PHENIX Collaboration

# Probing gluon spin-momentum correlations in transversely polarized protons
through midrapidity isolated direct photons in $p^{\uparrow}+p$ collisions at
$\sqrt{s}=200$ GeV

U.A. Acharya  Georgia State University, Atlanta, Georgia 30303, USA  C. Aidala
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040,
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  V. Andrieux  Department of Physics, University of Michigan, Ann Arbor,
Michigan 48109-1040, USA  N. Apadula  Iowa State University, Ames, Iowa 50011,
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  N.S.
Bandara  Department of Physics, University of Massachusetts, Amherst,
Massachusetts 01003-9337, 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  R. Belmont  University of Colorado, Boulder,
Colorado 80309, 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  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  V. Canoa Roman  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
R. Cervantes  Department of Physics and Astronomy, Stony Brook University,
SUNY, Stony Brook, New York 11794-3800, USA  C.Y. Chi  Columbia University,
New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533,
USA  M. Chiu  Physics Department, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  I.J. Choi  University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA  J.B. Choi  Deceased  Jeonbuk National University,
Jeonju, 54896, Korea  Z. Citron  Weizmann Institute, Rehovot 76100, Israel  M.
Connors  Georgia State University, Atlanta, Georgia 30303, USA  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
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  N. Cronin  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  M. Csanád  ELTE, Eötvös Loránd University, H-1117 Budapest,
Pázmány P. s. 1/A, Hungary  T. Csörgő  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
T.W. Danley  Department of Physics and Astronomy, Ohio University, Athens,
Ohio 45701, 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  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  D. Dixit
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  J.H. Do  Yonsei University, IPAP, Seoul
120-749, Korea  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  H. En’yo  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  R. Esha  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  S. Esumi  Tomonaga Center for the History of the Universe,
University of Tsukuba, Tsukuba, Ibaraki 305, Japan  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
Charles University, Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic  M.
Finger, Jr  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  Y. Fukuda  Tomonaga Center for the History of the Universe,
University of Tsukuba, Tsukuba, Ibaraki 305, Japan  C. Gal  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  P. Gallus  Czech Technical University, Zikova 4, 166 36
Prague 6, Czech Republic  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  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  H. Guragain  Georgia State
University, Atlanta, Georgia 30303, 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  S.Y. Han  Ewha Womans
University, Seoul 120-750, Korea  Korea University, Seoul 02841, Korea  J.
Hanks  Department of Physics and Astronomy, Stony Brook University, SUNY,
Stony Brook, New York 11794-3800, USA  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  X. He  Georgia State University, Atlanta, Georgia 30303, USA  T.K.
Hemmick  Department of Physics and Astronomy, Stony Brook University, SUNY,
Stony Brook, New York 11794-3800, USA  J.C. Hill  Iowa State University, Ames,
Iowa 50011, USA  K. Hill  University of Colorado, Boulder, Colorado 80309, 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  S. Huang  Vanderbilt
University, Nashville, Tennessee 37235, 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  Z. Ji  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  X. Jiang  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, 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  J.H. Kang
Yonsei University, IPAP, Seoul 120-749, Korea  D. Kapukchyan  University of
California-Riverside, Riverside, California 92521, USA  S. Karthas  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  D. Kawall  Department of Physics, University of
Massachusetts, Amherst, Massachusetts 01003-9337, USA  A.V. Kazantsev
National Research Center “Kurchatov Institute”, Moscow, 123098 Russia  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
C. Kim  University of California-Riverside, Riverside, California 92521, USA
Korea University, Seoul 02841, Korea  E.-J. Kim  Jeonbuk National University,
Jeonju, 54896, Korea  M. Kim  Department of Physics and Astronomy, Seoul
National University, Seoul 151-742, 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  J. Klatsky  Florida State
University, Tallahassee, Florida 32306, 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  D.
Kotov  PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region,
188300, Russia  Saint Petersburg State Polytechnic University, St. Petersburg,
195251 Russia  S. Kudo  Tomonaga Center for the History of the Universe,
University of Tsukuba, Tsukuba, Ibaraki 305, Japan  B. Kurgyis  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  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  Y.H. Leung
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  N.A. Lewis  Department of Physics, University
of Michigan, Ann Arbor, Michigan 48109-1040, USA  X. Li  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  S.H. Lim  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  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  V.-R.
Loggins  University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,
USA  S. Lökös  ELTE, Eötvös Loránd University, H-1117 Budapest, Pázmány P. s.
1/A, Hungary  D.A. Loomis  Department of Physics, University of Michigan, Ann
Arbor, Michigan 48109-1040, USA  K. Lovasz  Debrecen University, H-4010
Debrecen, Egyetem tér 1, Hungary  D. Lynch  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  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  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  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  Iu. Mitrankov  Saint Petersburg State
Polytechnic University, St. Petersburg, 195251 Russia  M. Mitrankova  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  G.
Mitsuka  KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki
305-0801, Japan  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  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  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
B. Mulilo  Korea University, Seoul 02841, Korea  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  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  K.
Nagai  Department of Physics, Tokyo Institute of Technology, Oh-okayama,
Meguro, Tokyo 152-8551, Japan  K. Nagashima  Hiroshima University, Kagamiyama,
Higashi-Hiroshima 739-8526, Japan  T. Nagashima  Physics Department, Rikkyo
University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, 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  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  T. Niida  Tomonaga Center for the
History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305, 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  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  N. Novitzky  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  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  G.J. Ottino  University of New Mexico, Albuquerque, New Mexico
87131, USA  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  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  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan  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  S.F. Pate  New
Mexico State University, Las Cruces, New Mexico 88003, USA  M. Patel  Iowa
State University, Ames, Iowa 50011, 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  M. Phipps  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  University of Illinois
at Urbana-Champaign, Urbana, Illinois 61801, USA  C. Pinkenburg  Physics
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, 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  N. Ramasubramanian  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  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.
Runchey  Iowa State University, Ames, Iowa 50011, USA  A.S. Safonov  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  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  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  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  T. Shioya  Tomonaga Center for the History of the Universe, University
of Tsukuba, Tsukuba, Ibaraki 305, Japan  P. Shukla  Bhabha Atomic Research
Centre, Bombay 400 085, India  A. Sickles  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  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
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  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  G. Tarnai  Debrecen University,
H-4010 Debrecen, Egyetem tér 1, Hungary  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  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.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  N. Vukman  Department of Physics, Faculty
of Science, University of Zagreb, Bijenička c. 32 HR-10002 Zagreb, Croatia
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.S. Watanabe  Center for Nuclear Study, Graduate School of
Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  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
C. Xu  New Mexico State University, Las Cruces, New Mexico 88003, USA  Q. Xu
Vanderbilt University, Nashville, Tennessee 37235, 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  Department of Physics and Astronomy, Stony Brook University,
SUNY, Stony Brook, New York 11794-3800, USA  H. Yamamoto  Tomonaga Center for
the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305,
Japan  A. Yanovich  IHEP Protvino, State Research Center of Russian
Federation, Institute for High Energy Physics, Protvino, 142281, Russia  J.H.
Yoo  Korea University, Seoul 02841, Korea  I. Yoon  Department of Physics and
Astronomy, Seoul National University, Seoul 151-742, Korea  H. Yu  New Mexico
State University, Las Cruces, New Mexico 88003, USA  Peking University,
Beijing 100871, People’s Republic of China  I.E. Yushmanov  National Research
Center “Kurchatov Institute”, Moscow, 123098 Russia  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. Zharko  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  L. Zou
University of California-Riverside, Riverside, California 92521, USA

