Strongly coupled quark-gluon plasma in heavy ion collisions

A decade ago, a brief summary of the field of the relativistic heavy ion physics could be formulated as the discovery of strongly coupled quark-gluon plasma, sQGP for short, a near-perfect fluid with surprisingly large entropy-density-to-viscosity ratio. Since 2010, the LHC heavy ion program added excellent new data and discoveries. Significant theoretical efforts have been made to understand these phenomena. Now there is a need to consolidate what we have learned and formulate a list of issues to be studied next. Studies of angular correlations of two and more secondaries reveal higher harmonics of flow, identified as the sound waves induced by the initial state perturbations. As in cosmology, detailed measurements and calculations of these correlations helped to make our knowledge of the explosion much more quantitative. In particular, their damping had quantified the viscosity. Other kinetic coefficients—the heavy-quark diffusion constants and the jet quenching parameters—also show enhancements near the critical point $T\approx T_c$. Since densities of QGP quarks and gluons strongly decrease at this point, these facts indicate large role of nonperturbative mechanisms, e.g., scattering on monopoles. New studies of the $pp$ and $pA$ collisions at high multiplicities reveal collective explosions similar to those in heavy ion $AA$ collisions. These “smallest drops of the sQGP” revived debates about the initial out-of-equilibrium stage of the collisions and mechanisms of subsequent equilibration.

Figure 1

Elliptic flow parameter $v_2$, the experimental data vs the hydrodynamical predictions for pions (upper plot) and protons (lower plot). From [3].

Figure 2

.The ratio $R_{AA}$ of the yield of high $p_{\perp }$ hadrons in $d\text{−}\mathrm{Au}$ and Au-Au collisions to the expected one according to parton model scaling. Strong deviations from one of this ration indicate the jet quenching phenomenon. From [3].

Figure 3

Upper plot: The entropy-density-to-viscosity ratio $s/\eta$ vs the temperature $T$ (GeV). The upper range of the plot $s/\eta =4\pi$ corresponds to the value in infinitely strongly coupled $\mathcal{N}=4$ plasma ([128]). The curve without points on the left side corresponds to pion rescattering according to chiral perturbation theory ([133]). The triangle (red) corresponds to the molecular dynamics study of classical strongly coupled colored plasma ([52]), and the square (black) corresponds to a numerical evaluation ([124]) on the lattice. The single point with error bar corresponds to the phenomenological value extracted from the data; see the text. The series of points connected by a line correspond to gluon-monopole scattering ([137]). Lower plot: The inverse ratio $\eta /s$ as a function of the temperature normalized to its critical value $T/T_c$. The solid line corresponds to gluon-monopole (gm) scattering ([137]), the same as in the upper plot, and the dashed line shows the perturbative gluon-gluon (gg) scattering.

Figure 4

Ratio of subsequent perturbative corrections to the first one $O(g^2)$: $g^3$, $g^4\log (g)$, $g^4$, $g^5$, and $g^6\log (g)$ are shown vs $g$ by the thick black, thick dashed (blue), dotted (red), and thin blue and brown dashed curves, respectively.

Figure 5

A schematic phase diagram on a (compactified) plane of temperature and baryonic chemical potential $T-\mu$. The shaded (blue) region shows the magnetically dominated region $g<e$, which includes the deconfined hadronic phase as well as a small part of the QGP domain. The unshaded region includes the electrically dominated part of QGP and the color superconducting (CS) region, which has $e$-charged diquark condensates and is therefore “magnetically confined.” The dashed line ($e=g$) is the line of electric-magnetic equilibrium. The solid lines indicate true phase transitions, while the dash-dotted line is a deconfinement crossover line. From [100].

Figure 6

Proton-to-pion and (rescaled) $\varphi /\pi$ ratios, as a function of transverse momentum $p_T$.

Figure 7

(a) Average initial stage deformations $⟨{\epsilon }_n⟩=⟨\cos (n\varphi )⟩$ for various centralities using the models indicated. From [67]. (b) Data points correspond to the event- by-event distribution of $v_3$ measured by the ATLAS Collaboration, compared to the distributions of initial eccentricities in the IP-GLASMA model and the distributions of $v_3$ from fluid dynamic evolution with IP-GLASMA initial conditions. From [143].

