PHENIX Collaboration

# Transverse single-spin asymmetry of midrapidity $\pi^{0}$ and $\eta$ mesons
in $p$$+$Au and $p$$+$Al collisions at $\sqrt{s_{{}_{NN}}}=$ 200 GeV

N.J. Abdulameer  Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary
U. 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  M. Csanád  ELTE, Eötvös Loránd
University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  T. Csörgő  MATE,
Laboratory of Femtoscopy, Károly Róbert Campus, H-3200 Gyöngyös, Mátraiút 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  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  V. Doomra  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  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
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, Faculty of Mathematics and Physics, 180 00
Troja, Prague, Czech Republic  M. Finger  Charles University, Faculty of
Mathematics and Physics, 180 00 Troja, Prague, Czech Republic  D. Firak
Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary  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  University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  R.S. Hollis
University of California-Riverside, Riverside, California 92521, USA  K. Homma
Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Higashi-Hiroshima, Hiroshima
739-8526, Japan  B. Hong  Korea University, Seoul 02841, Korea  T. Hoshino
Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Higashi-Hiroshima, 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  B. Kurgyis  ELTE, Eötvös Loránd University,
H-1117 Budapest, Pázmány P. s. 1/A, Hungary  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
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
A. Muhammad  Mississippi State University, Mississippi State, Mississippi
39762, 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  Physics Program and International
Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2),
Hiroshima University, Higashi-Hiroshima, 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  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  T. Novák  MATE, Laboratory of Femtoscopy, Károly
Róbert Campus, H-3200 Gyöngyös, Mátraiút 36, Hungary  Institute for Particle
and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of
Sciences (Wigner RCP, RMKI) H-1525 Budapest 114, POBox 49, Budapest, Hungary
G. Nukazuka  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  A.S. Nyanin  National Research Center
“Kurchatov Institute”, Moscow, 123098 Russia  E. O’Brien  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  C.A. Ogilvie
Iowa State University, Ames, Iowa 50011, USA  J. Oh  Pusan National
University, Pusan 46241, Korea  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  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  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  Mississippi State University,
Mississippi State, Mississippi 39762, USA  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  M. Shibata  Nara Women’s University, Kita-uoya Nishi-machi
Nara 630-8506, Japan  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
Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Higashi-Hiroshima, 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  Z. Shi  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  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, Faculty of Mathematics and Physics, 180 00
Troja, 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  Physics Program and
International Institute for Sustainability with Knotted Chiral Meta Matter
(SKCM2), Hiroshima University, Higashi-Hiroshima, 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  R. Takahama  Nara Women’s University, Kita-
uoya Nishi-machi Nara 630-8506, Japan  K. Tanida  Advanced Science Research
Center, Japan Atomic Energy Agency, 2-4 Shirakata Shirane, Tokai-mura, Naka-
gun, Ibaraki-ken 319-1195, Japan  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  Department of Physics
and Astronomy, Seoul National University, Seoul 151-742, Korea  M.J.
Tannenbaum  Physics Department, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  S. Tarafdar  Vanderbilt University, Nashville, Tennessee
37235, USA  Weizmann Institute, Rehovot 76100, Israel  A. Taranenko  National
Research Nuclear University, MEPhI, Moscow Engineering Physics Institute,
Moscow, 115409, Russia  Chemistry Department, Stony Brook University, SUNY,
Stony Brook, New York 11794-3400, USA  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  Physics Program and
International Institute for Sustainability with Knotted Chiral Meta Matter
(SKCM2), Hiroshima University, Higashi-Hiroshima, 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  Z. Wang  Baruch College, City University of New York, New
York, New York, 10010 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

Presented are the first measurements of the transverse single-spin asymmetries
($A_{N}$) for neutral pions and eta mesons in $p$$+$Au and $p$$+$Al collisions
at $\sqrt{s_{{}_{NN}}}=200$ GeV in the pseudorapidity range $|\eta|<$0.35 with
the PHENIX detector at the Relativistic Heavy Ion Collider. The asymmetries
are consistent with zero, similar to those for midrapidity neutral pions and
eta mesons produced in $p$$+$$p$ collisions. These measurements show no
evidence of additional effects that could potentially arise from the more
complex partonic environment present in proton-nucleus collisions.

