QCD thermalization: Ab initio approaches and interdisciplinary connections

Heavy-ion collisions at BNL’s Relativistic Heavy Ion Collider and CERN’s Large Hadron Collider provide strong evidence for the formation of a quark-gluon plasma, with temperatures extracted from relativistic viscous hydrodynamic simulations shown to be well above the transition temperature from hadron matter. Outstanding problems in QCD include how the strongly correlated quark-gluon matter forms in a heavy-ion collision, its properties off equilibrium, and the thermalization process in the plasma. The theoretical progress in this field in weak-coupling QCD effective field theories and in strong-coupling holographic approaches based on gauge-gravity duality is reviewed. The interdisciplinary connections of different stages of the thermalization process to nonequilibrium dynamics in other systems across energy scales ranging from inflationary cosmology to strong-field QED to ultracold atomic gases are outlined, with emphasis placed on the universal dynamics of nonthermal and hydrodynamic attractors. Measurements in heavy-ion collisions are surveyed that are sensitive to the early nonequilibrium stages of the collision and the potential for future measurements is discussed. The current state of the art in thermalization studies is summarized and promising avenues for further progress are identified.

Figure 1

Transverse hadron profile resolved in scattering with fixed squared momentum transfer $Q^2$ and increasing center-of-mass energy $\sqrt{s}$. The requirement for proliferating soft gluons to have maximal occupancy $1/{\alpha }_S$ generates the close packing saturation scale $Q_S$. Adapted from [416].

Figure 2

DIS in the dipole picture. The virtual photon emitted by the electron splits into a $q\overline{q}$ dipole that scatters off dynamical small $x$ gauge fields coupled to the static large-$x$ light cone sources. From [416].

Figure 3

Unintegrated gluon distribution (in units of the transverse area) vs the squared transverse momentum (normalized to its value at the peak of the initial condition curve) from the solution of the Balitsky-Kovchegov equation. The different curves represent increasing rapidities (left to right) for fixed and running coupling. From [282].

Figure 4

Solution of the JIMWLK equation for the correlator of Wilson lines $V(x_{\perp })V^{†}(y_{\perp })$ probed by the DIS dipole ([274]). As the nucleus is boosted from low energy (or rapidity) to high energy, the regions with large values of these correlator shrink spatially, corresponding to larger values of $Q_S$.

Figure 5

Top panel: multiparticle production from cut “vacuum-vacuum” graphs connecting time-dependent sources of the two nuclei after the collision. From [329]. Bottom panel: Schwinger-Keldysh closed time contour on which the sources and fields are defined.

Figure 6

Spacetime diagram of gauge field configurations. Before the collision, the gauge fields are pure gauge solutions with zero field strength. (In the text, the pure gauge solution of the right moving nucleus is denoted by $A_{1,\mathrm{cl}}^i$, and that of the left moving nucleus is denoted by $A_{2,\mathrm{cl}}^i$.) After the collision, the gauge field solution (${\mathcal{A}}^{i,\eta }$ in the text) corresponds to finite field strengths in the glasma. From [516].

Figure 7

Glasma flux tubes: boost-invariant LO glasma configurations of transverse size $1/Q_S$ at $\tau =0^{+}$ with parallel $E_{\eta }$ and $B_{\eta }$ corresponding to finite Chern-Simons charge. Such configurations decay rapidly and are unstable to quantum fluctuations. From [273].

Figure 8

Top panel: collisions of nuclei with subnucleon color charge fluctuations determined by the IP-Sat model. Bottom panel: LO energy density in the glasma at $\tau =1/Q_S$. From [657].

Figure 9

Growth of the maximally unstable Fourier mode of the longitudinal pressure $P_L={\tau }^2{T}^{\eta \eta }$. Note that since $g^2\mu \propto Q_S$, the results are in units of $Q_S^3/g^2$, with $g\sim {10}^{-5}$ and $L_{\eta }=1.6$. From [641].

Figure 10

Time evolution of the gluon distribution at early times $0\lesssim Q_S\tau \lesssim {\log }^2({\alpha }_S^{-1})$ from next-to-leading-order CGC initial conditions ([299]) at weak coupling (${\alpha }_S\sim {10}^{-6}$). From [136].

Figure 11

Time evolution of the longitudinal pressure-pressure correlator for different rapidity wave numbers $\nu$ with the parameters described by [137]. Once the initial fluctuations have grown larger, one observes the emergence of secondary instabilities at larger $\nu$ with enhanced growth rates.

Figure 12

Evolution in the occupancy-anisotropy plane. Indicated are the thermalization scenarios proposed by BMSS ([79]), BD ([169]), KM ([493]), and BGLMV ([159]). The blue lines show the results of classical-statistical simulations for different initial conditions ([117]; [136]). From [117].

Figure 13

Left panel: moments of the single-particle distribution function as a function of longitudinal momenta. The longitudinal spectra are evaluated at transverse momentum $p\simeq Q$. The different curves correspond to different times of the evolution: $Q\tau =750$, 1000, 1500, 2000, 3000 (from top to bottom). Right panel: the rescaled moments of the distribution function are found to collapse onto a single curve when plotted as a function of the rescaled longitudinal momentum variable. From [117].

Figure 14

The normalized distribution for the scalar theory $f_{\varphi }$ as a function of the rescaled longitudinal momentum at different times in the self-similar regime compared to the gauge theory $f_g$. From [121].

