Superconductivity from perturbative one-gluon exchange in high density quark matter

Frank Wilczek (IAS; Princeton); Thomas Schaefer

REVIEW

Superconductivity from perturbative one-gluon exchange in high density quark matter

Thomas Schaefer, Frank Wilczek (IAS, Princeton)

High Energy Physics - Phenomenology · Submitted 25 Jun 1999 · 15 references

Pith's verdict: A careful, honest derivation that turns Son's exponent into a usable formula — the g⁻⁵ prefactor is real, but its coefficient is only pinned to order one, and the authors say so.

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arXiv IDhep-ph/9906512 (version 3)

Pith IDNF35M7I7, Pith's permanent record of this paper and its review

CitationFrank Wilczek (IAS, Princeton), Thomas Schaefer. "Superconductivity from perturbative one-gluon exchange in high density quark matter." arXiv:hep-ph/9906512 (1999). https://pith.science/paper/NF35M7I7

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Editor's note

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Worth your time, with one caveat about what exactly is nailed down. The new content is not the exponent — that's Son's, and the abstract says so plainly (abstract:s3). What Schaefer and Wilczek add is a derivation of the Eliashberg equation from the Dyson-Schwinger equation in the weak-coupling limit, an explicit demonstration that the gap is gauge-parameter independent once you keep only the piece that gaps the Fermi surface, and a numerical solution with the full hard-dense-loop propagator including both magnetic and electric exchange. The g⁻⁵ prefactor falls out, and the analytic rescaling argument in Eq. (20) is checked against the numerics rather than fitted to it. That's a clean piece of work, and the gauge-independence argument in particular is the kind of thing people had been hand-waving about. The paper is also refreshingly direct about its own limits: §5 says the coefficient is correct only up to a factor of order one, §4 says the Meissner effect will shift the numerical coefficient, and the running-coupling scale at pF = μ is chosen by hand because, as they put it, without a higher-order calculation one cannot fix it. The 100 MeV figure for μ < 1 GeV is flagged as quantitatively unreliable in the abstract itself (abstract:s4), so don't cite it as a prediction; cite it as an order-of-magnitude consistency check against the instanton-based estimates, which is how they present it. The stress-test concern about the coefficient is correct but bounded: the analytic 2.5·10⁴ g⁻⁵ versus numerical 1.4·10⁴ g⁻⁵ gap is a ~45% effect they attribute to dropping cosθ and the retardation terms, and the scaling form survives both routes. One open issue they raise themselves and don't resolve: they restrict to J = 0 pairing while noting that nearly collinear scattering might make higher partial waves relevant. That's a genuine loose end, not a flaw in what they did. My one real reservation is that the vertex-correction argument around Eq. (24) shows the asymptotic form is untouched but never quantifies the coefficient shift, so the prefactor's numerical value should be treated as indicative. Bottom line: this is the paper to cite for the parametric form of the gap and for the DS derivation. It deserves referees, and I'd bring it to reading group.

Referee report

Significance

This is a careful and timely derivation that puts Son's exponent on a firmer footing by obtaining it directly from the Dyson-Schwinger equation rather than from an on-shell scattering amplitude, and by exhibiting the gauge-parameter cancellation explicitly. The numerical solution of the full Eliashberg equation, including both magnetic and electric exchange with hard-dense-loop screening, is a genuine advance over schematic treatments, and the resulting g^-5 prefactor with its numerical coefficient is a falsifiable prediction that later work could check. The paper is also candid about its limitations: the truncation to a free vertex and no wave-function renormalization, the hand-picked running-coupling scale, and the unreliability of the mu < 1 GeV extrapolation are all stated in the text. The phenomenological estimate of a ~100 MeV gap, while hedged, is likely to be widely quoted and is the paper's most consequential number for neutron-star and heavy-ion applications.

