Intro. Relativistic Heavy Ion Collisions

1. Intro. Relativistic Heavy Ion Collisions Cross Sections and Collision Geometry

2. The cross section: Experimental Meaning • Scattering Experiment • Monoenergetic particle beam • Beam impinges on a target • Particles are scattered by target • Final state particles are observed by detector at q.

3. Beam characteristics: Flux • Flux : • Number of particles/ unit area / unit time • Area: perpendicular to beam • For a uniform beam: • particle density • Number of particles / unit volume • Consider box in Figure. • Box has cross sectional area a. • Particles move at speed v with respect to target. • Make length of box • a particle entering left face just manages to cross right face in time Dt. • Volume of Box: • So Flux

4. Target: Number of Scattering Centers • How many targets are illuminated by the beam? • Multiple nuclear targets within area a • Target Density, • Number of targets per kg: • Recall: 1 mol of a nuclear species A will weigh A grams. i.e. the atomic mass unit and Avogadro’s number are inverses: • (NA x u) = 1 g/mol • So: Areaa L Densityr

5. Incident Flux and Scattering Rate • Scattered rate: • Proportional to • Incident Flux, Nt • size (and position) of detector • For a perfect detector : • Constant of proportionality: • Dimensional analysis: • Must have units of Area. Cross Section Areaa L Densityr

6. Differential Scattering cross section • For a detector subtending solid angle dW • If the detector is at an angle q from the beam, with the origin at the target location:

7. Physical Meaning of stot. • Compute: • Fraction of particles that are scattered • Area a contains Nt scattering centers • Total number of incident particles (per unit time) • Ni=Fa • Total number of scattered particles (per unit time) • Ns=F Ntstot • So Fraction of particles scattered is: • Ns/Ni=F Ntstot/ (F a) = Ntstot/ a • Cross section: effective area of scattering • Lorentz invariant: it is the same in CM or Lab. • For colliders, Luminosity: • Rate:

8. Interaction Cross Section: Theory • Quantum Mechanics: Fermi’s (2nd) Golden Rule • Calculation of transition rates • In simplest form: QM perturbation theory • Golden Rule: particles from an initial state ascatter to a final state b due to an interaction Hamiltonian Hint with a rate given by:

9. Quantum Case: Yukawa Potential • Quantum theory of interaction between nucleons • 1949 Nobel Prize • Limit m → ∞. • Treat scattering of particle as interaction with static potential. • Interaction is spin dependent • First, simple case: spin-0 boson exchange • Klein-Gordon Equation • Static case (time-independent):

10. Observables: From theory to experiment • Steps to calculating and observable: • Amplitude: f = • Probability ~ |f|2 . • Example: • Non-Relativistic quantum mechanics • Assume a is small. • Perturbative expansion in powers of a. • Problem: Find the amplitude for a particle in state with momentum qi to be scattered to final state with momentum qf by a potential Hint(x)=V(x).

11. Propagator: Origins of QFT. q = momentum transfer • q = qi - qf

12. Structure of propagator • QFT case, recover similar form of propagator! • Applies to single particle exchange • Lowest order in perturbation theory. • Additional orders: additional powers of a. • Numerator: • product of the couplings at each vertex. • g2, or a. • Denominator: • Mass of exchanged particle. • Momentum transfer squared: q2. • In relativistic case: 4-momentum transfer squared qmqm=q2. • Plug into Fermi’s 2nd Golden Rule: • Obtain cross sections

13. Cross Section in Nuclear Collisions • Nuclear forces are short range • Range for Yukawa Potential R~1/Mx • Exchanged particles are pions: R~1/(140 MeV)~1.4 fm • Nuclei interact when their edges are ~ 1fm apart • 0th Order: Hard sphere • Bradt & Peters formula • b decreases with increasing Amin • J.P. Vary’s formula: • Last term: curvature effects on nuclear surfaces R2 R1

14. Cross Sections at Bevalac • So: • Bevalac Data • Fixed Target • Beam: ~few Gev/A • AGS, SPS: works too • Bonus question: • Intercept: 7mb½ • What is r0? • Hints: 1 b = 100 fm2, √0.1=0.316, √π=1.772

15. Colliders: Van der Meer Scan • Vernier Scan • Invented by S. van der Meer • Sweep the beams across each other, monitor the counting rate • Obtain a Gaussian curve, peak at smallest displacement • Doing horizontal and vertical sweeps: • zero-in on maximum rate at zero displacement • Luminosity for two beams with Gaussian profile • 1,2 : blue, yellow beam • Ni: number of particles per bunch • Assumes all bunches have equal intensity • Exponential: Applies when beams are displaced by d

16. RHIC Results: BBC X-section • van der Meer Scan. • A. Drees et al., Conf.Proc. C030512 (2003) 1688 • Cross Section: • STAR:

17. Total and Elastic Cross Sections • World Data on pp total and elastic cross section • PDG: http://pdg.arsip.lipi.go.id/2009/hadronic-xsections/hadron.html • RHIC, 200 GeV • tot~50 mb • el~8 mb • nsd=42 mb • LHC, 7 TeV • tot=98.3±2.8 mb • el=24.8±1.2 mb • nsd=73.5 +1.8 – 1.3 mb (TOTEM, Europhys.Lett. 96 (2011) 21002) CERN-HERA Parameterization

18. Important Facts on Cross Sections • Froissart Bound, • Phys. Rev. 123, 1053–1057 (1961) • Marcel Froissart: • Unitarity, Analiticity • require the strong interaction cross sections to grow at most as for • Particles and Antiparticles • Cross sections converge for • Simple relation between pion-nucleon and nucleon-nucleon cross sections

19. Nuclear Cross Sections: Glauber Model

20. Nuclear Charge Densities • Charge densities: similar to a hard sphere. • Edge is “fuzzy”.

21. For the Pb nucleus (used at LHC) • Woods-Saxon density: • R = 1.07 fm * A 1/3 • a =0.54 fm • A = 208 nucleons • Probability :

22. Nuclei: A bunch of nucleons • Each nucleon is distributed with: • Angular probabilities: • Flat in f, flat in cos(q).

23. Impact parameter distribution • Like hitting a target: • Rings have more area • Area of ring of radius b, thickness db: • Area proportional to probability

24. Collision: • 2 Nuclei colliding • Red: nucleons from nucleus A • Blue: nucleons from nucleus B M.L.Miller, et al. Annu. Rev. Nucl. Part. Sci. 2007.57:205-243

25. Interaction Probability vs. Impact Parameter, b • After 100,000 events • Beyond b~2R Nuclei miss each other • Note fuzzy edge • Largest probability: • Collision at b~12-14 fm • Head on collisions: • b~0: Small probability

26. Binary Collisions, Number of participants • If two nucleons get closer than d<s/p they collide. • Each colliding nucleon is a “participant” (Dark colors) • Count number of binary collisions. • Count number of participants

27. Find Npart, Ncoll, b distributions • Nuclear Collisions

28. From Glauber to Measurements • Multiplicty Distributions in STAR MCBS, Ph.D Thesis Phys.Rev.Lett. 87 (2001) 112303

29. Comparing to Experimental data:CMS example • Each nucleon-nucleon collision produces particles. • Particle production: negative binomial distribution. • Particles can be measured: tracks, energy in a detector. • CMS: Energy deposited by Hadrons in “Forward” region

30. Centrality Table in CMS • From CMS MC Glauber model. • CMS: HIN-10-001, • JHEP 08 (2011) 141