QCD Plasma Instabilities and Bottom-Up Thermalization

Guy D. Moore; Jonathan Lenaghan; Peter Arnold

REVIEW

QCD Plasma Instabilities and Bottom-Up Thermalization

Peter Arnold, Jonathan Lenaghan, Guy D. Moore

High Energy Physics - Phenomenology · Submitted 25 Jul 2003 · 40 references

Pith's verdict: The paper that killed the first stage of bottom-up thermalization; the central argument holds, but Section IV is void and the authors' own erratum says so.

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arXiv IDhep-ph/0307325 (version 4)

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CitationGuy D. Moore, Jonathan Lenaghan, Peter Arnold. "QCD Plasma Instabilities and Bottom-Up Thermalization." arXiv:hep-ph/0307325 (2003). https://pith.science/paper/YNX5M5DI

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The one thing to know: this is the paper that killed the first stage of bottom-up thermalization, and the argument holds up. The second thing: Section IV is wrong, and the authors say so in an appended erratum. Both facts matter for how you read it. What's actually new. Mrowczynski had argued for years that anisotropic plasmas go unstable. What this paper adds is (a) the parametric argument that the growth time 1/|m_soft| ~ sqrt(tau/Qs) beats the expansion time tau for Qs tau >> 1, so the first stage of Baier-Mueller-Schiff-Son cannot be right even briefly; (b) Condition 1-b in Appendix B, a clean proof that anisotropic M(p-hat) is necessary and sufficient for magnetic instability in the ultra-relativistic limit; (c) explicit closed-form growth rates for the planar and line distributions, with dispersion relations you can actually solve. The Appendix B proof is the durable piece. It turns 'anisotropy causes instabilities' from folklore into a theorem, and it is the reason the paper keeps getting cited even by people who never touch the bottom-up story. Section II C's trapped-versus-untrapped picture is also the best qualitative account in the QCD literature. Where it's soft. The abstract's 'must drastically modify' [abstract:s1] rests on an existence argument for the realistic composite distribution, not a computed rate. The explicit growth rates are for delta(p_z) and delta^2(p_perp). For the actual mix of hard planar gluons plus softer, mildly anisotropic gluons, the paper asserts unstable modes at q ~ m_soft and moves on. That is probably fine, but it is asserted. The authors know: the conclusion says they cannot yet give detailed parametric estimates. So it is an open item, not a hidden one. Section IV is the real casualty. The erratum concedes the static non-Abelian solution is not a solution, from an error in eq. (4.2). The erratum also walks back the saturation energy estimate: for planar distributions q_max >> m_soft, so A ~ q_max/g and the field energy ~ q_max^4/g^2, which can exceed the hard-particle energy density. That is a substantive correction to the saturation picture, not a typo. One thing the reader flagged that I would push back on: the collisionless approximation is fine. t_col/t_growth ~ (Qs tau)^{1/6}, so it is well-defended. Who should read it: anyone working on early-time dynamics or anisotropic hydrodynamics. The instability criteria and the growth-rate calculations are worth the time; Section IV is not. Recommendation: send it to referees. The main claim survives the erratum.

Referee report

Significance

This is a foundational paper for the 'unstable plasma' picture of the earliest stage of a heavy-ion collision, and the general instability criteria it establishes, especially Condition 1-b, are used well beyond the specific distributions solved here. The derivations are parameter-free: m_∞ is an integral over f (Eq. 2.14), not a fit, and the growth rates follow from the linearized Vlasov self-energy (Eq. 2.8) with no adjustable input. The paper is also candid about what it does not deliver, stating explicitly that it cannot yet give parametric estimates of the equilibration time to replace the α^{-13/5}Q_s^{-1} of Baier et al. The closed-form dispersion relations (3.8), (3.23), (C9)-(C12) and the growth-rate figures are a durable reference, and the qualitative review of Sec. II C is genuinely useful.

Major objections (3)