###### Abstract

Studying spin-momentum correlations in hadronic collisions offers a glimpse
into a three-dimensional picture of proton structure. The transverse single-
spin asymmetry for midrapidity isolated direct photons in $p^{\uparrow}+p$
collisions at $\sqrt{s}=200$ GeV is measured with the PHENIX detector at the
Relativistic Heavy Ion Collider (RHIC). Because direct photons in particular
are produced from the hard scattering and do not interact via the strong
force, this measurement is a clean probe of initial-state spin-momentum
correlations inside the proton and is in particular sensitive to gluon
interference effects within the proton. This is the first time direct photons
have been used as a probe of spin-momentum correlations at RHIC. The
uncertainties on the results are a fifty-fold improvement with respect to
those of the one prior measurement for the same observable, from the Fermilab
E704 experiment. These results constrain gluon spin-momentum correlations in
transversely polarized protons.

Unlike lepton-hadron scattering, proton-proton collisions are sensitive to
gluon scattering at leading order. Direct photons are produced directly in the
hard scattering of partons and, because they do not interact via the strong
force, are a phenomenologically clean probe of the structure of the proton. At
large transverse momentum, direct photons are produced at leading order via
the quantum chromodynamics (QCD) 2-to-2 hard scattering subprocesses quark-
gluon Compton scattering ($g+q\rightarrow\gamma+q$) and quark-antiquark
annihilation ($\bar{q}+q\rightarrow\gamma+g$). Compton scattering dominates at
midrapidity Adare  _et al._ (2010) because the proton is being probed at
moderate longitudinal momentum fraction, $x$, where gluons are the primary
constituents of the proton. Thus midrapidity direct photon measurements are a
clean probe of gluon structure within the proton.

Transverse single-spin asymmetries (TSSAs) in hadronic collisions are
sensitive to various spin-momentum correlations, i.e. correlations between the
directions of the spin and momentum of partons and/or hadrons involved in a
scattering event. In collisions between one transversely polarized proton and
one unpolarized proton, the TSSA describes the azimuthal-angular dependence of
particle production relative to the transverse polarization direction. TSSAs
have been measured to be as large as 40% in forward charged pion production
Klem  _et al._ (1976); Adams  _et al._ (1991); Allgower  _et al._ (2002);
Arsene  _et al._ (2008) and significantly nonzero forward neutral pion
asymmetries have been measured with transverse momentum up to $p_{T}\approx
7~{}{\rm GeV}/c$ Adam  _et al._ (2021). In this context, $p_{T}$ serves as
proxy for a hard-scattering energy ($Q$) that is well into the perturbative
regime of QCD. Next-to-leading-order perturbative QCD calculations, which only
include effects from high energy parton scattering predict that these
asymmetries should be small and fall off as $m_{q}/Q$ Kane  _et al._ (1978),
where $m_{q}$ is the bare mass of the quark. Thus, to explain these large
TSSAs, they must be considered in the context of the dynamics present in
proton-proton collisions that cannot be calculated perturbatively, namely
dynamics describing proton structure and/or the process of hadronization.

