EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-114 LHCb-PAPER-2013-028 September 9, 2013

Measurement of the relative rate of prompt $\chi_{c0}$, $\chi_{c1}$ and
$\chi_{c2}$ production at $\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$

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

Prompt production of charmonium $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$
mesons is studied using proton-proton collisions at the LHC at a centre-of-
mass energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The $\chi_{c}$
mesons are identified through their decay to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$, with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$ $\mu^{+}\mu^{-}$
using photons that converted in the detector. A data sample, corresponding to
an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected by the LHCb
detector, is used to measure the relative prompt production rate of
$\chi_{c1}$ and $\chi_{c2}$ in the rapidity range $2.0<y<4.5$ as a function of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum from 3
to 20 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. First evidence for $\chi_{c0}$
meson production at a high-energy hadron collider is also presented.

Submitted to JHEP.

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, V.
Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-
Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F.
Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M.
Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53,
T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W. Hulsbergen40,
P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V.
Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40,
P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C.
Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M.
Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7,
I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40,
L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F.
Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N.
La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W.
Lambert41, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J.
Maratas5, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, B. Mc Skelly51, J. McCarthy44, A. McNab53, R. McNulty12, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A.
Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40, F. Muheim49, K. Müller39,
R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N. Neufeld37, A.D.
Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-
Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R.
Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A.
Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37, A.
Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G. Passaleva17,
G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C. Pavel-
Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i,
A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S.
Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E.
Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54,
J. Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig
Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38,
M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M.
Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
V. Salustino Guimaraes2, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i,
R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,
P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva Coutinho47, M. Sirendi46,
T. Skwarnicki58, N.A. Smith51, E. Smith54,48, J. Smith46, M. Smith53, M.D.
Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B.
Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U.
Straumann39, V.K. Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M.
Szczekowski27, P. Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7,
E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N.
Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A.
Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27,
D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez
Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M.
Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39,
D. Volyanskyy10, D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H.
Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58,
D.R. Ward46, N.K. Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J.
Wicht37, J. Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, M.
Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z.
Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, 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
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

The study of charmonium production provides an important test of the
underlying mechanisms described by quantum chromodynamics (QCD). In $pp$
collisions charmonia can be produced directly, or indirectly via the decay of
higher excited states (feed-down) or via the decay of $b$ hadrons. The first
two are referred to as prompt production. The mechanism for the production of
the prompt component is not yet fully understood, and none of the available
models adequately predicts both the transverse momentum spectrum and the
polarization of the promptly produced charmonium states [1].

At the LHC, $c\overline{}c$ pairs are expected to be produced at leading order
(LO) through gluon-gluon interactions, followed by the formation of bound
charmonium states. The production of the $c\overline{}c$ pair is described by
perturbative QCD while non-perturbative QCD is needed for the description of
the evolution of the $c\overline{}c$ pair to the bound state. Several models
have been developed for the non-perturbative part, such as the Colour Singlet
(CS) model [2, 3, 4] and the non-relativistic QCD (NRQCD) model [5]. The CS
model assumes the $c\overline{}c$ pair is created in a hard scattering
reaction as a colour singlet with the same quantum numbers as the final
charmonium state. The NRQCD model includes, in addition to the colour singlet
mechanism, the production of $c\overline{}c$ pairs as colour octets (CO) (in
this case the CO state evolves to the final charmonium state via soft gluon
emission). These two models predict different ratios of the $\chi_{c2}$ to
$\chi_{c1}$ production cross-sections.

The study of the production of $\chi_{c}$ states is also important since these
resonances give a substantial feed-down contribution to prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production [6] through their
radiative decay $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ and can have a significant impact on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization measurement [7].
Measurements of $\chi_{c1}$ and $\chi_{c2}$ production cross-section for
various particle beams and energies have been reported in Refs. [8, 9, 10, 11,
12].

In this paper we report a measurement of the ratio of prompt $\chi_{c2}$ to
$\chi_{c1}$ production cross-sections
$\sigma(pp\rightarrow\chi_{c2}X)/\sigma(pp\rightarrow\chi_{c1}X)$ at a centre-
of-mass energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ in the rapidity
range $2.0<y<4.5$ as a function of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ transverse momentum ($p_{\rm T}$) from 3 to
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The data sample corresponds to an
integrated luminosity of 1.0 fb-1 collected during 2011 by the LHCb detector.
The radiative decay $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ is used, where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ is reconstructed in the dimuon final state and only photons that
convert in the detector material are used. The converted photons are
reconstructed using $e^{+}$ and $e^{-}$ tracks, which allows a clean
separation of the $\chi_{c1}$ and $\chi_{c2}$ peaks, due to a better energy
resolution of converted photons than for those that are identified with the
calorimeter (referred to as calorimetric photons in the following).

The measurement performed by LHCb using calorimetric photons with 2010 data
[12] was limited by the fact that the two $\chi_{c}$ peaks were not well
separated. The measurements with calorimetric [12] and converted (as presented
in this study) photons are largely uncorrelated since the photon
reconstruction is based on different subdetectors. Furthermore, this is the
first measurement using converted photons in LHCb. The $\chi_{c0}$ state has
been previously observed in ${p}\overline{}p$ collisions at threshold [13],
but this letter reports the first evidence at high-energy hadron colliders.
Its production rate relative to that of the $\chi_{c2}$ is also reported.

##  2 The LHCb detector and dataset

The LHCb detector [14] is a single-arm forward spectrometer covering the
pseudorapidity 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 (VELO) 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 provides a momentum measurement with relative
uncertainty 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 impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad (SPD) and preshower detectors, an
electromagnetic calorimeter (ECAL) and a hadronic calorimeter. The SPD and
preshower are designed to distinguish between signals from photons and
electrons. The ECAL is constructed from scintillating tiles interleaved with
lead tiles. The reconstruction of converted photons that are used in this
analysis is described in Sec. 3. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers. The total
radiation length before the first tracking station is about 0.25$X_{0}$ [14].

