Collective Modes of an Anisotropic Quark-Gluon Plasma

Michael Strickland; Paul Romatschke

REVIEW

Collective Modes of an Anisotropic Quark-Gluon Plasma

Paul Romatschke, Michael Strickland

High Energy Physics - Phenomenology · Submitted 9 Apr 2003 · 11 references

Pith's verdict: The tensor decomposition and structure functions are the durable contribution; the unstable-mode count rests on an undocumented numerical scan that a referee should ask about.

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arXiv IDhep-ph/0304092 (version 2)

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CitationMichael Strickland, Paul Romatschke. "Collective Modes of an Anisotropic Quark-Gluon Plasma." arXiv:hep-ph/0304092 (2003). https://pith.science/paper/YC4XUGPB

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Romatschke and Strickland do the bookkeeping the earlier instability papers skipped: a full tensor basis for the anisotropic gluon self-energy, four structure functions as explicit angular integrals, and the resulting mode census. That is the contribution and it is real. The isotropic limit falls back on the HTL Π_T and Π_L (Eqs. 28–30), which is the check you want, and the static factorization in Eqs. 37–41 is clean. The small-ξ expansions in Sec. VII are the part I would actually use in my own work; they turn a numerical problem into algebra for anyone near isotropy. The chain from the Vlasov equations through the induced current to the self-energy is standard and correctly assembled, and nothing is fitted to a target — the dispersion relations come from zeros of the inverse propagator, so the circularity worry does not apply here. The soft spot is the unstable-mode count. The stable side is analytic: the discriminant in Eq. 43 is manifestly non-negative for real ω > k, so three stable modes is a theorem. The unstable side is not. The claim that Δ_G contributes at most one root rests on “numerically we find that Ω²_+ > 0 for all Γ > 0” (Sec. VI.B), and footnote 2's “we have checked for poles at complex ω numerically but found none” is the same kind of statement. No grid, no stated range of ξ, θ_n, or k, and the figures show only ξ = 10 and ξ = −0.9. The integrands carry (1 + ξ cos²θ)^{-2} weights that grow at both ends of ξ, which is exactly where the scan is thinnest. If Ω²_+ dips negative somewhere, “one or two” becomes “one, two, or three.” I do not think this threatens the qualitative point — instabilities exist and grow — but the counting headline is only as good as the scan behind it, and the paper should say what was scanned. Two smaller things. Eq. 22's one-parameter family is scoped in the abstract, so I will not call it a hidden assumption, but the mode count is a property of that family and the text never says so out loud. The thermalization sentence is speculation; Sec. VIII concedes as much, so it is honest hedging rather than a claim. Bibliographically, ref 5 is an unpublished preprint carrying the phenomenological motivation, and ref 8's volume and year do not line up. Minor. Worth a referee. The tensor machinery is the durable part and people will cite it.

Referee report

Significance

If the census holds, this is a useful and reusable piece of formalism: the tensor basis and inversion formula, the four structure functions with the azimuthal integral performed in closed form, the static mass scales and their factorization into m_+ and m_-, and the small-xi expansions are all results later work on anisotropic plasmas can build on directly. The paper also contains a genuine internal consistency check that is easy to verify: the xi -> 0 limit of alpha and beta reproduces the standard isotropic hard-thermal-loop Pi_T and Pi_L (Eqs. 28-30), and the corresponding limit of the static masses gives m_alpha -> 0, m_beta -> m_D^2, m_+ -> m_D. The identification of an unstable branch for both signs of xi, with the count depending on the sign, is a falsifiable statement about the spectrum of the stated distribution family. The paper is honest about scope: it is a linear-response, leading-order-in-g calculation with no collision term, and Sec. VIII states plainly that the linear approximation will break down quickly and that no phenomenological growth rate is extracted. No code or numerical data are provided.

