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Heavy Quark Radiative Energy Loss in QCD Matter

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Magdalena Djordjevic, Miklos Gyulassy

Nuclear Theory · Submitted 28 Oct 2003 · 39 references

Pith's verdict: The universal mass shift of the GLV opacity series is a real and clean result; the numerical claim that heavy-quark loss sits near the incoherent limit is softer than the abstract implies.

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arXiv IDnucl-th/0310076 (version 2)
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CitationMagdalena Djordjevic, Miklos Gyulassy. "Heavy Quark Radiative Energy Loss in QCD Matter." arXiv:nucl-th/0310076 (2003). https://pith.science/paper/5OUUIRAF
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The paper

The paper claims

Heavy-quark radiative energy loss in a quark-gluon plasma follows the same opacity series as the massless case, with every energy denominator shifted by one universal mass term.

The reading

When a fast quark or gluon is produced inside the hot matter made in a nuclear collision, it loses energy by radiating gluons whose emission is repeatedly interrupted by scattering in the medium. The standard bookkeeping for that process, an expansion in the number of scatterings (the opacity expansion), had been worked out only for massless quanta. This paper asks whether the same series survives when the radiating particle is a heavy quark with mass $M$ and the radiated gluon has a plasma-induced mass $m_g$, and answers that it does, provided every frequency in the series is shifted by $(m_g^2+x^2M^2)/(2xE)$. If that is right, a large body of massless-quark results transfers to charm and bottom with a single substitution, and the practical question becomes quantitative: how much energy do heavy quarks actually lose? The numerical part of the paper says charm loses roughly $0.15$ of its energy at RHIC energies and bottom about half that, both close to the incoherent, non-interfering limit, and that the distance over which a heavy quark radiates is comparable to or larger than a nucleus.

What carries the argument

The load-bearing device is the reaction-operator method: instead of squaring and summing Feynman diagrams order by order, one applies a recursive operator that adds one scattering center at a time, so the $n$-th order distribution is generated from the $(n-1)$-th. The paper's job is to show that the massive case differs from the massless one only through the substitutions in Eqs. (15) and (16), which it establishes by explicit diagram-by-diagram computation in Appendices B through G, extracting the poles of the longitudinal momentum integrals and reading off the residues. The residues supply the shifted phases $exp(iOmega Delta z)$ and the modified currents, and the shifted phases are what drive the interference cosines toward zero, pushing heavy quarks closer to the incoherent limit. The dead cone appears here in generalized form, at angles $theta < sqrt(m_g^2+x^2M^2)/(xE)$, so that the plasmon mass and the quark mass enter the same place.

Extended reading notes

Core claim

The paper's central claim is that the all-orders-in-opacity radiative energy loss series derived for massless partons carries over to massive quarks with two replacements and nothing else. First, every energy denominator (inverse formation time) is shifted by a momentum-transfer-independent term, $omega_{(m..n)}=(k-q_m-...-q_n)^2/2xE rightarrow Omega_{(m..n)}=omega_{(m..n)}+(m_g^2+x^2M^2)/(2xE)$, where $x$ is the gluon's energy fraction, $k$ its transverse momentum, and $M$, $m_g$ the quark and asymptotic plasmon masses. Second, the Hard, Gunion-Bertsch and Cascade amplitude currents are replaced by their massive versions $tilde H = k/(k^2+m_g^2+M^2x^2)$ and $tilde C_{(i_1...i_m)}=(k-q_{i_1}-...-q_{i_m})/((k-q_{i_1}-...-q_{i_m})^2+m_g^2+M^2x^2)$. Because the shift does not depend on the individual momentum transfers, the reaction-operator recursion of the massless case goes through unchanged, giving the full opacity series in Eq. (17). The authors also derive the incoherent short-formation-time limit, sum it to all orders into a closed form, and evaluate the first order numerically, finding charm and bottom energy losses much nearer the linear-in-$L$ Bethe-Heitler-like limit than light partons, with radiation lengths comparable to or exceeding nuclear radii.

Load-bearing premise

The derivation assumes that the triple-gluon vertices are unchanged by the quark mass whenever $M^2/p_z^2$ is small, and it is exactly this that makes the mass enter as one universal shift rather than as a momentum-transfer-dependent correction.

