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Deconfinement in dense 2-color QCD

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Simon Hands, Seyong Kim, Jon-Ivar Skullerud

High Energy Physics - Lattice · Submitted 4 Apr 2006 · 38 references

Pith's verdict: A careful, honest exploratory scan of dense two-color QCD; the three-regime picture holds, but the magnetic-screening headline is measured at one diquark source and visibly shrinks as that source goes to zero.

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arXiv IDhep-lat/0604004 (version 2)
Pith ID725UF573, Pith's permanent record of this paper and its review
CitationJon-Ivar Skullerud, Seyong Kim, Simon Hands. "Deconfinement in dense 2-color QCD." arXiv:hep-lat/0604004 (2006). https://pith.science/paper/725UF573
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The paper

The paper claims

Two-color QCD with two Wilson fermions at low temperature shows a vacuum phase, a diquark-condensed superfluid, and a deconfined quark-matter phase above mu about 600 MeV, with a minimum in energy per baryon.

The reading

Ordinary QCD cannot be simulated on a lattice at high baryon density, because the fermion determinant becomes complex and can no longer be used as a probability weight. Two-color QCD with an even number of flavors escapes this: a symmetry between quarks and antiquarks keeps the determinant real and positive, so the same physics questions can be attacked numerically. This paper does that with two flavors of Wilson fermion on an 8^3 x 16 lattice at a lattice spacing of about 0.22 fm, scanning the chemical potential mu from 0 up to 2.5 in lattice units while holding the temperature low and the diquark source small (j = 0.04). Three regimes appear. Below mu about m_pi/2 nothing changes: baryon density stays zero and the static quark potential is untouched. Between roughly mu = 0.4 and 0.65 a diquark condensate turns on, the baryon density rises, and the static potential gets screened, which the authors read as Bose-Einstein condensation of tightly bound scalar diquarks. Above mu about 0.65 the Polyakov loop rises, the magnetic gluon propagator acquires infrared screening even in the static limit (something perturbation theory does not predict), and thermodynamic quantities scale like their free-gas counterparts but with a constant factor near two. The ratio of quark energy density to quark number density has a shallow minimum around mu = 0.8, which the authors use to argue that hypothetical stars made of this matter would be mostly quark matter with a thin diquark atmosphere.
Extended reading notes

Core claim

Two-color QCD with two Wilson fermions at low temperature has three regimes along the chemical potential: a vacuum phase below mu about m_pi/2, a confined superfluid with a nonzero diquark condensate and screened static potential, and a deconfined quark-matter phase above mu_d about 0.65 a^-1 (about 600 MeV) where the Polyakov loop rises, the magnetic gluon propagator is screened in the static limit, and the energy per baryon has a minimum. If correct, this establishes a BEC-to-BCS crossover picture for a sign-problem-free gauge theory and gives a non-perturbative handle on magnetic screening at high density.

Load-bearing premise

That the smooth rise of the Polyakov loop and the change in the magnetic gluon propagator around mu a = 0.65 mark a genuine deconfinement transition, rather than a crossover or a finite-volume and lattice-spacing artifact. The Polyakov loop is not a strict order parameter once dynamical quarks break the Z2 center symmetry, and the paper itself notes the magnetic screening weakens as j goes to zero (section 4.6, figure 19) and that 'simulations at smaller j in the deconfinement phase will be necessary to firm up the picture.' All of this rests on one 8^3 x 16 lattice at a = 0.223 fm with j = 0.04.

