広島大学学位請求論文

Experimental study on QCD phase transition
by measuring charged particle density correlations
in Au+Au collisions at s NN =200GeV(核子対あたりの重心系衝突エネルギー200GeV の金・金原子核衝突における荷電粒子密度相関の測定による QCD 相転移の実験的研究)

２００９年
広島大学大学院理学研究科
物理科学専攻中村 智昭（理化学研究所 仁科加速器研究センター 延與放射線研究室）

目 次
１．主論文
Experimental study on QCD phase transition by measuring charged particle
density correlations in Au+Au collisions at s NN =200GeV(核子対あたりの重心系衝突エネルギー200GeV の金・金原子核衝突における荷電粒子密度相関の測定による QCD 相転移の実験的研究)
中村 智昭
２．公表論文(1) Measurement of density correlations in pseudorapidity via charged
particle multiplicity fluctuations in Au+Au collisions at s NN =200 GeV
S. S. Adler et al., (別紙、共著者リスト①参照)
Physical Review C 76, 034903 (2007).
３．参考論文(1) Systematic studies of the centrality and s NN dependence of thedET / dη and dN ch / dη in heavy ion collisions at midrapidity
S. S. Adler et al., (別紙、共著者リスト②参照)
Physical Review C 71, 034908 (2005).
(2) Charged hadron multiplicity fluctuations in Au+Au and Cu+Cu collisions
from s NN =22.5 to 200GeV
A. Adare et al., (別紙、共著者リスト③参照)
Physical Review C 78, 044902 (2008).
(3) PHENIX inner detectors
M. Allen et al., (別紙、共著者リスト④参照)
Nuclear Instruments and Methods in Physics Research A 499, 549 (2003).

別紙

共著者リスト①

S. S. Adler, S. Afanasiev, C. Aidala, N. N. Ajitanand, Y. Akiba, J. Alexander, R. Amirikas, L. Aphecetche,S. H. Aronson, R. Averbeck, T. C. Awes, R. Azmoun, V. Babintsev, A. Baldisseri, K. N. Barish,
P. D. Barnes, B. Bassalleck, S. Bathe, S. Batsouli, V. Baublis, A. Bazilevsky, S. Belikov, Y. Berdnikov,
S. Bhagavatula, J. G. Boissevain, H. Borel, S. Borenstein, M. L. Brooks, D. S. Brown, N. Bruner,
D. Bucher, H. Buesching, V. Bumazhnov, G. Bunce, J. M. Burward-Hoy, S. Butsyk, X. Camard, J.-S. Chai,
P. Chand, W. C. Chang, S. Chernichenko, C. Y. Chi, J. Chiba, M. Chiu, I. J. Choi, J. Choi,
R. K. Choudhury, T. Chujo, V. Cianciolo, Y. Cobigo, B. A. Cole, P. Constantin, D. dEnterria, G. David,
H. Delagrange, A. Denisov, A. Deshpande, E. J. Desmond, A. Devismes, O. Dietzsch, O. Drapier, A. Drees,
K. A. Drees, A. Durum, D. Dutta, Y. V. Efremenko, K. El Chenawi, A. Enokizono, H. Enyo, S. Esumi,
L. Ewell, D. E. Fields, F. Fleuret, S. L. Fokin, B. D. Fox, Z. Fraenkel, J. E. Frantz, A. Franz,
A. D. Frawley, S.-Y. Fung, S. Garpman, T. K. Ghosh, A. Glenn, G. Gogiberidze, M. Gonin, J. Gosset,
Y. Goto, R. Granier de Cassagnac, N. Grau, S. V. Greene, M. Grosse Perdekamp, W. Guryn,
H.-A°. Gustafsson, T. Hachiya, J. S. Haggerty, H. Hamagaki, A. G. Hansen, E. P. Hartouni, M. Harvey,
R. Hayano, N. Hayashi, X. He, H. W. van Hecke, M. Heffner, T. K. Hemmick, J. M. Heuser, M. Hibino,
J. C. Hill, W. Holzmann, K. Homma, B. Hong, A. Hoover, T. Ichihara, V. V. Ikonnikov, K. Imai,
D. Isenhower, M. Ishihara, M. Issah, A. Isupov, B. V. Jacak, W. Y. Jang, Y. Jeong, J. Jia, O. Jinnouchi,
B. M. Johnson, S. C. Johnson, K. S. Joo, D. Jouan, S. Kametani, N. Kamihara, J. H. Kang, S. S. Kapoor,
K. Katou, S. Kelly, B. Khachaturov, A. Khanzadeev, J. Kikuchi, D. H. Kim, D. J. Kim, D. W. Kim, E. Kim,
G.-B. Kim, H. J. Kim, E. Kistenev, A. Kiyomichi, K. Kiyoyama, C. Klein-Boesing, H. Kobayashi,
L. Kochenda, V. Kochetkov, D. Koehler, T. Kohama, M. Kopytine, D. Kotchetkov, A. Kozlov, P. J. Kroon,
C. H. Kuberg, K. Kurita, Y. Kuroki, M. J. Kweon, Y. Kwon, G. S. Kyle, R. Lacey, V. Ladygin, J. G. Lajoie,
A. Lebedev, S. Leckey, D. M. Lee, S. Lee, M. J. Leitch, X. H. Li, H. Lim, A. Litvinenko, M. X. Liu, Y. Liu,
C. F. Maguire, Y. I. Makdisi, A. Malakhov, V. I. Manko, Y. Mao, G. Martinez, M. D. Marx, H. Masui,
F. Matathias, T. Matsumoto, P. L. McGaughey, E. Melnikov, F. Messer, Y. Miake, J. Milan, T. E. Miller,
A. Milov, S. Mioduszewski, R. E. Mischke, G. C. Mishra, J. T. Mitchell, A. K. Mohanty, D. P. Morrison,
J. M. Moss, F. M¨uhlbacher, D. Mukhopadhyay, M. Muniruzzaman, J. Murata, S. Nagamiya, J. L. Nagle,
T. Nakamura, B. K. Nandi, M. Nara, J. Newby, P. Nilsson, A. S. Nyanin, J. Nystrand, E. OBrien,
C. A. Ogilvie, H. Ohnishi, I. D. Ojha, K. Okada, M. Ono, V. Onuchin, A. Oskarsson, I. Otterlund,
K. Oyama, K. Ozawa, D. Pal, A. P. T. Palounek, V. Pantuev, V. Papavassiliou, J. Park, A. Parmar,
S. F. Pate, T. Peitzmann, J.-C. Peng, V. Peresedov, C. Pinkenburg, R. P. Pisani, F. Plasil, M. L. Purschke,
A. K. Purwar, J. Rak, I. Ravinovich, K. F. Read, M. Reuter, K. Reygers, V. Riabov, Y. Riabov, R. du Rietz,
G. Roche, A. Romana, M. Rosati, P. Rosnet, S. S. Ryu, M. E. Sadler, N. Saito, T. Sakaguchi, M. Sakai,
S. Sakai, V. Samsonov, L. Sanfratello, R. Santo, H. D. Sato, S. Sato, S. Sawada, Y. Schutz, V. Semenov,
R. Seto, M. R. Shaw, T. K. Shea, T.-A. Shibata, K. Shigaki, T. Shiina, C. L. Silva, D. Silvermyr, K. S. Sim,C. P. Singh, V. Singh, M. Sivertz, A. Soldatov, R. A. Soltz, W. E. Sondheim, S. P. Sorensen, I. V. Sourikova,
F. Staley, P. W. Stankus, E. Stenlund, M. Stepanov, A. Ster, S. P. Stoll, T. Sugitate, J. P. Sullivan,
E. M. Takagui, A. Taketani, M. Tamai, K. H. Tanaka, Y. Tanaka, K. Tanida, M. J. Tannenbaum,
P. Tarj´an, J. D. Tepe, T. L. Thomas, A. Toia, J. Tojo, H. Torii, R. S. Towell, I. Tserruya, H. Tsuruoka,
S. K. Tuli, H. Tydesj¨o, N. Tyurin, J. Velkovska, M. Velkovsky, V. Veszpr´emi, L. Villatte, A. A. Vinogradov,
M. A. Volkov, E. Vznuzdaev, X. R.Wang, Y.Watanabe, S. N. White, F. K.Wohn, C. L.Woody, W. Xie, Y. Yang,
A. Yanovich, S. Yokkaichi, G. R. Young, I. E. Yushmanov, W. A. Zajc, C. Zhang, S. Zhou, S. J. Zhou,
L. Zolin.

共著者リスト②

S. S. Adler, S. Afanasiev, C. Aidala, N. N. Ajitanand, Y. Akiba, J. Alexander, R. Amirikas, L. Aphecetche,S. H. Aronson, R. Averbeck, T. C. Awes, R. Azmoun, V. Babintsev, A. Baldisseri, K. N. Barish,
P. D. Barnes, B. Bassalleck, S. Bathe, S. Batsouli, V. Baublis, A. Bazilevsky, S. Belikov, Y. Berdnikov,
S. Bhagavatula, J. G. Boissevain, H. Borel, S. Borenstein, M. L. Brooks, D. S. Brown, N. Bruner,
D. Bucher, H. Buesching, V. Bumazhnov, G. Bunce, J. M. Burward-Hoy, S. Butsyk, X. Camard, J.-S. Chai,
P. Chand, W. C. Chang, S. Chernichenko, C. Y. Chi, J. Chiba, M. Chiu, I. J. Choi, J. Choi,
R. K. Choudhury, T. Chujo, V. Cianciolo, Y. Cobigo, B. A. Cole, P. Constantin, D. G. dEnterria, G. David,
H. Delagrange, A. Denisov, A. Deshpande, E. J. Desmond, O. Dietzsch, O. Drapier, A. Drees, R. du Rietz,
A. Durum, D. Dutta, Y. V. Efremenko, K. El Chenawi, A. Enokizono, H. Enyo, S. Esumi, L. Ewell,
D. E. Fields, F. Fleuret, S. L. Fokin, B. D. Fox, Z. Fraenkel, J. E. Frantz, A. Franz, A. D. Frawley,
S.-Y. Fung, S. Garpman, T. K. Ghosh, A. Glenn, G. Gogiberidze, M. Gonin, J. Gosset, Y. Goto,
R. Granier de Cassagnac, N. Grau, S. V. Greene, M. Grosse Perdekamp, W. Guryn, H.-Å. Gustafsson,
T. Hachiya, J. S. Haggerty, H. Hamagaki, A. G. Hansen, E. P. Hartouni, M. Harvey, R. Hayano, X. He,
M. Heffner, T. K. Hemmick, J. M. Heuser, M. Hibino, J. C. Hill, W. Holzmann, K. Homma, B. Hong,
A. Hoover, T. Ichihara, V. V. Ikonnikov, K. Imai, D. Isenhower, M. Ishihara, M. Issah, A. Isupov,
B. V. Jacak, W. Y. Jang, Y. Jeong, J. Jia, O. Jinnouchi, B. M. Johnson, S. C. Johnson, K. S. Joo, D. Jouan,
S. Kametani, N. Kamihara, J. H. Kang, S. S. Kapoor, K. Katou, S. Kelly, B. Khachaturov, A. Khanzadeev,
J. Kikuchi, D. H. Kim, D. J. Kim, D. W. Kim, E. Kim, G.-B. Kim, H. J. Kim, E. Kistenev, A. Kiyomichi,
K. Kiyoyama, C. Klein-Boesing, H. Kobayashi, L. Kochenda, V. Kochetkov, D. Koehler, T. Kohama,
M. Kopytine, D. Kotchetkov, A. Kozlov, P. J. Kroon, C. H. Kuberg, K. Kurita, Y. Kuroki, M. J. Kweon,
Y. Kwon, G. S. Kyle, R. Lacey, V. Ladygin, J. G. Lajoie, A. Lebedev, S. Leckey, D. M. Lee, S. Lee,
M. J. Leitch, X. H. Li, H. Lim, A. Litvinenko, M. X. Liu, Y. Liu, C. F. Maguire, Y. I. Makdisi, A. Malakhov,
V. I. Manko, Y. Mao, G. Martinez, M. D. Marx, H. Masui, F. Matathias, T. Matsumoto, P. L. McGaughey,
E. Melnikov, M. Mendenhall, F. Messer, Y. Miake, J. Milan, T. E. Miller, A. Milov, S. Mioduszewski,
R. E. Mischke, G. C. Mishra, J. T. Mitchell, A. K. Mohanty, D. P. Morrison, J. M. Moss, F. M¨uhlbacher,
D. Mukhopadhyay, M. Muniruzzaman, J. Murata, S. Nagamiya, J. L. Nagle, T. Nakamura, B. K. Nandi,
M. Nara, J. Newby, P. Nilsson, A. S. Nyanin, J. Nystrand, E. OBrien, C. A. Ogilvie, H. Ohnishi, I. D. Ojha,
K. Okada, M. Ono, V. Onuchin, A. Oskarsson, I. Otterlund, K. Oyama, K. Ozawa, D. Pal,
A. P. T. Palounek, V. S. Pantuev, V. Papavassiliou, J. Park, A. Parmar, S. F. Pate, T. Peitzmann,
J.-C. Peng, V. Peresedov, C. Pinkenburg, R. P. Pisani, F. Plasil, M. L. Purschke, A. K. Purwar, J. Rak,
I. Ravinovich, K. F. Read, M. Reuter, K. Reygers, V. Riabov, Y. Riabov, G. Roche, A. Romana, M. Rosati,
P. Rosnet, S. S. Ryu, M. E. Sadler, N. Saito, T. Sakaguchi, M. Sakai, S. Sakai, V. Samsonov,
L. Sanfratello, R. Santo, H. D. Sato, S. Sato, S. Sawada, Y. Schutz, V. Semenov, R. Seto, M. R. Shaw,
T. K. Shea, T.-A. Shibata, K. Shigaki, T. Shiina, C. L. Silva, D. Silvermyr, K. S. Sim, C. P. Singh, V. Singh,
M. Sivertz, A. Soldatov, R. A. Soltz, W. E. Sondheim, S. P. Sorensen, I. V. Sourikova, F. Staley,
P. W. Stankus, E. Stenlund, M. Stepanov, A. Ster, S. P. Stoll, T. Sugitate, J. P. Sullivan, E. M. Takagui,
A. Taketani, M. Tamai, K. H. Tanaka, Y. Tanaka, K. Tanida, M. J. Tannenbaum, P. Tarj´an, J. D. Tepe,
T. L. Thomas, J. Tojo, H. Torii, R. S. Towell, I. Tserruya, H. Tsuruoka, S. K. Tuli, H. Tydesj¨o, N. Tyurin,
H. W. van Hecke, J. Velkovska, M. Velkovsky, L. Villatte, A. A. Vinogradov, M. A. Volkov, E. Vznuzdaev,
X. R. Wang, Y. Watanabe, S. N. White, F. K. Wohn, C. L. Woody, W. Xie, Y. Yang, A. Yanovich, S. Yokkaichi,
G. R. Young, I. E. Yushmanov, W. A. Zajc, C. Zhang, S. Zhou, S. J. Zhou, L. Zolin.

共著者リスト③

A. Adare, S. S. Adler, S. Afanasiev, C. Aidala, N. N. Ajitanand, Y. Akiba, H. Al-Bataineh, J. Alexander,
A. Al-Jamel, K. Aoki, L. Aphecetche, R. Armendariz, S. H. Aronson, J. Asai, E. T. Atomssa, R. Averbeck,
T. C. Awes, B. Azmoun, V. Babintsev, G. Baksay, L. Baksay, A. Baldisseri, K. N. Barish, P. D. Barnes,
B. Bassalleck, S. Bathe, S. Batsouli, V. Baublis, F. Bauer, A. Bazilevsky, S. Belikov, R. Bennett,
Y. Berdnikov, A. A. Bickley, M. T. Bjorndal, J. G. Boissevain, H. Borel, K. Boyle, M. L. Brooks,
D. S. Brown, N. Bruner, D. Bucher, H. Buesching, V. Bumazhnov, G. Bunce, J. M. Burward-Hoy,
S. Butsyk, X. Camard, S. Campbell, J.-S. Chai, P. Chand, B. S. Chang, W. C. Chang, J.-L. Charvet,
S. Chernichenko, J. Chiba, C. Y. Chi, M. Chiu, I. J. Choi, R. K. Choudhury, T. Chujo, P. Chung, A. Churyn,
V. Cianciolo, C. R. Cleven, Y. Cobigo, B. A. Cole, M. P. Comets, P. Constantin, M. Csan´ad, T. Cs¨orgo,
J. P. Cussonneau, T. Dahms, K. Das, G. David, F. De´ak, M. B. Deaton, K. Dehmelt, H. Delagrange,
A. Denisov, D. dEnterria, A. Deshpande, E. J. Desmond, A. Devismes, O. Dietzsch, A. Dion, M. Donadelli,
J. L. Drachenberg, O. Drapier, A. Drees, A. K. Dubey, A. Durum, D. Dutta, V. Dzhordzhadze,
Y. V. Efremenko, J. Egdemir, F. Ellinghaus, W. S. Emam, A. Enokizono, H. Enyo, B. Espagnon, S. Esumi,
K. O. Eyser, D. E. Fields, C. Finck, M. Finger Jr., M. Finger, F. Fleuret, S. L. Fokin, B. Forestier,B. D. Fox, Z. Fraenkel, J. E. Frantz, A. Franz, A. D. Frawley, K. Fujiwara, Y. Fukao, S.-Y. Fung,
T. Fusayasu, S. Gadrat, I. Garishvili, F. Gastineau, M. Germain, A. Glenn, H. Gong, M. Gonin, J. Gosset,
Y. Goto, R. Granier de Cassagnac, N. Grau, S. V. Greene, M. Grosse Perdekamp, T. Gunji,
H.-A°. Gustafsson, T. Hachiya, A. Hadj Henni, C. Haegemann, J. S. Haggerty, M. N. Hagiwara,
H. Hamagaki, R. Han, A. G. Hansen, H. Harada, E. P. Hartouni, K. Haruna, M. Harvey, E. Haslum,
K. Hasuko, R. Hayano, M. Heffner, T. K. Hemmick, T. Hester, J. M. Heuser, X. He, P. Hidas, H. Hiejima,
J. C. Hill, R. Hobbs, M. Hohlmann, M. Holmes, W. Holzmann, K. Homma, B. Hong, A. Hoover,T. Horaguchi, D. Hornback, M. G. Hur, T. Ichihara, V. V. Ikonnikov, K. Imai, M. Inaba, Y. Inoue,
M. Inuzuka, D. Isenhower, L. Isenhower, M. Ishihara, T. Isobe, M. Issah, A. Isupov, B. V. Jacak, J. Jia,
J. Jin, O. Jinnouchi, B. M. Johnson, S. C. Johnson, K. S. Joo, D. Jouan, F. Kajihara, S. Kametani,
N. Kamihara, J. Kamin, M. Kaneta, J. H. Kang, H. Kanou, K. Katou, T. Kawabata, T. Kawagishi,
D. Kawall, A. V. Kazantsev, S. Kelly, B. Khachaturov, A. Khanzadeev, J. Kikuchi, D. H. Kim, D. J. Kim,
E. Kim, G.-B. Kim, H. J. Kim, Y.-S. Kim, E. Kinney, A. Kiss, E. Kistenev, A. Kiyomichi, J. Klay,
C. Klein-Boesing, H. Kobayashi, L. Kochenda, V. Kochetkov, R. Kohara, B. Komkov, M. Konno,
D. Kotchetkov, A. Kozlov, A. Kr´al, A. Kravitz, P. J. Kroon, J. Kubart, C. H. Kuberg, G. J. Kunde,N. Kurihara, K. Kurita, M. J. Kweon, Y. Kwon, G. S. Kyle, R. Lacey, Y.-S. Lai, J. G. Lajoie, A. Lebedev,
Y. Le Bornec, S. Leckey, D. M. Lee, M. K. Lee, T. Lee, M. J. Leitch, M. A. L. Leite, B. Lenzi, H. Lim,
T. Liska, A. Litvinenko, M. X. Liu, X. Li, X. H. Li, B. Love, D. Lynch, C. F. Maguire, Y. I. Makdisi,A. Malakhov, M. D. Malik, V. I. Manko, Y. Mao, G. Martinez, L. Masek, H. Masui, F. Matathias,
T. Matsumoto, M. C. McCain, M. McCumber, P. L. McGaughey, Y. Miake, P. Mikes, K. Miki, T. E. Miller,
A. Milov, S. Mioduszewski, G. C. Mishra, M. Mishra, J. T. Mitchell, M. Mitrovski, A. K. Mohanty,
A. Morreale, D. P. Morrison, J. M. Moss, T. V. Moukhanova, D. Mukhopadhyay, M. Muniruzzaman,
J. Murata, S. Nagamiya, Y. Nagata, J. L. Nagle, M. Naglis, I. Nakagawa, Y. Nakamiya, T. Nakamura,
K. Nakano, J. Newby, M. Nguyen, B. E. Norman, A. S. Nyanin, J. Nystrand, E. OBrien, S. X. Oda,
C. A. Ogilvie, H. Ohnishi, I. D. Ojha, H. Okada, K. Okada, M. Oka, O. O. Omiwade, A. Oskarsson,
I. Otterlund, M. Ouchida, K. Oyama, K. Ozawa, R. Pak, D. Pal, A. P. T. Palounek, V. Pantuev,
V. Papavassiliou, J. Park, W. J. Park, S. F. Pate, H. Pei, V. Penev, J.-C. Peng, H. Pereira, V. Peresedov,
D. Yu. Peressounko, A. Pierson, C. Pinkenburg, R. P. Pisani, M. L. Purschke, A. K. Purwar, J. M. Qualls,
H. Qu, J. Rak, A. Rakotozafindrabe, I. Ravinovich, K. F. Read, S. Rembeczki, M. Reuter, K. Reygers,
V. Riabov, Y. Riabov, G. Roche, A. Romana, M. Rosati, S. S. E. Rosendahl, P. Rosnet, P. Rukoyatkin,
V. L. Rykov, S. S. Ryu, B. Sahlmueller, N. Saito, T. Sakaguchi, S. Sakai, H. Sakata, V. Samsonov,
L. Sanfratello, R. Santo, H. D. Sato, S. Sato, S. Sawada, Y. Schutz, J. Seele, R. Seidl, V. Semenov, R. Seto,
D. Sharma, T. K. Shea, I. Shein, A. Shevel, T.-A. Shibata, K. Shigaki, M. Shimomura, T. Shohjoh, K. Shoji,
A. Sickles, C. L. Silva, D. Silvermyr, C. Silvestre, K. S. Sim, C. P. Singh, V. Singh, S. Skutnik,

M. Slunecka, W. C. Smith, A. Soldatov, R. A. Soltz, W. E. Sondheim, S. P. Sorensen, I. V. Sourikova,
F. Staley, P. W. Stankus, E. Stenlund, M. Stepanov, A. Ster, S. P. Stoll, T. Sugitate, C. Suire, J. P. Sullivan,
J. Sziklai, T. Tabaru, S. Takagi, E. M. Takagui, A. Taketani, K. H. Tanaka, Y. Tanaka, K. Tanida,
M. J. Tannenbaum, A. Taranenko, P. Tarj´an, T. L. Thomas, M. Togawa, A. Toia, J. Tojo, L. Tom´asek,
H. Torii, R. S. Towell, V.-N. Tram, I. Tserruya, Y. Tsuchimoto, S. K. Tuli, H. Tydesj¨o, N. Tyurin,
T. J. Uam, C. Vale, H. Valle, H. W. van Hecke, J. Velkovska, M. Velkovsky, R. Vertesi, V. Veszpr´emi,
A. A. Vinogradov, M. Virius, M. A. Volkov, V. Vrba, E. Vznuzdaev, M. Wagner, D. Walker, X. R. Wang,
Y. Watanabe, J. Wessels, S. N. White, N. Willis, D. Winter, F. K. Wohn, C. L. Woody, M. Wysocki, W. Xie,
Y. L. Yamaguchi, A. Yanovich, Z. Yasin, J. Ying, S. Yokkaichi, G. R. Young, I. Younus, I. E. Yushmanov,
W. A. Zajc, O. Zaudtke, C. Zhang, S. Zhou, J. Zim´anyi, L. Zolin, X. Zong.

共著者リスト④

M. Allen, M. J. Bennett, M. Bobrek, J. B. Boissevain, S. Boose, E. Bosze, C. Britton, J. Chang, C. Y. Chi,
M. Chiu, R. Conway, R. Cunningham, A. Denisov, A. Deshpande, M. S. Emery, A. Enokizono, N. Ericson,
B. Fox, S.-Y. Fung, P. Giannotti, T. Hachiya, A. G. Hansen, K. Homma, B. V. Jacak, D. Jaffe, J. H. Kang,
J. Kapustinsky, S. Y. Kim, Y. G. Kim, T. Kohama, P. J. Kroon, W. Lenz, N. Longbotham, M. Musrock,
T. Nakamura, H. Ohnishi, S. S. Ryu, A. Sakaguchi, R. Seto, T. Shiina, M. Simpson, J. Simon-Gillo,
W. E. Sondheim, T. Sugitate, J. P. Sullivan, H. W. van Hecke, J. W. Walker, S. N. White, P. Willis, N. Xu.

主論文

Experimental study on QCD phase transition
by measuring charged particle density correlations√
in Au+Au collisions at sNN = 200 GeV

Tomoaki Nakamura

Department of Physical Science, Graduate School of Science
Hiroshima University
Kagamiyama 1-3-1, Higashi-Hiroshima, 739-8526, Japan
January 2009

Abstract
Theoretical studies of Quantum Chromodynamics (QCD), which is the underlying theory of strong interactions, indicate that QCD matter has rich phasestructures in non-perturbative regimes. The phase diagram can be parametrized
by temperature T and baryo-chemical potential µB . Based on the phase diagram,one can obtain perspectives on how the vacuum structure of the early universe
evolved in extremely high temperature states after the Big Bang. We expect thephase diagram to exhibit at least two distinct regions. One is the quark-gluonphase, and the other is the hadron phase. There is a first order phase boundaryat µB > 0 and T = 0. At µB = 0 and T > 0, a smooth crossover transitionis expected due to finite masses of quarks. Logically we can then expect thata critical end-point (CEP) exists at the end of the first order phase transitionline. The location of the CEP would be a landmark in understanding the whole
structure of the phase diagram. Although numerical calculations using latticegauge theory as well as model calculations predict the existence of the CEP, noneof them have reached a quantitative agreement on the location at the present
precision. Therefore experimental investigations are indispensable to pin down
the location, and to understand properties of the phase point based on fundamental observables. The high energy heavy ion experiment at the Relativistic
Heavy Ion Collider (RHIC) at Brookheaven National Laboratory (BNL) is oneof the unique opportunities to access such a high temperature state as well as
studying the properties of the QCD matter.
In this thesis, an experimental result on longitudinal density correlations of√
produced matter in Au+Au collisions at sN N = 200 GeV has been reported tosearch for the critical phase boundary of QCD. The density correlation, which isone of the basic observables in the most simplest thermodynamical treatment of
the matter based on Ginzburg-Landau phenomenology, is extracted from inclusive charged particle multiplicity distributions in various pseudorapidity interval
sizes δη measured by the PHENIX detector within the total pseudorapidity coverage of |η| < 0.35. The measured multiplicity distributions are fit by Negative
Binomial Distributions (NBD), and the NBD parameters µ (mean) and k −1 (deviation from a Poissonian width) are determined in various collision centralities.
The corresponding range in the average number of participant nucleons (Npart )at the nucleus-nucleus collision is from 29.8 to 352.2. The product of the correlation strength α and the correlation length ξ, which has a monotonic correspondence to a susceptibility, is extracted from a known relation between the
NBD k parameter and αξ as a function of δη. A local maximum of αξ is seen at
Npart ∼ 90. The corresponding energy density based on the Bjorken picture isǫBj τ ∼ 2.4 GeV/(fm2 · c) with a transverse area size of 60 fm2 . This behavior may
suggest a critical phase boundary based on the Ginzburg-Landau framework.

Contents
1 Introduction
1.1 QCD phase transition . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Theoretical approaches on the QCD phase diagram . . . . . . . .
1.3 High energy heavy ion experiment . . . . . . . . . . . . . . . . . .
1.4 Motivation of this study . . . . . . . . . . . . . . . . . . . . . . .

1
1
5
7
11

2 Experimental observable
2.1 Thermodynamic singularity . . . . . . . . . . . . . . . . . . . . .
2.2 Density fluctuation and susceptibility . . . . . . . . . . . . . . . .
2.3 Application to the phase transition . . . . . . . . . . . . . . . . .
2.4 Correlation and fluctuation . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Multiplicity distribution . . . . . . . . . . . . . . . . . . .
2.4.2 Extraction of correlation . . . . . . . . . . . . . . . . . . .

13
13
15
19
21
21
24

3 Experimental setup
3.1 Relativistic heavy ion collider (RHIC) . . . . . . . . . . . . . . . .
3.2 PHENIX detector overview . . . . . . . . . . . . . . . . . . . . .
3.3 Global detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Zero degree calorimeter (ZDC) . . . . . . . . . . . . . . . .
3.3.2 Beam-beam counter (BBC) . . . . . . . . . . . . . . . . .
3.4 Central arm spectrometer . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Drift chamber (DC) . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Pad chamber (PC) . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Time of flight counter (TOF) . . . . . . . . . . . . . . . .
3.4.4 Electro-magnetic calorimeter (EMCal) . . . . . . . . . . .
3.5 Data acquisition system (DAQ) . . . . . . . . . . . . . . . . . . .

27
27
27
31
31
33
34
35
36
37
39
40

4 Analysis
4.1 Event selections . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Data set . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Trigger selection . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Collision centrality and number of participant nucleons . .
4.1.4 Energy density estimation . . . . . . . . . . . . . . . . . .

44
44
44
44
48
54

i

4.1.5 Data quality assurance . . . . . . . . . . . . . . . . . . . .
Measurement of charged particles . . . . . . . . . . . . . . . . . .
4.2.1 Track selection criteria . . . . . . . . . . . . . . . . . . . .
4.2.2 Magnetic field condition . . . . . . . . . . . . . . . . . . .
Measurement of multiplicity distributions . . . . . . . . . . . . . .
Corrections and error estimations . . . . . . . . . . . . . . . . . .
4.4.1 Correction method . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Statistical and systematic errors . . . . . . . . . . . . . . .

59
59
59
63
64
68
68
69

5 Results
5.1 Multiplicity distributions and NBD fits . . . . . . . . . . . . . . .
5.2 δη dependence of NBD k . . . . . . . . . . . . . . . . . . . . . . .
5.3 Npart dependence of β and αξ . . . . . . . . . . . . . . . . . . . .

71
71
72
73

6 Discussions
6.1 Absolute scale of the correlation length . . . . . . . . . . . . . . .
6.1.1 Small correlation length . . . . . . . . . . . . . . . . . . .
6.1.2 Relation with Bose-Einstein correlation . . . . . . . . . . .
6.2 Pseudorapidity independent correlation . . . . . . . . . . . . . . .
6.3 Other correlation sources . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Contributions from weak decay particles . . . . . . . . . .
6.3.2 Contributions from detector materials . . . . . . . . . . . .
6.4 Evaluation of the non-monotonic αξ . . . . . . . . . . . . . . . . .
6.4.1 Significance of local maximum . . . . . . . . . . . . . . . .
6.4.2 Assumption of the one correlation length . . . . . . . . . .
6.5 Physical interpretation of the results . . . . . . . . . . . . . . . .

77
77
77
79
80
81
81
85
86
86
88
89

7 Conclusion

91

A Tables of NBD fit results

95

4.2

4.3
4.4

ii

Chapter 1
Introduction
1.1

QCD phase transition

One of the most important subjects in physics is to investigate the origin andevolution of our universe. According to the Big Bang cosmology [1], it is considered that the universe was expanding with decreasing temperature and changing
the state of matter characterized by each energy scale i.e. phase transitions. In
the process of the evolution of the early universe, elementary particles; quarks,
leptons and massive gauge bosons obtained their masses by the spontaneous
symmetry breaking of scalar potential via Higgs mechanism [2] at a critical temperature of electro-weak energy scale around 250 GeV followed by the energy
scale of Grand Unified Theory (GUT) immediately after the inflation [3]. As aresult of further expansion of the universe, hadrons are produced at the criticaltemperature Tc < 200 MeV and obtain those effective masses by the spontaneouschiral symmetry breaking with formations of the quark pair condensate hq̄qi.
Figure 1.1 shows a conceptual drawing of the matter consisting of quarks andgluons, which is the gauge particle of the strong interaction between quarks andgluons, at higher and lower temperature compared to the Tc .
In other words, the state of matter can be separated by the Tc or the corresponding singularity on some proper thermodynamic variables. In fact, various singular effects are often observed as critical phenomena when matter passthrough phase boundaries like critical opalescence at the phase transition to superconductivity phases in the field of condensed matter physics at low temperature [4]. These phenomena have been understood by the phenomenological theory
of V. L. Ginzburg and L. D. Landau [5] based on the thermodynamical treatmentof matter. Therefore, those theories developed in the low temperature physics
will be of assistance to understand the state of matter even in higher temperature, if we apply the concept of the phase transitions to the process of evolutionof the universe.
If one focuses on the phase transitions from the quarks and gluons (called
1

2

CHAPTER 1. INTRODUCTIONa) quark and gluon phase
T > Tcb) hadron phase
T < Tc
meson

quark

gluon
baryonFigure 1.1: Conceptual drawing of matter that consist of quarks and gluons.
a) Matter is in T > Tc . b) Matter is in T < Tc , quarks are confined in hadrons.
together partons) phase to the hadrons phase, it should be treated based on thequantum chromodynamics (QCD) [6], which describes the strong interaction between the quarks and gluons. The QCD is the first principle theory for the strong
interaction, and it well describes many experimental results of the particle production and the structure of the proton by using the perturbative approach. On
the other hand, the phenomena regarding to the phase transition are often related with low momentum (soft) particles. This means the QCD phase transition
belongs to the non-perturbative regime. However, since QCD is a non-Abeliangauge theory, it is difficult to derive the correct answers for the non-perturbative
regions analytically without introducing any phenomenological models.
Y. Nambu and G. Jona-Lasinio (NJL) introduced a model for a dynamicalmechanism of the chiral phase transition inspired by the phase transition of thesuperconductivity [7, 8]. It is expressed by a Lagrangian as
1
L = q̄iγ · ∂q + g[(q̄q)2 + (q̄iγ 5 τ q)2 ],
2

(1.1)

where γ 5 is the chirality operator and the first and second term correspond to
kinetic energy and potential term, respectively. For simplicity, considering onlyu and d quarks with zero masses, this Lagrangian can be rewritten by the linear

1.1. QCD PHASE TRANSITION

3 









Figure 1.2: Potential of the linear sigma model in Equation (1.2). a) QCD
vacuum in T > Tc . b) QCD vacuum in T < Tc .
sigma model (LSM) which is expressed as
1
L = [(∂µ σ)2 + (∂µ π)2 ] + V (σ 2 + π 2 ),
2

(1.2)

where σ and π meson’s field are defined by
σ = q̄q,
π = q̄iγ 5 τ q,

(1.3)

respectively. Figure 1.2 indicates the potential shape in Equation (1.2) and theposition of the QCD vacuum. The chiral symmetry is broken in the case ofchoosing a direction of the QCD vacuum hq̄qi for π = 0 at T < Tc as shownin Figure 1.2 b). Since the QCD vacuum has a finite vacuum expectation value
at T < Tc , a meson obtains the large effective mass as compared to the quark
current mass. Masses of dynamical particles are described by the excitationof the vacuum as shown in the arrows inside the figure. Since the arrow in
π direction indicates zero potential gradient, π meson is created as zero mass
particles named as Nambu-Goldstone boson as a result of this spontaneous chiralsymmetry breaking. It is supported by the small mass of π meson as compared tothe other hadrons. Thus, the hq̄qi condensation become a proper order parameterin the case of the chiral phase transition. Many phenomenological models of the

4

CHAPTER 1. INTRODUCTION

Figure 1.3: Theoretical QCD phase diagram for two massless quarks, up anddown, as a function of temperature T and baryo-chemical potential µ [14]. Solidand dashed line corresponds to the first and second order phase boundary, respectively. The phase tagged with 2SC is a color superconductor phase in which
up and down quarks with two out of three colors pair and form a condensate.
QCD phase transition have been developed based on the NJL model in modern
nuclear physics.
There is the another theoretical approaches for the QCD phase transition. It
is considered that two phases can exist in the many-body system separated by
the critical temperature Tc . One is the hadronic phase and the other is quark
deconfined phase namely quark-gluon-plasma (QGP) [9–11]. This concept has
been developed as an extension of hadron bag models [12]. The hadron bag
models are phenomenological theories to describe the quark confinement in a
hadron by introducing tensions from color electric flux between quarks as a bagpressure of the hadron [13]. In this concept, quarks are free from the confinedstate with increasing temperature or baryo-chemical potential of the system asshown in Figure 1.1 a).
The lattice gauge theory is the only way to calculate quantitative value ofthe transition temperature based on the first principle of QCD. Figure 1.3 shows
the QCD phase diagram for two massless quarks as a function of temperature T
and baryo-chemical potential µ [14]. Since the µ corresponds to the energy ofthe system per baryon number, the normal nuclear matter at zero temperature
indicates 1 GeV. According to the numerical lattice gauge simulation, two phases

1.2. THEORETICAL APPROACHES ON THE QCD PHASE DIAGRAM

5in Figure 1.3 are separated by the line from 150 < T < 200 GeV at µ = 0 [15].
Although the chiral phase transition and the phase transition to the quark deconfined state are different concepts in the QCD energy scale, the lattice gauge
simulation indicates the coincidence of the transition temperature at µ = 0 in
both cases [16]. However, it is known that there are several difficulties to obtain
the accurate values for the region of the large finite µ in the numerical calculations [17]. Therefore, it is crucial for both theoretical and experimental studies
to develop new methods to understand the phase structure quantitatively.
The motivation of this study is to investigate the QCD phase transition andthe structure of the phase boundary by creating hot and dense QCD matter
experimentally.

1.2

Theoretical approaches on the QCD phase
diagram

Before explaining experimental methods for the study of QCD matter, several
theoretical knowledge of the QCD phase diagram are reviewed in this section. The
theoretical studies of QCD in the non-perturbative regimes indicate that QCD
matter has the rich phase structure. The phase diagram can be parametrized bytemperature T and baryo-chemical potential µ.
The second order phase transition is predicted at µ = 0 and T > 0 by thelattice QCD calculation [15] at zero mass, m = 0. If one added one more dimension of quark mass m to the phase diagram, it can be depicted as Figure 1.4.
Figure 1.4 shows the three dimensional QCD phase diagram based on the NJL
model [18]. The first order phase boundaries extend to the mass direction ashatched area in the figure. Since actual quarks have been thought having afinite current mass, a smooth crossover transition is expected due to the finite
masses of quarks [19] at µ = 0 and T > 0 and m 6= 0. The crossover transitionis also predicted by the lattice QCD calculation in µ = 0 [20]. Thus, logically
one can expect that a critical end-point (CEP) exists at the end of the first order phase transition line [21]. According this picture, the CEP is defined by
the connecting point of the two second order lines and one first order line in
m = 0 plane. Consequently, the location of the CEP would be the landmark inunderstanding the whole structure of the phase diagram. Figure 1.5 shows thepredictions of the CEP locations based on the lattice QCD calculations, the NJL
models, the linear sigma model and the other phenomenological models. Dashedline is the expected second order phase boundary obtained by connecting two
CEP in the different lattice QCD calculations and the transition temperature atµ = 0. Although numerical calculations using the lattice gauge theory as well as
model calculations predict the existence of the CEP, none of them have reacheda quantitative agreement on the location at the present precision [22].

6

CHAPTER 1. INTRODUCTION

T/Λ

0.4
0.3
0.2
0.1
-0.1
-0.05

0
0.2

0.4

µ/Λ

0
0.6

0.05
0.8

0.1

m/Λ

Figure 1.4: Three dimensional QCD phase diagram based on the NJL model asa function of temperature T , baryon chemical potential µ and quark mass min [18].

The phase transition to the quark-gluon deconfinement phase derived by theseveral hadron bag models is the different concept from the QCD chiral phasetransition as explained in Section 1.1. Since the bag models does not assume thesecond order phase transition, the first order line obtained by this model does notinclude the CEP as shown in the solid line in Figure 1.5. However, those models
do not indicate any quantitative phase boundaries.
Therefore, experimental investigations are indispensable to pin down the location of the CEP and phase boundary to investigate properties of the phasestricture in the QCD energy scale based on fundamental observables.

T [MeV]1.3. HIGH ENERGY HEAVY ION EXPERIMENT

7

200

PHM4

LAT2

second order line
based on latticecalcurations

LAT1
150

NJL3

100

PHM2
PHM3

LSM

first order line of
MIT bag model

PHM1
NJL4

50

NJL2
NJL1

0
0

200

400

600

800

1000

1200

1400

1600

[MeV]

Figure 1.5: Theoretical predictions of the QCD phase boundary and critical endpoint summarized in [22]. Dashed line is expected second order phase boundary
connecting end-points of lattice QCD calculations in [15,23,24]. Markers indicate
critical end-points obtained by several theoretical calculations; filled circles arelattice QCD calculations (LAT1 in [23] and LAT2 in [24]), the open circle isbased on Linear Sigma Model (LSM in [25]), open diamonds are based on Nambu
and Jona-Lasinio model (NJL1 , NJL2 in [26], NJL3 in [27], NJL4 in [25]), andthe filled diamonds are based on other phenomenological models (PHM1 in [28],PHM2 in [29], PHM3 in [30], PHM4 in [31]). The solid line corresponds to thefirst order phase boundary by the MIT bag model [13].

1.3High energy heavy ion experiment

It is necessary to create the extremely hot and dense matter in order to verify
whether the QCD phase transition can occur and the new state of matter QGP
is formed in nature or not. In this case the hot and dense matter must have large
energy density beyond the normal nucleon energy density. One of the methods
to make it is colliding accelerated nucleus with high momentum to gain the energy density. For the purpose of producing a large size of the interaction region,
heavy ion collisions, A + A is effective compared to proton (anti-proton)-proton,

8

CHAPTER 1. INTRODUCTIONp (p̄) + p collisions, because former one has a larger radius and can collide many
incoming number of nucleons at the same time. There are many experiments tosearch for the QGP by using heavy ion accelerators in the past two decades asfollows. Firstly the beam energy of 2.1 GeV per nucleons (A GeV) was used in
Bevalac at Lawrence Berkeley Laboratory (LBL, 1975-1985), 15 A GeV heavy ion
beam became available by Alternating Gradient Synchrotron (AGS, 1987-1995) at
Brookhaven National Laboratory (BNL), and then the beam energy was achieved
at 400 A GeV in Super Proton Synchrotron (SPS, 1987-present) at European Organization for Nuclear Research (CERN). These were all fixed target experiments
at the extraction beam line. Relativistic Heavy Ion Collider (RHIC, 2000-present)
at BNL has accomplished to collide 100 A GeV of gold ions (Au) by using two
opposed circular beam line as the world’s first heavy ion collider experiment.
RHIC enabled us to study the QCD matter by the circumstance of center of√mass energy of 200 GeV per colliding nucleus pairs denoted as sN N .
It was already considered that the high temperature and high energy density
matter, like a fire ball, could be produced through the high energy nucleusnucleus collisions which explains observed multiple hadron productions in cosmic
ray experiments by E. Fermi and L. D. Landau around in 1950. J. D. Bjorken
has introduced a picture of the evolving matter in the intervening region of theincoming nuclei to estimate its energy density, which is independent of inertial
reference frame [32]. Figure 1.6 shows the schematic diagram of the head-on
colliding nuclei with respect to before and after the collisions based on the Bjorken
picture. The longitudinal width of the accelerated nuclei near the light velocity
in the beam direction z become less than 1 fm by the Lorentz contraction before
the collision as shown in Figure 1.6 a). The volume of the region of interest forthe produced matter after the collision in Figure 1.6 b) can be obtained as πR2 dz,
where R and dz is the transverse radius of the nucleus and the longitudinal gap
between the outgoing nuclei, respectively. The dz can be substituted with τf dy
by introducing a concept of a proper formation time τf for the produced matter,
where y is longitudinal rapidity expressed as!

1
E + qLy = ln,2
E − qL(1.4)

where E is energy and qL is longitudinal momentum for the accelerated nucleus.
The relations of the E, qL and y are
E = mT · cosh(y),qL = mT · sinh(y),(1.5)

where mT is the transverse mass defined by m2T ≡ m2 + qT2 = E 2 − qL2 . Althoughpartons start interactions immediately after the collisions, a finite time is needed
for the achievement of the thermal equilibrium. The volume of the system withthis concept of the proper formation time τf is appropriate for the estimation of1.3. HIGH ENERGY HEAVY ION EXPERIMENT

9a) before the collision
nucleusnucleuscollision vertex
partons(quarks and gluons)< 1 fmb) after the collision

region of interest

πR 2

leading partonsdz = τ f dy
Figure 1.6: Schematic diagram of the head-on colliding nuclei. a) The widthof accelerated nucleus becomes < 1 fm by the Lorentz contraction before thecollision. b) The region of interest for the particle production is indicated by thefilled area after the collision.
the energy density, if QGP is defined as a thermalized and partonic deconfinedphase. The rapidity y after the collision can be represented by a time t and alongitudinal space point z as
1t+z
y = ln
,
(1.6)
2
t−z
√
where a proper time τ is expressed as τ = t2 − z 2 . According to the relation, tand z becomet = τ · cosh(y),z = τ · sinh(y),

(1.7)

10

CHAPTER 1. INTRODUCTION

  

 !
"!#$%




Figure 1.7: Longitudinal Space (z) vs. time (t) diagram of a nucleus-nucleus collision. Produced matter is thermalized at the proper time τf plane. Hadornizations
occur at the τc plane.
where the longitudinal velocity β = z/τ in the laboratory frame is same as thecenter of mass frame. Figure 1.7 shows the space time evolution after the nucleusnucleus collision as a function of t and z. The beam of colliding nuclei, which areaccelerated near the light velocity, is indicated by the light cone. The hypersurfaces at τf (dashed line) and τc (dotted line) correspond to the proper formation
time and the hadornization time, respectively. The system achieves thermal equilibrium above the hypersurface of τf followed by parton-parton interactions after
the nucleus-nucleus collision. The temperature decreases with the expansion of
the system, and then the thermal hadornization occurs at the critical hypersurface of τc . The area surrounded by those two hypersurfaces corresponds to the
expected QGP phase. Since this phase is considered to achieve the thermalization, hydrodynamical treatments are possible. The filled area in the Figure 1.7
indicates a hydrodynamical volume element. According to this estimation of thevolume on the region of interest and an assumption that N particles producedat τf as hmT i = (qL2 + m2 )1/2 contribute to the energy production, the energy
density can be obtained as a function of τf ,hε(τf )i =hmT idN(τf )hmT i dN(τf )
=
.
2
πR dz
πR2 τf dy(1.8)

1.4. MOTIVATION OF THIS STUDY

11

Table 1.1: Bjorken energy density εBj at τf = 1 fm/c.
Accelerator
AGS
SPS
RHIC

Colliding nucleus
Au + Au
Pb + Pb
Au + Au√sN N (GeV)
5
17
200

εBj (GeV/fm3 )
1.5
2.9
5.4

Reference[33][34–36][35, 37]

With the relation ofdN
dET= hmT i
,
dy
dy

(1.9)

where ET is the total transverse energy of produced particles, the energy density
is also expressed ashε(τf )i =1 dET (τf ).
πR2 τfdy(1.10)

This is referred to as the Bjorken energy density εBj [32]. The εBj can be obtainedin each accelerator beam energy as shown in Table 1.1, if one puts τf = 1 fm/c
into the Equation (1.10), which is not based on any particular justifications but
based on Bjorken’s order estimate with respect to the particle production [32],and measured final state dET /dy by the experiments [33–37]. These estimations
contain some ambiguities due to the several assumptions, for instance, the assumption of τf and using the measured value of ET /dy in the final state, which is
not in the formation stage. It will be discussed in Section 4.1.4. However, as longas one refers to the Bjorken estimation of the energy density as a lower limit, thematter with a sufficiently high energy density for the QCD phase transition are
produced in the RHIC energy compared to the normal nuclear mater.

1.4

Motivation of this study

The direct indication of the dynamical chiral symmetry breaking is differenceof masses between light mesons e.g. σ (scalar) and π (pseudoscalar) as a resultof nondegeneracy [11]. Therefore, observations of those meson’s mass shift areconsidered to be a signature of the chiral phase transition [38]. However, it isexpected that the life time of those mesons are too long as compared to the
produced matter in heavy ion collisions. Since those mesons decay after thecooling of produced mater, their masses are not affected by the QCD medium.
Low mass vector mesons, ρ, ω and φ, have a possibility to keep the value ofthe shifted mass in the medium at the final state, because they have short life
times (∼ 10−22 second). However, measurements of the low mass vector mesons

12

CHAPTER 1. INTRODUCTION

are experimentally difficult due to the huge background of decay products fromthe other particles.
On the other hand, the suppression of the J/ψ production cross section, which
is a bound state of charm quarks, has been predicted as a direct signature on thephase transition to the quark deconfined state [39]. This is based on a mechanismof the Debye screening of the quark color charge. In this model, the screening radius becomes smaller than the binding radius between two quarks, and it resultsin the J/ψ suppression. However, this model has ambiguities on the unknown
production mechanism of the normal J/ψ production as a reference of the production cross section.
Therefore, it is necessary to investigate the QCD phase transition by theother complementary observables in a basic thermodynamic treatment of thematter. In this study, an analysis result of density correlations measured by
the PHENIX [40] detector at RHIC, which is potentially sensitive to both chiraland deconfinement phase transitions as explained in Chapter 2, will be discussed
under the experimental situation described in Section 1.3.

Chapter 2
Experimental observable
2.1

Thermodynamic singularity

In general, phase transitions are discussed with order parameters which characterize phases in many body systems. The transition point is defined by discontinuities of an extensive variables Φ as a function of intensive variables h, forexample temperature T , in thermodynamics. In this case, both Φ and h are thermodynamic variables. This Φ is defined by partial differential coefficient of Gibbs
free energy G with respect to the h. For instance, entropy S and specific heat Ch
correspond to the first order and second order derivative of G as indicated in
Equation (2.1) and Equation (2.2), respectively. The second order derivatives,
isothermal compressibility [41] and correlation length, are called susceptibility χcollectively.
∂GΦ(T, h) = −
∂h
χ=

∂Φ
∂h

!

T

!

∂2G
= −
∂h2

,
T

!

T

∂G
S=−
∂T
, Ch = −T

!

.

(2.1)

h

∂2G
∂T 2

!

.

(2.2)

h

Thus, the discontinuous or divergent behavior of the derivatives of G as functionsof thermodynamic variables determines not only the transition point but also thetransition order [4]. This is the thermodynamic singularity.
Here is one of the most obvious examples of phase transitions. Figure 2.1
shows the phase diagram of He4 as a function of pressure and temperature [42].
The phase indicated by ”Liquid He II” is the known super fluidity phase. Figure 2.2
shows the measured specific heat of the liquid He4 as a function of T − Tλ , where
Tλ is the critical temperature to the super fluidity phase [43, 44]. This figure ismade by changing the resolutions to Tc . The specific heat in Figure 2.2, whichcorresponds to the second order derivative as noted in Equation (2.2), indicatesthe divergence at T = Tλ . Therefore, the line with triangles in Figure 2.1 corresponds to the second order phase transition line.
13

14

CHAPTER 2. EXPERIMENTAL OBSERVABLE

Figure 2.1: The phase diagram of He4 as a function of pressure and temperature [42]. Liquid He II indicated inside the figure is the super fluidity phase. Theline with triangles corresponds to the second order transition line.

Figure 2.2: The specific heat of liquid He4 as a function of T − Tλ measured
by various resolutions [43, 44], where Tλ is the critical temperature for the super
fluidity phase.

2.2. DENSITY FLUCTUATION AND SUSCEPTIBILITY

15

The singular behaviors in the derivatives of G as in the specific heat of He4
are often seen when matter go through the phase boundary. That is to say, thecritical phenomena are not dependent on details of the local interactions among
the constituent particles in the matter but dependent on the global property ofthe matter such as susceptibility of the thermodynamic variables. Several global
properties; the number of dimensions of a system or the number of componentsof the order parameter and so on, can be considered to characterize the phasetransition. It is referred to as the Universality Hypothesis [45–47], which wasproven by using the renormalization group theory by K. G. Wilson [48,49] in thespecial case of the second order phase transition. Therefore, the thermodynamic
fluctuations of an order parameter or the singular behaviors of the susceptibility
become signatures of the phase transition even in the case of QCD phase transition. In Section 2.2, relations between an order parameter and the susceptibility
are explained based on the Ginzburg-Landau framework [5].

2.2

Density fluctuation and susceptibility

The relation between the thermodynamic fluctuations of order parameter andthe susceptibility is expressed by the Fluctuation-Dissipation theorem [4] more
precisely. In this section, however, it is explained based on the phenomenological Ginzburg-Landau (GL) [5, 50] framework for a scalar order parameter tounderstand its essence easily.
GL describes the relation between a free energy density f and an order parameter φ as a function of the system temperature T . By adding a spatiallyinhomogeneous term A(T )(∇φ)2 and an external field h, the general form is described as follows
1
f (T, φ, h) = f0 (T ) + A(T )(∇φ)2 +
2
1
1 4
2
a(T )φ + bφ + · · · − hφ,
2
4

(2.3)

where f0 is the equilibrium value of the free energy density, terms with odd
powers are neglected due to the symmetry of the order parameter introduced inthis analysis, and the sign of b is used to classify the transition orders: b < 0 for
the first order, b > 0 for the second order and b = 0 at the critical point. Since
the order parameter should vanish above a critical temperature Tc , it is naturalfor the coefficient a(T ) to be expressed as a(T ) = a0 |T − Tc |, while b is usuallyassumed to be constant in the vicinity of Tc . Figure 2.3 shows the free energy
density up to the fourth order therm as a function of the order parameter φwith respect to the different signs of a, where the sign of a > 0, a = 0 anda < 0 corresponds to the case at T > Tc , T = Tc and T < Tc , respectively. In thefollowing, the higher order terms beyond the second order term is neglected. This

16

CHAPTER 2. EXPERIMENTAL OBSERVABLE'(')*+,*-,*.,&

Figure 2.3: The free energy density as a function of scalar order parameter φwith the different signs of a(T ) in Equation (2.3). Lines with a > 0, a = 0 anda < 0 corresponds to T > Tc , T = Tc and T < Tc , respectively.
approximation corresponds to a picture where a system approaches to the phaseboundary far from the boundary, since φ introduced in the following analysis
is close to zero in the regions far from Tc . In this sense, the approximation isinsensitive to the details of the phase transition order, i.e. higher order terms,
but only sensitive to the behaviors near the Tc .
The order parameter in this analysis corresponds to the density fluctuation
as a function of spacial point r, which is defined as
φ(r) = ρ(r) − hρ(r)i,(2.4)

where the pair of brackets indicates an operator to take the average. With theP
Fourier expansion of the density fluctuation in r, φ(r) = k φk eikr where k isa wave number, one can express the deviation of the free energy density due tospatial fluctuations from the equilibrium value f0
∆F/Y

1 Z
=
(f − f0 )drY
1X|φk |2 (a(T ) + A(T )k 2 ),
=
2 k

(2.5)

where Y is the total range corresponding to a volume and up to the second orderterms are taken into account as an approximation in the vicinity of the critical2.2. DENSITY FLUCTUATION AND SUSCEPTIBILITY

17

point in Equation (2.3). Given the free energy deviation, one can obtain thestatistical weight W for fluctuation φ(r) to occur in a given temperature T as
W (φ(r)) = Ne−∆F/T .

(2.6)

Therefore the statistical average of the square of the density fluctuation with thewave number k is described as2

h|φk | i =
=

Z +∞
−∞

2

|φk | W

X

ikr

φk e

k

!

dφk

1
NT
.
Y a(T ) + A(T )k 2

(2.7)

An experimentally observable two point density correlation function can be
related to the statistical average of the square of the density fluctuation. With adensity ρ(ri ) for a given sub-volume dri , the two point density correlation G2 isexpressed in the case of hρ(r1 )i = hρ(r2 )i = hρi as
G2 (r1 , r2 ) = h(ρ(r1 ) − hρi)(ρ(r2 ) − hρi)i,(2.8)

where case 1 coinciding with case 2 is excluded to simplify the following discussion. Multiplying both sides of Equation (2.8) by e−ikr ≡ e−ik(r2 −r1 ) andintegrating over sub-volume dr1 and dr2 givesYZZ

G2 (r)e−ikr dr = h| (ρ(r) − hρi)e−ikr dr|2i= h|φk |2 i.

(2.9)

From Equation (2.7) and Equation (2.9), G2 can be related to the inverse Fouriertransformation of the statistical average of |φk |2 . This integral can be evaluatedby the Ornstein-Zernike formula [51, 52] as
G2 (r) ∝(|r|−(d−1)/2 e−|r|/ξ
|r|−d+2(T > Tc ),(T = Tc )(2.10)

where d corresponds to the number of dimensions of the system. Therefore in
the one dimensional case G2 is described as
G2 (r) =NTξ(T )e−|r|/ξ(T ),2
2Y A(T )(2.11)

where the correlation length ξ(T ) is introduced, which is defined as
ξ(T )2 =A(T ).
a0 |T − Tc |(2.12)

In general, a singular behavior of ξ(T ) as a function of T indicates the criticaltemperature of the phase transition.

18

CHAPTER 2. EXPERIMENTAL OBSERVABLE/01213601713401513Figure 2.4: The coordination of spin obtained by a simulation of two dimensional
Ising model [53] in a) T < Tc , b) T = Tc , and c) T > Tc . Black and whitecorresponds to upward and downward spin, respectively.
The wave number dependent susceptibility can also be defined from the expansion of the GL free energy density based on Equation (2.3) and Equation (2.5)
as follows,
χk

∂2f
= −
∂h2

!

=

∂h
∂φk

!−1

!−1

∂ 2 (∆F/Y )
=
∂φ2k
1
.
=
a0 |T − Tc |(1 + k 2 ξ(T )2)(2.13)In the case of the long wavelength limit, which correspond to k = 0, the susceptibility can be expressed asχk=0 =

1
2Y 2=
ξ(T )G2 (0).
a0 |T − Tc |NT

(2.14)In this framework, the ξ and χk=0 diverge at the same temperature. This is therelation between the fluctuations of order parameter (density fluctuation) andthe correlation length ξ and the susceptibility χk=0 .
The correlation length ξ is useful to understand the order of the phase intuitively. It also have an advantage for the application to the correlations among
dynamically produced particles. Figure 2.4 show coordinations of spin in a twodimensional Ising model simulation [53]. Figure 2.4 a) indicates the ordered statein T < Tc . Most of the directions of spins are lined up to upward in this phase.
A typical size of the domains itself corresponds to the correlation length ξ. Figure 2.4 b) is the coordination at T = Tc . One can find various sizes of domains

2.3. APPLICATION TO THE PHASE TRANSITION

19

exist together and the typical scale in T < Tc vanishes at T = Tc . This is theorigin of the divergence of ξ. Figure 2.4 c) shows the spin coordinate of disorderedphase in the T > Tc . In this case, the ξ decreases with increasing T .

2.3

Application to the phase transition

The first attempt to apply the free energy discussion to A + A collisions canbe found in [54], and the application to the case of the QCD phase transition
is indicated in [55]. Theoretically, further prospects have been suggested withrespect to derivations of the critical exponents [55] and to detecting the criticalopalescence [56] at the phase transition of the QCD matter based on the chiralcondensate.
The GL framework is applicable to the density correlations in the longitudinal space coordinate z in the heavy ion collisions based on the picture thatthe system dynamically evolves. A common proper time frame should be introduced as shown in Figure 1.7 for each hydrodynamical sub volume element in thisstudy for the purpose of the application. Therefore, the differential length dz between the neighboring hydrodynamical volume elements at the common proper3ξ3 (0.015)
ξ3 (0.01)

2.5

ξ3 (0.008)
ξ3eq (0.015)
ξ3eq (0.01)
ξ3eq (0.008)

ξ 3 [fm 3]

2
1.5
1
0.5
0
200

180

160

140

120

100
T [MeV]

Figure 2.5: Cube of the equilibrium correlation length (thin lines) and nonequilibrium correlation length (thick lines) as a function of temperature [60].

20

CHAPTER 2. EXPERIMENTAL OBSERVABLE√
time τ = t2 − z 2 is expressed as dz = τ · cosh(y)dy, where y is rapidity, as
introduced in [57]. The matter produced in the collision expands longitudinallyfrom its earliest time, which leads to cooling after the initial thermalization. If
the system’s evolution takes it near a critical phase boundary as it cools, thenthe large correlated density fluctuations will appear as T approaches Tc fromabove. If the expansion after that point is rapid enough then these fluctuations
can potentially survive into the final state [58]. One might have a question on
a possible incompleteness of the thermal equilibrium for the produced matterin heavy ion collisions. Nevertheless, it is indicated that the universal singulareffects can survive on the final state observables such as correlation length andfluctuations [59] originating from the critical phase boundary according to a hydrodynamical calculation [60]. Figure 2.5 shows the correlation length for theequilibrated fluid (thin line) and non-equilibrated fluid (thick line) as a functionof critical temperature in the hydrodynamical calculation.
Experimentally, spatial density fluctuations in longitudinal space z in the=;<

ABHF?>@ABCD>?@ABCDEFB>CGEG?>LFGEGLBM @ABCDN?O>HBFP

IJK @ABCD
89 :Figure 2.6: Schematic diagram of susceptibility χ or correlation length ξ as afunction of temperature T or energy density ε of the system. Perpendicular
dashed line indicates the hypothetical QCD phase boundary.

2.4. CORRELATION AND FLUCTUATION

21

early stage of an A + A collision can be measured in rapidity or pseudorapidity
space in the final state. The order parameter, in this case, corresponds to themultiplicity density fluctuation for identified or inclusive particles from the meandensity. If the study was limited to only a narrow region around mid-rapidity,
then dz ∼ τ dy is valid with the approximation of cosh(y) ∼ 1. Therefore one canobserve density fluctuations in the z coordinate as being mapped onto density
fluctuations in rapidity space. In this study, a survey on the susceptibility χ andcorrelation length ξ have been reported as a function of initial energy density ε√of produced matter in Au + Au collisions at sN N = 200 GeV. It is the study
to search for the phase boundary between the hadrons and QCD matter by targeting the thermodynamic singularity. Figure 2.6 shows a qualitatively expectedbehavior of ξ and χ as a function of temperature T or energy density ε of thesystem. Basically, ξ and χ are expected to smoothly decrease as power law withincreasing T or ε as described in Equation (2.12) and Equation (2.14) in T ≫ Tc ,respectively. Therefore, if no phase boundary is contained in the surveying range,
there is no divergent behavior as shown in solid line of Figure 2.6. However, if
there is transition point in the surveying range, χ and ξ could diverge as shown indashed and dotted line in Figure 2.6. Although the χ and ξ in the QCD matterare expected to decrease as a function of T and ε in T > Tc side, there is no
reason to expect the same values on them over partonic and hadronic matter
phases. In Section 2.4, the concrete analysis method will be explained for the
extraction of χ and ξ from the multiplicity fluctuations.

2.4

Correlation and fluctuation

In this study, the density fluctuation will be discussed by measuring inclusive
charged particle multiplicity distributions as a function of the various pseudorapidity η window size of δη. In the region around mid-rapidity used in this analysis,
one can approximate rapidity y by pseudorapidity η for the inclusive charged par√ticles, since the mean transverse momentum hpT i observed in sN N = 200 GeV
collisions is so high (hpT i = 0.57 GeV/c ≫ mπ ) as explained in Section 4.2.

2.4.1

Multiplicity distribution

It is known that the charged particle multiplicity distributions in hadronic processes are well described by the Negative Binomial Distribution (NBD) [61] empirically and it is confirmed in p (p̄) + p [62], A + A [63] collisions and evenin e+ e− annihilations [64] as shown in Figure 2.7 through Figure 2.9.
The
distribution is expressed as
P (n) =

Γ(n + k)(µ/k)nΓ(n + 1)Γ(k) (1 + µ/k)n+k(2.15)

22

CHAPTER 2. EXPERIMENTAL OBSERVABLE

Figure2.7: Charged particle multiplicity distributions in p̄ + p collisions at√
s = 540 GeV and NBD fits (solid line) [62]. Pseudorapidity intervals ηc are
indicated inside the figure.

Figure 2.8: Charged particle multiplicity distributions in central O + Cu collisions at 14.6 A GeV/c and NBD fits (solid line) [63]. Markers inside the figure
correspond to the differences of pseudorapidity η interval.

2.4. CORRELATION AND FLUCTUATION

23

Figure 2.9: Charged particle multiplicity distributions in hadronic Z 0 decays at
e+ + e− annihilations for various rapidity gaps |y| [64]. a) |y| < 0.5, b) |y| < 1.0,c) |y| < 1.5, d) |y| < 2.0. Markers inside the figures correspond to the number ofrequired jets. Solid lines indicate the NBD fits.
where µ is the mean of the distribution i.e. µ = hni, and k corresponds to thedifference between its width and that of a Poisson with that mean.
Poissonian : P (n) = e−µ µn /Γ(n − 1).
BED : P (n) = µn /(1 + µ)n+1.

(2.16)(2.17)

Thus the NBD coincides with the Poisson distribution in Equation (2.16) andthe Bose-Einstein distribution (BED) in Equation (2.17), in the case of k = ∞and k = 1, respectively. It should be notified that in all cases of above distributions, gamma functions are used instead of factorial functions for the statistical

24

CHAPTER 2. EXPERIMENTAL OBSERVABLE

data analysis. This is proven by using generating function [65] and familiar in
Quantum Optics, but one can easily understand by looking the relation between
the k parameter and standard deviation σ of the NBD,
1 1
σ2
= + .
2
µ
µ k

(2.18)

This is the reason why NBD is called as the generalized Bose-Einstein distribution. Hence multiple BED are convoluted in NBD and the multiplicity of BED
is expressed by the k parameter.
Here, considering the normalized factorial moment (NFM) with respect to a
distribution is convenient to figure out the property of the multiplicity distribution [66–69]. The q-th order NFM (Fq ), which is defined as
Fq ≡hn(n − 1) · · · (n − q + 1)i,hniq(2.19)

represents the strength of correlations among q particles. The second order NFM
can be deformed with the standard deviation σ 2 ≡ hn2 i − hni2 ashn(n − 1)i
hn2 i − hni
σ 2 + hni2 − hni
=
=
hni2
hni2
hni2
1
σ2
= 1+ 2 − .
µ
µ

F2 =

(2.20)As a bottom line, the relation between F2 and NBD k can be obtained with
Equation (2.18) as
k −1 = F2 − 1.

(2.21)

Thus the inverse of k parameter in NBD, which corresponds to called as cumulant
moment [70, 71] of the second order, directly reflects the degree of correlation
between any two particles produced into an experimental window. In other words,
NBD is good approximation for the multiplicity distribution in the case of two
particle correlation is dominant in the distribution. This is why the multiplicity
distribution is agree with NBD in the past high energy experiments.

2.4.2

Extraction of correlation

At the first setout, the normalized factorial moment (NFM) is transformed forthe purpose of extractions of density correlation from multiplicity fluctuations byintroducing q particle density ρq in the given pseudo rapidity window size δη asM
1 X
hnm (nm − 1) · · · (nm − q + 1)i
Fq (δη) ≡M m=1
hnm iq2.4. CORRELATION AND FLUCTUATION
M
1 X
=
M m=1
M
1 X
=
M m=1

R δη
R δη

ρq (η1 , · · ·, ηq )Πqi=1 dηiR( δη ρ(η)dη)qρq (η1 , · · ·, ηq )Πqi=1 dηi,(ρ̄m )q25(2.22)

where the full pseudo rapidity interval ∆Y is divided into M equal bins i.e. ∆Y =Mδη. Using this bin divided F2 (δη), the mathematical connection between
the F2 (δη) and the two particle correlation function C2 [72–74] can be expressedashn(n − 1)i
F2 (δη) =
=
hni2

RR δη

ρ2 (η1 , η2 )dη1 dη2{ ρ1 (η)dη}2RR δη
C2 (η1 , η2 ) + ρ1 (η1 )ρ1 (η2 )dη1 dη2
1
=
2
(δη)
ρ¯1 2
Z Z δη
C2 (η1 , η2 )1dη1 dη2 + 1
=
2
(δη)
ρ¯1 2
R δη

(2.23)

where n is the number of produced particles and δη is the pseudorapidity window
size inside which the multiplicities are measured. In Equation (2.23) one and twoparticle inclusive multiplicity densities ρ1 and ρ2 will be introduced based on
the inclusive differential cross section relative to the total inelastic cross section
σinelastic [61] as follows
1dσ = ρ1 (η)dη,
σinelastic
1d2 σ = ρ2 (η1 , η2 )dη1 dη2 .
σinelastic(2.24)

Here ρ¯1 is the average density per unit length within δη which is defined as
1
ρ¯1 =
δη

Z δη

ρ1 (η)dη.

(2.25)

With these densities, the two particle density correlation function is defined as
C2 (η1 , η2 ) ≡ ρ2 (η1 , η2 ) − ρ1 (η1 )ρ1 (η2 ).

(2.26)

Instead of measuring C2 or F2 directly, in this analysis the NBD k parameter as ameasure of particle correlations over η are extracted. This is partly for historicalreasons [63], but also because, as shown in Section 4.4, the measured k can becorrected for the detector imperfections in a very robust way by using a statisticalproperty of NBD, while the same correction made at the level of F2 would require
additional information on the parent distribution.

26

CHAPTER 2. EXPERIMENTAL OBSERVABLE

The normalized two particle correlation function C2 in the experiment can beparametrized based on the one dimensional functional form obtained in the GL
framework (see Equation (2.11)) asC2 (η1 , η2 )= αe−|η1 −η2 |/ξ + β,2
ρ¯1

(2.27)

where ρ¯1 is proportional to the mean multiplicity, and the scale factor α is thestrength of the correlations at the zero separation. The constant term β arises
from any kind of experimental and physical correlations which are independent of
the pseudorapidity separation, such as the residual effect on the finite resolution
for event characterizations in heavy ion collisions. One has to take into accountthe fact that the damping behavior in Equation (2.11) is caused only by thespatial inhomogeneity of the system at a fixed temperature. In realistic heavy
ion collisions and event samples, there is no single relevant temperature.
Finally, the relation between the NBD k parameter and the pseudorapidity
window size δη can be obtained by the substitution of Equation (2.27) into Equation (2.23) and the integration over the range 0 < η1 < δη and 0 < η2 < δη as
k −1 (δη) = F2 − 1 =2αξ 2(δη/ξ − 1 + e−δη/ξ )
+ β.
δη 2

(2.28)In the limit of ξ ≪ δη, Equation (2.28) can be approximated ask(δη) =1(ξ ≪ δη),2αξ/δη + β(2.29)

where experimentally it can not be resolved α and ξ separately, but the productαξ can be directly determined. The product is related to the susceptibility inthe long wavelength limit, χk=0 ∝ |T − Tc |−1 for a given temperature T based on
Equation (2.14). Combined with the parametrization in Equation (2.27), the αξ
product should then followαξ ∝ ρ¯1 −2

1.
|1 − Tc /T |(2.30)

Since ρ¯1 can be expected as a monotonic function of T in the limit of T far from
Tc , αξ should vary monotonically as a function of T . However, if T approaches
Tc , the αξ product will show a singular behavior. Therefore, any non-monotonic
increase of αξ could be an indication of T ∼ Tc near a critical point. If theexperimental bias term β is excluded in Equation (2.29), the slope in k vs. δηthus contains crucial information on the phase transition.
It is worth mentioning that in this method, correlations on scales even smaller
than the minimum δη window can be meaningfully discussed based on the differences of the NBD k as a function of δη window sizes, since the correlations arealways integrated from the limit of the detector resolution to δη window size.

Chapter 3
Experimental setup
3.1

Relativistic heavy ion collider (RHIC)

The relativistic heavy ion collider (RHIC), which is started the operation in 2000,
at Brookhaven National Laboratory is the world first collider that can accelerate
heavy ions such as gold ion to one hundred GeV/c order of the beam momentum.
The heavy ion sources are made at the Tandem Van de Graaff facility. The
facility consists of two 15 million volt electrostatic accelerators. It can provide
more than 40 different types of ion beams. The produced heavy ions are sent
to the Booster synchrotron and accumulated to increase the beam intensity, andthen the beam is injected to the Alternating Gradient Synchrotron (AGS). At the
AGS, the heavy ion beam is further accelerated to 9 GeV/c to inject the RHIC
ring. After the injection to the RHIC ring, the beam is finally accelerated to the
maximum energy of 100 GeV/c.
The RHIC ring has 3.82 km length of two beam pipes, which is arranged toaccelerate the beam in the opposite direction each other. The beam in each ringcan be collided at 6 collision points. Since at the maximum 120 beam bunch canbe stored at the same time, its designed luminosity corresponds to 2×1026 cm−2 s−2
for gold ions. Figure 3.1 shows the path of the heavy ion beam from the Tandem
Van de Graaff facility to the RHIC.

3.2

PHENIX detector overview

The Pioneering High Energy Nuclear Interaction eXperiment (PHENIX) is oneof the four experiments operating at RHIC [40]. The detector of PHENIX is amulti purpose detector, which is constructed to address for the measurements ofa lot of possible observables in the heavy ion physics and spin physics. Therefore,
the PHENIX detector consists of more than 15 sub systems. Figure 3.2 shows
the schematic view of PHENIX detector.
The PHENIX sub systems can be categorized as follows. The first category is27

28

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.1: Schematic diagram for path of heavy ion beam at RHIC.
the global detectors, which are placed at the forward region from the intersecting
point of nucleus beams for the purpose of beam measurements and triggers of
nucleus-nucleus collision. There are two types of detectors named as the zero degree calorimeter (ZDC) and the beam-beam counter (BBC) [75]. The ZDC and
BBC are designed to measure the number of spectator neutrons in the colliding nucleus and particles from the collision participant, respectively. A detailed
descriptions of the global detectors can be found in Section 3.3.
The second category is the central arm spectrometer for the measurement of
produced particles by the nucleus-nucleus collisions at the midrapidity region.
The PHENIX detector has two central spectrometer arms, denoted as East andWest. Each central arm covers the pseudorapidity range |η| < 0.35 and sub-3.2. PHENIX DETECTOR OVERVIEW

29

Figure 3.2: Schematic view of PHENIX detector.

tends an azimuthal angle range ∆φ of π/2 around the beam axis (z direction).
The nominal integrated magnetic field of 0.7 T·m can be applied for the gap
region between the beam pipe and the central arm spectrometer by the central
magnet system (CM) [76]. It enables the momentum measurement of charged
particles and particle identifications (PID) with the following tracking detectors and PID detectors. A drift chamber (DC) and multi wire chambers with
pad readout (PC) [77] are used for the charged particle tracking and momentum
measurement. The particle identification with respect to the charged hadrons canbe done by the time of flight counter in wide momentum range [78]. At the endof the central arm spectrometer, two type of electro magnetic calorimeter (EMCal) [79] are placed to observe neutral particles and for the energy measurementof neutral and charged particles. Details of these detectors will be explained in
Section 3.4. There are the other types of PID detectors in the central arm spectrometer. Electrons and positrons are identified by the ring imaging Čerenkov
counter (RICH) [78]. The time expansion chamber (TEC) is placed at the backward of RICH from the beam pipe for the purpose of high pT electron positron

30

CHAPTER 3. EXPERIMENTAL SETUP

identifications by using the energy loss information (dE/dx). Since this analysis
only focuses on the inclusive charged particles, mainly charged hadron, the RICH
and TEC were not used. Figure 3.3 shows the layout of these detectors above.

PHENIX Detector
PC3
PC2

PbSc

PC3
TEC

CentralMagnet

PbSc

PbSc

PbSc
DC

DC
RICH

RICH
BB

PbSc

PbGl
MVD

PC1

PbSc

PC1

PbGl

TOF

Beam View

West

East
et

So

uth

M

uo

Central Magnet
nM

nM

M
th

ag
n

n
ag

uo

r
No

et

BB
ZDC SouthZDC North

MuID

MuID
MVD
MuTr

South

Side View

North

Figure 3.3: PHENIX detector layout in beam view (top) and side view (bottom).

