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

qq

n

qq

HADRONnEQUARK

n

qq

qq

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