Principal component analysis of the nonlinear coupling of harmonic modes in heavy-ion collisions

The principal component analysis of flow correlations in heavy-ion collisions is studied. The correlation matrix of harmonic flow is generalized to correlations involving several different flow vectors. The method can be applied to study the nonlinear coupling between different harmonic modes in a double differential way in transverse momentum or pseudorapidity. The procedure is illustrated with results from the hydrodynamic model applied to Pb + Pb collisions at $\sqrt{s_{NN}}=2760$ GeV. Three examples of generalized correlations matrices in transverse momentum are constructed corresponding to the coupling of $v_2^2$ and $v_4$, of $v_2{v}_3$ and $v_5$, or of $v_2^3,v_3^3$, and $v_6$. The principal component decomposition is applied to the correlation matrices and the dominant modes are calculated.

Figure 1

First (solid line) and second (dotted line) scaled eigenvectors $V_n^{(\alpha =1,2)}\left(p\right)/(dN/dp)$ of the correlation matrix $C_{n\mathrm{\Delta }}(p_1,p_2)$ for the second (a) and third (b) order flow in Pb + Pb collisions at $\sqrt{s_{NN}}=2760$ GeV and centrality $0\text{–}5\%$. The dashed and dash-dotted lines represents the corresponding eigenvectors of the correlation matrix $c_{n\mathrm{\Delta }}(p_1,p_2)$.

Figure 2

Same as in Fig. 1 but for Pb + Pb collisions with centrality $30\text{–}40\%$.

Figure 3

Eigenvectors of the correlation matrix $c_{2;2|4}$ [Eq. (11)]. (a) The two components of the first eigenvector; (b) the two components of the second and third eigenvectors; Pb+Pb collisions with centrality $0\text{–}5\%$.

Figure 4

Same as in Fig. 3 but for collisions with centrality $30\text{–}40\%$.

Figure 5

Same as in Fig. 4 for the correlation matrix $c_{23|5}$.

Figure 6

Eigenvectors of the correlation matrix $c_{2;3|3;2|6}$. In panels (a) through (d) are shown the components of the eigenvectors corresponding to the first four eigenvalues. In all panels, three components of the eigenvectors are shown, I (solid lines), II (dotted lines), and III (dashed lines), corresponding to the sectors $q_2{\left(p\right)}^2,q_3{\left(p\right)}^3$, and $q_6\left(p\right)$ in the correlation matrix.