Exploring the beam-energy dependence of flow-like signatures in small-system $d+\mathrm{Au}$ collisions

Recent analyses of small collision systems, namely $p+p$ and $p+\mathrm{Pb}$ at the LHC, and $p+\mathrm{Au}$, $d+\mathrm{Au}$, and ${}^3\mathrm{He}+\mathrm{Au}$ at RHIC, have revealed azimuthal momentum anisotropies commonly associated with collective flow in larger systems. Viscous hydrodynamics and parton cascade calculations have proven successful at describing some flow-like observables in these systems. These two classes of calculations also confirm these observables to be directly related to the initial geometry of the created system. Describing data at the highest RHIC and LHC energies requires a quark-gluon plasma or partonic rescattering stage, which raises the question of how small and low in energy can one push the system before only hadronic interactions are required for a full description. Hence, a beam-energy scan of small systems—that amounts to varying the initial temperature and the lifetime of the medium—can provide valuable information to shed light on these issues. In this paper, we present predictions from viscous hydrodynamics (sonic and supersonic), and partonic (ampt) and hadronic (urmqd) cascade calculations for elliptic $\left(v_2\right)$ and triangular $\left(v_3\right)$ anisotropy coefficients in $d$ + Au collisions at $\sqrt{s_{{}_{NN}}}=7.7$, 20, 39, 62.4, 200 GeV and 5.02 TeV.

Figure 1

Transverse momentum dependence of $v_2$ in $d$ + Au collisions at $\sqrt{s_{{}_{NN}}}=7.7,20.0,39.6,62.4,200.0$ GeV and in $d$ + Pb collisions at $\sqrt{s_{{}_{NN}}}=5.02$ TeV. Results are shown from viscous hydrodynamic calculations (sonic and with pre-equilibrium dynamics supersonic) as well as the parton cascade model ampt (with and without a partonic scattering stage), and the purely hadronic cascade model (urqmd). Published experimental data for $v_2$ in $d$ + Au at $\sqrt{s_{{}_{NN}}}=200$ GeV is also shown.

Figure 2

Transverse momentum dependence of $v_3$ in $d$ + Au collisions at $\sqrt{s_{{}_{NN}}}=7.7,20.0,39.6,62.4,200.0$ GeV and in $d$ + Pb collisions at $\sqrt{s_{{}_{NN}}}=5.02$ TeV. Results are shown from viscous hydrodynamic calculations (sonic and with pre-equilibrium dynamics supersonic) as well as the parton cascade model ampt (with and without a partonic scattering stage).

Figure 3

Transverse momentum dependence of $v_2$ in $p$ + Pb and $d$ + Pb at $\sqrt{s_{{}_{NN}}}=5.02$ TeV from the sonic hydrodynamic model. CMS measurements in central $p$ + Pb events are shown for comparison.

Figure 4

(a) and (b) The ratio of $v_2 \left(v_3\right)$ as a function of $p_T$ at different energies to $d$ + Au $v_2$ at $\sqrt{s_{{}_{NN}}}=200$ GeV from the supersonic hydrodynamic calculation.

Figure 5

Collision energy dependence of the summed space-time volume element in the QGP in viscous hydrodynamics.

Figure 6

Ratio of $v_2\left(p_T\right)$ in (a) and $v_3\left(p_T\right)$ in (b) from sonic to supersonic for $d$ + Au at all collision energies.

Figure 7

(a) and (b) The ratio of $v_2$ and $v_3$, respectively, as a function of $p_T$ at different energies to $d$ + Au $v_2$ and $v_3$ at $\sqrt{s_{{}_{NN}}}=200$ GeV from AMPT.

Figure 8

Probability for a parton in ampt to undergo $N$ scattering events prior to hadronization, for $d$ + Au and $d$ + Pb collisions at a variety of collision energies.

Figure 9

Transverse momentum dependence of $v_2$ from the URQMD model at variety of collision energies, computed with two-particle cumulants and with respect to the participant plane.