The LHCb coordinate system is defined to be right-handed with its origin at
the nominal interaction point, the $z$ axis aligned along the beam line
towards the magnet and the $y$ axis pointing upwards. The magnetic field is
oriented along the $y$ axis.

The trigger [15] 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. Candidate events used in this analysis are first
required to pass a hardware trigger, which selects muons with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or dimuon candidates with a
product of their $p_{\rm T}$ larger than
$1.68~{}({\mathrm{Ge\kern-1.00006ptV\\!/}c})^{2}$. In the subsequent software
trigger, both muons are required to have $\mbox{$p_{\rm
T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, total momentum
$\mbox{$p$}>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and dimuon invariant mass
greater than $2.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

In the simulation, $pp$ collisions are generated using Pythia 6.4 [16] with a
specific LHCb configuration [17]. The NRQCD matrix elements are used in Pythia
6.4. Decays of hadronic particles are described by EvtGen [18], in which final
state radiation is generated using Photos [19]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [20, *Agostinelli:2002hh] as described in Ref. [22]. The
simulated samples consist of events in which at least one
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay
takes place. In a first sample used for background studies there is no
constraint on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production
mechanism. In the second sample used for the estimation of signal efficiencies
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is required to originate
from a $\chi_{c}$ meson.

##  3 Event reconstruction and selection

Photons that convert in the detector material are reconstructed from a pair of
oppositely charged electron candidates. Since photons that have converted in
the VELO have lower acceptance and worse energy resolution, only
$\gamma\rightarrow e^{+}e^{-}$ candidates without VELO hits are considered.
This selection strongly favours conversions that occur between the downstream
end of the VELO and the first tracking station upstream of the magnet.

Candidate $e^{+}e^{-}$ pairs are required to be within the ECAL acceptance and
produce electromagnetic clusters that have compatible $y$ positions. A
bremsstrahlung correction is applied to each electron track: any photon whose
position in the ECAL is compatible with a straight line extrapolation of the
electron track from the first tracking stations is selected and its energy is
added to the electron energy from the reconstructed track. If the same
bremsstrahlung candidate is found for both the $e^{+}$ and the $e^{-}$ of the
pair, the photon energy is added randomly to one of the tracks. The $e^{+}$
and $e^{-}$ tracks (corrected for bremsstrahlung) are then extrapolated
backward in order to determine the conversion point and a vertex fit is
performed to reconstruct the photon. The photon’s invariant mass is required
to be less than 100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Combinatorial
background is suppressed by applying a cut on the $e^{+}e^{-}$ invariant mass
($M_{e^{+}e^{-}}$) such that $M_{e^{+}e^{-}}<0.04\times z_{\rm
vtx}+20\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ where $z_{\rm vtx}$ is the
$z$ coordinate of the conversion in mm. Converted photons are required to have
transverse momentum ($p_{\rm T}^{\gamma}$) greater than
0.6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate is reconstructed
in its decay to $\mu^{+}\mu^{-}$. Each track must be identified as a muon with
$\mbox{$p_{\rm T}$}>0.65{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\mbox{$p$}>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a track fit
$\chi^{2}/\rm{ndf}$ smaller than 5, where ndf is the number of degrees of
freedom. The two muons must originate from a common vertex with vertex fit
$\chi^{2}_{\rm{vtx}}/\rm{ndf}$ smaller than 20. In addition the
$\mu^{+}\mu^{-}$ invariant mass is required to be in the range
3058–3138${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\gamma$ candidates are
associated with the primary vertex (PV) to which they have the smallest impact
parameter. These ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and photon
candidates are combined to form a $\chi_{c}$ candidate. Loose requirements are
applied in order to reject combinatorial background and poorly reconstructed
candidates using the following variables: the difference in $z$-positions of
the primary vertices associated with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\gamma$, the $\chi^{2}$ of
the $\chi_{c}$ candidate vertex fit and the difference between the $\chi^{2}$
of the PV reconstructed with and without the $\chi_{c}$ candidate. These cuts
remove about $20\%$ of the background and $5\%$ of the signal. Contributions
from $b\rightarrow\chi_{c}X$ are suppressed by requiring that the $\chi_{c}$
decay time is smaller than 0.15 ps. This removes about 85% of non-prompt
events and 0.5% of the prompt $\chi_{c}$ signal. Figure 1 shows the
distribution of the difference in the invariant masses of the $\chi_{c}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ selected candidates $\Delta
M\equiv M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})$ for candidates with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum ($p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$) in the range
3–20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

Figure 1:  Distribution of the mass difference $\Delta M\equiv
M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})$ for $\chi_{c}$ candidates with
$3<\mbox{$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$.

##  4 Determination of the ratio of cross-sections

The production cross-section ratio of the $\chi_{c2}$ and $\chi_{c1}$ mesons
is measured in ten $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
bins of different width (the bin limits are given in Table 1) with

$\frac{\sigma(\chi_{c2})}{\sigma(\chi_{c1})}=\frac{N_{\chi_{c2}}}{N_{\chi_{c1}}}\frac{\varepsilon_{\chi_{c1}}}{\varepsilon_{\chi_{c2}}}\frac{{\cal
B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{{\cal
B}(\chi_{c2}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)},$
(1)

where $\sigma(\chi_{cJ})$ is the prompt $\chi_{cJ}$ production cross-section,
$N_{\chi_{cJ}}$ is the prompt $\chi_{cJ}$ yield ($J=1,2$), and ${\cal
B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=(34.4\pm 1.5)\%$ and ${\cal
B}(\chi_{c2}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=(19.5\pm 0.8)\%$ [23] are the known branching fractions. The
efficiency ratio is expressed as

