EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-PH-EP-2012-334LHCb-PUB-2012-006, LHCb-PAPER-2012-031

May 1, 2013

Implications of LHCb measurementsand future prospects

The LHCb collaboration†

and

A. Bharuchaa, I.I. Bigib, C. Bobethc, M. Bobrowskid, J. Brode, A.J. Burasf , C.T.H. Daviesg,A. Dattah, C. Delaunayi, S. Descotes-Genonj , J. Ellisi,k, T. Feldmannl, R. Fleischerm,n,

O. Gedaliao, J. Girrbachf , D. Guadagnolip, G. Hillerq, Y. Hochbergo, T. Hurthr, G. Isidorii,s,S. Jagert, M. Jungq, A. Kagane, J.F. Kameniku,v, A. Lenzi,w, Z. Ligetix, D. Londony,

F. Mahmoudii,z, J. Matiasaa, S. Nandil, Y. Niro, P. Paradisii, G. Perezi,o, A.A. Petrovab,ac,R. Rattazziad, S.R. Sharpeae, L. Silvestriniaf , A. Soniag, D.M. Straubah, D. van Dykq,

J. Virtoaa, Y.-M. Wangl, A. Weilerai, J. Zupane

aInstitut fur Theoretische Physik, University of Hamburg, Hamburg, GermanybDepartment of Physics, University of Notre Dame du Lac, Notre Dame, USAcTechnical University Munich, Excellence Cluster Universe, Garching, Germany

dKarlsruhe Institute of Technology, Institut fur Theoretische Teilchenphysik, Karlsruhe,Germany

eDepartment of Physics, University of Cincinnati, Cincinnati, USAfTUM-Institute for Advanced Study, Garching, Germany

gSchool of Physics and Astronomy, University of Glasgow, Glasgow, United KingdomhDepartment of Physics and Astronomy, University of Mississippi, Oxford, USAiEuropean Organization for Nuclear Research (CERN), Geneva, Switzerland

jLaboratoire de Physique Theorique, CNRS/Univ. Paris-Sud 11, Orsay, FrancekPhysics Department, Kings College London, London, UK

lTheoretische Elementarteilchenphysik, Naturwissenschaftlich Techn. Fakultat, UniversitatSiegen, Siegen, Germany

mNikhef, Amsterdam, The NetherlandsnDepartment of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam, The

NetherlandsoDepartment of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot,

Israel

†Authors are listed on the following pages.

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pLAPTh, Universite de Savoie, CNRS/IN2P3, Annecy-le-Vieux, FranceqInstitut fur Physik, Technische Universitat Dortmund, Dortmund, Germany

rInstitute for Physics, Johannes Gutenberg University, Mainz, GermanysLaboratori Nazionali dell’INFN di Frascati, Frascati, Italy

tDepartment of Physics & Astronomy, University of Sussex, Brighton, UKuJ. Stefan Institute, Ljubljana, Slovenia

vDepartment of Physics, University of Ljubljana, Ljubljana, SloveniawInstitute for Particle Physics Phenomenology, Durham University, Durham, UK

xErnest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley,USA

yPhysique des Particules, Universite de Montreal, Montreal, CanadazClermont Universite, Universite Blaise Pascal, CNRS/IN2P3, Clermont-Ferrand, France

aaUniversitat Autonoma de Barcelona, Barcelona, SpainabDepartment of Physics and Astronomy, Wayne State University, Detroit, USA

acMichigan Center for Theoretical Physics, University of Michigan, Ann Arbor, USAadInstitut de Theorie des Phenomenes Physiques, EPFL, Lausanne, Switzerland

aePhysics Department, University of Washington, Seattle, USAaf INFN, Sezione di Roma, Roma, Italy

agDepartment of Physics, Brookhaven National Laboratory, Upton, USAahScuola Normale Superiore and INFN, Pisa, Italy

aiDESY, Hamburg, Germany

Abstract

During 2011 the LHCb experiment at CERN collected 1.0 fb−1 of√s = 7 TeV pp

collisions. Due to the large heavy quark production cross-sections, these data provideunprecedented samples of heavy flavoured hadrons. The first results from LHCbhave made a significant impact on the flavour physics landscape and have definitivelyproved the concept of a dedicated experiment in the forward region at a hadroncollider. This document discusses the implications of these first measurements onclasses of extensions to the Standard Model, bearing in mind the interplay with theresults of searches for on-shell production of new particles at ATLAS and CMS. Thephysics potential of an upgrade to the LHCb detector, which would allow an orderof magnitude more data to be collected, is emphasised.

Submitted to Eur. Phys. J. C

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

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LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C. Adrover6,A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S. Ali38, G. Alkhazov27,P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, L. Anderlini17,f , J. Anderson37,R. Andreassen57, M. Anelli18, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35,A. Artamonov 32, M. Artuso53, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45,C. Baesso54, W. Baldini16, H. Band38, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44,A. Bates48, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, C. Beigbeder-Beau7,S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18,S. Benson47, J. Benton43, A. Berezhnoy29, F. Bernard36, R. Bernet37, M.-O. Bettler44,M. van Beuzekom38, V. van Beveren38, A. Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h,P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31,V. Bocci22, B. Bochin27, H. Boer Rookhuizen38, G. Bogdanova29, E. Bonaccorsi35, A. Bondar31,N. Bondar27, W. Bonivento15, S. Borghi51,48, A. Borgia53, T.J.V. Bowcock49, E. Bowen37,C. Bozzi16, T. Brambach9, J. van den Brand39, L. Brarda35, J. Bressieux36, D. Brett51,M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Buchler-Germann37, I. Burducea26,A. Bursche37, J. Buytaert35, T. Caceres7, J.-P. Cachemiche6, S. Cadeddu15, O. Callot7,M. Calvi20,j , M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c,G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia47, L. Carson50,K. Carvalho Akiba2, A. Casajus Ramo33, G. Casse49, M. Cattaneo35, Ch. Cauet9, L. Ceelie38,B. Chadaj35, H. Chanal5, M. Charles52, D. Charlet7, Ph. Charpentier35, M. Chebbi35,P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, P. Ciambrone18, K. Ciba35, X. Cid Vidal34,G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38,J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu15,52, A. Cook43,M. Coombes43, B. Corajod35, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45,S. Cunliffe50, R. Currie47, C. D’Ambrosio35, I. D’Antone14, P. David8, P.N.Y. David38,I. De Bonis4, K. De Bruyn38, S. De Capua51, M. De Cian37, P. De Groen38, J.M. De Miranda1,L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, G. Decreuse35, H. Degaudenzi36,35,L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11,J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F. Domingo Bonal33,n, M. Domke9,S. Donleavy49, F. Dordei11, A. Dosil Suarez34, D. Dossett45, A. Dovbnya40, C. Drancourt4,O. Duarte7, R. Dumps35, F. Dupertuis36, P.-Y. Duval6, R. Dzhelyadin32, A. Dziurda23,A. Dzyuba27, S. Easo46,35, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42,F. Evangelisti16, A. Falabella14,e, C. Farber11, G. Fardell47, C. Farinelli38, S. Farry12,P.J.W. Faulkner42, V. Fave36, G. Felici18, V. Fernandez Albor34, F. Ferreira Rodrigues1,M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, C. Fohr10, M. Fontana10, F. Fontanelli19,i,R. Forty35, C. Fournier35, O. Francisco2, M. Frank35, C. Frei35, R. Frei36, M. Frosini17,f ,H. Fuchs10, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52,Y. Gao3, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35,R. Gauld52, E. Gersabeck11, M. Gersabeck51, T. Gershon45,35, S. Gets27, Ph. Ghez4,A. Giachero20, V. Gibson44, V.V. Gligorov35, C. Gobel54, V. Golovtsov27, D. Golubkov28,A. Golutvin50,28,35, A. Gomes2, G. Gong3, H. Gong3, H. Gordon52, C. Gotti20,M. Grabalosa Gandara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Grauges33,G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, V. Gromov38, O. Grunberg55, B. Gui53,

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E. Gushchin30, Yu. Guz32, Z. Guzik25, T. Gys35, F. Hachon6, C. Hadjivasiliou53, G. Haefeli36,C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52,S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, B. van der Heijden38,V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35,E. Hicks49, D. Hill52, M. Hoballah5, W. Hofmann10, C. Hombach51, P. Hopchev4,W. Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48,V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, O. Jamet35, E. Jans38,F. Jansen38, L. Jansen38, P. Jansweijer38, P. Jaton36, F. Jing3, M. John52, D. Johnson52,C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, O. Karavichev30,T.M. Karbach35, A. Kashchuk27, T. Kechadi12, I.R. Kenyon42, U. Kerzel35, T. Ketel39,A. Keune36, B. Khanji20, T. Kihm10, R. Kluit38, O. Kochebina7, V. Komarov36,29,R.F. Koopman39, P. Koppenburg38, M. Korolev29, J. Kos39, A. Kozlinskiy38, L. Kravchuk30,K. Kreplin11, M. Kreps45, R. Kristic35, G. Krocker11, P. Krokovny31, F. Kruse9,M. Kucharczyk20,23,j , Y. Kudenko30, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35,L. Landi16,e, G. Lanfranchi18,35, C. Langenbruch35, S. Laptev30, T. Latham45, I. Lax14,C. Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefevre5, A. Leflat29,35,J. Lefrancois7, O. Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, A. Likhoded32, M. Liles49,R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,N. Lopez-March36, H. Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7,I.V. Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, M. Maino20, S. Malde52, G. Manca15,d,G. Mancinelli6, N. Mangiafave44, U. Marconi14, R. Marki36, J. Marks11, G. Martellotti22,A. Martens8, A. Martın Sanchez7, M. Martinelli38, D. Martinez Santos34, D. Martins Tostes2,A. Massafferri1, R. Matev35, Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6,J. Mauricio33, A. Mazurov16,30,35,e, J. McCarthy42, R. McNulty12, B. Meadows57, M. Meissner11,H. Mejia47, V. Mendez-Munoz33,o, M. Merk38, D.A. Milanes13, M.-N. Minard4,J. Molina Rodriguez54, S. Monteil5, D. Moran51, P. Morawski23, R. Mountain53, I. Mous38,F. Muheim47, F. Mul39, K. Muller37, B. Munneke38, R. Muresan26, B. Muryn24, B. Muster36,P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, A. Nawrot25, M. Needham47,N. Neufeld35, A.D. Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,p, M. Nicol7, V. Niess5,N. Nikitin29, T. Nikodem11, Y. Nikolaiko41, S. Nisar56, A. Nomerotski52,35, A. Novoselov32,A. Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41,R. Oldeman15,d,35, M. Orlandea26, A. Ostankov32, J.M. Otalora Goicochea2, M. van Overbeek38,P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46,M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50,G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38,G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j , E. Perez Trigo34,A. Perez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K. Petridis50,A. Petrolini19,i, O. van Petten38, A. Phan53, E. Picatoste Olloqui33, D. Piedigrossi35,B. Pietrzyk4, T. Pilar45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8, G. Polok23,A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52,J. Prisciandaro36, M. Pugatch41, V. Pugatch41, A. Puig Navarro36, W. Qian4,J.H. Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N. Rauschmayr35,G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1, F. Rethore6, S. Ricciardi46,A. Richards50, K. Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7,E. Rodrigues51,48, P. Rodriguez Perez34, E. Roeland38, G.J. Rogers44, S. Roiser35,

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V. Romanovsky32, A. Romero Vidal34, K. de Roo38, J. Rouvinet36, L. Roy35, K. Rudloff9,T. Ruf35, H. Ruiz33, G. Sabatino22,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48,B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, R. Santacesaria22, C. Santamarina Rios34,E. Santovetti21,k, S. Saornil Gamarra37, M. Sapunov6, A. Saputi18, A. Sarti18,l, C. Satriano22,m,A. Satta21, T. Savidge50, M. Savrie16,e, P. Schaack50, M. Schiller39, A. Schimmel38,H. Schindler35, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36,T. Schneider35, A. Schopper35, H. Schuijlenburg38, M.-H. Schune7, R. Schwemmer35,B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50,N. Serra37, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin32, I. Shapoval40,35, P. Shatalov28,Y. Shcheglov27, T. Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50,S. Sigurdsson44, R. Silva Coutinho45, T. Skwarnicki53, M.W. Slater42, T. Sluijk38, N.A. Smith49,E. Smith52,46, M. Smith51, K. Sobczak5, M.D. Sokoloff57, F.J.P. Soler48, F. Soomro18,35,D. Souza43, B. Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, S. Squerzanti16,F. Stagni35, S. Stahl11, O. Steinkamp37, O. Stenyakin32, S. Stoica26, S. Stone53, B. Storaci38,M. Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35,C. Thomas52, E. Thomas35, A. Tikhonov30, J. van Tilburg11, V. Tisserand4, M. Tobin37,V. Tocut7, S. Tolk39, D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50,S. Tourneur36, M.T. Tran36, M. Tresch37, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38,M. Ubeda Garcia35, A. Ukleja25, O. Ullaland35, D. Urner51, U. Uwer11, V. Vagnoni14,G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43,M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, D. Vieira2, X. Vilasis-Cardona33,n,W. Vink38, S. Volkov27, V. Volkov29, A. Vollhardt37, D. Volyanskyy10, D. Voong43,A. Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, G. Vouters4, R. Waldi55, R. Wallace12,S. Wandernoth11, J. Wang53, D.R. Ward44, K. Warda9, N.K. Watson42, A.D. Webber51,D. Websdale50, P. Wenerke38, M. Whitehead45, J. Wicht35, D. Wiedner11, L. Wiggers38,G. Wilkinson52, M.P. Williams45,46, M. Williams50,q, F.F. Wilson46, J. Wishahi9, M. Witek23,W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47,35, Z. Xing53,T. Xue3, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, F. Zappon38,M. Zavertyaev10,a, M. Zeng3, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,L. Zhong3, E. Zverev29, A. Zvyagin35, A. Zwart38.

1Centro Brasileiro de Pesquisas Fısicas (CBPF), Rio de Janeiro, Brazil2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil3Center for High Energy Physics, Tsinghua University, Beijing, China4LAPP, Universite de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France5Clermont Universite, Universite Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France6CPPM, Aix-Marseille Universite, CNRS/IN2P3, Marseille, France7LAL, Universite Paris-Sud, CNRS/IN2P3, Orsay, France8LPNHE, Universite Pierre et Marie Curie, Universite Paris Diderot, CNRS/IN2P3, Paris, France9Fakultat Physik, Technische Universitat Dortmund, Dortmund, Germany10Max-Planck-Institut fur Kernphysik (MPIK), Heidelberg, Germany11Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg, Heidelberg, Germany12School of Physics, University College Dublin, Dublin, Ireland13Sezione INFN di Bari, Bari, Italy14Sezione INFN di Bologna, Bologna, Italy15Sezione INFN di Cagliari, Cagliari, Italy16Sezione INFN di Ferrara, Ferrara, Italy

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17Sezione INFN di Firenze, Firenze, Italy18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy19Sezione INFN di Genova, Genova, Italy20Sezione INFN di Milano Bicocca, Milano, Italy21Sezione INFN di Roma Tor Vergata, Roma, Italy22Sezione INFN di Roma La Sapienza, Roma, Italy23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland24AGH University of Science and Technology, Krakow, Poland25National Center for Nuclear Research (NCBJ), Warsaw, Poland26Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia32Institute for High Energy Physics (IHEP), Protvino, Russia33Universitat de Barcelona, Barcelona, Spain34Universidad de Santiago de Compostela, Santiago de Compostela, Spain35European Organization for Nuclear Research (CERN), Geneva, Switzerland36Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland37Physik-Institut, Universitat Zurich, Zurich, Switzerland38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands39Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, TheNetherlands40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine42University of Birmingham, Birmingham, United Kingdom43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom45Department of Physics, University of Warwick, Coventry, United Kingdom46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom48School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom50Imperial College London, London, United Kingdom51School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom52Department of Physics, University of Oxford, Oxford, United Kingdom53Syracuse University, Syracuse, NY, United States54Pontifıcia Universidade Catolica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2

55Institut fur Physik, Universitat Rostock, Rostock, Germany, associated to 11

56Institute of Information Technology, COMSATS, Lahore, Pakistan, associated to 53

57University of Cincinnati, Cincinnati, OH, United States, associated to 53

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, RussiabUniversita di Bari, Bari, ItalycUniversita di Bologna, Bologna, ItalydUniversita di Cagliari, Cagliari, ItalyeUniversita di Ferrara, Ferrara, ItalyfUniversita di Firenze, Firenze, ItalygUniversita di Urbino, Urbino, ItalyhUniversita di Modena e Reggio Emilia, Modena, ItalyiUniversita di Genova, Genova, Italy

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jUniversita di Milano Bicocca, Milano, ItalykUniversita di Roma Tor Vergata, Roma, ItalylUniversita di Roma La Sapienza, Roma, ItalymUniversita della Basilicata, Potenza, ItalynLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, SpainoPort d’Informacio Cientıfica (PIC), Barcelona, SpainpHanoi University of Science, Hanoi, Viet NamqMassachusetts Institute of Technology, Cambridge, MA, United States

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Contents

1 Introduction 11.1 Current LHCb detector and performance . . . . . . . . . . . . . . . . . . . 21.2 Assumptions for LHCb upgrade performance . . . . . . . . . . . . . . . . . 3

2 Rare decays 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Model-independent analysis of new physics contributions to leptonic,

semileptonic and radiative decays . . . . . . . . . . . . . . . . . . . . . . . 42.3 Rare semileptonic B decays . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Theoretical treatment of rare semileptonic B→M`+`− decays . . . 62.3.2 Angular distribution of B0→ K∗0µ+µ− and B0

s→ φµ+µ− decays . . 72.3.3 Strategies for analysis of B0 → K∗0`+`− decays . . . . . . . . . . . 82.3.4 Theoretically clean observables in B0 → K∗0`+`− decays . . . . . . 112.3.5 B+→ K+µ+µ− and B+→ K+e+e− . . . . . . . . . . . . . . . . . . 122.3.6 Rare semileptonic b→ d`+`− decays . . . . . . . . . . . . . . . . . 132.3.7 Isospin asymmetry of B0(+)→ K0(+)µ+µ− and B0(+)→ K∗0(+)µ+µ−

decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Radiative B decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Experimental status and outlook for rare radiative decays . . . . . . 152.5 Leptonic B decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.1 B0s→ µ+µ− and B0→ µ+µ− . . . . . . . . . . . . . . . . . . . . . . 16

2.5.2 B0s → τ+τ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Model-independent constraints . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Interplay with direct searches and model-dependent constraints . . . . . . 212.8 Rare charm decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8.1 Search for D0 → µ+µ− . . . . . . . . . . . . . . . . . . . . . . . . . 252.8.2 Search for D+

(s) → h+µ+µ− and D0 → hh′µ+µ− . . . . . . . . . . . 26

2.9 Rare kaon decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.10 Lepton flavour and lepton number violation . . . . . . . . . . . . . . . . . 27

2.10.1 Lepton flavour violation . . . . . . . . . . . . . . . . . . . . . . . . 272.10.2 Lepton number violation . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 Search for NP in other rare decays . . . . . . . . . . . . . . . . . . . . . . 28

3 CP violation in the B system 303.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 B0

(s) mixing measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 B0(s)–B

0(s) mixing observables . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Current experimental status and outlook . . . . . . . . . . . . . . . 323.2.3 Model independent constraints on new physics in B mixing . . . . . 373.2.4 CKM unitarity fits in SM and beyond . . . . . . . . . . . . . . . . . 403.2.5 Penguin pollution in b→ ccs decays . . . . . . . . . . . . . . . . . 42

vi

3.2.6 Future prospects with LHCb upgrade . . . . . . . . . . . . . . . . . 443.3 CP violation measurements with hadronic b→ s penguins . . . . . . . . . 45

3.3.1 Probes for new physics in penguin-only b→ sqq decays . . . . . . . 453.3.2 Current status and outlook of LHCb measurements . . . . . . . . . 473.3.3 Future prospects with LHCb upgrade . . . . . . . . . . . . . . . . . 48

3.4 Measurements of the CKM angle gamma . . . . . . . . . . . . . . . . . . . 503.4.1 Measurements of γ using tree-mediated decays . . . . . . . . . . . . 503.4.2 Theoretical cleanliness of γ from B → DK decays . . . . . . . . . . 553.4.3 Current LHCb experimental situation . . . . . . . . . . . . . . . . . 573.4.4 Measurements of γ using loop-mediated two-body B decays . . . . 603.4.5 Studies of CP violation in multibody charmless b hadron decays . . 653.4.6 Prospects of future LHCb measurements . . . . . . . . . . . . . . . 66

4 Mixing and CP violation in the charm sector 684.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Key observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.2 Status and near-term future of LHCb measurements . . . . . . . . . 704.1.3 Experimental aspects of ∆ACP and related measurements . . . . . . 74

4.2 Theory status of mixing and indirect CP violation . . . . . . . . . . . . . . 764.2.1 Theoretical predictions for ∆ΓD, ∆mD and indirect CP violation in

the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.2 New physics in indirect CP violation . . . . . . . . . . . . . . . . . 79

4.3 The status of calculations of ∆ACP in the Standard Model . . . . . . . . . 814.4 ∆ACP in the light of physics beyond the Standard Model . . . . . . . . . . 84

4.4.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.2 Universality of CP violation in flavour-changing decay processes . . 854.4.3 Explanations of ∆ACP within NP models . . . . . . . . . . . . . . . 864.4.4 Shedding light on direct CP violation via D→ V γ decays . . . . . . 874.4.5 Testing for CP -violating new physics in the ∆I = 3/2 amplitudes . 89

4.5 Potential for lattice computations of direct CP violation and mixing in theD0–D0 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.6 Interplay of ∆ACP with non-flavour observables . . . . . . . . . . . . . . . 934.6.1 Direct CP violation in charm and hadronic electric dipole moments 934.6.2 Interplay of collider physics and a new physics origin for ∆ACP . . 96

4.7 Future potential of LHCb measurements . . . . . . . . . . . . . . . . . . . 964.7.1 Requirements on experimental precision . . . . . . . . . . . . . . . 964.7.2 Prospects of future LHCb measurements . . . . . . . . . . . . . . . 97

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 The LHCb upgrade as a general purpose detector in the forward region1035.1 Quarkonia and multi-parton scattering . . . . . . . . . . . . . . . . . . . . 1035.2 Exotic meson spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 Precision measurements of b- and c-hadron properties . . . . . . . . . . . . 107

vii

5.4 Measurements with electroweak gauge bosons . . . . . . . . . . . . . . . . 1105.4.1 sin2 θlept

eff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.4.2 mW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.3 tt production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5 Searches for exotic particles with displaced vertices . . . . . . . . . . . . . 1135.6 Central exclusive production . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Summary 1186.1 Highlights of LHCb measurements and their implications . . . . . . . . . . 118

6.1.1 Rare decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.1.2 CP violation in the B sector . . . . . . . . . . . . . . . . . . . . . . 1196.1.3 Charm mixing and CP violation . . . . . . . . . . . . . . . . . . . . 1206.1.4 Measurements exploiting the unique kinematic acceptance of LHCb 120

6.2 Sensitivity of the upgraded LHCb experiment to key observables . . . . . . 1216.3 Importance of the LHCb upgrade . . . . . . . . . . . . . . . . . . . . . . . 124

References 126

viii

1 Introduction

During 2011 the LHCb experiment [1] at CERN collected 1.0 fb−1 of√s = 7 TeV pp

collisions. Due to the large production cross-section, σ(pp→ bbX) = (89.6± 6.4± 15.5)µbin the LHCb acceptance [2], with the comparable number for charm production about 20times larger [3, 4], these data provide unprecedented samples of heavy flavoured hadrons.The first results from LHCb have made a significant impact on the flavour physics landscapeand have definitively proved the concept of a flavour physics experiment in the forwardregion at a hadron collider.

