Geometrical Scaling of Direct-Photon Production in Hadron Collisions from RHIC to the LHC

We consider pp, dAu and AuAu production of photons at RHIC energies, and PbPb collisions at LHC energy. We show that the inclusive spectrum of photons in the transverse momentum range of 1 GeV<pT<= 4 GeV satisfies geometric scaling. Geometric scaling is a property of hadronic interactions predicted by theories of gluon saturation, and expresses rates in terms of dimensionless ratios of the transverse momentum to saturation momentum. We show excellent agreement with geometric scaling with the only input being the previously measured dependence of the saturation momentum upon Bjorken x and centrality.


10
The phenomenon of gluon saturation arises at high energies when the density of gluons per unit area in a hadron is large [1,2,3,4]. Saturation implies the existence of a saturation momentum scale; where R is the hadron size, dN/dy is the gluon density per unit rapidity, and κ is a constant of order 1. Up to effects of a running coupling constant, at very large 15 Q sat , the saturation momentum is the only scale for physical processes. This implies scaling relations for physical processes. In particular, geometric scaling was first discovered in deep inelastic scattering [5,6]. It was later applied to high Email address: mclerran@bnl.gov (Larry McLerran) energy particle production in hadron-hadron scattering and explains features of pp and pA scattering as a function of multiplicity, as well as particle production 20 in heavy ion collisions for fixed centrality as a function of energy [7,8,9,10,11] In this paper, we intend to apply geometric scaling to photon production in hadron-hadron scattering at RHIC and LHC energies ( √ s N N = 200 GeV and 2760 GeV). This is an extension of work where geometric scaling was applied to reproduce the multiplicity dependence of photon production data for AuAu 25 collisions at RHIC [12]. This paper considers in addition pp and dAu collisions at RHIC energy and P bP b collisions at LHC energy. The obtained agreement with experimental data indicates that geometric scaling works well for photon production.

30
Geometric scaling is a property of particle densities. In the theory of the Color Glass Condensate, one computes these densities from an underlying theory, and in the absence of the effect of running coupling, this theory is controlled by only one scale, the saturation momentum. Therefore, in a collision with overlap area πR 2 , for the production of a photon of momentum p T : The transverse overlap area πR 2 can be estimated for symmetric systems to be proportional to N 2/3 part . The saturation scale is given by [7]: with δ in the range of 0.22 to 0.28 and E the center of mass energy √ s N N . This parameterization is consistent with fits to deep inelastic scattering[13].
The scaling relationship above will work for any function. It is convenient 40 for us however to parameterize the functional form of the photon distribution as a power law in p T . For the finite range of momenta involved, roughly 1 − 4 GeV, such a parameterization of the data is quite good. We use The geometric scaling assumption can then be tested via rescaling the invariant yield only. Figure 1 shows a collection of data on direct photon production in 45 nuclear collisions taken from [14,15,16]. All data have been fit to a power law A · p −n T and different slopes are extracted for the various systems, between ≈ 5.2 − 6.9. In the following a slope of n = 6.1 will be used.
The knowledge of the slope fixes the only unknown a, when combining Equation (3) and (4) to extract the N part dependence of the spectrum at a fixed p T : so the invariant yield roughly changes as This estimate is close to the observed centrality dependence of N 1.48 part at RHIC energies [17]. A general scaling relation between different centralities and collision energies is given by the factor This relation has been used in Figure 2 to rescale the direct photon produc- we found that the scaled direct photon data from [18,19] is also close to the universal curve in Figure 2.
For asymmetric systems, such as dAu, the scaling relation is more complicated, since one cannot use N part any more as a proxy for the geometry. E.g.
in Equation (2)  as calculated in [20] is used so πR 2 ∝ 3.2 2/3 . Similarly, in Equation (3) the saturation scales of the individual partners need to be considered for d and Au.
Here we have assumed that the saturation momentum for the asymmetric dA collision is This is the case for an emission energy of the photon large compared to the saturation momentum, which should be the case for dAu collisions at RHIC 75 energies.

Summary and Conclusions
Geometric scaling provides a good description of the centrality and energy dependence of nucleus-nucleus collisions. It also includes pp and dAu scattering. This is quite remarkable since this involves an extrapolation over several orders 80 of magnitude in the number of nucleon participants, and because the scaling law for the saturation momentum in dA collisions is different in terms of the number of nucleon participants than it is in symmetric collisions.
But how can geometric scaling work so well? It is a property of particle emission that ignores final state interactions, but one expects that the pho-85 tons emitted from quarks and gluons arise from quarks and gluons that have undergone interactions. On the other hand, if there is scale invariance of the expansion, the saturation momentum will remain the only scale in the problem.
Thus until particle masses are important, or until the size of the system in the transverse direction becomes important, geometric scaling should be preserved.

90
The former would be true if the system produces photons at an energy scale large compared to meson masses, which might be possible. The latter is more difficult, since flow measurements for photons demonstrate [21,22] Figure 2: Geometrically scaled invariant yields of direct photon production below p T = 5 GeV/c, the assumed common power law shape p −6.1 T has been fit to the PHENIX AuAu data.