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### Pion InterferometryandRHIC Physics

John G. Cramer

Department of PhysicsUniversity of WashingtonSeattle, Washington, USA

Invited Talk presented at

IX Mexican Workshop on Particles and Fields

Physics Beyond the Standard Model

Universidad de Colima, Colima, Mexico

November 19, 2003

Part 1

About RHIC

The Relativistic Heavy Ion Collider

and STAR

Solenoidal Tracker At RHIC

at BNL

Brookhaven NationalLaboratory

John G. Cramer

Brookhaven/RHIC/STAR Overview

Systems:

Au + Au

CM Energies:

130 GeV/A

200 GeV/A1st Collisions:

06/13/2000 Location:

BrookhavenNationalLaboratory,

Long Island,NY

TandemVan de Graaff

AGS

Yellow Ring

Blue Ring

RHIC

Booster

Ring

John G. Cramer

What does RHIC do?

RHIC accelerates gold nuclei in two

beams to about 100 Gev/nucleon each

(i.e., to kinetic energies that are over

100 times their rest mass-energy)

and brings these beams into a

200 GeV/nucleon collision.

Four experiments, STAR,

PHENIX, PHOBOS, and

BRAHMS study these collisions.

In the year 2000 run, RHIC

operated at a collision energy

of 130 Gev/nucleon.

In 2001-2 it operated at 200 GeV/nucleon.

John G. Cramer

y1

Time Projection Chamber

24 sectors x 5692 rf pads x 350 t bins= 47,812,800 pixels

2 m

FTPCs

ZDC

ZDC

Vertex Position Scintillators (TOF)

Endcap EMC

4 m

Trigger Barrel(TOF)

Barrel EMC

Magnet B= 0.5 T

Silicon Vertex Tracker

RICH

John G. Cramer

Central Au +Au Collision at sNN = 130 GeV

Run: 1186017, Event: 32, central

colors ~ momentum: low-- -high

John G. Cramer

A Metaphor for RHIC Physics Understanding

John G. Cramer

Surprises from RHIC

- The “Hydro Paradox”: Relativistic hydrodynamic calculations work surprisingly well, while cascade string-breaking models have problems.
- Strong absorption of high pT pions: There is evidence for strong “quenching” of high momentum pions.
- The “HBT Puzzle”: The ratio of the source radii Rout/Rside is ~1, while the closest model predicts 1.2, and most models predict 4 or more. RLong is smaller than is consistent with boost invariance. In essence, all models on the market have been falsified by HBT.

In the remainder of this talk we will focus on theRHIC HBT Puzzle.

John G. Cramer

In Search of the Quark-Gluon Plasma (QGP)

A pion gas should have few degrees of freedom.

A quark-gluon plasma should have many degrees of freedom and high entropy.

Entropy should be roughly conserved during the fireball’s evolution.

Hence, look in phase space for evidence of:

Large source size,

Long emission lifetime,

Extended expansion,

Large net entropy …

John G. Cramer

A Happy Coincidence of Scales

For the Hanbury-Brown Twiss Effect to work, we must have ab/lL » 1, wherea = size of object,b = separation of detectorsl = wavelength of correlated particlesL = object-detector distance

Stars:a = 2 Rsun = 1.5 x 109 mL = 10 light years = 1017 m l = 500 nm = 5 x 10-7 m

Therefore, need b = lL/a = 33 m (OK!)

Pions:a = 10 fmL = 1 m l = 4.4 fm

Therefore, need b = lL/a = 44 cm (OK!)

So the same technique can be used on stars and on RHIC collision fireballs!

John G. Cramer

The Hanbury-Brown-Twiss Effect

Coherent interference between incoherent sources!

S(x,p)=S(x)S(p)

For non-interacting identical bosons:

The “bump” results fromthe Bose-Einstein statistics ofidentical pions (Jp=0-).

Width of the bump in theith momentum direction isproportional to 1/Ri.

