Particle spectra

1
Power-law tailed spectra from equilibrium
T.S.Biró and G.Purcsel (KFKI RMKI,
Giessen) A.Jakovác (BMGE), G.Györgyi (ELTE),
Zs.Schram (DE)

• Particle spectra
• Non-extensive Boltzmann eq. (NEBE)
• Particular distributions
• Parton cascade demo
• Lattice strategy

Quark Matter, Budapest, 3-8 August, 2005
2
Hadron statistics
hep-ph/0409157 JPG311, 2005

• Gibbs thermodynamics exponential
• Non-extensive thermodynamics power-law (pQCD
  jets are angle correlated)
• Collective flow effects scaling breakdown
• low pt and high pt connected?

3
Particle spectra and Eq. of State
????
3
3
(2?h) d N
?
d? ????????, k) f(?/T)
V dk
3
?
?
Spectrum
Spectral function
thermodynamics
??????????????
Gibbs Tsallis . . .
Peak particle bgd. field
Shifted peak quasiparticle
4
Experimental spectra pp
t
t

• 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
tt

tt
tt
t
tt
Gazdiczki Gorenstein (hep-ph / 0103010)
5
Experimental spectra AuAu
t
t

• pi, K, p, 200 GeV, m -scaling (i.e. E m
  )
• v 16.3
• (E 2.71 GeV, T 177 MeV)

t
t
t

c
t
Schaffner-Bielich, McLerran, Kharezeev
(NPA 705,
494, 2002)
6
RHIC Au Au heavy ion collision 200 GeV
0
? from AuAu at 200 GeV (PHENIX)
pions
1E-0
min. bias
1E-1
1E-2
v 9.527 0.181 E 1.008 0.0973 GeV
1E-3
2
c
d ?
2?p dp dy
1E-4
q 1.11727 T 118 9 MeV
t
t
1E-5
1E-6
1E-7
1E-8
T 364 18 MeV
1E-9
0 2 4 6 8 10 12 14
p (GeV)
t
7
0
Central 5 ? transverse spectrum
8
Experimental spectra cosmic rays
t
t

• 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

c
c
Ch. Beck cond-mat / 0301354
9
Experimental spectra e-beam
t
t

• 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

c
c
c
t
c
Bediaga et.al. hep-ph / 9905255
10
General Langevin problem
TSBGGyAJGP, JPG31, 2005
lt z(t) gt 0
.
?E
p z - G(E)
lt z(t)z(t') gt 2 D(E) ?(t-t')
?p
In the Fokker Planck equation
K (p) D(E)
2
?E

K (p) -G(E)
1
?p
Stationary distribution
(
)
dE
A
?
f(p) exp - G(E)
D(E)
D(E)
11
General inverse slope
Stationary distribution
(
)
dE
?
f(p) A exp -
T(E)
1) Gibbs T(E) T ? exp(-E/T)
2) Tsallis T(E) T/q (1-1/q) E ?
-q /(q-1)
( 1 (q-1) E / T)
T( T ) T a fixed point of the sliding
slope
12
Fluctuation Dissipation
theorem
)
(
D (E) T(E)
G (E)
D' (E)
ij
ij
ij
(Hamiltonian eom does not change energy E!)
.
p -G ? E z
i
j
ij
i
8
1
?
G (x) f(x) dx
D (E)
ij
ij
f(E)
E
with f(E) stationary distribution
13
Test v 1 E / T
c
Gaussian fit to parton distribution
lt p gt D / G 1 ... 1.5 GeV
Power-tail in ee- experiment (ZEUS)
v 5.8 0.5 -gt G / C 9.6 1
Derived inclination point at p
v D / C 3 ... 4 GeV.
2
t
?
c
14
0
Central 5 ? transverse slope
D(E)
T(E)
'
G(E) D (E)
15
All central transverse slopes
All central transverse slopes
Flow
16
Transverse flow correction
Energy in flowing cell
?
E u p ??(m cosh(y-?) - v p cos(?-?) )
?
T
T
Most detected forward flying (blue shifted) at
?? y, ?? ?.
E ? (m - v p )

T
T
Spectrum ?d? d? f(E)
17
Transverse flow corrected slopes
forward blast wave!
18
Average transverse momentum
R.Witt
19
Average transverse momentum
20
Hadronization dynamics

• Parton kinetics recombination (MFBN)
• Colored molecular dynamics (TBM)
• Color confinement as 1/density (ZBL)
• Multpilicative noise in quark matter (JB)
• Non-extensive Boltzmann equation (BP)

PRC591620, 1999
JPG27439, 2001
PRL94132302, 2005
hep-ph/0503204
21
Colored Molecular Dynamics
22
Non-Extensive Boltzmann Equation
hep-ph/0503204

