Emergence, Formation and Dynamics of Hot QCD Matter

Bruno Sebastian Scheihing-Hitschfeld

REVIEW 3 major objections 5 minor 300 references

Emergence, Formation and Dynamics of Hot QCD Matter

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The thesis pinpoints the QCD correlation functions that govern quarkonium dissociation and recombination in quark-gluon plasma, and shows that hydrodynamization in QCD kinetic theory is an adiabatic approach to a single evolving ground…

desk verdict Two solid, genuinely new calculations for quarkonium transport sit alongside an honest but scoped demonstration of Adiabatic Hydrodynamization in a truncated kinetic theory. read the letter →

arxiv 2412.20759 v1 pith:ACQLHUVJ submitted 2024-12-30 hep-ph hep-thnucl-th

Bruno Sebastian Scheihing-Hitschfeld This is my paper · ORCID

classification hep-phhep-thnucl-th

keywords quarkoniumtransportquark-gluonplasmageneralizedgluondistributionspotentialnon-relativisticQCDopenquantumsystemsAdS/CFTcorrespondencehydrodynamizationkinetictheoryattractors

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quarkonium suppression in heavy-ion collisions has long been described by screened-potential models, but the thesis argues the real quantities are Generalized Gluon Distributions (GGDs): thermal correlation functions of two chromoelectric fields joined by an adjoint Wilson line. It claims to be the first to formulate these non-perturbative objects for quarkonium transport, to compute the weak-coupling QCD GGD at next-to-leading order, to compute the strong-coupling version in N=4 supersymmetric Yang-Mills theory by holography, and to give the Euclidean form needed for lattice QCD. On the thermalization side, it claims that in the small-angle approximation to QCD kinetic theory, hydrodynamization follows the Adiabatic Hydrodynamization scenario: a shrinking set of low-energy eigenstates of an effective Hamiltonian dominates, an energy gap opens to mark each stage, and the hydrodynamic attractor is the single remaining ground state. A sympathetic reader would care because both claims replace phenomenological models with objects defined directly in QCD, opening a path from data to the QCD Lagrangian.

What carries the argument

The carrying object for the quarkonium half is the Generalized Gluon Distribution (GGD), a two-point function of chromoelectric fields dressed with a timelike adjoint Wilson line; it is the effective distribution of quasi-gluons that a heavy pair absorbs or emits. Its Euclidean version, together with Kubo-Martin-Schwinger relations connecting the thermal orderings, is what makes lattice QCD contact possible. The carrying object for the hydrodynamization half is the effective Hamiltonian of the kinetic equation in an adiabatic frame, built by rescaling the momentum variables of the small-angle Fokker-Planck collision kernel; its instantaneous eigenstates organize the system's evolution, with energy gaps between clustered levels separating stages of thermalization. Supporting machinery includes the Schwinger-Keldysh contour, Hard Thermal Loop resummation, Rξ gauge checks, and potential non-relativistic QCD (pNRQCD), the effective theory whose multipole expansion links the correlators to quarkonium wavefunctions.

What would settle it

Include the omitted Ibf2 term in the small-angle QCD kinetic equation: if the overlap of the true distribution with the instantaneous ground state of the effective Hamiltonian stops being monotonically dominant, or if a real attractor appears that is not an eigenstate, the hydrodynamization claim fails.

Watch

Extended reading notes

Core claim

The central claim is factorization plus adiabaticity. For a small-size heavy-quark pair in a QGP, the dissociation and recombination rates factorize into quarkonium wavefunctions and two Generalized Gluon Distributions, defined in Eqs. (3.28)-(3.31) as Wightman (real-time) correlation functions of chromoelectric fields connected by a timelike adjoint Wilson line. The thesis computes the weakly coupled QCD version to next-to-leading order in an Rξ gauge, demonstrating gauge independence, infrared and collinear safety, and a spectral function whose renormalization is governed by the QCD beta function. At strong coupling it computes the same object in N=4 supersymmetric Yang-Mills theory via AdS/CFT, finding that the leading Markovian contributions to singlet-octet transitions vanish, and that the QCD result approaches the strong-coupling curve as the coupling grows. The hydrodynamization claim is that in a simplified small-angle QCD kinetic theory, a time-dependent frame exists in which the distribution function evolves adiabatically: the state remains close to the instantaneous ground state of the effective Hamiltonian, higher eigenstates decay, and the opening of an energy gap above the ground state marks the onset of each stage of thermalization, ending with a single ground state whose adiabatic evolution is the hydrodynamic attractor.