Major objections (2)

[Sec. VI.B and footnote 2] The unstable-mode count is the headline result, and half of it rests on two numerical assertions that are stated without any supporting detail: 'numerically we find that Omega^2_+ > 0 for all Gamma > 0' (Sec. VI.B) and 'We have checked for poles at complex omega numerically but found none' (footnote 2). The stable-mode count, by contrast, is analytic: the discriminant in Eq. (43) equals (alpha - beta + gamma + k^2)^2 + 4 k^2 ntilde^2 delta^2, which is non-negative for real omega > k, so at most two stable modes follow from Delta_G without any scanning. The unstable count does not have that protection. A second root of Delta_G on the imaginary axis exists precisely when Omega^2_+(i Gamma) dips negative for some Gamma > 0, and the paper excludes this only by assertion. Please state the ranges of xi, theta_n, and k that were scanned, the grid resolution in Gamma, and whether the scan approached the two singular corners of the parameter space, xi -> -1 and large positive xi, where the weights (1 + xi cos^2 theta)^{-2} in Eqs. (B.1)-(B.4) become large and the sampling is presumably coarsest. If an analytic argument is available, it would be stronger than a scan: for instance, whether beta(i Gamma) > 0 (asserted in Sec. VII.B) together with the sign structure of alpha + gamma + k^2 at omega = i Gamma suffices to keep Omega^2_+ positive. As written, the 'one or two' count is only as exhaustive as an undocumented scan, and the abstract sentence [abstract:s3] states it without qualification.
[Abstract [abstract:s4] and Sec. VIII] The abstract's closing claim, that the unstable modes 'can lead to a more rapid thermalization and isotropization of the soft modes ... and therefore may play an important role in the dynamical evolution', is a statement about dynamics that nothing in the paper demonstrates. What is shown is that exponentially growing linear modes exist for the family in Eq. (22); whether they isotropize the plasma depends on their saturation, on the collision term that is absent from Eqs. (2)-(3), and on back-reaction, none of which is treated. The authors acknowledge exactly this in Sec. VIII, where they note that the linear approximation 'will break down very quickly' and that they 'have made no attempt to discuss the phenomenological rate for instability growth'. The acknowledgment is appropriate and I do not ask for the dynamical calculation to be added; the request is that the abstract carry the same qualification, for example by phrasing the final sentence as a possibility contingent on the nonlinear evolution rather than as an expected consequence.

Minor objections (11)

[Fig. 3 caption] The caption refers to 'm_g = m_D/sqrt(3)', but m_g is not defined anywhere in the text or in Appendix A; presumably a gluon mass introduced only for the plot. Please define it or remove it.
[Eq. (20)] The inversion formula as typeset is ambiguous: the final term reads '- dD b(a+c) - ntilde^2 k^2 d^2', and it is not clear whether the denominator is b(a+c) - ntilde^2 k^2 d^2 or whether the whole expression is a single fraction. Since Eq. (32) and everything downstream depend on this formula, please display it unambiguously.
[Eq. (36) and footnote 1] The definition m_delta^2 = lim_{omega -> 0} ntilde k^2 Im(delta)/omega deserves a word on its dimensions and on why the imaginary part is the relevant quantity, and footnote 1's remark about 'a suitable redeclaration of m_delta^2' when approaching along the imaginary axis should be made explicit, since the static factorization in Eqs. (38)-(40) relies on it.
[Sec. VI.A] The statement that there are three stable modes should be qualified for the isotropic limit: at xi = 0 the roots of Eqs. (45) and (46) coincide, omega_alpha = omega_+ = omega_T, so the count of three reduces to the two distinct transverse and longitudinal hard-thermal-loop modes. A sentence noting this degeneracy would prevent the 'three stable modes' phrasing from being read as a discontinuity at xi = 0.
[Eqs. (B.7)-(B.8)] As a check on the closed-form static masses, please state their xi -> 0 limits explicitly. Eq. (B.8) should reduce to m_beta^2 -> m_D^2 and Eq. (B.7) to m_alpha^2 -> 0, but both expressions are of the form 0/0 at xi = 0 and the limits are not transparent as written.
[Eqs. (7), (9), (21)] The notation partial_beta(p) f(p) for the momentum derivative is easy to misread as a spacetime derivative; writing it as partial f / partial p^beta, or defining it once in Appendix A, would help.
[Eqs. (43) and (47)] The symbols Omega_pm are used for two different objects, evaluated at real omega > k in Eq. (43) and at omega = i Gamma in Eq. (47). A distinct notation for the imaginary-axis quantities would make the relation between the two factorizations easier to follow.
[Abstract [abstract:s3]] The abstract says 'either one or two unstable modes depending on whether the distribution function is stretched or squeezed' without saying which sign of xi gives which count. Since the body of the paper is explicit (two for xi > 0, one for xi < 0), the abstract could be equally explicit at no cost in length.
[PACS numbers, title page] PACS 04.25.Nx appears to belong to gravitational physics rather than to this subject; please check the PACS assignments.
[Sec. VI.B, Figs. 4a and 4b] The growth rates are shown for one angle each (theta_n = pi/8 and theta_n = pi/2). Since the phenomenologically relevant quantity is the maximum growth rate and the wavenumber at which it occurs, a short statement of Gamma_max as a function of theta_n and xi, even in a table, would make the results considerably more usable than the two sample curves.
[Sec. IV, Eq. (22)] Two typographical slips in this section: 'a adjustable anisotropy parameter' and 'we can chose to run'. Also, the statement that xi > 0 corresponds to a contraction along n is worth one clause of justification, since the sign convention is easy to invert when comparing with other work.

