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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 3 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 6 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”! 7 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 8 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 9 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) 10 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? 11 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 ik rA1 I P1 | e I P1 I P1 ik rB1 2 e L >> d | 2(1 cos[k (rB1 rA1 )]) “source geometry” is related to interference pattern (brackets indicate time average -- which is what is usually measured) 12 “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 ik rA1 i A (t ) I P1 | e ik rB1 iB (t ) 2 e | 2(1 cos[k (rB1 rA1 ) (B A )]) I P1 2 (brackets again indicate time average) 13 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 )]) 14 HBT Example (incoherent sources) P1 rA1 A As before... ik rA1 i A (t ) P1 ik rA 2 i A (t ) P2 | e I I | e rB1 ik rB1 iB (t ) 2 e | rA2 ik rB 2 i B (t ) 2 e R | d P2 rB2 I P2 2 I P1 2 B But if we take the product before time averaging... I P1I P2 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! 15 Time average of the product C I1I 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 16 What does C mean? C If we independently monitor the intensity as a function of time at two points on the screen... I1I 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 17 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 / 18 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 19 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? 20 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 21 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 22 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 24 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 25 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) 26 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) 27 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 28