Load-bearing premise

The conclusions rest on two approximations: for the kinetic theory, small-angle scattering with part of the collision kernel omitted; for quarkonium, a wide separation between the heavy-quark mass, the pair size, and the plasma temperature, together with an initially factorized heavy-pair and plasma state.

Editorial extensions

If this is right

Quarkonium suppression in heavy-ion collisions can be interpreted directly in terms of QCD-level transport coefficients, rather than screened-potential models whose parameters are not fixed by the QCD Lagrangian.
Lattice QCD can in principle compute the Euclidean GGDs, providing a non-perturbative determination of the same objects that enter the dissociation and recombination rates.
At strong coupling, leading Markovian dissociation and recombination rates vanish, so quarkonium dynamics in strongly coupled QGP must be treated with non-Markovian or time-dependent perturbation theory.
In small-angle QCD kinetic theory, the pre-hydrodynamic attractor and the approach to local thermal equilibrium are described by the adiabatic evolution of a single instantaneous ground state of an effective Hamiltonian.
Loss of memory of the initial condition is explained by the decay of excited eigenstates and the opening of energy gaps, so the appearance of attractors is a consequence of adiabaticity rather than a separate assumption.

Reading between the lines

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

Beyond the paper, the GGD framework could be used to define in-medium color screening and gluon distributions in QGP beyond quarkonium, connecting quarkonium suppression to other hard probes.
Beyond the paper, the vanishing of leading Markovian rates at strong coupling suggests that existing Lindblad-equation simulations of quarkonium suppression may need memory-kernel or non-Markovian upgrades before being confronted with data.
Beyond the paper, the adiabatic eigenstate description of hydrodynamization may reduce the cost of pre-equilibrium modeling: once the gap opens, evolving a handful of instantaneous eigenstates should reproduce the attractor without solving the full kinetic equation.
Beyond the paper, because the underlying effective field theory is representation-independent, the GGD formalism should extend to any short-distance dipole in a thermal bath, such as heavy dark-matter co-annihilation pairs in the early universe.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core of this thesis is the GGD formalism for quarkonium transport and its first computations: a complete NLO chromoelectric correlator in weakly coupled QCD with explicit R_xi gauge invariance checks, an AdS/CFT strong-coupling result, and a Euclidean formulation that opens a lattice path. These are parameter-free field-theoretic calculations, cross-checked against the heavy-quark diffusion limit and against known zero-temperature limits. That part is solid and deserves a serious referee.

The Adiabatic Hydrodynamization chapter is more delicate. The claim is honestly restricted: the demonstration is in a small-angle Fokker-Planck truncation of QCD kinetic theory with the Ibf2 collision term omitted, and Appendix C.4 is devoted to that omission. The stress-test concern is legitimate: Ibf2 enters at the same parametric order in the overoccupied regime, so it can shift the eigenvalues and gap-opening times that define the attractor. The mechanism may survive, but it is not yet shown to survive. What the thesis does well is to make this boundary visible rather than hiding it, and the eigenstate decomposition plus the demonstration that two scaling frames give the same physical dynamics are real evidence for the truncated model.

The soft spots are proportionate: the hydrodynamization result is a model result, not a full-QCD result, and the abstract is careful to say “simplified description.” The pNRQCD hierarchy and the factorized initial state are load-bearing but standard and discussed. Lack of code and data for the numerical figures is a minor reproducibility issue, not a correctness one.

