PHENIX Collaboration

#  Transverse single-spin asymmetries of midrapidity $\pi^{0}$ and $\eta$
mesons in polarized $p$$+$$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  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  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. Firak  Debrecen University, H-4010 Debrecen, Egyetem tér 1,
Hungary  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  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  M. 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  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  W.J.
Metzger  Eszterházy Károly University, Károly Róbert Campus, H-3200 Gyöngyös,
Mátrai út 36, Hungary  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  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  S.I. Morrow  Vanderbilt
University, Nashville, Tennessee 37235, 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  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  X. Sun  Georgia State University, Atlanta,
Georgia 30303, 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  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 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
Y. Wu  University of California-Riverside, Riverside, California 92521, 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  Y. Zhai  Iowa State
University, Ames, Iowa 50011, USA  S. Zharko  Saint Petersburg State
Polytechnic University, St. Petersburg, 195251 Russia  L. Zou  University of
California-Riverside, Riverside, California 92521, USA

###### Abstract

We present a measurement of the transverse single-spin asymmetry for $\pi^{0}$
and $\eta$ mesons in $p^{\uparrow}$$+$$p$ collisions in the pseudorapidity
range $|\eta|<0.35$ and at a center-of-mass energy of 200 GeV with the PHENIX
detector at the Relativistic Heavy Ion Collider. In comparison with previous
measurements in this kinematic region, these results have factor-of-three-
smaller uncertainties. As hadrons, $\pi^{0}$ and $\eta$ mesons are sensitive
to both initial- and final-state nonperturbative effects for a mix of parton
flavors. Comparisons of the differences in their transverse single-spin
asymmetries have the potential to disentangle the possible effects of
strangeness, isospin, or mass. These results can constrain the twist-3
trigluon collinear correlation function as well as the gluon Sivers function.

##  I Introduction

Spin-momentum correlations in hadronic collisions have attracted increasing
experimental and theoretical interest in the past two decades. In particular,
transverse single-spin asymmetries (TSSAs) have been one of the primary means
to probe transverse partonic dynamics in the nucleon. In the context of
proton-proton collisions, one transversely-polarized proton collides with
another unpolarized proton and the TSSA measures the asymmetry in yields of
particles that travel to the left versus the right of the polarized-proton-
going direction. Large azimuthal asymmetries of up to $\approx\,$40% have been
observed from transversely polarized $p^{\uparrow}+p$ collisions in light
meson production at large Feynman-x ($x_{F}=2p_{L}/\sqrt{s}$), from center of
mass energies of $\sqrt{s}=4.9~{}{\rm GeV}$ up to $500~{}~{}{\rm GeV}$ Klem
_et al._ (1976); Adams  _et al._ (1991); Abelev  _et al._ (2008); Arsene  _et
al._ (2008); Adare  _et al._ (2014a); Adam  _et al._ (2020a). Next-to-leading
order perturbative quantum chromodynamics (QCD) calculations that only include
spin-momentum correlations from parton scattering predict small asymmetries on
the order of $m_{q}/Q$ Kane  _et al._ (1978), where $m_{q}$ is the bare quark
mass and $Q$ is the hard scale, indicating that significant nonperturbative
effects must dominate the large measured asymmetries. Two different approaches
have been proposed to describe the large asymmetries observed in hadronic
interactions.

In the first approach, nonperturbative parton distribution functions (PDFs)
and fragmentation functions (FFs) are explicitly dependent on transverse
momentum in the transverse-momentum-dependent (TMD) framework. These functions
depend on a soft ($k_{T}$) and hard ($Q$) momentum scale such that
$\Lambda_{QCD}\lesssim k_{T}\ll Q$. One possible origin of the large TSSAs is
the Sivers TMD PDF Sivers (1990), which correlates the nucleon transverse spin
with the parton transverse momentum, $k_{T}$. Another possible origin of the
TSSA is the Collins TMD FF Collins (1993), which correlates the transverse
polarization of a fragmenting quark to the angular distribution of hadrons.

