Systematic Studies of Elliptic Flow Measurements in Au+Au Collisions at \sqrt{s_{{}_{NN}}} = 200 GeV

Systematic Studies of Elliptic Flow Measurements
in Au+Au Collisions at = 200 GeV

S. Afanasiev Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    C. Aidala Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    N.N. Ajitanand Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    Y. Akiba RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    J. Alexander Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    A. Al-Jamel New Mexico State University, Las Cruces, NM 88003, U.S.    K. Aoki Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    L. Aphecetche SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Université de Nantes) BP 20722 - 44307, Nantes, France    R. Armendariz New Mexico State University, Las Cruces, NM 88003, U.S.    S.H. Aronson Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    R. Averbeck Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    T.C. Awes Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    B. Azmoun Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    V. Babintsev IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    A. Baldisseri Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    K.N. Barish University of California - Riverside, Riverside, CA 92521, U.S.    P.D. Barnes Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    B. Bassalleck University of New Mexico, Albuquerque, NM 87131, U.S.    S. Bathe University of California - Riverside, Riverside, CA 92521, U.S.    S. Batsouli Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    V. Baublis PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    F. Bauer University of California - Riverside, Riverside, CA 92521, U.S.    A. Bazilevsky Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    S. Belikov Deceased Brookhaven National Laboratory, Upton, NY 11973-5000, U.S. Iowa State University, Ames, IA 50011, U.S.    R. Bennett Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    Y. Berdnikov Saint Petersburg State Polytechnic University, St. Petersburg, Russia    M.T. Bjorndal Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    J.G. Boissevain Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    H. Borel Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    K. Boyle Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    M.L. Brooks Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    D.S. Brown New Mexico State University, Las Cruces, NM 88003, U.S.    D. Bucher Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    H. Buesching Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    V. Bumazhnov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    G. Bunce Brookhaven National Laboratory, Upton, NY 11973-5000, U.S. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    J.M. Burward-Hoy Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    S. Butsyk Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    S. Campbell Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    J.-S. Chai KAERI, Cyclotron Application Laboratory, Seoul, Korea    S. Chernichenko IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    J. Chiba KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan    C.Y. Chi Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    M. Chiu Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    I.J. Choi Yonsei University, IPAP, Seoul 120-749, Korea    T. Chujo Vanderbilt University, Nashville, TN 37235, U.S.    V. Cianciolo Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    C.R. Cleven Georgia State University, Atlanta, GA 30303, U.S.    Y. Cobigo Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    B.A. Cole Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    M.P. Comets IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    P. Constantin Iowa State University, Ames, IA 50011, U.S.    M. Csanád ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary    T. Csörgő KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences (MTA KFKI RMKI), H-1525 Budapest 114, POBox 49, Budapest, Hungary    T. Dahms Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    K. Das Florida State University, Tallahassee, FL 32306, U.S.    G. David Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    H. Delagrange SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Université de Nantes) BP 20722 - 44307, Nantes, France    A. Denisov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    D. d’Enterria Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    A. Deshpande RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S. Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    E.J. Desmond Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    O. Dietzsch Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil    A. Dion Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    J.L. Drachenberg Abilene Christian University, Abilene, TX 79699, U.S.    O. Drapier Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    A. Drees Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    A.K. Dubey Weizmann Institute, Rehovot 76100, Israel    A. Durum IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    V. Dzhordzhadze University of Tennessee, Knoxville, TN 37996, U.S.    Y.V. Efremenko Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    J. Egdemir Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    A. Enokizono Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    H. En’yo RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    B. Espagnon IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    S. Esumi Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    D.E. Fields University of New Mexico, Albuquerque, NM 87131, U.S. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    F. Fleuret Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    S.L. Fokin Russian Research Center “Kurchatov Institute”, Moscow, Russia    B. Forestier LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France    Z. Fraenkel Deceased Weizmann Institute, Rehovot 76100, Israel    J.E. Frantz Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    A. Franz Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A.D. Frawley Florida State University, Tallahassee, FL 32306, U.S.    Y. Fukao Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S.-Y. Fung University of California - Riverside, Riverside, CA 92521, U.S.    S. Gadrat LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France    F. Gastineau SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Université de Nantes) BP 20722 - 44307, Nantes, France    M. Germain SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Université de Nantes) BP 20722 - 44307, Nantes, France    A. Glenn University of Tennessee, Knoxville, TN 37996, U.S.    M. Gonin Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    J. Gosset Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    Y. Goto RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    R. Granier de Cassagnac Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    N. Grau Iowa State University, Ames, IA 50011, U.S.    S.V. Greene Vanderbilt University, Nashville, TN 37235, U.S.    M. Grosse Perdekamp University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    T. Gunji Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    H.-Å. Gustafsson Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    T. Hachiya Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    A. Hadj Henni SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Université de Nantes) BP 20722 - 44307, Nantes, France    J.S. Haggerty Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    M.N. Hagiwara Abilene Christian University, Abilene, TX 79699, U.S.    H. Hamagaki Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    H. Harada Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    E.P. Hartouni Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.    K. Haruna Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    M. Harvey Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    E. Haslum Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    K. Hasuko RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    R. Hayano Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    M. Heffner Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.    T.K. Hemmick Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    J.M. Heuser RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    X. He Georgia State University, Atlanta, GA 30303, U.S.    H. Hiejima University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.    J.C. Hill Iowa State University, Ames, IA 50011, U.S.    R. Hobbs University of New Mexico, Albuquerque, NM 87131, U.S.    M. Holmes Vanderbilt University, Nashville, TN 37235, U.S.    W. Holzmann Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    K. Homma Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    B. Hong Korea University, Seoul, 136-701, Korea    T. Horaguchi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan    M.G. Hur KAERI, Cyclotron Application Laboratory, Seoul, Korea    T. Ichihara RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    K. Imai Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    M. Inaba Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    D. Isenhower Abilene Christian University, Abilene, TX 79699, U.S.    L. Isenhower Abilene Christian University, Abilene, TX 79699, U.S.    M. Ishihara RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    T. Isobe Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    M. Issah Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    A. Isupov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    B.V. Jacak jacak@skipper.physics.sunysb.edu Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    J. Jia Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    J. Jin Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    O. Jinnouchi RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    B.M. Johnson Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    K.S. Joo Myongji University, Yongin, Kyonggido 449-728, Korea    D. Jouan IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    F. Kajihara Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S. Kametani Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Waseda University, Advanced Research Institute for Science and Engineering, 17 Kikui-cho, Shinjuku-ku, Tokyo 162-0044, Japan    N. Kamihara RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan    M. Kaneta RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    J.H. Kang Yonsei University, IPAP, Seoul 120-749, Korea    T. Kawagishi Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    A.V. Kazantsev Russian Research Center “Kurchatov Institute”, Moscow, Russia    S. Kelly University of Colorado, Boulder, CO 80309, U.S.    A. Khanzadeev PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    D.J. Kim Yonsei University, IPAP, Seoul 120-749, Korea    E. Kim System Electronics Laboratory, Seoul National University, Seoul, Korea    Y.-S. Kim KAERI, Cyclotron Application Laboratory, Seoul, Korea    E. Kinney University of Colorado, Boulder, CO 80309, U.S.    A. Kiss ELTE, Eötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, Hungary    E. Kistenev Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A. Kiyomichi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    C. Klein-Boesing Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    L. Kochenda PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    V. Kochetkov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    B. Komkov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    M. Konno Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    D. Kotchetkov University of California - Riverside, Riverside, CA 92521, U.S.    A. Kozlov Weizmann Institute, Rehovot 76100, Israel    P.J. Kroon Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    G.J. Kunde Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    N. Kurihara Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    K. Kurita Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    M.J. Kweon Korea University, Seoul, 136-701, Korea    Y. Kwon Yonsei University, IPAP, Seoul 120-749, Korea    G.S. Kyle New Mexico State University, Las Cruces, NM 88003, U.S.    R. Lacey Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    J.G. Lajoie Iowa State University, Ames, IA 50011, U.S.    A. Lebedev Iowa State University, Ames, IA 50011, U.S.    Y. Le Bornec IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    S. Leckey Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    D.M. Lee Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    M.K. Lee Yonsei University, IPAP, Seoul 120-749, Korea    M.J. Leitch Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    M.A.L. Leite Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil    H. Lim System Electronics Laboratory, Seoul National University, Seoul, Korea    A. Litvinenko Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    M.X. Liu Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    X.H. Li University of California - Riverside, Riverside, CA 92521, U.S.    C.F. Maguire Vanderbilt University, Nashville, TN 37235, U.S.    Y.I. Makdisi Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A. Malakhov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    M.D. Malik University of New Mexico, Albuquerque, NM 87131, U.S.    V.I. Manko Russian Research Center “Kurchatov Institute”, Moscow, Russia    H. Masui Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    F. Matathias Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    M.C. McCain University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.    P.L. McGaughey Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    Y. Miake Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    A. Mignerey University of Maryland, College Park, MD 20742, U.S.    T.E. Miller Vanderbilt University, Nashville, TN 37235, U.S.    A. Milov Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    S. Mioduszewski Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    G.C. Mishra Georgia State University, Atlanta, GA 30303, U.S.    J.T. Mitchell Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    D.P. Morrison Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    J.M. Moss Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    T.V. Moukhanova Russian Research Center “Kurchatov Institute”, Moscow, Russia    D. Mukhopadhyay Vanderbilt University, Nashville, TN 37235, U.S.    J. Murata Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S. Nagamiya KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan    Y. Nagata Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    J.L. Nagle University of Colorado, Boulder, CO 80309, U.S.    M. Naglis Weizmann Institute, Rehovot 76100, Israel    T. Nakamura Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    J. Newby Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.    M. Nguyen Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    B.E. Norman Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    R. Nouicer Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A.S. Nyanin Russian Research Center “Kurchatov Institute”, Moscow, Russia    J. Nystrand Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    E. O’Brien Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    C.A. Ogilvie Iowa State University, Ames, IA 50011, U.S.    H. Ohnishi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    I.D. Ojha Vanderbilt University, Nashville, TN 37235, U.S.    H. Okada Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    K. Okada RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    O.O. Omiwade Abilene Christian University, Abilene, TX 79699, U.S.    A. Oskarsson Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    I. Otterlund Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    K. Ozawa Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan    R. Pak Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    D. Pal Vanderbilt University, Nashville, TN 37235, U.S.    A.P.T. Palounek Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    V. Pantuev Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    V. Papavassiliou New Mexico State University, Las Cruces, NM 88003, U.S.    J. Park System Electronics Laboratory, Seoul National University, Seoul, Korea    W.J. Park Korea University, Seoul, 136-701, Korea    S.F. Pate New Mexico State University, Las Cruces, NM 88003, U.S.    H. Pei Iowa State University, Ames, IA 50011, U.S.    J.-C. Peng University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.    H. Pereira Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    V. Peresedov Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    D.Yu. Peressounko Russian Research Center “Kurchatov Institute”, Moscow, Russia    C. Pinkenburg Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    R.P. Pisani Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    M.L. Purschke Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A.K. Purwar Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    H. Qu Georgia State University, Atlanta, GA 30303, U.S.    J. Rak Iowa State University, Ames, IA 50011, U.S.    I. Ravinovich Weizmann Institute, Rehovot 76100, Israel    K.F. Read Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S. University of Tennessee, Knoxville, TN 37996, U.S.    M. Reuter Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    K. Reygers Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    V. Riabov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    Y. Riabov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    G. Roche LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France    A. Romana Deceased Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    M. Rosati Iowa State University, Ames, IA 50011, U.S.    S.S.E. Rosendahl Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    P. Rosnet LPC, Université Blaise Pascal, CNRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France    P. Rukoyatkin Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia    V.L. Rykov RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S.S. Ryu Yonsei University, IPAP, Seoul 120-749, Korea    B. Sahlmueller Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    N. Saito Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    T. Sakaguchi Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Waseda University, Advanced Research Institute for Science and Engineering, 17 Kikui-cho, Shinjuku-ku, Tokyo 162-0044, Japan    S. Sakai Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    V. Samsonov PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    H.D. Sato Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S. Sato Brookhaven National Laboratory, Upton, NY 11973-5000, U.S. KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    S. Sawada KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan    V. Semenov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    R. Seto University of California - Riverside, Riverside, CA 92521, U.S.    D. Sharma Weizmann Institute, Rehovot 76100, Israel    T.K. Shea Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    I. Shein IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    T.-A. Shibata RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan    K. Shigaki Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    M. Shimomura Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    T. Shohjoh Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    K. Shoji Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    A. Sickles Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, U.S.    C.L. Silva Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil    D. Silvermyr Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    K.S. Sim Korea University, Seoul, 136-701, Korea    C.P. Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India    V. Singh Department of Physics, Banaras Hindu University, Varanasi 221005, India    S. Skutnik Iowa State University, Ames, IA 50011, U.S.    W.C. Smith Abilene Christian University, Abilene, TX 79699, U.S.    A. Soldatov IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    R.A. Soltz Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.    W.E. Sondheim Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    S.P. Sorensen University of Tennessee, Knoxville, TN 37996, U.S.    I.V. Sourikova Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    F. Staley Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France    P.W. Stankus Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    E. Stenlund Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    M. Stepanov New Mexico State University, Las Cruces, NM 88003, U.S.    A. Ster KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences (MTA KFKI RMKI), H-1525 Budapest 114, POBox 49, Budapest, Hungary    S.P. Stoll Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    T. Sugitate Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan    C. Suire IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    J.P. Sullivan Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    J. Sziklai KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences (MTA KFKI RMKI), H-1525 Budapest 114, POBox 49, Budapest, Hungary    T. Tabaru RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    S. Takagi Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan    E.M. Takagui Universidade de São Paulo, Instituto de Física, Caixa Postal 66318, São Paulo CEP05315-970, Brazil    A. Taketani RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    K.H. Tanaka KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan    Y. Tanaka Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan    K. Tanida RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    M.J. Tannenbaum Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A. Taranenko Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, U.S.    P. Tarján Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary    T.L. Thomas University of New Mexico, Albuquerque, NM 87131, U.S.    M. Togawa Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    J. Tojo RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    H. Torii RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    R.S. Towell Abilene Christian University, Abilene, TX 79699, U.S.    V-N. Tram Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France    I. Tserruya Weizmann Institute, Rehovot 76100, Israel    Y. Tsuchimoto Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    S.K. Tuli Department of Physics, Banaras Hindu University, Varanasi 221005, India    H. Tydesjö Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden    N. Tyurin IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    C. Vale Iowa State University, Ames, IA 50011, U.S.    H. Valle Vanderbilt University, Nashville, TN 37235, U.S.    H.W. van Hecke Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.    J. Velkovska Vanderbilt University, Nashville, TN 37235, U.S.    R. Vertesi Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary    A.A. Vinogradov Russian Research Center “Kurchatov Institute”, Moscow, Russia    E. Vznuzdaev PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia    M. Wagner Kyoto University, Kyoto 606-8502, Japan RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN    X.R. Wang New Mexico State University, Las Cruces, NM 88003, U.S.    Y. Watanabe RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    J. Wessels Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    S.N. White Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    N. Willis IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France    D. Winter Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    C.L. Woody Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    M. Wysocki University of Colorado, Boulder, CO 80309, U.S.    W. Xie University of California - Riverside, Riverside, CA 92521, U.S. RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    A. Yanovich IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia    S. Yokkaichi RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, JAPAN RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, U.S.    G.R. Young Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.    I. Younus University of New Mexico, Albuquerque, NM 87131, U.S.    I.E. Yushmanov Russian Research Center “Kurchatov Institute”, Moscow, Russia    W.A. Zajc Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    O. Zaudtke Institut für Kernphysik, University of Muenster, D-48149 Muenster, Germany    C. Zhang Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, U.S.    J. Zimányi Deceased KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences (MTA KFKI RMKI), H-1525 Budapest 114, POBox 49, Budapest, Hungary    L. Zolin Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Abstract

We present inclusive charged hadron elliptic flow () measured over the pseudorapidity range 0.35 in Au+Au collisions at = 200 GeV. Results for are presented over a broad range of transverse momentum ( = 0.2–8.0 GeV/) and centrality (0–60%). In order to study non-flow effects that are not correlated with the reaction plane, as well as the fluctuations of , we compare two different analysis methods: (1) event plane method from two independent sub-detectors at forward ( = 3.1–3.9) and beam () pseudorapidities and (2) two-particle cumulant method extracted using correlations between particles detected at midrapidity. The two event-plane results are consistent within systematic uncertainties over the measured and in centrality 0–40%. There is at most 20% difference of the between the two event plane methods in peripheral (40–60%) collisions. The comparisons between the two-particle cumulant results and the standard event plane measurements are discussed.

pacs:
25.75.Ld

PHENIX Collaboration

I Introduction

Collisions of Au+Au nuclei at the Relativistic Heavy Ion Collider (RHIC) produce matter at very high energy density Arsene et al. (2005); Adcox et al. (2005); Back et al. (2005); Adams et al. (2005a). The dynamical evolution of this hot and dense medium reflects its state and the degrees of freedom that govern the different stages it undergoes Gyulassy and McLerran (2005); Muller (); Shuryak (2005). Azimuthal anisotropy measurements serve as a probe of the degree of thermalization, transport coefficients and the equation of state (EOS) Ollitrault (1992); Kolb et al. (2001); Hirano and Nara (2004) of the produced medium.

Azimuthal correlation measurements in Au+Au collisions at RHIC have been shown to consist of a mixture of jet and harmonic contributions Ajitanand (2003); Chiu (2003); Adler et al. (2003a, 2005). Jet contributions are found to be relatively small for 2.0 GeV/, with away-side jet yields strongly suppressed Adler et al. (2003a). Significant modifications to the away-side jet topology have also been reported Adams et al. (2005b); Adler et al. (2006a); Adare et al. (2008). The harmonic contributions are typically characterized by the Fourier coefficients,

(1)

where represents the azimuthal emission angle of a charged hadron and is the azimuth of the reaction plane defined as containing both the direction of the impact parameter vector and the beam axis. The brackets denote statistical averaging over particles and events. The first two harmonics and are referred to as directed and elliptic flow, respectively.

It has been found that at low (  GeV/) the magnitude and trends of are under-predicted by hadronic cascade models supplemented with string dynamics Bleicher and Stoecker (2002), but are well reproduced by models which either incorporate hydrodynamic flow Shuryak (2005); Kolb et al. (2001) with a first order phase transition and rapid thermalization, fm/c Adler et al. (2003b), or use a quasi-particle ansatz but include more than just 2-to-2 interactions Xu et al. (2008).

The mass dependence of as a function of has been studied using identified baryons and mesons Adler et al. (2003b); Adams et al. (2004a) and empirical scaling of elliptic flow per constituent quark was observed when the signal and the of the hadron were divided by the number of constituent quarks ( = 2 for mesons, 3 for baryons). This scaling is most clearly observed by plotting the data as a function of transverse kinetic energy   Adare et al. (2007), where and denote the transverse mass and mass of the particle, respectively. A recent study Huang (2008) finds that the constituent quark scaling holds up to   GeV. This indicates partonic, rather than hadronic flow, and suggests that the bulk matter collectivity develops before hadronization takes place Molnár and Voloshin (2003); Fries et al. (2003); Greco et al. (2003). Results for the of the meson further validate the observation of partonic collectivity. The is not expected to be affected by hadronic interactions in the late stages of the medium evolution, due to its small interaction cross section with non-strange hadrons  Shor (1985).

All of the measurements referenced above were performed using the event plane method Poskanzer and Voloshin (1998). In PHENIX the event plane was determined at forward and backward pseudorapidities ( = 3.1–3.9) with the assumption that correlations induced by elliptic flow dominate over all other non-flow correlations Adler et al. (2003b). Non-flow correlations are those which are not correlated with the reaction plane. Common sources of non-flow correlations include jets, the near-side ridge, quantum correlations and resonance decays. Simulation studies Adler et al. (2003b); Jia (2007) have shown that the correlations from jets and dijets become negligible when the rapidity separation between the particles and the event plane is greater than three units. Thus we expect that the event plane at forward pseudorapidities = 3.1–3.9 in the PHENIX experiment would not have significant jet-correlation with particles measured within the PHENIX central arm spectrometer covering the pseudorapidity window . PHOBOS has observed that in central Au+Au collisions there is a ridge of particles Alver et al. () that are correlated in azimuthal angle with a high- particle and that this ridge of particles extends to (for mid-rapidity triggers). The ridge could produce a non-flow correlation that we can provide information by using our measurements that are made with different techniques and at different rapidities.

Event-by-event flow fluctuations can also affect the magnitude of the extracted flow signal Sorensen (2008). This occurs because the event plane at forward pseudorapidities is reconstructed using particles from participant nucleons whose positions fluctuate event-by-event. Assuming that fluctuates according to a Gaussian distribution, the fluctuation is proportional to the fluctuation of the initial geometry. This effect scales as , where denotes the number of participant nucleons. The difference between values obtained from different methods can be quantitatively understood in terms of non-flow and fluctuation effects Ollitrault et al. ().

Hence in this paper we will compare the results from the event plane determined at two different pseudorapidities with the goal to investigate the sensitivity of to non-flow and fluctuation effects. Additionally, we extract the elliptic flow with the two-particle cumulant method, which is expected to have higher sensitivity to non-flow contributions to .

In this paper, we describe the PHENIX measurements of elliptic flow () at midrapidity () in Au+Au collisions at = 200 GeV obtained from a cumulant analysis of two-particle azimuthal correlations and the event plane method over a broad range of ( = 0.2–8 GeV/) and centrality (0–60%). The paper is organized as follows: Section II describes the PHENIX apparatus, with an emphasis on the detectors relevant to the presented results, as well as the track selections used in the analysis. Section III gives details of the event-plane and the cumulant methods as applied in PHENIX. Section IV discusses the systematic uncertainties of the event-plane and cumulant methods. The results from the two methods are reported in Section V. Section VI presents a comparison of results across different experiments and discussion. The values obtained from the different methods are tabulated in the Appendix.

Ii Experimental Analysis

ii.1 The PHENIX detector

Figure 1: PHENIX experimental layout in 2004. The top panel shows the PHENIX central arm spectrometers viewed along the beam axis. The bottom panel shows a side view of the PHENIX muon arm spectrometers.

The PHENIX detector consists of two central spectrometer arms at midrapidity that are designated East and West for their location relative to the interaction region, and two muon spectrometers at forward rapidity, similarly called North and South. A detailed description of the PHENIX detector can be found in Ref. Adcox et al. (2003). The layout of the PHENIX detector during data taking in 2004 is shown in Fig. 1. Each central spectrometer arm covers a pseudorapidity range of subtending degrees in azimuth and is designed to detect electrons, photons and charged hadrons. Charged particles are tracked by drift chambers (DC) positioned between 2.0 m and 2.4 m radially outward from the beam axis and layers of multi-wire proportional chambers with pad readout (two in the east arm and three in the west arm) PC1, PC2 and PC3 located at a radial distance of 2.4 m, 4.2 m and 5 m, respectively. Particle identification is provided by Ring Imaging erenkov counters (RICH), a time-of-flight scintillator wall (TOF), and two types of electromagnetic calorimeters (EMCAL), the lead scintillator (PbSc) and lead glass (PbGl).

The detectors used to characterize each event are the beam-beam counters (BBCs) Allen et al. (2003) and the zero-degree calorimeters (ZDCs) Adler et al. (2003c). These detectors are used to determine the time of the collision, the position of the collision vertex along the beam axis and the collision centrality and also provide the event trigger. In this analysis the BBCs are also used to determine the event plane. Each BBC is composed of 64 elements and a single BBC element consists of a one-inch diameter mesh dynode photomultiplier tube (PMT) mounted on a 3 cm long quartz radiator. The BBCs are installed on the north and south sides of the collision point along the beam axis at a distance of 144 cm from the center of the interaction region and surround the beam pipe. The BBC acceptance covers the pseudorapidity range and the full range of azimuthal angles.

The ZDCs are hadronic calorimeters located on both sides of the PHENIX detector. Each ZDC is mechanically subdivided into 3 identical modules of two interaction lengths. They cover a pseudorapidity range of and measure the energy of the spectator neutrons with a 20 GeV energy resolution Adler et al. (2003c). The shower maximum detectors (ZDC-SMDs) are scintillator strip hodoscopes between the first and second ZDC modules. This location approximately corresponds to the maximum of the hadronic shower. The horizontal coordinate is sampled by 7 scintillator strips of 15 mm width, while the vertical coordinate is sampled by 8 strips of 20 mm width. The active area of a ZDC-SMD is 105 mm 110 mm (horizontal vertical dimension). Scintillation light is delivered to a multichannel PMT M16 by wavelength-shifter fibers. The ZDC-SMD position resolution depends on the energy deposited in the scintillator. It varies from 3 mm when the number of particles exceeds 100, to 10 mm for a smaller number of particles.

ii.2 Event selection

Figure 2: The correlation between ZDC energy and BBC charge sum for Au + Au collisions at = 200 GeV. Solid lines represent the corresponding centrality boundaries up to 60% centrality bin.

For the analyses presented here we used approximately 850 10 minimum-bias triggered events. The minimum-bias trigger was defined by a coincidence between North and South BBC signals and an energy threshold of one neutron in the ZDCs. The events are selected offline to be within a -vertex of less than 30 cm from the nominal center of the PHENIX spectrometer. This selection corresponds to % of the 6.9 barn Au+Au inelastic cross section at = 200 GeV Miller et al. (2007). The event centrality was determined by correlating the charge detected in the BBCs with the energy measured in the ZDCs, as shown in Fig. 2.

A Glauber model Monte-Carlo simulation Glauber and Matthiae (1970); Adcox et al. (2001); Adler et al. (2004) that includes the responses of BBC and ZDC gives an estimate of the average number of participating nucleons for each centrality class. The simulation did not include fluctuations in the positions of the nucleons which give rise to eccentricity fluctuations. Table 1 lists the calculated values of for each centrality class.

Centrality
0–10%
10–20%
20–30%
30–40%
40–50%
50–60%
Table 1: Centrality classes and average number of participant nucleons obtained from a Glauber Monte-Carlo simulation of the BBC and ZDC responses for Au+Au collision at = 200 GeV. Each centrality class is expressed as a percentage of = 6.9 b inelastic cross section. Errors denote systematic uncertainties from the Glauber MC simulation.

ii.3 Track selection

Charged particle tracks are measured using information from the DC, PC1 and PC3 detectors and the -vertex from the BBC. The DC has 12 wire planes which are spaced at 0.6 cm intervals along the radial direction from the beam axis. Each wire provides a track position measurement, with better than 150 m spatial resolution in the azimuthal () direction. The PC1 provides a space point in the and beam directions, albeit with lower resolution. This space point and the vertex position help determine the three-dimensional momentum vector by providing the polar angle for charged tracks at the exit of the DC. Trajectories are confirmed by requiring matching hits at PC3 to reduce secondary background. Tracks are then projected back to the collision vertex through the magnetic field to determine the momentum  Mitchell et al. (2002). The momentum resolution is (GeV/). The momentum scale is known to 0.7%, as determined from the reconstructed proton mass using the TOF detector. Further details on track reconstruction and momentum determination can be found in Refs. Mitchell et al. (2002); Adler et al. (2004).

The tracks reconstructed by the DC which do not originate from the event vertex have been investigated as potential background to the charged particle measurement. The main background sources include secondary particles from decays and pairs from the conversion of photons in the material between the vertex and the DC Adler et al. (2004). Tracks are required to have a hit in the PC3, as well as in the EMCAL, within at most 2 of the expected hit location in both the azimuthal and beam directions. This cut reduces the background not originating in the direction of the vertex. In order to reduce the conversion background we further require tracks to have , where denotes the energy deposited in the EMCAL and is the transverse momentum of particles measured in the DC. Since most of the electrons from photon conversion are genuine low particles that were reconstructed as high particles, requiring a large deposit of energy in the EMCAL suppresses the electron background Adler et al. (2006b). We also require that there are no associated hits in the RICH. The RICH is filled with CO gas at atmospheric pressure and has a charged particle threshold to emit erenkov photons.

Iii Methods of azimuthal anisotropy measurement

In this section we introduce the methods for azimuthal anisotropy measurements as used in the PHENIX experiment. Section III.A describes the event plane method using the BBCs and ZDC-SMDs detectors and Sec. III.B describes the two-particle cumulant method.

iii.1 Event plane method

The event plane method Poskanzer and Voloshin (1998) uses the azimuthal anisotropy signal to estimate the angle of the reaction plane. The estimated reaction plane is called the “event plane” and is determined for each harmonic of the Fourier expansion of the azimuthal distribution. The event flow vector and azimuth of the event plane for -th harmonic of the azimuthal anisotropy can be expressed as

(2)
(3)
(4)

where denotes the number of particles used to determine the event plane, is the azimuthal angle of each particle, and is the weight chosen to optimize the event plane resolution. Once the event plane is determined, the elliptic flow can be extracted by correlating the azimuthal angle of emitted particles with the event plane

(5)

where is the azimuthal angle of tracks in the laboratory frame, is the -th order event plane and the brackets denote an average over all charged tracks and events. The denominator Res{} is the event plane resolution that corrects for the difference between the estimated event plane and true reaction plane .

In this paper the second-harmonic event planes were independently determined with two BBCs located at forward (BBC South, referred to as BBCS) and backward (BBC North, referred to as BBCN) pseudorapidities = 3.1–3.9 Adler et al. (2003b). The difference between the two independent event planes was used to estimate the event plane resolution. The planes were also combined to determine the event plane for the full event. A large pseudorapidity gap between the charged particles detected in the central arms and the event plane at the BBCs reduces the effect of possible non-flow contributions, especially those from dijets Jia (2007). The measured of hadrons in the central arms with respect to the combined second-harmonic BBC event plane will be denoted throughout this paper as {BBC}.

Two first-harmonic event planes were also determined using spectator neutrons at the two shower maximum detectors (ZDC-SMDs) that are sandwiched between the first and second modules of the ZDCs. Forward (ZDCS) and backward (ZDCN) SMDs which cover pseudorapidity 6.5 were used. The measured of hadrons in the central arms determined with respect to the first-harmonic ZDC-SMD event plane will be denoted as {ZDC-SMD}.

The pseudorapidity gap between the hadrons measured in the central arms and the ZDC-SMDs is larger than that for the BBCs which could cause a further reduction of non-flow contributions on {ZDC-SMD}. Since the ZDC-SMD measures spectator neutrons, the ZDC-SMD event plane should be insensitive to fluctuations in the participant event plane. Hence fluctuations in {ZDC-SMD} should be suppressed up to fluctuations in the spectator positions.

For completeness, two further event planes are defined 1) a combined event plane defined by the weighted average of event planes at the forward and backward pseudorapidities for both BBCs and ZDC-SMDs, and 2) an event plane found using tracks in the central arm. The event plane at the central arms (CNT) is only used to estimate the resolution of BBC and ZDC-SMD event planes by using three subevents combination of the ZDC-SMD, BBC and CNT.

iii.1.1 Event plane determination

To determine an event plane the contribution at each azimuthal angle needs to be appropriately weighted depending on the detector used. For the BBC we chose the weights to be the number of particles detected in each phototube, while for the ZDC-SMD the weights were based on the energy deposited in each of the SMD strips. For the CNT event plane the weight was taken to be proportional to up to 2 GeV/ and constant for 2 GeV/. For the CNT event plane we also adopted a unit weight () and found that the resulting CNT event plane resolution extracted by comparing the CNT event plane with the BBC and ZDC-SMD planes was nearly identical when using the -dependent or unit weights.

Corrections were performed to remove possible biases from the finite acceptance of the BBC and ZDC-SMD. In this analysis, we applied two corrections called the re-centering and shift methods. In the re-centering method, event flow vectors are shifted and normalized to a Gaussian distribution by using the mean and width of flow vectors;

(6)

This correction reduces the dependence of the event plane resolution on the laboratory angle. Most acceptance effects were removed by the application of the re-centering method. However, remaining small corrections were applied after re-centering using the shift method Poskanzer and Voloshin (1998), in which the reaction plane is shifted by defined by

(7)

where = 8 in this analysis. The shift ensures that is isotropic. When was reduced to , the difference in the extracted was negligible and thus we include no systematic uncertainty due to the choice of in our results.

Independent corrections were applied to each centrality selection in 5% increments and in 20 cm steps in -vertex in order to optimize the event plane resolution. The corrections were also done for each experimental run (the duration of a run is typically 1-3 hours) to minimize the possible time-dependent response of detectors.

Figure 3: Event plane distributions after applying all corrections for the ZDC-SMD (circles), BBC (triangles) and CNT (squares). The statistical error bars are smaller than the symbols. The distributions for the BBC and CNT event planes are scaled by 3/4 and 1/2 to improve visibility.

Figure 3 shows event plane distributions for a sub-sample of the entire data set. After all corrections are applied the event plane distributions are isotropic.

iii.1.2 Event plane resolution

The event plane resolution for was evaluated by both the two-subevents and three-subevents methods. In the two-subevents method the event plane resolution Poskanzer and Voloshin (1998) is expressed as

(8)

where , is the number of particles used to determine the event plane , is the modified Bessel function of the first kind and = 1 for the second harmonic BBC event plane. For the ZDC-SMD event plane the resolution is estimated with both = 1 or 2 in Eq. (8). We will discuss the difference between these estimates in Sec. IV.1.

To determine the event plane resolution we need to determine . Since the North and South BBCs have approximately the same coverage, the event plane resolution of each sub-detector is expected to be the same. The same is true for the North and South ZDC-SMDs. Thus, the subevent resolution for South and North event planes can be expressed as

(9)

where denotes the event plane determined by the South (North) BBC or ZDC-SMD. Once the subevent resolution is obtained from Eq. (9), one can calculate using Eq. (8). The for the full event can then be estimated by . This is then substituted into Eq. (8) to give the full event resolution. Since the multiplicity of the full event is twice as large as that of the subevent, is proportional to .

In the three-subevents method the resolution of each subevent is calculated by adding a reference event plane in Eq. (9):

(10)

where are the harmonics of the event plane for subevent A, B and C, respectively. The multiplicity of each subevent is not necessarily the same in Eq. (10).

The resolution of each sub-detector for the BBC and ZDC-SMD can be evaluated with the three-subevents method. For the BBC event plane the reference event plane is chosen to be the ZDC-SMD event plane and vice versa. We found that the agreement of the event plane resolutions for BBCS and BBCN is much better than 1%, while the ZDCS and ZDCN resolutions are comparable with each other within 2%.

Figure 4: Full-event resolutions for the ZDC-SMD (filled circles) and BBC (open diamonds) from the two-subevents method, Eq. (8), as a function of centrality in Au+Au at = 200 GeV. The dashed lines represent resolutions from the three-subevents method with the CNT event plane as a reference. Statistical errors are smaller than the symbols.

Figure 4 shows the full-event resolution as a function of centrality. The resolution of ZDC-SMD is much smaller than that of BBC because the resolution of the first-harmonic event plane is proportional to . The dashed lines are the resolutions obtained from the three-subevents method with the CNT event plane as the reference plane. For example, the BBC event plane resolution is estimated by substituting , , and in Eq. (10). By including the CNT event plane, the BBC resolution increases by about 3% compared to that of the two-subevents method. For the ZDC-SMD we observe the opposite effect, namely the resolution decreases by about 8%. In Sec. VI the resulting {BBC} and {ZDC-SMD}, corrected by the resolution obtained using the ZDC-BBC-CNT combination, will be compared to those with the resolution determined from South-North subevents. Table 2 summarizes the event plane resolutions.

Res{}
Centrality S-N ZDC-BBC-CNT
0–10% 0.2637 0.0003 0.272 0.003
10–20% 0.3809 0.0002 0.394 0.001
20–30% 0.3990 0.0002 0.4106 0.0008
30–40% 0.3634 0.0002 0.3759 0.0007
40–50% 0.2943 0.0003 0.3067 0.0007
50–60% 0.2106 0.0004 0.2240 0.0009
Res{}
Centrality S-N ZDC-BBC-CNT
0–10% 0.02 0.01 0.0223 0.0003
10–20% 0.059 0.003 0.0574 0.0002
20–30% 0.087 0.002 0.0818 0.0002
30–40% 0.100 0.002 0.0928 0.0002
40–50% 0.102 0.002 0.0920 0.0002
50–60% 0.100 0.002 0.0798 0.0003
Table 2: Event plane resolutions for centrality 0–60% at = 200 GeV. S-N denotes the resolutions estimated from South and North correlation of BBC and ZDC-SMD using Eq. (8) and (9), and resolutions for ZDC-BBC-CNT are estimated from Eq. (10). The errors are statistical only.

iii.1.3 Correlation of event planes

Figure 5: (a) Correlation of first harmonic event planes between forward and backward ZDC-SMDs (filled circles) and BBCs (open diamonds) as a function of centrality. (b) Correlation of first harmonic event planes between ZDC-SMDs and BBCs as a function of centrality, where filled (open) squares are the correlation for opposite side (same side) of subevents. Statistical errors are smaller than the symbols.

Figure 5 shows the correlation of two different event planes as a function of centrality. The first harmonic event plane correlation for South-North detector combinations is negative both for the ZDC-SMDs and the BBCs over all centrality bins, as shown in Fig. 5(a). This is due to the fact that is an odd function of . The magnitude of the ZDC-SMDs correlation is about a factor of two larger than that of the BBCs for midcentral collisions. This indicates that the magnitude of and/or the subevent multiplicity at higher pseudorapidities are larger compared to that at the BBC location, since the magnitude of the correlation is proportional to . Fig. 5(b) shows the correlation of the first harmonic event planes between BBC and ZDC-SMD. The same-side correlation is negative while the opposite-side correlation is positive, which shows that the particles detected at the BBCs (dominantly charged pions emitted from participant nucleons) have the opposite sign of compared to the spectator neutrons detected at the ZDCs-SMDs.

Figure 6: The correlation between the first harmonic ZDC-SMD and the second harmonic BBC event planes as a function of centrality. The dashed line shows the result obtained using Eq. (11). Statistical errors are smaller than the data symbols.

The correlation of the mixed harmonic event planes provides the sign of since the correlation is given by the expression Poskanzer and Voloshin (1998)

(11)

Three assumptions were made to obtain Eq. (11): (1) the BBC and ZDC-SMD are statistically independent, (2) the weak flow limit is applicable, and (3) the subevent multiplicity is equal in the North-South direction for the same detector type. Thus the sign of the correlation of the mixed harmonic event planes in Eq. (11) is determined by the term Res, which in turn determines the sign of measured at the BBC.

Figure 6 shows the mixed harmonic correlation of the ZDC-SMD and BBC event planes as a function of centrality. The approximations in Eq. (11) provide a good description of the magnitude of the measured correlation as shown by the dashed line. The correlation is positive over all centrality bins. This result indicates that the sign of at the BBC is positive.

iii.2 Cumulant method

In this section, we present the application of the cumulant method for azimuthal anisotropy measurements in PHENIX. This method uses cumulants of multiparticle correlations Borghini et al. (2001a, b) to extract the azimuthal anisotropy. The cumulant method has been successfully applied in several heavy-ion experiments utilizing detectors with full azimuthal coverage (NA49, STAR) Alt et al. (2003); Adler et al. (2002). Here, we describe the first application of the method for a detector with only partial azimuthal coverage. The cumulant method does not require the measurement of the reaction plane, instead the cumulants of multi-particle azimuthal correlations are related to the flow harmonics , where is the harmonic being evaluated. The cumulants can be constructed in increasing order according to the number of particles that are correlated with each other. Since PHENIX has partial azimuthal coverage, reliable extraction of azimuthal anisotropy requires the choice of a fixed number of particles from each event in order to avoid additional numerical errors Borghini et al. (2001a).

Particles in an event are selected over a fixed range where there is sufficient multiplicity. These particles (called “integral particles” hereafter) are used to determine integrated flow, that is flow measured over a large (,) bin. For differential flow measurement, we select particles (called “differential” particles) over small (,) bins, from which the integral particles are excluded so as to avoid autocorrelations. For each event a fixed number of particles, chosen at random among the integral particles in the event, are used to reconstruct the integrated flow through the generating function defined by:

(12)

where is the weight, chosen to be equal to 1 in our analysis, is the azimuth of the detected particles, and is the multiplicity chosen for the integrated flow reconstruction. is a function of the complex variable . The average of over events is then expanded in a power series to generate multi-particle azimuthal correlations. The generating function of the cumulants, defined by

(13)

generates cumulants of azimuthal correlations to all orders, the lowest being the second order, as detailed in Section II.B of Ref. Borghini et al. (2001a). The formulas used to compute the cumulants from which the is computed are given in Appendix B of Ref. Borghini et al. (2001a). In the case of a perfect acceptance the relations between the anisotropy parameter and the lowest order cumulants are

(14)
(15)

for the integrated anisotropy. Here and are the second and fourth order , respectively; whereas, and are the second and fourth order cumulants. Because the typical multiplicity of charged hadrons in PHENIX did not allow a reliable calculation of {4}, we report here only the {2} results.

