Ultra-relativistic collisions

1. Ultra-relativistic collisions

2. Multiple collisions and nuclear stopping If multiple collisions occur in a nucleus-nucleus collision, matter at high density and high temperature may in principle be formed. By multiple collisions, a nucleon from one nucleus may collide with several nucleons from the other nucleus, and may deposit a large amount of energy in the collision region. The number of collisions depend on the size of the target nucleus. The greater the radius of the target nucleus, the greater will be the loss of energy and momentum

3. Multiple collisions and nuclear stopping Experimental evidence of the occurrence of multiple collisions in the process p+A X is a relativistic invariant measure of the fraction of the forward light-cone momentum retained by the particle after the collision. Except for x close to 1, the distribution is nearly flat for pp collisions. However, for larger systems, the distribution is shifted to smaller values of x. This means that the probability to find the nucleon with a smaller value of x increases with the size of the system. There is then a slowing down (stopping) effect.

4. Glauber model of nucleus-nucleus collisions The Glauber model provides a description of the geometrical configuration of two nuclei when they collide. It is based on the concept of mean free path with an elementary baryon-baryon cross section (for simplicity assumed to be constant throughout the passage of the baryon inside the nucleus). Define t(b)db as the probability to have a baryon-baryon collision within the transverse area db, when the impact parameter of the nucleon is b. t(b) = baryon-baryon thickness function

5. The total probability of a collision integrated over all impact parameters is normalized For baryons not polarized, the collision does not depend on the orientation of b, hence t(b)= t(b) The probability to have a baryon-baryon (inelastic) collision is t(b)σin For simplicity it is assumed that all collisions are non-diffractive inelastic collisions, with a cross section of about 30 mbarn.

6. Consider the collision between 2 complex nuclei A and B. The probability to find a baryon in the volume element dbBdzB at the position (bB,zB) is with normalization

7. By analogy is the probability to find a baryon in the nucleus A, with normalization The probability element dP to have a collision is the product

8. The total probability for the occurrence of a baryon-baryon collision is given by integration And the thickness function for the AB collision is with normalization

9. The thickness function may be rewritten as by introducing the two normalized thickness functions for the nuclei A and B For nuclei not oriented and not polarized, no vector dependence is assumed

10. The probability to have n inelastic collisions with impact parameter b is: No.of combinations for n collisions out of AB possible NN collisions Probability to have exactly AB-n non-collisions Probability to have exactly n collisions The total probability of an inelastic event is

11. By integration, one gets the total inelastic cross section See Exercise 12.1 in Wong and the average number of collisions is given by: with standard deviation: where

12. In case of nucleon-nucleus collisions, B=1 and P(n,b) reduces to: T(b)= thickness function of A which can be approximated to:

13. In other terms, the distribution of the number of collisions with impact parameter b is a Poisson distribution Average number of collisions

14. The average number of baryon-baryon collisions, with the additional condition of an inelastic collisions is given by: and averaging over the impact parameter Weight factor Average number of baryon-baryon collisions in a nucleus-nucleus collision

15. From one gets the mean number of baryon-baryon collisions in an inelastic collision of nuclei A and B Further details for the interested readers in Wong, Chapter 12

16. The energy lost by the colliding nucleons and deposited close to the c.m. of the system may produce regions of very high energy densities. • The Bjorken estimate of the energy density allows to evaluate whether such values are high enough to permit the formation of quak-gluon-plasma. • In a qualitative way, heavy ion collisions may be divided into • baryon-free quark-gluon-plasma region (√s > 100 A GeV) • baryon-rich quark-gluon-plasma (√s = 5-10 A GeV)

17. To have a quantitative estimate of the baryon stopping, it can be shown that the average rapidity loss in the n-th collision is given by <yn-1> - <yn> = 1/α For a single nucleon-nucleon collision, α=1(one unit of rapidity is lost in each collision). For a collision of a baryon with a succession of nucleons, α is estimated to be either 1 for all collisions OR 1 for one collision and 3 for the other collisions On average, in n collisions: n < α < 1+(n-1)/3

18. The mean free path of a baryon is 2.5 fm. For a nucleon on Au (radius=7 fm), an average of 6 collisions are expected if it crosses all the nucleus diameter, and about 4 collisions if an average over all impact parameters is taken. Then, 2 < α < 4 is expected (from 2 to 4 units of rapidity loss ). For a collision between heavy nuclei at a few A GeV in the c.m. system, the separation between target rapidity and projectile rapidity is 3-4, of the same order of the baryon rapidity loss in a central collision.

19. Bjorken estimate of the energy density Consider the head-on collisions of 2 equal nuclei in the c.m. system. Due to Lorentz contraction in the longitudinal direction, the nuclei may be represented by 2 thin discs. Projectile nucleus B comes from z=-∞ and meets the nucleus A, coming from z=+∞, at z=0 and t=0. All the nucleons are considered to have the same longitudinal coordinates at t=0.

20. Space-time diagram of the collision Just after the collision at (z,t)=(0,0) the energy density may be sufficiently high to form QGP. This is not in equilibrium at the beginning, but equilibration may occur in a time t0. The plasma then evolves by hydrodynamics, the temperature goes down and hadronization occurs. Finally we have freeze-out of hadrons.

21. Estimate of energy density = Deposited energy / Volume Deposited energy: may be estimated from the produced hadrons. Most of them are pions. Assuming an average transverse momentum 0.35 GeV/c, then the average transverse mass is √0.35*0.35+0.14*0.14 ≈ 0.38 GeV/c Recalling the longitudinal momentum and the energy of a particle = the longitudinal velocity of the particle is

22. The emitted particle from (z,t) = (0,0) has velocity The space-time position of a particle is related to the rapidity y by Proper time and the rapidity is related to (z,t) by

23. Through the relations which link the spatial distribution to the rapidity distribution, one may extract the initial energy density: Transverse overlapping area A Longitudinal length dz Volume element: A dz The density in such volume element at z=0 and t=t0 is

24. Inserting dy/dz: The energy of a particle of rapidity y is Number of particles Then the initial energy density is Volume element Particle energy

25. The energy density may then be written as where the proper time t0 (the time at which the plasma is produced) is unknown. Bjorken estimate is about 1 fm/c Assuming A = π (1.2)2 A2/3 fm2 and dN/dy = 3/2 dNch/dy one can estimate the energy density for different systems and bombarding energies.

26. Typical energy densities with SPS light ions 32S+Al 1 GeV/fm3 32S+Au 2.1 GeV/fm3 16O+Au 2.3 GeV/fm3

27. Pb+Pb collisions at SPS full energy (160 A GeV, corresponding to about 17 A GeV c.m. energy) give an estimate of the energy density around 3 GeV/fm3 Au+Au collisions at RHIC (@130 A GeV c.m. energy) give 4.6 GeV/fm3 Pb+Pb central collisions @LHC could result in energy densities > 100 GeV/fm3 Recall: QCD Lattice calculations predict a deconfinement at energy densities in the order of 1 GeV/fm3