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Photon Emission from Quark-Gluon Plasma: Complete Leading Order Results

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

High Energy Physics - Phenomenology · Submitted 9 Nov 2001 · 32 references

Pith's verdict: The standard leading-order QGP photon rate, done properly: LPM resummation solved numerically with controlled convergence, and the earlier 2-to-2 result recovered as a limit rather than replaced.

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arXiv IDhep-ph/0111107 (version 1)
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CitationGuy D. Moore, Laurence G. Yaffe, Peter Arnold. "Photon Emission from Quark-Gluon Plasma: Complete Leading Order Results." arXiv:hep-ph/0111107 (2001). https://pith.science/paper/JQRJL4S7
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The paper

The paper claims

The photon emission rate from a hot quark-gluon plasma is now complete at leading order in the strong coupling, once multiple soft scatterings during emission are accounted for.

The reading

A thermalized plasma of quarks and gluons should glow, and the brightness of that glow is a measurable prediction of quantum chromodynamics at weak coupling. Earlier calculations captured only the two-to-two processes — Compton scattering and quark-antiquark annihilation — and missed the near-collinear bremsstrahlung and inelastic pair annihilation channels, which contribute at the same order $O(alpha_{EM} alpha_s)$ because their rates are enhanced by collinear singularities. Those channels are in turn suppressed by the Landau-Pomeranchuk-Migdal effect: a photon cannot be emitted coherently over a distance longer than the time between soft scatterings of the emitting quark. The paper assembles all of these pieces into a single closed expression, Eq. (1.7), $nu_e(k) = A(k)[ln(T/m_infty) + C_{tot}(k/T)]$, with the constant under the log computed numerically for arbitrary photon energy. The payoff is a rate that can be quoted without a leftover logarithm of the coupling, plus compact fitting formulas that make the result usable in heavy-ion phenomenology.

What carries the argument

The load-bearing object is an integral equation, Eq. (2.2), for a transverse-momentum-dependent function $f(p_perp, p_parallel, k)$ whose real part is proportional to the squared amplitude for a quark of momentum $p+k$ to emit a photon of momentum $k$. It reads $2 p_perp = i delta E(p_perp) f(p_perp) + int d^2 q_perp/(2 pi)^2 C(q_perp)[f(p_perp) - f(p_perp + q_perp)]$, where $delta E$ is the on-shell energy mismatch between the emitting and post-emission states and $C(q_perp)$ is a collision kernel built from the longitudinal and transverse hard-thermal-loop gluon self-energies with dynamical screening. The imaginary term encodes the formation time of the photon and is the source of LPM suppression; the collision term encodes how soft scatterings destroy coherence between emission at successive scattering events. This is the Migdal equation of 1950s electromagnetic energy-loss theory, rederived here for a genuine thermal plasma of dynamical colored charges rather than a model of static scattering centers. Because the kernel is rotationally invariant, $f = p_perp chi(p_perp^2)$, and the equation reduces to a scalar problem. The authors solve it by a variational ansatz — the functional $Q$ of Eq. (4.13) is stationary at the solution and its extremal value is exactly half the quantity appearing in the rate — with basis functions of Eqs. (4.20)-(4.21) and a scale $A$ chosen adaptively; the extremum is a saddle, so no bound is guaranteed, but the basis is flexible enough that raising the number of trial functions from 10-16 to 6-10 shifts the answer by under 50 parts per million.

Extended reading notes

Core claim

The paper claims the first complete leading-order photon emission rate for an equilibrated, zero-chemical-potential quark-gluon plasma. The answer is written as $nu_e(k) = A(k)[ln(T/m_infty) + C_{tot}(k/T)]$, where $A(k) = 2 alpha_{EM} [d_F sum_s q_s^2] m_infty^2 n_f(k)/k$ is the leading-log coefficient of the older literature and $C_{tot}$ is a coupling-independent function of $k/T$ built from three pieces: the full non-logarithmic two-to-two contribution $C_{2 leftrightarrow 2}(k/T)$, evaluated for general $k/T$ rather than only in the $k gg T$ limit where it was previously known, and the near-collinear bremsstrahlung and inelastic pair annihilation contributions $C_{brem}$ and $C_{annih}$, computed with LPM suppression included. The authors report that the older two-to-two-only results are within a factor of two of the correct rate only for $2.5 < k/T < 10$ at $alpha_s = 0.2$, that bremsstrahlung dominates below that window, and that inelastic pair annihilation dominates well above it. They also report a change in the parametric shape of the spectrum at both ends: $dGamma_gamma propto k^{-1/2} dk$ for soft photons instead of $k^{-1} dk$, and $k^{3/2} e^{-k/T} dk$ instead of $k^2 e^{-k/T} dk$ for very hard ones.

Load-bearing premise

The word complete in the title rests on the claim, argued in the authors' earlier paper, that no diagram class other than the summed ladders and the two-to-two processes contributes at order $alpha_{EM} alpha_s$, together with the near-collinear kinematics used to collapse those ladders into a single integral equation.

Editorial extensions

If this is right

  • The photon spectrum of a weakly coupled quark-gluon plasma can be quoted without an uncalculated constant under the logarithm, so the leading-order prediction is now a definite number at each photon energy rather than a leading-logarithmic shape.
  • For $alpha_s = 0.2$, bremsstrahlung dominates the rate for photons softer than about $2T$, two-to-two processes dominate the easily detectable window around $4T$ and above, and inelastic pair annihilation takes over only for $k gg 10T$, where the exponential population factor makes the yield negligible.
  • The LPM suppression is modest — 35 percent or less — over most of the range, but is significant for bremsstrahlung below $2T$ and for very hard inelastic pair annihilation, so the effect cannot be folded into a single overall factor.
  • The low-energy spectrum is less infrared singular than previously reported: $dGamma_gamma propto k^{-1/2} dk$ rather than $k^{-1} dk$, which changes the predicted soft-photon yield from a plasma.
  • The companion QED results, including the photon dispersion correction $m_gamma^2/(2k)$ in $delta E$, give the emissivity of an electromagnetic plasma on the same footing.