##  I Introduction

Transverse single-spin asymmetries (TSSAs) in particle production for hadronic
collisions involving a transversely polarized proton result from
nonperturbative spin-momentum correlations in the proton and/or the process of
hadronization [TSSA_review]. For recent discussions of TSSAs measured in
polarized $p$$+$$p$ collisions at the Relativistic Heavy Ion Collider (RHIC)
and the possible mechanisms contributing to them, see Refs. [PPG246, PPG247,
PPG235, PPG234, PPG238, STAR:2022hqg, STAR:2020nnl, STAR:2017wsi,
STAR:2017akg].

In hadronic collisions involving a nucleus, the underlying partonic origins of
the asymmetries could be affected by the presence of more complex quantum-
chromodynamics environments. For example, relations between TSSAs and the
physics of small parton momentum fractions have been proposed, in particular
how comparisons of asymmetries measured in $p^{\uparrow}$$+$$p$ and
$p^{\uparrow}$$+$$A$ collisions for forward hadron production could be used to
probe gluon saturation effects in the nucleus [Kang:2011ni]. Further
theoretical works have explored these ideas [Kovchegov:2012ga,
Schafer:2014xpa, Schafer:2014zea, Kovchegov:2015zha, Zhou:2015ima,
Hatta:2016wjz, Hatta:2016khv, Zhou:2017sdx, Benic:2018amn, Kovchegov:2020kxg,
Benic:2022qzv]. At RHIC, TSSA measurements for proton-nucleus collisions have
been performed for forward charged hadrons [PPG215, PPG226] and forward
$J/\psi$ mesons [PPG211] by PHENIX, and for forward $\pi^{0}$ production by
STAR [STAR:2020grs], revealing some nuclear dependencies that remain to be
understood in detail. PHENIX has additionally measured the TSSAs for far
forward neutron production, with the observed nuclear dependence of the
asymmetries understood to be due to the interplay of hadronic and
electromagnetic interactions in ultra-peripheral collisions [PPG203, PPG244].
No experimental measurements exist and very little theoretical work has been
done to explore possible nuclear effects for midrapidity TSSA observables,
which can only be studied at RHIC.

##  II Analysis

Figure 1: Invariant mass distributions around the
$\pi^{0}\rightarrow\gamma\gamma$ peak in (a) $p^{\uparrow}$$+$Au collisions
and (b) $p^{\uparrow}$$+$Al collisions and around the
$\eta\rightarrow\gamma\gamma$ peak in (c) $p^{\uparrow}$$+$Au collisions and
(d) $p^{\uparrow}$$+$Al collisions for photon pairs within $4<p_{T}~{}[{\rm
GeV}/c]<5$ in the west central-arm spectrometer. The [blue] leftward-hatched
regions are the signal peaks, used for quantifying yields for the $A_{N}$
calculations, the [red] rightward-hatched regions are the side bands, used to
quantify yields for the $A_{N}^{BG}$ calculations, and the [green] solid
curves correspond to fits to the combinatorial background, used in calculating
the background fractions.

This brief article reports the first measurement of the TSSAs of neutral pions
and eta mesons in proton-Gold ($p^{\uparrow}$$+$Au) and proton-Aluminum
($p^{\uparrow}$$+$Al) collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV at
midrapidity ($|\eta|<$0.35). The data were taken in 2015 at RHIC and total
integrated luminosities of approximately 202 and 690 nb-1, respectively, were
collected.

Measurements were performed with collisions of a vertically polarized proton
beam on an ion beam (Au or Al). The proton or ion bunches are separated by
$106$ ns in the RHIC rings. Each polarized proton bunch is assigned a
polarization direction, either up or down, so that measurements with both spin
directions can be performed nearly simultaneously. This significantly reduces
any possible systematic uncertainties related to the detector performance with
time. The average proton beam polarization was $0.60$ and $0.57$ in
$p^{\uparrow}$$+$Au and $p^{\uparrow}$$+$Al collisions respectively
[RHIC_polarimetry], with a relative uncertainty of $3\%$ due to uncertainty in
polarization normalization.

