We calculate the probability distribution $\textit{P}$( ) for a heavy quark with velocity $\textit{v}$ propagating through strongly coupled $$ \mathcal{N} $$ = 4 SYM plasma in the ’t Hooft limit ($\textit{N}$ → ∞, $\textit{λ}$ = $\textit{g}$ $\textit{N}$ → ∞) at a temperature $\textit{T}$ to acquire a momentum due to interactions with the plasma. This distribution encodes the well-known drag coefficient $\textit{η}$ and the transverse and longitudinal momentum diffusion coefficients $\textit{κ}$ and $\textit{κ}$ . The jet quenching parameter $$ \hat{q} $$ can be extracted from $\textit{P}$( ) for $\textit{v}$ = 1. Going beyond these known Gaussian characteristics of $\textit{P}$( ), our calculation determines all of the higher order and mixed moments to leading order in 1/ $$ \sqrt{\lambda } $$ for the first time. These non-Gaussian features of $\textit{P}$( ) include qualitatively novel correlations between longitudinal energy loss and transverse momentum broadening at nonzero $\textit{v}$. We show that all higher moments scale characteristically with an effective temperature of the boosted plasma in the heavy quark rest frame, and we demonstrate that these non-Gaussian characteristics can be sizable in magnitude and even dominant in physically relevant situations. We use these results to derive a Kolmogorov equation for the evolution of the probability distribution for the total momentum of a heavy quark that propagates through strongly coupled plasma. This evolution equation accounts for all higher order correlations between transverse momentum broadening and longitudinal energy loss, which we have calculated from first principles. It reduces to a Fokker-Planck equation when truncated to only include the effects of $\textit{η}$ , $\textit{κ}$ and $\textit{κ}$ . Remarkably, while heavy quarks do not reach kinetic equilibrium with the plasma if evolved with this Fokker-Planck equation, by showing that the Boltzmann distribution is a static solution of the all-order Kolmogorov equation that we have derived we demonstrate that heavy quarks do reach kinetic equilibrium if evolved with this equation. Our results thus provide a dynamically complete framework for understanding the thermalization of a heavy quark that may be initially far from equilibrium in the strongly coupled $$ \mathcal{N} $$ = 4 SYM plasma — as well as new insight into heavy quark transport and equilibration in quark-gluon plasma.

United States

Krishna Rajagopal

Bruno Scheihing-Hitschfeld

Switzerland

Urs Wiedemann

Dynamics of heavy quarks in strongly coupled $$ \mathcal{N} $$ = 4 SYM plasma

“Attractor” solutions for the pre-hydrodynamic, far-from-equilibrium, evolution of the matter produced in relativistic heavy ion collisions have emerged as crucial descriptors of the rapid hydrodynamization of quark-gluon plasma (QGP). Adiabatic Hydrodynamization (AH) has been proposed as a framework with which to describe, explain, and predict attractor behavior that draws upon an analogy to the adiabatic approximation in quantum mechanics. In this work, we systematize the description of pre-hydrodynamic attractors in kinetic theory by showing how to use the AH framework to identify these long-lived solutions to which varied initial conditions rapidly evolve, demonstrating the robustness of this framework. In a simplified QCD kinetic theory in the small-angle scattering limit, we use AH to explain both the early- and late-time scaling behavior of a longitudinally expanding gluon gas in a unified framework. In this context, we show that AH provides a unified description of, and intuition for, all the stages of what in QCD would be bottom-up thermalization, starting from a pre-hydrodynamic attractor and ending with hydrodynamization. We additionally discuss the connection between the notions of scaling behavior and adiabaticity and the crucial role of time-dependent coordinate redefinitions in identifying the degrees of freedom of kinetic theories that give rise to attractor solutions. The tools we present open a path to the intuitive explanation of how attractor behavior arises and how the attractor evolves in all stages of the hydrodynamization of QGP in heavy ion collisions.

Rachel Steinhorst

Adiabatic Hydrodynamization and the emergence of attractors: a unified description of hydrodynamization in kinetic theory

