Gluonic hot spot initial conditions in heavy-ion collisions

The initial conditions in heavy-ion collisions are calculated in many different frameworks. The importance of nucleon position fluctuations within the nucleus and subnucleon structure has been established when modeling initial conditions for input to hydrodynamic calculations. However, there remain outstanding puzzles regarding these initial conditions, including the measurement of the near equivalence of the elliptical $v_2$ and triangular $v_3$ flow coefficients in ultracentral 0–1% $\mathrm{Pb}+\mathrm{Pb}$ collisions at the CERN Large Hadron Collider. Recently a calculation termed magma incorporating gluonic hot spots via two-point correlators in the color glass condensate framework, and no nucleons, provided a simultaneous match to these flow coefficients measured by the ATLAS experiment, including in ultracentral 0–1% collisions. Our calculations reveal that the magma initial conditions do not describe the experimental data when run through full hydrodynamic sonic simulations or when the hot spots from one nucleus resolve hot spots from the other nucleus, as predicted in the color glass condensate framework. We also explore alternative initial condition calculations and discuss their implications.

Figure 1

A schematic display of the initial energy deposition calculation. The hot spots in nucleus A are multiplied by the smooth distribution of nucleus B, labeled as $B_{WS}$ and summed with the inverse, hot spots in nucleus B multiplied by the smooth distribution of nucleus A. The resulting energy spatial distribution is shown on the top right plot. In contrast, the bottom right plot shows the $A\times B$ calculation using the modified magma code.

Figure 2

Initial condition calculation with magma. (Top) The total energy deposit distribution over one million events. Solid (dashed) vertical lines correspond to 10% (top 1%) percentiles. (Bottom) The ${\varepsilon }_2\left\{2\right\}$, ${\varepsilon }_2\left\{4\right\}$, and ${\varepsilon }_3\left\{2\right\}$ values as a function of centrality selection, scaled up by the respective ${\kappa }_{2,3}$ values. In comparison, ATLAS experiment data are shown for $v_2\left\{2\right\}$, $v_2\left\{4\right\}$, and $v_3\left\{2\right\}$.

Figure 3

Ratio of $v_3\left\{2\right\}/v_2\left\{2\right\}$ as a function of $\mathrm{Pb}+\mathrm{Pb}$ collision centrality as measured by the ATLAS experiment. Multiple theoretical calculations are also shown.

Figure 4

Time evolution event display for a $\mathrm{Pb}+\mathrm{Pb}$ 0–1% most central collision using magma initial conditions run through sonic viscous hydrodynamics. The time snapshots show the temperature in the transverse ($x,y$) plane and are at $t=0.4,1.8,3.2,6.0$ fm/$c$.

Figure 5

Results from sonic hydrodynamic calculations for magma initial condition 0–1% central $\mathrm{Pb}+\mathrm{Pb}$ collisions. Shown are results in the upper (lower) panels from individual events for $v_2$ ($v_3$) versus the initial geometric ${\varepsilon }_2$ (${\varepsilon }_3$). The linear fits represent the $\kappa$ response coefficients. Also shown are the Pearson coefficients indicating the degree of linear correlation.

Figure 6

Results from sonic hydrodynamic calculations for magma initial condition 25–26% central $\mathrm{Pb}+\mathrm{Pb}$ collisions. Shown are results in the upper (lower) panels from individual events for $v_2$ ($v_3$) versus the initial geometric ${\varepsilon }_2$ (${\varepsilon }_3$). The linear fits represent the $\kappa$ response coefficients. Also shown are the Pearson coefficients indicating the degree of linear correlation.

Figure 7

Results from sonic hydrodynamic calculations for magma initial condition 0–1% (left) and 25–26% (right) $\mathrm{Pb}+\mathrm{Pb}$ collisions. Shown are results in the upper (middle) panels of the ${\kappa }_2$ (${\kappa }_3$) values as a function of charged hadron $p_T$. The horizontal line represented the $p_T$-integrated value over the range $0.5\text{–}3.0$ GeV. The lower panel shows the ratio of ${\kappa }_3/{\kappa }_2$ as a function of $p_T$. The dashed line is set at one for reference.

