The pressure of hot QCD up to g^6 ln(1/g)

K. Kajantie; K. Rummukainen; M. Laine; Y. Schroder

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The pressure of hot QCD up to g^6 ln(1/g)

K. Kajantie, M. Laine, K. Rummukainen, Y. Schroder

High Energy Physics - Phenomenology · Submitted 20 Nov 2002 · 29 references

Pith's verdict: The g^6 ln(1/g) coefficient of the QCD pressure is a real, well-structured result, but its two numerical inputs live in companion papers still listed as in preparation.

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arXiv IDhep-ph/0211321 (version 2)

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CitationK. Kajantie, K. Rummukainen, M. Laine, Y. Schroder. "The pressure of hot QCD up to g^6 ln(1/g)." arXiv:hep-ph/0211321 (2002). https://pith.science/paper/AT4NHKS5

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This is the paper that closes the perturbative expansion of the hot-QCD pressure at the last order perturbation theory can reach. Eq. (6.3) gives the g^6 ln(1/g) coefficient analytically, assembled from four-loop vacuum energies of the two 3d effective theories. That is the new content, and it is worth having: it is a permanent reference number that any resummed or lattice treatment has to reproduce, and it fixes the curvature of p/p_SB at small T/Λ_MS that the earlier lattice analysis in [14] got wrong because it missed a hard-scale logarithm. The remark in Sec. VII about that missing term is a genuine correction to the literature, not a cosmetic one, and the authors deserve credit for tracking it down rather than quietly ignoring the discrepancy with the old data. The internal checks are the right ones and they pass: the 1/ε poles cancel between p_E and p_M at O(g^4), the m_G-regulated computation is gauge-parameter independent and its higher-order poles cancel, and the O(g^5) result of [7] is reproduced from the Appendix A coefficients. On the reader's flagged weak point, I think the reader is wrong and the stress-test is right. Putting a hand mass m_G into the gauge and ghost propagators to separate IR from UV in a super-renormalizable theory is standard, and the 1/ε coefficient is IR-insensitive by construction; the gauge-independence check is exactly the confirmation you would want. That is not where the soft spot is. The soft spot is that α_M ≈ 0.555 and α_G ≈ 0.196, the numerical heart of the ln(1/g) term, come from [9] and [10], both listed as in preparation. The structural derivation here is self-contained, but the numbers are not verifiable from this paper, and if either is off, Eq. (6.3) is off. That is a real caveat, not a fatal one, and it resolves itself once the companions appear. The numerical section is explicitly conjectural and the authors say so repeatedly; do not hold the low-T lattice match against them. Read it if you work on thermal field theory or on the equation of state for heavy-ion phenomenology. It deserves referees, and the referee report should ask for the companion papers' status.

Referee report

Significance

If correct, this is the completion of the perturbatively determinable part of the QCD pressure: g^6 ln(1/g) is the last term that perturbation theory can deliver, since the constant beneath the logarithm receives a non-perturbative contribution β_G from the magnetically screened scale g^2T. The result therefore fixes a benchmark that any non-perturbative or resummed treatment of the hot QCD equation of state must reproduce, and it has direct bearing on the interpretation of dimensional-reduction lattice programmes, where the matching between the 3d and 4d theories is exactly the quantity computed here. The paper is also methodologically careful in ways worth naming: the m_G-regulated computation of α_G is validated by gauge-parameter independence and by cancellation of the 1/ε^2 poles and of the degree-nine polynomial in the gauge parameter; the O(g^4) 1/ε cancellation between p_E and p_M in Eq. (6.1) is exhibited explicitly; the O(g^5) truncation of Eq. (6.1) is checked to reproduce the known result of [7]; and the infrared insensitivity of p_M is verified by an independent mass-subtraction procedure. The numerical discussion in Sec. VII is candid about what is and is not under控制,

Major objections (2)

[Secs. III–IV, Eqs. (3.3) and (4.2)] The two numbers that carry the entire numerical content of Eq. (6.3), α_G ≈ 0.195715 and α_M ≈ 0.555017, are quoted from refs. [9] and [10], both listed as 'in preparation'. The structural derivation in this manuscript is self-contained, but the coefficient of the ln(1/g) term is 8C_A^2(α_M + 2α_G) plus the (α_E6 + α_E4 α_E7) piece, so an error in either companion number propagates directly into the main result and into every entry of p_4…p_6 in Eqs. (6.9)–(6.11). As it stands a reader cannot check these inputs from the present paper. I would ask either that the companion papers be made available to the referees and their publication coordinated with this one, or that Secs. III and IV be expanded enough to show the diagram topologies, the master integrals, and the renormalisation prescription by which the 1/ε coefficients were extracted, so that the numbers are at least in principle reproducible from this text. I want to be clear that I do not doubt the computation; the issue is verifiability of the load-bearing inputs, not the logic of the derivation.
[Sec. VII, Eq. (7.1) and Fig. 1] The second sentence of the abstract, that the new terms 'could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures', rests on Fig. 1 evaluated at a specific value δ = 0.7, described in the caption as 'an optimal constant'. The criterion by which δ = 0.7 is optimal is never stated: it is not said whether this is a least-squares fit to the lattice data of [22], a minimisation over a stated temperature window, or a value chosen by inspection. Since the visual impression of agreement in the left panel of Fig. 1 is the sole support for the abstract's claim, the selection rule should be given explicitly, together with the temperature range over which it was applied and the residual at that δ. The authors are commendably explicit that this 'cannot really be considered a prediction' and that dimensional reduction is expected to fail in the same region, and I do not read the section as overclaiming; but the abstract sentence is stronger than what a tuned constant can support, and either the criterion should be stated or the abstract wording softened to match Sec. VII.