Figure 15

Self-similar behavior of the spatial Wilson loop as a function of the time-rescaled area $\sim t^{-\zeta }A$ with universal scaling exponent $\zeta$ for gauge groups $\mathrm{SU}(N_c)$ with $N_c=2$ (circles) and $N_c=3$ (triangles). From [131].

Figure 16

Time evolution of the hard scale (${\mathrm{\Lambda }}^2$), the electric screening scale ($m_D^2$), and the spatial string tension ($\sigma$). Symbols for $m_D^2$ and $\sigma$ denote lattice results for the different lattice spacings employed, while the solid green line represents a continuum extrapolation for the hard scale. The dotted lines represent perturbative estimates for the hard and Debye scales and a fit to the lattice results for the nonperturbative string tension scale. From [540].

Figure 17

Illustration of rescaled classical three and four vertices that are independent of the coupling. From [440].

Figure 18

Illustration of rescaled quantum three and four vertices that are $\sim g^4$. From [440].

Figure 19

Momentum differentiated plots of the (left panel) energy density, (center panel) number density, and (right panel) screening mass at different stages of bottom-up thermalization. All curves are normalized by the instantaneous number density $n$ and lines correspond to different times ($\tau Q_S={10}^0,{10}^1,{10}^3$).

Figure 20

Gluon kinetic theory equilibration in the anisotropy-occupancy plane for initial anisotropy ${\xi }_0=10$ and different values of the coupling constant. Times corresponding to $\tau Q_S={10}^0,{10}^1,{10}^3$ are indicated by black symbols. Simulations with the smaller initial anisotropy ${\xi }_0=4$ are shown as dashed curves. Adapted from [501].

Figure 21

Time evolution of instantaneous scaling exponents extracted from different sets of integral moments of the distribution. Horizontal lines indicate possible asymptotic values. From [552].

Figure 22

The shear viscosity over entropy ratio as a function of temperature at leading-order (LO) and (nearly) next-to-leading-order (NLO) thermal QCD. The bands correspond to the scale variation of running coupling prescriptions. From [345].

Figure 23

Pressure anisotropy evolution in expanding geometry. Gluon kinetic theory simulations $\lambda =1,\dots ,10$ are compared to a supersymmetric Yang-Mills holographic model ($\lambda =\infty$ ). Note that here $T_i$ is the initial temperature, so at late times $(\eta /s{)}^{4/3}{T}_{i}t\approx 32{\tilde{w}}^{3/2}$. Adapted from [445].

Figure 24

Hydrodynamic attractors for preequilibrium evolution of energy density for different microscopic theories. From [349].

Figure 25

Energy density evolution in a chemically equilibrating quark-gluon plasma. The vertical lines indicate the times of approximate hydrodynamic, chemical, and thermal equilibriums. From [488].

Figure 26

Penrose diagrams dual to far-from-equilibrium states in strongly coupled QFTs. (a) The system starts in the vacuum with known bulk geometry and is perturbed by a nontrivial source, which appears as an asymptotic boundary condition in gravity. After the source is turned off, the QFT is in a nonequilibrium state modeled by a time-dependent geometry. (b) The sources are always off, but one instead specifies nontrivial initial conditions for the bulk metric. Adapted from [396].

Figure 27

Evolution of the effective temperature as a function of time for three different states with nonzero initial energy density. The gray curves denote the far-from-equilibrium regime. The blue dashed area extending indefinitely to the right mark the applicability of viscous hydrodynamic relations truncated at the third order in derivatives (122). The red dotted curves denote the series in Eq. (120) extracted using the method of [146]. Adapted from [391].

Figure 28

$⟨T^{\mu \nu }⟩$ in a holographic CFT as a function of the dimensionless clock variable $w$ for 29 different initial states (gray curves). Magenta, blue, and green curves denote predictions of hydrodynamic constitutive relations truncated, respectively, at first, second, and third order [Eq. (122)]. The orange curve is the hydrodynamic attractor ([639]). Adapted from [391], [392], and [639].

Figure 29

Hydrodynamization of states whose gravity dual initially has support close to the boundary (dashed curves) or deep in the bulk (solid curves) initialized at different times (different colors); see the text for details. Adapted from [496].

Figure 30

$⟨T^{\mu \nu }⟩$ resulting from a collision of thin planar shocks with $\varrho d=0.08$ (124). Top panel: lab-frame energy density as a function of time $x^0$ and longitudinal position $x^3$. Between the remnants and the central rapidity region, there are small regions of negative energy density. Bottom-left panel: at midrapidity, the transverse and longitudinal pressure after the collision are consistent with $⟨T^{00}⟩\sim {\tau }^2$ in Eq. (120). Bottom-right panel: the color encoding denotes deviations from constitutive relations and points to the applicability of hydrodynamics. The postcollision $⟨T^{\mu \nu }⟩$ does not have a rest frame in the gray region ([53]). Adapted from [213].

Figure 31

Energy density in holographic heavy-ion collisions with transverse dynamics. Top panels: off-central collision with modest elliptic flow. Bottom panels: proton-nucleus collision as modeled with a shock wave with a small Gaussian extent in the transverse plane (left projectile) and a planar shock (right projectile). The smaller projectile punches out a hole in the larger projectile and excites matter at midrapidity, leading to substantial radial flow. Adapted from [224], and [237].

Figure 32

(a) Absorption images of different magnetic hyperfine states of a spin-1 Bose gas with the extracted transversal spin (solid lines) for three different far-from-equilibrium initial conditions. (b) All initial conditions lead to the same universal scaling behavior, such that all data points collapse onto a single curve after rescaling with time using the universal exponents $\alpha$ and $\beta$. From [624].