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. Ürmössy MTA 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 472 p

Collision energy dependence in ALCOR

leading rapidity Stopped per cent of baryons AGS SPS RHIC LHC

Collision energy dependence in ALCOR leading rapidity Newly produced light dN/dy AGS SPS RHIC LHC

Collision energy dependence in ALCOR leading rapidity K+ / pi+ ratio AGS SPS RHIC LHC

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 E E uniform random: Y(E ) = (  h/  y) dx ∫ 0 E 3 3 E E h=const

Massless kinematics Tsallis rule

A special pair-energy: E = E + E + E E / E 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 + D 121 color state 2 Singlet channel: hadronization color state D + 8 D = 0 singletoctet Octet channel: parton distribution E = E + E - D singlet E = E + E + D / octet

Semiclassical binding: E = E + E - D = E + E - D Zero mass kinematics (for small  angle): Octet channel: Tsallis distribution singlettotrel kin rel kin E = 4 sin (  / 2) E E E + E constant? 4 / E c Singlet channel: convolution of Tsallis distributions - D / 2 virial Coulomb for

Coalescence cross section a: Bohr radius in Coulomb potential Pick-up reaction in non-relativistic potential

Limiting temperature with Tsallis distribution N = E – j T TE T = E / d  ; c c j=1 d c H Massless particles, d-dim. momenta, N-fold For N  2: Tsallis partons  Hagedorn hadrons ( with A. Peshier, Giessen ) hep-ph/

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/ claim: (e-3p) ~ T

High-T behavior of lattice eos SU(3)

High-T behavior of lattice eos hep-ph/ Fig.2 8 × 32 ³

High-T behavior of lattice eos

From pressure to mass Pressure of relativistic ideal gas of massive particles: Lattice QCD data from Budapest-Wuppertal: JHEP 0601, 089, Bielefeld: NPB 469, 419, 1996.

Lattice QCD eos + fit Peshier et.al. Biro et.al.

Quasiparticle mass distribution by 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

(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 Budapest

July 22, 2006, Budapest