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High Energy Physics - Phenomenology · arXiv:hep-ph/0404164 v2 · submitted 20 Apr 2004 · 130 references

Resummation in Hot Field Theories

Jens O. Andersen, M. Strickland

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“There has been significant progress in our understanding of finite-temperature field theory over the past decade. In this paper, we review the progress in perturbative thermal field theory focusing on thermodynamic quantities.”

more of the abstract

We first discuss the breakdown of naive perturbation theory at finite temperature and the need for an effective expansion that resums an infinite class of diagrams in the perturbative expansion. This effective expansion which is due to Braaten and Pisarski, can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of g. However, it turns that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small. We critically discuss various ways of reorganizing the perturbative series for thermal field theories in order to improve its convergence. These include screened perturbation theory (SPT), hard-thermal-loop perturbation theory (HTLPT), the Phi-derivable approach, dimensionally reduced (DR) SPT, and the DR Phi-derivable approach.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up. A useful, honest 2004 map of thermal resummation methods; the only overreach is the abstract's blanket 'these improve convergence' framing, which the body itself qualifies for DRSPT. This line is wrong

2 major objections · 8 minor

How often objections from reviews like this one are right: not yet measured

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Not checked: figures, code, what the 130 references say.

The paper says, at Sec. 10 vs. Sec. 8.4 and Fig. 19

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The concluding summary states that 'two reorganizations of the soft-sector computation, the DRSPT and DR Phi-derivable approach, both gave better convergence of the successive approximations to the pressure at large coupling.' For DR Phi-derivable this is supported by Fig. 20.

more of objection 1

For DRSPT it is only half-supported: Sec. 8.4 and Fig. 19 state that 'although DRSPT improves the situation in terms of the convergence of successive approximations, the result is still very sensitive to the renormalization scale having a larger variation than even the perturbative result.' Since the renormalization-scale band is the paper's principal diagnostic everywhere else, a reader who reads only Sec. 10 will form the wrong impression of the DRSPT entry. The conclusion should carry the scale-sensitivity caveat explicitly, or the sentence should be narrowed to the metric (successive-approximation stability) for which DRSPT does improve.

The paper says, at Sec. 9.2, Fig. 20, and Sec. 10

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The cross-method ranking in Fig. 20 and in the closing paragraph of Sec. 9.2 ('the final scale variation of the DRPHI result is smaller than the SPT result') compares approximants that are not of equal quality, and the text says so only in passing.

more of objection 2

The three-loop SPT curve is evaluated with a two-loop gap equation because the three-loop variational gap equation has no solution (stated in Sec. 9.2 and in the footnote in Sec. 8.4), while the DRSPT four-loop curve sets an unknown order-alpha^3 hard coefficient to zero, which the authors themselves call 'highly improbable' (Sec. 8.4). The comparison is therefore indicative rather than quantitative, and the manuscript should say this where the comparison is drawn, not only where the individual curves are constructed. This does not undermine the qualitative conclusion that reorganization helps, but it does bound how much weight the ranking can bear.

8 minor objections

The paper says, at Sec. 6.1, Eqs. (159) and (161)

Pith says: minor objection 1 of 8

Eq. (159) contains a term -(15/2) mhat_D^2 and a term +30 mhat_D^3, whereas Eq. (161) has no mhat_D^2 term at all and a coefficient -15 for mhat_D^3. If Eq. (161) is the complete leading-plus-next-to-leading coefficient, the text should say so explicitly, because as written it reads as a correction to Eq. (159) and the disappearance of the mhat_D^2 term is then unexplained.

The paper says, at Sec. 8, Eqs. (222), (224), (225) and Sec. 9, Eq. (237)

Pith says: minor objection 2 of 8

The definition of alpha used in the dimensionally reduced sections should be stated once, explicitly. With the standard three-dimensional normalization alpha = lambda/(4 pi), the one-loop tadpole gives (1/2) lambda times the integral in Eq. (B.3), i.e. -lambda m/(8 pi) = -alpha m/2, which does not reproduce the -2 pi alpha m T appearing in Eqs.

more of objection 2

(225) and (237); the same factor question enters the first term of Eq. (224). Either a nonstandard alpha is in use or a factor of 4 pi T has been absorbed; a one-line definition would settle it.

show 6 more

The paper says, at Sec. 8, Eq. (222)

Pith says: minor objection 3 of 8

Eq. (222) reads lambda = g^2 - 3 g^4 T/(4 pi)^2 (log(mu/4 pi T) + gamma), but Eq. (85) fixes lambda = g^2 T at tree level, so the first term of Eq. (222) appears to be missing a factor of T. Please check the normalization of this next-to-leading-order expression.