The physics objectives of the first phase of LHCb were set out prior to the commence-ment of data taking in the “roadmap document” [5]. They centred on six main areas, inall of which LHCb has by now published its first results: (i) the tree-level determination ofγ [6, 7], (ii) charmless two-body B decays [8, 9], (iii) the measurement of mixing-inducedCP violation in B0

s → J/ψφ [10], (iv) analysis of the decay B0s → µ+µ− [11–14], (v)

analysis of the decay B0 → K∗0µ+µ− [15], (vi) analysis of B0s → φγ and other radiative B

decays [16,17].1 In addition, the search for CP violation in the charm sector was establishedas a priority, and interesting results in this area have also been published [18,19].

The results demonstrate the capability of LHCb to test the Standard Model (SM) and,potentially, to reveal new physics (NP) effects in the flavour sector. This approach tosearch for NP is complementary to that used by the ATLAS and CMS experiments. Whilethe high-pT experiments search for on-shell production of new particles, LHCb can look fortheir effects in processes that are precisely predicted in the SM. In particular, the SM hasa highly distinctive flavour structure, with no tree-level flavour-changing neutral currents,and quark mixing described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix [20] whichhas a single source of CP violation. This structure is not necessarily replicated in extendedmodels. Historically, new particles have first been seen through their virtual effects sincethis approach allows one to probe mass scales beyond the energy frontier. For example,the observation of CP violation in the kaon system [21] was, in hindsight, the discoveryof the third family of quarks, well before the observations of the bottom and top quarks.Crucially, measurements of both high-pT and flavour observables are necessary in order todecipher the nature of NP.

The early data also illustrated the potential for LHCb to expand its physics programmebeyond these “core” measurements. In particular, the development of trigger algorithmsthat select events inclusively based on properties of b-hadron decays [22,23] facilitates amuch broader output than previously foreseen. On the other hand, limitations imposedby the hardware trigger lead to a maximum instantaneous luminosity at which data canmost effectively be collected (higher luminosity requires tighter trigger thresholds, so thatthere is no gain in yields, at least for channels that do not involve muons). To overcomethis limitation, an upgrade of the LHCb experiment has been proposed to be installedduring the long shutdown of the LHC planned for 2018. The upgraded detector will beread out at the maximum LHC bunch-crossing frequency of 40 MHz so that the trigger

1 Throughout the document, the inclusion of charge conjugated modes is implied unless explicitlystated.

1

can be fully implemented in software. With such a flexible trigger strategy, the upgradedLHCb experiment can be considered as a general purpose detector in the forward region.

The Letter of Intent for the LHCb upgrade [24], containing a detailed physics case, wassubmitted to the LHCC in March 2011 and was subsequently endorsed. Indeed, the LHCCviewed the physics case as “compelling.” Nevertheless, the LHCb collaboration continuesto consider further possibilities to enhance the physics reach. Moreover, given the strongmotivation to exploit fully the flavour physics potential of the LHC, it is timely to updatethe estimated sensitivities for various key observables based on the latest available data.These studies are described in this paper, and summarised in the framework technicaldesign report for the LHCb upgrade [25], submitted to the LHCC in June 2012 andendorsed in September 2012.

In the remainder of this introduction, a brief summary of the current LHCb detector isgiven, together with the common assumptions made to estimate the sensitivity achievableby the upgraded experiment. Thereafter, the sections of the paper discuss rare charmand beauty decays in Sec. 2, CP violation in the B system in Sec. 3 and mixing and CPviolation in the charm sector in Sec. 4. There are several other important topics, notcovered in any of these sections, that can be studied at LHCb and its upgrade, and theseare discussed in Sec. 5. A summary is given in Sec. 6.

1.1 Current LHCb detector and performance

The LHCb detector [1] is a single-arm forward spectrometer covering the pseudorapidityrange 2 < η < 5, designed for the study of particles containing b or c quarks. Thedetector includes a high precision tracking system consisting of a silicon-strip vertexdetector surrounding the pp interaction region, a large-area silicon-strip detector locatedupstream of a dipole magnet with a bending power of about 4 Tm, and three stations ofsilicon-strip detectors and straw drift tubes placed downstream. The combined trackingsystem has a momentum resolution ∆p/p that varies from 0.4 % at 5 GeV/c to 0.6 % at100 GeV/c, and an impact parameter resolution of 20µm for tracks with high transversemomentum. Charged hadrons are identified using two ring-imaging Cherenkov detectors.Photon, electron and hadron candidates are identified by a calorimeter system consisting ofscintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadroniccalorimeter. Muons are identified by a system composed of alternating layers of iron andmultiwire proportional chambers. The trigger consists of a hardware stage, based oninformation from the calorimeter and muon systems, followed by a software stage whichapplies a full event reconstruction.

During 2011, the LHCb experiment collected 1.0 fb−1 of integrated luminosity duringthe LHC pp run at a centre-of-mass energy

√s = 7 TeV. The majority of the data was

recorded at an instantaneous luminosity of Linst = 3.5 × 1032 cm−2 s−1, nearly a factorof two above the LHCb design value, and with a pile-up rate (average number of visibleinteractions per crossing) of µ ∼ 1.5 (four times the nominal value, but below the ratesof up to µ ∼ 2.5 seen in 2010). A luminosity levelling procedure, where the beams aredisplaced at the LHCb interaction region, allows LHCb to maintain an approximately

2

constant luminosity throughout each LHC fill. This procedure permitted reliable operationof the experiment and a stable trigger configuration throughout 2011. The hardware stageof the trigger produced output at around 800 kHz, close to the nominal 1 MHz, whilethe output of the software stage was around 3 kHz, above the nominal 2 kHz, dividedroughly equally between channels with muons, b decays to hadrons and charm decays.During data taking, the magnet polarity was flipped at a frequency of about one cycleper month in order to collect equal sized data samples of both polarities for periods ofstable running conditions. Thanks to the excellent performance of the LHCb detector, theoverall data taking efficiency exceeded 90 %.

1.2 Assumptions for LHCb upgrade performance

In the upgrade era, several important improvements compared to the current detectorperformance can be expected, as detailed in the framework TDR. However, to be conser-vative, the sensitivity studies reported in this paper all assume detector performance asachieved during 2011 data taking. The exception is in the trigger efficiency, where channelsselected at hardware level by hadron, photon or electron triggers are expected to havetheir efficiencies double (channels selected by muon triggers are expected to have marginalgains, that have not been included in the extrapolations). Several other assumptions aremade:

• LHC collisions will be at√s = 14 TeV, with heavy flavour production cross-sections

scaling linearly with√s;

• the instantaneous luminosity2 in LHCb will be Linst = 1033 cm−2 s−1: this will beachieved with 25 ns bunch crossings (compared to 50 ns in 2011) and µ = 2;

• LHCb will change the polarity of its dipole magnet with similar frequency as in2011/12 data taking, to approximately equalise the amount of data taken with eachpolarity for better control of certain potential systematic biases;

• the integrated luminosity will be Lint = 5 fb−1 per year, and the experiment will runfor 10 years to give a total sample of 50 fb−1.

2 It is anticipated that any detectors that need replacement for the LHCb upgrade will be designedsuch that they can sustain a luminosity of Linst = 2 × 1033 cm−2 s−1 [25]. Operation at instantaneousluminosities higher than the nominal value assumed for the estimations will allow the total data set to beaccumulated in a shorter time.

3

2 Rare decays

2.1 Introduction

The term rare decay is used within this document to refer loosely to two classes of decays:

• flavour-changing neutral current (FCNC) processes that are mediated by electroweakbox and penguin type diagrams in the SM;

• more exotic decays, including searches for lepton flavour or number violating decaysof B or D mesons and for light scalar particles.

The first broad class of decays includes the rare radiative process B0s → φγ and rare

leptonic and semileptonic decays B0(s)→ µ+µ− and B0→ K∗0µ+µ−. These were listed

as priorities for the first phase of the LHCb experiment in the roadmap document [5].In many well motivated new physics models, new particles at the TeV scale can enterin diagrams that compete with the SM processes, leading to modifications of branchingfractions or angular distributions of the daughter particles in these decays.

For the second class of decay, there is either no SM contribution or the SM contributionis vanishingly small and any signal would indicate evidence for physics beyond the SM.Grouped in this class of decay are searches for GeV scale new particles that might bedirectly produced in B or D meson decays. This includes searches for light scalar particlesand for B meson decays to pairs of same-charge leptons that can arise, for example, inmodels containing Majorana neutrinos [26–28].

The focus of this section is on rare decays involving leptons or photons in the finalstates. There are also several interesting rare decays involving hadronic final states thatcan be pursued at LHCb, such as B+ → K−π+π+, B+ → K+K+π− [29, 30], B0

s → φπ0

and B0s → φρ0 [31]; however, these are not discussed in this document.

Section 2.2 introduces the theoretical framework (the operator product expansion)that is used when discussing rare electroweak penguin processes. The observables andexperimental constraints coming from rare semileptonic, radiative and leptonic B decays arethen discussed in Secs. 2.3, 2.4 and 2.5 respectively. The implications of these experimentalconstraints for NP contributions are discussed in Secs. 2.6 and 2.7. Possibilities with rarecharm decays are then discussed in Sec. 2.8, and the potential of LHCb to search for rarekaon decays, lepton number and flavour violating decays, and for new light scalar particlesis summarised in Secs. 2.9, 2.10 and 2.11 respectively.

2.2 Model-independent analysis of new physics contributions toleptonic, semileptonic and radiative decays

Contributions from physics beyond the SM to the observables in rare radiative, semileptonicand leptonic B decays can be described by the modification of Wilson coefficients C

(′)i of

4

local operators in an effective Hamiltonian of the form

Heff = −4GF√2VtbV

∗tq

e2

16π2

∑i

(CiOi + C ′iO′i) + h.c. , (1)

where q = d, s, and where the primed operators indicate right-handed couplings. Thisframework is known as the operator product expansion, and is described in more detail in,e.g., Refs. [32, 33]. In many concrete models, the operators that are most sensitive to NPare a subset of

O(′)7 =

mb

e(qσµνPR(L)b)F

µν , O(′)8 =

gmb

e2(qσµνT

aPR(L)b)Gµν a,

O(′)9 = (qγµPL(R)b)(¯γµ`) , O

(′)10 = (qγµPL(R)b)(¯γµγ5`) ,

O(′)S =

mb

mBq

(qPR(L)b)(¯ ) , O(′)P =

mb

mBq

(qPR(L)b)(¯γ5`) , (2)

which are customarily denoted as magnetic (O(′)7 ), chromomagnetic (O

(′)8 ), semileptonic

(O(′)9 and O

(′)10), pseudoscalar (O

(′)P ) and scalar (O

(′)S ) operators.3 While the radiative b→ qγ

decays are sensitive only to the magnetic and chromomagnetic operators, semileptonicb→ q`+`− decays are, in principle, sensitive to all these operators.4

In the SM, models with minimal flavour violation (MFV) [34,35] and models with aflavour symmetry relating the first two generations [36], the Wilson coefficients appearingin Eq. (1) are equal for q = d or s and the ratio of amplitudes for b→ d relative to b→ stransitions is suppressed by |Vtd/Vts|. Due to this suppression, at the current level ofexperimental precision, constraints on decays with a b → d transition are much weakerthan those on decays with a b→ s transition for constraining C

(′)i . In the future, precise

measurements of b→ d transitions will allow powerful tests to be made of this universalitywhich could be violated by NP.

The dependence on the Wilson coefficients, and the set of operators that can contribute,is different for different rare B decays. In order to put the strongest constraints on theWilson coefficients and to determine the room left for NP, it is therefore desirable toperform a combined analysis of all the available data on rare leptonic, semileptonic andradiative B decays. A number of such analyses have recently been carried out for subsetsof the Wilson coefficients [37–42].

The theoretically cleanest branching ratios probing the b → s transition are theinclusive decays B → Xsγ and B → Xs`

+`−. In the former case, both the experimentalmeasurement of the branching ratio and the SM expectation have uncertainties of about7 % [43,44]. In the latter case, semi-inclusive measurements at the B factories still haveerrors at the 30 % level [43]. At hadron colliders, the most promising modes to constrainNP are exclusive decays. In spite of the larger theory uncertainties on the branching

3 In principle there are also tensor operators, OT (5) = (qσµνb)(¯σµν(γ5)`), which are relevant for someobservables.

4 In radiative and semileptonic decays, the chromomagnetic operator O8 enters at higher order in thestrong coupling αS.

5

fractions as compared to inclusive decays, the attainable experimental precision can leadto stringent constraints on the Wilson coefficients. Moreover, beyond simple branchingfraction measurements, exclusive decays offer powerful probes of C

(′)7 , C

(′)9 and C

(′)10 through

angular and CP -violating observables. The exclusive decays most sensitive to NP in b→ stransitions are B → K∗γ, B0

s → µ+µ−, B → Kµ+µ− and B → K∗µ+µ−. These decaysare discussed in more detail below.

2.3 Rare semileptonic B decays

The richest set of observables sensitive to NP are accessible through rare semileptonicdecays of B mesons to a vector or pseudoscalar meson and a pair of leptons. In particularthe angular distribution of B → K∗µ+µ− decays, discussed in Sec. 2.3.2, provides strongconstraints on C

(′)7 , C

(′)9 and C

(′)10 .

2.3.1 Theoretical treatment of rare semileptonic B→M`+`− decays

The theoretical treatment of exclusive rare semileptonic decays of the type B→M`+`− ispossible in two kinematic regimes for the meson M : large recoil (corresponding to lowdilepton invariant mass squared, q2) and small recoil (high q2). Calculations are difficultoutside these regimes, in particular in the q2 region close to the narrow cc resonances (theJ/ψ and ψ(2S) states).

In the low q2 region, these decays can be described by QCD-improved factorisation(QCDF) [45,46] and the field theory formulation of soft-collinear effective theory (SCET) [47,48]. The combined limit of a heavy b-quark and an energetic meson M , leads to theschematic form of the decay amplitude [49,50]:

T = C ξ + φB ⊗ T ⊗ φM +O(ΛQCD/mb) . (3)

which is accurate to leading order in ΛQCD/mb and to all orders in αS. It factorises thecalculation into process-independent non-perturbative quantities, B →M form factors, ξ,and light cone distribution amplitudes (LCDAs), φB(M), of the heavy (light) mesons, andperturbatively calculable quantities, C and T which are known to O(α1

S) [49, 50]. Further,in the case that M is a vector V (pseudoscalar P ), the seven (three) a priori independentB → V (B → P ) form factors reduce to two (one) universal soft form factors ξ⊥,‖ (ξP ) inQCDF/SCET [51]. The factorisation formula Eq. (3) applies well in the dilepton massrange, 1 < q2 < 6 GeV2.5

For B→ K∗`+`−, the three K∗ spin amplitudes, corresponding to longitudinal andtransverse polarisations of the K∗, are linear in the soft form factors ξ⊥,‖,

AL,R⊥,‖ ∝ CL,R⊥ ξ⊥, AL,R0 ∝ CL,R

‖ ξ‖, (4)

5 Light resonances at q2 below 1 GeV2 cannot be treated within QCDF, and their effects have to beestimated using other approaches. In addition, the longitudinal amplitude in the QCDF/SCET approachgenerates a logarithmic divergence in the limit q2 → 0, indicating problems in the description below1 GeV2 [49].

6

at leading order in ΛQCD/mb and αS. The CL,R⊥,‖ are combinations of the Wilson coefficients

C7,9,10 and the L and R indices refer to the chirality of the leptonic current. Symmetrybreaking corrections to these relationships of order αS are known [49,50]. This simplificationof the amplitudes as linear combinations of CL,R

⊥,‖ and form factors, makes it possible todesign a set of optimised observables in which any soft form factor dependence cancels outfor all low dilepton masses q2 at leading order in αS and ΛQCD/mb [52–54], as discussedbelow in Sec. 2.3.2.

Within the QCDF/SCET approach, a general, quantitative method to estimate theimportant ΛQCD/mb corrections to the heavy quark limit is missing. In semileptonicdecays, a simple dimensional estimate of 10 % is often used, largely from matching of thesoft form factors to the full-QCD form factors (see also Ref. [55]).

The high q2 (low hadronic recoil) region, corresponds to dilepton invariant massesabove the two narrow resonances of J/ψ and ψ(2S), with q2 >∼ (14 − 15) GeV2. In thisregion, broad cc-resonances are treated using a local operator product expansion [56,57].The operator product expansion (OPE) predicts small sub-leading corrections which aresuppressed by either (ΛQCD/mb)

2 [57] or αSΛQCD/mb [56] (depending on whether full QCDor subsequent matching on heavy quark effective theory in combination with form factorsymmetries [58] is adopted). The sub-leading corrections to the amplitude have beenestimated to be below 2 % [57] and those due to form factor relations are suppressednumerically by C7/C9 ∼ O(0.1). Moreover, duality violating effects have been estimatedwithin a model of resonances and found to be at the level of 2 % of the rate, if sufficientlylarge bins in q2 are chosen [57]. Consequently, like the low q2 region, this region istheoretically well under control.

At high q2 the heavy-to-light form factors are known only as extrapolations from lightcone sum rules (LCSR) calculations at low q2. Results based on lattice calculations arebeing derived [59], and may play an important role in the near future in reducing the formfactor uncertainties.

2.3.2 Angular distribution of B0→ K∗0µ+µ− and B0s→ φµ+µ− decays

The physics opportunities of B → V `+`− (` = e, µ, V = K∗, φ, ρ) can be maximisedthrough measurements of the angular distribution of the decay. Using the decay B →K∗(→ Kπ)`+`−, with K∗ on the mass shell, as an example, the angular distribution hasthe differential form [60,61]

d4Γ[B → K∗(→ Kπ)`+`−]

dq2 d cos θl d cos θK dφ=

9

32π

∑i

Ji(q2) gi(θl, θK , φ) , (5)

with respect to q2 and three decay angles θl, θK , and φ. For the B0 (B0), θl is the anglebetween the µ+ (µ−) and the opposite of the B0 (B0) direction in the dimuon rest frame,θK is the angle between the kaon and the direction opposite to the B meson in the K∗0

rest frame, and φ is the angle between the µ+µ− and K+π− decay planes in the B restframe. There are twelve angular terms appearing in the distribution and it is a long-term

7

experimental goal to measure the coefficient functions Ji(q2) associated with these twelve

terms, from which all other B → K(∗)`+`− observables can be derived.In the SM, with massless leptons, the Ji depend on bilinear products of six complex

K∗ spin amplitudes AL,R⊥,‖,0,6 such as

J1s =3

4

[|AL⊥|2 + |AL‖ |2 + |AR⊥|2 + |AR‖ |2

]. (6)

The expressions for the eleven other Ji terms are given for example in Refs. [53, 62].Depending on the number of operators that are taken into account in the analysis, it ispossible to relate some of the Ji terms. The full derivation of these symmetries can befound in Ref. [53].

