High baryon densities achievable in the fragmentation regions at RHIC and LHC
We use the McLerran-Venugopalan model of the glasma energy-momentum tensor to compute the rapidity loss and excitation energy of the colliding nuclei in the fragmentation regions followed by a space-time picture to obtain their energy and baryon densities. At the top RHIC energy we find baryon densities up to 3 baryons/fm, which is 20 times that of atomic nuclei. Assuming the formation of quark-gluon plasma, we find initial temperatures of 200 to 300 MeV and baryon chemical potentials of order 1 GeV. Assuming a roughly adiabatic expansion it would imply trajectories in the plane which would straddle a possible critical point.
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 USA
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For more than three decades the high energy heavy ion community has been focussed on the central rapidity region following the seminal work of Bjorken [BJ1983]. The main reasons are (i) the energy density is expected to be higher there, (ii) the matter is nearly baryon-free making it relevant to the type of matter that existed in the early universe, and (iii) detectors in a collider can more easily measure particle production and correlations in a few units of rapidity around the center-of-momentum. The earlier work of Anishetty, Koehler and McLerran [AKM1980], which found that nuclei were compressed by a factor of 3.5 and excited to an energy density of about 2 GeV/fm when they collide at extreme relativistic energies, was pursued only sporadically over the years [C1984, GC1986, MK2002, FS2003, ML2007]. Here we briefly report on our recent and ongoing work on this subject. We focus on the baryonic fireballs which emerge at large rapidity in the center-of-momentum frame. Our calculations are not relevant or applicable to the lower energy beam scans at RHIC, nor to future experiments at NICA and FAIR [CBM].
Consider central collisions of equal mass nuclei. We neglect transverse motion, which should not be important during the fraction of a fm/c time interval of relevance. Then the collision can be thought of as a sum of independent slab collisions each taking place at a particular value of the transverse coordinate with the beam along the -axis. The projectile slab has a 4-momentum per unit area in the center-of-momentum frame denoted by . Each slab loses energy and momentum to the classical color electric and magnetic fields produced in the region between the two receding nuclei, referred to as the glasma. This loss is determined by where is the infinitesimal four-vector perpendicular to the hypersurface spanned by , , and unit transverse area. The energy-momentum of the glasma has the form [CFKL2015, LK2016]
The and are known analytical [LK2016] and numerical (for SU(2)) [Gelis:2013rba] functions of proper time , while the dependence on space-time rapidity follows from the fact that is a second-rank tensor in a boost-invariant setting. The longitudinal position at each is a function of time. The is related to the time via the velocity , where is the momentum-space rapidity of the slab. The Lorentz invariant effective mass per unit area is defined via the relations and . The pair of differential equations describe both the loss of kinetic energy of the projectile nucleus and the internal excitation energy imparted to it during the collision. Thus is not constant but increases with time, unlike the case of the string model [MK2002].
Initial conditions are needed to solve the equations of motion. For that we turn to hydrodynamical descriptions of collisions at the top RHIC energy of GeV. Reference [SBH2011] assumed that viscous hydrodynamics became applicable at fm/c with GeV/fm. Extrapolating back to using the results in [LK2016] gives 123, 142, and 158 GeV/fm for UV cutoffs of 3, 4 and 5 GeV, respectively. We use a Wood-Saxon nuclear density distribution and assume as usual that the initial energy density is proportional to the square of the thickness function
Figure LABEL:F1 shows the momentum-space rapidity of the central core of a gold nucleus as a function of proper time . The central core loses about 3 units of rapidity within the first 0.1 to 0.2 fm/c; this is a robust result, insensitive to the value of . When averaged over the whole nucleus the baryon rapidity loss is about 2.4. For 0-10% centrality BRAHMS [BRAHMS2004, BRAHMS2009] found an average rapidity loss of about . This is consistent with our result, especially since we focus on 0% centrality for illustration. Figure LABEL:F2 shows the excitation energy per baryon in units of the nucleon mass as a function of proper time. There is a slow but monotonic increase, unlike the rapidity loss whose asymptotic limit is reached within a few tenths of a fm/c.