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Measuring the Size of Proton-Proton Collisions
Thomas D. Gutierrez
University of California, Davis
March 14, 2002
Department of Physics
Sonoma State University
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http://particleadventure.org
Hadrons = Made of quarks
Meson = qq+ = udK+ = us
“A neutron is a dud…”
Baryon = qqqp = uudn = dud
Particle Physics at a Glance
Free quarks havenever been observed!
This is interesting and strange…
Quarks knocked loose during a collision quickly form bound states through a process
called “hadronization”...
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“Hadronization of the universe” occurred here
Particle Accelerators allow us to study
aspects of the early universe in the lab
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Perspectives on Temperature
~10-6 K
~3 K
~300 K
~6000 K
~106 K
~1012 K
~ 120 MeV
Trapped Ions
Cosmic Microwave Background
Room Temperature ~ 1/40 eV
Solar Surface
Solar Interior
~109 K Neutron Star Thermonuclear Explosion
~10-10 K Rhodium metal spin cooling (2000)
(Low-T World Record!)
(Terrestrial Nuclear explosions)~107 K
Graphic courtesy JLK
Nucleus-Nucleus collisions
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Nuclear Collisions in Action
“Projectile”
“Target”
Baryons (p,n,,,…)
Mesons (,K,,,…)
Particle Key
Note the length contraction of the nucleialong the direction of motion!
This is because v~c
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Proton-proton (pp) collisions are the simplest case of nucleus-nucleus (AA) collisions...
This is akin to colliding blocks
of ice to study the phase diagram of water!
pp collisions form the “baseline”for AA collisions
“AA” is used to evoke the image of “Atomic Number”
…and by colliding nuclei, the bulk properties of nuclear matter can be studied under extreme conditions...
Collisions fling normal nuclear matter into exotic states
“material science”
Density of the system compared tonormal nuclear density (0.13/fm3)
High energy pp collisions tend to be somewhere in here
Why study proton-proton and nucleus-nucleus collisions at all?
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Why collide protons at all?
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”!
But why is that?Let’s look at two situations
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1. Space-Time Evolution of High Energy Nucleus-Nucleus Collision
Quark Formation & creation ~ 1fm/c
QGP
P T
Mixed Phase
Hadron Gas
N K
Thermal Freeze-out
z
t
Projectile Projectile Fragmentation Fragmentation RegionRegion
Lots of stuff happens between
when the hadrons are formed and when they fly off
to be detected
Hadronization
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2. Space-Time Evolution of proton-proton Collision
Quark scattering and creation
P T
z
t
N K
Because the system size is so small,there are very few interactions from
the moment of impactto when particles are
free-streaming towards the detector
That’s why pp collisions area cleaner probe of what is going
on during hadronization
Measuring the extent of this “space-time surface
of hadronization” is what is meant by the “size of the collision”
Hadronization
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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 hadronswere created -- this gives us insight into the mysterious
process of “hadronization”
Source sizes are measured using a technique called Hanbury-Brown Twiss
Intensity Interferometry (or just HBT for short)
HOW do you measure the size?
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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 just this January)
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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L >> d
Monochromatic Source
Plane wave
d
Two slit interference (between coherent sources at A and B)
A
B
rA1
rB1
P1
sin11 drr AB
)])(cos[1(2|| 112
111
ABrkirki
P rrkeeI BA
“source geometry” is related to interference pattern
11 PP II (brackets indicate time average -- which is what is usually measured)
2
k
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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 manyfluctuations
)])()(cos[1(2|| 112)()(
111
ABABtirkitirki
P rrkeeI BBAA
L >> d
A
B
rA1
rB1
P1
21 PI (brackets again indicate time average)
“Two slit interference” (between incoherent sources at A and B)
d
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Average of I over a very short time
What does <I> mean?
Average of I over a medium timeAverage of I over a
fairly long time
)])()(cos[1(2 11 ABAB rrkI
Long/Short compared to what?The time scale of the random fluctuations
Position on the screen in radians (for small angles)
21 PI
For very long time averages we get
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2)()(1 || 11 tirkitirki
PBBAA eeI
2)()(2 || 22 tirkitirki
PBBAA eeI
21 PI 22 PI
As before...
HBT Example (incoherent sources)
)](cos[24 2121 rrkII PP But if we take the product before time averaging...
)( 221121 BABA rrrrrr where
A
B
P2
P1
L >> (d & R)
d
R
rA1
rB1
rA2
rB2
Important: The random phase terms completely dropped outand left us with a non-constant expression!
