the rhic hbt puzzle, chiral symmetry restoration, and pion opacity john g. cramer (with gerald a....

The RHIC HBT Puzzle,Chiral Symmetry

Restoration, and Pion Opacity

The RHIC HBT Puzzle,Chiral Symmetry

Restoration, and Pion Opacity

John G. Cramer (with Gerald A. Miller)University of WashingtonSeattle, Washington, USA

John G. Cramer (with Gerald A. Miller)University of WashingtonSeattle, Washington, USA

ISMD 2005Kromeriz, Czech Republic

August 11, 2005

August 11, 2005 ISMD 2005, Kromeriz 2

Primer 1: HBT InterferometryPrimer 1: HBT Interferometry

1.Two identical pions will havea Bose-Einstein enhancementwhen their relative momentum(q) is small.

2.The 3-D momentum width of the BE enhancement the 3-D size (R) of the pion “fireball” source.

3.Assuming complete incoherence, the “height” of the BE bump tells us the fraction (½) of pions participating in the BE enhancement.

4.The “out” radius of the source requires a pion energy difference E related to (emission duration).

0 50 100 150 200

0

0.2

0.4

0.6

0.8

1

C(q

)-1

q (MeV/c)

1/RBEenhancement

C(p1,p2) = (p1,p2)/(p1)(p2)

August 11, 2005 ISMD 2005, Kromeriz 3beam dire

ction

HBT Momentum Geometry

HBT Momentum Geometry

Relative momentum between pions is a vector can extract 3D shape information

1 2q p p

p2

p1

q

11 22

K p p R lon

g

RsideRout

Rlong – along beam direction

Rout – along “line of sight”, includes time/energy information.

Rside – “line of sight”, no time/energy information.

Pre-RHIC expectations:

(1) Large Ro,s,l (~12-20 fm)

(2) Rout > Rside by ~4 fm.


The Featureless HBT Landscape

The Featureless HBT Landscape

The source radii, as inferred from HBT interferometry, are very similar over almost two orders of magnitude in collision energy.

The ratio of Ro/Rs is near 1 at all energies, which naively implies a “hard” equation of state and explosive emission behavior (~0).

AGS CERN RHIC

The RHIC HBT Puzzle

The RHIC HBT Puzzle


Primer 2:The Nuclear Optical Model

Primer 2:The Nuclear Optical Model

1. Divide the pions into “channels” and focus on pions (Channel 1) that participate in the BE correlation (about 60% of the spectrum pions). Omit “halo” and “resonance” pions and those converted to other particles (Channels 2, 3, etc.).

2. Solve the time-independent Klein-Gordon equation for the wave functions of Channel 1 pions, using a complex potential U. Im(U) accounts for those pions removed from Channel 1.

3. The complex optical potential U does several things:(a) absorbs pions (opacity);(b) deflects pion trajectories (refraction, demagnification);(c) steals kinetic energy from the emerging pions;(d) produces Ramsauer-type resonances in the well, which can modulate apparent source size and emission intensity.


Optical Wave Functions [||2(b)]Optical Wave Functions [||2(b)]Full

Calculation

KT =197 MeV/c

KT =592 MeV/c

KT =25 MeV/c

ImaginaryOnly

EikonalApprox.

Observer


1 2

41 14 ' ' ( ) ( )*

0 4 2 2

'( , ) ' ( , ') ( ') ( ')

(2 )iK x

p p

d xS x K d K S x K e x x x x

The DWEF FormalismThe DWEF Formalism

We use the Wigner distribution of the pion source current density matrix S0(x,K) (“the emission function”).

The pions interact with the dense medium, producing S(x,K), the distorted wave emission function (DWEF):

The s are distorted (not plane) wave solutions of: , where U is the optical potential.

Gyulassy et al., ‘79

Note: assumes chaotic pion sources.

2( )m U J 2

4

4 41 2

( , , )( , ) 1

( , ) ( , )

d x S x K qC K q

d x S x p d x S x p

Correlationfunction:

Distorted Waves


The “Hydro-Inspired”Emission Function

The “Hydro-Inspired”Emission Function

30 0( , ) ( , ) ( , ) /(2 )TS x k B b K S

2 20

0 2 22

( )cosh( , ) exp

2 22 ( )

S

1( , ) ( )

exp 1T TB b K M b

K u

T

2 2 2t z

1

2lnt z

t z

particle momentum 4-vector

trasverse flow 4-vector

K

u

(Bose-Einstein thermal function)

(medium density)


Primer 3: Chiral Symmetry

Primer 3: 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 small that quantum-uncertainty zitterbewegung gives both quarks large average momenta. 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

externalcolor 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 usually 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 that may be 140 MeV deep.

medium

vacuum


The Chiral Symmetry The Chiral Symmetry Potential;Potential;

[Son & Stephanov (2002)][Son & Stephanov (2002)]

The Chiral Symmetry The Chiral Symmetry Potential;Potential;

[Son & Stephanov (2002)][Son & Stephanov (2002)]

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

2 2 2 2ˆ[ ( )]v p m T

2 2 2 2 2ˆ( ( )) (1 )U m v m T v p

screening mass “velocity”Both v2 and v2m2

(T) 0 near T=Tc.

Dispersion relation forpions in nuclear matter.


Parameters of the ModelParameters of the Model

Thermal: T0 (MeV), (MeV)Space: RWS (fm), aWS (fm)Time: (MeV/c), (MeV/c)Flow: f (#)Optical Pot.: Re(w0) (fm-2), Re(w2) (#), Im(w2) (#)Wave Eqn.: (Kisslinger turned off for now)

Total number of parameters: 11 (+1)

Varied for theCu+Cu prediction.


