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• 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
Valence Quark Model of Hadrons
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
Valence Quark Model of Hadrons
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.
*
Valence Quark Model of Hadrons
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.
Valence Quark Model of Hadrons
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)
Valence Quark Model of Hadrons
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
Colored Molecular Dynamics
Colored Molecular Dynamics
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+ - ∫
Color confinement as 1/density
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
Color confinement as 1/density
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
Snapshot: Tsallis a = -0.2
Snapshot: Tsallis a = -0.2
Snapshot: Tsallis a = -0.2
Snapshot: Tsallis a = -0.2
Snapshot: Tsallis a = -0.2
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
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