[PDF] - Strongly Intensive Measures for Multiplicity Fluctuations V.V. Begun, PDF Document

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Strongly Intensive Measures for Multiplicity Fluctuations

V.V. Begun,1 V.P. Konchakovski,1, 2 M.I. Gorenstein,1, 3 and E. Bratkovskaya4

1Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine 2Institut f¨

ur Theoretische Physik, Universit¨ at Giessen, Germany

3Frankfurt Institute for Advanced Studies, Frankfurt,Germany 4Institut f¨

ur Theoretische Physik, Universit¨ at Frankfurt, Germany

Abstract

PACS numbers: 12.40.-y, 12.40.Ee Keywords: event-by-event fluctuations

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∆Kπ = 1 K + π

π ωK − K ωπ
,

(1) ΣKπ = 1 K + π

π ωK + K ωπ − 2 (Kπ − Kπ)
,

(2) where ωK ≡ K2 − K2 K , ωπ = π2 − π2 π (3)

I. MODEL OF INDEPENDENT SOURCES

K = K1 + K2 + . . . + KNpart , π = π1 + π2 + . . . + πNpart . (4) K1 = K2 = . . . = KNpart ≡ nK , (5) π1 = π2 = . . . = πNpart ≡ nπ . (6) K = nK Npart , π = nπ Npart , (7) K2 = K21 Npart + n2

K

N 2

part − Npart

,

(8) π2 = π21 Npart + n2

π

N 2

part − Npart

,

(9) Kπ = Kπ1 Npart + nK nπ

N 2

part − Npart

.

(10)

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K21 = K22 = . . . = K2Npart , (11) π21 = π22 = . . . = π2Npart , (12) Kπ1 = Kπ2 = . . . = KπNpart . (13) ωK = ω∗

K + nK ωpart ,

ωπ = ω∗

π + nπ ωpart ,

(14) where ω∗

K and ω∗ π are respectively the scaled variances of kaons and

pions from one source, ω∗

K = K21 − K2 1

K1 , ω∗

π = π21 − π2 1

π1 , (15) ωpart = N 2

part − Npart2

Npart . (16) K π − K π K + π = ρ∗

Kπ

nK + nπ + nK nπ nK + nπ ωpart , (17) where ρ∗

Kπ ≡ K π1 − K1 π1

(18) describes the correlations between K and π numbers in one source.

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Inelastic p+p collisions should be understood within MIS as the sys- tem with Npart = 2 and ωpart = 0. It then follows: nK ∼ = 1 2 Kpp , nπ ∼ = 1 2 πpp , (19) ρ∗

Kπ ∼

= 1 2

K πpp − Kpp πpp
,

(20) ω∗

K ∼

= K2pp − K2

pp

Kpp , ω∗

π ∼

= π2pp − π2

pp

πpp , (21) It can be easily shown that both measures ∆ (1) and Σ (2) are strongly intensive quantities, i.e. they are independent of Npart and

f ωpart :

∆Kπ = 1 nK + nπ [ nπ ω∗

K − nK ω∗ π ] ,

(22) ΣKπ = 1 nK + nπ [ nπ ω∗

K + nK ω∗ π − 2ρ∗ Kπ ] .

(23) Another interpretation of the model of independent sources can be

btained in terms of the statistical mechanics.

One finds ωpart ≪ 1 in Pb+Pb (or Au+Au) collisions with impact parameter equal to zero, b = 0. Therefore, one may define the param- eters of MIS as: nK = Kb=0 Npartb=0 , nπ = πb=0 Npartb=0 , (24) ρ∗

Kπ ∼

= K πb=0 − Kb=0 πb=0 Npartb=0 , (25) ω∗

π ∼

= ωπ(b = 0) , ω∗

K ∼

= ωK(b = 0) , (26)

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b [fm]

2 4 6 8 10 12 14

>

part

> / <N π >, <

part

<K> / <N

0.0 0.5 1.0 1.5 2.0 2.5 3.0 = 7.7 GeV

NN

s Au + Au @ π K (x7)

b [fm]