One approach toward explaining the large measured TSSAs is through transverse-
momentum-dependent (TMD) functions. These functions depend on the soft-scale-
parton transverse momentum, $k_{T}$, in addition to the partonic longitudinal
momentum fraction $x$ and $Q$, where $k_{T}\ll Q$. TMD functions can be
directly extracted from measurements that are sensitive to two momentum
scales, such as semi-inclusive deep-inelastic scattering (SIDIS) where the
angle and energy of the scattered electron can be used to directly measure the
hard-scale $Q$ and the transverse momentum of the measured hadron relates to
the soft scales $k_{T}$ of TMD parton distribution functions (PDFs) and
fragmentation functions. The Sivers function is a PDF that describes the
structure of the transversely polarized proton and correlates the transverse
spin of the proton and $k_{T}$ of the parton within it Sivers (1990). The
quark Sivers function has been extracted through polarized SIDIS measurements,
but the gluon Sivers function has remained comparatively less constrained
because SIDIS is not sensitive to gluons at leading order Adolph  _et al._
(2017). The direct photon TSSA in proton-proton collisions has been shown to
be sensitive to the gluon Sivers function Godbole  _et al._ (2019), but the
$k_{T}$ moment of TMD functions must be used to apply these functions to the
single-scale inclusive TSSAs measured in proton-proton collisions.

Twist-3 correlation functions are another approach toward describing TSSAs.
Unlike TMD functions, collinear twist-3 correlation functions depend only on a
single scale, the hard scale $Q$. Twist-3 functions describe spin-momentum
correlations generated by the quantum mechanical interference between
scattering off of one parton versus scattering off of two. There are two
different types: the quark-gluon-quark (qgq) correlation functions and the
trigluon (ggg) correlation function. In the context of proton structure, qgq
correlation functions describe the interference between scattering off of a
single quark in the proton versus scattering off of a quark, which carries the
same flavor and the same momentum fraction and an additional gluon.
Analogously, the trigluon correlation describes the interference between
scattering off of one gluon in the proton versus scattering off of two.
Additional twist-3 collinear correlation functions describing spin-momentum
correlations in the process of hadronization also exist, but are not relevant
to the production of direct photons. Collinear twist-3 functions have been
shown to be related to the $k_{T}$ moment of TMD functions Boer  _et al._
(2003); Ji  _et al._ (2006). For example, the Efremov-Teryaev-Qiu-Sterman
(ETQS) function is a qgq correlation function for the polarized proton Efremov
and Teryaev (1985); Qiu and Sterman (1992, 1999) that is related to the
$k_{T}$ moment of the Sivers TMD PDF. The ETQS function has also been
extracted from fits to inclusive TSSAs in proton-proton collisions Kanazawa
_et al._ (2014); Cammarota  _et al._ (2020), and the forward direct photon
TSSA has been suggested to be dominated by this ETQS function Kanazawa  _et
al._ (2015). The fact that both TMD and collinear twist-3 functions are
nonzero reflects that scattering partons do in fact interact with the color
fields present inside the proton, which goes beyond traditional assumptions
present in hadronic collision studies.

Multiple observables can provide sensitivity to the ggg correlation function.
Midrapidity inclusive hadron TSSA measurements are sensitive to gluon spin-
momentum correlations in the proton but also include potential effects from
hadronization and final-state color interactions. Heavy flavor production at
the Relativistic Heavy Ion Collider (RHIC) is dominated by gluon-gluon fusion
and thus particularly sensitive to gluons in the proton. A heavy flavor hadron
TSSA measurement Aidala  _et al._ (2017) has been used to estimate the
trigluon correlation function in the transversely polarized proton assuming no
effects from hadronization or final-state color interactions Koike and Yoshida
(2011). The midrapidity isolated direct photon TSSA is instead a clean probe
of the trigluon correlation function because it is insensitive to
hadronization effects as well as final-state color interactions Koike and
Yoshida (2012).

The only previously published direct photon TSSA measurement is the Fermilab
E704 result, which used a $200~{}{\rm GeV}/c$ polarized proton beam on an
unpolarized proton target ($\sqrt{s}=19.4$ GeV). It was found to be consistent
with zero to within 20% for $2.5<p_{T}^{\gamma}<3.1~{}{\rm GeV}/c$ Adams  _et
al._ (1995). The PHENIX results presented in this Letter measure photons with
$p_{T}^{\gamma}>5~{}{\rm GeV}/c$ with total uncertainties up to a factor of 50
smaller than the E704 measurements. This measurement will constrain trigluon
correlations in transversely polarized protons.

