PHENIX Collaboration

#  $J/\psi$ and $\psi(2S)$ production at forward rapidity in $p$$+$$p$
collisions at $\sqrt{s}=510$ GeV

U.A. Acharya  Georgia State University, Atlanta, Georgia 30303, USA  A. Adare
University of Colorado, Boulder, Colorado 80309, USA  C. Aidala  Department of
Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA  N.N.
Ajitanand  Deceased  Chemistry Department, Stony Brook University, SUNY, Stony
Brook, New York 11794-3400, USA  Y. Akiba<EMAIL_ADDRESS>RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  R. Akimoto  Center for Nuclear Study, Graduate School of
Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  M.
Alfred  Department of Physics and Astronomy, Howard University, Washington, DC
20059, USA  N. Apadula  Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  Y. Aramaki  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  H. Asano  Kyoto
University, Kyoto 606-8502, Japan  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  E.T. Atomssa  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  T.C. Awes  Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
B. Azmoun  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  V. Babintsev  IHEP Protvino, State Research Center of Russian
Federation, Institute for High Energy Physics, Protvino, 142281, Russia  M.
Bai  Collider-Accelerator Department, Brookhaven National Laboratory, Upton,
New York 11973-5000, USA  N.S. Bandara  Department of Physics, University of
Massachusetts, Amherst, Massachusetts 01003-9337, USA  B. Bannier  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  K.N. Barish  University of California-Riverside, Riverside,
California 92521, USA  S. Bathe  Baruch College, City University of New York,
New York, New York, 10010 USA  RIKEN BNL Research Center, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  A. Bazilevsky  Physics
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
M. Beaumier  University of California-Riverside, Riverside, California 92521,
USA  S. Beckman  University of Colorado, Boulder, Colorado 80309, USA  R.
Belmont  University of Colorado, Boulder, Colorado 80309, USA  Department of
Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA  Physics
and Astronomy Department, University of North Carolina at Greensboro,
Greensboro, North Carolina 27412, USA  A. Berdnikov  Saint Petersburg State
Polytechnic University, St. Petersburg, 195251 Russia  Y. Berdnikov  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  D.
Black  University of California-Riverside, Riverside, California 92521, USA
J.S. Bok  New Mexico State University, Las Cruces, New Mexico 88003, USA  K.
Boyle  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  M.L. Brooks  Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  J. Bryslawskyj  Baruch College, City University of New
York, New York, New York, 10010 USA  University of California-Riverside,
Riverside, California 92521, USA  H. Buesching  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  V. Bumazhnov  IHEP
Protvino, State Research Center of Russian Federation, Institute for High
Energy Physics, Protvino, 142281, Russia  S. Campbell  Columbia University,
New York, New York 10027 and Nevis Laboratories, Irvington, New York 10533,
USA  Iowa State University, Ames, Iowa 50011, USA  V. Canoa Roman  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  C.-H. Chen  RIKEN BNL Research Center, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  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  T. Chujo  Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  Z. Citron
Weizmann Institute, Rehovot 76100, Israel  M. Connors  Georgia State
University, Atlanta, Georgia 30303, USA  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, 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  A. Datta  University
of New Mexico, Albuquerque, New Mexico 87131, USA  M.S. Daugherity  Abilene
Christian University, Abilene, Texas 79699, USA  G. David  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  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
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  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  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  L. Ding  Iowa State
University, Ames, Iowa 50011, USA  A. Dion  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
J.H. Do  Yonsei University, IPAP, Seoul 120-749, Korea  A. Drees  Department
of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, New York
11794-3800, USA  K.A. Drees  Collider-Accelerator Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  J.M. Durham  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  A. Durum  IHEP
Protvino, State Research Center of Russian Federation, Institute for High
Energy Physics, Protvino, 142281, Russia  A. Enokizono  RIKEN Nishina Center
for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  Physics
Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo
171-8501, Japan  H. En’yo  RIKEN Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan  R. Esha  Department of Physics and Astronomy,
Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA  S. Esumi
Tomonaga Center for the History of the Universe, University of Tsukuba,
Tsukuba, Ibaraki 305, Japan  B. Fadem  Muhlenberg College, Allentown,
Pennsylvania 18104-5586, USA  W. Fan  Department of Physics and Astronomy,
Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA  N. Feege
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  D.E. Fields  University of New Mexico,
Albuquerque, New Mexico 87131, USA  M. Finger  Charles University, Ovocný trh
5, Praha 1, 116 36, Prague, Czech Republic  M. Finger, Jr  Charles University,
Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic  D. Fitzgerald
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040,
USA  S.L. Fokin  National Research Center “Kurchatov Institute”, Moscow,
123098 Russia  J.E. Frantz  Department of Physics and Astronomy, Ohio
University, Athens, Ohio 45701, USA  A. Franz  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  A.D. Frawley  Florida
State University, Tallahassee, Florida 32306, USA  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  E.A. Gamez  Department of Physics, University of
Michigan, Ann Arbor, Michigan 48109-1040, 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  F. Giordano  University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA  A. Glenn  Lawrence Livermore
National Laboratory, Livermore, California 94550, USA  Y. Goto  RIKEN Nishina
Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
USA  N. Grau  Department of Physics, Augustana University, Sioux Falls, South
Dakota 57197, USA  S.V. Greene  Vanderbilt University, Nashville, Tennessee
37235, USA  M. Grosse Perdekamp  University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA  Y. Gu  Chemistry Department, Stony Brook
University, SUNY, Stony Brook, New York 11794-3400, 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  S.Y. Han  Ewha Womans University, Seoul
120-750, Korea  Korea University, Seoul, 02841  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  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  T. Hoshino  Hiroshima University, Kagamiyama,
Higashi-Hiroshima 739-8526, Japan  J. Huang  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  S. Huang  Vanderbilt
University, Nashville, Tennessee 37235, USA  Y. Ikeda  RIKEN Nishina Center
for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  K. Imai
Advanced Science Research Center, Japan Atomic Energy Agency, 2-4 Shirakata
Shirane, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan  Y. Imazu  RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, 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  S. Ishimaru  Nara Women’s University,
Kita-uoya Nishi-machi Nara 630-8506, Japan  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  S.J. Jeon  Myongji University, Yongin,
Kyonggido 449-728, Korea  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  J. Jia  Physics
Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York
11794-3400, 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  E. Joo  Korea University, Seoul, 02841  K.S. Joo
Myongji University, Yongin, Kyonggido 449-728, Korea  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
J.S. Kang  Hanyang University, Seoul 133-792, Korea  D. Kawall  Department of
Physics, University of Massachusetts, Amherst, Massachusetts 01003-9337, USA
A.V. Kazantsev  National Research Center “Kurchatov Institute”, Moscow, 123098
Russia  J.A. Key  University of New Mexico, Albuquerque, New Mexico 87131, USA
V. Khachatryan  Department of Physics and Astronomy, Stony Brook University,
SUNY, Stony Brook, New York 11794-3800, USA  A. Khanzadeev  PNPI, Petersburg
Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia  A.
Khatiwada  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
K. Kihara  Tomonaga Center for the History of the Universe, University of
Tsukuba, Tsukuba, Ibaraki 305, Japan  C. Kim  Korea University, Seoul, 02841
D.H. Kim  Ewha Womans University, Seoul 120-750, Korea  D.J. Kim  Helsinki
Institute of Physics and University of Jyväskylä, P.O.Box 35, FI-40014
Jyväskylä, Finland  E.-J. Kim  Jeonbuk National University, Jeonju, 54896,
Korea  H.-J. Kim  Yonsei University, IPAP, Seoul 120-749, Korea  M. Kim  RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
Department of Physics and Astronomy, Seoul National University, Seoul 151-742,
Korea  Y.K. Kim  Hanyang University, Seoul 133-792, Korea  D. Kincses  ELTE,
Eötvös Loránd University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  E.
Kistenev  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  J. Klatsky  Florida State University, Tallahassee, Florida
32306, USA  D. Kleinjan  University of California-Riverside, Riverside,
California 92521, USA  P. Kline  Department of Physics and Astronomy, Stony
Brook University, SUNY, Stony Brook, New York 11794-3800, USA  T. Koblesky
University of Colorado, Boulder, Colorado 80309, USA  M. Kofarago  ELTE,
Eötvös Loránd University, H-1117 Budapest, Pázmány P. s. 1/A, 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  J. Koster  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  D. Kotov  PNPI,
Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300,
Russia  Saint Petersburg State Polytechnic University, St. Petersburg, 195251
Russia  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  M. Kurosawa  RIKEN Nishina
Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
USA  Y. Kwon  Yonsei University, IPAP, Seoul 120-749, Korea  R. Lacey
Chemistry Department, Stony Brook University, SUNY, Stony Brook, New York
11794-3400, USA  J.G. Lajoie  Iowa State University, Ames, Iowa 50011, USA  A.
Lebedev  Iowa State University, Ames, Iowa 50011, USA  K.B. Lee  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  S.H. Lee  Iowa State
University, Ames, Iowa 50011, 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  M. Leitgab
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, 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  University
of Colorado, Boulder, Colorado 80309, USA  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Pusan National University, Busan, 46241, South
Korea  Yonsei University, IPAP, Seoul 120-749, Korea  M.X. Liu  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  S. Lökös  ELTE, Eötvös
Loránd University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  Eszterházy
Károly University, Károly Róbert Campus, H-3200 Gyöngyös, Mátrai út 36,
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
Weizmann Institute, Rehovot 76100, Israel  Department of Physics, Faculty of
Science, University of Zagreb, Bijenička c. 32 HR-10002 Zagreb, Croatia  A.
Manion  Department of Physics and Astronomy, Stony Brook University, SUNY,
Stony Brook, New York 11794-3800, USA  V.I. Manko  National Research Center
“Kurchatov Institute”, Moscow, 123098 Russia  E. Mannel  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  M. McCumber
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  P.L.
McGaughey  Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
D. McGlinchey  University of Colorado, Boulder, Colorado 80309, USA  Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  C. McKinney
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  A.
Meles  New Mexico State University, Las Cruces, New Mexico 88003, USA  M.
Mendoza  University of California-Riverside, Riverside, California 92521, USA
B. Meredith  Columbia University, New York, New York 10027 and Nevis
Laboratories, Irvington, New York 10533, USA  W.J. Metzger  Eszterházy Károly
University, Károly Róbert Campus, H-3200 Gyöngyös, Mátrai út 36, Hungary  Y.
Miake  Tomonaga Center for the History of the Universe, University of Tsukuba,
Tsukuba, Ibaraki 305, Japan  A.C. Mignerey  University of Maryland, College
Park, Maryland 20742, USA  A.J. Miller  Abilene Christian University, Abilene,
Texas 79699, 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 Nishina Center
for Accelerator-Based Science, Wako, Saitama 351-0198, Japan  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  P. Montuenga  University of Illinois at Urbana-Champaign, Urbana,
Illinois 61801, USA  T. Moon  Korea University, Seoul, 02841  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  T.V. Moukhanova
National Research Center “Kurchatov Institute”, Moscow, 123098 Russia  B.
Mulilo  Korea University, Seoul, 02841  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  A.
Mwai  Chemistry Department, Stony Brook University, SUNY, Stony Brook, New
York 11794-3400, USA  S. Nagamiya  KEK, High Energy Accelerator Research
Organization, Tsukuba, Ibaraki 305-0801, Japan  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  K. Nagashima
Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan  RIKEN
Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan
J.L. Nagle  University of Colorado, Boulder, Colorado 80309, USA  M.I. Nagy
ELTE, Eötvös Loránd University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary
I. Nakagawa  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan  RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  H. Nakagomi  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  Tomonaga Center for
the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305,
Japan  K. Nakano  RIKEN Nishina Center for Accelerator-Based Science, Wako,
Saitama 351-0198, Japan  Department of Physics, Tokyo Institute of Technology,
Oh-okayama, Meguro, Tokyo 152-8551, Japan  C. Nattrass  University of
Tennessee, Knoxville, Tennessee 37996, USA  S. Nelson  Florida A&M University,
Tallahassee, FL 32307, USA  P.K. Netrakanti  Bhabha Atomic Research Centre,
Bombay 400 085, India  M. Nihashi  Hiroshima University, Kagamiyama, Higashi-
Hiroshima 739-8526, Japan  RIKEN Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan  T. Niida  Tomonaga Center for the History of
the Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  R. Nishitani
Nara Women’s University, Kita-uoya Nishi-machi Nara 630-8506, 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  Helsinki Institute of Physics and University
of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  Tomonaga Center for the History of the Universe, University of Tsukuba,
Tsukuba, Ibaraki 305, Japan  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  A. Oskarsson  Department of
Physics, Lund University, Box 118, SE-221 00 Lund, Sweden  K. Ozawa  KEK, High
Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
Tomonaga Center for the History of the Universe, University of Tsukuba,
Tsukuba, Ibaraki 305, Japan  R. Pak  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  V. Pantuev  Institute for Nuclear
Research of the Russian Academy of Sciences, prospekt 60-letiya Oktyabrya 7a,
Moscow 117312, Russia  V. Papavassiliou  New Mexico State University, Las
Cruces, New Mexico 88003, USA  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  L. Patel  Georgia State University, Atlanta, Georgia 30303,
USA  M. Patel  Iowa State University, Ames, Iowa 50011, USA  J.-C. Peng
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  W.
Peng  Vanderbilt University, Nashville, Tennessee 37235, USA  D.V. Perepelitsa
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  University of Colorado, Boulder, Colorado 80309, USA
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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
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Petti  Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  Department of Physics and Astronomy, Stony Brook University,
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Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
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National Laboratory, Upton, New York 11973-5000, USA  P.V. Radzevich  Saint
Petersburg State Polytechnic University, St. Petersburg, 195251 Russia  J. Rak
Helsinki Institute of Physics and University of Jyväskylä, P.O.Box 35,
FI-40014 Jyväskylä, Finland  N. Ramasubramanian  Department of Physics and
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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,
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Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia  Saint
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Saito  KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki
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Organization, Tsukuba, Ibaraki 305-0801, Japan  C.Y. Scarlett  Florida A&M
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USA  I. Shein  IHEP Protvino, State Research Center of Russian Federation,
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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,
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Sickles  Physics Department, Brookhaven National Laboratory, Upton, New York
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87545, USA  D. Silvermyr  Department of Physics, Lund University, Box 118,
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Ovocný trh 5, Praha 1, 116 36, Prague, Czech Republic  K.L. Smith  Florida
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New York 11973-5000, USA  T. Sumita  RIKEN Nishina Center for Accelerator-
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and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  X. Sun  Georgia State University, Atlanta, Georgia 30303, USA  Z. Sun
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York 11973-5000, USA  S. Tarafdar  Vanderbilt University, Nashville, Tennessee
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Lyon, Université Lyon 1, F-69622, Villeurbanne, France  A. Timilsina  Iowa
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Towell  Abilene Christian University, Abilene, Texas 79699, USA  R.S. Towell
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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  M. Vargyas  ELTE, Eötvös Loránd
University, H-1117 Budapest, Pázmány P. s. 1/A, 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  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  E. Vznuzdaev  PNPI,
Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300,
Russia  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  D. Watanabe  Hiroshima University, Kagamiyama,
Higashi-Hiroshima 739-8526, Japan  Y. Watanabe  RIKEN Nishina Center for
Accelerator-Based Science, Wako, Saitama 351-0198, Japan  RIKEN BNL Research
Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  Y.S.
Watanabe  Center for Nuclear Study, Graduate School of Science, University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  KEK, High Energy
Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan  F. Wei
New Mexico State University, Las Cruces, New Mexico 88003, USA  S. Whitaker
Iowa State University, Ames, Iowa 50011, USA  S. Wolin  University of Illinois
at Urbana-Champaign, Urbana, Illinois 61801, USA  C.P. Wong  Georgia State
University, Atlanta, Georgia 30303, 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  M.
Wysocki  Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  B.
Xia  Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701,
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  Center for Nuclear Study, Graduate School of
Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
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  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  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
USA  I. Yoon  Department of Physics and Astronomy, Seoul National University,
Seoul 151-742, Korea  I. Younus  Physics Department, Lahore University of
Management Sciences, Lahore 54792, Pakistan  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

The PHENIX experiment at the Relativistic Heavy Ion Collider has measured the
differential cross section, mean transverse momentum, mean transverse momentum
squared of inclusive $J/\psi$ and cross-section ratio of $\psi(2S)$ to
$J/\psi$ at forward rapidity in $p$+$p$ collisions at $\sqrt{s}$ = 510 GeV via
the dimuon decay channel. Comparison is made to inclusive $J/\psi$ cross
sections measured at $\sqrt{s}$ = 200 GeV and 2.76–13 TeV. The result is also
compared to leading-order nonrelativistic QCD calculations coupled to a color-
glass-condensate description of the low-$x$ gluons in the proton at low
transverse momentum ($p_{T}$) and to next-to-leading order nonrelativistic QCD
calculations for the rest of the $p_{T}$ range. These calculations
overestimate the data at low $p_{T}$. While consistent with the data within
uncertainties above $\approx 3$ GeV/$c$, the calculations are systematically
below the data. The total cross section times the branching ratio is BR
$d\sigma^{J/\psi}_{pp}/dy(1.2<|y|<2.2,0<\mbox{$p_{T}$}<10~{}\mbox{GeV/$c$})=$
54.3 $\pm$ 0.5 (stat) $\pm$ 5.5 (syst) nb.

##  I Introduction

Charmonium states such as $J/\psi$ and $\psi(2S)$ mesons are bound states of a
charm and anti-charm quark ($c\bar{c}$). At the Relativistic Heavy Ion
Collider (RHIC) energies, they are produced mostly from hard scattering of two
gluons into a $c\bar{c}$ pair followed by the evolution of this pair through a
hadronization process to form a physical charmonium. Despite several decades
of extensive studies Abe  _et al._ (1992); Adare  _et al._ (2007); Aamodt  _et
al._ (2011); Abelev  _et al._ (2012a); Adam  _et al._ (2016); Acharya  _et
al._ (2017); Adam  _et al._ (2019); Ma  _et al._ (2011); Ma and Venugopalan
(2014) since the discovery of $J/\psi$, we still have very limited knowledge
about the $J/\psi$ production mechanism and hadronization. Therefore, carrying
out as many charmonium measurements as possible in $p$+$p$ collisions over a
wide range of transverse momentum ($p_{T}$) and of rapidity ($y$) at different
energies is essential to understanding production mechanisms. These
measurements over a wide range of $p_{T}$ (down to zero $p_{T}$) and rapidity
allow calculating quantities, such as the mean transverse momentum $\langle
p_{T}\rangle$, the mean transverse momentum squared $\langle
p_{\mathrm{\textsc{t}}}^{2}\rangle$, and the $p_{T}$-integrated cross section
$d\sigma/dy$. The collision energy dependence of these quantities can put
stringent constraints on the different theoretical approaches that are used to
describe the hadronic production of $J/\psi$. These approaches include the
color-evaporation model (CEM) Fritzsch (1977); Amundson  _et al._ (1997), the
color-singlet model (CSM) R. Baier and R. Rückl (1981) and the nonrelativistic
quantum chromodynamics formalism (NRQCD) Bodwin  _et al._ (1995). In this
work, we compare the data to NRQCD, an effective field theory derived from QCD
and valid for heavy-quark pairs with low relative velocity, where a $J/\psi$
can be formed from $c\bar{c}$ pair produced in a color-singlet or a color-
octet state.

In this paper, we present the inclusive $J/\psi$ production cross section and
the ratio of $\psi(2S)$ to $J/\psi$ production cross sections at forward
rapidity ($1.2<|y|<2.2$) measured in $p$+$p$ collisions at center of mass
energy $\sqrt{s}$ = 510 GeV. These mesons are measured in the dimuon decay
channel. The $J/\psi$ inclusive differential cross sections are obtained as a
function of $p_{T}$ and $y$ over a wide range of $p_{T}$. The $J/\psi$ and
$\psi(2S)$ results at $\sqrt{s}$ = 510 GeV are the first measurements at this
rapidity. Comparisons to similar PHENIX measurements performed at $\sqrt{s}$ =
200 GeV Adare  _et al._ (2007) and Large Hadron Collider (LHC) measurements at
$\sqrt{s}$ = 2.76, 5.02, 7, 8 and 13 TeV Aamodt  _et al._ (2011); Abelev  _et
al._ (2012a); Adam  _et al._ (2016); Acharya  _et al._ (2017) allow studying
the variations of $\langle p_{T}\rangle$, $\langle
p_{\mathrm{\textsc{t}}}^{2}\rangle$ and $d\sigma/dy$ as a function of
$\sqrt{s}$. The results are also compared to next-to-leading order (NLO) NRQCD
calculations Ma  _et al._ (2011).

The paper is organized as follows: the PHENIX apparatus is described in Sec.
II, the data samples used for this analysis and the analysis procedure are
discussed in Sec. III, while the results are presented and compared to
measurements at different $\sqrt{s}$ as well as to models in Sec. IV.

##  II Experimental Setup

A complete description of the PHENIX detector can be found in Ref. Adcox  _et
al._ (2003). Only the detector systems relevant to this measurement are
briefly described here.

Figure 1:  A side view of the PHENIX detector, concentrating on the muon arm
instrumentation.

The PHENIX muon spectrometers, see Fig. 1, cover the full aziumth and the
north (south) arm cover forward (backward) rapidity, $1.2<y<2.2$
$(-2.2<y<-1.2)$. Each muon spectrometer comprises a hadronic absorber, a
magnet, a muon tracker (MuTr), and a muon identifier (MuID). The absorbers
comprise layers of copper, iron and stainless steel and have about 7.2
interaction lengths. Following the absorber in each muon arm is the MuTr,
which comprises three stations of cathode strip chambers in a radial magnetic
field with an integrated bending power of 0.8 T$\cdot$m. The MuID comprises
five alternating steel absorbers and Iarocci tubes. The composite momentum
resolution, $\delta p/p$, of particles in the analyzed momentum range is about
5%, independent of momentum and dominated by multiple scattering. Muon
candidates are identified by reconstructed tracks in the MuTr matched to MuID
tracks that penetrate through to the last MuID plane.

Since 2012 the PHENIX detector had a new forward vertex detector (FVTX) Aidala
_et al._ (2014), which comprises four planes of silicon strip detectors,
finely segmented in radius and coarsely segmented in azimuth. For the subset
of muon candidate tracks passing several of these detector planes, this
additional information was used to improve mass resolution by a factor of 1.5
for studying $\psi(2S)$.

Another detector system relevant to this analysis is the beam-beam counter
(BBC), comprising two arrays of 64 Čerenkov counters, located on both sides of
the interaction point and covering the pseudorapidity range $3.1<|\eta|<3.9$.
The BBC system was used to measure the $p$+$p$ collision vertex position along
the beam axis ($z_{\rm vtx}$), with 2 cm resolution, and initial collision
time. It was also used to measure the beam luminosity and form a minimum bias
(MB) trigger.

##  III Data analysis

The results presented here are based on the data sample collected by PHENIX
during the 2013 $p$+$p$ run at $\sqrt{s}$ = 510 GeV. The BBC counters provided
the MB trigger, which required at least one hit in each of the BBCs. Events,
in coincidence with the MB trigger, containing a muon pair within the
acceptance of the spectrometer are selected by the level-1 dimuon trigger
(MuIDLL1-2D) requiring that at least two tracks penetrate through the MuID to
its last layer. The data sample, used in this analysis, corresponds to
$3.02\times 10^{12}$ MB events or to an integrated luminosity of 94.4 pb-1.

###  III.1 Raw yield extraction

A set of quality cuts is applied to the data to select good $p$+$p$ events and
good muon candidates as well as to improve the signal-to-background ratio.
Good $p$+$p$ events are selected by requiring that the collision occurs in the
fiducial interaction region $|z_{\rm vtx}|<30$ cm as measured by the BBC. Each
reconstructed muon candidate comprises a combination of reconstructed muon
tracks in the MuTr and in the MuID. The MuTr track is required to have more
than 9 hits out of the maximum possible of 16 while the MuID track is required
to have more than 6 hits out of the maximum possible of 10. In addition, a cut
on individual MuTr track $\chi^{2}$ of 23 is applied. The MuTr track
$\chi^{2}$ is calculated from the difference between the measured hit
positions of the track and the subsequent fit for each MuTr track. The MuTr
tracks are then matched to the MuID tracks at the first MuID layer by applying
cuts on maximum position and angle differences. Furthermore, there is a
minimum allowed single muon momentum along the beam axis, $p_{z}$, which is
reconstructed and energy-loss corrected at the collision vertex, of 3.0
GeV/$c$ corresponding to the momentum cut effectively imposed by the
absorbers. Finally, a cut on the $\chi^{2}$ of the fit of the two muon tracks
to the common vertex of the two candidate tracks near the interaction point
was applied.

The invariant mass distribution is formed by combining muon candidate tracks
of opposite charges (unlike-sign). In addition to the charmonium signal, the
resulting unlike-sign dimuon spectrum includes correlated and uncorrelated
pairs. In the $J/\psi$ and $\psi(2S)$ region, the correlated pairs arise from
correlated semi-muonic decays of charmed hadrons, beauty and the Drell-Yan
process as well as light hadron decays. The uncorrelated pairs are mainly
coming from the decays of light hadrons ($\pi^{\pm}$, $K^{\pm}$ and $K^{0}$)
which decay before or after passing through the absorber, and form the
combinatorial background.

The combinatorial background is estimated using two methods: The first one
derives the combinatorial background from the mass distribution of the same
sign (like-sign) pairs of muon candidates within the same event. The second
method derives the combinatorial background from the mass distribution of the
unlike-sign pairs of muon candidates from different events (mixed-event) of
z-vertex position within 2 cm. The normalization of the mass distribution of
the combinatorial background from the like-sign dimuon distributions ($N_{++}$
and $N_{--}$) is calculated as: $N_{\rm CB}=2\sqrt{N_{++}N_{--}}$. The mixed-
event like-sign dimuon mass distribution is normalized to the same-event like-
sign combinatorial background distribution in the invariant mass range
$2.0-4.5$ GeV/$c^{2}$. This factor is then used to normalize the mixed-event
unlike-sign dimuon mass distribution.

Figure 2: Raw unlike-sign dimuon spectra (closed [black] circles) along with
normalized like-sign background (upward [red] triangles) and normalized mixed-
event background (inverted [blue] triangles) for (a) one $p_{T}$ bin and (b)
one rapidity bin. Panels (c) and (d) show the background-subtracted spectra
fitted as described in the text for (c) one $p_{T}$ bin and (d) one rapidity
bin.

Figure 2 shows the unlike-sign dimuon spectrum together with the combinatorial
background estimated by both methods. Both background distributions from the
mixed-event and like-sign methods are consistent, however, the mixed-event
background is more statistically stable, because we mix each event with the
previous four events. Therefore, the mixed-event background was used to
subtract the uncorrelated background from the unlike-sign dimuon spectrum.

After subtracting the uncorrelated background, the unlike-sign spectra
including the correlated background are fitted by the following function,

$\begin{split}f(m_{\mu\mu})=p_{0}[\frac{(1-p_{3})}{\sqrt{2\pi}p_{2}}\mbox{exp}(-\frac{1}{2}\frac{(m_{\mu\mu}-p_{1})^{2}}{p^{2}_{2}})+\\\
\frac{p_{3}}{\sqrt{2\pi}p_{4}}\mbox{exp}(-\frac{1}{2}\frac{(m_{\mu\mu}-p_{1})^{2}}{p^{2}_{4}})]+\\\
p_{5}\mbox{exp}(p_{6}+p_{7}m_{\mu\mu}),\end{split}$ (1)

where $p_{0}-p_{7}$ are free parameters and $m_{\mu\mu}$ is the unlike-sign
dimuon mass. The $J/\psi$ shape is better described with two Gaussian
distributions, corresponding to the first two terms in Eq. 1, one for the
$J/\psi$ peak and a second one with larger width to account for the wider
tails, which occurs due to limitations in MuTr resolution, as discussed in
sec. II. The peak also includes contribution from $\psi(2S)$, which is not
resolved. An exponential is used to describe the continuum contributions from
correlated backgrounds. Panels (a) and (b) of Fig. 2 show the raw spectra for
selected $p_{T}$ and rapidity bins and panels (c) and (d) show the spectra
after subtracting the combinatorial background fitted with the function
described above for those selected bins.

To extract the $\psi(2S)$ signal we improve the mass resolution of the muon
tracking systems by utilizing the FVTX. The FVTX being located before the
absorber allows measuring the dimuon opening angle before any multiple
scattering occurs in the absorber Aidala  _et al._ (2014). Using this
additional tracking information gives a more precise measurement of the dimuon
opening angle and thereby a more precise measurement of the pair mass, as well
as rejecting backgrounds from decay muons that emerge from the absorber.
However, these additional requirements on the dimuon tracks that are necessary
to separate the $J/\psi$ and $\psi(2S)$ peaks also reduce the statistics by a
factor of 6 due to the geometric acceptance of FVTX, therefore, we study the
dimuon mass spectra in each arm integrated over $p_{T}$ and rapidity within
each arm. The dimuon mass spectrum extracted including the FVTX after
subtracting the mixed-event background is shown in Fig. 3.

Figure 3: Raw unlike-sign dimuon spectrum summed over $p_{T}$ and the whole
backward rapidity range, $-2.2<y<-1.2$.

Given the resolution enhancement, the sum of a Gaussian and a crystal-ball
function Gaiser (1983); Adare  _et al._ (2017), rather than a double Gaussian,
was used for each of $J/\psi$ and $\psi(2S)$ peaks to fit the dimuon mass
spectrum. The $\psi(2S)$ peak is expected to be wider than the $J/\psi$ peak,
due to the fact that the higher mass and harder $p_{T}$ spectrum of the
$\psi(2S)$ state will produce higher momentum decay muons which have larger
uncertainty in their reconstructed momentum in the spectrometer due to a
smaller bend in the magnetic field. By selecting only poorly reconstructed
tracks, we found a $J/\psi$ width of $\approx 200$ MeV/c2, therefore, the
width of the second Gaussian in the fit to the entire sample of tracks is set
to 200 MeV/c2. The ratio of widths of the $\psi(2S)$ to $J/\psi$ is set to
1.15, following expectations of the performance of the muon tracking system
Akikawa  _et al._ (2003). The difference between the centroids of the
$\psi(2S)$ and $J/\psi$ peaks is set to the Particle Data Group value of 589
MeV/c2 Olive  _et al._ (2014). The relative normalization of the second
Gaussian is fixed to be the same for both resonances, as are the parameters
for the crystal-ball line shape.

###  III.2 Detector acceptance and reconstruction efficiency

The acceptance and reconstruction efficiency ($A\varepsilon_{\mathrm{rec}}$)
of the muon spectrometers, including the MuID trigger efficiency, is
determined by running a pythia111We used pythia6.421, with parton distribution
functions given by CTEQ6LL. The following parameters were modified: MSEL = 0,
MSUB(86) = 1, PARP(91) = 2.1, MSTP(51) = 10041, MDME(858,1) = 0, MDME(859,1) =
1, MDME(860,1) = 0, and Tune A. Sj$\ddot{o}$strand  _et al._ (2001) generated
$J/\psi$ signal through a geant4-based full detector simulation Agostinelli
_et al._ (2003) of PHENIX. The simulation tuned the detector response to a set
of characteristics (dead and hot channel maps, gains, noise, etc.) that
described the performance of each detector subsystem. The simulated vertex
distribution was tuned to match that of the 2013 data. The simulated events
are reconstructed in the same manner as the data and the same cuts are applied
as in the real data analysis.

Figure 4:  $A\varepsilon_{\rm rec}$ as a function of $p_{T}$ for $1.2<y<2.2$
(closed [red] circles) and $-2.2<y<-1.2$ (open [blue] circles).

Figure 4 shows $A\varepsilon_{\rm rec}$ as a function of $p_{T}$ and rapidity
for $J/\psi$. The relative difference in $A\varepsilon_{\rm rec}$ between the
two spectrometers is due to different detection efficiencies of the MuTr and
MuID systems and different amount of absorber material.

In the case of $\psi(2S)$, we are interested in the ratio of its differential
cross section to that of $J/\psi$, therefore, we extract the ratio of
$A\varepsilon_{\rm rec}$ for $\psi(2S)$ and $J/\psi$ with addition of the FVTX
information in analyzing the simulation to match that of the data analysis. A
factor of 0.77 (0.69) is applied to the $\psi(2S)$/$J/\psi$ ratio extracted
from the fit to the invariant mass spectrum to account for differences in
acceptance, efficiency, and dimuon trigger efficiencies between the north
(south) arm of the muon spectrometer.

###  III.3 Differential cross section

The differential cross section is evaluated according to the following
relation:

$\frac{d^{2}\sigma_{\psi}}{dydp_{T}}=\frac{1}{\Delta y\Delta
p_{T}}\frac{N_{\psi}}{A\varepsilon_{\rm rec}BR}\frac{\sigma_{\rm
BBC}}{\varepsilon_{\rm BBC}N^{\rm BBC}_{\rm MB}}$ (2)

where $N_{\psi}$ is the extracted $J/\psi$ or $\psi(2S)$ yield in $y$ and
$p_{T}$ bins with $\Delta y$ and $\Delta$$p_{T}$ widths, respectively. $BR$ is
the branching ratio where $BR_{J/\psi\rightarrow\mu^{+}\mu^{-}}=(5.93\pm
0.06)\times 10^{-2}$ and
$BR_{\mbox{$\psi(2S)$}\rightarrow\mu^{+}\mu^{-}}=(7.9\pm 0.9)\times 10^{-3}$
Olive  _et al._ (2014). $A\varepsilon_{\rm rec}$ is the product of the
acceptance and reconstruction efficiency. $N^{\rm BBC}_{\rm MB}$ $=3.02\times
10^{12}$ is the number of MB events and $\varepsilon_{\rm BBC}$ $=0.91\pm
0.04$ is the efficiency of the MB trigger for events containing a hard
scattering Adare  _et al._ (2016). $\sigma_{\rm BBC}$ is the PHENIX BBC cross
section, $32.5\pm 3.2$ mb at $\sqrt{s}=510$ GeV, which is determined from the
van der Meer scan technique Adare  _et al._ (2011).

###  III.4 Systematic uncertainties

All systematic uncertainties are evaluated as standard deviations and are
summarized in Tables 1 and 2. They are divided into three categories based
upon the effect each source has on the measured results:

Type-A

Point-to-point uncorrelated uncertainties allow the data points to move
independently with respect to one another and are added in quadrature with
statistical uncertainties; however, no systematic uncertainties of this type
are associated with this measurement.

Type-B

Point-to-point correlated uncertainties which allow the data points to move
coherently within the quoted range to some degree. These systematic
uncertainties include a 4% uncertainty from MuID tube efficiency and an 8.2%
(2.8%) from MuTr overall efficiency for the north (south) arm. A 3.9% signal
extraction uncertainty is assigned to account for the yield variations when
using different functions, i.e., second, third and fourth order polynomials,
to fit the correlated background and $\approx 3\%$ uncertainty is assigned to
account for the $\psi(2S)$ contribution. The systematic uncertainty associated
with $A\varepsilon_{\rm rec}$ includes the uncertainty on the input $p_{T}$
and rapidity distributions which is extracted by varying these distributions
over the range of the statistical uncertainty of the data, yielding 4.4%
(5.0%) for the north (south) arm. Additional 11.2% (8.8%) systematic effect
for the north (south) arm was also considered to account for the azimuthal
angle distribution difference between data and simulation. To be consistent
with the real data analysis, a trigger emulator was used to match the level-1
dimuon trigger for the data. The efficiency of the trigger emulator was
studied by applying it to the data and comparing the resulting mass spectrum
to the mass spectrum when applying the level-1 dimuon trigger which resulted
in a 1.5% (2%) uncertainty for the north (south) arm. Type-B systematic
uncertainties are added in quadrature and amount to 16.0% (12.4%) for the
north (south) arm. They are shown as shaded bands on the associated data
points.

Type-C

An overall normalization uncertainty of 10% was assigned for the BBC cross
section and efficiency uncertainties Adler  _et al._ (2003) that allow the
data points to move together by a common multiplicative factor.

Table 1:  Systematic uncertainties associated with $J/\psi$ differential cross section calculation in the north (south) arm. Type- | Origin | north (south)
---|---|---
B | MuID hit efficiency | 4.0% (4.0%)
B | MuTr hit efficiency | 8.2% (2.8%)
B | Signal extraction | 3.9% (3.9%)
B |  $\psi(2S)$ contribution | 3.0% (3.0%)
B |  $A\varepsilon_{\rm rec}$ $p_{T}$ and $y$ input distributions | 4.4% (5.0%)
B |  $A\varepsilon_{\rm rec}$ $\phi$ distribution | 11.2% (8.8%)
B |  $A\varepsilon_{\rm rec}$ trigger emulator | 1.5% (2.0%)
B | Quadratic sum | 16.0% (12.4)%
C | MB trigger efficiency | 10%

In the measurement of the $\psi(2S)$ to $J/\psi$ ratio, most of the mentioned
systematic uncertainties cancel out. However, the fit that was used to extract
the yields is more complex and additional systematic uncertainties arose from
the constraints applied during the fitting process.

A systematic uncertainty from constraining the normalization factor is
determined by varying the mass range over which the factor is calculated and a
3% systematic uncertainty is assigned for both arms. Systematic uncertainty of
3% (7%) was assigned to the north (south) arm on the fit range by varying the
range around the nominal values, 2–5 GeV/$c^{2}$. The effect of constraining
the second Gaussian peak width to 200 MeV/$c^{2}$ was studied by varying the
width between 175 and 225 MeV/$c^{2}$, resulting in a systematic uncertainty
of 12% (10%) in the north (south) arm.

The systematic uncertainty component on $A\varepsilon_{\rm rec}$ that survived
the ratio amounts to 2.7% (4.1%) in the north (south) arm. The systematic
uncertainties associated with the ratio measurement are summarized in Table 2.

Table 2:  Systematic uncertainties associated with the differential cross section ratio of $\psi(2S)$ to $J/\psi$ in the north (south) arm. Type | Origin | north (south)
---|---|---
B |  $\sigma(2s)/\sigma(1s)$ constraint | 3% (3%)
B | Background fit mass range | 3% (7%)
B | Second Gaussian width constraint | 12% (10%)
B | $A\varepsilon_{\rm rec}$ | 2.7% (4.1%)
B | Quadratic sum | 13% (13%)

##  IV Results

The inclusive $J/\psi$ differential cross section as a function of $p_{T}$ is
calculated independently for each muon arm, then the results are combined
using the best-linear-unbiased-estimate method Nisius (2014). Results obtained
using the two muon spectrometers are consistent within statistical
uncertainties. The combined inclusive $J/\psi$ differential cross section is
shown in Fig. 5 and listed in Table 3. The gray shaded bands represent the
weighted average of the quadratic sum of type-B systematic uncertainties of
the north and south arms, $\approx 10.1\%$. The average is weighted based on
the statistical uncertainties of each arm.

Figure 5: The inclusive $J/\psi$ differential cross section as a function of $p_{T}$ at $1.2<|y|<2.2$ at 510 GeV (closed [red] circles) and at 200 GeV (open [blue] squares). The error bars represent the statistical uncertainties, and gray shaded bands (although too small to appear on the data points) are added representing the quadratic sum of type-B systematic uncertainties. NRQCD calculations at 510 GeV Ma  _et al._ (2011) are also shown. Table 3: Differential cross sections in nb/(GeV/$c$)2 and $p_{T}$ in (GeV/$c$) of inclusive $J/\psi$ with statistical and type-B systematic uncertainties. $p^{\rm min}_{T}$ | $p^{\rm max}_{T}$ |  $\frac{BR}{2\pi p_{T}}\frac{d^{2}\sigma}{dydp_{T}}$ |
---|---|---|---
0.00 |  0.25 |  $(5.04\pm 0.23\pm 0.51)\times 10^{0}$ |
0.25 |  0.50 |  $(4.85\pm 0.17\pm 0.49)\times 10^{0}$ |
0.50 |  0.75 |  $(4.42\pm 0.15\pm 0.45)\times 10^{0}$ |
0.75 |  1.00 |  $(3.73\pm 0.13\pm 0.38)\times 10^{0}$ |
1.00 |  1.25 |  $(3.16\pm 0.11\pm 0.32)\times 10^{0}$ |
1.25 |  1.50 |  $(2.47\pm 0.08\pm 0.25)\times 10^{0}$ |
1.50 |  1.75 |  $(2.00\pm 0.07\pm 0.20)\times 10^{0}$ |
1.75 |  2.00 |  $(1.52\pm 0.05\pm 0.15)\times 10^{0}$ |
2.00 |  2.25 |  $(1.18\pm 0.04\pm 0.12)\times 10^{0}$ |
2.25 |  2.50 |  $(8.45\pm 0.30\pm 0.85)\times 10^{-1}$ |
2.50 |  2.75 |  $(6.44\pm 0.23\pm 0.65)\times 10^{-1}$ |
2.75 |  3.00 |  $(4.90\pm 0.18\pm 0.50)\times 10^{-1}$ |
3.00 |  3.25 |  $(3.69\pm 0.14\pm 0.37)\times 10^{-1}$ |
3.25 |  3.50 |  $(2.74\pm 0.10\pm 0.28)\times 10^{-1}$ |
3.50 |  3.75 |  $(1.99\pm 0.08\pm 0.20)\times 10^{-1}$ |
3.75 |  4.00 |  $(1.44\pm 0.06\pm 0.15)\times 10^{-1}$ |
4.00 |  4.50 |  $(9.53\pm 0.36\pm 0.96)\times 10^{-2}$ |
4.50 |  5.00 |  $(5.16\pm 0.21\pm 0.52)\times 10^{-2}$ |
5.00 |  6.00 |  $(2.31\pm 0.09\pm 0.23)\times 10^{-2}$ |
6.00 |  7.00 |  $(7.17\pm 0.34\pm 0.72)\times 10^{-3}$ |
7.00 |  8.00 |  $(2.05\pm 0.15\pm 0.21)\times 10^{-3}$ |
8.00 |  10.00 |  $(5.18\pm 0.44\pm 0.52)\times 10^{-4}$ |

