Transcript ppt - Quark Matter 2005

The RHIC HBT Puzzle,
Chiral Symmetry Restoration,
and Pion Opacity
John G. Cramer (with Gerald A. Miller)
University of Washington
Seattle, Washington, USA
Quark Matter 2005
Budapest
August 5, 2005
The Featureless HBT Landscape
The source radii, as
inferred from HBT
interferometry, are very
similar over almost two
orders of magnitude in
collision energy.
AGS
CERN
RHIC
The ratio of Ro/Rs is
near 1 at all energies,
which naively implies a
“hard” equation of state
and explosive emission
behavior.
August 5, 2005
Quark Matter 2005, Budapest
2
HBT Momentum Geometry
Relative momentum between pions is a vector q  p1  p 2
 can extract 3D shape information
Rlong – along beam direction
Rout – along “line of sight”, includes time/energy information.
Rside –  “line of sight”, no time/energy information.
K  12  p1  p2 
p1
q
Rside
Rout
p2
August 5, 2005
Quark Matter 2005, Budapest
3
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 chiralsymmetry breaking) as they pass from the hot, dense collision medium
[m(p)0]) to the outside vacuum [m(p)140 MeV]. This is
accomplished by solving the Klein-Gordon equation with an optical
potential, the real part of which is a deep, attractive, momentumdependent “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 (pp <
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.).
August 5, 2005
Quark Matter 2005, Budapest
4
About Chiral Symmetry
Question 1: The up and down “current” quarks have masses of 5 to 10 MeV.
The p (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 p 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.
August 5, 2005
Quark Matter 2005, Budapest
vacuum
medium
5
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 timedependent 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.
August 5, 2005
Quark Matter 2005, Budapest
6
DWEF 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. This DWEF model uses
8 pion source parameters and 3 optical potential parameters, for a
total of 11 parameters in the model. The correlation function C
(not the 2nd moment of C) is calculated.
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 LevenbergMarquardt fitting algorithm. In the resulting fit, the c2 per data
point is ~2.2 and the c2 per degree of freedom is ~3.3. Only
statistical (not systematic) errors are used in calculating c2.
We remove long-lived “halo” resonance contributions to the
spectrum (which are not included in the model) by multiplying the
uncorrected spectrum by l½ (the HBT parameter) before fitting,
then “un-correcting” the predicted spectrum with l½.
August 5, 2005
Quark Matter 2005, Budapest
7
DWEF Fits to STAR 200 GeV
Pion HBT Radii
Full
Calculation
U=0
Boltzmann
Re[U]=0
Non-solid curves
show the effects of
turning off various
parts of the
calculation.
No flow
August 5, 2005
Quark Matter 2005, Budapest
8
DWEF Fit to STAR 200 GeV
Pion Spectrum
Raw Fit
Non-solid curves show
the effects of turning off
various parts of the
calculation
Full
Calculation
U=0
Boltzmann
Re[U]=0
No flow
August 5, 2005
Quark Matter 2005, Budapest
9
Meaning of the Parameters
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  lmfp  8 fm @ KT=1 fm-1  strong absorption 
high density
 Pion chemical potential: mp=124 MeV, slightly less than mass(p)








