Virtual Journal on QCD Matter

Over the past 1-2 years some confusion has entered the field as to what exactly constitutes the elliptic flow scaling law predicted by the parton recombination models.

The development of the parton recombination model was driven by a series of observations, made by the RHIC experiments, which lacked a consistent explanation in the prevalent physics picture at that time. These observations were:

• the two component (thermal + power-law) shape of the hadronic transverse momentum spectra;
• the strikingly different behavior of the nuclear suppression factors for baryons and mesons at intermediate transverse momentum;
• the unusually large baryon-to -meson ratios at intermediate/high transverse momenta;
• the different transverse momentum dependence of the elliptic flow for baryons and mesons.

All of these phenomena could be explained by postulating that the formation of hadrons at intermediate transverse momenta (approx. 2 – 5 GeV/c) occurs by recombination of valence (constituent) quarks. Mathematically, the recombination probability depends on the overlap of a quark-antiquark or three quark state (which is usually assumed to be a product state, neglecting interactions among the quarks) with the meson or baryon state, respectively. A more extensive review of the parton recombination model will follow in a separate post; here we refer the reader to the following publications:

• Hadron production in heavy ion collisions: Fragmentation and recombination from a dense parton phase
• Parton coalescence at RHIC
• Scaling behavior at high p(T) and the p / pi ratio

Returning to the observation of species-dependent elliptic flow: by assuming that the elliptic flow is carried by constituent quarks at the time of hadron formation and parameterizing their momentum distribution accordingly, one can derive an expression for the elliptic flow of a hadron in terms of the elliptic flow of its constituent quarks:

Neglecting quadratic and cubic terms, this reduces to the well-known scaling law:

One needs to emphasize that this scaling law is derived for the transverse momentum, not the transverse mass or energy of the hadrons – the momentum is additive in recombination, not the energy! The scaling law exhibits its predictive and analytic power in that p_T domain, in which the elliptic flow saturates, i.e. where v₂ is flat as a function of p_T. It is trivially fulfilled when v₂ rises linearly with p_T, since rescaling of both axes by a constant will preserve the slope of the rise and project the curve v₂(p_T) upon itself.

Sergey Voloshin was the first to mention the approximate version of this scaling law in his talk at QM2002. After the recombination model was introduced by the groups at Duke, Texas A&M and Oregon, Voloshin and D. Molnar subsequently followed up Voloshin’s QM contribution with a more detailed publication. The first time the scaling law was demonstrated experimentally at an international conference was in Paul Sorensen’s talk at SQM2003.

Scaling with kE_T=m_T-m₀:

Recently it has become popular to plot the elliptic flow coefficient v₂ as a function of transverse kinetic energy kE_T and scale both axis by the number of constituent quarks contained in the hadron, see e.g. here. This scaling, which works extremely well on a purely phenomenological basis, is now often referred to as reco scaling. However, it is important to keep in mind that the kE_T scaling is not predicted by the parton recombination model, has not been derived from it, and thus should not be called “reco scaling”. Truth be told, the kE_T scaling was motivated by the reco scaling law and constitutes and ingenious way of combining the reco scaling with approximate hydrodynamic scaling, but in its core it is approximative and entirely phenomenological.

So why does this scaling work so well?

Let us divide the v₂ vs. p_T or kE_T curve into two domains. The domain in which the elliptic flow coefficient rises as a function of p_T or kE_T, we shall call the hydrodynamic domain. The other domain, in which v₂ is approximately flat as a function of p_T or kE_T, we shall denote as recombination domain (we neglect the very high p_T domain, which is dominated by jet energy loss and fragmentation).

In the recombination domain we are already at sufficiently high p_T that the difference between p_T and kE_T becomes small – therefore it is no surprise that using kE_T instead of p_T in this domain yields the same results.

The effects in the hydrodynamic domain are somewhat more complex: recall that the reco scaling law is trivially fulfilled during the (approximately) linear rise of v₂ with p_T, since rescaling of both axis by any constant will preserve the slope of the rise and project the curve upon itself. So apart from changing the saturation level, the rescaling by the constituent quark number does not do anything in the hydrodynamic domain! However, here is where the change from p_T to kE_T has its effect: as a function of p_T, the v₂ curves for different hadron species exhibit a very characteristic mass-ordering, which has been predicted by hydrodynamics. When plotting the same curves as a function of kE_T, they collapse approximately into one curve, but only in the hydrodynamic domain (this was already implicit in Paul Sorensen’s talk at SQM2003, even though he never combined the p_T to kE_T mapping with the scaling with constituent quark number. One should also note that initially the kE_T was not considered important and did not make it into the printed version of the proceedings, to be found here. Three years later, when the significance of the kE_T was realized an amended version of the preprint became available on the preprint server – see here). The reason for the kE_T scaling behavior was once explained to me by Ulrich Heinz:

The reason is that elliptic flow rides on the single particle spectra (it measures their anisotropy), and the single particle spectra for hadrons with different masses are much more similar at low p_T when plotted against m_T-m₀ than against p_T. Without radial flow, they would actually be identical when plotted against m_T (up to normalization, of course). So the difference in v₂(p_T) at fixed p_T for hadrons of different mass is first of all a kinematic effect, related to using a non-scaling variable (p_T). As you, of course, know well, radial flow breaks m_T-scaling, so v₂ doesn’t scale perfectly with m_T-m₀ either – the v₂ curves for different mass hadrons don’t line up perfectly when plotted against m_T-m₀, only much better than when plotted against p_T .

So, to return to the original question, the bottom line is that kE_T scaling is roughly equivalent to reco scaling in the recombination domain, but provides an additional approximate scaling in the hydrodynamic domain, which is left invariant by the rescaling of both axes with the number of constituent quarks.

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