REVIEW
Radiative energy loss of high energy quarks in finite-size nuclear matter and quark-gluon plasma
High Energy Physics - Phenomenology · Submitted 7 Apr 1997 · 17 references
Pith's verdict: The finite-size L² law and the suppression factor S(η,l) are the durable content; the numerical coefficients, especially the incident-quark headline number, are softer than the paper's blanket factor-of-2 caveat suggests.
T0 · AI review
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Referee report
The paper
The paper claims
A fast quark crossing dense matter loses energy as the square of the distance it travels, not linearly, provided it starts inside the medium.
The reading
What carries the argument
The calculation reduces the radiation rate to a two-dimensional Schrodinger equation in impact-parameter space, with the longitudinal coordinate $z$ playing the role of time and the medium entering as an imaginary potential $v(rho,z) = -i n(z) sigma_3(rho,x)/2$. Here $sigma_3$ is the cross section of a three-body $qbar{q}g$ colour-singlet system on a medium constituent, expressible through the colour-dipole cross section $sigma_2$ via $sigma_3 = (9/8)[sigma_2(rho)+sigma_2((1-x)rho)] - (1/8)sigma_2(xrho)$. The tractable step is the oscillator approximation: because radiation is dominated by $rho < 1/m_g$, $sigma_2(rho) = C_2 rho^2$ is frozen at $C_2(1/m_g)$, the Hamiltonian becomes harmonic with complex frequency $Omega$, and the Green's function is known in closed form. This yields the rate as a Bethe-Heitler cross section times a suppression factor $S(eta,l)$ depending only on $eta = L_f|Omega|$ and $l = L/L_f$, where $L_f$ is the gluon formation length. The small-$L$ behaviour $S approx -l^2 log l$ is the quantitative expression of the 'lost gluon cloud' picture, and the large-$L$ limits reproduce the infinite-medium suppression found in the author's earlier work.
Extended reading notes
Core claim
Zakharov computes the induced gluon radiation of a high-energy quark in a uniform medium of finite length $L$, treating the quark as produced at $z=0$ by a hard mechanism and propagating through matter of density $n(z)$. The central claim is that for a sufficiently energetic quark the radiative energy loss scales as $Delta E_q propto L^2$, with only a weak dependence on the initial quark energy $E_q$, and that this $L^2$ behaviour crosses over to $L^1$ as $E_q$ decreases. The author is explicit that this is not a Landau-Pomeranchuk-Migdal story: the leading contributions come from the narrow kinematic regions $x < delta_g$ and $1-x < delta_q$, where finite-size effects are marginal, and the quadratic law survives even though the Bethe-Heitler cross section diverges as $1/m_g^2$. For a quark incident on a nucleus the prediction is $Delta E_q approx 0.1 E_q (L/10~fm)^beta$ with $beta approx 0.9$-$1$ below 50 GeV and $0.85$-$0.9$ above 200 GeV, which the author contrasts with the much larger $0.25 (L/1~fm)$ GeV of the Brodsky-Hoyer estimate. Numerical results are quoted for cold nuclear matter at $n = 0.15~fm^{-3}$, $alpha_s = 1/2$, and for a quark-gluon plasma at $T = 250$ MeV with $alpha_s = 1/3$, where the losses are several times larger and the approach to $L^2$ sets in at higher energy.
Load-bearing premise
The closed-form and numerical results rest on freezing the dipole cross section at $C_2(1/m_g)$ so that $sigma_3 = C_3(x)rho^2$ and the Hamiltonian becomes an exactly solvable oscillator, with the normalization chosen by hand as $C_2(1/m_g) = 2$ out of an admitted range of 1.3 to 4; the author acknowledges that the predictions are rough estimates with uncertainties of at least a factor of two in either direction.
Editorial extensions
If this is right
- The radiative energy loss of an energetic quark in a finite medium is governed by a simple power law in path length, $Delta E_q propto L^2$, with weak dependence on the quark energy, so the stopping power is largely set by the geometry of the medium rather than by the beam energy.
- The $L^2$ law crosses over to $L^1$ as the quark energy decreases, giving a definite energy at which the quadratic regime is reached: the maximum medium-modified formation length must exceed the medium size by roughly a factor of two.
- For a quark incident on a nucleus the loss is about $0.1 E_q (L/10~fm)^beta$ with $beta$ near unity, roughly an order of magnitude below the Brodsky-Hoyer estimate of $0.25 (L/1~fm)$ GeV.
- In a quark-gluon plasma at $T = 250$ MeV the predicted losses are several times larger than in cold nuclear matter, which the author reads as support for jet quenching being a usable probe of deconfinement in heavy-ion collisions.
Reading between the lines
- Because the quadratic law comes from the two narrow endpoint regions $x < delta_g$ and $1-x < delta_q$ rather than from suppression of the whole spectrum, the effect should be visible as a hardening or softening of the leading-hadron fragmentation pattern at small and large $x$ separately, not as a uniform rescaling of the whole fragmentation function.
- The 'lost gluon cloud' picture makes the result depend on how the quark was produced: a quark created in a hard collision and one entering from outside the medium are treated as genuinely different initial conditions, which suggests that the measured energy loss in $pA$ versus $AA$ collisions should differ by more than a density rescaling.
- Since the quoted coefficients carry an admitted factor-of-two uncertainty dominated by the frozen dipole normalization $C_2(1/m_g)$, a measurement that fixes the overall coefficient could be used in reverse to constrain the perturbative component of the dipole cross section at transverse separations near $1/m_g$.
- The contrast with the Brodsky-Hoyer estimate is sharp enough to be settled by the same dijet longitudinal energy-flow data the author cites, provided the analysis separates energy loss from the purely geometric $L$-dependence of the production point distribution.
Original abstract
The induced gluon radiation of a high energy quark in a finite-size QCD medium is studied. For a sufficiently energetic quark produced inside a medium we find the radiative energy loss $\Delta E_{q}\propto L^{2}$, where L is the distance passed by quark in the medium. It has a weak dependence on the initial quark energy $E_{q}$. The $L^{2}$ dependence turns to $L^{1}$ as the quark energy decreases. Numerical calculations are performed for a cold nuclear matter and a hot quark-gluon plasma. For a quark incident on a nucleus we predict $\Delta E_{q}\approx 0.1 E_{q}(L/10 fm)^{\beta}$, with $\beta$ close to unity.
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.
The abstract, sentence by sentence
The induced gluon radiation of a high energy quark in a finite-size QCD medium is studied. For a sufficiently energetic quark produced inside a medium we find the radiative energy loss $\Delta E_{q}\propto L^{2}$, where L is the distance passed by quark in the medium. It has a weak dependence on the initial quark energy $E_{q}$. The $L^{2}$ dependence turns to $L^{1}$ as the quark energy decreases. Numerical calculations are performed for a cold nuclear matter and a hot quark-gluon plasma. For a quark incident on a nucleus we predict $\Delta E_{q}\approx 0.1 E_{q}(L/10 fm)^{\beta}$, with $\beta$ close to unity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The induced gluon radiation of a high energy quark in a finite-size QCD medium is studied. | contribution 0.98 | 0.88 | 0.05 | asserted 0.90 |
| s1 | For a sufficiently energetic quark produced inside a medium we find the radiative energy loss $\Delta E_{q}\propto L^{2}$, where L is the distance passed by quark in the medium. | result 1.00 | 0.95 | 0.73 | asserted 1.00 |
| s2 | It has a weak dependence on the initial quark energy $E_{q}$. | result 1.00 | 0.86 | 0.08 | asserted 1.00 |
| s3 | The $L^{2}$ dependence turns to $L^{1}$ as the quark energy decreases. | result 1.00 | 0.88 | 0.15 | asserted 1.00 |
| s4 | Numerical calculations are performed for a cold nuclear matter and a hot quark-gluon plasma. | method 0.91 | 0.93 | 0.03 | asserted 0.99 |
| s5 | For a quark incident on a nucleus we predict $\Delta E_{q}\approx 0.1 E_{q}(L/10 fm)^{\beta}$, with $\beta$ close to unity. | result 1.00 | 0.89 | 0.24 | asserted 0.69 |
What was examined, and what was not
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References
How later papers used it
174 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 19 papers with this label
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Followed this paper's formulation to give its rate schematically in a figure.
