PHENIX Collaboration

# Improving constraints on gluon spin-momentum correlations in transversely
polarized protons via midrapidity open-heavy-flavor electrons 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  V. Borisov  Saint Petersburg
State Polytechnic University, St. Petersburg, 195251 Russia  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  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  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  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  M. Csanád  ELTE, Eötvös Loránd University, H-1117
Budapest, Pázmány P. s. 1/A, 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  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  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  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  H. En’yo  RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
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
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, 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. Firak  Department of
Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  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  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  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  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
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  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  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  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  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  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  L. Kovacs  ELTE, Eötvös Loránd University, H-1117 Budapest, Pázmány P.
s. 1/A, Hungary  S. Kudo  Tomonaga Center for the History of the Universe,
University of Tsukuba, Tsukuba, Ibaraki 305, Japan  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  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
X. Li  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
V.-R. Loggins  University of Illinois at Urbana-Champaign, Urbana, Illinois
61801, USA  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  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  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  M. Mitrankova  Saint Petersburg State
Polytechnic University, St. Petersburg, 195251 Russia  Iu. Mitrankov  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  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  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  M.
Orosz  Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary  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  M. Patel  Iowa
State University, Ames, Iowa 50011, USA  S.F. Pate  New Mexico State
University, Las Cruces, New Mexico 88003, 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  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 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.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
L. Xue  Georgia State University, Atlanta, Georgia 30303, USA  C. Xu  New
Mexico State University, Las Cruces, New Mexico 88003, USA  Q. Xu  Vanderbilt
University, Nashville, Tennessee 37235, 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  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  L. Zou  University of California-Riverside, Riverside,
California 92521, USA

###### Abstract

Polarized proton-proton collisions provide leading-order access to gluons,
presenting an opportunity to constrain gluon spin-momentum correlations within
transversely polarized protons and enhance our understanding of the three-
dimensional structure of the proton. Midrapidity open-heavy-flavor production
at $\sqrt{s}=200$ GeV is dominated by gluon-gluon fusion, providing heightened
sensitivity to gluon dynamics relative to other production channels.
Transverse single-spin asymmetries of positrons and electrons from heavy-
flavor hadron decays are measured at midrapidity using the PHENIX detector at
the Relativistic Heavy Ion Collider. These charge-separated measurements are
sensitive to gluon correlators that can in principle be related to gluon
orbital angular momentum via model calculations. Explicit constraints on gluon
correlators are extracted for two separate models, one of which had not been
constrained previously.

Polarized proton-proton collisions provide a unique opportunity to improve our
understanding of gluon contributions to the spin structure of the proton, as
they are accessible at leading order, which is not the case in lepton-hadron
scattering. The complex spin structure of the proton leads to emergent
properties such as spin-momentum and spin-spin correlations analogous to those
in atomic systems that are experimentally accessible through observables known
as transverse single-spin asymmetries (TSSAs). TSSAs quantify azimuthal
modulations of particle production in collisions of transversely polarized
nucleons with unpolarized particles, and have been measured to reach
magnitudes up to 40% in hadron-hadron collisions [1, 2, 3, 4]. Calculations
based on perturbative quantum chromodynamics (pQCD) had predicted negligible
contributions to TSSAs ($<1$%) [5], while recent calculations suggest the
possibility of pQCD contributions [6].

Typically considered are modifications to the nonperturbative elements of
factorized cross sections, for which there are two complementary theoretical
frameworks. The first has explicit dependence on the transverse momentum
$k_{T}$ of partons within hadrons in addition to the longitudinal momentum
fraction $x$. In this apporoach, standard collinear parton distribution
functions (PDFs) and fragmentation functions (FFs) are replaced with
transverse-momentum-dependent (TMD) functions. The second approach considers
higher-twist effects, or power-suppressed terms with respect to the hard-
scattering energy scale $Q$ in the collinear factorization expansion.
Constraining TMD functions experimentally requires access to both a hard-
scattering energy scale $Q$ and soft scale $k_{T}$ sensitive to partonic
transverse momentum in the proton or the process of hadronization, with $Q\gg
k_{T}$. Alternatively, the higher-twist formalism only requires access to a
hard scale that is represented by the measured particle transverse momentum
$p_{T}$. At twist-3, quantum interference between standard $2\rightarrow 2$
QCD processes and those involving an extra gluon must be considered, which
introduces additional terms to cross-section calculations depending on the
number of colliding or produced hadrons. These terms encode quantum
interference in twist-3 correlation functions convoluted with standard
collinear PDFs and FFs. TSSAs are defined in Eq. (6), leading to the following
proportionality at twist-3 [7]:

$\displaystyle\small\begin{aligned}
A_{N}\propto\sum_{a,b,c}\phi^{(3)}_{a/A}(x_{1},x_{2},\vec{s}_{\perp})\otimes\phi_{b/B}(x^{\prime})\otimes\hat{\sigma}\otimes
D_{c\rightarrow C}(z)\\\ +\sum_{a,b,c}\delta
q_{a/A}(x,\vec{s}_{\perp})\otimes\phi^{(3)}_{b/B}(x^{\prime}_{1},x^{\prime}_{2})\otimes\hat{\sigma}^{\prime}\otimes
D_{c\rightarrow C}(z)\\\ +\sum_{a,b,c}\delta
q_{a/A}(x,\vec{s}_{\perp})\otimes\phi_{b/B}(x^{\prime})\otimes\hat{\sigma}^{\prime\prime}\otimes
D^{(3)}_{c\rightarrow C}(z_{1},z_{2})\end{aligned}$ (1)

Each term with a superscript $(3)$ corresponds to a twist-3 correlation
function, the rest are at leading twist (twist-2). The lower case subscripts
represent the parton type, the uppercase subscripts represent the parent
hadron, $\phi$ and $D$ denote PDFs and FFs respectively, and $\delta
q_{x/X}(x,\vec{s}_{\perp})$ is the transversity distribution, a spin-spin
correlation of transversely polarized quarks in transversely polarized hadrons
[8]. The twist-3 correlators have more intuitive physical meaning through
their relation to corresponding TMD PDFs [9, 10, 11, 12].

In $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV, open-heavy-flavor (OHF)
production at midrapidity is dominated by gluon-gluon fusion, receiving only a
small contribution from quark-antiquark annihilation. In gluon-gluon fusion
events, only the first term in Eq. (1) is relevant (as the gluon does not have
a transversity distribution) providing sensitivity to the trigluon correlation
functions in polarized protons. The relevant twist-3 correlators for quark-
antiquark annihilation and gluon-gluon fusion are the Efremov-Teryaev-Qiu-
Sterman ($qgq$) correlator [13, 14] and the trigluon ($ggg$) correlators [15,
16, 17, 18, 19] respectively. Note that the trigluon correlators were
introduced in Ref. [15] as a single function, and subsequently clarified to be
two independent functions [16, 17, 18, 19]. The $qgq$ correlator has been
experimentally constrained from global fits, discussed in Ref. [20], while the
$ggg$ correlators have received less attention, with only few measurements
capable of providing constraints [21, 22].

The TSSA for open-charm production in $p^{\uparrow}+p$ collisions at
$\sqrt{s}=200$ GeV was calculated in Refs. [23] and [24]. The trigluon
correlation functions are defined in Ref. [23] as $T_{G}^{(f)}(x,x)$
(antisymmetric) and $T_{G}^{(d)}(x,x)$ (symmetric), where the $(f)$ and $(d)$
superscripts represent three gluon-field color indices contracting with
antisymmetric or symmetric structure tensors. These functions can be
understood via normalization parameters $\lambda_{f}$ and $\lambda_{d}$ to the
unpolarized gluon PDF, introduced in Ref. [23]:

$T_{G}^{(f)}(x,x)=\lambda_{f}G(x),\hskip
30.00005ptT_{G}^{(d)}(x,x)=\lambda_{d}G(x).$ (2)

The trigluon correlation functions in Ref. [24] are defined as
$N(x_{1},x_{2})$ (antisymmetric), and $O(x_{1},x_{2})$ (symmetric). There are
four independent contributions to the TSSA from these functions,
$\\{N(x,x),N(x,0),O(x,x),O(x,0)\\}$, but as shown in Ref. [24], at
$\sqrt{s}=200$ GeV the asymmetries depend on effective trigluon correlation
functions $N(x,x)-N(x,0)$ and $O(x,x)+O(x,0)$. These combinations of $N$ and
$O$ functions are directly compared to the trigluon correlation functions
$T_{G}^{(f)}$ and $T_{G}^{(d)}$ in Ref. [18]. Reference [24] introduces the
following assumptions to constrain the $N$ and $O$ functions with data
relative to the unpolarized gluon PDF:

$\displaystyle O(x,x)$ $\displaystyle=O(x,0)=N(x,x)=-N(x,0)$ (3)
$\displaystyle[{\rm Model1}]\ \ O(x,x)$ $\displaystyle=K_{G}xG(x)$ (4)
$\displaystyle[{\rm Model2}]\ \ O(x,x)$
$\displaystyle=K_{G}^{\prime}\sqrt{x}G(x)$ (5)

Note that the assumptions on the trigluon correlation functions in Eqs. 2, 4,
and 5 (e.g., the functional dependence on $x$ and the proportionality to the
unpolarized gluon PDF) are oversimplified. For this reason, it is advantageous
to compare to different models with various $x$ dependences. The results
presented in this paper place direct constraints on
$\lambda_{f},\lambda_{d},K_{G}$ and $K_{G}^{\prime}$.

Open-charm production at the Relativistic Heavy Ion Collider (RHIC) has also
been investigated with the TMD factorization approach and proposed as a means
of constraining the gluon Sivers PDF in Ref. [25]. The measurements presented
here will be useful in providing constraints to the gluon Sivers TMD PDF both
directly through the phenomenological arguments shown in [25], and indirectly
through constraining the twist-3 trigluon correlators, which are related to
$k_{T}$ moments of the gluon Sivers PDF [9].

The asymmetry measurements presented here utilize data recorded in 2015 by the
PHENIX experiment at RHIC with collisions of transversely polarized protons on
transversely polarized protons at $\sqrt{s}=200$ GeV. Events with a collision
z-vertex within $\pm 10$ cm of the interaction point were considered for
analysis, yielding a total of approximately 23 pb-1 of integrated luminosity
used for the measurement. The polarization of each beam for Run-15 at RHIC is
measured to be $0.58\pm 0.02$ for the clockwise beam and $0.60\pm 0.02$ for
the counterclockwise beam, with transverse polarization direction aligned
vertically to the accelerator plane [26]. The polarization direction varies
from bunch to bunch in order to reduce systematics related to detector
coverage and performance, and to allow for the polarization of a single beam
to be considered at a time by averaging over the polarization directions of
the opposing beam. This yields two independent data sets from which the
transverse single-spin asymmetries are extracted, validated for consistency,
and averaged to obtain the final result.

The PHENIX detector is described in detail in Ref. [27]. Detector subsystems
used for midrapidity charged-particle detection comprise two central-arm
spectrometers oriented to the left and right of the beam axis, each with
acceptance $|\eta|<0.35$ and $\Delta\phi=0.5\pi$, and a silicon vertex
detector (VTX) [28, 29] with acceptance of $|\eta|<1$ and $\Delta\phi\approx
0.8\pi$ per arm. The central arms contain drift and pad chambers for tracking
[30], electromagnetic calorimeters (EMCal) to measure energy deposition of
charged particles and photons [31], and a ring-imaging Čerenkov (RICH)
detector for particle identification with $e/\pi$ separation up to $5$ GeV/$c$
[32].

Reconstructed tracks are projected to the RICH, EMCal, and VTX. The RICH
photomultiplier tubes are required to register at least one hit, and track
projections to the EMCal are required to have a ratio of energy deposited over
track momentum $E/p\approx 1$. A hit is required in both of the inner two
layers of the VTX to veto conversion electrons created by photons interacting
with detector material, and an additional hit is required in either of the
outer layers of the VTX. The narrow opening angle between $e^{+}e^{-}$ from
photonic conversions is exploited to further reduce background from
conversions; more details for this and the VTX detector can be found in Ref.
[33]. The details of curating an electron-candidate sample, described in the
supplemental material [34], follow the same procedure as in Ref. [33]. An
additional requirement was placed on the number of live trigger counts per
bunch crossing because the asymmetry analysis is performed bunch-by-bunch.

