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

pdf

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.