QCD phase diagram from the gluon propagator at finite temperature and density

Fabio Siringo; Giorgio Comitini

REVIEW 3 major objections 5 minor 1 cited by

QCD phase diagram from the gluon propagator at finite temperature and density

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The temperature at which the longitudinal gluon propagator peaks traces the QCD deconfinement boundary from about 77 MeV at zero density to zero at the light quark mass.

desk verdict A careful one-loop extension of the screened massive expansion to finite density QCD with quark loops, whose phase boundary rests on an imported criterion that the paper itself flags as an assumption. read the letter →

arxiv 2412.15414 v2 pith:3FYA2BAE submitted 2024-12-19 hep-th

Giorgio Comitini , Fabio Siringo This is my paper · ORCID

classification hep-th MSC 81V0581T28 PACS 12.38.-t11.10.Wx

keywords QCDphasediagramgluonpropagatorscreenedmassiveexpansiondeconfinementfinitetemperaturefieldtheorychemicalpotentialLandaugaugechiralsymmetryrestoration

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

Using a one-loop version of the screened massive expansion, with light quarks modeled as fields of fixed infrared mass, the paper argues that the QCD deconfinement boundary can be read off the temperature at which the zero-frequency longitudinal gluon propagator, at vanishing spatial momentum, reaches its maximum. At zero chemical potential that maximum sits at Tmax(0)≈77 MeV, roughly half the usually quoted crossover temperature, and the ratio of pure-glue to full-QCD critical temperatures comes out close to the Polyakov-loop ratio. As chemical potential grows below the lightest quark mass M1=350 MeV, Tmax falls monotonically to zero at μ=M1, making the infrared quark mass the critical chemical potential. Above μ=M1 the propagator ceases to be non-monotonic once quark masses drop as expected with chiral symmetry restoration. The shape of the resulting Tc(μ) curve reproduces the expected topology of the QCD phase diagram and is qualitatively insensitive to the expansion parameters.

What carries the argument

The machinery is the screened massive expansion: ordinary QCD perturbation theory reorganized by giving transverse gluons a tree-level mass and quarks fixed infrared masses, with a compensating two-gluon mass counterterm. To one loop the gluon polarization receives gluon, ghost, and quark-loop contributions, and the quark loop at finite temperature and density is evaluated by shifting the fermionic Matsubara frequencies by iμ, leaving closed expressions up to a one-dimensional momentum integral. The order parameter used here is the inverse of the longitudinal propagator at zero Matsubara frequency and vanishing spatial momentum, essentially a Debye mass; its first decrease and then growth with T defines Tmax.

What would settle it

A lattice calculation of the unquenched zero-frequency longitudinal gluon propagator in 2+1-flavor QCD at vanishing momentum that finds no maximum near T≈77 MeV at μ=0, or a gap-equation computation in which Tmax(μ) does not vanish at μ=M1, would disprove the claim.

Watch

Extended reading notes

Core claim

The central claim is that the deconfinement transition in full QCD is encoded in the non-monotonic temperature dependence of the longitudinal Landau-gauge gluon propagator at zero Matsubara frequency: the temperature Tmax at which ΔL(0,|p|→0) is maximal is the (pseudo)critical temperature Tc(μ). At one loop with effective quark masses M1=350 MeV and M2=450 MeV and gluon mass m0=656 MeV, the paper finds Tmax(0)≈0.117 m0≈77 MeV, a steady decrease with μ for μ<M1, and Tmax→0 as μ→M1. It also finds that the normalized Tmax(μ)/Tmax(0) curve is practically independent of the chosen quark and gluon masses when plotted against μ/M1, and that the two humps appearing for μ between M1 and M2 disappear if the effective quark masses drop to about half their values above the Tmax curve, as chiral symmetry restoration would suggest.

