### Scaling and Non-Scaling of Elliptic Flow

The detailed dependence of the elliptic flow parameter *v _{2}* on the particle masses and transverse momentum has provided strong evidence for the applicability of hydrodynamics for describing the bulk motion of final-state particles at RHIC. (For a partial list of references, see the Hydro Primer elsewhere in this journal.) Equally intriguing have been various scaling patterns observed in the flow data: The separate curves

*v*for different mass particles are substantially more alike when studied versus “transvese kinetic energy”

_{2}(p_{T},m)*KE*. Plotting

_{T}= m_{T}-m*v*as as a function of

_{2}*KE*collapses the data into one curve for

_{T}*KE*< 1 GeV, while showing two distinct branches for mesons and baryons for larger values of

_{T}*KE*. These two branches collapse into one curve when both

_{T}*v*and

_{2}*KE*are divided by the quark content

_{T}*n*(2 for mesons, 3 for baryons). These observations are discussed in

_{q}Scaling properties of azimuthal anisotropy in Au+Au and Cu+Cu collisions at √s_{NN} = 200 GeV

by the PHENIX Collaboration, who also observed that the *v _{2}* data for different centralities and for different species (Au+Au and Cu+Cu) could also be cast into a universal distribution when scaled by the

*p*-integrated

_{T}*v*for each centrality and/or species. Since the integrated

_{2}*v*at a given centrality is expected to be proportional to the initial eccentricity ε, the implication was that

_{2}*v*scaled by ε scaled as expected from hydrodynamic simulations.

_{2}This last observation has been called into question by a recent STAR publication

which does not observe a scaling of *v _{2}* with an explicitly calculated value of the ‘participant eccentricity’ ε

_{P}. This is not in direct contradiction with the PHENIX paper- when performing the identical scaling of charged hadron

*v*with the

_{2}*p*-integrated

_{T}*v*, a scaling behavior is observed by both collaborations. Nonetheless, there are least two puzzles here that require resolution:

_{2 }- Is there a substantial non-proportionality between
*p*-integrated_{T}*v*and participant eccentricity ε_{2}_{P }? If so, what is the physical interpretation of the observed scaling pattern? - Why does the observed scaling behavior of
*v*with_{2 }*p*-integrated_{T}*v*fail for identified hadrons? Is the observed scaling simply an ‘accident’ when summing over particle species?_{2}

As an aside, note that the STAR paper shows, and states in the conclusions but not the abstract “For the most central collisions (0-10%), negative values of *v _{2}* at the lowest

*p*studied have been observed for both K

_{T}^{0}

_{s }and Λ . This is the first time one found a negative

*v*in Au + Au collisions at RHIC. It is consistent with the strong expansion observed in hadron spectra analysis.”

_{2 }On the general issue of empirical scaling trends in data that are stronger than those suggested by the underlying theory, see the analysis in

Quasi-Particle Degrees of Freedom versus the Perfect Fluid as Descriptors of the Quark-Gluon Plasma

which examines the extent to which the *n _{q}* scaling is supported by hydrodynamic calculations, contaminated by resonances, and reconciliable (or not) with interpretations in terms of quasi-particles.