Major objections (2)

[§5 and Eq. (20), compared with Figs. 5-6] The analytic rescaling argument of Eq. (20) gives c = 256 pi^4 g^-5 ~ 2.5 x 10^4 g^-5, while the numerical solution of the full Eq. (14) gives c ~ 1.4 x 10^4 g^-5, a discrepancy of roughly 45% that the text attributes to dropping cos(theta) and (q0-p0)^2 in passing from Eq. (14) to Eq. (19). Because the abstract advertises the g^-5 prefactor as a result, the reader needs to know which approximations control the power of g and which affect only the coefficient. Please state explicitly whether the g^-5 power survives the truncations listed in §4 (wave-function renormalization, vertex corrections, Meissner effect), or whether it too is only established up to the same order-one uncertainties as the coefficient.
[§3 and abstract:s4] The estimate that the gap can be as large as 100 MeV for mu < 1 GeV rests on a one-loop running coupling evaluated at the hand-picked scale pF = mu, in a regime where g is approximately 2 to 4. The paper acknowledges both the scale ambiguity ('without a higher order calculation one cannot fix the scale') and the breakdown of perturbation theory, but the abstract states the 100 MeV figure without an accompanying uncertainty. A scale-variation band, for example evaluating the coupling at mu/2 and 2mu, would quantify the truncation error and make the phenomenological claim meaningful. As written, the number carries an unstated order-one uncertainty that readers outside the immediate subfield are likely to overlook.

Minor objections (9)

[Eq. (10) and the following sentence] The denominator of the transverse and longitudinal propagators in Eq. (10) contains G, but the text immediately below says 'where D and F are functions of q0 and |q|'. The symbol should be G throughout, or the equation and the text should be reconciled; as it stands a reader cannot tell whether D and G are the same object.
[§2, paragraph following Eq. (13)] The sentence 'nonstatic modes are modes are dynamically screened due to Landau damping' contains a duplicated phrase and should be corrected.
[§2, paragraph following Eq. (13)] The sentence 'Only the first term term in (13) has a singularity on the Fermi surface' contains a duplicated word and should be corrected.
[Title and author line] The title is split as 'i n' and the author name appears as 'Sch¨ afer'; these are typesetting artifacts that should be fixed in the proofs.
[Fig. 2 and surrounding discussion] The text notes that the k = 0 curve provides an excellent fit even at small chemical potentials, which is surprising given that the analytic result carries a g^-5 prefactor. A sentence explaining that at large mu the g^-5 factor varies slowly and the k-fits are nearly degenerate would clarify what range of g the fit actually discriminates.
[Eq. (2) and the sentence following it] The statement that one can replace alpha . q-hat by the unit matrix using the equations of motion is what reduces the two-gap system to a single equation, so the equation-of-motion argument deserves one explicit sentence rather than a citation to the weak-coupling limit alone.
[Reference [14]] The title of the cited book contains the typo 'Phsysics' and should read 'Physics'.
[Fig. 1 caption] The caption reads 'the dashed lines shows', which should be 'the dashed lines show'.
[§4, Meissner-effect paragraph] The discussion of the Meissner effect is qualitative. A sentence on whether the Meissner mass enters at the same order as the dynamic screening scale qM in the Nf = 2 case, where only one gluon is Higgsed, would help the reader judge the claim that it affects only the numerical coefficient.

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The paper

The paper claims

Deriving the superconducting gap of dense quark matter from one-gluon exchange gives $g^{-5}e^{-3π²/(√2g)}$

The reading

At asymptotically high baryon density, quarks form a Fermi sea whose surface is unstable to pairing, and the paper asks what sets the size of the resulting color-superconducting gap. The authors derive the Eliashberg gap equation directly from the Dyson-Schwinger equation, keeping the frequency dependence of the gap, and show that the answer is controlled by magnetic gluon exchange whose infrared singularity is cut off by Landau damping rather than by any hand-inserted cutoff. Solving the equation numerically, they find $Δ ∼ g^{-5} e^{-3π^{2}/(√2 g)}$: the exponent reproduces the result Son obtained from an on-shell scattering amplitude, and the calculation goes further by fixing the $g^{-5}$ prefactor and its numerical coefficient (about $1.4×10^{4}g^{-5}$ when electric exchange is included). The result matters because it upgrades a scaling argument into a controlled weak-coupling calculation, and because the same machinery lets the authors quote a number: for chemical potentials below about 1 GeV the gap comes out near 100 MeV. The authors are explicit that at those densities the coupling is of order 2 to 4 and the calculation is no longer quantitatively reliable, so the 100 MeV figure is an extrapolation rather than a prediction.