The paper says, at Sec. 4.1.1, Eq. (74)

Pith says: minor objection 4 of 8

The interaction Lagrangian in Eq. (74) lists (1/2)(nabla phi)^2 as an interaction term, but the same operator already appears in the free Lagrangian of Eq. (73). Presumably the intended term is the mass term; as printed the decomposition is inconsistent.

The paper says, at Sec. 5.5.4

Pith says: minor objection 5 of 8

The statement that 'the two-loop tadpole mass coincides with the one-loop tadpole mass' is a non-trivial cancellation and deserves one sentence of explanation, since it is the reason the tadpole prescription is preferred over the screening and variational ones.

The paper says, at Abstract and Sec. 1

Pith says: minor objection 6 of 8

The abstract sentence 'it turns that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small' needs 'turns out' and a singular verb ('is useless'). The same sentence in the abstract is the one most likely to be quoted out of context, so it is worth making it grammatically clean.

The paper says, at References

Pith says: minor objection 7 of 8

Several reference entries need correction: Ref. [3] reads 'Relastivistic', Ref. [12] reads 'Prarmana', Ref. [39] reads 'Mod. Rev. Phys.' where 'Rev. Mod. Phys.' is meant, Ref. [76] gives 'Phys. D65' where 'Phys. Rev. D65' is meant, and Refs. [95], [107] and [108] give the initial 'Y. Schroeder' where 'Y. Schroeder' should presumably be 'Y. Schroeder' (Schroeder). Refs. [115], [117] and [118] do not appear to be cited anywhere in the text and should either be cited or removed.

The paper says, at Sec. 4.3

Pith says: minor objection 8 of 8

The phrase 'the most severe divergences from a given infro ared divergent loop diagram' contains a typographical corruption of 'infrared'. Eq. (136) contains 'm41' where m^4 is presumably intended.

What would settle it

Recompute the three-loop DRSPT pressure band of Fig. 19(b) using the three-loop gap equation (229) in place of the four-loop gap equation (233) that was actually used, and compare the resulting mu-variation band width against the g^5 perturbative band in the same figure. If the DRSPT band is then narrower than the perturbative band, the 'improves convergence' claim survives for DRSPT and the abstract's framing is defensible; if it remains wider, the abstract should be qualified to state that DRSPT improves the convergence of the successive DRSPT approximants without improving on the scale sensitivity of the weak-coupling result.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up confidence high
  2. Skeptic agrees 50 s · 52,043 tokens read, 2,699 written
  3. Editor wrote the verdict above 19 s · 53,103 tokens read, 1,027 written
  4. Referee minor revision: 2 major objections, 8 minor 3 min 34 s · 52,489 tokens read, 11,628 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

This is a review, and it should be read as one. There is no new derivation, no new number, no new scheme. What it does is put the weak-coupling expansion, dimensional reduction, SPT, HTLPT, the Phi-derivable approach, DRSPT, and DR Phi-derivable side by side, with the actual formulas and the sum-integral appendices, so you can see what each method costs and where each one breaks. That is worth having, and it is done carefully. The order-g5 results are transcribed correctly as far as I can check, and the appendices make the calculations traceable rather than gestural. For a graduate student walking into this subfield, or a phenomenologist who needs to know which reorganization to reach for, this is a good first stop and a good citation for the state of play circa 2004. The authors are also candid about their own weak points: Sec. 3.1.4 says the mu-variation band is not necessarily a good error measure, and Sec. 5.5.4 shows the mass prescription can fail outright. That kind of honesty is not universal in reviews of this kind. The one soft spot is the abstract's unqualified claim that these reorganizations improve convergence. The body does not quite say that. Sec. 8.4 and Fig. 19 state plainly that DRSPT has a larger scale variation than the perturbative result it is meant to improve on, that the four-loop curve depends on an order-alpha3 hard coefficient set to zero (which the authors call 'highly improbable'), and that the three-loop gap equation has no solution so the three-loop pressure was evaluated with the four-loop gap equation. So the DRSPT entry is not on the same footing as the others, and the composite claim is really method-by-method. That is a framing issue, not a correctness error, and the paper flags it. The Fig. 20 SPT-versus-DR-Phi comparison is similarly confounded by the different gap-equation orders, which the paper also notes. Neither of these sinks the review. It is a solid, useful synthesis with one overreaching sentence in the abstract. I would send it to referees, mainly to check the formula transcriptions and the fairness of the method-by-method comparisons, not because I expect the physics to be wrong.

