EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-051 LHCb-PAPER-2011-036

Measurement of $\varUpsilon$ production in $pp$ collisions at
$\sqrt{s}=7~{}\mathrm{\,Te\kern-2.07413ptV}$

The LHCb collaboration 111Authors are listed on the following pages.

The production of $\varUpsilon(1S)$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$
mesons in proton-proton collisions at the centre-of-mass energy of
${\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}}$ is studied with the LHCb
detector. The analysis is based on a data sample of $25~{}\mbox{\,pb}^{-1}$
collected at the Large Hadron Collider. The $\varUpsilon$ mesons are
reconstructed in the decay mode
$\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$ and the signal yields are
extracted from a fit to the $\mu^{+}\mu^{-}$ invariant mass distributions. The
differential production cross-sections times dimuon branching fractions are
measured as a function of the $\varUpsilon$ transverse momentum $p_{\rm T}$
and rapidity $y$, over the range $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$. The cross-
sections times branching fractions, integrated over these kinematic ranges,
are measured to be

$\displaystyle\sigma(pp\rightarrow\varUpsilon(1S)\,X)\times\mathcal{B}(\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-})=2.29\phantom{0}\pm
0.01\phantom{0}\pm 0.10\phantom{0}\,\,_{-0.37}^{+0.19}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(2S)\,X)\times\mathcal{B}(\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-})=0.562\pm
0.007\pm 0.023\,_{-0.092}^{+0.048}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(3S)\,X)\times\mathcal{B}(\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-})=0.283\pm
0.005\pm 0.012\,_{-0.048}^{+0.025}~{}{\rm nb},$

where the first uncertainty is statistical, the second systematic and the
third is due to the unknown polarisation of the three $\varUpsilon$ states.

Published in Eur. Phys. J. C volume 72,6 (June 2012)

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S.
Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, E. Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38,
S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30,
C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2,
M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44,
V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V.
Niess5, N. Nikitin29, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, K. Pal53,
J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M.
Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M.
Patel50, S.K. Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-
Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M.
Schiller39, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O.
Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A.
Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N.
Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P.
Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31, O. Shevchenko40, V.
Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49,
E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam

##  1 Introduction

The measurement of heavy quark production in hadron collisions probes the
dynamics of the colliding partons. The study of heavy quark-antiquark
resonances, such as the $b\overline{b}$ bound states $\varUpsilon(1S)$,
$\varUpsilon(2S)$ and $\varUpsilon(3S)$ (indicated generically as
$\varUpsilon$ in the following) is of interest as these mesons have large
production cross-sections and can be produced in different spin
configurations. In addition, the thorough understanding of these states is the
first step towards the study of recently discovered new states in the
$b\bar{b}$ system [1, 2, 3, 4]. Although $\varUpsilon$ production was studied
by several experiments in the past, the underlying production mechanism is
still not well understood. Several models exist but fail to reproduce both the
cross-section and the polarisation measurements at the Tevatron [5, 6, 7].
Among these are the Colour Singlet Model (CSM) [8, 9, 10], recently improved
by adding higher order contributions (NLO CSM), the standard truncation of the
nonrelativistic QCD expansion (NRQCD) [11], which includes contributions from
the Colour Octet Mechanism [12, 13], and the Colour Evaporation Model (CEM)
[14]. Although the disagreement of the theory with the data is less pronounced
for bottomonium than for charmonium, the measurement of $\varUpsilon$
production is important as the theoretical calculations are more robust due to
the heavier bottom quark.

There are two major sources of $\varUpsilon$ production in $pp$ collisions:
direct production and feed-down from the decay of heavier prompt bottomonium
states, like $\chi_{b}$, or higher-mass $\varUpsilon$ states. This study
presents measurements of the individual inclusive production cross-sections of
the three $\varUpsilon$ mesons decaying into a pair of muons. The measurements
are performed in $7~{}\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass $pp$
collisions as a function of the $\varUpsilon$ transverse momentum
($\mbox{$p_{\rm T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and rapidity
($2<y<4.5$), in 15 bins of $p_{\rm T}$ and five bins of $y$. This analysis is
complementary to those recently presented by the ATLAS collaboration, who
measured the $\varUpsilon(1S)$ cross section for $|y|<2.4$ [15], and the CMS
collaboration, who measured the $\varUpsilon(1S),\varUpsilon(2S)$ and
$\varUpsilon(3S)$ cross sections in the rapidity region $|y|<2.0$ [16].

##  2 The LHCb detector and data

The results presented here are based on a dataset of $25.0\pm
0.9~{}\mbox{\,pb}^{-1}$ collected at the Large Hadron Collider (LHC) in 2010
with the LHCb detector at a centre-of-mass energy of 7
$\mathrm{\,Te\kern-1.00006ptV}$.

The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage which applies a full event
reconstruction. This analysis uses events triggered by one or two muons. At
the hardware level one or two muon candidates are required with $p_{\rm T}$
larger than 1.4 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for one muon, and 0.56
and 0.48 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for two muons. At the software
level, the combined dimuon mass is required to be greater than 2.9
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and both the tracks and the vertex
have to be of good quality. To avoid the possibility that a few events with a
high occupancy dominate the trigger processing time, a set of global event
selection requirements based on hit multiplicities is applied.

The Monte Carlo samples used are based on the Pythia 6.4 generator [18], with
a choice of parameters specifically configured for LHCb [19]. The EvtGen
package [20] describes the decay of the $\varUpsilon$ resonances, and the
Geant4 package [21] simulates the detector response. The prompt bottomonium
production processes activated in Pythia are those from the leading-order
colour-singlet and colour-octet mechanisms for the $\varUpsilon(1S)$, and
colour-singlet only for the $\varUpsilon(2S)$ and the $\varUpsilon(3S)$. QED
radiative corrections to the decay
$\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$ are generated with the Photos
package [22].

##  3 Cross-section determination

The double differential cross-section for the inclusive $\varUpsilon$
production of the three different states is computed as

$\frac{{\rm d}^{2}\sigma^{iS}}{{\rm d}\mbox{$p_{\rm T}$}{\rm
d}y}\,\times\,\mathcal{B}^{iS}=\frac{N^{iS}}{\mathcal{L}\times\varepsilon^{iS}\times\Delta
y\times\Delta\mbox{$p_{\rm T}$}},\quad i=1,2,3;$ (1)

where $\sigma^{iS}$ is the inclusive cross section
$\sigma(pp\rightarrow\varUpsilon(iS)X)$, $\mathcal{B}^{iS}$ is the dimuon
branching fraction $\mathcal{B}(\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-})$,
$N^{iS}$ is the number of observed $\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$
decays in a given bin of $p_{\rm T}$ and $y$, $\varepsilon^{iS}$ is the
$\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$ total detection efficiency
including acceptance effects, $\mathcal{L}$ is the integrated luminosity and
$\Delta y=0.5$ and $\Delta\mbox{$p_{\rm T}$}=1~{}{\rm GeV}/c$ are the rapidity
and $p_{\rm T}$ bin sizes, respectively. In order to estimate $N^{iS}$, a fit
to the reconstructed invariant mass distribution is performed in each of the
15 $\mbox{$p_{\rm T}$}\times 5~{}y$ bins. $\varUpsilon$ candidates are formed
from pairs of oppositely charged muon tracks which traverse the full
spectrometer and satisfy the trigger requirements. Each track must have
$\mbox{$p_{\rm T}$}>1~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, be identified
as a muon and have a good quality of the track fit. The two muons are required
to originate from a common vertex with a good $\chi^{2}$ probability. The
three $\varUpsilon$ signal yields are determined from a fit to the
reconstructed invariant mass $m$ of the selected $\varUpsilon$ candidates in
the interval 8.9–10.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The mass
distribution is described by a sum of three Crystal Ball functions [23] for
the $\varUpsilon(1S)$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$ signals and an
exponential function for the combinatorial background. The Crystal Ball
function is defined as

$f_{\mathrm{CB}}=\begin{dcases}\frac{\Big{(}\frac{n}{|a|}\Big{)}^{n}e^{-\frac{1}{2}a^{2}}}{\Big{(}\frac{n}{|a|}-|a|-\frac{m-M}{\sigma}\Big{)}^{n}}&{\mathrm{if}}\,\,\,\frac{m-M}{\sigma}<-|a|\\\
\exp\Bigg{(}-\frac{1}{2}\Big{(}\frac{m-M}{\sigma}\Big{)}^{2}\Bigg{)}&{\mathrm{otherwise}},\end{dcases}$
(2)

with $f_{\mathrm{CB}}=f_{\mathrm{CB}}(m;M,\sigma,a,n)$, where $M$ and $\sigma$
are the mean and width of the gaussian. The parameters $a$ and $n$ describing
the radiative tail of the $\varUpsilon$ mass distribution are fixed to
describe a tail dominated by QED photon emission, as confirmed by simulation.
The distribution in Fig. 1 shows the results of the fit performed in the full
range of $p_{\rm T}$ and $y$.

Figure 1: Invariant mass distribution of the selected
$\varUpsilon\rightarrow\mu^{+}\mu^{-}$ candidates in the range $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and $2.0<y<4.5$. The three
peaks correspond to the $\varUpsilon(1S)$, $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ signals (from left to right). The superimposed curves are
the result of the fit as described in the text.