Minor objections (10)

[Sec. III, Eq. (3.2)] The concern that the hand-introduced regulator m_G might contaminate the ultraviolet 1/ε coefficient does not, on reading, land: in a super-renormalisable theory the ultraviolet divergence is local and infrared-insensitive, and the paper supplies the two checks that would expose a contamination, namely gauge-parameter independence of α_G and cancellation of the 1/ε^2 poles and of the degree-nine polynomial in ξ. I would only suggest that this argument be stated in one sentence in the text, since the identification of a regulated 1/ε coefficient with the physical α_G of Eq. (3.1) is the step a careful reader will want justified rather than merely performed.
[Sec. VI, Eq. (6.3)] It would help to say explicitly that Eq. (6.3) fixes not only the coefficient of ln(1/g) but also the constants beneath the logarithms, ln α_E4^{1/2} and ln C_A^{1/2}, and that this is why the coefficient of ln[α_s/π (1 + N_f/6)] in p_6 of Eq. (6.11) is determined even though q_c(N_f) is not. As written, a reader may take the 'g^6 ln(1/g)' label to mean that only the leading logarithm is fixed.
[Sec. VI, Eq. (6.4)] Please state the renormalisation scheme and scale convention for α_s(μ̄) used in Eqs. (6.4)–(6.13); the logarithms of μ̄/(2πT) suggest MS-bar, but this should be said, together with the convention relating μ̄ to μ in Eq. (2.6).
[Sec. VII, Fig. 2] The vertical-axis label reads 'δ|p_M+p_G|/(T g_E^6)', where the leading δ appears to be a stray symbol, since the plotted quantities are the individual terms of the expansion normalised by T g_E^6 and no δ enters their definition. Please correct the label.
[Sec. VII, paragraph beginning 'Finally, it is perhaps interesting to remark'] The statement that the omission of the logarithmic term in the earlier analysis 'led to the wrong curvature of the pressure seen at small T/Λ_MS' is a retrospective claim about published work [14]. It would be worth naming the specific figure or observable in [14] to which this refers, and confirming that the sign reversal is independent of the scale choice μ̄ = μ̄_E = g_E^2 adopted there.
[Sec. VII, paragraph on resummation] The text states that the practical conclusions were checked to be unchanged when plotting Eqs. (6.4)–(6.11) directly rather than the un-expanded form, but no such comparison is shown. Either include it as an appendix figure or drop the remark, since an unshown check carries little weight.
[Sec. VII, paragraph on dimensional reduction] The assertion that dimensional reduction 'is known to break down at about this point' is given without a reference; please cite the lattice or perturbative study that establishes the breakdown temperature.
[Sec. IV, Eq. (4.1)] The λ_E^(2) term carries the group-theory factor d_A(2d_A − 1)/N_c, which depends on the convention adopted for the two quartic invariants (Tr A_0^2)^2 and Tr A_0^4 and on how they degenerate at N_c = 3. Since this factor feeds the numerical coefficients of p_4…p_6, please state the normalisation convention explicitly, and confirm the factor for N_c = 3.
[Sec. VI, Eq. (6.1)] The argument of the α_G logarithm is written as g2T C_A, whereas m_M was defined in Eq. (3.1) as C_A g2_M; a one-line reminder that g2_M = g2_E + O(g2_E/m_E) and that g2_E ≈ gT would make the identification transparent at the point of use.
[General] The copy supplied has broken word spacing throughout (for example 'tempe ratures', 'conc lusions', 'alt hough', 'U SA', and the author name 'Schr¨ oder'). If these are present in the source files rather than being an artefact of the copy I received, they should be corrected before publication.

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The paper

The paper claims

The pressure of hot QCD is now known through its last perturbatively computable term, the four-loop logarithm $g^6$.

The reading

The pressure of a quark-gluon plasma, at temperatures of a few hundred MeV and above, can in principle be expanded in powers of the strong coupling $g$. This paper asks for the last term in that series that perturbation theory can ever deliver: the coefficient of $g^6$ times the logarithm of $1/g$, which appears because the theory carries three widely separated momentum scales, $2πT$, $gT$ and $g^2T$. The authors obtain it by matching two three-dimensional effective theories onto the four-dimensional theory and adding their four-loop vacuum energies, arriving at the closed expression in Eq. (6.3). The stake is that the plain $g^6$ term is not perturbatively computable at all, so this logarithm is the final analytic anchor; the paper argues that, once it is combined with a plausible value for the unknown constant, the series tracks lattice data down to temperatures of only a few times $Λ_{MS}$.

What carries the argument

The engine is dimensional reduction in two steps. The four-dimensional pressure is split as $p_{QCD} = p_E + p_M + p_G$, where $p_E$ collects everything from the hard scale $2πT$, $p_M$ comes from integrating out the colour-electric scale $gT$ and yields a three-dimensional SU($N_c$) plus adjoint-Higgs theory, and $p_G$ comes from the colour-magnetic scale $g^2T$ and yields three-dimensional pure Yang-Mills. The $g^2T$ piece is non-perturbative, but its four-loop ultraviolet divergence is not: the authors regulate it with a hand-inserted mass $m_G$ in the gauge and ghost propagators and read off the coefficient of the $1/ε$ pole, which is the $α_G$ of Eq. (3.3). The $gT$ piece is infrared insensitive in strict dimensional regularisation and gives $α_M$ of Eq. (4.2) from the four-loop vacuum energy of the adjoint-Higgs theory. Because the logarithm in the four-dimensional pressure can only come from a ratio of two scales, these two coefficients plus the running of the lower orders fix it completely, even though the constant under the logarithm is not known.

