Quark-Gluon Plasma Physics
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1 Quark-Gluon Plasma Physics 7. Hanbury Brown Twiss correlations Prof. Dr. Klaus Reygers Heidelberg University SS 017
2 Momentum correlation of identical bosons emitted from two point sources ~xi : Single-particle state with exactly defined position (in position space) Representation in momentum space: ( ~ k)=h ~ k ~xi = 1 p V e i~ k~x Two-particle wave function in momentum representation: Momentum undefined in this case: P( ~ k)= ( ~ k) = 1 V = const. ( ~ k 1, ~ k )=h ~ k 1, ~ k ~x 1,~x i = 1 p V h e i~ k 1 ~x 1 e i~ k ~x + e i~ k 1 ~x e i~ k ~x 1 i symmetrization of the two-particle wave function, bosons: "+" P( ~ k 1, ~ k )= ( ~ k 1, ~ k ) = 1 V (1 + cos( ~ k ~x)) ~ k 0 enhanced due to Bose-Einstein statistics ~ k = ~ k1 ~ k, ~x = ~x 1 ~x
3 Spatially extended static particle source Extended distribution (~x) Z Single particle: P( ~ k)= of incoherent particle sources: d 3 x (~x) ( ~ k) Two particles: P( ~ k 1, ~ k )= 1 Z d 3 x 1 d 3 x (~x 1 ) (~x ) ( ~ k 1, ~ k ) Two-particle correlation function: C = d 6 N/(d 3 k 1 d 3 k ) (d 3 N/d 3 k 1 )(d 3 N/d 3 k ) = P(~ k 1, ~ k ) P( ~ k 1 )P( ~ k ) =1+ ( ~ k) Gaussian source: (~x) / exp 1 ~x /R! C =1+ exp Fourier transform of (~x) normalization (0) = 1 ~ k R with 3
4 Width of the correlation function is a measures the source size heavy ion collisions: typical dimensions 1 10 fm interference at momentum differences of 0 00 MeV/c 4
5 Where the name comes from: Stellar intensity interferometry Michelson stellar interferometry (measures spatial coherence of star light): direction of the star M 1 M L when the distance d of the pinholes is increased, interference fringes disappear when d = 1. λ/δθ Δθ requires distances of several meters: experimentally challenging Robert Hanbury Brown conceived a method based on the correlations of intensity fluctuations (less sensitive to vibrations and atmospheric fluctuations): d Robert Hanbury Brown (link) "As an engineer my education in physics had stopped far short of the quantum theory. Perhaps just as well... ignorance is sometimes a bliss in science" 5
6 Angular diameter of Sirius from HBT Correlations 1 I1 I /( I1 I ) d I 1 I C hi 1 I i Nature, Nov. 10, 1956, Vol. 178 Angular diameter of Sirius from intensity interferometry: rad 6
7 Hanbury Brown and Twiss tested their technique in the laboratory thermal photon source ("chaotic") half-silvered mirror t g () ( ) = hi (t)i (t + )i hi (t)iihi (t + )i t+τ (e.g. laser) coherence time P. Dirac (1958): "Each photon interferes only with itself; interference between different photons never occurs" No! Applies to conventional interference experiments, but not to HBT Hanbury Brown-Twiss experiment: milestone for the field of "quantum optics" 7
8 Back to heavy ions: Bertsch-Pratt variables: qout, qside, qlong C (~q, ~ K) ~q = ~p 1 ~p ~K = ~p 1 + ~p projection onto the transverse plane: ~q T, ~ K T y ~p 1 x ~p z y ~K T x ~K y transverse plane ~q T q side ~e out = ~ K ~ K z q out ~K T x ~e side = ~e out ~e z q side = ~q e side q out = ~q ~e out q long = ~q ~e z C (q out, q side, q long ) G. Bertsch, Phys. Rev. C37 (1988)
9 Two-Pion Bose- Einstein correlations π π correlation function in central Pb-Pb collisions at snn =.76 TeV projection on qa (a = out, side, long) axis was done for 30 MeV < qb, qc < 30 MeV characteristic width: MeV/c 9
