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A L C O R. From quark combinatorics to spectral coalescence. T.S. Bíró , J. Zimányi † , P. Lévai, T. Csörgő, K. Ürmössy MTA KFKI RMKI Budapest, Hungary. History of the idea Extreme relativistic kinematics Hadrons from quasiparticles Spectral coalescence. A L C O R: the history.
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A L C O R From quark combinatorics to spectral coalescence T.S. Bíró, J. Zimányi †, P. Lévai, T. Csörgő, K. Ürmössy MTA KFKI RMKI Budapest, Hungary History of the idea Extreme relativistic kinematics Hadrons from quasiparticles Spectral coalescence
A L C O R: the history Algebraic combinatoric rehadronization Nonlinear vs linear coalescence Transchemistry Recombination vs fragmentation Spectral coalescence
Robust ratios for competing channels PLB 472 p. 243 2000
Collision energy dependence in ALCOR 100 AGS Stopped per cent of baryons SPS 10 RHIC LHC leading rapidity 0 2 4 6 8 10
Collision energy dependence in ALCOR 200 AGS Newly produced light dN/dy 100 SPS RHIC LHC leading rapidity 0 2 4 6 8 10
Collision energy dependence in ALCOR 0.2 AGS 0.1 K+ / pi+ ratio SPS RHIC LHC leading rapidity 0 2 4 6 8 10
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 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 h=const 0
Massless kinematics Tsallis rule
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
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
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
Coalescence cross section a: Bohr radius in Coulomb potential Pick-up reaction in non-relativistic potential
Limiting temperature with Tsallis distribution ( with A. Peshier, Giessen ) hep-ph/0506132 Massless particles, d-dim. momenta, N-fold d <X(E)> TE c ; = T = E / d c H j=1 E – j T c N For N 2: Tsallis partons Hagedorn hadrons
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 hep-ph/0608234 Fig.2 8× 32 ³
Lattice QCD eos + fit Biro et.al. Peshier 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!
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!
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? 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 31:711,2005 w ( m ) is not constant zero probability for zero mass Conditions:
