Bulk Observables within a Hybrid Approach for Heavy Ion Collisions with SMASH Afterburner ^†

Abstract

We present a model of the dynamical evolution of relativistic heavy ion collisions, which combines second-order viscous hydrodynamics and microscopic transport. In particular, we present a hybrid approach with MUSIC hydrodynamics, and SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) afterburner. In this work, we focus on low-$p_T$ hadronic observables—identified hadron $p_T$ spectra and anisotropic flow coefficients. We also demonstrate how the hadronic chemistry is altered by the hadronic non-equilibrium dynamics, for example by baryon-antibaryon annihilation. The new MUSIC + SMASH hybrid approach is also compared to existing MUSIC + UrQMD results.

1. Introduction

For a few decades, high-energy nucleus-nucleus collisions have been carried out as a way to explore QCD matter at high temperature. One of the most important findings is that quark-gluon plasma (QGP) is created and behaves like a perfect fluid [1,2,3,4,5,6,7]. The hybrid framework, which combines viscous hydrodynamics and microscopic transport, has been successful in describing the low-$p_T$ hadronic observables in heavy ion collisions [8,9,10]. It has been consensus that it is necessary to have hadronic and electromagnetic degrees of freedom within a single framework of microscopic transport. In addition, given that resonance excitations and decays dominate the low-$p_T$ bulk dynamics in hadronic re-scattering, we are motivated to take the resonance mass distribution into account at the point of transition from hydrodynamics to microscopic transport. In this work, we present a hybrid approach with the IP-Glasma pre-thermalization dynamics [11], MUSIC hydrodynamics [12,13] with shear and bulk viscosities, Cooper-Frye particlization [14], and SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) [15] afterburner. In addition to the resonance mass sampling at particlization, we focus on how the hadronic chemistry is altered by the hadronic non-equilibrium dynamics, for example by baryon-antibaryon annihilation.

2. Materials and Methods

The pre-equilibrium dynamics in this work is described by the IP-Glasma model [11], where the low-x gluons with large multiplicity are characterized by a gluon field and their time evolution is governed by the classical Yang-Mills dynamics with color charges carried by colliding nuclei. The system is assumed to reach local thermal equilibrium at ${\tau }_0=0.4\mathrm{fm}$, until which the gluon field is evolved by the classical Yang-Mills equation. Once the energy-momentum tensor of gluon field is specified, the initial energy density and flow velocity for hydrodynamics are given by the eigenvalue equation $T_{\phantom{\mu }\nu }^{\mu }{u}^{\nu }=ϵu^{\mu }$.

Time evolution of the fireball is handled by MUSIC viscous hydrodynamics [12,13] with shear and bulk viscosities. In addition to the energy-momentum conservation ${\partial }_{\mu }{T}^{\mu \nu }=0$, the following equations determine time evolution of the shear stress tensor ${\pi }^{\mu \nu }$ and bulk pressure $\Pi$.

$$\begin{array}{ccc}\hfill {\tau }_{\Pi }\dot{\Pi }+\Pi & =& -\zeta \theta -{\delta }_{\Pi \Pi }\Pi \theta +{\lambda }_{\Pi \pi }{\pi }^{\mu \nu }{\sigma }_{\mu \nu }\hfill \\ \hfill {\tau }_{\pi }{\dot{\pi }}^{\langle \mu \nu \rangle }+{\pi }^{\mu \nu }& =& 2\eta {\sigma }^{\mu \nu }-{\delta }_{\pi \pi }{\pi }^{\mu \nu }\theta +{\phi }_7{\pi }_{\alpha }^{\langle \mu }{\pi }^{\nu \rangle \alpha }\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & -{\tau }_{\pi \pi }{\pi }_{\alpha }^{\langle \mu }{\sigma }^{\nu \rangle \alpha }+{\lambda }_{\pi \Pi }\Pi {\sigma }^{\mu \nu }\hfill \end{array}$$

(2)

where we have introduced the transport coefficients [16,17], which quantify how the system responds to spatial anisotropy and inhomogeneity. The equation of state, which provides a connection between energy density and pressure, has a cross-over phase transition between the hadronic resonance gas and lattice QCD [18].

