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Pion and kaon spectra from distributed mass quark matter

Károly Ürmössy and Tamás S. Bíró

KFKI Res.Inst.Part.Nucl.Phys. Budapest

• Hadronization by coalescence

• Quasiparticle mass and QCD eos

• Mass gap estimates due to Markov inequality

• Pion p spectra directly and from rho decay

Further collaborators

• József Zimányi

• Péter Lévai

• Péter Ván

• Gábor Purcsel

hep-ph / 0607079, 0606076, 0605274, 0612085

Hadronization by

coalescence

Entropy vs lattice eos (PLB 650, 193, 2007)

Lattice QCD eos: normalized pressure vs. temperature

Aoki, Fodor, Katz, SzaboJHEP 0601:089, 2006

Boltzmann mixtures

Tci

i

ii

i

Tci

ii

ii

eTVT

cN

TS

Vp

eTV

/

/

),(

,

),(

Boyle-Mariotte law

i i

TcTiTeff

effi

ic

iiii

i

eNSS

pVTNNT

/

1

Perfect fluid expands so that locally Seff is constant.

Can Neff and T be reduced by that?

N / S = pV / TS effective number / entropy

4

1

S

N

m

T

S

N

Cooling vs expansion (S = const.)

Number reduction (coalescence)

What do we conclude?

• Adiabatic cooling with number reduction to its 1 / 2 . . . 1 / 3

• Most of the reduction and cooling happens relatively short, the volume grows with a factor of 3 . . . 30

• N / S is constant for an ideal gas eos of type p ~ Tª

• Lower pressure can be achieved by higher mass ideal gas

Is high-T quark matter

perturbative?

Thermal probability of Q² values for massless partons

22 TQx

)(2)()( 22

13

641

2

2

xKxxKxPT

Q

< x >

On the average yes, but

watch for IR unsafe

quantities!

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)

Quasiparticle mass and

QCD eos

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)


hep-ph/0608234 Fig.2 8 × 32 ³


+ Gribov-Zwanziger dispersion

+ 1/m² ideal

constant m ideal

Mass dependence of the

relativistic pressure

xKxx 22

21)(

Boltzmann vs. Bose and Fermi

Fodor et.al.

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!

Mass gap estimates due

to Markov inequality

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

Particular inequalities

Chebyshev

Markov

Particular inequalities

Minimize the log of this upper bound in λ,

get the best estimate!

Chernoff

General Markov inequality

Extreme value probability estimate (upper bound)

with variable substitutionOriginal Markov: g=id, f=id

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


Upper bound for the low-mass part of the mass distribution.

I M D F


T and µ dependent w(m) requires mean field term,

but this is cancelled in (e+p) eos data!

g( ) = ( )

Boltzmann scaling functions


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

Hadron spectra from quarks

Tsallis fit to hadron spectra

Fit parameters at large p

Coalescence from Tsallis

distributed quark matter

What if a gap shows in the mass-distributions?

• Both distributions reproduce pion or kaon spectra fairly well at RHIC energies but the one without a gap (magenta) can not be fitted to lattice-QCD data.

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

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

Summary

• Spectral coalescence of Tsallis distributions

• Distributed mass quasi quarks from QCD eos

• Restrictions from inequalities on the pressure

• Pion spectra feel the power (q-1)

• Antiproton spectra feel the transverse flow

• Kaon spectra are insensitive but to T

• What about hyperons / antihyperons?