PHENIX Collaboration

# Nonprompt direct-photon production in Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV

U.A. Acharya  Georgia State University, Atlanta, Georgia 30303, USA  A. Adare
University of Colorado, Boulder, Colorado 80309, USA  C. Aidala  Department of
Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA  N.N.
Ajitanand  Deceased  Chemistry Department, Stony Brook University, SUNY, Stony
Brook, New York 11794-3400, USA  Y. Akiba<EMAIL_ADDRESS>RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  M. Alfred  Department of Physics and Astronomy, Howard
University, Washington, DC 20059, USA  N. Apadula  Iowa State University,
Ames, Iowa 50011, USA  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  H. Asano  Kyoto
University, Kyoto 606-8502, Japan  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  B. Azmoun  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  V. Babintsev
IHEP Protvino, State Research Center of Russian Federation, Institute for High
Energy Physics, Protvino, 142281, Russia  M. Bai  Collider-Accelerator
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
N.S. Bandara  Department of Physics, University of Massachusetts, Amherst,
Massachusetts 01003-9337, USA  B. Bannier  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
K.N. Barish  University of California-Riverside, Riverside, California 92521,
USA  S. Bathe  Baruch College, City University of New York, New York, New
York, 10010 USA  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  A. Bazilevsky  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  M. Beaumier  University
of California-Riverside, Riverside, California 92521, USA  S. Beckman
University of Colorado, Boulder, Colorado 80309, USA  R. Belmont  University
of Colorado, Boulder, Colorado 80309, USA  Department of Physics, University
of Michigan, Ann Arbor, Michigan 48109-1040, USA  Physics and Astronomy
Department, University of North Carolina at Greensboro, Greensboro, North
Carolina 27412, USA  A. Berdnikov  Saint Petersburg State Polytechnic
University, St. Petersburg, 195251 Russia  Y. Berdnikov  Saint Petersburg
State Polytechnic University, St. Petersburg, 195251 Russia  L. Bichon
Vanderbilt University, Nashville, Tennessee 37235, USA  B. Blankenship
Vanderbilt University, Nashville, Tennessee 37235, USA  D.S. Blau  National
Research Center “Kurchatov Institute”, Moscow, 123098 Russia  National
Research Nuclear University, MEPhI, Moscow Engineering Physics Institute,
Moscow, 115409, Russia  J.S. Bok  New Mexico State University, Las Cruces, New
Mexico 88003, USA  V. Borisov  Saint Petersburg State Polytechnic University,
St. Petersburg, 195251 Russia  K. Boyle  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  M.L. Brooks  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  J. Bryslawskyj  Baruch
College, City University of New York, New York, New York, 10010 USA
University of California-Riverside, Riverside, California 92521, USA  V.
Bumazhnov  IHEP Protvino, State Research Center of Russian Federation,
Institute for High Energy Physics, Protvino, 142281, Russia  S. Campbell
Columbia University, New York, New York 10027 and Nevis Laboratories,
Irvington, New York 10533, USA  Iowa State University, Ames, Iowa 50011, USA
V. Canoa Roman  Department of Physics and Astronomy, Stony Brook University,
SUNY, Stony Brook, New York 11794-3800, USA  C.-H. Chen  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  M.
Chiu  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  C.Y. Chi  Columbia University, New York, New York 10027 and
Nevis Laboratories, Irvington, New York 10533, USA  I.J. Choi  University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  J.B. Choi  Deceased
Jeonbuk National University, Jeonju, 54896, Korea  T. Chujo  Tomonaga Center
for the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305,
Japan  Z. Citron  Weizmann Institute, Rehovot 76100, Israel  M. Connors
Georgia State University, Atlanta, Georgia 30303, USA  R. Corliss  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  Y. Corrales Morales  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  M. Csanád  ELTE, Eötvös Loránd University,
H-1117 Budapest, Pázmány P. s. 1/A, Hungary  T. Csörgő  Institute for Particle
and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of
Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary
T.W. Danley  Department of Physics and Astronomy, Ohio University, Athens,
Ohio 45701, USA  A. Datta  University of New Mexico, Albuquerque, New Mexico
87131, USA  M.S. Daugherity  Abilene Christian University, Abilene, Texas
79699, USA  G. David  Physics Department, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  C.T. Dean  Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  K. DeBlasio
University of New Mexico, Albuquerque, New Mexico 87131, USA  K. Dehmelt
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  A. Denisov  IHEP Protvino, State Research
Center of Russian Federation, Institute for High Energy Physics, Protvino,
142281, Russia  A. Deshpande  RIKEN BNL Research Center, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
E.J. Desmond  Physics Department, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  A. Dion  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  P.B. Diss
University of Maryland, College Park, Maryland 20742, USA  J.H. Do  Yonsei
University, IPAP, Seoul 120-749, Korea  V. Doomra  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
A. Drees  Department of Physics and Astronomy, Stony Brook University, SUNY,
Stony Brook, New York 11794-3800, USA  K.A. Drees  Collider-Accelerator
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
J.M. Durham  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
A. Durum  IHEP Protvino, State Research Center of Russian Federation,
Institute for High Energy Physics, Protvino, 142281, Russia  A. Enokizono
RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198,
Japan  Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima,
Tokyo 171-8501, Japan  R. Esha  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  B. Fadem
Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA  W. Fan
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  N. Feege  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
D.E. Fields  University of New Mexico, Albuquerque, New Mexico 87131, USA  M.
Finger, Jr  Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech
Republic  M. Finger  Charles University, Ovocný trh 5, Praha 1, 116 36,
Prague, Czech Republic  D. Fitzgerald  Department of Physics, University of
Michigan, Ann Arbor, Michigan 48109-1040, USA  S.L. Fokin  National Research
Center “Kurchatov Institute”, Moscow, 123098 Russia  J.E. Frantz  Department
of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA  A. Franz
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  A.D. Frawley  Florida State University, Tallahassee, Florida
32306, USA  P. Gallus  Czech Technical University, Zikova 4, 166 36 Prague 6,
Czech Republic  C. Gal  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  P. Garg  Department
of Physics, Banaras Hindu University, Varanasi 221005, India  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  H. Ge  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  M. Giles  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  F. Giordano  University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA  A. Glenn  Lawrence Livermore National Laboratory,
Livermore, California 94550, USA  Y. Goto  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  N.
Grau  Department of Physics, Augustana University, Sioux Falls, South Dakota
57197, USA  S.V. Greene  Vanderbilt University, Nashville, Tennessee 37235,
USA  M. Grosse Perdekamp  University of Illinois at Urbana-Champaign, Urbana,
Illinois 61801, USA  T. Gunji  Center for Nuclear Study, Graduate School of
Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  T.
Guo  Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  T. Hachiya  Nara Women’s University, Kita-
uoya Nishi-machi Nara 630-8506, Japan  RIKEN Nishina Center for Accelerator-
Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research Center,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  J.S. Haggerty
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  K.I. Hahn  Ewha Womans University, Seoul 120-750, Korea  H.
Hamagaki  Center for Nuclear Study, Graduate School of Science, University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  H.F. Hamilton  Abilene
Christian University, Abilene, Texas 79699, USA  J. Hanks  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  S.Y. Han  Ewha Womans University, Seoul 120-750, Korea  Korea
University, Seoul 02841, Korea  M. Harvey  Texas Southern University, Houston,
TX 77004, USA  S. Hasegawa  Advanced Science Research Center, Japan Atomic
Energy Agency, 2-4 Shirakata Shirane, Tokai-mura, Naka-gun, Ibaraki-ken
319-1195, Japan  T.O.S. Haseler  Georgia State University, Atlanta, Georgia
30303, USA  K. Hashimoto  RIKEN Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan  Physics Department, Rikkyo University, 3-34-1
Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan  T.K. Hemmick  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  X. He  Georgia State University, Atlanta, Georgia 30303, USA
J.C. Hill  Iowa State University, Ames, Iowa 50011, USA  A. Hodges  Georgia
State University, Atlanta, Georgia 30303, USA  R.S. Hollis  University of
California-Riverside, Riverside, California 92521, USA  K. Homma  Hiroshima
University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan  B. Hong  Korea
University, Seoul 02841, Korea  T. Hoshino  Hiroshima University, Kagamiyama,
Higashi-Hiroshima 739-8526, Japan  N. Hotvedt  Iowa State University, Ames,
Iowa 50011, USA  J. Huang  Physics Department, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  K. Imai  Advanced Science Research Center,
Japan Atomic Energy Agency, 2-4 Shirakata Shirane, Tokai-mura, Naka-gun,
Ibaraki-ken 319-1195, Japan  M. Inaba  Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  A. Iordanova
University of California-Riverside, Riverside, California 92521, USA  D.
Isenhower  Abilene Christian University, Abilene, Texas 79699, USA  D.
Ivanishchev  PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad
region, 188300, Russia  B.V. Jacak  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  M. Jezghani
Georgia State University, Atlanta, Georgia 30303, USA  X. Jiang  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  Z. Ji  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  B.M. Johnson  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  Georgia State University,
Atlanta, Georgia 30303, USA  D. Jouan  IPN-Orsay, Univ. Paris-Sud, CNRS/IN2P3,
Université Paris-Saclay, BP1, F-91406, Orsay, France  D.S. Jumper  University
of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  S. Kanda  Center
for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033, Japan  J.H. Kang  Yonsei University, IPAP,
Seoul 120-749, Korea  D. Kawall  Department of Physics, University of
Massachusetts, Amherst, Massachusetts 01003-9337, USA  A.V. Kazantsev
National Research Center “Kurchatov Institute”, Moscow, 123098 Russia  J.A.
Key  University of New Mexico, Albuquerque, New Mexico 87131, USA  V.
Khachatryan  Department of Physics and Astronomy, Stony Brook University,
SUNY, Stony Brook, New York 11794-3800, USA  A. Khanzadeev  PNPI, Petersburg
Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia  A.
Khatiwada  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
B. Kimelman  Muhlenberg College, Allentown, Pennsylvania 18104-5586, USA  C.
Kim  Korea University, Seoul 02841, Korea  D.J. Kim  Helsinki Institute of
Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland
E.-J. Kim  Jeonbuk National University, Jeonju, 54896, Korea  G.W. Kim  Ewha
Womans University, Seoul 120-750, Korea  M. Kim  Department of Physics and
Astronomy, Seoul National University, Seoul 151-742, Korea  T. Kim  Ewha
Womans University, Seoul 120-750, Korea  D. Kincses  ELTE, Eötvös Loránd
University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  A. Kingan  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  E. Kistenev  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  R. Kitamura  Center for Nuclear
Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033, Japan  J. Klatsky  Florida State University, Tallahassee,
Florida 32306, USA  D. Kleinjan  University of California-Riverside,
Riverside, California 92521, USA  P. Kline  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
T. Koblesky  University of Colorado, Boulder, Colorado 80309, USA  B. Komkov
PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region,
188300, Russia  D. Kotov  PNPI, Petersburg Nuclear Physics Institute,
Gatchina, Leningrad region, 188300, Russia  Saint Petersburg State Polytechnic
University, St. Petersburg, 195251 Russia  L. Kovacs  ELTE, Eötvös Loránd
University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  K. Kurita  Physics
Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo
171-8501, Japan  M. Kurosawa  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  Y. Kwon  Yonsei
University, IPAP, Seoul 120-749, Korea  J.G. Lajoie  Iowa State University,
Ames, Iowa 50011, USA  D. Larionova  Saint Petersburg State Polytechnic
University, St. Petersburg, 195251 Russia  A. Lebedev  Iowa State University,
Ames, Iowa 50011, USA  S. Lee  Yonsei University, IPAP, Seoul 120-749, Korea
S.H. Lee  Iowa State University, Ames, Iowa 50011, USA  Department of Physics,
University of Michigan, Ann Arbor, Michigan 48109-1040, USA  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  M.J. Leitch  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  N.A. Lewis  Department of Physics, University of Michigan,
Ann Arbor, Michigan 48109-1040, USA  S.H. Lim  Pusan National University,
Pusan 46241, Korea  Yonsei University, IPAP, Seoul 120-749, Korea  M.X. Liu
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  X. Li
Science and Technology on Nuclear Data Laboratory, China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China  X. Li  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  D.A. Loomis  Department of
Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA  D. Lynch
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  S. Lökös  ELTE, Eötvös Loránd University, H-1117 Budapest,
Pázmány P. s. 1/A, Hungary  T. Majoros  Debrecen University, H-4010 Debrecen,
Egyetem tér 1, Hungary  Y.I. Makdisi  Collider-Accelerator Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  M. Makek
Department of Physics, Faculty of Science, University of Zagreb, Bijenička c.
32 HR-10002 Zagreb, Croatia  A. Manion  Department of Physics and Astronomy,
Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA  V.I.
Manko  National Research Center “Kurchatov Institute”, Moscow, 123098 Russia
E. Mannel  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  M. McCumber  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  P.L. McGaughey  Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  D. McGlinchey  University of Colorado, Boulder,
Colorado 80309, USA  Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  C. McKinney  University of Illinois at Urbana-Champaign, Urbana,
Illinois 61801, USA  A. Meles  New Mexico State University, Las Cruces, New
Mexico 88003, USA  M. Mendoza  University of California-Riverside, Riverside,
California 92521, USA  A.C. Mignerey  University of Maryland, College Park,
Maryland 20742, USA  A. Milov  Weizmann Institute, Rehovot 76100, Israel  D.K.
Mishra  Bhabha Atomic Research Centre, Bombay 400 085, India  J.T. Mitchell
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  M. Mitrankova  Saint Petersburg State Polytechnic University,
St. Petersburg, 195251 Russia  Iu. Mitrankov  Saint Petersburg State
Polytechnic University, St. Petersburg, 195251 Russia  S. Miyasaka  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  S. Mizuno  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  A.K. Mohanty
Bhabha Atomic Research Centre, Bombay 400 085, India  M.M. Mondal  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  P. Montuenga  University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA  T. Moon  Korea University, Seoul 02841, Korea
Yonsei University, IPAP, Seoul 120-749, Korea  D.P. Morrison  Physics
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
T.V. Moukhanova  National Research Center “Kurchatov Institute”, Moscow,
123098 Russia  B. Mulilo  Korea University, Seoul 02841, Korea  RIKEN Nishina
Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
Department of Physics, School of Natural Sciences, University of Zambia, Great
East Road Campus, Box 32379, Lusaka, Zambia  T. Murakami  Kyoto University,
Kyoto 606-8502, Japan  RIKEN Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan  J. Murata  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  Physics Department,
Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan  A.
Mwai  Chemistry Department, Stony Brook University, SUNY, Stony Brook, New
York 11794-3400, USA  K. Nagashima  Hiroshima University, Kagamiyama, Higashi-
Hiroshima 739-8526, Japan  J.L. Nagle  University of Colorado, Boulder,
Colorado 80309, USA  M.I. Nagy  ELTE, Eötvös Loránd University, H-1117
Budapest, Pázmány P. s. 1/A, Hungary  I. Nakagawa  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  H.
Nakagomi  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan  Tomonaga Center for the History of the Universe, University
of Tsukuba, Tsukuba, Ibaraki 305, Japan  K. Nakano  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  C. Nattrass  University of Tennessee, Knoxville, Tennessee 37996, USA
S. Nelson  Florida A&M University, Tallahassee, FL 32307, USA  P.K. Netrakanti
Bhabha Atomic Research Centre, Bombay 400 085, India  T. Niida  Tomonaga
Center for the History of the Universe, University of Tsukuba, Tsukuba,
Ibaraki 305, Japan  S. Nishimura  Center for Nuclear Study, Graduate School of
Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  R.
Nouicer  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  N. Novitzky  Helsinki Institute of Physics
and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  T. Novák
Eszterházy Károly University, Károly Róbert Campus, H-3200 Gyöngyös, Mátrai út
36, Hungary  Institute for Particle and Nuclear Physics, Wigner Research
Centre for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525
Budapest 114, POBox 49, Budapest, Hungary  G. Nukazuka  RIKEN Nishina Center
for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
USA  A.S. Nyanin  National Research Center “Kurchatov Institute”, Moscow,
123098 Russia  E. O’Brien  Physics Department, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  C.A. Ogilvie  Iowa State University, Ames,
Iowa 50011, USA  J.D. Orjuela Koop  University of Colorado, Boulder, Colorado
80309, USA  J.D. Osborn  Department of Physics, University of Michigan, Ann
Arbor, Michigan 48109-1040, USA  Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  A. Oskarsson  Department of Physics, Lund University,
Box 118, SE-221 00 Lund, Sweden  K. Ozawa  KEK, High Energy Accelerator
Research Organization, Tsukuba, Ibaraki 305-0801, Japan  Tomonaga Center for
the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305,
Japan  R. Pak  Physics Department, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  V. Pantuev  Institute for Nuclear Research of the
Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a, Moscow 117312,
Russia  V. Papavassiliou  New Mexico State University, Las Cruces, New Mexico
88003, USA  J.S. Park  Department of Physics and Astronomy, Seoul National
University, Seoul 151-742, Korea  S. Park  Department of Physics and
Astronomy, Seoul National University, Seoul 151-742, Korea  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  M. Patel  Iowa State University, Ames, Iowa 50011, USA  S.F.
Pate  New Mexico State University, Las Cruces, New Mexico 88003, USA  J.-C.
Peng  University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
W. Peng  Vanderbilt University, Nashville, Tennessee 37235, USA  D.V.
Perepelitsa  Physics Department, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  University of Colorado, Boulder, Colorado 80309, USA
G.D.N. Perera  New Mexico State University, Las Cruces, New Mexico 88003, USA
D.Yu. Peressounko  National Research Center “Kurchatov Institute”, Moscow,
123098 Russia  C.E. PerezLara  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  J. Perry  Iowa
State University, Ames, Iowa 50011, USA  R. Petti  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  C. Pinkenburg  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  R. Pinson  Abilene Christian
University, Abilene, Texas 79699, USA  R.P. Pisani  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  M. Potekhin
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  A. Pun  Department of Physics and Astronomy, Ohio University,
Athens, Ohio 45701, USA  M.L. Purschke  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  P.V. Radzevich  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  J. Rak
Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35,
FI-40014 Jyväskylä, Finland  N. Ramasubramanian  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
B.J. Ramson  Department of Physics, University of Michigan, Ann Arbor,
Michigan 48109-1040, USA  I. Ravinovich  Weizmann Institute, Rehovot 76100,
Israel  K.F. Read  Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831,
USA  University of Tennessee, Knoxville, Tennessee 37996, USA  D. Reynolds
Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York
11794-3400, USA  V. Riabov  National Research Nuclear University, MEPhI,
Moscow Engineering Physics Institute, Moscow, 115409, Russia  PNPI, Petersburg
Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia  Y.
Riabov  PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad
region, 188300, Russia  Saint Petersburg State Polytechnic University, St.
Petersburg, 195251 Russia  D. Richford  Baruch College, City University of New
York, New York, New York, 10010 USA  T. Rinn  University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA  Iowa State University, Ames,
Iowa 50011, USA  S.D. Rolnick  University of California-Riverside, Riverside,
California 92521, USA  M. Rosati  Iowa State University, Ames, Iowa 50011, USA
Z. Rowan  Baruch College, City University of New York, New York, New York,
10010 USA  J.G. Rubin  Department of Physics, University of Michigan, Ann
Arbor, Michigan 48109-1040, USA  J. Runchey  Iowa State University, Ames, Iowa
50011, USA  B. Sahlmueller  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  N. Saito  KEK, High
Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan  T.
Sakaguchi  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  H. Sako  Advanced Science Research Center, Japan Atomic
Energy Agency, 2-4 Shirakata Shirane, Tokai-mura, Naka-gun, Ibaraki-ken
319-1195, Japan  V. Samsonov  National Research Nuclear University, MEPhI,
Moscow Engineering Physics Institute, Moscow, 115409, Russia  PNPI, Petersburg
Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia  M.
Sarsour  Georgia State University, Atlanta, Georgia 30303, USA  S. Sato
Advanced Science Research Center, Japan Atomic Energy Agency, 2-4 Shirakata
Shirane, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan  B. Schaefer
Vanderbilt University, Nashville, Tennessee 37235, USA  B.K. Schmoll
University of Tennessee, Knoxville, Tennessee 37996, USA  K. Sedgwick
University of California-Riverside, Riverside, California 92521, USA  R. Seidl
RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198,
Japan  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  A. Sen  Iowa State University, Ames, Iowa 50011, USA
University of Tennessee, Knoxville, Tennessee 37996, USA  R. Seto  University
of California-Riverside, Riverside, California 92521, USA  P. Sett  Bhabha
Atomic Research Centre, Bombay 400 085, India  A. Sexton  University of
Maryland, College Park, Maryland 20742, USA  D. Sharma  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  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  Iowa State University, Ames,
Iowa 50011, USA  Nara Women’s University, Kita-uoya Nishi-machi Nara 630-8506,
Japan  P. Shukla  Bhabha Atomic Research Centre, Bombay 400 085, India  A.
Sickles  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  University of Illinois at Urbana-Champaign, Urbana, Illinois
61801, USA  C.L. Silva  Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  D. Silvermyr  Department of Physics, Lund University, Box 118,
SE-221 00 Lund, Sweden  Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  B.K. Singh  Department of Physics, Banaras Hindu University,
Varanasi 221005, India  C.P. Singh  Department of Physics, Banaras Hindu
University, Varanasi 221005, India  V. Singh  Department of Physics, Banaras
Hindu University, Varanasi 221005, India  M. Slunečka  Charles University,
Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic  K.L. Smith  Florida
State University, Tallahassee, Florida 32306, USA  M. Snowball  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  R.A. Soltz  Lawrence
Livermore National Laboratory, Livermore, California 94550, USA  W.E. Sondheim
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  S.P.
Sorensen  University of Tennessee, Knoxville, Tennessee 37996, USA  I.V.
Sourikova  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  P.W. Stankus  Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  M. Stepanov  Deceased  Department of Physics, University
of Massachusetts, Amherst, Massachusetts 01003-9337, USA  S.P. Stoll  Physics
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
T. Sugitate  Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526,
Japan  A. Sukhanov  Physics Department, Brookhaven National Laboratory, Upton,
New York 11973-5000, USA  T. Sumita  RIKEN Nishina Center for Accelerator-
Based Science, Wako, Saitama 351-0198, Japan  J. Sun  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  Z. Sun  Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary  J.
Sziklai  Institute for Particle and Nuclear Physics, Wigner Research Centre
for Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest
114, POBox 49, Budapest, Hungary  A. Taketani  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  K.
Tanida  Advanced Science Research Center, Japan Atomic Energy Agency, 2-4
Shirakata Shirane, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan  RIKEN
BNL Research Center, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  Department of Physics and Astronomy, Seoul National
University, Seoul 151-742, Korea  M.J. Tannenbaum  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  S. Tarafdar
Vanderbilt University, Nashville, Tennessee 37235, USA  Weizmann Institute,
Rehovot 76100, Israel  A. Taranenko  National Research Nuclear University,
MEPhI, Moscow Engineering Physics Institute, Moscow, 115409, Russia  Chemistry
Department, Stony Brook University, SUNY, Stony Brook, New York 11794-3400,
USA  R. Tieulent  Georgia State University, Atlanta, Georgia 30303, USA  IPNL,
CNRS/IN2P3, Univ Lyon, Université Lyon 1, F-69622, Villeurbanne, France  A.
Timilsina  Iowa State University, Ames, Iowa 50011, USA  T. Todoroki  RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  Tomonaga Center for the History of the Universe, University
of Tsukuba, Tsukuba, Ibaraki 305, Japan  M. Tomášek  Czech Technical
University, Zikova 4, 166 36 Prague 6, Czech Republic  C.L. Towell  Abilene
Christian University, Abilene, Texas 79699, USA  R. Towell  Abilene Christian
University, Abilene, Texas 79699, USA  R.S. Towell  Abilene Christian
University, Abilene, Texas 79699, USA  I. Tserruya  Weizmann Institute,
Rehovot 76100, Israel  Y. Ueda  Hiroshima University, Kagamiyama, Higashi-
Hiroshima 739-8526, Japan  B. Ujvari  Debrecen University, H-4010 Debrecen,
Egyetem tér 1, Hungary  H.W. van Hecke  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  J. Velkovska  Vanderbilt University, Nashville,
Tennessee 37235, USA  M. Virius  Czech Technical University, Zikova 4, 166 36
Prague 6, Czech Republic  V. Vrba  Czech Technical University, Zikova 4, 166
36 Prague 6, Czech Republic  Institute of Physics, Academy of Sciences of the
Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic  X.R. Wang  New
Mexico State University, Las Cruces, New Mexico 88003, USA  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  Y.
Watanabe  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  Y.S. Watanabe  Center for Nuclear Study,
Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan  KEK, High Energy Accelerator Research Organization, Tsukuba,
Ibaraki 305-0801, Japan  F. Wei  New Mexico State University, Las Cruces, New
Mexico 88003, USA  A.S. White  Department of Physics, University of Michigan,
Ann Arbor, Michigan 48109-1040, USA  C.P. Wong  Georgia State University,
Atlanta, Georgia 30303, USA  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  C.L. Woody  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  M. Wysocki  Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  B. Xia  Department of Physics and
Astronomy, Ohio University, Athens, Ohio 45701, USA  L. Xue  Georgia State
University, Atlanta, Georgia 30303, USA  S. Yalcin  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
Y.L. Yamaguchi  Center for Nuclear Study, Graduate School of Science,
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  A. Yanovich  IHEP Protvino, State Research Center of Russian
Federation, Institute for High Energy Physics, Protvino, 142281, Russia  I.
Yoon  Department of Physics and Astronomy, Seoul National University, Seoul
151-742, Korea  J.H. Yoo  Korea University, Seoul 02841, Korea  I.E. Yushmanov
National Research Center “Kurchatov Institute”, Moscow, 123098 Russia  H. Yu
New Mexico State University, Las Cruces, New Mexico 88003, USA  Peking
University, Beijing 100871, People’s Republic of China  W.A. Zajc  Columbia
University, New York, New York 10027 and Nevis Laboratories, Irvington, New
York 10533, USA  A. Zelenski  Collider-Accelerator Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  S. Zhou  Science and
Technology on Nuclear Data Laboratory, China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China  L. Zou  University of California-
Riverside, Riverside, California 92521, USA