The strongest objection

The central claim [abstract:s5] is that SPT, HTLPT, the Phi-derivable approach, DRSPT, and DR Phi-derivable each yield more stable successive approximations than naive weak-coupling thermodynamics. For SPT (Fig. 11), the Phi-derivable approach (Fig. 16), and DR Phi-derivable (Fig. 20) the evidence is a visibly narrowed mu-variation band, and the paper is candid that the band is heuristic (Sec. 3.1.4) and that the mass prescription matters (Sec. 5.5.4). Those caveats are acknowledged and I do not re-raise them as new findings. The residual concern is narrower and sits in Sec. 8.4/Fig. 19: the paper states that 'although DRSPT improves the situation in terms of the convergence of successive approximations, the result is still very sensitive to the renormalization scale having a larger variation than even the perturbative result.' That is a direct counterexample to the abstract's unqualified framing, and it is compounded by two further concessions in the same section: the four-loop DRSPT curve departs from the two- and three-loop curves because the order-alpha^3 hard coefficient was set to zero, which the authors call 'highly improbable,' and the three-loop DRSPT gap equation has no solution for g != 0, so the three-loop pressure was evaluated with the four-loop gap equation (233). The result is that the DRSPT entry in the review's five-method comparison is not on the same footing as the others: its band is not a clean measure of convergence, and its central value depends on an uncomputed coefficient. This does not make any equation wrong; it means the composite claim 'these reorganizations improve convergence' is heterogeneous across the five methods and would be more accurate stated method-by-method. A secondary, related point: the SPT-versus-DR-Phi comparison in Fig. 20 is confounded because three-loop SPT is forced to use a two-loop gap equation while DR-Phi uses a three-loop one, a difference the paper itself flags. Neither point is a correctness error; both bear,

The reader's weakest_assumption is correctly identified: the convergence comparisons rest on a factor-of-two mu variation treated as an error band, plus a choice among gap-equation mass prescriptions. The paper flags both (Sec. 3.1.4: the band 'is not necessarily a good measure of the error'; Sec. 5.5.4: the two-loop screening and variational masses are ill-behaved and the variational gap equation has no solution for finite g). Because these are acknowledged, I do not treat them as a new finding, and I do not think they move the verdict. What I add is a narrower observation the reader did not isolate: the abstract's claim is a composite over five methods, and the paper's own Sec. 8.4 and Fig. 19 supply a counterexample within that composite, since DRSPT's scale variation is stated to be larger than the perturbative result's. That is a framing imprecision in a review, not an error in any derivation, and the review's substantive content -- the order-g^5 weak-coupling series, the dimensional-reduction matching, the SPT and Phi-derivable gap equations, and the DR constructions -- is internally consistent and traceable to cited published calculations. The paper also does the honest thing at several points: it notes that the 3d lattice results of Ref. [95] used in Fig. 14 are now known to be incorrect because beta_E1 was assumed zero, and it flags the unresolved tension between Braaten-Petitgirard's uncancellable order-g^6 poles and the renormalizability claims of van Hees-Knoll and Blaizot-Iancu-Reinosa, while noting the difference does not enter at the order g^5 presented. Those are the marks of a careful review. My recommendation is therefore UNCHANGED: ACCEPT, with the concrete test above offered as a way to decide whether the abstract's blanket convergence claim should be narrowed to a per-method statement. Confidence in this assessment is moderate-to-high; the concern is about framing rather than about any equation in the paper.

The referee's report

If the survey is accurate — and on the formulas I checked it is — the paper is a genuinely useful reference. It collects in one place the explicit order-g^5 free energies for the scalar theory and for QCD (Eqs. (61) and (65)), the EQCD matching coefficients including the order-epsilon pieces of the Debye mass (Eqs. (110), (114), (115), (116), (117)), and the loop expansions of SPT, HTLPT and the Phi-derivable approach (Eqs. (137)-(139), (159)-(161), (170)-(198)), together with a consistent set of sum-integral and three-dimensional integral tables in the appendices. The comparative figures are the part most likely to be cited by non-specialists, and the paper is unusually candid about their weaknesses: Sec. 3.1.4 states that the factor-of-two band 'is not necessarily a good measure of the error', Sec. 5.5.4 reports that two of the three mass prescriptions fail at two loops, and Sec. 8.4 states outright that the DRSPT scale variation is larger than the perturbative one it is meant to improve on. That candour is a strength of the manuscript and should be preserved in any revision. The paper does not ship machine-checked proofs or reproducible code, and it makes no falsifiable newpred

Pith's reading of the paper (a machine's summary, not the authors')

This review asks why the standard weak-coupling expansion of thermodynamic quantities in hot field theories fails at the couplings actually reached in heavy-ion collisions, and what can be done about it. Naive perturbation theory breaks down at finite temperature because the massless zero-Matsubara-frequency modes produce infrared divergences that are cured only by summing an infinite class of diagrams; the Braaten-Pisarski program does this systematically and correctly, but the resulting series for the pressure still converges only for $g(2πT)$ well below 1. The authors walk through five reorganizations — screened perturbation theory, hard-thermal-loop perturbation theory, the $Φ$-derivable approach, dimensionally reduced SPT, and the dimensionally reduced $Φ$-derivable approach — each of which trades the strict coupling expansion for a variational mass or a three-dimensional effective theory. The value of the review is comparative: it puts the same benchmark, successive approximations to the pressure with renormalization-scale bands, against each method, and reports where each produces stable sequences and where it stalls.