Figure 8

(a) ATLAS data for $v_2$ and $v_3$ vs $N_{\text{part}}$. (b) $\ln (v_n/{\epsilon }_n)$ vs $1/R$ for the same data. From [94].

Figure 9

(a) ATLAS data $\ln (v_n/{\epsilon }_n)$ vs $n^2$ from viscous hydrodynamical calculations for three values of specific shear viscosity as indicated. (b) $\ln (v_n/{\epsilon }_n)$ vs $n^2$ for $\mathrm{Pb}+\mathrm{Pb}$ data. The $p_{\perp }$-integrated $v_n$ results in (a) and (b) are from ATLAS 0.1% central $\mathrm{Pb}+\mathrm{Pb}$ collisions at nucleon-nucleon $s_{NN}=2.76\mathrm{TeV}$; the curves are linear fits. (c) Exponent vs viscosity-to-entropy ratio $4\pi /s$ for curves shown in (a) and (b). From [94].

Figure 10

The perturbation is shown by a small (blue) circle at point O: its time evolution to points $x$ and $y$ is described by the Green’s function of linearized hydrodynamics shown by two lines. The perturbed region, shown by the gray circle, is inside the sound horizon. The sound wave effect is maximal at the intersection points of this area with the fireball boundary: $\mathrm{\Delta }\varphi$ angle is the value at which the peak in the two-body correlation function is to be found. Shifting the location of the perturbation, from (a) to (b), results in a rather small shift in $\mathrm{\Delta }\varphi$.

Figure 11

(a) Calculated two-pion distribution as a function of the azimuthal angle difference $\mathrm{\Delta }\varphi$, for viscosity-to-entropy ratios $\eta /s=0.134$. From [163]. (b) From ATLAS, [77]. (c) From ALICE, [2]. All for ultracentral collisions.

Figure 12

A power spectrum of cosmic microwave background radiation measured by the Planck Collaboration. From [4].

Figure 13

(a) The lines are from analytic linearized hydrodynamic calculations of the correlation function harmonics $v_m^2$, based on a Green’s function from a point source, for four values of viscosity $4\pi \eta /s=0$, 1, 1.68, and 2 (top to bottom at the right). The closed circles are the ATLAS data for the ultracentral bin. From [163]. (b) Calculation of harmonic flow spectra from analytic linearized hydrodynamics at $p_{\perp }=1\mathrm{GeV}$ demonstrates the dependence of the minimum on the location of the perturbation $r_0=7.5$, 8.0, 8.5, and 9 fm, by the circles (blue), squares (red), diamonds (brown), and triangles (green), respectively. From [55].

Figure 14

(a) $v_n\{2\}$ vs $n^2$. The solid (blue) circles correspond to viscous event-by-event hydrodynamics, in the “$\mathrm{IP}\text{GLASMA}+\text{Music}$” model, with viscosity value $\eta /s=0.14$. The straight line, shown to guide the eye, demonstrates that acoustic systematics does in fact describe the results of this heavy calculation quite accurately. The CMS data for the 0%–1% centrality bin, shown by the squares (red), in fact display larger deviations, perhaps oscillatory ones. From [141]. (b) The harmonics induced by a fixed perturbation at $r_0=4\mathrm{fm}$ for variable viscosity. From [55].

Figure 15

The leading (left) and subleading (right) harmonics of triangular flow. From [118].

Figure 16

The log-log plane proper time $\tau$, sound momentum $k$. The solid curve indicates the amplitude damping by a factor of $e$: only small-$k$ sounds thus survive until freezeout. The shadowed region on the right corresponds to that in which the magnetosonoluminescence effect may produce extra dileptons.

Figure 17

Distributions over participant (a) nucleons and participant (b) quarks. From [5] and [168].

Figure 18

The normalized temperature for $4\pi \eta /s=1$, 100, upper, and lower plots, respectively. The meaning of the different curves is explained in the upper plot. From [42].

Figure 19

The shear stress ${\pi }_{\xi }^{\xi }$ vs coordinate $\rho$, for viscosity $4\pi \eta /s=1$, 100, the upper, and the lower plots, respectively. From [42].

Figure 20

(Upper) Alice data on the femtoscopy radii and (lower) “coherence parameter” as a function of multiplicity, for $pp$, $p\mathrm{P}\mathrm{b}$, PbPb collisions. From [58].