The macroscopic degrees of freedom in hydrodynamics must be transformed into particles to simulate hadronic re-scatterings with microscopic transport. Based on the Cooper-Frye formalism [14], this transition is performed such that energy and momentum are conserved. The momentum-space distribution of particles is given by

$$\frac{dN}{d^3\mathbf{p}}=\frac{g}{{\left(2\pi \right)}^3}{\int }_{\Sigma }[f_0(x,\mathbf{p})+\delta f_{\mathrm{shear}}+\delta f_{\mathrm{bulk}}]\frac{p^{\mu }{d}^3{\Sigma }_{\mu }}{E_{\mathbf{p}}}$$

(3)

where we have the shear [19] and bulk [20] viscous corrections to the distribution function. They are linearly proportional to ${\pi }^{\mu \nu }$ and $\Pi$, respectively. The isothermal hypersurface with temperature $165\mathrm{MeV}$ is chosen as the hypersurface $\Sigma$, at which particlization occurs. Introducing resonance mass sampling at particlization is work in progress. It must be pointed out that, in addition to the mass sampling of individual particle, the multiplicity integrations for sampling also have to take the mass distribution into account. Figure 1 shows the spectral functions being considered. It is expected that hadronic chemistry as well as kinematics will be altered by mass sampling. If higher masses are favored due to the asymmetric spectral functions, the $p_T$ spectra will be shifted toward higher $p_T$ and the multiplicity will be reduced.

The hadronic re-scatterings are simulated by SMASH 1.5 [15], which is a microscopic transport approach incorporating hadronic degrees of freedom and a geometric collision criterion. The most significant hadronic interactions include elastic scatterings, resonance excitation and decay and baryon-antibaryon annihilations.

3. Results

In this work, we focus on the single-particle distribution, which is well captured by the identified hadron $p_T$-spectra and differential elliptic flow. They are presented in Figure 2 putting an emphasis on effects of hadronic re-scatterings. For the $p_T$-spectra, the PHENIX data [23] are shown for comparison. For the $p_T$-differential elliptic flows, the PHENIX [24] and STAR [25] data are shown. Note that in this calculation, all resonances are sampled with pole masses to allow for an apple-to-apple comparison to the existing MUSIC+UrQMD results [10].

The proton $p_T$ spectrum and elliptic flow is shifted toward higher $p_T$, due to scatterings with lighter hadrons. The proton yield is also reduced by annihilation processes. While protons are affected in the similar way as in MUSIC + UrQMD hybrid, pions are less accelerated during the re-scattering phase. This is partially due to the difference between SMASH and UrQMD [26,27] in terms of pion-pion elastic scatterings.

4. Discussion

By comparing our simulations with and without collisions, it can be shown that protons are accelerated as a consequence of collisions with lighter hadrons, which are flowing faster in the radial direction. This observation is well consistent with the previous work based on a hybrid approach [10]. In addition, the presence of the baryon-antibaryon annihilation process reduces proton multiplicity. As discussed in Refs. [10,28], one can argue that the higher masses of baryons, in conjunction with the expansion of the system, make baryon-antibaryon creation less significant compared to annihilations. Therefore, our current treatment of baryon-antibaryon annihilation is still a good approximation to deal with the baryonic chemistry.

As mentioned earlier, the resonance excitation and decay are dominant processes in the hadronic re-scattering phase, and therefore the resonance mass sampling at the particlization is a crucial step. Future works include investigation of uncertainties such as different formulation or difference between vacuum and in-medium spectral functions. Lastly, even though we have been focusing on the single-particle distributions so far, multi-particle distributions and correlations certainly deserve attention in a study with the hybrid framework. In this case, we expect that conservation laws at the particlization play important roles.