We solve a random matrix model for QCD at finite chemical potential, obtained by generalizing the Stephanov model by modifying the random-matrix integration measure with a one-parameter trace deformation. This allows one to check how important the integration measure is for the qualitative features of random matrix models, as well as to test the robustness and universality of the qualitative picture of the original model. While for a small trace deformation the phase diagram is identical to that of the Stephanov model, for a large deformation an exotic phase with spontaneous charge-conjugation breaking appears.

Hungary

György Baranka

Matteo Giordano

Phase diagram of a generalized Stephanov model for finite-density QCD

We consider nonequilibrium evolution of non-Gaussian fluctuations within relativistic hydrodynamics relevant for the QCD critical point search in heavy-ion collision experiments. We rely on the hierarchy of relaxation timescales, which emerges in the hydrodynamic regime near the critical point, to focus on the slowest mode such as the fluctuations of specific entropy, whose equilibrium magnitude, non-Gaussianity, and typical relaxation time are increasing as the critical point is approached. We derive evolution equations for the non-Gaussian correlators of this diffusive mode in an arbitrary relativistic hydrodynamic flow. We compare with the simpler case of the stochastic diffusion on a static homogeneous background and identify terms which are specific to the case of the full hydrodynamics with pressure fluctuations and flow.

United States

Poland

Xin An

Gökçe Başar

Mikhail Stephanov

Ho-Ung Yee

Non-Gaussian fluctuation dynamics in relativistic fluids

We propose a general approach to freezing out fluctuations in heavy-ion collisions using the principle of maximum entropy. We find the results naturally expressed as a direct relationship between the irreducible relative correlators quantifying the deviations of hydrodynamic as well as hadron gas fluctuations from the ideal hadron gas baseline. The method also allows us to determine heretofore unknown parameters crucial for the freeze-out of fluctuations near the QCD critical point in terms of the QCD equation of state.

Maneesha Sushama Pradeep

Maximum Entropy Freeze-Out of Hydrodynamic Fluctuations

We compute the relaxation rate of the spin density of heavy quarks in a perturbative QCD plasma to leading-log order in the coupling constant g. The spin relaxation rate Γ s  in spin hydrodynamics is shown to be Γ s  ~ g 4 log(1/g)T(T/M)2 in the heavy-quark limit T/M ≪ 1, which is smaller than the relaxation rate of other non- hydrodynamic modes by additional powers of T/M. We demonstrate three different methods to evaluate the spin relaxation rate: 1) the Green-Kubo formula in the spin hydrodynamic regime, 2) the spin density correlation function in the strict hydrodynamic limit, and 3) quantum kinetic theory of the spin distribution function in momentum space. We highlight the interesting differences between these methods, while they are ultimately connected to each other by the underlying Ward-Takahashi identity for the non-conserved spin density.

Japan

Masaru Hongo

China

Xu-Guang Huang

Matthias Kaminski

Spin relaxation rate for heavy quarks in weakly coupled QCD plasma

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

Krishna Rajagopal

Yi Yin

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

Using the second law of local thermodynamics and the first-order Palatini formalism, we formulate relativistic spin hydrodynamics for quantum field theories with Dirac fermions, such as QED and QCD, in a torsionful curved background. We work in a regime where spin density, which is assumed to relax much slower than other non-hydrodynamic modes, is treated as an independent degree of freedom in an extended hydrodynamic description. Spin hydrodynamics in our approach contains only three non-hydrodynamic modes corresponding to a spin vector, whose relaxation time is controlled by a new transport coefficient: the rotational viscosity. We study linear response theory and observe an interesting mode mixing phenomenon between the transverse shear and the spin density modes. We propose several field-theoretical ways to compute the spin relaxation time and the rotational viscosity, via the Green-Kubo formula based on retarded correlation functions.

Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation

In the context of the search for the QCD critical point using non-Gaussian fluctuations, we obtain the evolution equations for non-Gaussian cumulants to the leading order of the systematic expansion in the magnitude of thermal fluctuations. We develop a diagrammatic technique in which the leading order contributions are given by tree diagrams. We introduce a Wigner transform for multipoint correlators and derive the evolution equations for three- and four-point Wigner functions for the problem of nonlinear stochastic diffusion with multiplicative noise.

Evolution of Non-Gaussian Hydrodynamic Fluctuations

To describe dynamics of bulk and fluctuations near the QCD critical point we develop general relativistic fluctuation formalism for a fluid carrying baryon charge. Feedback of fluctuations modifies hydrodynamic coefficients including bulk viscosity and conductivity and introduces nonlocal and noninstantaneous terms in constitutive equations. We perform necessary ultraviolet (short-distance) renormalization to obtain cutoff-independent deterministic equations suitable for numerical implementation. We use the equations to calculate the universal nonanalytic small-frequency dependence of transport coefficients due to fluctuations (long-time tails). Focusing on the critical mode we show how this general formalism matches existing <math><mtext>Hydro+</mtext></math> description of fluctuations near the QCD critical point and nontrivially extends it inside and outside of the critical region.

Fluctuation dynamics in a relativistic fluid with a critical point

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

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

We present a general systematic formalism for describing dynamics of fluctuations in an arbitrary relativistic hydrodynamic flow, including their feedback (known as long-time hydrodynamic tails). The fluctuations are described by two-point equal-time correlation functions. We introduce a definition of equal time in a situation where the local rest frame is determined by the local flow velocity and a method of taking derivatives and Wigner transforms of such equal-time correlation functions, which we call confluent. We find that not only do the equations for confluent Wigner functions resemble kinetic equations but also the kinetic equation for phonons propagating on an arbitrary background nontrivially matches the equations for Wigner functions, including relativistic inertial and Coriolis forces due to acceleration and vorticity of the flow. We also describe the procedure of renormalization of short-distance singularities which eliminates cutoff dependence, allowing efficient numerical implementation of these equations.