$\frac{\varepsilon_{\chi_{c1}}}{\varepsilon_{\chi_{c2}}}=\frac{\varepsilon^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}_{\chi_{c1}}}{\varepsilon^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}_{\chi_{c2}}}\frac{\varepsilon^{\gamma}_{\chi_{c1}}}{\varepsilon^{\gamma}_{\chi_{c2}}},$
(2)

where $\varepsilon^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}_{\chi_{cJ}}$
is the efficiency to trigger, detect, reconstruct and select a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from a $\chi_{cJ}$ decay and
$\varepsilon^{\gamma}_{\chi_{cJ}}$ is the efficiency to detect, reconstruct
and select a photon from a $\chi_{cJ}$ decay once the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ has been selected and then to
select the $\chi_{cJ}$ meson. The efficiency
$\varepsilon^{\gamma}_{\chi_{cJ}}$ includes the probability for a photon to
convert upstream of the first tracking station (about $20\%$).

The ratio $\sigma(\chi_{c0})/\sigma(\chi_{c2})$ is also measured with
appropriate substitutions in Eqs. 1 and 2 and using the known value ${\cal
B}(\chi_{c0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=(1.17\pm 0.08)\%$ [23]. Due to this small branching fraction,
the number of reconstructed $\chi_{c0}$ mesons is also small and therefore the
ratio of production cross-sections is only measured in one wide $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin, 4–20
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\chi_{c0}$ cross-section is
measured relative to the $\chi_{c2}$ cross-section rather than to the
$\chi_{c1}$ cross-section because the $p_{\rm T}$ dependence is expected to be
similar inside this $p_{\rm T}$ range for $\chi_{c0}$ and $\chi_{c2}$ [24].

###  4.1 Background studies

There are two sources of background: a peaking component from non-prompt
$\chi_{c}$ (from $b$-hadron decays) production and a non-peaking combinatorial
contribution.

The peaking background is estimated by fitting the decay time distribution of
the $\chi_{c}$ candidates with decay time larger than 0.3 ps with an
exponential shape and extrapolating into the signal region ($0-0.15$ ps). The
combinatorial background from $b$-hadron decays lying under the peak is
evaluated using the lower or upper mass sidebands. The two estimates agree and
the average is used to subtract its contribution. The simulation predicts that
$\chi_{c}$ mesons from $b$-hadron decays tend to be more energetic than prompt
$\chi_{c}$ mesons. The fraction of peaking background is therefore estimated
in two regions of $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
below and above 9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the maximum
deviation from the mean value inside each range (as predicted by simulation)
is taken as a systematic uncertainty. For the $\chi_{c1}$ meson the remaining
peaking background is $(0.9\pm 0.3)\%$ of the signal for $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ below
9${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $(1.8\pm 0.4)\%$ above this value.
As expected [23, 25] the number of non-prompt $\chi_{c2}$ candidates is
smaller. The relative yield of non-prompt $\chi_{c2}$ and $\chi_{c1}$ mesons
is obtained from a fit to the $\Delta M$ distribution of the events rejected
by the cut on the $\chi_{c}$ decay time (using the method described in Sec.
4.3). The ratio of branching fractions is determined to be

$\frac{{\cal B}(b\rightarrow\chi_{c2})\times{\cal
B}(\chi_{c2}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{{\cal B}(b\rightarrow\chi_{c1})\times{\cal
B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}=0.184\pm 0.025\,\mathrm{(stat)}\pm 0.015\,\mathrm{(syst)},$

where the systematic uncertainty is obtained by varying the fit function
parameters. The remaining number of non-prompt $\chi_{c2}$ candidates is then
determined as the number of remaining non-prompt $\chi_{c1}$ mesons multiplied
by this ratio of branching fractions. For the $\chi_{c0}$ peak it is not
possible to estimate the non-prompt contribution from the data but this is
expected to be at most $2\%$. This assertion is based on the similar values
for ${\cal B}(b\rightarrow\chi_{c1}X)$ and ${\cal B}(b\rightarrow\chi_{c0}X)$
[23] and the small contamination of $b\rightarrow\chi_{c1}X$ decays as shown
above. Another peaking background arises from the decay of prompt $\psi{(2S)}$
to a $\chi_{c}$ meson. According to simulation and cross-section measurements
[26] this background can be safely neglected.

The shape of the combinatorial background is estimated using the selected data
sample by generating “fake photons” to mimic the candidate photon spectra in
data. For each $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ candidate, two fake photons are generated: one where the photon
energy is set equal to twice the $e^{-}$ energy, and a second where twice the
$e^{+}$ energy is used. In this way, a spread of fake photon energies are
produced, all with the same angular distribution as the candidate photons in
the data. Each of these photons is then combined with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate to form the fake
$\chi_{c}$ candidate. The contribution from the $\chi_{c}$ signal region is
normalized to the estimated background contribution in the same invariant mass
region (this procedure converges with few iterations). The procedure was
tested on simulated events and reproduces the $\Delta M$ distribution of the
combinatorial background in the region of the $\chi_{c1}$ and $\chi_{c2}$
signal peaks.

###  4.2 Efficiency corrections

The ratio of the overall efficiencies for the detection of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons originating from the
decay of a $\chi_{c1}$ meson compared to a $\chi_{c2}$ meson,
$\varepsilon^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}_{\chi_{c1}}/\varepsilon^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}_{\chi_{c2}}$, is estimated from simulation and is compatible with
unity for all $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins.