We have evidence suggesting a CHIRAL PHASE TRANSITION!
August 5, 2005
Quark Matter 2005, Budapest
10
Low pT Ramsauer Resonances
14
16
12
14
RS 12
(fm)
10
RO 8
(fm)
No flow
10
6
Boltzmann
8
4
2
6
KT (MeV/c)
10
20
30
40
50
60
70
KT (MeV/c)
10
20
30
40
50
Re[U]=0
70
Pion Spectrum
Full Calculation
|y(q, b)|2 r(b) at
KT = 49.3 MeV/c
60
U=0
1.5
1
0.5
1
0
1
0.5
Raw Fit
0.5
0
For fit, would need l=0.411
KT (MeV/c)
0
-0.5
-0.5
-1
August 5, 2005
Phobos 0-6%
(preliminary)
-1
Quark Matter 2005, Budapest
11
Potential-Off Radius Fits
Full
Calculation
Out
Side
No Real
Non-solid curves
show the effects of
refitting.
STAR Blast Wave
RO/RS Ratio
Long
No Optical
No Chemical or
Optical Pot.
August 5, 2005
Quark Matter 2005, Budapest
12
Potential-Off Spectrum Fits
Model
No Optical
No Real
Chi^2
Full Calculation
Chi^2/#data
Chi^2/#dof
69.19
2.16
3.29
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
No Real Potential
Raw Fit
Non-solid curves show
the effects of potentialoff refits.
Full
Calculation
STAR Blast Wave
No Chemical
or Optical Pot.
August 5, 2005
Quark Matter 2005, Budapest
13
200 GeV Cu+Cu Predictions
5.5
Scale RWS, t by Relect
5
5
4.5
4
Scale RWS, t by
Scale RWS, AWS, t, Dt by Relect
A1/3
RS fm
RO fm
5.5
3.5
3
4.5
4
3.5
3
Scale RWS, AWS, t, Dt by A1/3
100 200 300 400 500 600
KT MeV c
7
1.3
6
1.2
RO RS
RL fm
100 200 300 400 500 600
KT MeV c
5
1.1
4
1
3
0.9
100 200 300 400 500 600
KT MeV c
August 5, 2005
Quark Matter 2005, Budapest
STAR
(preliminary)
100 200 300 400 500 600
KT MeV c
14
Summary
 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 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 accompanied by a chiral phase
transition at about the same critical temperature.
August 5, 2005
Quark Matter 2005, Budapest
15
Outlook
 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 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)
August 5, 2005
Quark Matter 2005, Budapest
16
The End
A 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
PRC; See nucl-th/0507004
Backup Slides
Formalism
• Wigner distribution of p source current
density matrix S0(x,K)
• Pions interact with dense medium
Gyulassy et al ‘79

is distorted (not plane) wave
chaotic sources
August 5, 2005
Quark Matter 2005, Budapest
19
Source Properties
S0 ( x, k )  S0 (t , ) B (b, KT ) /(2p )3
2
2

(t  t 0 )
cosh 
 
S 0 (t , ) 
exp  

2
2
2
2 D 
2p (Dt )
 2 Dt
(“hydrodynamics inspired” source function of Heinz & collaborators)
1
B (b, KT )  M T
r (b)
 K  u  p  (medium density)
exp 
1

T
2
2
2


t t z
(Bose-Einstein thermal function)
tz
1
  2 ln 

tz
August 5, 2005
K  particle momentum 4-vector
u  trasverse flow 4-vector
Quark Matter 2005, Budapest
20
Wave Equation Solutions
We 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
August 5, 2005
Quark Matter 2005, Budapest
21
The Meaning of U
Im (U) : Opacity, Re (U) :Refraction
pions lose energy and flux
Im[U0]=-p r0,
 1 mb, r0 = 1fm-3,
Im[U0] = .15 fm-2, l = 7 fm
Re(U) must exist:
very strong attraction
chiral phase transition
August 5, 2005
Quark Matter 2005, Budapest
22
Son & Stephanov 2002
v2, v2 m2p Tapproach 0 near T = Tc
Both terms of U are negative (attractive)
U(b)=-(w0+w2p2)r(b), w0=real, w2=complex
August 5, 2005
Quark Matter 2005, Budapest
23
Compute Correlation Function
C ( K , q)  1 
d
d
4
4
2
x S ( x, K , q )
x S ( x, p1 )  d x S ( x, p2 )
4
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.
August 5, 2005
Quark Matter 2005, Budapest
24
Semi-Classical Eikonal Opacity
b
X
l
+
R
Heiselberg and Vischer
August 5, 2005
Quark Matter 2005, Budapest
25
Influence of the Real Potential
in the Eikonal Approximation
Factors of  cancel out in the product y (-) ( p1 , b)y *(-) ( p2 , b).
Therefore the real part of U, no matter
how large, has no influence here.
August 5, 2005
Quark Matter 2005, Budapest
26
Source De-magnification
by the Real Potential Well
Velocity
in well
n=1.33
n=1.00
Velocity
in vacuum
Vcsr = (120 MeV)2
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.
August 5, 2005
Quark Matter 2005, Budapest
Rays bend
closer to
radii
A Fly in a Bubble
27
|y(q, b)|2rb) at
KT = 1.000 fm-1 = 197 MeV/c
Observer
1
0.75
0.5
0.25
1
0
1
0.5
0.5
0
0
1
-0.5
0.5
1
-0.5
-1
Imaginary Only
-1
0
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
Wave Function of Full Calculation
Eikonal
August 5, 2005
Quark Matter 2005, Budapest
28