What carries the argument. The load-bearing device is hard-thermal-loop resummation: loop corrections whose internal momenta are of order $T$ are not small when the external momentum is of order $gT$, so effective propagators and vertices must replace the bare ones before any soft observable is computed. The paper then uses that structure in two ways. In SPT and HTLPT a single variational mass $m$ is added to and subtracted from the Lagrangian, the loop expansion becomes an expansion in a bookkeeping parameter $δ$, and $m$ is fixed by a gap equation; the paper compares Debye-mass, tadpole-mass, and variational-mass prescriptions and finds the tadpole one best behaved. In the dimensionally reduced schemes, the nonzero Matsubara modes and the fermions are integrated out into the coefficients of a three-dimensional Lagrangian, the hard part is kept as strict perturbation theory, and the soft part is computed with SPT or with a two-particle-irreducible $Φ$-derivable truncation, so that the $g^3$ and higher terms are retained in full rather than expanded.

Core claim. The central claim is that the weak-coupling expansion of thermodynamic quantities in hot field theories is not merely slowly converging but practically useless at the couplings of interest, and that this can be repaired by reorganizing the series around a resummed mass or a dimensionally reduced effective theory. The evidence for the first half is the explicit order-$g^5$ pressure for massless $φ^4$ theory, QED, and QCD: as each order is added, the renormalization-scale dependence grows rather than shrinks, and the order-$g^3$ term in QCD is smaller than the order-$g^2$ term only for $α_s ≤ 1/20$, corresponding to temperatures of order $10^5$ GeV. For the second half, the paper shows that screened perturbation theory with a gap-equation mass, HTL perturbation theory, and the two dimensionally reduced schemes all yield sequences of approximants whose bands overlap and whose scale variation is smaller than the strict-expansion bands, both in the scalar toy model and in pure-glue QCD. The review is explicit that the improvement is not uniform: two-loop HTLPT lies above the available lattice pressure for $T < 5T_c$, and several gap equations cease to have solutions beyond a critical coupling.

The premise it rests on. The comparison between methods leans on treating a factor-of-two variation of the renormalization scale, together with the choice of gap-equation mass prescription, as a stand-in for the theoretical error, and the paper itself says in Sec. 3.1.4 that this band is not necessarily a good measure of the error.

If the paper is right (Pith's inferences, not the authors')

The order-$g^6\log g$ coefficient of the QCD pressure can be assembled from a four-loop calculation in electrostatic QCD plus a nonperturbative coefficient from the magnetic scale, and the review reports both pieces.

If the dimensionally reduced $Φ$-derivable approach is as economical as the scalar-model results suggest, the QCD pressure becomes computable with a three-dimensional two-particle-irreducible truncation instead of a full four-dimensional one.

SPT and HTLPT are gauge invariant order by order and apply to real-time quantities, unlike dimensional reduction, so the same reorganization can in principle be carried to transport coefficients and damping rates.

The failure of several gap equations to admit solutions beyond a critical coupling sets a practical ceiling on how far these variational schemes can be pushed toward $T_c$.

Because the two-loop $Φ$-derivable entropy reduces to the one-loop expression, the entropy is a cleaner benchmark than the pressure for comparing reorganizations, since it avoids the integration constant the pressure requires.

The pattern in the figures — bands that overlap for the reorganizations but fan out for the strict expansion — is what one expects when a resummation has captured the correct infrared physics; on that reading, the residual scale dependence measures missing higher-order hard contributions rather than a flaw in the method.

The comparison suggests a natural division of labour: dimensional reduction where only static quantities are wanted, HTLPT or the $Φ$-derivable approach where real-time observables are needed, with the two cross-checked against each other on the pressure.

The Polyakov-loop extension raised near the end of Sec. 6.1 is where the missing confinement-scale physics near $T_c$ would enter; a version of HTLPT built on a Polyakov-loop background would be the sharpest test of whether the lattice discrepancy is a truncation artifact.

Since the review's own figures show the four-loop DRSPT result deviating because the order-$α^3$ hard coefficient was set to zero, computing that single coefficient is the cheapest way to decide whether the dimensional reduction route or the four-dimensional variational route is the better long-term investment.