Load-bearing premise

The result hinges on treating the light quark effective masses as constant inputs, M1=350 MeV and M2=450 MeV, that abruptly drop above the Tmax curve, and on assuming that the longitudinal gluon propagator's maximum marks deconfinement in full QCD exactly as it does in quenched lattice Yang-Mills.

Editorial extensions

If this is right

At μ=0 the full-QCD deconfinement temperature is predicted at about 77 MeV, roughly half the phenomenological 155–175 MeV, and the ratio T_c^YM/T_c^QCD≈1.57 is close to the Polyakov-loop based ratio 1.54.
For μ<M1 the critical temperature decreases monotonically with baryon chemical potential, with a change in concavity near (μ,T)=(0.46M1,0.62Tc) that may signal a change in the nature of the transition.
The critical chemical potential at T=0 equals the lightest infrared quark mass M1≈350 MeV; beyond it the longitudinal propagator is a decreasing function of temperature once quark masses are reduced above the boundary.
Adding a charm-like quark of mass ~1.2 GeV leaves Tc(μ) essentially unchanged for μ<M1, while dropping the strange quark raises Tc(μ) by 4–9% in the same region.

Reading between the lines

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

If the effective quark masses fall continuously rather than abruptly above Tc(μ), the phase boundary would bend smoothly and the endpoint would sit above μ=M1; this is testable by coupling the expansion to a gap equation for the quark mass.
Because the crossover temperature is gauge- and observable-dependent, a gauge-independent check would be to test whether the longitudinal-propagator maximum in a first-order region of the phase diagram coincides across covariant gauges; the paper leaves this as future work.
The curvature-change point near (μB,T)≈(0.48 GeV, 0.62Tc) could be compared with lattice Taylor-expansion and imaginary-chemical-potential determinations of the crossover line curvature.
The criterion could be extended to gluon spectral functions or transverse-sector observables, which would connect the propagator-maximum method to transport coefficients of the quark-gluon plasma.

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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

The new thing here is the finite-density extension of the screened massive expansion to full QCD with quark loops, and the phase diagram is read off from the maximum of the longitudinal gluon propagator at zero Matsubara frequency. The calculation is explicit, the analytic expressions are reduced to a one-dimensional integral, and the stability analysis across the free parameters is genuinely thorough. The authors also correct an error in their earlier code, which is a good sign of care.

What the paper does well: the normalized curve Tc(mu)/Tc(0) is robust across mass configurations, the Yang-Mills to full-QCD transition-temperature ratio comes out close to the lattice result, and the qualitative shape—monotonic decrease with mu and endpoint at the lightest quark mass—matches expectations. The authors are candid about the absolute scale being roughly half the known crossover temperature and about the fragility of the humps in Tmax(mu).

The soft spots are real but mostly acknowledged. The load-bearing premise is that the longitudinal propagator maximum marks deconfinement in full QCD; that is extrapolated from quenched lattice Yang-Mills, and the paper explicitly says it is an assumption. In full QCD at physical quark masses the transition is a crossover, so the correspondence is not automatic. The endpoint mu_c = M1 is literally an input, and the mass-drop procedure uses the Tmax curve itself to define the confined region before imposing lighter masses outside it. So the phase boundary is contingent on the modeling, not a first-principles output. The authors say so themselves, which keeps this from being misleading.

Is the central argument holding up? As a derived quantitative prediction, no; as a qualitative analytic estimate of the phase boundary shape and a ratio, yes. The calculation is sound within its one-loop framework, and the weakness is the interpretive link, not the algebra.

Who benefits: people working on massive expansion or Curci-Ferrari-type approaches, and anyone wanting an analytic route to the qualitative QCD phase boundary with a clear statement of its assumptions. It deserves a serious referee, not a desk reject; a referee should press on the Tmax criterion in full QCD and on the dynamical treatment of quark masses, but the paper already frames those as open problems.

