Intro. Relativistic Heavy
Ion Collisions
Cross Sections and Collision Geometry
Manuel Calderón de la Barca Sánchez
Scattering Experiment
Monoenergetic particle beam
Beam impinges on a target
Particles are scattered by target
Final state particles are observed by detector at q.
2
Flux :
N
aDt
Number of particles/ unit area / unit time
Area: perpendicular to beam
For a uniform beam:
particle density
ni =
N
V
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 Dx = vDt
a particle entering left face just manages to cross right
face in time Dt.
Volume of Box:
So Flux
V = aDx = avDt
N
Nv
=
=
= ni v
aDt V
How many targets are
illuminated by the beam?
Area a
L
Multiple nuclear targets
within area a
Target Density, r = m /V
Number of targets per kg:
N A / (A ×10-3 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:
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æ NA ö
Nt = ç
÷ r La
-3
è A ×10 kg ø
Density r
4
Scattered rate:
Area a
Proportional to
L
Incident Flux, Nt
size (and position) of detector
For a perfect detector :
N s æ Ni ö
µç
÷ Nt
Dt è aDt ø
Constant of proportionality:
Dimensional analysis:
Must have units of Area.
Density r
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For a detector subtending solid angle dW
æ Ni ö
dRs µ ç
÷ N t dW
è aDt ø
If the detector is at an angle q from the beam, with
the origin at the target location:
æN ö
dRs = s (q ) ç i ÷ N t dW
è aDt ø
ds (q ) º s (q )dW
6
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 Nt stot
So Fraction of particles scattered is:
Ns/Ni =F Nt stot / (F a) = Nt stot / a
Cross section: effective area of scattering
Lorentz invariant: it is the same in CM or Lab.
For colliders, Luminosity:
dN s
Rate:
= Ls
dt
7
Quantum Mechanics: Fermi’s (2nd) Golden Rule
Calculation of transition rates
In simplest form: QM perturbation theory
dN s
 Ls
dt
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:
dN s 2p
=
b H int a
dt
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2
r ( E)
8
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
-
( ) =-
2
¶
f x,t
2
¶t
2
( )
Static case (time-independent):
()
Ñ 2f x =
()
M x2c 2f x
2
( )
c Ñ2f x,t + M x2c 4f x,t
2 2
-
r
R
-
g2 e
e
V (r) = -gf (r) = = -a
4p c r
r
r
R
Steps to calculating and observable:
Amplitude: f =
Probability ~ |f|2 .
Example:
q f H int (x) qi
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).
q = momentum transfer
q = qi - qf
-i
q f ×x
q f V (x) qi =
3
d
ò xe
q f V (x) qi =
3
d
ò x V (x) e
V (x) e
i
q×x
Use V(x) = Yukawa Potential
2 2
-g
f (q2 ) = 2
q + M X2 c 2
i
qi ×x
= f (q2 )
QFT case, recover similar form of
propagator!
Applies to single particle exchange
-g
f (q ) = 2
2 2
q + MXc
2 2
2
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.
Plug
In relativistic case: 4-momentum transfer
squared qmqm=q2.
into Fermi’s 2nd Golden Rule:
Obtain cross sections
dN s 2p
=
b H int a
dt
2
r ( E)
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
s geom = p ( R1 + R2 ) = p r ( A + A
2
r0 =1.2 fm
2
0
1/3
1
)
1/3 2
2
Bradt & Peters formula
2
2
1/3
1/3
s geom = p r0 ( A1 + A2 - b)
b decreases with increasing Amin
J.P. Vary’s formula:
2
1/3
1/3
-1/3
-1/3 2
s geom = p r0 ( A1 + A2 - b0 (A1 + A2 ))
Last term: curvature effects on nuclear surfaces
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s geom = p r ( A + A - b0 (A
2
0
s
So:
A11/3 + A21/3
1/3
1
1/3
2
-1/3
1
-1/3
2
+A
))
2
A1-1/3 + A2-1/3
= p r0 - p r0b0 1/3 1/3
A1 + A2
Bevalac Data
Fixed Target
Beam: ~few Gev/A
AGS, SPS: works too
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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
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Exponential:
Applies when beams are displaced by d
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van der Meer Scan.
A. Drees et al., Conf.Proc. C030512 (2003) 1688
Cross Section:
s BBC = 2p Rmax (s x12+ s 2x2 )(s y12+ s 2y2 )kb / frev
STAR: s = 26.1± 0.2 ±1.0 ± 0.8 mb
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BBC
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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
s tot = 48 + 0.522 ln2 p +(-1.85)ln p
CERN-HERA Parameterization
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)
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Froissart Bound,
Phys. Rev. 123, 1053–1057 (1961)
Marcel Froissart:
Unitarity, Analiticity
require the strong interaction cross sections to grow at
most as ln 2 s for s ®¥
Particles and Antiparticles
Cross sections converge for
s ®¥
Simple relation between pion-nucleon and nucleonnucleon cross sections
s pN
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2
» s NN
3
18
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Charge densities: similar to a hard sphere.
Edge is “fuzzy”.
20
Woods-Saxon density:
R = 1.07 fm * A 1/3
a =0.54 fm
A = 208 nucleons
Probability :
r (r) =
r0
1+ e
r-R
a
µ r 2 r (r)
21
Each nucleon is distributed with:
P(r,q , f ) = r(r)dV = r(r)r 2 drd(cosq )df
Angular probabilities:
Flat in f, flat in cos(q).
22
Like hitting a target:
Rings have more area
Area of ring of radius b, thickness db:
Area proportional to probability
2p bdb
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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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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
25
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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Nuclear Collisions
27
ds /dNh- (b)
Multiplicty Distributions in STAR
Au+Au, ÖsNN = 130 GeV
STAR, p^ > 100 MeV/c, |h| < 0.5
-1
10
Hijing 1.35 (default settings)
STAR, 5% most central
-2
10
-3
10
-4
10
-5
10
0
50
100
150
200
250
300
350
MCBS, Ph.D Thesis
Phys.Rev.Lett. 87 (2001) 112303
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400
Nh-
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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
29
From CMS MC
Glauber model.
CMS: HIN-10-001,
JHEP 08 (2011) 141
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