The data analysis procedure follows almost exactly from the recent TSSA
measurement for $\pi^{0}$ and $\eta$ mesons in $\sqrt{s}=200$ GeV polarized
$p$$+$$p$ collisions [PPG234], with the distinction that only the proton beam
is polarized in $p^{\uparrow}$$+$$A$ collisions. Only events with a collision
z-vertex within $\pm 30~{}{\rm cm}$ from the nominal collision point were
selected. The collision or minimum-bias trigger, as well as vertex position,
were determined by two beam-beam counters (BBC) located at $\pm 144~{}{\rm
cm}$ from the nominal collision point along the beam line, and covering the
pseudorapidity range $3.1<|\eta|<3.9$ with full azimuthal coverage.

Neutral pions and eta mesons were reconstructed through their two-photon decay
in the electromagnetic calorimeters (EMCal) of PHENIX. The EMCal is located in
two nearly back-to-back central arm spectrometers (west and east), each
covering $\Delta\phi=\pi/2$ in azimuth and $\pm 0.35$ in pseudorapidity. The
EMCal comprises two types of calorimeters, six sectors of sampling lead-
scintillator (PbSc) calorimeter and 2 sectors of $\check{\rm C}$erenkov lead-
glass (PbGl) calorimeter [Aphecetche:2003zr]. The two calorimeter systems have
different granularity ($\Delta\phi\times\Delta\eta=0.011\times 0.011$ in PbSc
and $0.008\times 0.008$ in PbGl) and also have a different response to charged
hadrons, which provides important systematic cross checks for the measurement.

The PHENIX EMCal was also used to generate a high-$p_{T}$ photon trigger to
tag events with a high-energy cluster in the EMCal. The high-$p_{T}$ photon
trigger (with an energy threshold of 1.5 GeV) in coincidence with a minimum-
bias trigger that requires charged particles in both BBC detectors was used to
collect the $\pi^{0}$ and $\eta$ statistics in this analysis. The efficiency
of such a trigger for $\pi^{0}$’s increased from 20% at
$\mbox{$p_{T}$}=3~{}{\rm GeV}/c$ to 90% at $\mbox{$p_{T}$}>6~{}{\rm GeV}/c$,
with the plateaued efficiency level defined by the acceptance of the live
trigger tiles.

Photons were identified in the EMCal by placing selection criteria on the
shower profile and time-of-flight (TOF) with $|{\rm TOF}|<5$ ns, and with a
minimum energy selection of 0.5 GeV to reduce the contribution from electronic
noise in the EMCal, and combinatorial background in $\pi^{0}$ and $\eta$
reconstruction. A charged track veto was also implemented to eliminate
clusters that are geometrically associated with a track and to suppress the
background from electrons and charged hadrons. Photon pairs were reconstructed
by finding a high-$p_{T}$ trigger photon, and pairing it with another photon
from the same event and spectrometer arm. Photon pairs passing an energy-
asymmetry requirement: $\alpha=|E_{1}-E_{2}|/(E_{1}+E_{2})<0.8$ were selected
for further analysis.

fig:invmass shows the two-photon invariant mass distributions around the
$\pi^{0}$ and $\eta$ peaks for photon pairs within $4<p_{T}<5$ GeV/$c$ in the
west central arm spectrometer for $p$$+$Au and $p$$+$Al collisions. The
$\pi^{0}$ and $\eta$ meson yields were defined to be within the signal-
invariant mass window ([blue] leftward-hatched regions in Fig. 1) of $\pm$ 25
and $\pm$ 70 MeV/$c^{2}$ from the $\pi^{0}$ and $\eta$ mass peaks,
respectively, in the two-photon invariant mass distribution for each $p_{T}$
bin. The side-band regions used to approximate the combinatorial background
under the signal peak ([red] rightward-hatched regions in Fig. 1) are defined
as 47–97 and 177–227 MeV/$c^{2}$ for the $\pi^{0}$ and 300–400 and 700–800
MeV/c2 for the $\eta$ mesons. The same signal and side-band regions were used
in a previous PHENIX analysis [PPG234]. The combinatorial background for the
two-photon invariant mass spectrum (described by a third-order polynomial and
shown as the [green] solid lines in Fig. 1) was used to quantify the fraction
of background existing under the signal peaks. For the $\pi^{0}$, this ranged
from 14% (13%) to 6% (6%) from the lowest to the highest $p_{T}$ bins in
$p$$+$Au ($p$$+$Al) collisions, while for the $\eta$, the combinatorial
background under the signal peak ranged from 79% (77%) to 48% (43%) in
$p$$+$Au ($p$$+$Al) collisions.

