Renormalization of the three-flavor quark-meson diquark model

Jens O. Andersen; Mathias P. N{\o}dtvedt

REVIEW 3 major objections 4 minor 2 cited by

Renormalization of the three-flavor quark-meson diquark model

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The quark-meson diquark model of three-flavor color superconductivity can be renormalized at one loop, with diquark-sector counterterms and renormalization-group running; in ideal CFL matter the pairing gap tends to a constant and the…

desk verdict Substantial three-flavor extension, but the load-bearing subtraction term Eq. (107) is asserted, not derived; send to review with a demand for derivation. read the letter →

arxiv 2506.02941 v2 pith:IMTSUK5Z submitted 2025-06-03 hep-ph nucl-th

Jens O. Andersen , Mathias P. N{o}dtvedt This is my paper

classification hep-phnucl-th

keywords colorsuperconductivityCFLphasequark-mesondiquarkmodelrenormalizationthermodynamicpotentialgrouppairinggapspeedofsound

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends a low-energy effective description of dense quark matter, in which the degrees of freedom are quarks, mesons, and diquark fields, and establishes that it can be renormalized at one loop in both the two-flavor color-superconducting (2SC) phase and the three-flavor color-flavor-locked (CFL) phase. The scalar-sector parameters are fixed by matching meson pole masses and decay constants to their observed values, while the diquark-sector couplings, unknown a priori, receive counterterms and renormalization-group equations. A careful reader should care because the standard Nambu-Jona-Lasinio treatment of color superconductivity needs a momentum cutoff and develops artifacts at large chemical potential, whereas here the ultraviolet divergences are subtracted in a controlled dimensional-regularization scheme. Applying the renormalized potential to ideal CFL matter, the authors show that the pairing gap approaches a constant as the baryon chemical potential goes to infinity and that the speed of sound approaches the conformal value from above.

What carries the argument

The load-bearing object is the one-loop quark contribution to the thermodynamic potential, written as a sum over determinants of Nambu-Gorkov matrices, with $6\times 6$ blocks for the two-flavor pairings and an additional $12\times 12$ matrix in the CFL phase, whose entries are quark masses, chemical potentials, and diquark gaps. Divergences are isolated by a subtraction term obtained by expanding the determinant integrand around zero chemical potentials; the divergent part reduces to two master integrals, and counterterms are read off by matching powers of the scalar and diquark expectation values. The renormalization-group equations follow from demanding that bare parameters be scale independent; for example $g_{\Delta,\mathrm{MS}}^2(\Lambda) = g_{\Delta,0}^2 / [1 - \frac{4g_{\Delta,0}^2}{(4\pi)^2}\log(\Lambda^2/\Lambda_0^2)]$. This running is what makes the final thermodynamic potential cutoff-free and scale-improved.

What would settle it

Compute the full one-loop quark thermodynamic potential in the CFL phase with nondegenerate quark masses and unequal gaps in $d = 4 - 2\epsilon$ dimensions, subtract Eq. (107), and check that all $1/\epsilon$ poles cancel and the finite remainder is independent of the auxiliary mass $M$ and of the renormalization scale $\Lambda_0$ once running couplings are inserted; an uncancelled pole or residual $M$-dependence would show the subtraction term misses a divergence.

Watch

Extended reading notes

Core claim

The paper's central claim is that the quark-meson diquark model remains a renormalizable effective theory of color superconductivity once diquark degrees of freedom are included. In the 2SC phase with two flavors and in the CFL phase with three, the one-loop quark contribution to the thermodynamic potential is brought to a finite form for arbitrary quark masses and gaps by adding counterterms for the diquark-sector couplings: Eqs. (56)-(61) in the two-flavor case and Eqs. (111)-(116) in the three-flavor case. The divergences are isolated through a subtraction term built by expanding the quark determinant around vanishing chemical potentials, and the finite remainder is evaluated numerically in three dimensions. For the idealized CFL ground state with three massless quarks, the renormalized potential gives a gap that approaches a constant at asymptotic densities, $\Delta_0 = 2^{-1/3}\Delta_{2\mathrm{SC},0}$, and a speed of sound $c_s^2 = \frac{1}{3}\left(1+\frac{4}{3}\frac{g_\Delta^2\Delta^2}{\bar\mu^2}\right)$, which relaxes to the conformal value $1/3$ from above.

