Solving the RHIC HBT Puzzle

1
Solving theRHIC HBT Puzzle

• John G. Cramer (with Gerald A. Miller)
• University of Washington
• Seattle, Washington

STAR Physics Analysis Meeting, BNL Plenary Talk
(20 minutes) December 5, 2004
2
The RHIC HBT Puzzle

• The 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 AuAu 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 AuAu 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
  AuAu 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 cant
  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!

3
About Chiral Symmetry

• Question 1 The up and down quarks have masses
  of 5 to 10 MeV.The p- (down anti-up) has a
  mass of 140 MeV. Where does the extra mass come
  from?
• Answer 1 The quark pair is tightly bound by the
  color force intoa particle so small that
  quantum-uncertainty zitterbewegung givesboth
  quarks large average momenta. Most of the p-
  mass comesfrom 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
• The binding is reduced and the pion system
  expands because of externalcolor forces,
  reducing the zitterbewegung and the pion mass.
• 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 mediumto vacuum?
• Answer 3 It must do work against an average
  attractive force, losingkinetic energy while
  gaining mass. In effect, it must climb out of
  apotential well 140 MeV deep.

vacuum
medium
4
Overview of Our Model

• 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 (pp lt 600 MeV/c) where quantum
  wave-mechanical effects should be important.
• 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(p)0) to the
  outside vacuum m(p)140 MeV. This is
  accomplished by solving the Klein-Gordon equation
  with a deep, attractive mass-type optical
  potential (real part).
• In other words, the authors have reverted to
  their low-energy nuclear physics backgrounds,
  dusted off a trusted old friend, the nuclear
  optical model, and put it to good use for RHIC
  physics.

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.)
• Any time-dependent effects are 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 resonances.
• 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.
• We also note that we do not need to specify a
  freeze-out hyper-surface and do not need to
  assume the (causality-violating) Cooper-Frye
  criterion.

6
Wave Equation Solutions

• We assume an infinitely long Bjorken tube and
  azimuthal symmetry, so that the (incoming) waves
  factorize3D 2D(distorted)1D(plane)

We solve the reduced Klein-Gordon wave equation
for yp
U(b) is the optical potential
Re(U)Refraction, Im(U)Opacity
This complex optical potential makes pions lose
both energy and flux.

• Re(U) must exist if a potential Im(U) is
  present.
• If a chiral phase transition occurs in the
  collision, we expect a very deep Re(U)
  potential well and very strong attraction.

7
The Optical Potential ofChiral Symmetry
Restoration
Reference D. T. Son and M. A. Stephanov,
PRL 88, 202302 (2002).
SS derived the dispersion relation
Both v (the velocity) and v mp(T) (the pion
pole mass) approach zero near T Tc.
This work implies an equivalent potential of the
form
constant term
p2 term
Both terms of U are negative, and therefore
attractive.
8
Fitting STAR Data
We have calculated pion wave functions in a
partial wave expansion, applied them to a
hydro-inspired pion source function that is the
Wigner transform of the T-matrix of the system,
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 . We have fitted STAR data at
ÖsNN200 GeV, simultaneously fitting Ro, Rs, Rl,
and dNp/dy (both magnitude and shape) at 8
momentum values (i.e., 32 data points), using a
Levenberg-Marquardt fitting algorithm. In the
resulting fit, the c2 per data point is 3.7 and
the c2 per degree of freedom is 5.6.
9
Rout
No flow
Boltzmann
FullCalculation
Prediction
U0
ReU0
KT (MeV/c)
10
Rside
No flow
Boltzmann
Prediction
FullCalculation
ReU0
U0
KT (MeV/c)
11
Rlong
Boltzmann
FullCalculation
No flow
ReU0
U0
12
Rout/Rside
No flow
U0
FullCalculation
ReU0
Boltzmann
13
p Momentum Spectra
Prediction
FullCalculation
Boltzmann
U0
ReU0
No flow
14
Meaning of the Parameters

• Temperature 173 MeV Chiral PT predicted at
  170 MeV
• Transverse flow rapidity 1.3 vmax0.85 c,
  vav0.6 c
• Mean expansion time 8.2 fm/c system expansion
  at 0.5 c
• Pion emission between 5.4 fm/c and 11.1 fm/c
  soft EOS .
• WS radius 11.7 fm R(Au) 4.3 fm gt R _at_ SPS
• WS diffuseness 0.72 fm (similar to Low Energy
  NP experience)
• Re(U) 0.137 0.582 p2 deep well strong
  attraction.
• Im(U) 0.121 p2 lmfp 8 fm _at_ KT1 fm-1
  strong absorption high density
• Pion chemical potential mp123 MeV mass(p)
• We have evidence for a CHIRAL PHASE TRANSITION!

15
Summary

• Quantum mechanics has solved the technical
  problems of applying opacity to HBT.
• We obtain excellent fits to STAR ÖsNN200 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.

16
Testing the Model?
Rout
Rside
Spectrum

1. Look at low-pT p data (HBT spectra) from
   Phobos (and perhaps Brahms).
2. Look into doing low-pT Klong-Klong HBT and
   spectrum using STAR Data Set IV.

17
Extra TreatHere are the optical model wave
functions used in these calculations. We plot

• y(q, b)2 r(b)

18
y(q, b)2 r(b) atKT 0.125 fm-1 24.6 MeV/c
Observer
Imaginary Only
Real Imaginary
Eikonal
19
y(q, b)2 r(b) atKT 0.250 fm-1 49.3 MeV/c
Imaginary Only
Real Imaginary
Eikonal
20
y(q, b)2 r(b) atKT 0.500 fm-1 98.5 MeV/c
Imaginary Only
Real Imaginary
Eikonal
21
y(q, b)2 r(b) atKT 1.000 fm-1 197 MeV/c
Imaginary Only
Real Imaginary
Eikonal
22
y(q, b)2 r(b) atKT 2.000 fm-1 394 MeV/c
Imaginary Only
Real Imaginary
Eikonal
23
y(q, b)2 r(b) atKT 4.000 fm-1 788 MeV/c
Imaginary Only
Real Imaginary
Eikonal
24
The End
A preprint describing this work is on the
ArXiv preprint server as
nucl-th/0411031