The presented direct photon measurement was performed with the PHENIX
experiment in the central rapidity region $|\eta|<0.35$, using
$p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ =200 GeV. The data set was
collected in 2015 and corresponds to an integrated luminosity of approximately
60 pb-1. Direct photons were reconstructed using similar techniques to a
previously published direct photon cross section result at $\sqrt{s}$ = 200
GeV Adare  _et al._ (2012). The asymmetry was measured with transversely
polarized proton beams at RHIC where the clockwise and counter-clockwise beams
had an average polarization of $0.58\pm 0.02$ and $0.60\pm 0.02$, respectively
Schmidke  _et al._ (2018). Collisions between bunches are spaced 106 ns apart
and the polarization direction changes bunch-to-bunch such that two
statistically independent asymmetries can be measured with the same particle
yields through sorting them by the polarization direction in one beam,
effectively averaging over the polarization in the other beam. These two
independent measurements serve as a cross check and are averaged together to
calculate the final asymmetry.

The PHENIX central detector comprises two nearly-back-to-back arms each
covering $\Delta\phi=\pi/2$ in azimuth and $|\eta|<0.35$ in pseudorapidity.
Photons are identified through clusters in the electromagnetic calorimeter
(EMCal), which has two detector arms: the west and the east. The west arm
comprises four sectors of sampling lead-scintillator (PbSc) calorimeters with
granularity $\delta\phi\times\delta\eta=0.011\times 0.011$ and the east arm
comprises two more PbSc sectors along with two sectors of Čerenkov lead-glass
(PbGl) calorimeters with granularity $\delta\phi\times\delta\eta=0.008\times
0.008$ Aphecetche  _et al._ (2003).

The PHENIX central tracking system uses pad chambers and a drift chamber to
measure the position of charged particle tracks Adcox  _et al._ (2003). The
beam-beam counters (BBC) are far-forward arrays of quartz Čerenkov radiators
that cover the full azimuth and $3.0<|\eta|<3.9$ Allen  _et al._ (2003). They
measure the position of the vertex in the beam direction, for which a 30 cm
vertex cut around the nominal collision point is applied. The minimum-bias
trigger fires on crossings where at least one charged particle is measured in
each arm of the BBC. Events with high-$p_{T}$ photons are selected through an
EMCal-based high-energy photon trigger that is taken in coincidence with this
minimum-bias trigger.

All photons used in the asymmetry calculation are required to pass the
following cuts. A shower shape cut selects clusters whose energy distribution
is consistent with a parameterized profile from a photon shower. This reduces
the contribution of clusters from hadrons along with merged photons from high
energy $\pi^{0}$ decays, which resolve as a single cluster in the EMCal. A
time-of-flight cut suppresses the contribution of EMCal noise, where the
timing of the cluster is measured by the EMCal and the time zero reference of
the event is provided by the BBC. A charged-track-veto cut eliminates clusters
that geometrically match with a charged track and uses the track position
measured directly in front of the EMCal. This cut reduces the background from
electrons as well as charged hadrons that were not eliminated by the shower
shape cut.

Direct photon candidates are also required to pass tagging cuts that reduce
the hadronic decay background by eliminating photons that are tagged as coming
from either $\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ or
$\eta\rightarrow\gamma\gamma$ decays. The candidate direct photon is matched
with a partner photon in the same event and same EMCal arm, which has passed a
minimum-energy cut of 0.5 GeV. A photon is considered tagged as coming from a
$\mbox{$\pi^{0}$}\rightarrow\gamma\gamma$ ($\eta\rightarrow\gamma\gamma$)
decay if it is matched into a photon pair with invariant mass
$105<M_{\gamma\gamma}<165~{}{\rm MeV}/c^{2}$ ($480<M_{\gamma\gamma}<620~{}{\rm
MeV}/c^{2}$), which corresponds roughly to a $\pm 2\sigma$ window around the
observed $\pi^{0}$ and $\eta$ peaks.

Additionally, direct photon candidates have to pass an isolation cut, which
further reduces the contribution of decay photons Adare  _et al._ (2012). Ref.
Adare  _et al._ (2010) estimates that the contribution of the next-to-leading-
order fragmentation photons to the isolated direct photon sample is less than
15% for photons with $p_{T}>5~{}{\rm GeV}/c$. The photon isolation cut
requires that the sum of the particles’ energy surrounding the photon in a
cone of radius $r=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<0.4$ radians be
less than 10% of the candidate photon’s energy: $E_{\rm cone}<E_{\gamma}\cdot
10\%$. To be included in the cone sum energy, $E_{\rm cone}$, an EMCal cluster
must have energy larger than $0.15~{}{\rm GeV}$ and a charged track needs to
have a momentum above $0.2~{}{\rm GeV}/c$. To provide a more inclusive sample
of the particles surrounding the photon, the clusters and tracks that are
included in the $E_{\rm cone}$ sum are only required to pass a minimum set of
quality cuts. The charged track veto cut is still used to ensure charged
particles are not double counted by the energy that they deposit in the EMCal.
The shower-shape cut is not applied to EMCal clusters to ensure that neutral
hadrons and charged hadrons that were not reconstructed as charged tracks can
still contribute to $E_{\rm cone}$.

