A L C O R

1
A L C O R
From quark combinatorics to spectral coalescence
T.S. Bíró, J. Zimányi , P. Lévai, T. Csörgo,
K. Ürmössy MTA KFKI RMKI Budapest, Hungary

• History of the idea
• Extreme relativistic kinematics
• Hadrons from quasiparticles
• Spectral coalescence

2
A L C O R the history

• Algebraic combinatoric rehadronization
• Nonlinear vs linear coalescence
• Transchemistry
• Recombination vs fragmentation
• Spectral coalescence

3
Quark recombination combinatoric rehadronization
1981
4
Quark recombination combinatoric rehadronization
5
Robust ratios for competing channels
PLB 472 p. 243 2000
6
Collision energy dependence in ALCOR
7
Collision energy dependence in ALCOR
100
AGS
Stopped per cent of baryons
SPS
10
RHIC
LHC
leading rapidity
0
2
4
6
8
10
8
Collision energy dependence in ALCOR
200
AGS
Newly produced light dN/dy
100
SPS
RHIC
LHC
leading rapidity
0
2
4
6
8
10
9
Collision energy dependence in ALCOR
0.2
AGS
0.1
K / pi ratio
SPS
RHIC
LHC
leading rapidity
0
2
4
6
8
10
10
A L C O R kinematics

• 2-particle Hamiltonian
• massless limit
• virial theorem
• coalescence cross section

11
A L C O R kinematics
Non-relativistic quantum mechanics problem
12
Virial theorem for Coulomb
Deformed energy addition rule
13
Test particle simulation
y
h(x,y) const.
E
E
2
E
4
E
x
E
3
E
1
E
3
-1
?
uniform random Y(E ) ( ? h/ ? y)
dx
3
hconst
0
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Massless kinematics
Tsallis rule
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A special pair-energy
E E E E E / E
1
2
c
12
1
2
(1 x / a) (1 y / a ) 1 ( x y xy /
a ) / a
Stationary distribution
- v
f ( E ) A ( 1 E / E )
c
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Color balanced pair interaction
color state
color state
E E E D
2
12
1
singlet
octet
D 8 D 0
Singlet channel hadronization
singlet
E E E - D
2
12
1
Octet channel parton distribution
octet
E E E D / 8
2
12
1
17
Semiclassical binding
- D / 2
virial
singlet
tot
rel
E E E - D E E
- D
for
12
1
2
kin
kin
Coulomb
Zero mass kinematics (for small f angle)
E E
rel
2
1
2
E 4 sin (f / 2)
kin
4 / E
E E
c
1
2
constant?
Octet channel Tsallis distribution
Singlet channel convolution of Tsallis
distributions
18
Coalescence cross section
a Bohr radius in Coulomb potential
Pick-up reaction in non-relativistic potential
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Limiting temperature with Tsallis distribution
( with A. Peshier, Giessen )
hep-ph/0506132
Massless particles, d-dim. momenta, N-fold
d
ltX(E)gt
TE
?
c


T E / d
c
H
j1
E j T
c
N
For N ? 2 Tsallis partons ? Hagedorn hadrons
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Temperature vs. energy
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Hadron mass spectrum from X(E)-folding of Tsallis
N 2 N 3
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A L C O R quasiparticles

• continous mass spectrum
• limiting temperature
• QCD eos ? quasiparticle masses
• Markov type inequalities

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High-T behavior of ideal gases
Pressure and energy density
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High-T behavior of a continous mass spectrum of
ideal gases
interaction measure
Boltzmann f exp(- ? / T) ? ?(x) ? x
K1(x)
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High-T behavior of a single mass ideal gas
interaction measure for a single mass M
Boltzmann f exp(- ? / T) ? ?(0) ?
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High-T behavior of a particular mass spectrum of
ideal gases
Example 1/m² tailed mass distribution
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High-T behavior of a continous mass spectrum of
ideal gases
High-T limit ( µ 0 )
Boltzmann c ?/2, Bose factor ??(5), Fermi
factor ?(5)
Zwanziger PRL, Miller hep-ph/0608234 claim
(e-3p) T
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High-T behavior of lattice eos
SU(3)
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High-T behavior of lattice eos
hep-ph/0608234 Fig.2 8 32 ³
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High-T behavior of lattice eos
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High-T behavior of lattice eos
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From pressure to mass
Pressure of relativistic ideal gas of massive
particles
Lattice QCD data from Budapest-Wuppertal JHEP
0601, 089, 2006. Bielefeld NPB 469, 419,
1996.
33
Lattice QCD eos fit
Biro et.al.
Peshier et.al.
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Quasiparticle mass distributionby inverting the
Boltzmann integral
Inverse of a Meijer trf. inverse imaging
problem!
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Bounds on integrated mdf

• Markov, Tshebysheff, Tshernoff, generalized
• Applied to w(m) bounds from p
• Applied to w(mµ,T) bounds from ep
• Boltzmann mass gap at T0
• Bose mass gap at T0
• Fermi no mass gap at T0
• Lattice data

36
Markov inequality and mass gap
T and µ dependent w(m) requires mean field
term, but this is cancelled in (ep) eos data!
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Boltzmann scaling functions
?
?
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General Markov inequality
Relies on the following property of the function
g(t)
i.e. g() is a positive, montonic growing
function.
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Markov inequality and mass gap
There is an upper bound on the integrated
probability P( M ) directly from (ep) eos
data!
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SU(3) LGT upper bounds
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21 QCD upper bounds
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A L C O R spectral coalescence

• p-relative ltlt p-common
• convolution of thermal distributions
• convolution of Tsallis distributions
• convolution with mass distributions

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Idea Continous mass distribution

• Quasiparticle picture has one definite mass,
  which is temperature dependent M(T)
• We look for a distribution w(m), which may be
  temperature dependent

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Why distributed mass?
c o a l e s c e n c e c o n v o l u t i o n
valence mass ? hadron mass ( half or
third)
w(m) w(had-m)
w(m)
Zimányi, Lévai, Bíró, JPG 31711,2005
w ( m ) is not constant zero probability
for zero mass
Conditions
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Coalescence from Tsallisdistributed quark matter
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Kaons
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Recombination of Tsallis spectra at high-pT
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(q-1) is a quark coalescence parameter
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Properties of quark matter from fitting
quark-recombined hadron spectra

• T (quark) 140 180 MeV
• q (quark) 1.22
• power 4.5 (same as for ee- spectra)
• v (quark) 0 0.5
• Pion near coalescence (q-1) value

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SQM 1996 Budapest
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SQM 1996 Budapest
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July 22, 2006, Budapest