Kinetics of hadron resonances during hadronic freeze-out

Inga KuznetsovaDepartment of Physics, University of Arizona

Workshop on Excited Hadronic States and the Deconfinement TransitionFebruary 23-25, 2011

Thomas Jefferson National Accelerator FacilityNewport News, VA

Work supported by a grant from: the U.S. Department of Energy DE-FG02-04ER4131

I. Kuznetsova and J. Rafelski, Phys. Lett. B, 668 105 (2008) [arXiv:0804.3352].I. Kuznetsova and J. Rafelski, Phys. Rev. C ,79, 014903 (2009) [arXiv:0811.1409]I. Kuznetsova and J. Rafelski Phys. Rev. C, 82, 035203 (2010) [arXiv:1002.0375 ].

Kinetics of hadron resonances during hadronic freeze-out

Inga Kuznetsova Workshop on Excited Hadronic States and Deconfinement transition

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Phases of RHI collision

QGP (deconfinement) phase; Chemical freeze-out (QGP hadronization), hadrons are formed;

(140 <T0 <180 MeV)

Hadronic gas (kinetic) phase, hadrons interact; Kinetic freeze-out : reactions between hadrons stop; Hadrons expand freely (without interactions, decaying only).

We study how strange and light resonance yields change during the kinetic phase. Final yields of ground state p, n, π, K, Λ do not changecompared to statistical hadronization model.


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We explain high ratio Σ(1385)/Λ0 reported at RHIC (S.Salur, J.Phys. G 32, S469 (2006))

and Λ(1520)/Λ0 suppression reported in both RHIC and SPS experiments. (J. Adams et al., Phys. Rev. Lett. 97, 132301 (2006)[arXiv:0604019]; C. Markert [STAR Collaboration], J. Phys. G 28, 1753 (2002) [arXiv:nucl-ex/0308028].).

We predict ∆(1232)/N ratio. We study φ meson production during kinetic phase in KK→ φ. By suppression (enhancement) here we mean the suppression (enhancement)

compared to scaled pp (or low number of participants) collisions, and to the chemical SHM (statistical hadronization model) without kinetic hadronic gas phase.

We study how non-equilibrium initial conditions after QGP hadronization influence the yield of resonances.

How does resonance yield depend on the difference between chemical freeze-out temperature (QGP hadronization temperature) and kinetic freeze-out temperature?

Motivation


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Kinetic phase

We assume that hadrons are in thermal equilibrium (except probably very high energy pions, which may escape).

Resonances have short lifespan (width Γ(1/τ) ≈ 10- 200 MeV)

Resonance yields can be produced in kinetic scattering phase.

M. Bleicher and J.Aichelin, Phys. Lett. B, 530 (2002) 81

M. Bleicher and H.Stoecker,J.Phys.G, 30, S111 (2004)

213:Reactions

1

23


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Observed yield, invariant mass method.

rescater

Resonance yield can be reconstructed by invariant mass method only after kinetic freeze-out, when decay products do not rescatter.

Chemical freeze-out

Kinetic freeze-out

The yields of ground state almost does not change. Everything decaysback to ground states.


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Dominant reactions

Σ(1385)↔Λπ ,width Γ∑(1385) ≈ 35 MeV (from PDG);

Σ* ↔ Λ(1520) π, Γ∑* ≈ 20-30 MeV > ΓΛ(1520) = 15.5 MeV (from PDG);

Σ* = Σ(1670), Σ(1750), Σ(1775), Σ(1940)) Δ(1232) ↔ Nπ, width Γ≈120 MeV (from PDG); φ↔KK (83%), φ↔ ρπ (15%), Г = 4.26 MeV,

Eth = mφ-2mK=30 MeV is relatively small.


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Influence of backward reaction also depends on Eth.

The smaller Eth is, the slower excited state decays back with cooling due expansion, larger higher mass resonance enhancement.

The larger Eth is, the less population of exited

state in equilibrium is, the less lower mass particles are needed to excite this state, the less lower mass resonance suppression is;

Λ(1520) is more suppressed by lower mass Σ* excitation.


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Reactions for Σ(1385) and Λ(1520).

Width of decay channel


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A second scenario

Normally all reactions go in both directions.

For the late stage of the expansion, at relatively low density this assumption may not be fully satisfied, in particular pions of high momentum could be escaping from the fireball.

Dead channels scenario: For dead channels resonances decay only.

