A L C O R
description
Transcript of A L C O R
A L C O R
• History of the idea
• Extreme relativistic kinematics
• Hadrons from quasiparticles
• Spectral coalescence
T.S. Bíró, J. Zimányi †, P. Lévai, T. Csörgő, K. ÜrmössyMTA KFKI RMKI Budapest, Hungary
From quark combinatorics to spectral coalescence
A L C O R: the history
• Algebraic combinatoric rehadronization
• Nonlinear vs linear coalescence
• Transchemistry
• Recombination vs fragmentation
• Spectral coalescence
Quark recombination : combinatoric rehadronization
1981
Quark recombination : combinatoric rehadronization
Robust ratios for competing channels
PLB 472p. 2432000
Collision energy dependence in ALCOR
Collision energy dependence in ALCOR
100
10
0 2 4 6 8 10 leading rapidity
Sto
pp
ed
pe
r c
ent
of
ba
ryo
ns
AGS
SPS
RHICLHC
Collision energy dependence in ALCOR
200
100
0 2 4 6 8 10 leading rapidity
Ne
wly
pro
du
ced
lig
ht
dN
/dy
AGS
SPS
RHICLHC
Collision energy dependence in ALCOR
0.2
0.1
0 2 4 6 8 10 leading rapidity
K+
/ p
i+
rati
o
AGS
SPS
RHICLHC
A L C O R: kinematics
• 2-particle Hamiltonian
• massless limit
• virial theorem
• coalescence cross section
A L C O R: kinematics
Non-relativistic quantum mechanics problem
Virial theorem for Coulomb
Deformed energy addition rule
Test particle simulation
x
y
h(x,y) = const.
E
E
EE
13
4
2
uniform random: Y(E ) = ( h/ y) dx-1
∫0
E3
3
E
E
h=const
Massless kinematicsTsallis rule
A special pair-energy:
E = E + E + E E / E12 1 2 1 2 c
(1 + x / a) * (1 + y / a ) = 1 + ( x + y + xy / a ) / a
Stationary distribution:
f ( E ) = A ( 1 + E / E )c
- v
Color balanced pair interaction
E = E + E + D12 1
color state
2
Singlet channel: hadronization
color state
D + 8 D = 0singlet octet
Octet channel: parton distribution
E = E + E - D12 1 2
singlet
E = E + E + D / 812 1 2
octet
Semiclassical binding:
E = E + E - D = E + E - D12 1 2
Zero mass kinematics (for small angle):
Octet channel: Tsallis distributionOctet channel: Tsallis distribution
singlet tot rel
kinkin
rel
kin E = 4 sin ( / 2)
E E
E + E
1
1 2
22
constant?
4 / E c
Singlet channel: convolution of Tsallis distributionsSinglet channel: convolution of Tsallis distributions
- D / 2virial
Coulomb
for
Coalescence cross section
222
2
)1( relpa
a
a: Bohr radius in Coulomb potential
Pick-up reaction in non-relativistic potential
Limiting temperature with Tsallis distribution
<X(E)>
N=
E – j T
TE T = E / d ;
c
cj=1
d
cH
Massless particles, d-dim. momenta, N-fold
For N 2: Tsallis partons Hagedorn hadrons
( with A. Peshier, Giessen ) hep-ph/0506132
Temperature vs. energy
Hadron mass spectrum from X(E)-folding of Tsallis
N = 2 N = 3
A L C O R: quasiparticles
• continous mass spectrum
• limiting temperature
• QCD eos quasiparticle masses
• Markov type inequalities
High-T behavior of ideal gases
Pressure and energy density
High-T behavior of a continous mass spectrum of ideal gases
„interaction measure”
Boltzmann: f = exp(- / T) (x) = x K1(x)
High-T behavior of a single mass ideal gas
„interaction measure” for a single mass M:
Boltzmann: f = exp(- / T) (0) =
High-T behavior of a particular mass spectrum of ideal gases
Example: 1/m² tailed mass distribution
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
High-T behavior of lattice eos
2
20
T
mSU(3)
High-T behavior of lattice eos
hep-ph/0608234 Fig.2 8 × 32 ³
High-T behavior of lattice eos
High-T behavior of lattice eos
Lattice QCD eos + fit
TT
baTT
bac
ce
e
e
/
/
1
1
cTT76.1ln
54.01 Peshier et.al.
Biro et.al.
Quasiparticle mass distributionby inverting the Boltzmann integral
Inverse of a Meijer trf.: inverse imaging problem!
Bounds on integrated mdf
• Markov, Tshebysheff, Tshernoff, generalized
• Applied to w(m): bounds from p
• Applied to w(m;µ,T): bounds from e+p– Boltzmann: mass gap at T=0– Bose: mass gap at T=0– Fermi: no mass gap at T=0
• Lattice data
Markov inequality and mass gap
T and µ dependent w(m) requires mean field term,
but this is cancelled in (e+p) eos data!
Boltzmann scaling functions
General Markov inequality
Relies on the following property of the
function g(t):
i.e.: g() is a positive, montonic growing function.
Markov inequality and mass gap
There is an upper bound on the integrated
probability P( M ) directly from (e+p) eos data!
SU(3) LGT upper bounds
2+1 QCD upper bounds
A L C O R: spectral coalescence
• p-relative << p-common
• convolution of thermal distributions
• convolution of Tsallis distributions
• convolution with mass distributions
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
Why distributed mass?
valence mass hadron mass ( half or third…)
c o a l e s c e n c e : c o n v o l u t i o n
Conditions: w ( m ) is not constant zero probability for zero mass
Zimányi, Lévai, Bíró, JPG 31:711,2005
w(m)w(m) w(had-m)
Coalescence from Tsallis
distributed quark matter
Kaons
Recombination of Tsallis spectra at high-pT
)1(1
)1(1
)1(1)1(1
)()(
31
21
1
11
QUARKBARYON
QUARKMESON
QUARKBARYONMESON
n
HADRONnEQUARK
n
TTT
T
Eq
nT
Eq
Eff
q
(q-1) is a quark coalescence
parameter
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 e+e- spectra)
• v (quark) = 0 … 0.5
• Pion: near coalescence (q-1) value
SQM 1996 BudapestSQM 1996 Budapest
SQM 1996 BudapestSQM 1996 Budapest
July 22, 2006, Budapest