Extended reading notes

Core claim

The paper determines the coefficient of $g^6$ times the logarithm of $1/g$ in the pressure of hot QCD, written in Eq. (6.3) as $g^6 d_A C_A/(4π)^4$ times a combination of matching coefficients carrying two distinct logarithms, one of the form $gα_{E4}^{1/2}$ and one of the form $gC_A^{1/2}$. The two numerical ingredients are $α_M = 0.555017$, coming from the four-loop vacuum energy density of the three-dimensional SU($N_c$) plus adjoint-Higgs theory, and $α_G = 0.195715$, coming from the logarithmic ultraviolet divergence of the four-loop vacuum energy of three-dimensional pure Yang-Mills theory. Assemblng these with the running of the lower-order terms reproduces the known result through order $g^5$ and yields, for $N_c=3$, the explicit polynomials $p_0$ through $p_6$ in Eq. (6.4), with two of the three $α_s$-independent constants in the $g^6$ coefficient fixed by scale independence and the third left unknown. The authors then observe that an earlier dimensional-reduction lattice study had set the four-dimensional coefficient $β_{E1}$ to zero and thereby missed a term of order $g^6$ times the logarithm of $1/g$; restoring it flips the sign of the total logarithm and produces the curvature actually seen in four-dimensional lattice data.

Load-bearing premise

The extraction of $α_G$ assumes that inserting a hand-chosen infrared mass $m_G$ into the gauge and ghost propagators leaves the coefficient of the $1/ε$ ultraviolet pole untouched, so that this regulated coefficient can be identified with the physical divergence of Eq. (3.1); the paper supports this only by gauge-parameter independence and the cancellation of higher-order poles.

Editorial extensions

If this is right

The coupling-constant series for the pressure is now complete through order $g^6\ln(1/g)$, so any non-perturbative or resummed determination of the equation of state has a fixed analytic reference it must reproduce in the weak-coupling limit.
The two coefficients $α_M = 0.555017$ and $α_G = 0.195715$ become concrete targets for independent three-dimensional lattice or stochastic-perturbation measurements of the same four-loop vacuum energies.
With one plausible value of the unknown $g^6$ constant, the resummed series matches four-dimensional lattice data down to $T/Λ_{MS}$ of roughly two to three, which the authors present as an observation rather than a prediction.
The previously missed logarithm, once restored with the scale choice used in earlier dimensional-reduction work, reverses the sign of the total $g^6\ln(1/g)$ contribution and reproduces the curvature of the lattice pressure at small $T/Λ_{MS}$.
The pattern of three distinct logarithm types is specific to non-Abelian gauge theory: the corresponding QED pressure has none of them, so the known QED result through order $g^5$ is unaffected.

Reading between the lines

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

If the agreement with lattice data at $δ \approx 0.7$ is not a coincidence, the perturbative pressure becomes usable at temperatures relevant to heavy-ion phenomenology, which would tighten the equation of state input to hydrodynamic models of the observed near-ideal flow.
The same two-step reduction, with the same four-loop coefficients, should fix the logarithmic terms in other thermodynamic quantities at this order, such as the energy density and the quark-number susceptibilities, since they share the same scale hierarchy.
The $α_G$ extraction is a template for reading a physical ultraviolet coefficient out of a theory whose infrared behaviour is non-perturbative; if the template is sound, it can be reused for any super-renormalisable effective theory whose constant term is inaccessible.
Conversely, if the tuned constant turns out to be far from the value that matches lattice data, the apparent agreement would stand as a caution about how much weight a single fitted unknown can carry in an asymptotic series.

Original abstract

The free energy density, or pressure, of QCD has at high temperatures an expansion in the coupling constant g, known so far up to order g^5. We compute here the last contribution which can be determined perturbatively, g^6 ln(1/g), by summing together results for the 4-loop vacuum energy densities of two different three-dimensional effective field theories. We also demonstrate that the inclusion of the new perturbative g^6 ln(1/g) terms, once they are summed together with the so far unknown perturbative and non-perturbative g^6 terms, could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures.

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

The free energy density, or pressure, of QCD has at high temperatures an expansion in the coupling constant g, known so far up to order g^5. We compute here the last contribution which can be determined perturbatively, g^6 ln(1/g), by summing together results for the 4-loop vacuum energy densities of two different three-dimensional effective field theories. We also demonstrate that the inclusion of the new perturbative g^6 ln(1/g) terms, once they are summed together with the so far unknown perturbative and non-perturbative g^6 terms, could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The free energy density, or pressure, of QCD has at high temperatures an expansion in the coupling constant g, known so far up to order g^5.	background 0.90	0.04	0.03	asserted 1.00
s1	We compute here the last contribution which can be determined perturbatively, g^6 ln(1/g), by summing together results for the 4-loop vacuum energy densities of two different three-dimensional effective field theories.	contribution 0.88	0.97	0.57	asserted 1.00
s2	We also demonstrate that the inclusion of the new perturbative g^6 ln(1/g) terms, once they are summed together with the so far unknown perturbative and non-perturbative g^6 terms, could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures.	interpretation 0.99	0.35	0.38	hedged possibility 0.94

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from simulation or computation 0.50.

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Figures

**Figure 1.** FIG. 1: Left: perturbative results at various orders (the pr

**Figure 2.** FIG. 2: The absolute values of the various terms of the slowly

Figures reproduced from arXiv:hep-ph/0211321 by the authors.

