{3\pi^2\over\sqrt{2}}{1\over g})$, where $g$ is the small gauge coupling.	result 1.00	0.96	0.22	asserted 1.00
s3	We discuss the possibility of breaking rotational symmetry by the formation of a condensate with nonzero angular momentum, as well as interesting parallels to some condensed matter systems with long-range forces.	interpretation 0.52	0.45	0.04	hedged possibility 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

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

References

Works cited (15)

[11]

R. Pisarski and D.H. Rischke, nucl-th/9811104

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J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34, 1353 (1975)

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D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. Politzer, Phys. Rev. Lett. 30, 1346 (1973)

View on Pith 1973
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D. Bailin and A. Love, Phys. Rept. 107, 325 (1984), and references there in

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M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B 422, 247 (1998)

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R. Rapp, T. Sch¨ afer, E.V. Shuryak, and M. Velkovsky, Phys. R ev. Lett. 81, 53 (1998)

View on Pith 1998
[6]

J. Berges and K. Rajagopal, hep-ph/9804233

View on Pith Pith review arXiv
[7]

M. Alford, K. Rajagopal, and F. Wilczek, hep-ph/9804403

View on Pith Pith review arXiv
[8]

T. Sch¨ afer and F. Wilczek, hep-ph/9811473

View on Pith Pith review arXiv
[9]

G. Benfatto and G. Galavotti, Renormalization Group , Princeton University Press, Princeton, 1995; R. Shankar, Rev. Mod. Phys. 66, 129 (1993); J. Polchinski, hep-th/9210046

View on Pith Pith review arXiv 1993
[10]

N. Evans, S.D.H. Hsu, and M. Schwetz, hep-ph/9808444, hep-ph/9810514; T. Sch¨ afer and F. Wilczek, hep-ph/9810509

View on Pith Pith review arXiv
[12]

T. Holstein, R.E. Norton, and P. Pincus, Phys. Rev. B 6, 2649 (1973); M.Yu. Reizer, Phys. Rev. B 40, 11571 (1988); M.Yu. Reizer, Phys. Rev. B 44, 5476 (1991)

View on Pith 1973
[13]

See, e.g., C. Nayak and F. Wilczek, Nucl. Phys. B 417, 359 (1994); J. Polchinski, Nucl. Phys. B 422, 617 (1994); B.L. Altshuler, L.B. Ioﬀe, and A.J. Millis, Phys. Rev. B 50, 14048 (1994)

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[14]

B. Normand, P.A. Lee, and N. Nagaosa, Physica (Amsterdam) 185-189C, 1479 (1991); D.V. Khveshchenko, Phys. Rev. B 47, 3446 (1993); N.E. Bonesteel, I.A. McDonald, and C. Nayak, Phys. Rev. Lett. 77, 3009 (1996)

View on Pith 1991
[15]

See, e.g., G. Mahan, Many-Particle Physics , 2nd Ed., Plenum, New York, 1990, Section 9.7. 13

View on Pith 1990

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127 later papers cite it

Evidence: some. 126 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 9 of the 234 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 9 papers with this label

Solved its equation following this paper, with a slightly different calculation shown in an appendix.

Sakai-Sugimoto Brane System at High Density (2007) · 0.88 · paraphrased: the licence does not allow quoting

How the literature uses it

80 arXiv papers cite this in 143 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 · 98 sentences without a confident label

Uses its method or tool 5 papers

For this we use the leading order expression for the isospin density as a function of the isospin chemical potential obtained within the $χ$PT [ref] , i.e., $n_I=f_π^2μ_I(1-m_π^4/μ_I^4)$, where $m_π=140$ MeV and $f_π=92$ MeV are mass and decay constant of pion in the vacuum fixed according to the Review of Particle Physics [ref] .

Quarkyonic picture of isospin QCD · Onset of quarkyonic matter · 0.90
We solve Equation ([ref]), following References [ref] .

Sakai-Sugimoto Brane System at High Density · Solving Gap Equation · 1.00
As first proposed by Son [ref] one approximates [equation] Rewriting Eq. ([ref]) in terms of the new variables [ref] [equation] and differentiating twice with respect to $x$ one can transform the gap equation into the form of Airy's differential equation.

Some aspects of color superconductivity: an introduction · Solving the gap equation · 0.99

Uses its data or values 5 papers

The dispersion law of phonons in the CFL phase has been derived in [ref] , and one has [equation] In the asymptotic high density limit, the gap, $Δ$, can be computed from QCD [ref] [equation] where $b_0 = 512 π^4 (2/3)^5/2 exp (-π^2+4/8 )$ and $g$ is the QCD gauge coupling constant.