The remaining differential particles in the same event are selected in different (, ) bins and the differential cumulants are calculated from the generating function

(16)

where denotes an average over all events, and is the azimuth of each differential particle. denotes the second order differential cumulant computed with respect to the second order integral cumulant.

The differential , the second order differential with respect to the second order integrated , is calculated from the relation

(17)

where is the second order differential cumulant. These relations have to be modified through acceptance corrections which are detailed below.

iii.2.1 Acceptance/efficiency corrections

The central arms detectors in PHENIX have only partial azimuthal coverage and the implementation of the cumulant method requires an additional acceptance correction. In order to correct for the influence of the detector acceptance on the raw anisotropy values, we apply a correction factor using the prescription described in Ref. Borghini et al. (2001a). The acceptance and efficiency of the detector is characterized by a function which is expressed in terms of the Fourier series

(18)

The Fourier coefficients for the detector acceptance were extracted from the fit of the respective azimuthal distributions of integral and differential particles. The coefficients resulting from such fits were then used to calculate the correction factor for the raw values of the following the procedure detailed in Appendix C of Ref. Borghini et al. (2001a).

Figure 7: Azimuthal angular distribution and corresponding Fourier fit for centrality 20–40% and = 1.2–1.4 GeV/.

Figure 7 shows a typical azimuthal angular distribution of differential particles detected in the PHENIX central arms and the corresponding Fourier fit used to correct for acceptance inhomogeneities. The Fourier fit reproduces well the overall features of the acceptance profile. This produces typical correction factors that are in the range 1.1–1.2 for the differential flow and depend very little on centrality and , as shown in Fig. 8.

Figure 8: (a) Acceptance correction factor for differential as a function of for centrality 10–20% (b) Acceptance correction factor as a function of centrality for range 0.4–0.5 GeV/ in Au+Au collisions at = 200 GeV.

iii.2.2 Simulations

While Fig. 7 shows that the uneven detector acceptance is reproduced by the Fourier fit, a better test of the cumulant method is to use Monte-Carlo simulations, as was done in Ref. Borghini et al. (2001a). For these tests events were generated with particles having a distribution of the form , with known integrated and differential azimuthal anisotropies. The anisotropy was introduced into the events by way of a Fourier weighted selection of the azimuthal angles followed by a random event rotation designed to simulate the random orientation of the reaction plane. The multiplicity of these events was chosen to reflect the typical multiplicity measured with the PHENIX detector and the angles were chosen from a filter that is representative of the PHENIX acceptance. We extracted Fourier components from these simulated results and applied these to extract corrected elliptic flow values.

Figure 9: Comparison of input and extracted differential values for a fixed integral of 8. The dotted line indicates the expectation if input and reconstructed values are the same.

Figure 9 shows selected results from these simulations. Corrected differential anisotropy values are compared for various input differential values, with the integral kept fixed. The dotted line shows the trend expected if the extracted is identical to the input value used to generate the events. The good agreement between the input and extracted attests to the reliability of the analysis method within the acceptance of the PHENIX central arms.

Iv Systematic Uncertainties

In this section, we present the systematic uncertainties on the from the event plane method (Sec. IV.1) and the two-particle cumulant method (Sec. IV.2). Table 3 lists the different sources of systematic errors for each method. The errors in Tab. 3 are categorized by type:

  • point-to-point error uncorrelated between bins,

  • correlated, all points move in the same direction but not by the same factor,

  • an overall normalization error in which all points move by the same factor independent of .

Error source Percentage error Type
{BBC}  {ZDC-SMD}
Background contribution 5% in 4 GeV/ B
5–30% in 4 GeV/ B
Event plane calibration 1–5% C
Event plane determination 1–4% 1–16% C
Acceptance effect 1% 1–25% C
on event planes
{2}
Fixed multiplicity 5% B
Integrated range 3–8% B
Background correction 6–10% B
Table 3: List of systematic uncertainties given in percent on the {ZDC-SMD}, {BBC} and {2} measurements. The ranges correspond to different systematic errors for different centrality bins.

iv.1 Event plane method

iv.1.1 Background contributions

In order to study the influence of background on our results, we varied one of the track selections while keeping other cuts fixed and investigated the effect on in the following two cases: (i) the PC3 and EMCAL matching cuts, 1.5 and 2.5 matching cuts and (ii) and . For both conditions, we found that the difference of the is 1–2% for 4 GeV/, and 5–20% for 4 GeV/ depending on and centrality.

Figure 10: The radial PC3 matching distribution for real (open circles) and random tracks (solid lines) for 6 8 GeV/ in centrality 0–60%.

The effect of the RICH veto cut has also been studied. Since the contribution of charged increases without the RICH veto cut, the ratio decreases at high . Thus, the for charged hadrons could be modified due to the difference of between protons and in the range 4 8 GeV/. We found that is 10–20% different without the RICH veto cut for 4–5 GeV/, where the charged starts firing the RICH.

One of the remaining sources of background contribution comes from the random tracks that are accidentally associated with the tracks in PC3. These random tracks have been estimated by swapping the z-coordinate of the PC3 hits and then by associating those hits with the real tracks. Figure 10 shows the comparison of the radial PC3 matching distribution between the real and random tracks for 6 8 GeV/. The signal to background ratio is evaluated in the window, and is 52 for 6 8 GeV/ in centrality 0–60%.

Figure 11: The ratio of real to random tracks as a function of in centrality 0–60%. Solid and open circles show the ratio with and without , respectively.

The ratio of real and random tracks with and without the cut is shown as a function of for centrality 0–60% in Fig. 11. The cut reduces the random tracks and improves the ratio by a factor of 10–24 for 4 GeV/. Since random tracks are not expected to be correlated with the event plane, we assume that their and evaluate the systematic uncertainty on to be less than 2% for 4 GeV/, increasing to 5% for 0.5 GeV/.

Figure 12: (a) Comparison of averaged over 0.2 8 GeV/ as a function of centrality for the BBC event planes. Open triangles and crosses represent the with respect to the event planes from South and North sub-detectors and filled circles show the from combined South-North event planes. Results from South and North event planes are shifted in the x-direction to improve visibility. (b) The same comparison for the ZDC-SMD event planes. Only statistical errors are shown and they are smaller than the symbols.

There is a finite residual background contribution even after the has been applied, as observed in Fig. 10. The residual backgrounds have been estimated by fitting the with a double Gaussian while requiring that the signal and residual background distribution have the same mean. For the highest bin, we found that the signal to background ratio is 5 for . The systematic error on is evaluated by comparing the measured with that of signal

(19)

where , and are respectively of signal, background estimated for , and measured within the 2 matching window. The systematic uncertainties are less than 5% for 4 GeV/, and 5–10% for higher . All the above systematic errors are added in quadrature and the overall systematic error from the background contribution is estimated to vary from 5% for 4 GeV/ to 30% for higher .

iv.1.2 Event plane calibrations

The procedures used in the determination and calibration of event planes are the dominant sources of systematic errors on and are discussed in the following sections.