John G. Cramer

A Bose-Einstein Correlation “Bump”

This 3D histogram is STARdata that has been corrected forCoulomb repulsion ofidentical p-p- pairs andis a projection slice nearqlong=0 .

The central “bump” resultsfrom Bose-Einstein statisticsof identical pions (Jp=0-).

John G. Cramer

HBT axis

STAR HBT Matrix (circa Nov. 2000)Goal: reconstruct complete picture with full systematics

Year 1

Year 1 ??

Year 2

Analysis

In progress

From the beginning - study

correlations of nonidentical particles

and resonance production

John G. Cramer

HBT axis

STAR HBT Matrix (circa 2003)Analysis

in progress

published

Not shown:

submitted

3p Correlations (accepted PRL)

asHBT

Phase space density

Correlations with Cascades

dAu, pp

Cascades

John G. Cramer

“Naïve” picture (no space-momentum correlations):

Rout2 = Rside2+(bpairt)2

One step further:

Hydro calculation of Rischke & Gyulassy expects Rout/Rside ~ 2->4 @ kt = 350 MeV.

Looking for a “soft spot”

Small Rout/Rsideonly forTQGP=Tf (unphysical)).

Pre-RHIC HBT PredictionsRside

Rout

John G. Cramer

The RHIC HBT Puzzle

- p-space observables well-understood within hydrodynamic framework
- → hope of understanding early stage
- x-space observables not well-reproduced
- correct dynamical signatures with incorrect dynamic evolution?

Heinz & Kolb, hep-ph/0204061

John G. Cramer

time

The RHIC HBT Puzzle- p-space observables well-understood within hydrodynamic framework
- → hope of understanding early stage
- x-space observables not well-reproduced
- correct dynamical signatures with incorrect dynamic evolution?
- Over-large timescales are modeled?
- emission/freezeout duration (RO/RS)
- evolution duration (RL)

Heinz & Kolb, hep-ph/0204061

John G. Cramer

λ

0.6

0.4

0.2

6

6

RO (fm)

RS (fm)

4

4

1.2

6

RL (fm)

RO / RS

1

4

0.8

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

<kT> GeV/c

HBT at 200 GeV- HBT radii increase with increasing centrality
- HBT radii decrease with kT (flow)
- RO / RS~ 1 (short emission time) problem persists

STAR PRELIMINARY

John G. Cramer

λ

0.6

0.4

0.2

6

6

RO (fm)

RS (fm)

4

4

1.2

6

RL (fm)

RO / RS

1

4

0.8

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

<kT> GeV/c

HBT at 200 GeV- HBT radii increase with increasing centrality
- HBT radii decrease with kT (flow)
- RO / RS~ 1 (short emission time) problem persists
- Longitudinal radius
- Modified Sinyukov fit
- M. Herrmann and G.F. Bertsch, Phys. Rev. C51 (1995) 328
- <tfo>central ≈ 9 fm/c
- <tfo>peripheral ≈ 7 fm/c
- Tfo = 90MeV/c (spectra)

STAR PRELIMINARY

John G. Cramer

HBT Source Radius Excitation Function

Source radii from HBT interferometry do not show a significant increase between CERN energies and RHIC energies.

However, we would still liketo fill the gapwith future RHIC runs.

John G. Cramer

Conclusions from HBT Analysis

- The pion-emissionsource size is smaller than expected, with little growth from a factor of 10 increase in collision energy from the CERN SPS.
- The time from initial collision to emission is also about the same as observed at the SPS, about 9 fm/c.
- The emission duration is also very short, at most 1-2 fm/c.
- These results imply an explosive system with a very hard equation of state.
- We were expecting to bring the nuclear liquid to a gentle boil.
- Instead, it is exploding in our face!