Load-bearing premise

Renormalizability hinges on the subtraction procedure that expands the quark-loop integrand around vanishing chemical potentials to isolate the divergences, and on the assertion that these subtractions add no new infrared divergences and are independent of the auxiliary mass $M$ in the three-flavor case; if that assertion fails, every counterterm determined from it is wrong.

Editorial extensions

If this is right

The two-flavor model can be extended with the $\eta$ and $\vec a$ fields, allowing unequal up and down quark masses or chemical potentials while remaining renormalizable, which is needed for charge-neutral 2SC matter.
In the CFL phase, the seven diquark-sector operators introduced by three flavors have determined counterterms, so computations no longer depend on how the ultraviolet cutoff is chosen.
In ideal CFL matter the octet pairing gap becomes constant at asymptotically large baryon chemical potential, and the relation $\Delta_0 = 2^{-1/3}\Delta_{2\mathrm{SC},0}$ connects the three-flavor result to the two-flavor one.
The speed of sound satisfies $c_s^2 > 1/3$ at finite density and approaches $1/3$ from above as $\mu_B \to \infty$, a behavior this model shares with its isospin-condensed counterparts.
With charge neutrality imposed, the renormalized potential gives pressure and energy density as functions of $\mu_B$ alone, so the model can be used to build hybrid-star equations of state.

Reading between the lines

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

A natural next test is to verify numerically, order by order, that observables computed from the renormalized potential are independent of the arbitrary subtraction mass $M$ and the reference scale $\Lambda_0$; the paper asserts this but does not display the cancellation.
The same subtraction-by-expansion method should transfer to the pion-condensed phase with isospin chemical potential, where the paper notes the same sound-speed behavior appears; a direct three-flavor calculation there would test the method's reach.
If the asymptotic speed of sound really exceeds the conformal value in this model, hybrid-star equations of state built from it will be stiffer at intermediate densities than conformal parametrizations, shifting predictions for neutron-star radii and tidal deformability; this consequence goes beyond what the paper computes.
Comparing the constant-gap limit with a cutoff-regulated NJL calculation would show which features of the high-density behavior are generic and which depend on renormalizability.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on 2506.02941. The genuinely new piece is the three-flavor QMD model with left- and right-handed diquarks, the one-loop counterterms for the diquark-sector operators, and the RG equations. The two-flavor extension with isospin breaking is a natural but useful addition. The asymptotic results—constant gap, cs^2 approaching 1/3 from above—are not new physics; they reproduce known model behavior and Schäfer's relation between the octet and 2SC gaps. The authors are clear that the diquark couplings are free parameters and that numerical fits to neutron-star observables are postponed.

The main risk is the three-flavor subtraction term, Eq. (107). The last term, with the arbitrary mass M, is asserted rather than derived. The sentence "The M-dependence drops out in the final result" may be true, but it is not demonstrated anywhere, because the finite remainder Omega_fin^CFL is never computed. If that term is incomplete—wrong coefficient or wrong flavor-color structure—the 1/epsilon poles don't all cancel and the renormalized potential and RG equations inherit the error. This is a correctness issue in the central claim, not a presentation issue. The two-flavor sign discrepancy between Eqs. (43) and (45) is real in appearance but reconciled by the overall minus in the definition of Omega_2SC,div; I agree with the stress-test note that the two-flavor case is not where the risk lives.

A second, smaller gap: the simplified potential used for the gap and speed-of-sound calculation, Eq. (130), appears without a derivation showing the reduction to the massless limit. That's not fatal—it's a straightforward limit if the machinery works—but it adds to the burden on the reader.

Who should read this? Researchers building effective models for color superconductivity and hybrid neutron-star equations of state. The model definition and the two-flavor renormalization are usable now; the three-flavor part is not fully established until Eq. (107) is derived and the M-independence shown. I would send it to peer review: the technical content is substantial and the issues are fixable. But I'd ask for the derivation before accepting.

I wouldn't cite the three-flavor renormalization yet. The model Lagrangian and the two-flavor part are citable.

Referee Report

3 major / 4 minor

Summary. The manuscript extends the two-flavor quark-meson diquark (QMD) model of Andersen and Nødtvedt to three flavors, with the goal of providing a renormalizable effective description of color superconductivity. For the 2SC phase the authors construct a one-loop subtraction term, fix MS counterterms in the diquark sector, derive renormalization-group equations, and present a renormalized thermodynamic potential. For the CFL phase they repeat the construction for arbitrary quark masses and gaps, obtain counterterms and RG equations for seven diquark-sector couplings, and give the renormalized potential. As an application, they derive the zero-temperature gap equation in the ideal massless CFL phase, finding a gap that tends to a constant at large baryon chemical potential and a speed of sound that approaches the conformal value 1/3 from above.