3.3. GLOBAL DETECTORS

31

The final category of the subsystems is the data acquisition system and online
trigger system as described in Section 3.5. One can find detailed descriptions
in [80, 81].
The detector subsystems relevant for this analysis will be briefly summarized
below. The Au + Au minimum bias trigger and collision centrality were provided
by combining information from both BBC and ZDC. The collision vertex point ineach event, which is measured by BBC, is used not only for the charged particle
tracking but also for the purpose of event selections due to the fiducial volume of
the central arm spectrometer. The tracks of charged particles are measured by
the DC by requiring the two PC (PC1 and PC3) hit association in the East arm.
Cluster positions in EMCal made by charged particles are partly used for theselection of charged tracks to optimize the several parameters, which are relatedwith track associations. The EMCal is also used for the measurement of totaltransverse energy ET in each collision centrality. Since this analysis has been
done at the condition of no magnetic filed to enhance the low pT statistics of the
inclusive charged particles as described in Section 4.2.2, CM and PID detectors
are not used directly. However, the measured pT spectra of charged hadrons byusing those detectors are used for the estimation of background contributions inthis analysis.

3.3

Global detectors

PHENIX has two types of global detectors, which are placed at the forward region
at the intersection point of the nucleus beams provided from RHIC, for the purpose of collision triggers and determinations of event characterization. One is thezero degree calorimeter (ZDC) and the other is the beam-beam Counter (BBC).
Both BBC and ZDC have the same type of front-end electronics (FEM) modules for their readout. The FEM consists of discriminators, shaping amplifiers,
time-to-voltage converters (TVC) and flash ADC’s (FADC). The time and pulse
height information are digitized at the FEM’s, and the digitized values are stored
in digital buffer memories for the time of 40 beam bunch crossings to wait for the
event trigger decisions. The basic specifications of them are introduced in thissection, and their actual performances will be explained in Section 4.1.

3.3.1

Zero degree calorimeter (ZDC)

The zero degree calorimeter (ZDC) [82] in the PHENIX detector is a pair ofhadron calorimeters that are placed at a distance of 18.25 m from the intersection point of nucleus beams. Figure 3.4 shows a schematic layout of the ZDC, (A)and (B) correspond to the top view and the view along with the beam direction,
respectively. Since the ZDC is mounted on the backward of the bending magnet between two RHIC beam pipes, almost of the charged particles produced by

32

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.4: Schematic view of the ZDC layout for the top view (A) and the viewalong with the beam direction (B).

nucleus-nucleus collisions are swept out before the ZDC. The ZDC is composedof the tungsten absorber and layers of optical fibers to collect Čerenkov light,
and its dynamic range of energy deposits is optimized for the neutron response
i.e. it is adjusted by the single neutron peak. Therefore, the signal can be used
for the minimum bias level 1 collision trigger by requiring the coincidence hits ofboth side of the ZDC without the beam-gas background events, where beam-gas
background means collisions between the nucleus in the beam and residual gasesin the RHIC beam pipe. A method of the estimation of the trigger efficiency usedin this analysis will be explained in Section 4.1.2. In fact, the ZDC can count the
number of spectator neutrons in nucleus, which are not come from the nucleusnucleus inelastic collision. Combined with the information from the beam-beam
Counter (see Section 3.3.2), event characters of heavy ion collisions related with
impact parameter is determined. More details of the collision centrality measurement will be described in Section 4.1.3.

3.3. GLOBAL DETECTORS

3.3.2

33

Beam-beam counter (BBC)The beam-beam counter (BBC) [75, 83], which is one of the timing detectors in
PHENIX, plays several roles to provide basic information of the nucleus-nucleus
collisions. The first role is to provide the minimum bias trigger of the collisions,second one is the determination of the collision centrality. The vertex point of
the nucleus-nucleus collision in beam direction Vz and its collision timing T0 canbe derived by the hit timing information of the BBC. The Vz is used for the track
association of the charged particles, which is reconstructed by the central arm
tracking detector (see Section 3.4.1). The T0 value is used for the start timing of
the time of flight measurement of the charged particle with the PHENIX time offlight counter (see Section 3.4.3). In this section, the method of the derivationof event information from the measured values is described with explanations of
the BBC components.
The pair of BBC’s is placed at a distance of 1.44 m from the nominal intersection point of the nucleus beam along with the beam axis i.e. forward andbackward regions. One is called as North side of the BBC and the other is called
as South side of the BBC. The pseudorapidity (η) coverage and acceptance inazimuth (φ) corresponds to 3.1 < |η| < 4.1 and 2 π rad, respectively. EachBBC assembly has 64 elements consisting three cm quartz Čerenkov radiatorand photo multiplier tube (PMT). Figure 3.5 shows the mount position of the
BBC in PHENIX and its element. Since the BBC is placed at the position within
the magnetic field, a mesh dynode PMT is chosen to be made less subject to theamplification of signals [83]. The Čerenkov quartz radiator is adopted to achievea good timing resolution, because the coincidence hits information are used forthe level 1 trigger. The intrinsic timing resolution is known as ∼ 40 psec by
the test beam experiment [83]. The method to make the minimum bias trigger
and haw to estimate the trigger efficiency will be explained in Section 4.1.2. The
actual dynamic range of the charge measurement in each element of the BBC
is adjusted to be able to measure at maximum 30 incident minimum ionizing
particles. Therefore, in total ∼ 4,000 particles from a nucleus-nucleus inelastic
collision can be measured by the BBC. Using this information the collision centrality is determined with count of spectator neutrons measured by the ZDC as
described in Section 4.1.3.
The vertex point Vz and collision timing T0 can be calculated by the average
hit timing over all elements in each side as follows
TS − TNVz =· c,2
TS + TN − 2L/c
T0 =
,
(3.1)
2
where TS and TN corresponds to the average hit timing in South and North side,
respectively. L is the distance between the two BBC sides. This primary collision point Vz is used for the charged particle tracking with central arm tracking

34

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.5: (a) One of the BBC elements consisting a mesh dynode PMT withthree cm quartz radiator. (b) The BBC assembly has 64 elements. (c) The BBC
is mounted on the PHENIX detector surrounding the RHIC beam pipe.
detectors (see Section 3.4.1). The resolution of Vz corresponds to 0.5 cm at the√central Au + Au collisions at sN N = 200 GeV, which was evaluated by thecollision data [75]. The longitudinal size of beam bunch at RHIC Au + Au collisions is designed to be 25 cm RMS. However, there is a finite fiducial range forthe collision vertex in beam direction for the central arm spectrometer. For thisreason, a finite range of acceptable Vz is also set for the level 1 trigger. The T0
value is used as the start timing of the time of flight measurement as describedin Section 3.4.3. The resolution of T0 corresponds to 20 psec in the case of the√central Au + Au collisions sN N = 200 GeV.

3.4

Central arm spectrometer

Several detectors in PHENIX central arm spectrometer, which is used in thisanalysis, are focused on in this section. This section is organized as follows;
tracking detectors for charged particles are described in Section 3.4.1 and 3.4.2,
a detector for charged hadron identification is explained in Section 3.4.3 anda electro-magnetic calorimeter for the total transverse energy measurement at
midrapidity is introduced in Section 3.4.4.

3.4. CENTRAL ARM SPECTROMETER

3.4.1

35

Drift chamber (DC)The PHENIX drift chambers (DC) [77] are placed in the region from 2 to 2.4 m
from the beam axis (z direction) and 2 m along the z direction as shown in Figure 3.3. Figure 3.6 shows the construction of the DC frame. Each DC measures

R
.02

=2
m

o
90
6m

2.5 m

R

.4
=2

Ti frame
mylar window

Figure 3.6: Construction of DC frame.
charged particle trajectories in the r - φ direction to determine the transverse
momentum pT of each particle. The single wire resolution is designed to better
than 150 µm and 2 mm in r - φ and z direction, respectively. Two tracks can beseparated whose distance at the reference plane is greater than 1.5 mm by thesingle wire with 99% efficiency.
The DC frame is divided in 20 equal sectors. Each sector has six types ofwire modules named as X1, U1, V1, X2, U2 and V2. Figure 3.7 shows the layoutof these wires in one sector. The X1 and X2 wires run in parallel to the beamaxis to determine the azimuthal angle of charged particles. Since U1, V1, U2

36

CHAPTER 3. EXPERIMENTAL SETUP

4.5 degr.

Wire orientations, top view
V2 − planes

U2 − planes

Termination wires

Back wire

V1 − planes~ 1800 mm

X2 − planes

Gate wires
U1 − planes
Potential wire

sense (anode)
X1 − planes

wire

anode planecathode plane

U

X

V

Sector, side view

Figure 3.7: DC wire components. The layout of the wire position for one sector
inside the anode plane (left). The schematic diagram of wires in top view (right).
and V2 wires are for the pattern recognition of the tracks and to measure the z
coordinate, they have stereo angle of 6◦ relative to the X wires.
The charged tracks selected in this analysis are reconstructed by all type of theDC wires. The DC track resolution of 2 mm in the z direction at a reference radiusof 220 cm from the beam axis corresponds to 1.0 × 10−3 in η. The resolution in
φ is 1 mrad, but the maximum drift length in DC corresponds to 0.012 rad. In
this analysis, the two track separation cut for the rejection of fake tracks at thecircumstance of high multiplicity is optimized based on these parameters.

3.4.2

Pad chamber (PC)The PHENIX pad chamber (PC) is a cluster of multi wire proportional chambers.
The PC consists of three layers depending on their positions in r direction. The
purpose of the PC is pattern recognitions in charged particle tracking with DC
vectors and determinations of the location of z coordinate for the tracks. There
are three layers in the West arm but only two layers in the East arm as shown in3.4. CENTRAL ARM SPECTROMETER

37

Anode wire
Field wire

8.4mm
Center pixel

1.5mm
2.7mm

Side pixel

8.2mm

Figure 3.8: The PC pixel geometry (left). A cell defined by three pixels (right).
Figure 3.3. Since selected tracks in this analysis are measured by the East arm,
only two layers are used in this analysis. The used layers are named as PC1 andPC3 placed at 2.5 m and 5 m from the beam axis, respectively.
A special pad design was invented to bring down the cost. Each layer contains
a single plane of wires inside a gas volume bounded by two cathode planes.
One cathode is finely segmented into an array of pixels. Figure 3.8 shows thegeometrical configuration of the pixels. Each cell contains three pixels and anavalanche must be sensed by all three pixels to form a valid hit in the cell, and nine
pixels are connected to a common readout channel. However, the hit information
can be broken down by the requirement of identifying pattern of triplets of the
pixels.
The PC1 and PC3 have the same solid angle each other, and these pixel sizes
are 8.4 mm and 14.7 mm, respectively. These pixel sizes are greater than therequirement of two-track separation cuts used in this analysis as described in
Section 4.2.1, and these resolutions are 1.7 mm and 3.6 mm for PC1 and PC3
respectively in z direction, and these values also correspond to 1.0 × 10−3 in η.

3.4.3

Time of flight counter (TOF)

One of the basic observables to examine the property of the QCD matter byproduced particles in the heavy ion collisions is transverse momentum spectrumwith particle identification. The particle identification for the charged hadrons
in PHENIX has been done [84, 85] by the combined information from the tracking detectors and timing detectors. The tracking detectors (see Section 3.4.1
and 3.4.2) provide the particle trajectory and momentum information for eachcharged particle. Time of flight information for each track can be measured as thetime difference between the BBC (see Section 3.3.2) start timing, and stop timing
measured by the PHENIX time of flight counter (TOF). Using those information,

38

CHAPTER 3. EXPERIMENTAL SETUP

mass square value for each particle species can be derived as


tT OF
p2
m2 = 2 
c
L/c

!2



− 1 ,(3.2)

where p is the momentum, tT OF is the time of flight and L is a flight path length.

200 cmPMT
Bleeder
Scintillator

50 cm

Carbon fiber sheetµ-metal

Aluminium frame

Light guide

Honeycomb structured paper

The TOF [78] is designed to have the timing resolution of ∼ 85 psec to achieve
pion/kaon and kaon/proton separation up to 2.4 GeV/c and 4.0 GeV/c with 4σ,
respectively. It is placed at a distance of 5.1 m from the collision vertex, in between the Pad Chamber (see 3.4.2) and the Electro-magnetic Calorimeter (see3.4.4) in the East arm of the PHENIX. The geometrical acceptance corresponds
to 70◦ ≤ θ ≤ 110◦ in the longitudinal direction and 30◦ in azimuth. The TOF

Figure 3.9: The TOF panel (left). A panel have 96 segments of plastic scintillator
modules (right).

3.4. CENTRAL ARM SPECTROMETER

39

counter consists of 10 panels of the TOF walls. One TOF wall, called panel, consists of 96 segments equipped with a plastic scintillator slat and photomultipliertube readouts at both ends. The slat is oriented along the r - φ direction andprovides time and longitudinal position information of particles that hit the slat.
Figure 3.9 shows the schematic view of one panel of the TOF counter. In total
10 TOF panels, 960 slats of scintillators and 1920 channels of PMT’s have beenused for the measurement of the time of flight.

3.4.4

Electro-magnetic calorimeter (EMCal)

Figure 3.10: A module of lead scintillator type electro-magnetic calorimeter.
The primary role of the PHENIX electro-magnetic calorimeter (EMCal) [79]is to provide measurements of energies and spatial positions of photons and electrons produced in the heavy ion collisions. The measurements of the neutral pion
production cross section have been done with the π 0 → 2γ decay channel measured by this EMCal system [86]. Although the charged hadron identification isavailable only up to 4 GeV/c in the transverse momentum by the TOF counter,the results of pT distribution of π 0 enable us to discuss behaviors of the hadronproductions in high pT regions greater than 4 GeV/c.
There are two kinds of calorimeters in the PHENIX detector. One is a shashlik type lead scintillator sampling calorimeter (PbSc) [87] and the other is aČerenkov lead glass calorimeter (PbGl). Six PbSc sectors and two PbGl sectors
have been installed and each sector covers 1/8 π rad in azimuth. They cover
the same acceptance as tracking detectors in PHENIX central arms as shown in
Figure 3.3. The total ET measurement [35] have been done by using the PbSc

40

CHAPTER 3. EXPERIMENTAL SETUP

type calorimeter, which can be used for the event characterization in heavy ion
collisions as described in Section 4.1.4. Therefore, the specifications of this PbSc
type EMCal is explained in this section. A sector of PbSc is composed of 18 supermodules and it covers a 2 m × 4 m plane area approximately which includes 108
PbSc super modules in total. Since a super module has 12 × 12 readout channels
by the PMT, there are 15552 PbSc readout channels in total. Figure 3.10 shows
the schematic view of a PbSc module, which has four readout
channels. The√
PbSc calorimeter has a nominal energy resolution of 8.1% E GeV ⊕ 2.1% and
an intrinsic timing resolution better than 200 ps for electro-magnetic showers.

3.5

Data acquisition system (DAQ)

The interaction rate at design luminosity of RHIC varies from a few kHz for
Au + Au central collisions to approximately 500 kHz for minimum bias p + p
collisions. The PHNEIX data acquisition system (DAQ) [80] was designed to accommodate improvements in the design luminosity. The DAQ consists of several
types of electronics modules. Since all of them are pipelined and synchronizedwith the RHIC beam clock, the system has deadtimeless features. Figure 3.11
shows the connections between all modules and detectors.
Figure 3.12 shows the block diagram of the PHENIX timing system. The
RHIC beam clocks are distributed to the PHENIX detector system via opticalserial links through a VME board called the v124 broad in the PHENIX counting house. We can set the global timing delay from the RHIC clock to all of
the PHENIX subsystems to adjust the collision timing at this board. The adjusted clock are transferred to the Master Timing Modules (MTM). The MTM
has two roles as follows. One is to provide the clock to the Granule Timing
Module (GTM), which is allocated for each detector subsystem and the trigger
system to absorb the difference of timing offsets among the subsystems and send
the clock to the Front End Module (FEM). The clocks are converted from a 20 bit
parallel stream being clocked at 9.4 MHz of RHIC clock rate up to 4 times thebeam clock rate and then serialized into a 900 MHz stream at the GTM. Second
is to provide the Global level-1 trigger (GL1) decisions synchronized with theclock. The definition of the minimum bias trigger at the GL1 for the Au + Au
collisions and its trigger efficiency are explained in Section 4.1.2.
The analog data taken by each detector are transferred by cables to the FEM,
which is located near the detectors in the beam intersecting region. The transferred data are buffered in each FEM for the time of 40 beam bunch crossings to
wait for the decision of GL1, where the beam crossing rate corresponds to 106 ns
at RHIC. There are two different types of buffering methods, one is buffering
the digitized data in each beam crossing, and the other is buffering the analog
data from detector in Analog Memory Unit (AMU) and then digitized when it
received the trigger decision. The ZDC, BBC, DC, PC and TEC adopt former

3.5. DATA ACQUISITION SYSTEM (DAQ)

41

type. The TOF, RICH and EMCal adopt the later type of the buffering method.
Figure 3.13 shows the block diagram for a typical FEM with AMU. What kindof data are buffered e.g. pulse height and timing, depends on the purpose of eachdetectors and more detailed descriptions are elsewhere in [40].
The triggered data are transferred to the Data Collection Modules (DCM) via
the optical fiber. The maximum average level-1 trigger (LVL1) rate is 25 KHz and
the RHIC beam crossing clock runs at 9.4 MHz. At the maximum LVL1 triggerrate, the FEM’s send over 100 GBytes of data per second. The DCM are designed
to receive this large uncompressed event fragment data volume. Firstly, the DCM
provide data buffering up to five LVL1 events, and perform zero suppression for
all detector subsystems. At the same time, the data are performed error checking
and formatting. The formatted data are compressed and send to the PHENIX
Event Builder (EvB).
The two primary functions of the PHENIX EvB are to perform the final stageof event assembly in the PHENIX DAQ and to provide an environment in whichlevel-2 (LVL2) trigger processing is performed. The EvB receives many parallel
data streams from the PHENIX DCM’s. The fragments from each data stream

Figure 3.11: Schematic diagram of the PHENIX data stream.

42

CHAPTER 3. EXPERIMENTAL SETUP

Figure 3.12: Block diagram of the PHENIX timing system.

Figure 3.13: Block diagram of the typical FEM with AMU.

3.5. DATA ACQUISITION SYSTEM (DAQ)43

are assembled into complete events and performed LVL2 trigger processing on
the events. The accepted events are transmitted to the PHENIX Online Control
System (ONCS) for logging and distribution to monitoring processes. Figure 3.14
shows the block diagram of PHENIX EvB. The fragmented data in each subsystems are once stored to sub event buffer (SEB), and they are distributed by the
Asynchronous Transfer Mode (ATM) technology to the Assembly/Trigger Processors. In the case of Au + Au collisions, the total data size of the event is
all different depending on the collision centrality and the number of produced
particles. Therefore, the order of event in real time dose not correspond to theorder in the data file by this parallel processing. The performance of the baseline
EvB were to process events at an input rate of 12.5 kHz and to handle aggregate
data rates as high as 500 MByte/s.

Figure 3.14: Block diagram of PHENIX Event builder.

Chapter 4
Analysis
4.1

Event selections

4.1.1

Data setThis analysis was performed using a data set collected by the PHENIX detector during the run period of gold ion beam operation with 100 GeV/c beam
momentum per nucleons from 2001 to 2002. Figure 4.1 shows the integrated
luminosity delivered by RHIC as a function of days in the run period. In thisperiod, 24 µb−1 were collected at the PHENIX experiment out of the delivered
integrated machine luminosity of 85 µb−1 . The data were collected by the minimum bias trigger selections as described in Section 4.1.2 with no magnetic filed
condition for the PHENIX central arm tracking detectors (see Section 4.2.1).
This trigger condition includes the collision vertex cuts of 30 cm from the center of the PHENIX detector. Out of this data set, this analysis was targeted in
total 1.5 million events. These event samples were imposed to pass the rigorous
data quality assurance with respect to the detector stability as described in the
Section 4.1.5.

4.1.2

Trigger selection

The heavy ion physics aims to investigate the production of the QGP by colliding
many nucleons as much as possible at high energy. Therefore, estimations of theminimum bias collision trigger efficiency for the total inelastic cross section in
the nucleus-nucleus collision is one of the most important issues, because almostof physics results focus on the event characteristics as compared to the single
nucleon collisions i.e. (p̄)p + p interactions.
The minimum bias collision trigger used in this analysis was made by referring
the two forward global detector’s information from the BBC and ZDC, and theglobal level 1 electric logic board. The definition of the minimum bias event totake the data demands coincident hits of two PMT’s for each side of the BBC44

4.1. EVENT SELECTIONS

45

RHIC ExperimentIntegrated Luminosity -- Gold Ion Operations
Run1 and Run2

90

70

Ê

b

-1

]

80

Luminosity [60

50

BRAHMSPHENIXPHOBOS
STAR

40

30

c

c

2000 (66 GeV/ )20

2001 - 2002 (100 GeV/ )

10

0
0

14

28

0

14

28

42

56

70

84

98

Days

Figure 4.1: Integrated luminosity delivered by RHIC as a function of days. The
RHIC was operated with gold ion beam in this term. The beam momentum is
66 GeV/c (left) per nucleons in 2000 and 100 GeV/c (right) per nucleons from2001 to 2002. Markers correspond to delivered luminosity for four experiments
indicated in the legend.
and coincident hits of single neutron energy deposit for each side of the ZDC
within beam bunch crossing time. The sides correspond to the forward andbackward from collision vertex along with the beam axis. The BBC measuresrelativistic charged particles from participating nucleons in a Au + Au collision,and the ZDC measures the spectator neutrons, which are not participating to
the Au + Au interaction. Therefore, these two detectors take complementaryroles to detect Au + Au collisions. It was also required that information ofcollision vertex positions in the minimum bias trigger definition is to optimize
the acceptance of the central arm tracking detectors. In this analysis, a collisionvertex cut of z < ± 5 cm was applied on the event selections at the offline
data analysis. It is to avoid discrepancies on the geometrical acceptance of the
PHENIX central arm in event-by-event basis. Total 256 k events in Au + Au√
collisions at sN N = 200 GeV were used to obtain the final results.
The flow to make trigger decisions is explained as follows. The hit time ineach PMT of the BBC is digitized in the front-end electronics module (FEM) andsent to the BBC local level 1 (BBCLL1) electric logic board with bit reductions
for the fast judgment of the BBCLL1 trigger decision. The valid range of the hit
time and the time offset for each PMT are applied to the FEM a priori. In the

46

CHAPTER 4. ANALYSISBBCLL1 logic board, the number of hit PMT’s and collision vertex positions arecalculated from the reduced bit time information, and then it is sent to the global
level 1 board. In the global level 1 broad, using the predefined valid range of thecollision vertex and the number of hit PMT’s of the BBC i.e. two coincident hits
in each side, the minimum bias trigger is made associated with ZDC coincidence
hits for each side. The trigger decisions are sent to the other subsystems to takethe data for the buffered data in each FEM.
The trigger efficiency of the BBCLL1 with respect to the total inelastic crosssection of Au + Au collisions was estimated by the GEANT [88] based Monte
Carlo simulation with respect to HIJING 1.35 [89] event generator input. In the
GEANT based MC simulation, all material information of the PHENIX detector
system are implemented to reflect the effects of realistic particle interactions.
The realistic BBC responses are applied with calibration constants obtained by
the collision data to reproduce the digitized value, for instance, charge and time
distributions and the threshold values for each PMT. An software, which is toemulate the BBCLL1 logic board, is applied to reproduce the digitized values toestimate the effects on the bit reduction and the online z vertex cut. According
to the simulation, the trigger efficiency of the BBCLL1 for the valid range of z
vertex is defined as
εBBCLL1 ≡Z z−zε(z) · w(z)dz,(4.1)

where the efficiency ε(z) as an input-output ratio and weight w(z) for the z vertex
distribution in each z vertex position are defined as
N(on & off) (zrec ),Ninp (ztrue )1/Ninp(ztrue )w(z) ≡ R z,−z 1/Ninp (ztrue )dz
ε(z) ≡(4.2)

respectively. N(on & off) is the number of reconstructed events by applying theonline and offline z vertex cut. The online z vertex cut in the BBCLL1 board
corresponds to 75 cm, and the offline z vertex cut in the reconstructed z is 30 cm
for the offline analysis. Ninp means the number of generated events within thegenerated z vertex position. Figure 4.2 shows the obtained BBC trigger efficiency
as a function of reconstructed z vertex positions. Several things have to be taken
into account here for the systematic error estimations in this method. Although
the number of produced particles as an input is adjusted to the measured value,
this method depends on the model of the event generator e.g. particle composition
and momentum distributions of produced particles. Furthermore, the obtained
trigger efficiency depends on the vertex distributions itself. As an instrumental
ambiguity, variations of applied threshold values for each FEM channel should
be reflected to the trigger efficiency. Combining all ambiguities mentioned above,

Trigger efficiency (BBC LL1)

4.1. EVENT SELECTIONS

47

1.2
1
0.8
0.6
0.4
0.2
0
-200

-150

-100

-50

0

50

100

150

200
Z [cm]

Figure 4.2: Trigger efficiency of BBC local level 1 trigger as a function of z vertex
measured by BBC. This trigger efficiency is defined for the offline analysis. Solidlines (|z| = 75 cm) and dashed lines (|z| = 30 cm) correspond to online and offlinevertex cuts, respectively.
the BBC trigger efficiency for the selected trigger condition is finally obtained as
εBBCLL1 = 92.3% ± 0.4% (statistical error) ± 1.6% (systematic error).

(4.3)

The trigger efficiency of ZDC is estimated by the relative trigger rate withrespect to the BBC triggered events represented as
R=

NBBCLL1 & NZDCNS,NBBCLL1(4.4)

where NZDCNS is the number of events with coincident hits in the North andSouth ZDC sides. Figure 4.3 shows a transit of the relative trigger rate, which is
obtained by requiring the BBC offline z vertex cut of 30 cm, for a sampled period.
The average value of the rate was obtained by the 97.5%. The dropping to thelower value is due to the increase of background events e.g. beam-gas collisionand beam halos. The BBCLL1 trigger rate as the denominator in Equation (4.4)
is increased by the increase of the accidental coincidence hits in this case. Since
40% of BBCLL1 triggered events withouout ZDCNS triggered events correspondto the ZDC inefficiency, eventually, the ZDC trigger efficiency with respect to the
BBCLL1 can be obtained asε(ZDCNS | BBCLL1) = 99.0% + 1.0% − 1.6% (systematic error).

(4.5)

CHAPTER 4. ANALYSIS
Ratio

48

Run number

Figure 4.3: Relative counts of the ZDC triggered events with respect to the
BBCLL1 triggered events as a function of the run number. The horizontal solidline corresponds to 97.5%.
Finally, the minimum bias collision trigger efficiency for the total inelastic√
cross section in Au + Au collision at sN N = 200 GeV in this trigger selection
can be obtained by Equation (4.3) and Equation (4.5) asε(BBCLL1 & ZDCNS) = εBBCLL1 × ε(ZDCNS | BBCLL1)= 91.4% + 2.5% − 3.0% (systematic error),(4.6)

with all of the systematic errors explained so far.

4.1.3

Collision centrality and number of participant nucleons

A +A collisions have event-by-event characteristics for the particle productions
depending on the impact parameter in each event. The parameter is defined by
the distance between the center of two colliding nuclei as shown in Figure 4.4. The
basic idea to classify the event characteristics is based on a simple geometrical
picture named as Glauber model [9]. In this model, nucleons in nucleus which
are distributed as black disks within the finite size of the nucleus are distributedwith the nuclear density profile. The profile is parametrized by a Woods-Saxon
function as a function of the distance from the center of the nucleus r as
ρ(r) = ρ0 ·1
1 + e(r−R)/a

,

(4.7)

4.1. EVENT SELECTIONS

49

yimpact parameter b

beam direction

beam directionxoverlap
regionnucleusparticipantnucleons
nucleusFigure 4.4: Collision geometry of two colliding nuclei. The impact parameter isdefined by the distance between the center of two colliding nuclei.
where R and a corresponds to the diameter of the nucleus and the diffusion
parameter for the outer surface of the nucleus, respectively. A collision for eachnucleon pair in the nucleus takes place if their distance is less than (σN N /π)1/2 ,where σN N is the nucleon-nucleon inelastic cross section at the collision energy.
According to this model, the total inelastic cross section with respect to the A + A
collisions can be calculated by integrating the individual inelastic cross section
over all of possible pairs for a given impact parameter, which can be obtained by aMonte Carlo (MC) simulation with several ambiguities of the model. In the case√of Au + Au collision at sN N = 200 GeV, the model parameters in Equation (4.7)
correspond to R = 6.38 fm, and a = 0.54 fm [90] and the nucleon-nucleon crosssection σN N = 42 mb as the central value for the MC simulation. Figure 4.5
shows the result of differential Au + Au inelastic cross section dσAu+Au /db at√
sN N = 200 GeV as a function of the impact parameter b obtained by the MC
simulation.

CHAPTER 4. ANALYSIS

dσ / db [fm]

50

400
350
300
250
200
150
100
50
0
0

2

4

6

8

10

12
14
16
18
impact parameter b [fm]Figure 4.5: Differential Au + Au inelastic cross section ata function of impact parameter b.

√sN N = 200 GeV as

Although the event characteristics in Au + Au collisions are clearly dependent
on the impact parameter b, the direct measurement of the impact parameter isvery difficult such as in the circumstance of collider experiment. Nevertheless, it
can be estimated indirectly as a collision centrality percentile by the multiplicity
or energy measurement for the produced particles, because a monotonic relation
between the impact parameter and total multiplicity or total energy can be expected owing to the result of the impact parameter dependence of differential
cross section in the Glauber model MC simulation as shown in Figure 4.5. In
the case of the PHENIX experiment, the collision centrality is determined in theanti-correlation of multiplicity distributions between the two forward detector’s
information (ZDC and BBC) because these two detectors take complementaryrole to detect the Au + Au collisions as described in Section 4.1.2. Figure 4.6
shows the reproduced anti-correlation between the normalized energy sum in the
ZDC and charge sum at the BBC obtainedby the MC simulation. The energy resq
olution of the ZDC (σE /E = 218%/ E/GeV ) is applied, and the charged particle
multiplicity distribution and fluctuation in the BBC are adjusted to calibrated
collision event by the parameters of Negative Binomial Distribution (NBD) for
the MC simulation. Areas filled by the different colors are obtained by choosing
the range of impact parameter so that each event fraction equals 5% out of all
generated events. Since the BBC measures the charged particle multiplicity fromthe collision, it is increased with increasing collision centrality. Meanwhile, theZDC energy sum decreases with increasing collision centrality because it measuresthe deposit energy of neutrons comes from the spectator part of hadronic inter-

Energy in ZDC [arb. unit]

4.1. EVENT SELECTIONS

51

1
0.9
0.8
0.7
0.6
0.5

10-15%

0.4
0.3

5-10%

0.2

0-5%

0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9
1Charge in BBC [arb. unit]

Figure 4.6: BBC charge vs. ZDC energy calculated by a Monte Carlo simulation
based on Glauber model [9]. Areas filled by colors are obtained by choosing
the range of impact parameter so that each event fraction equals 5% out of all
generated events. The solid line indicates the limit of the most peripheral sample
used for this analysis.
action i.e. they do not participated in the Au + Au collisions. This is the reason
why Figure 4.6 indicates the anti-correlation in the region of the BBC charge
sum above around 0.2. For the low central region, which is called as peripheral
collision, the correlation between the BBC and ZDC can be understood as thedominant effect of mutual Coulomb dissociation [91] in place of hadronic interaction. The number of produced neutrons increases in this collision centrality (BBCcharge sum < 0.1) with increasing the centrality due to the increase of electromagnetically dissociated neutrons from both colliding gold ions as compared tothe decrease of neutrons in hadronic process. According to the event classification
based on the impact parameter in Glauber model MC simulation (see Figure 4.6),
the real event samples are separated by the lines from 0.15 in the BBC charge
and 0 in ZDC energy in a radial pattern. Figure 4.7 shows the definition of
the event classification with respect to the collision centrality in the real event
samples with 5% event fraction. The solid line indicates the limit of the mostperipheral event sample used for this analysis, and most right hand side region
tagged as 5% inside the figure corresponds to the most central 5% event samples.
The number of participant nucleons Npart at A + A collisions, which almost
corresponds to the number of colliding nucleons existed interior side of the overlap
region in Figure 4.4, is often used to represent the collision centrality classification
instead of the impact parameter b. This representation is useful in the case that

CHAPTER 4. ANALYSIS

Energy in ZDC [arb. unit]

52

1
0.9
0.8
0.7
0.6
0.5

10-15%

0.4
0.3

5-10%

0.2

0-5%

0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6
0.7
0.8
0.9
1
Charge in BBC [arb. unit]

Figure 4.7: Definition of collision centrality, BBC charges versus ZDC energy.
Event samples with 5% bin width are plotted from 0 - 5% (central) to 60 65% (peripheral). The solid line indicates the limit of the most peripheral sample
used for this analysis.
one wants to compare the events with p(p̄) + p collision events at the same
collision energy. It can be also extracted from the MC simulation in eventby-event basis and average values of hNpart i are calculated with respect to thecorresponding centrality classes defined in Figure 4.7. The obtained values are
summarized in Table 4.1.
At the same time, several systematic error sources on the estimation of thehNpart i in the MC simulation have been taken into account for both ambiguities
originating from the model parameters and adjustment with experimental data
due to the centrality classification as follows. Since the most dominant sourcesof the systematic errors are ambiguities on the nucleus-nucleus cross section andWoods-Saxon parameters, σN N , R and a are varied by 7% from the central values
at maximum in the simulation. In general, nuclei have a hard core component,
it means there is restricted area for the nucleus overlapping around the centerof nucleus. This effects is estimated by changing requirement of minimum distance between the center of two nucleons at the nucleon-nucleon collisions as
greater than 0.4 fm. Due to the formation of the deuteron by the coalescenceprocess [92] in the forward region, a fraction of detectable number of neutrons
is reduced compared to the number of spectator neutrons. It is described bya neutron loss function in the MC simulation and also evaluated for the systematic error estimation. The assumption of black disk is evaluated by varying
the interaction probability distribution as the following two perspectives. In thiscase, the black disk corresponds to 100% necleon-nucleon interaction probability

4.1. EVENT SELECTIONS

53

Table 4.1: Average of the number of participant nucleon hNpart i with the impact parameter hbi for each collision centrality class by the Glauber model MCsimulation.
Centrality class(%)
0-5
5 - 10
10 - 15
15 - 20
20 - 25
30 - 35
35 - 40
40 - 45
45 - 50
50 - 55
55 - 60
60 - 65

hbi systematic error(fm)
of hbi
2.3
0.1
4.1
0.2
5.2
0.3
6.2
0.2
7.0
0.4
8.4
0.3
9.1
0.4
9.7
0.4
10.2
0.4
10.7
0.4
11.2
0.4
11.7
0.5

hNpart i systematic errorof hNpart i
351.4
2.9
299.0
3.8
253.9
4.3
215.3
5.3
181.6
5.6
125.7
4.9
102.7
4.3
82.9
4.3
65.9
3.4
51.6
3.2
39.4
3.5
29.8
4.1

independent of the distance between the nucleons. One is using a reduced interaction probability (gray disk) by taking the ratio between the total and inelastic
nucleon-nucleon cross section as σinelastic /σtotal . The other is assuming the interaction probability as Gauss distribution depending on the nucleon-nucleon impact
parameter, which is same method as in an event generator FRITOF 7.0 [93].
For the estimation of systematic errors of hNpart i, which depend on the experimental conditions, following items are considered. The charged particle multiplicity fluctuation in the BBC are varied to evaluate the smearing effect on thedetector resolutions. Basically, the anti-correlation between the BBC charge and
ZDC energy indicates a linear relation for the central event samples in the MC
simulation. However, Figure 4.7 shows bending behavior due to the saturation
effect of the BBC charge in each PMT for the small fraction of events. In fact,
Figure 4.6 is made by the MC simulation with this artificial saturation effect. It
affects the centrality classification defined by the radial lines as shown in Figure 4.7. This effect is verified by changing the central point of the radial lines
from 0.5 to 2.0. Since the event classification depends on the event fraction, errorsof the minimum bias trigger efficiency (see Section 4.1.2) affects the differenceof selected event fraction between the MC simulation and real event samples. It
has been considered by changing the value of the trigger efficiency within thesystematic error of it. The statistics of the real event samples also affects theevent classifications. It is evaluated as the systematic error by looking at thestability of the fraction in the real data samples taken at the different run range.

54

CHAPTER 4. ANALYSIS

Figure 4.8: Summary of the verification on model ambiguities described in thelegend for the systematic error estimation of the hNpart i. Shaded bands indicate
the total systematic errors.
Figure 4.8 shows the summary of the systematic error sources and their values
explained above as a function of the Npart . The total systematic errors for the
hNpart i are obtained by the quadratic sum of all in above sources as shown inthe shaded band in Figure 4.8 and the values are listed in Table 4.1. Finally,√
the total inelastic cross section for Au + Au collision at sN N = 200 GeV was
obtained with the systematic error as
σAu+Au = 6847 ± 542 mb.

(4.8)

An analysis on the measurement of the density correlations was performed byusing this event sample with the collision centrality classification. Furthermore
detailed event selection criteria, which are related to the charged track reconstructions and multiplicity fluctuation measurements, will be explained in Section 4.2.