As the partons in a high energy jet propagate through the droplet of quark-gluon plasma (QGP) produced in a heavy-ion collision they lose energy to, kick, and are kicked by the medium. The resulting modifications to the parton shower encode information about the microscopic nature of QGP. A direct consequence, however, is that the momentum and energy lost by the parton shower are gained by the medium and, since QGP is a strongly coupled liquid, this means that the jet excites a wake in the droplet of QGP. After freezeout, this wake becomes soft hadrons with net momentum in the jet direction meaning that what an experimentalist later reconstructs as a jet includes hadrons originating from both the modified parton shower and its wake. This has made it challenging to find experimental observables that provide an unambiguous view of the dynamical response of a droplet of QGP to a jet shooting through it. Recent years have seen significant substantial advances in the theoretical and experimental understanding of the substructure of jets, in particular, using correlation functions,   <math> <mfenced> <mrow> <mi>E</mi> <mfenced> <msub> <mover> <mi>n</mi> <mo>→</mo> </mover> <mn>1</mn> </msub> </mfenced> <mo>⋯</mo> <mi>E</mi> <mfenced> <msub> <mover> <mi>n</mi> <mo>→</mo> </mover> <mi>k</mi> </msub> </mfenced> </mrow> </mfenced> </math>  $$ \left\langle \mathcal{E}\left({\overrightarrow{n}}_1\right)\cdots \mathcal{E}\left({\overrightarrow{n}}_k\right)\right\rangle $$ , of the energy flux operator in proton-proton collisions and, recently, in heavy-ion collisions. So far, such studies have focused primarily on the two-point correlator, which allows for the identification of the angular scale of the underlying dynamics. Higher-point correlators hold the promise of mapping out the dynamics themselves. In this paper we perform the first study of the shape-dependent three-point energy-energy-energy correlator in heavy-ion collisions. Using the Hybrid Model to simulate the interactions of high energy jets with the QGP medium, we show that the three-point correlator presents us with a striking new opportunity. We find that hadrons originating from wakes are the dominant contribution to the three-point correlator in the kinematic regime in which the three points are well-separated in angle, forming a roughly equilateral triangle. This equilateral region of the correlator is far from the region populated by collinear vacuum emissions, making it a canvas on which jet wakes are laid out, where experimentalists can map their shapes. Our work provides a key step towards the systematic use of energy correlators to image and unravel the dynamical response of a droplet of QGP that has been probed by a passing jet, and motivates numerous experimental and theoretical studies.

Hannah Bossi

United Kingdom

Arjun Kudinoor

Ian Moult

Spain

Daniel Pablos

Ananya Rai

Imaging the wakes of jets with energy-energy-energy correlators

We introduce a freeze-out procedure to convert the critical fluctuations in a droplet of quark-gluon plasma (QGP) that has, as it expanded and cooled, passed close to a posited critical point on the phase diagram into cumulants of hadron multiplicities that can subsequently be measured. The procedure connects the out-of-equilibrium critical fluctuations described in concert with the hydrodynamic evolution of the droplet of QGP by extended hydrodynamics, known as <math><mrow><mtext>Hydro</mtext><mo>+</mo></mrow></math>, with the subsequent kinetic description in terms of observable hadrons. We introduce a critical scalar isoscalar field sigma whose fluctuations cause correlations between observed hadrons due to the couplings of the sigma field to the hadrons via their masses. We match the QGP fluctuations obtained by solving the <math><mrow><mtext>Hydro</mtext><mo>+</mo></mrow></math> equations describing the evolution of critical fluctuations before freeze-out to the correlations of the sigma field. In turn, these are imprinted onto correlations and fluctuations in the multiplicity of hadrons, most importantly protons, after freeze-out via the generalization of the familiar half-century-old Cooper-Frye freeze-out prescription which we introduce. The proposed framework allows us to study the effects of critical slowing down and the consequent deviation of the observable predictions from equilibrium expectations quantitatively. We also quantify the suppression of cumulants due to conservation of baryon number. We demonstrate the procedure in practice by freezing out a <math><mrow><mtext>Hydro</mtext><mo>+</mo></mrow></math> simulation in an azimuthally symmetric and boost invariant background that includes radial flow discussed in Rajagopal et al. [Phys. Rev. D 102, 094025 (2020)].

Maneesha Pradeep

Mikhail Stephanov

China

Yi Yin

Freezing out fluctuations in <math><mrow><mtext>Hydro</mtext><mo>+</mo></mrow></math> near the QCD critical point

The selection of jets in heavy-ion collisions based on their p  T  after jet quenching is known to bias towards jets that lost little energy in the quark-gluon plasma. In this work, we study and quantify the impact of this selection bias on jet substructure observables so as to isolate effects caused by the modification of the substructure of jets by quenching. We do so at first in a simplified Monte Carlo study in which it is possible to identify the same jet before and after quenching. We show explicitly that jets selected based on their quenched (i.e. observable) p  T  have substantially smaller fractional energy loss than those selected based on the p  T  that they would have had in the absence of any quenching. This selection bias has a large impact on jet structure and substructure observables. As an example, we consider the angular separation ∆R of the hardest splitting in each jet, and find that the ∆R distribution of the (biased) sample of jets selected based upon their quenched p  T  is almost unmodified by quenching. In contrast, quenching causes dramatic modifications to the ∆R distribution of a sample of jets selected based upon their unquenched p  T , with a significant enhancement at larger ∆R coming from the soft particles originating from the wake of the jet in the quark-gluon plasma. The jets which contribute to this enhancement are those which have lost the most energy and which were, therefore, left out of the sample selected after quenching. In a more realistic study, we then show that the same qualitative effects can all be observed in Z+jet events. Selecting jets in such events based on either the jet p  T  or the Z-boson p  T  provides an experimentally accessible way to quantify the effects of selection biases in jet observables and separate them from the modification of jet substructure caused by quenching. Selecting Z+jet events based upon the jet p  T  yields a ∆R distribution that appears almost unmodified whereas selecting Z+jet events based upon the Z-boson p  T  reveals a significant modification to the ∆R-distribution caused by quenching, once again arising from the wakes of those jets that lose more energy.