Since the kinetic energy released in the $\chi_{c1}$ decay ($Q$-value) is
smaller than that of the $\chi_{c2}$ decay, the photon $p_{\rm T}$ spectrum
differs for the two decays. As a result, the photon $p_{\rm T}$ requirement
($\mbox{$p_{\rm T}^{\gamma}$}>0.6$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) has
a lower efficiency for the $\chi_{c1}$ decay. Moreover the reconstruction
efficiency of the converted photon decreases as the photon $p_{\rm T}$
decreases. This is due to the fact that low energy electrons escape the
detector before reaching the calorimeter and are therefore not identified as
electrons. Thus, the efficiency ratio is expected to be smaller than unity.
The value obtained from simulation is
$\varepsilon^{\gamma}_{\chi_{c1}}/\varepsilon^{\gamma}_{\chi_{c2}}=0.95\pm
0.01$ and shows no significant dependence on $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$.

The conversion probability and total efficiency for converted photons is
cross-checked using $\pi^{0}$ mesons, reconstructed either with two
calorimetric photons or with one calorimetric photon and one converted photon.
The ratio of efficiencies of converted photons to calorimetric photons is
measured in data and simulation as a function of $p_{\rm T}^{\gamma}$ and is
shown in Fig. 2(a). The total efficiency for calorimetric photons is described
well by simulation [25] therefore these measurements give a direct comparison
of the converted photon efficiency in data and simulation. The efficiency with
which converted photons are reconstructed in simulation is consistent with
data (within about $15\%$). The results obtained from this study are used to
correct the simulation. The corrected
$\varepsilon^{\gamma}_{\chi_{c1}}/\varepsilon^{\gamma}_{\chi_{c2}}$ ratio is
shown as a function of $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ in Fig. 2(b). This ratio is still compatible with a constant:
$\varepsilon^{\gamma}_{\chi_{c1}}/\varepsilon^{\gamma}_{\chi_{c2}}=0.96\pm
0.01$.

For the $\chi_{c0}$ to $\chi_{c2}$ ratio the corrected efficiency ratio is
$\varepsilon_{\chi_{c2}}/\varepsilon_{\chi_{c0}}=1.69\pm 0.18$. The departure
from unity is due to the different $Q$-values of the two decays, as discussed
above.

|
---|---
Figure 2: (a) Efficiency of converted photon reconstruction and selection
relative to the calorimetric photon efficiency for data (red circles) and
simulated events (blue triangles) as a function of $p_{\rm T}^{\gamma}$. (b)
Ratio of photon efficiencies
$\varepsilon^{\gamma}_{\chi_{c1}}/\varepsilon^{\gamma}_{\chi_{c2}}$ as a
function of $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ from
simulation (blue triangles) and after correcting the simulation for the
converted photon efficiency measured in data (red circles) taken from plot
(a).

###  4.3 Determination of the yield ratios

The $\Delta M$ spectrum is fitted to determine the signal yields. The
$\chi_{c1}$ and $\chi_{c2}$ signal peaks are each parametrized with a double-
sided Crystal Ball (CB) function [27]

$\displaystyle f_{i}(x)\propto$ $\displaystyle{\rm
exp}(-\frac{1}{2}(\frac{x-\Delta M_{i}}{\sigma_{i}})^{2})$
$\displaystyle~{}\mbox{ for }~{}-\alpha_{L}<\frac{x-\Delta
M_{i}}{\sigma_{i}}<\alpha_{R}$ $\displaystyle f_{i}(x)\propto$
$\displaystyle\frac{(n_{L}/\alpha_{L})^{n_{L}}{\rm
exp}(-\frac{1}{2}\alpha_{L}^{2})}{(n_{L}/\alpha_{L}-\alpha_{L}-(x-\Delta
M_{i})/\sigma_{i})^{n_{L}}}$ $\displaystyle~{}\mbox{ for }~{}\frac{x-\Delta
M_{i}}{\sigma_{i}}<-\alpha_{L}$ (3) $\displaystyle f_{i}(x)\propto$
$\displaystyle\frac{(n_{R}/\alpha_{R})^{n_{R}}{\rm
exp}(-\frac{1}{2}\alpha_{R}^{2})}{(n_{R}/\alpha_{R}-\alpha_{R}+(x-\Delta
M_{i})/\sigma_{i})^{n_{R}}}$ $\displaystyle~{}\mbox{ for }~{}\frac{x-\Delta
M_{i}}{\sigma_{i}}>\alpha_{R},$

where the index $i=1$ (2) refers to the $\chi_{c1}$ ($\chi_{c2}$) CB function.
The left tail accounts for events with unobserved bremsstrahlung photon(s)
while the right tail accounts for events reconstructed with background
photons. Simulation shows that the same $\alpha$ and $n$ parameters can be
used for both the $\chi_{c1}$ and $\chi_{c2}$ peaks and that the $\chi_{c2}$
mass resolution, $\sigma_{2}$, is $10\%$ larger than the $\chi_{c1}$ mass
resolution, $\sigma_{1}$. These constraints are used in all the fits. A
$\chi_{c0}$ contribution is also included and is modelled by the convolution
of a CB and a Breit-Wigner distribution with the width set to the $\chi_{c0}$
natural width ($10.4\pm 0.6$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [23])
and with the peak position fixed from simulation. For the $\chi_{c0}$ CB
shape, the same tail parameters are used as for the $\chi_{c1}$ and
$\chi_{c2}$ CB functions.

The full data sample ($3<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) after background
subtraction is fitted with the sum of these three functions. The peak
positions $\Delta M_{1}$ and $\Delta M_{2}$, the $\chi_{c1}$ resolution
$\sigma_{1}$ and the CB $n$ parameters obtained from this fit are then used
for the individual fits in each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. The same fit is
performed on simulated $\chi_{c}$ events (without background) and the value of
the $n$ parameter is found compatible with the data for the left tail while
slightly smaller for the right tail. These values are used when studying
systematic effects. The $\chi_{c}$ mass resolution is also found to be
significantly smaller in simulation due to better energy resolution in the
reconstruction of converted photons.