###### Abstract

The measurement of the direct-photon spectrum from Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV is presented by the PHENIX collaboration using
the external-photon-conversion technique for 0%–93% central collisions in a
transverse-momentum ($p_{T}$) range of 0.8–10 GeV/$c$. An excess of direct
photons, above prompt-photon production from hard-scattering processes, is
observed for $p_{T}<6$ GeV/$c$. Nonprompt direct photons are measured by
subtracting the prompt component, which is estimated as $N_{\rm coll}$-scaled
direct photons from $p$$+$$p$ collisions at 200 GeV, from the direct-photon
spectrum. Results are obtained for $0.8<p_{T}<6.0$ GeV/$c$ and suggest that
the spectrum has an increasing inverse slope from ${\approx}0.2$ to 0.4
GeV/$c$ with increasing $p_{T}$, which indicates a possible sensitivity of the
measurement to photons from earlier stages of the evolution of the collision.
In addition, like the direct-photon production, the $p_{T}$-integrated
nonprompt direct-photon yields also follow a power-law scaling behavior as a
function of collision-system size. The exponent, $\alpha$, for the nonprompt
component is found to be consistent with 1.1 with no apparent $p_{T}$
dependence.

##  I Introduction

Direct photons, defined as those not coming from hadron decays, have long been
considered a golden probe towards our understanding of the evolution of
relativistic heavy-ion collisions – from the quark-gluon plasma (QGP) phase to
the hadron-gas (HG) phase [1]. Unlike strongly interacting probes, such as
identified particles and jets, direct photons traverse the medium unmodified
due to the small cross section of electromagnetic interaction. These
penetrating photons encode information about the environment in which they
were created, including the temperature and the collective motion of the
medium. While the direct photons at high transverse momentum, $p_{T}$, are
dominated by photons created from hard-scattering processes, such as quark-
gluon Compton scattering, in the low-$p_{T}$ regime, they were initially
predicted to be of a thermal origin, being emitted from the QGP and HG phase
(see Ref. [2] for a recent review).

The $p_{T}$ spectrum of low-$p_{T}$ direct photons from Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV, first measured by PHENIX [3], shows a clear
excess above the hard-scattering contribution estimated from $p$$+$$p$
measurements for $p_{T}$ below 3 GeV/$c$. Followup measurements by PHENIX have
established that low-$p_{T}$ direct-photon emission also shows a large
anisotropy with respect to the reaction plane [4, 5], and that the yield
increases faster than $N_{\rm part}$ or $dN_{\rm ch}/d\eta$ as a function of
the centrality of the collision [6]. Low-$p_{T}$ direct photons in Au$+$Au
collisions at 200 GeV have also been measured by STAR [7] using the same basic
method as [3], but different detection techniques. Quantitatively, STAR
results appear to be a factor 3 smaller than those from PHENIX. This tension
has not yet been resolved. Furthermore, low $p_{T}$ photons have been measured
in Au$+$Au at lower $\sqrt{s_{{}_{NN}}}$ of 39 GeV and 62.9 GeV by PHENIX [8],
and in Pb$+$Pb at $\sqrt{s_{{}_{NN}}}=2760$ GeV by ALICE [9].

The excess of direct photons in A$+$A collisions, in the low-$p_{T}$ regime,
is usually interpreted as the contribution of thermal radiation emitted from
the expanding and cooling QGP and HG phase. Due to the rapid anisotropic
expansion of the system, the radiation is Doppler shifted. Over the years,
several theoretical models have been developed and refined to describe the
production rates and space-time evolution of thermal photons in relativistic
heavy-ion collisions [10, 11, 12, 13, 14, 15, 16, 17]. While most of these
state-of-the-art models describe the data qualitatively, they fall short of
simultaneously describing all the features of the data quantitatively. To
describe the large yield, early emission at high temperatures is favored,
while sufficient build up of collective motion is required to explain the
large anisotropy, thereby favoring late-stage emission. This tension, often
termed as the “direct-photon puzzle”, hints at an incomplete understanding of
the different sources and mechanisms of direct-photon production. This has
triggered more thoughts on other unconventional photon sources, such as
emission from the pre-equilibrium stage, strong magnetic field effects, etc.
[18, 19, 20, 21, 22, 23, 24, 10]. For that very reason this paper refers to
the low-$p_{T}$-excess direct photons as “nonprompt” instead of “thermal”.

To provide new insights and further understandings, the PHENIX collaboration
presents results from the high-statistics 2014 Au$+$Au data at
$\sqrt{s_{{}_{NN}}}=200$ GeV. With a 10-fold increase in statistics compared
to previously published results, differential direct-photon measurements as
functions of $p_{T}$ and system size over a broad $p_{T}$ range from 0.8–10
GeV/$c$ and in 10% centrality classes are discussed. A new algorithm, which
utilizes the silicon-vertex detector (VTX) as the conversion material for
photons, is developed for this analysis.