Reading between the lines

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

  • Because the suppression depends on the Debye mass through the collision kernel, the inelastic contribution relative to two-to-two grows roughly as the square root of the number of species; a plasma with more flavors or with additional light colored degrees of freedom should therefore be a proportionally brighter soft-photon source.
  • The same integral-equation machinery, with the photon replaced by a gluon and the appropriate color factors, is the natural route to the leading-order gluon emission rate, which would let the photon channel be cross-checked against the plasma's total radiative energy loss.
  • A lattice or resummed weak-coupling calculation of the photon rate at small but nonzero $g_s$ would test the completeness claim directly: agreement in the coefficient of $ln(1/g_s)$ alone would not, since that coefficient was already known, but agreement in the constant would.
  • The fit of Eq. (1.10) is stated to hold for $0.2 le k/T le 50$ and $N_f$ from 2 to 6; anyone using it outside that window, particularly for $k/T$ below $0.2$ where the photon thermal mass cuts off the calculation, should expect it to fail rather than extrapolate.
Original abstract

We compute the photon emission rate of an equilibrated, hot QCD plasma at zero chemical potential, to leading order in both alpha_{EM} and the QCD coupling g_s(T). This requires inclusion of near-collinear bremsstrahlung and inelastic pair annihilation contributions, and correct incorporation of Landau-Pomeranchuk-Migdal suppression effects for these processes. Analogous results for a QED plasma are also included.

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, 2,947 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We compute the photon emission rate of an equilibrated, hot QCD plasma at zero chemical potential, to leading order in both alpha_{EM} and the QCD coupling g_s(T). This requires inclusion of near-collinear bremsstrahlung and inelastic pair annihilation contributions, and correct incorporation of Landau-Pomeranchuk-Migdal suppression effects for these processes. Analogous results for a QED plasma are also included.

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 compute the photon emission rate of an equilibrated, hot QCD plasma at zero chemical potential, to leading order in both alpha_{EM} and the QCD coupling g_s(T). contribution 0.98 0.96 0.39 asserted 1.00
s1 This requires inclusion of near-collinear bremsstrahlung and inelastic pair annihilation contributions, and correct incorporation of Landau-Pomeranchuk-Migdal suppression effects for these processes. method 0.94 0.63 0.09 asserted 1.00
s2 Analogous results for a QED plasma are also included. contribution 0.87 0.77 0.04 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.77.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
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  • 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 which generate the leading
Figure 2. FIG. 2. Inelastic processes which, because of near-collinear singu
Figure 3. FIG. 3. Typical ladder diagram contributing to the electromagnetic
Figure 4. FIG. 4. Cut ladder diagram with
Figure 5. FIG. 5. Contribution
Figure 6. FIG. 6. Left panel: Contributions
Figure 7. FIG. 7. Relative importance of the LPM effect. The figure shows the
Figure 8. FIG. 8. Left panel: Total constant under the log, equal to the su
Figure 9. FIG. 9. Total photon emission rate, together with the bremsstra

Figures reproduced from arXiv:hep-ph/0111107 by the authors.

References
How later papers used it

136 later papers cite it

Evidence: substantial. 132 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 230 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

Compares with it 3 of 8 papers with this label
How the literature uses it

135 arXiv papers cite this in 229 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 · 135 sentences without a confident label

Uses its method or tool 18 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

Uses its data or values 2 papers
Compares with it 7 papers
  • We compare our results with the moments obtained by integrating the full leading-order weak-coupling spectral function derived by Arnold, Moore and Yaffe (AMY) from kinetic theory [ref] .

    Probing how bright the quark-gluon plasma glows in lattice QCD · Introduction · 1.00

  • In Fig. [ref] we compare our hadronic rates with the complete leading order quark-gluon plasma rates [ref] for a broad range of photon energies at the highest hadronic temperature $T=190$ MeV and for a substantial coupling $α_s=3/4$.

    Dileptons and Photons at RHIC · Hadronic Photon Rates · 1.00

  • We observe that for the third momentum group, containing momenta above 1.0 GeV, the values of $D_eff(k)·GeV$ cover the interval $[0, 0.7]$ and are thus compatible both with the leading-order weak-coupling prediction [ref] and the strongly-coupled SYM prediction [ref] , which lie between 0.3 and 0.5.

    The rate of photon production in the quark-gluon plasma from lattice QCD · Analysis of the spectral function · 0.97

Cites it as support 1 paper
Cites it as background 56 papers
  • Speficically, we will consider a weakly-coupled, infinitely extended, static and equilibrated medium and, after briefly summarizing the leading-order perturbative result [ref] [Partial LO results [ref] for non-equilibrated, anisotropic media have been presented at this conference in [ref] .

    Next-to-leading order thermal photon production in a weakly-coupled plasma · Introduction · 0.90

  • The computations of the spectral function in the weakly coupled regime has been done using perturbative SYM theory [ref] , also previous results for perturbative QCD were obtained in [ref] .

    Plasma photoemission from string theory · Weakly coupled SYM plasma photoemission rates · 0.89

  • 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

Discussion

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