Kinetics of hadron
resonances during hadronic
freeze-out
Inga Kuznetsova
Department of Physics, University of Arizona
Workshop on Excited Hadronic States and the Deconfinement Transition
February 23-25, 2011
Thomas Jefferson National Accelerator Facility
Newport News, VA
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 ].
Work supported by a grant from: the U.S. Department of Energy DE-FG02-04ER4131
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 change
compared to statistical hadronization model.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
2
Motivation
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?
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
3
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.
2
3
1
M. Bleicher and J.Aichelin, Phys. Lett. B, 530 (2002) 81
M. Bleicher and H.Stoecker,J.Phys.G, 30, S111 (2004)
Reactions : 3  1  2
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Workshop on Excited Hadronic States and
Deconfinement transition
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Observed yield, invariant mass method.
Chemical freeze-out
rescater
Kinetic freeze-out
Resonance yield can be reconstructed by invariant mass method
only after kinetic freeze-out, when decay products do not rescatter.
The yields of ground state almost does not change. Everything decays
back to ground states.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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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.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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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.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Reactions for Σ(1385) and Λ(1520).
Width of
decay channel
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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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.
Eth  m3  (m1  m2 )  300MeV
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
9
Fugacity definition
Reactions : 3  1  2
f2 
1
, meson(bose);
21 (t ) exp((u  p2 )  (t ))  1
fi 
1
, i  1, 3, baryons(fermi);
1
i (t ) exp((u  pi )  (t ))  1

u  pi  Ei for u  (1,0) in the rest frame of heat bath
We assume chemical potential μ=0, particle-antiparticle symmetry
Multiplicity of resonance (when ‘1’ in fi is negligible):
2
T 3  mi 
m 
Ni  i 2 gi   K2  i V
2
T 
T 
where K2(x) is Bessel function; gi is particle i degeneracy;
Υi is particle fugacity, i =1, 2, 3;
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Time evolution equations
Reactions : 3  1  2
j
1 dN3
dW1i23
dW3
1 2


V dt
dtdV
dtdV
i
j
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)
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Lorentz invariant rates
dW3i1 2
d 3 p3 d 3 p2 d 3 p1 4
1
i


(
p

p

p
)
p
M
p1 p2
1
2
3 
3
5



dtdV
8(2 ) (1  I ) E3
E2
E1
spin
2

 (1  f1 )(1  f 2 ) f3
dW1i 23
d 3 p3 d 3 p2 d 3 p1 4
1
i


(
p

p

p
)
p
p
M
p3
1
2
3 
1 2
5



dtdV
(2 ) (1  I ) E3
E2
E1
spin
 f1 f 2 (1  f3 )
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Workshop on Excited Hadronic States and
Deconfinement transition
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2

Detailed balance condition
 Bose enhancement factor: f 2  1  21(t ) exp((u  p2 ) (t )) f 2
 Fermi blocking factor: fi  1  i1(t ) exp((u  pi ) (t )) f i
 using energy conservation and time reversal symmetry:
p1 p2 M p3
2
 p3 M p1 p2
2
 we obtained detailed balance condition:
1 dW3i1 2
1 dW1i 23
i


R
3i dtdV
1i 2i dtdV
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Workshop on Excited Hadronic States and
Deconfinement transition
13
Fugacity (Υ) computation
1 dN3
i
Relaxation time:  3 
3
V d3
dW3i12
dtdV
1 1
d3
1
i i 1
  1 2 i  3     j 
d
3
i
j 3 
T  S
τ is time in fluid element co-moving frame.
d ln(x2 K2 ( x )) dT

,
T
dT
d
1
the entropy is
d ln(VT 3 )

0
conserved
S
d
1
We solve system of equations numerically, using classical forth order
Runge-Kutta method
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
14
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:
K0  γq γs ;
  γ q ;   γq ;
0
2
 Strangeness conservation:
Entropy conservation: S
In QGP γqQGP = 1 .
Inga Kuznetsova
HG
3
0
N
  γ q γs ;
N sHG  N sQGP
S
QGP
0
Y
2
fixes γs .
fixes γq>1 at T < 180 MeV.
Workshop on Excited Hadronic States and
Deconfinement transition
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Initial and Equilibrium Conditions
γq > 1, for T0 < 180 MeV; for strange baryons:
   ,     ;
  