[Sec. IV A and appended erratum] The erratum states that the static non-Abelian configuration of Sec. IV A 'is not in fact a solution to the equations of motion', because Eq. (4.2) is missing the term +gε^{abc}A^b_ν(∂^νA^{cμ} − ∂^μA^{cν}) and has a sign error in Eq. (4.1). The body of Sec. IV A is nevertheless unchanged: it still derives A² = −[q² + Π_⊥(0,q e_z)]/2g² from Eq. (4.5) and concludes that the saturation amplitude scales as q/g. Since the erratum is part of the manuscript the record is corrected, but a reader of the body alone is misled. Please amend Sec. IV A in the body (or mark it explicitly as withdrawn) and point to Ref. [40] for the corrected static solutions, rather than leaving the correction to a note appended after the reference list.
[Sec. II D, Eq. (2.24), and last paragraph of Sec. IV A] The erratum's second correction states that the last paragraph of Sec. IV A 'is misleadingly general' and applies only when the unstable momenta are of order m_∞, whereas for the approximately planar distributions that dominate the paper the unstable scale is q_max ≫ m_∞ (Eq. 3.13). That correction is not propagated to the saturation estimates. The energy density quoted at the end of Sec. IV A as ~ m_∞⁴/g² and the amplitude estimate (2.24) both depend on which q is inserted, and for a planar distribution with angular width Δθ the answer changes by (q_max/m_∞)⁴ ~ (Δθ)⁻⁴. Please state in Sec. II D which momentum scale enters (2.24) in each regime, and give the corrected energy-density estimate for the planar case.
[Sec. I A and abstract:s1] The step from 'the distribution is anisotropic' to 'the growth rate exceeds 1/τ' is computed only for the two δ-function limits. For the realistic composite distribution of Sec. I A, which superposes hard planar gluons with softer gluons that are 'anisotropic but not extreme', the text asserts unstable modes 'with momenta of order m_soft' but does not show that their growth rate still beats the expansion. This is the weakest link in the central claim, and it is worth closing explicitly, because the margin is large: for the hard component the angular width is Δθ ~ (Q_sτ)^{-1/3}, so Eq. (3.13) gives q_max ~ m_∞(Q_sτ)^{-1/3}, and since γ ~ q for q ≪ m_∞ (Fig. 9) one has γτ ~ (Q_sτ)^{1/6} ≫ 1. A short parametric argument of this kind, covering both the hard and the soft components, would convert the 'must' of abstract:s1 from an assertion into a demonstrated result; I would ask for it to be added.

Minor objections (7)

[Eq. (3.4b)] The printed result Π_n̂n̂(0,q̂) = m_∞² sin²θ does not follow from the integrand displayed on the same line, which evaluates to m_∞²(1 + cot²θ) = m_∞²/sin²θ. The latter is required by the trace identity Π_ii = m_∞² that follows from (B4) for the planar distribution, and it is also what Eq. (3.21) and Fig. 10 use. Please correct the equation, and add a sentence noting that Π_n̂n̂ diverges as θ → 0 just as Π_yy does, since Fig. 10 shows both.
[Eqs. (1.6) and (1.7)] As displayed, m_soft² ~ Q_sτ and t_growth ~ √(τQ_s) are inconsistent with the surrounding argument, which requires m_soft² ~ Q_s/τ and t_growth ~ √(τ/Q_s) in order that t_growth ≪ τ for Q_sτ ≫ 1. Please check the typesetting of these two scalings.
[Eq. (1.8)] The estimate σ ~ α²m_soft² has dimensions of (length)⁴ rather than (length)²; presumably the intended form is σ ~ α²/m_soft². Please check.
[Sec. I A, paragraph beginning 'The extreme planarity'] The phrase 'whenever there is significant deviation from anisotropy' should read 'deviation from isotropy'.
[Sentence following Eq. (3.11)] The erratum corrects the location of the maximum growth rate to q ≃ m_∞[2/(3 sin²θ)]^{1/4}. Please make this correction in the body text as well, so that the body is self-contained and does not require the erratum to be read alongside it.
[Figs. 9b and 11b] Please state in the captions that the small dashed line is the sinθ → 0 limit of Eq. (3.12) and Eq. (3.24) respectively, so that the reader can identify it without consulting the text.
[Sec. III A 1, footnote 15] The statement that the possibility of instability in this mode 'was overlooked in Ref. [6]' is a priority claim about another paper. Consider phrasing it as a description of what the present analysis adds, which is the explicit growth rate and the θ-dependence of q_max.

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The paper

The paper claims

Anisotropic momentum distributions in an early quark-gluon plasma drive Weibel and two-stream instabilities that grow faster than the plasma expands, invalidating the first stage of bottom-up thermalization.

The reading

When two heavy nuclei collide at high energy, they briefly create a quark-gluon plasma, a hot soup of quarks and gluons. A leading theory of how this plasma thermalizes, called bottom-up thermalization, assumed that hard gluons scatter off each other and gradually spread out in momentum. This paper argues that picture is qualitatively wrong from the very first moments. The reason is plasma instabilities. In the early plasma the particle momenta are highly anisotropic: most particles move transverse to the beam direction, with little momentum along it. The authors show that such an anisotropic distribution makes the soft gauge fields unstable. Small magnetic and electric fluctuations grow exponentially, on a timescale that scales as the square root of the expansion time divided by the saturation scale. That growth time is much shorter than the expansion time, so the soft fields become large and non-perturbative almost immediately. The paper gives general criteria for when an anisotropic distribution is unstable, reviews the physical mechanism in terms of particles trapped by the fields versus particles that pass through, and discusses how the growth eventually saturates. It then solves explicitly for the instability growth rates in two idealized cases: a flat planar distribution and a line distribution. The authors are careful to say they cannot yet give a detailed new thermalization timeline, only that the original bottom-up timeline cannot be right.

Extended reading notes

Core claim

The first stage of the bottom-up thermalization scenario is drastically modified by QCD plasma instabilities, which grow on a timescale t_growth ~ 1/|m_soft| ~ sqrt(tau/Qs) that is parametrically shorter than the expansion time tau for Qs tau >> 1. The paper states that 'the first stage of the original bottom-up thermalization scenario cannot, even briefly, be qualitatively accurate.' If correct, the standard picture of how a quark-gluon plasma equilibrates after a heavy-ion collision is wrong at its earliest stage.

