Quark Coalescence and Hadron Statistics

• Valence quark model of hadrons

• Quark recombination

• Hadronization dynamics

• Hadron statistics

Quark Coalescence and Hadron Statistics

T.S.Biró (RMKI Budapest, Univ. Giessen)

School of Collective Dynamics in High-Energy Collisions, Berkeley, 19-26 May, 2005

Collaborators• József Zimányi, KFKI RMKI Budapest• Péter Lévai, KFKI• Tamás Csörgő, KFKI• Berndt Müller, Duke Univ. NC USA• Christoph Traxler• Gábor Purcsel, KFKI• Antal Jakovác, BMGE (TU) Budapest• Géza Györgyi, ELTE Budapest• Zsolt Schram, DE Debrecen

Valence Quark Model of Hadrons

1. Mass formulas (flavor dependence)

2. Spin dependence

3. Alternatives: partons, strings, ...

Basic cross sections e p : e = 3 : 2


Quark masses: M = (u,d) m, (s) ms

Quark hypercharges: Y = (u,d) 1/3, (s) -2/3

Naive quark mass formula:

M = M - M Y0 1

with M = (2m + m ) / 3 and M = m - mss0 1

M ≠ 0 breaks SU(3) flavor symmetry1


More terms: M = a + bY + c T(T+1) + d Y2

Test on baryon decuplet masses with last 2 terms linear (like x + y Y)

(3/2, 1): 15c/4 + d = x + y

(1/2,-1): 3c/4 + d = x - y

( 0,-2): 4d = x - 2y

Solution: x = 2c, y = 3c/2 d = -c/4.

*


M = a + bY + c (T(T+1) - Y / 4)2

Gell-Mann Okubo mass formula:

N (qqq: ½, +1) a + b + c / 2 (qqs: 1, 0) a + 2 c (qqs: 0, 0) a (qss: ½, -1) a – b + c / 2

Check 3M( ) + M( ) = 2M(N) + 2M( ) : difference 8 MeV/ptl.


M = a + bY + c (T(T+1) - Y / 4) + d S(S+1)2

Gürsey - Radicati mass formula:

SU(6) quark model: (flavor SU(3), spin SU(2))

1. quark: [6] = [3,2]

2. meson: (3,2)×(3,2) = (1,1)+(8,1)+(1,3)+(8,3)

3. baryon: 6×6×6 = 20+56+70+70 (only 56 is color singlet)


Fit to 56-plet masses: a = 1066.6 MeV, b = -196.1 MeV c = 38.8 MeV, d = 65.3 MeV

More success: magnetic moments

No hint for formation probability

Linear dominance!

additive mass hadronization

Quark Recombination

1. (Non)Linear coalescence (Bialas, ZLB)

2. ALCOR (Zimanyi, Levai, Biro)

3. Distributed mass quarks (ZLB)

hep-ph/9904501

PLB347:6,1995PLB472:243,2000

nucl-th/0502060

Quark Recombination

Linear vs nonlinear coalescence

meson[ij] = a q[i] q[j]

baryon[ijk] = b q[i] q[j] q[k]

With lowest multiplets: quarks are redistributed in a few mesons and baryons # counting all flavors

q = + K + 3N + 2Y + Xq = + K + 3N + 2Y + Xs = + K + Y + 2X + 3s = + K + Y + 2X + 3

coalesced numbers

N = C b q 3 3

qN

et cetera

Quark Recombination

Q = b qqA simple example: q, q , N, N

_

_ _

q = C Q Q + 3C Q = + 3 NN

3

q = C Q Q + 3C Q = + 3 NN

3

_ __ _

N / C * N / C = ( / C )3NN

_

(r ) = (q - ) ( q - ) with r = (3C ) / CN3 2/3_

Quark Recombination

(q + q ) / 2

(q q )_

qq_

small r limit:

N = r q / 3(q – q)3 3__ _

N = (q – q)/3 + N

= q - 3 N

_ _

__

Features: N ≠ …q , ≠ … q q3

q > q_RHSLHS

_

Quark Recombination

Note: ≠ possible due to S ≠ S while s = s

Key: b is sensitive to the q – q inbalance!s

ratios of ratios and their powers are testable!d(K) = K/K = 1.80 ± 0.2d(Y) = (Y/Y) / (N/N) = 1.9 ± 0.3d(X) = (X/X) / (N/N) = 1.89 ± 0.15d() = (/) / (N/N) = 1.76 ±0.15

CERN SPS data

1/2 1/2

1/31/3

Quark Recombination

ALCOR: 2Nflavor parameters = Nf che-mical potentials + Nffugacities

this is just not grandcanonical, but explicitin the particle numbers.