Who gets value: quarkonium phenomenologists, lattice practitioners, and kinetic-theory people who want a concrete adiabatic mechanism. The GGD results alone justify a full peer review; the hydrodynamization claim needs a clear scope caveat but is worth referee time. No red flags otherwise.

Referee Report

3 major / 5 minor

Summary. This PhD thesis advances hot-QCD theory in two directions. First, it formulates Generalized Gluon Distributions (GGDs), defined as gauge-invariant chromoelectric field correlators connected by adjoint Wilson lines (Eqs. (3.28)-(3.31)), as the non-perturbative objects controlling quarkonium dissociation and recombination in quark-gluon plasma. It computes these at NLO in weakly coupled QCD (Section 3.3), at strong coupling in N=4 SYM via AdS/CFT (Section 3.4), and provides Euclidean versions for lattice QCD (Section 3.5). Second, it develops the Adiabatic Hydrodynamization (AH) scenario, showing that in a small-angle Fokker-Planck approximation to QCD kinetic theory the hydrodynamization of a longitudinally expanding gluon gas is governed by a monotonically shrinking set of low-energy eigenstates of an effective Hamiltonian, with the hydrodynamic attractor reached when a single ground state remains (Chapter 4).

Significance. If the central claims hold, the GGD results provide first-principles QCD-based transport coefficients for quarkonium, opening a path from quarkonium suppression data to QCD parameters, and the AH picture offers a mechanistic explanation of attractor behavior in kinetic theory. The manuscript has notable strengths: the NLO weak-coupling calculation includes a detailed proof of gauge invariance in R_xi gauge (Section 3.3.2), a careful discussion of infrared and collinear safety (Section 3.3.4), and consistency checks with the heavy-quark diffusion limit (Section 3.3.6). The strong-coupling calculation includes mode analysis and Euclidean numerical checks in appendices. The hydrodynamization analysis compares eigenstate decompositions with direct numerical solutions of the kinetic equation and shows that two different scaling-frame choices reproduce the same physical dynamics (Figs. 4.9-4.13). These are concrete, reproducible technical achievements. However, the hydrodynamization claim is established only in a truncated kinetic model, and the abstract overstates the scope. That limitation is acknowledged in the text but is load-bearing for the paper's central claim.

major comments (3)

[§4.1.1.2, §4.2.1.1, App. C.4] The Adiabatic Hydrodynamization mechanism is demonstrated in a small-angle Fokker-Planck truncation of the QCD Boltzmann collision kernel, with the Ibf2 term omitted. The text itself identifies this as a simplified description (Chapter 1, p. 46, and Appendix C.4). The eigenvalue gap-opening and ground-state dominance that define AH are properties of the retained collision operator; the omitted Ibf2 term contributes at the same parametric order in the overoccupied regime and can shift the eigenvalues and eigenvectors that set the gap-opening times and the attractor trajectory. The thesis is honest about this scope, but the central conclusion that hydrodynamization in QCD kinetic theory follows AH is not yet supported. Please provide either (a) a numerical eigenmode decomposition that includes the Ibf2 term for at least one representative initial condition, or (b) a parametric argument that the omitted term is subleading for the specific quantities (gap opening, ground-state dominance, Teff leveling off) that define the attractor.
[Abstract and Chapter 1 (pp. 3, 46)] The abstract and introductory summary state the result as 'hydrodynamization in QCD kinetic theory' and claim the thesis demonstrates 'that hydrodynamization is adiabatic' without the qualifier that only the small-angle subset of QCD scattering mechanisms is included. The body of the thesis is appropriately careful, but the abstract overstates the validated scope. Please revise the abstract and Chapter 1 to state explicitly that the AH mechanism is established for a simplified small-angle kinetic theory, and clarify what would be required to extend it to the full QCD collision kernel. This is not merely cosmetic, because the unsupported full-QCD reading is the one that would justify the title's claim about QCD matter.
[§3.2.2 and §3.3.5.2 (Figs. 3.9, 3.21)] The GGDs in Eqs. (3.28)-(3.31) involve adjoint Wilson lines extending to t = ±∞. As explained in Section 3.3.1.2, a regulator ε is needed to define these lines, and the text then sends t0 → -∞. The derivation of the KMS relations (3.33)-(3.35) and the spectral representation (3.43)-(3.45) should state explicitly how the ε → 0 limit and the t0 → -∞ limit commute with the operator orderings in the Wightman correlators. If this was addressed in the appendices, please add a pointer at the point of use; otherwise, this is a gap in the definition of the central objects of Chapter 3.

minor comments (5)

[Chapter 1, p. 46 and §3.2.3] The claim of 'rigorously formulat[ing] for the first time the non-perturbative objects' should be qualified: Refs. [167] and [242] already introduced chromoelectric correlators with adjoint Wilson lines as the quantities governing quarkonium dynamics. The novelty here is the systematic GGD framework, the NLO and strong-coupling calculations, and the Euclidean formulation; please adjust the novelty wording to avoid overclaiming.
[Fig. 3.24 caption] The caption contains the LaTeX artifact 'footnote434443'; this should be corrected to a proper reference or removed.
[Fig. 4.25 caption] The phrase '(the here quite early time at which the) gap' is awkward and partially parenthetical; rephrase for clarity.
[§3.3.5 and Figs. 3.9, 3.21] The plots of the NLO spectral function use a specific renormalization-scale choice μ(ω,T) from Ref. [248]. This is a scheme choice, not a parameter of QCD; please state in the caption that the scale choice is arbitrary and estimate the scale uncertainty, e.g., by varying μ by a factor of 2 around the chosen value.
[Appendix C.4] The discussion of the omitted Ibf2 term is relegated to an appendix. Given its importance to the scope of the Chapter 4 claim, consider moving at least a summary of this discussion into Section 4.2.1.1, where the approximations are introduced, so that the limitation is visible at the point of use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GGD results are independent field-theoretic calculations, and the Adiabatic Hydrodynamization claim is checked against direct numerical solutions with the frame choice shown not to alter the physical dynamics.

full rationale

The thesis's two central claims are not circular. In Chapter 3, the Generalized Gluon Distributions are defined as specific thermal correlation functions of chromoelectric fields dressed by adjoint Wilson lines, Eqs. (3.28)-(3.31). They are not fit to quarkonium observables and are not defined in terms of the transport coefficients they later determine. The weak-coupling NLO calculation in Section 3.3 is a parameter-free perturbative evaluation in QCD, the strong-coupling calculation in Section 3.4 is an independent AdS/CFT computation, and the Euclidean formulation in Section 3.5 is a proposal for a future lattice calculation. None of these steps reduces to the quantity they are meant to predict. In Chapter 4, the Adiabatic Hydrodynamization scenario is demonstrated in a simplified small-angle kinetic theory, with the omitted Ibf2 term acknowledged in Appendix C.4 and in the Chapter 1 abstract. This is an honest scope limitation, not a circularity. The eigenstate decomposition is validated by comparing with direct numerical solutions of the kinetic equation (e.g., Figs. 4.10, 4.11, 4.14, and 4.16), and Fig. 4.9 explicitly shows that an adiabaticity-maximizing frame and a constant frame produce identical physical dynamics, so the frame choice is interpretive rather than constitutive. The thesis does cite the author's prior work, but the cited results are parameter-free calculations or numerical demonstrations, not unpublished uniqueness theorems imported to force the conclusion. No load-bearing step reduces by definition or by fitted input to its own output.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central GGD derivations are parameter-free field-theory calculations; the only hand-chosen inputs are scheme and modeling choices in plots and in the illustrative Upsilon calculation. The hydrodynamization result depends on an explicitly simplified kinetic theory, which is the main axiom burden. No new particles, forces, or conserved quantities are introduced.

free parameters (2)

