{"id":"0bf9d9b1-124b-49ab-a8ae-d1ad2bf41a90","arxiv_id":"hep-ph/0404164","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"At experimentally relevant couplings, the standard weak-coupling expansion for hot-field thermodynamics is poorly convergent, and resummation schemes such as SPT, HTLPT, and Phi-derivable approaches give more stable approximations.","lead":"This review explains why ordinary perturbation theory fails for hot quantum fields and compares five resummation schemes that aim to restore convergence. It is a useful map for physicists working on quark-gluon plasma thermodynamics.","discovery_kind":"review","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The review's blanket claim that reorganizations improve convergence is not uniformly borne out by its own figures: Fig. 19 shows DRSPT with a larger renormalization-scale variation than the g^5 perturbative result it is meant to improve on.","rationale":"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.","tokens_in":52043,"tokens_out":2699,"duration_ms":49976,"concrete_test":"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.","verdict_should_be":"UNCHANGED","load_bearing_attack":"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,","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"This manuscript is a review of perturbative thermal field theory with emphasis on thermodynamic quantities. It is organized as a sequence of reorganizations of the weak-coupling expansion: the breakdown of naive perturbation theory and the Braaten-Pisarski resummation program (Sec. 2), the weak-coupling expansion of the free energy through order g^5 for massless phi^4 theory, QED and QCD (Sec. 3), dimensional reduction and the EQCD/MQCD effective-field-theory construction (Sec. 4), and then five reorganization schemes — screened perturbation theory (Sec. 5), hard-thermal-loop perturbation theory (Sec. 6), the Phi-derivable approach (Sec. 7), dimensionally reduced SPT (Sec. 8), and the dimensionally reduced Phi-derivable approach (Sec. 9) — followed by a summary (Sec. 10) and three appendices collecting the sum-integrals and three-dimensional integrals used throughout. The recurring quantitative diagnostic is the variation of a given approximant under a factor-of-two change of the renormalization or factorization scale, displayed as shaded bands in Figs. 6, 7, 8, 11, 14, 16, 19 and 20. The paper claims no new derivation; the explicit formulas it presents are quoted from the cited literature, which is the normal mode of operation for a review.","tokens_in":52489,"tokens_out":11628,"duration_ms":214339,"significance":"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 ","major_comments":[{"comment":"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. 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.","section":"Sec. 10 vs. Sec. 8.4 and Fig. 19","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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. 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.","section":"Sec. 9.2, Fig. 20, and Sec. 10","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"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.","section":"Sec. 6.1, Eqs. (159) and (161)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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. (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.","section":"Sec. 8, Eqs. (222), (224), (225) and Sec. 9, Eq. (237)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Sec. 8, Eq. (222)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Sec. 4.1.1, Eq. (74)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Sec. 5.5.4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Abstract and Sec. 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Sec. 4.3","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision"},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"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-alpha^3 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.","headline":"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.","tokens_in":53103,"tokens_out":1027,"duration_ms":18955},"pith_extraction":{"msc":[],"pacs":["12.38.Mh","11.10.Wx","12.38.Bx","05.70.Ce"],"model":"selfhosted/deepseek-v4.1-flash","headline":"The weak-coupling expansion of the pressure in hot QCD is useless at the couplings reached in heavy-ion collisions, and the paper compares five resummation schemes that make the successive approximations behave.","keywords":["quark-gluon plasma","finite-temperature field theory","resummation","hard thermal loops","screened perturbation theory","dimensional reduction","Phi-derivable approximation","QCD pressure"],"falsifier":"A continuum-extrapolated four-dimensional lattice calculation of the pressure in the window $T_c < T < 5T_c$, compared against the next-to-leading-order HTLPT band: the review already notes that HTLPT sits above the available lattice results there, so a precise lattice curve would decide whether the disagreement is a truncation artifact or a genuine failure of that reorganization.","tokens_in":53462,"feed_emoji":"🌡️","tokens_out":4418,"duration_ms":81085,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hot QCD's pressure series fails; five resummations compared","feed_subtitle":"At the couplings reached in heavy-ion collisions, adding perturbative orders makes the prediction worse, not better.","key_machinery":"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_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Braaten and Pisarski's resummation program, which defines the infinite class of diagrams that must be summed and underpins every method reviewed.","marker":"[28]"},{"why":"Introduces screened perturbation