For each $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin the
combinatorial background shape is determined using the candidates
reconstructed with the fake photons. The $\Delta M$ distribution of these
candidates is fitted with an empirical function

$f_{\rm bkg}(\Delta M)\propto{\arctan}\biggl{(}\frac{\Delta
M-m_{0}}{c}\biggr{)}+b\biggl{(}\frac{\Delta M}{m_{0}}-1\biggr{)}+a,$ (4)

where $m_{0}$, $a$, $b$ and $c$ are free parameters. This function is then
used to parametrize the combinatorial background with all parameters fixed
except for the normalization. In total there are six free parameters for each
fit: the CB function $\alpha$ parameters (left and right tails), the height of
the $\chi_{c1}$ and $\chi_{c0}$ peaks, the ratio of $\chi_{c2}$ to $\chi_{c1}$
heights and the background normalization. Figure 3 shows the $\Delta M$
distribution and the fit results for two ranges: $4<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$}<5$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $11<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$}<13$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

The $\chi_{c0}$ yield is not significant in the individual bins and is
therefore only measured over the integrated range $4<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The region 3–4
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is excluded because for this particular
$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin the background
is high and not well modelled below
300${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, close to the $\chi_{c0}$ peak.
Figure 4 (a) shows the total $\Delta M$ distribution superimposed with the
background estimate using the fake photons and the fit to this background
distribution. The $\chi_{c0}$ contribution is visible just above
$300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Figure 4 (b) shows the result
of the fit for $4<\mbox{$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ after background
subtraction.

|
---|---
Figure 3:  Distribution of $\Delta M=M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})$ for $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range (a) 4–5 ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and (b) 11–13 ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$. The results of the fit are also shown, with the total fitted function (blue solid curve), the $\chi_{c1}$ signal (green dashed curve), the $\chi_{c2}$ signal (red dot-dashed curve) and the $\chi_{c0}$ signal (purple long-dashed curve).  |
---|---
Figure 4:  Distribution of $\Delta
M=M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})$ (blue histogram) for
$4<\mbox{$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$. (a) The background
estimated using fake photons (green) is superimposed on the $\Delta M$
distribution, together with the function used to parametrize it (black solid
line). (b) The same $\Delta M$ distribution after background subtraction
(using the shape shown in (a) and its fitted normalization): total fitted
function (blue solid curve), $\chi_{c1}$ signal (green dashed curve),
$\chi_{c2}$ signal (red dot-dashed curve) and $\chi_{c0}$ signal (purple long-
dashed curve).

##  5 Systematic uncertainties

The fit is performed for each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin as explained in Sec. 4.
The $\chi_{c1}$ and $\chi_{c2}$ peak positions, the CB width and the left and
right tail $n$ parameters are fixed to those found in the fit to the whole
dataset. In order to assess the stability, the fit is also performed with all
parameters left free except for the peak positions or using the $n$ parameters
obtained with simulated events. The fit is also repeated in a smaller range
($\Delta M>290{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) in order to assess
the uncertainty coming from the imperfect modelling of the background at small
$\Delta M$. It is also repeated on the distribution with the background
subtracted. The largest variation from these alternative fits is taken as a
systematic uncertainty. The fit quality is usually good (the $p$-values of the
fits are greater than $1\%$) except for the first $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin where the background is
not well modelled for low $\Delta M$. However the ratio of $\chi_{c2}$ and
$\chi_{c1}$ yields is stable, indicating it is relatively insensitive to the
modelling in this low $\Delta M$ region. For the $\chi_{c0}$ yield this
systematic uncertainty is $20\%$ and is dominated by the variation of the
$n_{L}$ parameter. This large uncertainty is incurred because the $\chi_{c0}$
lies in the low mass tail of the $\chi_{c1}$ mass spectrum, and is sensitive
to the modelling of the $\chi_{c1}$ signal shape.

The bias due to the fitting procedure is studied using simulated events. This
study indicates a bias of $(-4.8\pm 1.8)\%$ and $(-2.4\pm 2.0)\%$ for the
first and second $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
bins, respectively, and therefore the data are corrected for these biases. The
other bins show no significant bias within the $3\%$ uncertainty of the test.
Conservatively, a systematic uncertainty of $3\%$ is assigned to all bins.

Imperfect modelling of the combinatorial background may introduce a bias. This
is studied with simulated events by comparing the results obtained using the
$\Delta M$ distribution of true background events and the distribution of the
background estimated with the fake photons. The bias is found to be within
$1\%$, which is assigned as a systematic uncertainty to all the bins. For the
$\chi_{c0}$ yield the impact of an imperfect modelling of the background can
be absorbed in the variation of the $n_{L}$ parameter of the $\chi_{c1}$ CB
function. This is therefore already accounted for in the fit systematic
uncertainty.

The peaking background ($\chi_{c}$ from $b$ hadrons) is estimated in Sec. 4.1
and is subtracted from the number of $\chi_{c1}$ candidates: $(0.9\pm 0.3)\%$
for $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ below 9
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $(1.8\pm 0.4)\%$ above. The number
of $\chi_{c2}$ candidates is $0.18\pm 0.03$ times the number of $\chi_{c1}$
candidates (see Sec. 4.1). The ratio of prompt $\chi_{c}$ mesons is corrected
for this background and a systematic uncertainty of $0.3\%$ ($0.4\%$) is
assigned for the $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
bins below (above) 9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. No peaking
background correction is applied for the ratio of $\chi_{c0}$ to $\chi_{c2}$
yields. This correction is estimated to be at most $2\%$ (see Sec. 4.1) which
is taken as the systematic uncertainty.