Quark Recombination

Distributed mass quarks form hadrons.

1.) assume hadronic wave packet is narrow in relative momentum p(a) = p(b) = p/22.) mass is nearly additive m = m(a)+m(b)3.) coalescence convolves phase space densities

F(m,p) = dm dm (m-m -m ) f(m ,p/2) f(m ,p/2)∫ a a a bbb∫0 0

The product f(x) f(m-x) is maximal at x = m /2 .

nucl-th/0502060

Quark Recombination

ln (m) = - (a/T) (a/m + m/a )½

f(m,p) = (m) exp ( - E(m,p) / T )

Quark Recombinationpion

Quark Recombinationproton

Quark Recombinationratio

Hadronization dynamics

1. Parton kinetics + recombination (MFBN)

2. Colored molecular dynamics (TBM)

3. Color confinement as 1/density (ZBL)

4. Multpilicative noise in quark matter (JB)

5. Non-extensive Boltzmann equation (BP)

PRC59:1620, 1999

JPG27:439, 2001

PRL94:132302, 2005

hep-ph/0503204

Colored Molecular Dynamics

g



Color confinement as 1/density

reaction A + B C

conserved: N + N = N (0), N + N = N (0)A A B BCC

rate eq.: N = -R ( N - N )(N - N )C C -+

resulted number:

C

N () = N N (1-K) / (N -KN )C ++ --

with K = exp(r (N - N )), r = R(t)dt+ - ∫


If A and B colored, C not: N () = N (0) = N (0) = N

limit: r(N - N ) 0, K exp linearized

N () = r N / ( 1 + rN ) (r is required!)

1-dim exp : r = v/V t ln ( t / t ) 3-dim exp : r = v/3V t ( 1 - (t /t ) )

conclusion: ~ t ~ 1 / density for all quarks to be hadronized

C A B 0

0

+ -

000

000 1

1 3

3

02


Additive and multiplicative noise

1. Langevin

p = - p = G = F

2C 2B 2D

2. Fokker Planck

∂f∂t

∂∂p

∂∂p

= ( K f ) - ( K f )1

2

2 2

K = F – Gp

K = D – 2Bp + Cp22

1

c c c

Equivalent descriptions: AJ+TSB, PRL 94, 2005

Exact stationary distribution:

f = f (D/K ) exp(- atan( ) ) 0

v2 D – Bp

p

with v = 1 + G/2C

= GB/C – F

= DC – B22

For F = 0 characteristic scale: p = D/C.c2

power

exponent

(small or large) parameter

Exact stationary distribution for F = 0, B = 0:

f = f ( 1 + ) 0

-(1+G/2C)2

D

C p

With E = p / 2m this is a Tsallis distribution!

f = f ( 1 + (q-1) ) 0

E

2

T

q

1 – q

Tsallis index: q = 1 + 2C / GTemperature: T = D / mG

Limits of the Tsallis distribution:

p p : Gauss

p p : Power-law

c

f ~ exp( - Gp /2D )

f ~ ( p / p )

2

c

-2vc

E E :

E E :

c f ~ exp( - E / T )

f ~ (E / E )-v

c c

Relation between slope, inflection and power !!

v = 1 + E / Tc

Energy distribution limits:

Stationary distributions

For F=0, B=0 the Tsallis distribution is the exact stationary solution

Gamma:p = 0.1 GeV F ≠ 0

Gauss: p = ∞

Zero:p = 10 GeV

B = D/C

Power: p = 1 GeV

F ≠ 0

c

c

c

c

2

Generalization

p = z - G(E) ∂E∂p

. < z(t) > = 0

< z(t)z(t') > = 2 D(E) (t-t')

In the Fokker – Planck equation:K (p) = D(E)

K (p) -G(E) ∂E∂p1

2

Stationary distribution:

f(p) = exp - G(E)∫ D(E)dE

D(E)A ( )

=

TSB+GGy+AJ+GP, JPG31, 2005

GeneralizationStationary distribution:

f(p) = A exp - ∫T(E)dE

1) Gibbs: T(E) = T exp(-E/T)

2) Tsallis: T(E) = T/q + (1-1/q) E

( 1 + (q-1) E / T) -q /(q-1)

( )

Inverse logarithmic slope temperature

T(E)1

= ln f (E)ddE

T (E) = D(E)

G(E) + D'(E)

T = D(0) / G(0) Gibbs

T = D(E) / G(E) Einstein

slope

c

T

T

E E

TEinstein

Gibbs

Walton – Rafelski ?

