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, 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) m s Quark hypercharges: Y = (u,d) 1/3, (s) -2/3 Naive quark mass formula: M = M - M Y 01 with M = (2m + m ) / 3 and M = m - m ss01 M ≠ 0 breaks SU(3) flavor symmetry 1

Valence Quark Model of Hadrons More terms: M = a + bY + c T(T+1) + d Y 2 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 = ( only 56 is color singlet )

Valence Quark Model of Hadrons Fit to 56-plet masses: a = MeV, b = 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/ PLB347:6,1995 PLB472:243,2000 nucl-th/

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 + X s =  + K + Y + 2X + 3  coalesced numbers N = C b q 33 qN et cetera

Quark Recombination Q = b q q A simple example: q, q  , N, N _ _ _ q = C Q Q + 3C Q =  + 3 N N 3  N 3  ____ N / C * N / C = (  / C ) 3 NN _  (r  ) = (q -  ) ( q -  ) with r = (3C ) / C  N 32/3 _

Quark Recombination  (q + q ) / 2 (q q ) _ qq _ small r limit: N = r q / 3(q – q) 33 _ _ _ N = (q – q)/3 + N  = q - 3 N _ _ _ _ Features: N ≠ …q,  ≠ … q q 3 q > q _ RHS LHS _

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 ± 0.2 d(Y) = (Y/Y) / (N/N) = 1.9 ± 0.3 d(X) = (X/X) / (N/N) = 1.89 ± 0.15 d(  ) = (  /  ) / (N/N) = 1.76 ±0.15 CERN SPS data 1/2 1/3

Quark Recombination ALCOR: 2Nflavor parameters = Nf che- mical potentials + Nf fugacities this is just not grand canonical, but explicit in 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/2 2.) 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) ∫ aaabbb ∫ 00  The product f(x) f(m-x) is maximal at x = m /2. nucl-th/

Quark Recombination ln  (m) = - (a/T) (a/m + m/a ) ½ f(m,p) =  (m) exp ( - E(m,p) / T )

Quark Recombination pion

Quark Recombination proton

Quark Recombination ratio

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/

Colored Molecular Dynamics g

Color confinement as 1/density reaction A + B  C conserved: N + N = N (0), N + N = N (0) AABBCC rate eq.: N = -R ( N - N )(N - N ) CC - +  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 CAB

Color confinement as 1/density

Additive and multiplicative noise 1. Langevin p =  -  p  = G  = F  2C  2B  2D 2. Fokker Planck ∂f ∂t ∂ ∂p ∂ = ( K f ) - ( K f ) K = F – Gp K = D – 2Bp + Cp c c c Equivalent descriptions: AJ+TSB, PRL 94, 2005

Exact stationary distribution: f = f (D/K ) exp(- atan( ) )  0 v 2  D – Bp  p with v = 1 + G/2C  = GB/C – F  = DC – B 22 For F = 0 characteristic scale: p = D/C. c 2 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 / G Temperature: 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 -2v c

E  E : E  E : c f ~ exp( - E / T ) f ~ (E / E ) -v c c Relation between slope, inflection and power !! v = 1 + E / T c 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. = 0 = 2 D(E)  (t-t') In the Fokker – Planck equation: K (p) = D(E) K (p) -G(E) ∂E ∂p 1 2 Stationary distribution: f(p) = exp - G(E) ∫ D(E) dE D(E) A () = TSB+GGy+AJ+GP, JPG31, 2005

Generalization Stationary 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) d dE 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 T Einstein Gibbs Walton – Rafelski ? T Gibbs T Einstein c E 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) dx ij D (E) = T(E) ij G (E) + ij D' (E) ij ( ) ( Hamiltonian eom does not change energy E!) p = -G  E + z i ij ij.

Fluctuation Dissipation theorem particular cases ( for constant G ): D = T ij G D (E) = T + (q-1) E ij G ( ) 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/

Non-extensive thermodynamics f = f f statistical independence E = h ( E, E ) non-extensive addition rule 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 ) ) 1,2 2,3 3 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 ∫ 412  f = w ( f f - f f ) w = M  ( p + p - p - p ) 1234   ( h( E, E ) - h( E, E ) ) 3 2

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

Recipe for binary events New energies: E = h_inv( h(E1,E2), 0 ) E3 = E * RND(0,1) reject if Y(E3) > Y(E) * RND(0,1) E4 = h_inv ( E, E3 ) New momenta: P[i] = p1[i] + p2[i], length P p3 = sqrt(E3*E3-m3*m3), p4 = sqrt(E4*E4-m4*m4) reject if P not in ( |p3-p4|, p3+p4) p3_x = (p3*p3 – p4*p4 + P*P)/(2*P), p3_t from p3 in-plane vec a[i]=P[i]/P, orthogonal in-plane vec b[i], p3[i] = p3_x * a[i] + p3_t * b[i], p4[i] = P[i] – p3[i].

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 ∫ ∫ 1234

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/aTq ( x + y ) E exp( - E / T) … Lévy q q qq 1/q x y ln E E f Rényi q 1- q 1 - 1/ (1-q) T h e r m o d y n a m i c s e s a 1

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-k1+k 22 -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 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/ JPG31:1, 2005

Particle spectra and Eq. Of State (2  h) d N 3 V dk 3 = d  , k) f(  /T)   3     Spectrum Spectral functionthermodynamics Gibbs Tsallis... Peak: particle bgd.: field Shifted peak: quasiparticle

Quasiparticle approximation:  k In this case: ~ f (  / T) k d N dk 3 3 T : parameter of environment  = b F ( k/b ) : result of interactions k 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 / ) t t t t

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 / tt 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 / c c c c

Gaussian fit to parton distribution: = D / G = 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 = GeV. t 2 c Test v = 1 + E / T ☺ c

pions RHIC Au Au heavy ion collision 200 GeV q = T = 118 ± 9 MeV v = ± E = ± GeV T = 364 ± 18 MeV  from AuAu at 200 GeV (PHENIX) 0 c 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 2  p dp dy t 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:   blue shifted Most detected: forward flying (blue shifted) at  = y,  = . E =  (m - v p ) T T T T Spectrum ~ ∫d  d  f(E)

Transverse flow corrected spectra forward flow !

E/N with Tsallis distribution Massless particles, d-dim. momenta, one ptl. average E = E c v – d – 1 d = 1 – d (q – 1) d T (Ito: =0) QGP E = ∫ dE E (1 + E / E ) d c ∫ d-1 c -v

E/N with Tsallis distribution Massive particles, 2-dim. momenta, one ptl. average energy E= a (2T + bm /(m+T) ) hadrons 2 with 1/a = 3 – 2q, b = 4q - 11q (BG: a=1, b=1) E > BG case for q > 1 _ _

Average transverse momentum R.Witt

Average transverse momentum

Limiting temperature with Tsallis distribution N = E – j T TE T = E / d Hagedorn  ; 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 )

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