EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-029 LHCb-PAPER-2013-067

Measurement of $\psi{(2S)}$ polarisation in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$

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

The polarisation of prompt $\psi{(2S)}$ mesons is measured by performing an
angular analysis of $\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays using
proton-proton collision data, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, collected by the LHCb detector at a centre-of-mass
energy of $7\mathrm{\,Te\kern-1.00006ptV}$. The polarisation is measured in
bins of transverse momentum $p_{\rm T}$ and rapidity $y$ in the kinematic
region $3.5<\mbox{$p_{\rm T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$, and is compared to theoretical models. No significant
polarisation is observed.

Submitted to Eur. Phys. J. C

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

LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel39, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30, W.
Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20,
L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells36, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C.
Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J.
Dalseno46, P. David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto11, H. Dijkstra38, S.
Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A.
Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch.
Elsasser40, S. Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C.
Farinelli41, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira
Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M.
Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F. Fontanelli19,j,
R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21,
E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, M. Gandelman2, P.
Gandini59, Y. Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38,
R. Gauld55, L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38,
C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C.
Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E.
Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62, J. He38, T. Head38, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, V.
Iakovenko44, M. Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J.
Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P.
Koppenburg41, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M.
Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31,38, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, T. Latham48, C. Lazzeroni45,
R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P.
Lowdon40, H. Lu3, D. Lucchesi22,q, J. Luisier39, H. Luo50, A. Lupato22, E.
Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O.
Maev30,38, S. Malde55, G. Manca15,e, G. Mancinelli6, M. Manzali16,f, J.
Maratas5, J.F. Marchand4, U. Marconi14, P. Marino23,s, R. Märki39, J. Marks11,
G. Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41, D.
Martinez Santos42, F. Martinez Vidal64, D. Martins Tostes2, A. Massafferri1,
R. Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,38,f, M. McCann53, J.
McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B. Meadows57,55, F.
Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez60, S. Monteil5, D. Moran54, M. Morandin22, P. Morawski26, A. Mordà6,
M.J. Morello23,s, J. Moron27, R. Mountain59, F. Muheim50, K. Müller40, R.
Muresan29, B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva1,
M. Needham50, N. Neri21, S. Neubert38, N. Neufeld38, M. Neuner11, A.D.
Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-Mucha27, V.
Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, G.
Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P. Owen53, A.
Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J. Parkinson9,
G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A. Pazos
Alvarez37, A. Pearce54, A. Pellegrino41, G. Penso25,m, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, G. Polok26, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat36, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,r, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Redford55, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S.
Ricciardi49, A. Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5,
P. Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38,
V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37,
N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez
Mayordomo64, B. Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37,
E. Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg41, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-
Cardona36,o, A. Vollhardt40, D. Volyanskyy10, D. Voong46, A. Vorobyev30, V.
Vorobyev34, C. Voß62, H. Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R.
Wallace12, J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45,
A.D. Webber54, D. Websdale53, M. Whitehead48, J. Wicht38, D. Wiedner11, L.
Wiggers41, G. Wilkinson55, M.P. Williams45, M. Williams56, F.F. Wilson49, J.
Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie61, Z. Xing59, Z. Xu39, Z.
Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3,
L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3,
A. Zvyagin38.

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 Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy

##  1 Introduction

Measurements of the heavy quarkonium production in hadron collisions can be
used to test predictions of quantum chromodynamics (QCD) in the perturbative
and non-perturbative regimes. Several theoretical models have been developed
within the framework of QCD to describe the quarkonium production cross-
section and polarisation as functions of the quarkonium transverse momentum,
$p_{\rm T}$, but none can simultaneously describe both of them [1]. Heavy
quarkonia can be produced in three ways in $pp$ collisions: directly in the
hard scattering, through feed-down from higher-mass quarkonia states, or via
the decay of $b$ hadrons, with the first two of these being referred to as
prompt production. In the case of $\psi{(2S)}$ mesons, the contribution from
feed-down is negligible, allowing a straightforward comparison between
measurements of prompt production and predictions for direct contributions.

The $\psi{(2S)}$ meson has spin, parity and charge-parity quantum numbers,
$J^{\mathrm{PC}}=1^{--}$ and its polarisation can be determined by studying
the angular distribution of muons in the
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays [2, 3]. The distribution is
described by

$\frac{d^{2}N}{d\cos\theta\;d\phi}(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\propto
1+\lambda_{\theta}\cos^{2}\\!\theta+\lambda_{\theta\phi}\sin
2\theta\cos\phi+\lambda_{\phi}\sin^{2}\\!\theta\cos 2\phi,$ (1)

where $\theta$ and $\phi$ are the polar and azimuthal angles of the $\mu^{+}$
direction in the rest frame of the $\psi{(2S)}$ meson, respectively, and
$\lambda_{\theta},\lambda_{\theta\phi}$ and $\lambda_{\phi}$ are the
polarisation parameters to be determined from the data. The case of
$(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})=(1,\,0,\,0)$ or
$(-1\,,0,\,0)$ corresponds to full transverse or longitudinal polarisation,
respectively, while
$(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})=(0,\,0,\,0)$
corresponds to the unpolarised state.111For a $\psi{(2S)}$ meson in a pure
spin state the three polarisation parameters cannot vanish simultaneously.  In
this study of the $\psi{(2S)}$ polarisation, two choices of polarisation frame
are used. These have a common definition of the $Y$-axis, taken to be the
normal to the production plane, which is formed by the momentum of the
$\psi{(2S)}$ meson and the beam axis in the rest frame of the colliding LHC
protons. The helicity frame [4] uses the $\psi{(2S)}$ momentum as the
$Z$-axis. In the Collins-Soper frame [5] the $Z$-axis is chosen to be the
bisector of the angle between the two incoming proton beams in the rest frame
of the $\psi{(2S)}$ meson. In both frames, the $X$-axis is defined to complete
a right-handed Cartesian coordinate system. The commonly used frame-invariant
variable $\lambda_{\mathrm{inv}}$ (see [6, 7]) is defined as

$\lambda_{\mathrm{inv}}=\frac{\lambda_{\theta}+3\lambda_{\phi}}{1-\lambda_{\phi}}.$
(2)

Two classes of theoretical models are compared with the measurements in this
paper: the colour-singlet model (CSM) [8] and non-relativistic QCD (NRQCD) [9,
*PhysRevD.55.5853, 11, 12, 13, 14], at next-to-leading order (NLO). In the
high-$p_{\rm T}$ region, where the quarkonium transverse momentum is much
larger than its mass (in natural units), the CSM underestimates significantly
the measured prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ production cross-sections [15, 16, 17], while the NRQCD model
provides a good description of the $p_{\rm T}$-dependent
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ cross-sections
measured by LHCb [16, 17] and CMS [18]. The CSM predicts large longitudinal
polarisation for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ mesons. On the other hand, in the NRQCD model, where quarkonium
production is dominated by the gluon fragmentation process in the high-$p_{\rm
T}$ region, the gluon is almost on-shell, leading to predictions of large
transverse polarisations [11]. Precise measurements of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation at both the
Tevatron [19] and the LHC [20, 21, 22], which show no significant longitudinal
or transverse polarisations, do not support either the CSM or NRQCD
predictions.

The prompt $\psi{(2S)}$ polarisation has been measured by the CDF experiment
[19] in $p\overline{p}$ collisions at
$\sqrt{s}=1.96\mathrm{\,Te\kern-1.00006ptV}$, and by the CMS experiment [21]
in $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, using the
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decay. The CDF (CMS) measurement used
$\psi{(2S)}$ mesons in the kinematic range $5<\mbox{$p_{\rm
T}$}<30{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ($14<\mbox{$p_{\rm
T}$}<50{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and rapidity $|y|<0.6$
($|y|<1.5$). The CDF result for $\mbox{$p_{\rm
T}$}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is in strong disagreement with
the NRQCD prediction of large transverse polarisation. At CMS, no evidence of
large transverse or longitudinal $\psi{(2S)}$ polarisation has been observed.

This paper presents the measurement of the prompt $\psi{(2S)}$ polarisation in
$pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, using data
corresponding to an integrated luminosity of 1$\mbox{\,fb}^{-1}$, from
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays. The $\psi{(2S)}$ polarisation
parameters are determined using unbinned maximum likelihood fits to the two-
dimensional angular distribution of the $\mu^{+}$ in the helicity and Collins-
Soper frames. The measurement is performed in the $\psi{(2S)}$ kinematic range
$3.5<\mbox{$p_{\rm T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$.

##  2 LHCb detector and data sample

The LHCb detector [23] 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 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 large transverse momentum.
Different types of charged hadrons are distinguished by information from two
ring-imaging Cherenkov detectors [24]. Photon, electron and hadron candidates
are identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers[25].

The trigger [26] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies full
event reconstruction. The hardware trigger requires the $p_{\rm T}$ of one
muon candidate to be larger than 1.48${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
or the product of the transverse momenta of two muon candidates to be larger
than $1.68{\,(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{2}}$. In a first stage of the
software trigger, two oppositely charged muon candidates with $\mbox{$p_{\rm
T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and momentum
$p>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are selected and their invariant
mass is required to be greater than
$2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In a second stage of the
software trigger, two muon candidates consistent with originating from a
$\psi{(2S)}$ decay are chosen and additional criteria are applied to refine
the sample of the $\psi{(2S)}$ candidates as follows. The invariant mass of
the candidate is required to be consistent with the known $\psi{(2S)}$ mass
[27], and for 0.7$\mbox{\,fb}^{-1}$ of data, the $p_{\rm T}$ of the candidate
is required to be greater than 3.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

In the simulation, $pp$ collisions are generated using Pythia [28] with a
specific LHCb configuration [29]. Decays of hadronic particles are described
by EvtGen [30], in which final state radiation is generated using Photos [31].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [32, *Agostinelli:2002hh] as
described in Ref. [34]. The prompt charmonium production is simulated in
Pythia according to the leading order colour-singlet and colour-octet
mechanisms [29, 35], and the charmonium is generated without polarisation.

##  3 Event selection

The $\psi{(2S)}$ candidates are reconstructed from pairs of good quality,
oppositely charged particles that originate from a common vertex. The
$\chi^{2}$ probability of the vertex fit must be larger than 0.5%. The
transverse momentum of each particle is required to be greater than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Both tracks must also be consistent
with the muon hypothesis. As in the measurement of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation [22], the
significance $S_{\tau}$, which is defined as the reconstructed pseudo-decay
time $\tau$ divided by its uncertainty, is used to distinguish between prompt
$\psi{(2S)}$ mesons and those from $b$-hadron decays. The pseudo-decay time
$\tau$ is defined as

$\tau\equiv\frac{(z_{\psi{(2S)}}-z_{\mathrm{PV}})\cdot{}M_{\psi{(2S)}}}{p_{z}},$
(3)

where $z_{\psi{(2S)}}$ ($z_{\mathrm{PV}})$ is the position of the $\psi{(2S)}$
decay vertex (the associated primary vertex) in the $z$-direction,
$M_{\psi{(2S)}}$ is the known $\psi{(2S)}$ mass, and $p_{z}$ is the measured
$z$ component of the $\psi{(2S)}$ momentum in the centre-of-mass frame of the
$pp$ collision. The $z$-axis of the LHCb coordinate system is defined as the
beam direction in the LHCb detector region. The $\psi{(2S)}$ mesons from
$b$-hadron decays tend to have large values of $S_{\tau}$. The requirement
$S_{\tau}<4$ reduces the fraction of the selected non-prompt $\psi{(2S)}$
mesons from about 20% to 3%.