MeV 300)( 213 mmmEth


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Fugacity definition

ii Epu )0,1(

u

VT

mK

T

mg

TN ii

iii

2

2

2

3

2

(fermi); baryons ,3 ,1,1))()exp(()(

1

;(bose) meson , 1))()exp(()(

1

1

21

22

itput

f

tputf

ii

i

for in the rest frame of heat bath

where K2(x) is Bessel function; gi is particle i degeneracy; Υi is particle fugacity, i =1, 2, 3;

Multiplicity of resonance (when ‘1’ in fi is negligible):

213:Reactions

We assume chemical potential μ=0, particle-antiparticle symmetry


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Time evolution equations

j

j

i

i

dtdV

dW

dtdV

dW

dt

dN

V21332131

Similar to 2-to-2 particles reactions: P.Koch, B.Muller and J.Rafelski Phys.Rept.142, 167 (1986); T.Matsui, B.Svetitsky and L.D. McLerran, Phys.Rev.D, 34, 783 (1986)

213:Reactions


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Lorentz invariant rates

2

2133214

1

13

2

23

3

33

5213 )(

)1()2(8

1

spin

ii

ppMppppE

pd

E

pd

E

pd

IdtdV

dW

( )( )1 11 2 3f f f

2

3213214

1

13

2

23

3

33

5321 )(

)1()2(

1

spin

ii

pMpppppE

pd

E

pd

E

pd

IdtdV

dW

f f f1 2 31( )


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Detailed balance condition

221

22 ))()exp(()(1 ftputf Bose enhancement factor: Fermi blocking factor: using energy conservation and time reversal symmetry:

we obtained detailed balance condition:

2

213

2

321 ppMppMpp

iiii ftputf ))()exp(()(1 1

ii

ii

i

iR

dtdV

dW

dtdV

dW

321

2

1

213

3

11


14

dtdVdW

ddN

V

ii 213

33

33

1

j

jSTi

iii

d

d

33

321

3 1111

Relaxation time:

,))(ln(1 2

2

d

dT

dT

xKxd

T

Fugacity (Υ) computation

0)ln(1 3

d

VTd

S

the entropy isconserved

τ is time in fluid element co-moving frame.

We solve system of equations numerically, using classical forth order Runge-Kutta method


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QGP Hadronization

We work in framework of fast hadronization to final state. Physical conditions (system volume, temperature) do not change. γq and γs are strange and light quarks fugacities:

Strangeness conservation: fixes γs .

Entropy conservation: fixes γq>1 at T < 180 MeV.

In QGP γqQGP = 1 .

QGPHG SS

QGPs

HGs NN

;γγ sq0 K ;γ 2

q0 ;γ 3

q0 N ;γγ s

2q

0 Y


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Initial and Equilibrium Conditions

eqeqeq321

02

01

03 reaction goes toward production of particle 3:

γq > 1, for T0 < 180 MeV; for strange baryons:

For one reaction equilibrium condition is:

; , 402

01

203 qsqs

If γq = 1 at hadronization, we have equilibrium. However withexpansion Υ3 increases faster than Υ1Υ2 and reaction would go towards resonance 3 decay:


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Expansion of hadronic phase

Growth of transverse dimension:

Taking

we obtain:

0

)()( 0 dvRR

constRTVT 233

1)/(2

3

1 Rv

Td

dT

)(v is expansion velocity


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Competition of two processes:

Non-equilibrium results towards heavier resonances production in backward reaction.

Cooling during expansion influence towards heavier states decay.


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The ratios NΔ/NΔ0, NN/NN

0 as a function of T

Υπ = const 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.

Δ(1232) ↔ Nπ


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∆(1232) enhancement

Δ(1232) ↔ N π, width Γ≈120 MeV;

Δ is enhanced whenN + π → Δ(1232) reaction dominates


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Resonances yields after kinetic phase:

Λ (1520) is suppressed due to Σ* excitation during kinetic phase.

∑(1385)/Λ is enhanced whenreaction Λπ →Σ(1385) dominates.


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Dead channels

In presence of dead channels the effect is amplified.∑* decays to ‘dead channels’ fast, the suppression of Λ(1520) by reaction Λ(1520)π→ ∑* increases.

∑*Λ(1520)

π

Λ, N, ∑

π, N, K


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Observable ratio Λ (1520)/Λ as a function of T

*00 )1193()1385(9.0 Ytot *

)1520()1520()1520( Yob

Λ (1520) is suppressed due to Σ*

excitation during kinetic phase.