Load-bearing premise

The collisionless approximation for the hard particles, justified in Section I A by the estimate t_col ~ (Qs tau)^{2/3}/Qs >> t_growth ~ sqrt(tau/Qs) for Qs tau >> 1 (eqs 1.8-1.9). If collisions damp the unstable modes faster than they grow, the central claim that instabilities dominate the first stage fails. The estimate is parametric and depends on the assumed scaling of the soft scale m_soft and the hard-particle density.

Original abstract

We study the role of QCD plasma instabilities in non-equilibrium quark-gluon plasmas. First, we argue that such instabilities must drastically modify the "bottom-up" thermalization scenario for heavy-ion collisions. Second, we discuss conditions for the existence of instabilities in a more general context than previously treated in the QCD literature. We also give a thorough qualitative review of the origin of instabilities. We discuss some mechanisms whereby the growth of plasma instabilities saturates. Finally, we solve explicitly for instabilities and their growth rates for two extreme cases of anisotropic non-equilibrium plasmas that can be treated relatively simply and analytically: $f(\p) = F(p_\perp) \delta(p_z)$ and $f(\p) = F(p_z) \delta^{(2)}(p_\perp)$, where $f(\p)$ is the distribution of particles in momentum space.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,981 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.41 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We study the role of QCD plasma instabilities in non-equilibrium quark-gluon plasmas. First, we argue that such instabilities must drastically modify the "bottom-up" thermalization scenario for heavy-ion collisions. Second, we discuss conditions for the existence of instabilities in a more general context than previously treated in the QCD literature. We also give a thorough qualitative review of the origin of instabilities. We discuss some mechanisms whereby the growth of plasma instabilities saturates. Finally, we solve explicitly for instabilities and their growth rates for two extreme cases of anisotropic non-equilibrium plasmas that can be treated relatively simply and analytically: $f(\p) = F(p_\perp) \delta(p_z)$ and $f(\p) = F(p_z) \delta^{(2)}(p_\perp)$, where $f(\p)$ is the distribution of particles in momentum space.

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	We study the role of QCD plasma instabilities in non-equilibrium quark-gluon plasmas.	contribution 0.81	0.83	0.05	unclear 0.57
s1	First, we argue that such instabilities must drastically modify the "bottom-up" thermalization scenario for heavy-ion collisions.	contribution 0.40	0.72	0.62	asserted 0.68
s2	Second, we discuss conditions for the existence of instabilities in a more general context than previously treated in the QCD literature.	contribution 0.99	0.82	0.05	asserted 0.69
s3	We also give a thorough qualitative review of the origin of instabilities.	contribution 0.63	0.70	0.03	asserted 0.71
s4	We discuss some mechanisms whereby the growth of plasma instabilities saturates.	contribution 0.97	0.76	0.05	asserted 0.56
s5	Finally, we solve explicitly for instabilities and their growth rates for two extreme cases of anisotropic non-equilibrium plasmas that can be treated relatively simply and analytically: $f(\p) = F(p_\perp) \delta(p_z)$ and $f(\p) = F(p_z) \delta^{(2)}(p_\perp)$, where $f(\p)$ is the distribution of particles in momentum space.	contribution 0.58	0.96	0.11	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.92.

What was examined, and what was not

sources read

arXiv listing abstract v4

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

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: The standard picture of lines of constant proper time

**Figure 2.** FIG. 2: Example of an effective potential

**Figure 3.** FIG. 3: (a) Parity opposites in a gas of free-streaming parti

**Figure 4.** FIG. 4: The same situation as in Fig. 3b but for particles with

**Figure 5.** FIG. 5: Some qualitatively different examples of anisotropic

**Figure 6.** FIG. 6: Qualitative picture of particle motion in a sinusoid

**Figure 7.** FIG. 7: Trajectories in an arbitrarily large magnetic field.

**Figure 8.** FIG. 8: Convention for labeling directions with respect to

**Figure 9.** FIG. 9: The growth rate vs

**Figure 10.** FIG. 10: Qualitative behavior of Π

**Figure 11.** FIG. 11: The electric instability growth rate vs.

**Figure 12.** FIG. 12: Examples of the continuity of eigenvalues of

**Figure 13.** FIG. 13: Examples of Nyquist contours

**Figure 14.** FIG. 14: An example (shown in the

Figures reproduced from arXiv:hep-ph/0307325 by the authors.