Figure 1:  Invariant mass distribution of $\psi{(2S)}$ candidates in the
kinematic region $5<\mbox{$p_{\rm T}$}<7{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$
and $2.5<y<3.0$. The solid blue line is the total fit function, the dot-dashed
green line represents the linear background function and the red dashed line
is the combination of the two CB functions.

The analysis is performed in five $p_{\rm T}$ and five $y$ bins of the
$\psi{(2S)}$ meson. As an example, the invariant mass distribution of
$\psi{(2S)}$ candidates for $5<\mbox{$p_{\rm
T}$}<7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $3.0<y<3.5$ is shown in Fig.
1. In each kinematic bin, the mass distribution is fitted with a combination
of two Crystal Ball (CB) functions [36] with a common peak position for the
signal and a linear function for the combinatorial background. The relative
fractions of the narrower and broader CB functions are fixed to 0.9 and 0.1,
respectively, determined from simulation.

Using the results of the fit to the mass distribution, the sWeight $w_{i}$ for
each candidate $i$ to be signal is computed by means of the sPlot technique
[37]. The correlation between the invariant mass of the $\psi{(2S)}$
candidates and the muon angular variables is found to be negligible, and the
sWeights are used to subtract the background from the angular distribution.

##  4 Polarisation fit

The polarisation parameters are determined from a fit to the
($\cos\theta,\phi$) angular distribution of the
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ signal candidates in each kinematic
bin of the $\psi{(2S)}$ meson independently. The angular distribution
described by Eq. 1 is modified by the detection efficiency $\epsilon$, which
varies as a function of the angular variables ($\cos\theta,\phi$). In each
kinematic bin, $\epsilon$ is obtained from a sample of simulated unpolarised
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decays, where $\cos\theta$ and $\phi$
are generated according to uniform distributions. As an example, Fig. 2 shows
the efficiency in the helicity frame for $\psi{(2S)}$ candidates in the
kinematic bin $5<\mbox{$p_{\rm T}$}<7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.5<y<3.0$. For smaller (larger) $p_{\rm T}$ and $y$ values, the coverage of
the reconstructed muon angular variables is narrower (broader). In the regions
$|\cos\theta|\approx 1$, and $|\phi|\approx 0$ or $\pi$, the efficiency is
lower because one of the two muons is likely to escape the LHCb detector
acceptance.

Figure 2: Detection efficiency in arbitrary units as a function of
$\cos\theta$ and $\phi$ in the helicity frame for $\psi{(2S)}$ mesons in the
range $5<\mbox{$p_{\rm T}$}<7{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and
$2.5<y<3.0$.

Combining the angular distribution given in Eq. 1 with the efficiency, the
logarithm of the likelihood function [38], in each $p_{\rm T}$ and $y$ bin, is
defined as

$\displaystyle\ln L$ $\displaystyle=$
$\displaystyle\alpha\sum^{N_{\mathrm{tot}}}_{i=1}w_{i}\times\ln\left[\frac{P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\;\epsilon(\cos\theta_{i},\phi_{i})}{N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})}\right],$
(4)

where
$P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\equiv
1+\lambda_{\theta}\cos^{2}\theta_{i}+\lambda_{\theta\phi}\sin
2\theta_{i}\cos\phi_{i}+\lambda_{\phi}\sin^{2}\theta_{i}\cos 2\phi_{i}$,
$w_{i}$ is the sWeight, and $N_{\mathrm{tot}}$ is the number of $\psi{(2S)}$
candidates in the data. The global factor
$\alpha\equiv\sum_{i=1}^{N_{\mathrm{tot}}}w_{i}/\sum_{i=1}^{N_{\mathrm{tot}}}w_{i}^{2}$
is introduced to estimate correctly the statistical uncertainty for the
weighted likelihood function. The normalisation
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$ is defined as

$\displaystyle N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$
$\displaystyle=$ $\displaystyle\int d\Omega
P(\cos\theta,\phi|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\times\epsilon(\cos\theta,\phi)$
(5) $\displaystyle=$ $\displaystyle
C\sum_{j=1}^{M_{\mathrm{tot}}}P(\cos\theta_{j},\phi_{j}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi}),$

where the sum extends over the $M_{\mathrm{tot}}$ candidates in the simulated
sample and $C$ is a constant factor. The last equality holds because the
$(\cos\theta,\phi)$ two-dimensional distribution for the fully simulated
unpolarised $\psi{(2S)}$ mesons is the same as the efficiency
$\epsilon(\cos\theta,\phi)$ up to a constant global factor.

The angular efficiency is validated in data by using muons from
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
Due to angular momentum conservation, the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson produced in this channel
is longitudinally polarised in the $B^{+}$ meson rest frame. After reweighting
the kinematic properties of the simulated $B^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons to reproduce the data,
the remaining differences of the angular distributions between the reweighted
simulation sample and the data are attributed to imperfections in the
modelling of the detector response. Figure 3 compares the $\cos\theta$
distributions in data for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates in the helicity frame with simulated data after reweighting. The
efficiency for simulated events is overestimated for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with
$|\cos\theta|>0.5$, therefore it is corrected as a function of $p_{\mu}$ and
$y_{\mu}$, the momentum and the rapidity of the muon in the centre-of-mass
frame of $pp$ collisions. The normalisation of Eq. 5 is calculated by
assigning a weight to each candidate as the product of the weights for
$\mu^{+}$ and $\mu^{-}$ depending on their $p_{\mu}$ and $y_{\mu}$ values.