The photon efficiency is discussed in Sec. 4.2: the simulation is corrected
using the efficiency measured using $\pi^{0}$ decays in data. The systematic
uncertainty is estimated by varying independently for each $p_{\rm
T}^{\gamma}$ bin the converted photon efficiency within the measurement
uncertainty and computing the corrected ratio of efficiency
$\varepsilon^{\gamma}_{\chi_{c1}}/\varepsilon^{\gamma}_{\chi_{c2}}$ for each
$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. The systematic
uncertainty is defined as the maximum variation observed. The correction and
the systematic uncertainty due to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ selection and reconstruction efficiency are found to be negligible.

The efficiency can be affected by the choice of the simulated $\chi_{c}$
$p_{\rm T}$ spectrum ($p_{\rm T}^{\chi_{c}}$): since the photon transverse
momentum is correlated with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
transverse momentum, the efficiency for each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin can vary depending on
the $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ spectrum inside
this bin. In order to assess the uncertainty due to the $p_{\rm T}$ spectrum
shape, the simulated $\chi_{c2}$ ($\chi_{c1}$) spectrum is changed to be
identical to the simulated $\chi_{c1}$ ($\chi_{c2}$) $p_{\rm T}$ spectrum. The
generated $\chi_{c2}$ and $\chi_{c0}$ decays have the same $p_{\rm T}$
dependence. For the ratio of $\chi_{c0}$ to $\chi_{c2}$ cross-sections the
systematic uncertainty is assessed using the $p_{\rm T}$ spectrum of the
$\chi_{c1}$ mesons instead (alternatively for $\chi_{c2}$ or $\chi_{c0}$
mesons): the efficiency ratio varies by $\pm 13\%$.

All of the systematic uncertainties are uncorrelated among bins, except those
related to the $p_{\rm T}$ spectrum shape. Table 1 summarises the systematic
uncertainties on the ratio of yields for each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin.

The ratio of cross-sections is also affected by the uncertainties on the
branching fraction of
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
leading to an additional systematic uncertainty of $6.0\%$ ($8.0\%$) on the
cross section ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$
($\sigma(\chi_{c0})/\sigma(\chi_{c2})$). For each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin the total systematic
uncertainty is defined as the quadratic sum of all the systematic
uncertainties detailed here.

Table 1: Systematic uncertainties on the ratio of $\chi_{c2}$ and $\chi_{c1}$ yields for each $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin (in percent). The total systematic uncertainty is defined as the quadratic sum of all the systematic uncertainties. $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin (${\mathrm{Ge\kern-1.00006ptV\\!/}c}$) | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 | 9-11 | 11-13 | 13-16 | 16-20 | 4-20
---|---|---|---|---|---|---|---|---|---|---|---
Fit bias | 1.8 | 2.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0
Fit | 2.6 | 4.0 | 2.2 | 2.0 | 2.0 | 2.2 | 2.0 | 2.8 | 5.5 | 4.0 | 2.0
Comb bkg | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Peaking bkg | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
Photon efficiency | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 2.0
$p_{\rm T}^{\chi_{c}}$ spectrum | 2.6 | 2.4 | 2.2 | 2.1 | 2.0 | 1.8 | 1.6 | 1.3 | 1.0 | 0.7 | 6.4
Total | 5.8 | 6.5 | 6.0 | 5.9 | 5.8 | 5.8 | 5.7 | 6.0 | 7.6 | 6.5 | 8.2

##  6 $\boldsymbol{\chi_{c}}$ polarization

The prompt $\chi_{c}$ polarization is unknown. The simulated $\chi_{c}$ mesons
are unpolarized and all the efficiencies given in the previous sections are
therefore determined under the assumption that the $\chi_{c1}$ and the
$\chi_{c2}$ mesons are produced unpolarized. The photon and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum distributions depend
on the polarization of the $\chi_{c}$ state and the same is true for the ratio
of efficiencies. The correction factors for the ratio of efficiencies under
other polarization scenarios are derived here.

The angular distribution of the
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay
is described by the angles $\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, $\theta_{\chi_{c}}$ and $\phi$ where:
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the angle between
the directions of the positive muon in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame;
$\theta_{\chi_{c}}$ is the angle between the directions of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame
and the $\chi_{c}$ in the laboratory frame; $\phi$ is the angle between the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay plane in the $\chi_{c}$
rest frame and the plane formed by the $\chi_{c}$ direction in the laboratory
frame and the direction of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
in the $\chi_{c}$ rest frame. The angular distributions of the $\chi_{c}$
states depend on $m_{\chi_{cJ}}$, which is the azimuthal angular momentum
quantum number of the $\chi_{cJ}$ state. The general expressions for the
angular distributions are independent of the choice of polarization axis (here
chosen as the direction of the $\chi_{c}$ in the laboratory frame) and are
detailed in Ref. [9]. For each simulated event in the unpolarized sample, a
weight is calculated from the values of
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, $\theta_{\chi_{c}}$
and $\phi$ in the various polarization hypotheses and the ratio of
efficiencies is deduced for each ($m_{\chi_{c1}}$,$m_{\chi_{c2}}$)
polarization combination. Table 2 gives the correction factors to apply to the
final $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ results for each
($m_{\chi_{c1}}$,$m_{\chi_{c2}}$) polarization combination.

These corrections are different from those found in the analysis using
calorimetric photons [12]. This is due to the fact that the acceptance
efficiency of converted photons highly depends on the polar angle of the
photon: for large angles there is a higher probability that one of the
electrons escapes the detector before the calorimeter. The systematic
uncertainties estimated in the case where both $\chi_{c1}$ and $\chi_{c2}$
mesons are produced unpolarized also apply to the other polarization
scenarios.