References

Works cited (40)

[40]

P. Arnold and J. Lenaghan, “The abelianization of QCD plasma inst abilities,” hep- ph/0408052. 39

View on Pith
[1]

S. Mr´ owczy´ nski, “Stream instabilities of the quark-g luon plasma,” Phys. Lett. B 214, 587 (1988)

View on Pith 1988
[2]

S. Mr´ owczy´ nski, “Plasma instability at the initial st age of ultrarelativistic heavy ion colli- sions,” Phys. Lett. B 314, 118 (1993)

View on Pith 1993
[3]

S. Mr´ owczy´ nski, “Color collective eﬀects at the early stage of ultrarelativistic heavy ion colli- sions,” Phys. Rev. C 49, 2191 (1994)

View on Pith 1994
[4]

S. Mr´ owczy´ nski, “Color ﬁlamentation in ultrarelativistic heavy-ion collisions,” Phys. Lett. B 393, 26 (1997) [hep-ph/9606442]

View on Pith Pith review arXiv 1997
[5]

S. Mr´ owczy´ nski and M. H. Thoma, “Hard loop approach to a nisotropic systems,” Phys. Rev. D 62, 036011 (2000) [hep-ph/0001164]

View on Pith Pith review arXiv 2000
[6]

J. Randrup and S. Mr´ owczy´ nski, “Chromodynamic Weibelinstabilities in relativistic nuclear collisions,” nucl-th/0303021

View on Pith Pith review arXiv
[7]

U. W. Heinz, “Quark-qluon transport theory,” Nucl. Phys . A 418, 603C (1984)

View on Pith 1984
[8]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in a qu ark-gluon plasma,” JETP Lett. 47, 12 (1988) [Pisma Zh. Eksp. Teor. Fiz. 47, 11 (1988)]

View on Pith 1988
[9]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in qua rk plasma at ﬁnite temperatures,” Sov. J. Nucl. Phys. 52, 146 (1990) [Yad. Fiz. 52, 229 (1990)]

View on Pith 1990
[10]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in th e quark-gluon plasma At ﬁnite temperatures,” Sov. J. Nucl. Phys. 52, 385 (1990) [Yad. Fiz. 52, 605 (1990)]

View on Pith 1990
[11]

O. P. Pavlenko, “Filamentation instability of hot quar k-gluon plasma with hard jet,” Sov. J. Nucl. Phys. 55, 1243 (1992) [Yad. Fiz. 55, 2239 (1992)]

View on Pith 1992
[12]

R. Baier, A. H. Mueller, D. Schiﬀ and D. T. Son, “’Bottom- up’ thermalization in heavy ion collisions,” Phys. Lett. B 502, 51 (2001) [hep-ph/0009237]

View on Pith Pith review arXiv 2001
[13]

L. V. Gribov, E. M. Levin and M. G. Ryskin, “Semihard Proc esses In QCD,” Phys. Rept. 100, 1 (1983)

View on Pith 1983
[14]

A. H. Mueller and J. w. Qiu, “Gluon Recombination And Sha dowing At Small Values Of x,” Nucl. Phys. B 268, 427 (1986)

View on Pith 1986
[15]

J. P. Blaizot and A. H. Mueller, “The Early Stage Of Ultra relativistic Heavy Ion Collisions,” Nucl. Phys. B 289, 847 (1987). 36

View on Pith 1987
[16]

L. D. McLerran and R. Venugopalan, “Computing quark and gluon distribution functions for very large nuclei,” Phys. Rev. D 49, 2233 (1994) [hep-ph/9309289]; “Green’s functions in the color ﬁeld of a large nucleus,” Phys. Rev. D 50, 2225 (1994) [hep-ph/9402335]

View on Pith Pith review arXiv 1994
[17]

P. Romatschke and M. Strickland, “Screening and antisc reening in anisotropic quark gluon plasma,” hep-ph/0304092

View on Pith Pith review arXiv
[18]

T.-Y. B. Yang, Y. Gallant, J. Arons, A. B. Langdon, “Weib el instability in relativistically hot magnetized electron-positron plasma,” Phys. Fluids B 5, 3369 (1993)

View on Pith 1993
[19]

E. S. Weibel, “Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution,” Phys. Rev. Lett. 2, 83 (1959)

View on Pith 1959
[20]

T. S. Biro, B. Muller and X. N. Wang, “Color screening in r elativistic heavy ion collisions,” Phys. Lett. B 283, 171 (1992)

View on Pith 1992
[21]

J. Bjoraker and R. Venugopalan, “From colored glass con densate to gluon plasma: Equilibra- tion in high energy heavy ion collisions,” Phys. Rev. C 63, 024609 (2001) [hep-ph/0008294]

View on Pith Pith review arXiv 2001
[22]

G. C. Nayak, A. Dumitru, L. D. McLerran and W. Greiner, “E quilibration of the gluon-minijet plasma at RHIC and LHC,” Nucl. Phys. A 687, 457 (2001) [hep-ph/0001202]

View on Pith Pith review arXiv 2001
[23]

U. Heinz, “Kinetic theory for nonabelian plasmas,” Phy s. Rev. Lett. 51, 351 (1983); “Quark- qluon transport theory. Part 1. The classical theory,” Anna ls Phys. 161, 48 (1985); “Quark- qluon transport theory. Part 2. Color response and color cor relations in a quark-gluon plasma,” ibid. 168, 148 (1986)

View on Pith 1983
[24]

M. C. Birse, C. W. Kao and G. C. Nayak, “Magnetic screenin g eﬀects in anisotropic QED and QCD plasmas,” hep-ph/0304209

View on Pith Pith review arXiv
[25]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, 1973)

View on Pith 1973
[26]

R. C. Davidson, “Kinetic waves and instabilites in a uni form plasma” in Basic Plasma Physics I (vol. 1 of Handbook of Plasma Physics ), eds. A. A. Galeev and R. N. Sudan (North-Holland, 1984)