Figure 3: (Left) Distributions of $\cos\theta$ in the helicity frame for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays in
data (filled triangles) and in the simulated sample (open circles) and (right)
their ratio after the weighting based on the $B^{+}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ kinematic properties.

##  5 Systematic uncertainties

Sources of systematic uncertainty are considered for each of the four
observables $\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$ and
$\lambda_{\mathrm{inv}}$ in both the Collins-Soper and helicity frames. In the
Collins-Soper frame, the overall systematic uncertainties are found to be
comparable for each of these observables in most kinematic bins, while for the
helicity frame the systematic uncertainties assigned to $\lambda_{\theta\phi}$
and $\lambda_{\phi}$ are typically a factor of 2–3 smaller than those
estimated for $\lambda_{\theta}$ and $\lambda_{\mathrm{inv}}$. For each of the
main sources of systematic uncertainty, Table 1 shows the range of values
assigned over all kinematic bins, and their average. The total systematic
uncertainties for each of the four observables can be found in Tables 2 and 3.

The dominant systematic uncertainty is due to the size of the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ control
sample. This leads to non-negligible statistical uncertainties in the
correction factors that are applied to simulated events in bins of $p_{\mu}$
and $y_{\mu}$. The uncertainty on a given correction factor is estimated by
varying it by one standard deviation of its statistical uncertainty, while
keeping all other factors at their central values. The polarisation parameters
are recalculated and the change relative to their default values is considered
as the contribution from this factor to the systematic uncertainty. This
procedure is repeated for all bins of $p_{\mu}$ and $y_{\mu}$, and the sum in
quadrature of all these independent contributions is taken as the total
systematic uncertainty.

The limited size of the sample of simulated events introduces an uncertainty
on the normalisation
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$, and this
uncertainty is propagated to the polarisation parameters.

The uncertainty of the sWeight of each candidate used for the background
subtraction is a source of uncertainty on the polarisation parameters. The
effect is studied by comparing the default polarisation parameters with those
obtained when varying the definition of the models used to fit the mass
distributions and re-evaluating the sWeight for each candidate. Several
alternative fitting models are studied, including an exponential function for
the background mass distribution, only one CB function for the signal mass
distribution, or shapes for signal and background mass distributions fixed to
those obtained from fits to the mass distributions in sub-regions of the
$(\cos\theta,\phi)$ distribution space. The largest variation with respect to
the default result is assigned as the systematic uncertainty.

In each kinematic bin, discrepancies between data and simulation in the
$\psi{(2S)}$ $p_{\rm T}$ and $y$ distributions introduce an additional
uncertainty. This is evaluated by comparing the default polarisation results
with those determined after the $\psi{(2S)}$ kinematic distribution in the
simulation is weighted to that in data. The difference between the two results
is quoted as a systematic uncertainty contribution.

The uncertainty due to the contamination of $\psi{(2S)}$ candidates from
$b$-hadron decays ($3\%$) is determined by relaxing the $S_{\tau}$ selection
and studying the variations of the polarisation parameters.

With the exception of the effects due to the differences in the $\psi{(2S)}$
kinematic spectrum and the size of the sample of simulated events,
correlations are expected among $\psi{(2S)}$ kinematic bins. The correlation
between these systematic uncertainties in adjacent bins could be as large as
50%, as the final state muons may have similar momentum and rapidity. For each
kinematic bin, the total systematic uncertainty is calculated as the quadratic
sum of the various sources of systematic uncertainties assuming no correlation
within each kinematic bin.

Table 1: Sources of systematic uncertainties on the polarisation parameter $\lambda_{\theta}$ in the helicity and Collins-Soper frames. For each type of uncertainty, the average and the range over all $\psi{(2S)}$ kinematic bins are shown. Source | Helicity frame | Collins-Soper frame
---|---|---
Average (range) | Average (range)
Efficiency correction | 0.055 (0.034 – 0.126) | 0.035 (0.019 – 0.078)
Simulation sample size | 0.034 (0.015 – 0.103) | 0.023 (0.010 – 0.094)
Fit to mass distribution | 0.008 (0.001 – 0.134) | 0.007 (0.001 – 0.188)
$\psi{(2S)}$ kinematic modelling | 0.018 (0.000 – 0.085) | 0.016 (0.000 – 0.074)
$b$-hadron contamination | 0.014 (0.002 – 0.035) | 0.013 (0.002 – 0.063)

##  6 Results

The results for the polarisation parameters $\lambda_{\theta}$,
$\lambda_{\theta\phi}$, $\lambda_{\phi}$ and $\lambda_{\mathrm{inv}}$, and
their uncertainties, in each $p_{\rm T}$ and $y$ bin of the prompt
$\psi{(2S)}$ meson sample, are reported in Tables 2 and 3 for the helicity and
the Collins-Soper frames, respectively. The systematic uncertainties are
similar in size to the statistical uncertainties. The parameters
$\lambda_{\theta}$ and $\lambda_{\mathrm{inv}}$ are also shown in Fig. 4 as
functions of the $p_{\rm T}$ of the $\psi{(2S)}$ mesons, for different $y$
bins.