Original abstract

We study SU(2) lattice gauge theory with two flavors of Wilson fermion at non-zero chemical potential mu and low temperature on a 8^3x16 system. We identify three regimes along the mu-axis. For mu<~m_pi/2 the system remains in the vacuum phase, and all physical observables considered remain essentially unchanged. The intermediate regime is characterised by a non-zero diquark condensate and an associated increase in the baryon density, consistent with what is expected for Bose-Einstein condensation of tightly bound diquarks. We also observe screening of the static quark potential here. In the high-density deconfined regime we find a non-zero Polyakov loop and a strong modification of the gluon propagator, including significant screening in the magnetic sector in the static limit, which must have a non-perturbative origin. The behaviour of thermodynamic observables and the superfluid order parameter are consistent with a Fermi surface disrupted by a BCS diquark condensate. The energy per baryon as a function of mu exhibits a minimum in the deconfined regime, implying that macroscopic objects such as stars formed in this theory are largely composed of quark matter.

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, 6,302 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.32 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We study SU(2) lattice gauge theory with two flavors of Wilson fermion at non-zero chemical potential mu and low temperature on a 8^3x16 system. We identify three regimes along the mu-axis. For mu<~m_pi/2 the system remains in the vacuum phase, and all physical observables considered remain essentially unchanged. The intermediate regime is characterised by a non-zero diquark condensate and an associated increase in the baryon density, consistent with what is expected for Bose-Einstein condensation of tightly bound diquarks. We also observe screening of the static quark potential here. In the high-density deconfined regime we find a non-zero Polyakov loop and a strong modification of the gluon propagator, including significant screening in the magnetic sector in the static limit, which must have a non-perturbative origin. The behaviour of thermodynamic observables and the superfluid order parameter are consistent with a Fermi surface disrupted by a BCS diquark condensate. The energy per baryon as a function of mu exhibits a minimum in the deconfined regime, implying that macroscopic objects such as stars formed in this theory are largely composed of quark matter.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study SU(2) lattice gauge theory with two flavors of Wilson fermion at non-zero chemical potential mu and low temperature on a 8^3x16 system. method 0.72 0.93 0.02 asserted 0.89
s1 We identify three regimes along the mu-axis. result 0.83 0.93 0.05 asserted 1.00
s2 For mu<~m_pi/2 the system remains in the vacuum phase, and all physical observables considered remain essentially unchanged. result 1.00 0.87 0.05 asserted 1.00
s3 The intermediate regime is characterised by a non-zero diquark condensate and an associated increase in the baryon density, consistent with what is expected for Bose-Einstein condensation of tightly bound diquarks. result 0.84 0.81 0.10 asserted 0.53
s4 We also observe screening of the static quark potential here. result 1.00 0.92 0.03 asserted 1.00
s5 In the high-density deconfined regime we find a non-zero Polyakov loop and a strong modification of the gluon propagator, including significant screening in the magnetic sector in the static limit, which must have a non-perturbative origin. result 0.98 0.96 0.21 asserted 0.99
s6 The behaviour of thermodynamic observables and the superfluid order parameter are consistent with a Fermi surface disrupted by a BCS diquark condensate. result 0.53 0.85 0.13 hedged possibility 0.95
s7 The energy per baryon as a function of mu exhibits a minimum in the deconfined regime, implying that macroscopic objects such as stars formed in this theory are largely composed of quark matter. interpretation 0.96 0.92 0.36 asserted 0.66