What carries the argument

The Eliashberg equation for the gap as a function of imaginary frequency, obtained by projecting the Dyson-Schwinger equation onto the color-antitriplet, spin-singlet, positive-parity channel and reducing the momentum integral to the vicinity of the Fermi surface. The load-bearing ingredient is the hard-dense-loop gluon propagator: a longitudinal part with Debye mass $F = 2m^{2}$ and a transverse part $G$ with Landau-damping imaginary part, which together replace the schematic infrared cutoff used in earlier work. The angular integral over scattering angle then yields a logarithm $g^{2}/(18π^{2}) ∫ dq_0 log(1 + 64πμ/(N_f g^{2}|p_0-q_0|))$, whose logarithmic dependence on frequency is what forces the gap to vary with $p_0$ and produces the double-logarithmic exponent rather than a simple BCS $e^{-1/g^{2}}$.

Extended reading notes

Core claim

The paper's central claim is that the color-superconducting gap of two-flavor quark matter at asymptotically large chemical potential is $Δ ∼ g^{-5} e^{-3π^{2}/(√2 g)}$, with the coefficient in the exponent verified against Son's earlier result and the prefactor determined here. The route is a Nambu-Gorkov gap equation in which the gluon propagator is dressed with hard-dense-loop self-energies: electric modes are Debye screened at a scale $m_D$, while transverse magnetic modes are only dynamically screened, at a much smaller scale set by the gap itself. Because the angular integral is dominated by nearly collinear scattering, this magnetic screening regulates the would-be infrared divergence and produces the exponential. The authors solve the resulting integral equation numerically, first with magnetic exchange alone and then with electric exchange added, and confirm that the solution has the form of Son's approximate answer rescaled by an overall constant, in the full case with $c ≃ 1.4×10^{4}g^{-5}$. They also argue that the gap is gauge-parameter independent in this approximation, because it is fixed by quarks that are almost on shell, and that vertex corrections from hard dense loops do not alter the asymptotic form.

Load-bearing premise

The calculation assumes that leading-order truncation is enough to fix the prefactor and its coefficient: a free quark-gluon vertex, no fermion wave-function renormalization, a one-loop hard-dense-loop gluon propagator, and a running coupling evaluated by hand at the Fermi momentum. The paper itself notes that the Meissner effect will shift the numerical coefficient and that without a higher-order calculation the renormalization scale cannot be fixed, so the exponent is on firmer ground than the number in front of it.

Editorial extensions

If this is right

The color-superconducting gap at asymptotically high density is parametrically larger than a BCS-style estimate would suggest, because the exponent is $1/g$ rather than $1/g^{2}$.
The result fixes the overall coefficient of the $g^{-5}$ prefactor up to a factor of order one, which is what makes a numerical estimate at accessible densities possible at all.
Because the gap grows relative to $Λ_{QCD}$ as $μ → ∞$, the magnetic screening scale stays perturbative and the weak-coupling treatment is self-consistent in the asymptotic regime.
Extrapolating to $μ < 1$ GeV gives gaps of order 100 MeV, comparable to earlier estimates built from instanton models and schematic interactions normalized at zero density.
The gap's dependence on frequency cannot be replaced by a constant: doing so preserves the double-logarithmic structure but gets the constant in the exponent wrong.

