enhancement of hadronic resonances in rhi collisions
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Enhancement of hadronic resonances in RHI collisions. Inga Kuznetsova and Johann Rafelski. We explain the relatively high yield of charged Σ ± (1385) reported by STAR - PowerPoint PPT PresentationTRANSCRIPT
Enhancement of hadronic resonances in RHI collisions
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.
Inga Kuznetsova and Johann RafelskiDepartment of Physics, University of Arizona
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.
dtdV
dW
dtdV
dW
dt
dN
VNN
1
dVdt
dW N anddVdt
dWN are Lorentz invariant rates
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 freeze-out temperature?
How this yield depends on degree of initial non-equilibrium?
Explain yields ratios observed in experiment .
Distribution functions
ii Epu )0,1(
u
,,,,1)exp(
1
;1)exp(
1
1
1
Nipu
f
puf
ii
i
for in the rest frame of heat bath
VxKxgT
N iiiii 22
2
3
2
where xi=mi /T; K2(x) is Bessel function; gi is particle i degeneracy;
Υi is particle fugacity, i = N, Δ, Σ, Λ;
Multiplicity of resonance:
Equations for Lorentz invariant rates
( )( )1 1 f fN
( )1 f
24
3
3
3
33
3
1)()2(
)2(2)2(22)2(
spinNN
N
NN
ppMpg
ppp
E
E
pd
E
pdg
dtdV
dW
24
3
3
3
33
3
1)()2(
)2(2)2(22)2(
spinN
NN
NN
NNN
pMppgg
ppp
E
pdff
E
pd
E
pdgg
dtdV
dW
dtdV
dW
dtdV
dW NNN
fpuf )exp(1 1 Bose enhancement factor:
Fermi blocking factor:
using energy conservation and time reversal symmetry:
we obtained:
22
spin
Nspin
N ppMppMpp
iiii fpuf )exp(1 1
dtdV
dW
dtdV
dW
dt
dN
VNN
1
We obtained:
I. Kuznetsova, T. Kodama and J. Rafelski, ``Chemical Equilibration Involving
Decaying Particles at Finite Temperature '' in preparation.
Equilibrium condition:
is global chemical equilibrium.
If in initial state then Δ production is dominant.
If in initial state then Δ decay is dominant.
eqeqeqN
dtdV
dW
dt
dN NN
1
1 N
000 N
000 N
dtdVdW
ddN
VN
1
dtdVdW
VN N
For Boltzmann distributions:
N
dt
dN N 1
We can write time evolution equation as
where is decay time in medium
We assume that no medium effects, τΔ≈τΔvac
We don’t know decay rate, we know decay width or decay timein vacuum τΔ
vac =1/Γ.
Model assumptions
Δ(1232) ↔ N π is fast. Other reactions do not influence Δ yield or
The same for Σ(1385)↔Λπ
Large multiplicity of pions does not change in reactions Most entropy is in pions and entropy is conserved during
expansion of hadrons as
constNNNNN totNN 000
constNNNN tot 000
.3 constdy
dVT
dy
dS
Equation for ΥΔ(Σ)
T
mx ,
)(1))(ln(11 3
32
2
d
VTd
VTd
dT
dT
xKxd
d
d
d
dN
N
ST
Nd
d
111
τ is time in fluid element comoving frame.
,))(ln(1 2
2
d
dT
dT
xKxd
T
0)ln(1 3
d
VTd
S
Expansion of hadronic phase
Growth of transverse dimension:
Taking
we obtain:
0
)()( 0 dvRR
constRTdy
dzRT
dy
dVT 23233
1)/(2
3
1 Rv
Td
dT
T
m
T
m
hTh
7.015.0
)/)(4arctan()( 0max rvv is expansion velocity
At hadronization time τh:
Solution for ΥΔ
),(~
q
d
d
,11
1)(~
TNN
N
where .1
)( 0
N
tot
N
Nq
hh h
dddq~
exp~
exp)( 0
Using particles number conservation: Δ + N = N0tot we obtain
equation :
The solution of equation is:
Non-equilibrium QGP hadronization
γq is light quark fugacity after hadronization
Entropy conservation fixes γq (≠1).
Strangeness conservation fixes γs (≠1).
γq is between 1.6 for T=140 MeV and 1 for T=180 MeV;
Initially and Δ (Σ) production is dominant
;30q ;30
qN ;20q
d S
d y
d S
d y
Q G P H G
;20qs ;20
qs
00)(
0)( N
Temperature as a function of time τ
The ratios NΔ/NΔ0, NN/NN
0 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
*0 )1193()1385( Ytot
29.0)1385()1385(
tottot
Experiment:
tottottot
)1385()1385(
2
3)1385(
J.Adams et al. Phys.Rev. Lett. 97, 132301 (2006)
S.Salur, J.Phys.G, 32, S469 (2006)
Observable:
Effect is smaller than for Δ because of smaller decay width
Future study
Γ≈ 150 MeV;
)892()1400(
)892(*
1
*
KK
KK
Γ≈50 MeV
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.