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Photon and Gluon Emission in Relativistic Plasmas

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Peter Arnold, Guy D. Moore, Laurence G. Yaffe

High Energy Physics - Phenomenology · Submitted 29 Apr 2002 · 43 references

Pith's verdict: The gluon half is the real contribution and it holds together, but the step from the two-variable equation (6.3) to the single-variable (6.6) is asserted in one sentence and never checked against anything.

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arXiv IDhep-ph/0204343 (version 2)
Pith IDLRWQCCFG, Pith's permanent record of this paper and its review
CitationGuy D. Moore, Laurence G. Yaffe, Peter Arnold. "Photon and Gluon Emission in Relativistic Plasmas." arXiv:hep-ph/0204343 (2002). https://pith.science/paper/LRWQCCFG
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The paper

The paper claims

The paper re-derives the LPM-suppressed photon emission rate in a hot plasma from a classical random background field, and extends the same ladder-resummation machinery to give a leading-order hard gluon emission rate.

The reading

In a plasma hotter than the QCD scale, a hard photon or gluon is emitted over a formation time comparable to the mean time between soft collisions, so the emission rate cannot be computed from any fixed order of perturbation theory — the Landau-Pomeranchuk-Migdal (LPM) effect. The authors' earlier diagrammatic derivation of the photon rate is here rebuilt from a simpler physical picture: the soft gluons of the plasma are treated as a random classical color field, the emitter propagates through it as a nearly on-shell particle, and the requirement that scattering events occur in a definite time order reduces the sum of interfering amplitudes to ladder diagrams. Those ladders collapse into a single linear integral equation for a function $f(p_⊥; p_∥, k)$, whose collision term is a difference of $f$ at shifted transverse momentum, so the infrared divergences of the width and the cross section cancel against each other. The new content is gluon emission: because the emitted gluon also carries color and also scatters, the ladder has three rails instead of two, and the collision term acquires three color-weighted difference terms with coefficients $C_R-C_A/2$, $C_A/2$, $C_A/2$. The payoff, if the derivation holds, is a leading-order gluon emission rate that correctly incorporates the dynamics of the moving plasma scatterers rather than static impurities.

What carries the argument

The load-bearing device is the relativistic one-particle Schrödinger equation for a nearly on-shell hard particle in a soft background field, $i∂_t ψ = E_p ψ$ with the interaction $i g v·A$ and $v = P/p_0$, which is spin-independent at leading order and so lets scalars, fermions and gauge bosons be handled by the same propagator. Combined with the pinching-pole approximation, which replaces the product of the two emitter propagators by $2πδ(p_0 - p_∥)θ(p_∥)/(iδE + Γ_{p+k} + Γ_p)$ with $δE = k(p_⊥^2+m_∞^2)/[2p_∥(k+p_∥)]$, the ladder sum becomes the integral equation of Eq. (3.34) for photons and Eq. (6.6b) for gluons. The second device is the rewriting of the thermal width $Γ_p$ in the same form as the collision term, Eq. (3.28), which turns the kernel into a difference $f(p_⊥) - f(p_⊥-q_⊥)$ and makes the equation manifestly infrared safe. The third is the color-singlet identity behind Eq. (6.9): because $T^a_{R1}+T^a_{R2}+T^a_{R3}$ annihilates the singlet, the gluon-exchange factor between any two rails is $½(C_{R3}-C_{R1}-C_{R2})$, which fixes the three coefficients without further diagrammatics.

Extended reading notes

Core claim

The paper's central claim is that the leading-order hard emission rate for both photons and gluons in an ultra-relativistic plasma can be reduced to a single linear integral equation in transverse momentum space, and that the gluon case is given by Eqs. (6.6a)–(6.6c): a rate formula containing the splitting weight $|J^{(s)}_{p_← p+k}|^2 k^{-3}$ and a three-rail integral equation whose collision kernel is built from the differences $F_s(h) - F_s(h - p_i q_⊥)$ weighted by $C_{R1}+C_{R2}-C_{R3}$-type color factors. The argument for the ladder structure is physical rather than diagrammatic: a wave packet of size $1/gT$ stays localized compared with the mean free path $1/g^2T$, so interactions with a given scatterer must appear consistently ordered on all rails, and only pairwise correlations of the background field survive the average. The authors state that this is the first treatment to give the full correct leading-order high-temperature gluon emission rate because it is the first to treat the scatterers' dynamics correctly, and they show that the photon result follows from the gluon equations by setting $C_A = 0$ and replacing $C_s$ with $q_s^2$.