TSSAs are typically calculated as amplitudes of sinusoidal modulations of
azimuthal particle production:

$A_{N}(\phi)=\frac{\sigma^{\uparrow}(\phi)-\sigma^{\downarrow}(\phi)}{\sigma^{\uparrow}(\phi)+\sigma^{\downarrow}(\phi)}=A_{N}\cos\phi,$
(6)

where $\sigma^{\uparrow,\downarrow}(\phi)$ correspond to transversely
polarized cross sections for different spin orientations. Due to the nature of
the $\phi$ acceptance of the PHENIX spectrometer arms, the measurement of
midrapidity TSSAs are integrated in $\phi$ for one arm at a time. This
necessitates division by an azimuthal correction factor
$\langle\mid\cos\phi\mid\rangle$. Equation (6) must also be corrected for the
polarization $P$. All of these corrections are applied as seen in the
“relative luminosity formula”, a well-established PHENIX method used in Refs.
[22, 35, 21, 36, 37] to extract TSSAs:

$A_{N}=\frac{1}{P\langle\mid\cos\phi\mid\rangle}\frac{N^{\uparrow}-\mathcal{R}N^{\downarrow}}{N^{\uparrow}+\mathcal{R}N^{\downarrow}}.$
(7)

In Eq. (7), $N^{\uparrow,\downarrow}$ are the spin-dependent yields for
collisions with $\uparrow,\downarrow$ polarized bunch crossings respectively,
and $\mathcal{R}=\mathcal{L}^{\uparrow}/\mathcal{L}^{\downarrow}$ is the
relative luminosity, defined as the ratio of luminosities for collisions with
oppositely oriented bunch crossing polarization. The azimuthal correction
factor $\langle\mid\cos\phi\mid\rangle$ is calculated in each transverse
momentum ($p_{T}$) bin for the electron candidate sample to account for
detector efficiency effects. To serve as a cross check to Eq. (7), the
asymmetries are also calculated with the “square root formula,” as shown in
Eq. (8). The difference in asymmetries calculated with the separate methods is
taken as a systematic uncertainty $\sigma^{\rm sys}_{\rm diff}$.

$A_{N}=\frac{1}{P\langle\mid\cos\phi\mid\rangle}\frac{\sqrt{N^{\uparrow}_{L}N^{\downarrow}_{R}}-\sqrt{N^{\downarrow}_{L}N^{\uparrow}_{R}}}{\sqrt{N^{\uparrow}_{L}N^{\downarrow}_{R}}+\sqrt{N^{\downarrow}_{L}N^{\uparrow}_{R}}}.$
(8)

The $L,R$ subscripts represent the left and right spectrometer arm with
respect to the polarized proton-going direction. The square root formula
cannot be used independently on each spectrometer arm, leading to only two
independent data sets for cross validation and averaging corresponding to the
two beams, rather than four independent data sets as is the case for the
relative luminosity formula, corresponding to the two beams and two
spectrometer arms.

Once $A_{N}$ is calculated for the electron candidate sample, it is corrected
for background in order to extract the asymmetry for OHF decay electrons. The
relevant background sources are electrons from other parent particles
($\pi^{0},\eta$, direct photons $\gamma,J/\psi,K^{0}_{S},K^{\pm}$) and charged
hadrons misidentified as electrons (primarily $\pi^{\pm}$). To calculate the
background-corrected asymmetry, the fraction of each background source present
in the data sample needs to be calculated and the background asymmetries need
to be measured. Equation (9) shows the formula for extracting the ${\rm
OHF}\rightarrow{e}$ asymmetry from the electron-candidate-sample asymmetry,

$A_{N}^{{\rm OHF}\rightarrow
e}=\frac{A_{N}-f_{h^{\pm}}A_{N}^{h^{\pm}}-f_{J/\psi}A_{N}^{J/\psi}}{1-f_{h^{\pm}}-f_{J/\psi}-f_{\pi^{0}}-f_{\eta}-f_{\gamma}},$
(9)

where $f_{i}$ represent the background fractions, $A_{N}$ is the asymmetry
calculated on the electron candidate sample, and $A_{N}^{i}$ are the
background asymmetries. The procedure to calculate the background fractions
and a more detailed description of background sources can be found in Ref.
[33]. This procedure is repeated in this analysis with the relevant $p_{T}$
bins [34], and uncertainties on calculated background fractions are propagated
through Eq. (9) to obtain systematic uncertainties $\sigma^{\rm
sys}_{f^{\pm}}$.