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The paper

Abstract

There has been significant progress in our understanding of finite-temperature field theory over the past decade. In this paper, we review the progress in perturbative thermal field theory focusing on thermodynamic quantities. We first discuss the breakdown of naive perturbation theory at finite temperature and the need for an effective expansion that resums an infinite class of diagrams in the perturbative expansion. This effective expansion which is due to Braaten and Pisarski, can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of g. However, it turns that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small. We critically discuss various ways of reorganizing the perturbative series for thermal field theories in order to improve its convergence. These include screened perturbation theory (SPT), hard-thermal-loop perturbation theory (HTLPT), the Phi-derivable approach, dimensionally reduced (DR) SPT, and the DR Phi-derivable approach.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.68

“However, it turns that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.63 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.26 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
There has been significant progress in our understanding of finite-temperature field theory over the past decade. background 1.00 0.04 0.02 asserted 1.00
In this paper, we review the progress in perturbative thermal field theory focusing on thermodynamic quantities. contribution 0.95 0.37 0.05 asserted 0.77
We first discuss the breakdown of naive perturbation theory at finite temperature and the need for an effective expansion that resums an infinite class of diagrams in the perturbative expansion. background 0.81 0.08 0.06 asserted 1.00
This effective expansion which is due to Braaten and Pisarski, can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of g. background 0.65 0.05 0.08 asserted 0.94
However, it turns that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small. limitation 0.63 0.27 0.18 asserted 1.00
We critically discuss various ways of reorganizing the perturbative series for thermal field theories in order to improve its convergence. contribution 0.87 0.43 0.07 asserted 0.88
These include screened perturbation theory (SPT), hard-thermal-loop perturbation theory (HTLPT), the Phi-derivable approach, dimensionally reduced (DR) SPT, and the DR Phi-derivable approach. method 0.49 0.10 0.04 asserted 0.96

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.98

What was examined, and what was not

Sources read

  • arXiv listing abstract v2

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

How the labels were made

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,618 input tokens, $0.0002 reported.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

Figures (20)
Figure 1 of the paper
Figure 1. One- and two-loop scalar self-energy graphs.
Figure 2 of the paper
Figure 2. Bubble diagrams contributing to the scalar self-ene
Figure 3 of the paper
Figure 3. Subleading three-loop self-energy diagram.
Figure 4 of the paper
Figure 4. One-loop photon self-energy diagram. is suppressed by g 2 and is thus a perturbative correction. However, for soft P, the hard thermal loop is O(1) and is therefore as important as the tree￾level contribution to the amplitude. These loop corrections are called “hard” because the relevant integrals are dominated by momenta of order T. Also note that the hard thermal loop in the two-point function is finite since it is exclusively due to thermal fluctuations. Quantum fluctuations do not enter. Both properties are shared by all hard thermal loops. What about higher-order n-point…
Figure 5 of the paper
Figure 5. Diagrams which contribute up to three-loop order in s
Figure 6 of the paper
Figure 6. Weak-coupling expansion for the pressure to orders
Figure 7 of the paper
Figure 7. Weak-coupling expansion for the pressure of
Figure 8 of the paper
Figure 8. It is interesting to note that the two-loop band is alm
Figure 9 of the paper
Figure 9. Diagrams which contribute up to three-loop order in s
Figure 10 of the paper
Figure 10. Solutions m∗(T) to the SPT one-loop gap equation (shaded bands) and two-loop gap equations (lines) as functions of g(2πT). 5.5.4 Comparison At one loop, the three different prescriptions give the same gap equation, Eq. (140). Moreover, it turns out that the two-loop tadpole mass coincides with the one-loop tadpole mass. However, at two-loops the screening and variational masses are ill-behaved [88]. The screening mass solution ceases to exist beyond g ∼ 2.6 and the variational gap equation has no solution for finite g. In
Figure 11 of the paper
Figure 11. One-, two-, and three-loop SPT-improved pressure a
Figure 12 of the paper
Figure 12. As can been seen from this Figure both modes approach
Figure 13 of the paper
Figure 13. Diagrams contributing through NLO in HTLPT. Shaded
Figure 14 of the paper
Figure 14. The LO and NLO results for the pressure in HTLPT compa
Figure 15 of the paper
Figure 15. The thermodynamic potential Ω2 is Ω2[D] = 1 2 P Z P log  P 2 + Π(P)  − 1 2 P Z P Π(P) P2 + Π(P) + 1 8 g 2
Figure 16 of the paper
Figure 16. Two- and three-loop Φ-derivable approximations to
Figure 17 of the paper
Figure 17. One-loop QCD self-energy graphs. The gap equations that are obtained by varying the thermodynamic po￾tential with respect to DT (P) and DL(P) are shown in
Figure 18 of the paper
Figure 18. Diagrams which contribute up to four-loop order in D
Figure 19 of the paper
Figure 19. (a) Two-, three-, and four-loop DRSPT results for th
Figure 20 of the paper
Figure 20. DR Φ-derivable approximations to the pressure divi

Figures are the authors', reproduced from the paper.

