Quark-Gluon Plasma Physics 7. Hanbury Brown Twiss correlations Prof. Dr. Klaus Reygers Heidelberg University SS 017
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
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
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
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 1916 00 (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
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: 3.1 10 8 rad 6
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
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) 1896 8
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: 30 40 MeV/c 9
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
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
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:101.4035 1
Energy dependence of Rout, Rside, and Rlong (fm) R out 10 8 6 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 0 8 6 KRAKOW HKM AZHYDRO HRM ALICE, arxiv:101.4035 c) 4 4 (fm) R side 0 8 STAR 6.4, 00 GeV PHOBOS 6.4, 00 GeV ALICE 760 GeV b) 0 0 4 6 8 10 1 14 1/3 dn /dη ch m) 6 4 0 0 4 6 8 10 1 14 1/3 dn /dη ch Significant increase at the LHC Reasonably well reproduced by hydro models 13
Energy dependence of Rout Rside Rlong ) 3 R out R side R long (fm 400 350 300 50 00 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:101.4035 150 100 50 0 0 500 1000 1500 000 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
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 1 10 8 6 4 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:101.4035 0 0 4 6 8 10 1 14 1/3 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
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 5 1 5 1 5 PHENIX AuAu @ 00 AGeV STAR AuAu @ 00 AGeV STAR CuCu @ 00 AGeV ALICE pp @ 7 TeV ALICE pp @ 0.9 TeV STAR pp @ 00 GeV STAR AuAu @ 6 AGeV STAR CuCu @ 6 AGeV CERES PbAu @ 17. AGeV ALICE, arxiv:1101.3665 a) b) initial geometry matters 1 4 6 8 1/3 dn /dη ch c) 16
Small systems (II): Evidence for radial flow ALICE pp @ 7 TeV 1-11 N ch 17-1-16 N ch 3-9 30-36 N ch N ch N ch ALICE, arxiv:1101.3665 N ch N ch N ch 37-44 45-58 59-140 (fm) R inv 1.5 1 0. 0.4 0.6 (GeV/c) k T 17
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