The second approach to describe the large asymmetries relies on collinear
higher-twist effects with multiparton correlations. In the twist-3 approach,
interference arises between scattering amplitudes with one and two collinear
partons, which leads to a nonzero TSSA. This approach applies to observables
in which only one sufficiently hard momentum scale is measured, such that
$Q\gg\Lambda_{QCD}$ Cammarota  _et al._ (2020). To keep the multiparton
correlation functions process independent, the initial- and final-state
interactions between the struck parton and the proton remnants are included in
the hard perturbative part of the twist-3 collinear factorization Aybat  _et
al._ (2012). Collinear twist-3 correlation functions are split into two types:
the quark-gluon-quark functions (qgq) and the trigluon functions (ggg). In the
context of initial-state effects, the qgq functions describe the interference
from scattering off of one quark versus scattering off of a gluon and a quark
of the same flavor, while the ggg functions capture the interference between
scattering off of one gluon versus scattering off of two. The twist-3 approach
is well suited to describe observed inclusive forward hadron asymmetries
because the observed hadron $p_{T}$ can be used as a proxy for the hard scale,
and unlike the TMD approach, these correlation functions do not explicitly
depend on a soft-scale transverse momentum. However, the twist-3 approach has
been related to $k_{T}$ moments of TMD PDFs and TMD FFs and has been shown to
be equivalent to the TMD approach in the overlapping kinematic regime Ji  _et
al._ (2006).

Because the Sivers function is PT-odd, in order to be nonzero it must include
a soft-gluon exchange with the proton remnant, which can occur before and/or
after the hard partonic scattering depending on the process Collins (2002).
Significant nonzero asymmetries due to the Sivers TMD PDF have been measured
in semi-inclusive deep-inelastic lepton-nucleon scattering (SIDIS) Airapetian
_et al._ (2005); Adolph  _et al._ (2015), where the soft-gluon exchange can
only happen in the final state.In hadronic interactions where at least one
final-state hadron is measured, both initial- and final-state interactions can
play competing roles in the measured asymmetries; here TMD-factorization
breaking has been predicted due to soft gluon exchanges that are possible in
both the initial and final states simultaneously Rogers and Mulders (2010).
Additional leading-power spin asymmetries have been predicted in hadronic
collisions due to this breakdown, without which these asymmetries would be
subleading Rogers (2013), but further work is needed to connect TMD-
factorization breaking to experimentally measured asymmetries. Note that
inclusive hadron TSSA measurements in hadronic collisions appear to plateau at
$p_{T}$ up to 5 GeV/c Adare  _et al._ (2014a); Abelev  _et al._ (2008) and
have been measured to be nonzero at up to $p_{T}\,\approx 7$ GeV/c Adam  _et
al._ (2020b). Recent studies in the twist-3 framework have successfully
described the $p_{T}$ dependence of these forward asymmetries by including
twist-3 effects in hadronization Kanazawa  _et al._ (2014). The twist-3
perturbative prediction is that the asymmetry should eventually decrease as
the hard scale $p_{T}$ continues increasing Rogers (2013).

Since the inception of the collinear twist-3 and TMD factorization pictures,
there has been theoretical evidence that they could combine to form a unified
picture of TSSAs in hard processes. This concept was recently tested with the
first simultaneous global analysis of TSSAs in SIDIS, Drell-Yan, $e^{+}e^{-}$
annihilation, and proton-proton collisions Cammarota  _et al._ (2020). This
study used quark TMD PDFs and FFs to describe the asymmetries in processes
that are sensitive to the soft-scale momentum, i.e. SIDIS, Drell-Yan, and
$e^{+}e^{-}$ annihilation. These TMD functions were also used to calculate
collinear twist-3 qgq correlation functions which were applied to inclusive
forward pion asymmetry measurements from the Relativistic Heavy Ion Collider
(RHIC). This simultaneous description of TSSAs across multiple collision
species indicates that all TSSAs have a common origin that is related to
multiparton correlations.