Significance. If the derivation is correct, the paper provides a substantial extension of a renormalizable low-energy model for dense QCD, with explicit counterterms, RG-improved thermodynamic potentials, and a nontrivial consistency check: Eq. (131) reproduces the known asymptotic relation between the CFL and 2SC gaps, Delta_CFL = 2^{-1/3} Delta_2SC, first derived in perturbative QCD by Schäfer. The prediction that the speed of sound approaches the conformal limit from above is a falsifiable model statement. The main weakness is that the three-flavor subtraction term, Eq. (107), is asserted rather than derived; because this term determines the 1/epsilon pole and the log M contribution in the renormalized CFL potential, the central renormalizability claim is not yet fully established.

major comments (3)

[Sec. V, Eq. (107)] The CFL subtraction term is asserted, not derived: the last term, -g_Delta^4 (Delta_ud^2 Delta_us^2 + Delta_ud^2 Delta_ds^2 + Delta_us^2 Delta_ds^2)/(p^2+M^2)^{3/2}, depends on an arbitrary mass M and is not obtained by expanding the determinant (98) around vanishing chemical potentials. This term controls both the 1/epsilon pole in Eq. (108) and the log M contribution in Eq. (123), so the counterterms (111)-(116) and the renormalized CFL potential inherit its correctness. The statement that "the M-dependence drops out in the final result" cannot be verified from the manuscript because Omega_CFL,fin is not computed and Eq. (123) explicitly contains log( Lambda_0^2 / M^2 ). Please provide the derivation of this subtraction term and explicitly demonstrate the cancellation of M; without this, the pole cancellation and the renormalized potential are not established.
[Sec. III, Eqs. (43) and (45)] The integrated result Eq. (45) is inconsistent with the subtraction integrand Eq. (43). In Eq. (43) the chemical-potential terms carry a factor g_Delta^2 Delta_ud^2 in the numerator, but Eq. (45) displays -2(mu_ur+mu_dg)^2 Delta_ud^2/(4 pi)^2 ... and -2(mu_ug+mu_dr)^2 Delta_ud^2/(4 pi)^2 ... without g_Delta^2. With the factor g_Delta^2 restored, the 1/epsilon poles cancel against delta Z_Delta and the final renormalized potential Eq. (76) is consistent; without it, the pole cancellation fails. Please correct Eq. (45), or state explicitly if a rescaling of Delta is intended.
[Secs. III and V, subtraction scheme] The subtraction scheme is defined by expanding the integrand around vanishing chemical potentials, and it is asserted that the subtraction terms introduce no additional infrared divergences. This assertion is load-bearing because Omega_fin = Omega_1 - Omega_div is meant to be evaluated numerically in d=3 dimensions, and the denominators in Eqs. (43) and (107) vanish at p=0 when the corresponding gap parameter vanishes. Please state the precise conditions under which the subtraction is infrared safe, and verify them for the CFL subtraction term (107), including the mixed-gap term with the arbitrary mass M.

minor comments (4)

[Eq. (123)] The shorthand "d->s+u->s" together with the remark that there is an additional factor of sqrt(2) in front of any phi_s under permutation is difficult to follow; please spell out the full permutation sum so the reader can reproduce the potential.
[Eqs. (107)-(108)] The phrase "with cyclic permutation over flavor in the first line and cyclic permutation over flavor and color in the second line" is ambiguous because the displayed expressions are not grouped by line in a way that makes the cyclic permutations explicit. Please define the permutation sums unambiguously.
[Sec. VI, first paragraph] The sentence "We have renormalized the thermodynamic potential ... and determined the counterterms of the couplings involving the diquark degrees of freedom. in the process." contains a punctuation error; the fragment "in the process" should be attached to the preceding sentence.
[References] The spelling of the second author's name is inconsistent: Ref. [20] uses "Nødetvedt" while the title page and Refs. [21] use "Nødtvedt"; please verify the bibliographic entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the one-loop renormalization and the asymptotic CFL results are self-contained, with external benchmarks; the open subtraction-term issue is a correctness risk, not a circularity.