Figure 8

Results from sonic hydrodynamic calculations for magma initial condition 0–30% $\mathrm{Pb}+\mathrm{Pb}$ collisions quantifying the ${\kappa }_2$ and ${\kappa }_3$ values both via the mean values and the Gaussian fit mean values. Also shown are values of ${\kappa }_n$ determined as the rms of $v_n$ divided by the rms of ${\varepsilon }_n$. An example fit is shown in the inset.

Figure 9

Results from sonic hydrodynamic calculations for magma initial conditions 0–1% for the ratio ${\kappa }_3/{\kappa }_2$ for two different values of $\eta /s$.

Figure 10

Initial condition calculation with magma but modified to run where the energy deposit is proportional to $A\times B$. The ${\varepsilon }_2\left\{2\right\}$, ${\varepsilon }_2\left\{4\right\}$, and ${\varepsilon }_3\left\{2\right\}$ values as a function of centrality selection are scaled up by the respective ${\kappa }_{2,3}$ values. In comparison, ATLAS experiment data are shown for $v_2\left\{2\right\}$, $v_2\left\{4\right\}$, and $v_3\left\{2\right\}$.

Figure 11

Number of magma color sources, or hot spots, for a Pb nucleus as a function of input parameters $Q_{s,0}$ and $m$.

Figure 12

Initial condition calculation with ip-jazma run where the energy deposit is proportional to $A\times B$, as is true in the ip-glasma model. The ${\varepsilon }_2\left\{2\right\}$, ${\varepsilon }_2\left\{4\right\}$, and ${\varepsilon }_3\left\{2\right\}$ values as a function of centrality selection, scaled up by the respective ${\kappa }_{2,3}$ values. In comparison, ATLAS experiment data are shown for $v_2\left\{2\right\}$, $v_2\left\{4\right\}$, and $v_3\left\{2\right\}$.

Figure 13

Initial condition calculation with ip-jazma run where the energy deposit is proportional to $\sqrt{A\times B}$ as is true in the trento model with parameter $p=0$. The ${\varepsilon }_2\left\{2\right\}$, ${\varepsilon }_2\left\{4\right\}$, and ${\varepsilon }_3\left\{2\right\}$ values as a function of centrality selection, scaled up by the respective ${\kappa }_{2,3}$ values. In comparison, ATLAS experiment data are shown for $v_2\left\{2\right\}$, $v_2\left\{4\right\}$, and $v_3\left\{2\right\}$.

Figure 14

The energy deposit distribution for minimum bias $\mathrm{Pb}+\mathrm{Pb}$ collisions is shown, with the integral for all models set to one and the mean energy deposit scaled to one.

Figure 15

The energy deposit spatial distribution in $b=0$ fm $\mathrm{Pb}+\mathrm{Pb}$ collisions is shown, with arbitrary units on the vertical axis.

Figure 16

Distributions of spatial eccentricities ${\varepsilon }_{2-6}$ from ip-glasma and ip-jazma for 1000 Monte Carlo Glauber $\mathrm{Au}+\mathrm{Au}$ events of impact parameter $b=0$.

Figure 17

An identical Monte Carlo Glauber $\mathrm{Au}+\mathrm{Au}$ event at $b=0$ fed through the ip-glasma (upper panel) and ip-jazma (lower panel) calculations. The energy density distributions are shown in the left panels, with a zoom inset for more detail. Also shown are the two-point energy-energy correlator averaged over many events in the ip-glasma (upper panel) and ip-jazma (lower panel) frameworks.

Figure 18

Distributions of spatial eccentricities ${\varepsilon }_{2-6}$ from ip-glasma and ip-jazma run with the trento-style energy deposit proportional to $\sqrt{T_A\times T_B}$ (i.e., $p=0$ trento mode) for 1000 Monte Carlo Glauber $\mathrm{Au}+\mathrm{Au}$ events of impact parameter $b=0$. ip-jazma results where the two-dimensional Gaussian width is adjusted by $\sqrt{2}$ are also shown.