Additional questions about the origin of the TSSAs in hadronic interactions
remain. Forward jet measurements indicate that the TSSA is significantly
smaller than neutral pion asymmetries at similar $x_{F}$ and $\sqrt{s}$ Bland
_et al._ (2015). Nonzero kaon and antiproton asymmetries observed at forward
rapidities show that the measured asymmetries cannot be due only to proton
valence quark contributions as naively predicted in a valence-like model,
where the Sivers effect from sea-quarks and/or gluons is ignored, and that the
fragmentation of quarks into hadrons in which they are not valence quarks
could play a role in the observed nonzero asymmetries Arsene  _et al._ (2008);
Lee and Videbaek (2009). Eta meson measurements, sensitive to potential
effects from strange quark contributions, isospin, and/or hadron mass show
forward asymmetries similar in magnitude to neutral pions Adare  _et al._
(2014b). At midrapidity at RHIC, nonzero TSSAs have been measured for charged
pion pair production Adamczyk  _et al._ (2015, 2018). Even four decades after
the initial discovery of large TSSAs in hadronic interactions Klem  _et al._
(1976), there remain many unresolved questions about their origin. Therefore
it is crucial to continue extending measurements to try to better understand
the nonperturbative dynamics which are responsible for the TSSAs in hadronic
collisions.

In this paper we report a measurement of the TSSA of $\pi^{0}$ and $\eta$
mesons in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ = 200 GeV in the
midrapidity region $|\eta|<$ 0.35. The data was taken during the 2015 RHIC run
and a total integrated luminosity of approximately 60 pb-1 was collected. This
measurement extends previous measurements from RHIC to higher $p_{T}$ and
reduces the statistical uncertainties by a factor of three in the overlapping
$p_{T}$ region.

##  II Analysis

The asymmetries are measured with transversely polarized proton beams where
the average polarization of the clockwise beam was $0.58\pm 0.02$ and that of
the counter-clockwise beam was $0.60\pm 0.02$ Schmidke  _et al._ (2018). The
direction of the beam polarization was found to be consistent with the
vertical within statistical uncertainties. The polarization direction of each
beam independently changes bunch to bunch which reduces systematic
uncertainties associated with variations in detector performance with time.
The relative luminosity is the ratio of the integrated luminosity for bunches
that were polarized in opposite directions. It is determined by the number of
times each crossing fires a minimum-bias (MB) trigger and is measured to
better than $10^{-4}$. The relative luminosity values for both the beams were
limited to the range of 0.91 to 1.09 for all beam fills used in these
measurements. The bunch-to-bunch changes in polarization direction also allow
for polarization-averaged measurements and, for a single-spin asymmetry
analysis, provide two ways to measure the TSSA with the same data set. This is
done by sorting the particle yields for the polarization directions of one
beam at a time, effectively averaging over the polarization of the other beam.
RHIC uses eight different spin patterns for sequential fills which are
carefully chosen to minimize potential effects from nonzero average
polarization of either beam. The statistically independent asymmetries
measured from the two beams are used to verify the analysis and are averaged
together for the final result.

The data analysis procedure is similar to our previous measurements Adare  _et
al._ (2014a). Neutral pion and eta mesons are reconstructed via their two-
photon decays by using the midrapidity electromagnetic calorimeter (EMCal).
The EMCal is located in two central arms, each covering $\Delta\phi=\pi/2$ in
azimuth and $|\eta|<0.35$ in pseudorapidity, centered at $\phi=\pi/16$ and
$15\pi/16$. The EMCal comprises two different types of calorimeters: six
sectors of sampling lead-scintillator calorimeters and two sectors of Čerenkov
lead-glass calorimeters Aphecetche  _et al._ (2003). The two calorimeter
systems have different granularity ($\delta\phi\times\delta\eta=0.011\times
0.011$ in the lead scintillator and $0.008\times 0.008$ in the lead glass) and
also different responses to charged hadrons, which provides important
systematic cross checks for these measurements. A tracking system includes a
drift chamber to measure track momentum and pad chamber stations to measure
the charged particle hit position Adcox  _et al._ (2003). The measurement of
the track positions in front of the calorimeter is used to veto charged
particles from the photon sample. The beam-beam counters (BBC) are arrays of
quartz Čerenkov radiators that surround the beam pipe and are placed $\pm 144$
cm away from the nominal collision point. The BBC covers full azimuth and
$3.0<|\eta|<3.9$ in pseudorapidity. They measure the z-vertex position; a
vertex cut of $\pm 30$ cm around the nominal collision point is used for this
analysis. The MB trigger requires at least one charged particle to be measured
in both sides of the BBC. This analysis is based on the data sample selected
with the EMCal-based high-energy-photon trigger with energy threshold of 1.5
GeV, which is taken in coincidence with the MB trigger.

