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 InstitutePylos, 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 formodels. However, from different analysis of available datathe 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:

(*) can be also fitted by:

(*) immediate conjecture: q fluctuations present in the system

Depth distributions of starting pointsof cascades in Pamir lead chamberCosmic ray experiment (WW, NPB (Proc.Suppl.) A75 (1999) 191

(*) WW, PRD50 (1994) 2318

2.02

22

3.1;)1(1

exp

1

1

qT

qconstdT

dN

Tconst

dT

dN

q

q – measure of fluctuations

2

22

1

11

11

q

(*) Parameter q is known in the literature as measure ofnonextensivity in the Tsallis statistics based on Tsallis entropy (a):

Sq = - (1-piq)/(1-q) => - pi ln pi for q1

(*) It can be shown to be a measure of fluctuations existing in thesystem (b):

(a) WW, Physica A305 (2002) 227(b) WW, PRL 84 (2000) 2770

q=1 q>1

NUWW PRD67 (2003) 114002

Inelasticity fromUA5 and similardata.....

q

ypydys

NE

s

NK

N

sypydy

yZdy

dN

Nyp

qmY

mY qTqq

qqq

qTmY

mY

Tqq

qq

3)(cosh

)(cosh

coshexp11

)(

(*) Input: s, T, Ncharged

(*) Fitted parameter: q, q-inelasticity q

(*) Inelasticity K:= fraction of the total energy s, which goes into observed secondaries produced in the central region of reactionvery important quantity in cosmic ray research and statistical models

NUWW PRD67 (2003) 114002

q

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’ TK s/N(*) q fluctuating Tfluctuating 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

nk

kn

)km(

km

!n

)1nk()1k(k)n(P

kn )1t(

k

m1t)n(P)t(F

k/mmDmn

22

generatingfunction:

average andvariance:

k = - Nbinomial

distribution N/mmD

mn22

N

)1t(N

m1)t(F

k = Poisson

distribution

)1t(mexp)t(F

mDn 2

Parameter q as measure of dynamical fluctuations in P(N)

(*) Experiment: P(N) is adequately described by NBD depending on <N> and k (k1) affecting its width:

(*) If 1/k is understood as measure of fluctuations of <N> then

with

(*) one expects: q=1+1/k what indeed is observed

NN

N

k

1)(12

2

nk

k

kkn

kn

nk

k

nn

n

nnndNP

1)()1(

)(

)(

)exp(

!

)exp()(

1

0

n

k 1)(

)(1

2

2

qn

nnD

k

(P.Carruthers,C.C.Shih,Int.J.Phys. A4 (1989)5587)

Multiplicity is important ...

Notice: there is remarkable linear relation between <Ncharged> and the corresponding cross section for pp and ppcollisions (cf. also: NP. in NC 63A (1981) 129 or Yokomi, PRL 36 (1976) 924)

V3/2

Fluctuations of multiplicity and should also be related ......

Multiplicity Distributions: (UA5, DELPHI, NA35)

Kodama et al..

e+e-

90GeVDelphi

SS(central)200GeV

<n> = 21.1; 21.2; 20.8D2 = <n2>-<n>2 = 112.7; 41.4; 25.7 Deviation from Poisson: 1/k

1/k = [D2-<n>]/<n2> = 0.21; 0.045; 0.011

0 10 20 30 40 50 60n

0.0001

0.001

0.01

0.1

1

Pn

C harged Partic le M ultip lic ity D istribution

U A5 s1/2 = 200 G eV

Poisson(Boltzm ann)

UA5200GeV

0 10 20 30 40 50 60n

0.0001

0.001

0.01

0.1

1

10

100

Pn

(%)

C harged Partic le M ultip lic ity D istributionD elphi 90 G eV

Po isson(Boltzm ann)

0 10 20 30 40n

0.0001

0.001

0.01

0.1

Pn

N egative Partic le M ultip lic ity D istributionN A35 S+S (centra l) 200 G eV/A

Po isson(Boltzm ann)

Parameter q as measure of dynamical fluctuations in P(N)

(*) Experiment: P(N) is adequately described by NBD depending on <N> and k (k1) affecting its width:

(*) If 1/k is understood as measure of fluctuations of <N> then

with

(*) one expects: q=1+1/k what indeed is observed

NN

N

k

1)(12

2

nk

k

kkn

kn

nk

k

nn

n

nnndNP

1)()1(

)(

)(

)exp(

!

)exp()(

1

0

n

k1

)()(

12

2

qn

nnD

k

Recent example from AA – (1) (MWW, APP B35 (2004) 819)

Dependence of the NBD parameter 1/k onthe number of participants for NA49 andPHENIX data

With increasing centralityfluctuations of the multiplicitybecome weaker and the respective multiplicitydistributions approachPoissonian form.

???

Perhaps: smaller NW smallervolume of interaction Vsmaller total heat capacity Cgreater q=1+1/C greater1/k = q-1

Recent example from AA – (2) (MWW, APP B35 (2004) 819)

Dependence of the NBD parameter 1/k onthe number of participants for NA49 andPHENIX data

It can be shown that

56.0/)(

/)(

33.0)1(

)1()()1()(1

22

22

2

2

EE

SSR

qR

qRNDqRN

ND

k

( Wróblewski law )

( for p/e=1/3)

in this case

q1.59

It (over)saturates therefore the limit imposed from Tsallisstatistics: q1.5 . 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 concerningfluctuations (MRW, nucl-th/0407012)

for AA collisions theusual superpositionmodel deos not workwhen applied to fluctuations (signalfor the phase transitionto Quark-Gluon-Plasmaphase of matter?...)

Limitations on fluctuation...

Notice: q 1.5 limit, if applied here,leads to saturation of fluctuations at energies s 33.32 TeV orELAB 0.5 1018 eV i.e., in the UHECR energy rangewhere effects of the GZK cut startsto be important

It is important for any analysis connectedwith 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/ ] expq[ -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