Collective non-Abelian instabilities in a melting Color Glass Condensate

Paul Romatschke; Raju Venugopalan

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Collective non-Abelian instabilities in a melting Color Glass Condensate

Paul Romatschke, Raju Venugopalan

High Energy Physics - Phenomenology · Submitted 10 Oct 2005 · 25 references

Pith's verdict: First 3+1-D CGC simulation with boost invariance relaxed; the √τ growth law is physically right, but the exponent discrimination is asserted rather than quantified.

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arXiv IDhep-ph/0510121 (version 2)

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CitationPaul Romatschke, Raju Venugopalan. "Collective non-Abelian instabilities in a melting Color Glass Condensate." arXiv:hep-ph/0510121 (2005). https://pith.science/paper/O5NG6L3J

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Editor's note

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Worth your time, with one caveat about the fit. The real result is the first 3+1-D SU(2) simulation of the expanding CGC with boost invariance relaxed, and the finding that soft modes grow as exp(Γ√(g²μτ)) rather than exp(Γτ). That is not cosmetic: it follows naturally from a plasmon mass falling as τ^{-1/2}, and the plasmon mass measurement in Fig. 3 is the solid part of the paper — κ0 ≈ 0.3, stable under refinement from 32² to 128². The √τ law is what you would expect for a system expanding at the speed of light, and the bottom-up literature needed exactly this. The HTL comparison in the table is a consistency check, not a prediction — κ0 and Γ_fit both come out of the same run — and the authors say so, including the factor-of-2 ambiguity in translating γ_stat τ into the expanding case. The abstract says 'we compare', which is honest. I would not hold the table against them. Where I would push is the exponent discrimination. The text reads as c3 fixed at 0.5, with no χ² or residual quoted, so 'exp(τ) is clearly ruled out' is a visual judgment over a long lever arm. In fairness the two fitted curves have different shapes and the exp(τ) fit visibly fails to track the data, so I believe the conclusion — but a free-c3 fit would cost nothing and settle it. Related: the fitted quantity is the maximum of τ²T^ηη over k_η, and Fig. 2 shows the maximizing mode drifting upward with time. So Γ is an envelope over a shifting mode, not a single unstable mode's rate. That is not discussed and it should be. Same for Δ-dependence: Δ runs 10⁻¹¹ to 10⁻⁸ but Γ(Δ) is never plotted, so the linear regime is not demonstrated. Minor: key numerical details are deferred to Ref. [18], listed as in preparation, and no code or data are shipped. That is a reproducibility cost, not a flaw in the physics. Who should read it: anyone working on CGC thermalization, Weibel instabilities, or anisotropic plasmas. It is a short, honest letter that opens a real direction. Send it to referees.

Referee report

Significance

If the result holds, it identifies a qualitatively new growth law for collective instabilities in an expanding non-Abelian plasma, distinct from the exp(γτ) behaviour established for static anisotropic plasmas, and it supplies a concrete, falsifiable relation Γ/κ₀ ≈ √3 between two independently extracted quantities. The paper is also the first to embed the CGC initial conditions of Refs. [5,6] in a 3+1-D simulation with rapidity-dependent fluctuations, which is a genuine methodological step. I give explicit credit for the κ₀ ∝ 1/√τ scaling in Fig. 3, which is shown to be stable across N⊥ = 32, 64, 128, and for the fact that the ratio Γ_fit/κ₀ is constant to roughly 2% across the three values of g²μL in the table (1.65, 1.66, 1.63); that constancy is real evidence for the plasmon-mass scaling of the growth rate, independent of the overall normalization.

Major objections (4)

[Fig. 1 and the sentence 'A best fit ... gives c2 = 0.427 ± 0.01 for c3 = 0.5'] The discrimination between exp(Γ√(g²μτ)) and exp(Γτ) is the headline of the paper, but the fit that supports it holds the exponent c3 fixed at 0.5 rather than fitting it. No chi-squared, residuals, or confidence interval on c3 is reported, and the competing exp(τ) curve in Fig. 1 uses a different amplitude normalization (0.00544 versus 0.427), so the two curves are not compared on equal footing. The statements that the exp(τ) form is 'clearly ruled out by the data' and that the √τ growth was demonstrated 'unambiguously' therefore rest on a visual judgment over a long lever arm rather than a quantified model comparison. Please report a fit with c3 left free, with its uncertainty, and a goodness-of-fit statistic for both functional forms; this is the single analysis that would convert the headline from an assertion into a measurement.
[Fig. 1 and Fig. 2, definition of the fitted quantity] The quantity fitted in Fig. 1 is the maximum of τ²T̃^{ηη} over k_η at each time, and Fig. 2 shows that the maximizing mode drifts upward from k_η ≈ 10 to k_η ≈ 110–120 over the fitted interval. For N_η = 64 and L_η = 1.6 the Nyquist mode of the η lattice is k_η^Nyq = πN_η/L_η ≈ 126, so the late-time growth being fitted occurs in modes close to the ultraviolet cutoff of the η direction. The extracted Γ is thus an envelope over a drifting mode that ends near the lattice cutoff, not the growth rate of a fixed soft mode, and it is not shown that this envelope is converged in N_η or L_η. Please extract Γ from a fixed low-k_η mode (or a fixed band well below the cutoff) and show that it agrees with the envelope fit, and/or demonstrate convergence of the late-time growth with N_η and L_η. The authors' remark that raising L_η to 3.2 lowers the onset time by 20% while leaving the growth rate unchanged is a useful partial check but does not address the ultraviolet end of the spectrum.
[Initial conditions, parameter Δ (text preceding Fig. 1)] The seed amplitude Δ is varied over 10⁻¹¹ to 10⁻⁸, but no result is shown for how the extracted Γ depends on Δ. The fit coefficients c0 and c1 are stated to be proportional to the seed, which is consistent with linear growth, but linearity in Δ is assumed rather than demonstrated. Since the physical interpretation of Γ as a growth rate requires the instability to be in a Δ-independent linear regime, please show Γ(Δ) for at least two or three values of Δ spanning the quoted range, and state explicitly that the quoted Γ is Δ-independent.
[Table and the paragraph beginning 'We now assume the growth rate ...'] The authors already acknowledge that a consistent treatment gives exp(2∫dτ'γ(τ')) and hence an additional factor of 2 in Γ_th relative to Γ_fit, and that it is not obvious that HTL relations generalize to CGC initial conditions; I do not restate that as a new finding. What remains unresolved is that the 4–6% agreement in the table depends on three separate inputs that are not independently tested: (i) κ₀ is extracted from a 2+1-D simulation for computational convenience (footnote [24]), while Γ_fit comes from the 3+1-D run, and the assertion that longitudinal fluctuations affect the transverse correlator only weakly is not tested; (ii) m∞² = (3/2)ω_pl² is stated for the specific anisotropic model of Refs. [14,23] and is applied here outside that setting; (iii) the identification of the equal-time correlator ratio in the ω(k⊥) definition with the HTL plasmon frequency is an assumption the authors say is under study. Given these, the agreement should be presented as a consistency check of the plasmon-mass scaling rather than as confirmation of the HTL prediction, and the abstract's phrasing should make clear that the comparison is to an extrapolated HTL relation, not to a first-principles expanding-plasma calculation.

Minor objections (10)

