Transcript Pion correlations in hydro-inspired models with resonances

Pion correlations in hydro-inspired
models with resonances
A. Kisiel1, W. Florkowski2,3, W. Broniowski2,3, J. Pluta1
(based on nucl-th/0602039, to be published in PRC)
1) Warsaw University of Technology, Warsaw
2) Akademia Świętokrzyska, Kielce
3) Institute of Nuclear Physics, Polish Academy of Sciences, Cracow
1. Hydro-inspired models
the measured particle spectra and correlations reflect properties of matter at the stage when
particles stop to interact, this moment is called the kinetic (thermal) freeze-out
hydro-inspired models use concepts borrowed from relativistic hydrodynamics but
they do not include the complete time evolution of the system, they help us to verify the
idea that matter, just before the kinetic freeze-out is locally thermalized and exhibits
collective behavior, the observables are expressed in terms of thermal (Bose-Einstein,
Fermi-Dirac) distributions convoluted with the collective expansion
CYM & LGT
PCM & clust. hadronization
freeze-out
NFD
NFD & hadronic TM
we assume one universal freeze-out for all processes (inealstic and elastic processes cease at
the same time, also emission of strange and ordinary hadrons happens at the same moment)
simplifying but very fruitful assumption, gives good description of particle yields,
transverse-momentum spectra, pion invariant-mass distributions, balance functions,
azimuthal asymmetry v2 series of papers by: W. Broniowski, WF, B. Hiller, P. Bożek, A.
Baran,
D. Prorok
talk tomorrow evening
consistent with sudden hadronization (explosion) scenario at RHIC, J.Rafelski and
J.Letessier, PRL 85 (2000) 4695
in the single-freeze-out model the thermodynamic parameters, such as temperature T and
baryon chemical potential μB, are obtained from the analysis of the hadron abundances
(ratios of the multiplicities)
in this talk the results obtained with the Monte-Carlo
version of the single-freeze-out model are presented
THERMINATOR (THERMal heavy-IoN generATOR),
A. Kisiel, T. Tałuć, W. Broniowski, and WF
Comp.
Phys. Comm. 174 (2006) 669
2. Freeze-out hypersurface and flow
Cracow single-freeze-out model
t 2  r2z   2   2,

 r  rx ry
v    ,
t t t
  const
,
rz
t



for boost-invariant and cylindrically
symmetric models the freeze-out
hypersurface is defined by the
freeze-out curve in Minkowski
space t - ρ (rz = 0)
  r 2 x  r 2 y , ~  t 2  r 2 z
2 geometric parameters: τ, ρmax
(generalized) blast-wave model
t 2  r 2 z    a  ,
2
 , a  const
 
~
~ rz 
v   v cos , v sin  , , v  const
t
t t 

3 geometric parameters: τ, a, ρmax
all these forms describe
well the transversemomentum spectra !!!
Cracow
Blast-wave
a=0.5
Blast-wave
a=0.0
Blast-wave
a=-0.5
3. Emission function
in our calculations all well established resonances are taken into account,
381 particle types with 1872 different decay modes are included
the Cracow and blast-wave
models are treated on the same
footing, the only important
difference resides in the definition
of the freeze-out hypersurface
THERMINATOR uses the same input as SHARE,
G. Torrieri, S. Steinke, W. Broniowski, WF, J. Letessier, J. Rafelski
Comput. Phys. Comm. 167 (2005) 229
the complete emission function is
obtained as the sum over all possible
decay channels
S ( x, p)   Sc ( x, p)
c


d 3 p2
p
S c ( x1 , p1 )  
B p2 , p1  d 2 2 e 2 2  d 4 x2 ( 4)  x2  2 2  x1   ...
E p2
m2




d 3 p N 1
p 

B p N 1 , p N  2  d N 1N 1e N 1 N 1  d 4 x N 1 ( 4 )  xN 1  N 1 N 1  x N  2 
E N 1
mN 1




d 3 pN
p 


B p N , p N 1  d N N e N N  d  ( x N ) p N  ( 4)  x N  N N  x N 1  f N  p N  u ( x N )
EN
mN


splitting functions in
momentum
freeze-out hypersurface
thermal distribution of
primordial particles
THERMINATOR generates events, sets of particles with the spacetime and momentum
distributions described by the emission function S(x,p)
4.1 Correlation Function – Basic Definitions
one-particle and two-particle pion distributions

dN
W1  p   E p 3 ,
d p
 
W2  p1 , p2   E p1 E p2
dN
d 3 p1d 3 p2
 
 
W2  p1 , p2 
the measured correlation function C  p1 , p2  


W1  p1 W1  p2 
model assumptions relate the correlation function to the emission function
 
 
C q, k 

 k ,r
*
*

d
4

x1S x1 , p1  d x2 S x2 , p2   k , r
4
*
4
4


d
x
S
x
,
p
d
 1 1 1  x2 S x2 , p2 
2
squared wave function of a pair
*

2
4.2 Monte-Carlo Method

 
 1
 
k  k0 , k  E p1  E p2 , p1  p2
average momentum of the pair
2

 
momentum difference q  q0 , q   E p1  E p2 , p1  p2



by definition of the Monte-Carlo method, the integration is replaced
by the summation over particles or pairs of particles
  