Figure 2: Transverse single-spin asymmetry for (a) $\pi^{0}$ and (b) $\eta$
mesons in $p^{\uparrow}$$+$Au collisions ([blue] circles), and
$p^{\uparrow}$$+$Al collisions ([green] squares) from this measurement, shown
alongside the same measurement in polarized $p$$+$$p$ collisions from Ref.
[PPG234] ([black] diamonds). The error bars represent the statistical
uncertainty ($\sigma^{\rm stat}$) while the boxes represent the total
systematic uncertainty ($\sigma^{\rm syst}$).

Table 1: Summary of final asymmetries with statistical and systematic uncertainties for $\pi^{0}$ and $\eta$ mesons in $p^{\uparrow}$$+$$A$ collisions. $\sigma^{\rm syst}$ corresponds to the systematic uncertainties, displayed by the shaded boxes in Fig. 2.  Meson | Collisions |  $p_{T}$ range [GeV/c] | $\langle\mbox{$p_{T}$}\rangle[{\rm GeV}/c]$ | $A_{N}$ | $\sigma^{\rm stat}$ | $\sigma^{\rm syst}$
---|---|---|---|---|---|---
$\pi^{0}$ | $p^{\uparrow}$$+$Au | 2–3 | 2.71 | 0.000818 | 0.000993 | 0.000569
|  | 3–4 | 3.73 | -0.000145 | 0.000701 | 0.000286
|  | 4–5 | 4.31 | -0.000135 | 0.000974 | 0.000257
|  | 5–6 | 5.4 | -0.00011 | 0.00164 | 0.00028
|  | 6–7 | 6.41 | 0.00097 | 0.00281 | 0.00024
|  | 7–8 | 7.42 | -0.00243 | 0.00464 | 0.00109
|  | 8–9 | 8.43 | 0.00179 | 0.00732 | 0.00055
|  | 9–10 | 9.44 | 0.0093 | 0.0106 | 0.0005
|  | 10–12 | 10.8 | -0.0072 | 0.0122 | 0.0014
|  | 12–20 | 13.5 | -0.0438 | 0.0198 | 0.0008
$\pi^{0}$ | $p^{\uparrow}$$+$Al | 2–3 | 2.67 | -0.00147 | 0.00163 | 0.00088
|  | 3–4 | 3.47 | 0.00056 | 0.00113 | 0.00006
|  | 4–5 | 4.41 | 0.00126 | 0.00153 | 0.00005
|  | 5–6 | 5.41 | -0.00018 | 0.00254 | 0.00051
|  | 6–7 | 6.42 | 0.00500 | 0.00429 | 0.00042
|  | 7–8 | 7.42 | -0.00809 | 0.00699 | 0.00060
|  | 8–9 | 8.43 | 0.0035 | 0.0109 | 0.0004
|  | 9–10 | 9.44 | 0.0058 | 0.0155 | 0.0010
|  | 10–12 | 10.8 | 0.0208 | 0.0181 | 0.0016
|  | 12–20 | 13.6 | -0.0099 | 0.0286 | 0.0011
$\eta$ | $p^{\uparrow}$$+$Au | 2–3 | 2.64 | 0.01279 | 0.00665 | 0.00306
|  | 3–4 | 3.44 | -0.00255 | 0.00377 | 0.00115
|  | 4–5 | 4.41 | -0.00168 | 0.00448 | 0.00117
|  | 5–6 | 5.4 | -0.00810 | 0.00667 | 0.00183
|  | 6–7 | 6.41 | 0.0064 | 0.0108 | 0.0029
|  | 7–8 | 7.42 | -0.0056 | 0.0170 | 0.0026
|  | 8–10 | 8.74 | -0.0122 | 0.0216 | 0.0002
|  | 10–20 | 11.7 | 0.0615 | 0.0351 | 0.0021
$\eta$ | $p^{\uparrow}$$+$Al | 2–3 | 2.64 | -0.0044 | 0.0107 | 0.0046
|  | 3–4 | 3.46 | 0.00043 | 0.00575 | 0.00219
|  | 4–5 | 4.42 | -0.00278 | 0.00686 | 0.00050
|  | 5–6 | 5.41 | -0.0022 | 0.0104 | 0.0031
|  | 6–7 | 6.42 | -0.0032 | 0.0163 | 0.0055
|  | 7–8 | 7.42 | 0.0468 | 0.0260 | 0.0004
|  | 8–10 | 8.74 | 0.0017 | 0.0318 | 0.0027
|  | 10–20 | 11.7 | 0.0395 | 0.0464 | 0.0133