Figure 21

CMS data for $v_2$ calculated using 2, 4, 6, and 8 particle correlations, as well as Lee-Yang zeros (basically all particles). Good agreement between those manifest collectivity of the phenomenon. From [41].

Figure 22

$v_n$ for $n=2$, 3, 4, 5 vs $p_{\perp }$ in GeV, for the high-multiplicity bin indicated in the figure. The points are from ATLAS, the lines from CMS (presentation at QM2015).

Figure 23

Temperature $T$ vs the fireball size $R$ plane. The solid blue line is the adiabatic $S=\text{const}$, approximately $TR=\text{const}$ for sQGP.

Figure 24

(a) Average transverse momentum of identified charged hadrons (left panel: pions, kaons, and protons) and ratios of particle yields (right panel) in the range $|y|<1$ as a function of the corrected track multiplicity for $|\eta |<2.4$, for $pp$ collisions (open symbols) at several energies, and for $p\mathrm{Pb}$ collisions (filled symbols) at $\sqrt{s_{NN}}=5.02\mathrm{TeV}$. (b) The slopes of the $m_{\perp }$ distribution $T^{\prime }$ (in GeV) as a function of the particle mass. The numbers on the right of the lines give the track multiplicity. From [34].

Figure 25

The freezeout surface in universal dimensionless time $t$ and radial distance $r$ coordinates. The thick solid (blue) line in the middle corresponds to central $AA$ (PbPb) collisions, and the thick solid (red) line on the top to the highest multiplicity $pp$. The two (black) thin lines correspond to the central $p\mathrm{Pb}$ case, before and after collapse compression, marked $p{A}_i$ and $p{A}_f$, respectively. The arrow connecting them indicates the effect of multistring collapse.

Figure 26

Sketch of the radial flow (arrows directed radially from the fireball center) explaining how it influences the HBT radii. At small $k_t$ the whole fireball (the large circle) contributes, but at larger $k_t$ one sees only the part of the fireball which is comoving in the same direction as the observed pair. This region, shown by the shaded ellipse, has a smaller radii and anisotropic shape, even for central collisions.

Figure 27

HBT radii vs the pair transverse momentum $k_T$, for various multiplicities of the $pp$ collisions, from ALICE. From [1].

Figure 28

HBT radii compared to ALICE data (solid circles), for solutions starting with different initial size of the fireball, indicated by the Gubser scale parameter $q$ (which is inversely proportional to the size). From [71].

Figure 29

Comparison of the flow profile, for hydrodynamics and free streaming. From [139].

Figure 30

Comparison of the flow harmonic for hydrodynamics and free streaming for PbPb collisions. From [139].

Figure 31

Comparison of the flow harmonic for hydrodynamics and free streaming in the $p\mathrm{Pb}$ central bin. From [139].

Figure 32

The integrated $v_2\{2\}$ for PbPb and $p\mathrm{Pb}$ vs multiplicity. Left: Original values. Right: The fluctuation dependent elliptic flow, with the geometrical part subtracted. This geometrical part was calculated using the Phobos Glauber model and is not a fit. From [15].

Figure 33

The CGC phase diagram: the saturation momentum $Q_s$ as a function of the fraction of momentum $x$ and the atomic number $A$. From [97].

Figure 34

(a) The upper green curve is $p_{\perp }/\epsilon$, and the lower red is $p_l/\epsilon$, as a function of time in units of saturation scale $\tau Q_s$ at $g=0.5$. From [45]. (b) The upper blue curve is $p_{\perp }/\epsilon$, and the lower red is $p_l/\epsilon$ at $g=0.3$. From [19].

Figure 35

Trajectories of the systems on the occupancy-anisotropy plane for various settings. The parameters $\lambda$ near the curves show the corresponding ’t Hooft coupling constant $\lambda =g^2{N}_c$. All solid lines originate from one initial distribution characterized by the anisotropy parameter $\xi =4$, and the dashed lines originate from a different point with $\xi =10$. From [93].

Figure 36

Transverse energy profile from the IP-GLASMA model for a semiperipheral ($b=8\mathrm{fm}$) $\mathrm{Au}+\mathrm{Au}$ collision at $s=200A\mathrm{GeV}$, at times $\tau =0.01$, 0.2, and $5.2\mathrm{fm}/c$. From $\tau =0.01$ to $0.2\mathrm{fm}/c$ the fireball evolves out of equilibrium according to the GLASMA model. From [143].