When combining B and B decays, it is possible to form both CP -averaged and CP -asymmetric quantities: Si = (Ji + Ji)/[d(Γ + Γ)/dq2] and Ai = (Ji − Ji)/[d(Γ + Γ)/dq2],from the Ji [52, 53,61–65]. The terms J5,6,8,9 in the angular distribution are CP -odd and,consequently, the associated CP -asymmetry, A5,6,8,9 can be extracted from an untaggedanalysis (making it possible for example to measure A5,6,8,9 in B0

s → φµ+µ− decays).Moreover, the terms J7,8,9 are T -odd and avoid the usual suppression of the correspondingCP -asymmetries by small strong phases [63]. The decay B0→ K∗0µ+µ−, where the K∗0

decays to K+π−, is self-tagging (the flavour of the initial B meson is determined from thedecay products) and it is therefore possible to measure both the Ai and Si for the twelveangular terms.

In addition, a measurement of the T -odd CP asymmetries, A7, A8 and A9, which arezero in the SM and are not suppressed by small strong phases in the presence of NP,would be useful to constrain non-standard CP violation. This is particularly true since thedirect CP asymmetry in the inclusive B → Xsγ decay is plagued by sizeable long-distancecontributions and is therefore not very useful as a constraint on NP [66].

2.3.3 Strategies for analysis of B0 → K∗0`+`− decays

In 1.0 fb−1 of integrated luminosity, LHCb has collected the world’s largest samples ofB0→ K∗0µ+µ− (with K∗0→ K+π−) and B0

s→ φµ+µ− decays, with around 900 and 80signal candidates respectively reported in preliminary analyses [67, 68]. These candidatesare however sub-divided into six q2 bins, following the binning scheme used in previousexperiments [69]. With the present statistics, the most populated q2 bin contains ∼ 300B0→ K∗0µ+µ− candidates which is not sufficient to perform a full angular analysis. Theanalyses are instead simplified by integrating over two of the three angles or by applyinga folding technique to the φ angle, φ→ φ+ π for φ < 0, to cancel terms in the angulardistribution.

In the case of massless leptons, one finds:

dΓ′

dφ=

Γ′

2π(1 + S3 cos 2φ+ A9 sin 2φ) , (7)

6 Further amplitudes contribute in principle, but they are either suppressed by small lepton masses ororiginate from non-standard scalar/tensor operators.

8

dΓ′

dθK=

3Γ′

4sin θK

(2FL cos2 θK + (1− FL) sin2 θK

), (8)

dΓ′

dθ`= Γ′

(3

4FL sin2 θ` +

3

8(1− FL)(1 + cos2 θ`) + AFB cos θ`

)sin θ`. (9)

where Γ′ = Γ + Γ. The observables appear linearly in the expressions. Experimentally, thefits are performed in bins of q2 and the measured observables are rate averaged over the q2

bin. The observables appearing in the angular projections are the fraction of longitudinalpolarisation of the K∗, FL, the lepton system forward-backward asymmetry, AFB, S3 andA9.

The differential branching ratio, AFB and FL have been measured by the B factories,CDF and LHCb [67,69,70]. The observable S3 is related to the asymmetry between theparallel and perpendicular K∗ spin amplitudes7 is sensitive to right-handed operators (C ′7)at low q2, and is negligibly small in the SM. In the future, the decay B0→ K∗0e+e− couldplay an important role in constraining C ′7 through S3 since it allows one to probe to smallervalues of q2 than the B0→ K∗0µ+µ− decay. First measurements have been performedby CDF and LHCb [67, 70].8 The current experimental status of these B0→ K∗0µ+µ−

angular observables at LHCb, the B factories and CDF is shown in Fig. 1. Improvedmeasurements of these quantities would be useful to constrain the chirality-flipped Wilsoncoefficients (C ′7, C ′9 and C ′10).

Whilst AFB is not free from form-factor uncertainties at low q2, the value of the dileptoninvariant mass q2

0, for which the differential forward-backward asymmetry AFB vanishes,can be predicted in a clean way.9 The zero crossing-point is highly sensitive to the ratio ofthe two Wilson coefficients C7 and C9. In particular the model-independent upper boundon |C9| implies q2

0 > 1.7 GeV2/c4, which improves to q20 > 2.6 GeV2/c4, assuming the sign

of C7 to be SM-like [39]. At next-to-leading order one finds [50]:10

q20[K∗0`+`−] = 4.36 +0.33

−0.31 GeV2/c4, q20[K∗+`+`−] = 4.15 +0.27

−0.27 GeV2/c4. (10)

where the first value is in good agreement with the recent preliminary result from LHCbof q2

0 = 4.9 +1.3−1.1 GeV2/c4 [67] for the B0→ K∗0µ+µ− decay.

7 The quantity S3 = (1− FL)/2×A(2)T (in the massless case) allows access to one of the theoretically

clean quantities, namely A(2)T . The observable A

(2)T is a theoretically cleaner observable than S3 due to

the cancellation of some of the form-factor dependence [71].8 Depending on the convention for the angle φ, dΓ′/dφ of Eq. (7) can also depend on S9, which is tiny

in the SM and beyond. Note that, due to different angular conventions, the quantity AIm reported inRef. [67] corresponds to S9, while AIm in Ref. [70] corresponds to A9.

9 In the QCDF approach at leading order in ΛQCD/mb, the value of q20 is free from hadronic uncertainties

at order α0s. A dependence on the soft form factor and on the light-cone wave functions of the B and K∗

mesons appears only at order α1s.

10 A recent determination of q20 in B0 decays gives 4.0 ± 0.3 GeV2/c4 [39]. The shift with respect to

Ref. [50] is of parametric origin and is driven in part by the choice of the renormalisation scale (µ = 4.2 GeVinstead of 4.8 GeV), but also due to differences in the implementation of higher O(αS) short-distancecontributions.

9

]4c/2 [GeV2q0 5 10 15 20

LF

0

0.2

0.4

0.6

0.8

1

PreliminaryLHCb

Theory Binned theoryLHCb CDF BELLE BaBar

(a)

]4c/2 [GeV2q0 5 10 15 20

FB

A

-1

-0.5

0

0.5

1

PreliminaryLHCb

Theory Binned theoryLHCb CDF BELLE BaBar

(b)

]4c/2 [GeV2q0 5 10 15 20

3S

-1

-0.5

0

0.5

1

Theory Binned theoryLHCb CDF

PreliminaryLHCb

(c)

]4c/2 [GeV2q0 5 10 15 20

9S

-1

-0.5

0

0.5

1LHCb

PreliminaryLHCb

(d)

Figure 1: Summary of recent measurements of the angular observables (a) FL, (b) AFB, (c) S3

and (d) S9 in B0→ K∗0µ+µ− decays at LHCb, CDF and the B factories [67]. Descriptions ofthese observables are provided in the text (see Eqs. (7), (8) and (9) and footnote 8). The theorypredictions at low- and high-dimuon invariant masses are indicated by the coloured bands andare also described in detail in the text.

It is possible to access information from other terms in the angular distribution byintegrating over one of the angles and making an appropriate folding of the remaining twoangles. From φ and θK only [72] it is possible to extract:

S5 = −4

3

[∫ 3π/2

π/2

−∫ π/2

0

−∫ 2π

3π/2

]dφ

[∫ 1

0

−∫ 0

−1

]d cos θK

d3(Γ− Γ)

dq2 d cos θKdφ

/d(Γ + Γ)

dq2.

(11)Analogously to AFB, the zero-crossing point of S5 has been shown to be theoretically clean.This observable is sensitive to the ratio of Wilson coefficients, (C7 +C ′7)/(C9 +mb(C7 +C ′7)),and if measured would add complementary information to AFB and S3 about new right-handed currents.

10

2.3.4 Theoretically clean observables in B0 → K∗0`+`− decays

By the time that 5 fb−1 of integrated luminosity is available at LHCb, it will be possibleto exploit the complete NP sensitivity of the B → K∗`+`− both in the low- and high-q2

regions, by performing a full angular analysis. The increasing size of the experimentalsamples makes it important to design optimised observables (by using specifically chosencombinations of the Ji) to reduce theoretical uncertainties. In the low q2 region, the lineardependence of the amplitudes on the soft form factors allows for a complete cancellationof the hadronic uncertainties due to the form factors at leading order. This consequentlyincreases the sensitivity to the structure of NP models [52,53].

In the low q2 region, the so-called transversity observables A(i)T , i = 2, 3, 4, 5 are an

example set of observables that are constructed such that the soft form factor dependencecancels out at leading order. They represent the complete set of angular observablesand are chosen to be highly sensitive to new right-handed currents via C ′7 [52, 53]. Asecond, complete, set of optimised angular observables was constructed (also in the cases ofnon-vanishing lepton masses and in the presence of scalar operators) in Ref. [54]. Recentlythe effect of binning in q2 on these observables has been considered [71]. In these sets ofobservables, the unknown ΛQCD/mb corrections are estimated to be of order 10 % on thelevel of the spin amplitudes and represent the dominant source of theory uncertainty.

In general, the angular observables are shown to offer high sensitivity to NP inthe Wilson coefficients of the operators O7, O9, and O10 and of the chirally flippedoperators [52, 53, 61, 63]. In particular, the observables S3, A9 and the CP -asymmetriesA7 and A8 vanish at leading order in ΛQCD/mb and αS in the SM operator basis [63].Importantly, this suppression is absent in extensions with non-vanishing chirality-flippedC ′7,9,10, giving rise to contributions proportional to Re(CiC

∗j′) or Im(CiC

∗j′) and making

these terms ideal probes of right-handed currents [52,53,61,63]. CP asymmetries are smallin the SM, because the only CP -violating phase affecting the decay is doubly Cabibbo-suppressed, but can be significantly enhanced by NP phases in C9,10 and C ′9,10, which atpresent are poorly constrained. In a full angular analysis it can also be shown that CP -conserving observables provide indirect constraints on CP -violating NP contributions [53].

At large q2, the dependence on the magnetic Wilson coefficients C(′)7 is suppressed,

allowing, in turn, a cleaner extraction of semileptonic coefficients (C(′)9 and C

(′)10 ). A set

of transversity observables H(i)T , i = 1, 2, 3 have been designed to exploit the features of

this kinematic region in order to have small hadronic uncertainties [64]. As a consequenceof symmetry relations of the OPE [39,64,65,73], at high q2, combinations of the angularobservables Ji can be formed within the SM operator basis (i.e. with C ′i = 0), whichdepend:

• only on short-distance quantities (e.g. H(2,3)T ) ;

• only on long-distance quantities (FL and low q2 optimised observables A(2,3)T ).

Deviations from these relations are due to small sub-leading corrections at order(ΛQCD/mb)

2 from the OPE.

11

In the SM operator basis it is interesting to note that A(2,3)T , which are highly sensitive to

short distance contributions (from C ′7) at low q2, instead become sensitive to long-distancequantities (the ratio of form factors) at high q2. The extraction of form factor ratios is

already possible with current data on S3 (A(2)T ) and FL and leads to a consistent picture

between LCSR calculations, lattice calculations and experimental data [40, 73]. In thepresence of chirality-flipped Wilson coefficients, these observables are no longer short-distance free, but are probes of right-handed currents [41]. At high q2, the OPE framework

predicts H(2)T = H

(3)T and J7 = J8 = J9 = 0. Any deviation from these relationships, would

indicate a problem with the OPE and the theoretical predictions in the high q2 region.

2.3.5 B+→ K+µ+µ− and B+→ K+e+e−

The branching fractions of B0(+) → K0(+)µ+µ− have been measured by BaBar, Belle andCDF [69,74,75]. In 1.0 fb−1 LHCb observes 1250 B+→ K+µ+µ− decays [76], and in thefuture will dominate measurements of these processes.

Since the B → K transition does not receive contributions from an axial vectorcurrent, the primed Wilson coefficients enter the B0(+) → K0(+)µ+µ− observables alwaysin conjunction with their unprimed counterparts as (Ci + C ′i). This is in contrast to theB → K∗µ+µ− decay and therefore provides complementary constraints on the Wilsoncoefficients and their chirality-flipped counterparts.

An angular analysis of the µ+µ− pair in the B0(+) → K0(+)µ+µ− decay would allowthe measurement of two further observables, the forward-backward asymmetry AFB andthe so-called flat term FH [77]. The angular distribution of a B meson decaying to apseudoscalar meson, P , and a pair of leptons involves just q2 and a single angle in thedilepton system, θl [77]

1

Γ`

dΓ`[B → P`+`−]

dcos θl=

3

4(1− FH)(1− cos2 θl) +

1

2FH + AFB cos θl . (12)

In the SM, the forward-backward asymmetry of the dilepton system is expected tobe zero. Any non-zero forward-backward asymmetry would point to a contribution fromnew particles that extend the SM operator basis. Allowing for generic (pseudo-)scalar andtensor couplings, there is sizeable room for NP contributions in the range |AFB| <∼ 15 %.The flat term, FH/2, that appears with AFB in the angular distribution, is non-zero, butsmall (for ` = e, µ) in the SM. This term can also see large enhancements in modelswith (pseudo-)scalar and tensor couplings of up to FH ∼ 0.5. Recent SM predictions atlow- and high-q2 can be seen in Refs. [39, 55, 77, 78]. The current experimental limitson B(B0

s → µ+µ−) now disfavour large CS and CP , and if NP is present only in tensoroperators then NP contributions are expected to be in the range |AFB| <∼ 5 % and FH <∼ 0.2.

In addition to AFB, FH and the differential branching fraction of the decays, it ispossible to probe the universality of lepton interactions by comparing the branchingfraction of decays B0(+)→ K0(+)`+`− with two different lepton flavours (e.g. electronsversus muons):

RK = Γµ/Γe (with the same q2 cuts). (13)

12

Lepton universality may be violated in extensions to the SM, such as R-parity-violatingSUSY models.11 In the SM, the ratio RSM

K is expected to be close to unity, RSMK =

1 +O(m2µ/m

2B) [82].

It is also interesting to note that at high q2 the differential decay rates and CPasymmetries of B0(+) → K0(+)`+`− and B0(+) → K∗0(+)`+`− (` = e, µ) are correlated [39]and exhibit the same short-distance dependence (in the SM operator basis). Any deviationwould point to a problem for the OPE used in the high q2 region.

2.3.6 Rare semileptonic b→ d`+`− decays

Rare b → d radiative decay processes, such as B → ργ, have been observed at the Bfactories [83,84]. In the 2011 data sample, the very rare decay B+→ π+µ+µ− was observedat the LHCb experiment (see Fig. 2). This is a rare b→ d`+`− transition, which in theSM is suppressed by loop and CKM factors proportional to |Vtd/Vts|. In the 1.0 fb−1

data sample, LHCb observes 25.3 +6.7−6.4 signal candidates corresponding to a branching

fraction of B(B+→ π+µ+µ−) = (2.4 ± 0.6 ± 0.2) × 10−8 [85]. This measurement is ingood agreement with the SM prediction, i.e. consistent with no large NP contribution tob→ d`+`− processes and with the MFV hypothesis.

The b → d transitions can show potentially larger CP - and isospin-violating effectsthan their b → s counterparts due to the different CKM hierarchy [50]. These studieswould need the large statistics provided by the future LHCb upgrade. A 50 fb−1 datasample will also enable a precision measurement of the ratio of the branching fractions ofB+ meson decays to π+µ+µ− and K+µ+µ−. This ratio would enable a useful comparisonof |Vtd/Vts| to be made using penguin processes (with form factors from lattice QCD) andbox processes (using ∆ms/∆md and bag-parameters from lattice QCD) and provide apowerful test of MFV.

2.3.7 Isospin asymmetry of B0(+)→ K0(+)µ+µ− and B0(+)→ K∗0(+)µ+µ− decays

Analyses at hadron colliders (at LHCb and CDF) have mainly focused on decay modes withcharged tracks in the final state. B meson decays involving K0 mesons are experimentallymuch more challenging due to the long lifetimes of K0

S and K0L mesons (the K0

L is notreconstructable within LHCb). Nevertheless, LHCb has been able to select 60 B0 →K0µ+µ− decays, reconstructed as K0

S→ π+π−, and 80 B+→ K∗+µ+µ−, reconstructed asK∗+→ K0

Sπ+, which are comparable in size to the samples that are available for these

modes in the full data sets of the B factories. The isolation of these rare decay modesenables a measurement of the isospin asymmetry of B→ K(∗)µ+µ− decays,

AI =B(B0 → K0µ+µ−)− (τB0/τB+)B(B+ → K+µ+µ−)

B(B0 → K0µ+µ−) + (τB0/τB+)B(B+ → K+µ+µ−). (14)

At leading order, isospin asymmetries (which involve the spectator quark) are expected tobe zero in the SM. Isospin-breaking effects are subleading in ΛQCD/mb, and are difficult

11 There are hints of lepton universality violation in recent measurements of B → D(∗)τν by BaBar [79]and Belle [80,81].

13

]2c [MeV/-µ +µ +π M5000 5500 6000 6500

)2 cC

andi

date

s / (

20 M

eV/

0

5

10

15

20

(a)

-µ +µ +π → +B

-µ +µ + K→ +B

-π +π +π → +B

Part. reco.

Combinatorial

LHCb

Figure 2: Invariant mass of selected B+ → π+µ+µ− candidates in 1.0 fb−1 of integratedluminosity [85]. In the legend, “part. reco.” and “combinatorial” refer to partially reconstructedand combinatorial backgrounds respectively.

to estimate due to unknown power corrections. Nevertheless isospin-breaking effects areexpected to be small and these observables may be useful in NP searches because theyoffer complementary information on specific Wilson coefficients [86].

The LHCb measurement of the K and K∗ isospin asymmetries in bins of q2 are shownin Fig. 3. For the K∗ modes AI is compatible with the SM expectation that ASM

I ' 0,but for the K+/K0 modes, AI is seen to be negative at low- and high-q2 [76]. This isconsistent with what has been seen at previous experiments, but is inconsistent with thenaıve expectation of ASM

I ∼ 0 at the 4σ level.12 Such a discrepancy would be hard toexplain in any model that is also consistent with other experimental results. Improvedmeasurements are needed to clarify the situation.

2.4 Radiative B decays

While the theoretical prediction of the branching ratio of the B → K∗γ decay is problematicdue to large form factor uncertainties, the mixing-induced asymmetry13 SK∗γ providesan important constraint due to its sensitivity to the chirality-flipped magnetic Wilsoncoefficient C ′7. At leading order it vanishes for C ′7 → 0, so the SM prediction is tiny andexperimental evidence for a large SK∗γ would be a clear indication of NP effects throughright-handed currents [88, 89]. Unfortunately it is experimentally very challenging tomeasure SK∗γ in a hadronic environment, requiring both flavour tagging and the abilityto reconstruct the K∗0 in the decay mode K∗0→ K0π0. However, the channel B0

s → φγ,which is much more attractive experimentally, offers the same physics opportunities, withadditional sensitivity due to the non-negligible width difference in the B0

s system. Moreover,

12 A calculation of ASMI (B → Kµ+µ−) has recently become available [87], giving values consistent with

the naıve expectation within 1 %.13 Note that the notation S used here and in the literature for mixing-induced asymmetries is not

related to the use of the notation in Sec. 2.3 for CP -averaged properties of the angular distributions.

14

]4c/2 [GeV2q0 5 10 15 20 25

IA

-1.5

-1

-0.5

0

0.5

1

LHCb-µ+µ K→B

(a)

]4c/2 [GeV2q0 5 10 15 20

IA

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

Theory Data

LHCb-µ+µ K →B *

(b)

Figure 3: (a) B → Kµ+µ− and (b) B → K∗µ+µ− isospin asymmetries in 1.0 fb−1 of datacollected by the LHCb collaboration in 2011 [76].

LHCb can study several other interesting radiative b-hadron decays.

2.4.1 Experimental status and outlook for rare radiative decays

In 1.0 fb−1 of integrated luminosity LHCb observes 5300 B0→ K∗0γ and 690 B0s→ φγ [17]

candidates. These are the largest samples of rare radiative B0 and B0s decays collected

by a single experiment. The large sample of B0→ K∗0γ decays has enabled LHCb tomake the world’s most precise measurement of the direct CP -asymmetry ACP (K∗γ) =0.8± 1.7± 0.9 %, compatible with zero as expected in the SM [17].

With larger data samples, it will be possible to add additional constraints on theC7 − C ′7 plane through measurements of b → sγ processes. These include results fromtime-dependent analysis of B0

s→ φγ [90], as described in detail in the LHCb roadmapdocument [5]. Furthermore, the large Λ0

b production cross-section will allow for measure-ments of the photon polarisation through the decays Λ0

b → Λ(∗)γ [91, 92]. In fact, thestudy of Λ0

b → Λ transitions is quite attractive from the theoretical point of view, sincethe hadronic uncertainties are under good control [93–95]. However, because the Λ0

b has

JP = 12

+and can be polarised at production, it will be important to measure first the Λ0

b

polarisation.B→ V Pγ decays with a photon, a vector and a pseudoscalar particle in the final state

can also provide sensitivity to C ′7 [96–99]. The decays B→ φKγ and B+ → K1(1270)+γhave been previously observed at the B factories [100, 101] and large samples will beavailable for the first time at LHCb.