full rationale

The paper's central chain is a direct one-loop calculation: the quark determinant is written in Gorkov form (Eqs. (98)-(106)), a subtraction term is added and subtracted (Eqs. (107)-(108)), counterterms are fixed by requiring exact cancellation of the 1/epsilon poles (Eqs. (111)-(116)), and the RG equations follow from the independence of bare parameters with respect to the renormalization scale (Eqs. (117)-(122)). None of the counterterms is fitted to a quantity that is later presented as a prediction. The asymptotic application solves the gap equation obtained from the renormalized potential (Eq. (130)) and yields Delta_0 = 2^{-1/3} Delta_2SC_0 and c_s^2 = (1/3)(1 + (4/3) g_Delta^2 Delta_0^2 / mu_bar^2); these are compared with, not taken from, the known perturbative-QCD and NJL results (Refs. [36-38]). The self-citations (Refs. [20,21]) supply previous two-flavor results and context but are not the argument for the three-flavor counterterms; the scalar-sector counterterms are also backed by external Refs. [30,44]. The place a reader should worry is Eq. (107), where the last subtraction term contains an arbitrary mass M and the statement that 'the M-dependence drops out in the final result' is asserted rather than demonstrated, while Omega_CFL,fin is not evaluated. That is a correctness and completeness risk in the claimed renormalization, not a circular reduction: Omega_CFL,fin is defined as the exact difference Omega_CFL,1 - Omega_CFL,div, so no prediction is being set equal to its input by construction. I therefore find no definable circular step.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The model introduces a diquark effective sector with eight free parameters (m_Delta^2, g_Delta, lambda3, lambda4, lambda5, lambda_Delta1, lambda_Delta2, lambda_Delta3). The scalar-sector parameters are matched to meson masses and decay constants in the on-shell scheme, as established in prior work, but the matching is not performed in this paper. The central renormalization procedure relies on the subtraction method described in the text.

free parameters (8)

m_Delta^2 (diquark mass parameter)
Free parameter of the diquark sector; enters the thermodynamic potential. Subdominant in the asymptotic gap equation.
g_Delta (quark-diquark coupling)
Free coupling; controls the gap and speed of sound in the asymptotic application.
lambda3 (scalar-diquark coupling)
Free coupling between mesons and diquarks; appears in the CFL tree potential and RG flow.
lambda4 (scalar-diquark coupling)
Free coupling; enters the CFL tree potential and counterterms.
lambda5 (scalar-diquark coupling)
Free coupling; appears in the CFL tree potential and counterterms.
lambda_Delta1 (diquark self-coupling)
Free quartic diquark self-coupling; enters the thermodynamic potential and RG equations.
lambda_Delta2 (diquark self-coupling)
Free quartic diquark self-coupling; enters the thermodynamic potential and RG equations.
lambda_Delta3 (diquark self-coupling)
Free quartic diquark self-coupling involving both left- and right-handed diquarks.

assumptions (5)

domain assumption The effective Lagrangian is constructed from symmetry and renormalizability, with diquark degrees of freedom as the correct low-energy description of color superconductivity.
Sec. I and Sec. IV state that the allowed terms are dictated by symmetry and renormalizability, and that the diquark couplings are unknown free parameters.
domain assumption The quark determinant's ultraviolet divergences are fully captured by expanding around vanishing chemical potentials, and the subtraction terms introduce no additional infrared divergences.
Sec. III before Eq. (43): 'We will construct subtraction terms that have the same ultraviolet behavior as the integrand... It is important that the subtraction term does not introduce any additional infrared divergences.'
domain assumption The one-loop approximation with mesons at tree level is sufficient for the renormalization and for the asymptotic gap and speed of sound.
Sec. III and Sec. V compute only quark-loop contributions; meson loops and higher-order effects are neglected.
standard math Standard dimensional regularization and the MS scheme are valid for the thermodynamic potential at finite chemical potential.
Used throughout Secs. III and V to regulate and renormalize the momentum integrals.
domain assumption Color superconductivity and pion condensation are mutually exclusive for two flavors, allowing a term to be dropped.
Sec. II, after Eq. (32), cites Ref. [26] for this exclusion.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Renormalization of the three-flavor quark-meson diquark model." pith.science (2026). https://pith.science/paper/IMTSUK5Z

