Strongly Intensive Measures for Multiplicity Fluctuations V.V. Begun, - PDF document

Strongly Intensive Measures for Multiplicity Fluctuations V.V. Begun, 1 V.P. Konchakovski, 1, 2 M.I. Gorenstein, 1, 3 and E. Bratkovskaya 4 1 Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine 2 Institut f¨ ur Theoretische Physik, Universit¨ at Giessen, Germany 3 Frankfurt Institute for Advanced Studies, Frankfurt,Germany 4 Institut f¨ ur Theoretische Physik, Universit¨ at Frankfurt, Germany Abstract PACS numbers: 12.40.-y, 12.40.Ee Keywords: event-by-event fluctuations 1

1 ∆ Kπ = � � � π � ω K − � K � ω π (1) , � K � + � π � 1 Σ Kπ = � � � π � ω K + � K � ω π − 2 ( � Kπ � − � K �� π � ) , � K � + � π � (2) where ω K ≡ � K 2 � − � K � 2 ω π = � π 2 � − � π � 2 (3) , � K � � π � I. MODEL OF INDEPENDENT SOURCES K = K 1 + K 2 + . . . + K N part , π = π 1 + π 2 + . . . + π N part . (4) � K � 1 = � K � 2 = . . . = � K � N part ≡ n K , (5) � π � 1 = � π � 2 = . . . = � π � N part ≡ n π . (6) � K � = n K � N part � , � π � = n π � N part � , (7) � K 2 � = � K 2 � 1 � N part � + n 2 � N 2 � � part � − � N part � , (8) K � π 2 � = � π 2 � 1 � N part � + n 2 � N 2 � � part � − � N part � (9) , π � N 2 � � � Kπ � = � Kπ � 1 � N part � + n K n π part � − � N part � (10) . 2

� K 2 � 1 = � K 2 � 2 = . . . = � K 2 � N part , (11) � π 2 � 1 = � π 2 � 2 = . . . = � π 2 � N part , (12) � Kπ � 1 = � Kπ � 2 = . . . = � Kπ � N part . (13) ω K = ω ∗ ω π = ω ∗ K + n K ω part , π + n π ω part , (14) where ω ∗ K and ω ∗ π are respectively the scaled variances of kaons and pions from one source, K = � K 2 � 1 − � K � 2 π = � π 2 � 1 − � π � 2 ω ∗ 1 ω ∗ 1 (15) , , � K � 1 � π � 1 ω part = � N 2 part � − � N part � 2 (16) . � N part � ρ ∗ � K π � − � K � � π � n K n π Kπ = + (17) ω part , � K + π � n K + n π n K + n π where ρ ∗ Kπ ≡ � K π � 1 − � K � 1 � π � 1 (18) describes the correlations between K and π numbers in one source. 3

Inelastic p+p collisions should be understood within MIS as the sys- tem with N part = 2 and ω part = 0. It then follows: = 1 = 1 n K ∼ n π ∼ 2 � K � pp , 2 � π � pp , (19) = 1 Kπ ∼ ρ ∗ � � � K π � pp − � K � pp � π � pp (20) , 2 � K 2 � pp − � K � 2 � π 2 � pp − � π � 2 K ∼ pp π ∼ pp ω ∗ ω ∗ = = (21) , , � K � pp � π � pp It can be easily shown that both measures ∆ (1) and Σ (2) are strongly intensive quantities, i.e. they are independent of � N part � and of ω part : 1 ∆ Kπ = [ n π ω ∗ K − n K ω ∗ π ] , (22) n K + n π 1 Σ Kπ = [ n π ω ∗ K + n K ω ∗ π − 2 ρ ∗ Kπ ] . (23) n K + n π Another interpretation of the model of independent sources can be obtained in terms of the statistical mechanics. One finds ω part ≪ 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: � K � b =0 � π � b =0 n K = n π = (24) , , � N part � b =0 � N part � b =0 = � K π � b =0 − � K � b =0 � π � b =0 Kπ ∼ ρ ∗ , (25) � N part � b =0 π ∼ K ∼ ω ∗ ω ∗ = ω π ( b = 0) , = ω K ( b = 0) , (26) 4

3.0 4 Au + Au @ s = 7.7 GeV NN > Au + Au @ s = 7.7 GeV part π NN 2.5 K (x7) > / <N 3 2.0 π >, < part 1.5 2 ω part 1.0 <K> / <N 1 0.5 0.0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 b [fm] b [fm] 2.0 8 Au + Au @ s = 7.7 GeV Au + Au @ s = 7.7 GeV NN NN ω ω π K 1.8 MIS MIS 6 1.6 K π ω 4 ω 1.4 2 1.2 0 1.0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 b [fm] b [fm] FIG. 1: The symbols correspond to the HSD results at different impact parameter b in Au+Au collisions at √ s NN = 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 � / � N part � is multiplied by a factor of 7. (b): The scaled variance ω part . (c): The scaled variance ω π . (d): The scaled variance ω K . 5

Au + Au @ s = 7.7 GeV NN 2.0 K π Σ K π ∆ MIS 1.5 π K Σ , 1.0 π K ∆ 0.5 0.0 0 2 4 6 8 10 12 14 b [fm] FIG. 2: The strongly intensive measures ∆ Kπ (1) and Σ Kπ (2). The symbols correspond to the HSD results for Au+Au collisions at √ s NN = 7 . 7 GeV. The horizontal lines show the results of MIS. 6

10 16 Au + Au @ s = 200 GeV NN > Au + Au @ s = 200 GeV part π NN 14 8 K (x7) > / <N 12 6 10 π >, < part 8 ω part 4 6 <K> / <N 4 2 2 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 b [fm] b [fm] 100 16 Au + Au @ s = 200 GeV Au + Au @ s = 200 GeV NN NN ω ω 14 π K 80 MIS MIS 12 60 10 K π ω 8 ω 40 6 4 20 2 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 b [fm] b [fm] FIG. 3: The same as in Fig. 1 but for √ s NN = 200 GeV. 7

Au + Au @ s = 200 GeV NN 2.0 K π Σ K π ∆ MIS 1.5 π K Σ , 1.0 π K ∆ 0.5 0.0 0 2 4 6 8 10 12 14 b [fm] FIG. 4: The same as in Fig. 2 but for √ s NN = 200 GeV. 4 Au + Au, b ≤ 4 fm 10 Au + Au, b 4 > ≤ part π > / <N K (x7) 8 3 π >, < 6 part 2 ω part 4 <K> / <N 1 2 0 0 10 10 2 10 10 2 s [GeV] s [GeV] NN NN 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 √ s NN . (a): The values of relative particle multiplicities per participating nucleon � K � / � N part � and � π � / � N part � . Note that � K � / � N part � is multiplied by a factor of 7. (b): The scaled variances ω part . 8

central Au + Au central Au + Au 2 10 b ≤ 4 fm b ≤ 4 fm 10 b = 0 b = 0 MIS MIS K π ω ω 10 1 1 2 2 10 10 10 10 s [GeV] s [GeV] NN NN FIG. 6: The HSD results for the scaled variances ω π (a) and ω K (b) in Au+Au collisions at b = 0 and at b ≤ 4 fm. The lines present the MIS results (see the text for details). central Au + Au 2.0 K π K π , b 4 fm Σ ≤ ∆ K π K π Σ , b = 0 ∆ 1.5 π K Σ , 1.0 π K ∆ 0.5 0.0 2 10 10 s [GeV] NN FIG. 7: The HSD results for the strongly intensive measures ∆ Kπ and Σ Kπ in Au+Au collisions at b = 0 and at b ≤ 4 fm. 9