(will be related to source and detector geometry)Difference of the path length differences
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II
IIC This quantity is known as a correlation function
It is important to note that for coherent sources (remembering in that case <I>=I)
2121 IIII
Time average of the product
Product of the time averages
soC=1
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What does C mean?
It’s not exactly the usual “statistical correlation function”…but it is related
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21
II
IIC
I1
I2
If I1 and I2 tend to increase togetherbeyond their averages
over the fluctuation times...This gives a big correlation
A plot of I1*I2
with the I’s treatedas variables
If we independently monitor the intensity as a function of time at two
points on the screen...
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
<I2>
<I1>
If I1 and I2 both tend to stick around their individual averages
over the fluctuation times…the correlation tends towards one
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For two incoherent point sources….
21 PI 22 PI
)](cos[24 2121 rrkII )](cos[
2
11 21 rrkC
If R>>d (like an elementary particle experiment):
2121ˆˆ~)( kkkdrrk
If d>>R (like an astronomy experiment):
Rkrrk ][~)( 21
Two interesting limits (with a “little” algebra)...
/ˆˆ21 kkkQ The momentum difference is called:
Recall
kh
p
2
k
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])cos[(2
11 RkC
]cos[2
11 QdC
Increasing angular size
Increasing source size d
Particle physics
Astronomy
Notice that the “widths” of these correlation functions are inversely related to the source geometry
For fixed k
A source can also be a continuous distributionrather than just points
Width wsource
Width ~1/wCorrelation function
The width of the correlation function will have a similar inverse relation to
the source size
I’ll drop
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Bosons and Fermions
Bosons are integer spin particles. Identical Bosons have a symmetric two particle wave function -- any number may occupy a given quantum state...
Fermions are half-integer spin particles. Identical Fermions have an antisymmetric wave function -- only one particle may occupy a quantum state
Photons and pions are examples of Bosons
Protons and electrons are examples of Fermions
The HBT effect at the quantum level is deeplyrelated to what kind of particle
we are working with
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More about Correlation functions
At the quantum levela non-constant C(Q) arises
because ofI) the symmetry of the two-
particle wave functionfor identical bosons or fermions
andII) the kind of “statistics”
particles of a particular type obey
22RQλe1C(Q)
The correlation function for Gaussian source distributions can be parameterized like:
1)0( QCChaoticity parameter
)2()1(
)2|1(
PP
PC
Joint probability of measuring a particle at both detectors 1 and 2
Probability of measurement at 1 timesprobability of a measurement at 2
At the quantum level:
A series of independent events should give C=1 (same as a coherent source)
Momentum difference
C
Q=|p1-p2|
1/R
Thermal Bosons
1
2
Partly coherent bosons+contamination
0Fermions
Coherent sources (like lasers) are flat for all Q
Fermions exhibit anticorrelation
1
0
1
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
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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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Real Data! 500k pp events from Experiment NA49 at CERN
GeV
GeV
GeV
1. signal
2. random
3. correlation
(preliminary analysis this year by TG)
I may be a theorist sortbut what can I say…real data is fun!
Q
Q
Q
1. Generate a cumulative signal histogram by taking the momentum difference Q between all combinations of pion pairs in one pp event; repeat this for all pp events2. Generate a random background histogram by taking the momentumdifference Q between pions pairs in different events3. Generate a correlation function by taking the ratio of signal/random
1/R=0.365 GeVR~2.74 GeV-1~ 0.55fm lam=0.358
GeVQ
Gaussian fit is only so-so for low Q
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C(Q)
Q (MeV/c)
NA44 at CERNNPA610 240 (96)
From Craig Ogilvie (2 Dec 1998)
Typical AA Data
This isn’t my analysis
C is narrower so R is bigger
Just for comparison...
R really increases with system size!
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My current research related to this work
• Analysis: HBT for pp collisions at NA49 (at CERN) and STAR (at RHIC)
• Evaluate phase space density of the pp system, extract temperature!
• Current pure theory project (mostly unrelated to particle physics): What are theoretical correlation functions for parastatistical particles and anyons?
• Lots of room for student involvement at various levels!
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What have we learned?
Boffin: A Personal Story of the Early Days of Radar, Radio Astronomy, and Quantum Optics R. Hanbury BrownIntensity Interferometry R. Hanbury-BrownQuantum Optics Scully and ZubairyQuantum Theory of Light LoudonTwo-Particle Correlations in Relativistic Heavy Ion Collisions Heinz and Jacak, nucl-th/9902020
More reading for the interested viewer...
Lots more interesting work to be done!
pp collisions are smaller than AA collisions!
HBT can be subtle and fun
Quark hadronization is complicated but studying the size of proton-proton collisions
using HBT may be able to tell us something about it
I guess we expected this :)