The signal yields obtained from the fit are $\varUpsilon(1S)=26\,410\pm 212$,
$\varUpsilon(2S)=6726\pm 142$ and $\varUpsilon(3S)=3260\pm 112$ events. The
mass resolution of the $\varUpsilon(1S)$ peak is $\sigma=53.9\pm
0.5~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The resolutions of the
$\varUpsilon(2S)$ and $\varUpsilon(3S)$ peaks are fixed to the resolution of
the $\varUpsilon(1S)$, scaled by the ratio of the masses, as expected from
resolution effects. The masses are allowed to vary in the fit and are measured
to be $M(\varUpsilon(1S))=9448.3\pm 0.5$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $M(\varUpsilon(2S))=10\,010.4\pm
1.4$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$M(\varUpsilon(3S))=10\,338.7\pm 2.6$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the quoted uncertainties are
statistical only. The fit is repeated independently for each of the bins in
$p_{\rm T}$ and $y$. When fitting the individual bins, due to the reduced
dataset, the masses and widths of the three $\varUpsilon$ states in the fit
are fixed to the values obtained when fitting the full range. Bins with fewer
than 36 entries are excluded from the analysis. The total efficiency
$\varepsilon$ entering the cross-section expression of Eq. (1) is the product
of the geometric acceptance, the reconstruction and selection efficiency and
the trigger efficiency. All efficiency terms have been evaluated using Monte
Carlo simulations in each ($\mbox{$p_{\rm T}$},y$) bin separately, with the
exception of that related to the global event selection which has been
determined from data. In the simulation the $\varUpsilon$ meson is produced in
an unpolarised state. The absolute luminosity scale was measured at specific
periods during the 2010 data taking using both van der Meer scans and a beam-
gas imaging method [24, 25]. The uncertainty on the integrated luminosity for
the analysed sample due to this method is estimated to be 3.5% [25]. The
knowledge of the absolute luminosity scale is used to calibrate the number of
tracks in the vertex detector, which is found to be stable throughout the
data-taking period and can therefore be used to monitor the instantaneous
luminosity of the entire data sample. The integrated luminosity of the data
sample used in this analysis is determined to be $25.0~{}\mbox{\,pb}^{-1}$.

##  4 Systematic uncertainties

Extensive studies on dimuon decays [26, 16, 15] have shown that the total
efficiency depends strongly on the initial polarisation state of the vector
meson. In this analysis, the influence of the unknown polarisation is studied
in the helicity frame [27] using Monte Carlo simulation. The angular
distribution of the muons from the $\varUpsilon$, ignoring the azimuthal part,
is

$\displaystyle\frac{{\rm d}N}{{\rm
d}\cos\theta}\,=\,\frac{1+\alpha\cos^{2}\theta}{2+2\alpha/3},$ (3)

where $\theta$ is the angle between the direction of the $\mu^{+}$ momentum in
the $\varUpsilon$ centre-of-mass frame and the direction of the $\varUpsilon$
momentum in the colliding proton centre-of-mass frame. The values
$\alpha=+1,-1,0$ correspond to fully transverse, fully longitudinal, and no
polarisation respectively. Figure 2 shows the $\varUpsilon(1S)$ total
efficiency for these three scenarios, and indicates that the polarisation
significantly affects the efficiencies and that the effect depends on $p_{\rm
T}$ and $y$. A similar behaviour is observed for the $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ efficiencies.

Figure 2:  Total efficiency $\varepsilon$ of the $\varUpsilon(1S)$ as a
function of (a) the $\varUpsilon(1S)$ transverse momentum and (b) rapidity,
estimated using the Monte Carlo simulation, for three different
$\varUpsilon(1S)$ polarisation scenarios, indicated by the parameter $\alpha$
described in the text.

Following this observation, in each $(\mbox{$p_{\rm T}$},y)$ bin the maximal
difference between the polarised scenarios ($\alpha=\pm 1$) and the
unpolarised scenario ($\alpha=0$) is taken as a systematic uncertainty on the
efficiency. This results in an uncertainty of up to $17\%$ on the integrated
cross-sections and of up to 40% in the individual bins. Several other sources
of possible systematic effects were studied. They are summarised in Table 1.

Table 1:  Summary of the relative systematic uncertainties on the cross-section measurements. Ranges indicate variations depending on the ($\mbox{$p_{\rm T}$},y$) bin and the $\varUpsilon$ state. All uncertainties are fully correlated among the bins.  Source | Uncertainty (%)
---|---
Unknown $\varUpsilon$ polarisation | 0.3–41.0
Trigger | 3.0
Track reconstruction | 2.4
Track quality requirement | 0.5
Vertexing requirement | 1.0
Muon identification | 1.1
Global event selection requirements | 0.6
$p_{\rm T}$ binning effect | 1.0
Fit function | 1.1–2.1
Luminosity | 3.5

The trigger efficiency is determined on data using an unbiased sample of
events that would trigger if the $\varUpsilon$ candidate were removed. The
efficiency obtained with this method is compared with the efficiency
determined in the simulation. The difference of 3.0% is assigned as a
systematic uncertainty.

The uncertainty on the muon track reconstruction efficiency has been estimated
using a data driven tag-and-probe approach based on partially reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
[28], and found to be 2.4% per muon pair. Additional uncertainties are
assigned, which account for the different behaviour in data and simulation of
the track and vertex quality requirements. The muon identification efficiency
is measured using a tag-and-probe approach, which gives an uncertainty on the
efficiency of 1.1% [26].

The measurement of the global event selection efficiency is taken as an
additional uncertainty associated with the trigger. An uncertainty of 1.0% is
considered to account for the difference in the $p_{\rm T}$ spectra in data
and Monte Carlo simulation for the three $\varUpsilon$ states, which might
have an effect on the correct bin assignment (“binning effect”).

The influence of the choice of the fit function describing the shape of the
invariant mass distribution includes two components. The uncertainty on the
shape of the background distribution is estimated using a different fit model
(1.0–1.5%). The systematic associated with fixing the parameters of the
Crystal Ball function is estimated by varying the central values within the
parameters uncertainties, obtained when leaving them free to vary in the fit
(0.5–1.4%).

##  5 Results

The double differential cross-sections as a function of $p_{\rm T}$ and $y$
are shown in Fig. 3 and Tables 2-4. The integrated cross-sections times
branching fractions in the range $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0~{}<~{}y~{}<~{}4.5$ are
measured to be

$\displaystyle\sigma(pp\rightarrow\varUpsilon(1S)\,X)\times\mathcal{B}^{1S}=2.29\phantom{0}\pm
0.01\phantom{0}\pm 0.10\phantom{0}\,\,_{-0.37}^{+0.19}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(2S)\,X)\times\mathcal{B}^{2S}=0.562\pm
0.007\pm 0.023\,_{-0.092}^{+0.048}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(3S)\,X)\times\mathcal{B}^{3S}=0.283\pm
0.005\pm 0.012\,_{-0.048}^{+0.025}~{}{\rm nb},$

where the first uncertainties are statistical, the second systematic and the
third are due to the unknown polarisation of the three $\varUpsilon$ states.
The integrated $\varUpsilon(1S)$ cross-section is about a factor one hundred
smaller than the integrated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
cross-section in the identical region of $p_{\rm T}$ and $y$ [26], and a
factor three smaller than the integrated $\varUpsilon(1S)$ cross-section in
the central region, as measured by CMS [16] and ATLAS [15].

Figure 3: Double differential $\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$
cross-sections times dimuon branching fractions as a function of $p_{\rm T}$
in bins of rapidity for (a) the $\varUpsilon(1S)$, (b) the $\varUpsilon(2S)$
and (c) the $\varUpsilon(3S)$. The error bars correspond to the total
uncertainty for each bin.

Figure 4 compares the LHCb measurement of the differential
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ production cross-section with
several theory predictions in the LHCb acceptance region. In Fig. 4(a) the
data are compared to direct production as calculated from a NNLO* colour-
singlet model [29, 30], where the notation NNLO* denotes an evaluation that is
not a complete next-to-next leading order computation and that can be affected
by logarithmic corrections, which are not easily quantifiable. Direct
production as calculated from NLO CSM is also represented. In Fig. 4(b) the
data are compared to two model predictions for the $\varUpsilon(1S)$
production: the calculation from NRQCD at NLO, including contributions from
$\chi_{b}$ and higher $\varUpsilon$ states decays, summing the colour-singlet
and colour-octet contributions [31], and the calculation from the NLO CEM,
including contributions from $\chi_{b}$ and higher $\varUpsilon$ states decays
[14]. Note that the NNLO* theoretical model computes the direct
$\varUpsilon(1S)$ production, whereas the LHCb measurement includes
$\varUpsilon(1S)$ from $\chi_{b}$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$
decays. However, taking into account the feed-down contribution, which has
been measured to be of the order of 50% [32], a satisfactory agreement is
found with the theoretical predictions.

Figure 4:  Differential $\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ production
cross-section times dimuon branching fraction as a function of $p_{\rm T}$
integrated over $y$ in the range 2.0–4.5, compared with the predictions from
(a) the NNLO* CSM [29] for direct production, and (b) the NLO NRQCD [31] and
CEM [14]. The error bars on the data correspond to the total uncertainties for
each bin, while the bands indicate the uncertainty on the theory prediction.

Figure 5 compares the LHCb measurement of the differential $\varUpsilon(2S)$
and $\varUpsilon(3S)$ production cross-sections times branching fraction with
the NNLO* theory predictions of direct production. It can be seen that the
agreement with the theory is better for the $\varUpsilon(3S)$, which is
expected to be less affected by feed-down. At present there is no measurement
of the contribution of feed-down to the $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ inclusive rate.

Figure 5: Differential (a) $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ and (b)
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ production cross-sections times
dimuon branching fractions as a function of $p_{\rm T}$ integrated over $y$ in
the range 2.0–4.5, compared with the predictions from the NNLO* CSM for direct
production [29]. The error bars on the data correspond to the total
uncertainties for each bin, while the bands indicate the uncertainty on the
theory prediction.

The cross-sections times the dimuon branching fractions for the three
$\varUpsilon$ states are compared in Fig. 6 as a function of rapidity and
transverse momentum.

Figure 6:  Differential cross-sections of $\varUpsilon(1S),\varUpsilon(2S)$
and $\varUpsilon(3S)$ times dimuon branching fractions as a function of (a)
$p_{\rm T}$ integrated over $y$ and (b) $y$ integrated over $p_{\rm T}$. The
error bars on the data correspond to the total uncertainties for each bin.

The cross-section results are used to evaluate the ratios $R^{iS/1S}$ of the
$\varUpsilon(2S)$ to $\varUpsilon(1S)$ and $\varUpsilon(3S)$ to
$\varUpsilon(1S)$ cross-sections times the dimuon branching fractions. Most of
the systematic uncertainties on the cross-sections cancel in the ratio, except
those due to the size of the data sample, the choice of fit function and the
unknown polarisation of the different states. The polarisation uncertainty has
been evaluated for the scenarios in which one of the two $\varUpsilon$ states
is completely polarised (either transversely or longitudinally) and the other
is not polarised. The maximum difference of these two cases ranges between 15%
and 26%. The ratios $R^{iS/1S},i=2,3,$ are given in Table 5 and shown in Fig.
7. The polarisation uncertainty is not included in these figures. The results
agree well with the corresponding ratio measurements from CMS [16] in the
$p_{\rm T}$ range common to both experiments.