0
3
0
1
0
3
2
s q
0
1
0
2
4
s q
0
2 reaction goes toward production of particle 3:
For one reaction equilibrium condition is:  eq eq   eq
1
2
3
If γq = 1 at hadronization, we have equilibrium. However with
expansion Υ3 increases faster than Υ1Υ2 and reaction would go
towards resonance 3 decay:
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Expansion of hadronic phase
 Growth of transverse dimension:

R ( )  R0   v ( )d 
0
v( ) is expansion velocity
 Taking
we obtain:
Inga Kuznetsova
T 3V  T 3 R2  const
dT
1  2(v / R )  1 
 

Td
3


Workshop on Excited Hadronic States and
Deconfinement transition
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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.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
18
The ratios NΔ/NΔ0, NN/NN0 as a function
of T
Δ(1232) ↔ Nπ
 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.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
19
∆(1232) enhancement
Δ(1232) ↔ N π, width Γ≈120 MeV;
Δ is enhanced when
N + π → Δ(1232) reaction dominates
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Resonances yields after kinetic phase:
Λ (1520)
is suppressed dueWorkshop
to Σ*
∑(1385)/Λ
on Excited
Hadronic States and is enhanced when
21
Inga Kuznetsova
Deconfinement transition
excitation during kinetic phase. reaction Λπ →Σ(1385) dominates.
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, N, ∑
Λ(1520)
∑*
π
Inga Kuznetsova
π, N, K
Workshop on Excited Hadronic States and
Deconfinement transition
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Observable ratio Λ (1520)/Λ as a
function of T
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
tot  0.9(1385)  0  0 (1193)  Y *
(1520)ob  (1520)  Y*(1520)
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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Observable ratio ∑(1385)/Λ as a
function of T
∑(1385)/Λ is enhanced when
reaction Λπ →Σ(1385) dominates.
The influence of reactions with
higher mass resonances is small.
(1385)ob  (1385)  Y
*
(1385)
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Workshop on Excited Hadronic States and
Deconfinement transition
24
Difference between Λ(1520) and Σ(1385).
 ΓΛ(1520) = 15.6 MeV; *  (1520 ) ( 20  30 MeV)   (1520 )
Eth for Λ(1520) production > Eth for Σ*s excitation
Λ(1520) + π → Σ* is dominant over 1 + 2 → Λ(1520)
 ΓΣ(1385) ≈ 36 MeV;
* (1385 ) ( 10 MeV)  (1385 )
Eth for Σ(1385) production < Eth for Σ*s excitation
Λ0 + π → Σ(1385) is dominant over Σ(1385) + π → Σ*
 mΣ(1385) < mΛ(1520) → nΣ(1385) > nΛ(1520)
 A lesser fraction of the lighter mass particle is needed to
equilibrate the higher mass particle.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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φ evolution (φ↔KK )
T, MeV γ
After non-equilibrium
hadronization production of φ
must be dominant over relatively
long period of time (small Eth)
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
Inga Kuznetsova
KK→φ and non-equilibrium
hadronization conditions can
noticeably change the result26
Workshop on Excited Hadronic States and
Deconfinement transition
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%
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
27
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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Workshop on Excited Hadronic States and
Deconfinement transition
28
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).
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
29
∑* evolution
∑(1775) is suppressed by
decay to channels with
lightest product, especially in
the case with ‘dead’ channels.
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Workshop on Excited Hadronic States and
Deconfinement transition
30
Calculation of particle 3 decay /
production rate
Particle 3 decay / production rate in a medium can be calculated,
using particle 3 decay time in the this particle rest frame.
dW31 2
g
m 1
1
 3 3  d 3 p3 f b, f 3 , p3  3  n3
dtdV
E3  3
3'
2 
Observer (heat bath) frame
Particle 3 rest frame
v

'
3
 3' 
 3 is particle3 lifespanin its rest framein medium of particles1 and 2
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
31
Temperature as a function of time τ
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Workshop on Excited Hadronic States and
Deconfinement transition
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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 :
R 
i
n3 / 3
 3'

n3 / 3
 3vac
1  f E ,

*

2
2
2
m

(
m

m
1
)
E*  3
is  energy in resonance3 rest frame
2m3
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
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∑(1385) decay\production relaxation
time in pion gas.
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
34
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
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
35
∑* reaction rates evolution (no dead
channels)
Larger difference m3-(m1+m2) sooner decay in this channel becomes
Workshop on Excited Hadronic States and
36
Inga Kuznetsova
dominant.
Deconfinement transition
Motivation
B.I.Abelev et al., Phys. Rev. C 78, 044906 (2008)
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
37
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
Inga Kuznetsova
Workshop on Excited Hadronic States and
Deconfinement transition
38