The paper is organized as follows: Section II presents the experimental setup
relevant to this measurement and the algorithm to reconstruct the conversion
photons. Section III describes the double-ratio method to determine the
direct-photon excess ratio, $R_{\gamma}$, and gives details of the
experimental measurement. Section IV investigates the systematic
uncertainties. Section V discusses the results. Section VI presents the
summary and conclusions. Finally, there are two appendices: Appendix A
discusses the event mixing procedures and their validity, while Appendix B
describes the Monte-Carlo (MC)-sampling method used to derive the final
systematic uncertainties on the direct-photon yield.

##  II Experimental setup and photon measurements

###  II.1 PHENIX 2014 Au$+$Au $\sqrt{s_{{}_{NN}}}=200$ GeV data set

In 2014, a total of 19 billion Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=200$
GeV were recorded by the PHENIX detector at the Relativistic Heavy Ion
Collider (RHIC) with a minimum-bias (MB) trigger, based on the response of two
beam-beam counters (BBC) [25]. The BBCs are located on either side of the
interaction point along the beam axis at $z={\pm}1.44$ m with a pseudorapidity
coverage of $3.1<|\eta|<3.9$ and full $2\pi$ azimuthal acceptance. The MB
trigger requires a coincident signal in both BBCs. Each BBC, comprising 64
Čerenkov counters, measures the total number of charged particles produced
during the collision within its acceptance. The charged-particle multiplicity
is used to divide the MB events into different centrality classes; 0%–10%
corresponds to the most central collisions which produces the largest number
of charged particles, while 80%–93% corresponds to peripheral collisions with
only a small number of charged particles. The BBCs also utilize the arrival
time of the produced particles on each side to determine the collision vertex
along the beam direction.

Figure 1: (a) The beam view of the PHENIX central-arm spectrometer for the
year 2014. (b) A magnified view of the silicon-vertex detector. The solid
curves correspond to the electron and positron tracks from photon conversion.

The direct-photon measurement, presented here, is based on the tracking and
identification of electrons and positrons from photon conversions in the
detector material and the direct calorimetric measurement of photons in the
two PHENIX central arm spectrometers shown in Fig. 1 [26].

The VTX [27] comprises four silicon layers at nominal radii of 2.6, 5.1, 11.8,
and 16.7 cm. In the beam direction, the active area covers approximately $\pm
11$ cm for the innermost layer and $\pm 19$ cm for the outer layer. The VTX is
not used as an active detector in the measurement. However, it acts as the
photon converter, which is critical for this analysis. The total material
thickness of the VTX in terms of radiation length, $X_{0}$, is
${\approx}13\%X_{0}$. Events are selected with a $z$ vertex within $\pm 10$ cm
of the nominal interaction point. After applying quality assurance criteria, a
total of $1.25\times 10^{10}$ events are analyzed.

The central-arm spectrometers have three major parts: A charged-particle
tracking system [28, 29], particle-identification detectors [30], and
electromagnetic calorimeters (EMCal) [31]. Each arm covers 90∘ in the
azimuthal direction with $|\eta|<0.35$. The tracking system is located
$\approx$2.2 m from the beam axis outside of an axial magnetic field. The main
tracking detectors are drift chambers (DC) and pad chambers (PC1). The DC
provides a precise measurement of the transverse momentum for charged
particles with $\mbox{$p_{T}$}>0.2$ GeV/$c$. The PC1 measures the momentum
along beam direction, $p_{z}$. The effective momentum resolution of the
central-arm tracking system, for this analysis, is
$\sigma_{p}/p=0.8\%{\oplus}2\%\,p$ [GeV/$c$], where $p$ is the transverse
momentum of the track.

Charged tracks are identified as electrons or positrons with a ring-imaging
Čerenkov detector (RICH). The RICH has a $CO_{2}$ gas radiator with a low
radiation threshold for electrons (0.018 GeV/$c$) and a relatively high
threshold for charged pions ($>4.87$ GeV/$c$). Requiring a signal in at least
two phototubes in the focal plane of the RICH at the expected ring location
effectively separates electrons below 5 GeV/$c$ from charged hadrons. A
further matching of the momentum, $p$, of the charged track to the energy,
$E$, as measured in the EMCal within $-2\sigma_{E/p}<E/p<5{\sigma_{E/p}}$
removes most hadrons remaining in the sample. Here $\sigma_{E/p}$ is the
momentum-dependent resolution of the energy to momentum ratio, $E/p$.

For the calorimetric identification of photons, two types of calorimeters are
used, lead-scintillator (PbSc) and lead-glass (PbGl). The PbSc EMCal, which
covers 3/4 of the acceptance, is a sandwich sampling detector, also referred
to as a Shashlik type calorimeter. Based on the widths of reconstructed
$\pi^{0}$ mass through the $\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ decay,
the effective photon-energy resolution in this analysis is
$\sigma_{E}/E=8.1\%/\sqrt{E~{}{\rm[GeV]}}{\oplus}5.0\%$. The remaining 1/4 of
the acceptance is covered by the PbGl EMCal, which is a homogeneous Čerenkov-
type detector with an effective resolution of
$\sigma_{E}/E=8.7\%/\sqrt{E~{}{\rm\,[GeV]}}{\oplus}5.8\%$. Nominal cuts on the
energy threshold ($E>500$ MeV) and shower shape ($\chi^{2}<3$) are applied to
identify photons.

###  II.2 External photon conversions in the VTX

Earlier measurements of direct photons from PHENIX are based on three
different strategies to measure photons in A$+$A collisions. The calorimeter
method is used to measure photons with $p_{T}$ of several GeV/$c$ via their
energy deposited in the EMCal [4]. To access lower $p_{T}$, $e^{+}e^{-}$ pairs
from photon conversions are reconstructed with the tracking system. These
$e^{+}e^{-}$ pairs are either from “internal” conversions of virtual photons
emitted from the collision [3] or “external” conversions of photons in the
detector material [6].

Figure 2:  Artificial $e^{+}e^{-}$ pair mass for external photon conversions.
Each curve corresponds to a different radius region, which roughly maps to the
locations of beam-pipe, layers 1 (B0) through 4 (B3) of the VTX, and the VTX
(CF) carbon-fiber enclosure.

Here, external photon conversions at the VTX detector are reconstructed from
$e^{+}e^{-}$ pairs. The VTX material is distributed between 2 and 25 cm along
the radial direction. Depending on the conversion point, a different amount of
magnetic field is traversed by the $e^{+}e^{-}$ pair. In the standard PHENIX
track-reconstruction algorithm, the tracking system measures a part of the
trajectory outside of the magnetic field at a radial position of $\approx$2.2
m. The momentum vector is determined by assuming that the particle originates
at the event vertex. This assumption is incorrect for the $e^{+}e^{-}$ pairs
from conversions in the VTX material. Both $e^{+}$ and $e^{-}$ traverse a
smaller $\int{Bdl}$ than tracks from the vertex and thus the azimuthal
component of the momentum vector is mismeasured in opposing directions,
leading to an artificial opening angle and mismeasured mass of the
$e^{+}e^{-}$ pair. Because the magnetic field in the region of the VTX
detector is approximately constant at 0.9 Tesla, the artificial mass acquired
is proportional to the radial location of the conversion point. Fig. 2 shows
the mass of $e^{+}e^{-}$ pairs simulated with the geant3 PHENIX-detector
simulation [32], different curves represent photon conversions in different
VTX layers. The $m_{e^{+}e^{-}}$ is larger for conversions at larger radii
with most conversions occurring in the third and fourth layers of the VTX,
where the material budget is the largest.

To correctly reconstruct and identify photon conversions at different VTX
layers, a new track-reconstruction algorithm is developed. The new algorithm
relies on the fact that the $e^{+}$ and $e^{-}$ from a conversion have the
same origin and that their momenta were initially parallel in radial
direction. This additional constraint eliminates the need to assume the origin
of the track.

Figure 3:  Schematic view of the conversion-reconstruction algorithm. The two
tracks are reconstructed to the same radius r. $\delta\phi$ is the azimuthal-
angular difference between the two tracks for a given reconstruction radius.
$\delta\phi$ is zero at the conversion point.

The algorithm is illustrated in Fig. 3. For all radii between 0 and 30 cm, all
possible momenta of the $e^{+}$ and $e^{-}$ are scanned to identify the
azimuthal location $\phi_{\pm}$ at which the track is perpendicular to the
circle of the given radius, or in other words points back radially to the
event vertex. The conversion point is determined by finding the radius for
which the difference of the azimuthal angles of the $e^{+}e^{-}$ pair,
$\delta\phi=\phi_{+}{-}\phi_{-}$, becomes zero. If such radius exists, the
pair is identified as a conversion candidate at the location $(\phi_{\rm
conv},r_{\rm conv})$, where $\phi_{\rm conv}$ is the azimuthal angle of the
conversion point, reconstructed with a resolution of $\approx$4 mrad, and
$r_{\rm conv}$ is the radial position reconstructed with a resolution of
$\approx$2 cm.

##  III Data Analysis

###  III.1 Double-ratio tagging method

The number of direct photons emitted in a Au$+$Au collision is small compared
to the number of photons from hadron decays. To make a precise measurement of
the direct-photon yield, a tagging method is employed [6], which measures the
ratio, $R_{\gamma}$, of all photons, referred to as inclusive photons,
$\gamma^{\rm incl}$, to the photons from hadron decays, $\gamma^{\rm hadr}$.
The ratio $R_{\gamma}$ is evaluated as double ratio, such that most systematic
uncertainties cancel explicitly. The $R_{\gamma}$ given in Eq. 1 features
three main terms:

$R_{\gamma}=\frac{\gamma^{\rm incl}}{\gamma^{\rm
hadr}}=\frac{\left(\frac{\gamma^{\rm
incl}}{\gamma^{\pi^{0}}}\right)}{\left(\frac{\gamma^{\rm
hadr}}{\gamma^{\pi^{0}}}\right)}=\frac{\langle\epsilon_{\gamma}f\rangle\left(\frac{N^{\rm
incl}_{\gamma}}{N_{\gamma}^{\pi^{0},\rm tag}}\right)_{\rm
Data}}{\left(\frac{\gamma^{\rm hadr}}{\gamma^{\pi^{0}}}\right)_{\rm Sim}}~{},$
(1)

  * (i)

The ratio of measured photon yields
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$ is the number of
measured conversion photons in a given $p_{T}$ bin, divided by the sub-sample
of those conversion photons that are tagged by a second photon as resulting
from a $\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ decay. This quantity is
measured in bins of fixed conversion photon $p_{T}$.

  * (ii)

The conditional acceptance and efficiency $\langle\epsilon_{\gamma}f\rangle$
is the conditional probability to detect and reconstruct the second $\pi^{0}$
decay photon with the EMCal, given that the first decay photon was
reconstructed as $e^{+}e^{-}$ pair from a photon conversion. The probability
is averaged over all parent $\pi^{0}$ $p_{T}$ that can contribute to the given
conversion photon $p_{T}$.

  * (iii)

The cocktail ratio $\gamma^{\rm hadr}$/$\gamma^{\rm{\pi^{0}}}$ is the ratio of
all photons from hadron decays over only those photons from $\pi^{0}$ decays.

The following sections discuss how each term is determined.

###  III.2 Ratio of the measured photon yields
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$

Electrons and positrons in a given event are combined to $e^{+}e^{-}$ pairs
and conversion candidates are selected with appropriate cuts, which results in
a foreground sample of $e^{+}e^{-}$ pair ${\rm FG}^{\rm{ee}}$. All conversion
candidates in a conversion photon $p_{T}$ bin, are combined with all photon
showers in the EMCal above an energy threshold, $E_{cut}$. The invariant mass
$m_{ee\gamma}$ is calculated and all combinations that lie in a mass window
around the $\pi^{0}$ mass are considered as candidates for tagged photons
${\rm FG}^{ee\gamma}$. Due to the large particle multiplicity in Au$+$Au
collisions, there are many false combinations where the electron, positron or
photon are not from the same source. These background pairs must be subtracted
statistically to obtain the signals of interest ${\rm SG}^{\rm{ee}}$ and ${\rm
SG}^{ee\gamma}$.

For $e^{+}e^{-}$ pairs, there are two possible combinations, signal pairs of
interest ${\rm SG}^{\rm{ee}}$ and uncorrelated background ${\rm BG}^{\rm{ee}}$
pairs where the electron and positron are from different sources. Their sum
constitutes the foreground ${\rm FG}^{\rm{ee}}$:

$\mbox{${\rm FG}^{\rm{ee}}$}=\mbox{${\rm SG}^{\rm{ee}}$}+\mbox{${\rm
BG}^{\rm{ee}}$}.$ (2)

When the $e^{+}e^{-}$ pairs are combined with photons to $e^{+}e^{-}\gamma$
combinations, both types of $e^{+}e^{-}$ pairs are combined with photons that
are either correlated or uncorrelated with the pair:

$\mbox{${\rm FG}^{ee\gamma}$}=\mbox{${\rm SG}^{ee\gamma}$}+\mbox{${\rm
BG}_{\rm uncorr}^{ee\gamma}$}+\mbox{${\rm BG}_{\rm corr}^{ee\gamma}$}.$ (3)

Introducing $i,j,k$ as the source of the positron, electron, and photon,
respectively, the terms in Eq. 3 are:

  * (1)

The first term is the signal of interest with positron, electron, and photon
from the same source ($i=j=k$).

  * (2)

The second term represents the cases where the $e^{+}e^{-}$ pair is combined
with uncorrelated photons. This includes the case ($i=j\neq k$), where the
$e^{+}e^{-}$ pair is correlated and randomly combined with a $\gamma$ as well
as the case ($i\neq j\neq k$) where all three are from different sources.

  * (3)

The third term represents cases ($(i\neq j=k)\vee(j\neq i=k)$), where the
$e^{+}e^{-}$ pair is not from the same source but the $\gamma$ is correlated
with either the $e^{+}$ or the $e^{-}$.

Each of the background terms is determined with different event-mixing
procedures, which were developed using the MC method. The event-mixing
procedures and their validity are discussed in detail in Appendix A.

####  III.2.1 Determination of the inclusive photon yield
$N_{\gamma}^{\rm{incl}}$

Photons that convert at the VTX detector are selected by pairing electron and
positron tracks to $e^{+}e^{-}$ pairs. All pairs are required to have a valid
conversion point at a radial location within the VTX detector, between 1 and
29 cm. In addition, both tracks need to match in the beam direction within
$|\Delta z|<4$ cm. The invariant mass distribution of the selected
$e^{+}e^{-}$ conversion pairs is shown in Fig. 4 for the $p_{T}$ range
$1.0<\mbox{$p_{T}$}<1.2$ GeV/$c$. The four panels correspond to four different
centrality selections. Each panel shows the same peak structure, which is
characteristic of the multilayer structure of the VTX detector.

Figure 4: Mass distribution, $m_{e^{+}e^{-}}$, of the $e^{+}e^{-}$ pairs after
conversion selection cuts are applied. All four panels are for the same
$p_{T}$ range $1.0<\mbox{$p_{T}$}<1.2$ GeV/$c$ for four different centrality
selections (a) 0%–20%, (b) 20%–40%, (c) 40%–60% and (d) 60%–93%. Shown are the
foreground ${\rm FG}^{\rm{ee}}$, background ${\rm BG}^{\rm{ee}}$ and signal
${\rm SG}^{\rm{ee}}$.