John G. Cramer

Phase Space Density: Definition & Expectations

- Phase Space Density - The phase space density f(p,x) plays a fundamental role in quantum statistical mechanics. The local phase space density is the number of pions occupying the phase space cell at (p,x) with 6-dimensional volume Dp3Dx3 = h3.
- The source-averaged phase space density is áf(p)ñ º ∫[f(p,x)]2 d3x / ∫f(p,x) d3x, i.e., the local phase space density averaged over thef-weighted source volume. Because of Liouville’s Theorem, for free-streaming particles áf(p)ñ is a conserved Lorentz scalar.
- At RHIC, with about the same HBT source size as at the CERN SPS but with more emitted pions, we expect an increase in the pion phase space density over that observed at the SPS.

John G. Cramer

Entropy: Calculation & Expectations

- Entropy – The pion entropy per particle Sp/Np and the total pion entropy at midrapidity dSp/dy can be calculated from áf(p)ñ. The entropy S of a colliding heavy ion system should be produced mainly during the parton phase and should grow only slowly as the system expands and cools.

A quark-gluon plasma has a large number of degrees of freedom. It should generate a relatively large entropy density, up to 12 to 16 times larger than that of a hadronic gas.

At RHIC, if a QGP phase grows with centrality we would expect the entropy to grow strongly with increasing centrality and participant number.

Entropy is conserved during hydrodynamic expansion and free-streaming. Thus, the entropy of the system after freeze-out should be close to the initial entropy and should provide a critical constraint on the early-stage processes of the system.

hep-ph/0212302

nucl-th/0104023

Can Entropy provide the QGP “Smoking Gun”??

John G. Cramer

Pion Phase Space Density at Midrapidity

The source-averaged phase space density áf(mT)ñ is the dimensionless number of pions per 6-dimensional phase space cell h3, as averaged over the source. At midrapidity áf(mT)ñ is given by the expression:

Average phasespace density

HBT “momentumvolume” Vp

PionPurityCorrection

Momentum Spectrum

Jacobianto make ita Lorentzscalar

John G. Cramer

RHIC Collisions as Functions of Centrality

Frequency of Charged Particlesproduced in RHIC Au+Au Collisions

At RHIC we can classifycollision events by impact parameter, based on charged particle production.

of sTotal

Participants

Binary Collisions

John G. Cramer

Corrected HBT Momentum Volume Vp /l½

50-80%

Centrality

40-50%

Peripheral

30-40%

Fits assuming:

Vpl-½=A0 mT3a

(Sinyukov)

20-30%

10-20%

5-10%

0-5%

Central

STAR Preliminary

mT - mp (GeV)

John G. Cramer

Global Fit to Pion Momentum Spectrum

- We make a global fit of the uncorrected pion spectrum vs. centrality by:
- Assuming that the spectrumhas the form of an effective-TBose-Einstein distribution:
- d2N/mTdmTdy=A/[Exp(E/T) –1]
- and
- Assuming that A and T have aquadratic dependence on thenumber of participants Np:A(p) = A0+A1Np+A2Np2T(p) = T0+T1Np+T2Np2

STAR Preliminary

John G. Cramer

Interpolated Pion Phase Space Density áfñat S½ = 130 GeV

HBT points with interpolated spectra

Note failure of “universal” PSDbetween CERN and RHIC.

}

NA49

Central

STAR Preliminary

Peripheral

John G. Cramer

Fits to Interpolated Pion Phase Space Density

HBT points using interpolated spectra fittedwith Blue-Shifted Bose Einstein function

Central

STAR Preliminary

Warning: PSD in the region measured contributes only about 60% to the average entropy per particle.

Peripheral

John G. Cramer

Converting Phase Space Density to Entropy per Particle (1)

Starting from quantum statistical mechanics, we define:

+0.2%

An estimate of the average pion entropy per particle áS/Nñ can be obtainedfrom a 6-dimensional space-momentum integral over the local phase spacedensity f(x,p):

O(f)

O(f3)

O(f4)

+0.1%

dS6(Series)/dS6

1.000

To perform the space integrals, we assume that f(x,p) = áf(p)ñg(x),where g(x) = Ö23 Exp[-x2/2Rx2-y2/2Ry2-z2/2Rz2], i.e., that the source hasa Gaussian shape based on HBT analysis of the system. Further, we make theSinyukov-inspired assumption that the three radii have a momentum dependenceproportional to mT-a. Then the space integrals can be performed analytically.This gives the numerator and denominator integrands of the above expressionfactors of RxRyRz = Reff3mT-3a.(For reference, a~½)