Dissipative superfluids, from cold atoms to quark matter · Quark matter at extremely high baryonic density · 0.80
The analytical results at high density contain the BCS gap $Δ$, which was computed for asymptotically high density in a weak-coupling approach in [ref] , as a parameter.

Exact results for two-color QCD at low and high density · Introduction · 0.93
In the asymptotic high density limit, the gap can be computed from QCD [ref] Δ≃μ_0 g^-5 exp(-3 π^2/√(2) g)b_0 , where $g$ is the QCD gauge coupling constant, and $b_0 = 512 π^4 (2/3)^5/2 exp (-π^2+4/8 )$ [ref] .

Bulk viscosities of a cold relativistic superfluid: color-flavor locked quark matter · Conformal limit · 0.85

Compares with it 6 papers

This behavior is clearly unphysical and contrasts with recent results available in the literature obtained from alternative approaches [ref] .

Dense and Cold Magnetized Quark Matter: A Review of Magnetic-Field-Independent Regularization and the Medium Separation Scheme · Medium Separation Scheme - MSS · 0.91
Ignoring the quark mass (which is legitimate at large-$μ_B$), the exponential scaling of the pairing gaps in the $1$ and $8$ of the unbroken $SU(3)_f$ symmetry takes the following form [ref] , [equation] which closely resembles [ref] upon setting $m_q = 0$ there.

Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases · High-Density QCD as a Gapless SPT · 0.89
With this choice of parameters we get $Δ(p=0)∼20-40$ MeV in the range of $μ$ considered here, in agreement with previous studies [ref] .

Topological Susceptibility in the Superconductive Phases of Quantum Chromodynamics: a Dyson-Schwinger Perspective · Solution of the Gap Equation for the CFL Phase · 0.97

Cites it as support 2 papers

Importantly, the gap size has a weaker exponential suppression because of the enhancement arising from the unscreened color-magnetic interaction [ref] .

Emergent QCD Kondo effect in two-flavor color superconducting phase · Coupled RG equations and RG-flow diagram · 0.98
Furthermore, we also emphasize that this conclusion is not modified by any possible color superconducting phases that may exist at high densities, since the corrections to bulk thermodynamics from these effects are exponentially suppressed $∼exp(-#/α_s^1/2)$ [ref] .

Equation of State of Cold Quark Matter to $O(\alpha_s^3 \ln \alpha_s)$ · Introduction · 0.97

Cites it as background 24 papers

In particular, as noted above, at asymptotically high densities $N_c=3$ QCD has a color-superconducting (`BCS') instability [ref] , and is expected to form a color superconductor with a non-trivial $⟨qq ⟩$ condensate.

Color Superconductivity at Large N: A New Hope · Introduction · 0.81
This estimation would be perturbatively complete up to higher-order corrections in $$ and nonperturbative effects such as pairing gaps [ref] .

Perturbative QCD meets phase quenching: The pressure of cold quark matter · Introduction · 0.81
Non-perturbative phenomenon in particular color superconductivity [ref] is believed to be active

Bayesian evidence for two peaks in the sound speed in cold dense QCD · Introduction · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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R. Pisarski and D.H. Rischke, nucl-th/9811104

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G. Benfatto and G. Galavotti, Renormalization Group , Princeton University Press, Princeton, 1995; R. Shankar, Rev. Mod. Phys. 66, 129 (1993); J. Polchinski, hep-th/9210046

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N. Evans, S.D.H. Hsu, and M. Schwetz, hep-ph/9808444, hep-ph/9810514; T. Sch¨ afer and F. Wilczek, hep-ph/9810509

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See, e.g., C. Nayak and F. Wilczek, Nucl. Phys. B 417, 359 (1994); J. Polchinski, Nucl. Phys. B 422, 617 (1994); B.L. Altshuler, L.B. Ioﬀe, and A.J. Millis, Phys. Rev. B 50, 14048 (1994)

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B. Normand, P.A. Lee, and N. Nagaosa, Physica (Amsterdam) 185-189C, 1479 (1991); D.V. Khveshchenko, Phys. Rev. B 47, 3446 (1993); N.E. Bonesteel, I.A. McDonald, and C. Nayak, Phys. Rev. Lett. 77, 3009 (1996)

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See, e.g., G. Mahan, Many-Particle Physics , 2nd Ed., Plenum, New York, 1990, Section 9.7. 13

View on Pith 1990