Referee Report

3 major / 5 minor

Summary. The manuscript extends the authors' screened massive expansion of the Landau-gauge gluon propagator from pure Yang-Mills theory to full QCD with 2+1 dynamical quarks at finite temperature T and baryon chemical potential μ. Quarks are modeled as massive fields with constant effective infrared masses M1=350 MeV and M2=450 MeV, and the one-loop polarization is evaluated analytically up to a one-dimensional integral. At zero Matsubara frequency and vanishing spatial momentum, the longitudinal propagator is found to be non-monotonic in T for μ<M1, with Tmax(0)≈0.117 m0≈77 MeV; Tmax decreases with μ and extrapolates to zero at μ→M1. Interpreting Tmax as the deconfinement temperature yields a phase diagram in the (T,μ) plane whose normalized shape is stable under parameter changes. The authors clearly state that the absolute scale is not quantitatively reliable.

Significance. The paper's technical core is sound: the finite-density extension of the quark-loop polarization is standard, the one-dimensional integral representation is useful, and the stability analysis in Sec. IIID covers the free parameters of the expansion. The main conceptual contribution is a falsifiable prediction for the shape Tc(μ)/Tc(0) for μ<M1, including a change of concavity that could hint at a critical endpoint, and a simple explanation of why the endpoint sits at the lightest infrared quark mass. The significance is conditional on the assumed correspondence between the longitudinal gluon propagator's maximum and deconfinement in full QCD; this correspondence is presently established only in pure Yang-Mills lattice simulations. The authors are honest about this limitation, and the analytic expressions and explicit caveats are strengths.

major comments (3)

[Sec. IVA, Eq. (38), Fig. 13] The phase boundary rests on the unvalidated assumption that the maximum of the zero-frequency longitudinal gluon propagator marks deconfinement in full QCD. In pure Yang-Mills the criterion is supported by lattice data [82], but in the presence of dynamical quarks the transition is a crossover and the gluon-propagator maximum has no independent lattice or functional check. The ratio T_YM/T_QCD ≈ 1.57 quoted in Sec. IVA is not a discriminating test: the one-loop calculation with temperature-independent parameters underestimates the pure-Yang-Mills Tc by about a factor of two (Sec. IIB), so a common systematic offset can produce the same ratio even if the criterion fails in full QCD. A concrete validation would be to compute Tmax from unquenched gluon-propagator data at μ=0 and compare it with the chiral-susceptibility or Polyakov-loop crossover temperature; until then, the claim should be presented as a model conjecture rather than a QCD phase-diagram prediction.
[Secs. IIIA and IVA, Eq. (43), Fig. 12] The endpoint of the phase boundary is an input, not a derived result. The effective quark masses M1=350 MeV and M2=450 MeV are free parameters, and Tmax(μ)→0 as μ→M1 follows from the Fermi-distribution threshold in Eq. (43) once M1 is chosen. Similarly, the deconfined-phase masses (M1',M2')=(125,225) MeV are imposed by hand in Sec. IVA, and the disappearance of the humps for μ>M1 in Fig. 13 is a consequence of that mass drop. The model therefore does not independently predict the absolute location μc; it predicts the normalized curve Tc(μ)/Tc(0) and its robustness. The authors should either implement the self-consistent determination of Mf(T,μ) they mention in Sec. IVA, or explicitly restrict the parametric predictions to μ<M1 and to statements about shape.
[Sec. IIIA, Sec. IVA] The absolute temperature scale Tc(0)=0.117 m0≈77 MeV is set by m0=656 MeV, which is taken from a fit to zero-temperature pure-Yang-Mills lattice data [13]. Because the calculation is one-loop and uses temperature-independent parameters, the analogous pure-Yang-Mills result is Tc≈121 MeV, about half of the physical 270 MeV (Sec. IIB). The authors acknowledge this in Sec. IVA, but the abstract and title still present the result as 'the QCD phase diagram'; the wording should make clear that only the normalized shape, and not the absolute position of the crossover line, is claimed.

minor comments (5)