 1   
*
*





q

p

p

k

p

p

k
,
r




i
j

i
j
2
 


i j i
C q, k 
  
 1   
  q  pi  p j    k   pi  p j 

2


i
j
 



 
1 if | px |  , | p y |  , | pz | 
   p  
2
2
2

0 otherwise



2
in the numerical
calculations Δ = 5 MeV
4.3 Reference Frames
for each pair the following transformations are made:
i) from the laboratory frame to the longitudinal co-moving system
(LCMS), using the Bertsch-Pratt decomposition, and subsequently
ii) from LCMS to the pair rest frame (PRF)
boost (in z -direction)
further boost to the
q
,
q
,
q
 
 

side
out
long
(around z -axis)
frame of a pair
p1 , p2  q, k rotation


rest

 q *
k , klong  0
in the pair-rest frame we calculate the relative distance and the generalized

momentum difference r* , k *
k q  k 
q~  q 
,
2
k


*
~
q  0, k in PRF
then one is able to
calculate the wave function
also in PRF !
the correlation function is a histogram of the squares of the wave function
calculated for each pair in PRF but tabulated in LCMS !
4.4 Wave Functions
we consider two options for the wave function:
1) The simplest wave function is taken into account which includes
symmetrization over the two identical pions but neglects all dynamical
interactions


 
1 ik* r*
ik * r *
 
e
e
,
2
Q

Q 2

* *
 1  cos 2k  r

2) The Coulomb interaction is included

QC
e




 
1 ik* r*

ik * r *
Ac ( )
e
F  i ,1, i  e
F  i ,1, i 
2
i c
 c  Coulombphaseshift, Ac  Gamow factor
* *
 k r k r

* *
  k *a  , a  Bohr radius, F  hypergeometricfunction
1

4.5 Fitting procedure
1) if the simple wave function is used, the 3D correlation function is fitted
with the standard gaussian formula

C  1   exp  R2out (k )q 2out  R2side (k )q 2side  R2long (k )q 2long

2) when the Coulomb wave function is used, the 3D correlation function is
fitted with the Bowler-Sinyukov formula

C  1      KCoul exp  R2out (k )q2out  R2side (k )q2side  R2long (k )q2long
here KCoul is the squared Coulomb wave function integrated over
a static gaussian source

5. Results
legend for the next plot:
resonances NOT included, only primordial pions,
simple wave function, gaussian fit
resonances included,
simple wave function, gaussian fit
resonances included, Coulomb wave function,
Bowler-Sinyukov fit
STAR experimental data
pions from weak decays included
Cracow
a=0.5
a=0.0
a=-0.5
decays of resonances increase the radii by about 1 fm (no van der Waals corrections)
All
pions
Primordial
pions
projections of the pion
correlation function for the
blast-wave model with
resonances, a = - 0.5
simple wave function is used
and the results are fitted with a
standard gaussian formula
Points:
projections
of 3D CF
|qx|<5 MeV
|qx|<10 MeV
|qx|<30 MeV
Lines:
projections
of 3D fit
0.25 GeV < kT < 0.35 GeV
the projections of the
correlation function (symbols)
and the projections of the 3D fit
(lines) are compared
deviations between the function
and the fit reflect the fact that
the underlying two-particle
distributions are not gaussian
projections lower the intercept
again projections of the pion
correlation function for the
blast-wave model with
resonances, a = - 0.5
but now the Coulomb wave
function is used and the results
are fitted with the BowlerSinyukov formula
0.25 GeV < kT < 0.35 GeV
the projections of the
correlation function (symbols)
and the projections of the 3D fit
(lines) are compared
Coulomb interactions dig holes
at low values of q, the BowlerSinyukov formula works very
well!
concepts to extract the
properties of the correlation
function from its behavior at
q=0 are useless
primordial pions
all pions
separation distributions of
pion pairs, blast-wave
model with resonances,
a=-0.5
the lines show the separation
distributions which are the
result of the fitting of the
corresponding correlations
function by a gaussian
parameterization
[CgaussSgauss  pair distr.]
the effect of the resonances
is visible in long-range tails
ω
other
ρ
primordial
resonance vivisection of the previous plot
for all pions
the pions are divided into four groups:
1) those coming from the decays of ρ
2) those coming from the decays of ω
3) other, coming from the decays of other
resonances than ρ or ω
4) primordial (primary)
in all three directions we observe longtails, „other” resonances give similar
effects as the rho meson
long tails in r give peaks for small values
of q, this effect leads to lowering of the
intercept
6. Conclusions
1) simulatanoues description of the transverse-momentum spectra and the
correlation radii is possible in the hydro-inspired models – special
choice of the freeze-out hypersurface must be made
2) our approach is as close as possible to the experimental treatment of
the correlations (two-particle method, Coulomb included)
3) the role of the resonances is analyzed in detail, some earlier
expectations were confirmed (decrease of intercept, the role of omega
meson), some not (increase of the radii due to the strong decays of
resonances)
4) future: connection to the advanced hydro evolution, Chojnacki’s talk