Figure 4: Polarisation parameters for prompt $\psi{(2S)}$ mesons as a function
of $p_{\rm T}$, in five rapidity intervals, (top left) $\lambda_{\theta}$ and
(bottom) $\lambda_{\mathrm{inv}}$, measured in the helicity frame, and (top
right) $\lambda_{\theta}$ in the Collins-Soper frame. The uncertainties on
data points are the sum in quadrature of statistical and systematic
uncertainties. The horizontal bars represent the width of the $p_{\rm T}$ bins
for the $\psi{(2S)}$ meson. The data points for each rapidity interval are
displaced horizontally to improve visibility.

The frame-invariant polarisation parameter $\lambda_{\mathrm{inv}}$ is
consistent with a negative polarisation with no strong dependence on the
$p_{\rm T}$ and $y$ of the $\psi{(2S)}$ meson. The values and uncertainties of
$\lambda_{\mathrm{inv}}$ that are measured in the helicity and the Collins-
Soper frames are in good agreement with each other, with differences much
smaller than the statistical uncertainties. In the Collins-Soper frame,
$\lambda_{\theta}$ takes small negative values especially in the low-$p_{\rm
T}$ region and increases with $p_{\rm T}$. This trend is more significant for
the extreme $y$ bins. In the helicity frame, the polarisation parameter
$\lambda_{\theta}$ is consistent with zero, with no significant dependence on
$p_{\rm T}$ or $y$ of the $\psi{(2S)}$ meson. The polarisation parameters
$\lambda_{\theta\phi}$ and $\lambda_{\phi}$ are consistent with zero in both
the helicity and Collins-Soper frames, and their absolute values are below 0.1
for most of the kinematic bins.

In Fig. 5, the measured values of $\lambda_{\theta}$ in the helicity frame as
a function of $p_{\rm T}$ of the $\psi{(2S)}$ meson, integrating over the
rapidity range $2.5<y<4.0$, are compared with the predictions of the CSM [39]
and NRQCD [39, 40, 41] models at NLO. Our results disfavour the CSM
calculations, in which the $\psi{(2S)}$ meson is significantly longitudinally
polarised. The three NRQCD calculations in Refs. [39, 40, 41], which use
different selections of experimental data to determine the non-perturbative
matrix elements, provide a good description of our measurements in the
low-$p_{\rm T}$ region. However, the prediction of increasing polarisation
with $p_{\rm T}$ in these models is not supported by the LHCb data.

Figure 5: Polarisation parameter $\lambda_{\theta}$ of the prompt $\psi{(2S)}$
meson in the helicity frame as a function of $p_{\rm T}$, in the rapidity
range $2.5<y<4$. The predictions of NLO CSM [39] and three NLO NRQCD models
(1) [39], (2) [40] and (3) [41] are also shown. Uncertainties on data are the
sum in quadrature of the statistical and systematic uncertainties. The
horizontal bars represent the width of $p_{\rm T}$ bins for the $\psi{(2S)}$
meson.

##  7 Conclusion

The polarisation of prompt $\psi{(2S)}$ mesons is measured as a function of
the $\psi{(2S)}$ $p_{\rm T}$ and $y$ in the range $3.5<\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$, in $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The analysis is
performed using data corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in 2011. The polarisation
parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$ and
$\lambda_{\mathrm{inv}}$ are determined in the helicity and Collins-Soper
frames by studying the angular distribution of the two muons produced in the
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ decay.

The frame-independent observable $\lambda_{\mathrm{inv}}$ is consistent with a
negative polarisation. The measured values of $\lambda_{\theta\phi}$ and
$\lambda_{\phi}$ are small over the accessible kinematic range. The
$\lambda_{\theta}$ distribution in the helicity frame shows that the
$\psi{(2S)}$ meson exhibits neither large transverse nor longitudinal
polarisation. Although a direct comparison with previous measurements by CMS
and CDF is not possible due to the different kinematic ranges, all results
disfavour large polarisation in the high-$p_{\rm T}$ region. The prompt
$\psi{(2S)}$ polarisation measured at LHCb disagrees with the CSM predictions
both in the size of the polarisation parameters and the $p_{\rm T}$
dependence. While the NRQCD models provide a good description of the LHCb data
in the low-$p_{\rm T}$ region, the predicted transverse polarisation at
high-$p_{\rm T}$ is not observed.

## Acknowledgements

We wish to thank M. Butenschön, B. Gong, H.-S. Shao and Y.-Q. Ma for providing
us with the theoretical calculations and helpful 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 EPLANET and 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 indebted
to the communities behind the multiple open source software packages on which
we depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).