The $e^{+}e^{-}$ pairs passing the conversion selection criteria contain
uncorrelated $e^{+}e^{-}$ pairs, where the $e^{+}$ and $e^{-}$ are from
different sources. These backgrounds are also shown in Fig. 4. Because of its
combinatorial nature, the background to foreground ratio increases towards
more central-event selections. An event-mixing technique is used to estimate
and subtract this background (see Appendix A for details). In this technique,
an $e^{+}$ from event A is paired with an $e^{-}$ from another event B to
produce the random $e^{+}e^{-}$ pair sample. To assure the events A and B have
similar topological characteristics, it is required that both events:

  * (a)

are from the same 10$\%$ centrality selection,

  * (b)

have their interaction vertex within $\Delta z=2.5$ cm,

  * (c)

have their reaction planes aligned within $\Delta\phi=\pi/6$.

After the background is subtracted, $N_{\gamma}^{\rm{incl}}$ is calculated by
integrating the counts in the mass range from 0.04 to 0.12 GeV/$c^{2}$,
corresponding to layers 3 and 4 of the VTX. The analysis is repeated for bins
in $p_{T}$ and in centrality.

####  III.2.2 Tagged photon raw yield $N_{\gamma}^{\rm{\pi^{0},tag}}$

Next, the subset of $e^{+}e^{-}$ pairs in the $N_{\gamma}^{\rm{incl}}$ sample
that can be tagged as photons from a $\pi^{0}$ decay,
$N_{\gamma}^{\rm{\pi^{0},tag}}$, is determined. For a given event, each
$e^{+}e^{-}$ conversion candidate, in the mass window in which
$N_{\gamma}^{\rm{incl}}$ is counted, is paired with all reconstructed showers
in the EMCal with shower shape $\chi^{2}<3$ and energy larger than
$E_{cut}=0.5$ GeV, excluding those matched to the $e^{+}e^{-}$ pair itself.
The energy cut, together with the $p_{T}$ cut of 0.2 GeV/$c$ on the $e^{+}$
and $e^{-}$, constitutes an implicit asymmetry cut on the $\pi^{0}$ decay
photons that depends on the $p_{T}$ of the $\pi^{0}$. For all
$e^{+}e^{-}\gamma$ combinations, the invariant mass $m_{ee\gamma}$ is
calculated. This constitutes the foreground ${\rm FG}^{ee\gamma}$, for which
an example is given in Fig. 6 for the $e^{+}e^{-}$ pair in the $p_{T}$ range
$1.0<\mbox{$p_{T}$}<1.2$ GeV/$c$. The four panels (a) to (d) correspond to
four centrality selections 0%–20%, 20%–40%, 40%–60%, and 60%–93%,
respectively.

Despite the large background, the signal, ${\rm SG}^{ee\gamma}$, is clearly
visible as a peak around the $\pi^{0}$ mass, even in panel (a), which is the
most central event selection. As discussed above, the background ${\rm
BG}^{ee\gamma}$ has two components:

$\mbox{${\rm BG}^{ee\gamma}$}=\mbox{${\rm BG}_{\rm
uncorr}^{ee\gamma}$}+\mbox{${\rm BG}_{\rm corr}^{ee\gamma}$},$ (4)

for which the shape and normalization are obtained from the event-mixing
procedures described in Appendix A. The results are also shown in Fig. 6. The
uncorrelated background, ${\rm BG}_{\rm uncorr}^{ee\gamma}$, is given in
panels (a) to (d). The much smaller correlated background, ${\rm BG}_{\rm
corr}^{ee\gamma}$, is only revealed after ${\rm BG}_{\rm uncorr}^{ee\gamma}$
is subtracted from the foreground, ${\rm FG}^{ee\gamma}$. The differences are
given in panels (e) to (h) for central to peripheral events, respectively.
Figure 6 indicates that the correlated background decreases with centrality
from $\mbox{${\rm BG}_{\rm corr}^{ee\gamma}$}/(\mbox{${\rm
FG}^{ee\gamma}$}-\mbox{${\rm BG}_{\rm uncorr}^{ee\gamma}$})=8.6\%$ in central
collisions to 0.5% in the most-peripheral collisions.

For the 0%–20% centrality selection, Fig. 6 shows the mass distributions
$m_{ee\gamma}$ for four different $e^{+}e^{-}$ pair $p_{T}$ ranges. The
representation is the same as for Fig. 6. Panels (a) through (d) all show a
clear peak around the $\pi^{0}$ mass. The backgrounds are the largest for low
$p_{T}$ and the most central events. As $p_{T}$ increases and the event
multiplicity decreases, the backgrounds are significantly reduced.

Figure 5: Mass distribution, $m_{ee\gamma}$, for $e^{+}e^{-}$ pairs with
$p_{T}$ from 1.0 to 1.2 GeV/$c$, for four centrality selection (a,e) 0%–20%,
(b,f) 20%–40% (c,g) 40%–60%, and (d,h) 60%–93%. Panels (a) through (d) show
the foreground ${\rm FG}^{ee\gamma}$ and the uncorrelated background ${\rm
BG}_{\rm uncorr}^{ee\gamma}$. Panels (e) through (h) show the difference
$\mbox{${\rm FG}^{ee\gamma}$}-\mbox{${\rm BG}_{\rm uncorr}^{ee\gamma}$}$,
together with the correlated background ${\rm BG}_{\rm corr}^{ee\gamma}$.

Figure 6: Mass distribution, $m_{ee\gamma}$, for $e^{+}e^{-}$ pairs from the
0%–20% centrality selection for four different $e^{+}e^{-}$ pair $p_{T}$
regions, (a,e) 0.8 to 1.0, (b,f) 1.4 to 1.6, (c,g) 2.0 to 2.5 and (d,h) 3.5 to
4 GeV/$c$. Panels (a) through (d) show the foreground ${\rm FG}^{ee\gamma}$
and the uncorrelated background ${\rm BG}_{\rm uncorr}^{ee\gamma}$. Panels (e)
through (h) show the difference $\mbox{${\rm FG}^{ee\gamma}$}-\mbox{${\rm
BG}_{\rm uncorr}^{ee\gamma}$}$, together with the correlated background ${\rm
BG}_{\rm corr}^{ee\gamma}$.

Because of the complexity of the particle correlations present in the real
Au$+$Au collision events, including effects of collective expansion, jet
production, hadron decays, etc., there is a small residual background that is
not captured by the event-mixing procedure. To remove this background, a low-
order polynomial, $f_{ee\gamma}$, is fitted to the ratio $(\mbox{${\rm
FG}^{ee\gamma}$}-\mbox{${\rm BG}^{ee\gamma}$})/\mbox{${\rm BG}_{\rm
uncorr}^{ee\gamma}$}$ in the mass range 0.05–0.08 and 0.23–0.45 GeV/$c^{2}$.
This function is used to correct ${\rm BG}_{\rm uncorr}^{ee\gamma}$ before it
is finally subtracted. Thus, the final distribution for
$N_{\gamma}^{\rm{\pi^{0},tag}}$ is:

$\mbox{$N_{\gamma}^{\rm{\pi^{0},tag}}$}=\mbox{${\rm
FG}^{ee\gamma}$}-\mbox{${\rm BG}_{\rm
corr}^{ee\gamma}$}-(1+f_{ee\gamma})\times\mbox{${\rm BG}_{\rm
uncorr}^{ee\gamma}$}.$ (5)

An example of the residual background is given in Fig. 7 for the $e^{+}e^{-}$
pair $p_{T}$ range of 1 to 1.2 GeV/$c$ and 0%–20% centrality selection. In
panel (a), ${\rm FG}^{ee\gamma}$ with all the background components are shown.
Panel (b) gives a second-order polynomial fit to the ratio $(\mbox{${\rm
FG}^{ee\gamma}$}-\mbox{${\rm BG}^{ee\gamma}$})/\mbox{${\rm BG}_{\rm
uncorr}^{ee\gamma}$}$ ratio, $f_{ee\gamma}$, which is used to determine the
residual background. Due to the unfavorably small signal-to-background ratio
in this case, the residual background in the $\pi^{0}$ mass region is
$\approx$9.4%. The residual background quickly drops with $p_{T}$ and
centrality bins, for example as $p_{T}$ increases to 3 GeV/$c$, the residual
background reduces to 2.7%. For each $p_{T}$-centrality bin combination,
$N_{\gamma}^{\rm{\pi^{0},tag}}$ is extracted by integrating the number of
entries in a window around the $\pi^{0}$ peak
($0.09<\mbox{$m_{ee\gamma}$}<0.19$) GeV/$c^{2}$ after all background
subtractions are applied.

Figure 7: (a) An example for ${\rm FG}^{ee\gamma}$ and the various background
components after normalization in the indicated regions. (b) The ratio
$(\mbox{${\rm FG}^{ee\gamma}$}-\mbox{${\rm BG}^{ee\gamma}$})/\mbox{${\rm
BG}_{\rm uncorr}^{ee\gamma}$}$ and the polynomial fit to determine the
residual-background correction $f_{ee\gamma}$.

###  III.3 Conditional probability $\langle\epsilon_{\gamma}f\rangle$

The probability, $\langle\epsilon_{\gamma}f\rangle$, that the second photon is
in the acceptance and is reconstructed, given a conversion $e^{+}e^{-}$ pair
from a $\pi^{0}$ decay, is extracted from the single $\pi^{0}$ simulation. In
this simulation, individual $\pi^{0}$ are tracked through the PHENIX MC-
simulation framework. The $\pi^{0}$ are generated with the published $p_{T}$
spectrum (see Sec. III.4) and uniform in pseudorapidity, $\eta$, and azimuthal
angle, $\phi$.

The energy scale and resolution of the EMCal in the MC simulation is tuned as
closely as possible to resemble the one observed in data by comparing the mean
and width of the measured and simulated $\pi^{0}$ mass distribution. The
$\pi^{0}$ are reconstructed through the
$\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ decay channel. For this purpose an
asymmetry of less than 20% between the energies of the two decay photons was
applied to keep the two-photon energies similar.

In the single $\pi^{0}$ MC simulation, $e^{+}e^{-}$ pairs in the mass window
$0.04<\mbox{$m_{e^{+}e^{-}}$}<0.12$ GeV/$c^{2}$ are counted to determine
$N_{ee}^{\rm{\pi^{0}}}$, the number of reconstructed $e^{+}e^{-}$ pairs in a
given $e^{+}e^{-}$ pair $p_{T}$ bin. The sub-sample for which the second
photon of the $\pi^{0}$ decay is reconstructed as a shower in the EMCal is
counted as $N_{ee}^{\rm{\pi^{0}},tag}$. The value of
$\langle\epsilon_{\gamma}f\rangle$ is then determined as:

$\mbox{$\langle\epsilon_{\gamma}f\rangle$}=\frac{\mbox{$N_{ee}^{\rm{\pi^{0}},tag}$}}{{N_{ee}^{\rm\pi^{0}}}}.$
(6)

For the extraction of $N_{ee}^{\rm{\pi^{0}},tag}$ the presence of other
showers in the EMCal needs to be taken into account. This is done by embedding
the showers from the simulated single $\pi^{0}$ into the EMCal response from
Au$+$Au collisions at the tower level. The combined EMCal information is then
reclustered to form new showers. All of the showers that contain energy
deposited by the embedded singe $\pi^{0}$ (identified by the MC ancestry
information) are combined with the $e^{+}e^{-}$ pair.

Figure 8: Conditional probability $\langle\epsilon_{\gamma}f\rangle$ as a
function of $p_{T}$ in 0%–20%, 20%–40%, 40%–60% and 60%–93% centrality
classes.

Similar to the $N_{\gamma}^{\rm{\pi^{0},tag}}$ extraction from data, a
residual background subtraction is applied. This eliminates any remaining
background inside the $\pi^{0}$ counting window. The residual background is
estimated by a second order polynomial function fit in the mass range
0.05–0.08 and 0.23–0.45 GeV/$c^{2}$. This residual background mainly comes
from events where both decay photon convert to $e^{+}e^{-}$ pairs, and the
reconstructed conversion photon gets paired with the EMCal cluster of the
$e^{+}$ or $e^{-}$ from the other conversion. The extracted
$\langle\epsilon_{\gamma}f\rangle$ is shown in Fig. 8 as a function of the
$e^{+}e^{-}$ pair $p_{T}$ for the four centrality selections.

The increasing trend of $\langle\epsilon_{\gamma}f\rangle$ with increasing
conversion photon $p_{T}$ is partly due to the decrease in the opening angle
between the conversion photon and the second photon so that the second photon
is more likely to fall into the acceptance of the EMCal. Another important
factor is that the average energy of the second photon increases with
increasing conversion photon $p_{T}$, and hence, the efficiency of the energy
threshold cut increases towards higher $p_{T}$. The difference in
$\langle\epsilon_{\gamma}f\rangle$ between different centrality classes is
mainly related to the shower shape ($\chi^{2}$) selection, because the showers
are more distorted in central Au$+$Au collisions due to the larger detector
occupancy, resulting in more accidental overlaps from the underlying event,
and the centrality dependent parent $\pi^{0}$ $p_{T}$ distributions.

Table 1: Parameters for the modified Hagedorn function Eq. 7 to PHENIX data [33, 34, 35] from Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV.  centrality | $A$ | $a$ | $b$ | $p_{0}$ | $n$
---|---|---|---|---|---
| c(GeV/$c$)-2 | (GeV/$c$)-1 | (GeV/$c$)-2 | GeV/$c$ |
min.bias | 504.5 | 0.5169 | 0.1626 | 0.7366 | 8.274
0%–10% | 1331.0 | 0.5654 | 0.1945 | 0.7429 | 8.361
10%–20% | 1001.0 | 0.5260 | 0.1628 | 0.7511 | 8.348
20%–30% | 750.7 | 0.4900 | 0.1506 | 0.7478 | 8.229
30%–40% | 535.3 | 0.4534 | 0.1325 | 0.7525 | 8.333
40%–50% | 364.5 | 0.4333 | 0.1221 | 0.7385 | 8.261
50%–60% | 231.2 | 0.4220 | 0.1027 | 0.7258 | 8.220
60%–70% | 118.1 | 0.4416 | 0.0559 | 0.7230 | 8.163
70%–80% | 69.2 | 0.2850 | 0.0347 | 0.7787 | 8.532
80%–93% | 51.1 | 0.2470 | 0.0619 | 0.7101 | 8.453

###  III.4 Cocktail ratio $\gamma^{\rm hadr}$/$\gamma^{\rm{\pi^{0}}}$

The last ingredient to calculate $R_{\gamma}$ is the cocktail ratio
$\gamma^{\rm hadr}$/$\gamma^{\rm{\pi^{0}}}$ of photons from $\pi^{0}$, $\eta$,
$\omega$, and $\eta^{\prime}$ decays over those from $\pi^{0}$ decays. The
cocktail ratio is obtained using the PHENIX meson decay generator EXODUS,
which simulates mesons according to given input $p_{T}$ spectra, decays them
based on the known decay kinematics and branching ratios, and aggregates the
decay photons in the PHENIX detector acceptance.

The photons from $\pi^{0}$ decays are generated from distributions obtained by
fitting a modified Hagedorn function (Eq. 7) to charged pion [33] and neutral
pion [34, 35] data measured by PHENIX.

