Baseline predictions of elliptic flow and fluctuations for the RHIC Beam Energy Scan using response coefficients

Currently the Relativistic Heavy Ion Collider (RHIC) Beam Energy Scan is exploring a new region of the quantum chromodynamic phase diagram at large baryon densities that approaches nuclear astrophysics regimes. This provides an opportunity to study relativistic hydrodynamics in a regime where the net conserved charges of baryon number, strangeness, and electric charge play a role, which will significantly change the theoretical approach to simulating the baryon-dense quark-gluon plasma. Here we detail many of the important changes needed to adapt both initial conditions and the medium to baryon-rich matter. Then, we make baseline predictions (i.e., assuming a high-energy approach) for the elliptical flow and fluctuations based on extrapolating the physics at Large Hadron Collider and top RHIC energies to support future analyses of where and how the new baryon-dense physics causes these extrapolations to break down. First we compare eccentricities across beam energies, exploring their underlying assumptions; we find the extrapolated initial state is predicted to be nearly identical to that at AuAu $\sqrt{s_{NN}}=200$ GeV. Then we make exploratory predictions of the final flow harmonic based on linear $+$ cubic response. We discuss preliminary STAR results in order to determine the implications that they have for linear $+$ cubic response coefficients at the lowest beam energy of AuAu $\sqrt{s_{NN}}=7$ GeV.

Figure 1

Comparison of $v_2\left\{4\right\}/v_2\left\{2\right\}$ STAR data to trento and mckln eccentricities and hydro calculations from trento $+$ v-usphydro, mckln $+$ v-usphydro, and IP-Glasma $+$ MUSIC. STAR data are taken from Ref. [96] and the $\mathrm{IP}\text{−}\mathrm{Glasma}+\mathrm{MUSIC}$ data are from Ref. [97].

Figure 2

Cartoon plot of Eq. (17) illustrating the comparison between the energy-independent cross section ${\sigma }_{\mathrm{eik}}$ and the energy-suppressed corrections ${\sigma }_{\mathrm{sub}\text{−}\mathrm{eik}}$ at high energies. Shaded regions sketch the kinematic regions of various collider programs, illustrating the dominance of ${\sigma }_{\mathrm{eik}}$ at LHC and top RHIC energies in contrast to the Beam Energy Scan (BES) where the new mechanisms contained within ${\sigma }_{\mathrm{sub}\text{−}\mathrm{eik}}$ can become very important.

Figure 3

Comparison of ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$ (left) and ${\varepsilon }_2\left\{2\right\}$ (right) for three descriptions of the initial state in trento: $p=-1$ (mckln-like), $p=0$ (IP-Glasma/EKRT-like), and $p=1$ (Glauber-like). The 200 GeV AuAu STAR data from Ref. [104] are shown in black. For the initial-state models, two different beam energies of AuAu, $\sqrt{s_{NN}}=200$ GeV and $\sqrt{s_{NN}}=7$ GeV, are considered.

Figure 4

Comparison of STAR and PHENIX experimental data for $v_2\left\{4\right\}/v_2\left\{2\right\}$ at RHIC AuAu $\sqrt{s_{NN}}=200$ GeV.

Figure 5

Comparison of PHENIX experimental data with and without subevents for $v_2\left\{4\right\}/v_2\left\{2\right\}$ at RHIC AuAu $\sqrt{s_{NN}}=200$ GeV.

Figure 6

Comparison of ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$ (left) and ${\varepsilon }_{\{}2\}$ (right) for varied $k$ fluctuations value in trento where $k=1.6$ is the standard value from Bayesian analysis. Calculations were done at AuAu $\sqrt{s_{NN}}=7$ GeV.

Figure 7

Comparison of ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$ (left) and ${\varepsilon }_{\{}2\}$ (right) for varied nucleon width, $\sigma$, in trento where $\sigma =0.51$ fm is the standard value from Bayesian analysis. Calculations were done at AuAu $\sqrt{s_{NN}}=7$ GeV. (Centrality given in %.)

Figure 8

Comparison of $v_n$ versus ${\varepsilon }_n$ relationship using either calculated ${\kappa }_{1,2}$ and ${\kappa }_{2,2}$ from Eq. (11) to extracted ones using numerical methods shown in mid-central collisions (left) and peripheral collisions (right) shown for trento $+$ v-usphydro at PbPb 2.76 TeV.