References

Works cited (29)

[14]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Phys. Rev. Lett. 86, 10 (2001) [hep-ph/0007109]

View on Pith Pith review arXiv 2001
[7]

C. Zhai and B. Kastening, Phys. Rev. D52, 7232 (1995) [hep-ph/9507380]

View on Pith Pith review arXiv 1995
[9]

Y. Schr¨ oder,Logarithmic divergence in the energy density of the three-d imensional Yang–Mills theory, in preparation

View on Pith
[10]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Four-loop vacuum energy density of the SU( Nc) + adjoint Higgs theory , in preparation

View on Pith
[22]

G. Boyd et al. , Nucl. Phys. B469, 419 (1996) [hep-lat/9602007]; A. Papa, Nucl. Phys. B478, 335 (1996) [hep-lat/9605004]; B. Beinlich, F. Karsch, E. La ermann and A. Peikert, Eur. Phys. J. C6, 133 (1999) [hep-lat/9707023]; M. Okamoto et al. [CP-PACS Collaboration], Phys. Rev. D60, 094510 (1999) [hep-lat/9905005]

View on Pith Pith review arXiv 1996
[1]

A.D. Linde, Phys. Lett. B96, 289 (1980)

View on Pith 1980
[2]

D.J. Gross, R.D. Pisarski and L.G. Yaﬀe, Rev. Mod. Phys. 53, 43 (1981)

View on Pith 1981
[3]

E.V. Shuryak, Sov. Phys. JETP 47, 212 (1978) [Zh. Eksp. Teor. Fiz. 74, 408 (1978)]; S.A. Chin, Phys. Lett. B78, 552 (1978)

View on Pith 1978
[4]

J.I. Kapusta, Nucl. Phys. B148, 461 (1979)

View on Pith 1979
[5]

T. Toimela, Phys. Lett. B124, 407 (1983)

View on Pith 1983
[6]

P. Arnold and C. Zhai, Phys. Rev. D50, 7603 (1994) [hep-ph/9408276]; ibid. D51, 1906 (1995) [hep-ph/9410360]

View on Pith Pith review arXiv 1994
[8]

G.D. Moore, JHEP 0210, 055 (2002) [hep-ph/0209190]; A. Ipp, G.D. Moore and A. Rebh an, hep-ph/0301057

View on Pith Pith review arXiv 2002
[11]

E. Braaten and A. Nieto, Phys. Rev. D53, 3421 (1996) [hep-ph/9510408]. 17

View on Pith Pith review arXiv 1996
[12]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnik ov, Nucl. Phys. B503, 357 (1997) [hep-ph/9704416]

View on Pith Pith review arXiv 1997
[13]

P. Ginsparg, Nucl. Phys. B170, 388 (1980); T. Appelquist and R.D. Pisarski, Phys. Rev. D23, 2305 (1981)

View on Pith 1980
[15]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnik ov, Nucl. Phys. B458, 90 (1996) [hep-ph/9508379]

View on Pith Pith review arXiv 1996
[16]

S. Chapman, Phys. Rev. D50, 5308 (1994) [hep-ph/9407313]

View on Pith Pith review arXiv 1994
[17]

Y. Schr¨ oder, hep-ph/0211288

View on Pith Pith review arXiv
[18]

J.A.M. Vermaseren, math-ph/0010025; http://www.nikhef.nl/~form/

View on Pith Pith review arXiv
[19]

K. Kajantie, M. Laine and Y. Schr¨ oder, Phys. Rev. D65, 045008 (2002) [hep-ph/0109100]

View on Pith Pith review arXiv 2002
[20]

K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shapos hnikov, Nucl. Phys. B425, 67 (1994) [hep-ph/9404201]

View on Pith Pith review arXiv 1994
[21]

S. Nadkarni, Phys. Rev. D38, 3287 (1988); N.P. Landsman, Nucl. Phys. B322, 498 (1989)

View on Pith 1988
[23]

R.R. Parwani, Phys. Lett. B334, 420 (1994); ibid. B342, 454 (1995) (E) [hep-ph/9406318]; R.R. Parwani and C. Corian` o, Nucl. Phys. B434, 56 (1995) [hep-ph/9409269]

View on Pith Pith review arXiv 1994
[24]

E. Braaten and A. Nieto, Phys. Rev. D51, 6990 (1995) [hep-ph/9501375]

View on Pith Pith review arXiv 1995
[25]

F. Karsch, M. L¨ utgemeier, A. Patk´ os and J. Rank, Phys. Lett. B390, 275 (1997) [hep-lat/9605031]

View on Pith Pith review arXiv 1997
[26]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Nucl. Phys. (Proc. Suppl.) B106, 525 (2002) [hep-lat/0110122]; hep-lat/0209072

View on Pith Pith review arXiv 2002
[27]

F. Di Renzo, E. Onofri, G. Marchesini and P. Marenzoni, N ucl. Phys. B426, 675 (1994) [hep-lat/9405019]

View on Pith Pith review arXiv 1994
[28]

S. Huang and M. Lissia, Nucl. Phys. B438, 54 (1995) [hep-ph/9411293]

View on Pith Pith review arXiv 1995
[29]

The constant part βG could be determined by measuring a suitable observable on th e lattice and converting then the result from lattice regularisation to the MS scheme by a perturbative 18 4-loop matching computation. 19

View on Pith

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Full leading-order shear viscosity, quark-number diffusion, and electrical conductivity are computed for hot QCD and QED, and the 1/log(1/g) expansion is shown to be a good but only asymptotic approximation.
Scalar induced gravitational waves as probes of dark QCD
hep-ph 2026-08 conditional novelty 6.0 of 10