Load-bearing premise

The random background color field is assumed to have only Gaussian two-point correlations, with higher connected correlations dropped, on the grounds that non-Gaussian and ultra-soft $g^2T$ physics do not affect the leading-order rate — a claim the paper asserts here and defers to its earlier work.

Editorial extensions

If this is right

  • The gluon emission rate becomes a computable leading-order ingredient for transport coefficients and for thermalization and jet-quenching phenomenology in a quark-gluon plasma.
  • The photon rate of the authors' earlier work is recovered as a limiting case, so the two derivations — diagrammatic and physical — agree by construction rather than by coincidence.
  • Because the collision kernel depends on the full frequency- and wavenumber-dependent HTL spectral density $\langle\langle A_+ A_+^*\rangle\rangle$ rather than on a static scattering potential, the result differs quantitatively from static-impurity LPM treatments.
  • The cancellation of the $q_⊥ \to 0$ divergence across all three rails shows that the hard emission rate is insensitive to non-perturbative ultra-soft magnetic physics at the $g^2T$ scale.

Reading between the lines

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

  • If the three-rail equation is solved numerically, the resulting gluon rate should differ measurably from static-scatterer LPM formulas at the same coupling, which gives a concrete way to discriminate the two pictures in phenomenological fits.
  • The same wave-packet localization argument should apply to any hard probe whose formation time matches the soft scattering time, which suggests the machinery transfers to e.g. heavy-quark energy loss once the emitter's spin-dependent vertices are restored.
  • The infrared-safety argument rests on the difference structure of the kernel; a truncated version of the same ladder sum, with external color currents, would retain the divergence, so the cancellation is a diagnostic that a given approximation has kept the right set of diagrams.
Original abstract

We recently derived, using diagrammatic methods, the leading-order hard photon emission rate in ultra-relativistic plasmas. This requires a correct treatment of multiple scattering effects which limit the coherence length of emitted radiation (the Landau-Pomeranchuk-Migdal effect). In this paper, we provide a more physical derivation of this result, and extend the treatment to the case of gluon radiation.

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, 2,909 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.64 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We recently derived, using diagrammatic methods, the leading-order hard photon emission rate in ultra-relativistic plasmas. This requires a correct treatment of multiple scattering effects which limit the coherence length of emitted radiation (the Landau-Pomeranchuk-Migdal effect). In this paper, we provide a more physical derivation of this result, and extend the treatment to the case of gluon radiation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We recently derived, using diagrammatic methods, the leading-order hard photon emission rate in ultra-relativistic plasmas. background 0.77 0.88 0.10 asserted 1.00
s1 This requires a correct treatment of multiple scattering effects which limit the coherence length of emitted radiation (the Landau-Pomeranchuk-Migdal effect). background 0.98 0.17 0.05 asserted 1.00
s2 In this paper, we provide a more physical derivation of this result, and extend the treatment to the case of gluon radiation. contribution 1.00 0.94 0.49 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 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
  • 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. Two-to-two particle processes contributing to the leading
Figure 2. FIG. 2. Bremsstrahlung and pair production contributions to phot
Figure 3. FIG. 3. An interference term, involving amplitudes for photon emiss
Figure 4. FIG. 4. Orders of magnitude of various momentum, distance, and a
Figure 5. FIG. 5. The perturbative bremsstrahlung and annihilation process
Figure 6. FIG. 6. Bremsstrahlung and annihilation processes including the mult
Figure 7. FIG. 7. Two diagrams, time ordered from left to right, whose interfe
Figure 8. FIG. 8. A time-ordered Z diagram
Figure 9. FIG. 9. A single diagram depicting the interference of the two diagra
Figure 10. FIG. 10. The emitter’s trajectory, thought of as approximately c
Figure 11. FIG. 11. Two amplitudes which do not interfere (at leading order) be
Figure 12. FIG. 12. Amplitudes for which the time and space ordering of interac
Figure 13. FIG. 13. The product of amplitudes, as in Fig. 9, now averaged over
Figure 14. FIG. 14. The resummation of self-energy insertions into the propa
Figure 15. FIG. 15. The leading-order contributions to the current-curren
Figure 16. FIG. 16. Two (time-ordered) diagrams whose interference contr
Figure 17. FIG. 17. Two diagrams, time ordered from left to right, whose inter
Figure 18. FIG. 18. A single diagram depicting the interference of the two gluon