15

2.5 Leptonic B decays

2.5.1 B0s→ µ+µ− and B0→ µ+µ−

The decays B0(s)→ µ+µ− are a special case amongst the electroweak penguin processes, as

they are chirality-suppressed in the SM and are most sensitive to scalar and pseudoscalaroperators. The branching fraction of B0

(s) → µ+µ− can be expressed as [102–105]:

B(B0q → µ+µ−) =

G2Fα

2

64π3f 2BqτBqm

3Bq |VtbV ∗tq|2

√1− 4m2

µ

m2Bq

(15)

×(

1− 4m2µ

m2Bq

)|CS − C ′S|2 +

∣∣∣∣(CP − C ′P ) + 2mµ

mBq

(C10 − C ′10)

∣∣∣∣2,

where q = s, d.Within the SM, CS and CP are negligibly small and the dominant contribution of C10

is helicity suppressed. The coefficients Ci are the same for B0s and B0 in any scenario

(SM or NP) that obeys MFV. The large suppression of B(B0 → µ+µ−) with respect toB(B0

s → µ+µ−) in MFV scenarios means that B0s→ µ+µ− is often of more interest than

B0 → µ+µ− for NP searches. The ratio B(B0s → µ+µ−)/B(B0 → µ+µ−) is however a very

useful probe of MFV.The SM branching fraction depends on the exact values of the input parameters: fBq ,

τBq and |VtbV ∗tq|2. The B0s decay constant, fBs , constitutes the main source of uncertainty

on B(B0s → µ+µ−). There has been significant progress in theoretical calculations of this

quantity in recent years. As of the year 2009 there were two unquenched lattice QCDcalculations of fBs , by the HPQCD [106] and FNAL/MILC [107] collaborations, which,when averaged, gave the value fBs = 238.8± 9.5 MeV [108]. The FNAL/MILC calculationwas updated in 2010 [109], and again in 2011 to give fBs = 242±9.5 MeV [110,111]. Also in2011, the ETM collaboration reported a value of fBs = 232± 10 MeV [112]. The HPQCDcollaboration presented in 2011 a result, fBs = 227 ± 10 MeV [113], which has recentlybeen improved upon with an independent calculation that gives fBs = 225± 4 MeV [114].

A weighted average of FNAL/MILC ’11 [110], HPQCD ’11 [113] and HPQCD ’12 [114]was presented recently [108], giving fBs = 227.6 ± 5.0 MeV. Using this value, the SMprediction for the branching ratio is [115]:

B(B0s → µ+µ−)SM = (3.1± 0.2)× 10−9 . (16)

This value is taken as the nominal B(B0s → µ+µ−)SM. Note that, in addition to fBs ,

other sources of uncertainty are due to the B0s lifetime, the CKM matrix element |Vts|,

the top mass mt, the electroweak corrections and scale variations. For a more detaileddiscussion of the SM prediction, see Ref. [116]. It is also possible to obtain predictionsfor B(B0

s → µ+µ−)SM with reduced sensitivity to the value of fBs using input from either∆ms [117] or from a full CKM fit [118].

16

Likewise for fBd , using the average of ETMC-11 (fBd = 195 ± 12 MeV) [112],FNAL/MILC-11 (fBd = 197±9 MeV) [110,111] and HPQCD-12 (fBd = 191±9 MeV) [114]results, which gives fBd = 194± 10 MeV [119], the branching ratio of B0 → µ+µ− is:

B(B0 → µ+µ−)SM = (1.1± 0.1)× 10−10 . (17)

NP models, especially those with an extended Higgs sector, can significantly enhancethe B0

(s) → µ+µ− branching fraction even in the presence of other existing constraints. In

particular, it has been emphasised in many works [120–127] that the decay B0s → µ+µ−

is very sensitive to the presence of SUSY particles. At large tan β – where tan β is theratio of vacuum expectation values of the Higgs doublets14 – the SUSY contribution tothis process is dominated by the exchange of neutral Higgs bosons, and both CS and CPcan receive large contributions from scalar exchange.

In constrained SUSY models such as the CMSSM and NUHM1 (see Sec. 2.7), predictionscan be made for B(B0

s → µ+µ−) that take into account the existing constraints from thegeneral purpose detectors. These models predict [128]:

1 <B(B0

s → µ+µ−)CMSSM

B(B0s → µ+µ−)SM

< 2 , 1 <B(B0

s → µ+µ−)NUHM1

B(B0s → µ+µ−)SM

< 3 . (18)

The LHCb [13] (and CMS [129]) measurements of B0s → µ+µ− have already excluded the

upper range of these predictions.Other NP models such as composite models (e.g. Littlest Higgs model with T -parity

or Topcolour-assisted Technicolor), models with extra dimensions (e.g. Randall Sundrummodels) or models with fourth generation fermions can modify B(B0

s → µ+µ−) [115,130–134]. The NP contributions from these models usually arise via (C10 − C ′10), and theyare therefore correlated with the constraints from other b → s`+`− processes, e.g. withB(B+→ K+µ+µ−) which depends on (C10 + C ′10). The term (CP − C ′P ) in the branchingfraction adds coherently with the SM contribution from (C10 − C ′10), and therefore canalso destructively interfere. In such cases, if (CS − C ′S) remains small, B(B0

s → µ+µ−)could be smaller than the SM prediction. A measurement of B(B0

s → µ+µ−) well belowthe SM prediction would be a clear indication of NP and would be symptomatic of amodel with a large non-degeneracy in the scalar sector (where C

(′)P is enhanced but C

(′)S is

not). If only C10 is modified, these constraints currently require the branching ratio to beabove 1.1× 10−10 [41]. In the presence of NP effects in both C10 and C ′10, even strongersuppression is possible in principle.

At the beginning of 2012, the LHCb experiment set the world best limits on theB(B0

(s) → µ+µ−) [13].15 At 95 % C.L.

B(B0s → µ+µ−) < 4.5× 10−9 ,

B(B0 → µ+µ−) < 1.0× 10−9 .

14 Note that elsewhere in this document the symbol β is used to denote an angle of the unitarity triangleof the CKM matrix.

15 Results on B(B0(s) → µ+µ−) presented at HCP2012 [14] are not included in this discussion.

17

Experimentally the measured branching fraction is the time-averaged (TA) branchingfraction, which differs from the theoretical value because of the sizeable width differencebetween the heavy and light B0

s mesons [135,136].16 In general,

B(B0s → µ+µ−)TH = [(1− y2

s)/(1 +A∆Γys)]× B(B0s → µ+µ−)TA (19)

where A∆Γ = +1 in the SM and ys = ∆Γs/(2Γs) = 0.088 ± 0.014 [138]. Thus theexperimental measurements have to be compared to the following SM prediction for thetime-averaged branching fraction:

B(B0s → µ+µ−)SM,TA = B(B0

s → µ+µ−)SM,TH/(1− ys) = (3.5± 0.2)× 10−9 . (20)

With 50 fb−1 of integrated luminosity, taken with an upgraded LHCb experiment,a precision better than 10 % can be achieved in B(B0

s → µ+µ−), and ∼ 35 % on theratio B(B0

s → µ+µ−)/B(B0 → µ+µ−). The dominant systematic uncertainty is likely tocome from knowledge of the ratio of fragmentation fractions, fd/fs, which is currentlyknown to a precision of 8 % from two independent determinations.17 One method [139]18

is based on hadronic B decays [141, 142], and relies on knowledge of the B(s) → D(s)

form factors from lattice QCD calculations [143]. The other [144] uses semileptonicdecays, exploiting the expected equality of the semileptonic widths [145,146]. However,the two methods have a common, and dominant, uncertainty which originates from themeasurement of B(D+

s → K+K−π+), which in the PDG is given to 4.9 % (coming from asingle measurement from CLEO [147]). A new preliminary result from Belle has recentlybeen presented [148] – inclusion of this measurement in the world average will improvethe uncertainty on B(D+

s → K+K+π+) to ∼ 3.5 %. With the samples available with theLHCb upgrade, it will be possible to go beyond branching fraction measurements andstudy the effective lifetime of B0

s → µ+µ−, that provides additional sensitivity to NP [135].In Sec. 2.7, the NP implications of the current measurements of B(B0

s → µ+µ−) andthe interplay with other observables, including results from direct searches, are discussedfor a selection of specific NP models. In general, the strong experimental constraintson B(B0

s → µ+µ−) [13, 129, 149, 150] largely preclude any visible effects from scalar orpseudoscalar operators in other b→ s`+`− decays.19

2.5.2 B0s → τ+τ−

The leptonic decay B0s → τ+τ− provides interesting information on the interaction of

the third generation quarks and leptons. In many NP models, contributions to third

16 This was previously observed in a different context [137].17 This value is valid for B mesons produced from

√s = 7 TeV pp collisions within the LHCb acceptance.

It will, in principle, need to be remeasured at each different LHC collision energy, and may depend onthe kinematic acceptance of the detector (i.e. on the transverse momentum and pseudorapidity of the Bmesons). However, once a suitable B0

s branching fraction, such as that for B0s → J/ψ φ or B0

s → K+K−,is known to good precision, normalisation can be carried out without direct need for an fd/fs value.

18 The results from Ref. [139] were updated at HCP2012 [140].19 Barring a sizeable, fortuitous cancellation among CS,P and C ′S,P [78].

18

generation quarks/leptons can be dramatically enhanced with respect to the first andsecond generation. This is true in, for example, scalar and pseudoscalar interactions insupersymmetric scenarios, for large values of tan β. Interestingly, there is also an interplaybetween b → sτ+τ− processes and the lifetime difference Γs12 in B0

s mixing (see Sec. 3).The correlation of both processes has been discussed model-independently [151,152] and inspecific scenarios, such as leptoquarks [153,154] or Z ′ models [155–157]. There are presentlyno experimental limits on B0

s → τ+τ−, however the interplay with Γs12, and the latestLHCb-measurement of Γd/Γs would imply a limit of B(B0

s → τ+τ−) < 3 % at 90 % C.L.Any improvement on this limit, which might be in reach with the existing LHCb data set,would yield strong constraints on models that couple strongly to third generation leptons.A large enhancement in b→ sτ+τ− could help to understand the anomaly observed bythe D0 experiment in their measurement of the inclusive dimuon asymmetry [158] andcould also reduce the tension that exists with other mixing observables [151,152].

The study of B0s → τ+τ− at LHCb presents significant challenges. The τ leptons must

be reconstructed in decays that involve at least one missing neutrino. Although it hasbeen demonstrated that the decay Z → τ+τ− can be separated from background at LHCb,using both leptonic and hadronic decay modes [159], at lower energies the backgroundsfrom semileptonic heavy flavour decays cause the use of the leptonic decay modes to bedisfavoured. However, in the case that “three-prong” τ decays are used, the vertices canbe reconstructed from the three hadron tracks. The analysis can then benefit from theexcellent vertexing capability of LHCb, and, due to the finite lifetime of the τ lepton,there are in principle sufficient kinematic constraints to reconstruct the decay. Work is inprogress to understand how effectively the different potential background sources can besuppressed, and hence how sensitive LHCb can be in this channel.

2.6 Model-independent constraints

Figure 4, taken from Ref. [41], shows the current constraints on the NP contributions to the

Wilson coefficients (defined in Eq. 1) C(′)7 , C

(′)9 and C

(′)10 , varying only one coefficient at a

time. The experimental constraints included here are: the branching fractions of B → Xsγ,B → Xs`

+`−, B → Kµ+µ− and B0s → µ+µ−, the mixing-induced asymmetries in

B → K∗γ and b→ sγ and the branching fraction and angular observables in B → K∗µ+µ−.One can make the following observations:

• At 95 % C.L., all Wilson coefficients are compatible with their SM values.

• For the coefficients present in the SM, i.e. C7, C9 and C10, the constraints on theimaginary part are looser than on the real part.

• For the Wilson coefficients C(′)10 , the constraint on B(B0

s → µ+µ−) is starting tobecome competitive with the constraints from the angular analysis of B → K(∗)µ+µ−.

• The constraints on C ′9 and C ′10 from B → Kµ+µ− and B → K∗µ+µ− are comple-mentary and lead to a more constrained region, and better agreement with the SM,than with B → K∗µ+µ− alone.

19

-0.5 0.0 0.5

-0.5

0.0

0.5

ReHC7NPL

ImHC 7

L

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

ReHC9NPL

ImHC 9

L

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

ReHC10NPL

ImHC 1

0L

-0.5 0.0 0.5

-0.5

0.0

0.5

ReHC7'L

ImHC 7

' L

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

ReHC9'L

ImHC 9

' L

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

ReHC10' L

ImHC 1

0'L

Figure 4: Individual 2σ constraints in the complex planes of Wilson coefficients, comingfrom B → Xs`

+`− (brown), B → Xsγ (yellow), ACP (b → sγ) (orange), B → K∗γ (purple),B → K∗µ+µ− (green), B → Kµ+µ− (blue) and B0

s → µ+µ− (grey), as well as combined 1 and2σ constraints (red) [41].

• A second allowed region in the C7-C′7 plane characterised by large positive con-

tributions to both coefficients, which was found previously to be allowed e.g. inRefs. [37, 38], is now disfavoured at 95 % C.L. by the new B → K∗µ+µ− data, inparticular the measurements of the forward-backward asymmetry from LHCb.

The second point above can be understood from the fact that for the branching fractionsand CP -averaged angular observables which give the strongest constraints, only NPcontributions aligned in phase with the SM can interfere with the SM contributions. As aconsequence, NP with non-standard CP violation is in fact constrained more weakly thanNP where CP violation stems only from the CKM phase. This highlights the need forimproved measurements of CP asymmetries directly sensitive to non-standard phases.20

Significant improvements of these constraints – or first hints for physics beyond theSM – can be obtained in the future by both improved measurements of the observablesdiscussed above and by improvements on the theoretical side. From the theory side, thereis scope for improving the estimates of the hadronic form factors from lattice calculations,

20 LHCb has presented results on ACP (B0 → K∗0µ+µ−) at CKM 2012 [160].

20

Figure 5: Constraints from flavour observables in CMSSM in the plane (m1/2,m0) with A0 = 0,for tanβ = (left) 50 and (right) 30 [161], using SuperIso [105,162]. The black line correspondsto the CMS exclusion limit with 1.1 fb−1 of data [163] and the red line to the CMS exclusionlimit with 4.4 fb−1 of data [164].

which will reduce the dominant source of uncertainty on the exclusive decays. On theexperimental side there are a large number of theoretically clean observables that can beextracted with a full angular analysis of B0→ K∗0µ+µ−, as discussed in Sec. 2.3.2.

2.7 Interplay with direct searches and model-dependent con-straints

The search for SUSY is the main focus of NP searches in ATLAS and CMS. Althoughthe results so far have not revealed a positive signal, they have put strong constraintson constrained SUSY scenarios. The understanding of the parameters of SUSY modelsalso depends on other measurements, such as the anomalous dipole moment of the muon,limits from direct dark matter searches, measurements of the dark matter relic density andvarious B physics observables. As discussed in Sec. 2.5, the rare decay channels studiedin LHCb, such as B0

(s) → µ+µ−, provide stringent tests of SUSY. In addition, the decays

B → K(∗)µ+µ− provide many complementary observables which are sensitive to differentsectors of the theory. In this section, the implications of the current LHCb measurementsin different SUSY models are explained, both in constrained scenarios and in a moregeneral case.

First consider the constrained minimal supersymmetric standard model (CMSSM)and a model with non-universal Higgs masses (NUHM1). The CMSSM is characterisedby the set of parameters m0,m1/2, A0, tan β, sgn(µ) and invokes unification boundaryconditions at a very high scale mGUT where the universal mass parameters are specified.The NUHM1 relaxes the universality condition for the Higgs bosons which are decoupledfrom the other scalars, adding then one extra parameter compared to the CMSSM.

Figure 5 shows the plane (m1/2,m0) for large and moderate values of tan β in theCMSSM where, for comparison, direct search limits from CMS are superimposed. It

21

Figure 6: SUSY spread of (top left) AFB(B → K∗µ+µ−) at low q2, (top right) q20(B → K∗µ+µ−)

and (bottom) FL(B → K∗µ+µ−) as a function of the lightest stop mass, for A0 = 0 and tanβ= 50 [119], using SuperIso [105,162]. The solid red lines correspond to the preliminary LHCbcentral value with 1.0 fb−1 [67], while the dashed and dotted lines represent the 1 and 2σ boundsrespectively, including both theoretical and experimental errors.

can be seen that, at large tan β, the constraints from flavour observables – in particularB(B0

s → µ+µ−)– are more constraining than those from direct searches. As soon as onegoes down to smaller values of tan β, the flavour observables start to lose importancecompared to direct searches. On the other hand, B → K∗µ+µ− related observables, inparticular the forward-backward asymmetry, lose less sensitivity and play a complementaryrole. To see better the effect of AFB(B → K∗µ+µ−) at low q2,21 the AFB zero-crossingpoint q2

0 and FL(B → K∗µ+µ−), in Fig. 6 their SUSY spread is shown as a function of thelightest stop mass for tan β = 50 [119]. As can be seen from the figure, small stop massesare excluded and in particular mt1 . 800 GeV is disfavoured by AFB at the 2σ level.

The impact of the recent B → K(∗)l+l− decay data on SUSY models beyond MFV(NMFV) with moderate tan β is shown in Fig. 7. The largest effect stems from left-rightmixing between top and charm super-partners. Due to the Z-penguin dominance of theSUSY-flavour contributions the constraints are most effective for the Wilson coefficientC10 (see Sec. 2.2). SUSY effects in C10 are reduced from about 50 % to 16 % (28 %) at68 (95) % C.L. by the recent data on the rare decay B0 → K∗0µ+µ− [166]. The constraints

21 The effect of SUSY models on AFB(B → K∗µ+µ−) is discussed in Ref. [165].

22

9C2.5 3 3.5 4 4.5 5 5.5 6 6.5

10C

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

95% CL

68% CL

Figure 7: SUSY spread in NMFV-models [166]. The light (dark) grey shaded areas are the95 % (68 %) confidence limit (C.L.) bounds from B → K(∗)l+l− data [39]. The red dotted linedenotes the Z-penguin correlation CZ−p

10 /CZ−p9 = 1/(4 sin2 θW − 1). The SM point (CSM

9 , CSM10 )

is marked by the red dot.

0 500 1000 1500 2000 2500 3000MA [GeV]

0

10

20

30

40

50

60

tan(β)

2.300

2.300

2.300

5.990

2.300

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0

10

20

30

40

50

60

tan(β)

2.300

2.300

2.300

5.990

5.990

Figure 8: Impact of the latest B0s → µ+µ− limits on the (MA , tanβ) plane in the (left) CMSSM

and (right) NUHM1 [167]. In each case, the full global fit is represented by an open green starand dashed blue and red lines for the 68 and 95 % C.L. contours, whilst the fits to the incompletedata sets are represented by closed stars and solid contours.

are relevant to flavour models based on radiative flavour violation (see, e.g., Ref. [168]),and exclude solutions to the flavour problem with flavour generation in the up-sector andsub-TeV spectra. The flavour constraints are stronger for lighter stops, hence there is animmediate interplay with direct searches.

23

Figure 9: Distribution of pMSSM points after the B0s → µ+µ− constraint projected on the MA

(left) and (MA, tanβ) plane (right) for all accepted pMSSM points (medium grey), points notexcluded by the combination of the 2010 LHCb and CMS analyses (dark grey) and the projectionfor the points compatible with the measurement of the SM expected branching fractions with a20 % total uncertainty (light grey) [171].

Figure 8 shows the (MA, tan β) plane from fits of the CMSSM and NUHM1 parameterspace to the current data from SUSY and Higgs searches in ATLAS and CMS, as well asdark matter relic density [128,169]. The study in constrained MSSM scenarios is illustrativebut not representative of the full MSSM. The strong constraints provided by the currentdata in the CMSSM are not necessarily reproduced in more general scenarios. To gobeyond the constrained scenarios, consider the phenomenological MSSM (pMSSM) [170].This model is the most general CP - and R-parity-conserving MSSM, assuming MFV atthe weak scale and the absence of FCNCs at tree level. It contains 19 free parameters: 10sfermion masses, 3 gaugino masses, 3 trilinear couplings and 3 Higgs masses.

To study the impact of the B0s → µ+µ− results on the pMSSM, the parameter space is

scanned and for each point in the space the consistency of the model with experimentalbounds is tested [171]. The left panel of Fig. 9 shows the density of points as a functionof MA before and after applying the combined 2010 LHCb and CMS B0

s → µ+µ− limit(1.1× 10−8 at 95 % C.L. [172]), as well as the projection for a SM–like measurement withan overall 20 % theoretical and experimental uncertainty. As can be seen the density of theallowed pMSSM points is reduced by a factor of 3, in the case of a SM–like measurement.The right panel shows the same distribution in the (MA, tan β) plane. Similar to theCMSSM case, the region with large tan β and small MA is most affected by the experimentalconstraints.

The interplay with Higgs boson searches can also be very illuminating as any viablemodel point has to be in agreement with all the direct and indirect limits. As an example,if a scalar Higgs boson is confirmed at ∼ 125 GeV,22 the MSSM scenarios in which theexcess would correspond to the heaviest CP -even Higgs (as opposed to the lightest Higgs)

22 At ICHEP 2012 the observation of a new particle consistent with the SM Higgs boson was reportedby ATLAS and CMS [173,174].