4.1.4

Energy density estimation

One of the methods to characterize events in A + A collisions without aid of
any models like the Glauber model described in Section 4.1.3 is to characterize
the events by the total measured energy for all produced particles. Although
PHENIX does not cover whole the acceptance (4 π) to measure the all produced
particles, the electro-magnetic calorimeter (EMCal [79]) in PHENIX has sufficientcapability to measure the total transverse energy in midrapidity region by the
sampled data in the limited acceptance. The total transverse energy ET is defined

55

1/event 1/ET

EMC

-1

[GeV ]

4.1. EVENT SELECTIONS-1

10200 GeV

-3

10

-5

10

-1

10

130 GeV

-3

10

-5

10

10

10

10

19.6 GeV

-1

-3

-5

0
0

20

40

60 E

TEMC [GeV]dET/dη [GeV]200

Figure 4.9: The raw ET distribution in two EMCal sectors measured in Au +Au
collisions at three beam energy described in the legend. The lowest axis correspond to the value of dET /dη for the midrapidity η =0. Distributions of four 5%
centrality bins, which are described in Section 4.1.3, are shown in each plot.
as
ET =

X

Ei sinθi ,

(4.9)

i

where θi is polar angle from the beam axis. The sum is taken over all particles emitted into a fixed solid angle per event. The ET measurement was performedq
by the PbSc type electro-magnetic calorimeter whose energy resolution isL
8.1% / E (GeV)
2.1% by the test beam electron as described in Section 3.4.4.
Figure 4.9 shows the uncorrected ET distributions in two EMCal sectors measured in Au +Au collisions at three beam energies of 200 GeV, 130 GeV and√19.6 GeV in sN N [35].
The correction factors for the raw ET values have been obtained by the
GEANT based MC simulation [88] with HIJING [89] inputs, which is tuned
to reproduce the measured data in PHENIX with respect to the particle compo-

CHAPTER 4. ANALYSIS

dET/dη/(0.5Np) [GeV]56

3200 GeV
130 GeV2
19.6 GeV
1
3

130 GeV2

1.5

19.6 GeV

1
0

100

200

300

Np

Figure 4.10: Corrected dET /dη divided by the number of participant
pairs (0.5 Npart ) as a function of Npart in Au +Au collisions at three beam energy
described in the legend. The vertical errors indicate the full systematic errors.
sition. The systematic error on the geometrical acceptance corrections has been
estimated as 2.0% by the MC simulation. At the same time, several study on thesystematic error estimations about the ET measurement for the Au + Au colli√
sions at sN N = 200GeV have been done, and these are summarized as follows.
It is known by the energy calibrations that there is 1.5% uncertaintiy in the absolute energy scale for the electro magnetic particles. Meanwhile, the uncertainty
in the EMCal response for hadrons give the 3% error to the total ET , because
this EMCal is too thin to detect the total energy deposit from hadrons. It is estimated by a comparison of the energy deposit values with respect to the hadrons
between the MC simulation and the test beam data [79]. This comparison wasmade by varying momenta of the hadrons. Since this estimation depends on the
input of MC simulation, an additional error of 1.3% comes from the ambiguitiesof the particle compositon and momentum distributions. The reconstructed energy deposit has finite energy loss at the edge of the EMCal detector, because foreach particle reconstructed by a clustering method [87]. The loss is estimated as

57

2∈Bjτ [GeV/fm /c]

4.1. EVENT SELECTIONS

4

2200 GeV
130 GeV19.6 GeV
0

100

200

300

Np

Figure 4.11: Bjorken energy density εBj · τ as a function of Npart in Au + Au
collisions at there energy as indicated in the legend.
1.5% of absolute uncertainty. The total systematic error on the EMCal response
for both electro magnetic particles and hadrons, which is obtained by taking thequadratic sum in above, corresponds to 3.9%. The systematic errors from background particles and noise in the detector readout system are estimated as from
0.2% to 6.0% with the real event samples in peripheral collisions by selecting
the events that have no activity in the midrapidity region corresponding to the
EMCal acceptance. Since the EMCal has a finite acceptance, the following two
things should be taken into account for the corrections. One is the lost energy
or outflow contributions for the total ET by missing particles due to the decay
products, which is originally in the EMCal acceptance at the primary collision
point. The other is the inflow contributions coming from the outside of the EMCal acceptance as a sum of the deposit energy from the decay products andthe background particles originating from the detector materials. The relative
amount of the outflow and inflow contributions with respect to the total ET are
estimated as ∼10% and ∼24%, respectively, with 3.0% systematic errors basedon the MC simulation. Figure 4.10 shows the corrected dET /dη divided by the
number of participant pairs (0.5 Npart ) as a function of Npart in Au +Au collisions√
at sN N equal 200 GeV, 130 GeV and 19.6 GeV with full systematic errors. It
is obtained by taking the quadratic sum of the errors described in above.
The event samples are classified by the collision centralities, and the definitionof the centrarities are based on the Glauber model as described in Section 4.1.3.
This is the reason why the collision centrality is often expressed by the average
number of participant nucleons hNpart i over an event sample in the experimentof heavy ion collisions. However, since the measured total ET values are also

58

CHAPTER 4. ANALYSIS

obtained by the event classification, the event classification can be represented
by the total ET instead of Npart . If one shows any results as a function of thetotal ET , one can obtain the dependencies of collision centrality only by themeasurement independently on any models. This is an advantage of the event
characterization by the total ET . However, this concept on the event characterization is implicitly based on an idea that the mechanism of particle productions
are linearly related with the total energy. It is a natural assumption, since theaim of the heavy ion experiment is to explore the property of the thermalized
QGP by the measurements of produced particles, event samples should be classified by values directly connected to the thermodynamic variables. In this sense,
the event samples should not be characterized by the geometrical values like theNpart , because the Npart is only regarded to the collision dynamics and depends
on the collision system and energy. The energy density is more meaningful rather
than the total energy, because the particle productions depend not only on the total energy but also on the system size. The energy density based on the Bjorken’s
picture εBj , which is described in Section 1.3, can be calculated through the ET
measurement as

εBj =

1 dET
,
A⊥ τ dy(4.10)

where τ and A⊥ corresponds to the formation time of the thermalized QGP and
the nuclei transverse overlap area, respectively. The transverse overlap area of twocolliding nuclei A⊥ is estimated by a MC simulation based on the Glauber model.
Comparing Equation (4.10) and Equation (1.10), the normalization to πR2 should
be considered, where R is in the Woods-Saxon parameter in Equation (4.7). Thisestimation has been done for the impact parameter b = 0 in the MC simulation.
Additionally, for the transformation from dET /dη|η=0 to dET /dy|y=0, which are
the values with respect to the midrapidity region, a scale factor of 1.25 ± 0.05
is applied. Figure 4.11 shows the obtained εBj · τ as a function of Npart in√
Au + Au collisions at sN N equal 200 GeV, 130 GeV and 19.6 GeV. Since
the formation time τ is unknown by the measurement, it is indicated by the
product of the εBj and τ . Although the observable in this analysis will be shown
as a function of Npart , it can be transformed by the energy density by using thisrelation between Npart and εBj ·τ . The εBj monotonically increases with increasing√
Npart in Au + Au collisions at sN N = 200 GeV as shown in Figure 4.11 in thecase of the formation time τ is not dependent on the Npart . Therefore, it can besaid that a scan for the energy density is possible by the scan for Npart . Thisimplies that the change of Npart at a fixed collision energy can provide a finer scanover the initial temperature compared to changing the collision energy density.

4.2. MEASUREMENT OF CHARGED PARTICLES

4.1.5

59

Data quality assurance

It is very important to guarantee that the detector conditions are surely stableto perform the analysis on the event-by-event fluctuation. The data quality assurance was made for the collected 1.5 million events under no magnetic filed
conditions by the minimum bias trigger selections, which is described in Section 4.1.2. The status and stability of the detectors is based on the following
points of view;• Whether large dead areas in the acceptance exist or not.
• Whether reasonable correlations on the multiplicity between the number ofcharged tracks reconstructed by DC and the number of photon like clusters
counted by EMCal are seen or not.
• Whether stable multiplicity are seen in the calibrated BBC charge sum, thenumber of DC tracks and the number of EMCal clusters or not.
These items were confirmed as a function of collected time for the subdivided data
segments, each of which contains typically 2,000 to 3,000 events. Figure 4.12 a)
and b) show the segment number dependence of the calibrated BBC charge sum a)and the number of reconstructed DC tracks in an event, respectively. The photon
like PbSc clusters c) and PbGl clusters d) were counted by requiring not toassociate with charged tracks with reasonable time of flight cut. The error bars
associated with the points are the statistical errors. All segments deviated fromthe band, which correspond to 2% fluctuation as indicated by the dashed lines
in the figure, were all rejected from the data analysis described in the following
sections.

4.2

Measurement of charged particles

4.2.1

Track selection criteria

In this analysis, charged tracks detected in the East arm (see Section 3.2), which
covers |η| < 0.35 and ∆φ < π/2, were used. As charged track selection criteria, we required that each straight-line track reconstructed by a DC hit patternassociated with a PC1 hit be aligned with a PC3 hit and the collision vertex
point measured by BBC. Figure 4.13 shows the positional relation for each detectors and alignment of the hit positions with a DC track. The DC has six
types of wire modules; two of them are used for the track reconstruction for theazimuthal angle and others are used for the pattern recognition. Selected tracks
were reconstructed by using all wire modules of DC. We required associations
between DC tracks and PC3 hits to be within 10 cm in the distance of closest
approach (DCA), which was determined to minimize the random associations

CHAPTER 4. ANALYSIS

BBC charge sum / event

60

a)
600
500
400
300
200
100
00

5

10

15

20

25

Segment numberb)
DC tracks / event

70
60
50
40
30
20
10
00

5

10

15

PbGl clusters / event

25

12
10
8
6
4
2
00

5

10

15

20

25

Segment numberd)
PbSc clusters / event

20

Segment number

c)

12
10
8
6
4
2
00

5

10

15

20

25

Segment number

Figure 4.12: Detector stability on the hit multiplicity. a) Average charge sum
in BBC. b) Average reconstructed DC tracks. c) Average number of photon likeclusters in PbSc. d) Average number of photon like clusters in PbGl.

4.2. MEASUREMENT OF CHARGED PARTICLES

61

without losing track detection efficiency unnecessarily. Figure 4.14 a) shows the
DCA from all DC tracks to the hit position of PC3. The Filled histogram in thisfigure was made by flipping the sign of z axis on PC3 hit position intentionallyto estimate the amount of random associations. By applying to the cut value,
90% of radamly associated DC tracks with PC3 hit positions can be removed.
Figure 4.14 b) shows the DCA from all DC tracks to the collision vertex measured
by BBC. Filled histogram was obtained after applying the association cut for theDC tracks and PC3 hit positions. Although almost of fake DC tracks, which is
not aligned with collision vertex, have already removed by the association cuts,
we have applied 28 cm of association cut with the collision vertex to reject obvious
fake track made by miss reconstructions. Since PbSc type calorimeter (Section
3.4.4) in PHENIX has sensitivity on the charged particles, clear peak made byminimum ionizing particles (MIP) can be seen in the cluster energy distributions.
The validity of the selected charged tracks with this association cut have been
confirmed by the MIP peak.
In addition to the single track selection, we required a minimum two-track separation in order to minimize effects from fake tracks and associated secondary par-`Yb ^_S\R`Yb qVp`Ya ^_S\R
XY ZR[RZR\]R ^_S\R`Ya qVpQRST SUVW

hZS]i
aj ]T

XY qVp ^SppRZ\
kl ]T

QQY mnoRZpRU

ccde cdfe gfde

Figure 4.13: Requirements of charged track association.

62

CHAPTER 4. ANALYSIS

Number of tracks

a)

16000
14000
12000
10000
8000
6000
4000
2000
0

0

10

20

30

40

50

Number of tracks

60

[cm]

DCA

b)
5

10

4

10

3

10

2

10

10
0

50

100

150

200
DCA

250
[cm]

Figure 4.14: a) Distance of closest approach from DC tracks to PC3 hit position.
Filled histogram was made by flipping the sign of z axis on PC3 hit position
intentionally to estimate the amount of random associations. b) Distance of
closest approach from DC tracks to the collision vertex measured by BBC. Filled
histogram was obtained after applying the association cut for the DC tracks andPC3 hit positions.
b)

0.04

0.03

0.02

0.01

c)
0.0034

Track pairs / total track pairs / events

0.05

Track pairs / total track pairs / eventsdφ [rad]a)0.00320.003
0.00280.00260.00240.00220.002
0.00180.0016

0.012

0.01

0.008

0.006

0.004

0.0014
0
0

0.01

0.02

0.03

0.04

0.05
dη

0

0.01

0.02

0.03

0.04

0.05
d φ [rad]

0

0.01

0.02

0.03

0.04

0.05
dη

Figure 4.15: a) Distance between any two tracks in η − φ plane. b) and c) is the
projection of the distance to the η and φ, respectively.

4.2. MEASUREMENT OF CHARGED PARTICLES

63

ticles. When we find tracks within the minimum separation window of δη < 0.001
and δφ < 0.012 rad, we count them as one track independent of the number ofreconstructed tracks in the window. These cut values were determined by lookingat δη and δφ distributions on the η − φ plane of any two track pairs in the real
data sample as shown in Figure 4.15. The DC track resolution of 2 mm in thez direction at a reference radius of 220 cm from the beam axis corresponds to
1.0 × 10−3 in η. PC1 and PC3, which are used for the track association, havethe same solid angle each other, and these pixel sizes are 8.4 mm and 14.7 mm,
respectively. These pixel sizes are greater than the requirement of two-track separation cuts, however, these resolutions are 1.7 mm and 3.6 mm for PC1 andPC3 respectively in z direction, and these values also corresponds to 1.0 × 10−3
in η. The resolution in φ is 1 mrad, but the maximum drift length in DC corresponds to 0.012 rad. Therefore the two-track separation window size in η and φis consistent with what is expected.

4.2.2

Magnetic field condition

In the case of normal magnetic field condition at the PHENIX detector, which
is used to identify the charged particles, the threshold transverse momenta pT
correspond to 0.2 GeV/c, 0.4 GeV/c and 0.6 GeV/c for charged pions π ± , charged
kaons K ± and protons p (p̄), respectively [84]. Figure 4.16 shows the measured√
pT spectra in Au + Au collisions at sN N = 200 GeV for these particles in the
0 - 5% (central) and 60 - 92% centrality bins.
Since this analysis used the data taken without magnetic field in order to optimize acceptance for the low pT charged particles, the threshold transverse momenta pT can be lowered. It is estimated by the GEANT based Monte Carlo (MC)simulation [88] by requiring the equivalent single track selection criteria, which
are described in Section 4.2.1. Empirical functions are used to make the input
pT distributions for the MC simulations based on the measured pT distributions.
Figure 4.17 shows the pT distributions for (π + + π − )/2, (K+ +K− )/2 and (p̄+p)/2
in 0 - 70% centrality bin. Curves in the figures are the fit results by the empiricalfunctions. Following fit functions are chosen as triple exponential, single exponential and exponential with the fourth order polynomial exponent for π, K andp, respectively. For the generation of those particles, the known ratio between
the different sign of charge, which is independent on the centrality [84] class, aretaken into account as 0.97, 0.93 and 0.75 for π − /π − , K− /K+ and p̄/p, respectively. Figure 4.18 shows the number of generated particles (black) based on theempirical functions and survived particle (red) with the track selection criteria
in real data analysis as a function of pT . Vertical lines indicate the pT thresholdfor each particle species as 0.1 GeV/c, 0.25 GeV/c and 0.35 GeV/c for π ± , K ±and p(p̄), respectively.
According to this MC simulation, the detectable charged particle compositionsin the no magnetic field condition with the same selection criteria as the real data

64

CHAPTER 4. ANALYSIS

analysis are estimated as 94% for charged pions, 4% for charged kaons and 2%
for proton and antiproton in 0 - 70% centrality. The average pT for the detected
inclusive charged particles are also estimated as 0.57 GeV/c in this MC study.
It means the difference of the rapidity and pseudorapidity is at most 3% at theedge of the PHENIX acceptance by assuming the mass of charged pion.

4.3

Measurement of multiplicity distributions2(1/2 π pT ) d N / dpT dy [c / GeV ] (1/2 π pT ) d N / dpT dy [c / GeV ]Multiplicity distributions of charged particles were measured while changing thepseudorapidity window size δη from 0.066 to 0.7 with a step size of 0.7/25 = 0.022.
For a given pseudorapidity window size, the window position in the pseudorapidity axis was shifted by a step of 0.7/28 = 0.0027 as long as the window isfully contained within the PHENIX acceptance of |η| < 0.35. For each window
position NBD fits were performed to the multiplicity distributions. Biases originating from inefficient detector areas were corrected with the procedure explained

2

10
10

π+
+
K
p

3

2

10

πK

p

2

1
10
10

-2

-3

Positive(0 - 5% central)

Negative(0 - 5% central)Positive(60 - 92% peripheral)

Negative(60 - 92% peripheral)

2

2

2

10

-1

10
1
10
10
10
10
10
10

-1

-2

-3

-4

-5

-6

0

0.5

1

1.5

2

2.5

3

3.5

4 4.5 0pT [GeV/c]

0.5

1

1.5

2

2.5

3

3.5

4 4.5
pT [GeV/c]

Figure 4.16: Differential cross sections as a function of pT for π + (circle), π − (diamond), K + (square), K − (cross), p+ (triangle), and p− (star). Upper and lower
figures correspond to 0 - 5% and 60 - 92% centrality bins, respectively.

4.3. MEASUREMENT OF MULTIPLICITY DISTRIBUTIONS

65

in Section 4.4.1. Since even corrected NBD k parameters are not necessarily equalin the case of extremely inefficient window positions, the window positions havebeen truncated where the reconstruction efficiency is below 50%. This truncation is mainly to exclude biases from the largest hole in the middle of the charged
particle detector as shown in Figure 4.19 (a) and (c). After the truncation, it isobtained that the weighted mean of corrected NBD parameters (hµc i, hkc i) for a+a) (π + + π - ) / 2

103

1

-1

102

2

102

10

2

10

2

2

10

(1/2π pT ) d N /dpT dy [c / GeV 2]

2

102

2

c) (p + p) / 2

310(1/2π pT ) d N /dpT dy [c / GeV 2]

10(1/2π pT ) d N /dpT dy [c / GeV 2]-b) (K + K ) / 2

3

1

-1

1

-1

10

10

10

10-2

10-2

10-2

10-3
0

0.5

1

1.5

2

2.5

3

10-3
0

3.5
4
pT [GeV/c]

0.5

1

1.5

2

2.5

3

10-3
0

3.5
4
pT [GeV/c]

0.5

1

1.5

2

2.5

3

3.5
4
pT [GeV/c]

Figure 4.17: Differential cross sections as a function of pT for (π + + π − )/2 a),(K+ +K− )/2 b) and (p̄+p)/2 c) in centrality 0 - 70%. Curves corresponds to thefits with several empirical functions for the input of MC simulation (see text).

+a) (π + + π - ) / 2

-

10-2

c) (p + p) / 2
Number of tracks / events

Number of tracks / events

Number of tracks / eventsb) (K + K ) / 2

10-2

-2

10

10-3

-3

10

0.01 GeV/c

10-40.25 GeV/c

0.35 GeV/c

-3

10

0

0.2

0.4

0.6

0.8
1
pT [GeV/c]

0

0.2

0.4

0.6

0.8
1
pT [GeV/c]

0

0.2

0.4

0.6

0.8
1
pT [GeV/c]

Figure 4.18: The number of generated particles (closed) and survived particle (open) as a function of pT in the MC simulation. a), b) and c) corresponds to(π + + π − )/2, (K+ +K− )/2 and (p̄+p)/2, respectively. Vertical lines indicate thepT threshold in the no magnetic field condition.

66

CHAPTER 4. ANALYSIS

φ (0 < φ < 1/2π) [rad]c)φ (0 < φ < 1/2π) [rad]

a)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
η (-0.35 < η < 0.35)

η (-0.35 < η < 0.35)

Number of bins

b)
4000

4σ

3500

3σ

3000

2σ

2500
2000
1500
1000
500
0
0

10

20

30

40

50

60

70

80

90
100
Number of hits

Figure 4.19: 2-dimensional dead map definitions. a) Track projection points ontothe η - φ plane in the data after all track selections. The scale is normalized
to the mean number of hits in the peak position in b). b) The number of bins
among subdivided 28 × 28 bins as a function of the accumulated number of hits
over the total event sample. c) Definition of the central dead map by excluding
the detector region below 3 σ, where black parts are identified as dead areas.

4.3. MEASUREMENT OF MULTIPLICITY DISTRIBUTIONS

67

centrality 30-40%

ÂÅ

)

40

NBD k(

35
30
25
20
15
10
5
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1ÂÅ

Figure 4.20: Uncorrected NBD k as a function of δη in the small δη regions ata typical centrality bin. Closed and open data points were obtained with andwithout rejection on the inefficient detector areas, respectively.
given window size, which are defined as
hµc i ≡
hkc i ≡nXδµc −2i µc i /i=1

δµc −2
i ,

i=1

i=1

n
X

n
X

δkc −2
i kc i /nXδkc −2i ,(4.11)

i=1

where n is the number of valid window positions after the truncation and δ indicates errors on fitting parameters by the MINUIT program [94] in each window
position i. This procedure have been performed in each centrality bin with 5%
and 10% centrality bin width, respectively.
The lower limit of 0.066 was determined so that small window sizes, where
corrected NBD k was seen to depend heavily on window position, are all excluded.
Figure 4.20 shows the uncorrected NBD k as a function of window size δη < 0.1
in a typical centrality bin. Closed and open data points were obtained with andwithout rejection on the bad window position, which includes large amount of
inefficient detector areas, respectively. One can find a discrepancy between the
two type of data points in the small δη regions. It is an effect on the fail of NBD

68

CHAPTER 4. ANALYSIS

fit due to the lack of the number of degree of freedom. Therefore, the lower limitof 0.066 was set to obtain the final results. The lower limit is common for all
centrality bins.

4.4

Corrections and error estimations

4.4.1

Correction method

Any dead or inefficient areas in the detector have been identified and the bias on
the NBD parameters has been corrected based on a suitable statistical property
of NBD. Maps of dead areas were produced from the track projection points ontothe η - φ plane in the data after the track selections, as shown in Figure 4.19 a),where the detector acceptance is divided into 28 × 28 bins in the η - φ plane.
The accumulated number of hits over the total event sample in each bin is shownby a gray scale reflecting the statistical weights. The scale is normalized to themean number of hits in the peak position shown in Figure 4.19 b). Figure 4.19 b)
shows the number of bins among subdivided 28 × 28 bins as a function of theaccumulated number of hits over the total event sample in each 1/28 × 1/28
acceptance. If there were no dead or inefficient area, a binomial distribution
is expected with a probability of 1/28 × 1/28 to the total acceptance. For thebinomial part, a ± 3 σ region are taken. On the other hand, if there are any
dead or inefficient areas they tend to contaminate the lower tail of the binomial
distribution. A central dead map are defined by excluding detector region below
3 σ as shown in Figure 4.19 c) where black indicates regions that are identified asdead areas. The fraction of good area corresponds to 78% of the total acceptance.
This map was used to completely suppress particles which hit the dead areas in
the real data.
As long as the baseline distribution is approximated as an NBD, which iscertainly true as observed in E802 experiment [63] and in the present analysis,
one can estimate the relation between true k values of the NBD and biased
k values due to dead or inefficient areas based on the convolution theorem of
NBD. For two independent NBD’s with (µ1 , k1 ) and (µ2 , k2 ), it is known that the
convolution of the two NBD’s is an NBD with (µc , kc ), which satisfies relationsaskc = k1 + k2 ,µc = µ1 /k1(k1 + k2 ),(4.12)

where µ1 /k1 = µ2 /k2 holds [65, 95]. Therefore the correction can be applied bymultiplying a ratio of the total number of η - φ bins in a given η window size tothe number of bins excluding dead area, as the geometrical acceptance corrections
can be applied.

4.4. CORRECTIONS AND ERROR ESTIMATIONS

69

Strictly speaking one can not completely reproduce the original k by thiscorrection, since NBD’s in different positions are not completely independent.
However, except for the large hole which is already excluded by the truncation,
small holes are scattered rather uniformly in azimuthal direction for any positionof the δη windows. As the simplest overall correction to each window position, theconvolution theorem [65, 95] was applied by assuming collection of independent
NBD sources. As long as the correction is applied in the same manner for all
the azimuthal holes, it does not greatly affect the differential measurement to
the pseudorapidity space. If the correction is accurate enough, one can expect aconstancy of the corrected k values which should be independent of the fractionof dead areas. Based on the degree of constancy of corrected k as a function of
the fraction of dead areas in each window position for a given δη window size,
the incompleteness of the correction in each window size has been checked. As
briefly mentioned in the last paragraph of Section 4.3, the window sizes to beanalyzed were determined so that systematic error bands on hkc i explained in
Section 4.4.2, can contain the most of the corrected k values independently of thefraction of dead areas in each window position.

4.4.2

Statistical and systematic errors

As a convolution of statistical errors, errors were adopted on weighted meanvalues (δhµc i, δhkc i) on corrected NBD parameters after the window truncation
mentioned in Section 4.3, which are defined as2δµ¯c i
,
nind
2
δk¯ci
≡
,
nind

δhµc i2 ≡δhkc i2(4.13)

P
P
where δµ¯c i and δk¯ci are respectively defined as ni=1 δµc /n and ni=1 δkc /n with
the number of valid window positions n after the truncation and nind ≡ 0.75/δη isthe number of statistically independent window positions for a given δη window
size. This statistical error on δhkc i is referred to as δhkc i (stat).
The dominant sources of systematic errors for the correlation length measurement are the correction procedure with dead maps and the two-track separation
cuts, since both introduce unphysical correlations. 2% fluctuation have been allowed on the average multiplcity of measured number of charged tracks as a run
selection. This fluctuation is also originated by the dead channels of the tracking detectors in Section 4.2.1. In order to estimate this, two more patterns ofdead maps were defined with the definition of 3 σ ± 0.5 σ as indicated in Figure 4.19 c). The deviation of hkc i from the central dead map definition is referred
to as δhkc i (dead), which corresponds to 3.4% typically.

70

CHAPTER 4. ANALYSIS

As for the two-track separation case, the main purpose is to reject fake trackeffects, which is dominantly observed in φ direction rather than in η, since the PC1
hit requirement can fix z positions along the beam axis as shown in Figure 4.15.
Therefore, the effect of the δφ cut was estimated as ± 0.002 rad around the central
cut value of 0.012 rad with a fixed cut value on δη of 0.001. The deviation of hkc ifrom the central definition on the fake track rejection is referred to as δhkc i (fake).
This systematic error increases at higher centrality bins, which was estimated as
5.8% and 0.3% at 0 - 5% and 60 - 65% centrality bins, respectively.
The hkc i (stat) is related to agreements between multiplicity distributions and
NBD. The hkc i (dead) and hkc i (fake) depends on the position of the window andthe average multiplicity in a selected centrality bin, respectively. By treatingthese contributions as independent systematic error sources, the total systematic
error δhkc i (total) on hkc i in each δη in each centrality, was obtained by thequadratic sum over δhkc i (stat), δhkc i (dead) and δhkc i (fake).

Chapter 5
Results
In this chapter the results of the NBD fits are first tabulated. Then the measured
NBD k as a function of the pseudorapidity window sizes in various centrality bins
are shown in Figure 5.3. Finally, the Npart dependences of extracted αξ productin Equation (2.29) are presented.

5.1

Multiplicity distributions and NBD fits

NBD fit results in all window sizes in all centrality bins are summarized in Appendix Table A.1 through Table A.25 where hµc i and hµi are weighted meansof corrected and uncorrected µ over all window positions respectively, hkc i andhki are weighted means of corrected and uncorrected k over all window positions,
respectively. The hµc i are corrected only for the effect of the detector dead areas
as described in Section 4.4.1. The mean multiplicities were confirmed to be consistent with the result of the independent analysis by the different method using
only PC1 and PC3 [35], after known additional correction factors were taken into
account. These consit of 15 % of correction for finite double hit resolutions in
PC1 and 24 % of correction for the energy loss in the detector materials withrespect to the hµc i. Statistical errors on weighted means δhkc i (stat) are obtained
as explained in Section 4.4.2. hχ2 /NDF i is the average of reduced χ2 of NBD
fits over all window positions. hNDF i is the average of the degree of freedomof NBD fits over all window positions, and the systematic errors δhkc i (dead),δhkc i (fake) and δhkc i (total) are already explained in Section 4.4.2. The mean
and r.m.s. of the reduced χ2 values in the NBD fit over all window positions andall δη sizes and all centrality bins were obtained as 0.75 and 0.33 respectively as
shown in Figure 5.1. The mean value corresponds to typically 80% confidence
level. Therefore, it is good enough to assume NBD as a baseline multiplicity
distribution to obtain the integrated correlation function via the k parameter.
As a demonstration to show how well the NBD fits work, Figure 5.2 shows
the charged particle multiplicity distributions in each pseudorapidity window71

72

CHAPTER 5. RESULTS

Mean
RMS

104

0.75150.3307

103

102

10

1
0

1

2

3

4

5

6

7

8
χ2/NDF

Figure 5.1: Reduced χ2 values in the NBD fit over all window positions and allδη sizes and all centrality bins.
size in 1/8 fractions of the full rapidity coverage of |η| < 0.35 with 0 - 10%
events in the collision centrality, where the uncorrected multiplicity distributions
within the total error bands on hkc i in Appendix Table A.1 are all merged. The
distributions are shown as a function of the number of tracks n normalized tothe mean multiplicity hni in each window. The error bars show the statisticalerrors on the merged distributions. The solid curves are fit results with NBD only
for the demonstration purpose. The fit results in Appendix Table A.1 throughTable A.25 are not obtained from these convoluted distributions whose accuracies
are degraded by the convolutions with different µ values due to different detector
biases depending on the window positions.

5.2

δη dependence of NBD k

Figure 5.3 a) and b) show hkc i as a function of pseudorapidity window size with
10% and 5% centrality bin width, respectively. Centrality classes are indicatedinside the figures. The error bars show δhkc i (total) defined in Section 4.4.2.
An upper limit on the absolute scale of ξ was obtained as ξ < 0.035 with respect to the all centrality bins by the free parameter fits based on Equation (2.28).
Since the upper limit of ξ is small enough compared to the fitting region ofδη (ξ ≪ δη), Equation (2.29) can be applied for the fits to the NBD k as afunction of δη. More discussions on the small ξ can be found in Section 6.1.1.
The solid lines in Figure 5.3 indicate the fit results based on Equation (2.29).
The fits were performed in the δη region from 0.066 to 0.7, which is explained inP(n)5.3. NP ART DEPENDENCE OF β AND αξ

73

107

1/8 × 107 (δ η=0.087)

106

2/8 × 106 (δ η=0.175)

105

3/8 × 105 (δ η=0.263)

104

4/8 × 104 (δ η=0.350)

103

5/8 × 103 (δ η=0.438)

102

6/8 × 102 (δ η=0.525)

10

7/8 × 101 (δ η=0.613)

1

8/8 (δ η=0.700)

10-1
10-2
10-3
10-4
10-5
10-6

0

0.5

1

1.5

2

2.5

3

3.5
4
n/<n>

Figure 5.2: Uncorrected charged particle multiplicity distributions in each pseudorapidity window size, as indicated in the legend, at 0 - 10% collision centrality.
The distributions are shown as a function of the number of tracks n normalized
to the mean multiplicity hni in each window. The error bars show the statisticalerrors. The solid curves are fit results of NBD.
Section 4.2.1. The typical χ2 /NDF in the fit based on Equation (2.29) is 0.132,
which corresponds to 99% confidence level.
As explained in Section 2.4 for Equation (2.29), in the limit of β = 0, theslopes in k vs. δη have crucial information on the phase transition. In Figure 5.3,
one can identify different behaviors in slopes around 40-50% centrality region
even without fit curves.

5.3

Npart dependence of β and αξFigure 5.4 a) and b) show the obtained fit parameters β and αξ with Equation (2.29) as a function of the Npart , where results for both the 5% and 10%
centrality bin width cases are plotted as filled and open circles, respectively. The
smooth solid and dotted curves are provided to guide the eye. The horizontal
error bars correspond to ambiguities on the mean value of Npart as explained

CHAPTER 5. RESULTS

0 - 10%a)

5 - 15%2

10

10 - 20%
15 - 25%
20 - 30%
25 - 35%k(δη) in δ φ = π/2 [rad]k(δη) in δ φ = π/2 [rad]74

0 - 5%b)

5 - 10%
10 - 15%
15 - 20%
20 - 25%
25 - 30%

30 - 40%

30 - 35%

35 - 45%
40 - 50%

35 - 40%2

40 - 45%10

45 - 55%

45 - 50%

50 - 60%

50 - 55%

55 - 65%

55 - 60%
60 - 65%

10
10

-1

10

1

δη

-1

10

1

δη

Figure 5.3: Weighted mean of corrected NBD k, hkc i as a function of pseudorapidity window size with a) 10% and b) 5% centrality bin widths. Centrality classes
are indicated in the figure legend. The error bars show δhkc i (total), as explainedin Section 4.4.2. The solid lines indicate the fit curves of Equation (2.29).
in Section 4.1.3. The vertical error bars are obtained from errors on the fitting
parameter by the MINUIT program [94]. Table 5.1 summarizes the fit results
where centrality bins, corresponding Npart , αξ, β and χ2 /NDF obtained by thefit with Equation (2.29) are shown for 10% and 5% centrality bin width cases,
respectively.
The β parameter is necessary to absorb any effects independent of pseudorapidity space correlations as explained in Section 2.4.2. For a wider centrality bin,β5.3. NP ART DEPENDENCE OF β AND αξ

75

0.09

a)

0.08
0.07

5% bin width
10% bin width

0.06
0.05
0.04
0.03
0.02
0.01
0
0

50

100

150

200

250

300

350

400

Npart
αξ

0.0014b)0.0012
5% bin width
10% bin width

0.001
0.00080.00060.00040.0002
0
0

50

100

150

200

250

300

350

400

Npart

Figure 5.4: Fit results based on Equation (2.29). a) is β and b) is products ofαξ as a function of Npart . The horizontal error bars correspond to ambiguities in
the mean value of Npart as explained in Section 4.1.3. The vertical error bars are
obtained from errors on the fitting parameter.
the width of multiplicity distribution becomes broader, since events with wider
range of centrality bins are included in the bin. As the result, the systematicdifference of β in the 5% and 10% centrality data sets is shown in Figure 5.4 a).
It will be discussed in more detail for the exploit of β term at Section 6.2.
The αξ product in the correlation function, which is monotonically related
to susceptibility in the long wavelength limit χk=0 as explained in Section 2.4.2,
indicates the non-monotonic behavior as a function of Npart . This is originatedin the different behaviors of slopes as shown in Figure 5.3 with respect to thecorresponding centrality bins i.e. 40 - 50% and 40 - 45% centrality bins. Since
this behavior is one of the main result of this analysis, its significance of the local

76

CHAPTER 5. RESULTS

Table 5.1: The αξ and β in Equation (2.29) obtained by the fits to hkc i vs. δη.
Upper and lower column corresponds to 10% and 5% centrality bin width cases,
respectively.
Centrality (%)

hNpart i

0 − 105 − 1510 − 2015 − 25
20 − 30
25 − 35
30 − 40
35 − 45
40 − 50
45 − 55
50 − 60
55 − 650−5
5 − 1010 − 1515 − 2020 − 25
25 − 30
30 − 35
35 − 40
40 − 45
45 − 50
50 − 55
55 − 60
60 − 65

325.2 ± 3.3
276.4 ± 4.0
234.6 ± 4.7
198.4 ± 5.4
166.6 ± 5.4
138.6 ± 4.9
114.2 ± 4.4
92.8 ± 4.3
74.4 ± 3.8
58.8 ± 3.3
45.5 ± 3.3
34.6 ± 3.8
351.4 ± 2.9
299.0 ± 3.8
253.9 ± 4.3
215.3 ± 5.3
181.6 ± 5.6
151.5 ± 4.9
125.7 ± 4.9
102.7 ± 4.3
82.9 ± 4.3
65.9 ± 3.4
51.6 ± 3.2
39.4 ± 3.5
29.8 ± 4.1αξ(∝ χk=0 )0.17 × 10−3 ± 0.03 × 10−3
0.19 × 10−3 ± 0.03 × 10−30.24 × 10−3 ± 0.04 × 10−30.36 × 10−3 ± 0.04 × 10−3
0.39 × 10−3 ± 0.05 × 10−3
0.46 × 10−3 ± 0.06 × 10−30.62 × 10−3 ± 0.06 × 10−30.71 × 10−3 ± 0.07 × 10−30.76 × 10−3 ± 0.09 × 10−3
0.54 × 10−3 ± 0.11 × 10−3
0.67 × 10−3 ± 0.14 × 10−30.69 × 10−3 ± 0.18 × 10−3
0.23 × 10−3 ± 0.03 × 10−3
0.20 × 10−3 ± 0.03 × 10−3
0.20 × 10−3 ± 0.04 × 10−30.36 × 10−3 ± 0.04 × 10−30.35 × 10−3 ± 0.04 × 10−30.45 × 10−3 ± 0.06 × 10−30.64 × 10−3 ± 0.08 × 10−30.82 × 10−3 ± 0.09 × 10−30.95 × 10−3 ± 0.11 × 10−30.68 × 10−3 ± 0.13 × 10−3
0.67 × 10−3 ± 0.18 × 10−3
1.02 × 10−3 ± 0.23 × 10−3
1.05 × 10−3 ± 0.29 × 10−3β0.80 × 10−2 ± 0.02 × 10−2
1.17 × 10−2 ± 0.02 × 10−2
1.51 × 10−2 ± 0.03 × 10−2
1.72 × 10−2 ± 0.03 × 10−2
1.96 × 10−2 ± 0.03 × 10−2
2.31 × 10−2 ± 0.04 × 10−2
2.77 × 10−2 ± 0.05 × 10−2
3.26 × 10−2 ± 0.05 × 10−2
3.96 × 10−2 ± 0.07 × 10−2
4.85 × 10−2 ± 0.08 × 10−2
6.22 × 10−2 ± 0.11 × 10−2
8.19 × 10−2 ± 0.14 × 10−2
0.19 × 10−2 ± 0.02 × 10−20.46 × 10−2 ± 0.02 × 10−20.75 × 10−2 ± 0.02 × 10−20.90 × 10−2 ± 0.03 × 10−2
1.08 × 10−2 ± 0.03 × 10−2
1.35 × 10−2 ± 0.04 × 10−2
1.55 × 10−2 ± 0.05 × 10−2
1.90 × 10−2 ± 0.05 × 10−22.40 × 10−2 ± 0.07 × 10−2
2.86 × 10−2 ± 0.08 × 10−2
3.72 × 10−2 ± 0.11 × 10−2
5.19 × 10−2 ± 0.16 × 10−2
6.64 × 10−2 ± 0.19 × 10−2

χ2 /N DF(N DF = 27)
0.24
0.16
0.14
0.26
0.09
0.09
0.13
0.14
0.14
0.05
0.11
0.05
0.18
0.27
0.17
0.18
0.32
0.02
0.09
0.08
0.06
0.08
0.11
0.06
0.08

maximum of αξ will be discussed at Section 6.4. One can find the absolute valueof αξ products were aliened to the same scale for the different centrality bin width
cases owing to the β parameter as described in above. The difference on the αξ
value between the 5% and 10% around the local maximum centrality bin width
cases can be understood as smearing effects around a peak-like shape, and it isconfirmed by a toy model MC calculation.