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The paper

The paper claims

An anisotropic quark-gluon plasma supports exponentially growing collective modes whose number depends on whether the gluon distribution is squeezed or stretched along one direction.

The reading

The paper asks what happens to the collective gluon modes of a hot quark-gluon plasma when the gluon momentum distribution is not isotropic, as it would be in the first moments of a heavy-ion collision. The authors take a one-parameter family of distributions obtained by rescaling an isotropic distribution along a single preferred direction, $f(p) = f_{iso}(p^2 + $ξ$(p·n)^2)$, and compute the hard-thermal-loop gluon self-energy for it. Because the anisotropy breaks rotational symmetry, the self-energy needs four structure functions instead of the usual two, and the authors derive all four, invert the propagator in a matching tensor basis, and solve the dispersion relations in both the stable and the unstable regions. The payoff is a counting statement: three stable modes for either sign of the anisotropy, plus two unstable modes when the distribution is squeezed and only one when it is stretched. If correct, this says the soft gluon field in an anisotropic plasma has channels that grow exponentially rather than oscillate, which would let the system redistribute momentum and approach isotropy faster than collisions alone would allow.

What carries the argument

The load-bearing device is a tensor decomposition adapted to a single preferred direction. The self-energy is symmetric and transverse, so its spatial part can be expanded in four tensors: $A_{ij} = δ_{ij} - k_i k_j/k^2$, $B_{ij} = k_i k_j/k^2$, $C_{ij} = ñ_i ñ_j/ñ^2$ with $ñ = A·n$, and $D_{ij} = k_i ñ_j + k_j ñ_i$, with coefficients $α$, $β$, $γ$, $δ$ fixed by four contractions of $Π_{ij}$. The same basis gives a closed inversion formula for the propagator, which is what turns the structure-function integrals into a solvable dispersion problem. The second device is the change of variables $p̃^2 = p^2(1 + $ξ$(v·n)^2)$, which lets the momentum magnitude be integrated out and leaves angular integrals over the direction of the parton velocity, so that the entire anisotropy dependence sits in those angles.

