Morphology of high-multiplicity events in heavy ion collisions

We discuss opportunities that may arise from subjecting high-multiplicity events in relativistic heavy ion collisions to an analysis similar to the one used in cosmology for the study of fluctuations of the cosmic microwave background (CMB). To this end, we discuss examples of how pertinent features of heavy ion collisions including global characteristics, signatures of collective flow, and event-wise fluctuations are visually represented in a Mollweide projection commonly used in CMB analysis, and how they are statistically analyzed in an expansion over spherical harmonic functions. If applied to the characterization of purely azimuthal dependent phenomena such as collective flow, the expansion coefficients of spherical harmonics are seen to contain redundancies compared to the set of harmonic flow coefficients commonly used in heavy ion collisions. Our exploratory study indicates, however, that these redundancies may offer novel opportunities for a detailed characterization of those event-wise fluctuations that remain after subtraction of the dominant collective flow signatures. By construction, the proposed approach allows also for the characterization of more complex collective phenomena like higher-order flow and other sources of fluctuations, and it may be extended to the characterization of phenomena of noncollective origin such as jets.

Figure 1

(a) Pseudorapidity density of charged particles simulated by HIJING for a single semicentral collision between two Pb nuclei at current LHC energies ($\sqrt{s_{nn}}=2.76\mathrm{TeV}$). (b) The same distribution as a function of the polar angle $\theta$ relative to the beam direction.

Figure 2

Density of charged particles as a function of the azimuthal angle $\phi$ in the plane transverse to the beam direction, simulated by HIJING and modified by an elliptic flow with $v_2=0.10$.

Figure 3

(a) Mollweide projection of a single HIJING event without flow. The azimuthal angle $\phi$ runs along the ‘equator', the polar angle $\theta$ runs from pole to pole. The color coding tracks deviations relative to the average level of counts. (b) The $m=0$ mode has been removed from the map. The angular resolution of the maps corresponds to $l_{\max }=100$ [see Eq. (1)]. The multiplicity of the event is 17000.

Figure 4

(a) Power spectra for the map from Fig. 3. The black line is the total power spectrum $C^S\left(l\right)$, the blue (gray) dots are for the $m=0$ mode only and the red (gray) line is for the $\left|m\right|⩾1$ modes. (b) The normalized distribution function $H\left(s\right)$ versus amplitude of fluctuations $s$ for the total signal (the black line) and for the signal with $\left|m\right|⩾1$ (the red line).

Figure 5

Schematic representation of the contribution of $v_2$ (elliptic) and $v_4$ flow to the various $m$ components of the $b_{l=4,m}$ coefficient. The $v_2$ modulation will change the amplitude and the phase of the $b_{4,2}$ coefficient without contributing to the other components. The $v_4$ modulation will only change the $b_{4,4}$ component.

Figure 6

The power spectrum for an event with elliptic flow $v_2=0.07$. The total power $C^S\left(l\right)$ is shown in black, the blue (gray) stars correspond to $C^S\left(l\right)-D\left(l\right)=C\left(l\right)$, and the $D\left(l\right)$ power spectrum is shown in red (gray) [see Eq.(3)].

Figure 7

Maps of the $|b_{22}|$ amplitude for input flow values of $v_2=0$ (a), $0.01$ (b), $0.05$ (c), $0.075$ (d), $0.1$ (e), $0.2$ (f), $0.3$ (g), and $0.5$ (h), respectively. For simplicity ${\Psi }_R^2=0$ in all maps. A different reaction plane angle would manifest itself as a phase shift in the ‘East-West' direction.

Figure 8

Estimator $D=|b_{2,2}|/g\left(2\right)|b_{2,0}|$ for the elliptic flow amplitude obtained for single events plotted versus the $v_2$ value used in the simulations. The error bars correspond to $68\%$ confidence level.

Figure 9

The reaction plane angle ${\Psi }_R$ (rad) for each of 202 HIJING events with flow = 0.07 versus the phase ${\phi }_{2,2}$ (rad) obtained from the analysis described in Eq. (9). The insert shows the distribution of the difference ${\Psi }_R-0.5{\phi }_{2,2}$ (rad).