A dark QCD crossover around 5.5 times the SM QCD scale imprints a frequency-shifted distortion on scalar induced gravitational wave spectra computed from rescaled effective degrees of freedom.
Thermal History of Non-equilibrated Scalars
hep-ph 2025-07 conditional novelty 6.0 of 10

A non-equilibrated scalar controlling a quartic coupling makes the coupling smaller at high temperature, strengthening the first-order dark Higgs phase transition.
A New State of Matter between the Hadronic Phase and the Quark-Gluon Plasma?
hep-ph 2025-05 conditional novelty 6.0 of 10

This paper proposes an intermediate SQGB phase in which quark degrees of freedom are liberated while gluons remain bound in glueballs until about 285 MeV.
Cold quark matter at N3LO: soft contributions
hep-ph 2021-03 accept novelty 6.0 of 10

The fully soft two-loop hard-thermal-loop contribution to the zero-temperature quark-matter pressure is evaluated in dimensional regularization, giving the coefficients p_{-2} = 11 pi^2/24, p_{-1} = 11.6840(15), and p...
Robust approach to thermal resummation: Standard Model meets a singlet
hep-ph 2021-02 conditional novelty 6.0 of 10

The authors derive the next-to-leading-order three-dimensional effective field theory for the Standard Model extended by a real singlet scalar, including two-loop matching relations, so that phase-transition thermodyn...
Real scalar phase transitions: a nonperturbative analysis
hep-ph 2021-01 accept novelty 6.0 of 10

Lattice Monte Carlo and three-loop renormalisation-group-improved perturbation theory agree to a few percent for the thermal first-order phase transition of a real scalar field, even where unimproved perturbation theo...
Gravitational wave complementarity and impact of NANOGrav data on gravitational leptogenesis: cosmic strings
hep-ph 2020-09 conditional novelty 6.0 of 10

Cosmic-string gravitational waves can probe gravitational leptogenesis, and if NANOGrav's signal is gravitational waves, it disfavors lightest neutrino masses above about 25 meV within this model.
Primordial gravitational waves, precisely: The role of thermodynamics in the Standard Model
hep-ph 2018-03 conditional novelty 6.0 of 10

Including Standard Model particle interactions in the cosmic equation of state raises the predicted primordial gravitational wave amplitude by 1 to 2 percent at DECIGO frequencies and adds new damping features from fr...
Standard Model thermodynamics across the electroweak crossover
hep-ph 2015-03 accept novelty 6.0 of 10

Combining three-loop perturbation theory with lattice input for the Higgs condensate yields a percent-level tabulated Standard Model equation of state between 100 and 300 GeV, with QCD corrections dominating the depar...
Equation of State of hot and dense QCD: Resummed perturbation theory confronts lattice data
hep-ph 2013-07 accept novelty 6.0 of 10

An exact one-loop hard-thermal-loop calculation confirms that the widely used mass-over-temperature expansion of the QCD pressure and quark number susceptibilities converges to roughly one percent.
A lattice study of the jet quenching parameter
hep-ph 2013-07 conditional novelty 6.0 of 10

Soft, non-perturbative contributions to the jet quenching parameter are large, and a lattice EQCD computation gives q-hat around 6 GeV squared per femtometer at T around 398 MeV.
Precision SU(3) lattice thermodynamics for a large temperature range
hep-lat 2012-04 conditional novelty 6.0 of 10

A high-precision lattice equation of state for pure SU(3) gauge theory spanning 0.7 to 1000 Tc, used to fit the g^6 perturbative coefficient and the HTL renormalization scale.
Three-loop HTL QCD thermodynamics
hep-ph 2011-03 conditional novelty 6.0 of 10

Three-loop hard-thermal-loop perturbation theory reproduces lattice QCD thermodynamics for the quark-gluon plasma down to temperatures of roughly two to three times the critical temperature.
Lightest sterile neutrino abundance within the nuMSM
hep-ph 2006-12 conditional novelty 6.0 of 10

A first-principles calculation of sterile neutrino production in the early universe caps the Dodelson-Widrow dark matter sterile neutrino at roughly 3.5 keV, and at 6 keV under the most conservative treatment of hadro...
Quark mass thresholds in QCD thermodynamics
hep-ph 2006-03 accept novelty 6.0 of 10

Finite quark masses change the QCD pressure at next-to-leading order by at most about five percent for three or four flavours, so charm quark effects appear at surprisingly low temperatures as leading-order estimates ...
QCD thermodynamics through the crossover as a gas of confining strings with repulsive interactions
hep-ph 2026-09 conditional novelty 5.0 of 10

An open-string gas with excluded-volume repulsion, fitted to lattice data, reproduces the QCD crossover equation of state with T_H ~ 300 MeV and indicates baryons need roughly four times stronger effective repulsion t...
Hadronic and partonic composition of QCD matter across the crossover
nucl-th 2026-07 conditional novelty 5.0 of 10

A three-parameter hadron–quark crossover equation of state fitted to lattice QCD data yields a switching temperature T₀ ≃ 216 MeV, implying hadronic degrees of freedom persist well above the chiral pseudocritical temperature.
Linear sigma model with quarks and Polyakov loop in rotation: phase diagrams, Tolman-Ehrenfest law and mechanical properties
nucl-th 2025-03 unverdicted novelty 5.0 of 10

Rotation lowers critical temperatures for chiral and deconfinement transitions in the Polyakov linear sigma model under causality constraints, with mechanical properties computed in the homogeneous limit.
Perturbative gravitational wave predictions for the real-scalar extended Standard Model
hep-ph 2024-11 conditional novelty 5.0 of 10