Figures reproduced from arXiv:hep-ph/0204343 by the authors.

References
How later papers used it

230 later papers cite it

Evidence: some. 217 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 7 of the 334 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

Uses its method or tool 2 of 22 papers with this label
How the literature uses it

223 arXiv papers cite this in 325 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 · 171 sentences without a confident label

Builds on it 2 papers
  • This polarization can be described by extending the framework developed in [ref] , see App. [ref].

    Polarized Photons from the Early Stages of Relativistic Heavy-Ion Collisions · Polarized photon emission · 0.97

  • The generalization of ([ref]) is [ For this form, see the discussion surrounding Eq. (6.11) and footnote 24 of Ref. [ref] . ] \begin{align} Γ_3(,t) = (C_R_2 + C_R_3 - C_R_1) Γ̅_2(x_1 ,t) + (C_R_3 + C_R_1 - C_R_2) Γ̅_2(x_2 ,t) + (C_R_1 + C_R_2 - C_R_3) Γ̅_2(x_3 ,t) \end{align} for a $R_1 →R_2 R_3$ splitting process with corresponding momentum fractions \begin{equation} x_1 = 1 , x_2 = x, x_3 = 1-x . \end{equation} For $s →gs$ processes, this gives ([ref]).

    Simple Formula for High-Energy Gluon Bremsstrahlung in a Finite, Expanding Medium · Pair Production $g→qq̅$ · 0.98

Uses its method or tool 21 papers
  • The formalism developed in Ref. [ref] is employed to account consistently for collisional and radiative energy loss of hard partons in the Arnold-Moore-Yaffe (AMY) approach [ref] .

    Jet energy loss and high $p_T$ photon production in hot quark-gluon plasma · Introduction · 1.00

  • Radiative energy loss is modeled using the Arnold-Moore-Yaffe approach, where the interference of bremsstrahlung gluons from multiple scatterings is taken into account with an LPM-like integral equation for the energy loss rate [ref] .

    Dijet asymmetry at the Large Hadron Collider · Transport of high-energy partons and martini · 0.99

  • Radiative energy loss is modeled using the Arnold-Moore-Yaffe approach, where the interference of bremsstrahlung gluons from multiple scatterings is taken into account with an LPM-like integral equation for the energy loss rate [ref] .

    Monte-Carlo simulation of jets in heavy-ion collisions · Transport of high-energy partons and martini · 1.00

Compares with it 3 papers
Cites it as background 119 papers
  • For the case of approximately on-shell massless particles traveling through an infinite medium, a complete leading-order analysis of such processes was carried out by Jeon and Moore [ref] using the formalism of Arnold, Moore, and Yaffe (AMY) [ref] .

    QCD Splitting/Joining Functions at Finite Temperature in the Deep LPM Regime · Introduction and Results · 0.81

  • This is often referred to as the scattering stage or the Baier-Dokshitzer-Mueller-Peigne-Schiff (BDMPS) stage [ref] or the Arnold-Moore-Yaffe (AMY) stage [ref] .

    The medium modification of high-virtuality partons · Introduction · 0.97

  • Formalisms based on Baier-Dokshitzer-Mueller-Peigne-Schiff- (BDMPS) [ref] , including Zakharov's [ref] contribution (BDMPS-Z), or the Arnold-Moore-Yaffe (AMY) [ref] approach, the Djordjevic-Gyulassy-Levai-Vitev [ref] prescription, as well as those inspired from higher twist [ref] have all be used in the past.

    Multi-scale evolution of charmed particles in a nuclear medium · Introduction · 0.93

Discussion

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