Figure 7:  Ratios of $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ and
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ with respect to
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of $p_{\rm T}$ of the
$\varUpsilon$ in the range $2.0<y<4.5$, assuming no polarisation. The error
bars on the data correspond to the total uncertainties for each bin except for
that due to the unknown polarisation, which ranges between 15% and 26% as
listed in Table 5.

##  6 Conclusions

The differential cross-sections $\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$,
for $i=1,2,3$, are measured as a function of the $\varUpsilon$ transverse
momentum and rapidity in the region $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, 2.0 $<y<$ 4.5 in the LHCb
experiment. The analysis is based on a data sample corresponding to an
integrated luminosity of 25 $\mbox{\,pb}^{-1}$ collected at the Large Hadron
Collider at a centre-of-mass energy of
$\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$. The results obtained are
compatible with previous measurements in $pp$ collisions at the same centre-
of-mass energy, performed by ATLAS and CMS in a different region of rapidity
[16, 15]. This is the first measurement of $\varUpsilon$ production in the
forward region at $\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$. A comparison
with theoretical models shows good agreement with the measured $\varUpsilon$
cross-sections. The measurement of the differential cross-sections is not
sufficient to discriminate amongst the various models, and studies of other
observables such as the $\varUpsilon$ polarisations will be necessary.

##  7 Acknowledgements

We thank P. Artoisenet, M. Butenschön, K.-T. Chao, B. Kniehl, J.-P. Lansberg
and R. Vogt for providing theoretical predictions of $\varUpsilon$ cross-
sections in the LHCb acceptance range. We express our gratitude to our
colleagues in the CERN accelerator departments for the excellent performance
of the LHC. We thank the technical and administrative staff at CERN and at the
LHCb institutes, and acknowledge support from the National Agencies: CAPES,
CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France);
BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO
(The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom
(Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS
Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the
support received from the ERC under FP7 and the Region Auvergne.

Table 2: Double differential cross-section
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(1S)$.

$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) |  |  |  |  |
0–1 | 53.1 $\pm$ 4.0 $\pm$ 2.5 ${}_{-17.3}^{+8.9}$ | 62.6 $\pm$ 3.0 $\pm$ 2.9 ${}_{-11.5}^{+6.1}$ | 48.0 $\pm$ 2.4 $\pm$ 2.2 ${}_{-5.8}^{+3.1}$ | 40.1 $\pm$ 2.4 $\pm$ 1.9 ${}_{-7.0}^{+3.9}$ | 22.9 $\pm$ 2.7 $\pm$ 1.1 ${}_{-5.9}^{+3.4}$
1–2 | 152.5 $\pm$ 6.8 $\pm$ 7.2 ${}_{-50.4}^{+25.7}$ | 148.8 $\pm$ 4.7 $\pm$ 7.0 ${}_{-27.5}^{+14.6}$ | 120.5 $\pm$ 3.8 $\pm$ 5.6 ${}_{-14.0}^{+7.5}$ | 93.3 $\pm$ 3.7 $\pm$ 4.3 ${}_{-14.8}^{+8.1}$ | 64.5 $\pm$ 4.5 $\pm$ 3.0 ${}_{-15.0}^{+8.7}$
2–3 | 211.0 $\pm$ 8.0 $\pm$ 10.0 ${}_{-67.2}^{+34.3}$ | 185.3 $\pm$ 5.2 $\pm$ 8.7 ${}_{-34.4}^{+18.1}$ | 150.0 $\pm$ 4.3 $\pm$ 7.0 ${}_{-17.4}^{+9.2}$ | 116.1 $\pm$ 4.1 $\pm$ 5.4 ${}_{-15.5}^{+8.4}$ | 69.8 $\pm$ 4.6 $\pm$ 3.3 ${}_{-14.6}^{+8.3}$
3–4 | 184.3 $\pm$ 7.3 $\pm$ 8.8 ${}_{-56.3}^{+28.8}$ | 167.7 $\pm$ 4.9 $\pm$ 7.9 ${}_{-29.3}^{+15.6}$ | 141.9 $\pm$ 4.2 $\pm$ 6.6 ${}_{-15.0}^{+8.0}$ | 109.7 $\pm$ 4.0 $\pm$ 5.1 ${}_{-11.9}^{+6.3}$ | 70.6 $\pm$ 4.6 $\pm$ 3.3 ${}_{-12.2}^{+6.7}$
4–5 | 187.3 $\pm$ 7.3 $\pm$ 8.9 ${}_{-54.8}^{+27.9}$ | 158.4 $\pm$ 4.8 $\pm$ 7.4 ${}_{-26.4}^{+14.0}$ | 120.9 $\pm$ 3.9 $\pm$ 5.7 ${}_{-11.3}^{+6.0}$ | 84.6 $\pm$ 3.5 $\pm$ 4.0 ${}_{-7.0}^{+3.7}$ | 50.4 $\pm$ 3.8 $\pm$ 2.4 ${}_{-7.0}^{+3.7}$
5–6 | 138.0 $\pm$ 6.2 $\pm$ 6.6 ${}_{-38.3}^{+19.4}$ | 134.5 $\pm$ 4.4 $\pm$ 6.3 ${}_{-20.8}^{+11.0}$ | 94.2 $\pm$ 3.5 $\pm$ 4.4 ${}_{-7.3}^{+3.8}$ | 70.6 $\pm$ 3.2 $\pm$ 3.3 ${}_{-4.0}^{+2.1}$ | 45.3 $\pm$ 3.6 $\pm$ 2.1 ${}_{-4.9}^{+2.5}$
6–7 | 105.3 $\pm$ 5.3 $\pm$ 5.0 ${}_{-27.6}^{+14.0}$ | 95.2 $\pm$ 3.7 $\pm$ 4.5 ${}_{-13.7}^{+7.2}$ | 73.5 $\pm$ 3.0 $\pm$ 3.5 ${}_{-4.6}^{+2.4}$ | 57.0 $\pm$ 2.9 $\pm$ 2.7 ${}_{-1.9}^{+1.0}$ | 29.5 $\pm$ 2.8 $\pm$ 1.4 ${}_{-2.5}^{+1.2}$
7–8 | 78.3 $\pm$ 4.5 $\pm$ 3.7 ${}_{-19.4}^{+9.8}$ | 72.9 $\pm$ 3.2 $\pm$ 3.4 ${}_{-9.6}^{+5.0}$ | 60.2 $\pm$ 2.7 $\pm$ 2.8 ${}_{-3.0}^{+1.6}$ | 38.3 $\pm$ 2.3 $\pm$ 1.8 ${}_{-0.8}^{+0.4}$ | 21.6 $\pm$ 2.4 $\pm$ 1.0 ${}_{-1.5}^{+0.7}$
8–9 | 63.5 $\pm$ 4.0 $\pm$ 3.0 ${}_{-14.8}^{+7.5}$ | 57.0 $\pm$ 2.8 $\pm$ 2.7 ${}_{-6.8}^{+3.6}$ | 43.3 $\pm$ 2.3 $\pm$ 2.0 ${}_{-1.9}^{+1.0}$ | 24.7 $\pm$ 1.9 $\pm$ 1.2 ${}_{-0.6}^{+0.3}$ | 13.6 $\pm$ 1.9 $\pm$ 0.6 ${}_{-0.8}^{+0.4}$
9–10 | 50.1 $\pm$ 3.5 $\pm$ 2.4 ${}_{-10.8}^{+5.5}$ | 43.2 $\pm$ 2.4 $\pm$ 2.0 ${}_{-5.0}^{+2.6}$ | 29.8 $\pm$ 1.9 $\pm$ 1.4 ${}_{-1.0}^{+0.5}$ | 19.4 $\pm$ 1.6 $\pm$ 0.9 ${}_{-0.6}^{+0.3}$ | 6.1 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.3}^{+0.1}$
10–11 | 35.4 $\pm$ 2.9 $\pm$ 1.7 ${}_{-7.3}^{+3.7}$ | 28.2 $\pm$ 1.9 $\pm$ 1.3 ${}_{-3.0}^{+1.6}$ | 23.9 $\pm$ 1.7 $\pm$ 1.1 ${}_{-0.8}^{+0.4}$ | 12.3 $\pm$ 1.3 $\pm$ 0.6 ${}_{-0.5}^{+0.2}$ | 6.8 $\pm$ 1.3 $\pm$ 0.3 ${}_{-0.4}^{+0.2}$
11–12 | 29.3 $\pm$ 2.6 $\pm$ 1.4 ${}_{-5.8}^{+2.9}$ | 19.4 $\pm$ 1.6 $\pm$ 0.9 ${}_{-1.9}^{+1.0}$ | 14.7 $\pm$ 1.3 $\pm$ 0.7 ${}_{-0.6}^{+0.3}$ | 6.7 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 4.3 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
12–13 | 20.3 $\pm$ 2.1 $\pm$ 1.0 ${}_{-3.7}^{+1.9}$ | 13.7 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.3}^{+0.7}$ | 10.3 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.3}^{+0.2}$ | 6.7 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.8 $\pm$ 0.8 $\pm$ 0.1 ${}_{-0.2}^{+0.1}$
13–14 | 10.4 $\pm$ 1.5 $\pm$ 0.5 ${}_{-1.9}^{+0.9}$ | 11.6 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.1}^{+0.6}$ | 8.6 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.2}^{+0.1}$ | 5.0 $\pm$ 0.8 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 0.8 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.1}^{+0.0}$
14–15 | 11.2 $\pm$ 1.5 $\pm$ 0.5 ${}_{-2.0}^{+1.0}$ | 8.9 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.8}^{+0.4}$ | 5.7 $\pm$ 0.8 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.2 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 1.8 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$

Table 3: Double differential cross-section
$\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(2S)$. Regions where the number of
events was not sufficient to perform a measurement are indicated with a dash.

$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) |  |  |  |  |
0–1 | 8.2 $\pm$ 1.7 $\pm$ 0.4 ${}_{-3.1}^{+1.5}$ | 15.8 $\pm$ 1.6 $\pm$ 0.7 ${}_{-2.8}^{+1.5}$ | 7.8 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.8}^{+0.4}$ | 8.6 $\pm$ 1.2 $\pm$ 0.4 ${}_{-1.5}^{+0.8}$ | -
1–2 | 25.8 $\pm$ 2.9 $\pm$ 1.2 ${}_{-9.2}^{+4.6}$ | 31.2 $\pm$ 2.2 $\pm$ 1.5 ${}_{-5.6}^{+3.1}$ | 23.0 $\pm$ 1.7 $\pm$ 1.1 ${}_{-2.9}^{+1.6}$ | 18.3 $\pm$ 1.6 $\pm$ 0.9 ${}_{-2.8}^{+1.6}$ | 10.4 $\pm$ 1.8 $\pm$ 0.5 ${}_{-2.3}^{+1.4}$
2–3 | 39.3 $\pm$ 3.6 $\pm$ 1.9 ${}_{-12.9}^{+6.4}$ | 45.7 $\pm$ 2.6 $\pm$ 2.1 ${}_{-8.2}^{+4.5}$ | 24.4 $\pm$ 1.8 $\pm$ 1.1 ${}_{-2.9}^{+1.5}$ | 26.3 $\pm$ 2.0 $\pm$ 1.2 ${}_{-3.4}^{+1.9}$ | 14.9 $\pm$ 2.2 $\pm$ 0.7 ${}_{-3.2}^{+1.8}$
3–4 | 55.8 $\pm$ 4.2 $\pm$ 2.6 ${}_{-17.4}^{+8.9}$ | 42.1 $\pm$ 2.5 $\pm$ 2.0 ${}_{-7.3}^{+3.8}$ | 37.8 $\pm$ 2.2 $\pm$ 1.8 ${}_{-4.3}^{+2.2}$ | 20.8 $\pm$ 1.8 $\pm$ 1.0 ${}_{-2.4}^{+1.3}$ | 11.9 $\pm$ 1.9 $\pm$ 0.6 ${}_{-2.1}^{+1.2}$
4–5 | 54.5 $\pm$ 4.1 $\pm$ 2.6 ${}_{-15.9}^{+8.2}$ | 39.2 $\pm$ 2.4 $\pm$ 1.8 ${}_{-6.7}^{+3.6}$ | 22.6 $\pm$ 1.7 $\pm$ 1.1 ${}_{-2.0}^{+1.1}$ | 18.3 $\pm$ 1.6 $\pm$ 0.9 ${}_{-1.6}^{+0.8}$ | 12.2 $\pm$ 1.9 $\pm$ 0.6 ${}_{-1.8}^{+1.0}$
5–6 | 39.1 $\pm$ 3.4 $\pm$ 1.9 ${}_{-10.3}^{+5.4}$ | 44.8 $\pm$ 2.6 $\pm$ 2.1 ${}_{-7.6}^{+3.9}$ | 32.8 $\pm$ 2.1 $\pm$ 1.5 ${}_{-2.8}^{+1.5}$ | 18.1 $\pm$ 1.6 $\pm$ 0.8 ${}_{-1.2}^{+0.6}$ | 7.8 $\pm$ 1.5 $\pm$ 0.4 ${}_{-0.9}^{+0.4}$
6–7 | 28.8 $\pm$ 2.9 $\pm$ 1.4 ${}_{-8.3}^{+4.1}$ | 25.1 $\pm$ 1.9 $\pm$ 1.2 ${}_{-3.9}^{+2.0}$ | 22.3 $\pm$ 1.7 $\pm$ 1.0 ${}_{-1.4}^{+0.7}$ | 11.6 $\pm$ 1.3 $\pm$ 0.5 ${}_{-0.5}^{+0.3}$ | 5.2 $\pm$ 1.2 $\pm$ 0.2 ${}_{-0.5}^{+0.2}$
7–8 | 21.9 $\pm$ 2.4 $\pm$ 1.0 ${}_{-5.4}^{+2.7}$ | 23.4 $\pm$ 1.9 $\pm$ 1.1 ${}_{-3.5}^{+1.8}$ | 16.3 $\pm$ 1.4 $\pm$ 0.8 ${}_{-0.9}^{+0.4}$ | 5.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.1}^{+0.1}$ | 5.4 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.4}^{+0.2}$
8–9 | 22.9 $\pm$ 2.4 $\pm$ 1.1 ${}_{-4.8}^{+2.6}$ | 17.1 $\pm$ 1.5 $\pm$ 0.8 ${}_{-2.0}^{+1.0}$ | 12.4 $\pm$ 1.2 $\pm$ 0.6 ${}_{-0.6}^{+0.3}$ | 7.6 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.2}^{+0.1}$ | 4.3 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
9–10 | 12.8 $\pm$ 1.8 $\pm$ 0.6 ${}_{-2.9}^{+1.5}$ | 12.9 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.2}^{+0.6}$ | 9.8 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.5}^{+0.2}$ | 7.0 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 1.2 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$
10–11 | 10.3 $\pm$ 1.6 $\pm$ 0.5 ${}_{-2.1}^{+1.1}$ | 9.5 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.9}^{+0.5}$ | 4.3 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 6.4 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.6 $\pm$ 0.8 $\pm$ 0.1 ${}_{-0.2}^{+0.1}$
11–12 | 8.6 $\pm$ 1.5 $\pm$ 0.4 ${}_{-2.4}^{+1.2}$ | 10.0 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.9}^{+0.5}$ | 4.4 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.0}$ | 1.2 $\pm$ 0.4 $\pm$ 0.1 ${}_{-0.0}^{+0.0}$ | -
12–13 | 5.8 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.9}^{+0.5}$ | 5.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.5}^{+0.3}$ | 4.1 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.0}$ | - | -
13–14 | 4.4 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.7}^{+0.4}$ | 1.7 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | 2.6 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | - | -
14–15 | 1.9 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.3}^{+0.2}$ | 4.9 $\pm$ 0.8 $\pm$ 0.2 ${}_{-0.5}^{+0.3}$ | 3.9 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$ | - | -

Table 4: Double differential cross-section
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(3S)$. Regions where the number of
events was not sufficient to perform a measurement are indicated with a dash.

$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) |  |  |  |  |
0–1 | 7.0 $\pm$ 1.5 $\pm$ 0.3 ${}_{-2.6}^{+1.3}$ | 6.3 $\pm$ 1.0 $\pm$ 0.3 ${}_{-1.0}^{+0.6}$ | 3.1 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.4}^{+0.2}$ | 5.0 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.9}^{+0.5}$ | -
1–2 | 14.1 $\pm$ 2.2 $\pm$ 0.7 ${}_{-5.3}^{+2.6}$ | 5.6 $\pm$ 0.9 $\pm$ 0.3 ${}_{-1.1}^{+0.6}$ | 11.6 $\pm$ 1.2 $\pm$ 0.6 ${}_{-1.3}^{+0.7}$ | 12.7 $\pm$ 1.4 $\pm$ 0.6 ${}_{-2.1}^{+1.2}$ | 10.2 $\pm$ 1.9 $\pm$ 0.5 ${}_{-2.6}^{+1.4}$
2–3 | 17.6 $\pm$ 2.3 $\pm$ 0.9 ${}_{-5.3}^{+2.7}$ | 22.3 $\pm$ 1.8 $\pm$ 1.1 ${}_{-4.1}^{+2.1}$ | 15.2 $\pm$ 1.4 $\pm$ 0.7 ${}_{-1.6}^{+0.8}$ | 6.7 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.9}^{+0.5}$ | 9.9 $\pm$ 1.7 $\pm$ 0.5 ${}_{-2.1}^{+1.2}$
3–4 | 24.9 $\pm$ 2.7 $\pm$ 1.2 ${}_{-7.7}^{+4.0}$ | 17.6 $\pm$ 1.6 $\pm$ 0.8 ${}_{-3.1}^{+1.6}$ | 13.5 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.6}^{+0.8}$ | 6.8 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.8}^{+0.4}$ | 7.5 $\pm$ 1.5 $\pm$ 0.4 ${}_{-1.3}^{+0.7}$
4–5 | 16.7 $\pm$ 2.2 $\pm$ 0.8 ${}_{-5.1}^{+2.6}$ | 17.5 $\pm$ 1.6 $\pm$ 0.8 ${}_{-3.0}^{+1.6}$ | 6.9 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.6}^{+0.3}$ | 6.1 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.5}^{+0.3}$ | 7.6 $\pm$ 1.5 $\pm$ 0.4 ${}_{-1.2}^{+0.6}$
5–6 | 16.6 $\pm$ 2.1 $\pm$ 0.8 ${}_{-4.6}^{+2.4}$ | 21.3 $\pm$ 1.8 $\pm$ 1.0 ${}_{-3.5}^{+1.8}$ | 12.1 $\pm$ 1.2 $\pm$ 0.6 ${}_{-1.1}^{+0.6}$ | 7.8 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.5}^{+0.3}$ | 7.6 $\pm$ 1.4 $\pm$ 0.4 ${}_{-0.9}^{+0.5}$
6–7 | 22.2 $\pm$ 2.5 $\pm$ 1.1 ${}_{-5.6}^{+3.0}$ | 19.1 $\pm$ 1.7 $\pm$ 0.9 ${}_{-3.0}^{+1.5}$ | 8.4 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.6}^{+0.3}$ | 7.1 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.3}^{+0.2}$ | 3.1 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
7–8 | 20.6 $\pm$ 2.4 $\pm$ 1.0 ${}_{-5.4}^{+2.7}$ | 10.5 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.6}^{+0.8}$ | 9.2 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.6}^{+0.3}$ | 5.2 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.1}^{+0.1}$ | 1.4 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$
8–9 | 13.7 $\pm$ 1.9 $\pm$ 0.7 ${}_{-3.3}^{+1.7}$ | 10.7 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.6}^{+0.8}$ | 6.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.3}^{+0.1}$ | 2.4 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 0.6 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.0}^{+0.0}$
9–10 | 11.3 $\pm$ 1.7 $\pm$ 0.5 ${}_{-2.5}^{+1.3}$ | 6.9 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.8}^{+0.4}$ | 5.7 $\pm$ 0.8 $\pm$ 0.3 ${}_{-0.3}^{+0.2}$ | 2.5 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 3.2 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.1}^{+0.1}$
10–11 | 8.4 $\pm$ 1.5 $\pm$ 0.4 ${}_{-2.0}^{+1.0}$ | 5.5 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.6}^{+0.3}$ | 4.3 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 2.6 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | -
11–12 | 8.7 $\pm$ 1.4 $\pm$ 0.4 ${}_{-1.7}^{+0.9}$ | 4.4 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.3}^{+0.2}$ | 3.2 $\pm$ 0.6 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 1.8 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | -
12–13 | 4.5 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.9}^{+0.4}$ | 3.2 $\pm$ 0.6 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$ | 3.5 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.1}$ | - | -
13–14 | 2.4 $\pm$ 0.7 $\pm$ 0.1 ${}_{-0.4}^{+0.2}$ | 0.7 $\pm$ 0.3 $\pm$ 0.0 ${}_{-0.1}^{+0.0}$ | 2.1 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | - | -
14–15 | 0.7 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.1}^{+0.1}$ | 1.5 $\pm$ 0.4 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | 0.9 $\pm$ 0.3 $\pm$ 0.0 ${}_{-0.0}^{+0.0}$ | - | -

Table 5:  Ratios of cross-sections $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$
and $\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ with respect to
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of $p_{\rm T}$ in the
range $2.0<y<4.5$, assuming no polarisation. The first uncertainty is
statistical, the second is systematic and the third is due to the unknown
polarisation of the three states.