$E\frac{d^{3}N}{dp^{3}}=A\
\Big{(}e^{-(ap_{T}+bp_{T}^{2})}+\frac{\mbox{$p_{T}$}}{p_{0}}\Big{)}^{-n}.$ (7)

The fit parameters are summarized in Table 1 for MB collisions, as well as for
nine centrality bins. The $\eta$ meson $p_{T}$ spectrum is obtained by
multiplying the $\pi^{0}$ spectrum with the $\eta/\mbox{$\pi^{0}$}$ ratio. The
ratio is extracted from an analysis of world data [36], which demonstrates a
universal value at high $p_{T}$ consistent with 0.487$\pm$0.024, independent
of collision energy, system size or centrality. The work takes into account
the fact that at lower $p_{T}$ the ratio deviates from $m_{T}$ scaling, and
that there are centrality dependent changes of $\eta/\mbox{$\pi^{0}$}$ due to
radial flow. The contribution from $\omega$ and $\eta^{\prime}$ decays are
based on $p_{T}$ distributions using the $\pi^{0}$ spectrum and replacing by
$f$($\sqrt{p_{T}+m^{2}_{\rm meson}-m^{2}_{\pi^{0}}}$). The normalization of
$\omega$ and $\eta^{\prime}$ are fixed at $p_{T}$ = 5 GeV/$c$ to $0.9\pm 0.06$
and $0.25\pm 0.075$, respectively [6]. The cocktail ratio $\gamma^{\rm
hadr}$/$\gamma^{\rm{\pi^{0}}}$ is shown in Fig. 9.

Figure 9: Cocktail ratio as a function of $p_{T}$ in the most central (0%–20%)
and the most peripheral (60%–93%) centrality classes.

##  IV Systematic uncertainties

This section describes the sources of systematic uncertainties for each of the
three components for the calculation of $R_{\gamma}$. The systematic
uncertainties are categorized into three types according to the correlation
between the measured data points:

  * •

Type A: No (or unknown) correlation between data points – uncertainties on the
individual data points can fluctuate independently, in the same way as the
statistical uncertainties.

  * •

Type B: The uncertainties are correlated between data points – the fluctuation
of each data point can be determined by the fluctuation of the neighboring
points.

  * •

Type C: A special form of type B uncertainty – every data point fluctuates
with the exact same fraction.

In the final results, type A systematic uncertainties are combined with the
statistical uncertainties and type B and C are combined to obtain the total
systematic uncertainty.

The following subsections discuss the major individual sources contributing to
the systematic uncertainties on $R_{\gamma}$ and on the direct-photon yield.
All contributions are summarized in Table 2 and depicted in Fig. 11 and Fig.
11 as functions of $p_{T}$ for $R_{\gamma}$ and $\gamma^{\rm dir}$. The final
systematic uncertainties on $\gamma^{\rm dir}$ and on all quantities derived
from $\gamma^{\rm dir}$ are determined using the error-sampling method
discussed in Appendix B

Figure 10: Systematic uncertainties of $R_{\gamma}$ as a function of
conversion photon $p_{T}$ in 0%–20%, 20%–40%, 40%–60% and 60%–93% centrality
bins. The black curve corresponds to total uncertainty, and colored curves
correspond to individual sources. The lines representing uncertainties from
the energy scale and the conversion loss overlap at 3%, so do the lines
representing uncertainties from the $\gamma$ reconstruction efficiency,
acceptance and input $p_{T}$ spectra.

Figure 11: Systematic uncertainties of $\gamma^{\rm hadr}$ as a function of
photon $p_{T}$.

Table 2: Summary of systematic uncertainties for $R_{\gamma}$ and $\gamma^{\rm dir}$. Uncertainties for which ranges are given vary with $p_{T}$. For details see Figs. 11 and 11. Observable | Factor | Source | correlation | correlation | 0%–20% | 20%–40% | 40%–60% | 60%–93%
---|---|---|---|---|---|---|---|---
|  |  | in $p_{T}$ | in centrality |  |  |  |
$R_{\gamma}$ |  $N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$ |  $N_{\gamma}^{\rm{incl}}$ purity | Type B | Type B | $<1\%$ | $<1\%$ | $<1\%$ | $<1\%$
|  |  $N_{\gamma}^{\rm{\pi^{0},tag}}$ residual background | Type A | Type A | 1.5%–4.5% | 0.5%–4% | 0.5%–4% | 0.5%–4%
|  |  $N_{\gamma}^{\rm{\pi^{0},tag}}$ event mixing | Type B | Type B | 1.5% | 1.5% | 1.5% | 1.5%
| $\langle\epsilon_{\gamma}f\rangle$ | energy scale | Type B | Type B | $3\%$ | $3\%$ | $3\%$ | $3\%$
|  | conversion loss | Type C | Type C | $3\%$ | $3\%$ | $3\%$ | $3\%$
|  |  $\gamma$ efficiency | Type B | Type A | $<1.4\%$ | $<1\%$ | $<1\%$ | $<1\%$
|  | active area & acceptance | Type C | Type C | $1\%$ | $1\%$ | $1\%$ | $1\%$
|  | input $\pi^{0}$ $p_{T}$ spectra | Type B | Type A | $1\%$ | $1\%$ | $1\%$ | $1\%$
| $\gamma^{hadr}/\gamma^{\pi^{0}}$ | $\eta/\pi^{0}$ | Type B | Type C | 1–1.5% | 1–1.5% | 1–1.5% | 1–1.5%
|  | $\omega,\eta^{\prime}$ | Type B | Type C | $<1\%$ | $<1\%$ | $<1\%$ | $<1\%$
| $\gamma^{hadr}$ | input $\pi^{0}$ $p_{T}$ spectrum | Type B | Type A | 10%–24% | 10%–24% | 10%–25% | 10%–24%

###  IV.1 Systematic uncertainties on
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$

####  IV.1.1 Purity of the conversion photon sample

Due to the high multiplicity of photons produced in Au$+$Au collisions, the
background in the conversion sample from uncorrelated $e^{+}e^{-}$ pairs can
be as large as 10% for the most central collisions and the lowest $p_{T}$ from
0.8 to 1.0 GeV/$c$. This background is subtracted statistically with a certain
accuracy. To estimate the effect on the final results, significantly more and
less stringent conversion selection cuts were applied, hence, increasing or
reducing the purity. The value of $\langle\epsilon_{\gamma}f\rangle$
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$, obtained from the
different cuts, varies by less than 1%. This range is quoted as systematic
uncertainty due to the limited purity of the conversion sample.

####  IV.1.2 $\pi^{0}$ yield extraction

One of the main sources of systematic uncertainty on the $R_{\gamma}$
measurement is the tagged photon or $\pi^{0}$ yield extraction. The
uncertainty of $\pi^{0}$ yield extraction arises from two sources: (i) from
the residual background subtraction, which is highly correlated with the
statistical accuracy of the mixed event background normalization, and (ii)
imperfect description of the large backgrounds using event-mixing techniques.

To evaluate the size of the uncertainty from the residual background
subtraction, different estimates are compared. These include using different
functional forms for the fit and different fit ranges to anchor the residual
background fit. In addition, the counting window for $\pi^{0}$ signal
extraction is varied. This gives a spread of
$\langle\epsilon_{\gamma}f\rangle$
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$ values in each
$p_{T}$ and centrality bin. The standard deviation of the spread is quoted as
the uncertainty. Due to the correlation with the statistical accuracy of the
foreground in the background region, this uncertainty depends on $p_{T}$ and
centrality.

To test the validity of the event-mixing techniques, an MC simulation with
high multiplicity $\pi^{0}$ events is performed. Details are discussed in
Appendix A. The simulation shows that
$N_{\gamma}^{\rm{\pi^{0},tag}}$/$\langle\epsilon_{\gamma}f\rangle$ can be
determined with the event-mixing technique to better than 1.5%.

###  IV.2 Systematic uncertainty on $\langle\epsilon_{\gamma}f\rangle$

####  IV.2.1 Energy Scale

The accuracy of the energy scale of the EMCal is the main source of systematic
uncertainties in the $\langle\epsilon_{\gamma}f\rangle$ evaluation. Because of
the energy threshold cut, the second photon is reconstructed only for
$\approx$25% of the $e^{+}e^{-}$ pairs with the lowest $p_{T}$, even though
the photon was in the EMCal acceptance. Any potential mismatch of the energy
scale between the simulation and real data will cause
$\langle\epsilon_{\gamma}f\rangle$ to be off; a higher (lower) energy scale in
simulation will lead to an underestimate (overestimate) of
$\langle\epsilon_{\gamma}f\rangle$. As mentioned earlier, to improve the
accuracy, the EMCal response in the simulation is carefully tuned to the data
using the $\pi^{0}$ mass measurement in the
$\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ decay channel. The tuning includes
scaling the MC energy scale by 0.3% and 2.2% for the PbSc and PbGl
calorimeters, respectively. In addition, the nonlinearity of the energy
response is adjusted by up to 5% at the lowest measured energies. After the
tuning, the $\pi^{0}$ peak positions in data and MC are consistent to better
than 1%. Considering the additional uncertainty due to the adjustment of the
nonlinearity, the energy scale is known to better than 2%. Changing the energy
scale by $\pm 2$% introduces a systematic uncertainty on
$\langle\epsilon_{\gamma}f\rangle$ of 3% at low $p_{T}$ and decrease towards
high $p_{T}$. The uncertainty on the energy resolution has a negligible
effect.

####  IV.2.2 Conversion Photon Loss

Another important source of systematic uncertainty on
$\langle\epsilon_{\gamma}f\rangle$ is related to the probability that the
second photon converts to an $e^{+}e^{-}$ pair before reaching the EMCal.
Depending on the location of the conversion point, the second photon may not
be properly reconstructed, thereby reducing
$\langle\epsilon_{\gamma}f\rangle$. To account for the “conversion loss”, the
material budget, i.e. thickness and location of material, implemented in the
simulation framework must accurately reflect reality. If there is a mismatch,
the probability for conversions to occur will be different and, hence,
$\langle\epsilon_{\gamma}f\rangle$ will by systematically off. As there is
essentially no magnetic field after the DC exit window, the $e^{+}e^{-}$ pair
from conversions between the DC and the EMCal will likely merge into one
shower in the EMCal. Therefore, the value of
$\langle\epsilon_{\gamma}f\rangle$ is most sensitive to differences in the
material budget of the VTX. Comparison of the available information about the
materials and their thickness for all detector subsystems, reveals that the
conversion probability in material within the magnetic field is known to
better than 3%, which directly translates into and uncertainty of $3\%$ on
$R_{\gamma}$.

####  IV.2.3 Photon Efficiency

An EMCal shower shape, $\chi^{2}$, cut is used to identify photon candidates
among the EMCal energy clusters and to reduce the number of hadrons in the
sample. Similar to the energy scale uncertainty, a difference between the
shower shape in simulation and the data will translate directly into a
systematic shift of $\langle\epsilon_{\gamma}f\rangle$. To evaluate this
uncertainty, the $\chi^{2}$ is varied simultaneously in both data and
simulation and $\langle\epsilon_{\gamma}f\rangle$
$N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$ is recalculated. It
changes by $1.4\%$ for 0.8–2 GeV/$c$ in the 0%–20% centrality bin and by less
than 1$\%$ for all the other cases.

####  IV.2.4 Active area and geometric acceptance

Due to the limited geometrical acceptance of EMCal and some inactive areas,
the second photon is registered only for $\approx$35% of the $e^{+}e^{-}$
pairs at the lowest $p_{T}$. Therefore, the accuracy with which the acceptance
and dead areas are known will contribute to the systematic uncertainties on
$\langle\epsilon_{\gamma}f\rangle$. The uncertainty of the acceptance is the
result of the accuracy with which the radial location of the EMCal sectors can
be determined. The possible remaining offset leads to $<0.3\%$ difference in
acceptance along $\phi$ direction and $<0.9\%$ in $z$ direction. The dead
areas in the real EMCal are carefully matched to the MC simulation and the
accuracy of the dead area determination is found to be better than 0.6$\%$. It
is due to the cases when a tower malfunctioned only in a very small number of
events, and not masked out in the simulation. Combining all these effects, the
systematic uncertainty on $R_{\gamma}$ from the acceptance is set to 1%.

####  IV.2.5 Input $\pi^{0}$ distribution

Because $\langle\epsilon_{\gamma}f\rangle$ is averaged over all parent
$\pi^{0}$ $p_{T}$ that contribute to a given $e^{+}e^{-}$ pair $p_{T}$ bin,
the $p_{T}$ dependence of $\langle\epsilon_{\gamma}f\rangle$ is sensitive to
the shape of the $\pi^{0}$ distribution. The $\pi^{0}$ parent distribution was
determined for each centrality selection by a fit to the best available data
from Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV measured by the same
experiment [33, 34, 35]. The remaining uncertainty on
$\langle\epsilon_{\gamma}f\rangle$ is smaller than 1%.

###  IV.3 Systematic uncertainty on $\gamma^{\rm
hadr}$/$\gamma^{\rm{\pi^{0}}}$

The cocktail $\gamma^{\rm hadr}$/$\gamma^{\rm{\pi^{0}}}$ accounts for photons
from hadron decays, other than those from $\pi^{0}$, which contributes
$\approx$23% of the decay photons at high $p_{T}$. Of the additional decay
photons more than 80% are from the $\eta\rightarrow\gamma\gamma$ decay, hence
the accuracy with which $\eta/\pi^{0}$ is known will determine the systematic
uncertainties on $R_{\gamma}$ from this source. The $p_{T}$ and centrality
dependent upper and lower bounds on $\eta/\pi^{0}$ for Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV are taken from [36]. Together with the much
smaller uncertainty on the contribution from $\omega$ and $\eta^{\prime}$
decays, the systematic uncertainty on $R_{\gamma}$ is below 2% for the entire
$p_{T}$ range.

###  IV.4 Systematic Uncertainties on $\gamma^{\rm dir}$

Once $R_{\gamma}$ is determined, the direct-photon yield $\gamma^{\rm dir}$ is
calculated as:

$\mbox{$\gamma^{\rm dir}$}=(\mbox{$R_{\gamma}$}-1)\ \mbox{$\gamma^{\rm
hadr}$}.$ (8)

In addition to the uncertainties on $R_{\gamma}$, the uncertainty on
$\gamma^{\rm hadr}$ needs to be determined. These systematic uncertainties
have been studied in detail in [6]. The main sources of uncertainty come from
the accuracy with which the $\pi^{0}$ $p_{T}$ spectrum can be determined.
These largely cancel in $R_{\gamma}$, but propagate directly to $\gamma^{\rm
hadr}$. The input $\pi^{0}$ spectrum is based on measurements of charged
pions, and $\pi^{0}$ from different data taking periods (see Sec. III.4). Each
data set comes with its own systematic uncertainties, and in addition, the
differences between different measurements are of the order of 10% [37]. The
latter is the dominant uncertainty. The uncertainty on the spectra of other
mesons ($\eta$, $\eta^{\prime}$, $\omega$) also contributes to the uncertainty
on $\gamma^{\rm hadr}$, but to a much smaller extent.

##  V Results

###  V.1 Direct photon $R_{\gamma}$

Figure 12 shows $R_{\gamma}$ as function of photon $p_{T}$ for every 20%
centrality class. The vertical error bar on each point corresponds to the
statistical uncertainty, while the box gives the systematic uncertainty. The
new results are compared with all other published PHENIX results for Au$+$Au
at $\sqrt{s_{{}_{NN}}}=200$ GeV; these were obtained with different methods
and have largely independent systematic uncertainties. The open circles were
determined using the external conversion method deploying the HBD detector as
converter [6], the full squares are from a virtual photon internal conversion
measurement [4], and the open squares were measured with the EMCal alone [38].
All measurements agree well within their independent systematic uncertainties.