Renormalization scale mu(omega,T) for weak-coupling spectral function plots = mu0 approximately 8.1 T; two-loop scale choice per text after Eq. (3.126)
Chosen for the NLO QCD plots in Fig. 3.9 and Section 3.3.5; not fit to data, but numerical results at finite coupling depend on this scheme choice.
Initial wavefunction width sigma0 and Bjorken temperature profile in the Upsilon formation illustration = sigma0 varied; Tf = 155 MeV, tau_i = 0.6 fm/c, tau_f = 10 fm/c
Inputs to the Section 3.6 illustrative calculation of Upsilon(1S) formation probability; not fit to data, and the thesis reports only truncation-error bands from wavefunction evolution, not systematic variation over initial conditions.

assumptions (6)

domain assumption The heavy quark mass M, inverse size 1/r, and binding energy satisfy M >> 1/r >> |Eb|, T, so pNRQCD and the dipole (multipole) expansion apply.
Invoked in Section 3.1 to write the pNREFT Lagrangian (3.1); phenomenological validity for Upsilon relies on this scale hierarchy.
domain assumption The QGP environment is in thermal equilibrium and the heavy-quark subsystem is weakly coupled to it, with factorized initial density matrix rho = rho_S x rho_E.
Used in Section 3.2, Eq. (3.18), to derive the density-matrix evolution and the GGDs; strongly-coupled plasma may violate this factorization, as the thesis itself notes in Section 3.6.
domain assumption AdS/CFT duality: N=4 SYM at large Nc and strong coupling is an acceptable analog for strongly coupled QGP.
Used throughout Section 3.4 to compute the GGD from classical gravity; this is a proxy theory, not QCD, and the thesis flags the difference.
ad hoc to paper The kinetic theory of the early plasma is governed by small-angle scattering (Fokker-Planck or Boltzmann with simplified kernel), excluding the Ibf2 term and other full QCD scattering mechanisms.
Sections 4.1.1.2 and 4.2.1.1; Appendix C.4 explicitly says the Ibf2 term is omitted. The hydrodynamization claim is for this restricted model.
domain assumption The plasma expansion is boost-invariant (Bjorken flow) with longitudinal scaling.
Section 4.1.1.1 sets up the time-dependent scaling; standard for early heavy-ion dynamics but restricts generality.
standard math Standard real-time thermal field theory, including KMS relations and the Schwinger-Keldysh contour, is valid.
Used in Sections 3.2.2 and 3.3.1 to define and manipulate the GGDs; no new axiom.

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Cite this review

Pith. "Pith review of Emergence, Formation and Dynamics of Hot QCD Matter." pith.science (2026). https://pith.science/paper/ACQLHUVJ

@misc{pith2026241220759,
  author       = {Pith},
  title        = {Pith review of: Emergence, Formation and Dynamics of Hot QCD Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACQLHUVJ}},
  note         = {Machine review of arXiv:2412.20759}
}

abstract

In this thesis, we make progress in two concrete directions in the vast landscape of hot QCD physics. The first one is quarkonium transport inside quark-gluon plasma (QGP), the high temperature phase of QCD. Over the past two decades it has been realized that a significant fraction of quarkonium suppression in high energy heavy ion collisions comes from dynamic dissociation and recombination processes, instead of static screening of the interaction potential as originally proposed by Matsui and Satz. Our contribution is the formulation of the precise correlation functions in QCD at finite temperature that describe the dissociation and recombination processes of heavy quarkonium in QGP, as well as their calculation in weakly coupled QCD and strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We also formulate the Euclidean version of these correlation functions so that they may be calculated using Lattice QCD techniques. The second contribution we make is the development of tools to understand the process of hydrodynamization in QCD kinetic theory and their application to a simplified description where only a subset of the QCD scattering mechanisms are included. By doing this, we learn that the process of hydrodynamization in this theory, and specifically, how memory of the initial condition is lost, follows the recently proposed Adiabatic Hydrodynamization scenario. Concretely, hydrodynamization proceeds through a sequential process in which a monotonously shrinking set of low-energy states dominate the dynamics, where the opening of an energy gap relative to the ground state(s) signals the start of each stage of this process. The hydrodynamic attractor is reached when only one low-energy state remains as the ground state, and the system approaches local thermal equilibrium following the adiabatic evolution of this low-energy state.