Photons are identified as clusters in the EMCal and are required to pass a
shower profile cut which suppresses clusters from hadrons. High-$p_{T}$
trigger photons are paired with another photon in the same event that is also
on the same side of the detector. A charged track veto cut eliminates clusters
that geometrically match with a measured charged track, reducing background
from electrons. The contribution of EMCal detector noise is reduced by a
minimum energy cut of 0.5 GeV and a time-of-flight cut of $|TOF|<5$ ns. The
timing of the cluster is measured by the EMCal and the time zero reference of
the event is provided by the BBC. Each photon pair is required to pass an
energy asymmetry cut: $\alpha=|E_{1}-E_{2}|/(E_{1}+E_{2})<0.8$. The $\pi^{0}$
yields comprise photon pairs with invariant mass in the signal region $\pm
25~{}{\rm MeV/c^{2}}$ from the $\pi^{0}$ mass peak and $\eta$ meson yields are
measured in the range $\pm 70~{}{\rm MeV}/c^{2}$ around the $\eta$ mass peak.

The transverse single-spin asymmetries are determined with 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)

which compares the yield of particles for when the beam was polarized up
versus down. Here $P$ is the beam polarization, $N$ refers to the meson yield,
the arrows refer to the up ($\uparrow$) or down ($\downarrow$) directions of
beam polarization, and
$\mathcal{R}=\mathcal{L}^{\uparrow}/\mathcal{L}^{\downarrow}$ is the relative
luminosity. The acceptance factor, $\left<\cos(\phi)\right>$, accounts for the
detector azimuthal coverage, where $\phi=0$ points $90^{\circ}$ from the
(vertical) spin axis. This correction is calculated as a function of photon
pair $p_{T}$ because the diphoton azimuthal acceptance depends heavily on the
decay angle and ranges from 0.95 at low $p_{T}$ to 0.89 at high $p_{T}$. The
asymmetry is calculated separately for the two detector arms and then the
average weighed by the statistical error is taken for the final result. As
written, Eq. (1) is for the arm to the left of the direction of travel of the
beam that is being taken as polarized. An overall minus sign is needed for the
asymmetry of particle yields in the arm to the right of the polarized-beam-
going direction.

An alternative method of calculating the asymmetry is the “square root”
formula

$A_{N}=\frac{1}{P\,\left<\cos(\phi)\right>}\frac{\sqrt{N_{L}^{\uparrow}N_{R}^{\downarrow}}-\sqrt{N_{L}^{\downarrow}N_{R}^{\uparrow}}}{\sqrt{N_{L}^{\uparrow}N_{R}^{\downarrow}}+\sqrt{N_{L}^{\downarrow}N_{R}^{\uparrow}}},$
(2)

which is used as a cross check. This formula combines data from the two arms
(left and right) and both beam polarization directions (up and down). The
subscripts in Eq. (2) refer to the yields to the left ($L$) and right ($R$)
side of the polarized-beam-going direction.

Figure 1: Example invariant mass distributions around the (a) $\pi^{0}$ and
(b) $\eta$ peak for photon pairs with ${4<p_{T}<5~{}{\rm GeV}/c}$ in one of
the detector arms. The minus-$45^{\circ}$ hatched [blue] region at the center
of each plot corresponds to the invariant mass region under the peak which is
used to calculate $A_{N}$ in Eq. (3) and the plus-$45^{\circ}$ [red] side band
regions correspond to the photon pairs that are used to calculate $A_{N}^{\rm
BG}$. The bold solid [green] curves correspond to the fit to the combinatorial
background, which is used to calculate the background fraction.