[Numerical procedure paragraph] The number of independent color-charge configurations averaged over is not stated, and no statistical error bars are shown on the data points of Fig. 1 or Fig. 2. Please state the ensemble size and, where feasible, the configuration-to-configuration spread.
[Fig. 1 caption and text] The two fit curves in Fig. 1 use different amplitude normalizations (0.427 and 0.00544). Please state in the caption that the comparison is of functional shapes and that the amplitudes are separately fitted, so that a reader does not read the two numbers as directly comparable.
[Fig. 2] Please mark the Nyquist mode k_η^Nyq = πN_η/L_η on Fig. 2 alongside k_min, so that the reader can see how close the maximally unstable mode comes to the ultraviolet cutoff of the η lattice.
[Fig. 4 caption] The caption notes that g²μa⊥ = 0.707 for all points except g²μL = 67.9, for which it is 1.06. Since κ₀ is being read off as a function of g²μL, please comment on whether the non-monotonic variation of κ₀ (0.304, 0.258, 0.283) could be partly a lattice-spacing effect rather than a physical g²μL dependence.
[Text following Eq. for δE_i, δE_η] The Gaussian white-noise correlator ⟨F(η)F(η')⟩ = Δ²δ(η−η') is combined with periodic boundary conditions in η. Please state how the delta function is discretized on the lattice so that the normalization of Δ is unambiguous.
[Paragraph on initial time τ₀] The statement that results are insensitive to variations of τ₀ by a factor of 2 is not backed by shown data. Either show the comparison or soften the claim.
[Paragraph on violations of boost invariance] The assertion that violations from finite-energy constraints are small is not supported in the text. A brief justification or a reference would help.
[Summary paragraph] The sentence 'the relevant time scales in Fig. 1, for RHIC collisions where g²μ ≈ 1 GeV, are too large to be interest' should read 'too large to be of interest'; the same paragraph would benefit from quoting the corresponding proper time (of order tens of fm/c) so the reader can judge the phenomenological gap directly.
[Text near the definition of m∞²] Typographical: 'anistropic' should be 'anisotropic'.
[References] Several extensions that bear on the central claims (larger Δ, the large-volume limit, the broader class of initial conditions, and the relation of the measured ω_pl to the HTL plasmon frequency) are deferred to Ref. [18], listed as 'in preparation'. This is acceptable for a Letter, but the dependence should be flagged in the text where the deferral occurs rather than only in the closing paragraph.

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The paper

The paper claims

In a 3+1-dimensional simulation of the gluon fields produced in a heavy-ion collision, small departures from boost invariance trigger a non-Abelian Weibel instability whose soft modes grow as $e^{0.43(0.5)}$ rather than as a simple $e^{c_2(

The reading

When two heavy nuclei collide at high energy, the gluon fields that fill the collision zone are usually treated as boost invariant: the same at every rapidity. This paper asks what happens when that assumption is relaxed by a tiny amount, and it runs the first fully three-dimensional (two transverse directions plus rapidity) lattice simulation of the expanding gluon matter to find out. The answer is that the small rapidity-dependent seeds are amplified by a non-Abelian Weibel instability, the same class of collective plasma effect that can make anisotropic plasmas unstable to magnetic fluctuations. The growth is not a simple exponential in proper time $e^{c_2(0.5)}$; it is $e^{c_2(0.5)}$ with $c_2(0.5)$ proportional to $c_2(0.5)$ in the exponent, and the rate is fixed by a plasmon mass that the collision itself generates and that falls off as $1/

What carries the argument

The load-bearing object is the plasmon mass generated by the collision, $c_2(0.5)$, with $c_2(0.5)$ extracted from the ratio of field-strength correlators to gauge-field correlators at zero transverse momentum. It does two jobs at once. First, it supplies the only dimensionful scale in the problem besides $c_2(0.5)$, so it converts the growth into the form $c_2(0.5)$ and explains why the exponent scales as the square root of proper time rather than linearly. Second, because the plasmon mass is measured independently of the growth rate, it turns the Hard Thermal Loop relation $c_2(0.5)$ into a parameter-free prediction $c_2(0.5)$ that can be checked against the fitted $c_2(0.5)$. The initial conditions are constructed so that Gauss's law holds exactly at the starting time: random transverse fields are multiplied by a Gaussian white-noise rapidity profile of amplitude $c_2(0.5)$, which keeps the violation of boost invariance small and periodic boundary conditions usable in rapidity.

