Solving the RHIC HBT Puzzle John G. Cramer and Gerald A. Miller University of Washington Seattle, Washington John G. Cramer and Gerald A. Miller University

Solving theRHIC HBT Puzzle

Solving theRHIC HBT Puzzle

John G. Cramer and Gerald A. MillerUniversity of Washington

Seattle, Washington

John G. Cramer and Gerald A. MillerUniversity of Washington

Seattle, Washington

Femtoscopy @ RHIC/AGS Users MeetingBrookhaven National Laboratory

June 21, 2005

June 21, 2005 RHIC/AGS Users Meeting 2

Outline of TalkOutline of TalkPart 1 – Introduction (Cramer)

RHIC Physics and the HBT Puzzle Overview of our model

Part 2 – Theory and Formalism (Miller) Distorted Waves and the Emission Function The Optical Potential and Chiral Symmetry Opacity and Refraction

Part 3 – Implementation and Fits to Data (Cramer) HBT Radius Fits Spectrum Fits Ramsauer Resonances Summary Outlook

Part 1Part 1IntroductionJohn G. Cramer

IntroductionJohn G. Cramer


The RHIC HBT PuzzleThe RHIC HBT PuzzleThe data from the first four years of RHIC operation paint a confusing picture. Some

evidence supports the presence of a QGP in the early stages of Au+Au collisions:There is evidence that relativistic hydrodynamics works very well in describing the

low and medium energy dynamics of the collision, suggesting a fluid-like medium.There is evidence from elliptic flow data of very high initial pressure and collective

behavior.There is evidence of strong suppression of the most energetic pions, those that

should be produced in the early stages of the collision.There is evidence of strong suppression of back-to-back jets.

BUT … a QGP-driven Au+Au system should expand to a fairly large size and should show a fairly long duration of pion emission. However, inteferometry says otherwise:HBT interferometry analysis indicates that the Au+Au collisions at RHIC seem to

be about the same size as collisions at much lower energies at the SPS and AGS.HBT interferometry analysis indicates that the emission of pions is of very short

duration , less than 1 fm/c, so short that a duration can’t be extracted from data. This explosive behavior would imply a very “hard” equation of state (EOS) for the system, while the QGP EOS is “soft” because of the many degrees of freedom.

That is the RHIC HBT Puzzle. Instead of bringing the nuclear liquid to a gentle boil and observing the steam of a QGP, the whole boiler seems to be exploding in our face!


Overview of Our ModelOverview of Our Model The medium is dense and strongly interacting, so the pions must “fight”

their way out to the vacuum. This modifies their wave functions, producing the distorted waves used in the model.

We explicitly treat the absorption of pions by inelastic processes (e.g., quark exchange and rearrangement) as they pass through the medium, as implemented with the imaginary part of an optical potential.

We explicitly treat the mass-change of pions due to chiral-symmetry breaking as they pass from the hot, dense collision medium [m()0]) to the outside vacuum [m()140 MeV]. This is accomplished by solving the Klein-Gordon equation with an optical potential, the real part of which is a deep, attractive “mass-type” potential.

We use relativistic quantum mechanics in a partial wave expansion to treat the behavior of the pions used in the HBT analysis.We note that most RHIC theories have been semi-classical, even though HBT analysis uses pions in the momentum region (p < 600 MeV/c) where quantum wave-mechanical effects should be important.

The model calculates only the spectrum of particles participating in the HBT correlation (not the spectrum from long-lived “halo” resonances).


qout

qside

qlong

Rsi

de

R long

Rout

p1

p2

p2

+

p2

p1

q

Quantum mechanical interference - space timeseparation of source.

q = p1p2

K = ½(p1p2)

C(q,K) 1

p1,p2p1p21

1 q2L R2

L q2S R2

S q2O R2

O …

HBT 2-Particle InterferometryHBT 2-Particle Interferometry

Hydrodynamics predicts big RO/RS,

Data says RO/RS about 1 HBT puzzle


About Chiral Symmetry

About Chiral Symmetry

Question 1: The up and down “current” quarks have masses of 5 to 10 MeV.The (a down + anti-up combination) has a mass of ~140 MeV. Where does the observed mass come from?

Answer 1: The quarks are more massive in vacuum due to “dressing”.Also the pair is tightly bound by the color force into a particle so smallthat quantum-uncertainty zitterbewegung gives both quarks large averagemomenta. Part of the mass comes from the kinetic energy of the constituent quarks .

Question 2: What happens when a pion is placed in a hot, dense medium?Answer 2: Two things happen:1. The binding is reduced and the pion system expands because of external

color forces, reducing the zitterbewegung and the pion mass.2. The quarks that were “dressed” in vacuum become “undressed” in medium, causing

up, down, and strange quarks to become more similar and closer to massless particles, an effect called “chiral symmetry restoration”. In many theoretical scenarios, chiral symmetry restoration and the quark-gluon plasma phase go together.

