Download Production and Decay of Hadronic Resonances after

Enhancement of hadronic resonances in
RHI collisions
Inga Kuznetsova and Johann Rafelski
Department of Physics, University of Arizona
 We explain the relatively high yield of charged Σ±(1385)
reported by STAR
 We show if we have initial hadrons multiplicity above
equilibrium the fractional yield of resonances A*/A
(A*→Aπ) can be considerably higher than expected in
SHM model of QGP hadronization.
 We study how non-equilibrium initial conditions after QGP
hadronization influence the yield of resonances.
Work supported by a grant from: the U.S. Department of Energy DE-FG02-04ER4131
Time evolution equation for NΔ, NΣ
 Δ(1232) ↔ Nπ, width Γ≈120 MeV (from PDG);
 Σ(1385)↔Λπ ,width Γ ≈ 35 MeV (from PDG).
Reactions are relatively fast. We assume that others
reactions don’t have influence on Δ (Σ) multiplicity.
1 dN  dWN  dW N


V dt
dtdV
dtdV
dW  N
dVdt
dWN  
and
are Lorentz invariant rates
dVdt
Phases of RHI collision
 QGP phase;
 Chemical freeze-out (QGP hadronization);
 We consider hadronic gas phase between chemical freeze-
out (QGP hadronization) and kinetic freeze-out; the
hadrons yields can be changed because of their
interactions;
M. Bleicher and J.Aichelin, Phys. Lett. B, 530 (2002) 81
M. Bleicher and H.Stoecker,J.Phys.G, 30, S111 (2004)
 Kinetic freeze-out : reactions between hadrons stop;
 Hadrons expand freely (without interactions).
Motivations
 How resonance yield depends on the difference
between chemical freeze-out temperature (QGP
hadronization temperature) and kinetic freezeout temperature?
 How this yield depends on degree of initial nonequilibrium?
 Explain yields ratios observed in experiment .
Distribution functions
1
f  1
;
 exp( u  p )  1
1
f i  1
, i  N , , , 
i exp( u  pi )  1
u  pi  Ei

for u  (1,0) in the rest frame of heat bath
3
Multiplicity of resonance:
T
2
N i  i 2 gi xi K 2 xi V
2
where xi=mi /T; K2(x) is Bessel function; gi is particle i degeneracy;
Υi is particle fugacity, i = N, Δ, Σ, Λ;
Equations for Lorentz invariant
rates
dW N
g
d 3 p
d 3 pN
d 3 p

f

3 
3 
3

dtdV
(2 ) 2 E 2 E N (2 )
2 E (2 )
1
 (2 )  ( pN  p  p )
g
4

p M pN p
2
(1  f N )(1  f  )
spin
dWN  g N g d 3 p
d 3 pN
d 3 p

f f

3 
3 N 
3

dtdV
(2 ) 2 E 2 EN (2 )
2 E (2 )
1
 (2 )  ( pN  p  p )
g N g
4

spin
2
pN p M p (1  f  )
 Bose enhancement factor:

f  1  1 exp( u  p ) f
1
f

1


Fermi blocking factor: i
i exp( u  pi ) f i
 using energy conservation and time reversal
symmetry:

spin

pN p M p
2
  p M pN p
spin
dW N
dWN 
we obtained: N 
 
dtdV
dtdV
1 dN  dWN  dW N


V dt
dtdV
dtdV
2
 We obtained:
 dWN
dN   N 
 
 1
dt
 
 dtdV
I. Kuznetsova, T. Kodama and J. Rafelski, ``Chemical Equilibration Involving
Decaying Particles at Finite Temperature '' in preparation.
 Equilibrium condition:

N      1
 If in initial state 
0

   
eq
N
eq
is global chemical equilibrium.
  
