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Non-perturbative computation of gluon mini-jet production in nuclear collisions at very high energies

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Alex Krasnitz (University of the Algarve), Raju Venugopalan (NBI)

High Energy Physics - Phenomenology · Submitted 17 Sep 1998 · 57 references

Pith's verdict: The lattice machinery and the Appendix B validation are the real contribution; the abstract's small-kt claim outruns what the gauge-fixed observable can support.

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arXiv IDhep-ph/9809433 (version 2)
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CitationAlex Krasnitz (University of the Algarve), Raju Venugopalan (NBI). "Non-perturbative computation of gluon mini-jet production in nuclear collisions at very high energies." arXiv:hep-ph/9809433 (1998). https://pith.science/paper/UATWCF4N
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The paper

The paper claims

Two colliding nuclei are evolved in real time on a lattice, and the gluon spectrum is read off

The reading

The paper asks whether the classical effective theory of QCD at small $x$ can be solved numerically rather than only in perturbation theory, and whether that solution reproduces the standard mini-jet picture where it should. The authors set up a real-time lattice simulation of the classical Yang-Mills fields produced when two highly Lorentz-contracted sheets of color charge pass through each other, with the pre-collision fields of each nucleus taken as pure gauges generated by Gaussian random color sources of variance set by $g^4mu^2$. Evolving forward in proper time $tau$ from matched initial conditions, they find that at large transverse momentum the field intensities and their Bessel-like time dependence agree with lattice perturbation theory, which is the lattice image of the perturbative mini-jet result. At momenta of order $alpha_S mu$ the agreement breaks down and the deviations grow with $mu$, so the infrared cutoff that perturbative mini-jet calculations must impose by hand is replaced by dynamics. Because the soft modes are off-shell, the authors argue that field intensities there are not gluon numbers, and they build gauge-invariant energy-energy correlators to estimate how energy is distributed in momentum as the system expands.

What carries the argument

The machinery is the Kogut-Susskind Hamiltonian in 2+1 dimensions coupled to an adjoint scalar, obtained by imposing boost invariance on the forward-light-cone Yang-Mills Hamiltonian in the $A_tau = 0$ gauge, so that only a transverse lattice is needed. The initial conditions are fixed by matching the singular terms of the lattice equations of motion on the light cone, giving $U|_{tau=0} = (U_1+U_2)(U_1^dagger+U_2^dagger)^{-1}$ for the transverse links and a commutator-like expression for the scalar momentum $p = 2alpha$, with $E_l = 0$ and $Phi = 0$. Evolution uses a leapfrog step adapted to the explicit $tau$ dependence, which preserves the Gauss constraints exactly. The second piece of machinery is the gauge-invariant energy-energy correlator: because each kinetic and potential term in the Hamiltonian is a squared quantity, the connected two-point correlator of, say, $B^2$ is used to define a momentum-space energy distribution $E_m(k_t)$, which stays meaningful where the gauge-fixed field intensity does not.

Extended reading notes

Core claim

The central claim is that the classical small-$x$ effective theory of nuclear collisions can be evolved non-perturbatively in real time, and that doing so reproduces perturbative mini-jet physics at large transverse momentum while showing new behaviour at $k_t sim alpha_S mu$. Concretely, the field intensity of a hard mode follows the predicted $J_0^2(omega tau)$ falloff with the lattice dispersion relation, and the intensity as a function of $k_t$ tracks the lattice perturbation theory expression down to small momenta when $mu$ is small. As $mu$ grows at fixed lattice size, the soft modes depart from that curve and the $1/k_t^4$ behaviour is softened, and the time history of the lowest nonzero mode stops following the Bessel form and appears to saturate. The authors then check whether the saturation is a gauge artifact, since the fields are only Coulomb-gauge-fixed at $tau=0$: using gauge-invariant energy-energy correlators, they find the ratio of energy in low-momentum modes to total energy flattens rather than grows, and the total energy density falls as $1/tau$ after a time of order $1/(g^2 mu)$. Their reading is that the soft-mode saturation in the gauge-fixed field intensity is not physically meaningful, while the departure from perturbation theory at $k_t sim alpha_S mu$ is.

Load-bearing premise

The load-bearing premise is that quantum fluctuations about the classical saddle point stay small at the soft scale $k_t sim alpha_S mu$ where the new behaviour is claimed, since that is exactly where the classical approximation is least under control.

Editorial extensions

If this is right

  • The infrared cutoff that perturbative mini-jet calculations must impose by hand is replaced by the nonlinear dynamics of the classical theory, so the sensitivity of mini-jet energy densities to that cutoff becomes a computable question rather than a parameter choice.
  • Because the classical theory is solved in real time, it supplies initial energy density and velocity profiles at a time of order $1/(g^2 mu)$ for a subsequent hydrodynamic or cascade treatment of the collision.
  • The scaling of the onset time of the $1/tau$ behaviour with $1/(g^2 mu)$ gives a concrete, if crude, estimate of when the produced gluon system starts to behave like a longitudinally expanding fluid.
  • Gauge-invariant energy-energy correlators offer a route from lattice configurations to semi-inclusive observables such as energy flux through a detector-like surface, without relying on a particle interpretation of soft modes.