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

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Figures
Figure 1. Pi and rho effective masses for µ = j = 0. Range mπ χ 2 /Ndf mρ χ 2 /Ndf mπ/mρ 3–4 0.809(2) 0.58 0.882(4) 0.71 0.917(3) 3–5 0.800(2) 0.97 0.870(3) 1.4 0.919(3) 3–6 0.795(2) 1.3 0.860(3) 2.5 0.924(3) 4–5 0.789(2) 0.81 0.854(4) 1.7 0.924(3) 4–6 0.786(2) 0.63 0.846(3) 1.9 0.929(3) 5–6 0.783(2) 0.80 0.836(4) 3.0 0.937(3)
Figure 2. Static quark potential for µ = j = 0, together with a fit to the Cornell potential. separation used was 2
Figure 3. Ratio of thermodynamic observables from χPT and free quarks given in (17) and (23). Since pχP T > pSB for µ < µd is required for thermodynamic stability, we find µd given by the largest positive real root of µ 3 d − 4πFr 3 Nc (µ 2 d − µ 2 o ) = 0. (24) This estimate takes no account of any non-Goldstone states in the hadron spectrum, or of any gluon degrees of free￾dom released at deconfinement. In fig. 3 we plot the ra￾tios nχP T /nSB, εχP T /εSB and pχP T /pSB as functions of µ/µo for the choice F 2 = Nc/6π 2 , corresponding to µd ≃ 2.288µo. Since nχP T < nSB at this p
Figure 4. n lat SB/µ3 for free massless Wilson fermions. for µ > 0.5. To correct for this lattice artefact, we have chosen to plot the ratio nq/nlat SB, shown as a function of µ in fig. 5. The free field density has been calculated for massless quarks, i.e. with κ = 1 8 . In order to assess the systematic error due to our lack of accurate knowledge of the non-zero quark mass, we repeated the free field calcu￾lation with κ = 0.120, corresponding to a conservatively large mqa = 0.167, and found the resulting n lat SB(µ) to have a similar shape, with a maximum departure over the
Figure 5. Ratio of thermodynamic observables to free field values versus µ for j = 0.04. Open symbols show extrapolations to j = 0. µ range of interest of O(20%). Where data are available we have plotted as open symbols the result of linearly ex￾trapolating j → 0; at µ = 0.3 this yields nq ≈ 0, consistent with being below the onset expected at µo = mπ/2. For µ = 0.5, 0.7 the extrapolation results in a downwards cor￾rection of O(20%). We also note that for µ & 0.5 the ratio is roughly constant and greater than one, which is plau￾sible if Stefan–Boltzmann scaling sets in at large µ, but with a…
Figure 7. Comparison of bare quark and gluon energy densities versus µ. density renormalisation factors prevents us from making any quantitative estimate of the gluon energy density as proportion of the total energy density. The quark and gluon contributions to the conformal anomaly (14,13) are plotted in fig. 8. As for energy den￾sity, the raw data requires a zero-µ subtraction. To extract δ, we also need information on the lattice beta-functions, which require an extensive simulation campaign and are not yet available. For this reason we restrict ourselves to qualitative remarks. The quark
Figure 8. Top: κ −1 Tr(1−hψψ¯ i), and bottom: 3β/Nc Trh✷t+✷si, as functions of µ at j = 0.04. ory, so Tr ✷(µ → ∞) < Tr ✷(µ = 0). Assuming a smooth passage to the limit we deduce δ < 0 at large µ. To summarise the thermodynamic information, we have been able to extract nq(µ) and p(µ) directly from the sim￾ulation — any remaining UV and IR artefacts in fig. 5 can be controlled by simulations closer to thermodynamic and continuum limits. The status of εq, εg is less secure, because these quantities require renormalisation by an as yet undetermined factor; similarly the trace of the…
Figure 9. Quark number density nq and Polyakov loop L vs. µ for values up to µ = 2.5. All lattice data are at j = 0.04. 4.3 The approach to saturation In figure 9 we plot nq(µ) including values of µ up to 2.5, as tabulated in tables 1, 3 and 4, together with both contin￾uum (22) and lattice (26) expectations for non-interacting quarks. On any system with a non-zero lattice spacing, there is a point at which every lattice site is occupied by the maximum value 2NcNf allowed by the Pauli exclu￾sion principle. For free Wilson fermions this occurs for µa ≈ 2.0; for the interacting quarks studied…
