Method for studying the rapidity fluctuation and de-correlation of harmonic flow in heavy-ion collisions

An “event-shape-twist” technique is proposed to study the longitudinal dynamics of harmonic flow, in particular the effects of rapidity fluctuation and event-plane de-correlation. This technique can distinguish between two types of rapidity de-correlation effects: a systematic rotation versus a random fluctuation of flow angles along the rapidity direction. The technique is demonstrated and the magnitude of the two de-correlation effects is predicted by using a multiphase transport model via a single-particle analysis and a two-particle correlation analysis. An observed de-correlation can be attributed to a systematic rotation of event-plane angle along the pseudorapidity, consistent with a collective response to an initial state twist of the fireball proposed by Bozek et al. This rotation is also observed for several higher-order harmonics with the same sign and similar magnitudes.

Figure 1

The two scenarios for the rapidity fluctuation of $v_2$: (a) the fluctuation arises from a systematic rotation as a function of $\eta$ [30], (b) the fluctuation is random between different rapidity ranges.

Figure 2

The $\eta$ ranges of the subevents for the event-shape twist ($S_{\mathrm{B}}$ and $S_{\mathrm{F}}$) and for calculating the reference event-plane angles via Eqs. (7)–(9) (A, B, and C). Note that subevent $S_{\mathrm{B}}$ (A) or $S_{\mathrm{F}}$ (C) contains half (a quarter) of the particles randomly selected from $-6<\eta <-3$ or $3<\eta <6$, and subevent B contains half of the particles randomly selected from $-1<\eta <1$. The subevents $S_{\mathrm{B}}$ and $S_{\mathrm{F}}$ together are also denoted as subevent S.

Figure 3

The ${\Psi }_m^{\mathrm{cut}}$ distribution in ten $q_m^{\mathrm{S}}$ bins with equal statistics for $m=2$ (left panel) and $m=3$ (right panel). The hashed area represents the 10% of the events with largest ${\Psi }_m^{\mathrm{cut}}$ in each $q_m^{\mathrm{S}}$ bin, and only these events are used in this paper.

Figure 4

The $v_n^{\mathrm{c},\mathrm{raw}}\left(\eta \right)$ (top row), $v_n^{\mathrm{s},\mathrm{raw}}\left(\eta \right)$ (second row), rotation angle $n\Delta {\Phi }_n^{\mathrm{rot}}$ (third row), and ${[{\left(v_n^{\mathrm{c},\mathrm{raw}}\right)}^2+{\left(v_n^{\mathrm{s},\mathrm{raw}}\right)}^2]}^{1/2}$ (bottom row) relative to reference EP angle calculated in subevents A, B, and C in Eqs. (7)–(9). They are obtained via Eq. (9) for events selected with largest ${\Psi }_m^{\mathrm{cut}}$ in the tenth $q_m^{\mathrm{S}}$ bin for $m=2$ (left column) and $m=3$ (right column).

Figure 5

The $v_n^{\mathrm{c},\mathrm{raw}}$ (top row), $v_n^{\mathrm{s},\mathrm{raw}}$ (middle row), and the rotation angle $n\Delta {\Phi }_n^{\mathrm{rot}}$ (bottom row) for events selected with largest ${\Psi }_m^{\mathrm{cut}}$ in the tenth $q_m^{\mathrm{S}}$ bin, where $m=2$ (left column) and $m=3$ (right column) and $n=2\text{to}5$. The reference event-plane angle used in Eqs. (7)–(9) is calculated in subevent B.

Figure 6

Left panel shows $\langle v_2^{\mathrm{c},\mathrm{raw}}\rangle$, middle panel shows $d\left(v_2^{\mathrm{s},\mathrm{raw}}\right)/d\eta$, and right panel shows $d\left(2\Delta {\Phi }_2^{\mathrm{rot}}\right)/d\eta$ as a function of the $q_2^{\mathrm{S}}$ bin defined in the hashed region in the left panel of Fig. 3. The reference event-plane angle used in Eqs. (7)–(9) is calculated in subevent B.

Figure 7

The $\eta$ range of the subevents for the event-shape twist ($S_{\mathrm{B}}$ and $S_{\mathrm{F}}$ over $-6<\eta <-4$ or $4<\eta <6$) and the particles used for two-particle correlation analysis ($-3<\eta <3$). The subevents $S_{\mathrm{B}}$ and $S_{\mathrm{F}}$ together are also denoted as subevent S.

Figure 8

(left panel) The two-dimensional correlation function, (middle panel) the one-dimensional correlation function in different $\Delta \eta$ slices, and (right panel) the extracted $n{\Phi }_n^{\mathrm{rot}}$ for $n=2--5$, for events selected in the fourth $q_2^{\mathrm{S}}$ bin with the largest ${\Psi }_2^{\mathrm{cut}}$ values.

Figure 9

(left panel) The 2D correlation function, (middle panel) 1D correlation function in different $\Delta \eta$ slices, and (right panel) the extracted $n{\Phi }_n^{\mathrm{rot}}$ for $n=2\text{to}5$ (right), for events selected in the tenth $q_2^{\mathrm{S}}$ bin with the largest ${\Psi }_2^{\mathrm{cut}}$ values.