[Sec. IIB, Table I] The caption says the starred value 10 MeV for the longitudinal m(T) at T=260 MeV is the lowest value that could be reached by the numerical routines, but the text says the routines could not reach lower values; clarify whether 10 MeV is a converged fit value or a numerical bound.
[Figs. 8 and 9] The panel labels use 'Tc(µ)' for the value Tmax(µ) before the notation is defined; define Tc(µ) in the captions or use Tmax consistently.
[Eq. (33)] The prefactor p²/p² in Eq. (33) is redundant and should be simplified to avoid confusion.
[Eq. (31)] The symbol μ is used both for the renormalization scale in Eq. (31) and for the chemical potential throughout the paper; use a distinct symbol such as μ_R for the renormalization scale.
[Sec. IVA, Sec. V] The phrase 'around twice as small as' should read 'about half of', and several garbled mathematical symbols appear in the text (e.g., '/greaterorapproxeql'); a careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 5.0 of 10

Fitted Yang-Mills benchmark and self-consistent mass-drop loop are partly circular, but the normalized Tc(µ)/Tc(0) shape remains an independently computed one-loop result.

fitted input called prediction [Sec. IIB, paragraph after Table I]
"Tab. I reports the corrected values of the parameters obtained from a fit of the lattice data... As for the value of the critical temperature itself, if we define Tc as the point at which the longitudinal propagator changes behavior with respect to T at fixed momentum, then the screened massive expansion trivially agrees with the lattice finding, Tc ≈ 270 MeV, given that at large enough momenta it is able to reproduce the lattice propagators for all temperatures."

The parameters m(T) and π0(T) were fitted independently and at each temperature to the lattice propagator data (Table I). The location of the maximum of the longitudinal propagator as a function of T is therefore inherited from the fitted data, not predicted from the expansion alone. The word 'trivially' in the paper concedes this: once the propagators are fitted at all temperatures, the extracted Tc is a restatement of the input. This step is used as a benchmark for the method, but it is not the central full-QCD prediction, which uses parameters fixed at T = 0.
self definitional [Sec. IVA, paragraph describing Fig. 13]
"In more detail, to obtain the curve in the figure, we first used our knowledge on Tmax(µ) – as evaluated under the assumption of constant quark masses over the whole (T, µ) plane – to identify the region of the phase diagram which we expect to be the confined phase. Then we decreased the quark masses from (M1, M2) = (350, 450) MeV to about half their value – (M1', M2') = (125, 225) MeV – outside of that region and recomputed the propagators and their maxima as a function of chemical potential."

The final phase boundary in Fig. 13 is the same Tmax(µ) curve that was used to define where the quark masses are dropped. For µ < M1 the maxima are explicitly reported to remain unchanged, so the boundary is the input curve itself. For µ > M1 the whole T–µ plane is declared to lie outside the confined phase, so the quark masses are set below the chemical potential; this guarantees that the quark loop is active at T = 0 and that the propagator becomes strictly decreasing, making Tmax = 0 by construction. The endpoint µc = M1 is thus inherited from the assumed phase boundary and from the input light-quark mass threshold, rather than being an independent output of the mass-drop procedure.

full rationale

The paper's central normalized result, Tc(µ)/Tc(0), is genuinely computed from the one-loop screened massive expansion with parameters held at their T = 0 values; it is not fitted to the QCD phase diagram. The absolute scale Tmax(0) ≈ 0.117 m0 is a derived consequence of the lattice-fitted gluon mass m0, and the authors explicitly disclaim its quantitative reliability. The equality µc = M1 is a model consequence of the Fermi-Dirac threshold controlled by the input effective quark mass, and the paper is transparent that M1 is chosen from lattice-inspired values rather than fitted to the endpoint. However, two steps do reduce by construction: the pure Yang-Mills Tc ≈ 270 MeV is obtained from per-temperature fits to the lattice propagators and then presented as 'trivially' agreeing with the lattice; and the mass-drop procedure of Sec. IVA uses the Tmax(µ) curve itself to define where the quark masses are decreased, so the resulting phase diagram for µ > M1 (and the disappearance of the humps) is imposed by the construction rather than independently predicted. These are real but partial circularities: the normalized shape, the stability against parameter changes, and the YM/full-QCD ratio retain independent content. Hence a score of 5 rather than 0 or 2.

