Transcript Prezentacja programu PowerPoint
Multiplicity fluctuations in high energy hadronic and nuclear collisions
M. Rybczyński
(a)
, G. Wilk
(b)
and Z. Włodarczyk
(a)
(a) Świętokrzyska Academy, Kielce, Poland (b) Soltan Institute for Nuclear Studies, Warsaw, Poland
XIII ISVHECRI – NESTOR Institute Pylos, Grece, 6-12 September 2004
(*) What counts most in the cosmic ray experiments in which such showers are observed?
(*) Cross-section of elementary interactions (or, rather, of hA and AA interactions):
(E) (*) Inelasticity K(E) defined as fraction of the actual energy used to produce secondaries and therefore lost for the subsequent interaction: Popular some time ago (
~1/
,K) have been exchanged for models. However, from different analysis of available data the picture seems to emerge that all successful models provide (almost...) the same (
,K ).
(*) We shall add here that important are also possible fluctuations in these variables and argue that they can be deduced from the measurements of multiplicities
(Problem that experimentally (
,K) are interrelated will not be discussed, see: SWWW, JPhG18 (1992) 1281)
Historical example: (*) observation of deviation from the expected exponential behaviour (*) successfully intrepreted (*) in terms of cross-section fluctuations :
2 2 2 0 .
2
(*) can be also fitted by:
dN dT
const
exp
T
dN dT
const
1 ( 1
q
)
T
1 1
q
;
q
1 .
3
Depth distributions of starting points of cascades in Pamir lead chamber Cosmic ray experiment (WW, NPB (Proc.Suppl.) A75 (1999) 191 (*) WW, PRD50 (1994) 2318 (*) immediate conjecture: q
fluctuations present in the system
q – measure of fluctuations (*) Parameter q is known in the literature as measure of nonextensivity in the Tsallis statistics based on Tsallis entropy (a) : S q = -
(1-p i q )/(1-q) => -
p i ln p i for q
1 (*) It can be shown to be a measure of fluctuations existing in the system (b):
2 2
q
1 1 1 2 1 1
(a) (b) WW, Physica A305 (2002) 227 WW, PRL 84 (2000) 2770
Inelasticity from UA5 and similar data.....
NUWW PRD67 (2003) 114002
q=1 q>1
(*) Input:
s,
T , N charged (*) Fitted parameter: q, q-inelasticity
q
NUWW PRD67 (2003) 114002 p q
(
y
) 1
N dN q dy
1
Z q
exp
q
q
T
cosh
y
Y m
Y m dy
T
cosh
K q
N s E q
y N s
p q
(
y
)
q
q N s
q
Y m
Y m dy
T
cosh
y
p q
(
y
) 3
q
q
(*) Inelasticity K:= fraction of the total energy
s, which goes into observed secondaries produced in the central region of reaction
very important quantity in cosmic ray research and statistical models
NUWW PRD67 (2003) 114002
NUWW PRD67 (2003) 114002
NUWW PRD67 (2003) 114002
NUWW PRD67 (2003) 114002
Possible meaning of parameter q in rapidity distributions
NUWW PRD67 (2003) 114002
(*) From fits to rapidity distribution data one gets systematically q>1 with some energy dependence (*) What is now behid this q?
(*) y-distributions
‘partition temperature’ T (*) q
K
s/N
fluctuating T
fluctuating N
(*) Conjecture: q-1 should measure amount of fluctuation in P(N) (*) It does so, indeed, see Fig. where data on q obtained from fits are superimposed with data on parameter k in Negative Binomial Distribution!
Negative-Binomial Distribution
P ( n )
k ( k
1 )
( k
n
1 ) n !
m n k k ( m
k ) k
n generating function: F ( t )
P ( n ) t n
1
m ( t k
1 )
k average and variance:
D 2 n
m m
m 2 / k k =
Poisson distribution F ( t )
exp
m ( t
1 )
n
D 2
m k = - N binomial distribution F ( t )
n
1
m m N ( t
1 )
N D 2
m
m 2 / N
Parameter q as measure of dynamical fluctuations in P(N) (*) Experiment: P(N) is adequately described by NBD depending on
1) affecting its width:
1
k
2 (
N
)
N
2 1
N
(*) If 1/k is understood as measure of fluctuations of
P
(
N
) 0
d n n n
exp(
n
)
n
!
k n k
1 exp(
n
) (
k
) ( (
k
1
n
)
n
) (
k
)
k
1
k
n (P.Carruthers,C.C.Shih, Int.J.Phys. A4 (1989)5587)
with
k
n
k
1
D
(
n
) 2 (
n
)
n
2
q
1
(*) one expects: q=1+1/k what indeed is observed
Multiplicity is important ...