Question 3: How can a pion regain its mass when it goes from medium to vacuum?Answer 3: It must do work against an average attractive force, losing kinetic energy

while gaining mass. In effect, it must climb out of a potential well ~140 MeV deep.

medium

vacuum

Part 2Part 2Formalism and Theory

Gerald A. Miller

Formalism and TheoryGerald A. Miller


FormalismFormalism

• Wigner distribution of source current density matrix S0(x,K)

• Pions interact with dense medium

is distorted (not plane) waveGyulassy et al ‘79

chaotic sources


Source PropertiesSource Properties3

0 0( , ) ( , ) ( , ) /(2 )TS x k B b K S2 2

00 2 22

( )cosh( , ) exp

2 22 ( )

S

1( , ) ( )

exp 1T TB b K M b

K u

T

2 2 2t z

1

2ln

t z

t z

particle momentum 4-vector

trasverse flow 4-vector

K

u

(“hydrodynamics inspired” source function of Heinz & collaborators)

(Bose-Einstein thermal function)

(medium density)


Wave Equation SolutionsWave Equation SolutionsWe assume an infinitely long Bjorken tube and azimuthal symmetry, so that the (incoming) waves factorize:

3D 2D(distorted)1D(plane)

We solve the reduced Klein-Gordon wave equation:

Partial wave expansion ! ordinary diff eq


The Meaning of U The Meaning of U

Im (U) : Opacity, Re (U) :Refractionpions lose energy and flux

Re(U) must exist:very strong attractionchiral phase transition

Im[U0]=-p 0, 1 mb, = 1fm-3,Im[U0] = .15 fm-2, = 7 fm


Son & Stephanov 2002Son & Stephanov 2002Son & Stephanov 2002Son & Stephanov 2002

v2, v2 m2approach near T = Tc

Both terms of U are negative (attractive)U(b)=-(w0+w2p2)(b), w0=real, w2=complex


Compute Correlation FunctionCompute Correlation Function

Correlation function is not Gaussian; we evaluate it near the q of experiment.

The R2 values are not the moments of the emission function S.

24

4 41 2

( , , )( , ) 1

( , ) ( , )

d x S x K qC K q

d x S x p d x S x p


Semi-Classical Eikonal OpacitySemi-Classical Eikonal Opacity

bl

R

Heiselberg and Vischer

X

+


Influence of the Real Potentialin the Eikonal Approximation

Influence of the Real Potentialin the Eikonal Approximation

Therefore the real part of U, no matter how large, has no influence here.

(-) *(-)1 2Factors of cancel out in the product ( , ) ( , ).p b p b


Source De-magnificationby the Real Potential Well Source De-magnification

by the Real Potential Well

Because of the mass loss in the potential well, the pions move faster there (red) than in vacuum (blue). This de-magnifies the image of the source, so that it will appear to be smaller in HBT measurements. This effect is largest at low momentum.

n=1.00

n=1.33

A Fly in a Bubble

Rays bendcloser to

radii

Vcsr = (120 MeV)2

Velocity in well

Velocityin vacuum

June 21, 2005 RHIC/AGS Users Meeting 24-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

10

0.2

0.4

0.6

0.8

-1

-0.5

0

0.5

1

|(, b)|b) atKT = 1.000 fm-1 = 197 MeV/c

|(, b)|b) atKT = 1.000 fm-1 = 197 MeV/c

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

10

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

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1

-1

-0.5

0

0.5

10

0.25

0.5

0.75

1

-1

-0.5

0

0.5

1

Wave Function of Full Calculation

Imaginary Only

Eikonal

Observer


Time-Independence,Resonances, and Freeze-Out

Time-Independence,Resonances, and Freeze-Out

We note that our use of a time-independent optical potential does not invoke the mean field approximation and is formally correct according to quantum scattering theory. (The semi-classical mind-set can be misleading.)

Sone time-dependent effects can be manifested in the energy-dependence of the optical potential. (Time and energy are conjugate quantum variables.)

The optical potential also includes the effects of resonances, including the heavy ones. Therefore, our present treatment implicitly includes those resonances produced inside the medium.

However, a more detailed coupled-channels calculation could be done, in which selected resonances were treated as explicit channels. Describing the present STAR data apparently does not require such an elaboration.

Part 3Part 3Implementation

John G. Cramer

ImplementationJohn G. Cramer


Fitting STAR DataFitting STAR Data We have calculated pion wave functions in a partial wave expansion, applied them to a “hydro-inspired” pion source function, and calculated the HBT radii and spectrum. The model uses 8 pion source parameters and 3 optical potential parameters, for a total of 11 parameters in the model.

We have fitted STAR data at sNN=200 GeV, simultaneously fitting Ro, Rs, Rl, and dNp/dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a Levenberg-Marquardt fitting algorithm. In the resulting fit, the 2 per data point is ~2.2 and the 2 per degree of freedom is ~3.3.

We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by ½ (the HBT parameter) before fitting.


