Forward-backward eccentricity and participant-plane angle fluctuations and their influences on longitudinal dynamics of collective flow

We argue that the transverse shape of the fireball created in the heavy-ion collision could be strongly influenced by event-by-event fluctuations of the eccentricity vectors for the forward-going and backward-going wounded nucleons: ${\vec{\epsilon }}_n^{\mathrm{F}}\equiv {\epsilon }_n^{\mathrm{F}}{e}^{in{\Phi }_n^{*\mathrm{F}}}$ and ${\vec{\epsilon }}_n^{\mathrm{B}}\equiv {\epsilon }_n^{\mathrm{B}}{e}^{in{\Phi }_n^{*\mathrm{B}}}$. Due to the asymmetric energy deposition of each wounded nucleon along its direction of motion, the eccentricity vector of the produced fireball is expected to interpolate between ${\vec{\epsilon }}_n^{\mathrm{F}}$ and ${\vec{\epsilon }}_n^{\mathrm{B}}$ along the pseudorapidity, and hence exhibits sizable forward-backward (FB) asymmetry (${\epsilon }_n^{\mathrm{B}}\ne {\epsilon }_n^{\mathrm{F}}$) and/or FB twist (${\Phi }_n^{*\mathrm{F}}\ne {\Phi }_n^{*\mathrm{B}}$). A transport model calculation shows that these initial-state longitudinal fluctuations for $n=2$ and 3 survive the collective expansion, and result in similar FB asymmetry and/or a twist in the final-state event-plane angles. These novel event-by-event longitudinal flow fluctuations should be accessible at RHIC and the LHC using the event-shape selection technique proposed in earlier papers. If these effects are observed experimentally, it could improve our understanding of the initial-state fluctuations, particle production, and collective expansion dynamics of the heavy-ion collision.

Figure 1

Schematic illustration of the forward-backward fluctuation of second-order eccentricity and participant plane, in (a) transverse plane and (b) along rapidity direction in A+A collisions. The dashed-lines indicate the particle production profiles for forward-going and backward-going participants, $f^{\mathrm{F}}\left(\eta \right)N_{\mathrm{part}}^{\mathrm{F}}$ and $f^{\mathrm{B}}\left(\eta \right)N_{\mathrm{part}}^{\mathrm{B}}$, respectively.

Figure 2

The (a) ellipticity and (b) triangularity calculated for all participating nucleons (filled circles) and using only forward-going participating nucleons (open circles). (c) shows the RMS width of the FB asymmetry parameters defined in Eqs. (4, 5) and (d) shows the RMS width of the twist angle between the eccentricity vectors of the two nuclei. The dotted-line indicate the value expected for flat twist distribution. All quantities are shown as a function of $N_{\mathrm{part}}$.

Figure 3

The $\eta$ ranges of three subevents (B, M, F) for calculating flow vector ${\vec{q}}_n=q_n{e}^{in{\Psi }_n}$ via Eq. (11). They cover the pseudorapidity ranges of $-6<\eta <-4$ (backward or B), $-1<\eta <1$ (middle or M), and $4<\eta <6$ (forward or F). The results in this section (Sec. 3) are obtained using all particles in their respective $\eta$ ranges. But event planes used in Sec. 5 are obtained using half of the particles, to allow the measurement of the flow harmonics over the full $\eta$ range.

Figure 4

Correlation of (a) ${\epsilon }_2^{\mathrm{B}}$ vs ${\epsilon }_2^{\mathrm{F}}$, (b) $2\Delta {\Phi }_2^{*\mathrm{FB}}$ vs ${\epsilon }_2$, (c) ${\epsilon }_2^{\mathrm{F}}$ vs $q_2^{\mathrm{F}}$, (d) ${\epsilon }_2^{\mathrm{F}}$ vs $q_2^{\mathrm{B}}$, (e) $2\Delta {\Phi }_2^{*\mathrm{FB}}$ vs $2\Delta {\Psi }_2^{\mathrm{FB}}$, and (f) ${\epsilon }_2^{\mathrm{F}}-{\epsilon }_2^{\mathrm{F}}$ vs $2\Delta {\Phi }_2^{*\mathrm{FB}}$ for AMPT Pb+Pb events with $b=8$ fm. The bars around the circles in (b) indicates the RMS width of $2\Delta {\Phi }_2^{*\mathrm{FB}}$ at given ${\epsilon }_2$ value, and the four regions delineated by the boxes in (f) indicate the cuts for the four event classes defined in Table 1. The numbers in some panels indicate the correlation coefficients of the distributions.