At next-to-next-to-leading order, gravitational wave predictions for the real-scalar singlet extension of the Standard Model appear to converge for typical parameters, but break down for the strongest, observable signals.
On the perturbative expansion at high temperature and implications for cosmological phase transitions
hep-ph 2021-04 accept novelty 5.0 of 10

At high temperature the widely used one-loop daisy-resummed effective potential carries an uncancelled renormalisation-scale dependence at O(g^4), removable only by two-loop terms, and this shifts predicted gravitatio...
Gravitational Waves as a Big Bang Thermometer
hep-ph 2020-11 conditional novelty 5.0 of 10

The thermal plasma of the early universe produces a guaranteed gravitational wave background peaking near 80 GHz whose amplitude scales linearly with the maximum temperature, making it a potential thermometer for the ...
Fluctuations and correlations in high temperature QCD
hep-lat 2015-07 conditional novelty 5.0 of 10

Continuum-extrapolated lattice QCD shows the hadron resonance gas describes charge fluctuations below the crossover and three-loop hard thermal loop perturbation theory describes them above about 250 MeV.
The Effects of QCD Equation of State on the Relic Density of WIMP Dark Matter
hep-ph 2015-03 accept novelty 5.0 of 10

A modern lattice QCD equation of state shifts the predicted thermal WIMP relic density by up to 9% (12%) for pure S-wave (P-wave) annihilation at WIMP masses of 3 to 15 GeV, and moves the required annihilation cross s...
Baryogenesis from Hawking Radiation
hep-ph 2014-04 conditional novelty 5.0 of 10

Black hole evaporation in a chemical potential generated by derivative couplings to the B-L current produces a baryon asymmetry, and the same mechanism can set the dark matter abundance.
Thermodynamics of the QCD plasma and the large-N limit
hep-lat 2009-07 conditional novelty 5.0 of 10

Per-gluon thermodynamics of the deconfined SU(N) gluon plasma is nearly N-independent from 0.8 to 3.4 Tc, supporting large-N and holographic descriptions of the QCD plasma.
What RHIC Experiments and Theory tell us about Properties of Quark-Gluon Plasma ?
hep-ph 2004-05 conditional novelty 5.0 of 10

RHIC data show the quark-gluon plasma is a strongly coupled liquid with viscosity near the AdS/CFT bound, and the author argues it hosts hundreds of mostly colored bound states that shape its thermodynamics and jet quenching.
Flipped Rotating Axion Non-minimally Coupled to Gravity: Baryogenesis and Dark Matter
hep-ph 2025-02 conditional novelty 4.0 of 10

A rotating axion, kicked into motion by a sign flip in its gravitational effective potential during kination, can co-generate baryon asymmetry and dark matter in a Majoron seesaw model.
Combining thermal resummation and gauge invariance for electroweak phase transition
hep-ph 2022-03 conditional novelty 4.0 of 10

A minimal 3d effective-field-theory setup combines a gauge-invariant expansion around leading-order minima with next-to-leading-order dimensional reduction, and applied to the complex-singlet Standard Model it shows t...
Precise WIMP Dark Matter Abundance and Standard Model Thermodynamics
hep-ph 2020-05 conditional novelty 4.0 of 10

Using an updated Standard Model equation of state, the predicted WIMP relic abundance shifts by about 5 to 10 percent relative to conventional estimates for dark matter masses of 1 to 1000 GeV.
Thermodynamics of QCD-inspired theories
hep-ph 2006-04 conditional novelty 4.0 of 10

A PhD thesis calculating the 1/N-corrected effective potential and pressure of QCD-inspired sigma and NJL models, finding that temperature-dependent divergences vanish at the minimum and that pseudoscalar condensation...
Emergent chiral spin symmetry, non-perturbative dynamics and thermoparticles in hot QCD
hep-ph 2025-12 unverdicted novelty 3.0 of 10

Lattice data indicate hot QCD features an intermediate phase with emergent chiral spin symmetry and thermoparticles as thermal constituents, differing from perturbative expectations.
Asymmetries from a charged memory-burdened PBH
hep-ph 2024-12 reject novelty 3.0 of 10

A parameter-space scan shows that very large curvature-current couplings can fit the baryon asymmetry and dark matter abundance, while the electric charge of the black hole plays no role in the mechanism.
Non-perturbative constraints on perturbation theory at finite temperature
hep-ph 2024-12 conditional novelty 3.0 of 10

Two-loop finite-temperature perturbation theory disagrees with lattice phi^4 data at high temperature, while thermoparticle spectral functions keep the temporal correlator consistent.
Thermodynamically Consistent Merging of Multidimensional QCD Equations of State
nucl-th 2026-08 conditional novelty 2.0 of 10

A statistical-mixture framework merges a hadron-resonance-gas model with a holographic quark-gluon plasma model into a single, thermodynamically consistent QCD equation of state.
Properties of hot and dense matter from relativistic heavy ion collisions
nucl-th 2015-10 accept novelty 2.0 of 10

Collisions of heavy nuclei at RHIC and the LHC produce a locally thermalized, nearly perfect fluid whose equation of state, transport coefficients, and freeze-out conditions can be read off bulk observables.
Strongly Correlated Quantum Fluids: Ultracold Quantum Gases, Quantum Chromodynamic Plasmas, and Holographic Duality
hep-th 2012-05 accept novelty 2.0 of 10

Ultracold Fermi gases and the quark-gluon plasma both flow as nearly perfect liquids, with shear viscosity to entropy density near hbar/(4 pi k_B), the value holographic duality predicts for strongly coupled fluids.
Gauge/String Duality, Hot QCD and Heavy Ion Collisions
hep-th 2011-01 accept novelty 2.0 of 10

The review synthesizes a decade of work showing that strongly coupled plasma calculations from gauge/string duality give a coherent, quantitatively useful picture of hot QCD matter and heavy ion observables.