Figure 9

Comparison of calculated ${\kappa }_{1,2}$ (left) and ${\kappa }_{2,2}$ (right) from Eq. (11) to extracted ones using numerical methods (linear regression) across centralities shown for trento $+$ v-usphydro at PbPb 2.76 TeV.

Figure 10

Linear coefficient ${\kappa }_{1,2}$ for elliptical flow across beam energy extracted using trento $+$v-usphydro. Extrapolated values for AuAu 7 GeV shown in brown dashed lines.

Figure 11

Cubic coefficient ${\kappa }_{2,2}$ for elliptical flow across beam energy extracted using trento $+$v-usphydro. Extrapolated values for AuAu 7 GeV shown in brown dashed lines.

Figure 12

Residual contribution to $\langle v_2^2\rangle$ for elliptical flow across beam energy extracted using trento $+$v-usphydro. Extrapolated values for AuAu 7 GeV shown in brown dashed lines.

Figure 13

Residual contribution to $\langle v_2^4\rangle$ for elliptical flow across beam energy extracted using trento and mckln $+$v-usphydro. Extrapolated values for AuAu 7 GeV shown in brown dashed lines.

Figure 14

Direct trento $+$ v-usphydro hydrodynamical calculations (solid black) versus the predicted $v_n\left\{2\right\}$ from linear $+$ cubic response (red long dashed) and the reconstructed $v_n\left\{2\right\}$ from linear $+$ cubic response $+$ residual (blue short dashed). Calculations are performed for PbPb $5.02\mathrm{TeV}$ with initial conditions using trento $p=0$.

Figure 15

The cumulant ratio $v_2\left\{4\right\}/v_2\left\{2\right\}$ calculated for the top RHIC and LHC energies using full hydrodynamics (black), linear response only (blue), and linear $+$ cubic response (red). Calculations are performed for AuAu $200\mathrm{GeV}$ collisions (top), PbPb $2.76\mathrm{TeV}$ collisions (middle), and PbPb $5.02\mathrm{TeV}$ collisions (bottom) with initial conditions using trento $p=0$.

Figure 16

Extrapolation of ${\kappa }_{1,2}$ down to lower beam energies using a linear fit either considering all three beam energies, linear fit with just the lowest two, or a fit with the format $a+blog\left[\sqrt{s_{NN}}\right]$ where $a$ and $b$ are constants. The calculations here use initial conditions generated by trento with $p=0$, and the plot is for the 10–15% centrality bin.

Figure 17

Direct trento $+$ v-usphydro hydrodynamic calculations for AuAu $200\mathrm{GeV}$ (solid black) versus the predicted $v_2\left\{2\right\}$ from extrapolating the linear $+$ cubic response coefficients and residual to lower energies. Here we compare a linear extrapolation to fits of all beam energies (left) versus the lowest two beam energies (right). Baseline predictions are made for $54\mathrm{GeV}$ (red long dashed), $27\mathrm{GeV}$ (blue dot dashed), and $7\mathrm{GeV}$ (brown short dashed).

Figure 18

Direct trento $+$ v-usphydro hydrodynamic calculations for AuAu $200\mathrm{GeV}$ (solid black) versus the predicted $v_2\left\{2\right\}$ (left) and $v_2\left\{4\right\}/v_2\left\{2\right\}$ (right) from extrapolating the linear $+$ cubic response coefficients and residual to lower energies. Here we assume a logarithmic extrapolation $a+blog\left[\sqrt{s_{NN}}\right]$ designed to mimic the onset of significant sub-eikonal physics at lower energies. Predictions are made for $54\mathrm{GeV}$ (red long dashed), $27\mathrm{GeV}$ (blue dot dashed), and $7\mathrm{GeV}$ (brown short dashed).

Figure 19

Direct trento $+$ v-usphydro hydrodynamic calculations (solid black) versus the predicted $v_n\left\{2\right\}$ from linear $+$ cubic response (red long dashed) and the reconstructed $v_n\left\{2\right\}$ from linear $+$ cubic response $+$ residual (blue short dashed). Calculations at PbPb 5.02 TeV.