Particle spectra and HBT radii for simulated central nuclear collisions of C+C, Al+Al, Cu+Cu, Au+Au, and Pb+Pb from Sqrt(s)=62.4-2760 GeV

We study the temperature profile, pion spectra and HBT radii in central symmetric and boost-invariant nuclear collisions using a super hybrid model for heavy-ion collisions (SONIC) combining pre-equilibrium flow with viscous hydrodynamics and late-stage hadronic rescatterings. In particular, we simulate Pb+Pb collisions at Sqrt(s)=2.76 TeV, Au+Au, Cu+Cu, Al+Al, and C+C collisions at Sqrt(s)=200 GeV and Au+Au, Cu+Cu collisions at Sqrt(s)=62.4 GeV. We find that SONIC provides a good match to the pion spectra and HBT radii for all collision systems and energies, confirming earlier work that a combination of pre-equilibrium flow, viscosity and QCD equation of state can resolve the so-called HBT puzzle. For reference, we also show p+p collisions at Sqrt(s)=7 TeV. We make tabulated data for the 2+1 dimensional temperature evolution of all systems publicly available for the use in future jet energy loss or similar studies.

We study the temperature profile, pion spectra and HBT radii in central symmetric and boostinvariant nuclear collisions using a super hybrid model for heavy-ion collisions (SONIC) combining pre-equilibrium flow with viscous hydrodynamics and late-stage hadronic rescatterings. In particular, we simulate Pb+Pb collisions at √ s = 2.76 TeV, Au+Au, Cu+Cu, Al+Al, and C+C collisions at √ s = 200 GeV and Au+Au, Cu+Cu collisions at √ s = 62.4 GeV. We find that SONIC provides a good match to the pion spectra and HBT radii for all collision systems and energies, confirming earlier work that a combination of pre-equilibrium flow, viscosity and QCD equation of state can resolve the so-called HBT puzzle. For reference, we also show p+p collisions at √ s = 7 TeV. We make tabulated data for the 2+1 dimensional temperature evolution of all systems publicly available for the use in future jet energy loss or similar studies.

I. INTRODUCTION
With the advent of gauge/gravity duality, it has become possible to effectively simulate far-from equilibrium thermalization in central (and smooth) nuclear collisions [1]. Combining this pre-equilibrium dynamics with hydrodynamics [2] and a late-stage hadronic cascade [3], one obtains a 'Super hybrid mOdel simulatioN for relativistic heavy-Ion Collisions' (SONIC for short) that effectively has only a limited number of parameters, namely those specifying the properties of the incoming nuclei, the speed of sound, and shear and bulk viscosities in the quark-gluon plasma. In this work, we use this model to study symmetric nuclear collisions of different nuclei (Pb,Au,Cu,Al,C) at collision energies ranging from √ s = 2.76 TeV to √ s = 62.4 GeV. We study the temperature evolution, pion spectra, and HBT radii for these different collision systems with the aim of both testing the model against experimental data where available and providing model predictions for the design of future experimental studies. In addition, we also show results for SONIC for central p+p collisions at √ s = 7 TeV energy, even though evidence for forming an equilibrated quark-gluon plasma in these systems is currently lacking.
A fundamental question regarding the quark-gluon plasma is at what temperature and what scale a strong coupling description is most appropriate versus weak coupling. Near-inviscid hydrodynamic modeling indicates strong coupling, though the exact sensitivity of final state hadrons to the temperature dependence of η/s is currently under investigation. There are experimental observables when compared to model calculations that are potential indicators of stronger coupling at temperatures near the transition point. Inclusion of this strongest coupling near the transition is proposed to help reconcile the full suite of jet quenching observables including the anisotropy in mid-central collisions [4,5], the larger than expected v 2 and v 3 of direct photons [6] and heavy quark observables [7]. An important motivation for the sPHENIX upgrade [8] is to answer the question regarding the underlying nature of the quarkgluon plasma near the point of strongest coupling. A key question is whether high statistics data sets in Au+Au collisions at √ s = 200 GeV at RHIC and Pb+Pb collisions at √ s = 2.76 TeV at the LHC are substantially augmented by hard process observables at lower RHIC energies and with different nuclear geometries for emphasizing emission and parton quenching interactions closer or further away from this transition temperature. In this work, we explore the temperature evolution of different systems and provide access to the space-time snapshots for utilization in jet quenching, photon emission, and heavy quark diffusion calculations.