-0.1%

O(f2)

-0.2%

f

John G. Cramer

Converting Phase Space Density to Entropy per Particle (2)

The entropy per particle áS/Nñ then reduces to a momentum integralof the form:

(6-D)

(3-D)

(1-D)

We obtain a from the momentum dependence of Vpl-1/2 and performthe momentum integrals numerically using momentum-dependent fits to áfñor fits to Vpl-1/2 and the spectra.

John G. Cramer

Entropy per Pion from Two Fit Methods

Peripheral

STAR Preliminary

Black = Combined fits to spectrum and Vp/l1/2

Red = BSBE1: Const

Green = BSBE2:~ bT

Blue = BSBE3: Odd 7th order Polynomial in bT

Central

John G. Cramer

Thermal Bose-Einstein Entropy per Particle

The thermal estimate of the p entropy per particle can beobtained by integrating a Bose-Einstein distribution over3D momentum:

mp/mp

T/mp

mp= 0

mp= mp

Note that the thermal-model entropy per particle usually decreases with increasing temperature T and chemical potential mp.

John G. Cramer

Entropy per Particle S/N with Thermal Estimates

STAR Preliminary

Peripheral

Solid line and points show S/Nfrom spectrum and Vp/l1/2 fits.

For T=110 MeV, S/N impliesa pion chemical potential ofmp=44.4 MeV.

Dashed line indicates systematicerror in extracting Vp from HBT.

Central

Dot-dash line shows S/N from BDBE2 fits to áfñ

John G. Cramer

Total Pion Entropy dSp/dy

STAR Preliminary

Dashed line indicates systematicerror in extracting Vp from HBT.

P&P

Why is dSp/dylinear with Np??

Solid line is a linear fit through (0,0)with slope = 6.58 entropy unitsper participant

Dot-dash line indicates dS/dy fromBSBEx fits to interpolated <f>.

P&P

Entropy content ofnucleons + antinucleons

John G. Cramer

Initial Entropy Density: ~(dSp/dy)/Overlap Area

Initial collision overlap area is roughlyproportional to Np2/3

Initial collision entropy is roughlyproportional to freeze-out dSp/dy.

Therefore, (dSp/dy)/Np2/3should be proportionalto initial entropydensity, a QGPsignal.

Solid envelope =Systematic errors in Np

STAR Preliminary

Data indicates that the initialentropy density does grow withcentrality, but not very rapidly.

Our QGP “smoking gun” seems to beinhaling the smoke!

John G. Cramer

Conclusions from PSD/Entropy Analysis

- The source-averaged pion phase space density áfñ is very high, in the low momentum region roughly 2´ that observed at the CERN SPS for Pb+Pb at ÖSnn=17 GeV.
- The pion entropy per particle Sp/Np is very low, implying a significant pion chemical potential (mp~44 MeV) at freeze out.
- The total pion entropy at midrapidity dSp/dy grows linearly with initial participant number Np, with a slope of ~6.6 entropy units per participant. (Why?? Is Nature telling us something?)
- For central collisions at midrapidity, the entropy content of all pions is ~5´ greater than that of all nucleons+antinucleons.
- The initial entropy density increases with centrality, but forms a convex curve that shows no indication of the dramatic increase in entropy density expected with the onset of a quark-gluon plasma.

John G. Cramer

Overall Conclusions

The useful theoretical models that has served us so well at the AGSand SPS for heavy ion studies have now been overloaded with a largevolume of puzzlingnew data from HBTanalysis at RHIC.

Things are a bitup in the air.

We need moretheoretical helpto meet the challengeof understandingwhat is going on inthe RHIC regime.

In any case, thisis a very excitingtime for the STARexperimentalistsworking at RHIC!

John G. Cramer

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