Measuring the Size of Subatomic
Collisions
Thomas D. Gutierrez
University of California, Davis
November 25, 2002
What Physicists Do
Department of Physics and Astronomy
Sonoma State University
1
Quarks knocked loose during a collision
quickly form bound states through a process
called “hadronization”...
Hadrons = Made of
quarks
Baryon = qqq
p = uud
n = dud
Meson = qq
p+ = ud
K+ = us
“A neutron is a dud…”
Free quarks have
never been observed!
This is interesting and
strange…
Particle Physics at a Glance
http://particleadventure.org
2
Particle Accelerators allow
us to study
aspects of the early
universe in the lab
“Hadronization of the universe” occurred here
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Graphic courtesy JLK
Perspectives on Temperature
~1012 K
~109 K
~107 K
Nucleus-Nucleus collisions
Neutron Star Thermonuclear Explosion
(Terrestrial Nuclear explosions)
~106 K
Solar Interior
~6000 K
~300 K
Solar Surface
~ 120 MeV
Room
Temperature
Cosmic
Microwave
Background
~3 K
~10-6 K
~10-10 K
~ 1/40 eV
Rhodium metal spin
cooling (2000)
(Low-T World Record!)
Trapped Ions
4
Nuclear Collisions in Action
Particle Key
“Projectile”
“Target”
Baryons
(p,n,,,…)
Note the length contraction of the nuclei
along the direction of motion!
This is because v~c
Mesons
(p,K,,,…)
5
Why study proton-proton and nucleus-nucleus collisions at all?
“AA” is used to evoke the image of “Atomic Number”
Proton-proton (pp) collisions are the simplest case
of nucleus-nucleus (AA) collisions...
…and by colliding nuclei, the bulk properties
of nuclear matter can be studied under extreme conditions...
“material science”
Density of the system compared to
normal nuclear density (0.13/fm3)
This is akin to
colliding blocks
of ice to study the
phase diagram of water!
pp collisions form the “baseline”
for AA collisions
High energy pp collisions
tend to be somewhere in here
Collisions fling normal nuclear matter into exotic states
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Why collide protons at all?
But why is that?
Let’s look at two situations
While AA collisions probe the
material science of nuclear matter (phase diagrams, etc.)
pp collisions more directly probe hadronization
The Relativistic Heavy Ion Collider (RHIC)
on Long Island, NY slams gold nuclei head-on at 0.99995c,
creating “little Big Bangs”!
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1. Space-Time Evolution of High
Energy Nucleus-Nucleus Collision
t
Thermal
Freeze-out
Lots of stuff happens
between
when the hadrons
are formed and when they fly off
to be detected
N p
K
p
p
Hadron Gas
Mixed Phase
QGP
Projectile
Fragmentation
Region
Quark Formation &
creation ~ 1fm/c
Hadronization
z
P
T
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2. Space-Time Evolution of
proton-proton Collision
t
Measuring the extent of this
“space-time surface
of hadronization” is what is meant
by the “size of the collision”
N p
K
That’s why pp collisions are
a cleaner probe of what is going
on during hadronization
p p
Because the system size is so small,
there are very few interactions from
the moment of impact
to when particles are
free-streaming towards the detector
Quark scattering and
creation
Hadronization
z
P
T
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Why measure the size of pp collisions?
Measuring the size of pp collisions gives information
about what the collision looked like when the hadrons
were created -- this gives us insight into the mysterious
process of “hadronization”
HOW do you measure the size?
Source sizes are measured using a technique called
Hanbury-Brown Twiss
Intensity Interferometry
(or just HBT for short)
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What is HBT?
The technique was originally developed by two English astronomers
Robert Hanbury-Brown and Richard Twiss (circa 1952)
(Sadly, RHB passed away in January of this year)
It’s form of “Intensity Interferometry”
-- as opposed to “regular” amplitude-level
(Young or Michelson) interferometry -and was used to measure the angular sizes of stars
A quantum treatment of HBT generated much controversy and
led to a revolution in quantum optics (photons can act strangely!)
Later it was used by high energy physicists to measure
source sizes of elementary particle or heavy ion collisions
But how does HBT work? And why use it instead of “regular” interferometry?
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Two slit interference (between coherent sources at A and B)
k
P1
rA1
2p