Editorial extensions

If this is right

  • Every massless-quark opacity result, including the closed-form all-orders sums, can be converted to the heavy-quark case by the single substitution of Eqs. (15) and (16), without redoing the diagrammatic algebra.
  • Charm quarks at RHIC energies lose a fraction of order $0.15$ of their energy at opacity $L/\lambda_g=4$, and bottom quarks roughly half that, with the induced loss scaling closer to $L^1$ than to the light-parton $L^2$.
  • Because the radiation length of heavy quarks is comparable to or larger than a nucleus, high-$p_T$ heavy-quark quenching is a volume effect rather than a surface effect, so surface-emission pictures that fit light-hadron suppression need not apply.
  • The Ter-Mikayelian plasmon effect is a small correction for heavy quarks but is what regularizes the $k\to 0$ divergence for massless ones, replacing the ad hoc momentum cutoff used in earlier treatments.
  • The net heavy-quark energy loss in medium, $\Delta E^{(0)}_{med}+\Delta E^{(1)}_{ind}$, exceeds the vacuum value by only about 5 to 10 percent of the quark energy at RHIC, and by about twice that at LHC, which the authors offer as a partial explanation of the apparent null result in single-electron data.

Reading between the lines

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

  • The universality of the shift is a statement about where mass can enter: it enters only through the on-shell relation of the radiating particle and the gluon, never through the scattering vertices. That suggests the same substitution should hold for any radiating probe whose mass enters only kinematically, for instance a heavy scalar or an off-shell quark propagating in a different medium model.
  • Because the shift grows with $x^2M^2$, the suppression is strongest at large gluon energy fraction, so heavy-quark quenching should show a distinctive hardening of the fragmentation function relative to light quarks at fixed $p_T$; this is a sharper experimental handle than the total $R_{AA}$ alone.
  • The authors note that the finite kinematic bound $|q_1|<\sqrt{6ET}$ changes the first-order answer by less than 10 percent, which hints that the dominant uncertainty in heavy-quark tomography is not the phase-space treatment but the plasma parameters $\mu$, $\lambda$ and their time evolution.
  • If the shift is exact to all orders, the mass dependence of the energy loss is controlled by a single dimensionless ratio $Mx/E$, so measurements at two beam energies with the same $x$ should collapse onto one curve after rescaling by the opacity.
Original abstract

Heavy quark medium induced radiative energy loss is derived to all orders in opacity, $(L/\lambda_g)^n$. The analytic expression generalizes the GLV opacity expansion for massless quanta to heavy quarks with mass $M$ in a QCD plasma with a gluon dispersion characterized by an asymptotic plasmon mass, $m_g=gT/\sqrt{2}$. Remarkably, we find that the general result is obtained by simply shifting all frequencies in the GLV series by $(m_g^2+x^2 M^2)/(2 x E)$. Numerical evaluation of the first order in opacity energy loss shows that both charm and bottom energy losses are much closer to the incoherent radiation limit than light partons in nuclear collisions at both RHIC and LHC energies. However, the radiation lengths of heavy quarks remain large compared to nuclear dimensions and hence high $p_T$ heavy quark production is volume rather than surface dominated.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,403 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.74 · human participants 0.01 · controlled comparison 0.46 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Heavy quark medium induced radiative energy loss is derived to all orders in opacity, $(L/\lambda_g)^n$. The analytic expression generalizes the GLV opacity expansion for massless quanta to heavy quarks with mass $M$ in a QCD plasma with a gluon dispersion characterized by an asymptotic plasmon mass, $m_g=gT/\sqrt{2}$. Remarkably, we find that the general result is obtained by simply shifting all frequencies in the GLV series by $(m_g^2+x^2 M^2)/(2 x E)$. Numerical evaluation of the first order in opacity energy loss shows that both charm and bottom energy losses are much closer to the incoherent radiation limit than light partons in nuclear collisions at both RHIC and LHC energies. However, the radiation lengths of heavy quarks remain large compared to nuclear dimensions and hence high $p_T$ heavy quark production is volume rather than surface dominated.