@misc{pith2026250602941,
  author       = {Pith},
  title        = {Pith review of: Renormalization of the three-flavor quark-meson diquark model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMTSUK5Z}},
  note         = {Machine review of arXiv:2506.02941}
}

abstract

We discuss the properties of the two- and three-flavor quark-meson diquark (QMD) model as a renormalizable low-energy effective model for color superconductivity in dense QCD. The effective degrees of freedom are scalars, pseudo-scalars, diquarks, and quarks. The parameters in the scalar/pseudo-scalar sector can be determined by matching the meson pole masses and decay constants to their observed values using the on-shell renormalization scheme. The remaining parameters are in the diquark sector and a priori unknown. In principle, they can be calculated from QCD, but we consider them free. We renormalize the thermodynamic potential in the 2SC phase for two flavors and in the color-flavor-locked (CFL) phase for three flavors, determining the counterterms of the couplings in the diquark sector. We derive a set of renormalization group equations for these couplings that are used to improve the thermodynamic potential. As an application, we calculate the gap and the speed of sound in the ideal CFL phase. It is shown that the gap approaches a constant as $\mu_B\rightarrow\infty$ and that the speed of sound relaxes to the conformal limit from above.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

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

Comparison of chiral limit studies in curvature mass versus on-shell renormalized quark-meson model using ChPT
hep-ph 2025-07 conditional novelty 6.0 of 10

Using ChPT scaling of f_pi, f_K, and M_eta^2, the QMVT model yields much smaller first-order regions in the Columbia plot than the on-shell renormalized RQM model, and moderately smaller than the FRG study.
Comparative analysis of critical regions: The renormalized quark-meson model under Polyakov loop, quark back-reaction, and vector interaction effects
hep-ph 2025-12 conditional novelty 4.0 of 10

On-shell renormalized quark-meson models give broad, roughly symmetric critical regions around the CEP; the Logarithmic Polyakov loop compresses their temperature width, while the PolyLog-glue back-reaction restores s...

Reference graph

Works this paper leans on

44 extracted references · 31 canonical work pages · cited by 2 Pith papers

[1]

B. A. Freedman and L. D. McLerran, Phys. Rev. D16, 1147 (1977)

1977
[2]

B. A. Freedman and L. D. McLerran, Phys. Rev. D16, 1169 (1977)

1977
[3]

Baluni, Phys

V. Baluni, Phys. Rev. D17, 2092 (1978)

work page 1978
[4]

Gorda, A

T. Gorda, A. Kurkela, P. Romatschke, S. S¨ appi, and A. Vuorinen, Phys. Rev. Lett.121, 202701 (2018)

work page 2018
[5]

Gorda, R

T. Gorda, R. Paatelainen, S. S¨ appi, and K. Sepp¨ anen, Phys. Rev. Lett.131, 181902 (2023)

work page 2023
[6]

Fernandez and J.-L

L. Fernandez and J.-L. Kneur, Phys. Rev. Lett.129, 212001 (2022); Phys. Rev. D111, 034020 (2025)

work page 2022
[7]

Annala, T

E. Annala, T. Gorda, J. Hirvonen, O. Komoltsev, A. Kurkela, J. N¨ attil¨ a, and A. Vuorinen, Nature Comm.14, 8451 (2023)

work page 2023
[8]

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

work page 1978

Show all 44 references

[9]

B. C. Barrois, PhD thesis(CALTECH,UMI 79-04847)
[10]

Bailin and A

D. Bailin and A. Love, J. Phys. A 12, L283 (1979), Phys. Rept.107, 325 (1984)

1979
[11]

Alford, K

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

1998
[12]

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

1998
[13]

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

1999
[14]

Kurkela, K

A. Kurkela, K. Rajagopal, and R. Steinhorst, Phys. Rev. Lett.132, 262701 (2024)

2024
[15]

Rajagopal and F

K. Rajagopal and F. Wilczek, At the frontier of parti- cle physics, Vol. 3 (World Scientific, Singapore, p 2061) (2001)

2001
[16]

M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨ afer, Rev. Mod. Phys.80, 1455 (2008)

2008
[17]

Fukushima and T

K. Fukushima and T. Hatsuda, Rept. Prog. Phys.74, 014001 (2011)

2011
[18]

G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, and T. Takatsuka, Rept. Prog. Phys.81, 056902 (2018)

2018
[19]

Gholami, M

H. Gholami, M. Hofmann, and M. Buballa, Phys. Rev. D111, 014006 (2025)

2025
[20]

J. O. Andersen and M. P. Nødetvedt, Phys. Rev. D111, 034031 (2025)

2025
[21]

J. O. Andersen and M. P. Nødetvedt, e-Print: 2502.10229 [hep-ph]

arXiv
[22]

Braun, M

J. Braun, M. Leonhardt, and J. M. Pawlowski, SciPost Phys. 6, 056 (2019)

2019
[23]

Braun and B

J. Braun and B. Schallmo, Phys. Rev. D 105 3 (2022)

2022
[24]

Gholami, L

H. Gholami, L. Kurth, U. Mire, M. Buballa, and Bernd- Jochen Sch¨ afer, e-Print: 2505.22542 [hep-ph]

arXiv
[25]

I. A. Shovkovy, Found. Phys.35, 1309 (2005)

2005
[26]

P. F. Bedaque, Nucl. Phys. A697, 569 (2002)

2002
[27]

Casalbuoni, Z

R. Casalbuoni, Z. Duan, and F. Sannino, Phys. Rev. D 62, 094004 (2000)

2000
[28]

S. B. R¨ uster, I. A. Shovkovy and D. H. Rischke Nucl. Phys. A743, 127 (2004)

2004
[29]

S. B. Ruester, V. Werth, M. Buballa, I. A. Shovkovy, D. H. Rischke, Phys. Rev. D72, 034004 (2005)

2005
[30]

S. K. Rai and V. K. Tiwari, Phys. Rev. D105, 094010 (2022)

2022
[31]

J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich, Phys. Rev. D62, 085008 (2000)

2000
[32]

Roder, J

D. Roder, J. Ruppert, and D. H. Rischke, Phys. Rev. D 68, 016003 (2003)

2003
[33]

D. T. Son and M. Stephanov, Phys. Rev. D61, 074012 (2000)

2000
[34]

M. G. Alford, M. Braby, and A. Schmitt, J. Phys. G 35, 025002 (2008)

2008
[35]

Schechter and Y

J. Schechter and Y. Ueda, Phys. Rev. D3, 168 (1971)

1971
[36]

Sch¨ afer, Nucl

T. Sch¨ afer, Nucl. Phys. B575, 269 (2000)

2000
[37]

Alford and K

M. Alford and K. Rajagopal, JHEP06, 031 (2002)

2002
[38]

A. W. Steiner, S. Reddy, and M. Prakash, Phys. Rev. D 66, 094007 (2002)

2002
[39]

Chiba and T

R. Chiba and T. Kojo, Phys. Rev. D109, 076006 (2024)

2024
[40]

Ayala, B

A. Ayala, B. S. Lopes, R. L. S. Farias, and L. C. Parra, Phys. Lett. B864(2025) 139396 (2025)

2025
[41]

Fukushima and S

K. Fukushima and S. Minato, Phys. Rev. D111, 094006 (2025)

2025
[42]

B. B. Brandt, V. Chelnokov, G. Endrodi, G. Marko, D. Scheid, and L. von Smekal, e-Print: 2502.04025 [hep-ph]

arXiv
[43]

Adhikari, J

P. Adhikari, J. O. Andersen, and P. Kneschke, Phys. Rev. D95, 036017 (2017)

2017
[44]

V. K. Tiwari, Phys. Rev. D108, 074002 (2023)

2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

B. A. Freedman and L. D. McLerran, Phys. Rev. D16, 1147 (1977)

1977

[2] [2]

B. A. Freedman and L. D. McLerran, Phys. Rev. D16, 1169 (1977)

1977

[3] [3]

Baluni, Phys

V. Baluni, Phys. Rev. D17, 2092 (1978)

work page 1978

[4] [4]

Gorda, A

T. Gorda, A. Kurkela, P. Romatschke, S. S¨ appi, and A. Vuorinen, Phys. Rev. Lett.121, 202701 (2018)

work page 2018

[5] [5]

Gorda, R

T. Gorda, R. Paatelainen, S. S¨ appi, and K. Sepp¨ anen, Phys. Rev. Lett.131, 181902 (2023)

work page 2023

[6] [6]

Fernandez and J.-L

L. Fernandez and J.-L. Kneur, Phys. Rev. Lett.129, 212001 (2022); Phys. Rev. D111, 034020 (2025)

work page 2022

[7] [7]

Annala, T

E. Annala, T. Gorda, J. Hirvonen, O. Komoltsev, A. Kurkela, J. N¨ attil¨ a, and A. Vuorinen, Nature Comm.14, 8451 (2023)

work page 2023

[8] [8]

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

work page 1978