$p_{\rm T}$ | $R^{2S/1S}$ | $R^{3S/1S}$
---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) |  |
0–1 | 0.202 $\pm$ 0.015 $\pm$ 0.006 $\pm$ 0.052 | 0.099 $\pm$ 0.010 $\pm$ 0.003 $\pm$ 0.025
1–2 | 0.192 $\pm$ 0.009 $\pm$ 0.005 $\pm$ 0.051 | 0.089 $\pm$ 0.006 $\pm$ 0.003 $\pm$ 0.024
2–3 | 0.207 $\pm$ 0.008 $\pm$ 0.006 $\pm$ 0.052 | 0.098 $\pm$ 0.005 $\pm$ 0.003 $\pm$ 0.025
3–4 | 0.247 $\pm$ 0.010 $\pm$ 0.007 $\pm$ 0.056 | 0.099 $\pm$ 0.006 $\pm$ 0.003 $\pm$ 0.023
4–5 | 0.234 $\pm$ 0.010 $\pm$ 0.007 $\pm$ 0.047 | 0.087 $\pm$ 0.005 $\pm$ 0.003 $\pm$ 0.017
5–6 | 0.305 $\pm$ 0.013 $\pm$ 0.009 $\pm$ 0.058 | 0.136 $\pm$ 0.007 $\pm$ 0.005 $\pm$ 0.023
6–7 | 0.260 $\pm$ 0.013 $\pm$ 0.007 $\pm$ 0.048 | 0.160 $\pm$ 0.009 $\pm$ 0.006 $\pm$ 0.027
7–8 | 0.268 $\pm$ 0.015 $\pm$ 0.008 $\pm$ 0.048 | 0.162 $\pm$ 0.011 $\pm$ 0.006 $\pm$ 0.027
8–9 | 0.309 $\pm$ 0.019 $\pm$ 0.009 $\pm$ 0.046 | 0.166 $\pm$ 0.013 $\pm$ 0.006 $\pm$ 0.028
9–10 | 0.303 $\pm$ 0.022 $\pm$ 0.009 $\pm$ 0.045 | 0.187 $\pm$ 0.016 $\pm$ 0.007 $\pm$ 0.032

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# Cross Chest Graph for Disease Diagnosis with Structural
Relational Reasoning

Gangming Zhao  Baolian Qi  Jinpeng Li  Gangming Zhao and Baolian Qi
contributed equally to this work.Gangming Zhao is with the Department of
Computer Science, The University of Hong Kong, Hong Kong.Qibao Lian and
Jinpeng Li are with University of Chinese Academy of Sciences, Beijing, China

###### Abstract

Locating lesions is important in the computer-aided diagnosis of X-ray images.
However, box-level annotation is time-consuming and laborious. How to locate
lesions accurately with few, or even without careful annotations is an urgent
problem. Although several works have approached this problem with weakly-
supervised methods, the performance needs to be improved. One obstacle is that
general weakly-supervised methods have failed to consider the characteristics
of X-ray images, such as the highly-structural attribute. We therefore propose
the Cross-chest Graph (CCG), which improves the performance of automatic
lesion detection by imitating doctor’s training and decision-making process.
CCG models the intra-image relationship between different anatomical areas by
leveraging the structural information to simulate the doctor’s habit of
observing different areas. Meanwhile, the relationship between any pair of
images is modeled by a knowledge-reasoning module to simulate the doctor’s
habit of comparing multiple images. We integrate intra-image and inter-image
information into a unified end-to-end framework. Experimental results on the
NIH Chest-14 database (112,120 frontal-view X-ray images with 14 diseases)
demonstrate that the proposed method achieves state-of-the-art performance in
weakly-supervised localization of lesions by absorbing professional knowledge
in the medical field.

##  1 Introduction

Chest radiographs are a type of medical images that can be conveniently
acquired for disease diagnosis. With the rapid development of deep learning,
automatic disease detection in chest X-ray images has become an important task
in the computer-aided diagnosis.

Figure 1: CCG network models the intra-image relationship between different
anatomical areas by leveraging the structural information to simulate the
doctor’s habit of observing different areas. Meanwhile, the relationship
between any pair of images is modeled by a knowledge-reasoning module to
simulate the doctor’s habit of comparing multiple images.

Deep convolutional neural networks (DCNN) have been widely applied in many
computer vision tasks, such as image classification [6, 18] , object detection
[3, 5, 16, 15, 10] and semantic segmentation [12, 17]. To achieve good
performance in these tasks, substantial images with careful annotations are
needed. Encouraged by the success of DCNN in computer vision, some researches
have directly applied DCNN models to analyze the medical images but cannot
achieve the same performance as in the natural images. The reasons lie in two
folds: 1. it is expensive to acquire accurate localization or classification
labels in chest X-ray images. 2. there exists much professional knowledge in
medical images that DCNN cannot exploit well. Therefore, how to exploit the
professional knowledge into DCNN models for solving these two questions still
opens a fully challenging problem. Our work transfers the knowledge into DCNN
models to reduce the problem of shortage of carefully annotated images.

Recent work paid much attention to utilize professional knowledge of chest
X-ray images into DCNN frameworks. However, they just proposed a simple fused
strategy to embed low-level information of chest X-ray into models, such as
Liu et al. [9] utilized contrastive learning to provide more localization
information with the help of healthy images. Zhao et al. [22] proposed to
exploit the contralateral information of chest X-ray via a simple fusion
module. These methods only exploit the apparent information of chest-Xray
images. They all overlooked the inner structure information of chest X-rays.
Therefore, they cannot apply their methods into real applications.

In this paper, we propose a Cross Chest Graph Network (CCG-Net) as shown in
Fig 1, which firstly utilizes deep expert knowledge to automatical detect
disease in chest X-ray images. We have known that medical experts have much
experience in finding out disease and how to treat patients. In fact, the
actions of medical experts consist of two phases: training and decision-making
processes. They pay much time to learn distinguish disease and embed their
experience into the decision process. During the training process, experts
would like to observe different areas and find out the relationship between
any pair of images. Our CCG-Net aims to model the observation way by a
knowledge-reasoning module to simulate the doctor’s habit of comparing
multiple images. Then we integrate intra-image and inter-image information
into a unified end-to-end framework.

Inspired from the experience of medical experts, our proposed CCG-Net consists
of four modules, 1. an end-to-end framework for deciding where and what is a
disease, 2. a inter-image relation module, which formulates the training
process of medical experts, to compare multiple images, 3. a intra-image
knowledge learning module, which builds the local relation graph for different
patches of chest X-ray images. Due to their highly structured property, every
chest X-ray image can be divided into several patches, we build a patch-wise
relation graph on them, 4. a knowledge reasoning module, which excavates the
inner knowledge from cross-image structural features. The last three
operations (2, 3, and 4) are similar to medical experts’ training process,
which learn intra-image and inter-image information to gain professional
knowledge. The first operation embeds the learned knowledge into DCNN
frameworks leading to better disease diagnosis models. Above all, our
contribution consists of three folds:

  * •

We propose CCG-Net, which is the first to formulate the medical experts’
training process by building relation graphs in the intra-image and inter-
image information of chest X-ray images. More generally, it provides
inspiration to address medical vision tasks with much professional knowledge
like in chest X-ray images.

  * •

We divide the experts’ professional actions into two stages including training
and decision-making processes. In addition, we utilize intra-image and inter-
image relation to learn much professional knowledge that would be embedded in
an end-to-end detection framework.

  * •

We achieve state-of-the-art results on the localization of NIH ChestX-ray14.

##  2 Related Work

###  2.1 Disease Detection

Object detection is one of the most important computer vision tasks, aiming to
localize and classify. Due to their strong feature representation ability,
DCNN achieved much progress in object detection tasks. For detection tasks,
DCNN methods consist of two style framework: 1. two-stage models, such as RCNN
series [16], 2. one-stage models, such as YOLO [15] and SSD [10]. However, for
disease detection, because of the shortage in careful annotations, traditional
detection framework cannot directly be applied in chest X-ray images. Besides,
since there is much distortion caused by other chest X-ray tissues, such low
contrast also causes the difficulty of disease finding.

Weakly supervised object detection (WSOD) can be considered as an effective
method to solve these problems. Based on CAM [23], researchers proposed many
techniques to use only image-level labels to detect objects. Although there is
no enough detection supervision, WSOD still achieved much progress. However,
researchers still face a big challenge when it comes to disease detection in
medical images. the existence of much professional knowledge greatly limits
the development of the applications of DCNN in medical fields. Therefore, in
this paper, we are inspired by the experts’ learning and decision processes to
propose CCG-Net, which not only exploits a larger amount of knowledge in chest
X-ray images but also builds a unified framework to detect disease in an end-
to-end style.

###  2.2 Knowledge-based Disease Diagnosis

Automatical disease diagnosis is a key problem in medical fields. However, due
to the shortage of careful annotations and the existence of much professional
knowledge, DCNN methods cannot achieve a good performance in medical tasks,
especially such a tough problem: disease detection in chest X-ray images. To
exploit medical knowledge and embed it into DCNN frameworks, researchers paid
much effort to utilize medical experts’ experience for disease diagnosis. Wang
et al. [20] firstly proposed a carefully annotated chest X-ray dataset and led
to a series of work that focuses on using image-level labels to localize the
disease. Li et al. [8] integrated classification and localization in a whole
framework with two multi instance-level losses and performed better. Liu et
al. [9] improved their work to propose contrastive learning of paired samples,
which utilizes healthy images to provide more localization information for
disease detection. Zhao et al. [22] proposed to utilize the symmetry
information in a chest X-ray to improve the disease localization performance.

Figure 2: The network consists of four modules: 1. an end-to-end framework for
disease detection under weakly-supervised settings, 2. the inter-image
relation module among different samples, 3. the intra-image knowledge learning
module based on the thoracic spatial structure, 4. the knowledge reasoning
module mining cross-image structural features. Our four modules are tightly
related and can be easily integrated into an end-to-end framework.

Besides, many works applied relation knowledge models to chest X-ray
diagnosis. Ypsilantis et al. [21], Pesce et al. [14], and Guan et al. [4]
proposed to build a relation attention model fusing DCNN models achieved much
progress. Li et al. [7] proposed a knowledge-graph based on medical reports
and images to determine the dependencies among chest X-ray images. Cheng et
al. [11] also proposed a new total strongly supervised dataset for
tuberculosis detection. However, they all overlooked the structural relation
among chest X-ray images. In this paper, we propose to build a structural,
relational graph for disease detection under weakly supervised scenarios in
chest X-ray images. Specifically, we build the global and local graph in chest
X-ray via three modules: 1. a inter-image relation module, 2. a intra-image
knowledge learning module, 3. knowledge reasoning module. Furthermore, we
integrate three modules into an end-to-end framework to jointly train our
network. Our proposed three relational modules provide better supervision
since we exploit the local structural knowledge and global relation among
different samples.

##  3 Method

###  3.1 Overview

Given the images $X=\\{x_{1},x_{2},...,x_{n}\\}$. Our proposed framework
consists of four modules:

  * •

The End to End Framework is to localize and classify the disease in chest
X-ray images. In our paper, we utilize the same multi-instance level losses
used in [9] and [8].

  * •

Inter-image Relation Module, which includes a learnable matrix $G\in
R^{n\times n}$. We also use a contrast-constrained loss to share similar
information of $X$ and exploit their contrasted structural knowledge. We build
a cross-sample graph for them to exploit the dependencies among different
samples. The graph $G\in R^{n\times n}$ is to build the inter-image relation
among sampled samples, which is a learnable matrix, and every element is
initialized by $\frac{1}{n}$. The element $g_{ij}$ of $G$,
$i,j\in\\{1,2,...,n\\}$, represents the similarity wight of images $x_{i}$ and
$x_{j}$.

  * •

Intra-image Knowledge Learning, which firstly acquires patch-wise features of
different images. Then the network can achieve a new image graph via building
a structural knowledge-based module. We denote this graph as $G_{k}\in
R^{n\times n}$. Assumed that the number of patches are $|p_{i}|$ and $|p_{j}|$
of images $x_{i}$ and $x_{j}$. The graph $G_{k}$ would be calculated on using
the graph $G_{l}\in R^{|p_{i}|\times|p_{j}|}$, which learns the relationship
between different paired patches of images.

  * •

Knowledge Reasoning Module, which is based on cross image structural
knowledge. When we get the whole structural information of different images,
we will utilize it to reason the inner structural dependencies among different
patches in different images.

###  3.2 End to End Framework

The end to end framework is to localize and classify the disease in chest
X-ray images in a coarse-grained style. More specifically, the input images
$X=\\{x_{1},x_{2},...,x_{n}\\}$ of the module are resized to $512\times 512$.
ResNet-50 pre-trained from the ImageNet dataset is adopted as the backbone for
this module. We use the feature map $F$ after C5 (last convolutional output of
5th-stage), which is 32 times down-sampled from the input image, and of size
$2048\times 16\times 16$. Each grid in the feature map denotes the existent
probability of disease. We pass $F$ through two $1\times 1$ convolutional
layers and a sigmoid layer to obtain the class-aware feature map P of size
$C\times H\times W$, where $C$ is the number of classes. Then we follow the
paradigm used in [9], computing losses and making predictions in each channel
for the corresponding class. For images with box-level annotations, if the
grid in the feature map overlaps with the projected ground truth box, we
assign label 1 to the grid. Otherwise, we assign 0 to it. Therefore, we use
the binary cross-entropy loss as used in [9] for each grid:

$L^{k}_{i}(\emph{P})=\sum_{j}-y_{ij}^{k}\log(p_{ij}^{k})-\sum_{j}(1-y_{ij}^{k})\log(1-p_{ij}^{k})$
(1)

where $k$, $i$, and $j$ are the index of classes, samples, and grids
respectively. $y^{k}_{ij}$ denotes the target label of the grid and
$p^{k}_{ij}$ denotes the predicted probability of the grid.

For images with only image-level annotations, we use the MIL loss used in [8].

$\begin{split}L^{k}_{i}(\emph{P})=-&y^{k}_{i}\log(1-\prod_{j}(1-p^{k}_{ij}))\\\
-&(1-y^{k}_{i})\log(\prod_{j}(1-p^{k}_{ij}))\end{split}$ (2)

where $y^{k}_{i}$ denotes the target label of the image. For this end to end
framework, the whole loss $L_{base}$ as shown in Fig. 2, is formulated as
follows.

$\begin{split}L_{base}=\sum_{i}\sum_{k}\lambda^{k}_{i}\beta_{B}L^{k}_{i}(\emph{P})+(1-\lambda^{k}_{i})L^{k}_{i}(\emph{P})\end{split}$
(3)

where $\lambda^{k}_{i}\in{0,1}$ denotes if the $k_{th}$ class in the $i_{th}$
sample has box annotation, and $\beta_{B}$ is the balance weight of the two
losses and is set to 4.

###  3.3 Inter-image Relation Module

Inter-image relation is formulated as a learnable matrix $G\in R^{n\times n}$.
A contrast-constrained loss is used to share similar information of $X$ and
exploit their contrasted structural knowledge, as following equation.

$\begin{split}L_{IR}=\frac{\sum_{(u,v)\in G}G(u,v)D(F_{u},F_{v})}{n\times
n}\end{split}$ (4)

$D(\cdot)$ is the distance metric function, where it is a Euclidean distance.
$F_{u}$ and $F_{v}$ means the feature map after C5 of the image $x_{u}$ and
$x_{v}$. We build a cross-sample graph for them to exploit the dependencies
among different samples. The graph $G\in R^{n\times n}$ is to build the inter-
image relation among sampled samples, which is a learnable matrix, and every
element is initialized by $\frac{1}{n}$. The element $g_{ij}$ of $G$,
$i,j\in\\{1,2,...,n\\}$, represents the similarity wight of images $x_{i}$ and
$x_{j}$. G is adaptively adjusted during training processes and changes with
diverse inputs to exploit the relationship fully.

###  3.4 Intra-image Knowledge Learning

Intra-image Knowledge Learning, which firstly utilizes Simple linear iterative
clustering (SLIC) [1], a super-pixel method to generate the patches for
different images. Assumed that the patches of the image $x_{i}$ is
$p_{i}={p^{i}_{1},p^{i}_{2},...,p^{i}_{m}}$. Then the network can achieve a
new image graph via building a structural knowledge-based module with the help
of $p_{i},i\in{1,2,...,n}$. We denote the graph as $G_{k}\in R^{n\times n}$,
which is the intra-image graph between paired images $x_{i}$ and $x_{j}$. The
graph $G_{k}$ is calculated on using the graph $G_{l}\in
R^{|p_{i}|\times|p_{j}|}$, which learns the dependencies among different
paired patches of images. Then the same contrast-constrained loss using this
graph to provide more structural knowledge for the whole framework.

$\begin{split}L_{IK}=\frac{\sum_{(u,v)\in
G_{k}}G_{k}(u,v)D(F_{u},F_{v})}{n\times n}\end{split}$ (5)
$\begin{split}G_{k}=W_{l}(G_{l})\end{split}$ (6)

Where, $W_{l}$ is a fully connected layer and

$\begin{split}G_{l}(l,p)=D^{{}^{\prime}}(H_{l},H^{{}^{\prime}}_{p}),l\in{1,2,...,|p_{i}|},p\in{1,2,...,|p_{j}|}\end{split}$
(7)

$H_{l}$ is the hash code [19] of the patch $p^{i}_{l}$ in the image $x_{i}$
and $H^{{}^{\prime}}_{p}$ is the hash code of the patch $p^{j}_{p}$ in the
image $x_{j}$. $D^{{}^{\prime}}(\cdot)$ is the Hamming distance.

###  3.5 Knowledge Reasoning Module

In addition to previous efforts to focus on information in a whole image, we
also explored the value of cross-image semantic relations in the medical
object. The correlations between patches across images are emphasized,
especially, the correlations between corresponding patches in two images.

Knowledge Reasoning Module focuses on the correlations of two images. After
getting the feature map $F_{u}$ and $F_{v}$ of the images, the affinity matrix
$P$ is firstly calculated between $F_{u}$ and $F_{v}$.

$P=F^{\mathrm{T}}_{u}W_{P}F_{v}\in\mathbb{R}^{HW\times HW}$

where the feature map $F_{u}\in\mathbb{R}^{C\times HW}$ and
$F_{v}\in\mathbb{R}^{C\times HW}$ are flattened into matrix formats, and
$W_{P}\in\mathbb{R}^{C\times C}$ is a learnable matrix. The affinity matrix
$P$ represents the similarity of all pairs of patches in $F_{u}$ and $F_{v}$.

Then $P$ is normalized column-wise to get the attention map of $F_{u}$ for
each patch in $F_{v}$ and row-wise to get the attention map of $F_{v}$ for
each patch in $F_{u}$.

$F^{{}^{\prime}}_{u}=F_{u}softmax(P)\in\mathbb{R}^{C\times HW}$
$F^{{}^{\prime}}_{v}=F_{v}softmax(P^{\mathrm{T}})\in\mathbb{R}^{C\times HW}$

where $softmax(P)$ and $softmax(P^{\mathrm{T}})$ pay attention to the similar
patches of the feature map $F_{u}$ and $F_{v}$ respectively. Therefore, they
can be used to enhance $F_{u}$ and $F_{v}$ respectively, so that similar
patches in $F_{u}$ and $F_{v}$ are highlighted.

The cross-image method can extract more contextual information between images
than using a single image. This module exploits the context of other related
images to improve the reasoning ability of the feature map, which is
beneficial to the localization and classification of disease in chest X-ray
images. Furthermore, we exploit the enhanced feature map to calculate the new
similarity between the paired images to gain a more strong supervisor.

$\begin{split}L_{KR}=\frac{\sum_{(u,v)\in
G_{k}^{{}^{\prime}}}G_{k}^{{}^{\prime}}(u,v)D(F^{{}^{\prime}}_{u},F^{{}^{\prime}}_{v})}{n\times
n}\end{split}$ (8)

T (IoU) | Models | Atelectasis | Cardiomegaly | Effusion | Infiltration | Mass | Nodule | Pneumonia | Pneumothorax | Mean
---|---|---|---|---|---|---|---|---|---|---
0.3 |  X, Wang [20] | 0.24 | 0.46 | 0.30 | 0.28 | 0.15 | 0.04 | 0.17 | 0.13 | 0.22
Z, Li [8] | 0.36 | 0.94 | 0.56 | 0.66 | 0.45 | 0.17 | 0.39 | 0.44 | 0.49
J, Liu [9] | 0.53 | 0.88 | 0.57 | 0.73 | 0.48 | 0.10 | 0.49 | 0.40 | 0.53
| Ours | 0.44 | 0.86 | 0.68 | 0.84 | 0.47 | 0.29 | 0.67 | 0.40 | 0.60
0.5 |  X, Wang [20] | 0.05 | 0.18 | 0.11 | 0.07 | 0.01 | 0.01 | 0.03 | 0.03 | 0.06
Z, Li [8] | 0.14 | 0.84 | 0.22 | 0.30 | 0.22 | 0.07 | 0.17 | 0.19 | 0.27
J, Liu [9] | 0.32 | 0.78 | 0.40 | 0.61 | 0.33 | 0.05 | 0.37 | 0.23 | 0.39
| Ours | 0.27 | 0.86 | 0.48 | 0.72 | 0.53 | 0.14 | 0.58 | 0.35 | 0.49
0.7 |  X, Wang [20] | 0.01 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 | 0.01
Z, Li [8] | 0.04 | 0.52 | 0.07 | 0.09 | 0.11 | 0.01 | 0.05 | 0.05 | 0.12
J, Liu [9] | 0.18 | 0.70 | 0.28 | 0.41 | 0.27 | 0.04 | 0.25 | 0.18 | 0.29
| Ours | 0.20 | 0.86 | 0.48 | 0.68 | 0.32 | 0.14 | 0.54 | 0.30 | 0.44

Table 1: The comparison results of disease localization among the models using
50% unannotated images and 80% annotated images. For each disease, the best
results are bolded.

T (IoU) | Models | Atelectasis | Cardiomegaly | Effusion | Infiltration | Mass | Nodule | Pneumonia | Pneumothorax | Mean
---|---|---|---|---|---|---|---|---|---|---
0.1 |  Z, Li [8] | 0.59 | 0.81 | 0.72 | 0.84 | 0.68 | 0.28 | 0.22 | 0.37 | 0.57
J, Liu [9] | 0.39 | 0.90 | 0.65 | 0.85 | 0.69 | 0.38 | 0.30 | 0.39 | 0.60
Ours | 0.66 | 0.88 | 0.79 | 0.85 | 0.69 | 0.28 | 0.40 | 0.47 | 0.63
0.3 |  J, Liu [9] | 0.34 | 0.71 | 0.39 | 0.65 | 0.48 | 0.09 | 0.16 | 0.20 | 0.38
Baseline | 0.36 | 0.69 | 0.35 | 0.64 | 0.44 | 0.08 | 0.02 | 0.23 | 0.35
Ours | 0.31 | 0.79 | 0.37 | 0.75 | 0.40 | 0.06 | 0.24 | 0.27 | 0.40
0.5 |  J, Liu [9] | 0.19 | 0.53 | 0.19 | 0.47 | 0.33 | 0.03 | 0.08 | 0.11 | 0.24
Baseline | 0.18 | 0.51 | 0.14 | 0.47 | 0.27 | 0.03 | 0.01 | 0.12 | 0.22
Ours | 0.19 | 0.71 | 0.14 | 0.52 | 0.31 | 0.08 | 0.05 | 0.13 | 0.27
0.7 |  J, Liu [9] | 0.08 | 0.30 | 0.09 | 0.25 | 0.19 | 0.01 | 0.04 | 0.07 | 0.13
Baseline | 0.11 | 0.34 | 0.06 | 0.32 | 0.20 | 0.01 | 0.00 | 0.06 | 0.14
Ours | 0.06 | 0.64 | 0.08 | 0.38 | 0.19 | 0.01 | 0.08 | 0.09 | 0.19

Table 2:  The comparison results of disease localization among the models
using 100% unannotated images and no any annotated images. For each disease,
the best results are bolded.

T (IoU) | Models | Atelectasis | Cardiomegaly | Effusion | Infiltration | Mass | Nodule | Pneumonia | Pneumothorax | Mean
---|---|---|---|---|---|---|---|---|---|---
0.3 |  J, Liu [9] | 0.55 | 0.73 | 0.55 | 0.76 | 0.48 | 0.22 | 0.39 | 0.30 | 0.50
Baseline | 0.47 | 0.84 | 0.65 | 0.82 | 0.33 | 0.04 | 0.57 | 0.29 | 0.50
Ours | 0.49 | 0.87 | 0.66 | 0.88 | 0.48 | 0.10 | 0.51 | 0.20 | 0.52
0.5 |  J, Liu [9] | 0.36 | 0.57 | 0.37 | 0.62 | 0.34 | 0.13 | 0.23 | 0.17 | 0.35
Baseline | 0.27 | 0.76 | 0.39 | 0.58 | 0.24 | 0.02 | 0.39 | 0.21 | 0.36
Ours | 0.26 | 0.80 | 0.41 | 0.67 | 0.15 | 0.06 | 0.42 | 0.18 | 0.37
0.7 |  J, Liu [9] | 0.19 | 0.47 | 0.20 | 0.41 | 0.22 | 0.06 | 0.12 | 0.11 | 0.22
Baseline | 0.14 | 0.62 | 0.20 | 0.42 | 0.07 | 0.00 | 0.23 | 0.08 | 0.22
Ours | 0.18 | 0.71 | 0.20 | 0.50 | 0.20 | 0.02 | 0.29 | 0.06 | 0.27

Table 3:  The comparison results of disease localization among the models
using 100% unannotated images and 40% annotated images. For each disease, the
best results are bolded.  Figure 3: Visualization of the predicted results on
both the baseline model and our method. The first column shows the original
images, the second and third columns show baseline and our method. The green
bounding box and red area mean the the ground truth and prediction.

The graph $G_{k}^{{}^{\prime}}$ is calculated on using the graph
$G^{{}^{\prime}}_{l}\in R^{|p_{i}|\times|p_{j}|}$.

$\begin{split}G_{k}^{{}^{\prime}}=W^{{}^{\prime}}_{l}(G^{{}^{\prime}}_{l})\end{split}$
(9)

where $W^{{}^{\prime}}_{l}$ is a fully connected layer and

$\begin{split}G_{l}^{{}^{\prime}}(l,p)=&D^{{}^{\prime}}(P_{l},P_{p}),\\\
&l\in\\{1,2,...,|p_{i}|\\},p\in\\{1,2,...,|p_{j}|\\}\end{split}$ (10)

$P_{l}$ is the $l$-th feature patch of $F^{{}^{\prime}}_{u}$ and $P_{p}$ is
the $p$-th feature patch of $F^{{}^{\prime}}_{v}$, respectively.

###  3.6 Training Loss

The overall loss function during the training is a weighted combination of
four loss functions,

$\displaystyle L_{all}=w_{1}L_{base}+w_{2}L_{IR}+w_{3}L_{IK}+w_{4}L_{KR}$ (11)

where $\sum^{4}_{i=1}w_{i}=1$. In our experiments, we always set
$w_{i}=0.25,i\in{1,2,..,4}$.

###  3.7 Training and Test

Figure 4: Visualization of the generated heatmap and ground truth of our
method, where the green bounding box means the ground truth.