CERN

Jasmine Brewer

Quinn Brodsky

Disentangling jet modification in jet simulations and in Z+jet data

We explore how to improve the hybrid model description of the particles originating from the wake that a jet produced in a heavy ion collision leaves in the droplet of quark-gluon plasma (QGP) through which it propagates, using linearized hydrodynamics on a background Bjorken flow. Jet energy and momentum loss described by the hybrid model become currents sourcing linearized hydrodynamics. By solving the linearized hydrodynamic equations numerically, we investigate the development of the wake in the dynamically evolving droplet of QGP, study the effect of viscosity, scrutinize energy-momentum conservation, and check the validity of the linear approximation. We find that linearized hydrodynamics works better in the viscous case because diffusive modes damp the energy-momentum perturbation produced by the jet. We calculate the distribution of particles produced from the jet wake by using the Cooper-Frye prescription and find that both the transverse momentum spectrum and the distribution of particles in azimuthal angle are similar in shape in linearized hydrodynamics and in the hybrid model. Their normalizations are different because the momentum-rapidity distribution in the linearized hydrodynamics analysis is more spread out, due to sound modes. Since the Bjorken flow has no transverse expansion, we explore the effect of transverse flow by using local boosts to add it into the Cooper-Frye formula. After including the effects of transverse flow in this way, the transverse momentum spectrum becomes harder: more particles with transverse momenta bigger than 2 GeV are produced than in the hybrid model. Although we defer implementing this analysis in a jet Monte Carlo, as would be needed to make quantitative comparisons to data, we gain a qualitative sense of how the jet wake may modify jet observables by computing proxies for two example observables: the lost energy recovered in a cone of varying open angle, and the fragmentation function. We find that linearized hydrodynamics with transverse flow effects added improves the description of the jet wake in the hybrid model in just the way that comparison to data indicates is needed. Our study illuminates a path to improving the description of the wake in the hybrid model, highlighting the need to take into account the effects of both transverse flow and the broadening of the energy-momentum perturbation in spacetime rapidity on particle production.

Jorge Casalderrey-Solana

Portugal

José Milhano

Norway

Xiaojun Yao

Jet wake from linearized hydrodynamics

Upcoming experimental programs, including the Beam Energy Scan at RHIC, will look for signatures of a possible critical point in the QCD phase diagram in fluctuation observables. To understand and predict these signatures, one must account for the fact that the dynamics of any critical fluctuations must be out-of-equilibrium: because of critical slowing down, the fluctuations cannot stay in equilibrium as the droplet of quark gluon plasma produced in a collision expands and cools. Furthermore, their out-of-equilibrium dynamics must also influence the hydrodynamic evolution of the cooling droplet. The recently developed Hydro+ formalism allows for a consistent description of both the hydrodynamics and the out-of-equilibrium fluctuations, including the feedback between them. We shall provide an explicit demonstration of how this works, setting up a Hydro+ simulation in a simplified setting: a rapidity-independent fireball undergoing radial flow with an equation of state in which we imagine a critical point close to the <math><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>0</mn></math> axis of the phase diagram. Within this setup, we show that we can quantitatively capture nonequilibrium phenomena, including critical fluctuations over a range of scales and memory effects. Furthermore, we illustrate the interplay between the dynamics of the fluctuations and the hydrodynamic flow of the fireball: as the fluid cools and flows, the dynamical fluctuations lag relative to how they would evolve if they stayed in equilibrium; there is then a backreaction on the flow itself due to the out-of-equilibrium fluctuations; and, in addition, the radial flow transports fluctuations outwards by advection. Within our model, we find that the backreaction from the out-of-equilibrium fluctuations does not yield dramatically large effects in the hydrodynamic variables. Further work will be needed in order to check this quantitative conclusion in other settings but, if it persists, this will considerably simplify future modeling.