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 Heavy quark medium induced radiative energy loss is derived to all orders in opacity, $(L/\lambda_g)^n$. contribution 0.83 0.94 0.17 asserted 1.00
s1 The analytic expression generalizes the GLV opacity expansion for massless quanta to heavy quarks with mass $M$ in a QCD plasma with a gluon dispersion characterized by an asymptotic plasmon mass, $m_g=gT/\sqrt{2}$. contribution 0.74 0.78 0.12 asserted 1.00
s2 Remarkably, we find that the general result is obtained by simply shifting all frequencies in the GLV series by $(m_g^2+x^2 M^2)/(2 x E)$. result 0.96 0.96 0.38 asserted 1.00
s3 Numerical evaluation of the first order in opacity energy loss shows that both charm and bottom energy losses are much closer to the incoherent radiation limit than light partons in nuclear collisions at both RHIC and LHC energies. result 1.00 0.93 0.34 asserted 1.00
s4 However, the radiation lengths of heavy quarks remain large compared to nuclear dimensions and hence high $p_T$ heavy quark production is volume rather than surface dominated. interpretation 0.79 0.71 0.19 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. The 1
Figure 2. shows the first order induced radiative energy loss for charm and bottom quarks as a function of opacity. We see that the induced contribution increases with L. Note that for heavy (charm and bottom) quarks the thickness dependence is closer to the linear Bethe-Heitler like form, L 1 , than the asymptotic energy quadratic form [14, 16]. 1 2 3 4 5 L @fmD 0 0.05 0.1 0.15 0.2 DEind H1L €€€€€€€€€€€€€€€ E c b FIG 2. The 1st order in opacity fractional energy loss for a 10 GeV charm and bottom quark is plotted versus the effective static thickness L of a plasma characterized by µ = 0.5…
Figure 3. FIG 3. Comparison of the incoherent limit (dashed curves) to
Figure 4. FIG 4. The fractional energy loss for a 10 GeV charm quark is pl
Figure 5. FIG 5. The 1
Figure 6. shows the first order induced radiative energy loss for charm and bottom quarks as a function of opacity. We see that, again, for bottom quarks the thickness dependence is still closer to the linear Bethe￾Heitler like form, L 1 , than the asymptotic energy quadratic form [14, 16]. However, for charm quark, the thickness dependence is somewhere between linear and quadratic form. 1 2 3 4 5 L @fmD 0 0.1 0.2 0.3 0.4 DEind H1L €€€€€€€€€€€€€€€ E c b FIG 6. The 1st order in opacity fractional energy loss for a 30 GeV charm and bottom quark is plotted versus the effective static thic
Figure 7. FIG 7. Comparison of the incoherent limit (dashed curves) to
Figure 8. FIG 8. The fractional energy loss for a 30 GeV charm quark is pl
Figure 9. FIG 9. The radiation length of charm quarks as a function of th
Figure 12. FIG 12. Diagrams

Showing 10 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:nucl-th/0310076 by the authors.

References
How later papers used it

76 later papers cite it

Evidence: substantial. 74 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 138 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 1 of 10 papers with this label

  • Calculated radiative energy loss according to DGLV with short path length corrections as derived in another work.

    Energy Loss in Small Collision Systems (2023) · 0.99 · paraphrased: the licence does not allow quoting

Compares with it 2 of 6 papers with this label
How the literature uses it

71 arXiv papers cite this in 136 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 · 84 sentences without a confident label

Contests it 1 paper
  • In [ref] we perform a thorough analysis of the various assumptions—including collinearity, softness, and large formation time—that are made by both DGLV [ref] and the correction to DGLV [ref] , where we find that the large formation time assumption is breaking down at $𝒪(30–50) GeV$.

    Energy Loss in Small Collision Systems · Energy Loss Framework and Results · contest 0.73

Builds on it 1 paper
Uses its method or tool 10 papers
Compares with it 5 papers
Cites it as background 29 papers
  • Roughly speaking, the opacity expansion is an expansion in the number of interactions $L/λ$, where $λ$ is the gluon mean free path, a hard parton has with the soft in-medium quasiparticles [ref] .

    Jet Broadening in the Opacity and Twist Expansions · Introduction · 0.85

  • 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

  • The majority of current approaches to the energy loss of light partons may be divided into four major schemes often referred to by the names of the original authors: * Higher Twist scheme (HT) [ref] * Path integral approach to the opacity expansion by Armesto, Salgado and Wiedemann, (BDMPS-Z/ASW) [ref] * Finite temperature field theory approach by Arnold, Moore and Yaffe (AMY) [ref] * Reaction Operator approach to the opacity expansion by Gyulassy, Levai and Vitev, (GLV) [ref] All these schemes utilize slightly different approximations regarding the various scales involved in the calculation and somewhat different quantitative pictures of the medium.

    Characterizations of the medium in jet quenching calculations · Introduction · 0.91

Discussion

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