How later papers used it

98 later papers cite it

Evidence: substantial. 92 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 133 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 used its data or values and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 1 paper · 1 by the same authors

By the same authors

Followed this paper's approach, referring to it for details and references.

Screened perturbation theory for 3d Yang-Mills theory and the magnetic modes of hot QCD (2009) (same authors) · 0.89 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 7 papers with this label

We thus consider a fit function with all the known perturbative coefficients, [equation] where $s_0,...,s_6$ are obtained from the perturbative expansion of the pressure [ref] .

The Equation of State of QCD up to very high temperatures (2025) · Discussion · 0.83
When we use these results, we use the same two-loop running coupling as used above in previous sections, and we take as the central value of the renormalization scale by $Λ̅= 0.723 ×4πT$, following [ref] , in place of the $2 μ_q$ used at $T = 0$.

Bayesian uncertainty quantification of perturbative QCD input to the neutron-star equation of state (2023) · pQCD terms at high T and mu = 0 · 0.93

Compares with it 1 of 4 papers with this label

Brought its own result to the O(alpha_s^3 ln alpha_s) level, on par with this paper's high-temperature counterpart from 2003.

Equation of State of Cold Quark Matter to $O(\alpha_s^3 \ln \alpha_s)$ (2023) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

91 arXiv papers cite this in 128 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 96 sentences without a confident label

Uses its method or tool 1 paper

Here we follow [ref] , to which we refer for details and references.

Screened perturbation theory for 3d Yang-Mills theory and the magnetic modes of hot QCD · High T effective theory from dimensional reduction · 0.89

Uses its data or values 6 papers

Whereas $α_G$ is known analytically [ref] , the infrared coefficient $β_G$ is receiving contributions from all loop levels and needs to be evaluated non-perturbatively, e.g. by lattice simulations of the effective theory [ref] , leading to a numerical value (with a certain error bar).

Screened perturbation theory for 3d Yang-Mills theory and the magnetic modes of hot QCD · High T effective theory from dimensional reduction · 0.81
We thus consider a fit function with all the known perturbative coefficients, [equation] where $s_0,...,s_6$ are obtained from the perturbative expansion of the pressure [ref] .

The Equation of State of QCD up to very high temperatures · Discussion · 0.83
Taking a reference value of $α_s(2 GeV) = 0.2994$ in the $N_f=3$ case [ref] , we then use three-loop running of the coupling to a lower scale $Λ̅ = X √((2 )^2 + (0.723 ×4 πT)^2)$, with $X ∈[1/2, 2]$ [ref] .

A new approach to determine the thermodynamics of deconfined matter to high accuracy · Introduction · 0.98

Compares with it 4 papers

Similarly if we compare the asymptotic behavior of the pressure [ref] eq:pasym p = π^3 ^3/16 G_5 T ^4 1-4/3β_0 α_h +2/94β_0^2-3β_1 α_h^2 = p_SB 1-2.33 α_h +1.86 α_h^2 , with the perturbative result from QCD [ref] eq:ppertqcd p =8π^2/45T^4 1-15/4 α/π +30 α/π^3/2 =p_SB 1-1.19 α+5.4α^3/2 , we see that not only the coefficients are different but also the power of $α$ in the next to leading order.

Trouble Finding the Optimal AdS/QCD · Introduction · 0.91
In this Letter, we compute the full EOS contributions from the screened gluonic sector at next-to-next-to-next-to-leading order (N3LO) in $α_s$ in unpaired cold QM, bringing the result to the $O(α_s^3 lnα_s)$ level and on par with its high-temperature counterpart from 2003 [ref] .

Equation of State of Cold Quark Matter to $O(\alpha_s^3 \ln \alpha_s)$ · Introduction · 0.97
In these figures, we show the usual renormalization-scale variation in $= X √((2μ_q)^2+ (0.723 ×4 πT)^2)$ [ref] , with $X ∈{1/2, 1, 2}$, with the central dashed or solid lines corresponding to $X = 1$.

Cool quark matter with perturbative quark masses · Results and Discussions · 0.99

Cites it as support 1 paper

Moreover, the perturbative series seems to be weakly convergent [ref] .

Thermodynamics of Quasi-Particles · Introduction · 0.98

Cites it as background 16 papers

This specific coefficient arises from the hard momentum scales and is sometimes referred to as $β_E1$ in the literature [ref] .

The $g^6$ pressure of hot Yang-Mills theory: Canonical form of the integrand · Setup · 0.83
Perturbative QCD (pQCD) calculations making use of this property have been applied to compute the pressure to partial next-to-next-to-next-to-leading order (N3LO) at $T>0,μ=0$ [ref] and at $T=0,μ>0$ [ref] , and to complete next-to-next-to-leading order (N2LO) at both $T>0,μ>0$ [ref] .

Cool quark matter with perturbative quark masses · Introduction · 0.88
As for the finite-temperature direction, the framework of electrostatic QCD (EQCD) [ref] has been established up to $𝒪(g^6)$ [ref] for massless quarks, while the massive quark effects have been accounted for up to $𝒪(g^2)$ [ref] , with the caveat of the still unknown nonperturbative contribution at $𝒪(g^6)$ [ref] which we will discuss more later.