10 Effect of collective expansion: Apparent reduction of the source size Only particles emitted from nearby space points have similar momenta, i.e., small momentum differences Space-momentum correlations from collective radial expansion lead to an apparent reduction of the source size β(r) static source: no dependence on pair momentum ~K radial position r Rnucleus expanding source: HBT radii depend on ~K 10
11 Gaussian HBT radii Rout, Rside, Rlong correction for Coulomb interaction C(~q) =N[(1 )+ K(q inv )(1 + G(~q))] G(~q) =exp[ (R outq out + R sideq side + R longq long)] 11
12 kt dependence of HBT radii: signature of radial flow 0-5% most central Pb-Pb Parameterization inspired by blast wave model R side R out R geom p 1+mT surf /T PHENIX, arxiv:nucl-ex/001008v3 HBT radii larger at the LHC: Effect of stronger radial flow? In some models prolonged lifetime of the source leads to R out /R side > 1 Not seen in data HBT radii larger than 1d rms radii of the nuclei: q q hrpb i/p 3 3. fm hrau i/p 3 3fm ALICE, arxiv:
13 Energy dependence of Rout, Rside, and Rlong (fm) R out E895.7, 3.3, 3.8, 4.3 GeV NA49 8.7, 1.5, 17.3 GeV CERES 17.3 GeV a) (fm) R long KRAKOW HKM AZHYDRO HRM ALICE, arxiv: c) 4 4 (fm) R side 0 8 STAR 6.4, 00 GeV PHOBOS 6.4, 00 GeV ALICE 760 GeV b) /3 dn /dη ch m) /3 dn /dη ch Significant increase at the LHC Reasonably well reproduced by hydro models 13
14 Energy dependence of Rout Rside Rlong ) 3 R out R side R long (fm E895.7, 3.3, 3.8, 4.3 GeV NA49 8.7, 1.5, 17.3 GeV CERES 17.3 GeV STAR 6.4, 00 GeV PHOBOS 6.4, 00 GeV ALICE 760 GeV ALICE, arxiv: dn /dη ch Freeze-out volume appears to scale linearly with dnch/dη Factor increase from RHIC to LHC: same factor as for dnch/dη 14
15 Rlong: longitudinal expansion of the fireball Bjorken expansion: v z = z/t Rlong determined by the distance one can move before the collective velocity overwhelms the thermal velocity: R long v therm dv z /dz Thermal velocity (non relativistic): This gives: v therm = p T /m T R long f p T /mt duration of emission (fm/c) τ f E895.7, 3.3, 3.8, 4.3 GeV NA49 8.7, 1.5, 17.3 GeV CERES 17.3 GeV STAR 6.4, 00 GeV PHOBOS 6.4, 00 GeV ALICE 760 GeV ALICE, arxiv: /3 dn /dη ch ALICE used: s R long (k T )= f T K (m T /T ) m T K 1 (m T /T ) Duration of particle emission at the LHC: about 10 fm/c 15
16 Small systems (I): HBT radii in pp HBT radii in pp smaller than in AA at same dnch/dη R (fm) R (fm) R (fm) G out G side G long PHENIX 00 AGeV STAR 00 AGeV STAR 00 AGeV ALICE 7 TeV ALICE 0.9 TeV STAR 00 GeV STAR 6 AGeV STAR 6 AGeV CERES 17. AGeV ALICE, arxiv: a) b) initial geometry matters /3 dn /dη ch c) 16
17 Small systems (II): Evidence for radial flow ALICE 7 TeV 1-11 N ch N ch N ch N ch N ch ALICE, arxiv: N ch N ch N ch (fm) R inv (GeV/c) k T 17
18 Summary HBT correlations: effect of Bose-Einstein statistics Name comes from stellar intensity interferometry Information about the freeze-out volume and lifetime of the source HBT radii depend on pair momentum kt: evidence for radial flow Freeze-out volume appears to scale linearly with dnch/dη: Approximately constant particle-density at freeze-out for different snn 18
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