There is additional suppression in observable ratio because Σ*s are suppressed at the end of kinetic phase and less of them decay back to Λ(1520) during free expansion.

Tk≈100 MeV; Th ≈ 140 MeV


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Observable ratio ∑(1385)/Λ as a function of T

*)1385()1385()1385( Yob

∑(1385)/Λ is enhanced whenreaction Λπ →Σ(1385) dominates.

The influence of reactions with higher mass resonances is small.


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Difference between Λ(1520) and Σ(1385).

ΓΛ(1520) = 15.6 MeV;

Eth for Λ(1520) production > Eth for Σ*s excitation

ΓΣ(1385) ≈ 36 MeV;

Eth for Σ(1385) production < Eth for Σ*s excitation

mΣ(1385) < mΛ(1520) → nΣ(1385) > nΛ(1520)

A lesser fraction of the lighter mass particle is needed to equilibrate the higher mass particle.

)1520()1520() MeV3020(*

)1385()1385() MeV10(*

Λ(1520) + π → Σ* is dominant over 1 + 2 → Λ(1520)

Λ0 + π → Σ(1385) is dominant over Σ(1385) + π → Σ*


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φ evolution (φ↔KK )

For comparison at equilibrium hadronization for φ decay only to KK, φ yield decreases by 7.5%; in inelastic scattering by 15%.Alvarez-Ruso and V.Koch, 2002

KK→φ and non-equilibrium hadronization conditions can noticeably change the result

After non-equilibrium hadronization production of φ must be dominant over relatively long period of time (small Eth)

T, MeV γ


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Summary

Λ(1520) yield is suppressed due to excitation of heavy Σ*s in the scattering process during kinetic phase and Σ*s preferable decay to ground states during kinetic phase.

Σ(1385) and Δ are enhanced due to Λ0 + π → Σ(1385) and N + π → Δ(1232) reactions for non

equilibrium initial conditions. We have shown that yields of Σ(1385) and Λ(1520) reported

in RHIC and SPS experiments are well explained by our considerations and hadronization at T=140 MeV is favored. Kinetic freeze-out is at T ≈ 100 MeV

For non-equilibrium hadronization φ yield can be enhanced by 6-7% by dominant KK→φ. For equilibrium hadronization φ yield suppression is about 4%


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Future research

ρ↔ππ, Г = 150 MeV ρ is much enhanced in pp collisions K* ↔ Kπ, Г = 50.8 MeV K* and ρ can participate in many other reactions.


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Difference between Σ(1385) and Λ(1520). Decay width for Σ(1385) to ground state is larger

than for Λ(1520). Decay widths of Σ*s to Σ(1385) is smaller than

those to Λ(1520). Eth for Σ(1385) excitation by ground states is

smaller than for Σ*s excitation by Σ(1385) and π fusion. Opposite situation is for Λ(1520).


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∑* evolution

∑(1775) is suppressed bydecay to channels with lightest product, especially in the case with ‘dead’ channels.


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Calculation of particle 3 decay / production rate

'3

333

333,3

33

3213 11,

2 n

E

mpfpd

g

dtdV

dWfb

'3 '

3

Particle 3 decay / production rate in a medium can be calculated, using particle 3 decay time in the this particle rest frame.

Particle 3 rest frame Observer (heat bath) frame

v

2 and 1 particles of mediumin framerest itsin lifespan 3 particle is 3


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Temperature as a function of time τ


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In medium effects for resonances

If particle 2 is pion (m2 = mπ) in medium effects may have influence. For heavy particle m3, m1 >> mπ :

, 1// *

3

33'3

33

Efnn

Rvac

i

framerest 3 resonance inenergy is 2

)(

3

221

23*

m

mmmE


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∑(1385) decay\production relaxation time in pion gas.


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Fugacity as a function of T(t)

If there are no reactions Ni = const, Υi is proportional to exp(mi/T)

for nonrelativistic Boltzmann distribution


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∑* reaction rates evolution (no dead channels)

Larger difference m3-(m1+m2) sooner decay in this channel becomes dominant.


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Motivation

B.I.Abelev et al., Phys. Rev. C 78, 044906 (2008)


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φ meson

Г = 4.26 MeV φ↔KK (83%), φ↔ ρπ (15%) Eth = mφ-2mK=30 MeV

After non-equilibrium hadronization production of φ must be dominant over relatively long period of time

Kinetics of hadron resonances during hadronic freeze-out

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