24

are ruled out by the B0s → µ+µ− limit, since they would lead to a too light pseudoscalar

Higgs.It is clear that with more precise measurements a large part of the supersymmetric

parameter space could be disfavoured. In particular the large tan β region is stronglyaffected by B0

s → µ+µ− as can be seen in Fig. 5. Also, a measurement of B(B0s → µ+µ−)

lower than the SM prediction would rule out a large variety of supersymmetric models.In addition, B → K∗µ+µ− observables play a complementary role especially for smallertan β values. With reduced theoretical and experimental errors, the exclusion bounds inFig. 6 and Fig. 7 for example would shrink leading to important consequences for SUSYparameters.

2.8 Rare charm decays

So far the focus of this chapter has been on rare B decays, but the charm sector alsoprovides excellent probes for NP in the form of very rare decays. Unlike the B decaysdescribed in the previous sections, the smallness of the d, s and b quark masses makes theGlashow-Iliopoulos-Maiani (GIM) cancellation in loop processes very effective. Branchingratios governed by FCNC are hence not expected to exceed O(10−10) in the SM. Theseprocesses can then receive contributions from NP scenarios which can be several orders ofmagnitude larger than the SM expectation.

2.8.1 Search for D0 → µ+µ−

The branching fraction of the D0 → µ+µ− decay is dominated in the SM by the long dis-tance contributions due to the two photon intermediate state, D0 → γγ. The experimentalupper limit on the two photon mode can be combined with theoretical predictions toconstrain B(D0 → µ+µ−) in the framework of the SM: B(D0 → µ+µ−) < 6×10−11 at 90 %C.L. [175]. Particular NP models where this decay is enhanced include supersymmetricmodels with R-parity violation (RPV), which provides tree-level contributions that wouldenhance the branching fraction. In such models, the branching fraction would be relatedto the D0–D0 mixing parameters. Once the experimental constraints on the mixingparameters are taken into account, the corresponding tree-level couplings can still giverise to B(D0 → µ+µ−) of up to O(10−9) [176].

Preliminary results from a search for these rare decays have been performed by theLHCb collaboration [177]. The upper limit obtained with 0.9 fb−1 of data taken in 2011 is:

B(D0 → µ+µ−) ≤ 1.3 (1.1)× 10−8 at 95 (90) % C.L. (21)

This upper limit on the branching fraction, already an improvement of an order ofmagnitude on previous results, is expected to improve down to 5× 10−9 by the end of thefirst data-taking phase of the LHCb experiment.

25

2.8.2 Search for D+(s) → h+µ+µ− and D0 → hh

′µ+µ−

The D+(s) → h+µ+µ− decay rate is dominated by long distance contributions from tree-

level D+(s) → h+V decays, where V is a light resonance (V = φ, ρ, ω). The long-distance

contributions have an effective branching fraction (with V → µ+µ−) above 10−6 in theSM. Large deviations in the total decay rate due to NP are therefore unlikely. However,the regions of the dimuon mass spectrum far from these resonances are interesting probes.Here, the SM contribution stems only from FCNC processes, that should yield no partialbranching ratio above 10−11 [178]. NP contributions could enhance the branching fractionaway from the resonances by several orders of magnitude: e.g. in the RPV model mentionedabove, or in models involving a fourth quark generation [178,179].

The LHCb experiment is well-suited to search for D+(s) → h∓µ+µ± decays. The long

distance contributions can be used to normalise the decays searched for at high and lowdimuon mass: their decay rate will be measured relative to that of D+

(s) → π+φ(µ+µ−).These resonant decays have a clean experimental signature and their final state onlydiffers from the signal in the kinematic distributions, which helps to reduce the systematicuncertainties. The sensitivity of the LHCb experiment can be estimated by comparingthe yields of D+

(s) → π+φ(µ+µ−) decays observed in LHCb with those obtained by the

D0 experiment, which established the best limit on these modes so far [180]. With anintegrated luminosity corresponding to 1.0 fb−1, upper limits on the D+ (D+

s ) modes areexpected close to 10−8 (10−7) at 90 % C.L.

In analogy to the B sector, there is a wealth of observables potentially available infour-body rare decays of D mesons. In the decays D0 → hh

′µ+µ− (with h(′) = K or π),

forward-backward asymmetries or asymmetries based on T -odd quantities could reveal NPeffects [178,181,182]. Clearly the first challenge is to observe the decays which, dependingon their branching fractions, may be possible with the 2011 data set. However, the 50 fb−1

collected by the upgraded LHCb detector will be necessary to exploit the full set ofobservables in these modes.

2.9 Rare kaon decays

The cross-section for K0S production at the LHC is such that ∼ 1012 K0

S → π+π− wouldbe reconstructed and selected in LHCb with a fully efficient trigger. This provides agood opportunity to search for rare K0

S decays in channels with high trigger efficiency, inparticular K0

S → µ+µ−.The decay K0

S → µ+µ− is a flavour-changing neutral current that has not yet beenobserved. This decay is strongly suppressed in the SM, with an expected branching fractionof [183,184]

B(K0S → µ+µ−) = (5.0± 1.5)× 10−12 , (22)

while the current experimental upper limit is 3.2 × 10−7 at 90 % C.L. [185]. The studyof K0

S → µ+µ− has been suggested as a possible way to look for new light scalars [183],and indeed NP contributions up to one order of magnitude above the SM expectationare allowed [184]. Enhancements above 10−10 are less likely. Bounds on B(K0

S → µ+µ−)

26

close to 10−11 could be useful to discriminate among NP scenarios if other modes, suchas K+ → π+νν, indicated a non-standard enhancement of the s→ dll transition. Firstresults from LHCb, B(K0

S → µ+µ−) < 9×10−9 at 90 % C.L. [186], have significantly bettersensitivity than the existing results. With improved triggers on low mass dimuons, LHCbcould reach branching fractions of O(10−11) or below with the luminosity of the upgrade.Decays of K0

L mesons into charged tracks can also be reconstructed, but with much less(∼ 1 %) efficiency compared to a similar decay coming from a K0

S meson. This is due to thelong distance of flight of the K0

L state, which tends to decay outside the tracking system.

2.10 Lepton flavour and lepton number violation

The experimental observation of neutrino oscillations provided the first signature of leptonflavour violation (LFV). The consequent addition of mass terms for the neutrinos in the SMimplies LFV also in the charged sector, but with branching fractions smaller than 10−40.NP could significantly enhance the rates but, despite steadily improving experimentalsensitivity, charged lepton flavour violating (cLFV) processes like µ− → e−γ, µ–N → e–N ,µ− → e+e−e−, τ− → `−γ and τ− → `+`−`− (with `− = e−, µ−) have not been observed.Numerous theories beyond the SM predict larger LFV effects in τ− decays than µ− decays,with branching fractions within experimental reach [187]. An observation of cLFV wouldthus be a clear sign for NP, while lowering the experimental upper limit will help to furtherconstrain theories [188].

Another approach to search for NP is via lepton number violation (LNV). Decayswith LNV are sensitive to Majorana neutrino masses — their discovery would answer thelong-standing question of whether neutrinos are Dirac or Majorana particles. The strongestconstraints on minimal models that introduce neutrino masses come from neutrinolessdouble beta decay processes, but searches in heavy flavour decays provide competitive andcomplementary limits in models with extended neutrino sectors.

In this section, LFV and LNV decays of τ leptons and B mesons with only chargedtracks in the final state are discussed.

2.10.1 Lepton flavour violation

The neutrinoless decay τ− → µ+µ−µ− is a particularly sensitive mode in which to searchfor LFV at LHCb as the inclusive τ− production cross-section at the LHC is large (∼ 80µb,coming mainly from D+

s decays23) and muon final states provide clean signatures in thedetector. This decay is experimentally favoured with respect to the decays τ− → µ−γand τ− → e+e−e− due to the considerably better particle identification of the muonsand better possibilities for background discrimination. LHCb has reported preliminaryresults from a search for the decay τ− → µ+µ−µ− using 1.0 fb−1 of data [190]. The upperlimit on the branching fraction was found to be B(τ− → µ+µ−µ−) < 7.8 (6.3)× 10−8 at95 % (90 %) C.L, to be compared with the current best experimental upper limit from

23 Calculated from the bb and cc cross-sections measured at the LHCb experiment and the inclusivebranching ratios b→ τ and c→ τ [189].

27

Belle: B(τ− → µ+µ−µ−) < 2.1× 10−8 at 90 % C.L. As the data sample increases this limitis expected to scale as the square root of the available statistics, with possible furtherreduction depending on improvements in the analysis. The large integrated luminositythat will be collected by the upgraded experiment will provide sensitivity correspondingto an upper limit of a few times 10−9. Searches will also be conducted in modes such asτ− → pµ+µ− or τ− → φµ−, where the existing limits are much weaker, and low backgroundcontamination is expected in the data sample.24

The pseudoscalar meson decays probe transitions of the type q → q′``′ and hence areparticularly sensitive to leptoquark-models and thus provide complementarity to leptonicdecay LFV processes [192, 193]. For the LHCb experiment, both decays from D and Bmesons are accessible. Sensitivity studies for the decays B0

(s) → e−µ+ and D0 → e−µ+ areongoing. Present estimates indicate that LHCb will be able to match the sensitivity of theexisting limits from the B factories and CDF in the near future.

2.10.2 Lepton number violation

In lepton number violating B and D meson decays a search can be made for Majorananeutrinos with a mass of O(1 GeV). These indirect searches are performed by analysingthe production of same sign charged leptons in D or B decays such as D+

s → π−µ+µ+ orB+ → π−µ+µ+ [27,194]. These same sign dileptonic decays can only occur via exchangeof heavy Majorana neutrinos. Resonant production may be possible if the heavy neutrinois kinematically accessible, which could put the rates of these decays within reach of thefuture LHCb luminosity. Non-observation of these LNV processes, together with lowenergy neutrino data, would lead to better constraints for neutrino masses and mixingparameters in models with extended neutrino sectors.

Using 0.4 fb−1 of integrated luminosity from LHCb, limits have been set on the branchingfraction of B+ → D−(s)µ

+µ+ decays at the level of a few times 10−7 and on B+→ π−µ+µ+

at the level of 1× 10−8 [195, 196]. These branching fraction limits imply a limit on, forexample, the coupling |Vµ4| between νµ and a Majorana neutrino with a mass in the range1 < mN < 4 GeV/c2 of |Vµ4|2 < 5× 10−5.

2.11 Search for NP in other rare decays

Many extensions of the SM predict weakly interacting particles with masses from a fewMeV to a few GeV [197] and there are some experimental hints for these particles fromastrophysical and collider experiments [198,199]. For example, the HyperCP collaborationhas reported an excess of Σ+ → pµ+µ− events with dimuon invariant masses around214 MeV/c2 [200]. These decays are consistent with the decay Σ+ → pX with the subsequentdecay X → µ+µ−. Phenomenologically, X can be interpreted as a pseudoscalar or axial-vector particle with lifetimes for the pseudoscalar case estimated to be about 10−14 s [201–203]. Such a particle could, for example, be interpreted as a pseudoscalar sgoldstino [202]or a light pseudoscalar Higgs boson [204].

24 Preliminary results on τ− → pµ+µ− and τ− → pµ−µ− were presented at TAU 2012 [191].

28

The LHCb experiment has recorded the world’s largest data sample of B and D mesonswhich provides a unique opportunity to search for these light particles. Preliminary resultsfrom a search for decays of B0

(s) → µ+µ−µ+µ− have been reported [205]. Such decays could

be mediated by sgoldstino pair production [206]. No excess has been found and limits of1.3 and 0.5× 10−8 at 95 % C.L. have been set for the B0

s and B0 modes respectively. Theanalysis can naturally be extended to D0 → µ+µ−µ+µ− decays, as well as B0

(s) → V 0µ+µ−

(V 0 = K(∗)0, ρ0, φ), where the dimuon mass spectrum can be searched for any resonantstructure. Such an analysis has been performed by the Belle collaboration [207]. Withthe larger data sample and flexible trigger of the LHCb upgrade, it will be possible toexploit several new approaches to search for exotic particles produced in decays of heavyflavoured hadrons (see, e.g. Ref. [208]).

29

3 CP violation in the B system

3.1 Introduction

CP violation, i.e. violation of the combined symmetry of charge conjugation and parity, isone of three necessary conditions to generate a baryon asymmetry in the Universe [209].Understanding the origin and mechanism of CP violation is a key question in physics.In the SM, CP violation is fully described by the CKM mechanism [20]. While thisparadigm has been successful in explaining the current experimental data, it is knownto generate insufficient CP violation to explain the observed baryon asymmetry of theUniverse. Therefore, additional sources of CP violation are required. Many extensions ofthe SM naturally contain new sources of CP violation.

The b hadron systems provide excellent laboratories to search for new sources of CPviolation, since new particles beyond the SM may enter loop-mediated processes such asb→ q FCNC transitions with q = s or d, leading to discrepancies between measurementsof CP asymmetries and their SM expectations. Two types of b→ q FCNC transitions areof special interest: neutral B meson mixing (∆B = 2) processes, and loop-mediated Bdecay (∆B = 1) processes.

The LHCb experiment exploits the large number of b hadrons, including the particularlyinteresting B0

s mesons, produced in proton-proton collisions at the LHC to search forCP -violating NP effects. Section 3.2 provides a review of the status and prospects inthe area of searches for NP in B0

(s) mixing, in particular through measurements of the

mixing phases φd(s) and the semileptonic asymmetries ad(s)sl . The LHCb efforts to search

for NP in hadronic b → s penguin decays, such as B0s → φφ, are discussed in Sec. 3.3.

Section 3.4 describes the LHCb programme to measure the angle γ of the CKM unitaritytriangle (UT) in decay processes described only by tree amplitudes, such as B± → DK±,B0 → DK∗0 and B0

s → D∓s K±. These measurements allow precise tests of the SM

description of quark-mixing via global fits to the parameters of the CKM matrix, as well asdirect comparisons with alternative determinations of γ in decay processes involving loopdiagrams, such as B0

s→ K+K−. At the end of each section, a brief summary of the mostpromising measurements with the upgraded LHCb detector and their expected/projectedsensitivities is provided.

3.2 B0(s) mixing measurements

3.2.1 B0(s)–B

0(s) mixing observables

The effective Hamiltonian of the B0q–B

0q (q = d, s) system can be written as

Hq =

(M q

11 M q12

M q∗12 M q

22

)− i

2

(Γq11 Γq12

Γq∗12 Γq22

). (23)

where M q11 = M q

22 and Γq11 = Γq22 hold under the assumption of CPT invariance. Theoff-diagonal elements M q

12 and Γq12 are responsible for B0q–B

0q mixing phenomena. The

30

“dispersive” part M q12 corresponds to virtual ∆B = 2 transitions dominated by heavy

internal particles (top quarks in the SM) while the “absorptive” part Γq12 arises fromon-shell transitions due to decay modes common to B0

q and B0q mesons. Diagonalising the

Hamiltonian matrix leads to the two mass eigenstates BqH,L (H and L denote heavy and

light, respectively), with mass M qH,L and decay width ΓqH,L, being linear combinations of

flavour eigenstates with complex coefficients25 p and q that satisfy |p|2 + |q|2 = 1,

|BqL,H〉 = p|B0

q 〉 ± q|B0q 〉 . (24)

The magnitudes of M q12 and Γq12 and their phase difference are physical observables

and can be determined from measurements of the following quantities (for more detailssee, e.g., Refs. [210,211]):

• the mass difference between the heavy and light mass eigenstates

∆mq ≡M qH −M q

L ≈ 2|M q12|(

1− |Γq12|28|M q

12|2sin2 φq12

), (25)

where φq12 = arg(−M q12/Γ

q12) is convention-independent;

• the decay width difference between the light and heavy mass eigenstates

∆Γq ≡ ΓqL − ΓqH ≈ 2|Γq12| cosφq12

(1 +

|Γq12|28|M q

12|2sin2 φq12

); (26)

• the flavour-specific asymmetry26

aqsl ≡|p/q|2 − |q/p|2

|p/q|2 + |q/p|2≈ |Γ

q12|

|M q12|

sinφq12 ≈∆Γq∆mq

tanφq12 . (27)

The correction terms in Eqs. (25) and (26) proportional to sin2 φq12 are tiny. In addition,the ratio of q and p can be written(

q

p

)= − ∆mq + i

2∆Γq

2(M q12 − i

2Γq12)

, (28)

and hence in both B0 and B0s systems one obtains, to a good approximation, a convention-

dependent expression (for an unobservable quantity) arg(−q/p) ≈ − arg(M q12). Since

B–B mixing is dominated by the box diagram with internal top quarks, this leads to anexpression in terms of CKM matrix elements arg(−q/p) = 2 arg(V ∗tbVtq).

25 Strictly, the coefficients p and q should also have subscripts q to indicate that they can be differentfor B0 and B0

s , but these are omitted to simplify the notation.26 The notation aqsl is used to denote flavour-specific asymmetries, reflecting the fact that the measure-

ments of these quantities use semileptonic decays.

31

Further information can be obtained by measuring the phase difference between theamplitude for a direct decay to a final state f and the amplitude for decay after oscillation.In the case that the decay is dominated by b→ ccs tree amplitudes, and where f is a CPeigenstate f with eigenvalue ηf ,

27 this phase difference is denoted as

φq ≡ − arg

(ηfq

p

AfAf

), (29)

where Af and Af are the decay amplitudes of B → f and B → f , respectively. In theabsence of direct CP violation Af/Af = ηf . With these approximations, the CP -violatingphases in B mixing give the unitarity triangle angles, φd ≈ 2β and φs ≈ −2βs,

28 wherethe angles are defined as [43]

β ≡ arg

(−VcdV

∗cb

VtdV ∗tb

), βs ≡ arg

(−VtsV

∗tb

VcsV ∗cb

). (30)

Clearly, if there is NP in M q12 or in the decay amplitudes, the measured value of φq can

differ from the true value of (−)2β(s). Similarly, NP in either M q12 or Γq12 can make the

observed value of aqsl differ from its SM prediction. Note, however, that even within theSM, there is a difference between φq and φq12 [212]. Nonetheless, the notations φd(s) andβ(s) are usually used interchangeably.

The φs notation has been used in the LHCb measurements of the CP -violating phase inB0s mixing, using J/ψ φ [10,138] and J/ψ f0(980) [213,214] final states. By using the same

notation for different decays, an assumption that arg(Af/Af ) is common for different finalstates is being made. This corresponds to an assumption that the penguin contributionsto these decays are negligible. Although this is reasonable with the current precision,as the measurements improve it will be necessary to remove such assumptions – severalmethods to test the contributions of penguin amplitudes are discussed below. These includemeasuring φq with different decay processes governed by different quark-level transitions.Previous experiments have used the notation 2βeff in particular for measurements based onb→ qqs (q = u, d, s) transitions; for symmetry the notation 2βeff

s is used in correspondingcases in the B0

s system, although the cancellation of the mixing and decay phases in B0s

decays governed by b→ qqs amplitudes is expected to lead to a vanishing CP violationeffect (within small theoretical uncertainties).

In the SM, the mixing observables can be predicted using CKM parameters froma global fit to other observables and hadronic parameters (decay constants and bagparameters) from lattice QCD calculation. These predictions can be compared to theirdirect measurements to test the SM and search for NP in neutral B mixing.

3.2.2 Current experimental status and outlook

The current measurements and SM predictions for the mixing observables are summarisedin Table 1.

27 The cases for more generic final-states can be found in the literature, e.g. Ref. [43].28 Note the conventional sign-flip between β and βs ensures that both are positive in the SM.

32

Table 1: Status of B mixing measurements and corresponding SM predictions. New resultspresented at ICHEP 2012 and later are not included. The inclusive same-sign dimuon asymmetryAbSL is defined below and in Ref. [158].

Observable Measurement Source SM prediction ReferencesB0s system

∆ms ( ps−1) 17.719± 0.043 HFAG 2012 [43] 17.3± 2.6 [215–217]17.725± 0.041± 0.026 LHCb (0.34 fb−1) [218]

∆Γs ( ps−1) 0.105± 0.015 HFAG 2012 [43] 0.087± 0.021 [215–217]0.116± 0.018± 0.006 LHCb (1.0 fb−1) [138]

φs (rad) −0.044+0.090−0.085 HFAG 2012 [43] −0.036± 0.002 [118,216,217]

−0.002± 0.083± 0.027 LHCb (1.0 fb−1) [138]assl (10−4) −17± 91 +14

−15 D0 (no AbSL) [219] 0.29+0.09−0.08 [118,216,217]

−105± 64 HFAG 2012 (including AbSL ) [43]Admixture of B0 and B0

s systemsAbSL (10−4) −78.7± 17.1± 9.3 D0 [158] −2.0± 0.3 [215–217]

B0 system∆md ( ps−1) 0.507± 0.004 HFAG 2012 [43] 0.543± 0.091 [211,216,217]

∆Γd/Γd 0.015± 0.018 HFAG 2012 [43] 0.0042± 0.0008 [215–217]sin 2β 0.679± 0.020 HFAG 2012 [43] 0.832 +0.013

−0.033 [118,216,217]adsl (10−4) −5± 56 HFAG 2012 [43] −6.5 +1.9

−1.7 [118,216,217]

The HFAG average of the B0s mass difference ∆ms in Table 1 is based on measurements

performed at CDF [220] and LHCb [218,221]. It is dominated by the preliminary LHCbresult obtained using 0.34 fb−1 of data [218], which is also given in Table 1. These areall consistent with the SM prediction. Improving the precision of the SM prediction isdesirable to further constrain NP in M s

12, and requires improving the accuracy of latticeQCD evaluations of the decay constant and bag parameter (see Ref. [211] and referencestherein).

