Anti - ) Deuteron Coalescence in Ultrarelativistic Heavy Ion Collisions

We assume that due to their small binding energy, deuterons from the midrapidity region cannot survive inside fireballs created in ultrarelativistic nuclear collisions. All observed deuterons are thus expected to be created from a neutron proton pair at freeze-out by coalescence. This coalescence process involves small relative momenta and is thus a non-relativistic phenomenon. Switching to center-of-mass and relative coordinates and momenta, the probability dN d (R, P d) for creating a deuteron with given momentum P d at space-time point R is most conveniently evaluated in the center-of-mass frame of this deuteron. The highly relativistic dynamics of the fireball needs to be fully taken into account only later on, when calculating the Lorentz invariant deuteron spectrum by integrating over the freeze-out hypersurface σ: E d d 3 N d dP 3 d = P d · d 3 σ(R) dN d (R, P d) (1) In non-relativistic statistical quantum mechanics dN d is calculated by projecting the deuteron density matrix on the density matrix of the fireball at freeze-out [1]. The latter contains, however, the relativistic dynamics of the heavy ion collision and is difficult to compute. This problem may be bypassed by switching to the equivalent Wigner function formalism: P. Danielewicz has shown in [2] how, starting from a complete quantum mechanical description in terms of density matrices, one can via a set of reasonable assumptions reduce the deuteron formation rate to an integral over a product of nucleon single particle Wigner functions (which we will approximate by hydrodynamical, local equilibrium distribution functions) and a quantum mechanical transition matrix element. The transition matrix element implements conservation of energy-momentum in deuteron formation via interactions with third particles in the fire-ball and determines the probability of deuteron formation. We derive the following expression for the number dN d of deuterons with four-momentum P d freezing out at space-time point R: where the first term accounts for spin degeneracy and phasespace, f d is the local thermal equilibrium distribution function for an elementary particle with deuteron quantum numbers, and H describes the dependence of the fugacity exp(µ/T) (and thus of the particle density) on the position in the fireball. A implements the quantum mechanics of the formation process and is determined by the internal structure of the deuteron and the variation of the nu-cleon single particle distribution functions f around the deuteron formation point R: