PHENIX Collaboration

# Production of $b\bar{b}$ at forward rapidity in $p$+$p$ collisions at
$\sqrt{s}=510$ GeV

U. 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  L.
Bichon  Vanderbilt University, Nashville, Tennessee 37235, USA  D. Black
University of California-Riverside, Riverside, California 92521, USA  B.
Blankenship  Vanderbilt University, Nashville, Tennessee 37235, USA  J.S. Bok
New Mexico State University, Las Cruces, New Mexico 88003, USA  V. Borisov
Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia
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  M. Csanád  ELTE, Eötvös Loránd
University, H-1117 Budapest, Pázmány P. s. 1/A, Hungary  T. Csörgő  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  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  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
K. DeBlasio  University of New Mexico, Albuquerque, New Mexico 87131, USA  K.
Dehmelt  Department of Physics and Astronomy, Stony Brook University, SUNY,
Stony Brook, New York 11794-3800, USA  A. Denisov  IHEP Protvino, State
Research Center of Russian Federation, Institute for High Energy Physics,
Protvino, 142281, Russia  A. Deshpande  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  E.J. Desmond  Physics Department, Brookhaven National Laboratory, Upton,
New York 11973-5000, USA  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. Firak  Debrecen University, H-4010 Debrecen,
Egyetem tér 1, Hungary  D. Fitzgerald  Department of Physics, University of
Michigan, Ann Arbor, Michigan 48109-1040, USA  S.L. Fokin  National Research
Center “Kurchatov Institute”, Moscow, 123098 Russia  J.E. Frantz  Department
of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA  A. Franz
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  A.D. Frawley  Florida State University, Tallahassee, Florida
32306, USA  C. Gal  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  P. Gallus  Czech
Technical University, Zikova 4, 166 36 Prague 6, Czech Republic  P. Garg
Department of Physics, Banaras Hindu University, Varanasi 221005, India
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  H. Ge  Department of Physics and Astronomy,
Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA  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, Korea  J. Hanks  Department of Physics and Astronomy,
Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA  S.
Hasegawa  Advanced Science Research Center, Japan Atomic Energy Agency, 2-4
Shirakata Shirane, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan  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  A. Hodges  Georgia State University, Atlanta, Georgia 30303, USA
R.S. Hollis  University of California-Riverside, Riverside, California 92521,
USA  K. Homma  Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526,
Japan  B. Hong  Korea University, Seoul 02841, Korea  T. Hoshino  Hiroshima
University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan  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  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, Korea  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, Korea  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  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  D. Larionova  Saint Petersburg
State Polytechnic University, St. Petersburg, 195251 Russia  M. Larionova
Saint Petersburg State Polytechnic University, St. Petersburg, 195251 Russia
A. Lebedev  Iowa State University, Ames, Iowa 50011, USA  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  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
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  S. Lökös  ELTE, Eötvös Loránd University, H-1117 Budapest,
Pázmány P. s. 1/A, 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  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, Korea  Yonsei University, IPAP, Seoul
120-749, Korea  D.P. Morrison  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  S.I. Morrow  Vanderbilt
University, Nashville, Tennessee 37235, USA  T.V. Moukhanova  National
Research Center “Kurchatov Institute”, Moscow, 123098 Russia  B. Mulilo  Korea
University, Seoul 02841, Korea  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  T. Murakami  Kyoto University, Kyoto
606-8502, Japan  RIKEN Nishina Center for Accelerator-Based Science, Wako,
Saitama 351-0198, Japan  J. Murata  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  Physics Department, Rikkyo University,
3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan  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  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. 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  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  Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, 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  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  Columbia University, New York, New
York 10027 and Nevis Laboratories, Irvington, New York 10533, 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  Department of Physics
and Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800,
USA  C. Pinkenburg  Physics Department, Brookhaven National Laboratory, Upton,
New York 11973-5000, USA  R. Pinson  Abilene Christian University, Abilene,
Texas 79699, 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  New
Mexico State University, Las Cruces, New Mexico 88003, USA  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  J. Rak  Helsinki Institute of
Physics and University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland
N. Ramasubramanian  Department of Physics and Astronomy, Stony Brook
University, SUNY, Stony Brook, New York 11794-3800, USA  I. Ravinovich
Weizmann Institute, Rehovot 76100, Israel  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  N. Riveli
Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
D. Roach  Vanderbilt University, Nashville, Tennessee 37235, 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.G.
Rubin  Department of Physics, University of Michigan, Ann Arbor, Michigan
48109-1040, USA  J. Runchey  Iowa State University, Ames, Iowa 50011, USA  N.
Saito  KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki
305-0801, Japan  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  S. Sawada  KEK, High Energy Accelerator Research
Organization, Tsukuba, Ibaraki 305-0801, 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  J. Seele  RIKEN BNL
Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000,
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  P. Sett  Bhabha Atomic Research Centre, Bombay 400 085, India  A. Sexton
University of Maryland, College Park, Maryland 20742, USA  D. Sharma
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony
Brook, New York 11794-3800, USA  I. Shein  IHEP Protvino, State Research
Center of Russian Federation, Institute for High Energy Physics, Protvino,
142281, Russia  T.-A. Shibata  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  Department of Physics, Tokyo Institute
of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan  K. Shigaki
Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan  M.
Shimomura  Iowa State University, Ames, Iowa 50011, USA  Nara Women’s
University, Kita-uoya Nishi-machi Nara 630-8506, Japan  P. Shukla  Bhabha
Atomic Research Centre, Bombay 400 085, India  A. Sickles  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  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  Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  B.K. Singh
Department of Physics, Banaras Hindu University, Varanasi 221005, India  C.P.
Singh  Department of Physics, Banaras Hindu University, Varanasi 221005, India
V. Singh  Department of Physics, Banaras Hindu University, Varanasi 221005,
India  M. Slunečka  Charles University, Ovocný trh 5, Praha 1, 116 36, Prague,
Czech Republic  K.L. Smith  Florida State University, Tallahassee, Florida
32306, USA  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  M. Stepanov
Deceased  Department of Physics, University of Massachusetts, Amherst,
Massachusetts 01003-9337, USA  S.P. Stoll  Physics Department, Brookhaven
National Laboratory, Upton, New York 11973-5000, USA  T. Sugitate  Hiroshima
University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan  A. Sukhanov
Physics Department, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  T. Sumita  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan  J. Sun  Department of Physics and
Astronomy, Stony Brook University, SUNY, Stony Brook, New York 11794-3800, USA
X. Sun  Georgia State University, Atlanta, Georgia 30303, USA  Z. Sun
Debrecen University, H-4010 Debrecen, Egyetem tér 1, Hungary  J. Sziklai
Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, Hungarian Academy of Sciences (Wigner RCP, RMKI) H-1525 Budapest 114,
POBox 49, Budapest, Hungary  A. Takahara  Center for Nuclear Study, Graduate
School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033,
Japan  A. Taketani  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. 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  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  H. Torii  Center for Nuclear Study, Graduate School of Science,
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  M. Towell
Abilene Christian University, Abilene, Texas 79699, USA  R. Towell  Abilene
Christian University, Abilene, Texas 79699, USA  R.S. Towell  Abilene
Christian University, Abilene, Texas 79699, USA  I. Tserruya  Weizmann
Institute, Rehovot 76100, Israel  Y. Ueda  Hiroshima University, Kagamiyama,
Higashi-Hiroshima 739-8526, Japan  B. Ujvari  Debrecen University, H-4010
Debrecen, Egyetem tér 1, Hungary  H.W. van Hecke  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  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  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  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  C.L. Woody  Physics Department, Brookhaven National
Laboratory, Upton, New York 11973-5000, USA  Y. Wu  University of California-
Riverside, Riverside, California 92521, USA  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  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  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  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 cross section of bottom quark-antiquark ($b\bar{b}$) production in $p$+$p$
collisions at $\sqrt{s}=510$ GeV is measured with the PHENIX detector at the
Relativistic Heavy Ion Collider. The results are based on the yield of high
mass, like-sign muon pairs measured within the PHENIX muon arm acceptance
($1.2<|y|<2.2$). The $b\bar{b}$ signal is extracted from like-sign dimuons by
utilizing the unique properties of neutral $B$ meson oscillation. We report a
differential cross section of
$d\sigma_{b\bar{b}\rightarrow\mu^{\pm}\mu^{\pm}}/dy=0.16\pm
0.01~{}(\mbox{stat})\pm 0.02~{}(\mbox{syst})\pm 0.02~{}(\mbox{global})$ nb for
like-sign muons in the rapidity and $p_{T}$ ranges $1.2<|y|<2.2$ and $p_{T}>1$
GeV/$c$, and dimuon mass of 5–10 GeV/$c^{2}$. The extrapolated total cross
section at this energy for $b\bar{b}$ production is $13.1\pm
0.6~{}(\mbox{stat})\pm 1.5~{}(\mbox{syst})\pm 2.7~{}(\mbox{global})~{}\mu$b.
The total cross section is compared to a perturbative quantum chromodynamics
calculation and is consistent within uncertainties. The azimuthal opening
angle between muon pairs from $b\bar{b}$ decays and their $p_{T}$
distributions are compared to distributions generated using ps pythia6, which
includes next-to-leading order processes. The azimuthal correlations and pair
$p_{T}$ distribution are not very well described by pythia calculations, but
are still consistent within uncertainties. Flavor creation and flavor
excitation subprocesses are favored over gluon splitting.

##  I Introduction

The bottom-quark production in hadron-hadron collisions is an important test
of perturbative quantum chromodynamics (pQCD) calculations. Because of its
large mass, $m_{b}\gg\Lambda_{QCD}$, the $b$-quark production cross section
can be reliably calculated by including next-to-leading order (NLO) processes,
especially at high center of mass energies Mangano  _et al._ (1992). The
measurement of the $b\bar{b}$ production cross section over a wide range of
colliding energies in hadron-hadron collisions provides a meaningful test of
pQCD theory calculations and a baseline measurement for studying modifications
of heavy quark production in heavy ion collisions.

Cross section measurements for bottom production in hadron-hadron collision
experiments have been made from lower energy fixed-target experiments
Alexopoulos  _et al._ (1999); Jansen  _et al._ (1995); Abt  _et al._ (2006)
($\sqrt{s}<45$ GeV) up to collider energies ($\sqrt{s}>100$ GeV). It was found
that pQCD predictions match experimental results well at energies greater than
$\sqrt{s}=1$ TeV Acosta  _et al._ (2005); Aaij  _et al._ (2010, ); Aad  _et
al._ (2012); Abelev  _et al._ (2013); Khachatryan  _et al._ (2011); Chatrchyan
_et al._ (2011, ), but less so at lower energies. Results at the wide range of
collision energies of the Relativistic Heavy Ion Collider explore an important
gap between the low-energy fixed-target and TeV-energy regimes.

Without displaced vertex $b$-tagging capability at PHENIX, $b$-quark
production has been studied using unlike-sign dileptons from heavy quark
decays Adare  _et al._ (2015). The PHENIX and STAR collaborations have
previously measured the bottom cross section in $p$+$p$ collisions at
$\sqrt{s}=200$ GeV using electron-hadron correlations Adare  _et al._ (2009a);
Aggarwal  _et al._ (2010) and using dilepton invariant mass and momentum
distributions Aidala  _et al._ (2017); Adare  _et al._ (2009b); Aidala  _et
al._ (2019).

Like-sign dimuons have previously been used to investigate the phenomenon of
neutral $B$ meson oscillations in $e^{+}e^{-}$ collisions by the CLEO
Collaboration Bean  _et al._ (1987), the ARGUS Collaboration Albrecht  _et
al._ (1987), the ALEPH Collaboration Buskulic  _et al._ (1994), and in
$p+\bar{p}$ collisions by the UA1 Collaboration Albajar  _et al._ (1987). In
this measurement, we use the yield of like-sign dimuons along with the
properties of neutral $B$ meson oscillation to determine the $b\bar{b}$ cross
section. The correlated like-sign pairs at high mass (5–10 GeV/$c^{2}$) are
dominated by the semileptonic decay of open bottom pairs and the other
correlated sources (i.e. dijets or punch-through hadrons) amount to less than
10%, and therefore provide a clean probe to study the $b\bar{b}$ production.

In the Standard Model, neutral $B$ meson oscillation is a result of higher
order weak interactions that transform a neutral $B$ meson into its
antiparticle: $B^{0}\rightarrow\bar{B}^{0}$ because the flavor eigenstates
differ from the physical mass eigenstates of the meson-antimeson system
Glashow (1961); Abe  _et al._ (1999). In the absence of oscillation as shown
in Fig. 1(a), primary-primary decays, where the lepton’s direct parent is the
$B$ meson, can only produce unlike-sign lepton pairs. For example
$b\rightarrow\bar{B}(B^{-},\bar{B}^{0},\bar{B}^{0}_{s},..)\rightarrow l^{-}$
and $\bar{b}\rightarrow B(B^{+},B^{0},B^{0}_{s},..)\rightarrow l^{+}$ while
like-sign lepton pairs can result from a mixture of primary and secondary
decays (decay chain).

Figure 1:  Example diagrams of lepton pair sources. (a) Like-sign primary-
secondary or unlike-sign primary-primary dileptons from $B$ decay chain. (b)
Primary-primary dileptons from neutral $B$ meson oscillation.

However, if oscillation occurs, as is the case for neutral $B$ mesons
($B^{0}_{d}$ and $B^{0}_{s}$), the $\bar{B}^{0}$ meson can spontaneously
change into a $B^{0}$ meson as shown in Fig. 1(b). Unless otherwise noted, we
denote $B$($\bar{B}$) as a generic admixture of bottom (antibottom) hadrons
with production ratios, from weak decays ($i.e.$ $Z\rightarrow b\bar{b}$) of:
$B^{+}(B^{-})=40.4\pm 0.9\%$, $B^{0}(\bar{B}^{0})=40.4\pm 0.9\%$,
$B^{0}_{s}(\bar{B}^{0}_{s})=10.3\pm 0.9\%$, and $b(\bar{b}$)-baryon $=8.9\pm
1.5\%$ Olive  _et al._ (2014). The $B_{c}$ production ratio is negligible
(0.2%) and less than the uncertainties associated with bottom hadrons listed
above. The time-integrated probability for a neutral $B$ meson to oscillate
before it decays is defined as

$\chi_{d/s}=\frac{1}{2}\frac{(\Delta m/\Gamma)^{2}}{1+(\Delta
m/\Gamma)^{2}}\;,$ (1)

where $\Delta m$ is the mass difference between heavy and light mass
eigenstates and $\Gamma$ is the decay rate of the weak eigenstates. These
values are found to be $\chi_{d}\approx 0.1874\pm 0.0018$ and $\chi_{s}\approx
0.499311\pm 0.000007$ for the $B^{0}_{d}$ and $B^{0}_{s}$ mesons, respectively
Olive  _et al._ (2014). This process can result in a like-sign dilepton event
from a primary-primary decay as sown in Fig. 1(b). Given the large branching
ratio of the $B\rightarrow\mu$ decay channel ($\approx 10.99\%$) Olive  _et
al._ (2014), the like-sign dilepton from a primary-primary decay provides a
unique opportunity for extracting the $b\bar{b}$ cross section.

In this paper, we present measurements of $b\bar{b}$ production cross section
through the like-sign dimuon decays and the azimuthal opening angle between
the muon pair and their $p_{T}$ distributions in $p$+$p$ collisions at
$\sqrt{s}=510$ GeV at forward ($1.2<\\!y<\\!2.2$) and backward
($-2.2\\!<y\\!<-1.2$) rapidities. The azimuthal opening angle and pair $p_{T}$
distributions are compared to distributions generated using pythia6 with
parton-shower (ps) model Sjostrand  _et al._ (2001). The model approximates
the correction to all higher orders (almost next-to-leading-log) for
$b\bar{b}$ production, which includes flavor creation, flavor excitation, and
gluon splitting. The extrapolated total cross section, using ps pythia6
Sjostrand  _et al._ (2001) and pythia8 Sj$\ddot{o}$bstrand  _et al._ (2015),
and MC$@$NLO Frixione and Webber  calculations, is also presented and compared
to pQCD calculation.

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
presented in Sec. III. The results are presented and discussed in Sec. IV. The
summary and conclusions are presented in Sec. V

##  II Experimental Setup

Figure 2:  A side view of the PHENIX detector, concentrating on the muon arm
instrumentation.

A complete description of the PHENIX detector can be found in Ref. Adcox  _et
al._ (2003). We briefly describe here only the detector subsystems used in
these measurements. The relevant systems, which are shown in Fig. 2, include
the PHENIX muon spectrometers covering forward and backward rapidities and the
full azimuth. 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 interactions
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 muon spectrometers.

Another detector system relevant to this analysis is the beam-beam counter
(BBC), consisting of 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 trigger (MB). The MB trigger requires at least one hit in
each BBC on the sides of the interaction point.

##  III Data Analysis

###  III.1 Data set and quality cuts

The data set for this analysis is collected by PHENIX during the 2013 $p$+$p$
run at $\sqrt{s}=510$ GeV. 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 two layers. After applying a vertex cut
of $|z_{\rm vtx}|<30$ cm and extensive quality assurance checks, the data
remaining correspond to $3.02\times 10^{12}$ MB events or to an integrated
luminosity of 94.4 pb-1.

A set of cuts was used to select good muon candidates and improve the signal-
to-background ratio. Hits in the MuTr are used to make MuTr tracks and hits in
the MuID are used to make MuID roads. The MuTr track is required to have more
than 9 hits out of the maximum possible of 16 while the MuID road is required
to have more than 6 hits out of the maximum possible of 10. Additional
$\chi^{2}$ cut is applied on MuTr track that is calculated from the difference
between the measured hit positions of the track and the subsequent fit. MuTr
tracks are then projected to the MuID at the first MuID gap and matched to
MuID roads by applying cuts on maximum position and angle differences.

Muon candidates are required to have a minimum $p_{T}$ greater than 1 GeV/$c$.
This cut improves the sample quality by reducing background from pions and
kaons. A minimum of 3.0 GeV/$c$ is applied to single muon momentum along the
beam axis, $p_{z}$, which is reconstructed and energy-loss corrected at the
collision vertex, corresponding to the momentum cut effectively imposed by the
absorbers. Muon candidates are further restricted to the rapidity range of
$-2.2<y<-1.2$ for the south muon arm and $1.2<y<2.2$ for the north muon arm.
Additionally, 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 is
applied.

###  III.2 Detector acceptance and reconstruction efficiency

The product of the acceptance and reconstruction efficiency ($A\epsilon$) is
determined using Monte Carlo (MC) simulation. The $A\epsilon$ is defined by
the number of dimuons reconstructed in the muon spectrometers with respect to
the number of dimuons generated in the same kinematic region. The kinematic
distributions of pythia111We used pythia6 (ver 6.421), 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. Field and Group
generated $p_{T}$, rapidity, and $b\bar{b}$ mass shape were used as input into
a full PHENIX geant4 simulation Agostinelli  _et al._ (2003).

The $p_{T}$ and rapidity distributions were tuned such that the reconstructed
distributions match those of 2013 data. Variations within the uncertainties of
data are taken as systematic uncertainty.

The detector response in the simulation is tuned to a set of characteristics
(dead and hot channel maps, gains, noise, etc.) that describes the performance
of each detector subsystem. The simulated vertex distribution is also tuned to
match that of the 2013 data. The simulated events are further embedded with
real data to account for the effects of detector noise and other background
tracks, and then are reconstructed in the same manner as the real data. A
final cross check was done on $J/\psi$ invariant yield after $A\epsilon$
correction, which matched very well within statistical uncertainties in all
$p_{T}$ and rapidity bins Acharya  _et al._ (2020).

Figure 3 shows the $A\epsilon$ as a function of (a) dimuon mass $m_{\mu\mu}$,
(b) dimuon opening angle $\delta\Phi$, and (c) dimuon $p_{T}$. The relative
difference in $A\epsilon$ between the two spectrometers is due to different
detection efficiencies of the MuTr and MuID systems and different amounts of
absorber material.

Figure 3:  $A\epsilon$ as a function of (a) invariant mass for like-sign
dimuons, (b) dimuon azimuthal opening angle, and (c) dimuon $p_{T}$. Shown are
the weighted averages of $\mu^{+}\mu^{+}$ and $\mu^{-}\mu^{-}$ distributions.

###  III.3 Raw yield extraction

We measure like-sign dimuons in the same muon arm that have an invariant mass
between 5 and 10 GeV/$c^{2}$. In this mass range, the correlated pairs in the
dimuon spectrum are dominated by the semileptonic decay of open bottom pairs
either from the primary-secondary decay chain as shown in Fig. 1(a) or from
the primary-primary pairs from neutral $B$ meson oscillation as shown in Fig.
1(b). Dileptons from the Drell-Yan process and quarkonia decays can only yield
unlike-sign pairs. $D$ mesons can produce like-sign pairs through their decay
chain. For example, $c\rightarrow D^{+}\rightarrow\mu^{+}+anything$ and the
other open charm decays as $\bar{c}\rightarrow D^{-}\rightarrow
K^{+}+anything\rightarrow\mu^{+}\nu_{\mu}$. However, in the mass range of
interest the like-sign pairs from $D$ mesons are negligible. The contribution
from neutral $D$ meson oscillation to the like-sign signal is expected to be
very small because the oscillation probability is $\mathcal{O}(<10^{-2})$ Aaij
_et al._ (2013); therefore, it is not included.

####  III.3.1 Correlated background

Additional contribution to the correlated pairs could originate from
correlated sources such as dijets or punch-through hadrons. Hadrons
(particularly $\pi^{\pm}$ and $K^{\pm}$) can punch through to the last gap of
the MuID or decay to muons creating a background to the correlated like-sign
signal. These contributions are estimated using MC simulation by determining
the $p_{T}$-dependent survival probability that a hadron will traverse the
muon arm detectors and applying it to pythia generated dihadron pairs to get
the yield expected at the back of the muon arm detectors. $\pi^{\pm}$ and
$K^{\pm}$ are generated with pythia222Non-default parameters used in
Multiparton Interaction (MPI)“Tune-A” ps pythia6 simulation for hadron and jet
production. The following parameters were modified:MSEL = 1, PMAS(5,1) = 4.1,
PYTUNE 100, and PARP(90) = 0.25 Field and Group ; Aidala  _et al._ (2019);
Adare  _et al._ (2011a) and then run through the PHENIX detector simulation
chain to determine a $p_{T}$-dependent probability that the hadrons penetrate
the last gap of the MuID.

To get a better estimate of the survival probability, the hadron simulation is
run using two different hadron interaction packages for geant: fluka and
geisha Brun  _et al._ ; Adare  _et al._ (2012). Figure 4 shows the simulated
invariant mass spectra from irreducible background are fitted with an
exponential function of the form exp($a+b\times m+c\times m^{2}$) between 5
and 10 GeV/$c^{2}$, where $m$ is the invariant mass and $a$, $b$ and $c$ are
fit parameters. The average of the indicated results from geisha and fluka is
used to subtract the hadronic background from like-sign pairs while the
difference is considered as a systematic uncertainty.

Figure 4:  Like-sign invariant mass distribution from jet background
simulation in the north and south arms. The solid lines are fits to the data
with an exponential function between 5 and 10 GeV/$c^{2}$ while the dashed
lines represent the averages of the resulting fits.

The invariant mass distribution for like-sign pairs is then constructed from
pythia generated dihadron pairs within the same event and from mixed events,
with each entry weighted by the survival probability. Event-mixing procedure
is discussed in the next section. Just as with data, the correlated like-sign
signal is obtained by subtracting the mixed event spectrum from the like-sign
spectrum, providing the correlated like-sign signal due to dijets or punch-
through hadrons. The sum of $\pi$ and $K$ correlated like-sign signals is
weighted based on their $p_{T}$-dependent cross sections Adare  _et al._
(2018); Albajar  _et al._ (1990).

Fake like-sign pairs due to charge misidentification and like-sign pairs from
Drell-Yan process or quarkonia decays and muon-decayed or punch-through
hadrons were also studied and found to be negligible.

####  III.3.2 Uncorrelated background

The uncorrelated pair contribution is estimated using event mixing technique
Crochet and Braun-Munzinger (2002), where like-sign pairs are constructed by
pairing muons in the current event with those of the same sign and same arm in
previous events of z-vertex position within 2 cm. The mixed event pairs
($N_{++}^{BG}$ and $N_{--}^{BG}$) form the uncorrelated background spectrum
which is normalized to the the foreground ($N_{++}^{FG}$ and $N_{--}^{FG}$)
using a normalization factor
($\sqrt{N_{++}^{FG}N_{--}^{FG}}/\sqrt{N_{++}^{BG}N_{--}^{BG}}\;$). The
normalization factor requires that the integrated counts from event mixing
equal those from the like-sign in the low mass region where the correlated
pairs are expected to be negligible Crochet and Braun-Munzinger (2002). The
normalized like-sign pairs from event mixing are given as:

$N_{\pm\pm}^{BG}=\left(N_{++}^{BG}+N_{--}^{BG}\right)\frac{\sqrt{N_{++}^{FG}N_{--}^{FG}}}{\sqrt{N_{++}^{BG}N_{--}^{BG}}}.$
(2)

However, the specific range where the signal of interest is negligible is not
well known, and the average of normalization factors over five mass ranges
(0.6–2.6 GeV/$c^{2}$, 1.0–2.0 GeV/$c^{2}$, 1.6–3.2 GeV/$c^{2}$, 2.6–3.8
GeV/$c^{2}$, and 0.6–4.2 GeV/$c^{2}$) is used. The correlated like-sign signal
($N_{\pm\pm}^{\rm cor}$) is then isolated by subtracting the mixed-event
spectrum ($N_{\pm\pm}^{BG}$) from the “foreground” like-sign pairs
($N_{\pm\pm}^{FG}$) according to the following,

$N_{\pm\pm}^{\rm cor}=N_{\pm\pm}^{FG}-N_{\pm\pm}^{BG}.$ (3)

To further improve the normalization process, the $b\bar{b}$ invariant mass
distribution shape from ps pythia6 simulation is utilized. This is done by
normalizing the integral of the ps pythia6 distribution to the result of Eq.
(3), over the signal mass range 5–10 GeV/$c^{2}$. The integral of the
normalized $b\bar{b}$ mass distribution is then subtracted from the background
distribution in Eq. (2) for each of the background ranges and the
normalization factor is recalculated. The second step is then repeated until
the value of the mixed-events normalization factor converges.

Figure 5:  Invariant mass spectra for like-sign pairs from the same event
($N_{\pm\pm}^{FG}$, solid black points), event-mixing ($N_{\pm\pm}^{BG}$, red
band), and the difference between the two ($N_{\pm\pm}^{\rm cor}$, empty blue
pluses) for the (a) north arm and (b) south arm. These distributions are
corrected with $A\epsilon$. The solid green triangles show pythia $b\bar{b}$
shape.

Figure 5 shows the resulting distributions of $N_{\pm\pm}^{FG}$,
$N_{\pm\pm}^{BG}$ and $N_{\pm\pm}^{\rm cor}$ as a function of the invariant
mass of the pairs. These distributions are corrected with $A\epsilon$. To
extract the $b\bar{b}$ distribution as a function of the azimuthal opening
angle between muon pairs ($\Delta\phi$) and their $p_{T}$, the normalization
factors obtained previously are used to normalize $\Delta\phi$ and $p_{T}$
mixed event distributions, which are then subtracted from $\Delta\phi$ and
$p_{T}$ foreground distributions, respectively.

###  III.4 Systematic uncertainties

Table 1 summarizes the systematic uncertainties. Evaluated as standard
deviations, they are divided into three categories based upon the effect each
source has on the measured results:

Type-A

Point-to-point uncorrelated uncertainties that 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. The systematic
uncertainty associated with $A\epsilon$ includes the uncertainties on the
input $p_{T}$ and rapidity distributions which are 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. To be consistent with the real
data analysis, a trigger emulator was used to match the MuIDLL1-2D trigger for
the data. The efficiency of the trigger emulator was studied by comparing the
dimuon mass spectrum requiring the dimuon passes the trigger emulator to the
dimuon mass spectrum requiring the dimuon passes the MuIDLL1-2D trigger, which
resulted in a 1.5% (2%) uncertainty 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.

The source of systematic uncertainty in signal extraction is the normalization
of mixed events which could come from the choice of the different
normalization ranges in the mixed events or $b\bar{b}$ shape from pythia used
to guide the signal extraction. A 1.9% uncertainty on the mixed events
normalization was observed from using each of the five normalization windows
by itself as well as the different combinations of these normalization
windows. pythia $b\bar{b}$ shape is the sum of three subprocesses: flavor
creation, flavor excitation and gluon splitting. A maximum variation of 3.1%
on the extracted signal was observed from choosing each of the subprocesses by
itself as the source of $b\bar{b}$ shape. Added in quadrature, they result in
a 3.6% uncertainty on signal extraction.

The systematic uncertainty associated with correlated backgrounds could come
from the input $p_{T}$ distribution, differences between geisha and fluka, and
differences between geant3 and geant4. pythia $p_{T}$ distributions of
$\pi^{\pm}$ and $K^{\pm}$ were compared separately to fits of UA1 data Adare
_et al._ (2018); Albajar  _et al._ (1990) and an overall difference of 18% was
observed. Differences of up to 30% and 20% between fluka and geisha, see Fig.
4, were observed in the north and south arms, respectively. Additional 15% was
considered to account for the difference between geant3 and geant4. Added in
quadrature, all three sources give an overall effect on the hadronic
background of 39% (31%) for the north (south) arm for the mass and
$\Delta\phi$ distributions. For $p_{T}$ distribution, a $p_{T}$-dependent
correction was used for the effect on the input $p_{T}$ spectra and the other
two sources gave an overall effect on the hadronic background of 34% (25%) for
the north (south) arm. To extract the systematic uncertainty associated with
the cross section (or invariant yields) for all distributions (mass,
$\Delta\phi$ and $p_{T}$), the hadronic background was varied between the
limits listed above which resulted in an overall systematic of 5.1% (4.5%) for
north (south) arm.

The Type-B systematic uncertainties are added in quadrature and amount to
16.0% (12.8%) for the north (south) arm. They are shown as shaded bands on the
associated data points.

Type-C

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

Table 1:  Systematic uncertainties associated with the 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 |  $A\epsilon$ $p_{T}$ and $y$ input distributions | 4.4% (5.0%)
B |  $A\epsilon$ trigger emulator | 1.5% (2.0%)
B |  $A\epsilon$ $\phi$ distribution | 11.2% (8.8%)
B | Signal extraction | 3.6% (3.6%)
B | Correlated background | 5.1% (4.5%)
B | Quadratic sum | 16.4% (12.8)%
C | MB trigger efficiency | 10%

##  IV Results and Discussion

###  IV.1 Differential cross section

The differential yield and cross section of $B$ meson pairs decaying into
like-sign dimuons as a function of mass are calculated according to the
following relations,

$\frac{d^{2}N}{dydm}=\frac{1}{\Delta y\Delta
m}\frac{N_{\mu\mu}}{A\epsilon(y,m)}\frac{\epsilon_{\rm BBC}^{\rm
MB}}{\epsilon^{\rm BBC}N_{\rm MB}}\;,$ (4)
$\frac{d^{2}\sigma}{dydm}=\frac{d^{2}N}{dydm}\frac{\sigma_{\rm
BBC}^{pp}}{\epsilon_{\rm BBC}^{\rm MB}}\;,$ (5)

where $N_{\mu\mu}/A\epsilon(y,m)$ is the yield of like-sign dimuons from $B$
meson decay normalized by $A\epsilon(y,m)$ in $y$ and $m$ bin with $\Delta y$
and $\Delta m$ widths, respectively. The yields of the north and south arms
are calculated independently and are consistent within statistical
uncertainties; therefore, the weighted average Nisius (2014) is used in the
differential yield calculation. $\sigma_{\rm BBC}^{pp}=32.5\pm 3.2$ mb is the
cross section as seen for the BBC in $p$+$p$ collisions at $\sqrt{s}=510$ GeV,
which is determined from the van der Meer scan technique Adare  _et al._
(2011b). $\epsilon_{\rm BBC}^{\rm MB}=0.53\pm 0.02$ is the fraction of
inelastic $p$+$p$ collisions recorded by the BBC Adare  _et al._ (2009c).
$\epsilon^{\rm BBC}=0.91\pm 0.04$ is the efficiency of the MB trigger for
events containing a hard scattering Acharya  _et al._ (2020). $N_{\rm MB}$ is
the number of MB events.

The differential cross section of like-sign dimuons from $B$ meson decay is
shown in Fig. 6. 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. In addition to type-B systematic uncertainties, we have a 10%
global systematic uncertainty for BBC cross section and efficiencies Adler
_et al._ (2003).

Figure 6:  Differential cross section of like-sign dimuons from $B$ meson
decay. The error bars represent the statistical uncertainties, and the gray
shaded bands represent the quadratic sum of type-B systematic uncertainties.

The total cross section, $d\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}}/dy$, within the mass range,
$5<m_{\mu^{\pm}\mu^{\pm}}<10$ GeV/$c^{2}$, and rapidity and $p_{T}$ ranges,
$1.2<|y|<2.2$ and $p_{T}$ $>$ 1 GeV/$c$, respectively, is extracted by
integrating $d^{2}\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}}/dydm$, which resulted
$d\sigma_{b\bar{b}\rightarrow\mu^{\pm}\mu^{\pm}}/dy~{}(1.2<|y|<2.2,~{}p_{T}>1~{}\mbox{GeV}/c,~{}5<m_{\mu^{\pm}\mu^{\pm}}<10~{}\mbox{GeV}/c^{2})=0.16\pm
0.01~{}\mbox{(stat)}\pm 0.02~{}\mbox{(type-B syst)}\pm 0.02~{}(\mbox{global
syst})$ nb.

To obtain the differential cross section of all $B$ meson pairs that decay
into dimuons, regardless of the muon pair charge, the differential cross
section of like-sign dimuons from $B$ meson decay is scaled by the ratio of
the total number of all $B$ meson pairs that decay into dimuons, regardless of
their sign, to those of like-sign. For clarification purposes, the process is
divided into two separate steps defined by two variables $\alpha(m)$ and
$\beta$, both of which depend on the signal from like-sign dimuons due to
oscillation.

The ratio of like-sign dimuons at mass $m$ and from primary-primary decays due
to $B^{0}$ oscillation to like-sign muon pairs resulting from primary-primary
or a mixture of primary-secondary decays is defined as:

$\alpha(m)=\frac{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}\mbox{ (osc)}}{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}},$ (6)

which is calculated in the mass range $5<m<10~{}\mbox{GeV}/c^{2}$ at
$1.2<|y|<2.2$ and $p_{T}$ $>$ 1 GeV/$c$ and extrapolates the correlated like-
sign signal to an open bottom signal from oscillation, $N_{\pm\pm}^{osc}$. The
$\alpha(m)$ is obtained using open bottom events from three model
calculations: mc@nlo (ver 4.10), ps pythia6 (ver 6.421) and pythia8 (ver
8.205) as shown in Fig. 8. The red line is a second-order polynomial fit with
$\chi^{2}/ndf$ of 3.8/4. The shaded boxes represent the uncertainty based on
the three model calculations.

Figure 7:  Fraction of like-sign dimuons from neutral $B$ meson oscillation
($\alpha(m)$) from mc@nlo (blue points) , ps pythia6 (magenta points) and
pythia8 (green points) within the PHENIX muon-arms acceptance. Cyan data
points are the RMS average of the three model calculations. The shaded boxes
are the associated errors based on the three model calculations. The red curve
is a second-order polynomial fit to the RMS data points.

Figure 8:  Differential cross section of all dimuons from $B$ meson decay. The
error bars represent the statistical uncertainties, and the gray shaded band
represents the quadratic sum of type-B systematic uncertainties.

Figure 9:  (a) Bottom cross section, $\sigma_{b\bar{b}}$ as a function of
$\sqrt{s}$. The curves are NLO pQCD calculation Vogt (2008) with the dashed
lines being error bands obtained by varying the renormalization scale,
factorization scale and bottom quark mass. (b) Ratio of data to NLO pQCD
calculation.

Figure 10:  Like-sign $\mu\mu$ yield as a function of the azimuthal opening
angle. The data are compared to the distributions calculated with ps pythia6.
The model calculations are normalized to the data. For ps pythia6 the $\mu\mu$
pair yield is broken down into contributions from flavor creation, flavor
excitation, and gluon splitting.

Figure 11:  Like-sign $\mu\mu$ yield as a function of the pair $p_{T}$. The
data are compared to the distributions calculated with ps pythia6. The model
calculations are normalized to the data. For ps pythia6 the $\mu\mu$ pair
yield is broken down into contributions from flavor creation, flavor
excitation, and gluon splitting.

$\beta$ is the ratio of primary-primary like-sign dimuons due to $B^{0}$
oscillation to all $B$ meson pairs that decay into primary-primary dimuons
with all possible muon charge pairs ($++,--$ and $+-$). $\beta$ converts the
number of muon pairs from oscillation into all $B$ meson pairs and is defined
as:

$\beta=\frac{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}\mbox{
(osc)}}{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu\mu}.$ (7)

The value of $\beta$ is $0.22\pm 0.01$ which is the calculated RMS value from
the three model simulations described above. The error of $\beta$ is the
standard deviation of the three model calculations which represents the model-
dependent uncertainty.

The differential cross section of all $B$ meson pairs that decay into a
primary-primary dimuon, regardless of the muon pair charge, is then calculated
as follows:

$\frac{d^{2}\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu\mu}}{dydm}=\frac{\alpha(m)}{\beta}\frac{d^{2}\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu^{\pm}\mu^{\pm}}}{dydm}.$ (8)

Figure 8 shows the differential cross section of all $B$ meson pairs that
decay into a primary-primary dimuon. Additional type-B systematic
uncertainties associated with this measurement due to $\alpha(m)$ and $\beta$
and amount to 1.9% and 4.5%, respectively, are included. This brings the
type-B systematic uncertainties on $d^{2}\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu\mu}/dydm$ to 11.2%.

The total cross section, $d\sigma_{b\bar{b}\rightarrow
B\bar{B}\rightarrow\mu\mu}/dy$, within the mass range, $5.0<m_{\mu\mu}<10.$
GeV/$c^{2}$, and rapidity and $p_{T}$ ranges, $1.2<|y|<2.2$ and $p_{T}$ $>$ 1
GeV/$c$, respectively, is extracted by integrating
$d^{2}\sigma_{b\bar{b}\rightarrow B\bar{B}\rightarrow\mu\mu}/dydm$, which
resulted
$d\sigma_{b\bar{b}\rightarrow\mu\mu}/dy~{}(1.2<|y|<2.2,~{}p_{T}>1~{}\mbox{GeV}/c,~{}5<m_{\mu\mu}<10~{}\mbox{GeV}/c^{2})=0.31\pm
0.01~{}\mbox{(stat)}\pm 0.04~{}\mbox{(type-B syst)}\pm 0.03~{}(\mbox{global
syst})$ nb.

###  IV.2 Total cross section

To extrapolate from the $b\bar{b}$ differential cross section in the muon
decay channel within the acceptance of muon arms to a total $b\bar{b}$ cross
section, the differential cross section is scaled by the ratio of $B$ pairs
that decay to dimuons within the measured region to those over the entire
kinematic range. This method is similar to that found in Ref. S.  _et al._ .

The total cross section, $\sigma_{b\bar{b}}$, is extrapolated and corrected
for the semileptonic branching ratio in the following manner:

$\sigma_{b\bar{b}}=\frac{d\sigma_{b\bar{b}\rightarrow\mu\mu}}{dy}\times\frac{1}{scale}\times\frac{1}{(BR_{B\rightarrow\mu})^{2}}\;,$
(9)

where $BR_{B\rightarrow\mu}$ is the branching ratio of $B$ to muon through the
primary decay channel (=10.99$\%$), and $scale$, defined as:

$scale=\frac{B\bar{B}\rightarrow\mu\mu(1.2<y<2.2;p_{T}>1;5<m_{\mu\mu}<10)}{B\bar{B}\rightarrow\mu\mu(all)},$
(10)

which is a factor used to convert from the visible kinematic region to full
phase space. The $scale$ factor is determined from pythia and mc@nlo
simulations. It is taken as the average value, $1.96\times 10^{-3}$, of ps
pythia6 (CTEQ6LL), ps pythia6 (CTEQ5M1), pythia8 (CTEQ6LL) and mc@nlo (CTEQ5M)
as listed in Table 2.

Table 2:  Values of the scale factor as found using ps pythia6 Sjostrand  _et al._ (2001), pythia8 Sj$\ddot{o}$bstrand  _et al._ (2015), and MC$@$NLO Frixione and Webber .  Simulation | Scale Factor
---|---
pythia8 (CTEQ6LL) | 0.00210
ps pythia6 (CTEQ6LL) | 0.00207
ps pythia6 (CTEQ5M1) | 0.00255
mc@nlo (CTEQ5M) | 0.00113
Average Value | 0.00196

The difference in the $scale$ factor due to the different models and parton
distribution functions is considered to be a global type-C uncertainty, which
amounts to 18.1%. This results in a total cross section of $13.1\pm
0.6~{}(\mbox{stat})\pm 1.5~{}(\mbox{type-B syst})\pm 2.7~{}(\mbox{global
syst})~{}\mu$b. Type-B systematic uncertainties are from the differential
cross section while global uncertainties are the quadrature sum of type-C from
the differential cross section and uncertainties arising from the
extrapolation.

The $\sigma_{b\bar{b}}$ measured at $\sqrt{s}=510$ GeV is shown in Fig. 9 and
compared to measurements from other experiments Adare  _et al._ (2009b);
Alexopoulos  _et al._ (1999); Jansen  _et al._ (1995); Abt  _et al._ (2006);
Albajar  _et al._ (1991); Abelev  _et al._ (2013). The solid line is the cross
section from NLO pQCD calculations Vogt (2008) and the dashed lines are error
bands, and they are obtained by varying the renormalization scale,
factorization scale and bottom quark mass. At $\sqrt{s}=510$ GeV, the NLO pQCD
calculation predicts $\sigma_{b\bar{b}}=11.5^{+6.5}_{-3.9}~{}\mu$b, which is
consistent with the extrapolated total cross section using the current dimuon
analysis within uncertainties. Figure 9 also shows the ratio of data to
theory.

###  IV.3 Azimuthal correlations and pair $p_{T}$

The like-sign $\mu\mu$ pair yield from $b\bar{b}$ decays is shown in Fig. 11
and Fig. 11 as a function of $\Delta\phi$ and pair $p_{T}$, respectively. The
spectra are compared to model calculations based on ps pythia6 that are
normalized by fitting the subprocesses sum to the data Aidala  _et al._
(2019). The generated pairs are filtered with the same kinematic cuts that are
applied in the data analysis.

The azimuthal opening angle distribution from ps pythia6 shows a similar
pattern to that of the data, an increase until $\approx$2.6 rad and then drop,
and it is consistent with the data with $\chi^{2}/ndf{\approx}27/28$, when
considering the quadrature sum of the statistical and systematic
uncertainties. The data show steeper $p_{T}$ dependence than that of ps
pythia6 but they are still consistent when considering the large statistical
and systematic uncertainties. We note that flavor creation fits the data much
better than any other subprocess with $\chi^{2}/ndf{\approx}8.4/7$. These
results show similar behavior to that observed at 200 GeV Aidala  _et al._
(2019) where the data favors a dominant mix of flavor creation and flavor
excitation subprocesses over gluon splitting.

##  V Summary and Conclusion

In summary, we presented first measurements of the differential cross section
for dimuons from bottom quark-antiquark production in $p$+$p$ collisions at
$\sqrt{s}=510$ GeV, which we found to be:
$d\sigma_{b\bar{b}\rightarrow\mu^{\pm}\mu^{\pm}}/dy=0.16\pm
0.01~{}(\mbox{stat})\pm 0.02~{}(\mbox{syst})\pm 0.02~{}(\mbox{global})$ nb.
The analysis technique is based on the yield of high-mass correlated like-sign
dimuons from parity-violating decays of $B$ meson pairs at forward and
backward rapidities. Using a model dependent extrapolation, the measured
differential cross section is converted to a total cross section of $13.1\pm
0.6~{}(\mbox{stat})\pm 1.5~{}(\mbox{syst})\pm 2.7~{}(\mbox{global})~{}\mu$b.
This extrapolated total cross section is consistent with NLO pQCD calculations
within uncertainties. This agreement with NLO pQCD calculations at
$\sqrt{s}=510$ GeV is better than what was observed at 200 GeV Aidala  _et
al._ (2019), possibly indicating a better match with NLO pQCD calculations at
higher energies. However, the measurement at $\sqrt{s}=200$ GeV used the
unlike-sign pairs method and could be impacted by the presence of Drell-Yan
process and resonances.

The azimuthal opening angle between the muons from $b\bar{b}$ decays and the
pair $p_{T}$ distributions are compared to distributions generated using ps
pythia6 Sjostrand  _et al._ (2001), which includes NLO processes. While the
data tend to have a wider azimuthal distribution than ps pythia6 calculations
and present a steeper $p_{T}$ distribution, both are still consistent within
uncertainties with ps pythia6, where flavor creation and flavor excitation
subprocesses are dominant. This is similar to what was observed at 200 GeV
Aidala  _et al._ (2019).

###### 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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x+h\rvert^{2}+(\gamma-1)\lvert x+h\rvert^{2}}{\lvert x+h\rvert}\leq
y-\frac{\lvert x\rvert^{2}+(\gamma-1)m^{2}}{m}-\frac{2xh}{m}.$ (7.4)

This is equivalent to

$\frac{\lvert h\rvert^{2}-\gamma\lvert x+h\rvert^{2}}{\lvert
x+h\rvert}\leq\frac{-\lvert x\rvert^{2}-(\gamma-1)m^{2}}{m}-\frac{2xh}{m}.$
(7.5)

The inequality (7.5) is equivalent to

$\frac{\lvert h\rvert^{2}m}{m^{2}\lvert x+h\rvert}-\gamma\frac{\lvert
x+h\rvert}{m}\leq\frac{-\lvert x+h\rvert^{2}+\lvert
h\rvert^{2}}{m^{2}}-(\gamma-1).$

Let $t:=\lvert x+h\rvert/m>1$ and $\tilde{t}:=\lvert h\rvert/m$. Note that
$\lvert t-\tilde{t}\rvert=\lvert\lvert x+h\rvert-\lvert
h\rvert\rvert/m\leq\lvert x\rvert/m\leq 1$. With this notation, (7.5) is
equivalent to

$\tilde{t}^{2}/t-\gamma t\leq-t^{2}+\tilde{t}^{2}-(\gamma-1),$

or

$\gamma\geq\frac{1}{t-1}\bigl{(}t^{2}-1-\tilde{t}^{2}(1-1/t)\bigr{)}=(t+1)-\tilde{t}^{2}/t.$

Hence, it suffices to ensure

$\gamma\geq\sup_{t>1,\lvert t-\tilde{t}\rvert\leq 1}(t+1)-\tilde{t}^{2}/t.$

The supremum in $\tilde{t}$ is assumed for $\tilde{t}=(t-1)$, so this
condition becomes

$\gamma\geq\sup_{t>1}t+1-(t-1)^{2}/t=\sup_{t>1}3-1/t=3.$

∎

###  7.2 Sharp constant in the Davis inequality for the square function

######  Proposition 7.3.

Let $f$ be a real-valued martingale. Then

$\mathbb{E}Sf\leq\sqrt{3}\mathbb{E}f^{*}.$

###### Proof.

By Hölder’s inequality and (7.1), we obtain

$\displaystyle\mathbb{E}Sf$
$\displaystyle\leq\mathbb{E}\bigl{(}(f^{*})^{1/2}\bigl{(}\sum_{n=1}^{N}\frac{\lvert
df_{n}\rvert^{2}}{f^{*}_{n}}\bigr{)}^{1/2}\bigr{)}$
$\displaystyle\leq\bigl{(}\mathbb{E}f^{*}\bigr{)}^{1/2}\bigl{(}\mathbb{E}\bigl{(}\sum_{n=1}^{N}\frac{\lvert
df_{n}\rvert^{2}}{f^{*}_{n}}\bigr{)}\bigr{)}^{1/2}$
$\displaystyle\leq\sqrt{3}\mathbb{E}f^{*}.\qed$

The next result shows that $\sqrt{3}$ is the best possible constant in
Proposition 7.3.