A

Plane wave
rB1
d
Monochromatic
Source
B
rB1  rA1  d sin 
L >> d
I P1 | e
I P1  I P1

ik rA1
e

ik rB1 2
|  2(1  cos[ k (rB1  rA1 )])
“source geometry” is related to interference pattern
(brackets indicate time average -- which is what is usually measured)
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“Two slit interference” (between incoherent sources at A and B)
A
Two monochromatic but incoherent
sources
(i.e.with random, time dependent phase)
produce no interference pattern
at the screen -assuming we time-average
our measurement over many
fluctuations
d
P1
rA1
rB1
B
L >> d
I P1 | e
 
ik rA1  i A ( t )
e
 
ik rB 1  i B ( t ) 2
|  2(1  cos[ k (rB1  rA1 )  ( B   A )])
I P1  2
(brackets again indicate time average)
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What does <I> mean?
Average of I over a very short time
Average of I over a medium time
Average of I over a
fairly long time
Position on the screen in radians (for small angles)
For very long time averages we get
I P1  2
Long/Short compared to what?
The time scale of the random fluctuations
I  2(1  cos[k (rB1  rA1 )  (B   A )])
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HBT Example (incoherent sources)
P1
rA1
A
As before...
I P1 | e
 
ik rA1  i A ( t )
I P 2 | e
 
ik rA 2  i A ( t )
e
rB1
 
ik rB 1  i B ( t ) 2
e
|
R
rA2
 
ik rB 2  i B ( t ) 2
|
d
P2
rB2
I P2  2
I P1  2
B
But if we take the product before time averaging...
I P1I P 2  4  2 cos[k (r1  r2 )]
where
L >> (d & R)
r1  r2  rA1  rB1  (rA2  rB 2 )
(will be related to source and detector geometry)
Difference of the path length differences
Important: The random phase terms completely dropped out
and left us with a non-constant expression!
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Time average of the product
C
I1 I 2
I1 I 2
This quantity is known as a correlation function
Product of the time averages
It is important to note that for coherent sources
(remembering in that case <I>=I)
I1I 2  I1 I 2
so
C=1
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What does C mean?
C
If we independently monitor the
intensity as a function of time at two
points on the screen...
I1 I 2
I1 I 2
If I1 and I2 tend to increase together
beyond their averages
over the fluctuation times...
This gives a big correlation
<I2>
I1
A plot of I1*I2
with the I’s treated
as variables
<I1>
If I1 and I2 both tend to stick around their
individual averages
over the fluctuation times…
the correlation tends towards one
I2
If either I1 or I2 (or both) tend to be below their
averages or are near zero
over the fluctuation times…
the correlation tends towards zero
It’s not exactly the usual “statistical correlation function”…
but it is related
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For two incoherent point sources….
I1I 2  4  2 cos[k (r1  r2 )]
I P1  2
I P2  2
1
C  1  cos[ k (r1  r2 )]
2
Two interesting limits (with a “little” algebra)...
If d>>R (like an astronomy experiment):
k (r1  r2 ) ~ [k ]R
If R>>d (like an elementary particle experiment):
k
2p

k ( r1  r2 ) ~ kd kˆ1  kˆ2
Recall
p
h

 k
The momentum difference is called:
Q  k kˆ1  kˆ2 / 
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Increasing angular size
Notice that the “widths” of these correlation functions are
inversely related to the source geometry
Astronomy
For fixed k
1
C  1  cos[( k ) R]
2
source
Width w
Increasing source size d
A source can also be a continuous distribution
rather than just points
Particle physics
The width of the correlation function
will have a similar inverse relation to
the source size
Correlation function
1
C  1  cos[Qd ]
2
Width ~1/w
I’ll drop