Universality (beyond leading log) of soft radiative corrections to $\hat q$ in $p_\perp$ broadening and energy loss (2021) · 0.99 · paraphrased: the licence does not allow quoting
How the literature uses it
163 arXiv papers cite this in 253 sentences
Papers citing it · 118 sentences without a confident label
Contests it 1 paper
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Thus, the law $ΔE(λ≪L ≪L^*) ∝L^2$ [ref] for the induced energy loss of a light quark produced in the medium (we will review this case in Section [ref]), is not valid for an asymptotic quark.
Energy losses in a hot plasma revisited · Light parton · contest 0.80
Builds on it 3 papers
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In Sec. [ref], we first give the general formalism, as a generalization of that by BDMPS-Z [ref] , for calculating parton energy loss due to multiple gluon emission.
Radiative energy loss and radiative p_T-broadening of high-energy partons in QCD matter · Introduction · 0.95
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Nevertheless, the original techniques in [ref] can be extended beyond the limit $x→0$.
Medium-induced emissions of hard gluons · Introduction · 0.84
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This approach has been originally developed [ref] for an isotropic plasma and for the branching of unpolarised partons, but here we shall provide its generalisation to an anisotropic medium and to gluons with definite polarisation states.
Jet polarisation in an anisotropic medium · Introduction · 0.94
Uses its method or tool 18 papers
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We compute the two-point correlator (EEC) on a quark jet which propagates through a static medium of finite length within the BDMPS-Z formalism [ref] .
Resolving the Scales of the Quark-Gluon Plasma with Energy Correlators · Introduction · 0.97
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First of all, we make use of a semi-hard implementation [ref] of the multiple scattering BDMPS-Z formalism [ref] for medium-induced radiation using two different parton-medium interaction models: a gaussian (also referred to as the Harmonic Oscillator) and a Yukawa (Gyulassy-Wang) model.
A Coherent View of the Quark-Gluon Plasma from Energy Correlators · Introduction · 1.00
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In this manuscript, we first derive the medium-induced radiation spectrum for the general case where $τ_m ≥τ_p$ within the BDMPS-Z framework [ref] which includes the resummation of multiple in-medium scatterings, and then isolate the contribution from allowing the initial hard parton to radiate before entering the medium.
Medium-induced radiation with vacuum propagation in the pre-hydrodynamics phase · Introduction · 0.83
Compares with it 5 papers
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We shall demonstrate in this appendix that the LIDO model is able to quantitatively reproduce the single parton emission rate as calculated in the full theory [ref] , where multiple collisions to the single parton emissions are resummed to all orders with full leading-order weakly-coupled jet-medium interaction.
QGP modification to single inclusive jets in a calibrated transport model · Comparisons of single-gluon emission rates · 0.88
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This is in contrast to the limit of medium-induced branching [ref] , where one investigates emissions with transverse momenta dominated by interactions with the medium, i.e. $p_⊥∼√(q̂ z E)$.
Mapping collinear in-medium parton splittings · Implementing the semi-classical limit · 0.94
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Thus, Eq. ([ref]) shows the same kind of suppression for the emission of highly energetic gluons due to interferences as the approach by Baier Dokshitzer Mueller Peigné, Schiff, and Zakharov (BDMPS-Z) [ref] .
QCD theory overview · Non-Gaussian broadening in jet quenching · 0.98
Cites it as background 93 papers
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The seminal analyses of radiative parton energy loss by Baier, Dokshitzer, Mueller, Peigné, Schiff [ref] and by Zakharov [ref] (BDMPS-Z) are based on this approximation.
Coherent Radiative Parton Energy Loss beyond the BDMPS-Z Limit · Introduction · 0.93
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Several groups have calculated the corresponding medium-induced gluon energy distribution $ωI/ω$ in the kinematical regime [ref]
A local Monte Carlo framework for coherent QCD parton energy loss · Time-scale for medium-induced interference in the opacity expansion · 0.86
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such that the parton may interact with several scattering centers in the plasma, the medium induced spectrum admits a full analytic treatment, captured by the Baier-Dokshitzer-Mueller-Peigné-Schiff-Zakharov (BDMPS-Z [As was later shown [ref] , this formalism also includes the regime of single hard scattering explored in the Gyulassy-Levai-Vitev (GLV) framework [ref] .]) formalism [ref] .
Improved opacity expansion at NNLO for medium induced gluon radiation · Introduction · 0.81
Discussion
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