Extended reading notes

Core claim

The paper reports that in an SU(2) Yang-Mills simulation of matter expanding into the vacuum after a collision, violations of boost invariance grow into a non-Abelian Weibel instability. The soft modes of the longitudinal pressure grow with proper time $c_2(0.5)$ as $c_2(0.5)$, with $c_2(0.5)$ fitted at $0.427(1)$ for $c_2(0.5)$; a fit with a simple exponential in $c_2(0.5)$ is ruled out by the data. The most unstable rapidity mode number $c_2(0.5)$ itself grows with time, so the instability migrates from the softest modes upward. Crucially, the collision dynamically generates a mass gap, identified with a plasmon mass $c_2(0.5)$ with $c_2(0.5)$, which is measured independently and is stable as the lattice spacing is reduced. The authors then assume the static Hard Thermal Loop relations $c_2(0.5)$ and $c_2(0.5)$ continue to hold in the expanding case, which predicts $c_2(0.5)$ with $c_2(0.5)$; the measured rates for three values of $c_2(0.5)$ agree with this to 4-6%. The authors state plainly that a more consistent treatment, $c_2(0.5)$, would introduce an extra factor of two and that it is not obvious the HTL relations generalize to these initial conditions.

Load-bearing premise

The comparison to the Hard Thermal Loop prediction assumes that the static relations between growth rate, asymptotic mass, and plasmon mass survive unchanged in a system expanding at the speed of light, and that the static rate should be translated into the expanding case in the particular way the authors choose rather than the more consistent double-integral form.

Editorial extensions

If this is right

The time for a soft mode to reach a given amplitude scales as the square of the logarithm of that amplitude, not linearly, so estimates of when instabilities become important in a melting Color Glass Condensate must be redone with a square-root clock.
Because the growth rate is tied to a plasmon mass that falls as $c_2(0.5)$, measuring the plasmon mass in a simulation or in an effective description fixes the instability rate with no extra free parameter.
The instability is a soft-mode phenomenon: hard modes with transverse momentum of order the saturation scale are unaffected, while the most unstable rapidity mode number grows with time, indicating a cascade toward harder rapidity modes.
If the growth is fast enough at larger seed amplitudes, classical collective instabilities could compete with, or replace, the perturbative scattering-based picture of how the gluon medium thermalizes.

Reading between the lines

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

If the square-root growth law is generic to expanding non-Abelian plasmas, the same scaling should appear wherever an anisotropic gauge plasma expands relativistically, for instance in the early universe; that would make the exponent a shared signature rather than a model detail.
The factor-of-two ambiguity between the naive and consistent translations of the static growth rate is not a small correction, and it sits exactly on the quantity being compared. Resolving it analytically, by computing the expanding Hard Thermal Loop self-energy rather than borrowing the static relation, would decide whether the 4-6% agreement is confirmation or coincidence.
The measured $c_2(0.5)$ is not monotonic in $c_2(0.5)$ across the three simulated volumes, so part of the quoted agreement may come from a quantity that is itself fitted rather than predicted; a continuum extrapolation in the rapidity direction on larger lattices would test this.
Because the seeds here are deliberately tiny, the reported onset times are far longer than a real collision allows; the physically relevant question is what happens at order-unity violations of boost invariance, which the authors defer to follow-up work.

Original abstract

We present first results for 3+1-D simulations of SU(2) Yang-Mills equations for matter expanding into the vacuum after a heavy ion collision. Violations of boost invariance cause a Weibel instability leading soft modes to grow with proper time $\tau$ as $\exp(\Gamma \sqrt{g^2\mu \tau})$, where $g^2\mu$ is a scale arising from the saturation of gluons in the nuclear wavefunction. The scale for the growth rate $\Gamma$ is set by a plasmon mass, defined as $\omega_{\rm pl}= \kappa_0 \sqrt{\frac{g^2\mu}{\tau}}$, generated dynamically in the collision. We compare the numerical ratio $\Gamma/\kappa_0$ to the corresponding value predicted by the Hard Thermal Loop formalism for anisotropic plasmas.

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, 3,785 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.71 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We present first results for 3+1-D simulations of SU(2) Yang-Mills equations for matter expanding into the vacuum after a heavy ion collision. Violations of boost invariance cause a Weibel instability leading soft modes to grow with proper time $\tau$ as $\exp(\Gamma \sqrt{g^2\mu \tau})$, where $g^2\mu$ is a scale arising from the saturation of gluons in the nuclear wavefunction. The scale for the growth rate $\Gamma$ is set by a plasmon mass, defined as $\omega_{\rm pl}= \kappa_0 \sqrt{\frac{g^2\mu}{\tau}}$, generated dynamically in the collision. We compare the numerical ratio $\Gamma/\kappa_0$ to the corresponding value predicted by the Hard Thermal Loop formalism for anisotropic plasmas.

