Cumulants and nonlinear response of high $p_T$ harmonic flow at $\sqrt{s_{\mathit{NN}}}=5.02$ TeV

Event-by-event fluctuations caused by quantum mechanical fluctuations in the wave function of colliding nuclei in ultrarelativistic heavy ion collisions were recently shown to be necessary for the simultaneous description of $R_{\text{AA}}$ as well as the elliptic and triangular flow harmonics at high $p_T$ in PbPb collisions at the Large Hadron Collider (LHC). In fact, the presence of a finite triangular flow as well as cumulants of the flow harmonic distribution that differ from the mean are only possible when these event-by-event fluctuations are considered. In this paper we combine event-by-event viscous hydrodynamics and jet quenching to make predictions for high $p_T{R}_{\text{AA}}, v_2\left\{2\right\}, v_3\left\{2\right\}$, and $v_2\left\{4\right\}$ in PbPb collisions at $\sqrt{s_{NN}}=5.02$ TeV. With an order of magnitude larger statistics we find that high $p_T$ elliptic flow does not scale linearly with the soft elliptical flow, as originally thought, but has deviations from perfectly linear scaling. A new experimental observable, which involves the difference between the ratio of harmonic flow cumulants at high and low $p_T$, is proposed to investigate the fluctuations of high $p_T$ flow harmonics and measure this nonlinear response. By varying the path length dependence of the energy loss and the viscosity of the evolving medium we find that $R_{AA}\left(p_T\right)$ and $v_2\left\{2\right\}\left(p_T\right)$ strongly depend on the choice for the path length dependence of the energy loss, which can be constrained using the new LHC run 2 data.

Figure 1

Model calculations for the soft sector $v_2\left\{2\right\}$ and $v_3\left\{2\right\}$ as a function of centrality for $0.2\le p_T\le 3$ GeV, computed using $\eta /s=0.05$ (black curves) and $\eta /s=0.12$ (red curves), and their comparison to ALICE $\sqrt{s_{NN}}=5.02$ TeV PbPb data [24].

Figure 2

Event-by-event scatter plot of $v_2^{\text{hard}}$ vs. $v_2^{\text{soft}}$ at $p_T=10$ GeV for $\eta /s=0.05$ in the $0–20\%$ centrality window.

Figure 3

Event-by-event scatter plot of $v_2^{\text{hard}}$ vs. $v_2^{\text{soft}}$ at $p_T=10$ GeV for $\eta /s=0.05$ in the $40–60\%$ centrality window.

Figure 4

Pearson Coefficient, $Q_n$ between $v_n^{\text{hard}}$ and $v_n^{\text{soft}}$, across all centralities, described by Eq. (30) for $p_T=10$ GeV.

Figure 5

Pearson coefficient, $Q_n$ between $v_n^{\text{hard}}$ and $v_n^{\text{soft}}$, across all centralities, described by Eq. (30) for $p_T=100$ GeV.

Figure 6

Pearson coefficient, $Q_n$ between $v_n^{\text{hard}}$ and ${\varepsilon }_n$ for $p_T=100$ GeV (left) and between $v_n^{\text{soft}}$ and ${\varepsilon }_n$ (right), across all centralities.

Figure 7

$R_{AA}\left(p_T\right)$ across centralities assuming linear path length dependence of the energy loss $dE/dL\propto L$, jet-medium decoupling temperature $T_d=160$ MeV, and $\eta /s=0.05$ across all centralities and plotted up to $p_T=100$ GeV. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 8

$v_2\left\{2\right\}\left(p_T\right)$ and $v_3\left\{2\right\}\left(p_T\right)$ across centralities computed assuming a linear path length dependence of the energy loss, $dE/dL\propto L$, and jet-medium decoupling temperature $T_d=160$ MeV. For the black curves $\eta /s=0.05$ while for the red curves $\eta /s=0.12$. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 9

Variation of $R_{AA}\left(p_T\right), v_2\left\{2\right\}\left(p_T\right)$, and $v_3\left\{2\right\}\left(p_T\right)$ with the path length dependence $dE/dL\propto L$ vs. $dE/dL\propto L^2$ and the jet-medium decoupling temperature $T_d=160$ MeV vs. $T_d=120$ MeV, keeping $\eta /s=0.05$. Only $0–5\%$ and $20–30\%$ centralities are shown. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 10

Cumulants $v_2\left\{2\right\}\left(p_T\right)$ and $v_2\left\{4\right\}\left(p_T\right)$ for $\eta /s=0.05$ and $\eta /s=0.12$ assuming a linear path length dependence $dE/dL\propto L$ and jet-medium decoupling temperature $T_d=160$ MeV across centralities. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 11

$v_2\left\{4\right\}\left(p_T\right)/v_2\left\{2\right\}\left(p_T\right)$ ratio across centralities for $\eta /s=0.05, dE/dL\propto L$, and jet-medium decoupling parameter $T_d=160$ MeV. The black band denotes the corresponding value of this ratio in the soft sector. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 12

$v_2\left\{4\right\}\left(p_T\right)/v_2\left\{2\right\}\left(p_T\right)$ ratio across centralities for $\eta /s=0.12, dE/dL\propto L$, and jet-medium decoupling parameter $T_d=160$ MeV. The red band denotes the corresponding value of this ratio in the soft sector. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 13

Variation in the $v_2\left\{4\right\}\left(p_T\right)/v_2\left\{2\right\}\left(p_T\right)$ ratio with changes in the path length dependence $dE/dL\propto L$ vs. $dE/dL\propto L^2$ as well as in the jet-medium decoupling temperature $T_d=160$ MeV vs. $T_d=120$ MeV, for $\eta /s=0.05$ and $\eta /s=0.12$. Only the $0–5\%$ centrality class is shown. All values are calculated for PbPb LHC collisions at $\sqrt{s_{NN}}=5.02$ TeV.

Figure 14

Difference between the soft and hard fluctuations, ${\mathrm{\Delta }}_2^{\text{SH}}$, defined in Eq. (33) for $\eta /s=0.05$ across all centralities and for three values of $p_T$.

Figure 15

Difference between the soft and hard fluctuations, ${\mathrm{\Delta }}_2^{\text{SH}}$, defined in Eq. (33) for $\eta /s=0.12$ across all centralities and for three values of $p_T$.

Figure 16

Distribution of the difference in the event plane angles at low and high $p_T, P\left(n[{\psi }_n^{\text{soft}}-{\psi }_n^{\text{hard}}\left(p_T\right)]\right)$, at $20–30\%$ centrality for LHC PbPb $\sqrt{s_{NN}}=5.02$ TeV.

Figure 17

$\langle cosn\left[{\psi }_n^{\text{soft}}-{\psi }_n^{\text{hard}}\left(p_T\right)\right]\rangle$, for $n=2$ and $n=3$, across all centralities for two different viscosities $\eta /s=0.05$ and $\eta /s=0.12$ for LHC PbPb collisions at $\sqrt{s_{NN}}=5.02$ TeV.