Table 2: Measured prompt $\psi{(2S)}$ polarisation parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$ and $\lambda_{\mathrm{inv}}$ in bins of $p_{\rm T}$ and $y$ in the helicity frame. The first uncertainty is statistical and the is second systematic. $p_{\rm T}$ ($\mathrm{Ge\kern-0.90005ptV\\!/}c$) | $\lambda$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---|---
3.5–4 | $\lambda_{\theta}$ |  $-$0.331$\pm$ |  0.174 $\pm$ | 0.142 |  $-$0.055$\pm$ |  0.052 $\pm$ | 0.056 |  0.028$\pm$ |  0.040 $\pm$ | 0.046 |  0.008$\pm$ |  0.040 $\pm$ | 0.050 |  $-$0.080$\pm$ |  0.063 $\pm$ | 0.092
$\lambda_{\theta\phi}$ |  $-$0.233$\pm$ |  0.076 $\pm$ | 0.086 |  $-$0.172$\pm$ |  0.021 $\pm$ | 0.026 |  $-$0.039$\pm$ |  0.020 $\pm$ | 0.023 |  0.007$\pm$ |  0.021 $\pm$ | 0.028 |  $-$0.048$\pm$ |  0.036 $\pm$ | 0.049
$\lambda_{\phi}$ |  $-$0.049$\pm$ |  0.036 $\pm$ | 0.037 |  $-$0.039$\pm$ |  0.017 $\pm$ | 0.024 |  $-$0.074$\pm$ |  0.018 $\pm$ | 0.022 |  $-$0.081$\pm$ |  0.022 $\pm$ | 0.027 |  $-$0.110$\pm$ |  0.043 $\pm$ | 0.047
$\lambda_{\mathrm{inv}}$ |  $-$0.456$\pm$ |  0.195 $\pm$ | 0.160 |  $-$0.165$\pm$ |  0.063 $\pm$ | 0.078 |  $-$0.180$\pm$ |  0.054 $\pm$ | 0.063 |  $-$0.217$\pm$ |  0.057 $\pm$ | 0.073 |  $-$0.371$\pm$ |  0.089 $\pm$ | 0.114
4–5 | $\lambda_{\theta}$ |  $-$0.194$\pm$ |  0.113 $\pm$ | 0.113 |  0.007$\pm$ |  0.038 $\pm$ | 0.052 |  $-$0.003$\pm$ |  0.028 $\pm$ | 0.052 |  $-$0.026$\pm$ |  0.029 $\pm$ | 0.052 |  0.007$\pm$ |  0.050 $\pm$ | 0.095
$\lambda_{\theta\phi}$ |  $-$0.238$\pm$ |  0.049 $\pm$ | 0.053 |  $-$0.086$\pm$ |  0.016 $\pm$ | 0.023 |  $-$0.026$\pm$ |  0.015 $\pm$ | 0.021 |  0.003$\pm$ |  0.017 $\pm$ | 0.025 |  0.023$\pm$ |  0.027 $\pm$ | 0.043
$\lambda_{\phi}$ |  $-$0.043$\pm$ |  0.023 $\pm$ | 0.024 |  $-$0.082$\pm$ |  0.012 $\pm$ | 0.014 |  $-$0.048$\pm$ |  0.012 $\pm$ | 0.023 |  $-$0.087$\pm$ |  0.016 $\pm$ | 0.025 |  $-$0.088$\pm$ |  0.033 $\pm$ | 0.035
$\lambda_{\mathrm{inv}}$ |  $-$0.309$\pm$ |  0.126 $\pm$ | 0.120 |  $-$0.222$\pm$ |  0.045 $\pm$ | 0.060 |  $-$0.140$\pm$ |  0.042 $\pm$ | 0.057 |  $-$0.265$\pm$ |  0.044 $\pm$ | 0.070 |  $-$0.237$\pm$ |  0.072 $\pm$ | 0.102
5–7 | $\lambda_{\theta}$ |  $-$0.198$\pm$ |  0.074 $\pm$ | 0.091 |  0.083$\pm$ |  0.030 $\pm$ | 0.051 |  0.003$\pm$ |  0.024 $\pm$ | 0.039 |  $-$0.088$\pm$ |  0.024 $\pm$ | 0.046 |  $-$0.189$\pm$ |  0.039 $\pm$ | 0.092
$\lambda_{\theta\phi}$ |  $-$0.164$\pm$ |  0.031 $\pm$ | 0.039 |  $-$0.072$\pm$ |  0.013 $\pm$ | 0.018 |  $-$0.026$\pm$ |  0.013 $\pm$ | 0.020 |  0.002$\pm$ |  0.015 $\pm$ | 0.026 |  0.044$\pm$ |  0.025 $\pm$ | 0.051
$\lambda_{\phi}$ |  $-$0.058$\pm$ |  0.014 $\pm$ | 0.021 |  $-$0.046$\pm$ |  0.009 $\pm$ | 0.013 |  $-$0.058$\pm$ |  0.009 $\pm$ | 0.018 |  $-$0.038$\pm$ |  0.012 $\pm$ | 0.019 |  $-$0.039$\pm$ |  0.025 $\pm$ | 0.028
$\lambda_{\mathrm{inv}}$ |  $-$0.352$\pm$ |  0.080 $\pm$ | 0.094 |  $-$0.054$\pm$ |  0.039 $\pm$ | 0.060 |  $-$0.162$\pm$ |  0.035 $\pm$ | 0.054 |  $-$0.195$\pm$ |  0.040 $\pm$ | 0.067 |  $-$0.294$\pm$ |  0.065 $\pm$ | 0.105
7–10 | $\lambda_{\theta}$ |  $-$0.142$\pm$ |  0.066 $\pm$ | 0.079 |  $-$0.064$\pm$ |  0.034 $\pm$ | 0.053 |  $-$0.001$\pm$ |  0.032 $\pm$ | 0.051 |  $-$0.196$\pm$ |  0.033 $\pm$ | 0.071 |  $-$0.159$\pm$ |  0.058 $\pm$ | 0.118
