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 ExperimentMonoenergetic particle beamBeam impinges on a targetParticles are scattered by targetFinal 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 volumeConsider 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 CentersHow 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 RateScattered 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 centersTotal 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 / aCross 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: TheoryQuantum Mechanics: Fermi’s (2nd) Golden RuleCalculation of transition rates

In 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:

sdN Ldt

s

Quantum Case: Yukawa PotentialQuantum theory of interaction between nucleons

1949 Nobel PrizeLimit 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 propagatorQFT 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 CollisionsNuclear 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 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 ScanVernier Scan

Invented by S. van der MeerSweep 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 intensityExponential: Applies when beams are displaced by d

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RHIC Results: BBC X-sectionvan der Meer Scan.

A. Drees et al., Conf.Proc. C030512 (2003) 1688

Cross Section: STAR:

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Total and Elastic Cross SectionsWorld Data on pp total and elastic cross section

PDG: http://pdg.arsip.lipi.go.id/2009/hadronic-xsections/hadron.html

RHIC, 200 GeVtot~50 mbel~8 mbnsd=42 mb

LHC, 7 TeVtot=98.3±2.8 mbel=24.8±1.2 mbnsd=73.5 +1.8 – 1.3 mb (TOTEM, Europhys.Lett. 96 (2011) 21002)

CERN-HERA Parameterization

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Important Facts on Cross SectionsFroissart 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 DensitiesCharge 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 distributionLike hitting a target:Rings have more areaArea of ring of radius b, thickness db:Area proportional to probability

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Collision:2 Nuclei collidingRed: nucleons from nucleus ABlue: 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 edgeLargest 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/π 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 MeasurementsMultiplicty 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