Figures

Figures reproduced from arXiv: 2412.20759 by the authors.

**Figure 1.1.** The Standard Model of Particle Physics, with all of the particles [PITH_FULL_IMAGE:figures/full_fig_p031_1_1.png] view at source ↗

**Figure 1.2.** Sketch of the QCD phase diagram, assuming a (conjectured) critical endpoint [PITH_FULL_IMAGE:figures/full_fig_p033_1_2.png] view at source ↗

**Figure 1.3.** Sketch of the stages that QCD matter undergoes in a Heavy-Ion collision. Figure [PITH_FULL_IMAGE:figures/full_fig_p038_1_3.png] view at source ↗

Figures from the paper (56 more)

**Figure 1.4.** Figure 1.4: Attractor curves for the ratio between longitudinal pressure and energy density [PITH_FULL_IMAGE:figures/full_fig_p039_1_4.png]

**Figure 1.5.** Figure 1.5: Top panels: nuclear modification factor RAA of Υ(1S) and Υ(2S) as a function of average number of participants. Middle panels: nuclear modification factor RAA of Υ(1S) and Υ(2S) as a function of rapidity. Bottom panel: nuclear modification factor RAA of Υ(1S) as a fu…

**Figure 1.6.** Figure 1.6: Left panels: nuclear modification factor [PITH_FULL_IMAGE:figures/full_fig_p045_1_6.png]

**Figure 1.7.** Figure 1.7: Left panel: nuclear modification factor RAA of Υ(1S), Υ(2S), and Υ(3S) as a function of average number of participants. Right panels: nuclear modification factor RAA of Υ(1S), Υ(2S), and Υ(3S) as a function of transverse momentum. Figures reproduced from [179]. QCD r…

**Figure 3.1.** Figure 3.1: A few perturbative Feynman diagrams for quarkonium dynamics (left) and its [PITH_FULL_IMAGE:figures/full_fig_p056_3_1.png]

**Figure 3.2.** Figure 3.2: Left: (Imaginary) Time integration contour in the imaginary-time formalism, [PITH_FULL_IMAGE:figures/full_fig_p070_3_2.png]

**Figure 3.3.** Figure 3.3: Feynman rules associated with different types of gauge boson propagators. [PITH_FULL_IMAGE:figures/full_fig_p074_3_3.png]

**Figure 3.4.** Figure 3.4: Feynman rules associated to the 3-gauge boson vertex, for both types of fields [PITH_FULL_IMAGE:figures/full_fig_p075_3_4.png]

**Figure 3.5.** Figure 3.5: Feynman rule associated with gauge boson insertions of Wilson lines. The color [PITH_FULL_IMAGE:figures/full_fig_p076_3_5.png]

**Figure 3.6.** Figure 3.6: List of all diagrams contributing to the electric field correlator ( [PITH_FULL_IMAGE:figures/full_fig_p078_3_6.png]

**Figure 3.7.** Figure 3.7: Diagrams with O((1−ξ) 2 ) gauge dependence, with V2 an arbitrary 2-gauge boson vertex to which the electric field, represented by the grey blob on the left, is connected in either of the two ways (a) and (b) shown. that are not directly connected to an electric field…

**Figure 3.8.** Figure 3.8: Decomposition of the imaginary part of the forward scattering amplitude of [PITH_FULL_IMAGE:figures/full_fig_p096_3_8.png]

**Figure 3.9.** Figure 3.9: Spectral function for quarkonium transport in weakly coupled QCD, as given [PITH_FULL_IMAGE:figures/full_fig_p100_3_9.png]

