The goal of this program was to deepen our theoretical understanding of the phase structure of QCD-in particular the de-confinement transition and possible critical point-from present experimental programs, and to provide theoretical guidance to upcoming programs at several facilities worldwide. All talks and discussions were centered around this goal and highlighted different facets of this overarching task. Even though many questions remain unanswered, the program helped coordinate the extensive theoretical (and experimental) activity going on in preparation for the upcoming programs at NICA, FAIR and especially the second phase of the beam energy scan at RHIC. It has also become clear, that progress in theoretical understanding is necessary within the next 2-3 years to guide the measurements.

In the following, the main findings - and newly sharpened questions - are summarized.

Initial conditions:
The initial conditions at low beam energies require a detailed three-dimensional picture of coordinate and momentum space information of energy, momentum and net baryon densities and possibly other conserved charge densities including their event-by-event fluctuations. One of the main open questions is how the stopping of baryons can be understood. The experimental observables that are sensitive to the initial state and the stopping are the net baryon rapidity spectra and correlations as well as HBT correlations between pions and protons. Directed flow and Lambda polarization give insights on the rotation and angular momentum of the system. Three different approaches are promising to pursue:

• Color Glass Condensate/Gluon Saturation: This approach is very successful at high beam energies and the main question is how to extend it to 3 dimensions and beam velocities that are less than the speed of light. Down to which beam energies are the assumptions for this approach fulfilled?

• Hadron-string transport approaches: At very low beam energies the hadronic interactions based on resonance dynamics provide a good description. At the energies relevant for the RHIC beam energy scan the initial pre-equilibrium dynamics is likely dominated by string excitation and fragmentation. Can the string model reproduce the stopping mechanism correctly?

• 3-fluid hydrodynamics: The projectile and target are baryon-rich fluids and the fireball fluid is created by source terms from the interaction of these two fluids. Can these source terms be used to provide an understanding of the stopping at low beam energies?

Hot and dense stage:
The hot and dense stage is described by approaches based on relativistic hydrodynamics that are very successful in describing the collective behavior of the system and the anisotropic flow coefficients, especially at high energies. Within hybrid approaches the beam energy dependence of elliptic flow is nicely understood with an increasing viscosity at lower beam energies. The triangular flow is more sensitive to viscous effects and can be a sign for the break-down of hydrodynamic behavior. One of the main questions regarding hydrodynamics is how to judge, if the theory is still valid at the lower beam energies of interest. What are the limits of hydrodynamics and how can they be probed experimentally.

The dependence of the various transport coefficients like the shear and bulk viscosity or the baryon diffusion on the temperature and net baryon chemical potential is currently under investigation. Depending on the universality class of the critical point they diverge or have a minimum at the transition from the quark-gluon plasma to the hadron gas.

Equation of state:
At zero chemical potential lattice QCD calculations provide a consistent picture of the cross-over transition between partonic and hadronic degrees of freedom. Recent advances based on the Taylor expansion and the analytic continuation from imaginary chemical potential lead to the conclusion that there is no critical point in the QCD phase diagram up to net baryon chemical potential of twice the temperature. In general, due to increasing computing power, lattice calculations have made enormous progress within the last 5-10 years and constitute important constraints for any phenomenological model on the thermodynamic quantities and their fluctuations, such as the susceptibilities.

On the other hand, there is rather firm knowledge about the properties of nuclear matter close to the ground state and the liquid-gas phase transition. Also, constraints from neutron star observations have been discussed. At high baryon chemical potentials, there is still a lot of freedom to model the equation of state and more effort is required towards implementing all the known constraints in a parameterization for the equation of state that can be used for example in hydrodynamic calculations.

Propagation of fluctuations:
The theoretical understanding on how to separate the hydrodynamic and the critical modes and how to dynamically evolve both of them in a meaningful way has recently advanced considerably. The first numerical implementations of propagating thermal fluctuations within relativistic hydrodynamics are becoming available. Therefore, one major item of discussion was how to include those fluctuations properly in the Cooper-Frye transition to a hadronic transport approach. The late stage non-equilibrium evolution is necessary to include detector effects, kinematic cuts and resonance decays. The usual particlization sampling algorithm introduces spurious Poissonian fluctuations, and it has to be clarified how this double counting can be avoided and how the fluctuations and correlations of interest on the hydrodynamic hypersurface can be preserved. Ideas include the use of test particles or to define correlation functions that can be calculated directly without the particlization step. Even though the propagation equations for hydrodynamics are based on conservation laws the hybrid framework of hydrodynamics + Cooper-Frye usually does not conserve the quantum numbers event-by-event.

Electromagnetic fields and anomalous transport:
There has been progress in implementing hydrodynamic evolution of the relevant conserved charges coupled to the magnetic field. A full 3+1 dimensional magnetohydrodynamics framework is still missing including a quantitative assessment of charge correlation observables that are associated with the chiral magnetic effect. The Lambda polarization measurement provides complementary information by offering the possibility to extract the final magnetic field in the late stages of interaction. In that sense, at least the order of magnitude of the magnetic field created in heavy ion collisions is accessible by a fully independent measurement.

The upper two figures indicate the kurtosis of the net-proton number in central gold-gold collisions measured for different beam energies corresponding to different baryon chemical potentials from the STAR collaboration. On the left hand side, the expectations for the kurtosis in the critical region in the phase diagram are depicted.

The lower two figures show charge correlation measurements with respect to the reaction plane, that are associated with the chiral magnetic effect in heavy ion collisions.

Promising observables:
The first stage of the beam energy scan program at RHIC and previous heavy ion programs at intermediate beam energies at the SPS have provided a wealth of experimental data. The most interesting features that have been observed are:

• the dip in the slope at midrapidity of the net proton directed flow as a possible indication of the first order phase transition

• the peak in the ratio of HBT radii R_O/R_S indicating a longer life-time associated with the mixed phase

• deviations from baseline expectations in the excitation functions of higher moments of the net proton distribution (kurtosis)

• global Lambda polarization has been observed for the first time indicating a finite angular momentum/magnetic field in heavy ion collisions

As an important cross-check the HADES collaboration has collected enough statistics in Au+Au collisions at very low beam energies of 1.23 AGeV to perform detailed fluctuation and correlation measurements, such as the kurtosis, and charge correlations with respect to the event plane associated with the chiral magnetic effect. First results of the cumulant analysis have been reported at the workshop and HADES sees enhancements of the kurtosis similar or even larger than that reported by STAR.

The directed flow and the HBT radii still await quantitative explanations by theoretical dynamical approaches. The analysis of higher moments is the most discussed signature for a critical endpoint. It requires a large amount of experimental scrutiny to provide final measurements of these fluctuations. All details are important ranging from kinematic cuts, via two-particle efficiencies to fluctuations of the number of participants in each centrality bin. Future measurements of interest include:

• published identified particle spectra as a function of transverse momentum and rapidity, since bulk observables are important to calibrate the basic parameters of dynamical approaches.

• the net proton rapidity spectra and correlations will allow insights into the stopping mechanism. First results from STAR on rapidity correlations have been presented at the workshop.

• HBT correlations of pions with protons can be useful for insights about the stopping mechanism as well.

• HBT lifetime measurements with other particle species beyond pions.

• more detailed charge correlation measurements to confirm or disprove the CME signals.

• higher statistics for the net proton higher moments analysis.

• proton-nucleus beam energy scan to understand initial conditions and to disentangle magnetic field effects as well as constrain the baryon stopping at low energies.

• N-particle correlation functions as a function of rapidity difference, since it seems that correlation functions are easier to interpret than cumulants.

• Low-mass dileptons, since they are changing with baryonic resonances and their behavior in turn depends on the baryon diffusion.

In general, there are great prospects for future measurements, but it is crucial that the theory community is prepared to interpret those measurements. Relying on existing observables for the bulk dynamics it is now the time to develop and refine the dynamical models including a phase transition or a critical point and make quantitative predictions. The final goal should be to provide exclusion plots on the phase diagram at the minimum or even better to indicate regions for the location of the critical point and the phase transition.