Reading between the lines

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

  • The natural next discriminator is the gauge group: repeating the simulation for SU(3) would show whether the softening of the $1/k_t^4$ spectrum at $k_t sim alpha_S mu$ is a generic feature of the classical theory or an artifact of the two-color truncation.
  • If the departure from perturbation theory is real, it should appear as an excess of energy at low transverse momentum relative to a mini-jet calculation with the same $mu$; that is a comparison a future simulation could make directly against a pQCD spectrum rather than against lattice perturbation theory.
  • The authors' own diagnosis of the soft-mode saturation as a gauge artifact is testable in a cheap way: re-fixing Coulomb gauge at each measurement time and re-plotting the field intensity would show whether the plateau disappears, which would settle the interpretation without needing new correlator statistics.
  • Because the classical approximation is least controlled exactly where the novel effects are claimed, a one-loop estimate of the size of quantum corrections to the soft-mode correlators would tell a reader how much of the low-momentum deviation could survive in the full theory.
Original abstract

At very high energies, in the infinite momentum frame and in light cone gauge, a hard scale proportional to the high parton density arises in QCD. In an effective theory of QCD at small $x$, this scale is of order $\alpha_S\mu$, where $\mu$ is simply related to the gluon density at higher rapidities. The ab initio real time evolution of small $x$ modes in a nuclear collision can be described consistently in the classical effective theory and various features of interest can be studied non-perturbatively. In this paper, we discuss results from a real time lattice computation of the production of gluon jets at very high energies. At very large transverse momenta, $k_t\geq \mu$, our results match the predictions from pQCD based mini-jet calculations. Novel non-perturbative behaviour of the small $x$ modes is seen at smaller momenta $k_t\sim \alpha_S\mu$. Gauge invariant energy-energy correlators are used to estimate energy distributions evolving in proper time.

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, 5,618 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

At very high energies, in the infinite momentum frame and in light cone gauge, a hard scale proportional to the high parton density arises in QCD. In an effective theory of QCD at small $x$, this scale is of order $\alpha_S\mu$, where $\mu$ is simply related to the gluon density at higher rapidities. The ab initio real time evolution of small $x$ modes in a nuclear collision can be described consistently in the classical effective theory and various features of interest can be studied non-perturbatively. In this paper, we discuss results from a real time lattice computation of the production of gluon jets at very high energies. At very large transverse momenta, $k_t\geq \mu$, our results match the predictions from pQCD based mini-jet calculations. Novel non-perturbative behaviour of the small $x$ modes is seen at smaller momenta $k_t\sim \alpha_S\mu$. Gauge invariant energy-energy correlators are used to estimate energy distributions evolving in proper time.

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 At very high energies, in the infinite momentum frame and in light cone gauge, a hard scale proportional to the high parton density arises in QCD. background 1.00 0.11 0.04 asserted 1.00
s1 In an effective theory of QCD at small $x$, this scale is of order $\alpha_S\mu$, where $\mu$ is simply related to the gluon density at higher rapidities. background 0.99 0.16 0.04 asserted 1.00
s2 The ab initio real time evolution of small $x$ modes in a nuclear collision can be described consistently in the classical effective theory and various features of interest can be studied non-perturbatively. method 0.50 0.16 0.10 asserted 0.83
s3 In this paper, we discuss results from a real time lattice computation of the production of gluon jets at very high energies. contribution 0.94 0.94 0.10 asserted 0.95
s4 At very large transverse momenta, $k_t\geq \mu$, our results match the predictions from pQCD based mini-jet calculations. result 0.99 0.91 0.25 asserted 1.00
s5 Novel non-perturbative behaviour of the small $x$ modes is seen at smaller momenta $k_t\sim \alpha_S\mu$. result 1.00 0.91 0.25 asserted 1.00
s6 Gauge invariant energy-energy correlators are used to estimate energy distributions evolving in proper time. method 1.00 0.84 0.06 asserted 0.98