View on Pith 1984
[27]

P. Arnold, G. D. Moore and L. G. Yaﬀe, “Eﬀective kinetic the ory for high temperature gauge theories,” JHEP 0301, 030 (2003) [hep-ph/0209353]

View on Pith Pith review arXiv 2003
[28]

O. Buneman, “Instability, turbulence, and conductivi ty in current-carrying plasma,” Phys. Rev. Lett. 1, 8 (1958)

View on Pith 1958
[29]

D. Bohm and E. P. Gross, “Theory of plasma oscillations. A. Origin of medium-like behavior,” Phys. Rev. 75, 1851 (1949)

View on Pith 1949
[30]

I. B. Bernstein, J. M. Greene, and M. D. Kruskal, “Exact n onlinear plasma oscillations,” Phys. Rev. 108, 546 (1957)

View on Pith 1957
[31]

R. J. Goldston and P.H. Rutherford, Introduction to Plasma Physics (IOP Publishing, 1995)

View on Pith 1995
[32]

R. L. Berger and R. C. Davidson, “Equilibrium and stabil ity of large-amplitude magnetic Bernstein-Greene-Kruskal waves,” Phys. Fluids 15, 2327 (1972)

View on Pith 1972
[33]

R. C. Davidson, D. A. Hammer, U. Haber, and C. E. Wagner, “ Nonlinear development of electromagnetic instabilities in anisotropic plasmas,” P hys. Fluids 15, 317 (1972)

View on Pith 1972
[34]

T.-Y. B. Yang, J. Arons, A. B. Langdon, “Evolution of the Weibel instability in relativistically hot electron-positron beams,” Phys. Plasmas 1, 3059 (1994)

View on Pith 1994
[35]

F. Califano, N. Attico, F. Pegoraro, G. Bertin, and S. V. Bulanov, “Fast formation of magnetic islands in a plasma in the presence of counterstreaming elec trons,” Phys. Rev. Lett. 86, 5293 (2001)

View on Pith 2001
[36]

J. P. Blaizot and E. Iancu, “Non-Abelian plane waves in t he quark-gluon plasma,” Phys. Lett. B 326, 138 (1994) [hep-ph/9401323]

View on Pith Pith review arXiv 1994
[37]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se ries, and Products, 4th edition 37 (Academic Press, 1980)

View on Pith 1980
[38]

O. K. Kalashnikov and V. V. Klimov, Sov. J. Nucl. Phys. 31, 699 (1980) [Yad. Fiz. 31, 1357 (1980)]

View on Pith 1980
[39]

H. A. Weldon, Phys. Rev. D26, 1394 (1982). 38 Erratum: QCD plasma instabilities and bottom-up thermaliz ation [JHEP 08 (2003) 002] Peter Arnold, Jonathan Lenaghan, and Guy D. Moore In the last sentence of the paragraph containing (3.11), q ≃ m2∞( 23 sin2 θ)1/4 should instead read q ≃ m∞[2/(3 sin2 θ)]1/4. Before eq. (4.1), there is a misleadingly general s...

View on Pith 1982

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Equilibration of a weakly coupled nonabelian plasma takes a time set by the coupling, the initial excitation momentum, and the final temperature, with a faster instability-driven route when the plasma is sufficiently ...
Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma
hep-th 2011-06 conditional novelty 6.0 of 10

A regular finite-temperature supergravity solution dual to an anisotropic N=4 plasma yields a conserved stress tensor with a conformal anomaly and a phase diagram in which the homogeneous anisotropic phase can be unst...
The evolving Glasma
hep-ph 2011-06 conditional novelty 6.0 of 10

Lattice SU(2) Yang-Mills simulations of the expanding Glasma show the rapidity-wavenumber energy spectrum approaching a -5/3 power law consistent with Kolmogorov turbulence.
Spectral densities for hot QCD plasmas in a leading log approximation
nucl-th 2010-03 accept novelty 6.0 of 10

Solving the leading-log Boltzmann equation numerically yields the full spectral densities of the stress tensor and current in a hot QCD plasma and determines its second-order transport coefficients.
A model of the effect of collisions on QCD plasma instabilities
hep-ph 2006-03 conditional novelty 6.0 of 10

A BGK collision term added to hard-loop transport damps Weibel-type instabilities in anisotropic QCD plasmas, with a critical collision rate above which no instability grows.
Collective modes of an Anisotropic Quark-Gluon Plasma II
hep-ph 2004-06 conditional novelty 6.0 of 10

Momentum anisotropy in a quark-gluon plasma creates poles on unphysical Riemann sheets that, in the infinite-anisotropy limit, migrate onto the physical sheet as real spacelike gluon modes.
Collective phenomena in chirally imbalanced medium
hep-ph 2025-06 conditional novelty 5.0 of 10

Chirally imbalanced quark-gluon plasma splits transverse gluon modes into circular polarizations, adds instabilities, and makes quarkonium melt faster via a larger Debye mass and decay width.
Equilibration of weakly coupled QCD plasmas
hep-ph 2020-12 conditional novelty 5.0 of 10