The asymmetry measurement is formed from photons that satisfy these criteria,
using similar techniques to previously published PHENIX TSSAs which include
Refs. Aidala  _et al._ (2017) and Acharya  _et al._ (2021). The TSSA is
determined using the relative luminosity formula:

$A_{N}=\frac{1}{P\,\left<\cos(\phi)\right>}\frac{{N^{\uparrow}}-\mathcal{R}{N^{\downarrow}}}{{N^{\uparrow}}+\mathcal{R}{N^{\downarrow}}},$
(1)

where $\mathcal{R}=\mathcal{L}^{\uparrow}/\mathcal{L}^{\downarrow}$ is the
relative luminosity of collisions for when the beam was polarized up versus
down. $P$ is the average polarization of the beam and
$\left<\cos(\phi)\right>$ is the acceptance factor accounting for the
azimuthal coverage of each detector arm. In Eq. (1), $N$ refers to the
particle yield and the up ($\uparrow$) or down ($\downarrow$) arrow
superscripts refer to the direction of the beam polarization. The asymmetries
are calculated separately for each arm of the detector and averaged together
for the final result, weighted by the statistical uncertainty.

The main source of direct-photon background comes from decay photons that were
not eliminated by the tagging cut because their partner photon was not
measured. This can occur because the partner photon was out of acceptance, hit
a dead area of the detector, or did not pass the minimum-energy cut. To
calculate the isolated direct-photon asymmetry, $A_{N}^{\rm dir}$, the
candidate isolated direct-photon asymmetry, $A_{N}^{\rm iso}$, must be
corrected for the contribution from background:

$A_{N}^{\rm dir}=\frac{A_{N}^{\rm iso}-r_{\pi^{0}}\,A_{N}^{{\rm
iso},\pi^{0}}-r_{\eta}\,A_{N}^{{\rm iso},\eta}}{1-r_{\pi^{0}}-r_{\eta}}.$ (2)

This expression removes the effects of background asymmetries from isolated
$\pi^{0}$ photons, $A_{N}^{{\rm iso},\pi^{0}}$, and isolated $\eta$ photons,
$A_{N}^{{\rm iso},\eta}$, where $r_{\pi^{0}}$ and $r_{\eta}$ are the
background fractions due to photons from $\pi^{0}$ and $\eta$ decays,
respectively. Because the midrapidity $\pi^{0}$ and $\eta$ TSSAs have been
measured to be consistent with zero to high statistical precision Acharya  _et
al._ (2021) and their isolated asymmetries were also confirmed to be
consistent with zero, $A_{N}^{\rm{iso},\pi^{0}}$ and $A_{N}^{\rm{iso},\eta}$
are set to zero in Eq. (2). The systematic uncertainty due to setting the
background asymmetries to zero dominates the total systematic uncertainty of
the direct-photon asymmetry for all $p_{T}$ bins. It is assigned by
integrating the inclusive midrapidity $\pi^{0}$ and $\eta$ TSSAs over photon
$p_{T}$ and propagating their uncertainties through Eq. (2).

The background fraction calculation is performed by taking the ratio of
measured photon yields: $N^{{\rm iso},h}_{\rm tag}/N^{\rm iso}$, where $N^{\rm
iso}$ is the isolated direct photon candidate sample. $N^{{\rm iso},h}_{\rm
tag}$ is the number of photons that were tagged as coming from a diphoton
decay of hadron $h$ and pass the photon pair isolation cut, $E_{\rm
cone}-E_{\rm partner}<E_{\gamma}\cdot 10\%$, which subtracts off the energy of
the partner photon, $E_{\rm partner}$. Tagged photons that pass this cut would
have been included in the isolated direct photon candidate sample had their
partner photon not been detected. Simulations are used to calculate how to
convert from the number of tagged decay photons to the number of decay photons
where the partner photon was missed. The background fraction, $r_{h}$, for
photons from $\pi^{0}$ and $\eta$ meson decays is calculated separately to
account for their differences in particle production and decay kinematics,

$r_{h}=R_{h}\frac{N^{{\rm iso},h}_{\rm tag}}{N^{\rm iso}},$ (3)

where $R_{h}$ is the one-miss ratio for the decay of hadron $h$. It is the
ratio in single particle Monte Carlo of the number of photons for which only
one of the simulated decay photons was reconstructed to the number of photons
in which both decay photons were reconstructed Adare  _et al._ (2012). These
simulations include the geometry, resolution, and configuration of the dead
areas of the EMCal and use the previously measured $\pi^{0}$ Adare  _et al._
(2007) and $\eta$ Adare  _et al._ (2011) cross sections. The background
fractions for photons from $\pi^{0}$ ($\eta$) decays are plotted in Fig. 1 and
are systematically larger in the east arm versus the west due to the PbGl
sectors having slightly more dead area compared to the PbSc sectors. The
contribution of decay photons from sources heavier than $\eta$ mesons is
estimated to be less than 3% with respect to the measured background and so an
even smaller percentage of the total direct photon sample. The uncertainty on
the background fraction is propagated through Eq. (2) to assign an additional
systematic uncertainty to the direct-photon asymmetry.