References (130)
  1. [1] D.J. Schwarz, Annalen. Phys. 12, 220 (2003) on Pith
  2. [2] C.-Y. Wong, Introduction to High-Energy Heavy-Ion Collisions , World Scientific Pub Cok, (1994) on Pith
  3. [3] J. Bartke, Relastivistic Heavy Ion Physics , Singapore : World Sci. (2003) on Pith
  4. [4] J. Letessier and J. Rafelski, “Hadrons and Quark-Gluon P lasma,” Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 18, 1 (2002) on Pith
  5. [5] U. Heinz, Nucl. Phys. A685, 414 (2001) on Pith
  6. [6] I. Tserruya, Pramana 60, 577 (2003); nucl-ex/0204012 on PithPith's reviewarXiv
  7. [7] H. Heiselberg and M. Hjorth-Jensen, Phys. Rept. 328, 237 (2000) on Pith
  8. [8] E.S. Fraga, R.D. Pisarski and J. Schaffner-Bielich, Phys . Rev. D63, 121702 (2001) on Pith
  9. [9] J.O. Andersen and M. Strickland, Phys. Rev. D66 105001, (2002) on Pith
  10. [10] U. Heinz, K. Kajantie and T. Toimela, Phys. Lett. 183B, 96 (1987); Ann. of Phys. (N.Y.) 176, 218 (1987) on Pith
  11. [11] K. Kajantie and J. Kapusta, Ann. of Phys. (N.Y.), 160, 477 (1985) on Pith
  12. [12] J.C. Parikh, P.J. Siemens and J.A. Lopez, Prarmana 32, 555 (1989) on Pith
  13. [13] T.H. Hansson and I. Zahed, Phys. Rev. Lett. 58, 2397 (1987); Nucl. Phys. B292, 725 (1987) on Pith
  14. [14] H.-Th. Elze, U. Heinz, K. Kajantie and T. Toimela, Z. Phy s. C37, 305 (1988) on Pith
  15. [15] H.-Th. Elze, K. Kajantie and T. Toimela, Z. Phys. C37, 601 (1988) on Pith
  16. [16] R. Kobes and G. Kunstatter, Phys. Rev. Lett. 61, 392 (1988) on Pith
  17. [17] S. Nadkarni, Phys. Rev. Lett. 61, 396 (1988) on Pith
  18. [18] M.E. Carrington, T.H. Hansson, H. Yamagishi and I. Zahe d, Ann. of Phys. (N.Y.) 190, 373 (1989) on Pith
  19. [19] J. Milana, Phys. Rev. D39, 2419 (1989) on Pith
  20. [20] G. Gatoff and J. Kapusta, Phys. Rev. D41, 611 (1990) on Pith
show 80 more
  1. [21] J. Kapusta and T. Toimela, Phys. Rev. D 39 , 3197 (1989) on Pith
  2. [22] R. Kobes, G. Kunstatter and K.W. Mak, Phys. Lett. 223B, 433 (1989) on Pith
  3. [23] S. Catani and E. d’Emilio, Phys. Lett. B238, 373 (1990) on Pith
  4. [24] U. Kraemmer, M. Kreuzer and A. Rebhan, Ann. of Phys. (N.Y .) 201, 223 (1990). 77 on Pith
  5. [25] V.V. Lebedev and A.V. Smilga, Annals Phys. 202, 229 (1990) on Pith
  6. [26] R.D. Pisarski, Phys. Rev. Lett. 63, 1129 (1989) on Pith
  7. [27] R.D. Pisarski, Nucl. Phys. A525, 175 (1991) on Pith
  8. [28] E. Braaten and R.D. Pisarski, Nucl. Phys. B337, 569 (1990) on Pith
  9. [29] J. Frenkel, A.V. Saa and J.C. Taylor, Phys. Rev. D46, 3670 (1992) on Pith
  10. [30] C. Corian` o and R.R. Parwani, Phys. Rev. Lett. 73, 2398 (1994) on Pith
  11. [31] P. Arnold and C.-X. Zhai, Phys. Rev. D50 7603, (1994); D51, 1906 (1995) on Pith
  12. [32] R.R. Parwani and H. Singh, Phys. Rev. D51, 4518 (1992) on Pith
  13. [33] E. Braaten and A. Nieto, Phys. Rev. D51, 6990 (1995) on Pith
  14. [34] R.R. Parwani, Phys. Lett. B334, 420 (1994); 342, 454 (E); R.R. Parwani and C. Corian` o, Nucl. Phys.B434, 56 (1995) on Pith
  15. [35] J.O. Andersen, Phys. Rev. D53, 7286 (1996) on Pith
  16. [36] C.-X. Zhai and B. Kastening, Phys. Rev. D52, 7232 (1995) on Pith
  17. [37] E. Braaten and A. Nieto, Phys. Rev. Lett. 76, 1417 (1996); Phys. Rev. D53, 3421 (1996) on Pith
  18. [38] P. Ginsparg, Nucl. Phys. B170, 388 (1980) on Pith
  19. [39] D. Gross, R.D. Pisarski and L.G. Yaffe, Mod. Rev. Phys. (1 981) on Pith