Reading between the lines

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

If the gap really sits near 100 MeV at neutron-star densities, the pairing scale is a substantial fraction of the Fermi energy, which would make the superconducting transition temperature and the resulting cooling behavior of young neutron stars sensitive to details the asymptotic calculation cannot resolve.
The structure of the argument suggests a natural diagnostic: the ratio of the gap to the screening scale, and the location of the crossover where $q_M$ approaches $Λ_{QCD}$, could be used to mark where weak-coupling reasoning stops being trustworthy, independently of the numerical value of the gap.
Because the Meissner mass enters at the same parametric order as dynamic screening at finite momentum, a calculation that treats the two together would test whether the coefficient quoted here survives, rather than only whether the exponent does.
The same Eliashberg machinery, applied to three flavors where color-flavor locking gaps all gluons, could be run with the full Higgsed gluon spectrum to see whether the exponent is unchanged and only the prefactor shifts.

Original abstract

We study color superconductivity in QCD at asymptotically large chemical potential. In this limit, pairing is dominated by perturbative one-gluon exchange. We derive the Eliashberg equation for the pairing gap and solve this equation numerically. Taking into account both magnetic and electric gluon exchanges, we find $\Delta\sim g^{-5}\exp(-c/g)$ with $c=3\pi^2/\sqrt{2}$, verifying a recent result by Son. For chemical potentials that are of physical interest, $\mu< 1$ GeV, the calculation ceases to be reliable quantitatively, but our results suggest that the gap can be as large as 100 MeV.

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, 4,273 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.98 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We study color superconductivity in QCD at asymptotically large chemical potential. In this limit, pairing is dominated by perturbative one-gluon exchange. We derive the Eliashberg equation for the pairing gap and solve this equation numerically. Taking into account both magnetic and electric gluon exchanges, we find $\Delta\sim g^{-5}\exp(-c/g)$ with $c=3\pi^2/\sqrt{2}$, verifying a recent result by Son. For chemical potentials that are of physical interest, $\mu< 1$ GeV, the calculation ceases to be reliable quantitatively, but our results suggest that the gap can be as large as 100 MeV.

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 study color superconductivity in QCD at asymptotically large chemical potential.	contribution 0.79	0.87	0.04	asserted 0.77
s1	In this limit, pairing is dominated by perturbative one-gluon exchange.	background 0.91	0.11	0.07	asserted 1.00
s2	We derive the Eliashberg equation for the pairing gap and solve this equation numerically.	method 0.96	0.97	0.06	asserted 1.00
s3	Taking into account both magnetic and electric gluon exchanges, we find $\Delta\sim g^{-5}\exp(-c/g)$ with $c=3\pi^2/\sqrt{2}$, verifying a recent result by Son.	result 1.00	0.92	0.73	asserted 1.00
s4	For chemical potentials that are of physical interest, $\mu< 1$ GeV, the calculation ceases to be reliable quantitatively, but our results suggest that the gap can be as large as 100 MeV.	limitation 0.95	0.77	0.18	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.61.

What was examined, and what was not

sources read

arXiv listing abstract v3

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. Solution of the Eliashberg equation (17) as a functio

**Figure 2.** FIG. 2. Dependence of the gap on the chemical potential for th

**Figure 3.** FIG. 3. Same as figure 1 for the solution of the Eliashberg equa

Showing 3 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9906512 by the authors.

References

Works cited (15)

[1]

S. C. Frautschi, Asymptotic freedom and color superconductiv ity in dense quark matter, in: Proceedings of the Workshop on Hadronic Matter at Extreme En ergy Density, N. Cabibbo, Editor, Erice, Italy (1978)

View on Pith 1978
[2]

F. Barrois, Nucl. Phys. B129, 390 (1977)

View on Pith 1977
[3]

M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B537, 443 (1999)

View on Pith 1999
[4]

T. Sch¨ afer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999)

View on Pith 1999
[5]

D. Bailin and A. Love, Phys. Rept. 107, 325 (1984)

View on Pith 1984
[6]

D. T. Son, Phys. Rev. D 59 , 094019 (1999); hep-ph/9812287

View on Pith Pith review arXiv 1999
[7]

T. Sch¨ afer and F. Wilczek, Phys. Lett. B450, 325 (1999); hep-ph/9810509

View on Pith Pith review arXiv 1999
[8]