The measured asymmetries are also corrected for background

$A_{N}^{\rm Sig}=\frac{A_{N}-r\cdot A_{N}^{\rm BG}}{1-r},$ (3)

where $r$ is the fractional contribution of photon pairs from combinatorial
background within the invariant mass peak. The background fraction is
calculated from fits to the invariant mass spectra where a Gaussian is used to
describe the invariant mass peak, and a third order polynomial is used to
describe the combinatorial background, as shown in the green curves in Fig. 1.
Using this method, the contribution of combinatorial background under the
$\pi^{0}$ peak is determined to vary from 10% in the lowest $p_{T}$ bin to 6%
in the highest. Under the $\eta$ meson invariant mass peak, the background
fraction varies from 71% to 47% in the lowest to highest $p_{T}$ bins. In Eq.
(3), the background asymmetry, $A_{N}^{\rm BG}$, is evaluated with photon
pairs in side band regions located on either side of the signal peak, as
represented in the red regions in Fig. 1. For the $\pi^{0}$ analysis these
side band regions are $47<M_{\gamma\gamma}<97~{}{\rm MeV}/c^{2}$ and
$177<M_{\gamma\gamma}<227~{}{\rm MeV}/c^{2}$, and for the $\eta$ meson
analysis these regions are $300<M_{\gamma\gamma}<400~{}{\rm MeV}/c^{2}$ and
$700<M_{\gamma\gamma}<800~{}{\rm MeV}/c^{2}$. These background regions match
the ranges that were used in previous results Adare  _et al._ (2014a) and are
chosen to approximate the behavior of the combinatorial background under the
invariant mass peak. They are selected to be close to the peak while still far
enough away to contain negligible contributions from signal photon pairs. The
background asymmetries are consistent with zero across all $p_{T}$ bins. The
background asymmetries between the low-mass and high-mass regions are also
consistent with zero and with each other.

Table 1:  The measured $A_{N}$ of $\pi^{0}$ in $p$$+$$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 shown. The total
$\sigma_{\rm syst}$ in the lowest $p_{T}$ bin includes an additional
systematic uncertainty of $1.06\times 10^{-4}$ from bunch shuffling.

$\langle\mbox{$p_{T}$}\rangle$ |  $p_{T}$ bin ranges | $A_{N}$ | $\sigma_{\rm stat}$ | $\sigma_{\rm syst}$ | $\sigma_{\rm syst}$ | $\sigma_{\rm syst}$
---|---|---|---|---|---|---
$({\rm GeV}/c)$ | $({\rm GeV}/c)$ |  |  | (rel. lumi. vs sqrt.) | (bg. fraction) | (total)
2.58 | 2–3 | $1.43\times 10^{-4}$ | $2.81\times 10^{-4}$ | $5.71\times 10^{-5}$ | $3.92\times 10^{-7}$ | $1.20\times 10^{-4}$
3.42 | 3–4 | $-3.43\times 10^{-4}$ | $3.21\times 10^{-4}$ | $1.73\times 10^{-5}$ | $3.92\times 10^{-6}$ | $1.77\times 10^{-5}$
4.40 | 4–5 | $3.35\times 10^{-4}$ | $5.71\times 10^{-4}$ | $6.56\times 10^{-5}$ | $1.91\times 10^{-6}$ | $6.57\times 10^{-5}$
5.40 | 5–6 | $2.33\times 10^{-3}$ | $1.06\times 10^{-3}$ | $9.61\times 10^{-5}$ | $6.68\times 10^{-7}$ | $9.61\times 10^{-5}$
6.41 | 6–7 | $-6.89\times 10^{-4}$ | $1.87\times 10^{-3}$ | $1.12\times 10^{-4}$ | $2.11\times 10^{-5}$ | $1.14\times 10^{-4}$
7.42 | 7–8 | $1.93\times 10^{-3}$ | $3.11\times 10^{-3}$ | $3.41\times 10^{-4}$ | $7.61\times 10^{-5}$ | $3.50\times 10^{-4}$
8.43 | 8–9 | $-2.38\times 10^{-3}$ | $4.88\times 10^{-3}$ | $2.45\times 10^{-4}$ | $3.99\times 10^{-4}$ | $4.69\times 10^{-4}$
9.43 | 9–10 | $4.04\times 10^{-4}$ | $7.03\times 10^{-3}$ | $3.31\times 10^{-4}$ | $1.16\times 10^{-4}$ | $3.51\times 10^{-4}$
10.79 | 10–12 | $7.34\times 10^{-3}$ | $7.99\times 10^{-3}$ | $9.71\times 10^{-5}$ | $3.13\times 10^{-4}$ | $3.28\times 10^{-4}$
13.53 | 10–20 | $-1.05\times 10^{-2}$ | $1.27\times 10^{-2}$ | $6.86\times 10^{-4}$ | $1.15\times 10^{-5}$ | $6.86\times 10^{-4}$