The observables φs and ∆Γs have been determined simultaneously from B0s → J/ψ φ

decays using time-dependent flavour tagged angular analyses [222,223]. The first LHCbtagged analysis using 0.34 fb−1 of data [10] already provided a significant constraint on φsand led to the first direct evidence for a non-zero value of ∆Γs. LHCb has also determinedthe sign of ∆Γs to be positive at 4.7σ confidence level [224] by exploiting the interferencebetween the K+K− S-wave and P-wave amplitudes in the φ(1020) mass region [225]. Thisresolved the two-fold ambiguity in the value of φs for the first time. LHCb has madea preliminary update of the B0

s → J/ψ φ analysis using the full data sample of 1.0 fb−1

collected in 2011 [138]. The results from this analysis,

φs = −0.001± 0.101± 0.027 rad , ∆Γs = 0.116± 0.018± 0.006 ps−1 , (31)

are shown in Fig. 10 (left), and are in good agreement with the SM expectations.LHCb has also studied the decay B0

s → J/ψ π+π−. This decay process is expected toproceed dominantly via b→ ccs (the ss produced in the decay rescatters to π+π− througheither a resonance such as f0(980) or a nonresonant process). Therefore, these eventscan be used to measure φs. The π+π− mass range 775–1550 MeV shown in Fig. 11 (left)

33

[rad]s

φ-0.4 -0.2 0 0.2 0.4

]-1

[p

ssΓ∆

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Conf. Levels

68% C.L.

90% C.L.

95% C.L.

Standard Model

LHCbPreliminary

Conf. Levels

68% C.L.

90% C.L.

95% C.L.

Standard Model

0.25

CDF

LHCb

ATLAS

Combined

SM

0.20

0.15

0.10

0.05

0-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

68% CL contours( )

HFAGFall 2012

LHCb 1.0 fb –1 + CDF 9.6 fb –1 + ATLAS 4.9 fb 1+ D 8 fb– –1

D

Figure 10: (Left) Preliminary LHCb measurement of φs and ∆Γs from B0s → J/ψ φ decays using

1.0 fb−1 [138]. (Right) HFAG 2012 combination of φs and ∆Γs results, where the 1σ confidenceregion is shown for each experiment and the combined result [43]. Note the different scales.

is used for the measurement. In contrast to B0s → J/ψ φ, no angular analysis is needed

to disentangle the CP eigenstates, since the final state is determined to be dominantlyCP -odd in this mass range [226]. On the other hand, ∆Γs cannot be determined in thisdecay channel alone.29 Using as input the value of ∆Γs obtained from B0

s → J/ψ φ, themeasurement from the analysis of B0

s → J/ψ π+π− with 1.0 fb−1 is [214]

φs = −0.019 +0.173−0.174

+0.004−0.003 rad . (32)

Figure 11 (right) shows the log-likelihood scan for the φs parameter for the B0s → J/ψ π+π−

analysis. The latest HFAG average in Table 1 combines the LHCb results with theB0s → J/ψ φ analysis results from CDF using 9.6 fb−1 [229] and D0 using 8.0 fb−1 [230].

The LHCb result dominates the combination, which is in good agreement with the SMpredictions, as seen in Fig. 10 (right).30

The LHCb B0s → J/ψ φ and B0

s → J/ψ π+π− analyses discussed above only usedopposite side flavour tagging [231, 232]. Future updates of these analyses will gain insensitivity by also using the same side kaon tagging information, which so far has been usedin a preliminary determination of ∆ms [218,233]. Currently, the systematic uncertaintyon φs is dominated by imperfect knowledge of the background, angular acceptance effectsand by neglecting potential contributions of direct CP violation. All of these uncertaintiesare expected to be reduced with more detailed understanding and some improvementsin the analysis. Therefore it is expected that the determination of φs will remain limitedby statistical uncertainties, even with the data samples available after the upgrade of theLHCb detector. In addition to B0

s → J/ψ φ and B0s → J/ψ π+π−, other b → ccs decay

modes of B0s mesons, such as J/ψη, J/ψη′ [234] and D+

s D−s [235] will be investigated.

These decays have been measured at LHCb [236,237].

29 The effective lifetime of B0s → J/ψ f0(980) is sensitive to ∆Γs and CP violation parameters [227] and

has been measured by LHCb [228].30 Results from ATLAS and CMS, presented at ICHEP2012 or later, are not included in this compilation.

34

) (MeV)π πm(500 1000 1500 2000

Eve

nts

/ 15

MeV

0

100

200

300

400

500

600

700

800 LHCb

+ -

↓ ↓ (rad)

sφ

-3 -2 -10

5

10

15

20

25

30

35 LHCb

Δln(L)

0 1 2 3

Figure 11: (Left) π+π− mass distribution of selected B0s → J/ψ π+π− candidates and range

used for the φs measurement. (Right) log-likelihood difference as a function φs [214].

The SM prediction φs = −0.036 ± 0.002 rad could receive a small correction fromdoubly CKM-suppressed penguin contributions in the decay. The value of this correctionis not precisely known, and may depend on the decay mode. Moreover, NP in the b→ ccsdecay may also affect the results. Although such effects are already constrained by resultsfrom B+ and B0 decays, NP in the decay amplitudes can lead to polarisation-dependentmixing-induced CP asymmetries and triple product asymmetries in B0

s → J/ψ φ [238].Such effects will be searched for in future analyses.

The flavour-specific asymmetries provide important complementary constraints on∆B = 2 processes. The D0 collaboration has performed a direct measurement of assl insemileptonic B0

s decays [219], which is only weakly constraining.31 However, a measurementof the inclusive same-sign dimuon asymmetry provides better precision, and shows evidenceof a large deviation from its SM prediction [158]. The inclusive measurement is sensitiveto a linear combination of the flavour-specific asymmetries, AbSL = Cd a

dsl + Cs a

ssl, where

Cq depend on the production fractions and mixing probabilities, and are determined to beCd = 0.594 ± 0.022, Cs = 0.406 ± 0.022 [158].32 As discussed in Sec. 3.2.3, the D0 AbSL

result is in tension with other ∆B = 2 observables. Improved measurements of assl and adslfrom LHCb are needed to solve this puzzle.

In LHCb, assl can be determined from the asymmetry between the time-integrateduntagged decay rates of B0

s decays to D+s µ−X and D−s µ

+X, with D±s → φπ±, φ→ K+K−

(or with the full D±s → K+K−π± Dalitz plot). Detector- and trigger-induced asymmetriescan be calibrated in control channels, and the fact that data is taken with both magnetdipole polarities can be used as a handle to reduce systematic uncertainties. The effect ofB0s production asymmetry is cancelled due to the fast oscillation, so the asymmetry in the

yields of D+s µ−X and D−s µ

+X decays is trivially related to assl. A first preliminary LHCb

31 An updated measurement has been presented by D0 at ICHEP 2012 [239].32 The factors Cd and Cs depend in principle on the collision environment and the kinematic acceptance,

though the dependence appears to be weak. Trigger requirements can also affect the values of theseparameters.

35

result on assl, based on 1.0 fb−1, has been reported at ICHEP 2012, and is the most precisemeasurement of this quantity to date [240],

assl = (−0.24± 0.54± 0.33) % . (33)

It will also be possible to measure adsl using D+µ−X final states with D+ → K−π+π+.In this case extra care must be taken to calibrate the difference between K+ and K−

detection efficiencies and an independent measurement of the B0 production asymmetryis needed as input. Moreover, the CP -symmetric background from charged B decays issignificant and must be accurately subtracted.

In the B0 system, ∆md and sinφd (i.e. sin 2β) have been measured precisely by the Bfactories [43]. The measurements of ∆Γd and adsl are consistent with their SM predictions,but their uncertainties are at least an order of magnitude larger than those of the predictions.Hence a large improvement in precision is needed to test the SM using these observables.In the B0 sector there has been for some time a tension between the measurements ofsin 2β [43] and the branching ratio B(B+ → τ+ν) [241, 242], as shown in Fig. 12,33 anddiscussed in Sec. 3.2.4. This motivates improved measurements of sin 2β and improvedunderstanding of the possible effects of penguin contributions to this observable.

`sin 20.5 0.6 0.7 0.8 0.9 1.0

)io A

BR

(B

0.00

0.05

0.10

0.15

0.20

0.25

0.30-310×

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0p-value

Winter 12

CKMf i t t e r

Figure 12: Comparison of direct and indirect determinations of sinφd ≡ sin 2β vs. B(B+ → τ+ν),from Ref. [244].

LHCb has already presented first results on ∆md [221,245] and sin 2β [246]. The ∆md

result is the world’s most precise single measurement of this quantity, while the sensitivityon sin 2β will be competitive with the B factory results using the data sample that willbe collected by the end of 2012. LHCb can also search for enhancements in the value of∆Γd above the tiny value expected in the SM, e.g. by comparing the effective lifetimes

33 An updated measurement of B(B+ → τ+ν) using the hadronic tag method was presented by Belle atICHEP 2012 [243]: this new result reduces, but does not completely remove, the tension in the fits. Theanalyses discussed here do not include this new result.

36

of B0 → J/ψK0S and B0 → J/ψK∗0 [247]. Significantly improving the precisions of the

B0 mixing observables is an important goal of the LHCb upgrade, as will be discussed inSec. 3.2.6.

The SM predictions of b-hadron lifetimes and ∆Γq are all obtained within the frameworkof the heavy quark expansion. LHCb is actively working on measurements of b-hadronlifetimes and lifetime ratios, which will be used to test these predictions. The knowledgeobtained from this work will allow to improve the SM predictions of ∆Γq for the purpose ofsearching for NP. Furthermore, a more precise measurement of the ratio ofB0

s toB0 lifetimescould either support or strongly constrain the existence of NP in Γs12 [151,152,211,215,248].

3.2.3 Model independent constraints on new physics in B mixing

Neutral Bq meson mixing is described in terms of the three parameters |M q12|, |Γq12| and

φq = arg(−M q12/Γ

q12) for each of the two systems q = d, s. In the context of model-

independent analyses, the NP contributions can be parametrised in the form of twocomplex quantities ∆q and Λq [152,249]

M q12 = M q,SM

12 |∆q| eiφ∆q , Γq12 = Γq,SM

12 |Λq| eiφΛq , (34)

i.e., 4 real degrees of freedom. The observables which depend on these parameters arethe mass and decay width differences and flavour-specific CP -asymmetries. They can beexpressed in terms of the SM predictions and NP parameters as

∆mq = (∆mq)SM |∆q| , ∆Γq = (∆Γq)SM |Λq|cos(φq,SM

12 + φ∆q − φΛ

q

)cosφq,SM

12

, (35)

aqsl = (aqsl)SM|Λq||∆q|

sin(φq,SM

12 + φ∆q − φΛ

q

)sinφq,SM

12

, (36)

up to corrections suppressed by tiny (Γq12/Mq12)2. Note that the expressions of Eqs. (35)

and (36) depend only on the difference (φ∆q − φΛ

q ). The SM predictions of ∆mq, ∆Γq andaqsl can be found in Table 1 and for φq12 [215]

φd,SM12 = (−0.075± 0.024) rad , φs,SM

12 = (0.0038± 0.0010) rad. (37)

The values of ∆mq have been precisely measured, giving rather strong constraints on|∆q| which are limited by the knowledge of hadronic matrix elements. The new ∆Γsmeasurement of LHCb starts to provide useful constraints. As discussed above, theCP -asymmetries aqsl are currently rather weakly constrained.

Further information can be extracted from the mixing-induced CP -asymmetries inB0 → J/ψK0

S and B0s → J/ψφ decays

φd = 2β + φ∆d − δd, φs = −2βs + φ∆

s − δs, (38)

37

where δd and δs denote shifts of φd and φs induced by either SM penguin diagrams orNP contributions in the decay process. In the SM φd and φs are related to the angles βand βs of the according unitarity triangles. When short-distance NP contributions areintroduced, φq depends on the phase φ∆

q of M q12, whereas the phase φΛ

q of Γq12 does notenter. The SM penguin pollution to δq is expected to be negligible for the current precisionof φq, and is discussed in detail in Sec. 3.2.5. Beyond the SM, NP can contribute to δqin principle in both the tree b → ccs decay and the penguin process. However, in themodel-independent analysis described here, NP contributions in the b → ccs decay areneglected and any observed deviation from the SM will be interpreted as effects of NP inneutral B meson mixing. When δq is neglected, Eqs. (35), (36) and (38) allow to determinethe NP parameters |∆q|, φ∆

q , |Λq| and φΛq .

The assumption of NP in M q12 only, or equivalently in ∆B = 2 processes only, implies

that there is no NP in ∆B = 1 processes which contribute to the absorptive part Γq12.Consequently, NP can only decrease ∆Γq (since cos(φq,SM

12 ) is maximal, see Eq. (35)) withrespect to the SM [223, 250]. This scenario has been studied in extensions of the CKMfit of the SM which includes ∆B = 2 measurements to constrain the CKM elementsVtq [248,251], in combination with many other flavour-changing processes. Including LHCbmeasurements [138,221]34 the SM point ∆d = ∆s = 1 is disfavoured by 2.4σ [248] (priorto the LHCb results being available, a similar analysis gave a discrepancy of 3.6σ drivenmainly by the anomalous dimuon asymmetry [251]). The analysis gives ∆s consistent withthe SM, within large uncertainties, whereas the more precise data in the B0 system hintat a deviation in ∆d (see Fig. 13). Moreover, NP effects up to 30–40 % are still allowed inboth systems at the 3 σ level. It should be noted, that the large deviations in the B0 sectorare not only due to AbSL, but also due to the tension between sinφd and B(B+ → τ+ν).

NP contributions to the absorptive part Γq12 of B mixing can enter through ∆B = 1decays b→ qX with light degrees of freedom X of total mass below mB. In some particularmodels such contributions can arise [153, 252] and interfere constructively or destructivelywith the SM contribution. The recent measurements of ∆Γq and of AbSL revived interest inthis possibility. Model-independent analyses have confirmed that the AbSL measurementcannot be accommodated within the SM [253,254]. A model-independent fit assuming NPin both M q

12 and Γq12 has been considered in the framework of an extended CKM fit [248].In this case, the experimental data can be accommodated, and the B0

s system remainsrather SM-like, but large NP contributions in the B0 system are required.

Model-independent analyses based on Eq. (34) are restricted to a particular set ofobservables, mainly those with ∆B = 2, since correlations with ∆B = 1 measurementsare difficult to quantify. Either additional assumptions on the nature of X in b→ qX orexplicit NP models will permit better exploitation of the wealth of future experimentalinformation. In fact, such analyses have found it difficult to accommodate the hypothesisof large NP in Γq12 with current ∆B = 1 measurements, therefore NP in Γq12 seems unlikelyto provide a full explanation of the measured value of AbSL. In the case of X = ff ,the ∆B = 1 operators b → (d, s)ff (f = q or `) are strongly constrained [151], with

34 But not including results shown for the first time at ICHEP 2012 or later.

38

)s(BSL

) & ad

(BSL & aSLA

FSso & sK 6

sm6 & dm6

s`-2s 6q

SM point

s6Re -2 -1 0 1 2 3

s6

Im

-2

-1

0

1

2excluded area has CL > 0.68

Winter 2012

CKMf i t t e r mixing sB - s New Physics in B

exp_

)s(BSL

) & ad

(BSL & aSLA

sm6 & dm6SM point

)d

`+2d 6qsin()>0

d`+2d 6qcos(

d6Re -2 -1 0 1 2 3

d6

Im

-2

-1

0

1

2excluded area has CL > 0.68

Winter 2012

CKMf i t t e r mixing dB -

d New Physics in B

Figure 13: Model-independent fit [248] in the scenario that NP affects M q12 separately. The

coloured areas represent regions with C.L. < 68.3 % for the individual constraints. The red areashows the region with C.L. < 68.3 % for the combined fit, with the two additional contoursdelimiting the regions with C.L. < 95.45 % and C.L. < 99.73 %.

the exception of b → scc and b → sτ+τ−. Currently, only a weak upper bound onB(B+ → K+τ+τ−) <∼ 3.3 ·10−3 at 90 % C.L. [255] exists whereas other decays B0

s → τ+τ−,B → Xsτ

+τ− might be indirectly constrained with additional assumptions (see also thediscussion in Sec. 2.5.2). As an example, the improved LHCb measurement of τB0

s/τB0

allowed the derivation of a stronger bound on B(B0s → τ+τ−). Still, a model-independent

analysis of the complete set of b→ sτ+τ− operators does not allow for deviations largerthan 35 % from the SM in Γs12 [152], which is much too small to resolve the tension withAbSL. For b→ dτ+τ− operators there exists a stronger constraint B(B0 → τ+τ−) <∼ 4 ·10−3

and even smaller NP effects are expected in Γd12. Other proposed solutions such as theexistence of new light spin-0 [256] or spin-1 [257] X states could be seriously challenged byimproved measurements of quantities, such as ratios of lifetimes, which are theoreticallyunder good control [215].

In summary, NP contributions to |∆q| are already quite constrained due to ∆mq mea-surements and theoretical progress is required in order to advance. Although the phasesφ∆q are constrained by the recent LHCb measurement of φs, and B factory measurements

of φd, there is a mild tension with the SM in model-independent fits of ∆B = 2 measure-ments [152, 248, 253, 254], especially when allowing for NP in Γq12. On the other hand,NP effects in Γq12 are expected to be limited when constraints from ∆B = 1 observablesare taken into account. Independent improved measurements of aqsl are needed in orderto resolve the nature of the current discrepancies between the ∆B = 2 observables withtheir SM expectations and other observables entering global CKM fits. Further, improved

39

measurements of Γq and ∆Γq, as well as of control channels, are needed to constrain NPin Γq12.

3.2.4 CKM unitarity fits in SM and beyond

This section presents the results of the unitarity triangle (UT) analysis performed bytwo groups: UTfit [258] and CKMfitter [244].35 The main aim of the UT analysis isthe determination of the values of the CKM parameters, by comparing experimentalmeasurements and theoretical predictions for several observables. The popular Wolfensteinparametrisation allows for a transparent expansion of the CKM matrix in terms of the sineof the small Cabibbo angle, λ, with the other three parameters being A, ρ and η. Assumingthe validity of the SM, one can perform a fit to the available measurements. LHCb resultsalready make important contributions to the constraints on γ and ∆ms. With morestatistics, LHCb results are expected to impact on other CKM fit inputs, including α andsin 2β. It is important to note the crucial role of lattice QCD calculations as input to theCKM fits. For example, the parameters fBs

√BBs and ξ enter the constraints on ∆ms and

∆md/∆ms. At the end of 2011, the precision of the calculations was at the level of 5.4 %and 2.6 %, respectively [108]. The necessary further progress to obtain the full benefit ofthe LHCb measurements appears to be in hand exploiting algorithmic advances as well asever increasing computing power for the lattice calculations.

The overall quality of the fit can be judged using the projection of the likelihoods on theρ, η plane. This projection is shown in Fig. 14. The fit can also be made removing oneof the inputs, giving a prediction for the removed parameter, which then can be comparedto the experimental value. The results of this study are presented in Table 2. Both groupsfind a tension between B(B → τν) and sin 2β, as can be seen in Fig. 12. (As discussed inSec. 3.2.2 this tension will be reduced once the latest Belle result on B(B+ → τ+ντ ) [243]is included in the fits.) Improved measurements of sin 2β can shed further light on thisproblem.

In order to estimate the origin of the tensions, the UTfit and CKMfitter groups haveperformed analyses including model-independent NP contributions to neutral meson mixingprocesses (see Refs. [248, 262] for details). The NP effects are introduced through the realvalued C and φ parameters (ANP = CeiφASM) in case of UTfit and the complex valued ∆parameter (ANP = ∆ASM) for CKMfitter. The parameters are added separately for theB0s and B0 sectors. In the absence of NP, the expected values are C = 1, φ = 0, and

∆ = 1. For the B0 sector the fits return C = 0.94 ± 0.14 and φ = (−3.6 ± 3.7), and∆ = (0.823 +0.143

−0.095) + i(−0.199 +0.062−0.048). The results for both groups show some disagreement

with the SM, driven by tensions in the input parameters mentioned above. In theB0s sector, on the other hand, the situation is much closer to the SM than before the

LHCb measurements were available: C = 1.02 ± 0.10 and φ = (−1.1 ± 2.2), and∆ = (0.92 +0.13

−0.08) + i(0.00± 0.10).The results of the studies by both groups point to the absence of big NP effects in

∆B = 2 processes. Nevertheless there is still significant room for NP in mixing in both

35 Similar approaches have been developed in Ref. [259,260].

40

ρ-1 -0.5 0 0.5 1

η

-1

-0.5

0

0.5

1γ

β

α

)γ+βsin(2

sm∆dm∆

dm∆

Kε

cbVubV

)ντ→BR(B

ICHEP12

SM fit

dmK

K

sm & dm

ubV

sin 2

(excl. at CL > 0.95) < 0sol. w/ cos 2

excluded at CL > 0.95

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5excluded area has CL > 0.95

Winter 12

CKMf i t t e r

Figure 14: Result of the UT fit within the SM: ρ, η plane obtained by (left) UTfit [258] and(right) CKMfitter [244]. The 95 % probability regions selected by the single constraints are alsoshown with various colours for the different constraints.