Extended reading notes

Core claim

For the rescaled distribution $f(p) = f_{iso}(p^2 + $ξ$(p·n)^2)$ with $ξ > -1$, the authors decompose the gluon self-energy on a four-tensor basis built from the momentum $k$, the anisotropy vector $n$, and its projection $ñ$ perpendicular to $k$, obtaining structure functions $α$, $β$, $γ$, $δ$ that reduce to the standard transverse and longitudinal hard-thermal-loop self-energies as $ξ → 0$. Inverting the propagator factorizes it into a piece $∆_A$ and a piece $∆_G$ that can be written as a product over two mode branches. Solving the resulting dispersion relations, they find three stable quasiparticle modes whose dispersion depends on the angle between $k$ and $n$, plus poles on the imaginary frequency axis, $ω = iΓ$: for $ξ > 0$ (a squeezing of the distribution along $n$) there are two growing modes, one from $∆_A$ and one from $∆_G$, while for $ξ < 0$ (a stretching) only the $∆_G$ mode survives because $β(iΓ) > 0$ for all $Γ > 0$. The static limit makes the mechanism visible: the screening mass $m_α^2$ goes negative for $ξ > 0$ at any angle away from $π/2$, so the transverse response is not screening at all but unstable.

Load-bearing premise

Everything rests on the one-parameter family $f(p) = f_{iso}(p^2 + $ξ$(p·n)^2)$ being a fair stand-in for the real anisotropy, and on the collisionless Vlasov limit used throughout, so the stable-versus-unstable mode count is a property of that family rather than of anisotropic plasmas in general.

Editorial extensions

If this is right

The soft gluon sector of an anisotropic plasma has modes with real exponential growth rates $Γ(k)$ rather than only oscillatory collective modes, so linear response to a coloured perturbation is not bounded.
The count of unstable branches is tied to the sign of the anisotropy: squeezing the distribution produces two, stretching produces one, and the unstable branch from $∆_A$ disappears entirely for $ξ < 0$.
Because the growth is exponential, the linearised treatment used to derive the self-energy must break down quickly, so the late-time behaviour of these modes cannot be read off from the dispersion relations alone.
Small-$ξ$ expansions of all four structure functions and of the static masses $m_±^2$ are given, giving a controlled entry point for treating weak anisotropy without the full angular integrals.

Reading between the lines

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

The instability is a QCD analogue of the Weibel/filamentation instability, so the natural next test is whether the growth rates here reproduce the electrodynamic Weibel rate in the appropriate limit; the paper does not perform that comparison explicitly.
Since the unstable modes are the fastest thing in the linearised system, a kinetic simulation that seeds a small anisotropy and follows the fields past the linear regime would show whether the growth actually drives the distribution toward isotropy or merely saturates into a different anisotropic state.
The same tensor-basis machinery should apply to any anisotropic distribution, not just the rescaled family, which suggests the one-versus-two count could be replaced by a classification in terms of the sign structure of the static masses.
If the growth rates are comparable to elastic scattering rates, as the cited phenomenological estimate suggests, then any hydrodynamic description of the early plasma that assumes local isotropy is being applied outside its domain of validity.

Original abstract

We analyze the collective modes of high-temperature QCD in the case when there is an anisotropy in the momentum-space distribution function for the gluons. We perform a tensor decomposition of the gluon self-energy and solve the dispersion relations for both stable and unstable modes. Results are presented for a class of anisotropic distribution functions which can be obtained by stretching or squeezing an isotropic distribution function along one direction in momentum space. We find that there are three stable modes and either one or two unstable modes depending on whether the distribution function is stretched or squeezed. The presence of unstable modes which have exponential growth can lead to a more rapid thermalization and isotropization of the soft modes in a quark gluon plasma and therefore may play an important role in the dynamical evolution.

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We analyze the collective modes of high-temperature QCD in the case when there is an anisotropy in the momentum-space distribution function for the gluons. We perform a tensor decomposition of the gluon self-energy and solve the dispersion relations for both stable and unstable modes. Results are presented for a class of anisotropic distribution functions which can be obtained by stretching or squeezing an isotropic distribution function along one direction in momentum space. We find that there are three stable modes and either one or two unstable modes depending on whether the distribution function is stretched or squeezed. The presence of unstable modes which have exponential growth can lead to a more rapid thermalization and isotropization of the soft modes in a quark gluon plasma and therefore may play an important role in the dynamical evolution.

