We present a new method for analyzing directed and elliptic flow in heavy-ion collisions. Unlike standard methods, it separates the contribution of flow to azimuthal correlations from contributions due to other effects. The separation relies on a cumulant expansion of multiparticle azimuthal correlations, and includes corrections for detector inefficiencies. This new method allows the measurement of the flow of identified particles in narrow phase-space regions, and can be used in every regime, from intermediate to ultrarelativistic energies.

The methods currently used to measure azimuthal distributions of particles in heavy-ion collisions assume that all azimuthal correlations between particles result from their correlation with the reaction plane. However, other correlations exist, and it is safe to neglect them only if azimuthal anisotropies are much larger than $1/\sqrt{N},$ with N the total number of particles emitted in the collision. This condition is not satisfied at ultrarelativistic energies. We propose a new method, based on a cumulant expansion of multiparticle azimuthal correlations, which allows measurements of much smaller values of azimuthal anisotropies, down to $1/N.$ It is simple to implement and can be used to measure both integrated and differential flow. Furthermore, this method automatically eliminates the major systematic errors, which are due to azimuthal asymmetries in the detector acceptance.

Directed and elliptic flow measurements for charged pions and protons are reported as a function of transverse momentum, rapidity, and centrality for $40A$ and $158A\mathrm{GeV}$ $\mathrm{Pb}+\mathrm{Pb}$ collisions, as recorded by the NA49 detector. Both the standard method of correlating particles with an event plane and the cumulant method of studying multiparticle correlations are used. In the standard method the directed flow is corrected for conservation of momentum. In the cumulant method elliptic flow is reconstructed from genuine four-, six-, and eight-particle correlations, showing the first unequivocal evidence for collective motion in $A+A$ collisions at SPS energies.

If the matter produced in ultrarelativistic heavy-ion collisions reaches thermal equilibrium, its subsequent evolution follows the laws of ideal fluid dynamics. We show that general predictions can be made on this basis alone, irrespective of the details of the hydrodynamical model. We derive several scaling rules for momentum spectra and anisotropic flow (in particular the elliptic flow, v2, and the hexadecupole flow, v4) of identified particles. Comparison with existing data is briefly discussed, and qualitative predictions are made for LHC