Full leading-order QCD kinetic theory with three light flavors shows kinetic and chemical equilibration of a quark-gluon plasma take roughly the same time, one to two relaxation times, and that a longitudinally expand...
The non-equilibrium attractor for kinetic theory in relaxation time approximation
nucl-th 2018-09 conditional novelty 5.0 of 10

In the 0+1d relaxation-time Boltzmann model, all moments of the one-particle distribution function and the distribution function itself collapse onto a universal attractor, except moments dominated by high-transverse-...
Non-boost-invariant anisotropic dynamics
nucl-th 2010-11 conditional novelty 5.0 of 10

Three coupled partial differential equations describe the non-boost-invariant evolution of an anisotropically expanding quark-gluon plasma, recovering Bjorken boost-invariant expansion as a limiting case.
Some Features of the Glasma
hep-ph 2006-02 conditional novelty 5.0 of 10

High-energy heavy-ion collisions create, immediately after impact, longitudinal color electric and magnetic fields, giving a large Chern-Simons charge density that decays as the Glasma turns into quark-gluon plasma.
Jets as differential probes of hot QCD matter
hep-ph 2026-10 accept novelty 4.0 of 10

A comprehensive theoretical framework for extracting directional and multi-scale properties of the quark-gluon plasma from differential jet substructure observables.
Magnetized bottom-up thermalization in heavy-ion collisions
hep-ph 2026-06 unverdicted novelty 4.0 of 10

Strong magnetic fields may accelerate early quark production via gluon decay in the bottom-up scenario when |eB| approaches Q_s^2, modifying pre-equilibrium chemical composition.
Anisotropic plasma at finite $U(1)$ chemical potential
hep-th 2014-04 conditional novelty 4.0 of 10

Adding a U(1) charge to the Mateos-Trancanelli anisotropic black brane yields a two-branch temperature-horizon relation whose small-radius branch has negative specific heat, an instability the authors argue is indepen...
Causal Viscous Hydrodynamics for Relativistic Heavy Ion Collisions
nucl-th 2009-08 conditional novelty 4.0 of 10

Causal viscous hydrodynamics in 2+1 dimensions with shear viscosity near the KSS bound suppresses elliptic flow by roughly 30%, making v2 a sensitive diagnostic of the quark-gluon plasma shear viscosity to entropy ratio.
QCD thermalization: Ab initio approaches and interdisciplinary connections
hep-th 2020-05 accept novelty 3.0 of 10

A comprehensive review of QCD thermalization that organizes weak- and strong-coupling descriptions around universal non-thermal and hydrodynamic attractors.
Relativistic anisotropic hydrodynamics
nucl-th 2017-12 accept novelty 3.0 of 10

Anisotropic hydrodynamics, which deforms the particle distribution function rather than linearizing around equilibrium, reproduces exact Boltzmann solutions better than second- and third-order viscous hydrodynamics in...
Observables in Strongly Coupled Anisotropic Theories
hep-th 2013-06 accept novelty 3.0 of 10

The axion-deformed anisotropic holographic plasma cannot be quantitatively matched to the observed quark-gluon plasma, because the anisotropy required by data pushes the model outside its small-anisotropy regime.
Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all
nucl-ex 2012-12 unverdicted novelty 3.0 of 10

A community white paper lays out the physics case, golden measurements, and machine and detector requirements for a proposed US Electron-Ion Collider aimed at gluon-dominated QCD.
Chiral effects in astrophysics and cosmology
astro-ph.CO 2022-07 accept novelty 2.0 of 10

Chirality of Standard Model fermions generates anomaly-driven transport that has been proposed as a mechanism for neutron-star kicks, magnetar fields, supernova turbulence, primordial magnetogenesis, baryogenesis, and...
Thermalization and isotropization in heavy-ion collisions
hep-ph 2013-12 accept novelty 2.0 of 10

The quark gluon plasma produced in ultrarelativistic heavy ion collisions thermalizes quickly but stays momentum-space anisotropic for several fm/c, so anisotropic parton distributions must be built into QGP phenomenology.
Jets: Hard-soft correlation -- Theory overview
hep-ph 2025-04 unverdicted

A review of recent theoretical progress on hard-soft correlations in jets, covering kinetic theory, x-scape-gim, TMD-based anisotropies, and color fluctuations.

How the literature uses it

50 arXiv papers cite this in 61 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We investigate the impact of plasma instabilities on the first stage of the ``bottom-up'' thermalization scenario in which such a two-component plasma emerges, and find that even in the case of non-abelian saturation instabilities qualitatively change the bottom-up picture.

The impact of QCD plasma instabilities on bottom-up thermalization · 0.87

Papers citing it · 41 sentences without a confident label

Uses its method or tool 1 paper

The non-equilibrium rates are enhanced using a bremsbremstrahlung ansatz analogous to the complete leading order (LO) thermal rate [ref] .

Non-equilibrium photons from the bottom-up thermalization scenario · Introduction · 0.91

Uses its data or values 1 paper

An estimate in [ref] is that the saturation value for mode number $q$ is $gB≃q^2$ (see however, [ref] ).