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 present first results for 3+1-D simulations of SU(2) Yang-Mills equations for matter expanding into the vacuum after a heavy ion collision.	contribution 0.91	0.94	0.04	asserted 0.96
s1	Violations of boost invariance cause a Weibel instability leading soft modes to grow with proper time $\tau$ as $\exp(\Gamma \sqrt{g^2\mu \tau})$, where $g^2\mu$ is a scale arising from the saturation of gluons in the nuclear wavefunction.	result 0.97	0.67	0.36	asserted 1.00
s2	The scale for the growth rate $\Gamma$ is set by a plasmon mass, defined as $\omega_{\rm pl}= \kappa_0 \sqrt{\frac{g^2\mu}{\tau}}$, generated dynamically in the collision.	result 0.88	0.75	0.14	asserted 1.00
s3	We compare the numerical ratio $\Gamma/\kappa_0$ to the corresponding value predicted by the Hard Thermal Loop formalism for anisotropic plasmas.	result 0.52	0.86	0.10	asserted 0.77

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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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 maximum Fourier mode amplitudes of

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References

Works cited (25)

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P. Romatschke and R. Venugopalan, in preparation

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The eﬀect of small longitudinal ﬂuctuations on transve rse quantities should be rather small. We therefore extract ω(k⊥), for computational convenience, from a 2+1-D simulation

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P. Arnold, J. Lenaghan, G.D. Moore, JHEP 0308, 002 (2003); P. Arnold, J. Lenaghan, G. D. Moore and L. G. Yaﬀe, Phys. Rev. Lett. 94, 072302 (2005)

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A. Rebhan, P. Romatschke, M. Strickland, Phys. Rev. Lett 94, 102303 (2005); JHEP 0509, 041 (2005). P. Arnold, G.D. Moore, L.G. Yaﬀe, Phys.Rev. D 72, 054003 (2005); for a related approach, see A. Dumitru and Y. Nara, Phys. Lett. B 621, 89 (2005)

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P. Arnold and G. D. Moore, arXiv:hep-ph/0509206; hep-ph/0509226

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[20]

It is essential to distinguish “strict” boost invarian ce of ﬁelds from that of single particle distributions. The latte r can be preserved even if the former is violated

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[21]

This procedure is similar to that of Ref. [6]. Further de - 5 tails will be presented in Ref. [18]

View on Pith
[22]

In the classical theory of the CGC, Q2s = g4µ2Nc/2π · ln(g2µ/Λ), where Λ is an infrared scale of order Λ QCD

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[23]

Averaging over longitudinal and transverse modes (see the 2nd paper by Rebhan et al. in Ref. [16]), one obtains ω2pl = m23√ξ arctan√ξ and m2∞ = m22√ξ arctan√ξ, where ξ denotes the strength of the anisotropy and m2 is a soft scale proportional to the Debye mass. Therefore, ω2pl = 2 3 m2∞

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[25]

The mass gap was noted in Ref. [6] in context of infrared ﬁnite number distributions. Interestingly, the behavior o f ωpl versus g2µL is similar to that of the gluon liberation factor fN

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Solving Einstein's equations for two colliding shock waves in AdS5 yields an analytic early-time boundary energy-momentum tensor that violates Bjorken boost invariance and suggests a thermalization time scaling as the...
Wilson line correlator in the MV model: relating the glasma to deep inelastic scattering
hep-ph 2007-11 accept novelty 6.0 of 10

The Wilson line correlator computed with the standard glasma lattice prescription gives Qs approximately 0.6 g squared mu, revising the gluon liberation coefficient to c approximately 1.1.
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-...
Universal attractor in a highly occupied non-Abelian plasma
hep-ph 2013-11 accept novelty 5.0 of 10