QCD Vacuum Energy and Its Implication for Quark Nugget Stability · Introduction · 0.93

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [14]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Phys. Rev. Lett. 86, 10 (2001) [hep-ph/0007109]

View on Pith Pith review arXiv 2001

[2] [7]

C. Zhai and B. Kastening, Phys. Rev. D52, 7232 (1995) [hep-ph/9507380]

View on Pith Pith review arXiv 1995

[3] [9]

Y. Schr¨ oder,Logarithmic divergence in the energy density of the three-d imensional Yang–Mills theory, in preparation

View on Pith

[4] [10]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Four-loop vacuum energy density of the SU( Nc) + adjoint Higgs theory , in preparation

View on Pith

[5] [22]

G. Boyd et al. , Nucl. Phys. B469, 419 (1996) [hep-lat/9602007]; A. Papa, Nucl. Phys. B478, 335 (1996) [hep-lat/9605004]; B. Beinlich, F. Karsch, E. La ermann and A. Peikert, Eur. Phys. J. C6, 133 (1999) [hep-lat/9707023]; M. Okamoto et al. [CP-PACS Collaboration], Phys. Rev. D60, 094510 (1999) [hep-lat/9905005]

View on Pith Pith review arXiv 1996

[6] [1]

A.D. Linde, Phys. Lett. B96, 289 (1980)

View on Pith 1980

[7] [2]

D.J. Gross, R.D. Pisarski and L.G. Yaﬀe, Rev. Mod. Phys. 53, 43 (1981)

View on Pith 1981

[8] [3]

E.V. Shuryak, Sov. Phys. JETP 47, 212 (1978) [Zh. Eksp. Teor. Fiz. 74, 408 (1978)]; S.A. Chin, Phys. Lett. B78, 552 (1978)

View on Pith 1978

[9] [4]

J.I. Kapusta, Nucl. Phys. B148, 461 (1979)

View on Pith 1979

[10] [5]

T. Toimela, Phys. Lett. B124, 407 (1983)

View on Pith 1983

[11] [6]

P. Arnold and C. Zhai, Phys. Rev. D50, 7603 (1994) [hep-ph/9408276]; ibid. D51, 1906 (1995) [hep-ph/9410360]

View on Pith Pith review arXiv 1994

[12] [8]

G.D. Moore, JHEP 0210, 055 (2002) [hep-ph/0209190]; A. Ipp, G.D. Moore and A. Rebh an, hep-ph/0301057

View on Pith Pith review arXiv 2002

[13] [11]

E. Braaten and A. Nieto, Phys. Rev. D53, 3421 (1996) [hep-ph/9510408]. 17

View on Pith Pith review arXiv 1996

[14] [12]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnik ov, Nucl. Phys. B503, 357 (1997) [hep-ph/9704416]

View on Pith Pith review arXiv 1997

[15] [13]

P. Ginsparg, Nucl. Phys. B170, 388 (1980); T. Appelquist and R.D. Pisarski, Phys. Rev. D23, 2305 (1981)

View on Pith 1980

[16] [15]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnik ov, Nucl. Phys. B458, 90 (1996) [hep-ph/9508379]

View on Pith Pith review arXiv 1996

[17] [16]

S. Chapman, Phys. Rev. D50, 5308 (1994) [hep-ph/9407313]

View on Pith Pith review arXiv 1994

[18] [17]

Y. Schr¨ oder, hep-ph/0211288

View on Pith Pith review arXiv

[19] [18]

J.A.M. Vermaseren, math-ph/0010025; http://www.nikhef.nl/~form/

View on Pith Pith review arXiv

[20] [19]

K. Kajantie, M. Laine and Y. Schr¨ oder, Phys. Rev. D65, 045008 (2002) [hep-ph/0109100]

View on Pith Pith review arXiv 2002

[21] [20]

K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shapos hnikov, Nucl. Phys. B425, 67 (1994) [hep-ph/9404201]

View on Pith Pith review arXiv 1994

[22] [21]

S. Nadkarni, Phys. Rev. D38, 3287 (1988); N.P. Landsman, Nucl. Phys. B322, 498 (1989)

View on Pith 1988

[23] [23]

R.R. Parwani, Phys. Lett. B334, 420 (1994); ibid. B342, 454 (1995) (E) [hep-ph/9406318]; R.R. Parwani and C. Corian` o, Nucl. Phys. B434, 56 (1995) [hep-ph/9409269]

View on Pith Pith review arXiv 1994

[24] [24]

E. Braaten and A. Nieto, Phys. Rev. D51, 6990 (1995) [hep-ph/9501375]

View on Pith Pith review arXiv 1995

[25] [25]

F. Karsch, M. L¨ utgemeier, A. Patk´ os and J. Rank, Phys. Lett. B390, 275 (1997) [hep-lat/9605031]

View on Pith Pith review arXiv 1997

[26] [26]

K. Kajantie, M. Laine, K. Rummukainen and Y. Schr¨ oder, Nucl. Phys. (Proc. Suppl.) B106, 525 (2002) [hep-lat/0110122]; hep-lat/0209072

View on Pith Pith review arXiv 2002

[27] [27]

F. Di Renzo, E. Onofri, G. Marchesini and P. Marenzoni, N ucl. Phys. B426, 675 (1994) [hep-lat/9405019]

View on Pith Pith review arXiv 1994

[28] [28]

S. Huang and M. Lissia, Nucl. Phys. B438, 54 (1995) [hep-ph/9411293]

View on Pith Pith review arXiv 1995

[29] [29]

The constant part βG could be determined by measuring a suitable observable on th e lattice and converting then the result from lattice regularisation to the MS scheme by a perturbative 18 4-loop matching computation. 19

View on Pith