Chapter 6
Discussions
6.1

Absolute scale of the correlation length

6.1.1

Small correlation length<χ 2/NDF> (NDF=28)

As long as reliable NBD k in δη ∼ 0 can be obtained, which corresponds toδη ≪ ξ, α can be treated as a free parameter by Equation (2.28) to extract
the absolute value of correlation length ξ in each centrality. If this is not thecase, the absolute ξ can not be obtained without any constraints due to the anticorrelations between α and ξ. Under this situation, the correlation lengths have
0.432
0.431

α=0.22

α=1.0

0.43
0.429
0.428
<χ2/NDF>=0.4278

0.427
0.426
0.425

<χ2/NDF>=0.4250

0.424
-1
10

1

10

α

Figure 6.1: Average χ2 /NDF (NDF = 28) over all centrality bins at the fits
to the integrated correlation function Equation (2.28) as a function of fixed α.
The minimum average χ2 /NDF (0.425) at α = 0.22 is indicated as solid line.
Average χ2 /NDF (0.428) obtained with α = 1.0 is indicated as dashed line.
77

78

CHAPTER 6. DISCUSSIONSβ (α=0.22)

been obtained by fixing the parameter α in Equation (2.28) to 0.22, which waschosen so that the average χ2 /NDF over all centrality bins is minimized in the
fit results for the relation between NBD k and δη. Figure 6.1 shows the average
χ2 /NDF (NDF = 28) over all centrality bins as a function of fixed α value. One
can find the minimum average χ2 /NDF (0.425) at α = 0.22 as indicated by thesolid line. Figure 6.2 shows fit parameters of β and ξ as a function of Npart withthis fixed parameter α = 0.22. A physical reasoning to motivate the parameter
α are fixed will be explained in Section 6.1.2.
The obtained ξ indicate very small value as shown in Figure 6.2, and themethod of extraction of ξ with fixed α can not be justified enough only by the

0.09

a)

0.08
5% bin width
10% bin width

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

50

100

150

200

250

300

350

400

ξ (α=0.22)

Npart
0.006

b)
0.005
5% bin width
10% bin width
0.004

0.003

0.002

0.001

0
0

50

100

150

200

250

300

350

400

Npart

Figure 6.2: Fit results based on Equation (2.28) with fixed α = 0.22. a) is β andb) is ξ as a function of Npart . The horizontal error bars correspond to ambiguities
in the mean value of Npart as explained in Section 4.1.3. The vertical error bars
are obtained from errors on the fitting parameter.

6.1. ABSOLUTE SCALE OF THE CORRELATION LENGTH

79

minimum reduced χ2 value in Figure 6.1. However, at least the upper limit of
the absolute scale of ξ was obtained as ξ < 0.035 by the free parameter fits inEquation (2.28). It is qualitatively consistent with expectation from numericalcalculations [96] that the correlation lengths become smaller in the RHIC energy
than the (p̄) p + p collisions [61] and low energy A + A collisions [63]. Since theupper limit of ξ is small enough compared to the fitting region of δη (ξ ≪ δη),
Equation (2.29) can be applied for the fits to the NBD k as a function of δη.
In this case, the αξ product, which is proportional to the susceptibility at thestatic limit as defined in Equation (2.14), can be obtained by the fits without any
physical assumptions. The typical χ2 /NDF in the fit based on Equation (2.29)
is 0.132, which corresponds to 99 % confidence level as described in Section 5.2.
Therefore, the small correlation length is confirmed as below the minimum δηwindow sizes of 0.066 based on the validity of the approximated function for thesmall ξ of Equation (2.29).

6.1.2

Relation with Bose-Einstein correlation

If the observed correlation were to originate only from the effect of Bose-Einsteincorrelation (HBT), then the α is expected to be directly related with the chaoticity parameter λ in HBT analysis which is measured in relative momentum space q.
A similar measurement in pseudorapidity space based on Equation (2.27) without
β term in low energy A + A collision systems [97] indicates the direct relation
between λ and α quantitatively. The observed two particle correlation strength
α in pseudorapidity space is weaker than λ measured in q space and essentiallybecomes zero for the particle pairs selected in the higher q region where HBTeffect also becomes zero. This indicates that the observed pseudorapidity correlations in the lower energy A + A collisions are essentially explained purely by
the HBT effect.
√
In Au + Au collisions at sN N = 200 GeV, measured λ shows the constantbehavior as a function of Npart within 12 % of its systematic errors [98, 99].
Figure 6.3 shows one dimensional HBT radius parameter Rinv (bottom) andchaoticity parameter λ (top) as a function of Npart with respect to the correlationsof identified pions in same charge. According to the correspondence between α
and λ, α can be rephrased as a fraction of correlated particles in the chaotic
source. Therefore, the maximum α of 1.0, which is indicated by the dashedline in Figure 6.1, is a physical limit. This means 100 % correlation for the
produced particles. In this sense, obtained α of 0.22 by the fit is not contrived
with the HBT measurement. This is the reason why the constant α was assumed
in Section 6.1.1.
However, obtained parameters for both αξ in Figure 5.4 and ξ in Figure 6.2
indicates the non-monotonic behavior as a function of Npart , where its nonmonotonicity will be discussed in Section 6.4, in spite of the fact that λ andRinv indicates the constant behavior. This implies that the non-monotonic be-

80

CHAPTER 6. DISCUSSIONS

π+π +

λ

0.8

π -π -

0.6
0.4
0.2

Rinv [fm]

0

6

4

2

0

100

200

300

400 0

N part

100

200

300

400

N partFigure 6.3: One dimensional HBT radius parameter Rinv (bottom) and chaoticity
parameter λ (top) as a function of Npart for π + π + (left) and π − π − (right)
correlations [99].
havior of the αξ product can not be explained solely as a result of the known
HBT effect, because α ∝ λ is expected to be constant for any Npart and ξ which
would be related with the HBT source radii is expected to be monotonic, if the
known HBT effect is the only source of the correlation.

6.2

Pseudorapidity independent correlation

It should be emphasized that the parametrization in Equation (2.27) is practically necessary. The β parameter can absorb any effects independent of pseudorapidity space correlations. For a wider centrality bin, the width of multiplicity
distribution becomes broader, since events with wider range of centrality bins areincluded in the bin. This causes the systematic difference of β in the 5% and
10% centrality data sets as shown in Figure 5.4 a). The systematic shift of βparameters to smaller values in the smaller centrality bin width, suggests that β
dominantly contains fluctuations on Npart .
The ambiguity on Npart measured by PHENIX is not large compared, for example, to NA49 experiment at SPS where a non-monotonic behavior on scaled6.3. OTHER CORRELATION SOURCES

81

variance of multiplicities was seen as a function of the number of projectile participant nucleons Nproj [100, 101]. In NA49, only spectators from the projectile
nucleus are measurable and it causes an increase of scaled variance of multiplicity
distributions in peripheral collisions due to dominantly large Nproj fluctuationsin the target nucleus [102]. This is due to the partial sampling with respect tothe total number of nucleons in two colliding nuclei. Since both projectile andtarget nuclei in both sides can be measured by BBC and ZDC at PHENIX, thiskind of large ambiguities on Npart is more suppressed even in peripheral collisions.
However, even though the Npart fluctuation is still remaining, β parameter canabsorb this kind of offset parts of fluctuations and the Npart fluctuation is notharmful for the measurement of the αξ products, since they are based on the
differential values of fluctuations for a given centrality bin.
In addition, β is expected to absorb effects from azimuthal correlations. Since
the PHENIX detector does not cover the full azimuthal range, fluctuations of low
pT particles caused by event reaction plane rotations and elliptic flow [103] should
contribute to the two particle correlation function even in the pseudorapidity
direction as an offset in principle. Owing to the β parameter, the non-monotonic
behavior of the measured αξ in the pseudorapidity direction cannot be biased byelliptic flow nor by initial geometrical biases, since the azimuthal correlations areconstant over the narrow pseudorapidity window of |η| < 0.35 [104, 105].

6.3

Other correlation sources

In this section, following two type of other sources on the correlations will beconcerned, which are not related to the density correlations but could affect themeasurement of the inclusive charged particle multiplicity fluctuations. The firstis charged track pairs from particle decays in flight. The second is backgroundcharged track pairs originating from secondary particle interactions in detectormaterials e.g. showers, conversion pairs and so on. These two sources have been
estimated as effects of contaminations to the inclusive charged particle multiplicity fluctuations by the GEANT based MC [88] simulations.

6.3.1

Contributions from weak decay particles

The statistically dominant weak decay particles which can contribute to the inclusive charged particle multiplicity are KS0 → π + π − and Λ (Λ̄) → p (p̄) π − (π + ).
The relative invariant yields of those particles to charged pions are 15% and 5%
for KS0 and Λ (Λ̄), respectively. It is calculated by using the measured produc√tion cross section in Au + Au collisions at sN N = 200 GeV for Λ (Λ̄) [106,107].
The production cross section of KS0 is assumed to be same as measured charged
kaons [84]. The detection efficiency of the charged track pairs from these weak
decay particles in the one arm acceptance of PHENIX detector (|η| < 0.35,

82

CHAPTER 6. DISCUSSIONS∆φ < π/2) are obtained by the GEANT based [88] MC simulation. It is estimated by using the pT spectra of charged kaons for KS0 as the most dominant
meson, and by using the pT spectra of charged pions with transverse mass scalingfor Λ (Λ̄) as the most dominant baryon, which contribute to the inclusive charged
particle multiplicity fluctuation. As the result, the ratios of charged track pairs
originating from those weak decay particles to the number of produced charged

Fraction of track pairs [%]

a)
2

Λ+Λ
K0s

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
Difference between two traks in η

Relative fraction of track pairs [%]

b)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

Λ+Λ
K0s

0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6
0.7
δη window size

Figure 6.4: a) Fraction of charged track pairs originated in single Ks0 (opencircle) and Λ + Λ̄ (closed circle) as a function of difference between two tracks
in η. b) Relative fraction of the track pairs as compared to charged π yield ineach δη window size. Statistical errors are included in each marker. Solid lines
indicate the minimum window size in this analysis. The dashed line and dottedline correspond to the yield with respect to the each parent particles normalized
by the charged π yield in the total acceptance.

6.3. OTHER CORRELATION SOURCES

83

pions per event are 0.7% and 0.9% for KS0 and Λ + Λ̄, respectively.

(k conv - 2k) / 2k [%]The effects of those correlations on NBD k were estimated as follows. Suppose
two independent NBD’s in different windows have the same NBD parameters ofµ and k for a given window size of δη/2. If there is no correlation between the two
windows, NBD in the δη window size becomes a convoluted distribution between
the two NBD’s. This is certainly true, since it is known the correlation length iswell below the minimum size of δη windows as already discussed in Section 6.1.
Based on the NBD convolution theorem [65,95], the convoluted NBD parameters,µconv and kconv are expressed as µconv = 2µ and kconv = 2k, respectively, in the case03a) K s

2

1

0

-1

-2

(k conv - 2k) / 2k [%]

0

0.1

0.2

0.3

0.4

0.5

3

0.6

0.7

δη

b) Λ + Λ

2

1

0

-1

-2
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

δη

Figure 6.5: Effects in NBD k estimated by (kconv − 2k)/2k as a function of δη
window size from a) Ks0 and b) Λ + Λ̄. Solid and dashed lines correspond toweighted mean value and corresponding one standard deviation, respectively, forover all δη.

84

CHAPTER 6. DISCUSSIONS

of no correlation. For the case where the correlated pairs are embedded, a fraction
f is defined for the number of correlated pairs with respected to µ. Then the meanvalue before the correlated pairs are embedded is expressed as µ(1 − f ) in theδη/2 window. The effect of the embedded correlation on kconv can be estimatedby adding the number of correlated pairs to both windows simultaneously withthe fraction of f . With µ(1 − f ) and k, one can generate NBD with a random
number generator in each window of δη/2 and convolute the two NBD’s. From
the NBD fit to the convoluted distribution, it can be obtained as kconv including
the effect of the correlated pairs. The ratio of the deviation of kconv is definedto the independent case, ∆k ≡ (kconv − 2k)/2k for KS0 and Λ + Λ̄, respectively.
In this method, the pair fraction f depends on δη window size, since weak decay
particles have their own correlation length due to the kinematical constraint
as shown in Figure 6.4 a). Figure 6.4 a) shows the pair fraction of charged
track pairs originated in single Ks0 (open circle) and Λ + Λ̄ (closed circle) as afunction of difference between two tracks in pseudorapidity η. It is also evaluatedfrom the GEANT based MC simulation with the actual track selection criteria.
The fraction f were obtained based on the two particle correlation of decayedpairs as a function of δη window size. It should be noted that the integrated
fractions correspond to the fractions mentioned above as 0.7% and 0.9% for KS0
and Λ + Λ̄, respectively. Figure 6.4 b) shows the relative fraction of the track
pairs as compared to the charged π yield in each δη window size. Statisticalerrors are included in each marker. Solid lines indicate the minimum window sizein this analysis. The dashed line and dotted line correspond to the pair yieldwith respect to the each parent particles normalized by the charged π yield in
the total acceptance.
For all observed (hµc i, hkc i) values as input values in all δη windows and in all
centrality bins, the ∆k have been estimated according to the estimated fraction
f for each parent particles. Figure 6.5 a) and Figure 6.5 b) shows the effects
in NBD k estimated by the (kconv − 2k)/2k as a function of δη window size fora) Ks0 and b) Λ + Λ̄, respectively. As a result, the average values of ∆k overall data points were estimated as + 0.27% ± 0.35% (standard deviation) and+ 0.40% ± 0.35% (standard deviation) for KS0 and Λ + Λ̄ decays, respectively.
The average values and errors are calculated by taking weighted mean and corresponding one standard deviation, respectively, over all points as shown in solidand dashed lines in Figure 6.5.
Meanwhile, the average value of relative errors, δhkc i(total)/hkc i in measured
k is ± 7.34% ± 3.29% (standard deviation). It is confirmed that the estimated∆k values are all included within the range of the relative errors on measured k.
Therefore, it can be concluded that the effect of the statistically dominant weak
decay pairs with a few percent level on the αξ product can not exceed the fullerror sizes of the αξ products in Table 5.1.

6.3. OTHER CORRELATION SOURCES

85

Table 6.1: Summary of detector materials in PHENIX [40] and photon conversion
probability in front of EMCal [79] detector. Since TOF is located in front of only
PbGl type detector, the total radiation length indicate the different value between
the PbSc type and PbGl.
Detector and materials
Beam pipe + MVD + (He) + DC [75]
PC1 [77]
RICH (CO2 ) [78]
TEC [77]
PC3 [77]
TOF [78]
Total
Origin particle
Single γ
Decay from π 0 (maximum)

6.3.2

Total radiation length (%)
1.3
1.2
2.1
0.8
2.4
0.0 (PbSc) or 6.0 (PbGl)
7.8 (PbSc) or 13.8 (PbGl)
Photon conversion probability (%)
5.9 (PbSc) or 10.1 (PbGl)
11.8 (PbSc) or 20.2 (PbGl)

Contributions from detector materials

The most significant process originating from the detector materials, which could
affect the charged particle multiplicity fluctuation, is photon conversions. Somefraction of photons produces electron-positron pairs in the detector materials
i.e. γ → e+ e− , depending on the total amount of the detector materials in frontof the tracking detectors for the charged particles. Almost 100% of photons up to√
4 GeV/c of pT in Au + Au collsions at sN N = 200 GeV are produced by decays
from π 0 [108,109]. Nevertheless, the same number of π 0 are produced as charged
pions at the collisions. Therefore, the effects of the photon conversion process
should be considered in the analysis of charged particle multiplicity fluctuation.
The total photon conversion probability in front of the EMCal is estimatedbased on the corresponding radiation lengths X0 for each detectors. It can beobtained with respect to single photon γ and neutral pion π 0 as
Pconv (γ) = 1 − e−7/9(x/X0 ) ,Pconv (π 0 ) = 2Pconv (γ),(6.1)

where Pconv (π 0 ) is maximum probability assmuming both decay photons convert. Table 6.1 indicates the summary of the radiation length in each sectionof the detector materials in PHENIX and the total photon conversion probability in front of the two type of the EMCal (see Section 3.4.4). Meanwhile, the86

CHAPTER 6. DISCUSSIONS

charged tracks used in this analysis are recontacted by the vector information in
DC (see Section 3.4.1) mainly, and the information of the PC1, PC3 and EMCal
are used only for the tracking association as described in Section 4.2.1. Therefore, only the amount of detector materials before and inside DC is important,and it correspond to 1.3% of a radiation length as indicated in Table 6.1. It
produces electron-positron pairs with 1.0% photon conversion probability. The
detection efficiency of electron-positron pairs which survive after the requirementof the charged track associations and two track separations in Section 4.2.1 isestimated as 0.22%. It was estimated by the MC simulations with flat pT distribution of photons. Since the opening angle of the conversion pairs are verysmall, these conversion electrons are strongly suppressed by the two track separation cuts. Consequently, electron-positron pairs of 2.2 × 10−3 % with respect tothe produced charged pions per event contribute to the multiplicity fluctuations.
The efficiency of charged track pairs, which is produced by the materials fromsingle charged hadrons as knock-on electrons (positrons), is estimated as less than
5.8 × 10−5 %. Since the total pair fractions are much smaller than that in weak
decays by several orders of magnitude, it can be concluded that the effect of thosesecondary particles on the αξ products are negligible.

6.4

Evaluation of the non-monotonic αξ

6.4.1

Significance of local maximum

The αξ product obtained by Equation (2.29) is related with susceptibility in thelong wavelength limit, χk=0 as described in Section 2.4.2. According to Equation (2.30), if the system temperature T is far from the critical temperature Tc ,αξ is expected to decrease monotonically with increasing T which is a monotonic
function of Npart as discussed in Section 4.1.4. Therefore, one can assume themonotonically decreasing function as a hypothesis of the baseline in T far from Tc .
As the baseline functional forms, following two cases have been considered. The
one is a power law function which is naturally expected from Equation (2.30),
and the other is a linear function as the simplest assumption. The power law
baseline and the linear baseline are parametrized asαξ(Npart ) = p1 (Npart )p2 ,(6.2)αξ(Npart) = p1 + p2 Npart ,

(6.3)

and

with fit parameter p1 and p2 , respectively. As a test hypothesis, a local maximum
on the monotonic baselines in αξ vs. Npart is assumed. Although the functional
form around the local maximum is not known a priori without introducing a physical model, at least one can discuss the significance of the local maximum from

6.4. EVALUATION OF THE NON-MONOTONIC αξαξ0.0014αξ0.0014a) 5% bin width0.00120.001

0.00080.00080.00060.00060.00040.00040.00020.0002

50

100

150

200

250

300

350

c) 5% bin widh

0.0012

0.001

0
0

87

0
0

400

50

100

150

200

250

300

350

400

Npart
αξ

0.0014αξ0.0014Npartb) 10% bin width0.0012
0.001

0.001

0.00080.00080.00060.00060.00040.00040.00020.0002

0
0

50

100

150

200

250

300

350

d) 10% bin width0.0012

0
0

400

50

100

150

Npart

200

250

300

350

400

Npart

Figure 6.6: αξ vs. Npart in Table 5.1 with fit curves. The dashed and solid curves
show the fit results with the baseline functions and with the composite functions,
respectively. a) and b) correspond to 5% and 10% bin width cases with the powerlaw baselines, respectively. c) and d) correspond to 5% and 10% bin width caseswith the linear baselines, respectively.
the monotonic baseline by introducing a Gaussian distribution. The composite
functions are defined asp2−αξ(Npart ) = p1 (Npart ) + ae

(Npart −m)2
2w2

(6.4)

and
αξ(Npart ) = p1 + p2 Npart + ae−(Npart −m)2
2w2

,

(6.5)

where a, m and w correspond to amplitude, mean and width of the Gaussiancomponent, respectively. Fits with the four functional forms were performedto αξ vs. Npart in 20 < Npart < 200. Figure 6.6 shows αξ vs. Npart in Table 5.1 with those fit curves. The dashed and solid curves show the fit resultswith the baseline functions and with the composite functions, respectively. Figure 6.6 a) and b) correspond to 5% and 10% bin width cases with the powerlaw baselines, respectively. Figure 6.6 c) and d) correspond to 5% and 10% bin

88

CHAPTER 6. DISCUSSIONS

Table 6.2: The fit parameters in Equation (6.2), Equation (6.3), Equation (6.4)
and Equation (6.5).
Functional form
Power law in Eq. (6.2)
Power law + Gaussian in Eq. (6.4)
Linear in Eq. (6.3)
Linear + Gaussian in Eq. (6.5)
Power law in Eq. (6.2)
Power law + Gaussian in Eq. (6.4)
Linear in Eq. (6.3)
Linear + Gaussian in Eq. (6.5)

Centrality bin
bin width (%)
5
5
5
5
10
10
10
10

χ2 /N DF(N DF )
2.76 (7)0.60 (4)1.23 (7)0.79 (4)2.10 (7)0.38 (4)1.09 (7)0.43 (4)a ± δaSignificance(a/δa)

0.37 × 103 ± 0.09 × 103

3.98

0.27 × 103 ± 0.21 × 103

1.24

0.27 × 103 ± 0.08 × 103

3.21

0.22 × 103 ± 0.13 × 103

1.69

width cases with the linear baselines, respectively. Table 6.2 summarizes all thefit results by the MINUIT program [94], where functional forms, centrality bin
widths, χ2 /NDF (NDF ), the Gaussian amplitude a with its error δa and thesignificance of the amplitude defined by a/δa. Although the significance of thelocal maximum with the linear baseline is smaller than that with the power law
baseline, it is mainly due to the larger errors on a in Equation (6.5) than errors inEquation (6.4). This is a reflection that the combination of Gaussian distribution
with the linear baseline is not as good as that with the power law baseline forthe given data points. The difference on the significance between 5% and 10%
centrality bin width cases is not attributed to the correlations with β parameter,
since β was introduced as a parameter independent of δη. This can be rather
understood as smearing effects of a peak-like shape due to the difference of centrality bin widths around the mean Npart . In any cases in Table 6.2, χ2 /NDF
indicate that all the composite functions are favored compared to the monotonic
functions. This result supports the non-monotonicity of αξ as a function of Npart .

6.4.2

Assumption of the one correlation length

Although there is a possibility that more than one correlation length scale isdynamically present, the functional form with one correlation length in Equation (2.29) can reasonably describe the region of 0.066 < δη < 0.7 with theaverage χ2 /NDF of 0.43 over all centrality bins. A further check on the Npartdependence of the αξ products have been performed by limiting the region of fit
to 0.066 < δη < 0.306 as shown in Figure 6.7. The characteristic behavior of αξ
still present at around Npart ∼ 90.

β6.5. PHYSICAL INTERPRETATION OF THE RESULTS

89

0.09

a)

0.08
5% bin width
10% bin width

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

50

100

150

200

250

300

350

400

αξ

Npart
0.002

0.0018b)0.00165% bin width
10% bin width0.00140.00120.001
0.00080.00060.00040.0002
0
0

50

100

150

200

250

300

350

400

Npart

Figure 6.7: Fit results based on Equation (2.29) by limiting the range of δη from
0.066 to 0.306. a) is β and b) is products of αξ as a function of Npart . The
horizontal error bars correspond to ambiguities in the mean value of Npart as
explained in Section 4.1.3. The vertical error bars are obtained from errors on
the fitting parameter.

6.5

Physical interpretation of the results

This analysis is predicated on an application of the most simplest model i.e.
Ginzburg-Landau thermodynamical theory, for the high-density matter createdby the high energy heavy ion collision. Therefore, it is important whether thesystem reached thermal equilibrium or not. According to the fruitful RHIC results already obtained, it is almost established that a strongly interacting, highdensity matter has been created in nucleus-nucleus collisions [37]. Especially,strong suppression of hadrons at high transverse momentum (pT ) observed in√central Au+Au collisions at sN N = 200GeV at RHIC indicates creation of the

90

CHAPTER 6. DISCUSSIONS

high density partonic matter [86, 112]. Additionally, strong elliptic flow, which
is one of the indicative observables on collective motions of the matter, indicatesthat the matter thermalizes rapidly and behaves like a fluid with very low viscosity [103, 110]. Furthermore, the valence quark number scaling of the elliptic
flow suggests that quark-like degrees of freedom are pertinent in the evolution of
the flow [111]. Those observations naturally lead us to the expectation that theinitial thermalized state of matter is at T > Tc in central Au+Au collisions, andpossibly at T < Tc in the most peripheral collisions. Therefore it can be considered that a system with initial T = Tc may exist somewhere between peripheral
and central collisions.
One can form a lot of hypothesis on the space-time evolution of the systembefore and after the thermalizaion. However, it is difficult to conclude the correct
path on the phase diagram only by the experimental observables exclusively,
because we always observe the final state particles. Even in this case, fluctuation
observables have possibilities to keep the information at the phase transition
as described in Section 2.3. According to the scenario, a singular behavior on
the thermodynamic quantity will be seen at one point of the scanning regions.
It corresponds to Npart in this analysis case. αξ obtained by this analysis as
a function of Npart indicates statistically non-monotonic part at Npart ∼ 90 asshown in Figure 5.4. This behavior is consistent with the scenario of rapid coolingof the system and surviving fluctuations.

Chapter 7
Conclusion√
The multiplicity distributions measured in Au+Au collisions at sN N = 200 GeV
are found to be well described by negative binomial distributions (NBD) in cases
where the collision centrality is greater than 65%.
The longitudinal density correlations have been measured via the functional
form for pseudorapidity (η) density fluctuations derived in the Ginzburg-Landau
thermodynamical theory, up to the second order term in the free energy, with ascalar order parameter defined as pseudorapidity dependent multiplicity fluctuations around the mean value. The functional form can reasonably fit NBD k
parameters as a function of pseudorapidity window sizes δη in the measured all
centrality (0 - 65%) at |η| < 0.35 region with one correlation length ξ assumption and the constant term β which is independent of δη. The β is found to benecessary to absorb residual effects of finite centrality binning.
The absolute scale of the correlation length ξ depends on the magnitude of
the correlation strength at zero distance α within the range of pseudorapidity
window sizes available in this analysis. However, according to the free parameter
fit results, the upper limit on ξ < 0.035 was obtained, and it was confirmed by
the accuracy of the fits with approximated integrated correlation function in thelimit of the small correlation length (ξ ≪ δη).
The αξ product in the correlation function, which is monotonically related
to susceptibility in the long wavelength limit χk=0 , was seen to exhibit a nonmonotonic behavior as a function of the number of participant nucleons Npart . A
possible indication of a local maximum is seen at Npart ∼ 90 and the corresponding
energy density based on the Bjorken picture is εBj · τ ∼ 2.4 GeV/fm2 · c with thetransverse area size of 60 fm2 .
Trivial particle correlations originating from charged track reconstructions intracking detectors have been suppressed in this analysis a priori. The ratio of
charged particles from statistically dominant weak decays and secondary particles
produced in the detector materials, which contribute as correlated pairs, arerespectively estimated below ∼ 1% and ∼ 10−3 % with respect to the total numberof charged pions in the PHENIX acceptance per event. It is estimated that those91

92

CHAPTER 7. CONCLUSION

effects on measured k values and the deviation due to the effect is well inside
the total error size of observed δhkc i. Therefore, it can be concluded that theircontributions are almost negligible to the observed behavior of the αξ products.
The behavior may be explained by the onset of a mixture of different types ofparticle production mechanisms which are not necessarily related with temperature nor density correlations. However, interpreted within the Ginzburg-Landauframework, the local maximum of the αξ product could be an indication of a
critical phase boundary.

Acknowledgment
I would like to thank Prof. T. Sugitate for the longstanding and continuous leading
as my supervisor at Hiroshima University and also his works as the leader of
PHENIX-BBC subsystem group.
I would like to express my gratitude to Dr. K. Homma. Since the main partof this work consists of unprecedented analysis in the field of high energy heavy
ion physics, there were a lot of difficulties. I have no doubt that this work neverhas been possible without Kensuke’s help.
I appreciate Dr. J. T. Mitchell, Dr. P. W. Stankus, Dr. T. Chujo and Dr. A. Toia
for their combined efforts as members of the paper preparation group and theinternal review committee for the PHENIX publication. I also would like tothank Dr. M. J. Tannenbaum and Dr. Y. Akiba for their useful comments andfeedback with deep knowledge on the physics and experiment. I wish to express
my thank to the conveners of Global/Hadron physics working group, Prof. S. Esumi, Prof. C. Ogilvie, Dr. D. P. Morrison, Dr. J. P. Sullivan, Dr. A. Franz andProf. R. A. Soltz for their arrangement of the meeting and helpful comments. I
would like to thank Dr. B. M. Johnson for his great editorial work of the paper.
Owing to their supports, we can publish the paper quite smoothly.
I am indebted to Prof. H. Hamagaki and Dr. K. Ozawa for their works on thefinancial supports of PHENIX-J to continue the study and enroll in the PHENIX
experiment. In particular, Kyoichiro’s help was indispensable during my stay at
BNL. I also would like to thank Prof. K. Shigaki for his coordinations of official
trip from Hiroshima University. Chief scientist Dr. H. En’yo and Prof. S. Sawada
gave a chance to continue the work by accepting me to RIKEN and KEK after I
left Hiroshima University. I could not continue the study without their supports.
I am proud of all the member of the PHENIX collaboration, and I have had a
lot of supports and encouragement from PHENIX spokesperson, Prof. W. A. Zajc
and Prof. B. V. Jacak. Dr. H. Ohnishi, Dr. T. Kohama and Mr. T. Hachiya
made substantial contributions for the start up of BBC operation, so I would
like to appreciate for their works. I would like particularly thank Dr. C. Y. Chi,
Dr. M. Chiu and Dr. S. Belikov for their maintenance works of the Front End
electronics of the BBC and global level-1 trigger. We could not provide correct
data by the BBC without their contributions. I appreciate Dr. E. O’Brien andProf. A. D. Frawley for their coordination as the leader of experimental organizing
93

94

CHAPTER 7. CONCLUSION

group and the PHENIX-Run2 coordinator. We received generous supports for the
BBC operations from DAQ group, especially I would like to thank Dr. J. S. Haggerty, Dr. C. Pinkenburg, Dr. M. L. Purschke. I think we could not take such afruitful physics data without their help.
I have greatly benefited from a lot of mentor, Dr. M. Kaneta, Dr. H. Torii,
Dr. A. Kiyomichi, Dr. A. Taketani, Dr. S. Sato, Dr. Y. Goto and Dr. K. Okada.
Particularly, Masashi gives me many constructive comments and discussions on
the heavy ion physics. I would like to great appreciate the member of CCJ,
Dr. T. Ichihara, Dr. Y. Watanabe, Dr. S. Yokkaichi and Dr. S. Kametani for
their supports on my works at RIKEN. I am happy to work together all themember of quark physics laboratory at Hiroshima University, Dr. A. Enokizono, Mr. T. Uzumaki, Mr. N. Sugita, Mr. Y. Tsuchimoto, Mr. R. Kohara,
Mr. H. Harada, Mr. K. Haruna, Mr. Y. Nakamiya, Mr. H. Sakata, Ms. M. Ouchida, Mr. K. Yamaura, Mr. D. Watanabe and Mr. K. Kijima. I also would liketo thank all Japanese colleague worked at BNL together, Dr. T. Horaguchi,
Dr. S. Sakai, Dr. H. Masui, Dr. F. Kajihara, Dr. Y. Fukao, Dr. M.Togawa,Dr. T. Gunji, Dr. T. Isobe, Ms. M. Shimomura, Mr. K. Aoki, Mr. Y. Morino,
Mr. Y. Yamaguchi, Mr. Y. Aramaki.
This analysis has been made by using great computing facilities, RCF at BNL
and CCJ at RIKEN. I would like to thank all the staff at the facilities for theirquick maintenance. I also would like to express my deepest appreciations to all
secretaries at Hiroshima University, KEK and RIKEN for their supports.
Finally, I would like to express my great thanks to my family, father Yoshiteru
and Mather Misae, and also my sister Mayumi for their kindly supports to continue this work. I could not finalize this thesis without my wife Sae’s understandings and supports on my works.

Appendix A
Tables of NBD fit results
NBD fit results in all window sizes in all centrality bins in Figure 5.3. In thefollowing Table A.1 through Table A.12 and Table A.13 through Table A.25
correspond to results in 10% and 5% bin width cases, respectively. hµc i and hµi
are weighted means of corrected and uncorrected µ over all window positions,
respectively. hkc i and hki are weighted means of corrected and uncorrected kover all window positions, respectively. Statistical errors on weighted meansδhkc i (stat) are obtained as explained in Section 4.4.2. hχ2 /NDF i is the averageof reduced χ2 of NBD fits over all window positions. hNDF i is the average ofthe degree of freedom of NBD fits over all window positions. Systematic errors
δhkc i (dead), δhkc i (fake) and δhkc i (total) are explained in Section 4.4.2.

