ICPAQGP, GOA, INDIA
6-10, Dec,2010
Kaon and Lambda productions
in Heavy Ion Collision
Outline of the talk ………………
 Brief introduction on strangeness
production in heavy ion collision
 Strangeness to equilibrium entropy:
K/pi in different models
 Strangeness production in different
scenarios: partonic & hadronic
 Strangeness calculation using
transport equation
 Kaon to Pion ratio and Lambda to Pion ratio
Jajati K. Nayak
 Summary and outlook
Variable Energy Cyclotron Center, Kolkata
ICPAQGP-2010, GOA, INDIA
PRC, 81,024911(2010), STAR collaboration
Strangeness enhancement
compared to p-p collision at
same energy
p-Be
p-Au
Si-Au
K+
p+
4%
10%
20%
K+
p+
At AGS energy
K
4% for all
p
Non-monotonic energy dependence in K+/p+ ratio
Monotonic energy dependence in K-/p- ratio
Introduction..
Variable Energy Cyclotron Center, Kolkata
K
p
ICPAQGP-2010, GOA, INDIA
Experiments
AGS
RHIC
SPS
Experiments
Ebeam (A GeV)
sNN (GeV)
System
Particles
AGS
E802, E866, E877,
E891, E895, E917
2-10.7
2.7-4.9
Au+Au
p, K, p, 
SPS
NA45, NA49, NA57,
(NA44, WA98)
20-158
6.3-17.3
Pb+Pb
p, K, p, , ,
, ,...
RHIC
STAR, PHENIX,
BRAHMS, PHOBOS
-
20.0-200.0
Au+Au
p, K, p, , ,
, , ...
Introduction..
Variable Energy Cyclotron Center, Kolkata
ICPAQGP-2010, GOA, INDIA
Models available for the explanation of the
horn
Statistical Hadron Model:
J Cleymans et al/Phys. Lett B 615(2005)
A. Andronic, P. Braun-Munzinger and
J. Stachel, arXiv:0812.1186 [nucl-th]
Model by Gorenstein and Gadzicki
Transport models (UrQMD, HSD):
Hadronic kinetic model:
E.L. Bratkovskaya et al.,
PRC 69 (2004), 054907
Boris Tomasik et al.
Eur.Phys.J.C49:115120,2007
Model by Rafelski
Model by Tawfik
HRG Model
HRG Model (vol. correction)
Chatterjee, Gupta & Godbole
Tiwari, Singh et al. BHU group
Model by us: J. K. Nayak, S. Banik and Jan-e Alam
Different models for ratio…..
Variable Energy Cyclotron Center, Kolkata
ICPAQGP-2010, GOA, INDIA
Study microscopically the strangeness productions
Study different possibilities to explain the data
Scenario-I
Initial hadronic state is assumed for all
s NN
Initial number density ni is assumed to be away from
equilibrium
Scenario-II
Initial partonic state is assumed beyond
sNN  7.6 GeV
Initial number density ni is assumed to be
away from equilibrium
Scenario-III
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed in complete equilibrium
Without secondary productions
Scenario-IV
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed in complete equilibrium
With secondary productions
Scenario-V
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed completely away from equilibrium
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Strangeness production
in a partonic scenario
In partonic scenario
In a hadronic scenario
Gluon gluon fusion
Quark and anti-quark
annihilation
gg  s s
qq  ss
Cross-section up to lowest order QCD
J. Rafelski & B. Muller,
P.R.L.48 (1982) 1066
Where,
s   p1 + p2 
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
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Strangeness production in hadronic scenario
Meson-meson, meson-baryon
and baryon-baryon interaction
MM  K K
and others…
MB  YK
G. E. Brown, C. M. Ko,
Z. G. Wu & L. H. Xia,
PRC 43 (1991) 1881
BB  BYK
1. MM  K K
K
K
r
K
K
K
*
LK *Kp  g K *Kp K * K  p     K p 
r


 
LrKK  g rKK K   K     K  K r 
Isospin averaged cross section
1 p'
 