Notice: there is remarkable linear relation between
pp
collisions
(cf. also: NP. in NC 63A (1981) 129 or Yokomi, PRL 36 (1976) 924)
V
3/2 Fluctuations of multiplicity and
should also be related ......
Charged Particle Multiplicity Distribution
Multiplicity Distributions: (UA5, DELPHI, NA35)
UA5 s1/2 = 200 GeV Delphi 90 GeV NA35 S+S (central) 200 GeV/A
Kodama et al..
1 Poisson (Boltzmann) 100 Poisson (Boltzmann) 0.1
Poisson (Boltzmann) 10 0.1
1 0.01
0.01
0.1
0.001
UA5 200GeV
0.01
e + e 90GeV Delphi
0.001
SS (central) 200GeV
0.001
0.0001
0 10 20 30 n 40 50 60 0.0001
0 10 20 30 n 40 50 60 0.0001
0 10
<n> = 21.1; 21.2; 20.8
D 2
=
Deviation from Poisson: 1/k
1/k = [D 2 -
20 n 30 40
Parameter q as measure of dynamical fluctuations in P(N) (*) Experiment: P(N) is adequately described by NBD depending on
1) affecting its width:
1
k
2
N
(
N
) 2 1
N
with
P
(
N
) 0
d n n n
exp(
n
)
n
!
k n k
1 exp(
n
) (
k
) ( (
k
1
n
)
n
) (
k
)
k
1
k
n
k n
1
k
D
(
n
) 2 (
n
)
n
2
q
1
Recent example from AA – (1)
(MWW, APP B35 (2004) 819)
Dependence of the NBD parameter 1/k on the number of participants for NA49 and PHENIX data With increasing centrality fluctuations of the multiplicity become weaker and the respective multiplicity distributions approach Poissonian form.
???
Perhaps: smaller N W
volume of interaction V smaller
smaller total heat capacity C
greater q=1+1/C
greater 1/k = q-1
Recent example from AA – (2)
(MWW, APP B35 (2004) 819)
It can be shown that
1
k
R
(
q D
2
N
1 ) (
N
) 2 0 .
33
R
(
q
1 )
(
D
(
N
)
R
(
q
1 )
Wróblewski law ) R
( (
S E
) 2 ) 2 /
S
2 /
E
2 0 .
56
(
for p/e=1/3)
in this case q
1.59
Dependence of the NBD parameter 1/k on the number of participants for NA49 and PHENIX data It (over)saturates therefore the limit imposed from Tsallis statistics: q
1.5 . For q=1.5 one has: 0.33
0.28 (in WL) or 1/3
0.23 (in EoS)
Potentially very important result from AA collisions concerning fluctuations
(MRW, nucl-th/0407012)
for AA collisions the usual superposition model deos not work when applied to fluctuations (signal for the phase transition to Quark-Gluon-Plasma phase of matter?...)
Limitations on fluctuation...
Notice: q
1.5 limit, if applied here, leads to saturation of fluctuations at energies
s
33.32 TeV or E LAB
0.5
10 18 eV i.e., in the UHECR energy range where effects of the GZK cut starts to be important It is important for any analysis connected with GZK to know the fluctuation pattern - it can be decisive factor here!
Summary (*) Inelasticity K and cross-section
seem still be main parameters influencing development of the cosmic ray cascades (*) In some recent analysis presenting cross section obtained from cosmic ray data it is not clear whether it was accounted for that in any single CR experiment K and
are measured in junction (*) Some data call for proper accounting of fluctuations, which can be most economically described by changing
exp[ -x/
]
exp q [ -x/
] = [1-(1-q) x/
] 1/(1-q)
with q being new parameter (reaction and energy dependent) (*) Fluctuations in
, K and multiplicity can substantially change the predicted (expected?) development of all kinds of CR cascades (*) The single parameter q seems to summarily account for all new effects, which can have different (mostly unknown yet) sources