Figure 37

Sketch of the initial state in central $pA$ collisions. (a) The IP-GLASMA model, with colored circles representing multiple gluons. (b) $N_p=16$ Pomerons, each represented by a pair of cold strings. The open circles are quarks and filled blue circles are diquarks.

Figure 38

Multiplicity dependence of the root-mean-square elliptic flow coefficient $v_2$ in $\mathrm{Pb}+\mathrm{Pb}$ (open symbols) and $p+\mathrm{Pb}$ collisions (solid symbols) from the IP-$\mathrm{G}\mathrm{L}\mathrm{A}\mathrm{S}\mathrm{M}\mathrm{A}+\text{music}$ model (connected triangles) compared to experimental data by the CMS Collaboration. From [143].

Figure 39

Rapidity distribution in $p\mathrm{P}\mathrm{b}$ collisions for different centrality classes, from “Measurement of the centrality dependence of the charged particle pseudorapidity distribution in proton-lead collisions,” ATLAS-CONF-2013-096, https://cds.cern.ch/record/1599773.

Figure 40

Oscillation of the energy density in simulations starting from “GLASMA”-like initial conditions. $k=5$ is the number of fluxes through the flux tubes. From [51].

Figure 41

(a) A static quark-antiquark pair is indicated by shaded circles: those are connected by the flux tube (QCD string). At distance $r$ from the tube the local value of the quark condensate $\overline{q}q(r)$ is measured. (b) Normalized chiral condensate as a function of the coordinate $r$ transverse to the QCD string (in lattice units). Points are from the lattice calculation ([75]). The curve is Eq. (44) with $C=0.26$, $s_{\text{string}}=0.176\mathrm{fm}$.

Figure 42

Instantaneous collective potential in units $2g_N{\sigma }_T$ for an $AA$ configuration with $b=11\mathrm{fm}$, $g_N{\sigma }_T=0.2$, and $N_s=50$ at the time $\tau =1\mathrm{fm}/c$. White regions correspond to the chirally restored phase.

Figure 43

A set of frequency modes, on the $\omega$ complex plane(. The dots are for a particular wave vector $k$, and the arrows indicate the direction of motion as $k$ increases. From [90].

Figure 44

Profile of a strong shock in QGP in its rest frame, according to the Navier-Stokes hydrodynamics, as a function of the coordinate normal to the shock front. The time goes right to left, so the left-hand side shows the final values of the observables, while the right-hand side shows the initial ones. The pressure, shown by the solid (blue) line, is in units of its asymptotic value: thus the curve jumps to 1 on the left side. The flow rapidity, shown by the dashed (black) line, is reduced. The process is thus a rapid formation of hotter denser QGP by an influx of a cooler and more dilute one. From [147].

Figure 45

(a) A sketch of a falling shell geometry. Its radius is $r(t)=1/z(t)$ used in the text. (b) Single-point observer $O_1$ and the two-point observers $O_2$ and $O_3$ near a collapsing shell shown by a circle.

Figure 46

(Left) The temperature evolution combination $d\log (w)/d\log \tau$ for different initial conditions (thin black curves) converging into a universal function of $w=T\tau$, compared to the hydrodynamical prediction. (Right) The pressure anisotropy for one of the evolutions compared to first- (NS), second-, and third-order hydrodynamics. From [68].

Figure 47

(a) A projection of the marginally trapped surface onto a fixed time slice of the AdS geometry. From [62]. (b) The area of the trapped surface vs the impact parameter, with the comparison of the numerical studies ([107]) shown by points and analytic curves. The vertical line shows the location of the critical impact parameter $b_c$ beyond which there is no trapped surface. From [63].

Figure 48

(a) An early time snapshot of a pair of strings created after one color exchange. The coordinates are explained on the right: the colliding objects move with a speed of light away from each other and strings are stretched. (b) Later time snapshot, in AdS/QCD background. After strings reach the “levitation surface, ” shown by a rectangular shape, they start to oscillate around it. (c) In the case of high density of many strings they can be approximated by a continuous membrane.

Figure 49

Glueball masses calculated on the lattice (diamonds) organized in Regge trajectories (lines). From [153].