It was determined through simulation that the kaon decay lepton fraction was
negligible and not considered in the background correction procedure. TSSAs
for each background source have been measured at PHENIX at midrapidity in
$p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV. The asymmetries for
photonic background sources $\pi^{0},\eta$, and $\gamma$ were all measured by
PHENIX to be consistent with zero using the same dataset as this measurement
[35, 22]. They are therefore set to zero in Eq. (9), with a systematic
uncertainty $\sigma^{\rm sys}_{A_{N}^{B}}$ assigned for setting
$A_{N}^{\pi^{0}}=0$ and $A_{N}^{\eta}=0$ based on propagating uncertainties
from the measurements, while the uncertainty associated with setting the
direct photon TSSA $A_{N}^{\gamma}=0$ is negligible since $f_{\gamma}$ is on
the order of $10^{-3}$ [34]. The TSSAs for $J/\psi$ [36] and charged hadrons
[37] were measured with previous PHENIX data sets. The TSSA for $J/\psi$ has a
large statistical uncertainty [36], and contributes significantly to the
statistical uncertainty of this measurement, especially at high $p_{T}$. This
is due to the azimuthal angle of the decay lepton becoming more strongly
correlated with the azimuthal angle of the $J/\psi$ at higher $p_{T}$.
Additionally, the statistical precision of $A_{N}(p^{\uparrow}+p\rightarrow
J/\psi+X)$ could not be improved upon in the Run-15 data given the high degree
of photonic electron background. The TSSA for midrapidity $J/\psi$ production
measured in Ref. [36] was recalculated as a function of decay lepton $p_{T}$
using pythia [38] decay simulations for the $J/\psi\rightarrow e^{+}e^{-}$
channel in order to apply Eq. (9).

The TSSAs for midrapidity open charm production predicted in Refs. [23] and
[24] were also recalculated as a function of decay lepton $p_{T}$ for all
possible semileptonic decay channels, with decay kinematics simulated in
pythia [38] to obtain correlations between $p_{T}$ and $\phi$ of the decay
lepton and $D$ meson. The $\phi^{e}$ distribution was then weighted in
accordance with $w(\phi^{e})=1+A_{N}^{D}(p_{T}^{D})\cos\phi^{D}$ in each
$p_{T}$ bin and then fit with $f(\phi)=N_{0}(1+A_{N}^{e}\cos\phi)$. $D^{0}$
and $\bar{D}^{0}$ production was considered for comparisons to results from
both papers, while $D^{+}$ and $D^{-}$ production was additionally considered
when comparing to results of Ref. [24]. OHF production is dominated by open
charm at the relevant kinematics, for which $D^{0},\bar{D}^{0},D^{+}$ and,
$D^{-}$ cover a significant fraction. The effect of including $D^{+}$ and
$D^{-}$ in comparing to results from Ref. [24] makes very little difference as
supported by our simulations, implying that comparing to $D^{0}$ and
$\bar{D}^{0}$ for Ref. [23] is sufficient. A scan over
$(\lambda_{f},\lambda_{d})$ parameter space as well as independent scans over
$K_{G}$ and $K_{G}^{\prime}$ were performed to generate a set of theory curves
to compare to, allowing for best-fit parameters and confidence intervals to be
determined from data.

Figure 1: $A_{N}(OHF\rightarrow{e^{\pm}})$ (red) circles and (blue) squares
for positrons and electrons, respectively. Also plotted are predictions of
$A_{N}(D^{0}/\bar{D^{0}}\rightarrow{e^{\pm}})$ from Ref. [23], and
$A_{N}((D^{0}/\bar{D^{0}}+D^{+/-})\rightarrow e^{\pm})$ from Ref. [24] for
best-fit trigluon-correlator-normalization parameters [(red/blue) solid,
dashed, and dotted lines]

Figure 2: Results of the statistical analysis performed to extract best-fit
parameters $\lambda_{f}$ and $\lambda_{d}$ by comparing data to theory [23].
$\chi^{2}(\lambda_{f},\lambda_{d})-\chi^{2}_{\rm min}$ is shown for (a)
$e^{+}$ and (b) $e^{-}$. Panel (c) shows the 1, 2, and 3$\sigma$ confidence
level regions, $\chi^{2}(\lambda_{f},\lambda_{d})-\chi^{2}_{\rm min}<n^{2}$
($n=1,2,3$).