Table 2:  The measured $A_{N}$ of $\eta$ mesons in $p$$+$$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 shown. The total
$\sigma_{\rm syst}$ in the lowest $p_{T}$ bin includes an additional
systematic uncertainty of $6.20\times 10^{-4}$ from bunch shuffling.

$\langle\mbox{$p_{T}$}\rangle$ |  $p_{T}$ bin edges | $A_{N}$ | $\sigma_{\rm stat}$ | $\sigma_{\rm syst}$ | $\sigma_{\rm syst}$ | $\sigma_{\rm syst}$
---|---|---|---|---|---|---
$({\rm GeV}/c)$ | $({\rm GeV}/c)$ |  |  | (rel. lumi. vs sqrt.) | (bg. fraction) | (total)
2.39 | 2–3 | $2.44\times 10^{-3}$ | $1.83\times 10^{-3}$ | $5.18\times 10^{-4}$ | $4.58\times 10^{-5}$ | $8.09\times 10^{-4}$
3.53 | 3–4 | $-1.99\times 10^{-3}$ | $1.59\times 10^{-3}$ | $8.36\times 10^{-5}$ | $3.31\times 10^{-5}$ | $8.99\times 10^{-5}$
4.39 | 4–5 | $-3.31\times 10^{-3}$ | $2.48\times 10^{-3}$ | $1.44\times 10^{-4}$ | $4.55\times 10^{-5}$ | $1.51\times 10^{-4}$
5.40 | 5–6 | $-1.39\times 10^{-3}$ | $4.21\times 10^{-3}$ | $2.41\times 10^{-4}$ | $3.59\times 10^{-5}$ | $2.44\times 10^{-4}$
6.41 | 6–7 | $2.22\times 10^{-3}$ | $7.09\times 10^{-3}$ | $1.12\times 10^{-3}$ | $6.35\times 10^{-6}$ | $1.12\times 10^{-3}$
7.42 | 7–8 | $1.03\times 10^{-2}$ | $1.15\times 10^{-2}$ | $7.03\times 10^{-4}$ | $1.60\times 10^{-4}$ | $7.20\times 10^{-4}$
8.75 | 8–10 | $7.90\times 10^{-3}$ | $1.37\times 10^{-2}$ | $1.24\times 10^{-3}$ | $1.88\times 10^{-4}$ | $1.25\times 10^{-3}$
11.76 | 10–20 | $1.68\times 10^{-2}$ | $2.19\times 10^{-2}$ | $4.25\times 10^{-3}$ | $3.70\times 10^{-4}$ | $4.26\times 10^{-3}$

Tables 2 and 2 show the asymmetries with statistical and systematic
uncertainties. The total systematic uncertainty is the sum of the three
sources of systematic uncertainty added in quadrature. The systematic
uncertainty on the asymmetry due to the background fraction in Eq. (3) is
determined by varying the fit ranges when computing $r$ and calculating how
much the background-corrected asymmetry changes. While the asymmetries
calculated with the “relative luminosity” [Eq. (1)] and the “square root” [Eq.
(2)] formulas were found to be statistically consistent, their difference was
assigned as a conservative systematic uncertainty due to possible variations
in detector performance and beam conditions. This dominates the total
systematic uncertainty for most $p_{T}$ bins.