TGibbsT

EinsteincE

111= +

Special case: both D(E) and G(E) are linear

Fluctuation Dissipation theorem

D (E) = 1

f(E)

with f(E) stationary distribution

∫E

∞

G (x) f(x) dxij ij

D (E) = T(E)ij

G (E) + ij

D' (E) ij ( )

(Hamiltonian eom does not change energy E!)

p = -G E + z i ijij.

Fluctuation Dissipation theorem

particular cases ( for constant G ):

D = Tij

G ij

D (E) = T + (q-1) Eij

G ij

( )

Gibbs:

Tsallis:

T. S. Bíró and G. Purcsel (University of Giessen, KFKI RMKI Budapest)

Non-Extensive Boltzmann Equation

• Non-extensive thermodynamics

• 2-body Boltzmann Equation + non-ext. rules

• Unconventional distributions

• H-theorem and non-extensive entropy

• Numerical simulation

hep-ph/0503204

Non-extensive thermodynamics

f = f f12 1 2 statistical independence

E = h ( E , E )12 1 2

non-extensive additionrule

non-extensive addition rules for energy, entropy, etc.

h ( x, y ) ≠ x + y

Sober addition rules

associativity:

h ( h ( x, y ) , z ) = h ( x, h ( y, z ) )

1,22,33

1

general math. solution: maps it to additivity

X ( h ) = X ( x ) + X ( y )

X( t ) is a strict monotonic, continous real function, X(0) = 0

Boltzmann equation

∫ 4 1 2

f = w ( f f - f f )1 1234

234

1

2

3 4 w = M ( p + p - p - p )

1234 1234

1 2 3 4 ( h( E , E ) - h( E , E ) )

3

2

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

Consequences

• canonical equilibrium: f ~ exp ( - X( E ) / T )

• 2-body collisions: X(E ) + X(E ) = X(E ) + X(E )

• non-extensive entropy density: s = df X ( - ln f )

• H-theorem for X( S ) = - f ln f tot

-1

∫∫

1 2 3 4

rule additive equilibrium entropy name

h ( x, y ) X ( E ) f ( E ) s [ f ] general

x + y E exp( - E / T) - f ln f Gibbs

x + y + a xy ln(1+aE) (1+aE) (f - f)/(q-1) Tsallis-1/aT q

( x + y ) E exp( - E / T) … Lévyqqqq 1/q

x y ln E E f Rényiq

1- q

1 - 1/ (1-q)

T h e r m o d y n a m i c s e sT h e r m o d y n a m i c s e s

a1

S o m e m o r e . . .

k-deformed statistics (G.Kaniadakis),

X( E ) = (T / k) asinh ( kE / T ), h( x, y ) = x sqrt( 1 + ( ky / T ) ) + y sqrt( 1 + ( kx / T ) )

s[ f ] = ( f /(1-k) - f /(1+k) ) / 2k

also gives a power-law tail: ~ (2kE/T)

1-k 1+k

2 2

-1/k

Cascade simulation

• Momenta and energies of N “test” particles

• Microevent: new random momenta, so that X(E1) + X(E2) = X(E1’) + X(E2’)

• Relative angle rejection or acceptance

• Initially momentum spheres, Lorentz-boosted

• Distribution of E is followed and plotted logarithmically

Movie: Boltzmann a = 0 proton y=2

Movie: Boltzmann a = 0 proton y=2

Movie: Boltzmann a = 0 pion y=2

Movie: Boltzmann a = 0 pion y=2

Snapshot: Tsallis a = -0.2







Non-extensive Boltzmann eq.

BG TS (a = 2)

Tsallis distribution

Hadron statistics

1. Gibbs thermodynamics: exponential

2. Non-extensive thermodynamics: power-law

3. Collective flow effects: scaling breakdown

4. low pt and high pt: connected?

hep-ph/0409157JPG31:1, 2005

Particle spectra and Eq. Of State

(2h) d N 3

V dk3= d , k) f(/T)

3

∫

Spectrum Spectral function thermodynamics

Gibbs

Tsallis

. . .