Figure 10. The Polyakov loop L and the superfluid diquark con￾densate hqqi/µ2 for j = 0.04 as function of µ. Open symbols show hqqi/µ2 extrapolated to j = 0. 0 0.02 0.04 0.06 0.08 j 0 0.05 0.1 µ=0.3 µ=0.5 µ=0.7
Figure 11. Superfluid condensate hqqi vs. j for various µ 4.4 Order parameters Might it be possible to reconcile the behaviour reported in section 4.2 with the models of section 4.1? To eluci￾date this question we next review order parameters for superfluidity and deconfinement. In fig. 10 we plot both the superfluid condensate hqqi given in (15), rescaled by a factor µ −2 , and the Polyakov loop L. Both show a marked change of behaviour at µ ≃ 0.6; for µ greater than this value hqqi/µ2 is approximately constant, while L rises from zero. To interpret the superfluid condensate, we first need
Figure 12. The static quark potential for various values of the chemical potential µ and diquark source j = 0.04, together with the µ = j = 0 potential shown in fig. 2. 1 2 3 4 R/a 0.6 0.8 1 V(R)/V0(R) µ=0.3 µ=0.4 µ=0.5 µ=0.6 µ=0.7 µ=0.8 µ=0.9 µ=1.0
Figure 13. The static quark potential for various values of the chemical potential µ and diquark source j = 0.04, divided the µ = j = 0 potential shown in fig. 2. 0 1 2 3 4 5 R/a 0.5 1 1.5 2 2.5 aV(R) j=0.02 j=0.04 j=0.06
Figure 14. The static quark potential at µ = 0.7 for different values of j. 0 2 4 6 8 q 2 a 2 0 1 2 3 4 DM DE
Figure 16. Gluon propagator at µ = 0.7, j = 0.04, together with the µ = j = 0 propagator for comparison. momentum |q| for different µ. We see that there is very lit￾tle change up to the deconfinement transition, after which there is a dramatic infrared suppression and clear ultravi￾olet enhancement of the propagator. These changes makes it look more like an ordinary massive boson propagator, i.e., the gluon has acquired a magnetic mass which grows as µ increases beyond µd. This effect does depend on the diquark source j, becoming weaker as j → 0, as figure 19 shows. Simulations at smaller j…
Figure 17. Magnetic gluon propagator for q0 = 0 (top) and aq0 = 2 sin π/16 (bottom) and diquark source j = 0.04, as a function of spatial momentum |q| for various values of the chemical potential. 5 Discussion and outlook Our results indicate that, at least for the (rather heavy) quark mass employed in this study, QC2D has three sep￾arate phases: 1. A vacuum phase, for µ < µo ≈ mπ/2, where the baryon density remains zero and all other physical quantities (with the likely exception of the hadron spec￾trum [23]) are unchanged. 2. A confined, bosonic superfluid phase, for µo < µ < µd, characteri
Figure 18. As fig. 17, for the electric gluon propagator. 0 0.5 1 1.5 2 2.5 |q|a 0 1 2 3 4 D M (q,q 0 ) j=0.06, n0 =0 n 0 =1 n 0 =2 j=0.04 j=0.02
Figure 19. Magnetic gluon propagator at µ = 0.7 for various values of the diquark source j. phase, as kF > EF . We observe Debye screening of the electric gluon propagator, as well as strong screening of both static and non-static magnetic gluon modes. The deconfinement transition occurs at aµd ≈ 0.65, which in physical units corresponds to µd ≈ 600 MeV. The corre￾sponding quark density may be estimated as nq ≈ 11fm−3 0.2 0.4 0.6 0.8 1 µ 0 1 2 3 ε q /n q raw data ε q /n q corrected data
Figure 20. The ratio εq/nq vs. µ, using both the raw data of table 4, and rescaled by a factor n lat SB/εlat SB directly from table 4, or from fig. 5 where lattice artefacts are taken into account, as ≈6.5fm−3 . Let us discuss some phenomenological implications of our results. First consider the nature of the deconfinement transition. Since both the low density “nuclear matter” and high density “quark matter” phases are characterised by a U(1)B breaking superfluid condensate, it is tempt￾ing to postulate the quark/hadron continuity proposed for QCD with Nf = 3 light flavors in [30]. In that

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57 later papers cite it

Evidence: substantial. 53 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

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