32p sp
K
p

+1
1
dx M ( s, x)
where s is the square of total c.m. energy, p & p’ are 3-momenta
of incoming meson & kaon in c.m. frame. M(s,x) is the isospin-averaged squared invariant amplitude.
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
p
ICPAQGP-2010, GOA, INDIA
2. MB  YK
Y
Y
K
N*
N*
N
N
p
K
r
Isospin averaged cross-section
 pN K
2 Ji + 1
 
i ( 2 s1 + 1)( 2 s2 + 1)
p
ki2
s
12
B ini B out i i2
 mi

2
+ i2 / 4
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
ICPAQGP-2010, GOA, INDIA
3. BB  BYK and others
Isospin averaged cross section
 NN  NK
2
n
3m

2 p 2 p2 s
Wmax
q+2
 

2
f
q2
2
2 pNN
2
2
2
dW
W
k
dq
F
q

W
;
q
0
2
2
2 2


m
(
q

m
2
p
p)
Wmin
q

Wmin  mK + m
N3
Wmax  s1 2  mN


12
K
Y
p
q2  2 mn2  2 E E '  2 p p '
while E  p 2 + mN2

& p are the energy & momentum
p
of nucleon N1 in the c.m. system while E' & p' are those of N 3 .
2  mp2
F 2
;  is the cut off parameter   1 GeV.
2
 q
A
N1
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Absorption reactions for For K- apart from production
In a baryon rich medium K- gets absorbed due to its interaction with baryons.
Rate calculations for strangeness production
 
 
a1  p1  + a2  p2   a3 p1' + a4 p2' , where a1  a2
d 3 p1
d 3 p2
R(T )  
F ( p1 ) 
F ( p2 ) vrel  ( M )
3
3
2p 
2p 
Where,
F ( p)  exp[  ( E   ) T ],

M 2  E 2  p2
 
vrel  v1  v2 ,
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
ICPAQGP-2010, GOA, INDIA
Rate of production of strangeness
MM interactions
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
BB interactions
MB interactions
Strange quarks
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ICPAQGP-2010, GOA, INDIA
Rate of kaon production from MM, MB, BB interaction
•Baryon dominance at lower
s NN
•At low temperature baryonic contribution is more
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Total K+ and K - production rate
Strange production in different scenarios…. Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Evolution of Strangeness
It is assumed that non-strange quarks and hadrons are in thermal equilibrium
Momentum integrated Boltzman equation is used to study the evolution
Partonic and
hadronic phase phase
In mixed phase
Where in the mixed phase,
J. Kaputsa & A. Mekjian,
PRD 33 (1986), 1304
Strangeness calculation….
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Space-time Evolution
The partonic/hadronic system evolves in space-time.
The net baryon number is conserved.
The baryon chemical potential at freeze out is taken from the
parameterization of
O. Ristea for BRAHMS collab. Romanian Reports in Physics, 56(2004) 659.
The initial baryonic chemical potential is obtained from net
baryon number conservation equation.
The initial temperature of the system formed after nuclear collisions
have been taken using measured hadronic multiplicity by using
2p 4
1
Ti 
45  3 p R 2  i
3
Strangeness calculation….
90
4p 2 g eff
dN
dy
Variable Energy Cyclotron Center, Kolkata
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Non-monotonic behaviour of K+/p+ ratio can be understood due to
K+/p+ ratio
K-/p- ratio
Ratio within the assumption of input
parameters considered for different
s NN
Horn in the Kaon to pion ratio….
K-/p- ratio seems to differentiate little
between the two scenarios. However, they
both saturate at higher
s NN
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Scenario-I
Initial hadronic state is assumed for all
s NN
Initial number density ni is assumed to be 20%
away from equilibrium
Scenario-II
sNN  7.6 GeV
Initial partonic state is assumed beyond
Initial number density ni is assumed to be 20%
away from equilibrium
Scenario-III
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed in complete equilibrium
Without secondary productions
Scenario-IV
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed in complete equilibrium
With secondary productions
Scenario-V
Initial partonic state is assumed beyond
sNN  7.6 GeV
System is assumed to be formed completely away from equilibrium
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
K+/p+ ratio
Horn in the Kaon to pion ratio….
K-/p- ratio
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Lambda production
Rate of production
Lambda production
Variable Energy Cyclotron Center, Kolkata
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Summary
•The momentum integrated Boltzmann Equation is used to study the
evolution of strangeness.
•We got a non-monotonic behavior for K+/p+ when a partonic state
is assumed at c. m. energy >7.6 GeV (within the ambit of assumed parametres).
•However a monotonic rise in K+/p+ is observed for a pure hadronic matter.
•K-/p- data is unable to differentiate between the two initial condition.
•
^/π is evaluated within the ambit of same model parametres
Collaborators: Sarmistha Banik and Jan-e Alam
Summary and outlook….
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Hadron gas
time
Mixed
phase
QGP fluid
1st order
collision axis
0
time
Continuous/
crossover
Hadron gas
QGP fluid
collision axis
0
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
p is the lightest hadron. It carries maximum entropy and is copiously produced in QGP phase.