Bunch shuffling is a technique used to investigate potential sources of
systematic uncertainty that could cause the measured asymmetry results to vary
from their true values beyond statistical fluctuations. Bunch shuffling
involves randomizing the assigned bunch-by-bunch polarization directions of
the beam such that the physical asymmetry disappears, thereby isolating the
statistical variations present in the data. All asymmetry values have bunch
shuffling results consistent with statistical variations except for the lowest
$p_{T}$ bin where there is 7% and 6% more variation beyond what is expected
from statistical fluctuations in the $\pi^{0}$ and $\eta$ meson analyses,
respectively. These values are used to assign additional systematic
uncertainties to the lowest $p_{T}$ bin of the $\pi^{0}$ and $\eta$ meson
asymmetries and dominate the total systematic uncertainty for those bins.

Additional cross checks included examining the asymmetries in the two arms
separately using Eq. (1) and measuring the asymmetry as an explicit function
of $\phi$. All checks were statistically consistent with the main asymmetry
results.

##  III Results and discussion

Figure 3 shows the $A_{N}$ of neutral pions at midrapidity in
$p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}$ =200 GeV, where the bands
represent the systematic uncertainty and the bars represent the statistical
uncertainty. The comparison to previous results Adare  _et al._ (2014a)
demonstrates the improvement in statistical precision. The inset in Fig. 3
shows a zoomed-in comparison at small $p_{T}$. The new measurement is
consistent with our previous measurement and improves the precision on average
by a factor of 3. The new measurement of $A_{N}$ of neutral pions is
consistent with zero in the entire $p_{T}$ range.

Figure 2: Transverse single-spin asymmetry of neutral pions measured at
$|\eta|<0.35$ in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}=200~{}{\rm
GeV}$. An additional scale uncertainty of 3.4% due to the polarization
uncertainty is not shown.

Figure 3: Transverse single-spin asymmetry of eta mesons measured at
$|\eta|<0.35$ in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}=200~{}{\rm
GeV}$. An additional scale uncertainty of 3.4% due to the polarization
uncertainty is not shown.

The measurement of $A_{N}$ of $\eta$ mesons in $p^{\uparrow}$$+$$p$ collisions
at $\sqrt{s}$ = 200 GeV is shown in Fig. 3. This measurement is also compared
to the previous result, similarly to Fig. 3. The new measurement is consistent
with the previous result and with zero across the entire $p_{T}$ range. In
principle, comparisons of $\pi^{0}$ and $\eta$ meson TSSAs may indicate
additional effects from strange quarks, isospin differences, or hadron mass.
At forward rapidity, existing measurements Adare  _et al._ (2014b); Adamczyk
_et al._ (2012) do not yet clearly resolve whether the $\eta$ meson asymmetry
is larger than the $\pi^{0}$ asymmetry as predicted in some models Kanazawa
and Koike (2011). At midrapidity, there is a larger contribution from gluon
dynamics and, as shown in Fig. 4, both asymmetries are consistent with zero
and therefore show no evidence for differences due to strangeness, isospin, or
mass.

Figure 4: Comparison of the $\pi^{0}$ and $\eta$ meson asymmetries measured at
$|\eta|<0.35$ in $p^{\uparrow}$$+$$p$ collisions at $\sqrt{s}=200~{}{\rm
GeV}$. An additional scale uncertainty of 3.4% due to the polarization
uncertainty is not shown. Figure 5: This $\pi^{0}$ asymmetry result plotted
with theory calculations for the asymmetry in both the collinear twist-3
Cammarota  _et al._ (2020) and TMD D’Alesio  _et al._ (2019) theoretical
frameworks. See text for details.

Figure 5 shows this $\pi^{0}$ TSSA result plotted with theoretical
predictions. The qgq curve shows the predicted contribution from collinear
twist-3 qgq functions from both the polarized proton and the process of
hadronization. This curve was calculated with fits that were published in Ref.
Cammarota  _et al._ (2020) and has been reevaluated in the rapidity range of
PHENIX. Midrapidity $\pi^{0}$ production includes a large fractional
contribution from gluons in the proton, so a complete collinear twist-3
description of the midrapidity $\pi^{0}$ TSSA would also need to include the
contribution from the trigluon correlation function. Given the small expected
contribution from the qgq correlation function, this measurement can constrain
future calculations of the ggg correlation function, such as those in Ref.
Beppu  _et al._ (2014).