Table 2:  Correction factors to be applied to the final $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ results for each $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin for different combinations of $\chi_{c1}$ and $\chi_{c2}$ polarization states $|J,m_{\chi_{cJ}}>$ with $|m_{\chi_{cJ}}|=0,...,J$ (“unpol” means the $\chi_{c}$ is unpolarized). The polarization axis is defined as the direction of the $\chi_{c}$ in the laboratory frame. |  $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ]
---|---
($|m_{\chi_{c1}}|$,$|m_{\chi_{c2}}|$) | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 | 9-11 | 11-13 | 13-16 | 16-20
(unpol,0) | 1.07 | 1.04 | 1.00 | 0.96 | 0.93 | 0.94 | 0.91 | 0.87 | 0.89 | 0.86
(unpol,1) | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | 0.97 | 0.96 | 0.95 | 0.98
(unpol,2) | 0.97 | 0.98 | 1.02 | 1.05 | 1.08 | 1.07 | 1.13 | 1.16 | 1.16 | 1.16
(0,unpol) | 1.03 | 1.01 | 0.98 | 0.97 | 0.94 | 0.92 | 0.94 | 0.91 | 0.89 | 0.90
(0,0) | 1.10 | 1.05 | 0.98 | 0.93 | 0.88 | 0.86 | 0.85 | 0.79 | 0.79 | 0.77
(0,1) | 1.02 | 1.00 | 0.96 | 0.95 | 0.92 | 0.90 | 0.90 | 0.88 | 0.84 | 0.88
(0,2) | 1.00 | 0.99 | 1.00 | 1.01 | 1.02 | 0.98 | 1.06 | 1.05 | 1.03 | 1.05
(1,unpol) | 1.00 | 1.01 | 1.02 | 1.02 | 1.03 | 1.03 | 1.04 | 1.06 | 1.05 | 1.07
(1,0) | 1.07 | 1.05 | 1.02 | 0.98 | 0.96 | 0.97 | 0.94 | 0.92 | 0.93 | 0.92
(1,1) | 0.99 | 1.00 | 1.00 | 1.00 | 1.01 | 1.01 | 1.00 | 1.02 | 1.00 | 1.05
(1,2) | 0.97 | 0.98 | 1.04 | 1.06 | 1.11 | 1.11 | 1.17 | 1.22 | 1.22 | 1.25

##  7 Results

For each $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin the
ratio of $\chi_{c2}$ to $\chi_{c1}$ yields, obtained from a least squares fit
described in Sec 4.3, is corrected for the peaking background (see Sec. 4.1),
by the efficiency ratio (see Sec. 4.2) and by the ratio of branching fractions
of $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
(see Sec. 4). Figure 5 (left) shows the ratio of the $\chi_{c2}$ to
$\chi_{c1}$ production cross-sections as a function of $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ under the assumption that
the $\chi_{c}$ mesons are produced unpolarized. The overall systematic
uncertainty ($6.0\%$) due to the branching fraction of
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ is not
shown here. Table 3 gives the ratio of cross-sections with their statistical
and systematic uncertainties for each $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin including that
originating from the unknown polarization of the $\chi_{c}$ states. Figure 5
(right) shows a comparison of this measurement with the next to leading order
(NLO) NRQCD calculation of Ref. [5] and with the LO NRQCD calculation of Ref.
[24].

A $\chi_{c0}$ signal is observed for $4<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ with a statistical
significance, determined from the ratio of the signal yield and its
uncertainty, of $4.3~{}\sigma$ and the extracted yield is $N(\chi_{c0})=705\pm
163$. The ratio of $\chi_{c0}$ and $\chi_{c2}$ yields obtained from the fit is
corrected by the efficiency ratio (see Sec. 4.2) and the ratio of branching
fractions in order to obtain the ratio of cross-sections (under the hypothesis
of unpolarized states) and integrated over $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$

$\sigma(\chi_{c0})/\sigma(\chi_{c2})=1.19\pm 0.27\,\mathrm{(stat)}\pm
0.29\,\mathrm{(syst)}\pm 0.16\,(\,\mbox{$p_{\rm T}$}\,\mathrm{model})\pm
0.09\,(\,{\cal B}),$

where the first uncertainty is statistical, the second is the systematic
uncertainty dominated by the photon efficiency, the $\chi_{c1}$ tail
parameters and background modelling, the third from the choice of $p_{\rm T}$
spectrum and the fourth from the branching fraction uncertainty. For
comparison, the ratio of $\chi_{c2}$ to $\chi_{c1}$ production cross-sections
for the same $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ range
is

$\sigma(\chi_{c2})/\sigma(\chi_{c1})=0.787\pm 0.014\,\mathrm{(stat)}\pm
0.034\,\mathrm{(syst)}\pm 0.051\,(\,\mbox{$p_{\rm T}$}\,\mathrm{model})\pm
0.047\,(\,{\cal B}).$

