<p>We present a detailed numerical and analytical study of the out-of-equilibrium dynamics of Model G, the dynamical universality class relevant to the chiral phase transition. We perform numerical 3D stochastic (Langevin) simulations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></math> critical point for large lattices in the chiral limit. We quench the system from the high-temperature unbroken phase to the broken phase and study the nonequilibrium dynamics of pion fields. Strikingly, the nonequilibrium evolution of the two-point functions exhibits a regime of growth, a parametrically large enhancement, and a subsequent slow relaxation to equilibrium. We analyze our numerical results using dynamic critical scaling and mean-field theory. The growth of the two-point functions is determined by the nonlinear dynamics of an ideal non-Abelian superfluid, which is a limit of Model G that reflects the broken chiral symmetry. We also relate the nonequilibrium two-point functions to a long-lived parametric enhancement of soft pion yields relative to thermal equilibrium following a quench.</p>

Germany

United States

Adrien Florio

Italy

Eduardo Grossi

Aleksas Mazeliauskas

Slovenia

Alexander Soloviev

Derek Teaney

Quenching through the QCD chiral phase transition

<p>We discuss a universal nonequilibrium enhancement of long-wavelength Goldstone bosons induced by quenches to the broken phase in Model G—the dynamical universality class of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> antiferromagnet and the chiral phase transition in QCD. Scaling arguments for the coarsening dynamics describing the formation of the chiral condensate predict a parametric enhancement in the infrared spectra of Goldstone bosons, a prediction confirmed by stochastic simulations of the transition. The details of the enhancement are determined by the nonlinear dynamics of a superfluid effective theory, which is a limit of Model G reflecting the broken <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> symmetry. Our results translate to a parametric enhancement of low-momentum pions in heavy-ion collisions at the LHC, which are underpredicted in current hydrodynamic models without critical dynamics.</p>

Supercooled Goldstone Bosons at the QCD Chiral Phase Transition

We present a detailed study of the finite momentum dynamics of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point of QCD, which lies in the dynamic universality class of “model G.” The critical scaling of the model is analyzed in multiple dynamical channels. For instance, the finite momentum analysis allows us to precisely extract the pion dispersion curve below the critical point. The pion velocity is in striking agreement with the predictions relation and static universality. The pion damping rate and velocity are both consistent with the dynamical critical exponent <math><mi>ζ</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> of model G. Similarly, although the critical amplitude for the diffusion coefficient of the conserved <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> charges is small, it is clearly visible both in the restored phase and with finite explicit symmetry breaking, and its dynamical scaling is again consistent with <math><mi>ζ</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math>. We determine a new set of universal dynamical critical amplitude ratios relating the diffusion coefficient to a suitably defined order parameter relaxation time. We also show that in a finite volume simulation, the chiral condensate diffuses on the coset manifold in a manner consistent with dynamical scaling, and with a diffusion coefficient that is determined by the transport coefficients of hydrodynamic pions. Finally, the amplitude ratios (together with other nonuniversal amplitudes also reported here) compile all relevant information for further studies of model G both in and out of equilibrium.

Dynamics of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point in QCD: Critical pions and diffusion in model G

We determine the kinetic coefficients of ultrasoft pions using chiral perturbation theory at finite temperature close to the chiral limit. This is used to compute the axial charge diffusion and damping coefficients in the hydrodynamic effective theory for these pion waves. We show that to provide a leading order answer for these coefficients one needs to explore the dynamics of hard, soft, and ultrasoft pion modes, which are represented microscopically by the appropriate kinetic and hydrodynamic descriptions.

Spain

Juan M. Torres-Rincon

Kinetics of hydrodynamic pions in chiral perturbation theory

We perform a Langevin simulation of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point, which lies in the dynamic universality class of “model G.” This is the dynamic universality class of the chiral phase transition in QCD with two massless flavors. The axial charge and the order parameter <math><msub><mi>ϕ</mi><mi>a</mi></msub><mo>=</mo><mo>(</mo><mi>σ</mi><mo>,</mo><mover><mi>π</mi><mo>→</mo></mover><mo>)</mo></math> exhibit a rich dynamical interplay, which reflects the qualitative differences in the hydrodynamic effective theories above and below <math><msub><mi>T</mi><mi>c</mi></msub></math>. From the axial charge correlators on the critical line we extract a dynamical critical exponent of <math><mrow><mi>ζ</mi><mo>=</mo><mn>1.47</mn><mo>±</mo><mn>0.01</mn><mo>(</mo><mtext>stat.</mtext><mo>)</mo></mrow></math>, which is compatible with the theoretical expectation of <math><mi>ζ</mi><mo>=</mo><mi>d</mi><mo>/</mo><mn>2</mn></math> (with <math><mi>d</mi><mo>=</mo><mn>3</mn></math>) when systematic errors are taken into account. At low temperatures, we quantitatively match the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> simulations to the superfluid effective theory of soft pions.

