Relation of hard and total cross sections to centrality

We compare the fractions of the hard and geometric cross sections as a function of impact parameter. For a given definition of central collisions, we calculate the corresponding impact parameter and the fraction of the hard cross section contained within this cut. We use charm quark production as a definite example. In this note, revised from Ref. [1], standard nuclear density distributions are described and the resulting geometrical overlap in nuclear collisions is calculated. We then compare the hard process cross section to the total geometric cross section as a function of impact parameter and discuss how they are related to the collision centrality. A three–parameter Woods–Saxon shape is used to describe the nuclear density distribution, ρA(r) = ρ0 1 + ω(r/RA) 2 1 + exp((r −RA)/z) , (1) where RA is the nuclear radius, z is the surface thickness, and ω allows for central irregularities. The electron scattering data of Ref. [2] is used where available for RA, z, and ω. When data is unavailable, the parameters ω = 0, z = 0.54 fm and RA = 1.19A 1/3 − 1.61A fm are used. The central density ρ0 is found from the normalization ∫ drρA(r) = A. For results with other nuclear shape parameterizations, see the appendix of Ref. [3]. The nuclear shape parameters are given in Table 1. See also Ref. [4] for more detailed discussion of nuclear density distributions. In minimum bias (impact parameter averaged) AB collisions we expect the production cross section for hard processes to grow approximately as σ AB = σ hard pp (AB) α , (2) 0231-4428/97/ $ 5.00 c ©1997 Akadémiai Kiadó, Budapest