The 2014 data presented here have smaller statistical uncertainties than in
previous publications at RHIC due to the increased luminosity and
significantly larger amount of conversion material. The new results provide a
continuous measurement across a wide range of $p_{T}$ from 0.8 to 10 GeV/$c$.
This range has previously been covered by measurements done with different
techniques with different systematics. Up to 3 to 4 GeV/$c$ internal or
external photon conversions to $e^{+}e^{-}$ pairs have been used, while above
4 GeV/$c$ photons were measured in the EMCal. For all centrality selections,
$R_{\gamma}$ shows a significant excess that is rather constant below
$\approx$3 GeV/$c$. Beyond that, $R_{\gamma}$ increases with $p_{T}$, the
increase being most pronounced for central collisions, and $R_{\gamma}$
continuously decreases towards more peripheral collisions. This is expected as
phenomena such as jet quenching reduce the number of decay photons from hadron
decays in more central collisions, which in turn increases $R_{\gamma}$ [34,
35].

The high statistics of the 2014 data set allows to divide the data sample into
nine centrality bins, from 0%–10% to 80%–93%, 10% bins each, except for the
last one which is slightly larger. The resulting $R_{\gamma}$ are shown in
Fig. 13. Up to 50%–60% centrality, data from the earlier calorimeter
measurement [38] are also shown.

For most bins the overall shape of $R_{\gamma}$ as a function of $p_{T}$ is
similar to what is observed in Fig. 12, with a notable difference for panel
(i), which is the most-peripheral centrality 80%–93%. Below 5 GeV/$c$, the
most-peripheral Au$+$Au data show no significant excess above unity and are
very consistent with the direct-photon result from $p$$+$$p$ collisions, which
is also shown in panel (i).

Figure 12: The ratio, $\mbox{$R_{\gamma}$}=\mbox{$\gamma^{\rm
incl}$}/\mbox{$\gamma^{\rm hadr}$}$, as a function of conversion photon
$p_{T}$ in 0%–20%, 20%–40%, 40%–60% and 60%–93% centrality bins. The 2014
Au$+$Au data at $\sqrt{s_{{}_{NN}}}=200$ GeV are compared to results from
previous PHENIX publications for the same system and $\sqrt{s_{{}_{NN}}}$.
Figure 13: $R_{\gamma}$ of direct photons as a function of conversion photon
$p_{T}$ in 0%–10% to 80%–93% centrality bins.

The MC sampling method is used to calculate both the statistical and
systematic uncertainties on $\gamma^{\rm dir}$ and all quantities derived from
the direct photon $p_{T}$ spectra. This method propagates the error correctly
in the presence of unphysical values of $\mbox{$R_{\gamma}$}<1$ and $p_{T}$
and centrality dependent correlations of uncertainties; it is discussed in
detail in Appendix B.

Figure 14: Invariant yield of direct photons as a function of conversion
photon $p_{T}$ in 0%–10% to 80%–93% centrality bins. Figure 15: Invariant
yield of direct photons as a function of conversion photon $p_{T}$ in (a)
0%–20%, (b) 20%–40%, (c) 40%–60% and (d) 60%–93% centrality bins.

###  V.2 Direct-photon invariant yield

The direct-photon spectra are calculated from $R_{\gamma}$ and $\gamma^{\rm
hadr}$ using Eq. 8. The results for all 10% centrality selections are given in
Fig. 14. 111As the yields in the most-peripheral bin, 80%–93%, are mostly
upper limits on the measurement, this bin will not be included for estimation
of any further derived quantities in every 10% centrality selection. Figure 15
compares the direct-photon spectra with previous measurements, as shown in
broader centrality bins (a) 0–20%, (b) 20–40%, (c) 40–60%, and (d) 60–93%.
Each panel also presents the $N_{\rm coll}$-scaled pQCD calculation [12] and a
fit to direct-photon data from $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV [39,
40, 41]. The $p$$+$$p$ fit is performed with a pQCD-inspired functional form
[42]:

$\frac{d^{3}N}{d^{2}\mbox{$p_{T}$}dy}=\frac{A_{pp}}{(1+(\frac{p_{T}}{p_{0}})^{2})^{n}},$
(9)

where the parameters are $A_{pp}=1.60\\!\cdot\\!10^{-4}$ (GeV/$c$)-2,
$p_{0}=1.45$ GeV/$c$ and $n=3.3$. The error band around the central fit
function represents the uncertainty propagated from both the data and the
unknown true functional form of the spectrum down to very low $p_{T}$. The
$p$$+$$p$ fit and the pQCD calculation agree well above 2 GeV/$c$, and can be
used as an estimate for the prompt-photon contribution.

Figure 15 also shows that the direct-photon yield for $p_{T}$ larger than 5
GeV/$c$ is well described by the $N_{\rm coll}$-scaled $p$$+$$p$ result and
pQCD calculations, which confirms that the high-$p_{T}$ direct photons are
predominately from initial hard-scattering processes. Below 4–5 GeV/$c$ a
clear direct-photon excess develops above the prompt component, gradually
becoming larger towards lower $p_{T}$.

Figure 16: nonprompt direct-photon yield as a function of conversion photon
$p_{T}$ in (a) 0%–20%, (b) 20%–40%, (c) 40%–60%, and (d) 60%–93% centrality
bins.

###  V.3 Nonprompt direct-photon excess

To extract the direct-photon excess above the prompt-photon contribution, the
$N_{\rm coll}$ scaled $p$$+$$p$ fit is subtracted from the Au$+$Au data. This
excess is thought to be mostly the radiation that is emitted during the
collision from the hot-expanding fireball, and will be referred to here as
nonprompt direct-photon spectra. Figure 16 compares the nonprompt direct-
photon spectra to previously published results from Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV [6], which had significantly lower statistics.
The new 2014 data extend the coverage, both in $p_{T}$ and centrality.

Table 3: Inverse slopes fitted to the direct-photon spectra in different $p_{T}$ ranges, and for different centrality selections. For each centrality range, $N_{\rm coll}$ and $dN_{\rm ch}/d\eta$ values are quoted, which are taken from previous work [43, 44], except for the $dN_{\rm ch}/d\eta$ values for the two most peripheral bins. Those were extrapolated using a fit of the form $\mbox{$dN_{\rm ch}/d\eta$}=B(\mbox{$N_{\rm coll}$})^{\beta}$.  centrality | $dN_{\rm ch}/d\eta$ | $N_{\rm coll}$ |  $T_{\rm eff}$ (GeV/$c$) |  $T_{\rm eff}$ (GeV/$c$)
---|---|---|---|---
|  |  |  $0.8<\mbox{$p_{T}$}<1.9$ GeV/$c$ | $2<\mbox{$p_{T}$}<4$
0%–20% | $519.2\pm 26.3$ | $770.6\pm 79.9$ | $0.277\pm 0.017\ ^{+0.036}_{-0.014}$ |  $0.428\pm 0.031\ ^{+0.031}_{-0.030}$
20%–40% | $225.4\pm 13.2$ | $241.1\pm 28.4$ | $0.264\pm 0.010\ ^{+0.014}_{-0.007}$ |  $0.354\pm 0.019\ ^{+0.020}_{-0.030}$
40%–60% | $85.5\pm 8.0$ | $82.6\pm 9.3$ | $0.247\pm 0.007\ ^{+0.005}_{-0.004}$ |  $0.392\pm 0.023\ ^{+0.022}_{-0.022}$
60%–93% | $16.4\pm 2.8$ | $12.1\pm 3.1$ | $0.253\pm 0.011\ ^{+0.012}_{-0.006}$ |  $0.331\pm 0.036\ ^{+0.031}_{-0.041}$
0%–10% | $623.9\pm 32.2$ | $951\pm 98.5$ | $0.268\pm 0.024\ ^{+0.026}_{-0.012}$ |  $0.514\pm 0.061\ ^{+0.066}_{-0.039}$
10%–20% | $414.2\pm 20.2$ | $590.1\pm 61.1$ | $0.303\pm 0.024\ ^{+0.062}_{-0.021}$ |  $0.358\pm 0.033\ ^{+0.024}_{-0.035}$
20%–30% | $274\pm 14.8$ | $357.2\pm 35.5$ | $0.263\pm 0.011\ ^{+0.014}_{-0.007}$ |  $0.351\pm 0.024\ ^{+0.020}_{-0.030}$
30%–40% | $176.8\pm 11.6$ | $207.5\pm 21.2$ | $0.256\pm 0.011\ ^{+0.009}_{-0.005}$ |  $0.333\pm 0.024\ ^{+0.020}_{-0.032}$
40%–50% | $109.4\pm 9.1$ | $111.1\pm 10.8$ | $0.244\pm 0.009\ ^{+0.003}_{-0.005}$ |  $0.389\pm 0.029\ ^{+0.020}_{-0.021}$
50%–60% | $61.6\pm 7.1$ | $54.1\pm 7.9$ | $0.246\pm 0.010\ ^{+0.005}_{-0.005}$ |  $0.345\pm 0.031\ ^{+0.019}_{-0.032}$
60%–70% | 32 $\pm$ 5 | $24\pm 6$ | $0.261\pm 0.015\ ^{+0.020}_{-0.008}$ |  $0.319\pm 0.049\ ^{+0.037}_{-0.042}$
70%–80% | 16 $\pm$ 4 | $10\pm 3$ | $0.263\pm 0.016\ ^{+0.016}_{-0.007}$ |  $0.335\pm 0.044\ ^{+0.020}_{-0.035}$
80%–93% | 7 $\pm$ 2 | $4\pm 1$ | $-$ |  $-$

The data are very consistent in the region of overlap. In the range 0.8 to 1.9
GeV/$c$, the data are fitted with an exponential function and the results are
also shown on the panels of Fig. 16. The slope values are given in Table 3.
All centrality selections are consistent with an average inverse slope,
$T_{\rm eff}$, of ${\approx}0.260{\pm}0.011$ GeV/$c$. However, it is evident
from Fig. 16 that the nonprompt direct-photon spectra are not described by a
single exponential but rather have a continually increasing with $p_{T}$
inverse slope, $T_{\rm eff}$. Figure 17 brings this out more clearly where
each nonprompt direct-photon spectrum is divided by a fit with a fixed slope,
$T_{\rm eff}$ = 0.260 GeV/$c$. All centrality selections follow the same
trend. Over the $p_{T}$ range of up to 2 GeV/$c$ the ratios are consistent
with unity, but above 2 GeV/$c$, they start to rise monotonically.

Figure 17: Ratio of the yield of nonprompt direct photons over the exponential
fit result ($T_{\rm eff}$ fixed to 0.26 GeV/$c$) as a function of photon
$p_{T}$. Figure 18: $T_{\rm eff}$ as a function of charged-particle
multiplicity at midrapidity.

To quantify this changing slope, the nonprompt direct-photon spectra are
fitted with a second exponential function in the $p_{T}$ range from 2 to 4
GeV/$c$; the results are also included in Fig. 16. All data are consistent
with an average inverse slope of $0.376\pm 0.037$ GeV/$c$, which is
significantly larger than the slope observed below $p_{T}$ = 2 GeV/$c$. Above
4 GeV/$c$, the statistical and systematic uncertainties from the prompt-photon
subtraction become too large for a detailed analysis.

To establish any dependence on the system size, the nonprompt direct photon
spectra are determined for each 10% centrality bin, and subsequently fitted by
two exponential functions in the $p_{T}$ ranges $0.8<\mbox{$p_{T}$}<1.9$
GeV/$c$ and $2<\mbox{$p_{T}$}<4$ GeV/$c$. The resulting $T_{\rm eff}$ values
are tabulated in Table 3 and depicted in Fig. 18 as a function of $dN_{\rm
ch}/d\eta$. The figure also shows the average of the inverse slope values from
fitting Fig. 16. The $T_{\rm eff}$ values are consistent with a constant
value, independent of $dN_{\rm ch}/d\eta$. However, given the uncertainties on
the data, a possible increase of $T_{\rm eff}$ with $dN_{\rm ch}/d\eta$ can
not be excluded.

In addition to investigating the $p_{T}$ and system-size dependence of the
shape of the nonprompt direct-photon spectra, one can also look at the
dependence of the yield on system size and $p_{T}$. As reported previously,
the integrated direct-photon yield scales with $dN_{\rm ch}/d\eta$ to a power
$\alpha$ [8]:

$\frac{dN_{\gamma}}{dy}=\int_{p_{T,\rm{min}}}^{p_{T,\rm{max}}}\frac{dN_{\gamma}^{\rm
dir}}{d\mbox{$p_{T}$}dy}d\mbox{$p_{T}$}=A\times\left(\frac{dN_{\rm{ch}}}{d\eta}\right)^{\alpha},$
(10)

where all rapidity densities are densities at midrapidity. The direct-photon
spectra shown in Fig. 14 are integrated from $\mbox{$p_{T}$}_{,\rm{min}}=1$
GeV/$c$ to $\mbox{$p_{T}$}_{,\rm{max}}=5$ GeV/$c$ and plotted as a function of
$dN_{\rm ch}/d\eta$ in Fig. 19. They are in reasonable agreement with a
compilation of other direct-photon results [8, 45], also shown in the figure.
All data follow a trend similar to the $N_{\rm coll}$ scaled $p$$+$$p$ fit,
shown as band, but at a roughly 10 times larger yield. Scaling with $N_{\rm
coll}$ corresponds to $\alpha=1.25$ $\pm$ 0.02 [8]. The current high
statistics data allow for finer centrality binning and changes this picture
somewhat at the lowest and highest $dN_{\rm ch}/d\eta$. Fitting only the new
results in Fig. 19 gives a value of $\alpha=1.11\pm 0.02(\rm{stat})\
_{-0.08}^{+0.09}(\rm{sys})$. This value is lower, but consistent within
systematic uncertainties, with $\alpha=1.23$ $\pm$ 0.06 $\pm$ 0.18, found by
fitting all previously published PHENIX A$+$A data [45].

Note that the previous PHENIX measurements obtained the $\eta$ spectrum by
$m_{T}$-scaling the $\pi^{0}$ spectrum, while in the current measurement the
$\eta$ spectrum is obtained from the $\eta/$$\pi^{0}$ ratio using the world
data. There are significant differences between the two approaches in the
low-$p_{T}$ region [36]. Because the integration range starts at low $p_{T}$
and is wide (1–5 GeV/$c$), the power $\alpha$ is smaller than previously
published values, but is consistent within stated systematic uncertainties.
However, it is also consistent with unity within uncertainties.