DWEF Fits to STAR DataDWEF Fits to 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 correlation function C is calculated at q=30 MeV/c, about half way down the BE bump. (We do not use the 2nd moment of C, which is unreliable.)

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. Only statistical (not systematic) errors are used in calculating 2.

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, then “un-correcting” the predicted spectrum with ½.


DWEF Fits to STAR 200 GeVPion HBT Radii

DWEF Fits to STAR 200 GeVPion HBT Radii

U=0

Re[U]=0

No flow

Boltzmann

FullCalculation

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


DWEF Fit to STAR 200 GeVPion Spectrum

DWEF Fit to STAR 200 GeVPion Spectrum

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


Temperature: 222 MeV Chiral PT predicted at ~ 193 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 suggesting a CHIRAL PHASE TRANSITION!


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


Potential-Off Spectrum Fits

Potential-Off Spectrum Fits

No Chemicalor Optical Pot.

No RealNo Optical

STAR Blast Wave

FullCalculation

Raw Fit

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

Model Chi^2 Chi^2/#data Chi^2/#dof

Full Calculation 69.19 2.16 3.29

No Real Potential 905.09 28.28 43.10

No Optical Potential 1003.44 31.36 47.78

No Opt/ Chem Potential 1416.66 44.27 67.46

August 11, 2005 ISMD 2005, Kromeriz 1810 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 0-6%(preliminary)

Raw Fit

U=0Re[U]=0

Boltzmann

No flow

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

For fit, would need =0.41


200 GeV Cu+Cu Predictions

200 GeV Cu+Cu Predictions

Scale RWS, AWS, , by A1/3

Scale RWS, by A1/3

Scale RWS, AWS, , by Relect

Scale RWS, by Relect

STAR (preliminary

)

Conclusion: Space-time parameters (4) scale as A1/3.


SummarySummary Quantum mechanics has solved the technical problems

of applying opacity to HBT.

We obtain excellent DWEF fits to STAR sNN=200 GeV data, simultaneously fitting three HBT radii and the pT spectrum. The key is the deep real optical potential.

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 suggesting a chiral phase transition in RHIC collisions.

We note that in most quark-matter scenarios, the QGP phase transition is usually 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 collision systems.

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 presumably go away.

We plan to try to replace the empirical emission function with a relativistic hydrodynamic calculation of the multidimensional phase space density.(DWEF DWRHD)

The EndThe End A short paper (with erratum) describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005); See ArXiv: nucl-th/0411031; A longer paper has been submitted to Phys. Rev. C; See ArXiv: nucl-th/0507004

Backup SlidesBackup Slides


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.)

While the optical potential is not time-dependent, some time-dependent effects can be manifested in the energy-dependence of the potential . (Time and energy are conjugate quantum variables.)

An optical potential can implicitly include the effects of resonances, including heavy ones. Therefore, our present treatment implicitly includes resonances produced within the hot, dense medium.

We note that more detailed quantum coupled-channels calculations could be done, in which selected resonances were treated as explicit channels coupled through interactions. Describing the present STAR data apparently does not require this kind of elaboration.




Temperature: 222 MeV Chiral PT predicted at ~ 193 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 suggesting a CHIRAL PHASE TRANSITION!


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


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


Overview of the DWEF Model

Overview of the DWEF 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 (e.g., 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, momentum-dependent “mass-type” potential.

We use relativistic quantum mechanics in a cylindrical geometry partial wave expansion to treat the behavior of pions producing Bose-Einstein correlations. We note that most RHIC theories are semi-classical, even though most HBT analyses use pions in the momentum region (p < 600 MeV/c) where quantum wave-mechanical effects should be important.

The model calculates only the spectrum of pions participating in the BE correlation (not those contributions to the spectrum from long-lived “halo” resonances, etc.).


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


Correlation Functions (linear)Correlation Functions (linear)

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Out Side LongKT =

100 MeV/c

KT =200 MeV/c

KT =400 MeV/c

KT =600 MeV/c


Correlation Functions (log)Correlation Functions (log)

0 20 40 60 80 100

0.0001

0.001

0.01

0.1

1

0 50 100 150 2001. 1014

1. 1011

1. 108

0.00001

0.01

0 100 200 300 400

1. 1040

1. 1033

1. 1026

1. 1019

1. 1012

0.00001

0 100 200 300 400 500 6001. 1072

1. 1059

1. 1046

1. 1033

1. 1020

1. 107

0 20 40 60 80 100

0.0001

0.001

0.01

0.1

1

0 50 100 150 200

1. 1013

1. 1010

1. 107

0.0001

0.1

0 100 200 300 400

1. 1033

1. 1026

1. 1019

1. 1012

0.00001

0 100 200 300 400 500 600

1. 1053

1. 1041

1. 1029

1. 1017

0.00001

0 20 40 60 80 100

1. 106

0.00001

0.0001

0.001

0.01

0.1

1

0 50 100 150 200

1. 1015

1. 1012

1. 109

1. 106

0.001

1

0 100 200 300 400

1. 1034

1. 1027

1. 1020

1. 1013

1. 106

0 100 200 300 400 500 6001. 1060

1. 1049

1. 1038

1. 1027

1. 1016

0.00001

Out Side LongKT =

100 MeV/c

KT =200 MeV/c

KT =400 MeV/c

KT =600 MeV/c

0 20 40 60 80 100

0.0001

0.001

0.01

0.1

1

August 11, 2005 ISMD 2005, Kromeriz 35-1

-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

0.5

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

the rhic hbt puzzle, chiral symmetry restoration, and pion opacity john g. cramer (with gerald a....

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