**Figure 3.10.** Figure 3.10: Diagrammatic representation of the chromoelectric field correlators for open [PITH_FULL_IMAGE:figures/full_fig_p104_3_10.png]

**Figure 3.11.** Figure 3.11: Feynman diagrams relevant for the difference between the chromoelectric field [PITH_FULL_IMAGE:figures/full_fig_p108_3_11.png]

**Figure 3.12.** Figure 3.12: Graphic representation of (3.188). Time increases toward the right of the figure. The contour γ µ starts at −(T /2)t µ , goes in a straight line to (T /2)t µ (in blue; this is the segment where − T 2 t µ < s < T 2 ), and then backtracks over itself (in red; this is …

**Figure 3.13.** Figure 3.13: Graphic representation of the antisymmetric deformations ( [PITH_FULL_IMAGE:figures/full_fig_p119_3_13.png]

**Figure 3.14.** Figure 3.14: The Schwinger-Keldysh contour, as discussed in this section, including the [PITH_FULL_IMAGE:figures/full_fig_p132_3_14.png]

**Figure 3.15.** Figure 3.15: Schematic representation of the chromoelectric field correlators relevant for 11 [PITH_FULL_IMAGE:figures/full_fig_p134_3_15.png]

**Figure 3.16.** Figure 3.16: Left: the relation ∆ϕ(πT zm) that determines the angular distance spanned on the S5 by the connected configuration that reaches a maximal AdS radial coordinate z = zm. Right: dimensionless configuration energy Ee for the extremal worldsheet that is described by a co…

**Figure 3.17.** Figure 3.17: Solid lines: real (blue) and imaginary (orange) parts of the mode solution [PITH_FULL_IMAGE:figures/full_fig_p144_3_17.png]

**Figure 3.18.** Figure 3.18: Real part (left) and imaginary part (right) of the non-Abelian electric field [PITH_FULL_IMAGE:figures/full_fig_p145_3_18.png]

**Figure 3.19.** Figure 3.19: Spectral function for quarkonium transport calculated in [PITH_FULL_IMAGE:figures/full_fig_p147_3_19.png]

**Figure 3.20.** Figure 3.20: Coupling dependence of the time-ordered chromoelectric correlator in vacuum [PITH_FULL_IMAGE:figures/full_fig_p156_3_20.png]

**Figure 3.21.** Figure 3.21: Spectral function for quarkonium transport in weakly coupled QCD with 2 [PITH_FULL_IMAGE:figures/full_fig_p158_3_21.png]

**Figure 3.22.** Figure 3.22: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p159_3_22.png]

**Figure 3.23.** Figure 3.23: Lattice discretization of the chromoelectric field correlator. The electric field [PITH_FULL_IMAGE:figures/full_fig_p167_3_23.png]

**Figure 3.24.** Figure 3.24: Regeneration/formation probability for an [PITH_FULL_IMAGE:figures/full_fig_p173_3_24.png]

**Figure 4.1.** Figure 4.1: We illustrate time-dependent scaling behavior and its connection to adiabaticity. [PITH_FULL_IMAGE:figures/full_fig_p181_4_1.png]

**Figure 4.2.** Figure 4.2: Evolution of time-dependent scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p187_4_2.png]

**Figure 4.3.** Figure 4.3: The rescaled distribution function w (4.1) for the numerical solutions of the Fokker-Planck equation with (gs, σ0) = (10−3 , 0.1). The left panel shows the results with A, B, C determined by the BMSS exponent while the right panel shows the same but with time-depende…

**Figure 4.4.** Figure 4.4: The same as Fig [PITH_FULL_IMAGE:figures/full_fig_p189_4_4.png]

**Figure 4.5.** Figure 4.5: Stream flow of the scaling exponents. Blue arrows represent the flow of the [PITH_FULL_IMAGE:figures/full_fig_p203_4_5.png]

**Figure 4.6.** Figure 4.6: Evolution of scaling exponents for solutions to Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p206_4_6.png]