Figure 1: The fractional contribution of photons from (a) $\pi^{0}$ and (b)
$\eta$ decays to the isolated direct photon candidate sample.

A similar method to Eq. (3) is used to find the contribution of merged
$\pi^{0}$ decay photons. The equivalent $R_{h}$ is calculated using simulated
$h\rightarrow\gamma\gamma$ decays, taking the ratio of the number of
reconstructed EMCal clusters produced by merged decay photons divided by the
number of reconstructed clusters associated with a single decay photon. The
contribution from merged photon clusters was found to be less than 0.2%, small
compared to the up to 50% background fraction due to the one-miss effects, and
the contribution from merged $\eta$ decays was confirmed to be negligible.

An additional systematic study is performed by calculating the asymmetry with
the square root formula:

$A_{N}=\frac{1}{P\,\left<\cos(\phi)\right>}\frac{\sqrt{N_{L}^{\uparrow}N_{R}^{\downarrow}}-\sqrt{N_{L}^{\downarrow}N_{R}^{\uparrow}}}{\sqrt{N_{L}^{\uparrow}N_{R}^{\downarrow}}+\sqrt{N_{L}^{\downarrow}N_{R}^{\uparrow}}},$
(4)

where the $L$ and $R$ subscripts refer to yields to the left and to the right
of the polarized-beam-going direction, respectively. This result is verified
to be consistent with the relative luminosity formula results from Eq. (1) and
the differences between these results are assigned as an additional systematic
uncertainty due to possible variations in detector performance and beam
conditions. The systematic uncertainty due to setting the background
asymmetries to zero dominates the total systematic uncertainty by an order of
magnitude for all $p_{T}$ bins except for the highest $p_{T}$ bin, where it is
only slightly larger than the difference between the square root formula and
relative luminosity formula. Another study using bunch shuffling found no
additional systematic effects. Bunch shuffling is a technique that randomizes
the bunch-by-bunch beam polarization directions to confirm that the variations
present in the data are consistent with what is expected by statistical
variation.

Figure 2: Transverse single-spin asymmetry of isolated direct photons measured at midrapidity $|\eta|<0.35$ in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ = 200 GeV. An additional scale uncertainty of 3.4% due to the polarization uncertainty is not shown.  Table 1:  The measured $A_{N}$ of isolated direct photons in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ =200 GeV as a function of $p_{T}$. An additional scale uncertainty of 3.4% due to the polarization uncertainty is not included.  $\langle\mbox{$p_{T}$}\rangle[{\rm GeV}/c]$ | $A_{N}^{\rm dir}$ | $\sigma_{\rm stat}$ | $\sigma_{\rm syst}$
---|---|---|---
5.39 | -0.000492 | 0.00299 | 0.00341
6.69 | 0.00247 | 0.00404 | 0.00252
8.77 | 0.00777 | 0.00814 | 0.00159
11.88 | 0.00278 | 0.0105 | 0.00106

The results for the $A_{N}$ of isolated direct photons, $A_{N}^{\rm dir}$, at
midrapidity in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ = 200 GeV are
shown in Table 1 and in Fig. 2, where the shaded [gray] bands represent the
systematic uncertainty and the vertical bars represent the statistical
uncertainty. The measurement is consistent with zero to within 1% across the
entire $p_{T}$ range. Figure 2 also shows predictions from collinear twist-3
correlation functions. The solid [green] curve shows the contribution of qgq
correlation functions to the direct-photon asymmetry which is calculated using
functions that were published in Ref. Kanazawa  _et al._ (2015) that are
integrated over the $|\eta|<0.35$ pseudorapidity range of the PHENIX central
arms. This calculation includes contributions from the qgq correlation
functions present in both the polarized and unpolarized proton, including the
ETQS function which is extracted from a global fit in Ref. Cammarota  _et al._
(2020). The error band plotted with the solid [green] curve in Fig. 2 includes
uncertainties propagated from fits to data, but does not include uncertainties
associated with assuming functional forms. Quark-flavor dependence is not
considered in these calculations, including qgq correlators. Direct-photon
production in $p$$+$$p$ collisions is four times more sensitive to the up
quark than the down quark in the proton because of the factor of electric
charge squared in the production cross section.

Given the small predicted contributions from qgq correlation functions to the
midrapidity direct photon TSSA, this measurement can provide a clean
extraction of the ggg function. The predicted ranges for the trigluon
correlation function’s contribution to the direct-photon asymmetry are also
plotted in Fig. 2. The dashed [blue] and dotted [red] curves use results that
were published in Ref. Koike and Yoshida (2011) and were reevaluated as a
function of photon $p_{T}$ for pseudorapidity $\eta=0$ 111The trigluon Model 1
and Model 2 curves in Fig. 2 were provided by S. Yoshida, while D. Pitonyak
provided the quark-gluon-quark curve.. Models 1 and 2 assume different
functional forms for the trigluon correlation function in terms of the
collinear leading-twist gluon PDF; no uncertainties are available for these
curves. As this figure shows, this measurement has the statistical precision,
especially at low $p_{T}$, to constrain the trigluon correlation function.