The data points are corrected to account for the finite width of the analyzed
$p_{T}$ bins Lafferty and Wyatt (1995). We compare the data to inclusive
$J/\psi$ data at 200 GeV Adare  _et al._ (2007) which show similar $p_{T}$
dependence. At low $p_{T}$, the data are compared to prompt $J/\psi$ leading-
order (LO) NRQCD calculations Ma  _et al._ (2011); Bodwin  _et al._ (1995)
coupled to a Color Glass Condensate (CGC) description of the low-$x$ gluons in
the proton Ma and Venugopalan (2014). For the rest of $p_{T}$ range, the data
are compared to prompt $J/\psi$ NLO NRQCD calculations Ma  _et al._ (2011);
Bodwin  _et al._ (1995). The LO-NRQCD+CGC calculations overestimate the data
at low $p_{T}$. The NLO-NRQCD calculations underestimate the data at high
$p_{T}$, while to some extent, are consistent with the data at intermediate
$p_{T}$, 3–5 GeV/$c$. It is important to stress that the nonprompt $J/\psi$
contribution (from excited charmonium states and from $B$-meson decays) is not
included in these calculations. This is expected to be a significant
contribution at high $p_{T}$; therefore, the addition of the nonprompt
$J/\psi$ contribution could account for the difference between the data and
calculations Aidala  _et al._ (2017); Cacciari  _et al._ (2005); Nelson  _et
al._ (2013). The $p_{T}$ coverage down to zero $p_{T}$ allows the extraction
of the $p_{T}$-integrated cross section, BR
$d\sigma^{J/\psi}_{pp}/dy(1.2<|y|<2.2,0<\mbox{$p_{T}$}<10~{}\mbox{GeV/$c$})=$
54.3 $\pm$ 0.5 (stat) $\pm$ 5.5 (syst) nb.

Figure 6: The inclusive $J/\psi$ differential cross section integrated over $0<p_{T}<10$ GeV/$c$ as a function of rapidity at 510 GeV (closed [red] circles) and at 200 GeV (open [blue] squares). The error bars represent the statistical uncertainties, and the gray shaded band represents the quadratic sum of type-B systematic uncertainties. NLO-NRQCD calculations Ma  _et al._ (2011) are also shown. Table 4: Differential cross sections in nb versus rapidity of inclusive $J/\psi$ over $0<p_{T}<10$ (GeV/$c$) with statistical and type-B systematic uncertainties. $y^{\rm min}$ | $y^{\rm max}$ |  $BR\frac{d\sigma}{dy}$ (nb) |
---|---|---|---
-2.20 |  -2.00 |  $27.6\pm 1.3\pm 4.4$ |
-2.00 |  -1.90 |  $37.7\pm 1.8\pm 6.0$ |
-1.90 |  -1.80 |  $47.0\pm 2.1\pm 7.5$ |
-1.80 |  -1.70 |  $57.6\pm 2.6\pm 9.2$ |
-1.70 |  -1.60 |  $65.2\pm 2.9\pm 10.4$ |
-1.60 |  -1.50 |  $71.5\pm 3.2\pm 11.4$ |
-1.50 |  -1.20 |  $75.9\pm 3.4\pm 12.1$ |
1.20 |  1.50 |  $72.0\pm 3.4\pm 11.5$ |
1.50 |  1.60 |  $69.1\pm 3.2\pm 11.0$ |
1.60 |  1.70 |  $65.5\pm 3.0\pm 10.4$ |
1.70 |  1.80 |  $52.3\pm 2.4\pm 8.3$ |
1.80 |  1.90 |  $46.7\pm 2.2\pm 7.5$ |
1.90 |  2.00 |  $38.4\pm 1.9\pm 6.1$ |
2.00 |  2.20 |  $27.8\pm 1.4\pm 4.4$ |

Inclusive $J/\psi$ differential cross section as a function of rapidity is
listed in Table 4 and shown in Fig. 6, which also includes PHENIX inclusive
$J/\psi$ data at 200 GeV Adare  _et al._ (2007) and NLO-NRQCD calculations Ma
_et al._ (2011). The 510 GeV data show a similar rapidity dependence pattern
to that of the 200 GeV data. NLO-NRQCD calculations overestimate the data, and
this is consistent with what was observed in the case of $p_{T}$-dependent
differential cross section (see Fig. 5) because the $y$-dependent differential
cross section is dominated by the low-$p_{T}$ region where NRQCD calculation
overestimates the data.

To quantify the feed-down contribution of excited charmonium states, the ratio
of the cross section of $\psi(2s)$ to $J/\psi$, multiplied by their respective
branching ratio to dimuons, is measured ($R=2.84{\pm}0.45$%) and shown in Fig.
7. This ratio is compared with other $p$$+$$p$ and $p$$+A$ systems at
different collision energies Alessandro  _et al._ (2006); Adare  _et al._
(2012); Antoniazzi  _et al._ (1993); Abreu  _et al._ (1998); Clark  _et al._
(1978); Albajar  _et al._ (1991); Abe  _et al._ (1997); Aaij  _et al._ (2013);
Abelev  _et al._ (2014); Adare  _et al._ (2017). The results are consistent
with world data within uncertainties with no significant dependence on
collision energy.

Figure 7: Comparison of world data on the ratio of $\psi(2s)$ to $J/\psi$
mesons in dilepton decays Alessandro  _et al._ (2006); Adare  _et al._ (2012);
Antoniazzi  _et al._ (1993); Abreu  _et al._ (1998); Clark  _et al._ (1978);
Albajar  _et al._ (1991); Abe  _et al._ (1997); Aaij  _et al._ (2013); Abelev
_et al._ (2014); Adare  _et al._ (2017); Adam  _et al._ (2019). The associated
uncertainties are the quadrature sum of the statistical and systematic
uncertainties.

To better understand the shape of the $p_{T}$ spectrum for $J/\psi$ at forward
rapidity and quantify its hardening at $\sqrt{s}$ = 510 GeV, we calculate the
corresponding mean transverse momentum $\langle p_{T}\rangle$ and mean
transverse momentum squared $\langle p^{2}_{T}\rangle$. This is done by
fitting the inclusive $J/\psi$ $p_{T}$-dependent differential cross sections
with the following function Adare  _et al._ (2007); Acharya  _et al._ (2017):

Figure 8:  $\langle p_{T}\rangle$ as a function of $\sqrt{s}$ for $J/\psi$
from this work at 510 GeV (closed [red] circle), from PHENIX at 200 GeV Adare
_et al._ (2007) at midrapidity (closed [blue] square) and forward rapidity
(open [blue] square), and from ALICE at different energies Acharya  _et al._
(2017) (closed [green] diamonds).

Figure 9:  $\langle p^{2}_{T}\rangle$ as a function of $\sqrt{s}$ for
$J/\psi$. The figure includes results from this work at 510 GeV (closed [red]
circle) and PHENIX results at 200 GeV Adare  _et al._ (2007) at midrapidity
(closed [blue] square) and forward rapidity (open [blue] square). Also shown
are higher-energy data from CDF Acosta  _et al._ (2005) (closed [cyan]
downward triangle) and ALICE Acharya  _et al._ (2017) (closed [green]
diamonds). Also shown at lower energies are data from NA38/NA50/NA51 Drapier
(1998); Adare  _et al._ (2007) (open [black] circles) and E789 Schub  _et al._
(1995) (closed [magenta] upward triangle).

$f(p_{T})=A\frac{p_{T}}{(1+(\frac{p_{T}}{B})^{2})^{n}}$ (3)

where $A$, $B$ and $n$ are free parameters and their values from the fit are
$54.6\pm 0.5$, $10.4\pm 0.4$ and $0.45\pm 0.06$, respectively, and $\langle
p_{T}\rangle$ and $\langle p^{2}_{T}\rangle$ are the first and second moments
of Eq. 3 in a given $p_{T}$ range. This fit results in a $\langle
p_{T}\rangle=1.90\pm 0.02\pm 0.30$ GeV/$c$ and $\langle
p^{2}_{T}\rangle=5.00\pm 0.06\pm 0.51$ (GeV/$c$)2.

The first error is statistical, and the second is the systematic uncertainty
from the maximum shape deviation permitted by the type-B correlated errors.

Figure 9 shows $\langle p_{T}\rangle$ as a function of $\sqrt{s}$ from this
measurement compared with results from 200 GeV PHENIX data at the same
rapidity range Adare  _et al._ (2007), and results from ALICE at different
$\sqrt{s}$ values and in the rapidity range, $2.5<y<4.0$ Aaij (2014). This
result follows the increasing pattern observed between PHENIX results at 200
GeV and ALICE results at 2.76–13 TeV.

Figure 9 shows $\langle p^{2}_{T}\rangle$ as a function of $\sqrt{s}$ from
this measurement compared with several other measurements Adare  _et al._
(2007); Acharya  _et al._ (2017); Abe  _et al._ (1992); Acosta  _et al._
(2002, 2005); Aaij (2014); Drapier (1998). Similar to $\langle p_{T}\rangle$,
$\langle p^{2}_{T}\rangle$ from this measurement follows the increasing
pattern versus $\sqrt{s}$ established by several sets of data over a wide
range of energies. Below $\sqrt{s}$ of 2 TeV, the trend is qualitatively
consistent with a linear fit of $\langle p^{2}_{T}\rangle$ versus the log of
the center of mass energy from Ref. Adare  _et al._ (2007). However, above
$\sqrt{s}$ of 2 TeV, the ALICE data indicate $\langle p^{2}_{T}\rangle$ grows
at an increased rate which is interpreted by authors of Ref. Acharya  _et al._
(2017) as due to the fact that ALICE data sets have different $p_{T}$ ranges.
The bottom cross section also increases with increasing $\sqrt{s}$, changing
the relative prompt and $B$-meson decay contributions to the inclusive
$J/\psi$ samples discussed here Abelev  _et al._ (2012b); Aidala  _et al._
(2017). This may also contribute to the observed differences in the measured
$\langle p^{2}_{T}\rangle$.

The $d\sigma^{J/\psi}_{pp}/dy$ measurement at $\sqrt{s}$ = 510 GeV offers an
opportunity to test the center-of-mass energy dependence of the
$p_{T}$-integrated cross section. Moreover, it bridges the gap between RHIC
data at 200 GeV and ALICE data starting at 2.76 TeV Aamodt  _et al._ (2011);
Abelev  _et al._ (2012a); Adam  _et al._ (2016); Acharya  _et al._ (2017).
However, ALICE data are collected at mid $(|y|<0.9)$ and forward $(2.5<y<4.0)$
rapidities and to have a proper comparison we interpolate the ALICE data to
the PHENIX forward rapidity range, $1.2<y<2.2$. This is done by fitting the
pythia generated $d\sigma/dy$ distribution at each energy to the data at the
same energy with only the normalization as a free parameter. An example is
shown in Fig. 11. We used several pythia Sj$\ddot{o}$strand  _et al._ (2006)
tunes including PHENIX default, tune-A, modified tune-A and atlas-csc
Sjöstrand and Skands (2005). After fitting each of these pythia tunes to the
data, we extracted $d\sigma/dy$ at $1.2<y<2.2$, from these fits. The rms value
of the extracted $d\sigma/dy$ from the different fits is used in the
comparison to RHIC data. The error on the rms value is the rms of the errors
associated with the fit results.

Figure 11 shows the results from this measurement, 200 GeV PHENIX data (closed
[blue] squares), ALICE data (open [green] circles), and interpolated ALICE
data (closed [red] circles) at several energies. Figure 11 shows that the data
are well described by a power law,
$d\sigma^{J/\psi}_{pp}/dy\propto(\sqrt{s})^{b}$, where the exponent is
$b=0.72\pm 0.03$.

Figure 10: Inclusive $J/\psi$ differential cross section as a function of
rapidity at $\sqrt{s}$ = 7 TeV Aamodt  _et al._ (2011) fitted with different
pythia tunes at the same energy.

Figure 11: Inclusive $J/\psi$ differential cross section, $d\sigma/dy$, as a
function of $\sqrt{s}$. The vertical errors correspond to the quadratic sum of
the statistical and type-B systematic uncertainties.

##  V Summary

We studied inclusive $J/\psi$ production in $p$+$p$ collisions at $\sqrt{s}$ =
510 GeV for $1.2<|y|<2.2$ and $0<p_{T}<10$ GeV/$c$, through the dimuon decay
channel. We measured inclusive $J/\psi$ differential cross sections as a
function of $p_{T}$ as well as a function of rapidity. The $p_{T}$ integrated
differential cross section multiplied by $J/\psi$ branching ratio to dimuons
is $BR$ $d\sigma^{J/\psi}_{pp}/dy$
$(1.2<|y|<2.2,0<\mbox{$p_{T}$}<10~{}\mbox{GeV/$c$})=$ 54.3 $\pm$ 0.5 (stat)
$\pm$ 5.5 (syst) nb. With these data measured over a wide $p_{T}$ range, we
calculated $\langle p_{T}\rangle$, $\langle p^{2}_{T}\rangle$ and
$d\sigma/dy$. The results were compared to similar quantities at different
energies from RHIC and LHC to study their $\sqrt{s}$ dependence. These new
measurements could put stringent constraints on $J/\psi$ production models.

The inclusive $J/\psi$ differential cross sections were compared to prompt
$J/\psi$ calculations. These calculations included LO-NRQCD+CGC at low $p_{T}$
and NLO-NRQCD for the rest of $p_{T}$ range. These model calculations
overestimated the data at low $p_{T}$ and underestimated the data at high
$p_{T}$. The nonprompt $J/\psi$ contribution was not included which could
account for the underestimation at high $p_{T}$ where the nonprompt processes
are significant.

In addition, we measured the ratio of the cross section of $\psi(2s)$ to
$J/\psi$, multiplied by their respective branching ratio to dimuons,
$R=2.84\pm 0.45$%. The result is consistent with world data within
uncertainties with no dependence on collision energy.

###### 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 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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