1 / 30

Intro. Relativistic Heavy Ion Collisions

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

alamea
Download Presentation

Intro. Relativistic Heavy Ion Collisions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related