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 analyze the collective modes of high-temperature QCD in the case when there is an anisotropy in the momentum-space distribution function for the gluons.	contribution 0.91	0.84	0.04	asserted 0.78
s1	We perform a tensor decomposition of the gluon self-energy and solve the dispersion relations for both stable and unstable modes.	method 0.98	0.97	0.04	asserted 0.99
s2	Results are presented for a class of anisotropic distribution functions which can be obtained by stretching or squeezing an isotropic distribution function along one direction in momentum space.	method 0.73	0.48	0.04	asserted 0.95
s3	We find that there are three stable modes and either one or two unstable modes depending on whether the distribution function is stretched or squeezed.	result 1.00	0.97	0.59	asserted 1.00
s4	The presence of unstable modes which have exponential growth can lead to a more rapid thermalization and isotropization of the soft modes in a quark gluon plasma and therefore may play an important role in the dynamical evolution.	interpretation 1.00	0.41	0.55	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Figures

**Figure 1.** FIG. 1: Real and imaginary parts of

**Figure 2.** FIG. 2: Angular dependence of

**Figure 3.** FIG. 3: Angular dependence of

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References

Works cited (11)

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E. S. Weibel, Phys. Rev. Lett. 2, 83 (1959)

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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-...
Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow
hep-th 2017-11 conditional novelty 5.0 of 10

Anisotropic hydrodynamics reproduces the exact kinetic-theory attractor of a Gubser-expanding plasma to within about 0.06% at late times, while Israel-Stewart and DNMR second-order hydrodynamics do not.
New theories of relativistic hydrodynamics in the LHC era
hep-ph 2017-07 accept novelty 5.0 of 10

The hydrodynamic gradient expansion is an asymptotic series whose large-order factorial growth encodes the transient, nonhydrodynamic modes of the underlying plasma.
Bulk viscous evolution within anisotropic hydrodynamics
hep-ph 2014-05 conditional novelty 5.0 of 10

Adding an explicit bulk-pressure degree of freedom to anisotropic hydrodynamics improves agreement with the exact Boltzmann solution for the bulk pressure of a massive gas, at a small cost to the pressure anisotropy.
Probing strongly coupled anisotropic plasma
hep-th 2012-02 conditional novelty 5.0 of 10

In a holographic anisotropic N=4 plasma, jet quenching and drag force are enhanced along the anisotropy direction, with the enhancement ordered by direction and growing with the anisotropy parameter.
Thermal Bottomonium Suppression at RHIC and LHC
nucl-th 2011-12 conditional novelty 5.0 of 10

Bottomonium suppression data from RHIC and LHC are consistent with a shear-viscosity-to-entropy ratio between 0.08 and 0.24, provided the heavy-quark potential is taken from the internal energy rather than the free energy.
Thermal Upsilon(1s) and chi_b1 suppression in sqrt(s_NN)=2.76 TeV Pb-Pb collisions at the LHC
hep-ph 2011-06 conditional novelty 5.0 of 10

Thermal suppression of Upsilon(1s) and chi_b1 in 2.76 TeV Pb-Pb collisions, computed with anisotropic hydrodynamics and a complex heavy-quark potential, reproduces preliminary CMS Upsilon(1s) R_AA across centrality an...
Quarkonium states in a complex-valued potential
hep-ph 2011-01 conditional novelty 5.0 of 10

Including the imaginary part of the heavy-quark potential raises the predicted quarkonium disassociation temperatures and produces a roughly 60 MeV anisotropy-induced splitting of the chi_b p-wave states.
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.
The anisotropic non-equilibrium hydrodynamic attractor
nucl-th 2017-09 conditional novelty 4.0 of 10

The aHydro attractor for a 0+1d conformal RTA plasma matches the exact Boltzmann attractor to 0.04 percent, which the authors attribute to an all-orders resummation in inverse Reynolds number.
Anisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions
nucl-th 2017-05 conditional novelty 4.0 of 10

Quasiparticle anisotropic hydrodynamics with three momentum-anisotropy parameters and anisotropic freeze-out describes ALICE Pb-Pb spectra, elliptic flow, and HBT radii using three fitted parameters.
Do nuclear collisions create a locally equilibrated quark-gluon plasma?
nucl-th 2016-09 conditional novelty 4.0 of 10

Hydrodynamic behavior in nuclear collisions requires neither local thermal equilibrium nor local isotropy, only that non-hydrodynamic modes be negligible; the breakdown scale is estimated at roughly seven times the te...
Exact solution of the (0+1)-dimensional Boltzmann equation for a massive gas
hep-ph 2014-02 conditional novelty 4.0 of 10

An exact integral-equation solution of the relaxation-time Boltzmann equation for a boost-invariant massive gas confirms standard shear and bulk viscosity formulas at late times but shows that second-order hydrodynami...
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...
Hydrodynamic attractors in heavy ion collisions: a review
hep-th 2021-09 accept novelty 2.0 of 10

Boost-invariant expanding plasmas in hydrodynamics, holography and kinetic theory all converge onto a common attractor solution that erases memory of initial conditions before thermalization, and this is the review's ...
Relativistic hydrodynamics in heavy-ion collisions: general aspects and recent developments
nucl-th 2016-05 accept novelty 2.0 of 10

Relativistic hydrodynamics with a small shear viscosity to entropy ratio quantitatively reproduces the collective flow observables of the quark-gluon plasma at RHIC and LHC, and the same framework is being extended to...
Anisotropic Hydrodynamics: Three lectures
nucl-th 2014-10 conditional novelty 2.0 of 10

Anisotropic hydrodynamics extends relativistic fluid modeling to far-from-equilibrium heavy-ion systems by promoting momentum-space anisotropy to a leading-order dynamical variable, and it outperforms standard second-...
Quarkonia in the Quark Gluon Plasma
hep-ph 2013-02 conditional novelty 2.0 of 10

Quarkonium melting in the quark-gluon plasma is governed by a complex-valued heavy-quark potential whose imaginary part produces in-medium widths of 20 to 100 MeV, blurring the sharp dissociation temperatures of the o...
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

123 later papers cite it

Evidence: substantial. 86 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 233 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: 1 used its method or tool, 1 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 2 of 32 papers with this label · 1 by the same authors

Assumed its anisotropic distribution to have the Romatschke-Strickland form of this paper, with time-dependent anisotropy and momentum-scale parameters.

On nonadditive anisotropic relativistic hydrodynamics (2021) · 0.81 · paraphrased: the licence does not allow quoting

By the same authors

Assumed that all its distribution functions are of this paper's Romatschke-Strickland form.

Anisotropic hydrodynamics with a scalar collisional kernel (2018) (same authors) · 0.83 · paraphrased: the licence does not allow quoting

Uses its data or values 1 paper

Quoted this paper's masses in the small-anisotropy limit scaled with respect to the Debye mass to sum the singular parts of the static gluon propagator.

Jet quenching and broadening: the transport coefficient $\hat{q}$ in an anisotropic plasma (2008) · 0.90 · paraphrased: the licence does not allow quoting

Compares with it 1 of 4 papers with this label

It should be mentioned here that in this case the growth rate of the unstable mode has amplitudes similar to the spheroidal case [ref] whereas, with positive anisotropy parameters, a several times larger growth rate can be observed for certain angular values as reported in Ref. [ref] .

Covariant formulation of gluon self-energy in presence of ellipsoidal anisotropy (2020) · Results · 1.00

How the literature uses it

111 arXiv papers cite this in 212 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

It is shown that systems with anisotropic momentum-space distributions contain unstable modes in addition to the stable quasiparticle modes, which may be of great importance for the dynamical evolution of an incompletely thermalized quark-gluon plasma.

Quasiparticle description of the hot and dense quark-gluon plasma · 0.91

Papers citing it · 133 sentences without a confident label

Contests it 1 paper

In this case, however, the Romatschke–Strickland distribution is not conformal [ref] :

Far-from-equilibrium hydrodynamic simulations of ultrarelativistic nuclear collisions · Anisotropic distribution function · contest 0.74

Builds on it 3 papers

In both cases, we generalize the Romatschke-Strickland form [ref] to include a dynamical chemical potential.