Table 2: Predictions for some parameters of the SM fit and their measurements as combined bythe UTfit and CKMfitter groups. Note that the two groups use different input values for someparameters. The lines marked with (*) are not used in the full fit. Details of the pull calculationcan be found in Refs. [251,261]. New results presented at ICHEP2012 and later are not includedin these analyses.

ParameterUTfit CKMfitter

prediction measurement pull prediction measurement pullα () 87.5± 3.8 91.4± 6.1 +0.5σ 95.9 +2.2

−5.6 88.7 +2.2−5.9 −1.0σ

sin 2β 0.809± 0.046 0.667± 0.024 −2.7σ 0.820 +0.024−0.028 0.679± 0.020 −2.6σ

γ () 67.8± 3.2 75.5± 10.5 +0.7σ 67.2 +4.4−4.6 66 +12

−12 −0.1σVub (10−3) 3.62± 0.14 3.82± 0.56 +0.3σ 3.55 +0.15

−0.14 3.92± 0.09± 0.45 0.0σVcb (10−3) 42.26± 0.89 41± 1 −0.9σ 41.3 +0.28

−0.11 40.89± 0.38± 0.59 0.0σεk (10−3) 1.96± 0.20 2.229± 0.010 +1.3σ 2.02 +0.53

−0.52 2.229± 0.010 0.0σ∆ms ( ps−1) 18.0± 1.3 17.69± 0.08 −0.2σ 17.0 +2.1

−1.5 17.731± 0.045 0.0σB(B → τν) (10−4) 0.821± 0.0077 1.67± 0.34 +2.5σ 0.733 +0.121

−0.073 1.68± 0.31 +2.8σβs rad (*) 0.01876± 0.0008 0.01822 +0.00082

−0.00080

B(B0s → µµ) (10−9) (*) 3.47± 0.27 3.64 +0.21

−0.32

41

B0 and B0s systems. More precise results, in particular from LHCb, can enable more

careful studies. Besides providing null tests of the SM hypothesis, improved φs and asslmeasurements are crucial to quantity effects of NP in mixing. In addition a precise γdetermination is essential, not only for a SM global consistency test, but also to fix theapex of the UT in the extended fits.

3.2.5 Penguin pollution in b→ ccs decays

In addition to the very clear experimental signature, precise determination of the B0 andB0s mixing phases is possible due to the fact that in the “golden modes”, B0 → J/ψ K0

S

and B0s → J/ψ φ, explicit calculation of the relevant matrix elements can be avoided,

once subleading doubly Cabibbo-suppressed and loop-suppressed terms are assumed tovanish [263]. Estimates yield corrections of the order O(10−3) only [264–266]; it is howevernotoriously difficult to actually calculate the relevant matrix elements, and non-perturbativeenhancements cannot be excluded. Given the future experimental precision for these andrelated modes, a critical reconsideration of this assumption is mandatory.

The main problem lies in the fact that once the assumption of negligible penguincontributions is dropped, the evaluation of hadronic matrix elements again becomesnecessary, which still does not seem feasible to an acceptable precision for the decays inquestion. To avoid explicit calculation, symmetry relations can be used, exploiting eitherflavour SU(3) or U-spin symmetry [267–273]. Without taking into account any QCDevaluation and only using control channels to estimate the size of the penguin amplitude,the analyses in Refs. [270,273] still allow a phase shift of up to a few degrees for φd, whichwould correspond to a very large non-perturbative enhancement of the penguin size. InRef. [270] a negative sign is preferred which (slightly) reduces the tension in the unitaritytriangle fit shown in Fig. 12. The reason for the large allowed range of the shift of φd isdue to the limited precision to which the corresponding control channels B0 → J/ψ π0 andB0s → J/ψ K0, which are Cabibbo-suppressed compared to the golden modes, are known.

For φs, an analogous analysis [269] cannot yet constrain the penguin contribution, due tothe lack of a B → J/ψ V control channel data for B0

s → J/ψ φ. However, in principle theeffects in the B → J/ψ V modes are expected to be of the same order of magnitude as inthe B → J/ψ P modes. The control channel B0

s → J/ψ K∗0 has already been observed atCDF [274] and LHCb [275], and work is ongoing to measure its decay rate, polarisationsand direct CP asymmetries. This will enable the first direct constraint on the shift of φsdue to penguin contributions in the decay B0

s → J/ψ φ.36 For B0s → J/ψ f0(980) there is

an additional complication due to the unknown hadronic structure of the f0(980) [227].In addition to insufficient data, there are, at present, theoretical aspects limiting the

precision of this method at present, the most important of which is the violation of SU(3)symmetry. Regarding the B0 mixing phase, a full SU(3) analysis can be performed [277](instead of using only one control channel) to be able to model-independently includeSU(3) breaking. The inclusion of SU(3)-breaking contributions is important: their neglectcan lead to an overestimation of the subleading effects. Including recent data for two of

36 Other data-driven methods to control penguin contributions to B0s → J/ψ φ have been proposed [276].

42

the relevant modes [278,279], the analysis shows that the data are at the moment actuallycompatible with vanishing penguin contributions, with SU(3)-breaking contributions of theorder 20 %. Including the penguin contributions, an upper limit on the shift of the mixing-induced CP asymmetry ∆S = sinφd− sin 2β is derived: |∆S| . 0.01, with a negative signfor ∆S slightly preferred.37 This is the most stringent limit available, despite the moregeneral treatment of SU(3) breaking. In this analysis still some (conservatively chosen)theoretical inputs are needed to exclude fine-tuned solutions: SU(3)-breaking effects havebeen restricted to at most 40 % for a few parameters which are not well determinedby the fit and also have only small influence on the CP violation observables, and thepenguin matrix elements are constrained to be at most 50 % of the leading contributions.Importantly, these theory inputs can be replaced by experimental measurements, namelyof the CP asymmetries in the decay B0

s → J/ψ K0S , the decay rate of which has already

been measured at LHCb [279] after its observation at CDF [274]. Furthermore, data fromall the corresponding modes (i.e. Bd,u,s → J/ψ P , with light pseudoscalar meson P = π, η(′)

or K) can be used to determine the shift more precisely, i.e. the related uncertainty is notirreducible, but can be reduced with coming data.

Turning to the second golden mode, B0s → J/ψ φ, in general, the absolute shift is

not expected to be larger than in the B0 case. At the moment the data are not yetavailable to make a comparable analysis. While the penguin decay mode B0

s → φφ is notrelated by symmetry with B0

s → J/ψφ, comparing their decay rates indicates that thepenguin contributions are small, and there are no huge enhancements to be expected forthe penguin matrix elements in question.

Nonetheless, a quantitative analysis will ultimately be warranted here as well. Inprinciple, these methods can be adapted to extract the B0

s mixing phase including penguincontributions and model-independent SU(3) breaking, thereby improving the methodproposed in Ref. [269]. The corresponding partners of the golden mode B0

s → J/ψ φ areall the decays Bu,d,s → J/ψ V , with the light vector mesons V = K∗, ρ, φ or ω. However,the complete analysis requires results on the polarisation fractions and CP asymmetriesfor each of these final states, and for some of them the experimental signature is quitechallenging. In addition, the φ meson is a superposition of octet and singlet, thereforethe “control channels” involving K∗ and ρ are not as simply related as in the case with apseudoscalar meson, but require the usage of nonet symmetry, whose precision has to beinvestigated in turn.

Nevertheless, significant progress can be expected. Several B → J/ψ V modes, includingB0

(s) → J/ψK∗0 [275], are being studied at LHCb. While measurements of the modesinvolving b→ d transitions are expected to exhibit rather large uncertainties at first, theadvantage of the proposed method is the long “lever arm” due to the relative enhancement∼ 1/λ2 in the control channels, so that even moderate precision will be very helpful.

37 Note the definition of ∆S here has a sign difference to that in Ref. [277].

43

3.2.6 Future prospects with LHCb upgrade

Current measurements of φs carried out by LHCb in the J/ψ φ and J/ψ π+π− final statesshow no deviation from the SM prediction within uncertainties [138,214], putting strongconstraints on NP in B0

s mixing, as discussed in Sec. 3.2.3. Table 3 shows the currentresults with 1.0 fb−1 and the projected precision for 50 fb−1 with the upgraded detector. Aprecision of < 10 mrad is expected for 50 fb−1 with the upgraded detector. It is expectedthat even with this data sample, the main limitation will be statistical: the largestsystematic uncertainties on the current measurement (background description, angularacceptance, effect of fixed physics parameters) [138] are expected to be removed with moresophisticated analyses or to scale with statistics. Thus changes as small as a factor oftwo with respect to the SM should be observable with 3σ significance. This precisionwill make it possible either to measure a significant deviation from the SM prediction orotherwise to place severe constraints on NP scenarios.

Table 3: LHCb measurements of φs. The quoted uncertainties are statistical and systematic,respectively.

Final State Current value (rad) with 1.0 fb−1 Projected uncertainty (50 fb−1)J/ψφ −0.001± 0.101± 0.027 0.008

J/ψπ+π− −0.019 +0.173−0.174

+0.004−0.003 0.014

Both −0.002± 0.083± 0.027 0.007

As discussed in Sec. 3.2.5, contributions from doubly CKM-suppressed SM penguindiagrams could have a non-negligible effect on the mixing-induced CP asymmetry and biasthe extracted value of φs. Naive estimates of the bias are of the orderO(10−3) only [264–266],but this must be examined with experimental data using flavour symmetries to exploitcontrol channels. LHCb can perform an SU(3) analysis using measurements of the decaysrates and CP asymmetries in B0

s → J/ψK∗0, B0 → J/ψρ0 and B0 → J/ψφ as controlchannels for B0

s → J/ψφ. The necessary high precision can only be reached using thelarge data sample that will be collected with the upgraded LHCb detector. The 50 fb−1

data sample will also allow to measure φs in the penguin-free (b → cus/ucs) B0s → Dφ

decay [280,281].Another important goal is a more precise determination of sin 2β in the B0 system,

motivated by the tension between the direct and indirect determinations of sin 2β seen byboth UTfit and CKMfitter groups, as shown in Table 2. With the upgraded detector, usingthe B0 → J/ψ K0

S final state alone, a statistical precision of ±0.006 is expected, to becompared to the current error from the B factories of ±0.023 [189]. Given experience withthe current detector it seems feasible to control the systematic uncertainties to a similarlevel. Such precision, together with better control of the penguin pollution, will allow usto pin down any NP effects in B0 mixing. In addition, the penguin-free (b → cud/ucd)B0 → Dρ0 channel can be used to get another handle on sin 2β [282,283].

The importance of improved measurements of ∆Γq has been emphasised in Sec-tions 3.2.1–3.2.3. LHCb has made a preliminary measurement of ∆Γs in B0

s → J/ψ φ

44

using a 1.0 fb−1 data sample [138]. The effective lifetime of B0s → J/ψ f0(980) [284] has

also been measured [228]. Based on this, the statistical precision on ∆Γs with 50 fb−1 isprojected to be ∼ 0.003 ps−1. It is hoped that the systematic uncertainty can be controlledto the same level.

A measurement of ∆Γd is of interest as any result larger than the tiny value expectedin the SM would clearly signal NP [153,247,285]. To determine this quantity, LHCb willcompare the effective lifetimes of the two decay modes B0 → J/ψ K0

S with B0 → J/ψ K∗0.The estimated precision for 1.0 fb−1 is ∼ 0.02 ps−1. With the upgraded detector and 50 fb−1

a statistical precision of ∼ 0.002 ps−1 on ∆Γd can be achieved. The systematic uncertaintyis under study.

The LHCb upgrade will also have sufficient statistics to make novel tests of CPTsymmetry. Any observation of CPT violation indicates physics beyond the SM. Anexample of a unique test in the B0 system uses B0 → J/ψ K0 and its charge-conjugatedecay, where the K0 decays semileptonically [286–288]. This measurement involves lookingat four separate decay paths that interfere. While several tests can be performed, oneparticular observable is the asymmetry Abk, that can be measured without the need offlavour tagging, where

Abk =Γ(B0 +B0 → J/ψ [π−µ+ν]− Γ(B0 +B0 → J/ψ [π+µ−ν])

Γ(B0 +B0 → J/ψ [π−µ+ν] + Γ(B0 +B0 → J/ψ [π+µ−ν]). (39)

In terms of the CPT violation parameter θ′, the kaon decay time tK , the B0 decay timetB, the B0 mass difference ∆md and CP -violating phase 2β, and kaon decay widths ΓKSand ΓKL , this can be expressed

Abk = Re(θ′)2e−

12(ΓKS +ΓKL )tK sin 2β (1− cos ∆mdtB)

e−ΓKS tK + e−ΓKL tK. (40)

A signature of CPT violation would be a 1 − cos ∆mBtB dependence of the decay rateafter integrating over kaon decay times. Roughly 5000 such decays can be expected withthe upgrade. It is possible to detect these decays with low background level, even withthe missing neutrino, using the measured B0 direction, the detected J/ψ four-momentum,and the kaon decay vertex. Other methods to test CPT symmetry (e.g. Ref. [289]) arealso under investigation.

3.3 CP violation measurements with hadronic b→ s penguins

3.3.1 Probes for new physics in penguin-only b→ sqq decays

The presence of physics beyond the SM can be detected by looking for its contribution tob→ sqq (q = s, d) decays,38 which in the SM can only proceed via FCNC loop diagramsthat are dynamically suppressed. These decays provide a rich set of observables that arerather precisely known in the SM but could potentially receive sizeable corrections fromnew heavy particles appearing in the loop.

38 Decays mediated primarily by b→ suu transitions are discussed in Secs. 3.4.4 and 3.4.5.

45

• Direct CP asymmetries. In the SM b→ sqq decays are dominated by the penguindiagram with an internal top quark. As a consequence, the direct CP asymmetry isexpected to be small. If there is a NP amplitude with comparable size interferingwith the SM amplitude, and it has different strong and weak phases than the SMamplitude, a much larger direct CP asymmetry can arise.

• Polarisation and triple product asymmetries. For B decays into two vectormesons V1 and V2, followed by vector to two pseudoscalar decays V1 → P1P

′1

and V2 → P2P′2, there are three transversity states, labelled “longitudinal” (0),

“perpendicular” (⊥) and “parallel” (‖). Measurements of the fractions of the totaldecay rate in each of these states, which correspond to determinations of thepolarisation in the final state, provide useful information about the chiral structureof the electroweak currents, as well about non-perturbative effects such as rescatteringand penguin annihilation. In the SM, the decay to each transversity state is dominatedby a single amplitude with magnitude |Aj|, weak phase Φj and strong phase δj . TheCP -violating observables Im(A⊥A

∗j − A⊥A∗j) are then

Im(A⊥A∗j − A⊥A∗j) = 2|A⊥||Aj| cos(δ⊥ − δj) sin(Φ⊥ − Φj), j = 0, ‖ . (41)

The values of these observables are tiny since in the SM the weak phases are thesame to a very good approximation, but Im(A⊥A

∗j − A⊥A∗j) can significantly differ

from zero if there is a sizeable CP -violating NP contribution in the loop.

These observables can be extracted from the differential distributions in terms ofthe angles θ1, θ2 and φ, where θ1 (θ2) is the polar angle of P1 (P2) in the rest frameof V1 (V2) with respect to the opposite of the direction of motion of the B meson,and φ is the angle between the decay planes of V1 → P1P

′1 and V2 → P2P

′2 in the

rest frame of the B meson. The two observables can also be related to two tripleproduct asymmetries for CP -averaged decays39 which are equal to asymmetriesbetween the number of events with positive and negative values of U = sin 2φ andV = sign(cos θ1 cos θ2) sinφ:

Im(A⊥A∗‖ − A⊥A∗‖) ∝ AU =

N(U > 0)−N(U < 0)

N(U > 0) +N(U < 0), (42)

Im(A⊥A∗0 − A⊥A∗0) ∝ AV =

N(V > 0)−N(V < 0)

N(V > 0) +N(V < 0). (43)

A review of this subject can be found in Ref. [290] and references therein.

• Mixing-induced CP asymmetries. Mixing-induced CP asymmetries in b→ sqqdecays of neutral B to CP eigenstates are precisely predicted. Due to the fact thatthe penguin diagram with an internal top quark is expected to dominate, the values

39 The triple product asymmetries in B0s → φφ and B0

s → K∗0K∗0 decay could in principle also receivecontribution from non-zero mixing-induced CP asymmetries arising from NP in B0

s mixing. However, thiscontribution is suppressed by ∆Γs/Γs and is already highly constrained.

46

of 2βeff determined using B0 → φK0S , B0 → η′K0

S , B0 → f0(980)K0S , etc., are all

expected to give ≈ 2β (see, e.g. Refs. [291, 292] and the discussion in Ref. [43]).Similarly, the values of 2βeff

s determined from B0s → φφ, B0

s → K∗0K∗0, etc., areexpected to vanish due to cancellation of weak phases between mixing (top box)and decay (top penguin) amplitudes. Higher order corrections from subleadingdiagrams are expected to be small compared to the precision that can be achievedin the near-term, but further theoretical studies will be needed as the upgrade eraapproaches. NP with a flavour structure different from the SM will alter these CPasymmetries through the decay amplitudes, even if there is no NP in B mixing. Anumber of quasi-two-body or three-body decay modes can be studied.

• Correlations between direct and mixing-induced asymmetries. Penguin-only decay modes are particularly interesting as the difference between formal “tree”and “penguin” contributions boils down to a difference in the quark-flavour runningin the loop of the penguins. This difference, dominated by short distances, can beassessed accurately using QCD factorisation, and it can be used to correlate thebranching ratio and the CP asymmetries of penguin-mediated modes. As discussedin Refs. [137,293,294], these observables can be correlated not only within the SM,but can also be used to extract the B0

s mixing phase even in the presence of NPaffecting only this phase.

3.3.2 Current status and outlook of LHCb measurements

LHCb published the first observation and measurement of the branching ratio and polari-sation amplitudes in the B0

s → K∗0K∗0 decay mode [295] using 35 pb−1 of data collectedin 2010. A clean mass peak corresponding to 50 ± 8 B0

s → (K+π−)(K+π+) decays isseen (Fig. 15 (left)), mostly from resonant B0

s → K∗0K∗0 decays. Using this signal thelongitudinal polarisation amplitude is measured to be fL = 0.31± 0.12(stat)± 0.04(syst)and the branching ratio to be B(B0

s → K∗0K∗0) = (2.81± 0.46(stat)± 0.45(syst))× 10−5.LHCb also published the measurement of the polarisation amplitudes and triple product

asymmetries in B0s → φφ [296] using the 2011 data set of 1.0 fb−1. In this data set 801±29

events are observed with excellent signal-to-background ratio (see Fig. 15 (right)). Thepolarisation amplitudes are measured to be

|A0|2 = 0.365± 0.022 (stat)± 0.012 (syst) ,|A⊥|2 = 0.291± 0.024 (stat)± 0.010 (syst) ,|A‖|2 = 0.344± 0.024 (stat)± 0.014 (syst) ,

(44)

where the sum of the square of the amplitudes is constrained to unity. The triple productasymmetries in this mode are measured to be

AU = −0.055± 0.036 (stat)± 0.018 (syst) ,AV = 0.010± 0.036 (stat)± 0.018 (syst) .

(45)

The results of this analysis are in agreement with, and more precise than, the previousmeasurement [297], and are also consistent with the SM.

47

)2c) (MeV/+π-K-π+M(K5000 5200 5400 5600 5800

)2 cE

ven

ts /

(30

MeV

/

0

5

10

15

20

25

30

35

40

LHCb

)2 (MeV/cKKKKm5100 5200 5300 5400 5500 5600

2C

andi

date

s / 5

MeV

/c

20

40

60

80

100

120 LHCb

Figure 15: (Left) fit of the K+π−K−π+ mass distribution for B0s → K∗0K∗0 candidates from

35 pb−1 [295]; (right) fit of the K+K−K−K+ mass distribution for B0s → φφ candidates from

1.0 fb−1 [296].

First measurements of CP asymmetries in these modes from time-dependent flavour-tagged angular analyses are expected to follow. With high statistics, it will be possible tomeasure polarisation-dependent direct and mixing-induced CP asymmetries, but for thefirst analysis it will be more convenient to determine a single complex observable commonto all polarisations (as done for B0

s → J/ψ φ)

λ = ηjq

p

AjAj

(46)

where j denotes one of the three transversity states, which are also CP eigenstates witheigenvalues ηj, and Aj (Aj) is the decay amplitude of B0

s (B0s) to the corresponding state.

With this approximation it will be possible to determine the magnitude |λ| and phaseφeffs ≡ − arg(λ). The SM expectation is |λ| ≈ 1 and φeff

s ≈ 0 due to the dominanceof the top-quark loop, and any observed deviation from these expectations would be asignature of NP. Since NP in B0

s mixing is already constrained by measurement of φs fromB0s → J/ψ φ, the main interest in these b → s penguin modes is to look for NP in the

decay processes. Based on simulation studies, a sensitivity on φeffs of 0.3–0.4 radians with

1.0 fb−1 is expected for both B0s → φφ and B0

s → K∗0K∗0.