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

The central claim rests on the fitted gluon mass m0, the additively renormalized constant π0, and the hand-chosen infrared quark masses M1 and M2. The mass-drop scenario adds another pair of free masses. No new particles or forces are introduced. The main non-dynamical element is the assumed correspondence between the longitudinal propagator maximum and the deconfinement temperature.

free parameters (5)

m0 = 656 MeV
Gluon mass parameter fixed from T = 0 Yang-Mills lattice data in previous work; sets the absolute scale of Tmax(0) ≈ 0.117 m0.
π0 = -0.876
Additive renormalization constant from the same T = 0 fit; shifts the propagator normalization but not the qualitative behavior.
M1 = 350 MeV
Lightest effective infrared quark mass, chosen by hand from lattice expectations; by construction the chemical potential at which Tc(μ) reaches zero.
M2 = 450 MeV
Effective mass of the heavier quark, chosen with an exaggerated splitting to distinguish flavor effects.
M1', M2' (deconfined masses) = 125 MeV, 225 MeV
Used in the mass-drop scenario for T above Tc(μ); the disappearance of the threshold humps depends on this choice.

assumptions (5)

domain assumption The screened massive expansion is perturbatively equivalent to ordinary pQCD (Sec. IIA, Ref. [40]).
Underpins the one-loop polarization calculation and the cancellation of spurious mass divergences from the gluon mass counterterm.
domain assumption The exact gluon polarization is transverse, so any longitudinal parts can be removed by an implicit resummation (Sec. IIA).
Used to write the polarization as Πμν = Π tμν in Eqs. (9)-(11), reducing the propagator to transverse and longitudinal scalar functions.
ad hoc to paper The maximum of the longitudinal gluon propagator at zero Matsubara frequency marks the deconfinement temperature in full QCD as it does in pure Yang-Mills lattice data (Secs. I and IVA).
This is the central interpretive assumption; the authors cite lattice evidence in pure Yang-Mills but explicitly assume the parallelism in full QCD.
domain assumption The Lagrangian of Eq. (23) with constant infrared quark masses Mf describes the nonperturbative quark dynamics (Sec. IIIA).
Replaces current masses by constituent masses based on known infrared enhancement, but the values are not derived in this paper.
domain assumption One-loop order with temperature-independent parameters retains the correct qualitative behavior of the propagators (Sec. IIID).
Needed to interpret the constant-parameter results as predictions; the paper shows insensitivity to parameter choices but not to higher orders.

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Pith. "Pith review of QCD phase diagram from the gluon propagator at finite temperature and density." pith.science (2026). https://pith.science/paper/3FYA2BAE

@misc{pith2026241215414,
  author       = {Pith},
  title        = {Pith review of: QCD phase diagram from the gluon propagator at finite temperature and density},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FYA2BAE}},
  note         = {Machine review of arXiv:2412.15414}
}

read the original abstract

The screened massive expansion of full QCD is used in conjunction with a model for infrared quark masses to compute the Landau-gauge gluon propagator at finite temperature and baryonic density. Analytic expressions up to a one-dimensional momentum integral are provided for the propagator, and its behavior is studied at zero Matsubara frequency with respect to temperature, chemical potential, and the parameters of the expansion. The phase diagram of QCD is explored under the assumption that the deconfinement temperature can be identified as the position of the maximum of the longitudinal gluon propagator at zero Matsubara frequency and fixed spatial momentum.

Figures

Figures reproduced from arXiv: 2412.15414 by the authors.