Different calibration procedures of the BBC event plane were extensively studied for previous Au + Au data sets Adler et al. (2003b). We followed the same procedure to study the systematic errors on the BBC and ZDC-SMD event planes. Systematic uncertainties from the shift methods on {BBC} are 1-5% depending on the centrality. The systematic errors on the {ZDC-SMD} are 1-2% larger than those on {BBC} for centrality 10–30% and 50–60%, although those are still less than 5%.

iv.1.3 Event plane determination

Figure 12 shows the comparison of for different sub-detectors with respect to the BBC and ZDC-SMD event planes as a function of centrality. Systematic errors are estimated by taking the maximum difference of the from the South and North event planes to that from the combined South-North event plane scaled by for each centrality. Systematic errors range from 1-4% for the BBC, and 1-16% for the ZDC-SMD event planes depending on the centrality bins.

iv.1.4 Effect of non-uniform acceptance on

In this subsection we discuss the effect of non-uniform acceptance on the measured . In practice, the imperfect azimuthal acceptance of the BBC or ZDC-SMD or the central arms could induce an azimuthal-dependent event plane resolution and/or smear the magnitude of . In order to study the possible effect of non-uniform acceptance, the measured is decomposed into X and Y components Selyuzhenkov and Voloshin (2008):

(20)

where denotes the azimuthal angle of hadrons measured in the central arms and are the acceptance correction factors of the measured in the central arms. The coefficient should be unity in the case of perfect azimuthal acceptance. Res and Res denote the event plane resolution for and respectively and are expressed as

(21)

where are the harmonics of event planes for subevents A, B, and C, respectively. Another acceptance effect from the difference between Res and Res is discussed below.

Figure 13: Acceptance correction factors in the central arms as a function of for centrality 0–60%. Correction factors become unity for a perfect azimuthal acceptance. Statistical errors are smaller than the symbols.

Figure 13 shows the acceptance correction factor as a function of in the central arms for centrality 0–60%. The dependence is parameterized by

(22)

where ( = 0,1,…,5) are free parameters. From the fit, we get , , , , and . There is no centrality dependence of the acceptance corrections in the measured centrality range and these same correction factors are applied for all centrality bins.

Figure 14: (a) Raw {BBC} without the acceptance correction as a function of in centrality 20–60% for (filled circles), (filled squares) with the South BBC event plane and or (open circles), (open squares) with the North BBC event plane. (b) The same comparison with the acceptance correction.

Figure 14 shows the raw {BBC} as a function of in 20–60% centrality bin. is systematically higher than for 1 GeV/ as shown in Fig. 14(a). Figure 14(b) shows that and agree with each other after dividing by , the remaining difference between them being accounted for as a systematic error. For the ZDC-SMD event plane we observed a similar trend for and .

A possible non-uniform acceptance of the BBC and ZDC-SMD could lead to the difference between Res and Res. If the azimuthal coverage of both detectors is perfect, Res and Res should be identical. Therefore, the effect of the acceptance of the detector on the event plane resolution can be assessed by comparing Res and Res.

Figure 15: (a) Comparison of Res (open squares) and Res (filled circles) with Res (dashed lines) for the BBC event plane ( = 2) as a function of centrality. The resolutions are calculated by using Eq. (21) with the ZDC-SMD, BBC and CNT event planes. Res is divided by in order to compare Res and Res. (b) The same comparison for the ZDC-SMD event plane ( = 1). Only statistical errors are shown and are smaller than symbols.

Figure 15 shows Res and Res of the BBC and ZDC-SMD as a function of centrality. The resolutions are calculated by using Eq. (21) with the ZDC-SMD, BBC and CNT event planes. Res was comparable with Res for both the BBC and ZDC-SMD event planes. They also agreed, within statistical errors, with the expected resolution, namely the full event resolution scaled by 1/. We also evaluated Res and Res of BBC and ZDC-SMD for the two-subevents method. Res was consistent with Res. However, for the ZDC-SMD event plane, Res (Res) was systematically higher (lower) by about 30% than the expected resolution when the resolutions were calculated with = 1 in Eq. (8). The difference between Res and Res for the two-subevents method is attributed to the non-uniform acceptance between horizontal (x) and vertical (y) directions of the ZDC-SMD. Those resolutions of the ZDC-SMD were consistent with each other using = 2. For = 2, the non-uniform acceptance in the azimuthal directions cancels out since Res contain both and terms. Thus, Res should be the same and consistent with that from the expected resolution.

Figure 16: (a) Comparison of (open diamonds) and (open crosses) with the total (filled circles) for the BBC event plane as a function of for the centrality bin 20–60%. Res and Res are calculated by the combination of the ZDC-SMD, BBC and CNT event planes. Acceptance corrections are included into and . Error bars denote statistical errors. (b) The same comparison for the ZDC-SMD event plane.

The comparison of and with with respect to the BBC and ZDC-SMD event planes is shown in Fig. 16. The maximum difference of and relative to {BBC} is about 2% for the centrality range 20–60% and is independent of centrality. Systematic uncertainties are evaluated by scaling the maximum difference by . The same comparison is also made for {ZDC-SMD} as shown in the bottom panel in Fig. 16. The systematic errors range from 1–25% and in this case strongly depend on the centrality, as well as on the corrections by the different event plane resolutions. and are 10–25% different from {ZDC-SMD} in the 0–20% centrality bin due to the very low resolution. This systematic uncertainty is denoted as “Acceptance effect on event planes” in Table 3.

iv.2 Cumulant method

The potential sources of systematic errors on the cumulant measurements are detailed below.

iv.2.1 Fixed multiplicity cut

Following Ref. Borghini et al. (2001a) a fixed multiplicity is used to reconstruct the integrated flow to avoid introducing additional errors arising from a fluctuating multiplicity. In our analysis the systematic errors were estimated by varying the fixed multiplicity cut used for the reconstruction of the integrated flow and studying its effect on the differential flow values.

Figure 17: (a) {2} as a function of for centrality 10–20 in Au+Au collisions at = 200 GeV for different fixed multiplicity cuts, corresponding to 60% (filled triangles), 70% (open circles) and 80% (open crosses) of the mean multiplicity. (b) The ratio of () for the two lowest multiplicity cuts to () for 80% of the mean multiplicity.

Figure 17(a) shows the variation of with for integral multiplicity cuts equal to 60%, 70%, and 80% of the mean multiplicity for the centrality bin 20–40%. The ratio of the differential values, shown in Fig. 17(b), is used to estimate the systematic error on our measurements, which is 5%.

iv.2.2 range for integrated flow

In order to assess the influence of the range used to estimate the integrated flow on the differential flow, we chose different ranges over which the integral particles were selected. Differential results were obtained for three ranges: 0.25 - 2.0 GeV/, 0.25 - 1.5 GeV/ and 0.3 - 1.5 GeV/. The systematic error from this source is estimated to be 3-8% depending on centrality and .

iv.2.3 Background contribution

The procedures followed for studying the background contribution to {2} were the same as for the event plane method. After background subtraction the systematic error is calculated by determining the difference between the obtained from using 2 and 3 association cuts. We determined that the overall systematic error due to these differences is 6–10% depending on and centrality.

V Results

v.1 dependence of

Figure 18: Charged hadron () in Au+Au collisions at = 200 GeV from the two-particle cumulant method (filled squares), the BBC event plane (filled triangles) and the ZDC-SMD event plane (filled circles) for centrality (a) 0–10%, (b) 10–20%, (c) 20–30%, (d) 30–40%, (e) 40–50%, and (f) 50–60%. Error bars denote statistical errors. The type B systematic uncertainties are represented by the open boxes for the {BBC} and {ZDC-SMD}, and by the solid lines for the {2}. The gray bands and blue boxes represent the type C systematic uncertainties on the {BBC} and {ZDC-SMD}, respectively.
Figure 19: The ratio of to {BBC} as a function of for six centrality bins over the range 0–60% in Au+Au collisions at = 200 GeV. Data symbols are the same as in the Fig. 18. Error bars denote statistical errors. The solid red lines represent the type B systematic errors on the {2}. The blue and yellow bands represent type C systematic uncertainties on {ZDC-SMD} and {2}.
Figure 20: Comparison of charged hadron at 1 1.2 GeV/ as a function of for {BBC} (filled triangles), {ZDC-SMD} (filled circles) and {2} (filled squares) in Au+Au at = 200 GeV. The error bars represent statistical errors. The open boxes represent type B systematic uncertainties on {BBC} and {ZDC-SMD}. Type B systematic uncertainties on {2} are represented by solid red lines. The gray and blue bands represent type C systematic errors on {BBC} and {ZDC-SMD}, respectively. {2} values are shifted in the x-axis to improve the plot.
Figure 21: Anisotropy parameter as a function of pseudorapidity within the PHENIX central arms using event planes from the BBC (filled circles), ZDC-SMD (filled squares), and from the two-particle cumulant method (open triangles) for centrality 20–40%. The results are shown for three bins, which are from top to bottom: 2.0–3.0, 1.2–1.4 and 0.6–0.8 GeV/. Only statistical errors are shown.