II. METHODOLOGY
We model heavy-ion collisions by using a super hybrid model which we call SONIC which combines pre-equilibrium flow with hydrodynamics and a late-stage hadronic afterburner. Introducing the radius r = x 2 + y 2 , the different nuclei are modeled by employing an overlap function with R, a the charge radius and skin depth parameters listed in Table I. 0 is an overall normalization constant that controls the total final multiplicity. The pre-equilibrium flow has been calculated numerically assuming an infinite number of colors and infinite coupling for central (and smooth) Pb-Pb collisions at √ s = 2.76 TeV in [1]. We reanalyzed the results from Ref.
[1], finding that after the system has thermalized, the velocity is consistent with the early-time analytic result derived in [9] up to an overall factor of two (see Figure 1). Therefore, in the following we will employ a pre-equilibrium radial flow velocity where τ = √ t 2 − z 2 , see Fig. 1. Using Eq.
(2) and an initial energy density profile given by (cf. [9]) we start the hydrodynamic evolution at a time τ sw . Following the observations in Ref. [1,10], τ sw has to be large enough such that a local rest-frame can be defined as (τ sw > ∼ 0.35 fm/c) and before non-linear effects prohibit the use of Eq. (2) (τ sw < ∼ 0.6 fm/c). Using the energy density from Eq. (3), the flow profile from Eq. (2), and setting the initial shear and bulk stresses to zero, we can solve the subsequent system evolution using the relativistic viscous hydrodynamics solver VH2+1 [2, 12], version 1.7. The fluid shear viscosity over entropy ratio is set to η/s = 0.08 and the bulk viscosity over entropy ratio is set to η/s = 0.01. The equation of state used is that from Ref. [11] which is consistent with lattice QCD data [13,14] at vanishing baryon density and matches a hadron resonance gas at low temperatures. We monitor the isothermal hypersurface defined by T S = 170 MeV throughout the system evolution until the last fluid cell has cooled below T S .
From the information about fluid temperature, velocity and dissipative stress components we generate hadrons with masses up to 2.2 GeV and follow their rescattering dynamics using the hadron cascade code B3D [3]. We generate 5000 B3D events for each hydro event. Once the particles have stopped interacting we collect information about the particle spectra and report total charged multiplicity dN ch dy , the mean pion transverse momentum < p T > and the pion HBT radii R out , R side , R long .
With the pre-equilibrium flow given by Eq.
(2), and adjusting 0 so that total multiplicity is constant, we find that the final particle < p T > and the extracted HBT radii are insensitive to the choice of τ sw , just as in the full gauge/gravity+hydro+cascade calculation (cf. Ref. [1]). Thus, τ sw is not a relevant parameter of SONIC. This leaves a total of 6 relevant parameters for the system evolution: three numbers (R, a, T S ) and three functions (the temperature dependent ratios η/s, ζ/s and the equation of state). Note that 0 is fixed by requiring the final charged multiplicity to match experimental data, wherever it is available [15][16][17], cf. Tab. II. For C+C and Al+Al collisions at √ s = 200 GeV, we employ the formula where dNpp dy is the charged multiplicity for nucleon-nucleon collisions at a given collision energy √ s, and α( √ s = 200 GeV) = 0.13 (cf. Ref. [18]).
We note that a full 2+1 dimensional simulation of a single symmetric nuclear collision can be executed on a modern desktop in a matter of minutes, which makes SONIC a viable tool to investigate collisions having granular initial  For reference, also p + p collisions at √ s = 7 TeV are shown, even though this system may not equilibrate at all. The "kink" at 2 fm/c in the temperature evolution in the p + p system around T = TS is due to the center r = 0 being cooler than the surrounding matter.
conditions on an event-by-event basis in the future.