Similar to the recent $p$$+$$p$ $\pi^{0}$ and $\eta$ TSSA analysis [PPG234],
the transverse single-spin asymmetry $A_{N}$ is determined with the “relative-
luminosity” formula, which is calculated separately for the two detector arms.
This yields measurements from two independent data sets that are verified for
consistency and then averaged to obtain the final result. The equation for the
relative-luminosity TSSA is

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

where $P$ is the beam polarization, and $\mathcal{R}$ is the relative
luminosity, defined as the ratio of integrated luminosities between the
bunches with $\uparrow$ and $\downarrow$ spin states and measured by the BBC
detectors. $\left<cos(\phi)\right>$ is the acceptance factor which reflects
the detector azimuthal coverage, calculated separately for each $p_{T}$ bin
and spectrometer arm, and $N$ refers to the yields, with the arrows referring
to the up ($\uparrow$) or down ($\downarrow$) polarization of the proton beam.

Another method to calculate the asymmetry is the “square-root” formula, which
is used as a cross check. The square-root formula is defined as

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

and it is used to calculate the asymmetry for both spectrometer arms
simultaneously, where the $L$ and $R$ subscripts of $N$ correspond to yields
measured in the left and right detector arms respectively, with respect to the
polarized-proton-going direction. This results in only one measurement of
$A_{N}$ that can be compared with the weighted average of the left and right
relative-luminosity asymmetry calculation. The comparison of results using Eq.
(1) and Eq. (2) was used as a cross check, with corresponding systematic
uncertainties discussed below.

As an additional cross check, the TSSA is calculated as a function of $\phi$,
in which case a cosine modulation is fit to extract the asymmetry. This was
found to be statistically consistent with the main asymmetry results.

The measured asymmetries were corrected for the background as follows,

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

where $r$ is the background fraction under the $\pi^{0}$ or $\eta$ peaks,
calculated from the background fits ([green] solid lines) shown in Fig. 1.
$A_{N}^{\rm BG}$ is the background asymmetry, which was evaluated from the
side bands on both sides of the $\pi^{0}$ and $\eta$ peaks, also shown in Fig.
1. The background asymmetry was consistent with zero in all $p_{T}$ bins and
all collision systems for both the $\pi^{0}$ and $\eta$ mesons.

Asymmetries were calculated separately for each accelerator fill, during which
the detector performance and beam conditions are considered to be relatively
stable. The final asymmetry was obtained from the weighted average over the
accelerator fills.

The possible sources of systematic uncertainties considered are 1) the
background contribution $r$ in Eq. (3), and 2) possible variation in detector
performance and beam conditions. A systematic uncertainty on the background
fraction is quantified by varying the fit ranges used to calculate $r$, and
computing how much the background-corrected asymmetry changes. The variations
in detector performance and beam conditions, including uncertainty on the
relative luminosity, were tested by comparing results calculated with the
“relative-luminosity formula” Eq. (1) and the “square-root formula” Eq. (2),
and with a technique known as “bunch shuffling” [bunchShuffling]. The
asymmetries calculated with the different formulas were found to be
statistically consistent when taking into account the correlation between
datasets. However, a conservative systematic uncertainty calculated as the
absolute value of the difference in central values is assigned. In the bunch-
shuffling procedure, the polarization of each bunch is randomly assigned to be
up or down, and a distribution of $A_{N}$ in each $p_{T}$ bin is obtained by
repeating the procedure 10,000 times [bunchShuffling]. While most $p_{T}$ bins
were found to be consistent with the statistical variation, some of the lower
$p_{T}$ bins for both the $\pi^{0}$ and the $\eta$ in both $p$$+$$A$ collision
systems included up to 15% variation beyond what was expected from statistical
fluctuations. To account for this effect, an additional systematic uncertainty
was assigned in any $p_{T}$ bin showing variations significantly beyond
expected statistical fluctuations.

##  III Results and discussion