$\lambda_{\theta\phi}$ |  0.044$\pm$ |  0.028 $\pm$ | 0.034 |  0.002$\pm$ |  0.014 $\pm$ | 0.021 |  0.008$\pm$ |  0.016 $\pm$ | 0.023 |  0.003$\pm$ |  0.019 $\pm$ | 0.031 |  0.124$\pm$ |  0.037 $\pm$ | 0.058
$\lambda_{\phi}$ |  $-$0.036$\pm$ |  0.014 $\pm$ | 0.017 |  $-$0.075$\pm$ |  0.010 $\pm$ | 0.012 |  $-$0.088$\pm$ |  0.011 $\pm$ | 0.012 |  $-$0.036$\pm$ |  0.014 $\pm$ | 0.018 |  0.007$\pm$ |  0.030 $\pm$ | 0.031
$\lambda_{\mathrm{inv}}$ |  $-$0.241$\pm$ |  0.079 $\pm$ | 0.095 |  $-$0.269$\pm$ |  0.043 $\pm$ | 0.064 |  $-$0.245$\pm$ |  0.043 $\pm$ | 0.062 |  $-$0.292$\pm$ |  0.052 $\pm$ | 0.100 |  $-$0.140$\pm$ |  0.101 $\pm$ | 0.138
10–15 | $\lambda_{\theta}$ |  $-$0.137$\pm$ |  0.080 $\pm$ | 0.123 |  $-$0.235$\pm$ |  0.047 $\pm$ | 0.075 |  $-$0.258$\pm$ |  0.048 $\pm$ | 0.073 |  $-$0.371$\pm$ |  0.059 $\pm$ | 0.135 |  $-$0.706$\pm$ |  0.081 $\pm$ | 0.161
$\lambda_{\theta\phi}$ |  0.157$\pm$ |  0.034 $\pm$ | 0.050 |  0.045$\pm$ |  0.020 $\pm$ | 0.026 |  0.094$\pm$ |  0.023 $\pm$ | 0.032 |  0.052$\pm$ |  0.031 $\pm$ | 0.054 |  0.104$\pm$ |  0.059 $\pm$ | 0.079
$\lambda_{\phi}$ |  $-$0.014$\pm$ |  0.021 $\pm$ | 0.022 |  $-$0.059$\pm$ |  0.017 $\pm$ | 0.011 |  $-$0.074$\pm$ |  0.020 $\pm$ | 0.014 |  $-$0.047$\pm$ |  0.027 $\pm$ | 0.020 |  0.044$\pm$ |  0.053 $\pm$ | 0.048
$\lambda_{\mathrm{inv}}$ |  $-$0.176$\pm$ |  0.103 $\pm$ | 0.164 |  $-$0.390$\pm$ |  0.062 $\pm$ | 0.086 |  $-$0.446$\pm$ |  0.067 $\pm$ | 0.096 |  $-$0.489$\pm$ |  0.089 $\pm$ | 0.150 |  $-$0.601$\pm$ |  0.162 $\pm$ | 0.136
Table 3: Measured prompt $\psi{(2S)}$ polarisation parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$ and $\lambda_{\mathrm{inv}}$ in bins of $p_{\rm T}$ and $y$ in the Collins-Soper frame. The first uncertainty is statistical and the second is systematic. $p_{\rm T}$ ($\mathrm{Ge\kern-0.90005ptV\\!/}c$) | $\lambda$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---|---
3.5–4 | $\lambda_{\theta}$ |  $-$0.457$\pm$ |  0.142 $\pm$ | 0.144 |  $-$0.282$\pm$ |  0.026 $\pm$ | 0.036 |  $-$0.105$\pm$ |  0.023 $\pm$ | 0.031 |  $-$0.047$\pm$ |  0.028 $\pm$ | 0.041 |  $-$0.168$\pm$ |  0.058 $\pm$ | 0.076
$\lambda_{\theta\phi}$ |  0.141$\pm$ |  0.088 $\pm$ | 0.065 |  0.018$\pm$ |  0.027 $\pm$ | 0.031 |  $-$0.043$\pm$ |  0.022 $\pm$ | 0.027 |  $-$0.038$\pm$ |  0.024 $\pm$ | 0.032 |  $-$0.010$\pm$ |  0.044 $\pm$ | 0.059
$\lambda_{\phi}$ |  $-$0.003$\pm$ |  0.039 $\pm$ | 0.028 |  0.040$\pm$ |  0.020 $\pm$ | 0.023 |  $-$0.027$\pm$ |  0.021 $\pm$ | 0.024 |  $-$0.061$\pm$ |  0.023 $\pm$ | 0.028 |  $-$0.076$\pm$ |  0.031 $\pm$ | 0.045
$\lambda_{\mathrm{inv}}$ |  $-$0.465$\pm$ |  0.194 $\pm$ | 0.179 |  $-$0.169$\pm$ |  0.062 $\pm$ | 0.068 |  $-$0.180$\pm$ |  0.054 $\pm$ | 0.062 |  $-$0.218$\pm$ |  0.057 $\pm$ | 0.076 |  $-$0.368$\pm$ |  0.089 $\pm$ | 0.118
4–5 | $\lambda_{\theta}$ |  $-$0.374$\pm$ |  0.077 $\pm$ | 0.086 |  $-$0.192$\pm$ |  0.019 $\pm$ | 0.032 |  $-$0.080$\pm$ |  0.017 $\pm$ | 0.030 |  $-$0.075$\pm$ |  0.020 $\pm$ | 0.035 |  $-$0.035$\pm$ |  0.042 $\pm$ | 0.056
$\lambda_{\theta\phi}$ |  0.103$\pm$ |  0.059 $\pm$ | 0.062 |  $-$0.020$\pm$ |  0.019 $\pm$ | 0.028 |  $-$0.010$\pm$ |  0.015 $\pm$ | 0.035 |  $-$0.027$\pm$ |  0.017 $\pm$ | 0.032 |  $-$0.047$\pm$ |  0.034 $\pm$ | 0.057