The ${\rm OHF}\rightarrow{e^{\pm}}$ TSSAs are plotted in Fig. 2 alongside
theoretical predictions of $A_{N}(D^{0}/\bar{D}^{0}+X)$ from Ref. [23] in
(red/blue) solid lines, and $A_{N}(D/\bar{D}+X)$ from Ref. [24] in (red/blue)
dashed and dotted lines, with $\lambda_{f}$, $\lambda_{d}$, $K_{G}$ and
$K_{G}^{\prime}$ chosen to best fit the data for the separate charges
simultaneously. The measurements are consistent with zero, and are
statistically more precise than previous measurements. The total systematic
uncertainties come from combining those associated with the background
fractions, background asymmetries, and the difference in calculating $A_{N}$
with Eqs. (7) and (8); there is no dominant source of systematic uncertainty
across charges and $p_{T}$ bins. The systematic uncertainty reaches at most
37% of the corresponding statistical uncertainty, while it is typically
suppressed by an order of magnitude or more [34]. The placement of the
theoretical curves in Fig. 2 differs for $e^{+}$ vs $e^{-}$ due to the
contribution of the symmetric trigluon correlator having opposing signs in
charm vs. anticharm production, leading to constructive vs destructive
interference with the antisymmetric trigluon correlator contribution for the
separate charges. This allows for significant constraining power of all
parameters.

To determine theoretical parameters that fit the data best,
$\chi^{2}(\lambda_{f},\lambda_{d})$, $\chi^{2}(K_{G})$, and
$\chi^{2}(K_{G}^{\prime})$ were calculated for the separate charges and summed
to extract minimum values. The results along with $1\sigma$ confidence
intervals are $\lambda_{f}=-0.01{\pm}0.03$ GeV and $\lambda_{d}=0.11{\pm}0.09$
GeV for those introduced Ref. [23], and $K_{G}=0.0006^{+0.0014}_{-0.0017}$,
and $K_{G}^{\prime}=0.00025{\pm}0.00022$ for those introduced in Ref. [24].
This corresponds to the first constraints on $(\lambda_{f},\lambda_{d})$, and
is in agreement with previous constraints on $K_{G}$ and $K_{G}^{\prime}$
derived in Ref. [24]. Figure 2 summarizes the results of the statistical
analysis performed to extract best-fit parameters $\lambda_{f}$ and
$\lambda_{d}$, in which case the theoretical asymmetries depend on both
parameters. Nicely illustrated are the constraining power of the individual
charges and the necessity of combining the charges in the statistical
analysis. Each individual charge predicts that contributions from trigluon
correlations are small, indicating that the best fit prefers $\lambda_{f}$ and
$\lambda_{d}$ values that result in cancellation of their contributions to the
asymmetry calculation.

In summary, the PHENIX experiment has measured the transverse single-spin
asymmetry of midrapidity open-heavy-flavor decay electrons and positrons as a
function of $p_{T}$ in $p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV.
Open-heavy-flavor production at RHIC is an ideal channel for probing trigluon
correlations in polarized protons because initial-state $qgq$ correlations in
the proton and final-state twist-3 correlations in hadronization contribute
negligibly. This measurement provides significant constraints for the
antisymmetric and symmetric trigluon correlation functions in transversely
polarized protons, including the first constraints on $\lambda_{f}$ and
$\lambda_{d}$ as $\lambda_{f}=-0.01{\pm}0.03$ GeV and
$\lambda_{d}=0.11{\pm}0.09$ GeV — a necessary step forward in our
understanding of proton structure through correlations between proton spin and
gluon momentum.

###### 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 Z. Kang and S.
Yoshida for providing $A_{N}^{D}(p_{T})$ calculations corresponding to the
models introduced in Refs. [23] and [24]. 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), 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.

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