Fits to 200 GeV Pion HBT Radii

Fits to 200 GeV Pion HBT Radii

100 200 300 400 500 600KTMeVc4

4.55

5.56

6.57

RLmf

100 200 300 400 500 600KTMeVc0.95

1

1.05

1.1

1.15

ROR S

100 200 300 400 500 600KTMeVc4

4.5

5

5.5

6

6.5

ROmf

100 200 300 400 500 600KTMeVc4

4.5

5

5.5

6

6.5

RSmfU=0

Re[U]=0

No flow

Boltzmann

FullCalculation

Non-solid curves show the effects of turning off various parts of the calculation.


Fit to 200 GeV Pion Spectrum

Fit to 200 GeV Pion Spectrum

100 200 300 400 500 600 700KTMeVc

50

100

200

500

Nd2 2M T

MdT

YdVeG

2

U=0

Re[U]=0

No flow

Boltzmann

FullCalculation

Raw Fit

Non-solid curves show the effects of turning off various parts of the calculation


Meaning of the Meaning of the ParametersParameters

Meaning of the Meaning of the ParametersParameters

Temperature: 222 MeV Chiral PT predicted at ~ 170 MeV Transverse flow rapidity: 1.6 vmax= 0.93 c, vav= 0.66 c Mean expansion time: 8.1 fm/c system expansion at ~ 0.5 c Pion emission between 5.5 fm/c and 10.8 fm/c soft EOS . WS radius: 12.0 fm = R(Au) + 4.6 fm > R @ SPS WS diffuseness: 0.72 fm (similar to Low Energy NP experience) Re(U): 0.113 + 0.725 p2 deep well strong attraction. Im(U): 0.128 p2 mfp 8 fm @ KT=1 fm-1 strong absorption

high density Pion chemical potential: m=124 MeV, slightly less than mass()

We have evidence for a CHIRAL PHASE TRANSITION!

June 21, 2005 RHIC/AGS Users Meeting 3110 20 30 40 50 60 70

500

700

1000

1500

2000

Low pT Ramsauer Resonances

Low pT Ramsauer Resonances

10 20 30 40 50 60 70

2

4

6

8

10

12

14

10 20 30 40 50 60 70

6

8

10

12

14

16

RO

(fm)

Pion Spectrum

KT (MeV/c) KT (MeV/c)

KT (MeV/c)-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

10

0.5

1

1.5

-1

-0.5

0

0.5

1

RS

(fm)

Phobos(corrected)

Raw Fit

U=0Re[U]=0

Boltzmann

No flow

Full Calculation|(q, b)|2 (b) atKT = 49.3 MeV/c


100 200 300 400 500 600KTMeVc4

4.55

5.56

6.57

RLmf

100 200 300 400 500 600KTMeVc0.95

1

1.05

1.1

1.15

ROR S

100 200 300 400 500 600KTMeVc4

4.5

5

5.5

6

6.5

ROmf

100 200 300 400 500 600KTMeVc4

4.5

5

5.5

6

6.5

RSmf

Potential-Off Radius FitsPotential-Off Radius Fits

No Chemical orOptical Pot.

No Optical

No Real

STAR Blast Wave

FullCalculation

Non-solid curves show the effects of refitting.

Out Side

Long RO/RS Ratio


100 200 300 400 500 600 700KTMeVc

50

100

200

500

Nd2 2M T

MdT

YdPotential-Off Spectrum

FitsPotential-Off Spectrum

Fits

No Chemicalor Optical Pot.

No Real

No Optical

STAR Blast Wave

FullCalculation

Raw Fit

Non-solid curves show the effects of potential-off refits.


SummarySummary Quantum mechanics has solved the technical problems

of applying opacity to HBT.

We obtain excellent fits to STAR sNN=200 GeV data, simultaneously fitting three HBT radii and the pT spectrum.

The fit parameters are reasonable and indicate strong collective flow, significant opacity, and huge attraction.

They describe pion emission in hot, highly dense matter with a soft pion equation of state .

We have replaced the RHIC HBT Puzzle with evidence for a chiral phase transition in RHIC collisions.

We note that in most quark-matter scenarios, the QGP phase transition is accompanied by a chiral phase transition at about the same critical temperature.


OutlookOutlook We have a new tool for investigating the

presence (or absence) of chiral phase transitions in heavy ion collisions.

Its use requires both high quality pion spectra and high quality HBT analysis over a region that extends to fairly low momenta (KT~150 MeV/c).

We are presently attempting to “track” the CPT phenomenon to lower collision energies, where the deep real potential should not be present.

A detailed paper for Phys. Rev. C describing this distorted wave emission function theory and its implementation will be placed on the ArXiv preprint server soon.

The EndThe End A paper describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005), and is on the ArXiv preprint server as nucl-th/0411031

Documents

Solving the RHIC HBT Puzzle John G. Cramer and Gerald A. Miller University of Washington Seattle, Washington John G. Cramer and Gerald A. Miller University