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

• Non-extensive thermodynamics
• 2-body Boltzmann Equation non-ext. rules
• Unconventional distributions quasi-energy
• H-theorem and non-extensive entropy
• Principles of the numerical simulation

23
Generalized sum
f f f
statistical independence
1
12
2
non-extensive addition rule
E h ( E , E )
1
12
2
non-extensive addition rules for energy, entropy,
etc. h ( x, y )
?????x y
24
Sober addition rules
associativity
3
1
h ( h ( x, y ) , z ) h ( x, h ( y, z ) )
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
25
Boltzmann equation
?
?
f w ( f f - f f )
4
1
2
3
1
1234
234

2
w M ? ( p p - p - p )
?
3
4
1234
1234
2
1

? ( h( E , E ) - h( E , E ) )
2
3
4
1
26
Consequences
1
2
3
4
-1
s
?
tot
s
27
NEBE control sums
Non-extensive entropy
28
Entropies
Boltzmann
Kaniadakis (NLBE)
Tsallis entropy
Rényi entropy
NEBE
29
T h e r m o d y n a m i c s e s
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
-1/aT
q
1
x y a xy ln(1aE) (1aE)
(f - f)/(q-1) Tsallis
a
q 1 - aT
b
b
1/b
incomplete gamma fct.
b
b
( x y ) (aE) / a
exp(-(aE) /aT) Lévy
q
1
- 1/ aT
1
a x y ln aE (aE)
ln ?f Rényi
a
1- q
q 1 - aT
30
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
hconst
0
31
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

32
Movie Tsallis a 0 y2
33
Movie Tsallis a 1 y2
34
Movie Tsallis a -0.25, y2
35
Snapshot Tsallis a -0.2
36
Snapshot Tsallis a -0.2
37
Snapshot Tsallis a -0.2
38
Snapshot Tsallis a -0.2
39
Snapshot Tsallis a -0.2
40
Snapshot Tsallis a -0.2
41
Snapshot Tsallis a -0.2
42
Tsallis distribution
43
Limiting temperature with Tsallis distribution
( with A. Peshier, Giessen )
Massless particles, d-dim. momenta, N-fold
Hagedorn
d
ltX(E)gt
TE
?
c


T E / d
c
H
j1
E j T
c
N
For N ? 2 Tsallis partons ? Hagedorn hadrons
44
Lattice Monte Carlo with fluctuating
(Gamma-distributed) temperature
(Research with Zs. Schramm, Debrecen)
G-distributed a /a ratio Average effective
action Large system limit Metropolis strategy
s
t
45
q 1 1 / c
Canonical distribution POWER LAW
TAILED
-(c1)
f ? exp( - X / T ) ( 1 E / cT )
This equals to Gamma distributed Gibbs factors
-(c1)
c
1
?
-t
-xt/c
( 1 x / c )
dt t e e
?(c1)
Interpretations fluctuating
temperature,
energy imbalance,
multiplicative additive noise,
. . .
46
Gamma distribution
max 1 1/c, mean 1, spread 1
/ v c
47
Lattice theory
Expectation values of observables
-S(t,U)
DU dt w (t) e t A(U)
?
?
v
c
?A?
-S(t,U)
DU dt w (t) e
?
?
c
Action S(t,U) a(U) t b(U) / t
t a / a asymmetry parameter
t
s
48
Lattice theory
Effective calculation
-S (U,v)
DU e A(U)
?
eff
?A?
-S (U,0)
?
DU e
eff
v0 Polyakov line, v1 ss Plaquettes,
v-1 ts Plaquettes
49
Lattice theory effective action
8
c
cv-1
c
-(ac)t - b/t
?
S
dt t e
- ln
G(c)
eff
0

Plaquette sums
space-space a ? (1 Re tr P ss)
space-time b ? (1 Re tr P ts)
Evaluation methods

• exact analytical
• saddle point
• numerical (Gauss-Laguerre)

50
Lattice theory effective action
(cv)/2
c

c

b
(
)
(
)
S
2K (2 ? b(ac) )
- ln
ac
cv
eff
G(c)
Asymptotics

• large a,b finite c 2 ? ab
• large a,b,c and ?a-b?ltlt (ab) a b

51
Order parameter Re Polyakov
52
Strong weak coupling transition
action MC ab, TS lt at b/t gt
53
Strong weak coupling transition
Polyakov line t-independent both for MC and TS
54
Summary

• Power-law distributions are predicted by
  non-extensive thermodynamics
• Multiplicative noise ? sliding slope T(E)
• NEBE also leads to power-law tails
• Tsallis distribution can be modelled by Gamma
  distributed inv. temperature
• Non-extensive statistical properties are worth to
  study