Data | Models | Atelectasis | Cardiomegaly | Effusion | Infiltration | Mass | Nodule | Pneumonia | Pneumothorax | Mean
---|---|---|---|---|---|---|---|---|---|---
0.5_0.8 |  X, Wang [20] | 0.01 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 | 0.01
Z, Li [8] | 0.04 | 0.52 | 0.07 | 0.09 | 0.11 | 0.01 | 0.05 | 0.05 | 0.12
J, Liu [9] | 0.18 | 0.70 | 0.28 | 0.41 | 0.27 | 0.04 | 0.25 | 0.18 | 0.29
Baseline | 0.34 | 1.00 | 0.40 | 0.68 | 0.11 | 0.14 | 0.65 | 0.00 | 0.41
IK | 0.22 | 0.82 | 0.36 | 0.56 | 0.32 | 0.14 | 0.25 | 0.35 | 0.38
IR | 0.24 | 0.82 | 0.40 | 0.56 | 0.32 | 0.07 | 0.38 | 0.30 | 0.39
KR | 0.24 | 0.89 | 0.32 | 0.68 | 0.26 | 0.14 | 0.21 | 0.30 | 0.38
(IR+IK) | 0.20 | 0.86 | 0.48 | 0.68 | 0.32 | 0.14 | 0.54 | 0.30 | 0.44
IR+IK+KR | 0.27 | 0.86 | 0.40 | 0.56 | 0.37 | 0.14 | 0.13 | 0.30 | 0.38
1.0_0.0 |  J, Liu [9] | 0.08 | 0.30 | 0.09 | 0.25 | 0.19 | 0.01 | 0.04 | 0.07 | 0.13
Baseline | 0.11 | 0.34 | 0.06 | 0.32 | 0.20 | 0.01 | 0.00 | 0.06 | 0.14
IK | 0.10 | 0.59 | 0.07 | 0.37 | 0.20 | 0.00 | 0.13 | 0.06 | 0.19
GR | 0.06 | 0.61 | 0.07 | 0.28 | 0.14 | 0.00 | 0.05 | 0.08 | 0.16
IK | 0.09 | 0.63 | 0.06 | 0.36 | 0.22 | 0.00 | 0.09 | 0.07 | 0.19
IR+IK | 0.06 | 0.64 | 0.08 | 0.38 | 0.19 | 0.01 | 0.08 | 0.09 | 0.19
IR+IK+KR | 0.12 | 0.51 | 0.07 | 0.36 | 0.22 | 0.03 | 0.02 | 0.07 | 0.17
1.0_0.4 |  J, Liu [9] | 0.19 | 0.47 | 0.20 | 0.41 | 0.22 | 0.06 | 0.12 | 0.11 | 0.22
Baseline | 0.14 | 0.62 | 0.20 | 0.42 | 0.07 | 0.00 | 0.23 | 0.08 | 0.22
IK | 0.14 | 0.66 | 0.09 | 0.47 | 0.15 | 0.00 | 0.30 | 0.06 | 0.23
GR | 0.14 | 0.75 | 0.24 | 0.42 | 0.11 | 0.00 | 0.26 | 0.12 | 0.25
KR | 0.13 | 0.68 | 0.20 | 0.47 | 0.19 | 0.06 | 0.17 | 0.08 | 0.25
IR+IK | 0.13 | 0.72 | 0.13 | 0.43 | 0.20 | 0.00 | 0.23 | 0.06 | 0.24
IR+IK+KR | 0.18 | 0.71 | 0.20 | 0.50 | 0.20 | 0.02 | 0.29 | 0.06 | 0.27

Table 4:  The comparison results of disease localization among the models
using three sets of data at T(IoU)=0.7, including 50% unannotated and 80%
annotated images (0.5_0.8), 100% unannotated and no any annotated images
(1.0_0.0), and 100% unannotated and 40% unannotated images (1.0_0.4). For each
disease, the best results are bolded.

Training All the models are trained on NIH chest X-ray dataset using the SGD
algorithm with the Nesterov momentum. With a total of 9 epochs, the learning
rate starts from 0.001 and decreases by 10 times after every 4 epochs.
Additionally, the weight decay and the momentum is 0.0001 and 0.9,
respectively. All the weights are initialized by pre-trained ResNet [6] models
on ImageNet [2]. The mini batch size is set to 2 with the NVIDIA 1080Ti GPU.
All models proposed in this paper are implemented based on PyTorch [13].

Testing We also use the threshold of 0.5 to distinguish positive grids from
negative grids in the class-wise feature map as described in [8] and [9]. All
test setting is same as [9], we also up-sampled the feature map before two
last fully convolutional layers to gain a more accurate localization result.

##  4 Experiments

###  4.1 Dataset and Evaluation Metrics

Dataset. NIH chest X-ray dataset [20] include 112,120 frontal-view X-ray
images of 14 classes of diseases. There are different diseases in each image.
Furthermore, the dataset contains 880 images with 984 labeled bounding boxes.
We follow the terms in [8] and [9] to call 880 images as “annotated” and the
remaining 111,240 images as “unannotated”. Following the setting in [9], we
also resize the original 3-channel images from resolution of $1024\times 1024$
to $512\times 512$ without any data augmentation techniques.

Evaluation Metrics. We follow the metrics used in [8]. The localization
accuracy is calculated by the IoU (Intersection over Union) between
predictions and ground truths. Since it is a coarse-grained task, our
localization predictions are discrete small rectangles. The eight diseases
with ground truth boxes is reported in our paper. The localization result is
regarded as correct when $IoU>T(IoU)$, where T(*) is the threshold.

###  4.2 Comparison with the State-of-the-art

In order to evaluate the effectiveness of our models for weakly supervised
disease detection, we design the experiments on three sets of data and conduct
a 5-fold cross-validation. In the first experiment, we use the 50% unannotated
images and 80% annotated images for training, and test the models with the
remaining 20% annotated images. In the second experiment, we use the 100%
unannotated images and no any annotated images for training, and test the
models with all annotated images. In the third experiment, we use the 100%
unannotated images and 40% annotated images for training, and test the models
with remaining 60% annotated images. Additionally, our experimental results
are mainly compared with four methods. The first method is X, Wang [20], which
proposes a carefully annotated chest X-ray dataset and a unified weakly
supervised multi-label image classification and disease localization
framework. The second method is Z, Li [8], which uses fully convolutional
neural network to localize and classify the disease in chest X-ray images. The
third method is J, Liu [9], which proposes contrastive learning of paired
samples to provide more localization information for disease detection. The
last method is our baseline model, which is a unified end-to-end framework
that doesn’t use our approach to locate and classify the disease.

In the first experiment, we compare the localization results of our model with
[20], [8] and [9]. We can observe that our model outperforms existing methods
in most cases, as shown in Table 1. Particularly, with the increase of T(IoU),
our model has greater advantages over the reference models. For example, when
T(IoU) is 0.3, the mean accuracy of our model is 0.60, and outperforms [20],
[8] and [9] by 0.38, 0.11 and 0.07 respectively. However, when T(IoU) is 0.7,
the mean accuracy of our model is 0.44, and outperforms [20], [8] and [9] by
0.43, 0.32 and 0.15 respectively. Overall, the experimental results shown in
Table 1 demonstrate that our method is more accurate for disease localization
and classification, which provides a great role for clinical practices.

In the second experiment, we train our model without any annotated images
comparing the first experiment. Since [8] only provides the results when
T(IoU) = 0.1, in order to better show the performance of our model, we add an
evaluation method of T(IoU) = 0.1. It can be seen that our model outperforms
[8] and [9] in most cases, as shown in Table 2. For example, when T(IoU) is
0.1, the mean accuracy of our models is 0.63, which is 0.06 higher than [8],
and 0.03 higher than [9]. Furthermore, when T(IoU) is 0.7, the mean
localization result of our model is 0.19, which is 0.06 higher than [8] and
0.05 higher than [9]. Compared with the baseline model, our approach performs
better in most classes except for “Atelectasis” and “Nodule”. The trend stays
the same that at higher T(IoU), our approach demonstrates more advantages over
baseline methods. The added unannotated training samples contribute more than
the removed annotated ones in those classes, which implies that our approach
can better utilize the unannotated samples. The overall results show that even
without annotated data used for training, our approach can achieve decent
localization results.

In the third experiment, we use more annotated images comparing the second
experiment. We compare the localization results of our model with [9] in same
data setting. It can be seen that our model outperforms [9] in most cases, as
shown in Table 3. With T(IoU) = 0.3 and 0.7, our model outperforms [9] by 0.02
and 0.05 respectively. Similar improvements are achieved comparing the second
experiment. Overall, the experimental results demonstrate that our method can
improve the performance of models with limited annotated images.

To better demonstrate the final effect of our approach on disease localization
and classification, we visualize some of typical predictions of both the
baseline model and our method, as shown in Figure 3. The first column shows
the original images, the second and third columns show baseline model and our
method. The green bounding box and red area mean the ground truth and
prediction. It can be seen that our models can predict more accurate in most
cases comparing the baseline model. For example, the class “Atelectasis” and
“Nodule”, the localization reslut of the baseline model is completely
inconsistent with the ground truth, but the localization reslut of our method
is consistent with the ground truth. It shows that using the structural
information of intra-image and inter-image can improve the performance of
automatic lesion detection. Additionally, we also visualize the generated
heatmap and ground truth of our model, as shown in Figure 4. It can be seen
that the proposed method can effectively locate and classify medical images.

###  4.3 Ablation Studies

In this section, we explore the influence of different modules on our method
for ablation studies. To evaluate our method more comprehensively, we build 6
models, including the model of the end to end framework (Baseline), the model
with the intra-image knowledge learning (IK), the model with the inter-image
relation module (IR), the model with the knowledge reasoning (KR), the model
combining the inter-image relation module and the intra-image knowledge
learning (IR+IK), the model combining the inter-image relation module, the
intra-image knowledge learning and the knowledge reasoning module (IR+IK+KR).

Table 4 shows the results of the three experiments mentioned in section 4.2 at
T(IOU)=0.7. It can be seen that our method performs better in most classes
except for “Atelectasis”, “Effusion” and “Mass” comparing [20], [8] and [9].
Furthermore, comparing the baseline model, it can be observed that the
performance of our other models are improved in most cases, which shows that
our method is effective for improving model performance. However, a model does
not always maintain the advantage in the three experiments, for example, the
model (IR+IK) achieves the best performance in the data (0.5_0.8), the model
(IK), the model (KR) and the model (IR+IK) achieve the best performance in the
data (1.0_0.0), and the model (IR+IK+KR) achieves the best performance in the
data (1.0_0.4). Overall, the experimental results demonstrate that using
structural relational information can improve the performance of models. For
different experimental data, our models can achieve different results. It is
difficult for us to determine which model is the best, but we can be sure that
our method is effective, because no matter what kind of data we use, our
models achieve great improvement. Particularly, the method can achieve good
localization results even without any annotation images for training.

##  5 Conclusion

By imitating doctor’s training and decision-making process, we propose the
Cross-chest Graph (CCG) to improve the performance of automatic lesion
detection under limited supervision. CCG models the intra-image relationship
between different anatomical areas by leveraging the structural information to
simulate the doctor’s habit of observing different areas. Meanwhile, the
relationship between any pair of images is modeled by a knowledge-reasoning
module to simulate the doctor’s habit of comparing multiple images. We
integrate intra-image and inter-image information into a unified end-to-end
framework. Experimental results on the NIH Chest-14 dataset demonstrate that
the proposed method achieves state-of-the-art performance in diverse
situations.

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