######  Proposition 7.4.

Let $\gamma\geq 0$. Suppose that the inequality

$\mathbb{E}Sf\leq\gamma\mathbb{E}f^{*}$ (7.6)

holds for all real-valued simple martingales $f$ with $f_{0}=0$. Then
$\gamma\geq\sqrt{3}$.

###### Proof.

Let $V:\mathcal{D}\to\mathbb{R}$ be given by

$V(x,t,z):=\sqrt{t}-\gamma z.$

Define $U:\mathcal{D}\to(-\infty,\infty]$ by

$U(x,t,z):=\sup\Big{\\{}\mathbb{E}V(f_{\infty},t+S^{2}(f),f^{*}\vee
z)\nonscript\>\Big{|}\allowbreak\nonscript\>\mathopen{}f_{0}=x\Big{\\}},$
(7.7)

where the supremum is taken over all simple martingales (that is, martingales
$f$ that take only finitely many values). This function will play the role of
the “best” Bellman function for the inequality (7.6).

Substituting the constant martingale into the definition (7.7), we see that,
for any $(x,t,z)\in\cal D$, we have

$V(x,t,z)\leq U(x,t,z),$ (7.8)

We claim that, for any simple measurable function $d:\Omega\to\mathbb{R}$ with
$\mathbb{E}d=0$, we have

$\mathbb{E}U(x+d,t+\lvert d\rvert^{2},\lvert x+d\rvert\vee z)\leq U(x,t,z).$
(7.9)

###### Proof of (7.9).

Fix $x\in\mathbb{R}$ with $\lvert x\rvert\leq z$. Let $d:\Omega\to\mathbb{R}$
be a simple function with $\mathbb{E}d=0$ and
$\mathbb{P}(d=s_{j})=p_{j}\in(0,1]$ for $1\leq j\leq m$, where
$\sum_{j=1}^{m}p_{j}=1$. Choose $b_{j}\in\mathbb{R}$ so that

$U\left(x+s_{j},t+\lvert s_{j}\rvert^{2},\lvert x+s_{j}\rvert\vee
z\right)>b_{j}.$

Then, by the definition of $U$, there exists a martingale $f^{j}$ with
$f^{j}_{0}=x+s_{j}$ satisfying

$\mathbb{E}V\bigl{(}f_{\infty}^{j},t+\lvert
s_{j}\rvert^{2}+S^{2}\left(f^{j}\right),(f^{j})^{*}\vee z\bigr{)}>b_{j}.$

Let $f$ be a martingale with $f_{1}=x+d$ which continues with the same
distribution as $f^{j}$ (rescaled by $p_{j}$ in measure) on the set
$\\{d=s_{j}\\}$. Because $\lvert x\rvert\leq z$, we have $f^{*}\vee
z=(f^{j})^{*}\vee z$ on $\\{d=s_{j}\\}$. By (7.7), we have

$\displaystyle U(x,t,z)$ $\displaystyle\geq
EV\left(f_{\infty},t+S^{2}(f),f^{*}\vee z\right)$
$\displaystyle=\sum_{j=1}^{m}\int_{\\{d=s_{j}\\}}V\left(f_{\infty},t+\lvert
s_{j}\rvert^{2}+\lvert df_{2}\rvert^{2}+\cdots,f^{*}\vee
z\right)\mathop{}\\!\mathrm{d}\mathbb{P}$
$\displaystyle=\sum_{j=1}^{m}p_{j}\mathbb{E}V\left(f_{\infty}^{j},t+\lvert
s_{j}\rvert^{2}+S^{2}(f^{j}),(f^{j})^{*}\vee z\right)$
$\displaystyle\geq\sum_{j=1}^{m}p_{j}b_{j}.$

Using the freedom in the choice of $b_{j}$’s, this implies (7.9). ∎

For every $\lambda>0$, we have

$V(x,t,z)=\lambda V(x/\lambda,t/\lambda^{2},z/\lambda),\quad U(x,t,z)=\lambda
U(x/\lambda,t/\lambda^{2},z/\lambda).$

Define $u,v:[-1,1]\times\mathbb{R}_{\geq 0}\to(-\infty,\infty]$ by

$v(x,t):=V(x,t,1),\quad u(x,t)=U(x,t,1).$

Since $t\mapsto V(x,t,z)$ is nondecreasing, the same holds for $t\mapsto
u(x,t)$, so left limits exist. We claim that

$u(1,1-)\geq u(0,2-)+u(1,1-).$ (7.10)

To see this, let $0<s<1<r$. Let $d$ be a random variable such that
$\mathbb{P}(d=-1)=r/(r+1)$ and $\mathbb{P}(d=r)=1/(r+1)$. Then, (7.9) and
scaling imply that

$\displaystyle u(1,s)$ $\displaystyle=U(1,s,1)$
$\displaystyle\geq\frac{r}{r+1}U(0,s+1,1)+\frac{1}{1+r}U(1+r,s+r^{2},1+r)$
$\displaystyle=\frac{r}{r+1}u(0,s+1)+u(1,(s+r^{2})/(1+r)^{2})$
$\displaystyle\geq\frac{r}{r+1}u(0,s+1)+u(1,r^{2}/(1+r)^{2}),$

where we used monotonicity of $u$ in the second variable in the last step.
Both summands on the RHS are increasing in both variables $s,r$. Taking
$r\to\infty$ and $s\to 1$, we obtain (7.10).

If $f$ is a simple martingale with $f_{0}=0$, then, by the hypothesis (7.6),
we have

$\mathbb{E}V(f_{\infty},0+S^{2}f,f^{*}\vee
1)=\mathbb{E}Sf-\gamma\mathbb{E}(f^{*}\vee
1)\leq\mathbb{E}Sf-\gamma\mathbb{E}f^{*}\leq 0.$

By the definition (7.7), this implies that $u(0,0)=U(0,0,1)\leq 0$.

Let $d$ be a random variable with $\mathbb{P}(d=1)=\mathbb{P}(d=-1)=1/2$.
Using (7.9) and the fact that $u(x,t)=u(-x,t)$, we obtain

$0\geq u(0,0)\geq\frac{1}{2}[u(1,1)+u(-1,1)]=u(1,1)\geq u(1,1-)\geq
v(1,1-)=1-\gamma.$

This implies that $u(1,1-)$ is finite. So (7.10) yields $u(0,2-)\leq 0$. Let
$d$ be a random variable with $\mathbb{P}(d=1)=\mathbb{P}(d=-1)=1/2$. By
(7.9), for every $\varepsilon>0$, we obtain

$0\geq u(0,2-)\geq
u(0,2-\varepsilon)\geq\frac{1}{2}[u(1,3-\varepsilon)+u(-1,3-\varepsilon)]\\\
=u(1,3-\varepsilon)\geq v(1,3-\varepsilon)=\sqrt{3-\varepsilon}-\gamma$

Therefore, $\gamma\geq\sqrt{3-\varepsilon}$. Since $\varepsilon>0$ was
arbitrary, this implies $\gamma\geq\sqrt{3}$. ∎

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