III. RESULTS
Our results for the multiplicity and mean pion transverse momentum in the different systems are reported in Table II alongside with experimental results where available. Since the experimental multiplicity is used to fix one of the model parameters ( 0 ), only the pion < p T > is a non-trivial model output. Comparing experimental measurements of < p T > with model output from SONIC, we find that with model parameter choices η/s = 0.08, ζ/s = 0.01, T S = 170 MeV and a QCD equation of state, there is good agreement with experimental data for all collision systems at all collision energies.
For reference, we also show SONIC runs for p+p collisions at √ s = 7 TeV collision energy, even though this system may not form an equilibrated state of matter (and thus SONIC would not be applicable in this case). Note that, nevertheless, the < p T > value for p+p collisions is not too far from the experimental value, which may just be a reflection of the fact that transverse flow is not an indicator of system equilibration (cf.[1]).
It should be noted that our current implementation of SONIC does not properly reproduce the experimentally measured proton spectra because the number of protons and antiprotons is too high. The reason for this has been identified to be the missing implementation of baryon-antibaryon annihilation [25,26]. For this reason, we currently are unable to report physically viable results for baryons.
The time evolution for the temperature in the center of the fireball (r = 0 fm) is reported in Fig. 2, where we distinguish between the evolution spent in the hydrodynamic phase (T > T S ) and the hadron gas phase at low temperature. Shown in Fig. 3 is the radial velocity profile for the different collision systems at τ = 2 fm/c inside the hydrodynamic phase. Not surprisingly, larger systems tend to build up smaller radial flow and tend to live longer than smaller systems. However, possibly interesting features for temperature evolution between different systems may also be identified in Fig. 3. For instance, note that Fig. 3  the Au+Au √ s = 200 GeV curve. We are making the full two-dimensional space-time evolution profiles available for potential use in studies of jet energy loss, direct photon emission and heavy quark diffusion [27].
Our results for pion HBT radii are calculated as described in [3] and the results are shown in Fig. 5 for the different collision systems. Despite some remaining discrepancies between our model results and experimental data, the overall agreement between SONIC and experiment for different collision energies and systems is striking, given that the inability of standard hydrodynamics to describe the data has been labeled the 'HBT puzzle' in the literature. As noted in Ref. [28], it is possible to resolve this 'puzzle' by a combination of different ingredients, notably pre- equilibrium flow, viscosity, and a QCD-like equation of state. Since all of these ingredients are naturally incorporated in SONIC, it is gratifying to observe that the HBT puzzle is no longer a puzzle but rather a (small) discrepancy in some of the data-model comparison.
In Figure 4, we show the pion transverse momentum spectra for the different collision systems. As remarked above, we do find that with constant values of η/s = 0.08, ζ/s = 0.01 and a QCD equation of state, SONIC provides a good overall description of the available experimental data. Note that the discrepancy in the pion spectra for Pb+Pb collisions at p T > 1.5 GeV was not observed in Ref.
[1]. The reason is that in Ref.
[1], the actual calculation erroneously used a model parameter value of R = 6.48 fm instead of R = 6.62 fm (cf. Tab. I) for Pb. Once correcting for this error, we do find slightly less transverse flow in Pb+Pb collsions, leading to the discrepancy observed in Fig. 4. However, it is expected that implementing more realistic granular initial conditions will lead to higher transverse flow velocities. This could help to improve the description of experimental data at p T > 1.5 GeV in SONIC in the future.

IV. CONCLUSIONS
We have presented SONIC, a new super hybrid model for heavy-ion collisions that combines pre-equilibrium flow, viscous hydrodynamics, and hadronic cascade dynamics into one package. SONIC was used to simulate boostinvariant, central symmetric collisions of smooth nuclei (Pb,Au,Cu,Al,C) at energies ranging from √ s = 62.4 GeV to √ s = 2.76 TeV. We found that for a QCD equation of state and a choice of QCD viscosity over entropy ratios of η/s = 0.08, ζ/s = 0.01, the particle spectra and pion HBT radii were in reasonable agreement with available experimental data. We also made predictions for pion mean transverse momentum and HBT radii for C+C and Al+Al collisions at √ s = 200 GeV. The 2+1 dimensional space-time evolution of the temperature obtained with SONIC are publicly available [27] in order to be of use in future studies of jet energy loss or photon emission.

ACKNOWLEDGMENTS
This work was supported by the Department of Energy, DOE award No. DE-SC0008027 and DE-FG02-00ER41244. We would like to thank the JET Collaboration and Chun Shen for useful discussions.