P


g
2p 3 
g
2p 
3
1
3
p 2 + m2 f d 3 p

p2
p 2 + m2
f d3 p
 +P N
p2
s 
4 
 g  T3
T
90
For massless particles
The g rises sharply in QGP phase
 Kaon/Pion ratio proposed as measure of strangeness to entropy ratio
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
Strangeness enhancement & K+ distillation
(high density region)
T 0
Strangeness
Enhancement
p
u ,d
F
s ms  200 MeV
u
d
mu , md  0 MeV
EFu ,d  pFu ,d
13
 3   r 
N

   pF  450 MeV  m s + m s
 2  3   r0 
13
Therefore, it is easier to create
s s than
pair
uu or q q
Variable Energy Cyclotron Center, Kolkata
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ICPAQGP-2010, GOA, INDIA
 ( s s) K (q s) K (qs)  (qss )  (qss )  ( sss )  ( sss )

K+ (or )
distillation
K+
u
d
s
s
+
 
u s : not easy
K u s : easy
K

u d s : easy
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Initial conditions…….
Strangeness calculation….
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Statistical Hadron Model
In the statistical Hadron model frame work, a rapid change is expected as the hadronic
gas under goes a transition from a baryon dominated to a meson dominated gas. The
transition occurs at temperature 140 MeV .i.e.,
sNN  8.2 GeV
 B  410 MeV
Cleymans et al.
PLB,615(2005)50
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Gadzicki Model
Gorenstein Model
Canonical phase space suppression: Statistical Model for Early Stage (SMES)
Large Entropy
Production Beyond
Threshold Energy
Gadzicki Acta Phys Pol, B, 35,187(2004);
Gorenstein JPG,28,1623(2002)
Es=Ratio of strangeness
to entropy
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Extended Hadron Resonance Gas Model
They investigated the kaon to pion ratio from the Hadronic Resonance Gas model considering
hadron masses up to 1.8 GeV (from particle data book )and higher hagedron resonances. The
dacay of the higher resonances to strange hadrons has been considered.
Chatterjee, Godbole, Gupta, PRC,88,044907(2010)
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Statistical Model: A Tawfik
The experimental curves for all the below ratios can be explained if the particle production is considered due
to non-equilibrium processes for both non-strange and strange quarks. That is, the statistical parameters  i
for both light quarks and strange quarks are less than 1. No phase transition ; parameters less than 1 .
No. density=darivative of pressure w.r.t
chem. Pot and for finite isospin fugacity
In a grand canonical treatment
Pressure : contributions from all hadron resonances
treated as free gas. Function of baryon chem. pot.,
Strange chem. pot. and iso spin chem. Pot.
i  exp( i ) Corresponding fugacity
 i  Saturation factor    q s
i
Daviation from
equilibrium
 1 For equilibrium
q  light flavors
I
q 1
Variable
s 1
to
Fails
q and  s
describe
u, d
A. Tawfik, Fizika B, 18, 141-150(2009),
arxiv:hep-ph/0508244(2006)
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Microscopic Evolution Model : J.K. Nayak, S. Banik & Jan-e Alam
K+/p+ ratio
K-/p- ratio
Evoluton : Boltzman equation
Rate of productions for K+ , K- and Lambda:
cross-sections from Lagrangian
We study the evolution with many possible scenarios
Scenario-II, with a partonic assumption beyond certain centre of mass enrgy explians better
Horn in the Kaon to pion ratio….
Variable Energy Cyclotron Center, Kolkata
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