The dependence of has been instrumental in revealing the hydrodynamic properties of the matter formed at RHIC Adler et al. (2003b); Adams et al. (2004a). In this context, it is important to compare the dependence of from different methods to establish the robustness of our measurements. This comparison is displayed in Fig. 18 which shows the differential charged hadron as a function of from the event plane and cumulant methods for different centrality bins in the range 0–60% in Au+Au at = 200 GeV. {2} increases up to 3 GeV/ and saturates at  0.1–0.25, depending on centrality, for higher . On the other hand, {BBC} and {ZDC-SMD} reach their maximum value at 3 GeV/, and decrease for higher .

The differences between {BBC} and {ZDC-SMD} are independent of within systematic errors in the measured centrality range. {ZDC-SMD} is consistent with {BBC} within systematic errors in the 0–40% centrality range, but is 10–20% smaller than {BBC} in the 40–60% centrality range. These results could indicate that the influence of non-flow effects on {BBC} is small and within the systematic errors, because non-flow effects are not expected to influence {ZDC-SMD}. The difference between {BBC} and {ZDC-SMD} in peripheral collisions could be attributed to non-flow contributions that might be proportionally larger in more peripheral collisions.

The cumulant and event plane agree well within systematic uncertainties in the centrality range 0–40%. In more peripheral collisions, there may be some differences developing above 4 GeV/. Correlations between particles from jets affect the cumulant results, but have less influence on {BBC}, as explained in Ref. Jia (2007), where it was shown that the smaller the rapidity gap between the leading particle from a jet and the event plane, the greater the of the leading particle of the jet.

In order to illustrate more clearly the differences between the different methods, Fig. 19 shows the ratio of {ZDC-SMD} and {2} to {BBC}. The results from the three methods are comparable in magnitude within systematic errors, except for the central and peripheral bins where the largest deviations occur. In addition, {2} and {ZDC-SMD} show different behaviors at 3 GeV/, with {2} being larger, and {ZDC-SMD}, smaller than {BBC}.

Figure 22: (a) Comparison of the {ZDC-SMD} from the S-N (filled circles) and ZDC-BBC-CNT subevents (filled squares) as a function of in the 20–60% centrality range. (b) The same comparison as (a) for the {BBC}, where filled triangles and open circles represent the from the S-N and ZDC-BBC-CNT subevents, respectively. (c) Comparison of between BBC (filled triangles) and ZDC-SMD event planes (filled circles) from the S-N subevent as a function of in centrality 20–60%. (d) The same comparison as (c) from the ZDC-BBC-CNT subevent, where filled squares and open circles represent the {ZDC-SMD} and {BBC}, respectively. Error bars denote statistical errors. Open boxes and shaded bands describe the quadratic sum of type B and C systematic uncertainties from the S-N and ZDC-BBC-CNT subevents, respectively.

v.2 Centrality dependence of

Figure 20 shows the dependence of from different methods for charged hadrons in the range 1.0 1.2 GeV/. is observed to increase with decreasing and then decrease slightly for 75. Note that values obtained with the different methods agree well within systematic errors for all centralities. This is dependent, as shown in Fig. 18.

v.3 Pseudorapidity dependence of v

Figure 21 compares the pseudorapidity dependence of the of charged hadrons within the range ( 0.35) of the PHENIX central arms for different selections. It can be observed that is constant over the coverage of the PHENIX detector and the constancy does not depend on . This is not the case when the is measured far from midrapidity where the PHOBOS and STAR collaborations observe a drop in for  Adams et al. (2005c); Alver et al. (2007).

Vi Discussion

vi.1 Effect of CNT event plane resolution

Figure 22 shows the comparison of {ZDC-SMD} and {BBC} as a function of corrected either by the resolution from South-North correlations from the same detectors or by the resolution from ZDC-SMD-CNT correlations in the 20–60% centrality bin. Figures 22(a) and (b) compare the obtained by using two different corrections from the South-North and ZDC-BBC-CNT subevents for the BBC (a) and ZDC-SMD event planes (b). The from the South-North subevent is consistent with that from the ZDC-BBC-CNT subevent, within systematic uncertainties. The small difference between South-North and ZDC-BBC-CNT subevents is attributed to the difference between the event plane resolution, as shown in Fig. 4. Figures 22(c) and (d) compare {ZDC-SMD} with {BBC} for the South-North (c) and ZDC-BBC-CNT subevent (d). The data points in Fig. 22(c) and (d) are the same as in Fig. 22(a) and (b). Figure 22(c) shows that {ZDC-SMD} is about 10% smaller than {BBC} for the South-North subevent. The ratio of {ZDC-SMD} to {BBC} is found to be independent of except for GeV/. If jets are the dominant source of non-flow, one expects its contribution to to become larger at higher . The constant ratio suggests that the non-flow contribution from jets is small and fluctuations may affect {BBC} below GeV/ since the effect of fluctuations is expected to be independent of . {ZDC-SMD} agrees with {BBC} within systematic uncertainties for the ZDC-BBC-CNT subevent as shown in Fig. 22(d). The event plane resolution from the ZDC-BBC-CNT subevents includes the effect of non-flow contributions and fluctuations since the CNT and BBC event planes are sensitive to both effects, though non-flow effects especially from jets could be negligible in the BBC event plane, as discussed earlier. The consistency between from the ZDC-SMD and BBC event planes may suggest that {ZDC-SMD} becomes sensitive to fluctuations by the inclusion of the BBC and CNT event planes to estimate the resolution.

vi.2 Comparison with other experiments

Figure 23: (a) Comparison of charged hadron {2} between PHENIX (filled squares) and STAR experiments (open stars) as a function of in centrality 20–60%. Solid lines represent the quadratic sum of type B and C systematic errors on the PHENIX {2}. (b) Comparison of charged hadron from four-particle cumulant {4} at STAR (open stars) with the PHENIX {BBC} (filled triangles) and {ZDC-SMD} (filled circles) as a function of in centrality 20–60%. Open boxes and shaded bands represent the quadratic sum of type B and C systematic errors on the {BBC} and {ZDC-SMD}, respectively. STAR results are taken from Ref. Adams et al. (2004b). Systematic errors on the STAR are not plotted, see text for more details.

It is instructive to compare measurements made by different experiments at RHIC. Figure 23 shows a comparison of the dependence of charged hadron in the 20–60% centrality range between PHENIX and STAR experiments Adams et al. (2004b). The relative systematic errors on the STAR {2} and {4} measurements range up to 10% for 1 GeV/, with the lowest bin having the largest error 10%, while they are of the order of 1% above 1 GeV/ Adams et al. (2004b). The {2} from PHENIX is lower than that from STAR, but they are comparable within systematic uncertainties, as shown in Fig. 23(a). Figure 23(b) compares {BBC} and {ZDC-SMD} with {4}, obtained from four particle cumulants, as measured in STAR. For 2 GeV/, the STAR {4} is systematically smaller than the PHENIX event plane , while {ZDC-SMD} is lower than {BBC}. However, the three set of measurements are consistent within systematic errors. The order of , {BBC} {ZDC-SMD} {4} could be explained by the effect of flow fluctuations Bhalerao and Ollitrault (2006); Miller and Snellings () if other non-flow contributions are small.

Figure 24: Comparison of the PHENIX {BBC} (filled triangles) and {ZDC-SMD} (filled circles) with the STAR from modified event plane method (open stars) for charged hadrons Adams et al. (2005c) as a function of in centrality (a) 10–20%, (b) 20–30%, and (c) 30–40%. Open boxes and shaded bands represent the quadratic sum of type B and C systematic errors on {BBC} and {ZDC-SMD}, respectively.

Figure 24 shows the comparison of our charged hadron from the BBC and ZDC-SMD event planes to from a modified event plane method Adams et al. (2005c), labelled {EP}, from the STAR experiment for three centrality bins in the range 10–40%. Particles within around the highest particle were excluded for the determination of the modified event plane in order to reduce some of the non-flow effects at high . We find that