N. Evans, S. Hsu, and M. Schwetz, Nucl. Phys. B551, 275 (1999) and Phys. Lett. B449, 281 (1999)

View on Pith 1999
[9]

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

View on Pith Pith review arXiv
[10]

M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998)

View on Pith 1998
[11]

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

View on Pith 1998
[12]

R. D. Pisarski, D. H. Rischke, preprint, nucl-th/9903023

View on Pith Pith review arXiv
[13]

D. K. Hong, preprints, hep-ph/9905523 and hep-ph/981251 0

View on Pith Pith review arXiv
[14]

A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Phsysics , Prentice Hall, Englewood Cliﬀs, New Jersey 1963

View on Pith 1963
[15]

M. LeBellac, Thermal Field Theory , Cambridge University Press, Cambridge 1996. 12 FIGURES FIG. 1. Solution of the Eliashberg equation (17) as a functio n of imaginary frequency q0. The upper and lower panels show the solution for µ = 400 MeV and µ = 10 10 MeV, respectively. The solid lines show the numerical solution and the dashed lines shows the approx...

View on Pith 1996

Forward citations

Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Inverse meson mass ordering in color-flavor-locking phase of high density QCD
hep-ph 1999-10 accept novelty 8.0 of 10

In the color-flavor-locking phase of dense QCD, meson masses depend only on bare quark masses and are partly inverted, with the eta-prime the lightest state.
Symmetric nuclear matter from the strong interaction
nucl-th 2019-07 conditional novelty 7.0 of 10

A QCD-based calculation of symmetric nuclear matter, joined to chiral effective field theory, predicts a speed-of-sound maximum above 1/3 that tracks the onset of a diquark gap.
Crystalline Color Superconductivity
hep-ph 2000-08 accept novelty 7.0 of 10

When up and down quark chemical potentials differ by a suitable amount, quark matter favors a crystalline color superconducting (LOFF) phase over both uniform BCS pairing and no pairing, and this phase necessarily con...
Renormalization group analysis of color superconductivity revisited
hep-ph 2025-08 conditional novelty 6.0 of 10

An RG treatment with self-energy corrections reproduces the known O(g^0) color-superconducting gap and claims to fix its overall coefficient, implying a factor-two reduction.
Towards holographic color superconductivity in QCD
hep-th 2025-05 conditional novelty 6.0 of 10

In the V-QCD holographic model, quark pairing transitions can occur with critical temperatures up to about 30 MeV, but spatially modulated instabilities grow faster and may dominate.

Show 6 more citing papers

High Density Quark Matter under Stress
hep-ph 2001-05 accept novelty 6.0 of 10

Flavor breaking in color-flavor-locked quark matter triggers kaon or pion condensation at a scale of order sqrt(m ms) times Delta over pF, parametrically much smaller than the gap Delta.
Fierz-complete NJL model study III: Emergence from quark-gluon dynamics
hep-ph 2019-09 conditional novelty 5.0 of 10

Starting from the QCD action with all four-quark couplings set to zero in the ultraviolet, gluon exchange dynamically generates a Fierz-complete set whose dominant channel switches from chiral to diquark as quark chem...
Crystalline color superconductors
hep-ph 2013-02 accept novelty 3.0 of 10

Crystalline color superconducting quark matter can be energetically favored over homogeneous phases in compact-star cores and is chromomagnetically stable, with a shear modulus 20 to 1000 times that of the neutron-star crust.
The phase diagram of dense QCD
hep-ph 2010-05 accept novelty 3.0 of 10

A structured survey of the QCD phase diagram that separates firmly established results (lattice QCD at small baryon density, weak-coupling colour superconductivity at asymptotically high density) from conjectured phas...
The CFL phase and m_s: An effective field theory approach
nucl-th 2006-02 unverdicted novelty 2.0 of 10

Increasing the strange quark mass drives color-flavor-locked quark matter through a kaon condensate into a gapless supercurrent phase, according to effective field theory.
Advances in perturbative thermal field theory
hep-ph 2003-10 accept novelty 2.0 of 10

A review arguing that effective field theory and hard-thermal-loop resummation restore predictive power to perturbative thermal field theory, including hot QCD down to a few times the deconfinement temperature.