Classical-statistical SU(2) Yang-Mills simulations show a universal self-similar attractor governing thermalization of highly occupied non-Abelian plasmas, with scaling exponents consistent with the bottom-up scenario.
On the gluon spectrum in the glasma
hep-ph 2010-05 conditional novelty 5.0 of 10

A linearized time evolution starting from exact Coulomb-gauge initial fields reproduces the full classical Yang-Mills gluon spectrum in nucleus-nucleus collisions to within about 15 percent at intermediate momenta, bu...
Energy density of the Glasma
hep-ph 2006-06 accept novelty 5.0 of 10

The initial Glasma energy density factorizes into a product of the colliding nuclei's integrated gluon distributions, and its logarithmic ultraviolet divergence is an artifact of the exact collision time that is gone ...
The Unstable Glasma
hep-ph 2006-05 conditional novelty 5.0 of 10

Simulating the expanding Glasma on 3+1-D lattices confirms a non-Abelian Weibel instability with a continuum-stable growth rate, but the isotropization it produces takes far longer than RHIC phenomenology requires.
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.
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 Fluid Dynamics In and Out of Equilibrium -- Ten Years of Progress in Theory and Numerical Simulations of Nuclear Collisions
nucl-th 2017-12 accept novelty 3.0 of 10

The gradient expansion of relativistic hydrodynamics diverges but is Borel-resummable, and the resulting non-equilibrium attractor explains why low-order viscous hydrodynamics describes heavy-ion collision data far be...
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...

How the literature uses it

53 arXiv papers cite this in 106 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 · 79 sentences without a confident label

Contests it 1 paper

It is likely that the numerical simulation in Ref. [ref] underestimates the instability strength in the same way as for the energy density.

Randomness in infinitesimal extent in the McLerran-Venugopalan model · Introduction · contest 0.80

Uses its method or tool 3 papers

In this work, we relax boost invariance in the IP-Glasma framework by providing longitudinal structure using the JIMWLK renormalization group equation [ref] , and solve the Classical Yang-Mills (CYM) equations on a 3-dimensional lattice [ref] .

The 3+1D initialization and evolution of the Glasma · Introduction · 0.92
To check if pile-up has occurred, we monitor the Fourier transform of the magnetic component of Hamiltonian, [equation] which has successfully been used as an indicator for this purpose in the past [ref] .

Real Time Quantum Gravity Dynamics from Classical Statistical Yang-Mills Simulations · UV Problem in Classical Statistical Lattice Simulations · 0.86
For this purpose, we choose the initial fluctuations according to the prescription used in Refs. [ref] and we fix the MV model parameter $g^2μ$ in the same way as in the boost-invariant situation.

The evolving Glasma · Breaking the Boost Invariance · 0.95

Compares with it 3 papers

We revealed that the characteristic features of the rapidity-dependent fluctuations found numerically in Ref. [ref] are qualitatively reproduced as the instability of the homogeneous color magnetic field à la Nielsen-Olesen [ref] .

Instabilities in non-expanding glasma · Introduction · 0.81
As the factor $(e^-Q_s^2 τ^2I_0(Q_s^2τ^2))^1/4$ decreases quite slowly, one may observe an approximate linear increase of $ν_max$ in $τ$, in accordance with the numerical result [ref] .

Expanding color flux tubes and instabilities · Time-dependent magnetic field · 0.91
For our purpose of studying the instability, we can nevertheless consider $<|P__L(ν)|>$ (i.e. we compute the modulus first, and we perform the average over initial conditions next) in order to confirm some known results reported previously [ref] .

The evolving Glasma · Dependence on the transverse grid size $N$ · 0.84

Cites it as support 2 papers

In fact, instability in the glasma is already found in the lattice numerical simulation [ref] .

Expanding color flux tubes and instabilities · Introduction · 0.98
However: [equation] Thus it appears inconsistent to set $$ and/or the dilaton to zero; in fact weak coupling analysis suggests that there are instabilities in expanding plasma generating VEVs of various operators [ref] , in particular $⟨F^2() ⟩$.

