Manuel Calderón de la Barca Sánchez Intro. Relativistic Heavy Ion Collisions Cross Sections and Collision Geometry
Manuel Calderón de la Barca Sánchez
Intro. Relativistic Heavy Ion Collisions
Cross Sections and Collision Geometry
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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.
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Beam characteristics: Flux
Flux :Number of particles/ unit area / unit timeArea: 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
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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:
L
Area a
Density r
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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
L
Area a
Density r
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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:
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Physical Meaning of stot.Compute:
Fraction of particles that are scatteredArea 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 Nt stot
So Fraction of particles scattered is:Ns/Ni =F Nt stot / (F a) = Nt stot / a
Cross section: effective area of scatteringLorentz invariant: it is the same in CM or Lab.
For colliders, Luminosity:Rate:
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Interaction Cross Section: Theory
Quantum Mechanics: Fermi’s (2nd) Golden Rule
Calculation of transition ratesIn simplest form: QM perturbation theory
Golden Rule: particles from an initial state a scatter to a final state b due to an interaction Hamiltonian Hint with a rate given by:
sdNL
dt
Quantum Case: Yukawa PotentialQuantum theory of interaction between nucleons
1949 Nobel Prize
Limit m → ∞.Treat scattering of particle as interaction with static potential.
Interaction is spin dependentFirst, simple case: spin-0 boson exchange
Klein-Gordon Equation
Static case (time-independent):
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Observables: From theory to experiment
Steps to calculating and observable:Amplitude: f = Probability ~ |f|2 .
Example:Non-Relativistic quantum mechanicsAssume 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).
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Propagator: Origins of QFT.
q = momentum transferq = qi - qf
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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
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Cross Section in Nuclear Collisions
Nuclear forces are short rangeRange for Yukawa Potential R~1/Mx
Exchanged particles are pions: R~1/(140 MeV)~1.4 fm
Nuclei interact when their edges are ~ 1fm apart0th Order: Hard sphere
Bradt & Peters formula b decreases with increasing Amin
J.P. Vary’s formula: Last term: curvature effects on nuclear surfaces
R2R1
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Cross Sections at Bevalac
So:
Bevalac DataFixed TargetBeam: ~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
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Colliders: Van der Meer Scan
Vernier ScanInvented by S. van der Meer
Sweep the beams across each other, monitor the counting rateObtain a Gaussian curve, peak at smallest displacementDoing horizontal and vertical sweeps:
zero-in on maximum rate at zero displacement
Luminosity for two beams with Gaussian profile
1,2 : blue, yellow beamNi: number of particles per bunch
Assumes all bunches have equal intensity
Exponential: Applies when beams are displaced by d
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RHIC Results: BBC X-section
van der Meer Scan. A. Drees et al., Conf.Proc. C030512 (2003) 1688
Cross Section:
STAR:
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Total and Elastic Cross Sections
World Data on pp total and elastic cross sectionPDG: 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
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Important Facts on Cross Sections
Froissart Bound, Phys. Rev. 123, 1053–1057 (1961)
Marcel Froissart: Unitarity, Analiticityrequire the strong interaction cross sections to grow at most as for
Particles and AntiparticlesCross sections converge for
Simple relation between pion-nucleon and nucleon-nucleon cross sections
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Nuclear Cross Sections: Glauber Model
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Nuclear Charge Densities
Charge densities: similar to a hard sphere.Edge is “fuzzy”.
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For the Pb nucleus (used at LHC)
Woods-Saxon density: R = 1.07 fm * A 1/3
a =0.54 fmA = 208 nucleons
Probability :
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Nuclei: A bunch of nucleons
Each nucleon is distributed with:
Angular probabilities:Flat in f, flat in cos(q).
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Impact parameter distribution
Like hitting a target:
Rings have more area
Area of ring of radius b, thickness db:
Area proportional to probability
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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
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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
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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
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Find Npart, Ncoll, b distributions
Nuclear Collisions
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From Glauber to Measurements
Multiplicty Distributions in STAR
MCBS, Ph.D ThesisPhys.Rev.Lett. 87 (2001) 112303
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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
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Centrality Table in CMS
From CMS MC Glauber model. CMS: HIN-10-001,
JHEP 08 (2011) 141