France

Dynamics of the <math><mrow><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math> critical point in QCD

An energetic parton traveling through a quark-gluon plasma loses energy via occasional hard scatterings and frequent softer interactions. Whether or not these interactions admit a perturbative description, the effect of the soft interactions can be factorized and encoded in a small number of transport coefficients. In this work, we present the numerical implementation of a hard-soft factorized parton energy loss model which combines a stochastic description of soft interactions and rate-based modeling of hard scatterings. We introduce a scale to estimate the regime of validity of the stochastic description, allowing for a better understanding of the model's applicability at small and large coupling. We study the energy and fermion-number cascade of energetic partons as an application of the model.

Tianyu Dai

Jean-François Paquet

Steffen A. Bass

Parton energy loss in a hard-soft factorized approach

We analyze the evolution of hydrodynamic fluctuations for QCD matter below <math><msub><mi>T</mi><mi>c</mi></msub></math> in the chiral limit, where the pions (the Goldstone modes) must be treated as additional non-Abelian superfluid degrees of freedom, reflecting the broken <math><mi>S</mi><msub><mi>U</mi><mi>L</mi></msub><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>S</mi><msub><mi>U</mi><mi>R</mi></msub><mo>(</mo><mn>2</mn><mo>)</mo></math> symmetry of the theory. In the presence of a finite pion mass <math><msub><mi>m</mi><mi>π</mi></msub></math>, the hydrodynamic theory is ordinary hydrodynamics at long distances, and superfluidlike at short distances. The presence of the superfluid degrees of freedom then gives specific contributions to the bulk viscosity, the shear viscosity, and diffusion coefficients of the ordinary theory at long distances which we compute. This determines, in some cases, the leading dependence of the transport parameters of QCD on the pion mass. We analyze the predictions of this computation, as the system approaches the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point.

Fanglida Yan

Transport and hydrodynamics in the chiral limit

In the context of ultra-relativistic nuclear collisions, we present a fast method for calculating the final particle spectra after the direct decay of resonances from a Cooper–Frye integral over the freeze-out surface. The method is based on identifying components of the final particle spectrum that transform in an irreducible way under rotations in the fluid-restframe. Corresponding distribution functions can be pre-computed including all resonance decays. Just a few of easily tabulated scalar functions then determine the Lorentz invariant decay spectrum from each space-time point, and simple integrals of these scalar functions over the freeze-out surface determine the final decay products. This by-passes numerically costly event-by-event calculations of the intermediate resonances. The method is of considerable practical use for making realistic data to model comparisons of the identified particle yields and flow harmonics, and for studying the viscous corrections to the freeze-out distribution function.

Stefan Floerchinger

Fast resonance decays in nuclear collisions

We develop a macroscopic description of the space-time evolution of the energy-momentum tensor during the pre-equilibrium stage of a high-energy heavy-ion collision. Based on a weak coupling effective kinetic description of the microscopic equilibration process (à la “bottom-up”), we calculate the nonequilibrium evolution of the local background energy-momentum tensor as well as the nonequilibrium linear response to transverse energy and momentum perturbations for realistic boost-invariant initial conditions for heavy-ion collisions. We demonstrate how this framework can be used on an event-by-event basis to propagate the energy-momentum tensor from far-from-equilibrium initial-state models to the time <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> when the system is well described by relativistic viscous hydrodynamics. The subsequent hydrodynamic evolution becomes essentially independent of the hydrodynamic initialization time <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> as long as <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> is chosen in an appropriate range where both kinetic and hydrodynamic descriptions overlap. We find that for <math><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>2.76</mn><mspace width="0.16em"></mspace><mspace width="0.16em"></mspace><mtext>TeV</mtext></mrow></math> central Pb-Pb collisions, the typical timescale when viscous hydrodynamics with shear viscosity over entropy ratio <math><mrow><mi>η</mi><mo>/</mo><mi>s</mi><mo>=</mo><mn>0.16</mn></mrow></math> becomes applicable is <math><mrow><msub><mi>τ</mi><mtext>hydro</mtext></msub><mo>∼</mo><mn>1</mn><mspace width="0.16em"></mspace><mrow><mi>fm</mi><mo>/</mo><mi>c</mi></mrow></mrow></math> after the collision.

CERN

Aleksi Kurkela

Sören Schlichting

Effective kinetic description of event-by-event pre-equilibrium dynamics in high-energy heavy-ion collisions

High-energy nuclear collisions produce a nonequilibrium plasma of quarks and gluons which thermalizes and exhibits hydrodynamic flow. There are currently no practical frameworks to connect the early particle production in classical field simulations to the subsequent hydrodynamic evolution. We build such a framework using nonequilibrium Green’s functions, calculated in QCD kinetic theory, to propagate the initial energy-momentum tensor to the hydrodynamic phase. We demonstrate that this approach can be easily incorporated into existing hydrodynamic simulations, leading to stronger constraints on the energy density at early times and the transport properties of the QCD medium. Based on (conformal) scaling properties of the Green’s functions, we further obtain pragmatic bounds for the applicability of hydrodynamics in nuclear collisions.