Figure 19: Integrated direct-photon yield (1–5 GeV$/c$) versus charged-particle multiplicity at midrapidity. The present data is compared to a previous compilation of data from [8, 45] and the $N_{\rm coll}$ scaled fit to $p$$+$$p$ data. Also given are fits with Eq. 10 to different data; the solid line is a fit to the present data resulting in $\alpha=1.11\pm 0.02(\rm{stat})\ _{-0.08}^{+0.09}(\rm{sys})$, and the dashed line is from fitting previously published PHENIX data [45] that gave $\alpha=1.23$ $\pm$ 0.06 $\pm$ 0.18.  Table 4:  Scaling power, $\alpha$, of the $dN_{\rm ch}/d\eta$ dependence of nonprompt and direct-photon yields in various integration ranges.  $p_{T}$ (GeV/$c$) | $\alpha(\mbox{$\gamma^{\rm{nonprompt}}$})$ | $\alpha(\mbox{$\gamma^{\rm dir}$})$
---|---|---
0.8–1.2 | $1.119\pm 0.038\ _{-0.094}^{+0.116}$ |  $1.124\pm 0.036\ _{-0.089}^{+0.121}$
1.2–1.6 | $1.107\pm 0.029\ _{-0.082}^{+0.108}$ |  $1.118\pm 0.027\ _{-0.073}^{+0.097}$
1.6–2.0 | $1.136\pm 0.034\ _{-0.091}^{+0.129}$ |  $1.152\pm 0.029\ _{-0.077}^{+0.113}$
2.0–3.0 | $1.087\pm 0.032\ _{-0.092}^{+0.108}$ |  $1.120\pm 0.025\ _{-0.065}^{+0.095}$
3.0–4.0 | $1.119\pm 0.078\ _{-0.134}^{+0.206}$ |  $1.171\pm 0.048\ _{-0.076}^{+0.114}$
4.0–5.0 | $0.950\pm 0.176\ _{-0.205}^{+0.315}$ |  $1.137\pm 0.077\ _{-0.082}^{+0.108}$
5.0–10.0 |  |  $1.296\pm 0.078\ _{-0.091}^{+0.129}$

To better understand the behavior of the scaling power, $\alpha$, in more
detail, the direct-photon yield and its nonprompt component are integrated for
six different nonoverlapping finer $p_{T}$ regions and for 10% centrality
classes. The integrated nonprompt yields are shown in Fig. 20. The $\alpha$
values are determined for each $p_{T}$ selection by fitting the data with Eq.
10. The fits are also shown in the figure. All $\alpha$ values, both for the
direct photon yield and the nonprompt component, are tabulated in Table 4 and
shown in Fig. 21. It is evident that the values for the direct component, for
higher $p_{T}$ ranges, are consistent with the prompt component,
$\alpha=1.25\pm 0.02$, corresponding to $N_{\rm coll}$ scaling. However, they
tend to be smaller, but still consistent within systematic uncertainties, with
previous measurements [8] for the lower $p_{T}$ ranges.

With increasing $p_{T}$, the $\alpha$ values for the nonprompt component are
slightly lower than those from direct photons. The systematic uncertainties
are larger due to the subtraction. The values of $\alpha$ for the nonprompt
component, as shown in Fig. 21, are remarkably constant with no evident
$p_{T}$ dependence.

Figure 20: Integrated nonprompt direct-photon yield versus charged particle
multiplicity at midrapidity for different $p_{T}$ integration ranges.  Figure
21: Scaling factors, $\alpha$, extracted from fitting Eq. 10 to integrated
direct and nonprompt-photon yields as a function of $dN_{\rm ch}/d\eta$.
Values were obtained for different $p_{T}$ integration ranges tabulated in
Table 4.

##  VI Concluding discussion of the results

The PHENIX collaboration has measured direct-photon production in Au$+$Au
collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV using photon conversions to
$e^{+}e^{-}$ pairs. A large yield of direct photons below a $p_{T}$ of 3
GeV/$c$ is observed for all centrality bins except for the most peripheral bin
of 80%–93% with $dN_{\rm ch}/d\eta$ = 7.4, where it seems to be consistent
with the prompt-photon production with little or no radiation from a fireball.
The next centrality bin from 70%–80% with $dN_{\rm ch}/d\eta$ = 15.5 already
shows a significant yield with properties very similar to that of the
radiation from the more central bins.

The nonprompt direct-photon spectra are isolated by subtracting the prompt-
photon contribution, which is estimated through a fit to the direct-photon
data from $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV, measured by PHENIX, and
scaled by $N_{\rm coll}$. Results are obtained for the $p_{T}$ range from 0.8
to 5 GeV/$c$ and for 0%–93% central collisions, covering a system size
spanning two orders of magnitude in $dN_{\rm ch}/d\eta$ from $\approx$7 to
620. The wealth of data enabled PHENIX to carry out double-differential
analyses of the shape of the momentum spectra and the rapidity density
$dN_{\gamma}/dy$ in $p_{T}$ and $dN_{\rm ch}/d\eta$.

For the centrality selections from 0%–10% to 70%–80%, all nonprompt direct-
photon spectra are very similar in shape, exhibiting increasing $T_{\rm eff}$
from 0.2 to 0.4 GeV/$c$ over the $p_{T}$ range from 0.8 to 4 GeV/$c$. The
changing $T_{\rm eff}$ is not surprising, because the spectra are time
integrated over the full evolution of the expanding fireball, from its
earliest pre-equilibrium state, through the QGP phase, crossing over to a HG,
and further expanding and cooling until hadrons eventually stop interacting.
Throughout the evolution the system cools, and thus earlier phases are
characterized by higher temperatures. In turn, the contributions from the
earliest times of the evolution are likely to dominate the emission at higher
$p_{T}$, consistent with the observation of an increasing $T_{\rm eff}$ with
$p_{T}$.

In the lower $p_{T}$ range from 0.8 to 1.9 GeV/$c$, the spectra are well
described by a $T_{\rm eff}$ = 0.26 GeV/$c$. This is consistent with what is
expected for radiation from the late QGP stage until freeze-out [14]. During
this period of the evolution, the temperature drops from $\approx$170 MeV near
the transition to $\approx$110 MeV when the system freezes out. At the same
time the system is rapidly expanding and thus, the radiation is blue shifted.
This compensates the temperature drop and results in an average $T_{\rm eff}$
= $\approx$0.26 GeV/$c$, with only minor variations with centrality of the
collision. In Ref. [14], a moderate increase of $T_{\rm eff}$ with centrality
was predicted. While the data favors a $T_{\rm eff}$ independent of
centrality, they are not precise enough to exclude a moderate change.

Above a $p_{T}$ of 2 GeV/$c$, the inverse slope of the spectra continues to
increase with $p_{T}$. Between $p_{T}$ = 2 and 4 GeV/$c$ the average inverse
slope is $T_{\rm eff}$ $\approx$0.376 GeV/$c$. This $T_{\rm eff}$ is larger
than what can be accommodated by a rapidly expanding HG, thus suggesting that
emissions from earlier times in the evolution starts to dominate the spectra.
Expected initial temperatures at RHIC are $\approx$375 MeV with maximum
$T_{\rm eff}$ in the range of 0.35 to 0.4 GeV/$c$, depending on viscosity
[14]. Thus, it is likely that an additional contribution from the pre-
equilibrium stage is needed to account for the measured $T_{\rm eff}$.

Figure 22: Nonprompt direct-photon yields for (a) 0%–20% and (b) 20%–40%
compared with model predictions from Ref. [10, 46]. (c,d) ratios of the yields
from data to the sum of yields from thermal and pre-equilibrium contributions.

In Fig. 22, the measured nonprompt direct-photon spectra are compared to a
recent calculation including contributions from the pre-equilibrium phase [10,
46]. These calculations predicted that the pre-equilibrium radiation becomes
the dominant source above a $p_{T}$ of 3 GeV/$c$. In the range
$2<\mbox{$p_{T}$}<4$ GeV/$c$, a fit of the thermal contribution with an
exponential function results in an inverse slope of $\approx$0.36 GeV/$c$,
while for the pre-equilibrium contribution a larger inverse slope of
$\approx$0.52 GeV/$c$ is found, for the more central collisions. Fitting the
same $p_{T}$ range for the combined thermal and pre-equilibrium spectra from
the model gives an inverse slope of $\approx$0.425 GeV/$c$. While the shape is
reproduced well, the overall yield predicted by the calculations falls short
compared to the data, in particular, below 2 GeV/$c$ where the nonprompt-
photon yield appears to be a factor of two to three larger.

The integrated nonprompt direct-photon yield exhibits a power-law relation
with $(\mbox{$dN_{\rm ch}/d\eta$})^{\alpha}$ [8]. Fitting the power $\alpha$
for multiple nonoverlapping $p_{T}$ ranges results in values consistent with
$\alpha=1.12\pm 0.06{\rm(\rm{stat})}\pm 0.14{\rm(\rm{sys})}$ with no apparent
dependence on $p_{T}$. The model calculations in [14] predict that the
radiation from the HG phase scale with $\alpha$ close to 1.2, while those from
the hot and dense QGP phase exhibit closer to a $(\mbox{$dN_{\rm
ch}/d\eta$})^{2}$ dependence. Because the QGP phase has a larger relative
contribution to the $p_{T}$ spectrum with increasing $p_{T}$, it is expected
that $\alpha$ increases with $p_{T}$. However, the $p_{T}$ dependence of
$\alpha$ from the pre-equilibrium phase needs further theoretical
understanding.

In conclusion, the 10-fold increase in statistics compared to previous samples
of Au$+$Au collisions recorded by PHENIX enabled detailed measurements of the
radiation from the hot and expanding fireball. The experimentally observed
inverse slopes of the $p_{T}$ spectra are qualitatively consistent with
predictions for thermal and pre-equilibrium radiation. However, there seems to
be more photons emitted from Au$+$Au collisions than can be accounted for in
model calculations. Furthermore, although this work presents no new data on
the azimuthal anisotropy, maximum anisotropy is observed for photons
$\approx$2–3 GeV/$c$. In this $p_{T}$ range, the yield is larger than what
would be expected from a rapidly but anisotropically expanding hadronic
fireball. Finally, the centrality dependence of the nonprompt direct-photon
yield, expressed in terms of the scaling power $\alpha(\mbox{$p_{T}$})$, shows
no indication of changing with $p_{T}$.

###### Acknowledgements.

We thank the staff of the Collider-Accelerator and Physics Departments at
Brookhaven National Laboratory and the staff of the other PHENIX participating
institutions for their vital contributions. We also thank J.F. Paquet for many
fruitful discussions and sharing additional information. We acknowledge
support from the Office of Nuclear Physics in the Office of Science of the
Department of Energy, the National Science Foundation, Abilene Christian
University Research Council, Research Foundation of SUNY, and Dean of the
College of Arts and Sciences, Vanderbilt University (USA), Ministry of
Education, Culture, Sports, Science, and Technology and the Japan Society for
the Promotion of Science (Japan), Natural Science Foundation of China
(People’s Republic of China), Croatian Science Foundation and Ministry of
Science and Education (Croatia), Ministry of Education, Youth and Sports
(Czech Republic), Centre National de la Recherche Scientifique, Commissariat à
l’Énergie Atomique, and Institut National de Physique Nucléaire et de Physique
des Particules (France), J. Bolyai Research Scholarship, EFOP, the New
National Excellence Program (ÚNKP), NKFIH, and OTKA (Hungary), Department of
Atomic Energy and Department of Science and Technology (India), Israel Science
Foundation (Israel), Basic Science Research and SRC(CENuM) Programs through
NRF funded by the Ministry of Education and the Ministry of Science and ICT
(Korea). Ministry of Education and Science, Russian Academy of Sciences,
Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation
(Sweden), University of Zambia, the Government of the Republic of Zambia
(Zambia), the U.S. Civilian Research and Development Foundation for the
Independent States of the Former Soviet Union, the Hungarian American
Enterprise Scholarship Fund, the US-Hungarian Fulbright Foundation, and the
US-Israel Binational Science Foundation.

##  Appendix A Event-mixing procedures and validation

In this analysis, $e^{+}e^{-}$ pairs and $e^{+}e^{-}\gamma$ combinations
result from combining positrons, electrons, and photons measured in the same
event. Given the large multiplicity of produced particles in Au$+$Au
collisions, the combinations include a significant background from particles
of different physical origin, for example different $\pi^{0}$ decays. For
$e^{+}e^{-}$ pairs there are two possible combinations: signal pairs, ${\rm
SG}^{\rm{ee}}$, that have the same source and background pairs, ${\rm
BG}^{\rm{ee}}$, that have different sources. Both types will be combined with
photons to get $e^{+}e^{-}\gamma$ combinations. There are three possibilities:
A correlated $e^{+}e^{-}$ pair is combined with a photon from the same source
(${\rm SG}^{ee\gamma}$); the $e^{+}e^{-}$ pair is not correlated, but the
photon is correlated to the $e^{+}$ or $e^{-}$ (${\rm BG}_{\rm
corr}^{ee\gamma}$); or the photon is uncorrelated to the $e^{+}e^{-}$ pair,
irrespective whether it is ${\rm SG}^{\rm{ee}}$ or ${\rm BG}^{\rm{ee}}$ (${\rm
BG}_{\rm uncorr}^{ee\gamma}$).

All backgrounds are determined using event-mixing techniques that were
developed and validated with MC studies of high-multiplicity events, for which
a large sample of simulated $\pi^{0}$ events was generated. These events serve
as pseudodata. The $\pi^{0}$ are generated according to the experimentally
observed $p_{T}$ spectrum, uniform in azimuthal angle, and with a constant
pseudorapidity density of 280 $\pi^{0}$, which corresponds to the typical
$\pi^{0}$ multiplicity in the most central Au$+$Au collisions at
$\sqrt{s_{{}_{NN}}}=200$ GeV.

From these pseudodata, $N_{\gamma}^{\rm{incl}}$ and
$N_{\gamma}^{\rm{\pi^{0},tag}}$ are extracted using the cuts and event-mixing
schemes developed for the analysis of real data. They are corrected by
$\langle\epsilon_{\gamma}f\rangle$, resulting in $R_{\gamma}$. Because in the
pseudodata there are no other hadronic decay channels contributing to
$\gamma^{\rm hadr}$ other than $\pi^{0}$, the $R_{\gamma}$ from this
pseudodata is given by:

$R_{\gamma}^{\rm pseudo}=\frac{N^{\rm incl}_{\gamma}}{N_{\gamma}^{\pi^{0},\rm
tag}}\times\mbox{$\langle\epsilon_{\gamma}f\rangle$}$ (11)

As there are no direct photons in the pseudodata, the expected result would be
$R_{\gamma}$ = 1, within the statistical uncertainties of the simulation. The
rest of this sections details each step of the $R_{\gamma}$ determination from
the pseudodata. The exact same procedure is also applied to the real data.

###  A.1 Determination of the inclusive photon yield $N_{\gamma}^{\rm{incl}}$

Photon conversion candidates are created by combining $e^{+}$ and $e^{-}$ from
the same pseudodata event by requiring a valid conversion point within
$1<R<29$ cm. This results in a foreground, ${\rm FG}^{\rm{ee}}$, containing a
signal, ${\rm SG}^{\rm{ee}}$, that is, conversions of $\pi^{0}$ decay photons,
and a background, ${\rm BG}^{\rm{ee}}$, where the $e^{+}$ and $e^{-}$ come
from conversion of two different $\pi^{0}$ decay photons. The background is
determined by combining electrons and positrons from different pseudodata
events, which are paired and subjected to the same cuts and conversion
selection criteria. The mixed event background thus obtained, ${\rm
MBG}^{\rm{ee}}$, is normalized to the foreground, ${\rm FG}^{\rm{ee}}$, in the
mass region $0.16<\mbox{$m_{e^{+}e^{-}}$}<$ 0.3 GeV/$c^{2}$, which does not
contain $e^{+}e^{-}$ pairs from conversions (see Fig. 2 for reference).

Figure 23: Invariant mass distributions of $e^{+}e^{-}$ pairs reconstructed
from the high-multiplicity $\pi^{0}$ pseudodata in the $p_{T}$ range
$0.8<\mbox{$p_{T}$}<1.0$ GeV/$c$. The least-restrictive conversion selection
cuts are applied, which only require that the reconstruction algorithm has
identified the $e^{+}e^{-}$ pair as a conversion candidate. Panel (a) compares
foreground, ${\rm FG}^{\rm{ee}}$, the true background, ${\rm BG}^{\rm{ee}}$,
and the background determined from the mixed event technique, ${\rm
MBG}^{\rm{ee}}$. Panel (b) gives the extracted conversion photon signal.