Figure 50

(a) Dipole-dipole scattering configuration in Euclidean space. The dipoles have size $a$ and are $b$ apart. The dipoles are tilted by $\pm \theta /2$ (Euclidean rapidity) in the longitudinal $x_0\text{−}{x}_L$ plane. (b) A sketch illustrating Pomeron exchange for baryon-baryon scattering: only one pair of quarks become wounded quarks.

Figure 51

(a) The solid line is the empirical LHC data parametrization. The dashed line is the shape corresponding to the “excited string” approximation for fixed sizes of the dipoles, while the circles correspond to the profile averaged over the fluctuating dipoles. (b) The corresponding elastic amplitude (the absolute value squared of the profile Bessel transform) as a function of the momentum transfer. Model prediction agrees with parametrization well at small $t$, up to the dip.

Figure 52

Illustrations of the “direct photon puzzle.” (a) The yield, (b) $v_2$, and (c) $v_3$ of the direct photons. Points are data and curves are from theory based on the hydrodynamical model, the reference indicated on the figure. From A. Drees, PHENIX presentation at QM2015.

Figure 53

The charm quark diffusion coefficients from quenched lattice QCD (circles, square, and triangle) compared to model calculations based on different elastic interactions in the QGP corresponding to the $A(p=0)$ limit: $T$-matrix calculations with either free (green band) or internal energy (red band) as potential, pQCD Born calculations from HTL and pQCD matching using a reduced Debye mass and running coupling (Nantes, pink dash-dotted line) or with perturbative Debye mass and fixed coupling (Torino, cyan band), as well as schematic leading order (LO) pQCD with fixed coupling and Debye mass $mD=gT$ (purple dash-dotted line). The blue dashed line below $T_c$ is a calculation of $D$-meson diffusion in hadronic matter from elastic scattering off various mesons, antibaryons, and baryons. From [135].

Figure 54

Free energy singlet potentials $F_1(T,r)$ (top plot) and the potential energy $U_1(T,r)$ below and above $T_c$. Note that the vertical scale is different. From [78].

Figure 55

One-dimensional Fokker-Planck equation for an interacting $\overline{c}c$ pair. The relaxation of the initial narrow Gaussian distribution is shown by curves (black, red, brown, green, blue, or top to bottom at $r=0$) corresponding to times $t=0$, 1, 5, and 10 fm, respectively. From [179].

Figure 56

Jet suppression and elliptic parameter $v_2$, data vs models. From [176].

Figure 57

The normalized dimensionless jet quenching parameter $\widehat{q}/T^3$ vs the temperature $T$ (GeV), for a 20 GeV jet. The lower curve marked pQCD + HTL is a standard perturbative contribution of quarks and gluons. Other curves with a maximum include contributions of scattering on monopoles. The horizontal line crudely corresponds to the viscosity/entropy ratio in the AdS/CFT approach.

Figure 58

ALICE data on particle yields compared to the thermal model. The main fit parameters are indicated in the upper plot. From [7].

Figure 59

(a) Freezeout parameters in the $T-{\mu }_B$ plane: comparison between the curve obtained in Cleymans et al. (red band) and the values obtained in the present analysis from a combined fit for net-electric charge and net protons (blue symbols). (b) The freezeout parameters fit from two different set of particle ratios.

Figure 60

The temperature of the kinetic freezeout vs mean velocity of the radial flow, fitted to ALICE spectra of the identified secondaries ($\pi ,K,p,\mathrm{\Lambda },\mathrm{\Xi },\mathrm{\Omega }$).

Figure 61

(a) Points show the ratio of PbPb to SS spectra, from NA44. The three curves are for pion chemical potential ${\mu }_{\pi }=60$, 80, and 100 MeV. From [72]. (b) Dots are ALICE transverse momentum spectra of pions in the low-$p_t$ region, compared to the model with (upper plot) and without (lower plot) pion chemical potential. From [17].

Figure 62

The enhanced contribution to a four-particle correlator. From [164].

Figure 63

The kurtosis, a four-particle correlator, in units of the width, as a function of the collision energy $\sqrt{s},\mathrm{GeV}$.

Figure 64

Continuum extrapolated results for pressure, energy density, and entropy density. Solid lines on the low temperature side correspond to results obtained from hadron resonance gas (HRG) model calculations. The (yellow) band marked $T_c$ indicates the phase transition region for deconfinement and chiral symmetry restoration.