**Figure 2.** Transverse component of the Landau-gauge Eu [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Transverse component of the Landau-gauge Eu [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (12 more)

**Figure 6.** Figure 6: Temperature dependence of the free parameters [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Full-QCD one-loop quark polarization diagram. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Transverse gluon propagator as a function of spati [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: Longitudinal gluon propagator as a function of spa [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: Transverse (left) and longitudinal (right) gluo [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: Transverse (left) and longitudinal (right) gluo [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: Temperature Tmax at which the zero-frequency (ω = 0) longitudinal gluon propagator ∆L(ω, p) attains a maximum in the |p| → 0 limit, as a function of chemical potential. Tmax(0) ≈ 0.117 m0 ≈ 77 MeV. happens at the µ = M2 threshold, causing the second hump to appear in …

**Figure 13.** Figure 13: Phase diagram of QCD obtained by decreasing the [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: Transverse gluon propagator as a function of spat [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

**Figure 15.** Figure 15: Longitudinal gluon propagator as a function of sp [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]

**Figure 16.** Figure 16: Zero-frequency longitudinal propagator’s [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 17.** Figure 17: shows the first portion of Tmax(µ) – which again we denote with Tc(µ) – for a number of different mass configurations, reported in Tab. II. As we can see, normalizing the chemical potential by M1 and Tc(µ) by Tc(0) yields curves whose dependence mostly comes from the…

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Reference graph

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In terms of a Debye mass deﬁned as mD = ∆ −1/2 L (ω = 0, |p| → 0), this translates to mD ﬁrst decreasing to very small values as T →Tc, and then growing with T for T >T c

we know that, below the critical temperature and for any given ﬁxed spatial momentum, the longitudinal gluon propagator evaluated at zero Matsubara frequency expe- riences a sudden increase with temperature which peaks atTc ≈ 270 MeV, to then become a decreasing function of T forT >Tc just like the transverse propagator. In terms of a Debye mass deﬁned as...
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Linea di intervento 1

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In terms of a Debye mass deﬁned as mD = ∆ −1/2 L (ω = 0, |p| → 0), this translates to mD ﬁrst decreasing to very small values as T →Tc, and then growing with T for T >T c

we know that, below the critical temperature and for any given ﬁxed spatial momentum, the longitudinal gluon propagator evaluated at zero Matsubara frequency expe- riences a sudden increase with temperature which peaks atTc ≈ 270 MeV, to then become a decreasing function of T forT >Tc just like the transverse propagator. In terms of a Debye mass deﬁned as...

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As discussed in Sec

= (125, 225) MeV beyond the ﬁrst segment of Tmax(µ) – which in what follows we will call Tc(µ) for simplicity, adopting its most obvious interpretation. As discussed in Sec. IV A, this procedure is motivated by the expected restoration of (the remnants of) chiral symmetry, and is carried out to test whether an abrupt decrease in the quark masses outside t...

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0 200 400 600 800 1000 0 100 200 300 400 500 Longitudinal Transverse m(T) (MeV) T (MeV) -8 -6 -4 -2 0 2 0 100 200 300 400 500 Longitudinal Transverse π0(T) T (MeV) Figure 6

and ( 7) for the gluons, while treating the newly introduced quark masses as free parameters 7 Or, to be more precise, if the current masses do not vanish, of what is left of the approximate chiral symmetry. 0 200 400 600 800 1000 0 100 200 300 400 500 Longitudinal Transverse m(T) (MeV) T (MeV) -8 -6 -4 -2 0 2 0 100 200 300 400 500 Longitudinal Transverse...

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5% – across all temperatures, chemical potentials and both the components. 13 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 µ = 500 MeV µ = 0 MeV m0 2∆ T(p) p/m0 T = 0.00 Tc T = 0.50 Tc T = 1.00 Tc T = 1.50 Tc 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 µ = 0 MeV µ = 500 MeV m0 2∆ L(p) p/m0 T = 0.00 Tc T = 0.50 Tc T = 1.00 Tc T = 1.50 Tc 0 0.5 1 1.5 2 2...