The other theory curves in Fig. 5 show predictions for the midrapidity
$\pi^{0}$ TSSA generated by the Sivers TMD PDF. These curves include
contributions from both the quark and gluon Sivers functions and have been
evaluated for $x_{F}=0$, which approximates the measured kinematics. These
calculations use the generalized parton model (GPM) which takes the first
$k_{T}$ moment of the Sivers function (e.g. $\int{k_{T}\cdot q(k_{T})}$) and
does not include next-to-leading-order interactions with the proton fragments.
The “GPM” curve uses the parameters stated in Eq. (32) of Ref. D’Alesio  _et
al._ (2019). The color-gauge-invariant generalized parton model (CGI-GPM)
expands on the GPM by including initial- and final-state interactions through
the one-gluon exchange approximation. This model has been shown to reproduce
the predicted sign change for the quark Sivers function in SIDIS and Drell-
Yan. The CGI-GPM curves plotted in Fig. 5 show two different scenarios for
this model, the specifics of which can be found in Eq. (34) of Ref. D’Alesio
_et al._ (2019). The values that are used for the Scenario 1 curve are chosen
to maximize the open heavy flavor TSSA generated by the gluon Sivers function
while still keeping this asymmetry within the statistical error bars of the
published result in Ref. Aidala  _et al._ (2017) and simultaneously describing
the previously published midrapidity $\pi^{0}$ TSSA from Ref. Adare  _et al._
(2014a). The values used in the Scenario 2 curve are similarly calculated,
except that they minimize the open heavy flavor TSSA within the range of the
published statistical error bars. As shown in the zoomed-in inset of Fig. 5,
this $\pi^{0}$ TSSA result has the statistical precision at low $p_{T}$ needed
to distinguish between the GPM and CGI-GPM frameworks, preferring CGI-GPM
Scenario 2.

Measurements of TSSAs in $p$$+$$p$ collisions are essential to understanding
the underlying nonperturbative processes which generate them. In particular,
further measurements are necessary to clarify certain questions in the
interpretations of the TSSAs. For example, the small forward jet asymmetries
measured in Ref. Bland  _et al._ (2015) have been interpreted as a
cancellation of up and down quark asymmetries, implying that the comparatively
forward large neutral pion asymmetries include significant contributions from
spin-momentum correlations in hadronization Kanazawa  _et al._ (2014).
Additionally, the $p_{T}$ dependence of these forward rapidity measurements
remains to be clearly understood; measurements of nonzero asymmetries out to
even higher $p_{T}$ would help confirm that these twist-3 observables
eventually fall off with increasing hard scale. While the midrapidity
measurements here are all consistent with zero, they still provide the highest
available statistical precision and $p_{T}$ reach available at the PHENIX
experiment. While forward rapidity light hadron TSSAs are dominated by valence
quark spin-momentum correlations in the polarized proton, these midrapidity
TSSA measurements are sensitive to both quark and gluon dynamics at leading
order. Thus these data also provide further constraints to gluon spin-momentum
correlations in transversely polarized protons Beppu  _et al._ (2014);
D’Alesio  _et al._ .

##  IV Summary

The measurements presented here were motivated by the outstanding questions
regarding the physical origin of TSSAs. The TSSAs of $\pi^{0}$ and $\eta$
mesons were measured at midrapidity in $p$$+$$p$ collisions at
$\mbox{$\sqrt{s}$}=200$ GeVby the PHENIX experiment. The measured $\pi^{0}$
($\eta$) meson asymmetry is consistent with zero in the presented $p_{T}$
range, up to precision of $3\times 10^{-4}$ $(2\times 10^{-3})$ in the lowest
$p_{T}$ bins. Both measurements have a significant reduction in uncertainty
from previous measurements at midrapidity at RHIC. These data extend previous
constraints to any presence of gluon spin-momentum correlations in
transversely polarized protons.

###### 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, S.
Yoshida, U. D’Alesio, F. Murgia and C. Pisano 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.

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