0
N
0
   
 If in initial state
0

eq

0
N
0
then Δ production is dominant.
then Δ decay is dominant.
We don’t know decay rate, we know decay width or decay time
in vacuum τΔvac =1/Γ.
We can write time evolution equation as
 N
dN   N 
 
 1
dt  
 
1 dN 
dW N
where   
is decay time in medium

V d
dtdV
N
For Boltzmann distributions:   
V
We assume that no medium effects, τΔ≈τΔvac
dW N
dtdV
Model assumptions
 Δ(1232) ↔ N π is fast. Other reactions do not influence Δ
yield or
N  NN  N  N  N
0

0
N
tot
0
 const
 The same for Σ(1385)↔Λπ
N   N   N  N    const
0

0

tot
0
 Large multiplicity of pions does not change in reactions
 Most entropy is in pions and entropy is conserved during
expansion of hadrons as
dS
3 dV
T
 const.
dy
dy
Equation for ΥΔ(Σ)
1 dN 
1 d d ln( x2 K2 ( x )) dT
1 d (VT 3 ) x  m


 3
,  T
N  d
 d
dT
d VT
d
d
1
1
1
  N       
d

T
S
τ is time in fluid element comoving frame.
d ln( x2 K2 ( x )) dT

,
T
dT
d
1
d ln( VT 3 )

0
S
d
1
Expansion of hadronic phase
 Growth of transverse dimension:

R ( )  R0   v( )d 
0
v( )  vmax arctan( 4(   0 ) /  r ) is expansion velocity
dV
3 2 dz
3 2
T

T
R

T
R  const
 Taking

dy
dy
dT
1  2( v / R )  1 
 

we obtain:
Td
3


3
At hadronization time τh:
0.5 m
1
0.7 m


h T
T
h T
Solution for ΥΔ
Using particles number conservation: Δ + N = N0tot we obtain
equation :
d
~
   q( ),
d

N  1 1
~
where ( )  1  
 ,

NN    T

N 0tot 1
q( )  
.
NN 
The solution of equation is:









~
~
0


 ( )     q exp   d  d  exp    d  

 
 



h
h
 
 h


Non-equilibrium QGP hadronization
 γq is light quark fugacity after hadronization
 q ;
 q ;
0
N
3
0

 Entropy conservation
   s q ;
2
0

3
   q ;
0
2
dS QGP dS HG fixes γ (≠1).

q
dy
dy
   s q ;
0

2
 Strangeness conservation fixes γs (≠1).
 γq is between 1.6 for T=140 MeV and 1 for T=180 MeV;
 Initially
0( )  N0 (  ) 0 and Δ (Σ) production is
dominant
Temperature as a function of time τ
The ratios NΔ/NΔ0, NN/NN0 as a function
of T
 NΔ increases during
expansion after
hadronization when γq>1
(ΥΔ < ΥNΥπ) until it reaches
equilibrium. After that it
decreases (delta decays)
because of expansion.
Opposite situation is with
NN.
If γq =1, there is no Δ
enhancement, Δ only decays
with expansion.
NΔ/Ntot ratio as a function of T.
 Ntot (observable) is total
multiplicity of resonances
which decay to N.
 Dot-dashed line is if we have
only QGP freeze-out.
 Doted line (SHARE) is
similar to dot-dashed line
with more precise decays
consideration.
 There is strong dependence
of resulting ratio on
hadronization temperature.
NΣ/Λtot ratio as a function of T
Observable:
 tot  (1385)    0 (1193)  Y *
Effect is smaller than for Δ because of
smaller decay width
Experiment:
  (1385)    (1385)
 0.29
 tot   tot
(1385) 3  (1385)    (1385)

 tot
2
 tot   tot
J.Adams et al. Phys.Rev. Lett. 97, 132301 (2006)
S.Salur, J.Phys.G, 32, S469 (2006)
Future study


  
Γ≈ 150 MeV;
K * (892)  K
Γ≈50 MeV
K1 (1400)  K * (892)
Γ≈170 MeV
Conclusions
 If we have initial hadrons multiplicity above equilibrium
the fractional yield of resonances A*/A can be
considerably higher than expected in SHM model of
QGP hadronization.
 Because of relatively strong temperature dependence,
Δ/Ntot can be used as a tool to distinguish the different
hadronization conditions as chemical non-equilibrium
vs chemical equilibrium;
 We have shown that the relatively high yield of
charged Σ±(1385) reported by STAR is well explained
by our considerations and hadronization at T=140
MeV is favored.