95

96

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.1: NBD fit results in centrality 0-10%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
77.535 (60.011) ± 0.108
75.445 (58.445) ± 0.106
72.977 (56.455) ± 0.103
70.485 (54.450) ± 0.101
67.998 (52.489) ± 0.099
65.530 (50.572) ± 0.096
63.050 (48.609) ± 0.094
60.569 (46.624) ± 0.091
58.100 (44.655) ± 0.089
55.637 (42.682) ± 0.086
53.164 (40.688) ± 0.084
50.682 (38.672) ± 0.081
48.209 (36.654) ± 0.079
45.743 (34.645) ± 0.076
43.283 (32.654) ± 0.074
40.838 (30.677) ± 0.071
38.424 (28.782) ± 0.049
36.034 (27.062) ± 0.047
33.665 (25.363) ± 0.045
31.288 (23.638) ± 0.043
28.916 (21.900) ± 0.041
26.542 (20.145) ± 0.039
24.155 (18.360) ± 0.030
21.758 (16.553) ± 0.028
19.355 (14.723) ± 0.023
16.948 (12.880) ± 0.021
14.536 (11.031) ± 0.017
12.119 (9.172) ± 0.014
9.695 (7.305) ± 0.011
7.308 (5.685) ± 0.008

hkc i(hki)114.28 (88.45) ± 3.21
113.11 (87.62) ± 3.29
114.40 (88.50) ± 3.38
114.02 (88.08) ± 3.42
114.44 (88.33) ± 3.52
114.28 (88.18) ± 3.60
114.62 (88.35) ± 3.71
114.27 (87.94) ± 3.80
114.38 (87.89) ± 3.92
114.36 (87.70) ± 4.03
114.41 (87.54) ± 4.17
115.19 (87.87) ± 4.35
114.89 (87.33) ± 4.51
115.05 (87.11) ± 4.71
114.86 (86.61) ± 4.90
115.20 (86.46) ± 5.17
115.87 (86.69) ± 3.88
115.35 (86.43) ± 4.04
114.46 (86.00) ± 4.21
113.91 (85.92) ± 4.39
111.53 (84.56) ± 4.51
109.53 (83.35) ± 4.67
107.67 (82.28) ± 4.03
105.84 (80.90) ± 4.29
102.63 (78.59) ± 3.96
97.91 (75.16) ± 4.19
93.93 (72.13) ± 4.01
87.92 (66.93) ± 3.81
78.94 (58.95) ± 3.35
65.53 (49.27) ± 2.87hχ2 /NDF i(hNDF i)
0.96 (71.0)
0.89 (69.7)
0.95 (68.2)
0.92 (66.7)
0.94 (65.3)
0.95 (63.9)
0.97 (62.3)
0.96 (60.8)
0.95 (59.7)
0.93 (57.8)
0.94 (56.3)
0.98 (54.2)
0.98 (52.4)
0.98 (50.3)
0.97 (48.3)
1.00 (46.2)
1.04 (44.5)
1.04 (42.9)
1.08 (41.2)
1.09 (39.9)
1.10 (37.9)
1.08 (36.3)
1.11 (34.4)
1.15 (32.4)
1.21 (30.1)
1.25 (27.6)
1.34 (24.9)
1.39 (21.9)
1.34 (18.7)
1.09 (15.4)

δhkc i(dead)
±2.89
±4.92
±4.33
±4.54
±3.79
±3.59
±3.88
±3.61
±3.79
±3.64
±4.24
±4.10
±4.49
±4.79
±4.37
±4.22
±3.81
±4.04
±4.28
±3.60
±3.75
±3.71
±3.66
±3.17
±3.59
±3.83
±3.77
±3.48
±3.48
±4.05

δhkc i(f ake)
±2.28
±3.31
±3.04
±3.01
±2.31
±2.62
±2.80
±2.98
±2.87
±3.17
±3.40
±3.14
±3.72
±3.49
±4.15
±4.10
±4.22
±4.58
±4.89
±5.13
±5.29
±5.40
±5.96
±6.12
±6.85
±6.93
±7.03
±7.30
±7.34
±6.83

δhkc i(total)
±4.89
±6.78
±6.28
±6.43
±5.66
±5.72
±6.05
±6.03
±6.16
±6.29
±6.85
±6.75
±7.37
±7.57
±7.77
±7.83
±6.88
±7.32
±7.74
±7.65
±7.90
±8.04
±8.07
±8.12
±8.69
±8.96
±8.93
±8.94
±8.79
±8.45

δhkc i(f ake)
±2.17
±1.53
±1.11
±1.57
±1.43
±1.50
±1.88
±1.68
±1.88
±1.91
±1.59
±1.70
±2.29
±2.03
±2.36
±2.29
±2.23
±2.71
±2.56
±2.87
±3.26
±3.39
±3.09
±3.67
±3.58
±3.97
±4.10
±4.28
±4.32
±4.18

δhkc i(total)
±3.60
±4.21
±3.22
±3.40
±3.72
±3.83
±3.83
±3.99
±4.07
±3.69
±3.71
±3.76
±4.46
±4.49
±4.88
±4.89
±4.48
±4.82
±4.73
±4.72
±5.12
±5.22
±4.64
±5.26
±5.06
±5.46
±5.48
±5.64
±5.43
±5.34

Table A.2: NBD fit results in centrality 5-15%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
66.445 (51.428) ± 0.098
64.658 (50.089) ± 0.097
62.535 (48.377) ± 0.094
60.379 (46.644) ± 0.092
58.258 (44.970) ± 0.090
56.156 (43.337) ± 0.088
54.043 (41.665) ± 0.085
51.923 (39.968) ± 0.083
49.809 (38.283) ± 0.081
47.691 (36.586) ± 0.078
45.573 (34.878) ± 0.076
43.446 (33.150) ± 0.073
41.327 (31.420) ± 0.071
39.211 (29.696) ± 0.069
37.107 (27.993) ± 0.067
35.007 (26.295) ± 0.064
32.938 (24.672) ± 0.044
30.889 (23.196) ± 0.042
28.861 (21.741) ± 0.040
26.832 (20.270) ± 0.038
24.795 (18.778) ± 0.036
22.758 (17.270) ± 0.035
20.708 (15.737) ± 0.026
18.654 (14.187) ± 0.025
16.591 (12.614) ± 0.020
14.526 (11.035) ± 0.019
12.458 (9.451) ± 0.015
10.385 (7.857) ± 0.013
8.306 (6.253) ± 0.010
6.262 (4.870) ± 0.007

hkc i(hki)80.80 (62.54) ± 1.92
80.40 (62.28) ± 1.95
81.17 (62.79) ± 1.99
81.10 (62.65) ± 2.02
80.61 (62.21) ± 2.06
79.66 (61.47) ± 2.08
79.26 (61.09) ± 2.12
78.98 (60.79) ± 2.17
79.14 (60.81) ± 2.23
79.62 (61.06) ± 2.32
79.66 (60.95) ± 2.39
80.40 (61.33) ± 2.48
79.99 (60.80) ± 2.56
80.34 (60.83) ± 2.67
80.40 (60.62) ± 2.79
80.45 (60.38) ± 2.92
80.58 (60.28) ± 2.17
80.09 (60.08) ± 2.25
79.49 (59.81) ± 2.33
79.11 (59.73) ± 2.44
78.06 (59.22) ± 2.55
77.16 (58.82) ± 2.68
76.33 (58.44) ± 2.32
75.26 (57.80) ± 2.47
73.89 (56.91) ± 2.30
71.68 (55.18) ± 2.44
69.08 (53.03) ± 2.31
65.32 (49.65) ± 2.23
59.45 (44.57) ± 2.03
51.33 (38.86) ± 1.88

hχ2 /NDF i(hNDF i)
1.20 (68.0)
1.15 (66.4)
1.20 (65.0)
1.23 (64.1)
1.21 (62.4)
1.12 (60.6)
1.07 (59.2)
1.09 (57.4)
1.11 (56.2)
1.13 (54.7)
1.13 (53.0)
1.18 (51.3)
1.15 (49.4)
1.18 (47.7)
1.15 (45.9)
1.18 (43.9)
1.21 (42.1)
1.22 (40.3)
1.22 (38.6)
1.24 (37.0)
1.23 (35.3)
1.22 (33.8)
1.27 (32.1)
1.29 (30.2)
1.36 (28.1)
1.41 (25.9)
1.47 (23.1)
1.46 (20.3)
1.29 (17.4)
0.89 (14.4)

δhkc i(dead)
±2.14
±3.40
±2.28
±2.24
±2.75
±2.85
±2.59
±2.90
±2.83
±2.15
±2.35
±2.25
±2.84
±2.98
±3.24
±3.19
±3.22
±3.29
±3.21
±2.85
±3.01
±2.93
±2.56
±2.84
±2.74
±2.84
±2.81
±2.92
±2.58
±2.75

97

Table A.3: NBD fit results in centrality 10-20%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
55.290 (42.794) ± 0.090
53.832 (41.701) ± 0.089
52.092 (40.299) ± 0.087
50.301 (38.858) ± 0.085
48.533 (37.463) ± 0.082
46.769 (36.093) ± 0.080
45.006 (34.698) ± 0.078
43.255 (33.296) ± 0.076
41.499 (31.895) ± 0.074
39.742 (30.488) ± 0.072
37.976 (29.063) ± 0.070
36.206 (27.625) ± 0.067
34.435 (26.179) ± 0.065
32.667 (24.739) ± 0.063
30.908 (23.315) ± 0.061
29.156 (21.898) ± 0.059
27.434 (20.547) ± 0.040
25.727 (19.316) ± 0.038
24.039 (18.105) ± 0.037
22.347 (16.876) ± 0.035
20.653 (15.635) ± 0.033
18.956 (14.380) ± 0.032
17.251 (13.105) ± 0.024
15.537 (11.813) ± 0.023
13.816 (10.501) ± 0.018
12.090 (9.179) ± 0.017
10.367 (7.858) ± 0.014
8.639 (6.528) ± 0.011
6.909 (5.194) ± 0.009
5.215 (4.052) ± 0.007

hkc i(hki)63.58 (49.21) ± 1.47
62.10 (48.11) ± 1.46
61.78 (47.79) ± 1.48
61.91 (47.82) ± 1.51
61.80 (47.70) ± 1.54
62.00 (47.85) ± 1.58
61.89 (47.71) ± 1.61
61.70 (47.49) ± 1.65
61.86 (47.55) ± 1.70
61.94 (47.51) ± 1.75
61.92 (47.38) ± 1.80
61.91 (47.24) ± 1.85
61.83 (47.01) ± 1.91
62.02 (46.98) ± 1.99
62.18 (46.92) ± 2.07
62.35 (46.85) ± 2.16
62.21 (46.61) ± 1.59
61.90 (46.54) ± 1.64
61.43 (46.42) ± 1.71
60.84 (46.18) ± 1.77
60.20 (45.91) ± 1.85
59.81 (45.79) ± 1.95
59.09 (45.44) ± 1.68
58.31 (44.99) ± 1.78
57.38 (44.38) ± 1.65
55.59 (43.04) ± 1.75
52.81 (40.76) ± 1.63
49.83 (38.38) ± 1.57
46.30 (35.28) ± 1.45
42.21 (32.18) ± 1.43hχ2 /NDF i(hNDF i)
1.14 (64.0)
0.93 (63.1)
0.88 (62.3)
0.90 (60.3)
0.88 (58.5)
0.90 (57.9)
0.87 (56.6)
0.88 (55.1)
0.89 (53.8)
0.90 (52.4)
0.90 (50.8)
0.91 (48.8)
0.91 (47.2)
0.95 (45.6)
1.00 (43.8)
1.02 (42.1)
1.02 (40.4)
1.06 (38.6)
1.09 (36.8)
1.10 (35.1)
1.12 (33.3)
1.19 (31.6)
1.22 (29.7)
1.25 (27.6)
1.31 (25.5)
1.33 (23.0)
1.27 (20.6)
1.14 (18.2)
0.97 (15.6)
0.75 (13.0)

δhkc i(dead)
±0.37
±2.48
±2.34
±2.06
±2.10
±1.74
±1.77
±1.80
±1.95
±1.89
±1.84
±1.84
±1.91
±1.96
±2.02
±2.14
±2.30
±2.05
±2.09
±2.30
±2.19
±2.27
±2.19
±2.23
±1.84
±1.77
±1.83
±1.85
±1.82
±2.04

δhkc i(f ake)
±0.57
±2.60
±1.54
±1.23
±1.09
±1.21
±1.06
±1.34
±1.21
±1.38
±1.26
±1.46
±1.55
±1.63
±1.58
±1.69
±1.77
±1.66
±1.68
±1.84
±1.93
±1.91
±2.04
±2.11
±2.17
±2.18
±2.31
±2.55
±2.85
±2.90

δhkc i(total)
±1.62
±3.88
±3.17
±2.83
±2.83
±2.64
±2.62
±2.78
±2.86
±2.92
±2.87
±2.99
±3.12
±3.23
±3.30
±3.48
±3.31
±3.11
±3.17
±3.44
±3.45
±3.54
±3.43
±3.55
±3.29
±3.31
±3.36
±3.52
±3.68
±3.82

δhkc i(f ake)
±1.78
±0.53
±0.71
±0.58
±1.12
±0.70
±0.82
±0.88
±0.76
±0.81
±0.66
±0.86
±0.87
±0.84
±1.07
±1.02
±1.07
±1.02
±1.09
±1.17
±1.17
±1.28
±1.32
±1.39
±1.30
±1.41
±1.55
±1.67
±1.77
±1.96

δhkc i(total)
±2.85
±2.04
±2.14
±2.05
±2.63
±2.33
±2.42
±2.47
±2.44
±2.31
±2.29
±2.44
±2.57
±2.61
±2.78
±2.77
±2.63
±2.50
±2.57
±2.66
±2.60
±2.65
±2.60
±2.53
±2.38
±2.36
±2.45
±2.45
±2.47
±2.63

Table A.4: NBD fit results in centrality 15-25%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
45.868 (35.502) ± 0.082
44.657 (34.594) ± 0.080
43.211 (33.428) ± 0.078
41.737 (32.242) ± 0.076
40.288 (31.098) ± 0.075
38.829 (29.965) ± 0.073
37.368 (28.808) ± 0.071
35.911 (27.642) ± 0.069
34.454 (26.481) ± 0.067
32.995 (25.311) ± 0.065
31.532 (24.131) ± 0.063
30.062 (22.936) ± 0.061
28.591 (21.736) ± 0.059
27.125 (20.541) ± 0.057
25.664 (19.358) ± 0.055
24.206 (18.179) ± 0.053
22.778 (17.058) ± 0.036
21.359 (16.035) ± 0.035
19.955 (15.027) ± 0.033
18.552 (14.008) ± 0.032
17.146 (12.979) ± 0.030
15.736 (11.936) ± 0.029
14.320 (10.876) ± 0.022
12.897 (9.802) ± 0.021
11.465 (8.711) ± 0.017
10.030 (7.612) ± 0.015
8.597 (6.513) ± 0.013
7.163 (5.409) ± 0.011
5.730 (4.304) ± 0.008
4.327 (3.360) ± 0.006

hkc i(hki)53.43 (41.35) ± 1.25
53.48 (41.43) ± 1.29
53.45 (41.35) ± 1.31
53.70 (41.48) ± 1.35
53.41 (41.23) ± 1.38
53.66 (41.42) ± 1.42
53.48 (41.23) ± 1.45
53.32 (41.04) ± 1.48
53.41 (41.05) ± 1.52
53.58 (41.11) ± 1.56
53.73 (41.13) ± 1.61
53.65 (40.95) ± 1.65
53.69 (40.84) ± 1.71
53.76 (40.75) ± 1.78
53.81 (40.64) ± 1.84
54.03 (40.64) ± 1.93
54.05 (40.55) ± 1.42
53.79 (40.53) ± 1.48
53.36 (40.43) ± 1.54
52.52 (40.03) ± 1.59
51.86 (39.73) ± 1.66
51.03 (39.31) ± 1.74
49.80 (38.58) ± 1.48
48.78 (37.96) ± 1.57
47.43 (37.04) ± 1.43
45.48 (35.65) ± 1.51
43.13 (33.84) ± 1.41
40.33 (31.64) ± 1.34
37.88 (29.64) ± 1.27
35.48 (27.59) ± 1.28hχ2 /NDF i(hNDF i)
1.06 (57.0)
0.95 (55.3)
0.98 (54.2)
0.96 (53.0)
0.92 (51.2)
0.96 (50.3)
0.94 (49.0)
0.91 (47.6)
0.95 (46.1)
1.02 (44.9)
1.02 (44.0)
1.02 (42.8)
1.03 (41.4)
1.04 (40.0)
1.03 (38.5)
1.05 (37.1)
1.09 (35.7)
1.12 (34.5)
1.14 (33.2)
1.16 (31.7)
1.19 (29.9)
1.25 (28.0)
1.26 (26.0)
1.26 (24.1)
1.26 (22.3)
1.21 (20.2)
1.06 (18.1)
0.79 (15.9)
0.64 (13.7)
0.54 (11.6)

δhkc i(dead)
±1.83
±1.49
±1.53
±1.43
±1.94
±1.71
±1.76
±1.77
±1.75
±1.50
±1.50
±1.57
±1.70
±1.71
±1.78
±1.71
±1.94
±1.74
±1.74
±1.78
±1.63
±1.53
±1.69
±1.43
±1.39
±1.13
±1.27
±1.21
±1.15
±1.20

98

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.5: NBD fit results in centrality 20-30%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
37.610 (29.110) ± 0.073
36.606 (28.358) ± 0.072
35.403 (27.388) ± 0.070
34.199 (26.419) ± 0.069
33.002 (25.474) ± 0.067
31.812 (24.550) ± 0.065
30.609 (23.598) ± 0.063
29.413 (22.641) ± 0.062
28.218 (21.688) ± 0.060
27.021 (20.729) ± 0.058
25.823 (19.762) ± 0.057
24.623 (18.787) ± 0.055
23.421 (17.806) ± 0.053
22.218 (16.826) ± 0.051
21.024 (15.858) ± 0.050
19.831 (14.894) ± 0.048
18.658 (13.973) ± 0.033
17.499 (13.138) ± 0.031
16.350 (12.314) ± 0.030
15.201 (11.480) ± 0.029
14.048 (10.636) ± 0.027
12.892 (9.781) ± 0.026
11.729 (8.913) ± 0.020
10.561 (8.032) ± 0.019
9.389 (7.139) ± 0.015
8.216 (6.239) ± 0.014
7.045 (5.342) ± 0.012
5.871 (4.439) ± 0.009
4.697 (3.532) ± 0.007
3.547 (2.756) ± 0.005

hkc i(hki)47.51 (36.77) ± 1.17
47.66 (36.92) ± 1.20
47.76 (36.94) ± 1.22
47.45 (36.66) ± 1.24
47.80 (36.90) ± 1.27
47.37 (36.56) ± 1.28
47.42 (36.56) ± 1.31
47.14 (36.29) ± 1.34
47.13 (36.23) ± 1.37
47.16 (36.19) ± 1.41
47.14 (36.08) ± 1.45
47.14 (35.98) ± 1.50
46.88 (35.66) ± 1.54
47.13 (35.72) ± 1.61
46.86 (35.38) ± 1.67
46.92 (35.28) ± 1.74
46.79 (35.08) ± 1.29
46.40 (34.90) ± 1.33
46.22 (34.90) ± 1.40
45.61 (34.60) ± 1.45
44.99 (34.29) ± 1.51
44.22 (33.84) ± 1.58
43.26 (33.22) ± 1.34
42.28 (32.55) ± 1.41
41.04 (31.68) ± 1.29
39.50 (30.60) ± 1.36
38.08 (29.47) ± 1.31
36.53 (28.20) ± 1.30
35.06 (27.06) ± 1.31
32.64 (25.34) ± 1.37hχ2 /NDF i(hNDF i)
0.82 (53.0)
0.86 (51.7)
0.83 (51.4)
0.83 (50.2)
0.97 (49.3)
0.96 (48.6)
0.93 (47.7)
0.92 (46.0)
0.94 (44.8)
0.98 (43.6)
0.98 (42.2)
0.98 (41.1)
0.95 (39.4)
1.02 (38.0)
0.99 (36.6)
1.00 (35.1)
1.02 (33.7)
1.02 (32.1)
1.07 (30.6)
1.06 (29.0)
1.06 (27.4)
1.07 (25.6)
1.05 (24.0)
1.05 (22.4)
0.96 (20.4)
0.83 (18.6)
0.68 (16.6)
0.54 (14.7)
0.51 (12.4)
0.46 (10.4)

δhkc i(dead)
±1.28
±1.61
±1.10
±1.54
±0.99
±1.29
±1.12
±1.26
±1.38
±1.29
±1.40
±1.37
±1.53
±1.42
±1.61
±1.60
±1.63
±1.63
±1.72
±1.56
±1.65
±1.54
±1.36
±1.30
±1.26
±1.18
±1.07
±1.08
±1.03
±1.02

δhkc i(f ake)
±0.76
±0.68
±0.43
±0.65
±0.68
±0.62
±0.51
±0.68
±0.66
±0.53
±0.52
±0.57
±0.68
±0.55
±0.68
±0.78
±0.83
±0.79
±0.73
±0.82
±0.90
±0.91
±0.93
±0.95
±0.96
±1.13
±1.20
±1.31
±1.57
±1.72

δhkc i(total)
±1.89
±2.12
±1.70
±2.08
±1.75
±1.92
±1.80
±1.96
±2.05
±1.98
±2.08
±2.11
±2.28
±2.21
±2.42
±2.49
±2.23
±2.25
±2.33
±2.28
±2.41
±2.39
±2.13
±2.14
±2.04
±2.13
±2.08
±2.14
±2.29
±2.42

δhkc i(f ake)
±0.38
±0.13
±0.55
±0.47
±0.55
±0.36
±0.53
±0.55
±0.44
±0.55
±0.48
±0.39
±0.42
±0.52
±0.50
±0.43
±0.51
±0.53
±0.56
±0.63
±0.59
±0.60
±0.67
±0.74
±0.74
±0.82
±0.85
±0.95
±1.11
±1.24

δhkc i(total)
±2.28
±1.87
±1.94
±1.81
±1.69
±1.71
±1.91
±1.87
±1.87
±2.05
±2.07
±2.09
±2.15
±2.25
±2.29
±2.27
±2.04
±2.00
±2.04
±2.02
±1.95
±1.99
±1.78
±1.81
±1.78
±1.82
±1.74
±1.82
±1.87
±1.98

Table A.6: NBD fit results in centrality 25-35%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
30.796 (23.836) ± 0.065
29.940 (23.193) ± 0.063
28.970 (22.412) ± 0.062
27.967 (21.605) ± 0.060
26.977 (20.823) ± 0.059
26.003 (20.068) ± 0.058
25.020 (19.289) ± 0.056
24.036 (18.502) ± 0.055
23.058 (17.722) ± 0.053
22.086 (16.943) ± 0.052
21.108 (16.155) ± 0.050
20.126 (15.356) ± 0.049
19.143 (14.554) ± 0.047
18.163 (13.755) ± 0.046
17.184 (12.962) ± 0.044
16.206 (12.172) ± 0.043
15.249 (11.420) ± 0.029
14.299 (10.736) ± 0.028
13.362 (10.063) ± 0.027
12.422 (9.381) ± 0.026
11.476 (8.689) ± 0.024
10.531 (7.990) ± 0.023
9.581 (7.281) ± 0.018
8.626 (6.560) ± 0.017
7.670 (5.832) ± 0.013
6.712 (5.099) ± 0.013
5.756 (4.365) ± 0.010
4.800 (3.629) ± 0.009
3.841 (2.888) ± 0.006
2.902 (2.255) ± 0.005

hkc i(hki)41.09 (31.80) ± 1.05
41.40 (32.07) ± 1.06
41.18 (31.85) ± 1.07
41.07 (31.73) ± 1.09
40.97 (31.62) ± 1.11
40.72 (31.42) ± 1.12
40.64 (31.33) ± 1.15
40.41 (31.11) ± 1.17
40.37 (31.03) ± 1.20
40.25 (30.88) ± 1.23
40.05 (30.66) ± 1.26
40.00 (30.53) ± 1.30
39.77 (30.26) ± 1.34
39.51 (29.96) ± 1.38
39.36 (29.73) ± 1.44
39.43 (29.67) ± 1.50
39.31 (29.50) ± 1.11
38.92 (29.31) ± 1.14
38.24 (28.93) ± 1.18
37.66 (28.62) ± 1.22
37.27 (28.42) ± 1.28
36.55 (27.96) ± 1.33
35.75 (27.42) ± 1.13
35.01 (26.95) ± 1.19
34.26 (26.41) ± 1.09
33.65 (25.96) ± 1.19
32.98 (25.42) ± 1.16
31.80 (24.50) ± 1.17
30.92 (23.78) ± 1.17
28.55 (22.23) ± 1.20

hχ2 /NDF i(hNDF i)
0.77 (48.0)
0.83 (48.9)
0.81 (47.9)
0.84 (46.8)
0.85 (45.4)
0.88 (44.1)
0.89 (42.6)
0.91 (41.1)
0.91 (40.0)
0.90 (38.8)
0.91 (37.2)
0.92 (35.9)
0.92 (34.6)
0.88 (33.3)
0.91 (31.8)
0.95 (30.6)
0.98 (29.2)
1.00 (27.9)
0.97 (26.5)
0.95 (25.2)
0.97 (23.9)
0.91 (22.4)
0.78 (21.0)
0.73 (19.3)
0.63 (17.6)
0.59 (16.0)
0.53 (14.3)
0.48 (12.4)
0.54 (10.7)
0.47 (9.0)δhkc i(dead)
±1.99
±1.53
±1.52
±1.36
±1.15
±1.23
±1.43
±1.35
±1.37
±1.54
±1.56
±1.59
±1.62
±1.70
±1.72
±1.64
±1.63
±1.55
±1.56
±1.48
±1.35
±1.35
±1.21
±1.15
±1.20
±1.12
±0.98
±1.01
±0.94
±0.98

99

Table A.7: NBD fit results in centrality 30-40%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
24.860 (19.241) ± 0.058
24.177 (18.729) ± 0.057
23.382 (18.088) ± 0.055
22.573 (17.438) ± 0.054
21.776 (16.808) ± 0.053
20.986 (16.195) ± 0.051
20.198 (15.571) ± 0.050
19.406 (14.937) ± 0.049
18.616 (14.307) ± 0.048
17.830 (13.678) ± 0.046
17.040 (13.040) ± 0.045
16.246 (12.395) ± 0.043
15.451 (11.746) ± 0.042
14.662 (11.103) ± 0.041
13.869 (10.461) ± 0.040
13.081 (9.824) ± 0.038
12.306 (9.216) ± 0.026
11.540 (8.663) ± 0.025
10.781 (8.119) ± 0.024
10.020 (7.565) ± 0.023
9.258 (7.008) ± 0.022
8.494 (6.443) ± 0.021
7.727 (5.870) ± 0.016
6.958 (5.290) ± 0.015
6.187 (4.702) ± 0.012
5.416 (4.112) ± 0.011
4.645 (3.520) ± 0.009
3.873 (2.926) ± 0.008
3.098 (2.328) ± 0.006
2.341 (1.819) ± 0.004

hkc i(hki)34.21 (26.48) ± 0.88
34.00 (26.34) ± 0.88
33.88 (26.21) ± 0.90
33.73 (26.06) ± 0.91
33.63 (25.96) ± 0.93
33.54 (25.89) ± 0.94
33.46 (25.80) ± 0.97
33.43 (25.74) ± 0.99
33.48 (25.74) ± 1.02
33.37 (25.61) ± 1.05
33.31 (25.51) ± 1.08
33.30 (25.43) ± 1.12
33.14 (25.23) ± 1.16
32.95 (24.99) ± 1.20
32.84 (24.83) ± 1.25
32.69 (24.61) ± 1.30
32.49 (24.40) ± 0.95
31.92 (24.08) ± 0.98
31.35 (23.78) ± 1.00
30.76 (23.49) ± 1.04
30.22 (23.20) ± 1.07
29.75 (22.94) ± 1.12
29.20 (22.62) ± 0.95
28.62 (22.29) ± 1.01
27.94 (21.88) ± 0.93
27.29 (21.40) ± 1.00
26.74 (20.98) ± 0.99
25.80 (20.22) ± 1.01
24.72 (19.40) ± 1.02
22.77 (17.95) ± 1.02

hχ2 /NDF i(hNDF i)
0.75 (47.0)
0.77 (45.9)
0.81 (44.2)
0.90 (42.5)
0.94 (40.9)
0.92 (39.8)
0.97 (38.4)
1.01 (37.2)
1.03 (36.0)
1.06 (34.7)
1.03 (33.4)
1.04 (32.2)
1.03 (31.0)
0.97 (29.7)
0.99 (28.4)
1.00 (26.8)
1.00 (25.5)
0.99 (24.2)
0.94 (23.1)
0.91 (22.0)
0.83 (20.8)
0.77 (19.5)
0.70 (18.3)
0.65 (17.0)
0.59 (15.5)
0.50 (14.0)
0.50 (12.5)
0.50 (10.9)
0.54 (9.4)
0.49 (8.0)δhkc i(dead)
±1.14
±1.12
±0.91
±1.10
±1.08
±0.97
±1.04
±1.04
±1.04
±1.07
±1.11
±1.03
±1.09
±1.18
±1.21
±1.08
±1.05
±1.05
±1.05
±1.01
±0.93
±0.91
±0.85
±0.76
±0.83
±0.84
±0.81
±0.86
±0.75
±0.64

δhkc i(f ake)
±0.33
±0.30
±0.29
±0.25
±0.27
±0.25
±0.26
±0.24
±0.25
±0.28
±0.28
±0.28
±0.34
±0.38
±0.33
±0.36
±0.35
±0.36
±0.36
±0.35
±0.36
±0.39
±0.37
±0.38
±0.46
±0.56
±0.60
±0.66
±0.74
±0.81

δhkc i(total)
±1.47
±1.46
±1.31
±1.45
±1.44
±1.38
±1.44
±1.45
±1.48
±1.52
±1.58
±1.55
±1.63
±1.73
±1.77
±1.73
±1.46
±1.48
±1.49
±1.49
±1.46
±1.49
±1.33
±1.32
±1.33
±1.42
±1.41
±1.48
±1.46
±1.45

δhkc i(f ake)
±0.09
±0.17
±0.16
±0.13
±0.10
±0.20
±0.18
±0.24
±0.21
±0.21
±0.20
±0.21
±0.23
±0.23
±0.26
±0.26
±0.21
±0.21
±0.16
±0.21
±0.22
±0.29
±0.29
±0.28
±0.30
±0.37
±0.38
±0.41
±0.47
±0.52

δhkc i(total)
±1.06
±1.01
±1.13
±1.12
±1.15
±1.23
±1.20
±1.30
±1.29
±1.26
±1.31
±1.38
±1.37
±1.40
±1.47
±1.44
±1.18
±1.15
±1.14
±1.18
±1.21
±1.29
±1.07
±1.11
±1.03
±1.13
±1.11
±1.10
±1.16
±1.27

Table A.8: NBD fit results in centrality 35-45%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
19.726 (15.268) ± 0.051
19.191 (14.867) ± 0.050
18.568 (14.364) ± 0.049
17.920 (13.844) ± 0.048
17.291 (13.348) ± 0.046
16.672 (12.867) ± 0.045
16.044 (12.369) ± 0.044
15.417 (11.867) ± 0.043
14.789 (11.366) ± 0.042
14.165 (10.866) ± 0.041
13.537 (10.360) ± 0.040
12.908 (9.849) ± 0.039
12.276 (9.334) ± 0.037
11.648 (8.822) ± 0.036
11.019 (8.313) ± 0.035
10.391 (7.805) ± 0.034
9.774 (7.322) ± 0.023
9.165 (6.883) ± 0.022
8.562 (6.451) ± 0.021
7.959 (6.013) ± 0.020
7.353 (5.569) ± 0.019
6.748 (5.122) ± 0.018
6.138 (4.666) ± 0.014
5.527 (4.205) ± 0.013
4.915 (3.739) ± 0.011
4.303 (3.270) ± 0.010
3.691 (2.801) ± 0.008
3.076 (2.328) ± 0.007
2.461 (1.851) ± 0.005
1.859 (1.445) ± 0.004

hkc i(hki)29.23 (22.62) ± 0.80
29.24 (22.65) ± 0.81
29.05 (22.47) ± 0.82
29.04 (22.43) ± 0.83
28.91 (22.31) ± 0.85
28.64 (22.10) ± 0.86
28.57 (22.02) ± 0.88
28.48 (21.92) ± 0.90
28.48 (21.89) ± 0.92
28.36 (21.76) ± 0.94
28.27 (21.64) ± 0.97
28.14 (21.47) ± 1.00
28.00 (21.28) ± 1.03
27.82 (21.07) ± 1.06
27.61 (20.83) ± 1.09
27.49 (20.65) ± 1.13
27.32 (20.46) ± 0.83
26.94 (20.23) ± 0.85
26.60 (20.05) ± 0.87
26.19 (19.83) ± 0.90
25.80 (19.62) ± 0.93
25.24 (19.28) ± 0.97
24.84 (19.05) ± 0.83
24.45 (18.82) ± 0.88
24.05 (18.53) ± 0.81
23.55 (18.15) ± 0.88
22.88 (17.57) ± 0.85
22.21 (16.99) ± 0.88
21.27 (16.15) ± 0.89
19.60 (14.74) ± 0.92hχ2 /NDF i(hNDF i)
1.12 (40.0)
1.02 (39.0)
0.98 (38.2)
1.09 (37.1)
1.06 (36.2)
1.04 (35.3)
1.06 (34.2)
1.07 (33.3)
1.12 (32.2)
1.10 (31.2)
1.11 (29.9)
1.11 (28.8)
1.10 (27.7)
1.08 (26.6)
1.04 (25.5)
0.98 (24.5)
0.92 (23.4)
0.89 (22.4)
0.86 (21.4)
0.82 (20.2)
0.75 (19.3)
0.65 (18.3)
0.63 (17.1)
0.61 (15.9)
0.62 (14.7)
0.57 (13.3)
0.53 (11.8)
0.56 (10.4)
0.61 (8.9)
0.59 (7.4)

δhkc i(dead)
±0.69
±0.58
±0.76
±0.73
±0.77
±0.86
±0.79
±0.91
±0.88
±0.82
±0.85
±0.92
±0.88
±0.89
±0.96
±0.85
±0.81
±0.75
±0.72
±0.73
±0.73
±0.80
±0.61
±0.62
±0.55
±0.60
±0.59
±0.51
±0.56
±0.70

100

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.9: NBD fit results in centrality 40-50%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
15.446 (11.955) ± 0.044
15.032 (11.644) ± 0.043
14.534 (11.243) ± 0.042
14.033 (10.841) ± 0.041
13.547 (10.457) ± 0.041
13.055 (10.075) ± 0.040
12.561 (9.684) ± 0.038
12.072 (9.293) ± 0.037
11.583 (8.903) ± 0.037
11.093 (8.510) ± 0.036
10.600 (8.112) ± 0.035
10.106 (7.711) ± 0.034
9.610 (7.306) ± 0.033
9.116 (6.904) ± 0.032
8.620 (6.503) ± 0.030
8.131 (6.107) ± 0.029
7.652 (5.731) ± 0.020
7.176 (5.388) ± 0.019
6.706 (5.050) ± 0.018
6.235 (4.708) ± 0.018
5.762 (4.362) ± 0.017
5.287 (4.011) ± 0.016
4.811 (3.656) ± 0.012
4.333 (3.295) ± 0.012
3.852 (2.929) ± 0.009
3.372 (2.561) ± 0.009
2.891 (2.192) ± 0.007
2.410 (1.821) ± 0.006
1.928 (1.449) ± 0.004
1.456 (1.131) ± 0.004

hkc i(hki)23.99 (18.57) ± 0.65
23.87 (18.49) ± 0.66
24.03 (18.59) ± 0.68
24.11 (18.63) ± 0.69
23.98 (18.51) ± 0.71
23.97 (18.50) ± 0.73
23.89 (18.42) ± 0.74
23.70 (18.24) ± 0.76
23.65 (18.17) ± 0.78
23.66 (18.15) ± 0.81
23.64 (18.09) ± 0.83
23.56 (17.98) ± 0.86
23.42 (17.81) ± 0.89
23.27 (17.62) ± 0.92
23.36 (17.62) ± 0.96
23.21 (17.44) ± 1.00
22.92 (17.18) ± 0.73
22.61 (17.05) ± 0.76
22.30 (16.92) ± 0.78
22.05 (16.82) ± 0.81
21.72 (16.65) ± 0.85
21.38 (16.47) ± 0.89
20.89 (16.16) ± 0.76
20.52 (15.95) ± 0.81
20.17 (15.72) ± 0.76
19.78 (15.44) ± 0.82
19.34 (15.10) ± 0.81
18.81 (14.69) ± 0.85
18.14 (14.19) ± 0.87
17.03 (13.51) ± 0.91hχ2 /NDF i(hNDF i)
1.07 (36.0)
1.08 (35.1)
1.24 (34.0)
1.25 (33.2)
1.18 (32.4)
1.15 (31.6)
1.16 (30.8)
1.09 (30.2)
1.02 (29.3)
1.05 (28.1)
1.00 (27.3)
0.95 (26.2)
0.86 (25.2)
0.80 (23.9)
0.81 (22.9)
0.72 (22.0)
0.64 (20.9)
0.63 (19.8)
0.62 (18.8)
0.60 (18.0)
0.59 (17.0)
0.60 (16.0)
0.57 (14.9)
0.53 (13.9)
0.54 (12.8)
0.51 (11.6)
0.49 (10.6)
0.46 (9.3)
0.48 (7.9)
0.48 (6.6)δhkc i(dead)
±1.02
±0.64
±0.61
±0.78
±0.77
±0.80
±0.68
±0.74
±0.63
±0.65
±0.66
±0.58
±0.60
±0.67
±0.55
±0.62
±0.77
±0.74
±0.70
±0.65
±0.61
±0.55
±0.58
±0.51
±0.49
±0.45
±0.43
±0.36
±0.38
±0.32

δhkc i(f ake)
±0.17
±0.18
±0.06
±0.16
±0.17
±0.16
±0.14
±0.14
±0.12
±0.15
±0.17
±0.16
±0.14
±0.18
±0.17
±0.18
±0.24
±0.21
±0.20
±0.20
±0.22
±0.23
±0.24
±0.25
±0.28
±0.28
±0.28
±0.33
±0.35
±0.35

δhkc i(total)
±1.22
±0.93
±0.91
±1.06
±1.06
±1.09
±1.02
±1.07
±1.01
±1.05
±1.08
±1.05
±1.08
±1.15
±1.12
±1.19
±1.08
±1.08
±1.07
±1.06
±1.06
±1.07
±0.98
±0.99
±0.94
±0.98
±0.96
±0.98
±1.01
±1.03

Table A.10: NBD fit results in centrality 45-55%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
11.838 (9.163) ± 0.038
11.517 (8.922) ± 0.037
11.142 (8.620) ± 0.036
10.759 (8.311) ± 0.035
10.380 (8.013) ± 0.035
10.005 (7.721) ± 0.034
9.629 (7.424) ± 0.033
9.254 (7.123) ± 0.032
8.879 (6.824) ± 0.031
8.505 (6.525) ± 0.030
8.130 (6.222) ± 0.030
7.751 (5.915) ± 0.029
7.370 (5.604) ± 0.028
6.991 (5.295) ± 0.027
6.612 (4.988) ± 0.026
6.236 (4.684) ± 0.025
5.868 (4.395) ± 0.017
5.502 (4.131) ± 0.017
5.141 (3.872) ± 0.016
4.780 (3.610) ± 0.015
4.418 (3.345) ± 0.015
4.056 (3.077) ± 0.014
3.689 (2.803) ± 0.011
3.321 (2.526) ± 0.010
2.953 (2.244) ± 0.008
2.583 (1.962) ± 0.007
2.215 (1.678) ± 0.006
1.846 (1.394) ± 0.005
1.477 (1.109) ± 0.004
1.117 (0.875) ± 0.003

hkc i(hki)20.15 (15.59) ± 0.61
20.06 (15.54) ± 0.61
19.94 (15.43) ± 0.62
19.94 (15.40) ± 0.64
19.99 (15.43) ± 0.65
19.95 (15.39) ± 0.67
19.85 (15.30) ± 0.68
19.76 (15.21) ± 0.70
19.73 (15.16) ± 0.72
19.69 (15.10) ± 0.74
19.63 (15.02) ± 0.76
19.62 (14.97) ± 0.79
19.66 (14.95) ± 0.82
19.74 (14.95) ± 0.85
19.80 (14.94) ± 0.89
19.79 (14.86) ± 0.93
19.74 (14.79) ± 0.69
19.58 (14.75) ± 0.72
19.30 (14.66) ± 0.74
18.97 (14.51) ± 0.78
18.76 (14.44) ± 0.82
18.53 (14.33) ± 0.87
18.38 (14.30) ± 0.76
18.29 (14.30) ± 0.83
18.03 (14.17) ± 0.79
17.80 (14.06) ± 0.89
17.55 (13.94) ± 0.92
17.10 (13.66) ± 0.98
16.55 (13.29) ± 1.05
16.05 (12.98) ± 1.09hχ2 /NDF i(hNDF i)
1.14 (28.0)
1.10 (27.9)
1.10 (26.6)
1.08 (25.8)
1.06 (24.9)
1.06 (24.1)
1.05 (23.2)
1.04 (22.5)
1.02 (21.8)
0.96 (21.4)
0.89 (20.7)
0.77 (20.1)
0.66 (19.6)
0.63 (18.8)
0.62 (18.1)
0.58 (17.3)
0.58 (16.7)
0.62 (16.1)
0.62 (15.5)
0.59 (14.9)
0.61 (14.2)
0.59 (13.6)
0.61 (13.0)
0.60 (12.2)
0.56 (11.4)
0.55 (10.4)
0.57 (9.3)
0.56 (8.2)
0.51 (7.0)
0.51 (5.8)δhkc i(dead)
±0.71
±0.60
±0.48
±0.47
±0.51
±0.41
±0.44
±0.47
±0.42
±0.42
±0.44
±0.49
±0.49
±0.49
±0.55
±0.56
±0.55
±0.52
±0.51
±0.47
±0.48
±0.48
±0.49
±0.44
±0.52
±0.46
±0.34
±0.34
±0.29
±0.26

δhkc i(f ake)
±0.06
±0.03
±0.08
±0.11
±0.08
±0.11
±0.10
±0.12
±0.11
±0.13
±0.14
±0.14
±0.14
±0.16
±0.17
±0.17
±0.20
±0.21
±0.19
±0.17
±0.21
±0.19
±0.19
±0.20
±0.20
±0.22
±0.24
±0.29
±0.27
±0.34

δhkc i(total)
±0.94
±0.86
±0.79
±0.80
±0.83
±0.79
±0.82
±0.85
±0.84
±0.86
±0.89
±0.94
±0.96
±1.00
±1.06
±1.10
±0.91
±0.91
±0.92
±0.93
±0.97
±1.01
±0.93
±0.96
±0.97
±1.02
±1.00
±1.08
±1.12
±1.17

101

Table A.11: NBD fit results in centrality 50-60%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
4.508 (3.489) ± 0.023
4.387 (3.398) ± 0.022
4.248 (3.286) ± 0.022
4.102 (3.169) ± 0.021
3.954 (3.052) ± 0.021
3.813 (2.942) ± 0.020
3.666 (2.827) ± 0.020
3.518 (2.708) ± 0.019
3.378 (2.596) ± 0.019
3.236 (2.482) ± 0.018
3.091 (2.366) ± 0.018
2.949 (2.250) ± 0.017
2.806 (2.133) ± 0.017
2.664 (2.017) ± 0.017
2.520 (1.901) ± 0.016
2.376 (1.785) ± 0.015
2.236 (1.675) ± 0.011
2.097 (1.575) ± 0.010
1.960 (1.476) ± 0.010
1.822 (1.376) ± 0.009
1.683 (1.274) ± 0.009
1.545 (1.172) ± 0.008
1.407 (1.069) ± 0.006
1.267 (0.963) ± 0.006
1.126 (0.856) ± 0.005
0.986 (0.749) ± 0.005
0.846 (0.641) ± 0.004
0.706 (0.534) ± 0.003
0.565 (0.424) ± 0.002
0.428 (0.339) ± 0.002

hkc i(hki)8.89 (6.88) ± 0.29
8.85 (6.85) ± 0.29
8.78 (6.79) ± 0.29
8.74 (6.75) ± 0.30
8.81 (6.80) ± 0.31
8.67 (6.69) ± 0.31
8.66 (6.67) ± 0.32
8.78 (6.76) ± 0.33
8.67 (6.66) ± 0.34
8.67 (6.65) ± 0.35
8.71 (6.67) ± 0.36
8.67 (6.62) ± 0.37
8.63 (6.56) ± 0.39
8.58 (6.50) ± 0.40
8.60 (6.49) ± 0.42
8.69 (6.54) ± 0.45
8.77 (6.57) ± 0.34
8.75 (6.58) ± 0.36
8.74 (6.61) ± 0.38
8.71 (6.63) ± 0.41
8.70 (6.67) ± 0.44
8.66 (6.67) ± 0.47
8.55 (6.62) ± 0.41
8.45 (6.57) ± 0.45
8.41 (6.59) ± 0.44
8.31 (6.54) ± 0.50
8.07 (6.36) ± 0.52
7.85 (6.21) ± 0.54
7.86 (6.23) ± 0.64
7.22 (5.77) ± 0.64hχ2 /NDF i(hNDF i)
0.90 (15.0)
0.90 (14.8)
0.76 (14.2)
0.76 (13.9)
1.14 (13.4)
0.91 (13.1)
0.96 (12.9)
1.19 (12.5)
0.93 (12.0)
0.86 (11.8)
0.88 (11.6)
0.80 (11.1)
0.76 (10.5)
0.65 (9.9)
0.64 (9.5)
0.75 (8.9)
0.73 (8.5)
0.69 (8.1)
0.74 (7.6)
0.75 (7.1)
0.78 (6.7)
0.75 (6.3)
0.70 (5.9)
0.59 (5.5)
0.59 (5.2)
0.58 (4.8)
0.52 (4.4)
0.40 (3.8)
0.58 (3.3)
0.44 (2.9)δhkc i(dead)
±0.21
±0.23
±0.25
±0.26
±0.23
±0.24
±0.27
±0.21
±0.21
±0.25
±0.23
±0.20
±0.20
±0.24
±0.23
±0.26
±0.25
±0.25
±0.22
±0.23
±0.21
±0.19
±0.18
±0.18
±0.15
±0.14
±0.16
±0.08
±0.06
±0.06

δhkc i(f ake)
±0.01
±0.01
±0.01
±0.01
±0.01
±0.01
±0.02
±0.01
±0.01
±0.01
±0.02
±0.02
±0.03
±0.03
±0.02
±0.02
±0.02
±0.02
±0.03
±0.03
±0.03
±0.03
±0.03
±0.03
±0.04
±0.05
±0.06
±0.07
±0.09
±0.07

δhkc i(total)
±0.35
±0.37
±0.39
±0.39
±0.38
±0.39
±0.42
±0.40
±0.40
±0.43
±0.43
±0.42
±0.44
±0.47
±0.48
±0.52
±0.43
±0.44
±0.44
±0.47
±0.49
±0.51
±0.45
±0.48
±0.47
±0.52
±0.55
±0.55
±0.65
±0.65

Table A.12: NBD fit results in centrality 55-65%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
3.086 (2.389) ± 0.019
3.000 (2.324) ± 0.019
2.902 (2.245) ± 0.018
2.803 (2.166) ± 0.018
2.703 (2.086) ± 0.017
2.607 (2.012) ± 0.017
2.508 (1.934) ± 0.017
2.410 (1.855) ± 0.016
2.313 (1.777) ± 0.016
2.215 (1.699) ± 0.015
2.117 (1.620) ± 0.015
2.018 (1.540) ± 0.015
1.919 (1.459) ± 0.014
1.820 (1.378) ± 0.014
1.721 (1.298) ± 0.013
1.623 (1.219) ± 0.013
1.527 (1.144) ± 0.009
1.432 (1.075) ± 0.008
1.339 (1.008) ± 0.008
1.244 (0.940) ± 0.008
1.150 (0.871) ± 0.007
1.056 (0.801) ± 0.007
0.961 (0.730) ± 0.005
0.866 (0.658) ± 0.005
0.770 (0.585) ± 0.004
0.674 (0.511) ± 0.004
0.578 (0.438) ± 0.003
0.482 (0.364) ± 0.003
0.386 (0.292) ± 0.002
0.292 (0.232) ± 0.002

hkc i(hki)5.90 (4.56) ± 0.20
5.88 (4.56) ± 0.20
5.85 (4.52) ± 0.20
5.78 (4.46) ± 0.20
5.80 (4.48) ± 0.21
5.76 (4.45) ± 0.21
5.76 (4.44) ± 0.22
5.75 (4.43) ± 0.23
5.75 (4.42) ± 0.23
5.76 (4.42) ± 0.24
5.79 (4.43) ± 0.25
5.78 (4.41) ± 0.26
5.79 (4.40) ± 0.27
5.81 (4.40) ± 0.28
5.84 (4.41) ± 0.30
5.84 (4.39) ± 0.32
5.87 (4.41) ± 0.24
5.86 (4.42) ± 0.25
5.85 (4.43) ± 0.27
5.85 (4.45) ± 0.28
5.85 (4.48) ± 0.31
5.83 (4.49) ± 0.33
5.81 (4.50) ± 0.29
5.73 (4.46) ± 0.32
5.72 (4.46) ± 0.32
5.67 (4.44) ± 0.36
5.50 (4.32) ± 0.36
5.31 (4.18) ± 0.39
4.90 (3.89) ± 0.43
4.56 (3.63) ± 0.44hχ2 /NDF i(hNDF i)
0.76 (13.0)
0.79 (12.7)
0.76 (12.5)
0.69 (12.2)
0.78 (12.1)
0.70 (11.3)
0.62 (10.9)
0.55 (10.3)
0.50 (9.9)
0.47 (9.6)
0.49 (9.2)
0.49 (8.9)
0.50 (8.8)
0.52 (8.6)
0.53 (8.4)
0.54 (7.8)
0.52 (7.6)
0.49 (7.4)
0.48 (7.2)
0.52 (6.9)
0.59 (6.5)
0.59 (6.2)
0.62 (5.9)
0.57 (5.5)
0.54 (5.2)
0.53 (4.8)
0.40 (4.5)
0.51 (4.0)
0.54 (3.6)
0.57 (3.1)

δhkc i(dead)
±0.25
±0.13
±0.14
±0.14
±0.07
±0.14
±0.13
±0.12
±0.16
±0.16
±0.16
±0.16
±0.17
±0.20
±0.15
±0.18
±0.21
±0.24
±0.21
±0.21
±0.14
±0.11
±0.10
±0.10
±0.10
±0.09
±0.05
±0.05
±0.05
±0.05

δhkc i(f ake)
±0.01
±0.01
±0.02
±0.02
±0.02
±0.02
±0.02
±0.04
±0.04
±0.01
±0.01
±0.01
±0.02
±0.03
±0.02
±0.01
±0.01
±0.02
±0.02
±0.02
±0.02
±0.02
±0.03
±0.03
±0.04
±0.06
±0.07
±0.10
±0.02
±0.05

δhkc i(total)
±0.32
±0.24
±0.24
±0.25
±0.22
±0.25
±0.25
±0.26
±0.28
±0.29
±0.30
±0.30
±0.32
±0.34
±0.34
±0.36
±0.32
±0.34
±0.34
±0.35
±0.34
±0.35
±0.31
±0.34
±0.33
±0.38
±0.37
±0.41
±0.43
±0.45

102

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.13: NBD fit results in centrality 0-5%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
84.371 (65.303) ± 0.148
82.149 (63.638) ± 0.144
79.451 (61.463) ± 0.141
76.740 (59.282) ± 0.138
74.027 (57.142) ± 0.135
71.331 (55.050) ± 0.133
68.629 (52.910) ± 0.130
65.925 (50.747) ± 0.127
63.249 (48.612) ± 0.124
60.566 (46.465) ± 0.121
57.877 (44.295) ± 0.118
55.171 (42.098) ± 0.115
52.479 (39.900) ± 0.112
49.791 (37.710) ± 0.109
47.118 (35.547) ± 0.106
44.456 (33.395) ± 0.103
41.832 (31.335) ± 0.070
39.229 (29.465) ± 0.068
36.653 (27.626) ± 0.065
34.073 (25.755) ± 0.062
31.493 (23.870) ± 0.060
28.903 (21.959) ± 0.057
26.305 (20.020) ± 0.044
23.677 (18.009) ± 0.042
21.066 (16.025) ± 0.034
18.425 (13.948) ± 0.032
15.807 (11.964) ± 0.026
13.174 (9.949) ± 0.022
10.522 (7.808) ± 0.017
7.947 (6.122) ± 0.013

hkc i(hki)341.22 (264.10) ± 31.75
359.73 (278.66) ± 35.51
369.96 (286.25) ± 37.82
370.46 (286.18) ± 38.62
372.37 (287.35) ± 40.79
374.07 (288.55) ± 42.84
374.10 (288.36) ± 44.31
367.65 (282.90) ± 44.42
368.12 (282.88) ± 46.12
364.03 (279.07) ± 46.64
362.25 (277.19) ± 48.15
363.40 (277.12) ± 50.80
355.07 (269.86) ± 51.95
352.80 (267.18) ± 54.68
349.73 (263.69) ± 57.42
347.01 (260.24) ± 64.15
340.53 (254.77) ± 52.65
340.79 (254.20) ± 55.01
332.14 (246.66) ± 60.16
333.98 (247.30) ± 64.64
313.43 (230.73) ± 62.92
294.86 (214.95) ± 59.84
278.11 (200.81) ± 51.12
253.36 (181.02) ± 56.29
232.35 (165.72) ± 50.48
199.21 (140.84) ± 48.99
181.55 (126.91) ± 37.35
157.69 (108.85) ± 28.73
125.65 (86.76) ± 21.21
97.74 (67.60) ± 13.55

hχ2 /NDF i(hNDF i)
0.54 (58.0)
0.58 (57.8)
0.66 (56.4)
0.63 (55.4)
0.62 (54.7)
0.64 (53.6)
0.64 (52.7)
0.64 (52.2)
0.68 (51.2)
0.65 (50.2)
0.65 (49.0)
0.66 (47.9)
0.64 (46.5)
0.66 (45.2)
0.64 (43.6)
0.65 (42.1)
0.67 (40.6)
0.65 (39.4)
0.68 (37.9)
0.69 (37.0)
0.69 (35.2)
0.68 (33.5)
0.69 (31.9)
0.71 (30.2)
0.73 (28.3)
0.75 (26.2)
0.80 (24.2)
0.83 (21.4)
0.80 (18.2)
0.73 (15.1)

δhkc i(dead)±54.10±29.67±13.85±19.09±13.54±13.41±18.44±16.76±16.19±15.42±12.93±13.82±9.95±14.68±13.29±3.82±11.74±2.10±12.90
±1.38
±1.26
±2.22
±3.21
±6.54
±3.92
±8.99
±10.43
±3.79
±3.14
±8.30

δhkc i(f ake)
±6.72
±26.46±14.18±14.98±15.22±17.15±19.49±24.95±24.02±23.48±24.16±20.65±25.57±26.89±32.16±30.09±34.67±27.89±31.13±33.33±26.90±32.24±29.02±23.20±24.00±17.20±17.64±14.86±11.02±9.98

δhkc i(total)±63.09±53.31±42.69±45.61±45.60±48.06±51.80±53.63±54.46±54.45±55.40±56.55±58.75±62.68±67.14±70.96±64.13±61.72±68.95±72.74±68.44±68.01±58.87±61.24±56.04±52.70±42.61±32.57±24.10±18.77

δhkc i(f ake)
±9.61
±6.69
±4.56
±6.48
±6.31
±5.32
±7.98
±8.57
±8.41
±7.23
±7.38
±7.36
±10.08
±7.74
±9.43
±9.93
±9.02
±12.28±11.88±13.82±15.75±15.73±12.91±15.13±13.32±15.35±14.45±14.31±10.51±8.08

δhkc i(total)±14.40±16.33±12.91±13.32±13.79±14.04±15.16±15.92±16.47±15.83±16.44±16.94±20.24±19.83±20.40±21.86±17.28±19.55±19.53±20.75±22.69±23.19±19.31±21.52±20.16±21.30±19.80±19.23±15.06±12.34

Table A.14: NBD fit results in centrality 5-10%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
72.246 (55.918) ± 0.126
70.291 (54.452) ± 0.124
67.996 (52.602) ± 0.121
65.656 (50.720) ± 0.119
63.346 (48.897) ± 0.116
61.055 (47.118) ± 0.114
58.752 (45.296) ± 0.111
56.443 (43.447) ± 0.108
54.144 (41.614) ± 0.106
51.839 (39.768) ± 0.103
49.534 (37.909) ± 0.100
47.220 (36.030) ± 0.097
44.921 (34.153) ± 0.095
42.618 (32.277) ± 0.092
40.328 (30.425) ± 0.089
38.053 (28.585) ± 0.086
35.798 (26.815) ± 0.059
33.572 (25.212) ± 0.057
31.368 (23.631) ± 0.055
29.158 (22.030) ± 0.052
26.943 (20.408) ± 0.050
24.731 (18.770) ± 0.048
22.501 (17.102) ± 0.037
20.271 (15.421) ± 0.035
18.030 (13.713) ± 0.028
15.787 (11.998) ± 0.026
13.540 (10.276) ± 0.022
11.289 (8.548) ± 0.018
9.029 (6.805) ± 0.014
6.807 (5.296) ± 0.011

hkc i(hki)187.62 (145.22) ± 10.56
185.02 (143.32) ± 10.61
188.09 (145.50) ± 11.00
188.72 (145.76) ± 11.29
186.50 (143.95) ± 11.39
185.01 (142.75) ± 11.53
184.61 (142.29) ± 11.90
183.38 (141.14) ± 12.28
183.73 (141.18) ± 12.87
186.84 (143.32) ± 13.65
187.11 (143.19) ± 14.16
189.72 (144.71) ± 15.07
187.66 (142.61) ± 15.56
188.72 (142.85) ± 16.50
186.94 (140.87) ± 17.11
186.28 (139.66) ± 18.10
187.25 (139.96) ± 13.64
184.92 (138.46) ± 14.19
182.12 (136.56) ± 14.59
178.29 (134.27) ± 14.70
173.07 (131.50) ± 15.26
169.01 (129.45) ± 15.73
167.76 (129.28) ± 13.97
160.25 (123.65) ± 14.29
152.49 (118.31) ± 13.95
144.71 (112.06) ± 13.58
137.12 (106.29) ± 12.40
122.24 (92.30) ± 10.92
99.52 (72.96) ± 9.30
75.37 (54.94) ± 6.89

hχ2 /NDF i(hNDF i)
0.80 (61.0)
0.68 (61.3)
0.71 (59.8)
0.76 (59.0)
0.79 (57.0)
0.76 (55.6)
0.76 (54.0)
0.78 (52.2)
0.78 (51.2)
0.81 (49.4)
0.84 (47.9)
0.88 (46.3)
0.85 (44.6)
0.86 (43.1)
0.82 (41.6)
0.85 (39.9)
0.89 (38.5)
0.89 (37.3)
0.91 (36.0)
0.89 (34.6)
0.91 (33.1)
0.88 (32.1)
0.94 (30.7)
0.94 (29.1)
0.99 (27.4)
1.00 (25.4)
1.05 (22.9)
1.09 (20.3)
1.03 (17.2)
0.81 (14.2)

δhkc i(dead)±1.88±10.46
±4.99
±2.81
±4.55
±6.00
±4.96
±5.40
±5.90
±3.47
±3.90
±2.39
±8.12
±7.80
±5.86
±7.17
±5.58
±5.46
±5.25
±4.86
±5.84
±6.53
±3.27
±5.43
±5.87
±5.81
±5.42
±6.77
±5.46
±6.29

103

Table A.15: NBD fit results in centrality 10-15%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
60.607 (46.909) ± 0.119
59.026 (45.725) ± 0.118
57.108 (44.179) ± 0.116
55.139 (42.596) ± 0.113
53.198 (41.064) ± 0.111
51.256 (39.556) ± 0.108
49.321 (38.025) ± 0.106
47.393 (36.482) ± 0.103
45.463 (34.943) ± 0.101
43.535 (33.398) ± 0.098
41.602 (31.839) ± 0.095
39.668 (30.267) ± 0.092
37.729 (28.684) ± 0.090
35.787 (27.104) ± 0.087
33.864 (25.547) ± 0.084
31.943 (23.994) ± 0.082
30.061 (22.517) ± 0.056
28.191 (21.172) ± 0.053
26.340 (19.845) ± 0.051
24.493 (18.509) ± 0.049
22.636 (17.150) ± 0.047
20.776 (15.774) ± 0.044
18.905 (14.375) ± 0.034
17.029 (12.958) ± 0.033
15.142 (11.519) ± 0.026
13.253 (10.073) ± 0.024
11.367 (8.626) ± 0.020
9.474 (7.170) ± 0.017
7.575 (5.704) ± 0.013
5.714 (4.443) ± 0.010

hkc i(hki)129.59 (100.31) ± 6.29
123.34 (95.54) ± 6.09
121.44 (93.94) ± 6.08
121.88 (94.14) ± 6.24
120.98 (93.37) ± 6.33
120.49 (92.96) ± 6.46
119.78 (92.31) ± 6.58
117.86 (90.68) ± 6.62
117.65 (90.38) ± 6.79
117.79 (90.32) ± 7.03
118.04 (90.30) ± 7.29
118.96 (90.76) ± 7.64
118.41 (90.03) ± 7.87
118.19 (89.52) ± 8.24
118.35 (89.25) ± 8.67
118.19 (88.71) ± 9.20
117.56 (87.92) ± 6.85
117.37 (87.84) ± 7.21
117.25 (87.95) ± 7.59
117.01 (87.82) ± 8.16
114.86 (86.40) ± 8.44
114.37 (86.11) ± 8.95
112.39 (84.83) ± 7.68
111.32 (84.34) ± 8.15
109.44 (83.15) ± 7.72
106.03 (80.26) ± 8.35
96.68 (72.34) ± 7.60
88.58 (65.38) ± 7.20
77.55 (55.69) ± 6.79
64.41 (47.02) ± 6.22hχ2 /NDF i(hNDF i)
0.80 (58.0)
0.76 (56.6)
0.70 (55.3)
0.70 (54.2)
0.69 (52.9)
0.71 (52.2)
0.70 (51.3)
0.71 (50.1)
0.71 (49.2)
0.71 (48.2)
0.71 (46.9)
0.73 (45.2)
0.72 (44.0)
0.74 (42.5)
0.76 (41.1)
0.75 (39.6)
0.75 (38.1)
0.78 (36.3)
0.78 (34.7)
0.82 (33.5)
0.80 (31.8)
0.82 (30.3)
0.82 (28.8)
0.83 (26.9)
0.88 (25.1)
0.94 (23.0)
0.94 (20.5)
0.87 (18.1)
0.81 (15.6)
0.63 (12.8)

δhkc i(dead)
±1.29
±4.71
±5.55
±4.92
±5.23
±3.80
±3.61
±3.12
±3.77
±3.47
±3.18
±3.46
±3.52
±4.23
±4.60
±4.92
±4.91
±5.14
±4.90
±4.41
±5.74
±5.34
±5.37
±6.15
±5.22
±4.18
±4.30
±3.97
±3.54
±4.49

δhkc i(f ake)
±1.51
±9.77
±4.90
±3.01
±2.63
±3.87
±2.65
±3.60
±3.44
±4.14
±3.55
±4.65
±4.95
±5.06
±4.92
±5.31
±5.85
±5.41
±5.07
±5.83
±5.96
±6.20
±7.32
±7.47
±7.32
±6.94
±7.09
±8.10
±7.64
±6.13

δhkc i(total)±6.60±12.44
±9.58
±8.50
±8.62
±8.44
±7.95
±8.15
±8.49
±8.87
±8.71
±9.59
±9.94
±10.56±10.98±11.70±10.26±10.38±10.36±10.96±11.82±12.13±11.89±12.66±11.85±11.64±11.25±11.54±10.82±9.82

Table A.16: NBD fit results in centrality 15-20%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
50.243 (38.888) ± 0.109
48.892 (37.875) ± 0.106
47.301 (36.592) ± 0.104
45.685 (35.292) ± 0.102
44.087 (34.031) ± 0.100
42.483 (32.785) ± 0.097
40.884 (31.519) ± 0.095
39.301 (30.251) ± 0.092
37.708 (28.981) ± 0.090
36.108 (27.700) ± 0.088
34.505 (26.406) ± 0.085
32.897 (25.100) ± 0.083
31.285 (23.784) ± 0.081
29.677 (22.474) ± 0.078
28.079 (21.180) ± 0.076
26.483 (19.889) ± 0.073
24.918 (18.661) ± 0.050
23.365 (17.541) ± 0.048
21.831 (16.441) ± 0.046
20.294 (15.325) ± 0.044
18.754 (14.199) ± 0.042
17.215 (13.060) ± 0.040
15.665 (11.902) ± 0.031
14.110 (10.728) ± 0.029
12.547 (9.540) ± 0.024
10.977 (8.337) ± 0.022
9.409 (7.133) ± 0.018
7.840 (5.924) ± 0.015
6.271 (4.713) ± 0.012
4.734 (3.678) ± 0.009

hkc i(hki)95.08 (73.59) ± 4.42
95.76 (74.18) ± 4.54
95.63 (73.98) ± 4.65
95.70 (73.94) ± 4.78
94.78 (73.17) ± 4.87
95.97 (74.07) ± 5.05
96.10 (74.10) ± 5.22
95.92 (73.85) ± 5.36
96.52 (74.20) ± 5.57
96.81 (74.28) ± 5.77
96.51 (73.88) ± 5.92
95.50 (72.89) ± 6.03
95.28 (72.47) ± 6.24
95.11 (72.09) ± 6.45
94.76 (71.57) ± 6.69
95.08 (71.55) ± 7.04
94.28 (70.78) ± 5.11
92.98 (70.16) ± 5.28
91.68 (69.59) ± 5.47
89.06 (68.15) ± 5.60
87.46 (67.23) ± 5.76
86.16 (66.47) ± 6.04
83.50 (64.86) ± 5.15
81.20 (63.31) ± 5.36
79.16 (61.58) ± 4.86
73.67 (57.42) ± 4.99
67.77 (52.73) ± 4.44
61.02 (47.54) ± 4.01
54.47 (42.05) ± 3.54
47.73 (35.64) ± 3.59hχ2 /NDF i(hNDF i)
0.79 (53.0)
0.67 (52.4)
0.75 (51.2)
0.71 (49.2)
0.71 (47.7)
0.70 (47.1)
0.69 (46.1)
0.70 (44.8)
0.71 (43.6)
0.75 (42.5)
0.73 (41.4)
0.73 (40.1)
0.77 (38.6)
0.78 (37.3)
0.78 (35.9)
0.81 (34.6)
0.80 (33.3)
0.82 (32.3)
0.84 (31.2)
0.85 (29.8)
0.85 (28.5)
0.88 (27.0)
0.90 (25.4)
0.88 (23.7)
0.93( 21.9)
0.91( 19.8)
0.83 (17.8)
0.74 (15.6)
0.63 (13.4)
0.58 (11.3)

δhkc i(dead)
±4.84
±3.20
±2.80
±2.09
±3.79
±2.95
±2.66
±3.31
±3.47
±2.84
±2.38
±2.67
±2.78
±3.02
±2.98
±2.59
±3.11
±2.56
±2.32
±2.62
±2.47
±2.30
±3.27
±2.47
±2.06
±1.63
±2.00
±1.99
±1.64
±2.06

δhkc i(f ake)
±6.50
±1.41
±1.66
±1.96
±3.17
±1.99
±2.10
±2.60
±1.98
±2.39
±1.88
±2.54
±2.27
±2.26
±3.07
±2.66
±2.66
±2.78
±3.06
±3.13
±2.87
±3.25
±3.46
±3.88
±3.41
±3.89
±3.98
±4.09
±3.81
±3.33

δhkc i(total)
±9.23
±5.73
±5.68
±5.57
±6.94
±6.18
±6.22
±6.82
±6.85
±6.86
±6.65
±7.07
±7.19
±7.47
±7.94
±7.96
±6.55
±6.50
±6.69
±6.93
±6.90
±7.23
±7.01
±7.06
±6.28
±6.53
±6.29
±6.06
±5.46
±5.31

104

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.17: NBD fit results in centrality 20-25%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
41.340 (31.997) ± 0.101
40.233 (31.167) ± 0.099
38.920 (30.109) ± 0.096
37.606 (29.051) ± 0.094
36.282 (28.006) ± 0.092
34.986 (27.000) ± 0.090
33.656 (25.947) ± 0.088
32.342 (24.895) ± 0.086
31.029 (23.848) ± 0.083
29.714 (22.794) ± 0.081
28.395 (21.731) ± 0.079
27.071 (20.654) ± 0.077
25.754 (19.579) ± 0.074
24.428 (18.499) ± 0.072
23.116 (17.436) ± 0.070
21.803 (16.374) ± 0.068
20.516 (15.364) ± 0.046
19.244 (14.448) ± 0.044
17.979 (13.542) ± 0.043
16.715 (12.625) ± 0.041
15.448 (11.697) ± 0.039
14.177 (10.756) ± 0.037
12.899 (9.802) ± 0.029
11.616 (8.833) ± 0.027
10.327 (7.850) ± 0.022
9.035 (6.859) ± 0.021
7.744 (5.870) ± 0.017
6.452 (4.877) ± 0.014
5.160 (3.881) ± 0.011
3.895 (3.027) ± 0.008

hkc i(hki)80.17 (62.05) ± 3.85
80.14 (62.08) ± 3.92
81.43 (63.00) ± 4.07
81.18 (62.71) ± 4.17
82.48 (63.67) ± 4.38
81.78 (63.12) ± 4.43
81.76 (63.04) ± 4.53
81.09 (62.42) ± 4.60
81.53 (62.67) ± 4.76
81.92 (62.85) ± 4.93
82.20 (62.93) ± 5.11
82.39 (62.89) ± 5.33
81.71 (62.18) ± 5.47
82.61 (62.66) ± 5.83
81.91 (61.93) ± 6.04
82.06 (61.77) ± 6.34
82.29 (61.74) ± 4.77
81.98 (61.71) ± 4.99
81.64 (61.65) ± 5.26
80.08 (60.81) ± 5.43
78.84 (60.25) ± 5.73
77.06 (59.22) ± 5.98
74.63 (57.61) ± 5.08
72.40 (56.12) ± 5.38
68.48 (53.34) ± 4.75
62.87 (49.41) ± 4.84
58.29 (45.68) ± 4.29
54.02 (41.89) ± 4.10
49.72 (38.24) ± 4.19
44.74 (34.54) ± 4.38

hχ2 /NDF i(hNDF i)
0.71 (48.0)
0.67 (47.2)
0.68 (46.9)
0.68 (45.8)
0.78 (44.7)
0.76 (44.2)
0.75 (43.4)
0.68 (42.5)
0.72 (41.5)
0.77 (40.5)
0.77 (39.3)
0.79 (38.3)
0.78 (37.1)
0.82 (35.8)
0.78 (34.6)
0.78 (33.3)
0.82 (32.1)
0.82 (30.8)
0.82 (29.7)
0.83 (28.4)
0.84 (26.9)
0.86 (25.2)
0.88 (23.8)
0.92 (22.1)
0.88 (20.2)
0.82 (18.3)
0.71 (16.3)
0.58 (14.5)
0.49 (12.2)
0.49 (10.2)

δhkc i(dead)
±1.34
±3.19
±1.52
±3.68
±1.90
±2.76
±1.38
±1.91
±1.98
±1.94
±2.21
±2.25
±2.79
±2.05
±2.84
±2.84
±3.10
±2.85
±3.02
±2.99
±3.19
±2.73
±2.84
±2.12
±2.15
±2.09
±2.05
±1.91
±1.74
±1.45

δhkc i(f ake)
±1.27
±1.69
±0.92
±1.51
±1.68
±1.37
±1.07
±1.56
±1.63
±1.10
±1.08
±1.31
±1.73
±1.10
±1.75
±1.94
±2.13
±2.02
±1.98
±2.30
±2.34
±2.06
±2.57
±2.05
±2.18
±2.55
±2.65
±2.54
±2.92
±3.09

δhkc i(total)
±4.27
±5.33
±4.44
±5.76
±5.07
±5.39
±4.86
±5.22
±5.40
±5.41
±5.67
±5.93
±6.38
±6.28
±6.90
±7.21
±6.08
±6.09
±6.38
±6.61
±6.96
±6.89
±6.36
±6.13
±5.65
±5.85
±5.45
±5.19
±5.40
±5.55

Table A.18: NBD fit results in centrality 25-30%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
33.991 (26.309) ± 0.090
33.067 (25.615) ± 0.088
31.992 (24.750) ± 0.086
30.890 (23.863) ± 0.084
29.811 (23.012) ± 0.082
28.732 (22.173) ± 0.081
27.643 (21.312) ± 0.079
26.558 (20.443) ± 0.077
25.479 (19.583) ± 0.075
24.403 (18.721) ± 0.073
23.325 (17.852) ± 0.071
22.243 (16.972) ± 0.069
21.155 (16.084) ± 0.067
20.074 (15.204) ± 0.065
18.991 (14.327) ± 0.063
17.912 (13.455) ± 0.061
16.855 (12.625) ± 0.042
15.805 (11.868) ± 0.040
14.769 (11.126) ± 0.038
13.731 (10.376) ± 0.037
12.685 (9.610) ± 0.035
11.640 (8.838) ± 0.033
10.591 (8.054) ± 0.026
9.535 (7.258) ± 0.024
8.479 (6.455) ± 0.020
7.420 (5.643) ± 0.018
6.363 (4.832) ± 0.015
5.306 (4.017) ± 0.013
4.245 (3.196) ± 0.010
3.205 (2.492) ± 0.007

hkc i(hki)67.99 (52.63) ± 3.18
68.77 (53.27) ± 3.29
67.53 (52.23) ± 3.30
67.26 (51.95) ± 3.34
67.40 (52.02) ± 3.41
66.18 (51.07) ± 3.41
66.17 (51.01) ± 3.50
65.89 (50.72) ± 3.57
65.64 (50.44) ± 3.66
65.21 (50.01) ± 3.75
64.30 (49.19) ± 3.80
64.65 (49.31) ± 3.94
63.75 (48.45) ± 4.03
63.34 (47.95) ± 4.15
62.47 (47.09) ± 4.26
62.61 (47.00) ± 4.49
62.05 (46.45) ± 3.30
61.38 (46.04) ± 3.39
60.26 (45.34) ± 3.48
59.70 (44.97) ± 3.62
59.20 (44.67) ± 3.84
57.91 (43.73) ± 4.00
56.00 (42.22) ± 3.34
54.60 (41.11) ± 3.51
52.59 (39.35) ± 3.21
51.06 (38.08) ± 3.48
49.16 (36.41) ± 3.38
45.94 (33.94) ± 3.32
43.38 (31.95) ± 3.30
38.41 (28.81) ± 3.58hχ2 /NDF i(hNDF i)
0.56 (44.0)
0.61 (44.0)
0.56 (43.6)
0.58 (42.8)
0.64 (41.9)
0.65 (41.3)
0.62 (40.2)
0.65 (38.9)
0.66 (37.9)
0.66 (37.1)
0.64 (35.9)
0.67 (35.0)
0.68 (33.4)
0.70 (32.4)
0.69 (31.0)
0.71 (29.9)
0.72 (28.7)
0.72 (27.5)
0.72 (26.3)
0.71 (25.1)
0.74 (23.7)
0.71 (22.3)
0.64 (21.0)
0.67 (19.3)
0.64 (17.6)
0.62 (16.0)
0.56 (14.3)
0.49 (12.3)
0.49 (10.5)
0.47 (8.8)δhkc i(dead)
±2.58
±2.56
±3.20
±2.31
±1.64
±2.18
±2.87
±2.32
±2.53
±3.12
±3.10
±2.62
±2.73
±2.72
±2.95
±2.80
±2.34
±2.67
±2.88
±2.44
±2.61
±2.54
±2.38
±2.10
±2.49
±1.76
±1.61
±1.57
±1.68
±1.46

δhkc i(f ake)
±0.80
±0.05
±1.31
±1.01
±1.27
±0.79
±1.24
±1.30
±0.86
±1.03
±0.97
±0.71
±0.89
±1.01
±1.01
±0.83
±0.81
±1.02
±1.01
±1.21
±1.06
±1.09
±1.26
±1.43
±1.42
±1.53
±1.59
±1.75
±2.00
±2.20

δhkc i(total)
±4.17
±4.17
±4.78
±4.19
±3.99
±4.12
±4.69
±4.45
±4.53
±4.99
±5.00
±4.78
±4.95
±5.06
±5.28
±5.35
±4.12
±4.43
±4.63
±4.54
±4.76
±4.86
±4.29
±4.33
±4.30
±4.19
±4.07
±4.07
±4.21
±4.45

105

Table A.19: NBD fit results in centrality 30-35%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
27.652 (21.402) ± 0.079
26.900 (20.838) ± 0.078
26.016 (20.126) ± 0.076
25.115 (19.401) ± 0.074
24.225 (18.699) ± 0.073
23.345 (18.015) ± 0.071
22.464 (17.318) ± 0.070
21.583 (16.613) ± 0.068
20.702 (15.911) ± 0.066
19.828 (15.210) ± 0.065
18.948 (14.500) ± 0.063
18.064 (13.781) ± 0.061
17.181 (13.060) ± 0.059
16.302 (12.344) ± 0.058
15.419 (11.629) ± 0.056
14.543 (10.920) ± 0.054
13.684 (10.246) ± 0.037
12.833 (9.633) ± 0.035
11.991 (9.029) ± 0.034
11.146 (8.415) ± 0.033
10.299 (7.796) ± 0.031
9.450 (7.169) ± 0.030
8.597 (6.532) ± 0.023
7.740 (5.885) ± 0.022
6.881 (5.231) ± 0.018
6.023 (4.574) ± 0.016
5.165 (3.914) ± 0.013
4.306 (3.253) ± 0.011
3.445 (2.587) ± 0.009
2.606 (2.037) ± 0.006

hkc i(hki)58.33 (45.15) ± 2.86
57.89 (44.85) ± 2.86
57.99 (44.86) ± 2.94
57.51 (44.43) ± 2.98
56.95 (43.97) ± 3.03
56.94 (43.95) ± 3.10
56.67 (43.71) ± 3.17
55.99 (43.12) ± 3.22
56.01 (43.07) ± 3.34
55.63 (42.71) ± 3.44
55.29 (42.37) ± 3.57
55.70 (42.57) ± 3.73
54.86 (41.80) ± 3.89
53.65 (40.75) ± 3.97
53.33 (40.40) ± 4.19
52.71 (39.80) ± 4.38
52.57 (39.60) ± 3.27
51.53 (38.93) ± 3.35
49.87 (37.95) ± 3.43
48.20 (36.98) ± 3.59
47.05 (36.22) ± 3.67
45.91 (35.45) ± 3.79
44.64 (34.66) ± 3.18
43.61 (34.05) ± 3.30
42.44 (33.32) ± 3.04
41.28 (32.47) ± 3.26
40.29 (31.79) ± 3.21
38.26 (30.17) ± 3.27
35.11 (27.79) ± 3.39
30.96 (24.26) ± 3.44hχ2 /NDF i(hNDF i)
0.67 (41.0)
0.60 (40.9)
0.63 (39.6)
0.66 (38.4)
0.67 (37.0)
0.70 (36.3)
0.74 (35.0)
0.71 (33.9)
0.73 (33.0)
0.76 (31.8)
0.72 (30.9)
0.77 (30.1)
0.79 (29.1)
0.68 (28.0)
0.74 (26.8)
0.75 (25.4)
0.78 (24.3)
0.81 (23.0)
0.77 (22.0)
0.76 (20.9)
0.71 (19.7)
0.69 (18.6)
0.62 (17.5)
0.57 (16.3)
0.50 (14.9)
0.45 (13.6)
0.45 (12.2)
0.46 (10.7)
0.51 (9.2)
0.54 (7.9)δhkc i(dead)
±3.24
±2.95
±2.17
±2.60
±2.60
±2.27
±1.78
±2.04
±2.00
±1.94
±2.25
±1.76
±2.03
±2.50
±2.48
±2.14
±2.15
±2.25
±2.29
±2.06
±1.60
±1.51
±1.43
±1.22
±1.36
±1.39
±1.33
±1.62
±1.18
±1.25