Figure 24: Invariant mass distributions of $e^{+}e^{-}$ pairs reconstructed
from the high-multiplicity $\pi^{0}$ pseudodata. Same as Fig. 24, but with an
additional constraint that the $e^{+}$ and $e^{-}$ match in beam direction.
Panel (a) compares foreground, ${\rm FG}^{\rm{ee}}$, the true background,
${\rm BG}^{\rm{ee}}$, and the background determined from the mixed event
technique, ${\rm MBG}^{\rm{ee}}$. Panel (b) gives the extracted conversion
photon signal.

Figure 25: Extracted $N_{\gamma}^{\rm incl}$ after the background subtraction,
as a function of conversion photon $p_{T}$. The diamonds are obtained by
subtracting the background from the mixed event technique; they are compared
to the open symbols for which the true background was subtracted. Panel (b)
shows the ratio of the event-mixing result over the true information result.
Figure 26: Invariant mass distributions of $e^{+}e^{-}\gamma$ pairs.

Figure 24(a) shows the background, ${\rm MBG}^{\rm{ee}}$, obtained from the
mixed event technique together with the true background, ${\rm BG}^{\rm{ee}}$,
which was obtained from the MC ancestry information. Figure 24(b) shows the
results (solid curve) after subtracting the mixed-event background from the
foreground and (open symbols) subtracting the true background. Note that the
two are practically indistinguishable, which means that ${\rm BG}^{\rm{ee}}$
is equal to ${\rm MBG}^{\rm{ee}}$.

Even though the background can be subtracted accurately with the mixed-event
technique to obtain $N_{\gamma}^{\rm{incl}}$, the subtraction can only be done
statistically. Thus in the next step, where conversion photons from $\pi^{0}$
decays are tagged, the background pairs also need to be matched with EMCal
showers. This substantially increases the background in the $m_{ee\gamma}$
distribution. To reduce this background, additional cuts are applied in the
conversion-photon selection.

The magnetic field deflects electrons and positrons in a plane perpendicular
to the beam direction ($z$). Thus, $e^{+}e^{-}$ pairs from a conversion can be
constrained by requiring a match in the beam direction using the PC1
information. A cut of $|\Delta z|<4$ cm is applied. Because the conversion
reconstruction algorithm uses the projection of the tracks in the plane
perpendicular to the beam axis, the additional match reduces the number of
possible random-track combinations significantly. The $z$ cut effectively
truncates the mass distribution as the $e^{+}e^{-}$ pairs are required to have
the possible conversion point at radii below 29 cm and only the pairs with an
opening angle in the beam direction will create larger masses. The background
rejection is clearly visible in Fig. 24. The background normalization for the
mixed events is given by the less-restrictive cuts shown in Fig. 24, and
applied here. For the lowest $p_{T}$ and the highest-multiplicity bin, the
background rejection is approximately a factor of eight with a signal
efficiency of more than 85%. The background to foreground ratio, $\mbox{${\rm
BG}^{\rm{ee}}$}/\mbox{${\rm FG}^{\rm{ee}}$}$, is 12.1%. As $p_{T}$ increases
the multiplicity decreases and the $\mbox{${\rm BG}^{\rm{ee}}$}/\mbox{${\rm
FG}^{\rm{ee}}$}$ ratio decreases to 0.3% at the $p_{T}$ above 7 GeV/$c$.

The analysis is repeated for the entire accessible $p_{T}$ range and
$N_{\gamma}^{\rm{incl}}$ is calculated in the mass range from 0.04 to 0.12
GeV/$c^{2}$ by subtracting the background obtained from the mixed-event
technique, ${\rm MBG}^{\rm{ee}}$, from the foreground, ${\rm FG}^{\rm{ee}}$.
The result is compared to the true number of photon conversions determined
from the MC-ancestry information in Fig. 25. Panel (b) shows that the
difference is less than 1% for all $p_{T}$.

###  A.2 The tagged photon yield $N_{\gamma}^{\rm{\pi^{0},tag}}$

Next, the subset $N_{\gamma}^{\rm{\pi^{0},tag}}$ of $e^{+}e^{-}$ pairs in the
$N_{\gamma}^{\rm{incl}}$ sample that can be tagged as photons from a $\pi^{0}$
decay is determined. For a given pseudodata event, each $e^{+}e^{-}$
conversion candidate is paired with all reconstructed showers in the EMCal,
excluding the showers matched to the $e^{+}e^{-}$ pair itself. For each
combination the invariant mass $m_{ee\gamma}$ is calculated. This constitutes
the foreground, ${\rm FG}^{ee\gamma}$, for which an example is given in panel
(a) of Fig. 26. Despite the large background the signal
$N_{\gamma}^{\rm{\pi^{0},tag}}$ is clearly visible as peak around the
$\pi^{0}$ mass. The background has two components: (i) combinations of
$e^{+}e^{-}$ pairs with an EMCal shower from another unrelated $\pi^{0}$ decay
denoted as ${\rm BG}_{\rm uncorr}^{ee\gamma}$, and (ii) a correlated
background, ${\rm BG}_{\rm corr}^{ee\gamma}$, where the shower in the EMCal
and the electron or positron are from the same $\pi^{0}$ decay, but the
$e^{+}e^{-}$ pair itself is a combination of an $e^{+}$ and $e^{-}$ from
different $\pi^{0}$ decay photons.

The uncorrelated background can be determined with a similar event mixing
technique as used for the extraction of $N_{\gamma}^{\rm{incl}}$; an
$e^{+}e^{-}$ pair from one event is mixed with the EMCal showers from a
different event resulting in mixed combinations, ${\rm MBG}_{\rm
uncorr}^{ee\gamma}$. These are normalized to the foreground, ${\rm
FG}^{ee\gamma}$, in the mass region from 0.25 to 0.45 GeV/$c^{2}$, where no
signal is expected. Figure 26(a) shows the corresponding distribution. There
is almost no visible difference between the mixed-event background, ${\rm
MBG}_{\rm uncorr}^{ee\gamma}$, and the true background, ${\rm BG}_{\rm
uncorr}^{ee\gamma}$, which is obtained using the MC-ancestry information.
Figure 26(b) shows the signal and remaining correlated background after the
uncorrelated mixed-event background is subtracted (${\rm FG}^{ee\gamma}$-${\rm
MBG}_{\rm uncorr}^{ee\gamma}$), as well as after subtracting the true
uncorrelated background (${\rm FG}^{ee\gamma}$-${\rm BG}_{\rm
uncorr}^{ee\gamma}$). Again they are indistinguishable.

The correlated background, ${\rm BG}_{\rm corr}^{ee\gamma}$, is determined
with a second event-mixing scheme. An $e^{+}$ from a given event is combined
with an $e^{-}$ from a different event, and the resulting $e^{+}e^{-}$ pair is
then combined with the showers in the EMCal from both events; again excluding
the showers from the $e^{+}$ and $e^{-}$. The $e^{+}e^{-}\gamma$ combinations
contain the correlated background, ${\rm MBG}_{\rm cor}^{ee\gamma}$, plus the
random background in which the $e^{+}$, $e^{-}$, and $\gamma$ are from three
different $\pi^{0}$ decays, ${\rm MBG}_{\rm comb}^{ee\gamma}$. The
normalization is per generated $e^{+}e^{-}$ pair, multiplied by ${\rm
FG}^{ee\gamma}$, i.e. the number of background pairs in the $e^{+}e^{-}$ pair
foreground.

Figure 27: Invariant mass distributions of $e^{+}e^{-}\gamma$ pairs from the
same event (FG) and different event-mixing setups.

The random background, ${\rm MBG}_{\rm comb}^{ee\gamma}$, can easily be
determined in a third event-mixing step, where $e^{+}$, $e^{-}$, and $\gamma$
are from three different events. The ${\rm MBG}_{\rm comb}^{ee\gamma}$ is
normalized to (${\rm MBG}_{\rm cor}^{ee\gamma}$ $+$${\rm MBG}_{\rm
comb}^{ee\gamma}$) in the mass range from 0.65 to 1.0 GeV/$c^{2}$ and
subtracted. Figure 27(a) shows the the result, ${\rm MBG}_{\rm
cor}^{ee\gamma}$, together with the foreground and the other background
components.

Last but not least, to account for any possible mismatch between the true
background and the one obtained from our multistep event-mixing procedure, the
ratio $(\mbox{${\rm FG}^{ee\gamma}$}-\mbox{${\rm MBG}_{\rm
cor}^{ee\gamma}$}-\mbox{${\rm MBG}_{\rm uncorr}^{ee\gamma}$})/\mbox{${\rm
MBG}_{\rm uncorr}^{ee\gamma}$}$ is fit with a second-order polynomial,
$f_{ee\gamma}$, excluding the $\pi^{0}$ peak regions. The fit result is shown
as a line on Fig. 27(b). This fit is used to correct ${\rm MBG}_{\rm
uncorr}^{ee\gamma}$ before subtraction. The final distribution for
$N_{\gamma}^{\rm{\pi^{0},tag}}$ is thus:

$\mbox{$N_{\gamma}^{\rm{\pi^{0},tag}}$}=\mbox{${\rm
FG}^{ee\gamma}$}-\mbox{${\rm MBG}_{\rm cor}^{ee\gamma}$}-(1+f_{ee\gamma})\
\mbox{${\rm MBG}_{\rm uncorr}^{ee\gamma}$}$ (12)

Figure 28: Extracted $N_{\gamma}^{\rm{\pi^{0},tag}}$ as a function of
conversion-photon $p_{T}$ using the (red) true information and (blue) event-
mixing technique. The bottom panel shows the ratio of the event-mixing result
over the true information result.

For each $p_{T}$ bin $N_{\gamma}^{\rm{\pi^{0},tag}}$ is extracted by counting
the number of entries in a window around the $\pi^{0}$ peak
($0.09<\mbox{$m_{ee\gamma}$}<0.19$) GeV/$c^{2}$. Figure 28 shows
$N_{\gamma}^{\rm{\pi^{0},tag}}$ as function of $p_{T}$ using the true MC-
ancestry information and the event-mixing technique. Overall the agreement is
very good, however, the result from the event-mixing technique is on average
lower. This mismatch is accounted for in the systematic uncertainties on
$R_{\gamma}$, which is discussed in more detail in the next section.

Figure 29: Average conditional probability $\langle\epsilon_{\gamma}f\rangle$
as a function of conversion photon $p_{T}$.  Figure 30: Ratio $R_{\gamma}^{\rm
pseudo}$ as a function of conversion photon $p_{T}$. The dashed line gives a
constant offset of 1.3% fit to the points, and the dashed band represents a
$\pm 1.5\%$ range around unity.

###  A.3 Completing the validation by determining $R_{\gamma}$

With $N_{\gamma}^{\rm{incl}}$ and $N_{\gamma}^{\rm{\pi^{0},tag}}$ established
from the pseudodata, the conditional probability
$\langle\epsilon_{\gamma}f\rangle$ remains to be determined to calculate
$R_{\gamma}$ and fully validate the background-subtraction procedure. In the
same way as for the data, a single $\pi^{0}$ simulation is embedded into
pseudodata events. The $e^{+}e^{-}$ pairs and $e^{+}e^{-}\gamma$ combinations
are reconstructed and counted as discussed in Sec. III.3. The extracted
$\langle\epsilon_{\gamma}f\rangle$ is shown in Fig. 29 as a function of the
conversion photon $p_{T}$.

With $N_{\gamma}^{\rm{incl}}$/$N_{\gamma}^{\rm{\pi^{0},tag}}$ from the
pseudodata and $\langle\epsilon_{\gamma}f\rangle$ from the embedded single
$\pi^{0}$ simulation in hand, $R_{\gamma}$ is calculated using Eq. A1. The
result is shown in Fig. 30, all points are close to unity indicating that the
analysis procedure is self consistent. There may be a 1.5% enhancement above
unity, which is consistent with the slightly lower-than-expected value found
for $N_{\gamma}^{\rm{\pi^{0},tag}}$. This difference is taken into account in
the estimate of the systematic uncertainty.

##  Appendix B Uncertainty propagation with a MC sampling method

The uncertainties on $\gamma^{\rm dir}$ and any other quantity derived from
$\gamma^{\rm dir}$, such as $T_{\rm eff}$ or $\alpha$, are determined using a
MC-sampling method, which allows taking into account the $p_{T}$ and
centrality dependent correlations of individual sources of systematic
uncertainties, as well as the fact that the region $R_{\gamma}$ $<1$ is
unphysical.

###  B.1 Systematic uncertainties

In the MC-sampling method, for each source of uncertainty, $i$, a variation
$\delta_{i}$ of $R_{\gamma}$ or $\gamma^{\rm hadr}$ is sampled from a Gaussian
distribution centered at zero with a width corresponding to the associated
uncertainty, $\sigma_{i}$. The size of $\delta_{i}$ depends not only on
$\sigma_{i}$, but also on whether the adjacent bins in $p_{T}$ and centrality
have uncorrelated (Type A) or correlated (Type B/C) uncertainties due to the
source $i$. The values of $\sigma_{i}$ and classification of each source is
summarized in Table 2.

If source $i$ is classified as uncorrelated, $\delta_{i}$ is calculated
independently for neighboring bins from Gaussian distributions of width
$\sigma_{i}$. For correlated uncertainties of Type C in $p_{T}$ or centrality,
$\delta_{i}$ is calculated with one common fraction $w$ so that
$\delta_{i}=w\sigma_{i}$ for all points. The fraction $w$ is determined
randomly from a Gaussian distribution of width 1. And finally, for Type B
uncertainties, $\delta_{i}$ is determined separately for the minimum and
maximum of the $p_{T}$ or centrality range using the same procedure as Type C.
All intermediate points are varied proportionally to create a smooth
transition from the minimum to the maximum of the range. Uncertainties on the
input $\pi^{0}$ $p_{T}$ distribution are a special case of Type B
uncertainties, as it is known that the systematic uncertainties move
simultaneously either up or down. In this case, $\delta_{i}$ at the minimum
and maximum of the range are chosen to have the same sign.

After applying all variations $\delta_{i}$ to recalculate $R_{\gamma}$ and
$\gamma^{\rm hadr}$, new values of $\gamma^{\rm dir}$, $T_{\rm eff}$, and
$\alpha$ are determined. This process is repeated multiple times, taking into
account the different sources of uncertainties, to obtain a distributions of
$\gamma^{\rm dir}$, $T_{\rm eff}$, and $\alpha$. The width of these
distribution is quoted as the systematic uncertainty. For individual
$\gamma^{\rm dir}$ points, it is possible that $\langle\mbox{$\gamma^{\rm
dir}$}\rangle-\sigma$ is less than 0, that is, unphysical. In such cases, an
upper limit of 90% confidence level (CL) is quoted based on the part of the
probability distribution in the physical region $\int_{0}^{\rm
upper}/\int_{0}^{+\infty}=90\%$.

###  B.2 Statistical uncertainties

The statistical uncertainties on $R_{\gamma}$ are assumed to have a Gaussian
probability distribution and for most cases the statistical uncertainty on
$\gamma^{\rm dir}$ can be calculated with the usual error propagation.
However, there are two cases that need to be treated separately:

  * •

$\mbox{$R_{\gamma}$}<1$: In this case $\gamma^{\rm dir}$ is unphysical, and
hence an upper limit at 90% CL is quoted, based on the physical part of the
probability distribution $\int_{0}^{\rm upper}/\int_{0}^{+\infty}=90\%$.

  * •

$\mbox{$R_{\gamma}$}-\sigma_{\rm stat}<1$: In this case $\gamma^{\rm dir}$ is
in the physical region, but consistent with zero within less than one standard
deviation. For these situations the central value is shown, but the
uncertainty is given as 90% CL, calculated as above.

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