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 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. The lattice size dependence of the scalar kinetic e
Figure 2. Normalized field intensity of a hard (kt = 2.16GeV) mode vs proper time τ in units of fm (diamonds). Solid line is the LPTh prediction. Next, we determine the relation between the continuum µ and its lattice coun￾27
Figure 3. Field intensity over µ 4 as a function of kt for µ = 200MeV (squares), µ = 100MeV (pluses), and µ = 50MeV (diamonds). Solid line is the LPTh prediction. The field intensity is in arbitrary units and kt is in GeV. In this work, we measure a variety of observables. On the one hand, extensive quantities like the energy, with well defined local gauge-invariant densities, provide useful information on the system and are easily measurable. On the other hand, in an experiment one usually deals with gluon fields with definite momenta. Such fields are not easy to define outside perturbation
Figure 4. Normalized field intensity of a soft (kt = 108MeV) mode vs proper time τ (in units of fm) for µ = 200MeV (squares), µ = 100MeV (pluses), and µ = 50MeV (diamonds). Solid line, nearly coinciding with the µ = 50MeV curve, is the LPTh prediction. case, nx = 20, ny = 0. The solid line in the figure is the square of the Bessel function J0(ωτ ) where ω = q 2(2 − cos(kx) − cos(ky)). Since the τ–direction in our simulation is continuous, the time dependence of the high transverse momentum perturbative modes should agree with the continuum perturbative result which predicts a time dependence…
Figure 5. Time history of the energy density in units of
Figure 6. Energy per mode k normalized to the total energy as a
Figure 7. Energy per mode k normalized to energy of zeroth mod

Figures reproduced from arXiv:hep-ph/9809433 by the authors.

References
How later papers used it

90 later papers cite it

Evidence: some. 73 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 156 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 24 papers with this label
How the literature uses it

79 arXiv papers cite this in 138 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

Papers citing it · 78 sentences without a confident label

Uses its method or tool 21 papers
  • The time evolution of the fields is obtained by solving the Yang-Mills equations numerically, using a real-time lattice gauge theory formulation of these equations [ref] that ensure a gauge-independent discretization.

    Heavy flavor angular correlations as a direct probe of the glasma · Introduction · 0.99

  • In order to answer the questions above, we shall compare approximate solutions with the full numerical solutions of the CYM equations, obtained using the method developed in Refs. [ref] .

    On the gluon spectrum in the glasma · Introduction · 0.98

  • In the numerical computation of the glasma fields [ref] there has been no longitudinal structure in the source, and the Wilson lines have been constructed simply as [equation] with the transverse charge densities depending on a single parameter $μ$, independent of $x^-$: [equation]

    Wilson line correlator in the MV model: relating the glasma to deep inelastic scattering · Wilson lines and glasma fields · 0.85

Uses its data or values 1 paper
Compares with it 2 papers
  • The boost-invariant initial conditions for the electric field at $τ=0^+$ created by the collision of charge densities of two nuclei $ρ̂_1(x_T)$ and $ρ̂_2(x_T)$ are given by [ref] [equation] where $α_1,2^i(x_T)$ is determined from the relations [equation] which are similar to our initial conditions in the laboratory frame: The first equation corresponds to Eq. ([ref]), and the second one to the Poisson equation ([ref]).

    Simulating collisions of thick nuclei in the color glass condensate framework · Comparison with boost-invariant initial conditions · 0.94

  • Our result favors a larger value for $c$ than the early numerical simulations [ref] , but is consistent with the analytical calculation $c≈2ln2≈1.4$ [ref] and agrees well with the latest numerical evaluation [ref] .

    Initial energy density and gluon distribution from the Glasma in heavy-ion collisions · DISCUSSIONS AND OUTLOOKS · 1.00

Cites it as support 1 paper
  • This is seen explicitly and analytically in the “pA” case [ref] and in numerical computations of the glasma fields in the fully nonlinear case [ref] .

    Glittering Glasmas · Calculation of the multiplicity distribution · 0.96

Cites it as background 25 papers
  • From the QCD phenomenological perspective, energy correlator observables were first proposed to study jets in HICs in [ref] .[See [ref] for an earlier proposal to use energy correlators to study many-body states in HICs.] From this perspective, energy correlator observables have so far focused primarily on its coupling to jet-like excitations produced in the same event, see [ref] for jet-quenching studies using energy-flow operators.

    Hydrodynamics and Energy Correlators · Introduction · 0.91

  • In particular, the pre-equilibrium evolution [ref] can be divided into the Glasma phase consisting of highly occupied QCD matter far from equilibrium [ref] that can be simulated using a classical-statistical approach [ref] , and the subsequent kinetic phase [ref] that is dominated by scatterings of quasi-particle excitations [ref] .

    Heavy-quark diffusion in 2+1D and Glasma-like plasmas: evidence of a transport peak · Introduction · 0.98

  • In this scenario, the quark-gluon system that begins in an initial state due to saturated gluon fields created in nuclear collisions (dominated by the classical gluon fields of the McLerran-Venugopalan (MV) model [ref] ) progresses to a thermalized isotropic medium due to $2 →2$, $2 →3$ and $3 →2$ rescatterings.

    Time-Dependent Observables in Heavy Ion Collisions II: in Search of Pressure Isotropization in the $\varphi^4$ Theory · Introduction · 0.98

Discussion

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