Spin polarization of heavy quarks in matter: predictions from effective field theories · Heavy Quark Effective Theory in matter · 0.83

Cites it as background 18 papers

In the case of a non-equilibrium QGP it is still possible to make studies of the collective modes of the system, however, the situation is complicated by the presence of unstable gluonic modes related to the chromo-Weibel instability [ref] .

Collective modes of a collisional anisotropic quark-gluon plasma · Introduction · 0.82
The identification of the deconfined phase of quarks and gluons at the Relativistic Heavy Ion Collider, a phase akin to a nonabelian plasma, has led to a number of investigations on instabilities in a nonabelian plasma [ref] .

A Note on Schwinger Mechanism and a Nonabelian Instability in a Nonabelian Plasma · Introduction · 0.99
There are important qualitative differences between QED and QCD plasma instabilities [ref] .

Non-abelian plasma instabilities for strong anisotropy · Introduction · 0.83

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] [40]

P. Arnold and J. Lenaghan, “The abelianization of QCD plasma inst abilities,” hep- ph/0408052. 39

View on Pith

[2] [1]

S. Mr´ owczy´ nski, “Stream instabilities of the quark-g luon plasma,” Phys. Lett. B 214, 587 (1988)

View on Pith 1988

[3] [2]

S. Mr´ owczy´ nski, “Plasma instability at the initial st age of ultrarelativistic heavy ion colli- sions,” Phys. Lett. B 314, 118 (1993)

View on Pith 1993

[4] [3]

S. Mr´ owczy´ nski, “Color collective eﬀects at the early stage of ultrarelativistic heavy ion colli- sions,” Phys. Rev. C 49, 2191 (1994)

View on Pith 1994

[5] [4]

S. Mr´ owczy´ nski, “Color ﬁlamentation in ultrarelativistic heavy-ion collisions,” Phys. Lett. B 393, 26 (1997) [hep-ph/9606442]

View on Pith Pith review arXiv 1997

[6] [5]

S. Mr´ owczy´ nski and M. H. Thoma, “Hard loop approach to a nisotropic systems,” Phys. Rev. D 62, 036011 (2000) [hep-ph/0001164]

View on Pith Pith review arXiv 2000

[7] [6]

J. Randrup and S. Mr´ owczy´ nski, “Chromodynamic Weibelinstabilities in relativistic nuclear collisions,” nucl-th/0303021

View on Pith Pith review arXiv

[8] [7]

U. W. Heinz, “Quark-qluon transport theory,” Nucl. Phys . A 418, 603C (1984)

View on Pith 1984

[9] [8]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in a qu ark-gluon plasma,” JETP Lett. 47, 12 (1988) [Pisma Zh. Eksp. Teor. Fiz. 47, 11 (1988)]

View on Pith 1988

[10] [9]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in qua rk plasma at ﬁnite temperatures,” Sov. J. Nucl. Phys. 52, 146 (1990) [Yad. Fiz. 52, 229 (1990)]

View on Pith 1990

[11] [10]

Y. E. Pokrovsky and A. V. Selikhov, “Filamentation in th e quark-gluon plasma At ﬁnite temperatures,” Sov. J. Nucl. Phys. 52, 385 (1990) [Yad. Fiz. 52, 605 (1990)]

View on Pith 1990

[12] [11]

O. P. Pavlenko, “Filamentation instability of hot quar k-gluon plasma with hard jet,” Sov. J. Nucl. Phys. 55, 1243 (1992) [Yad. Fiz. 55, 2239 (1992)]

View on Pith 1992

[13] [12]

R. Baier, A. H. Mueller, D. Schiﬀ and D. T. Son, “’Bottom- up’ thermalization in heavy ion collisions,” Phys. Lett. B 502, 51 (2001) [hep-ph/0009237]

View on Pith Pith review arXiv 2001

[14] [13]

L. V. Gribov, E. M. Levin and M. G. Ryskin, “Semihard Proc esses In QCD,” Phys. Rept. 100, 1 (1983)

View on Pith 1983

[15] [14]

A. H. Mueller and J. w. Qiu, “Gluon Recombination And Sha dowing At Small Values Of x,” Nucl. Phys. B 268, 427 (1986)

View on Pith 1986

[16] [15]

J. P. Blaizot and A. H. Mueller, “The Early Stage Of Ultra relativistic Heavy Ion Collisions,” Nucl. Phys. B 289, 847 (1987). 36

View on Pith 1987

[17] [16]

L. D. McLerran and R. Venugopalan, “Computing quark and gluon distribution functions for very large nuclei,” Phys. Rev. D 49, 2233 (1994) [hep-ph/9309289]; “Green’s functions in the color ﬁeld of a large nucleus,” Phys. Rev. D 50, 2225 (1994) [hep-ph/9402335]

View on Pith Pith review arXiv 1994

[18] [17]

P. Romatschke and M. Strickland, “Screening and antisc reening in anisotropic quark gluon plasma,” hep-ph/0304092

View on Pith Pith review arXiv

[19] [18]

T.-Y. B. Yang, Y. Gallant, J. Arons, A. B. Langdon, “Weib el instability in relativistically hot magnetized electron-positron plasma,” Phys. Fluids B 5, 3369 (1993)