On SUGRA description of boost-invariant conformal plasma at strong coupling · III: Janik-Peschanski proposal for string theory dual to boost invariant expansion · 0.98

Cites it as background 13 papers

The instability of an expanding glasma numerically found in Ref. [ref] was previously interpreted as the Nielsen-Olesen instability of the longitudinally uniform color magnetic fields produced in between two sheets of the Color Glass Condensates [ref] .

Instabilities in non-expanding glasma · Summary and discussions · 0.93
Then, converting the coordinates from the light-cone variables $x^±$ to the proper-time $τ$ and the rapidity $η$ which are related by $x^±=(τ/√(2))e^±η$, we can express the initial fields right after two nuclei collide as [ref] [equation] and the transverse fields ($E^i$, $B^i$) are zero [ref] , under the assumption that $ρ^(1)(,x^-)$ and $ρ^(2)(,x^+)$ behave like $δ(x^-)$ and $δ(x^+)$ respectively near the light cone [ref] .

Randomness in infinitesimal extent in the McLerran-Venugopalan model · McLerran-Venugopalan Model · 0.85
The contribution of plasma instabilities to isotropization has also been studied using purely classical field simulations [ref] without performing the Hard Loop approximation.

Time evolution of linearized gauge field fluctuations on a real-time lattice · Introduction · 0.88

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.

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P. Romatschke and R. Venugopalan, in preparation

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The eﬀect of small longitudinal ﬂuctuations on transve rse quantities should be rather small. We therefore extract ω(k⊥), for computational convenience, from a 2+1-D simulation

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P. Huovinen, arXiv:nucl-th/0305064; in Hwa, R.C. (ed.) et al.: in QGP3 (World Scientiﬁc Publishers)

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A. H. Mueller, Nucl. Phys. B 572, 227 (2000); Phys. Lett. B 475, 220 (2000); J. Bjoraker and R. Venugopalan, Phys. Rev. C 63, 024609 (2001)

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P. Arnold, J. Lenaghan, G.D. Moore, JHEP 0308, 002 (2003); P. Arnold, J. Lenaghan, G. D. Moore and L. G. Yaﬀe, Phys. Rev. Lett. 94, 072302 (2005)

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A.H. Mueller, A.I. Shoshi, S.M.H. Wong, arXiv: hep-ph/0505164; D. B¨ odeker, arXiv:hep-ph/0508223

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P. Romatschke, M. Strickland, Phys. Rev. D 68 , 036004 (2003); Phys. Rev. D 70 , 116006 (2004)

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A. Rebhan, P. Romatschke, M. Strickland, Phys. Rev. Lett 94, 102303 (2005); JHEP 0509, 041 (2005). P. Arnold, G.D. Moore, L.G. Yaﬀe, Phys.Rev. D 72, 054003 (2005); for a related approach, see A. Dumitru and Y. Nara, Phys. Lett. B 621, 89 (2005)

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P. Arnold and G. D. Moore, arXiv:hep-ph/0509206; hep-ph/0509226

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Y. V. Kovchegov, arXiv:hep-ph/0507134; hep-ph/0503038; D. Kharzeev and K. Tuchin, Nucl. Phys. A 753, 316 (2005)

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It is essential to distinguish “strict” boost invarian ce of ﬁelds from that of single particle distributions. The latte r can be preserved even if the former is violated

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This procedure is similar to that of Ref. [6]. Further de - 5 tails will be presented in Ref. [18]

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In the classical theory of the CGC, Q2s = g4µ2Nc/2π · ln(g2µ/Λ), where Λ is an infrared scale of order Λ QCD

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[24] [23]

Averaging over longitudinal and transverse modes (see the 2nd paper by Rebhan et al. in Ref. [16]), one obtains ω2pl = m23√ξ arctan√ξ and m2∞ = m22√ξ arctan√ξ, where ξ denotes the strength of the anisotropy and m2 is a soft scale proportional to the Debye mass. Therefore, ω2pl = 2 3 m2∞

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[25] [25]

The mass gap was noted in Ref. [6] in context of infrared ﬁnite number distributions. Interestingly, the behavior o f ωpl versus g2µL is similar to that of the gluon liberation factor fN

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