Gregory W. Ridgway

Ryan Weller

Understanding the out-of-equilibrium dynamics near a critical point in the QCD phase diagram

We construct a family of equations of state for QCD in the temperature range <math><mrow><mn>30</mn><mspace width="0.16em"></mspace><mi>MeV</mi><mo>≤</mo><mi>T</mi><mo>≤</mo><mn>800</mn><mspace width="0.16em"></mspace><mi>MeV</mi></mrow></math> and in the chemical potential range <math><mrow><mn>0</mn><mo>≤</mo><msub><mi>μ</mi><mi>B</mi></msub><mo>≤</mo><mn>450</mn><mspace width="0.16em"></mspace><mi>MeV</mi></mrow></math>. These equations of state match available lattice QCD results up to <math><mrow><mi>O</mi><mo>(</mo><msubsup><mi>μ</mi><mi>B</mi><mn>4</mn></msubsup><mo>)</mo></mrow></math> and in each of them we place a critical point in the three-dimensional (3D) Ising model universality class. The position of this critical point can be chosen in the range of chemical potentials covered by the second Beam Energy Scan at the Relativistic Heavy Ion Collider. We discuss possible choices for the free parameters, which arise from mapping the Ising model onto QCD. Our results for the pressure, entropy density, baryon density, energy density, and speed of sound can be used as inputs in the hydrodynamical simulations of the fireball created in heavy ion collisions. We also show our result for the second cumulant of the baryon number in thermal equilibrium, displaying its divergence at the critical point. In the future, comparisons between RHIC data and the output of the hydrodynamic simulations, including calculations of fluctuation observables, built upon the model equations of state that we have constructed may be used to locate the critical point in the QCD phase diagram, if there is one to be found.

Germany

Paolo Parotto

France

Poland

Marcus Bluhm

Debora Mroczek

Marlene Nahrgang

J. Noronha-Hostler

Claudia Ratti

Thomas Schäfer

QCD equation of state matched to lattice data and exhibiting a critical point singularity

Sufficiently energetic collisions of heavy ions result in the formation of a droplet of a strongly coupled liquid state of QCD matter known as quark-gluon plasma. By using gauge-gravity duality (holography), a model of a rapidly hydrodynamizing and thermal- izing process like this can be constructed by colliding sheets of energy density moving at the speed of light and tracking the subsequent evolution. In this work, we consider the dual gravitational description of such collisions in the most general bulk theory with a four-derivative gravitational action containing a dynamical metric and a gauge field in five dimensions. Introducing the bulk gauge field enables the analysis of collisions of sheets which carry nonzero “baryon” number density in addition to energy density. Introducing the four-derivative terms enables consideration of such collisions in a gauge theory with finite gauge coupling, working perturbatively in the inverse coupling. While the dynamics of energy and momentum in the presence of perturbative inverse-coupling corrections has been analyzed previously, here we are able to determine the effect of such finite coupling corrections on the dynamics of the density of a conserved global charge, which we take as a model for the dynamics of nonzero baryon number density. In accordance with expec- tations, as the coupling is reduced we observe that after the collisions less baryon density ends up stopped at mid-rapidity and more of it ends up moving near the lightcone.

Asmund Folkestad

Sašo Grozdanov

Netherlands

Wilke Schee

Coupling constant corrections in a holographic model of heavy ion collisions with nonzero baryon number density

By finding rare (but not exponentially rare) large-angle deflections of partons within a jet produced in a heavy ion collision, or of such a jet itself, experimentalists can find the weakly coupled short-distance quark and gluon particles (scatterers) within the strongly coupled liquid quark-gluon plasma (QGP) produced in heavy ion collisions. This is the closest one can come to probing QGP via a scattering experiment and hence is the best available path toward learning how a strongly coupled liquid emerges from an asymptotically free gauge theory. The short-distance, particulate, structure of liquid QGP can be revealed in events in which a jet parton resolves, and scatters off, a parton from the droplet of QGP. The probability for picking up significant transverse momentum via a single scattering was calculated previously, but only in the limit of infinite parton energy which means zero angle scattering. Here, we provide a leading order perturbative QCD calculation of the Molière scattering probability for incident partons with finite energy, scattering at a large angle. We set up a thought experiment in which an incident parton with a finite energy scatters off a parton constituent within a “brick” of QGP, which we treat as if it were weakly coupled, as appropriate for scattering with large momentum transfer, and compute the probability for a parton to show up at a nonzero angle with some energy. We include all relevant channels, including those in which the parton that shows up at a large angle was kicked out of the medium as well as the Rutherford-like channel in which what is seen is the scattered incident parton. The results that we obtain will serve as inputs to future jet Monte Carlo calculations and can provide qualitative guidance for how to use future precise, high statistics, suitably differential measurements of jet modification in heavy ion collisions to find the scatterers within the QGP liquid.

Italy

Francesco D’Eramo

Molière scattering in quark-gluon plasma: finding point-like scatterers in a liquid