View on Pith 1993

[20] [19]

E. S. Weibel, “Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution,” Phys. Rev. Lett. 2, 83 (1959)

View on Pith 1959

[21] [20]

T. S. Biro, B. Muller and X. N. Wang, “Color screening in r elativistic heavy ion collisions,” Phys. Lett. B 283, 171 (1992)

View on Pith 1992

[22] [21]

J. Bjoraker and R. Venugopalan, “From colored glass con densate to gluon plasma: Equilibra- tion in high energy heavy ion collisions,” Phys. Rev. C 63, 024609 (2001) [hep-ph/0008294]

View on Pith Pith review arXiv 2001

[23] [22]

G. C. Nayak, A. Dumitru, L. D. McLerran and W. Greiner, “E quilibration of the gluon-minijet plasma at RHIC and LHC,” Nucl. Phys. A 687, 457 (2001) [hep-ph/0001202]

View on Pith Pith review arXiv 2001

[24] [23]

U. Heinz, “Kinetic theory for nonabelian plasmas,” Phy s. Rev. Lett. 51, 351 (1983); “Quark- qluon transport theory. Part 1. The classical theory,” Anna ls Phys. 161, 48 (1985); “Quark- qluon transport theory. Part 2. Color response and color cor relations in a quark-gluon plasma,” ibid. 168, 148 (1986)

View on Pith 1983

[25] [24]

M. C. Birse, C. W. Kao and G. C. Nayak, “Magnetic screenin g eﬀects in anisotropic QED and QCD plasmas,” hep-ph/0304209

View on Pith Pith review arXiv

[26] [25]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, 1973)

View on Pith 1973

[27] [26]

R. C. Davidson, “Kinetic waves and instabilites in a uni form plasma” in Basic Plasma Physics I (vol. 1 of Handbook of Plasma Physics ), eds. A. A. Galeev and R. N. Sudan (North-Holland, 1984)

View on Pith 1984

[28] [27]

P. Arnold, G. D. Moore and L. G. Yaﬀe, “Eﬀective kinetic the ory for high temperature gauge theories,” JHEP 0301, 030 (2003) [hep-ph/0209353]

View on Pith Pith review arXiv 2003

[29] [28]

O. Buneman, “Instability, turbulence, and conductivi ty in current-carrying plasma,” Phys. Rev. Lett. 1, 8 (1958)

View on Pith 1958

[30] [29]

D. Bohm and E. P. Gross, “Theory of plasma oscillations. A. Origin of medium-like behavior,” Phys. Rev. 75, 1851 (1949)

View on Pith 1949

[31] [30]

I. B. Bernstein, J. M. Greene, and M. D. Kruskal, “Exact n onlinear plasma oscillations,” Phys. Rev. 108, 546 (1957)

View on Pith 1957

[32] [31]

R. J. Goldston and P.H. Rutherford, Introduction to Plasma Physics (IOP Publishing, 1995)

View on Pith 1995

[33] [32]

R. L. Berger and R. C. Davidson, “Equilibrium and stabil ity of large-amplitude magnetic Bernstein-Greene-Kruskal waves,” Phys. Fluids 15, 2327 (1972)

View on Pith 1972

[34] [33]

R. C. Davidson, D. A. Hammer, U. Haber, and C. E. Wagner, “ Nonlinear development of electromagnetic instabilities in anisotropic plasmas,” P hys. Fluids 15, 317 (1972)

View on Pith 1972

[35] [34]

T.-Y. B. Yang, J. Arons, A. B. Langdon, “Evolution of the Weibel instability in relativistically hot electron-positron beams,” Phys. Plasmas 1, 3059 (1994)

View on Pith 1994

[36] [35]

F. Califano, N. Attico, F. Pegoraro, G. Bertin, and S. V. Bulanov, “Fast formation of magnetic islands in a plasma in the presence of counterstreaming elec trons,” Phys. Rev. Lett. 86, 5293 (2001)

View on Pith 2001

[37] [36]

J. P. Blaizot and E. Iancu, “Non-Abelian plane waves in t he quark-gluon plasma,” Phys. Lett. B 326, 138 (1994) [hep-ph/9401323]

View on Pith Pith review arXiv 1994

[38] [37]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se ries, and Products, 4th edition 37 (Academic Press, 1980)

View on Pith 1980

[39] [38]

O. K. Kalashnikov and V. V. Klimov, Sov. J. Nucl. Phys. 31, 699 (1980) [Yad. Fiz. 31, 1357 (1980)]

View on Pith 1980

[40] [39]

H. A. Weldon, Phys. Rev. D26, 1394 (1982). 38 Erratum: QCD plasma instabilities and bottom-up thermaliz ation [JHEP 08 (2003) 002] Peter Arnold, Jonathan Lenaghan, and Guy D. Moore In the last sentence of the paragraph containing (3.11), q ≃ m2∞( 23 sin2 θ)1/4 should instead read q ≃ m∞[2/(3 sin2 θ)]1/4. Before eq. (4.1), there is a misleadingly general s...

View on Pith 1982