Anisotropic hydrodynamics with number-conserving kernels · Introduction · 0.91
Here, we extend the tensor basis used in [ref] to a four-tensor basis appropriate for use in general covariant gauges.

The heavy-quark potential in an anisotropic plasma · Hard-Thermal-Loop self-energy in an anisotropic plasma · 1.00
We further apply the same method to also modify a Romatschke–Strickland anisotropic distribution [ref] at leading order (i.e. with only nonzero diagonal elements in the momentum deformation matrix $A_ij$) in such a way that it can approximately capture the smaller off-diagonal components of the energy-momentum tensor.

Modified equilibrium distributions for Cooper--Frye particlization · Introduction · 0.89

Uses its method or tool 31 papers

In this paper we assume that, when written in terms of LRF momentum, the one-particle distribution function is of spheroidal (Romatschke-Strickland) form [ref] ,

Photon production from a non-equilibrium quark-gluon plasma · Photon rate in anisotropic plasma · 0.85
The resulting dynamical equations describe the full spatiotemporal evolution of the transverse temperature $Λ$ and spheroidal momentum-space anisotropy parameter $ξ$ [ref] .

Dilepton production from the quark-gluon plasma using (3+1)-dimensional anisotropic dissipative hydrodynamics · Introduction · 0.88
In order to take into account the largest corrections to ideal hydrodynamic behavior which arise from the rapid longitudinal expansion of the plasma at early times, the one-particle distribution function for the quarks and anti-quarks can be described at leading order by the following spheroidal form [ref]

Photon and dilepton production from a non-equilibrium quark-gluon plasma · Dilepton Rate · 0.87

Uses its data or values 1 paper

A short way to see the anomalous contribution is to sum the singular parts in the gluon propagator for the static case in terms of the masses, here explicitly quoted from [ref] in the limit of small anisotropy $ξ→0$ scaled with respect to the Debye mass Eq. ([ref]), i.e. $m̂ = m/m_D$,

Jet quenching and broadening: the transport coefficient $\hat{q}$ in an anisotropic plasma · anisotropic case · 0.90

Compares with it 4 papers

It should be mentioned here that in this case the growth rate of the unstable mode has amplitudes similar to the spheroidal case [ref] whereas, with positive anisotropy parameters, a several times larger growth rate can be observed for certain angular values as reported in Ref. [ref] .

Covariant formulation of gluon self-energy in presence of ellipsoidal anisotropy · Results · 1.00
As compared to the tensor basis used in Refs. [ref] , the only difference is that the structure functions for $ñ^i ^j$ and $^i ñ^j$ are not identical any more.

Collective modes of a collisional anisotropic quark-gluon plasma · The gluon self-energy from kinetic theory · 0.87
The first two agree with the isotropic results given by Eqs. (20) and (19) of Ref. [ref] for $Π^T_F$ and $Π^L_F$, respectively, upon using $β_F = (ω^2/p^2) Π^L_F$ [ref] and $m^2_D$ is replaced by its QED definition, i.e. Eq. ([ref]).

The static hard-loop gluon propagator to all orders in anisotropy · Feynman gluon self-energy · 1.00

Cites it as background 27 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
Within the original formulation of anisotropic hydrodynamics [ref] , one assumes that the non-equilibrium distribution function $f$ is well approximated by the spheroidal Romatschke-Strickland form [ref] ,

Gradient expansion for anisotropic hydrodynamics · Anisotropic distribution functions · 0.89
Anisotropic hydrodynamics considers an expansion around a distribution function which is anisotropic in momentum-space [For a system where the chemical potential vanishes exactly, this distribution function is assumed to have the Romatschke-Strickland form [ref] which in the local rest frame looks like $f (x^μ,p_i)=f(√(p_T^2+(1+ξ)p_z^2),Λ)$ where $Λ$ is a particular momentum scale which is identified with the temperature by matching the energy density [ref] .].

Viscous anisotropic hydrodynamics for the Gubser flow · Introduction · 0.87

Discussion

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