δhkc i(f ake)
±0.86
±0.85
±0.88
±0.72
±0.86
±0.77
±0.77
±0.69
±0.70
±0.84
±0.85
±0.78
±1.04
±1.08
±0.99
±0.93
±0.93
±0.95
±0.94
±0.89
±0.93
±0.93
±1.00
±1.01
±1.10
±1.27
±1.30
±1.28
±1.41
±1.14

δhkc i(total)
±4.41
±4.20
±3.75
±4.02
±4.09
±3.92
±3.72
±3.87
±3.96
±4.03
±4.30
±4.20
±4.51
±4.82
±4.97
±4.96
±4.02
±4.14
±4.23
±4.23
±4.11
±4.18
±3.62
±3.66
±3.51
±3.76
±3.71
±3.87
±3.86
±3.83

Table A.20: NBD fit results in centrality 35-40%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
21.970 (17.005) ± 0.071
21.372 (16.556) ± 0.070
20.684 (16.001) ± 0.068
19.964 (15.423) ± 0.067
19.253 (14.862) ± 0.065
18.572 (14.333) ± 0.064
17.877 (13.783) ± 0.063
17.176 (13.221) ± 0.061
16.475 (12.662) ± 0.059
15.778 (12.104) ± 0.058
15.082 (11.542) ± 0.057
14.380 (10.972) ± 0.055
13.678 (10.399) ± 0.053
12.978 (9.829) ± 0.052
12.280 (9.264) ± 0.050
11.582 (8.700) ± 0.049
10.894 (8.161) ± 0.033
10.213 (7.670) ± 0.032
9.540 (7.188) ± 0.031
8.867 (6.699) ± 0.029
8.191 (6.204) ± 0.028
7.517 (5.707) ± 0.027
6.837 (5.198) ± 0.021
6.155 (4.685) ± 0.019
5.474 (4.166) ± 0.016
4.792 (3.644) ± 0.015
4.111 (3.122) ± 0.012
3.426 (2.594) ± 0.010
2.742 (2.069) ± 0.008
2.070 (1.609) ± 0.006

hkc i(hki)47.25 (36.57) ± 2.31
47.47 (36.77) ± 2.32
46.97 (36.33) ± 2.33
46.50 (35.92) ± 2.35
46.36 (35.78) ± 2.40
45.46 (35.08) ± 2.41
45.55 (35.11) ± 2.48
45.52 (35.05) ± 2.54
45.45 (34.94) ± 2.63
45.41 (34.85) ± 2.70
44.90 (34.38) ± 2.77
44.50 (33.97) ± 2.83
43.95 (33.42) ± 2.91
43.68 (33.09) ± 2.99
43.03 (32.45) ± 3.06
42.60 (31.98) ± 3.17
42.04 (31.45) ± 2.30
41.13 (30.87) ± 2.34
40.19 (30.29) ± 2.39
39.08 (29.61) ± 2.46
38.50 (29.35) ± 2.57
37.29 (28.54) ± 2.64
36.65 (28.17) ± 2.29
35.93 (27.66) ± 2.40
34.69 (26.74) ± 2.20
33.23 (25.56) ± 2.37
31.76 (24.13) ± 2.33
30.11 (22.76) ± 2.52
28.05 (20.82) ± 2.59
24.89 (18.20) ± 2.64hχ2 /NDF i(hNDF i)
0.79 (38.0)
0.69 (37.7)
0.71 (37.1)
0.79 (36.0)
0.86 (34.6)
0.82 (33.4)
0.83 (32.5)
0.83 (31.7)
0.87 (30.7)
0.87 (29.8)
0.84 (28.6)
0.81 (27.6)
0.82 (26.3)
0.81 (25.4)
0.79 (24.2)
0.77 (23.1)
0.74 (22.0)
0.72 (21.0)
0.70 (20.0)
0.69 (18.9)
0.67 (18.0)
0.57 (17.1)
0.55 (16.1)
0.55 (15.0)
0.52 (13.7)
0.50 (12.4)
0.48 (11.1)
0.54 (9.8)
0.55 (8.3)
0.54 (7.0)δhkc i(dead)
±0.68
±0.44
±0.92
±0.86
±0.91
±1.16
±1.22
±1.26
±1.09
±0.92
±1.02
±1.03
±1.15
±1.16
±1.23
±1.08
±1.00
±0.90
±0.86
±0.92
±0.97
±1.25
±0.92
±0.77
±0.77
±0.86
±0.88
±0.65
±0.86
±0.94

δhkc i(f ake)
±0.60
±0.02
±0.23
±0.30
±0.15
±0.36
±0.37
±0.44
±0.34
±0.22
±0.36
±0.31
±0.39
±0.33
±0.41
±0.45
±0.34
±0.35
±0.19
±0.28
±0.34
±0.50
±0.47
±0.51
±0.65
±0.72
±0.72
±0.72
±0.84
±0.95

δhkc i(total)
±2.48
±2.36
±2.51
±2.52
±2.57
±2.70
±2.79
±2.87
±2.87
±2.86
±2.97
±3.03
±3.15
±3.22
±3.32
±3.38
±2.53
±2.54
±2.55
±2.64
±2.77
±2.97
±2.51
±2.58
±2.43
±2.62
±2.59
±2.69
±2.85
±2.96

106

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.21: NBD fit results in centrality 40-45%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
17.453 (13.508) ± 0.063
16.988 (13.160) ± 0.062
16.429 (12.709) ± 0.061
15.863 (12.255) ± 0.059
15.315 (11.822) ± 0.058
14.758 (11.389) ± 0.057
14.201 (10.948) ± 0.056
13.652 (10.508) ± 0.054
13.096 (10.065) ± 0.053
12.537 (9.618) ± 0.051
11.980 (9.169) ± 0.050
11.422 (8.715) ± 0.049
10.861 (8.258) ± 0.048
10.302 (7.803) ± 0.046
9.742 (7.349) ± 0.045
9.189 (6.902) ± 0.043
8.649 (6.478) ± 0.030
8.110 (6.090) ± 0.029
7.578 (5.708) ± 0.027
7.044 (5.321) ± 0.026
6.509 (4.930) ± 0.025
5.973 (4.533) ± 0.024
5.435 (4.131) ± 0.018
4.894 (3.723) ± 0.017
4.353 (3.311) ± 0.014
3.811 (2.896) ± 0.013
3.267 (2.478) ± 0.011
2.724 (2.060) ± 0.009
2.179 (1.639) ± 0.007
1.646 (1.279) ± 0.005

hkc i(hki)37.64 (29.13) ± 1.83
37.36 (28.94) ± 1.85
37.57 (29.06) ± 1.91
37.55 (29.01) ± 1.96
36.88 (28.47) ± 1.96
36.84 (28.43) ± 2.00
36.74 (28.32) ± 2.04
36.19 (27.85) ± 2.07
36.01 (27.67) ± 2.12
36.07 (27.66) ± 2.21
35.96 (27.51) ± 2.28
35.74 (27.26) ± 2.36
35.46 (26.95) ± 2.43
35.19 (26.64) ± 2.51
35.42 (26.71) ± 2.63
35.02 (26.31) ± 2.70
34.15 (25.59) ± 1.94
33.55 (25.21) ± 1.99
33.04 (24.89) ± 2.06
32.48 (24.52) ± 2.12
31.66 (23.97) ± 2.17
30.82 (23.44) ± 2.22
29.59 (22.59) ± 1.88
28.75 (22.00) ± 2.00
28.04 (21.41) ± 1.89
26.88 (20.50) ± 2.02
25.91 (19.73) ± 2.01
24.44 (18.61) ± 2.05
22.93 (17.48) ± 2.12
20.46 (15.59) ± 2.26hχ2 /NDF i(hNDF i)
0.74 (35.0)
0.74 (34.1)
0.83 (33.0)
0.82 (32.2)
0.76 (31.4)
0.83 (30.6)
0.88 (29.8)
0.85 (29.2)
0.83 (28.3)
0.89 (27.1)
0.91 (26.3)
0.91 (25.2)
0.87 (24.2)
0.89 (22.9)
0.92 (22.0)
0.83 (21.1)
0.73 (19.9)
0.71 (19.0)
0.70 (18.0)
0.67 (17.0)
0.63 (16.1)
0.61 (15.1)
0.58 (14.0)
0.54 (13.1)
0.56 (12.1)
0.50 (10.9)
0.52 (9.9)
0.47 (8.7)
0.55 (7.4)
0.54 (6.3)

δhkc i(dead)
±0.40
±1.08
±0.91
±1.54
±1.75
±1.76
±1.32
±1.54
±1.30
±1.25
±1.27
±1.12
±1.08
±1.10
±0.69
±0.86
±1.22
±1.19
±1.22
±1.18
±1.12
±1.00
±1.09
±0.83
±0.90
±0.74
±0.63
±0.38
±0.38
±0.59

δhkc i(f ake)
±0.98
±0.27
±0.00
±0.31
±0.34
±0.32
±0.26
±0.32
±0.22
±0.28
±0.33
±0.33
±0.28
±0.38
±0.34
±0.36
±0.51
±0.45
±0.41
±0.43
±0.45
±0.46
±0.45
±0.49
±0.54
±0.51
±0.54
±0.61
±0.55
±0.51

δhkc i(total)
±2.12
±2.16
±2.12
±2.51
±2.65
±2.68
±2.45
±2.60
±2.50
±2.55
±2.63
±2.63
±2.67
±2.77
±2.74
±2.86
±2.35
±2.36
±2.43
±2.47
±2.48
±2.48
±2.23
±2.22
±2.16
±2.21
±2.18
±2.18
±2.22
±2.39

Table A.22: NBD fit results in centrality 45-50%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
13.493 (10.444) ± 0.054
13.123 (10.166) ± 0.053
12.698 (9.823) ± 0.052
12.261 (9.472) ± 0.051
11.829 (9.131) ± 0.050
11.403 (8.800) ± 0.049
10.976 (8.462) ± 0.048
10.547 (8.119) ± 0.046
10.120 (7.778) ± 0.045
9.694 (7.437) ± 0.044
9.267 (7.093) ± 0.043
8.836 (6.742) ± 0.042
8.403 (6.389) ± 0.041
7.970 (6.037) ± 0.039
7.537 (5.686) ± 0.038
7.108 (5.339) ± 0.037
6.688 (5.009) ± 0.025
6.271 (4.709) ± 0.024
5.860 (4.413) ± 0.023
5.450 (4.115) ± 0.022
5.037 (3.813) ± 0.021
4.623 (3.508) ± 0.020
4.205 (3.196) ± 0.016
3.785 (2.879) ± 0.015
3.364 (2.558) ± 0.012
2.943 (2.236) ± 0.011
2.523 (1.913) ± 0.009
2.104 (1.592) ± 0.008
1.692 (1.306) ± 0.006
1.277 (1.014) ± 0.004

hkc i(hki)32.22 (24.94) ± 1.72
32.44 (25.13) ± 1.74
32.30 (24.99) ± 1.77
32.54 (25.14) ± 1.85
32.98 (25.46) ± 1.94
33.01 (25.47) ± 2.00
32.73 (25.23) ± 2.04
32.55 (25.05) ± 2.09
32.54 (25.00) ± 2.16
32.48 (24.91) ± 2.23
32.14 (24.60) ± 2.28
31.98 (24.40) ± 2.37
31.90 (24.26) ± 2.46
31.80 (24.10) ± 2.56
31.73 (23.97) ± 2.67
31.47 (23.68) ± 2.78
31.14 (23.38) ± 2.05
30.59 (23.16) ± 2.12
29.82 (22.83) ± 2.20
29.04 (22.41) ± 2.27
28.60 (22.23) ± 2.41
28.11 (21.89) ± 2.48
27.55 (21.51) ± 2.15
27.20 (21.34) ± 2.35
26.76 (21.15) ± 2.25
26.24 (20.74) ± 2.47
25.81 (20.49) ± 2.54
25.12 (19.94) ± 2.71
23.73 (18.71) ± 2.55
21.69 (16.94) ± 2.72hχ2 /NDF i(hNDF i)
0.77 (27.0)
0.79 (26.9)
0.81 (25.6)
0.89 (24.6)
0.89 (23.9)
0.88 (23.1)
0.84 (22.3)
0.84 (21.7)
0.80 (21.0)
0.76 (20.5)
0.67 (19.8)
0.60 (19.1)
0.55 (18.5)
0.52 (17.7)
0.54 (17.1)
0.53 (16.4)
0.54 (15.6)
0.55 (15.0)
0.55 (14.4)
0.54 (13.7)
0.57 (13.1)
0.55 (12.4)
0.55 (11.8)
0.56 (11.0)
0.58 (10.3)
0.57 (9.4)
0.53 (8.5)
0.48 (7.6)
0.45 (6.6)
0.49 (5.4)

δhkc i(dead)
±1.57
±1.12
±0.70
±0.78
±0.81
±0.56
±0.56
±0.52
±0.30
±0.39
±0.44
±0.57
±0.56
±0.58
±0.79
±0.83
±0.84
±0.77
±0.72
±0.57
±0.44
±0.47
±0.45
±0.42
±0.45
±0.16
±0.12
±0.09
±0.14
±0.51

δhkc i(f ake)
±0.13
±0.09
±0.22
±0.32
±0.20
±0.28
±0.24
±0.29
±0.28
±0.31
±0.30
±0.34
±0.36
±0.38
±0.42
±0.37
±0.38
±0.37
±0.35
±0.36
±0.43
±0.42
±0.38
±0.41
±0.40
±0.40
±0.42
±0.52
±0.38
±0.45

δhkc i(total)
±2.34
±2.07
±1.92
±2.04
±2.11
±2.10
±2.13
±2.18
±2.20
±2.28
±2.34
±2.46
±2.54
±2.65
±2.82
±2.92
±2.25
±2.29
±2.34
±2.37
±2.49
±2.56
±2.23
±2.42
±2.33
±2.50
±2.57
±2.76
±2.58
±2.80

107

Table A.23: NBD fit results in centrality 50-55%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
10.177 (7.877) ± 0.046
9.898 (7.667) ± 0.045
9.584 (7.414) ± 0.044
9.257 (7.151) ± 0.044
8.935 (6.897) ± 0.043
8.613 (6.647) ± 0.042
8.287 (6.389) ± 0.041
7.964 (6.130) ± 0.040
7.640 (5.872) ± 0.039
7.318 (5.614) ± 0.038
6.991 (5.351) ± 0.037
6.664 (5.085) ± 0.036
6.338 (4.818) ± 0.035
6.013 (4.554) ± 0.034
5.689 (4.292) ± 0.033
5.365 (4.031) ± 0.032
5.047 (3.781) ± 0.022
4.731 (3.554) ± 0.021
4.421 (3.331) ± 0.020
4.110 (3.106) ± 0.019
3.798 (2.877) ± 0.018
3.487 (2.646) ± 0.018
3.173 (2.412) ± 0.013
2.858 (2.174) ± 0.013
2.541 (1.932) ± 0.010
2.224 (1.691) ± 0.010
1.906 (1.447) ± 0.008
1.593 (1.215) ± 0.006
1.279 (0.982) ± 0.005
0.969 (0.770) ± 0.004

hkc i(hki)27.08 (20.96) ± 1.52
26.98 (20.90) ± 1.53
26.02 (20.12) ± 1.50
25.65 (19.81) ± 1.51
25.16 (19.42) ± 1.52
24.75 (19.10) ± 1.53
24.69 (19.03) ± 1.58
24.46 (18.82) ± 1.62
24.42 (18.76) ± 1.67
24.25 (18.59) ± 1.72
24.35 (18.62) ± 1.80
24.37 (18.58) ± 1.89
24.32 (18.48) ± 1.96
24.34 (18.42) ± 2.05
24.43 (18.41) ± 2.14
24.59 (18.43) ± 2.28
24.82 (18.54) ± 1.71
24.90 (18.65) ± 1.81
24.53 (18.49) ± 1.90
24.09 (18.29) ± 2.00
23.76 (18.20) ± 2.19
23.58 (18.20) ± 2.34
23.10 (17.98) ± 2.13
22.73 (17.85) ± 2.36
22.47 (17.74) ± 2.36
22.22 (17.61) ± 2.64
21.76 (17.39) ± 2.93
20.48 (16.47) ± 3.02
19.00 (15.26) ± 2.67
16.64 (13.48) ± 2.31hχ2 /NDF i(hNDF i)
1.18 (24.0)
1.14 (23.0)
1.07 (21.9)
0.97 (21.2)
0.85 (20.8)
0.82 (20.0)
0.80 (19.6)
0.79 (19.1)
0.79 (18.5)
0.74 (18.0)
0.74 (17.5)
0.70 (16.9)
0.61 (16.3)
0.55 (15.7)
0.52 (15.0)
0.52 (14.3)
0.54 (13.8)
0.62 (13.4)
0.62 (12.9)
0.58 (12.3)
0.61 (11.6)
0.59 (11.1)
0.56 (10.5)
0.53 (9.8)
0.54 (9.0)
0.57 (8.2)
0.64 (7.4)
0.59 (6.6)
0.54 (5.5)
0.52 (4.5)δhkc i(dead)
±1.23
±0.62
±0.94
±0.43
±0.63
±0.77
±0.54
±0.70
±0.67
±0.67
±0.57
±0.79
±0.93
±0.93
±1.01
±0.85
±0.85
±0.58
±0.47
±0.62
±0.57
±0.59
±0.77
±0.74
±0.95
±0.89
±0.75
±0.69
±0.67
±0.59

δhkc i(f ake)
±0.17
±0.26
±0.23
±0.21
±0.16
±0.21
±0.10
±0.22
±0.28
±0.25
±0.26
±0.24
±0.26
±0.39
±0.22
±0.35
±0.37
±0.26
±0.35
±0.20
±0.11
±0.23
±0.24
±0.25
±0.27
±0.29
±0.36
±0.35
±0.47
±0.54

δhkc i(total)
±1.96
±1.67
±1.79
±1.58
±1.66
±1.73
±1.68
±1.77
±1.81
±1.86
±1.91
±2.06
±2.19
±2.28
±2.38
±2.46
±1.94
±1.92
±1.99
±2.10
±2.27
±2.43
±2.28
±2.49
±2.56
±2.80
±3.05
±3.11
±2.80
±2.45

Table A.24: NBD fit results in centrality 55-60%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
7.465 (5.778) ± 0.041
7.258 (5.622) ± 0.040
7.021 (5.431) ± 0.039
6.781 (5.238) ± 0.038
6.547 (5.054) ± 0.038
6.307 (4.867) ± 0.037
6.070 (4.680) ± 0.036
5.833 (4.490) ± 0.035
5.596 (4.301) ± 0.034
5.360 (4.112) ± 0.033
5.122 (3.920) ± 0.033
4.885 (3.728) ± 0.032
4.648 (3.534) ± 0.031
4.411 (3.340) ± 0.030
4.173 (3.148) ± 0.029
3.935 (2.955) ± 0.028
3.701 (2.772) ± 0.019
3.472 (2.607) ± 0.018
3.244 (2.444) ± 0.018
3.016 (2.279) ± 0.017
2.787 (2.111) ± 0.016
2.558 (1.942) ± 0.015
2.326 (1.769) ± 0.012
2.095 (1.595) ± 0.011
1.861 (1.417) ± 0.009
1.630 (1.240) ± 0.009
1.399 (1.062) ± 0.007
1.166 (0.882) ± 0.006
0.935 (0.714) ± 0.004
0.703 (0.548) ± 0.003

hkc i(hki)18.75 (14.52) ± 1.12
19.02 (14.74) ± 1.14
18.80 (14.54) ± 1.14
18.41 (14.22) ± 1.14
18.01 (13.89) ± 1.13
17.98 (13.87) ± 1.15
17.84 (13.75) ± 1.15
17.66 (13.59) ± 1.17
17.52 (13.46) ± 1.19
17.52 (13.43) ± 1.22
17.49 (13.38) ± 1.26
17.37 (13.24) ± 1.30
17.22 (13.08) ± 1.33
17.20 (13.01) ± 1.39
17.18 (12.95) ± 1.44
17.34 (13.00) ± 1.54
17.31 (12.94) ± 1.15
17.17 (12.87) ± 1.19
17.09 (12.85) ± 1.25
16.91 (12.74) ± 1.32
16.61 (12.58) ± 1.40
16.53 (12.57) ± 1.50
16.34 (12.46) ± 1.35
16.13 (12.30) ± 1.44
16.07 (12.24) ± 1.45
15.67 (11.91) ± 1.54
14.90 (11.39) ± 1.50
14.25 (10.82) ± 1.56
13.41 (10.20) ± 1.63
11.85 (9.19) ± 1.63hχ2 /NDF i(hNDF i)
0.71 (16.0)
0.76 (16.0)
0.73 (15.8)
0.57 (15.4)
0.51 (14.7)
0.56 (14.4)
0.51 (14.1)
0.47 (13.7)
0.44 (13.4)
0.43 (13.2)
0.47 (13.1)
0.49 (12.6)
0.51 (12.1)
0.58 (11.5)
0.62 (10.9)
0.63 (10.4)
0.65 (9.9)
0.66 (9.5)
0.68 (9.2)
0.67 (8.6)
0.62 (8.1)
0.61 (7.6)
0.66 (7.1)
0.62 (6.7)
0.72 (6.3)
0.59 (5.7)
0.47 (5.1)
0.42 (4.5)
0.46 (4.0)
0.53 (3.3)δhkc i(dead)
±0.59
±0.60
±0.61
±0.57
±0.70
±0.70
±0.69
±0.63
±0.60
±0.67
±0.66
±0.69
±0.74
±0.62
±0.78
±0.84
±0.67
±0.71
±0.64
±0.53
±0.56
±0.52
±0.49
±0.48
±0.34
±0.17
±0.35
±0.37
±0.30
±0.35

δhkc i(f ake)
±0.13
±0.08
±0.09
±0.09
±0.07
±0.07
±0.07
±0.07
±0.08
±0.09
±0.09
±0.09
±0.10
±0.10
±0.12
±0.13
±0.08
±0.10
±0.09
±0.08
±0.09
±0.11
±0.10
±0.11
±0.12
±0.13
±0.15
±0.17
±0.15
±0.17

δhkc i(total)
±1.27
±1.29
±1.29
±1.28
±1.33
±1.34
±1.35
±1.33
±1.34
±1.40
±1.43
±1.47
±1.53
±1.52
±1.64
±1.76
±1.33
±1.39
±1.41
±1.42
±1.51
±1.59
±1.44
±1.52
±1.49
±1.56
±1.54
±1.61
±1.66
±1.67

108

APPENDIX A. TABLES OF NBD FIT RESULTS

Table A.25: NBD fit results in centrality 60-65%.
δη
0.700
0.678
0.656
0.634
0.613
0.591
0.569
0.547
0.525
0.503
0.481
0.459
0.438
0.416
0.394
0.372
0.350
0.328
0.306
0.284
0.263
0.241
0.219
0.197
0.175
0.153
0.131
0.109
0.087
0.066

hµc i(hµi)
5.337 (4.131) ± 0.033
5.197 (4.026) ± 0.033
5.031 (3.892) ± 0.032
4.861 (3.755) ± 0.032
4.686 (3.617) ± 0.031
4.519 (3.487) ± 0.030
4.344 (3.349) ± 0.029
4.169 (3.209) ± 0.029
4.003 (3.077) ± 0.028
3.835 (2.942) ± 0.027
3.665 (2.805) ± 0.027
3.495 (2.667) ± 0.026
3.326 (2.529) ± 0.025
3.159 (2.393) ± 0.025
2.989 (2.255) ± 0.024
2.818 (2.117) ± 0.023
2.652 (1.986) ± 0.016
2.487 (1.867) ± 0.015
2.324 (1.750) ± 0.015
2.160 (1.631) ± 0.014
1.996 (1.511) ± 0.013
1.832 (1.390) ± 0.013
1.667 (1.267) ± 0.010
1.502 (1.143) ± 0.009
1.335 (1.015) ± 0.007
1.168 (0.888) ± 0.007
1.005 (0.768) ± 0.006
0.838 (0.639) ± 0.005
0.673 (0.524) ± 0.004
0.507 (0.405) ± 0.003

hkc i(hki)14.40 (11.14) ± 0.81
14.25 (11.04) ± 0.82
14.37 (11.12) ± 0.86
14.35 (11.09) ± 0.88
14.41 (11.13) ± 0.93
13.99 (10.80) ± 0.90
14.01 (10.80) ± 0.92
14.23 (10.96) ± 0.97
13.97 (10.74) ± 0.97
13.88 (10.65) ± 1.00
13.89 (10.63) ± 1.04
13.85 (10.57) ± 1.07
13.78 (10.48) ± 1.11
13.65 (10.35) ± 1.15
13.71 (10.35) ± 1.23
13.95 (10.48) ± 1.33
14.10 (10.57) ± 1.00
14.06 (10.58) ± 1.06
14.16 (10.71) ± 1.14
14.12 (10.76) ± 1.22
14.01 (10.76) ± 1.31
13.79 (10.64) ± 1.39
13.49 (10.50) ± 1.24
13.23 (10.37) ± 1.35
13.17 (10.38) ± 1.32
12.96 (10.25) ± 1.55
12.13 (9.56) ± 1.46
11.68 (9.21) ± 1.66
10.96 (8.64) ± 1.65
9.15 (7.32) ± 1.29

hχ2 /NDF i(hNDF i)
0.79 (15.0)
0.77 (14.8)
0.61 (14.2)
0.53 (13.8)
0.78 (13.1)
0.63 (12.9)
0.73 (12.7)
0.82 (12.5)
0.64 (12.0)
0.59 (11.8)
0.60 (11.6)
0.59 (11.1)
0.56 (10.5)
0.48 (9.9)
0.46 (9.5)
0.52 (8.9)
0.54 (8.5)
0.56 (8.1)
0.66 (7.6)
0.66 (7.1)
0.63 (6.6)
0.53 (6.3)
0.51 (5.9)
0.47 (5.5)
0.50 (5.1)
0.56 (4.7)
0.45 (4.3)
0.44 (3.8)
0.54 (3.3)
0.41 (2.7)δhkc i(dead)
±0.33
±0.45
±0.46
±0.47
±0.44
±0.53
±0.51
±0.41
±0.41
±0.45
±0.43
±0.34
±0.33
±0.38
±0.37
±0.40
±0.41
±0.41
±0.34
±0.38
±0.36
±0.40
±0.40
±0.46
±0.47
±0.38
±0.29
±0.15
±0.14
±0.16

δhkc i(f ake)
±0.04
±0.04
±0.03
±0.04
±0.04
±0.04
±0.05
±0.04
±0.04
±0.04
±0.04
±0.05
±0.06
±0.06
±0.05
±0.05
±0.06
±0.06
±0.06
±0.06
±0.06
±0.06
±0.07
±0.07
±0.08
±0.10
±0.10
±0.12
±0.14
±0.09

δhkc i(total)
±0.88
±0.94
±0.97
±0.99
±1.03
±1.05
±1.06
±1.05
±1.06
±1.10
±1.12
±1.12
±1.16
±1.22
±1.28
±1.39
±1.09
±1.15
±1.19
±1.28
±1.36
±1.45
±1.30
±1.43
±1.40
±1.60
±1.50
±1.67
±1.66
±1.31

Bibliography[1] See, e.g. J. A. Peacock, “Cosmological physics,” Cambridge, Univ. Pr. (1999).
[2] See, e.g. W. N. Cottingham and D. A. Greenwood, “AN INTRODUCTIONTO NUCLEAR PHYSICS,” Cambridge Univ. Pr. (1986).
[3] See, e.g. Y. Nagashima, “PROGRESSIONS OF HIGH ENERGY PHYSICS,”
Asakura Shoten (1999).
[4] See, e.g. W. Gebhardt and U. Krey, “Phase transition and critical phenomena,” Friedr. Vieweg & Shon, Braunschweig/Wiesbaden (1980). S. Komura,
Yoshioka Shoten (1992).
[5] V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
[6] See, e.g. T. Muta, “Foundations of quantum chromodynamics. Second edition,” World Sci. Lect. Notes Phys. 57, 1 (1998).
[7] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).
[8] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961).
[9] See, e.g. C. Y. Wong, “Introduction to high-energy heavy ion collisions,” Singapore, World Sci. (1994).
[10] See, e.g. T. Kanki, “Quark Gluon Plasma,” Maruzen (1992).
[11] See, e.g. K. Yagi, T. Hatsuda and Y. Miake, “Quark-Gluon Plasma: From
Big Bang To Little Bang,” Cambridge, Univ. Pr. (2005).
[12] N. Cabibbo and G. Parisi, Phys. Lett. B 59, 67 (1975).
[13] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys.
Rev. D 9, 3471 (1974).
[14] K. Rajagopal, Acta Phys. Polon. B 31, 3021 (2000).
[15] F. Karsch, Lect. Notes Phys. 583, 209 (2002).
109

110

BIBLIOGRAPHY[16] J. B. Kogut, M. Stone, H. W. Wyld, W. R. Gibbs, J. Shigemitsu,
S. H. Shenker and D. K. Sinclair, Phys. Rev. Lett. 50, 393 (1983).
[17] S. Muroya, A. Nakamura, C. Nonaka and T. Takaishi, Prog. Theor. Phys.
110, 615 (2003).
[18] H. Fujii, Phys. Rev. D 67, 094018 (2003).
[19] F. R. Brown et al., Phys. Rev. Lett. 65, 2491 (1990).
[20] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B 643, 46 (2006).
[21] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. Lett. 81,
4816 (1998).
[22] M. A. Stephanov, Prog. Theor. Phys. Suppl. 153, 139 (2004) [Int. J. Mod.
Phys. A 20, 4387 (2005)].
[23] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002).
[24] S. Ejiri, C. R. Allton, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann
and C. Schmidt, Prog. Theor. Phys. Suppl. 153, 118 (2004).
[25] O. Scavenius, A. Mocsy, I. N. Mishustin and D. H. Rischke, Phys. Rev. C
64, 045202 (2001).
[26] M. Asakawa and K. Yazaki, Nucl. Phys. A 504, 668 (1989).
[27] J. Berges and K. Rajagopal, Nucl. Phys. B 538, 215 (1999).
[28] A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto and G. Pettini, Phys.
Rev. D 41, 1610 (1990).
[29] M. A. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov and J. J. M. Verbaarschot, Phys. Rev. D 58, 096007 (1998).
[30] Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003).
[31] N. G. Antoniou and A. S. Kapoyannis, Phys. Lett. B 563, 165 (2003).
[32] J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
[33] L. Ahle et al. [E-802 Collaboration], Phys. Lett. B 332, 258 (1994).
[34] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 87, 052301 (2001).
[35] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 71, 034908 (2005).
[36] S. Margetis et al. [NA49 Collaboration], Phys. Rev. Lett. 75 (1995) 3814.

BIBLIOGRAPHY

111[37] K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005).
[38] R. S. Hayano and T. Hatsuda, arXiv:0812.1702 [nucl-ex].
[39] T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986).
[40] K. Adcox et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 469(2003).
[41] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 78, 044902 (2008).
[42] J. H. Vignos and H. A. Fairbank, Phys. Rev. Lett. 6, 265 (1961).
[43] W. M. Fairbank, M. J. Buckingam, Int. Conf. on Low Temp. Phys. (1957).
[44] See, e.g. W. M. Fairbank, C. K. Kellers, “Critical Phenimena,” National
Bureau of Standards, Washington D. C. (1965).
[45] M. E. Fisher, Phys. Rev. Lett. 16, 11 (1966).
[46] R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).
[47] R. B. Griffiths, Phys. Rev. Lett. 24, 1479 (1970).
[48] K. G. Wilson, Phys. Rev. B 4, 3174 (1971).
[49] K. G. Wilson, Phys. Rev. B 4, 3184 (1971).
[50] See, e.g. L. D. Landau, E. M. Lifshitz, ”Vol. 5, Statisitical Physics, Third
edition, Part 1,” Pegamon Press, Oxford (1980).
[51] L. S. Ornstein and F. Zernike, Proc. Akad. Sci. 15, 54 (1912).
[52] L. S. Ornstein and F. Zernike, Proc. Akad. Sci. 17, 793 (1914).
[53] See, e.g. H. Nishimori, “Statistical phycsics of phase transition and criticalphenomena,” Baifukan, (2005).
[54] D. J. Scalapino and R. L. Sugar, Phys. Rev. D 8, 2284 (1973).
[55] N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos, A. I. Karanikas and
C. N. Ktorides, Nucl. Phys. A 693, 799 (2001).
[56] N. G. Antoniou, F. K. Diakonos, A. S. Kapoyannis and K. S. Kousouris,
Phys. Rev. Lett. 97, 032002 (2006).
[57] N. G. Antoniou, C. N. Ktorides, I. S. Mistakidis and F. K. Diakonos, Eur.
Phys. J. C 4, 513 (1998).

112

BIBLIOGRAPHY[58] N. Sasaki, O. Miyamura, S. Muroya and C. Nonaka, Europhys. Lett. 54, 38(2001).
[59] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D 60, 114028(1999).
[60] C. Nonaka and M. Asakawa, Phys. Rev. C 71, 044904 (2005).
[61] E. A. De Wolf, I. M. Dremin and W. Kittel, Phys. Rept. 270, 1 (1996).
[62] G. J. Alner et al. [UA5 Collaboration], Phys. Lett. B 160, 193 (1985).
[63] T. Abbott et al. [E-802 Collaboration], Phys. Rev. C 52, 2663 (1995).
[64] P. Abreu et al. [DELPHI Collaboration], Z. Phys. C 56, 63 (1992).
[65] P. Carruthers and C. C. Shih, Phys. Lett. B 165, 209 (1985).
[66] A. Bialas and R. Peschanski, Nucl. Phys. B 273, 703 (1986).
[67] W. A. Zajc, Phys. Lett. B 175, 219 (1986).
[68] A. Bialas and R. Peschanski, Nucl. Phys. B 308, 857 (1988).
[69] P. Carruthers, E. M. Friedlander, C. C. Shih and R. M. Weiner, Phys. Lett.
B 222, 487 (1989).
[70] P. Carruthers and I. Sarcevic, Phys. Rev. Lett. 63, 1562 (1989).
[71] P. Carruthers, H. C. Eggers and I. Sarcevic, Phys. Rev. C 44, 1629 (1991).
[72] A. Giovannini and L. Van Hove, Z. Phys. C 30, 391 (1986).
[73] M. J. Tannenbaum and J. Kang [E802 Collaboration.], Nucl. Phys. A 566,
423C (1994).
[74] H. Nakamura and R. Seki, Phys. Rev. C 66, 024902 (2002).
[75] M. Allen et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 549(2003).
[76] S. H. Aronson et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499,
480 (2003).
[77] K. Adcox et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 489(2003).
[78] M. Aizawa et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 508(2003).

BIBLIOGRAPHY

113[79] L. Aphecetche et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499,
521 (2003).
[80] S. S. Adler et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 560(2003).
[81] S. S. Adler et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 593(2003).
[82] C. Adler, A. Denisov, E. Garcia, M. J. Murray, H. Strobele and S. White,
Nucl. Instrum. Meth. A 470, 488 (2001).
[83] K. Ikematsu et al., Nucl. Instrum. Meth. A 411, 238 (1998)[84] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 69, 034909 (2004).
[85] A. Kiyomichi, Ph. D. thesis, Department of Physics, University of Tsukuba(2005).
[86] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072301(2003).
[87] H. Torii, Ph. D. thesis, Department of Physics, Kyoto University (2004).
[88] Computer code GEANT 3.21, CERN, Genva, (1993).
[89] X. N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 (1991).
[90] H. R. Collard, L. R. B. Elton, R. Hofstadter, “Nuclear Radii” LandoltBrnstein, Vol. 2, Springer-Verlag, (1967).
[91] M. Chiu, A. Denisov, E. Garcia, J. Katzy and S. White, Phys. Rev. Lett.
89, 12302 (2002).
[92] K. Tokushuku et al., Phys. Lett. B 235, 245 (1990).
[93] H. Pi, Comput. Phys. Commun. 71, 173 (1992).
[94] F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).
[95] M. J. Tannenbaum, E802 Internal Rept. No. E-802-MEM-58 (1993).
[96] G. Baym, B. Blattel, L. L. Frankfurt, H. Heiselberg and M. Strikman, Phys.
Rev. C 52, 1604 (1995).
[97] Y. Akiba et al. [E802 Collaboration], Phys. Rev. C 56, 1544 (1997).
[98] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 93, 152302(2004).

114

BIBLIOGRAPHY[99] A. Enokizono, Ph. D. thesis, Department of Physical Science, Hiroshima
University (2004).
[100] M. Rybczynski et al. [NA49 Collaboration], J. Phys. Conf. Ser. 5, 74 (2005).
[101] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 75, 064904 (2007).
[102] V. P. Konchakovski, S. Haussler, M. I. Gorenstein, E. L. Bratkovskaya,
M. Bleicher and H. Stoecker, Phys. Rev. C 73, 034902 (2006).
[103] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 182301(2003).
[104] B. B. Back et al. [PHOBOS Collaboration], Phys. Rev. C 72, 051901 (2005).
[105] E. B. Johnson, AIP Conf. Proc. 842, 137 (2006).
[106] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 98, 062301 (2007).
[107] J. Adams et al. [STAR Collaborations], arXiv:nucl-ex/0601042.
[108] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 94, 232301(2005).
[109] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 96, 032301(2006).
[110] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 98, 172301(2007).
[111] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 98, 162301(2007).
[112] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. C 69, 034910 (2004).

公表論文(1) Measurement of density correlations in pseudorapidity via charged
particle multiplicity fluctuations in Au+Au collisions at s NN =200 GeV
S. S. Adler et al., (別紙、共著者リスト①参照)
Physical Review C 76, 034903 (2007).

参考論文(1) Systematic studies of the centrality and s NN dependence of thedET / dη and dN ch / dη in heavy ion collisions at midrapidity
S. S. Adler et al, (別紙、共著者リスト②参照)
Physical Review C 71, 034908 (2005).

(2) Charged hadron multiplicity fluctuations in Au+Au and Cu+Cu collisions
from s NN =22.5 to 200GeV
A. Adare et al, (別紙、共著者リスト③参照)
Physical Review C 78, 044902 (2008).

(3) PHENIX inner detectors
M. Allen et al., (別紙、共著者リスト④参照)
Nuclear Instruments and Methods in Physics Research A 499, 549 (2003).