  20. [40] T. Appelquist and R.D. Pisarski, Phys. Rev. D23, 2305 (1981) on Pith
  21. [41] S. Nadkarni, Phys. Rev. D27, 917 (1983); Phys. Rev. D38, 3287 (1988); Phys. Rev. Lett. 60, 491 (1988) on Pith
  22. [42] N.P. Landsman, Nucl. Phys. B322, 498 (1989) on Pith
  23. [43] G.P. Lepage, in From Actions to Answers , edited by T. DeGrand and D. Toussaint, World Scientific, Singapore (1989) on Pith
  24. [44] T. Appelquist and J. Carazzone, Phys. Rev. D11, 2856 (1975) on Pith
  25. [45] K. Farakos, K. Kajantie, K. Rummukainen, M. E. Shaposhn ikov, Phys. Lett. B336, 494 (1994); Nucl. Phys. B425, 67 (1994); Nucl. Phys. B442, 317 (1995) on Pith
  26. [46] T. Hatsuda, Phys. Rev. D 56 (1997) 8111; B. Kastening, Phys. Rev. D 56 (1997) 8107 on Pith
  27. [47] R.R. Parwani, Phys. Rev. D 63, 054014 (2001); Phys. Rev. D 64, 025002 (2001) on Pith
  28. [48] V. I. Yukalov and E. P. Yukalova, in ‘Relativistic Nucle ar Physics and Quantum Chromodynamics’ ed. by A.M. Baldin and V.V. Burov (Joint Ins t. Nucl. Res., Dubna, 2000); arXiv:hep-ph/0010028 on PithPith's reviewarXiv
  29. [49] G. Cvetiˇ c and R. K¨ ogerler, Phys. Rev. D66, 105009 (2002). 78 on Pith
  30. [50] F. Karsch, Lectures given at 40th Internationale Unive rsitatswochen fu¨ r Theoretische Physik: Dense Matter (International Univers ity School of Theoretical Physics) (IUKT40), Schladming, Styria, Austr ia, 3-10 Mar 2001. Lect.Notes Phys. 583 209 (2002); Also in *Schladming 20... on Pith
  31. [51] G. Boyd et al., Phys. Rev. Lett. 75, 4169 (1995); Nucl. Phys. B469, 419 (1996) on Pith
  32. [52] S. Gottlieb et al., Phys. Rev. D55, 6852 (1997); C. Bernard et al., Phys. Rev. D55, 6861 (1997); J. Engels et al., Phys. Lett. B396, 210 (1997) on Pith
  33. [53] Z. Fodor and S.D. Katz, Phys. Lett. B534, 87 (2002) on Pith
  34. [54] Z. Fodor and S.D. Katz, JHEP 0203, 014 (2002) on Pith
  35. [55] P. de Forcrand and O. Philipsen, Nucl. Phys. B642, 290 (2002) on Pith
  36. [56] C.R. Allton, S. Ejiri, S.J. Hands,O. Kaczmarek, F. Kars ch, E. Laermann, C. Schmidt and L. Scorzato, Phys. Rev. D66, 074507 (2002) on Pith
  37. [57] Z. Fodor Z, S.D. Katz, and K.K. Szab´ o, Phys. Lett. B568, 73 (2003) on Pith
  38. [58] M.D. Elia and M.P. Lombardo, Phys. Rev. D67, 014505 (2003) on Pith
  39. [59] R.V. Gavai and S. Gupta, Phys. Rev. D68, 034506 (2003) on Pith
  40. [60] C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Kars ch, E. Laermann, and C. Schmidt, Phys. Rev. D68, 014507 (2003) on Pith
  41. [61] Z. Fodor and S. D. Katz, arXiv:hep-lat/0402006 on PithPith's reviewarXiv
  42. [62] T.S. Biro, P. Levai and B.M¨ uller, Phys. Rev. D42, 3078 (1990) on Pith
  43. [63] A. Peshier, B. K¨ ampfer, O.P. Pavlenko and G. Soff, Phys. Rev. D54, 2399 (1996) on Pith
  44. [64] P. L´ evai and U. Heinz, Phys. Rev. C57, 1879 (1998) on Pith
  45. [65] A. Peshier, B. Kampfer and G. Soff, Phys. Rev. C61, 045203 (2000); Phys. Rev. D66, 094003 (2002) on Pith
  46. [66] K. K. Szabo and A. I. Toth, JHEP 0306, 008 (2003) on Pith
  47. [67] J.M. Luttinger and J.C. Ward, Phys. Rev. 118, 1417 (1960) on Pith
  48. [68] G. Baym, Phys. Rev. 127, 1391 (1962); Phys. Rev. D63, 065003 (2001) on Pith
  49. [69] J. M. Cornwall, R. Jackiw and E. Tomboulis, Phys.Rev. D10, 2428 (1974) on Pith
  50. [70] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Lett. B470, 181 (1999) on Pith