How the literature uses it

32 arXiv papers cite this in 58 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 · 38 sentences without a confident label

Uses its method or tool 1 paper

To resolve this, one may integrate the momenta about the Fermi surface as often done in QCD studies at high densities in the presence of a Cooper instability, see, e.g., Refs. [ref] .

Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order · Two-loop Contribution to the Effective Potential · 0.84

Uses its data or values 1 paper

The dependence of $Δ$ on $g$ has been computed as $Δ∼μg^-5 ^-c/g$ with $c=3π^2/√(2)$ for $N_c=3$ and $c=2π^2$ for $N_c=2$ [ref] .

Overscreened Kondo effect, (color) superconductivity and Shiba states in Dirac metals and quark matter · Taxonomy of impurity states in multichannel (color) superconductors · 0.93

Compares with it 2 papers

It is interesting to compare this results to the one gluon exchange perturbative calculations [ref] .

Asymptotically conformal CFL quark matter within a nonlocal chiral quark model · Nonlocal NJL model at high densities · 1.00
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 background 15 papers

We also apply the model to modelling the color-superconducting color-flavor-locked (CFL) phase of quark matter [ref] , which is expected to be the ground state of dense QCD [ref] .

Asymptotically conformal CFL quark matter within a nonlocal chiral quark model · Introduction · 0.80
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
In this work, we compute $χ$ in superdense phases of QCD, in which the system could be in a color-superconductive phase [ref] .

Topological Susceptibility in the Superconductive Phases of Quantum Chromodynamics: a Dyson-Schwinger Perspective · Introduction · 0.92

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.

[1] [1]

S. C. Frautschi, Asymptotic freedom and color superconductiv ity in dense quark matter, in: Proceedings of the Workshop on Hadronic Matter at Extreme En ergy Density, N. Cabibbo, Editor, Erice, Italy (1978)

View on Pith 1978

[2] [2]

F. Barrois, Nucl. Phys. B129, 390 (1977)

View on Pith 1977

[3] [3]

M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B537, 443 (1999)

View on Pith 1999

[4] [4]

T. Sch¨ afer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999)

View on Pith 1999

[5] [5]

D. Bailin and A. Love, Phys. Rept. 107, 325 (1984)

View on Pith 1984

[6] [6]

D. T. Son, Phys. Rev. D 59 , 094019 (1999); hep-ph/9812287

View on Pith Pith review arXiv 1999

[7] [7]

T. Sch¨ afer and F. Wilczek, Phys. Lett. B450, 325 (1999); hep-ph/9810509

View on Pith Pith review arXiv 1999

[8] [8]

N. Evans, S. Hsu, and M. Schwetz, Nucl. Phys. B551, 275 (1999) and Phys. Lett. B449, 281 (1999)

View on Pith 1999

[9] [9]

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

View on Pith Pith review arXiv

[10] [10]

M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998)

View on Pith 1998

[11] [11]

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

View on Pith 1998

[12] [12]

R. D. Pisarski, D. H. Rischke, preprint, nucl-th/9903023

View on Pith Pith review arXiv

[13] [13]

D. K. Hong, preprints, hep-ph/9905523 and hep-ph/981251 0

View on Pith Pith review arXiv

[14] [14]

A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Phsysics , Prentice Hall, Englewood Cliﬀs, New Jersey 1963

View on Pith 1963

[15] [15]

M. LeBellac, Thermal Field Theory , Cambridge University Press, Cambridge 1996. 12 FIGURES FIG. 1. Solution of the Eliashberg equation (17) as a functio n of imaginary frequency q0. The upper and lower panels show the solution for µ = 400 MeV and µ = 10 10 MeV, respectively. The solid lines show the numerical solution and the dashed lines shows the approx...

View on Pith 1996