theory for the thermodynamics of massless $φ^4$ theory, the prototype the gauge-theory methods generalize.","marker":"[86]"},{"why":"Three-loop SPT calculation for the scalar pressure and the comparison of mass prescriptions that fixes the tadpole gap equation as the usable one.","marker":"[88]"},{"why":"Two-loop HTL perturbation theory free energy for pure-glue QCD, the number compared against lattice data in Fig. 14.","marker":"[92]"},{"why":"Braaten and Nieto's construction of electrostatic and magnetostatic QCD and the order-$g^5$ free energy via effective field theory.","marker":"[37]"},{"why":"Arnold and Zhai's order-$g^4$ free energy of QCD, the first order at which renormalization effects enter the weak-coupling series.","marker":"[31]"},{"why":"Braaten and Petitgirard's strategy for solving the $n$-loop $Φ$-derivable approximation by expanding in $m/T$ and $g^2$, and the three-loop scalar result.","marker":"[80]"},{"why":"Blaizot, Iancu, and Rebhan's approximate two-loop $Φ$-derivable treatment of QCD using hard-thermal-loop self-energies as gap-equation solutions.","marker":"[72]"},{"why":"Blaizot, Iancu, and Rebhan's dimensionally reduced SPT applied to electrostatic QCD, the scheme the paper's own DRSPT section follows.","marker":"[130]"},{"why":"Boyd et al. lattice thermodynamics of pure-glue QCD, the benchmark against which the HTLPT pressure bands are judged.","marker":"[51]"}],"fun_headline_variants":["Hot QCD pressure series worsens with each order at heavy-ion couplings","Five resummations overlap in bands but HTLPT overshoots lattice below 5Tc","Gap equations stop having solutions beyond a critical coupling in hot QCD","Weak-coupling pressure expansion is useless until alpha_s is about 1/20","Resummed mass and dimensionally reduced schemes shrink scale bands in hot QCD"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hot QCD pressure series worsens with each order at heavy-ion couplings","Five resummations overlap in bands but HTLPT overshoots lattice below 5Tc","Gap equations stop having solutions beyond a critical coupling in hot QCD","Weak-coupling pressure expansion is useless until alpha_s is about 1/20","Resummed mass and dimensionally reduced schemes shrink scale bands in hot QCD"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2702,"prompt_tokens":955,"completion_tokens":1747,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":1655}},"tokens_in":955,"tokens_out":1747,"duration_ms":34560,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":1655,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-13T08:45:08.773127+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"pipeline_version":"pith-pipeline@v1.5.0-archive-selfhost","input_kind":null,"input_chars":null,"input_total_chars":null,"passes_missing":null,"published_standing":null,"published_confidence":null,"objection_audit":null,"run_record":null,"basis":{"kind":"not_recorded","source":null,"chars_given":null,"chars_total":null},"receipt":null,"falsifier":"A continuum-extrapolated four-dimensional lattice calculation of the pressure in the window $T_c < T < 5T_c$, compared against the next-to-leading-order HTLPT band: the review already notes that HTLPT sits above the available lattice results there, so a precise lattice curve would decide whether the disagreement is a truncation artifact or a genuine failure of that reorganization.","supporting_citations":[{"cited_title":"Phys. Rev. D65, 085039 (2002); Phys.Rev. D65, 041701","cited_arxiv_id":null,"evidence_quote":"Braaten and Petitgirard's strategy for solving the $n$-loop $Φ$-derivable approximation by expanding in $m/T$ and $g^2$, and the three-loop scalar result."},{"cited_title":"Nucl. Phys. B337, 569","cited_arxiv_id":null,"evidence_quote":"Braaten and Pisarski's resummation program, which defines the infinite class of diagrams that must be summed and underpins every method reviewed."},{"cited_title":"A. Patk´ os, and P. Petreczky, Phys. Lett. B401, 69","cited_arxiv_id":null,"evidence_quote":"Introduces screened perturbation theory for the thermodynamics of massless $φ^4$ theory, the prototype the gauge-theory methods generalize."},{"cited_title":"Phys. Rev. D63, 105008","cited_arxiv_id":null,"evidence_quote":"Three-loop SPT calculation for the scalar pressure and the comparison of mass prescriptions that fixes the tadpole gap equation as the usable one."},{"cited_title":"Phys. Rev. D66, 085016","cited_arxiv_id":null,"evidence_quote":"Two-loop HTL perturbation theory free energy for pure-glue QCD, the number compared against lattice data in Fig. 14."},{"cited_title":"Phys. Rev. Lett. 76, 1417 (1996); Phys. Rev. D53, 3421","cited_arxiv_id":null,"evidence_quote":"Braaten and Nieto's construction of electrostatic and magnetostatic QCD and the order-$g^5$ free energy via effective field theory."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Arnold and Zhai's order-$g^4$ free energy of QCD, the first order at which renormalization effects enter the weak-coupling series."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Blaizot, Iancu, and Rebhan's approximate two-loop $Φ$-derivable treatment of QCD using hard-thermal-loop self-energies as gap-equation solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Blaizot, Iancu, and Rebhan's dimensionally reduced SPT applied to electrostatic QCD, the scheme the paper's own DRSPT section follows."},{"cited_title":"Boyd et al., Phys","cited_arxiv_id":null,"evidence_quote":"Boyd et al. lattice thermodynamics of pure-glue QCD, the benchmark against which the HTLPT pressure bands are judged."}],"lane_accuracy":{"status":"not yet measured","lane":{"model":"selfhosted/deepseek-v4.1-flash","input_kind":null},"url":"https://pith.science/accuracy"},"review_version":1}