2 4 6 8 10 12 14

part

ω

1 2 3 4 = 7.7 GeV

NN

s Au + Au @

b [fm]

2 4 6 8 10 12 14

π

ω

2 4 6 8 = 7.7 GeV

NN

s Au + Au @

π

ω MIS

b [fm]

2 4 6 8 10 12 14

K

ω

1.0 1.2 1.4 1.6 1.8 2.0 = 7.7 GeV

NN

s Au + Au @

K

ω MIS

FIG. 1: The symbols correspond to the HSD results at different impact parameter b in Au+Au

collisions at √sNN = 7.7 GeV. The lines show the results calculated in MIS. (a): The HSD ratio of pion and kaon multiplicities to the average number of participants. Note that K/Npart is multiplied by a factor of 7. (b): The scaled variance ωpart. (c): The scaled variance ωπ. (d): The scaled variance ωK.

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b [fm]

2 4 6 8 10 12 14

π K

Σ ,

π K

∆

0.0 0.5 1.0 1.5 2.0 = 7.7 GeV

NN

s Au + Au @

π K

Σ

π K

∆ MIS

FIG. 2: The strongly intensive measures ∆Kπ (1) and ΣKπ (2). The symbols correspond to the HSD

results for Au+Au collisions at √sNN = 7.7 GeV. The horizontal lines show the results of MIS.

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b [fm]

2 4 6 8 10 12 14

>

part

> / <N π >, <

part

<K> / <N

2 4 6 8 10 12 14 16 = 200 GeV

NN

s Au + Au @ π K (x7)

b [fm]

2 4 6 8 10 12 14

part

ω

2 4 6 8 10 = 200 GeV

NN

s Au + Au @

b [fm]

2 4 6 8 10 12 14

π

ω

20 40 60 80 100 = 200 GeV

NN

s Au + Au @

π

ω MIS

b [fm]

2 4 6 8 10 12 14

K

ω

2 4 6 8 10 12 14 16 = 200 GeV

NN

s Au + Au @

K

ω MIS

FIG. 3: The same as in Fig. 1 but for √sNN = 200 GeV.

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b [fm]

2 4 6 8 10 12 14

π K

Σ ,

π K

∆

0.0 0.5 1.0 1.5 2.0 = 200 GeV

NN

s Au + Au @

π K

Σ

π K

∆ MIS

FIG. 4: The same as in Fig. 2 but for √sNN = 200 GeV.

[GeV]

NN

s

10

2

10

>

part

> / <N π >, <

part

<K> / <N

2 4 6 8 10 4 fm ≤ Au + Au, b π K (x7)

[GeV]

NN

s

10

2

10

part

ω

1 2 3 4 4 ≤ Au + Au, b

FIG. 5: The HSD results in Au+Au collisions at b ≤ 4 fm as the functions of the center of mass energy

per nucleon pair √sNN. (a): The values of relative particle multiplicities per participating nucleon K/Npart and π/Npart. Note that K/Npart is multiplied by a factor of 7. (b): The scaled variances ωpart.

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[GeV]

NN

s

10

2

10

π

ω

1 10

2

10 central Au + Au 4 fm ≤ b b = 0 MIS

[GeV]

NN

s

10

2

10

K

ω

1 10 central Au + Au 4 fm ≤ b b = 0 MIS

FIG. 6: The HSD results for the scaled variances ωπ (a) and ωK (b) in Au+Au collisions at b = 0 and

at b ≤ 4 fm. The lines present the MIS results (see the text for details).

[GeV]

NN

s

10

2

10

π K

Σ ,

π K

∆

0.0 0.5 1.0 1.5 2.0 central Au + Au

π K

∆ 4 fm ≤ , b

π K

Σ

π K

∆ , b = 0

π K

Σ

FIG. 7: The HSD results for the strongly intensive measures ∆Kπ and ΣKπ in Au+Au collisions at