  51. [71] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. 83, 2906 (1999) on Pith
  52. [72] J.-P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. D 63, 065003 (2001) on Pith
  53. [73] A. Peshier, Phys. Rev. D63, 105004 (2001). 79 on Pith
  54. [74] P. Romatschke, arXiv:hep-ph/0210331 on PithPith's reviewarXiv
  55. [75] A. Rebhan and P. Romatschke, Phys. Rev. D68, 025022 (2003) on Pith
  56. [76] H. Van Hees and J. Knoll, Phys. D65, 025010 (2002); D65, 105005 (2002) on Pith
  57. [77] J.-P. Blaizot, E. Iancu and U. Reinosa, Phys. Lett. B568, 160 (2003); Nucl. Phys. A736, 149 (2004) on Pith
  58. [78] A. Arrizabalaga and J. Smit, Phys. Rev. D66, 065014 (2002) on Pith
  59. [79] M. E. Carrington, G. Kunstatter and H. Zaraket, arXiv:h ep-ph/0309084 on Pith
  60. [80] E. Braaten and E. Petitgirard, Phys. Rev. D65, 085039 (2002); Phys.Rev. D65, 041701 (2002) on Pith
  61. [81] J.-P. Blaizot, E. Iancu and A. Rebhan, “Thermodynamics of the high- temperature quark gluon plasma,” in ’Quark Gluon Plasma 3’, eds. R.C. Hwa and X.N. Wang, World Scientific (Singapore); arXiv:hep-ph/ 0303185 on Pith
  62. [82] U. Kraemer and A. Rebhan, Rept. Prog. Phys. 67, 351 (2004) on Pith
  63. [83] P. M. Stevenson, Phys. Rev. D23, 2916 (1981) on Pith
  64. [84] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polym er Physics, 2nd edition, World Scientific Publishing Co., Singapore, ( 1995); A. N. Sisakian, I. L. Solovtsov, O. Shevchenko, Int. J. Mod. P hys. A 9, 1929 (1994); W. Janke and H. Kleinert, quant-ph/9502019 on PithPith's reviewarXiv
  65. [85] A. Duncan and M. Moshe, Phys. Lett. 215B, 352 (1988); A. Duncan, Phys. Rev.D47, 2560, (1993) on Pith
  66. [86] F. Karsch, A. Patk´ os, and P. Petreczky, Phys. Lett. B401, 69 (1997) on Pith
  67. [87] S. Chiku and T. Hatsuda, Phys. Rev. D58, 076001 (1998) on Pith
  68. [88] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D63, 105008 (2001) on Pith
  69. [89] J.O. Andersen and M. Strickland, Phys. Rev. D64, 105012 (2001) on Pith
  70. [90] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83, 2139 (1999); Phys. Rev. D61, 014017 (2000) on Pith
  71. [91] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D61, 074016 (2000) on Pith
  72. [92] J.O. Andersen, E. Braaten, E. Petitgirard and M. Strick land, Phys. Rev. D66, 085016 (2002) on Pith
  73. [93] J.O. Andersen, E. Petitgirard and M. Strickland, Phys. Rev. D 70 , 024004 (2003) on Pith
  74. [94] K. Kajantie et al., Phys. Rev. Lett. 79, 3130 (1997) on Pith
  75. [95] K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Phys. Rev. Lett. 86, 10 (2001). 80 on Pith
  76. [96] A. Vuorinen, Phys. Rev. D68, 054017 (2003) on Pith
  77. [97] J.I. Kapusta, Finite-Temperature Field Theory , Cambridge University Press (1989) on Pith
  78. [98] M. le Bellac Thermal Field Theory , Cambridge University Press (1996) on Pith
  79. [99] E. Braaten and R.D. Pisarski, Phys. Rev. D45, 1827 (1992) on Pith
  80. [100] J.C. Taylor and S.M.H. Wong, Nucl. Phys. B346, 115 (1990) on Pith

The first 100 of 130 references are listed here. All of them are in this paper's citation record.

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  24. Physics of Strongly coupled Quark-Gluon Plasma hep-ph · Jul 2008
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