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What I just described was HBT
for classical waves
In this sense, you can do HBT with sound waves,
water waves, and radio waves
But how to interpret the HBT result for particles
where the waves involved are quantum mechanical
probability amplitudes?
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Bosons and Fermions
The HBT effect at the quantum level is deeply
related to what kind of particle
we are working with
Bosons are integer spin particles.
Identical Bosons have a symmetric two particle wave function -any number may occupy a given quantum state...
Photons and pions are examples of Bosons
Fermions are half-integer spin particles.
Identical Fermions have an antisymmetric wave function -only one particle may occupy a quantum state
Protons and electrons are examples of Fermions
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More about
Correlation
functions
At the quantum level:
Joint probability of measuring a
particle at both detectors 1 and 2
P(1 | 2)
C
P(1) P(2)
A series of independent events should give C=1 (same as a coherent source)
The correlation function for Gaussian source distributions
can be parameterized like:
C(Q)  1  λe
 Q2 R 2
Probability of measurement at 1 times
probability of a measurement at 2
C
Thermal Bosons   1
2
Partly coherent bosons+contamination
Chaoticity parameter   C (Q  0)  1
At the quantum level
a non-constant C(Q) arises
because of
I) the symmetry of the twoparticle wave function
for identical bosons or fermions
and
II) the kind of “statistics”
particles of a particular type
obey
Coherent sources (like lasers)
are flat for all Q
 1
~1/R
 0
1
Q=|p1-p2|
Momentum difference
Fermions exhibit anticorrelation
0
Fermions
  1
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HBT Summary and Observations
• The correlation function contains information about the source
geometry
• The width of the correlation function goes like 1/(source width)
• The HBT correlation function is insensitive to random phases that
would normally destroy “regular” interference patterns
23
Back to pp Collisions
• Pions (also bosons) are used in the HBT rather than photons
• Basic idea is the same: Correlation function contains information
about pion emission source size in the collision and may give clues
about the nature of hadronization
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Practicalities of HBT Interferomertry using particles
•
Compares relative momenta of identical particles to determine information
about space-time geometry of source.
• Experimentally, 1D Qinv correlation functions are created by comparing
relative 4-momenta of pairs from a “real” event signal to pairs from
“mixed” events. The mixed background presumably has no HBT signal.
STAR Preliminary
STAR Preliminary
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More HBT practicalities
•The correlation function, “C2”, is created by dividing the “real” pairs by
“mixed” pairs. The histogram is then normalized to the baseline.
•The data are fit to a Gaussian C2g = 1 + λexp(-qinv2Rinv2) or an exponential
C2e = 1 + λexp(-qinvRinv) to extract fit parameters Rinv and λ.
lambda_g=0.397 +/- 0.013;
STAR Preliminary
R_g=1.16 fm +/- 0.032;
lambda_e=0.749 +/- 0.030;
~λ
R_e=1.94 fm +/- 0.071
The Coulomb
repulsion experienced
by charged pairs tends to deplete
the correlation function at low Q
-- this can be corrected
~1/R
Both fits are to the
Coulomb corrected
data (dark blue)
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NA44 at CERN
NPA610 240 (96)
R really increases with system size!
Just for comparison...
C(Q)
C is narrower so R is bigger
Typical AA Data
This isn’t my analysis
Q (MeV/c)
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From Craig Ogilvie (2 Dec 1998)
What have we learned?
Measuring the size of subatomic and nuclear collisions using
HBT can be subtle and fun and interesting
Quark hadronization is complicated but
studying the size of proton-proton collisions
using HBT may be able to tell us something about it;
Still a lot to learn == perhaps the subject of another talk
I guess we expected this :)
pp collisions are smaller than AA collisions!
Lots more interesting work to be done!
More reading for the interested viewer...
Boffin: A Personal Story of the Early Days of Radar, Radio Astronomy, and Quantum Optics R. Hanbury Brown
Intensity Interferometry R. Hanbury-Brown
Quantum Optics Scully and Zubairy
Quantum Theory of Light Loudon
Two-Particle Correlations in Relativistic Heavy Ion Collisions Heinz and Jacak, nucl-th/9902020
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