In summary, the TSSA of midrapidity isolated direct photons was measured by
the PHENIX experiment to be consistent with zero in the presented $p_{T}$
range, with uncertainties as low as 0.4% in the lowest $p_{T}$ bins. This is
the first time direct photons have been used to probe transversely polarized
proton collisions at RHIC and the first measurement of this TSSA in almost 30
years, with significantly higher $p_{T}$ reach and up to a fifty-fold
improvement in uncertainty. Direct photons are a clean probe of proton
structure with no contributions from final-state QCD effects and at
midrapidity are particularly sensitive to gluon dynamics. When included in the
global analysis of world TSSA data, this measurement will constrain gluon
spin-momentum correlations in the transversely polarized proton for $x\approx
x_{T}=0.05-0.18$, marking an important step toward creating a more three-
dimensional picture of proton structure.

###### 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 D. Pitonyak and S.
Yoshida for helpful discussions. 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 (U.S.A), Ministry of Education, Culture, Sports,
Science, and Technology and the Japan Society for the Promotion of Science
(Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and
Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), 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), Bundesministerium für Bildung und
Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt
Stiftung (Germany), 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). Physics Department, Lahore University of Management Sciences
(Pakistan), Ministry of Education and Science, Russian Academy of Sciences,
Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation
(Sweden), 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.