**Figure 4.7.** Figure 4.7: We compare the evolution of the scaling exponents from Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p207_4_7.png]

**Figure 4.8.** Figure 4.8: The comparison between the evolution of the scaling exponents from Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p208_4_8.png]

**Figure 4.9.** Figure 4.9: Eigenstate coefficients ai (bottom panels) ordered by color, and effective temperature Teff = Ia/Ib (top panels) as a function of rescaled time τ˜ = λ0ℓCbτ . In the left panels, the adiabaticity-maximizing choice of scaling D(τ ) described in the text is used, while…

**Figure 4.10.** Figure 4.10: A comparison of the time evolution of the extracted scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p231_4_10.png]

**Figure 4.11.** Figure 4.11: A comparison of scaling exponents as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p232_4_11.png]

**Figure 4.12.** Figure 4.12: Eigenvalues corresponding to the time-dependent eigenstates of the effective [PITH_FULL_IMAGE:figures/full_fig_p233_4_12.png]

**Figure 4.13.** Figure 4.13: Sums over each energy “band” of eigenstate coefficients [PITH_FULL_IMAGE:figures/full_fig_p234_4_13.png]

**Figure 4.14.** Figure 4.14: Evolution of the typical momentum scales encoded in the scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p238_4_14.png]

**Figure 4.15.** Figure 4.15: Plot of the eigenvalues of the effective Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p240_4_15.png]

**Figure 4.16.** Figure 4.16: Plot of the coefficients in the H eigenstate decomposition of the distribution function for gs = 1 and the initial condition given in Eq. (4.161). The coefficients are normalized relative to the occupation of the zero-energy-eigenvalue state that carries the particl…

**Figure 4.17.** Figure 4.17: Plot of the eigenvalues of the effective Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p241_4_17.png]

**Figure 4.18.** Figure 4.18: Plot of the coefficients in the H eigenstate decomposition of the distribution function for gs = 2.6 and the initial condition given in Eq. (4.161). The coefficients are normalized relative to the occupation of the zero-energy-eigenvalue state that carries the parti…

**Figure 4.19.** Figure 4.19: Evolution of the typical momentum scales encoded in the scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p246_4_19.png]

**Figure 4.20.** Figure 4.20: Evolution of the typical momentum scales encoded in the scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p247_4_20.png]

**Figure 4.21.** Figure 4.21: Left: evolution of the pressure anisotropy as a function of the occupancy [PITH_FULL_IMAGE:figures/full_fig_p249_4_21.png]

**Figure 4.22.** Figure 4.22: Left: evolution of the pressure anisotropy as a function of the occupancy [PITH_FULL_IMAGE:figures/full_fig_p250_4_22.png]

**Figure 4.23.** Figure 4.23: Plot of the instantaenous eigenvalues of the effective Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p252_4_23.png]

**Figure 4.24.** Figure 4.24: Plot of the coefficients in the Heff (4.172) eigenstate decomposition of the distribution function for gs = 10−3 and the initial condition given in Eq. (4.180), grouped by bands of nearly degenerate eigenvalues (see [PITH_FULL_IMAGE:figures/full_fig_p253_4_24.png]

**Figure 4.25.** Figure 4.25: Plot of the instantaneous eigenvalues of the effective Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p254_4_25.png]

**Figure 4.26.** Figure 4.26: Plot of the coefficients in the Heff (4.172) eigenstate decomposition of the distribution function for gs = 1 and the initial condition given in Eq. (4.180). The coefficients are normalized relative to the occupation of the zero-energy-eigenvalue state that carries …

**Figure 4.27.** Figure 4.27: Plot of the eigenvalues of the effective Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p256_4_27.png]

**Figure 4.28.** Figure 4.28: Plot of the coefficients in the Heff (4.172) eigenstate decomposition of the distribution function for gs = 2.6 and the initial condition given in Eq. (4.180). The coefficients are normalized relative to the occupation of the zero-energy-eigenvalue state that carri…

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