|
---|---
Figure 5:  (left) Ratio of $\chi_{c2}$ to $\chi_{c1}$ cross-sections at $\sqrt{s}=7$ $\mathrm{\,Te\kern-0.90005ptV}$ for $2.0<y<4.5$. The statistical uncertainty is shown with a red error bar and the systematic uncertainty with a hashed rectangle. (right) Comparison of the LHCb results (with total uncertainty) with the NLO NRQCD calculation from Ref. [5] (blue shading) and the LO NRQCD calculation of Ref. [24] (solid green). The LHCb results are obtained assuming the $\chi_{c}$ mesons are produced unpolarized.  |
---|---
Figure 6:  Comparison of the ratio of $\chi_{c2}$ to $\chi_{c1}$ cross-sections obtained by LHCb using calorimetric photons [12] (green open squares), CMS result [11] (blue filled squares), CDF result (purple filled triangles) [10] and the result presented here (red open circles) under the assumption (left) of unpolarized states and (right) under the assumption $(m_{\chi_{c1}},m_{\chi_{c2}})=(0,0)$ in the helicity frame. The uncertainty due to the limited knowledge of the branching fractions of $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$, which is common to all the measurements, is not included here. Table 3: Measurements of the ratio of $\chi_{c2}$ to $\chi_{c1}$ production cross-sections for the given $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ range assuming unpolarized $\chi_{c}$ production. The first uncertainty is statistical, the second is systematic, the third is from the branching fractions used and the last gives the maximum correction due to the unknown polarization.  $p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ [${\mathrm{Ge\kern-1.00006ptV\\!/}c}$ ] | $\sigma(\chi_{c2})/\sigma(\chi_{c1})$
---|---
$3-4$ | $1.037\pm 0.033\mathrm{(stat)}\pm 0.060\mathrm{(syst)}\pm 0.062\,({\cal B})\,^{+0.10}_{-0.03}({\rm pol})$
$4-5$ | $0.923\pm 0.029\mathrm{(stat)}\pm 0.060\mathrm{(syst)}\pm 0.055\,({\cal B})\,^{+0.05}_{-0.02}({\rm pol})$
$5-6$ | $0.795\pm 0.028\mathrm{(stat)}\pm 0.048\mathrm{(syst)}\pm 0.048\,({\cal B})\,^{+0.03}_{-0.03}({\rm pol})$
$6-7$ | $0.746\pm 0.032\mathrm{(stat)}\pm 0.044\mathrm{(syst)}\pm 0.045\,({\cal B})\,^{+0.05}_{-0.05}({\rm pol})$
$7-8$ | $0.692\pm 0.039\mathrm{(stat)}\pm 0.040\mathrm{(syst)}\pm 0.042\,({\cal B})\,^{+0.08}_{-0.08}({\rm pol})$
$8-9$ | $0.699\pm 0.044\mathrm{(stat)}\pm 0.041\mathrm{(syst)}\pm 0.042\,({\cal B})\,^{+0.08}_{-0.10}({\rm pol})$
$\,\,9-11$ | $0.625\pm 0.035\mathrm{(stat)}\pm 0.036\mathrm{(syst)}\pm 0.038\,({\cal B})\,^{+0.11}_{-0.09}({\rm pol})$
$11-13$ | $0.600\pm 0.057\mathrm{(stat)}\pm 0.036\mathrm{(syst)}\pm 0.036\,({\cal B})\,^{+0.13}_{-0.13}({\rm pol})$
$13-16$ | $0.675\pm 0.067\mathrm{(stat)}\pm 0.051\mathrm{(syst)}\pm 0.040\,({\cal B})\,^{+0.15}_{-0.15}({\rm pol})$
$16-20$ | $0.581\pm 0.096\mathrm{(stat)}\pm 0.038\mathrm{(syst)}\pm 0.035\,({\cal B})\,^{+0.15}_{-0.15}({\rm pol})$

##  8 Conclusion

The ratio of prompt production cross-sections of $\chi_{c2}$ and $\chi_{c1}$
is measured in a rapidity range $2.0<y<4.5$ as a function of $p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ from 3 to 20
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$ using the decays
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ where
the photon converts in the detector material.

This ratio was also measured by LHCb using calorimetric photons [12], by the
CMS experiment [11] in the rapidity range $|y|<1$ using converted photons at
$\sqrt{s}=7$ $\mathrm{\,Te\kern-1.00006ptV}$ and by CDF [10] using converted
photons at $\sqrt{s}=1.96$ $\mathrm{\,Te\kern-1.00006ptV}$ in the range
$|\eta({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})|<1$ and $\mbox{$p_{\rm
T}$}(\gamma)>1.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. These measurements
are compared in Fig. 6. The ratios are expected to be similar for $pp$ and
${p}\overline{}p$ collisions since $\chi_{c}$ mesons are produced
predominantly via gluon-gluon interactions and depend only weakly on the
centre-of-mass energy and $y$ coverage [5, 28]. The results from this analysis
are compatible with the CMS and CDF results. The statistical and systematic
uncertainties can be safely assumed to be uncorrelated between the analysis
presented here and the LHCb analysis using calorimetric photons, since the
data samples are different, the photon reconstruction is based on different
subdetectors (calorimeter or tracker) and the background modelling is
performed in a different way. The measurements are in agreement but the
results of the analysis using converted photons are systematically lower. As
underlined in Sec. 6 analysis-dependent corrections have to be applied to
these ratios depending on the polarization hypothesis (see Table 2). When
correcting the results assuming the $\chi_{c}$ states are polarized with
$(m_{\chi_{c1}},m_{\chi_{c2}})=(0,0)$, all the results are in better agreement
as shown in Fig. 6 (right).

The $\chi_{c0}$ meson prompt production is also studied and its production
cross section ratio relative to the $\chi_{c2}$ meson is measured in the range
$4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. This is the first evidence
for $\chi_{c0}$ meson production at a hadron collider. Our result is in
agreement with the NLO NRQCD prediction of
$\sigma(\chi_{c0})/\sigma(\chi_{c2})=0.62\pm 0.10$ ($4<\mbox{$p_{\rm
T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) [5] and with the LO NRCQD
prediction of $\sigma(\chi_{c0})/\sigma(\chi_{c2})=0.53\pm 0.02$
($4<\mbox{$p_{\rm T}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) [24].

## Acknowledgements

We thank Y.Q. Ma for providing the NLO NRQCD predictions. We also thank A.
Luchinsky and A. Likhoded for providing the LO NRQCD predictions and for
interesting discussions. 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 the LHCb institutes. We
acknowledge support from CERN and from the national agencies: CAPES, CNPq,
FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne
(France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM
and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom,
RFBR and NRC “Kurchatov Institute” (Russia); MinECo, 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. The Tier1 computing centres are supported by IN2P3 (France), KIT
and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.

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