Peak: particle

bgd.: field

Shifted peak: quasiparticle

Quasiparticle approximation: k

In this case: ~ f ( / T)k

d Ndk 3

3

T : parameter of environment

= b F ( k/b ) : result of interactionsk

Modified quark matter dispersion: change F( x )

Modified thermodynamics: change f( x )

Experimental spectra: pp

mesons, 30 GeV, p -tail v = 10.1 ± 0.3

pions, 30 GeV, m -tail v = 9.8 ± 0.1

pions, 540 GeV, m -tail v = 8.1 ± 0.1

quarkonia, 1.8 TeV, m -tail v = 7.7 ± 0.4

t

t

t

t

Gazdiczki + Gorenstein (hep-ph / 0103010)

tt

tt

tt

tt

Experimental spectra: AuAu

pi, K, p, 200 GeV, m -scaling (i.e. E = m )

v = 16.3

(E = 2.71 GeV, T = 177 MeV)

t

t

t

t

Schaffner-Bielich, McLerran, Kharezeev (NPA 705, 494, 2002)

t t

c

Experimental spectra: cosmic rays

before knee, m -scaling (i.e. E = m )

v = 5.65 (E = 0.50 GeV, T = 107 MeV)

in ankle,

v = 5.50 (E = 0.48 GeV, T = 107 MeV)

t

t

t

Ch. Beck cond-mat / 0301354

t t

c

c

Experimental spectra: e-beam

integral over longitudinal momenta

TASSO 14 GeV v = 51 (E = 6.6 GeV)

TASSO 34 GeV v = 9.16 (E = 0.94 GeV)

DELPHI 91 GeV v = 5.50 (E = 0.56 GeV)

DELPHI 161 GeV v = 5.65 (E = 0.51 GeV)

t

t

t

t

Bediaga et.al. hep-ph / 9905255

c

c

c

c

Gaussian fit to parton distribution: < p > = D / G = 1 ... 1.5 GeV Power-tail in e+e- experiment (ZEUS): v = 5.8 ± 0.5 -> G / C = 9.6 ± 1

Derived inclination point at

p = √ D / C = 3 ... 4 GeV.

t2

c

Test v = 1 + E / T

☺

c

pions

RHIC Au Au heavy ion collision 200 GeV

q = 1.11727 T = 118 ± 9 MeV

v = 9.527 ± 0.181E = 1.008 ± 0.0973 GeV

T = 364 ± 18 MeV

from AuAu at 200 GeV (PHENIX) 0

c

0 2 4 6 8 10 12 14 p (GeV) t

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E-0

d 2

2p dp dyt t

min. bias

Central 5% transverse spectrum 0

Central 5% transverse slope 0

D(E) T(E) =

G(E) + D (E) '

All central transverse slopes

Flow

All central transverse slopes

Transverse flow correction

E = u p = (m cosh(y-) - v p cos(-) )

Energy in flowing cell:

Most detected: forward flying (blue shiftedblue shifted) at = y, = .

E = (m - v p )

TT

TT

Spectrum ~ ∫d d f(E)

Transverse flow corrected spectra

forward flow !

E/N with Tsallis distributionMassless particles, d-dim. momenta, one ptl. average

E = Ec v – d – 1

d=

1 – d (q – 1)

d T

(Ito: =0)

QGP

E =∫ dE E (1 + E / E )

dc

∫ dE E (1 + E / E )d-1

c

-v

-v

E/N with Tsallis distributionMassive particles, 2-dim. momenta, one ptl. average energy

E = a (2T + bm /(m+T) )

hadrons

2

with 1/a = 3 – 2q, b = 4q - 11q + 82

(BG: a=1, b=1)E > BG case for q > 1E > BG case for q > 1

_

_

Average transverse momentumR.Witt

Average transverse momentum

Limiting temperature with Tsallis distribution

<E>N

=E – j T

TE T = E / d

Hagedorn

;c

cj=1

d

cH

Massless particles, d-dim. momenta, N-fold

For N 2: Tsallis partons Hagedorn hadrons

( with A. Peshier, Giessen )

Summary

• Basis of coalescence: valence quark model• ALCOR: microcanonical nonlinear, non-eq.• Mol.dyn.: nice spectra, but too slow• Power-tailed stationary distributions from• a) multiplicative noise• b) Non-extensive Boltzmann-Equation• Simple relation: v = 1 + E / T.• Limiting temperature, m-scaling of exp.values

c

Documents

Quark Coalescence and Hadron Statistics