Kinetics of hadron resonances during hadronic freeze-out
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Transcript of Kinetics of hadron resonances during hadronic freeze-out
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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
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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
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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