$\lambda_{\phi}$ |  0.032$\pm$ |  0.029 $\pm$ | 0.027 |  $-$0.011$\pm$ |  0.017 $\pm$ | 0.025 |  $-$0.021$\pm$ |  0.017 $\pm$ | 0.024 |  $-$0.069$\pm$ |  0.019 $\pm$ | 0.028 |  $-$0.073$\pm$ |  0.027 $\pm$ | 0.041
$\lambda_{\mathrm{inv}}$ |  $-$0.288$\pm$ |  0.125 $\pm$ | 0.123 |  $-$0.221$\pm$ |  0.045 $\pm$ | 0.061 |  $-$0.141$\pm$ |  0.042 $\pm$ | 0.058 |  $-$0.264$\pm$ |  0.044 $\pm$ | 0.071 |  $-$0.237$\pm$ |  0.072 $\pm$ | 0.096
5–7 | $\lambda_{\theta}$ |  $-$0.265$\pm$ |  0.040 $\pm$ | 0.062 |  $-$0.147$\pm$ |  0.014 $\pm$ | 0.024 |  $-$0.095$\pm$ |  0.015 $\pm$ | 0.023 |  $-$0.029$\pm$ |  0.019 $\pm$ | 0.030 |  0.038$\pm$ |  0.037 $\pm$ | 0.067
$\lambda_{\theta\phi}$ |  0.123$\pm$ |  0.041 $\pm$ | 0.051 |  $-$0.022$\pm$ |  0.013 $\pm$ | 0.026 |  $-$0.013$\pm$ |  0.011 $\pm$ | 0.025 |  0.026$\pm$ |  0.013 $\pm$ | 0.028 |  0.050$\pm$ |  0.029 $\pm$ | 0.053
$\lambda_{\phi}$ |  $-$0.024$\pm$ |  0.026 $\pm$ | 0.032 |  0.033$\pm$ |  0.014 $\pm$ | 0.024 |  $-$0.024$\pm$ |  0.015 $\pm$ | 0.024 |  $-$0.060$\pm$ |  0.018 $\pm$ | 0.030 |  $-$0.125$\pm$ |  0.029 $\pm$ | 0.051
$\lambda_{\mathrm{inv}}$ |  $-$0.330$\pm$ |  0.080 $\pm$ | 0.098 |  $-$0.049$\pm$ |  0.040 $\pm$ | 0.059 |  $-$0.163$\pm$ |  0.035 $\pm$ | 0.056 |  $-$0.198$\pm$ |  0.040 $\pm$ | 0.067 |  $-$0.299$\pm$ |  0.066 $\pm$ | 0.106
7–10 | $\lambda_{\theta}$ |  0.035$\pm$ |  0.039 $\pm$ | 0.044 |  $-$0.078$\pm$ |  0.019 $\pm$ | 0.028 |  $-$0.098$\pm$ |  0.020 $\pm$ | 0.031 |  0.008$\pm$ |  0.028 $\pm$ | 0.044 |  0.225$\pm$ |  0.061 $\pm$ | 0.082
$\lambda_{\theta\phi}$ |  0.006$\pm$ |  0.038 $\pm$ | 0.046 |  $-$0.002$\pm$ |  0.014 $\pm$ | 0.023 |  $-$0.034$\pm$ |  0.013 $\pm$ | 0.021 |  0.065$\pm$ |  0.017 $\pm$ | 0.031 |  $-$0.017$\pm$ |  0.040 $\pm$ | 0.058
$\lambda_{\phi}$ |  $-$0.096$\pm$ |  0.032 $\pm$ | 0.037 |  $-$0.070$\pm$ |  0.019 $\pm$ | 0.031 |  $-$0.053$\pm$ |  0.019 $\pm$ | 0.031 |  $-$0.111$\pm$ |  0.025 $\pm$ | 0.040 |  $-$0.131$\pm$ |  0.045 $\pm$ | 0.065
$\lambda_{\mathrm{inv}}$ |  $-$0.230$\pm$ |  0.079 $\pm$ | 0.093 |  $-$0.269$\pm$ |  0.043 $\pm$ | 0.066 |  $-$0.244$\pm$ |  0.043 $\pm$ | 0.062 |  $-$0.293$\pm$ |  0.052 $\pm$ | 0.081 |  $-$0.149$\pm$ |  0.101 $\pm$ | 0.137
10–15 | $\lambda_{\theta}$ |  0.163$\pm$ |  0.055 $\pm$ | 0.055 |  0.042$\pm$ |  0.037 $\pm$ | 0.042 |  0.087$\pm$ |  0.045 $\pm$ | 0.052 |  0.138$\pm$ |  0.063 $\pm$ | 0.089 |  0.675$\pm$ |  0.175 $\pm$ | 0.222
$\lambda_{\theta\phi}$ |  $-$0.103$\pm$ |  0.043 $\pm$ | 0.065 |  0.015$\pm$ |  0.022 $\pm$ | 0.026 |  $-$0.024$\pm$ |  0.024 $\pm$ | 0.023 |  0.062$\pm$ |  0.034 $\pm$ | 0.046 |  0.221$\pm$ |  0.090 $\pm$ | 0.075
$\lambda_{\phi}$ |  $-$0.117$\pm$ |  0.044 $\pm$ | 0.068 |  $-$0.163$\pm$ |  0.032 $\pm$ | 0.045 |  $-$0.211$\pm$ |  0.036 $\pm$ | 0.045 |  $-$0.251$\pm$ |  0.050 $\pm$ | 0.080 |  $-$0.539$\pm$ |  0.117 $\pm$ | 0.133
$\lambda_{\mathrm{inv}}$ |  $-$0.168$\pm$ |  0.103 $\pm$ | 0.162 |  $-$0.385$\pm$ |  0.063 $\pm$ | 0.088 |  $-$0.450$\pm$ |  0.067 $\pm$ | 0.086 |  $-$0.492$\pm$ |  0.089 $\pm$ | 0.149 |  $-$0.613$\pm$ |  0.161 $\pm$ | 0.130

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