Figure 5

Correlation of (a) ${\epsilon }_3^{\mathrm{B}}$ vs ${\epsilon }_3^{\mathrm{F}}$, (b) $3\Delta {\Phi }_3^{*\mathrm{FB}}$ vs ${\epsilon }_3$, (c) ${\epsilon }_3^{\mathrm{F}}$ vs $q_3^{\mathrm{F}}$, (d) ${\epsilon }_3^{\mathrm{F}}$ vs $q_3^{\mathrm{B}}$, (e) $3\Delta {\Phi }_3^{*\mathrm{FB}}$ vs $3\Delta {\Psi }_3^{\mathrm{FB}}$, and (f) ${\epsilon }_3^{\mathrm{F}}-{\epsilon }_3^{\mathrm{F}}$ vs $3\Delta {\Phi }_3^{*\mathrm{FB}}$ for AMPT Pb+Pb events with $b=8$ fm. The bars around the circles in (b) indicates the RMS width of $3\Delta {\Phi }_3^{*\mathrm{FB}}$ at given ${\epsilon }_3$ value, and the four regions delineated by the boxes in (f) indicate the cuts for the four event classes defined in Table 1. The numbers in some panels indicate the correlation coefficients of the distributions.

Figure 6

The $v_n^{\mathrm{c}}\left(\eta \right)$ (top row) and $v_n^{\mathrm{s}}\left(\eta \right)$ (second row) relative to the reference angle taken as one of the three participant planes. They are obtained for type 1 events for $n=2$ (left column) and $n=3$ (right column).

Figure 7

The $v_n^{\mathrm{c}}\left(\eta \right)$ (top row) and $v_n^{\mathrm{s}}\left(\eta \right)$ (second row) relative to the reference angle taken as one of the three participant planes. They are obtained for type 2 events for $n=2$ (left column) and $n=3$ (right column).

Figure 8

The $v_n^{\mathrm{c}}\left(\eta \right)$ (top row), $v_n^{\mathrm{s}}\left(\eta \right)$ (second row), rotation angle $n\Delta {\Phi }_n^{\mathrm{rot}}$ (third row) and $v_n=\sqrt{{\left(v_n^{\mathrm{c}}\right)}^2+{\left(v_n^{\mathrm{s}}\right)}^2}$ (bottom row) relative to the reference angle taken as one of the three participant planes. They are obtained via Eq. (15) for type 3 events for $n=2$ (left column) and $n=3$ (right column).

Figure 9

The $v_n^{\mathrm{c}}\left(\eta \right)$ (top row), $v_n^{\mathrm{s}}\left(\eta \right)$ (second row), rotation angle $n\Delta {\Phi }_n^{\mathrm{rot}}$ (third row) and $v_n=\sqrt{{\left(v_n^{\mathrm{c}}\right)}^2+{\left(v_n^{\mathrm{s}}\right)}^2}$ (bottom row) relative to the reference angle taken as one of the three participant planes. They are obtained via Eq. (15) for type 4 events for $n=2$ (left column) and $n=3$ (right column).

Figure 10

The $v_n^{\mathrm{c}}\left(\eta \right)$ (top row) and $v_n^{\mathrm{s}}\left(\eta \right)$ (second row) relative to the reference angle taken as one of the three participant planes for $n=2$ (left column) and $n=3$ (right column). No selection cuts have been applied for these events.

Figure 11

The $v_n^{\mathrm{c}}\left(\eta \right)$ obtained with the three raw event planes (see Fig. 3) for the type 1 events. They are compared with those obtained from the participant plane for all wounded nucleons, ${\Phi }_n^{*}$, from Fig. 6.

Figure 12

The $v_n^{\mathrm{c}}\left(\eta \right)$ obtained with the three EPs for the type 2 events. They are compared with those obtained from the participant plane for all wounded nucleons, ${\Phi }_n^{*}$, from Fig. 7.

Figure 13

The twist angles obtained with the three EPs for the type 3 events. They are compared with those obtained from the three participant planes from Fig. 8.

Figure 14

The twist angles obtained with the three EPs for the type 4 events. They are compared with those obtained from the three participant planes for all wounded nucleons, ${\Phi }_n^{*}$, from Fig. 9.

Figure 15

Schematic illustration of the origin for the shift of the center of mass between the two nuclei in the overlap region. Due to the almond shape, ${\vec{\epsilon }}_2^{\mathrm{F}/\mathrm{B}}$ strongly aligns with the ${\vec{r}}_0^{\mathrm{F}/\mathrm{B}}$.

Figure 16

(a) The magnitude of the shift of the center of mass of wounded nucleons in one nucleus from center of mass of all wounded nucleons in the collisions. The bars represent the RMS values of the shift for events with the same $N_{\mathrm{part}}$. (b) The ${\epsilon }_2^{\mathrm{F}}-{\epsilon }_2$ is compared with the estimate obtained form the shift [Eq. (A7)].