## References

  * Adare  _et al._ (2010) A. Adare _et al._ (PHENIX Collaboration), “High $p_{T}$ direct photon and $\pi^{0}$ triggered azimuthal jet correlations and measurement of $k_{T}$ for isolated direct photons in $p+p$ collisions at $\sqrt{s}=200$ GeV,” Phys. Rev. D 82, 072001 (2010).
  * Klem  _et al._ (1976) R. D. Klem, J. E. Bowers, H. W. Courant, H. Kagan, M. L. Marshak, E. A. Peterson, K. Ruddick, W. H. Dragoset, and J. B. Roberts, “Measurement of Asymmetries of Inclusive Pion Production in Proton Proton Interactions at 6-GeV/c and 11.8-GeV/c,” Phys. Rev. Lett. 36, 929 (1976).
  * Adams  _et al._ (1991) D. L. Adams _et al._ (FNAL-E704 Collaboration), “Analyzing power in inclusive $\pi^{+}$ and $\pi^{-}$ production at high $x_{F}$ with a 200-GeV polarized proton beam,” Phys. Lett. 264, 462 (1991).
  * Allgower  _et al._ (2002) C.E. Allgower _et al._ , “Measurement of analyzing powers of $\pi^{+}$ and $\pi^{-}$ produced on a hydrogen and a carbon target with a 22-GeV/c incident polarized proton beam,” Phys. Rev. D 65, 092008 (2002).
  * Arsene  _et al._ (2008) I. Arsene _et al._ (BRAHMS Collaboration), “Single Transverse Spin Asymmetries of Identified Charged Hadrons in Polarized $p+p$ Collisions at $\sqrt{s}$ = 62.4 GeV,” Phys. Rev. Lett. 101, 042001 (2008).
  * Adam  _et al._ (2021) Jaroslav Adam _et al._ (STAR), “Comparison of transverse single-spin asymmetries for forward $\pi^{0}$ production in polarized $pp$, $p\rm{Al}$ and $p\rm{Au}$ collisions at nucleon pair c.m. energy $\sqrt{s_{\mathrm{NN}}}=200$ GeV,” Phys. Rev. D 103, 072005 (2021).
  * Kane  _et al._ (1978) G. L. Kane, J. Pumplin, and W. Repko, “Transverse Quark Polarization in Large $p_{T}$ Reactions, $e^{+}e^{-}$ Jets, and Leptoproduction: A Test of QCD,” Phys. Rev. Lett. 41, 1689 (1978).
  * Sivers (1990) D. W. Sivers, “Single Spin Production Asymmetries from the Hard Scattering of Point-Like Constituents,” Phys. Rev. D 41, 83 (1990).
  * Adolph  _et al._ (2017) C. Adolph _et al._ (COMPASS Collaboration), “First measurement of the Sivers asymmetry for gluons using SIDIS data,” Phys. Lett. B 772, 854 (2017).
  * Godbole  _et al._ (2019) R. M. Godbole, A. Kaushik, A. Misra, and S. Padval, “Probing the Gluon Sivers Function through direct photon production at RHIC,” Phys. Rev. D 99, 014003 (2019).
  * Boer  _et al._ (2003) D. Boer, P.J. Mulders, and F. Pijlman, “Universality of T odd effects in single spin and azimuthal asymmetries,” Nucl. Phys. B667, 201 (2003).
  * Ji  _et al._ (2006) X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, “A Unified picture for single transverse-spin asymmetries in hard processes,” Phys. Rev. Lett. 97, 082002 (2006).
  * Efremov and Teryaev (1985) A. V. Efremov and O. V. Teryaev, “QCD Asymmetry and Polarized Hadron Structure Functions,” Phys. Lett. B 150, 383 (1985).
  * Qiu and Sterman (1992) J. Qiu and G. F. Sterman, “Single transverse spin asymmetries in direct photon production,” Nucl. Phys. B378, 52 (1992).
  * Qiu and Sterman (1999) J. Qiu and G. F. Sterman, “Single transverse spin asymmetries in hadronic pion production,” Phys. Rev. D 59, 014004 (1999).
  * Kanazawa  _et al._ (2014) K. Kanazawa, Y. Koike, A. Metz, and D. Pitonyak, “Towards an explanation of transverse single-spin asymmetries in proton-proton collisions: the role of fragmentation in collinear factorization,” Phys. Rev. D 89, 111501 (2014).
  * Cammarota  _et al._ (2020) J. Cammarota, L. Gamberg, Z.-B. Kang, J. A. Miller, D. Pitonyak, A. Prokudin, T. C. Rogers, and N. Sato (Jefferson Lab Angular Momentum Collaboration), “Origin of single transverse-spin asymmetries in high-energy collisions,” Phys. Rev. D 102, 054002 (2020).
  * Kanazawa  _et al._ (2015) K. Kanazawa, Y. Koike, A. Metz, and D. Pitonyak, “Transverse single-spin asymmetries in $p^{\uparrow}p\to\gamma X$ from quark-gluon-quark correlations in the proton,” Phys. Rev. D 91, 014013 (2015).
  * Aidala  _et al._ (2017) C. Aidala _et al._ (PHENIX Collaboration), “Cross section and transverse single-spin asymmetry of muons from open heavy-flavor decays in polarized $p$+$p$ collisions at $\sqrt{s}=200$ GeV,” Phys. Rev. D 95, 112001 (2017).
  * Koike and Yoshida (2011) Y. Koike and S. Yoshida, “Probing the three-gluon correlation functions by the single spin asymmetry in $p^{\uparrow}p\rightarrow DX$,” Phys. Rev. D 84, 014026 (2011).
  * Koike and Yoshida (2012) Y. Koike and S. Yoshida, “Three-gluon contribution to the single spin asymmetry in Drell-Yan and direct-photon processes,” Phys. Rev. D 85, 034030 (2012).
  * Adams  _et al._ (1995) D.L. Adams _et al._ (E704 Collaboration), “Measurement of single spin asymmetry for direct photon production in p p collisions at 200-GeV/c,” Phys. Lett. B 345, 569 (1995).
  * Adare  _et al._ (2012) A. Adare _et al._ (PHENIX Collaboration), “Direct-Photon Production in $p+p$ Collisions at $\sqrt{s}=200$ GeV at Midrapidity,” Phys. Rev. D 86, 072008 (2012).
  * Schmidke  _et al._ (2018) W. D. Schmidke _et al._ (The RHIC Polarimetry Group), “RHIC polarization for Runs 9–17,” https://technotes.bnl.gov/Home/ViewTechNote/209057 (2018).
  * Aphecetche  _et al._ (2003) L. Aphecetche _et al._ (PHENIX Collaboration), “PHENIX calorimeter,” Nucl. Instrum. Methods Phys. Res., Sec. A 499, 521 (2003).
  * Adcox  _et al._ (2003) K. Adcox _et al._ (PHENIX Collaboration), “PHENIX central arm tracking detectors,” Nucl. Instrum. Methods Phys. Res., Sec. A 499, 489 (2003).
  * Allen  _et al._ (2003) M. Allen _et al._ , “PHENIX inner detectors,” Nucl. Instrum. Methods Phys. Res., Sec. A 499, 549 (2003).
  * Acharya  _et al._ (2021) U. A. Acharya _et al._ (PHENIX), “Transverse single-spin asymmetries of midrapidity $\pi^{0}$ and $\eta$ mesons in polarized $p+p$ collisions at $\sqrt{s}=200$ GeV,” Phys. Rev. D 103, 052009 (2021).
  * Adare  _et al._ (2007) A. Adare _et al._ (PHENIX Collaboration), “Inclusive cross-section and double helicity asymmetry for $\pi^{0}$ production in $p+p$ collisions at $\sqrt{s}=$ 200 GeV: Implications for the polarized gluon distribution in the proton,” Phys. Rev. D 76, 051106 (2007).
  * Adare  _et al._ (2011) A. Adare _et al._ (PHENIX Collaboration), “Cross section and double helicity asymmetry for $\eta$ mesons and their comparison to neutral pion production in p+p collisions at $\sqrt{s}=200$ GeV,” Phys. Rev. D 83, 032001 (2011).
  * Note (1) The trigluon Model 1 and Model 2 curves in Fig. 2 were provided by S. Yoshida, while D. Pitonyak provided the quark-gluon-quark curve.