3.3.3 Future prospects with LHCb upgrade

The latest results on mixing-induced CP violation in b→ s transitions show no significantdeviation from the SM, as seen in Fig. 16, which compares the mixing-induced CP violationparameter sin 2βeff measured in penguin-dominated b→ s decays with the value of sin 2βmeasured in the tree-dominated b→ ccs decays. In the absence of NP these observablesshould only differ by small amounts. Due to these results, large NP contributions inb → sqq decays are unlikely but further tests with higher precision remain interesting.LHCb will be able to make competitive measurements of sin 2βeff in B0 → φK0

S and several

48

other b→ sqq decays, but a significant improvement in precision requires the 50 fb−1 ofthe LHCb upgrade. The improved trigger efficiency in the LHCb upgrade is particularlyimportant for these decays, which have only hadrons in the final state. With the upgradedata sample, the statistical error of sin 2βeff(B0 → φK0

S ) is estimated to be roughly 0.06,which is still above the SM uncertainty of ∼ 0.02 [298].

There are several more NP probes in b→ sqq decays that can be exploited at LHCband its upgrade, such as mixing-induced CP asymmetries and triple product asymmetriesin both B0

s → φφ and B0s → K∗0K∗0 decays. The statistical precision of φeff

s with eachchannel is estimated to be 0.3–0.4 rad for 1.0 fb−1. The projected precision for 50 fb−1 isabout 0.03 rad each. This can be compared with the uncertainties of their SM predictionsof about 0.02 rad. It is also possible to perform a combined analysis of B0

s → K∗0K∗0

and its U-spin related channel B0 → K∗0K∗0, which will put strong constraint on thesubleading penguin diagrams in B0

s → K∗0K∗0, thus further reducing the theoreticaluncertainty in the measurement of φeff

s [299,300]. The statistical precision of AU and AVis estimated to be about 0.004, compared with an upper bound of 0.02 on their possiblesizes in the SM [290].

sin(2`eff) > sin(2qe1ff)

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orio

nd 2

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orio

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Mor

iond

201

2

HFA

G

Mor

iond

201

2

HFA

G

Mor

iond

201

2H

FAG

Mor

iond

201

2H

FAG

Mor

iond

201

2H

FAG

Mor

iond

201

2

bAccs

q K0

dv K

0

K S K S

K S

/0 K0

l0 KS

t K

Sf 0 K

S

K+ K- K

0

K+ K- K

0

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

World Average 0.68 ± 0.02BaBar 0.66 ± 0.17 ± 0.07Belle 0.90 +-0

0..01

99

Average 0.77 +-00

.

.11

02

BaBar 0.57 ± 0.08 ± 0.02Belle 0.64 ± 0.10 ± 0.04Average 0.59 ± 0.07BaBar 0.94 +-0

0..22

14 ± 0.06

Belle 0.30 ± 0.32 ± 0.08Average 0.72 ± 0.19BaBar 0.55 ± 0.20 ± 0.03Belle 0.67 ± 0.31 ± 0.08Average 0.57 ± 0.17BaBar 0.35 +-0

0..23

61 ± 0.06 ± 0.03

Belle 0.64 +-00

.

.12

95 ± 0.09 ± 0.10

Average 0.54 +-00

.

.12

81

BaBar 0.55 +-00

.

.22

69 ± 0.02

Belle 0.11 ± 0.46 ± 0.07Average 0.45 ± 0.24BaBar 0.74 +-0

0..11

25

Belle 0.63 +-00

.

.11

69

Average 0.69 +-00

.

.11

02

BaBar 0.65 ± 0.12 ± 0.03Belle 0.76 +-0

0..11

48

Average 0.70 +-00

.

.01

90

Average 0.69 ± 0.07

H F A GH F A GMoriond 2012PRELIMINARY

Figure 16: HFAG compilation of results for sin 2βeff in b→ sqq decays [43].

In summary, the LHCb upgrade will allow the exploitation of the full potential ofthe NP probes in b → sqq decays. Table 4 compares the current and projected (LHCbupgrade, i.e. 50 fb−1) precisions of the key observables with the theory uncertainties oftheir SM predictions.

49

Table 4: Current and projected precisions of the key observables in b→ sqq decays.

Observable Current LHCb upgrade (50 fb−1) Theory uncertainty

AU, V (B0s → φφ) 0.04 (LHCb 1.0 fb−1) 0.004 0.02 [301]

φeffs (B0

s → φφ) – 0.03 0.02 [298]φeffs (B0

s → K∗0K∗0) – 0.03 0.02 [298]sin 2βeff(B0 → φK0

S ) 0.12 (B factories) 0.06 0.02 [178]

3.4 Measurements of the CKM angle γ

3.4.1 Measurements of γ using tree-mediated decays

The CKM angle γ, defined as the phase γ = arg [−VudV ∗ub/(VcdV ∗cb)], is one of the anglesof the unitarity triangle formed from the hermitian product of the first (d) and third(b) columns of the CKM matrix V . It is one of the least well known parameters of thequark mixing matrix. However, since it can be determined entirely through decays ofthe type B → DK 40 that involve only tree amplitudes — an unusual, even unique,property amongst all CP violation parameters — it provides a benchmark measurement.The determination from tree level decays has essentially negligible theoretical uncertainty,at the level of δγ/γ = O(10−6), as will be shown in the next section. This makes γ avery appealing “standard candle” of the CKM sector. It serves as a reference point forcomparison with γ values measured from loop decays (see Sec. 3.4.4).

Moreover, the determination of γ is crucial to improve the precision of the globalCKM fits, and resulting limits on (or evidence for) NP contributions (see Sec. 3.2.4). Inparticular, the measurement of ∆md and the oscillation phase sin 2β in B0–B0 mixing canbe converted to a measurement of γ (in the SM). This can be compared to the referencevalue from B → DK — their consistency verifies that the Kobayashi-Maskawa mechanismof CP violation is the dominant source in quark flavour-changing processes. Existingmeasurements provide tests at the level of O(10 %), but improving the precision to searchfor smaller effects of NP is well motivated.

Several established methods to measure γ in tree decays exploit the B− → D(∗)K(∗)−

decays. They are based on the interference between the b→ u and b→ c tree amplitudes,which arises when the neutral D meson is reconstructed in a final state accessible to bothD0 and D0 decays. The interference between the amplitudes results in observables thatdepend on their relative weak phase γ. Besides γ they also depend on hadronic parameters,namely the ratio of magnitudes of amplitudes rB ≡ |A(b→ u)/A(b→ c)| and the relativestrong phase δB between the two amplitudes. These hadronic parameters depend on the Bdecay under investigation. They can not be precisely calculated from theory (see, however,Ref. [302]), but can be extracted directly from data by simultaneously reconstructingseveral different D final states.

40 By B → DK all related tree-dominated decay processes are implicitly included, including B+ → DK+,B0 → DK∗0, B0

s → Dφ, B0s → D∓s K

∓ and B0 → D(∗)∓π±. In these specific decay processes, the notationD refers to a neutral D meson that is an admixture of D0 and D0 states.

50

The various methods differ by the D(∗) final state that is used. The three maincategories of D decays considered so far by the B factories BaBar and Belle, and by CDF,are:

• CP eigenstates (the GLW method [303,304]),

• doubly Cabibbo-suppressed (DCS) decays (the ADS method [305,306]),

• three-body, self-conjugate final states (the GGSZ or “Dalitz” method [307]).

An additional category has not been possible to pursue at previous experiments due tolimited event sample sizes:

• singly Cabibbo-suppressed (SCS) decays (the GLS method [308]).

In practise, except for the case of two-body decays, there is often no clear distinctionbetween the different methods.

The best sensitivity to γ obviously comes from combining the results of all differentanalyses. This not only improves the precision on γ, but provides additional constraintson the hadronic parameters. It also allows one to overcome the fact that CP -odd finalstates such as K0

Sπ0 are not easily accessible in LHCb’s hadronic environment.

A brief review of the main ideas of the different methods follows. The amplitudes ofthe B− → D0K− and B− → D0K− processes are written as:

A(B− → D0K−) = Aceiδc , A(D0 → f) = Afe

iδf , (47)

A(B− → D0K−) = Auei(δu−γ) , A(D0 → f) = Afe

iδf ,

where Ac, Au, Af and Af are real and positive (and CP violation in D0 decays has beenneglected). The subscripts c and u refer to the b→ c and b→ u transitions, respectively.The amplitudes for the D0 decay can generally include the case where the D0 decays toa three-body final state. In this case, Af , Af , δf and δf are functions of the Dalitz plotcoordinates. The amplitude of the process B− → D[→ f ]K− can be written, neglectingD0–D0 mixing, as

A(B− → D[→ f ]K−) = AcAfei(δc+δf ) + AuAfe

i(δu+δf−γ) , (48)

and the rate is given by

Γ(B− → D[→ f ]K−) ∝ A2cA

2f + A2

uA2f + 2AcAfAuAfRe(ei(δB+δD−γ))

∝ A2c

(A2f + r2

BA2f + 2rBAfAfRe(ei(δB+δD−γ))

), (49)

where rB = Au/Ac, δB = δu − δc and δD = δf − δf . The rate for the charge-conjugatedmode (still neglecting CP violation in D0 decays) is obtained by exchanging γ → −γ.Taking into account CKM factors and, in the case of charged B decays, colour suppressionof the b→ u amplitude, rB is expected to be around 0.1 for B− decays and around 0.3 for

51

B0 decays. From Eq. (49) all the relevant formulae of the GLW, ADS and GGSZ methodscan be derived.

In the GLW analysis, the neutral D mesons are selected in CP eigenstates fCP± suchas D → K−K+ (CP = +1) or D → K0

Sπ0 (CP = −1). Thus Af/Af = 1 and δD = 0, π for

CP = ±1. Eq. (49) becomes:

Γ(B− → D[→ fCP±]K−) ∝ A2c(1 + r2

B ± 2rB cos(δB − γ)) . (50)

The B− → DK− decays, where the D decays to Cabibbo-favoured (CF) final states (e.g.D0 → K−π+) can be used to normalise the rates in order to construct observables thatminimise the systematic uncertainties. For those decays, to a good approximation,

Γ(B− → D[→ K−π+]K−) = Γ(B+ → D[→ K+π−]K+) ∝ A2c . (51)

From Eqs. (50) and (51) and their CP conjugates the usual GLW observables follow:

RCP± =2[Γ(B− → DCP±K

−) + Γ(B+ → DCP±K+)]

Γ(B− → D0K−) + Γ(B+ → D0K+)(52)

ACP± =Γ(B− → DCP±K

−)− Γ(B+ → DCP±K+)

Γ(B− → DCP±K−) + Γ(B+ → DCP±K+). (53)

Eqs. (52) and (53) provide a set of four observables that are connected to the threeunknowns γ, rB and δB through

RCP± = 1 + r2B ± 2rB cos δB cos γ (54)

ACP± =±2rB sin δB sin γ

RCP±. (55)

However, only three of these equations are independent since, from Eq. (55), RCP+ACP+ =−RCP−ACP−. Analogous relations hold for B → D∗CPK and B → DCPK

∗ decays, withdifferent values of the hadronic parameters characterising the B decay. However, in theB → D∗CPK case one has to take into account a CP flip due to the different chargeconjugation quantum numbers of the π0 and the photon from the D∗ decay [309]: D∗CP± →DCP±π

0, but D∗CP± → DCP∓γ. For analysis of B → DCPK∗ the finite width of the K∗

resonance must be taken into account [310]. There are related important consequences forthe ADS and GGSZ analyses of B → D∗K and B → DK∗ decays.

In the ADS analysis, the neutral D mesons are selected in CF and DCS decays, suchas D0 → K−π+ and D0 → π−K+, respectively. The B decay rate is the result of theinterference of the colour allowed B− → D0K− decay followed by the DCS D0 → π−K+

decay and the colour suppressed B− → D0K− decay followed by the CF D0 → K−π+

decay. As a consequence, the interfering amplitudes are of similar magnitude and hencelarge interference effects can occur. From Eq. (49) one finds

Γ(B∓ → D[→ K±π∓]K∓) ∝ r2B + r2

D ± 2rBrD cos(δB + δD ∓ γ) (56)

52

where both rD = Af/Af = |A(D0 → π−K+)/A(D0 → K−π+)| and the phase differenceδD are measured in charm decays. The value of δD can be determined directly using datacollected from e+e− collisions at the ψ(3770) resonance, as has been done by CLEO [311,312], but the most precise value comes from a global fit including charm mixing parameters.The results provided by HFAG [43] from a combination with CP violation in charm allowedare rD = 0.0575± 0.0007, δD =

(202 +10

−11

). Defining RADS and AADS as

RADS =Γ(B− → D[→ π−K+]K−) + Γ(B+ → D[→ π+K−]K+)

Γ(B− → D[→ K−π+]K−) + Γ(B+ → D[→ K+π−]K+), (57)

AADS =Γ(B− → D[→ π−K+]K−)− Γ(B+ → D[→ π+K−]K+)

Γ(B− → D[→ π−K+]K−) + Γ(B+ → D[→ π+K−]K+), (58)

and using Eqs. (51) and (56) gives

RADS = r2B + r2

D + 2rB rD cos γ cos(δB + δD) , (59)

AADS = 2rB rD sin γ sin(δB + δD)/RADS . (60)

It has been noted that for the extraction of γ it can be more convenient to replace thepair of observables RADS, AADS with a second pair, R+, R−, defined as:

R± ≡Γ(B± → [K∓π±]DK

±)

Γ(B± → [K±π∓]DK±)= r2

B + r2D + 2rBrD cos(δB + δD ± γ) (61)

Unlike RADS, AADS, the two quantities R+, R− are statistically independent. The ADS de-cay chain B± → [π±K∓]DK

± has been observed for the first time by LHCb [6], confirmingthe evidence that had begun to accumulate in previous measurements [313–315].

In the GGSZ analysis, the neutral D mesons are selected in three-body self-conjugatefinal states. The channel that has been used most to date is D → K0

Sπ+π−, though

first results have also been presented with D → K0SK

+K− and other channels are underconsideration. For concreteness, consider D → K0

Sπ+π−, with Afe

iδf = f(m2−,m

2+) and

Afeiδf = f(m2

+,m2−), where m2

− and m2+ are the squared masses of the K0

Sπ− and K0

Sπ+

combinations. The rate in Eq. (49) can be re-written as:

Γ(B∓ →D[→ K0Sπ−π+]K∓) ∝ |f(m2

∓,m2±)|2 + r2

B|f(m2±,m

2∓)|2 (62)

+ 2rB|f(m2∓,m

2±)||f(m2

±,m2∓)| cos(δB + δD(m2

∓,m2±)∓ γ) ,

where δD(m2∓,m

2±) is the strong phase difference between f(m2

±,m2∓) and f(m2

∓,m2±). Due

to the fact that rB is required to be positive, the direct extraction of rB, δB and γ can bebiased. To avoid these biases, the “Cartesian coordinates” have been introduced [316]

x± = Re[rBei(δB±γ)] , y± = Im[rBe

i(δB±γ)] , (63)

allowing Eq. (62) to be rewritten as

Γ(B∓ → D[→ K0Sπ

+π−]K∓) ∝ |f∓|2 + r2B|f±|2 + 2

[x∓Re[f∓f

∗±] + y∓Im[f∓f

∗±]]. (64)

53

Here the notation has been simplified using f± = f(m2±,m

2∓). This Dalitz plot-based

method can be implemented in a model-dependent way by parametrising the amplitudeas a function of the Dalitz plot of the three-body state, or in a model-independent wayby dividing the Dalitz plot into bins and making use of external measurements of the Ddecay strong phase differences within these bins [307,317,318].41

Besides the established methods based on direct CP violation in B → DK decays,it is also possible to measure γ using time-dependent analyses of neutral B0 and B0

s

tree decays [320–322]. The method still relies on the interference of b → u and b → camplitudes, but interference is achieved through B0 (B0

s ) mixing. Thus one measures thesum of γ and the mixing phase, namely γ + 2β and γ − 2βs in the B0 and B0

s systems,respectively. Since both sin 2β and βs are becoming increasingly well measured, thesemeasurements provide sensitivity to γ.

Pioneering time-dependent measurements using the B0 → D(∗)∓π± decays have beenperformed by both BaBar [323,324] and Belle [325,326]. In these decays the amplituderatios rDπ = |A(B0 → D(∗)+π−)/A(B0 → D(∗)−π+)| are expected to be small, rDπ . 0.02,limiting the sensitivity. In the decays B0

s → D∓s K±, however, both b → c and b → u

amplitudes are of same order in the Wolfenstein parameter λ, O(λ3), so that the interferenceeffects are expected to be large. In addition, the decay width difference in the B0

s system,∆Γs, is non-zero, which adds sensitivity to the weak phase through the hyperbolic termsin the time evolution (see also Ref. [327]). The time-dependent decay rates of the initiallyproduced flavour eigenstates are given by the decay equations

dΓB0s (B0

s)→f (t)

dt e−Γst=

1

2|Af |2(1 + |λf |2)

×[cosh

(∆Γst

2

)−Df sinh

(∆Γst

2

)± Cf cos (∆mst)∓ Sf sin (∆mst)

],

(65)

where Γs, ∆Γs, ∆ms are the usual mixing parameters of the B0s system and |q/p| = 1 has

been assumed. The top (bottom) of the ± and ∓ signs is used when the initial particle istagged as a B0

s (B0s) meson. In Eq. (65), Af is the decay amplitude for a B0

s meson todecay to a final state f , and λf = (q/p)(Af/Af) where Af is the amplitude for a B0

s todecay into f . Similar equations hold for the charge conjugate processes replacing Af byAf , λf by λf = (p/q)(Af/Af ), and with a separate set of coefficients Cf , Sf and Df . As

each decay is dominated by a single diagram, |λf | = |λf |. The CP asymmetry observablesare then given by

Cf = Cf =1− |λf |21 + |λf |2

, Sf =2Im(λf )

1 + |λf |2, Df =

2Re(λf )

1 + |λf |2,

Sf =2Im(λf )

1 + |λf |2, Df =

2Re(λf )

1 + |λf |2. (66)

41 As for δD in the ADS method, the strong phase differences can be determined directly fromψ(3770)→ DD data, which has been done by CLEO [319]. In future, it is expected that the most precisevalue will come from a global fit including results of time-dependent analyses of multibody charm decays.

54

The equality Cf = Cf results from |q/p| = 1 and |λf | = |λf |. The term λf is connected tothe weak phase by

λf =(qp

)AfAf

=(V ∗tbVtsVtbV ∗ts

)(VubV ∗csV ∗cbVus

)∣∣∣A2

A1

∣∣∣ei∆ = |λf |ei(∆−(γ−2βs)) , (67)

where |A2/A1| is the ratio of the hadronic amplitudes between B0s → D−s K

+ and B0s →

D+s K

−, ∆ is their strong phase difference, and γ − 2βs is the weak phase difference. Ananalogous relation exists for λf , λf = |λf |ei(∆+(γ−2βs)). Thus one obtains five observablesfrom Eq. (66) and solves for |λf |, ∆, and (γ − 2βs).

The LHCb experiment has the necessary decay time resolution, tagging power andaccess to large enough signal yields to perform this time-dependent CP measurement.42

The signal yields can be seen from the measurement of B(B0s → D∓s K

±) [139] (see Sec. 3.4.3below). The identification of the initial flavour of the signal B0

s candidate can be donecombining both the responses of opposite-side and same-side kaon tagging algorithms, asis planned for other measurements of mixing-induced CP -violation in B0

s decays, and hasalready been implemented in the preliminary analysis of B0

s → D−s π+ decays [218].

3.4.2 Theoretical cleanliness of γ from B → DK decays

The answer to the question of why it is interesting to measure γ precisely depends on theexperimental precision that can be achieved. In the era of LHCb, the main motivation isthe theoretically clean measurement of the SM CKM phase. The search for NP can thusbe performed by comparing the extracted value of γ to other observables, for example inthe CKM fit (see Sec. 3.2.4). However, one can also cross-check for the presence of NP inB → DK channels themselves. One way is to test that the values of γ determined fromthe many different B → DK type channels all coincide. Another is automatically built into the method for γ extraction in the GGSZ analysis. Consider the case where the decayamplitudes get modified by an extra contribution with a new strong phase δ′B and a weakphase γ′. Then instead of the decay amplitudes in Eq. (48) one finds

A(B± → fDK±) ∝ 1 + rDe

iδD(rBei(δB±γ) + r′Be

i(δ′B±γ′)) . (68)

This means that for B+ and B− decays the rB ratios are different

rB+ → |rBei(δB+γ) + r′Bei(δ′B+γ′)| , rB− → |rBei(δB−γ) + r′Be

i(δ′B−γ′)| . (69)

Discovering that rB− 6= rB+ would signal a CP -violating NP contribution to the B → DKamplitude. One signature of NP would then be x2

+ + y2+ 6= x2

− + y2−, though it is also

possible that the equality could be satisfied even in the presence of NP: in this case therecan be a shift in the extracted value of γ.

Existing measurements place strong constraints on tree-level NP effects, yet thepossibility of discoveries in this sector in the near term is not ruled out. In the far future,

42 Preliminary results have been presented at CKM 2012 [328].

55

with much larger statistics, the measurement of γ is well suited to search for high scaleNP since it is theoretically very clean. For example, NP with contributions of differentchirality could give different shifts in γ, so the above test is meaningful.

A useful question to ask is, what is the energy scale that could be probed in principle?To answer this, the irreducible theoretical uncertainty in the determination of γ must beestimated. There are several sources that can induce a bias in the determination of γ fromB → DK decays. However, most of these can be avoided, either (i) with more statistics(for example, the Dalitz plot model uncertainty where a switch to a model-independentmethod is possible), or (ii) by modifying the equations used to determine γ (an exampleis to correct for effects of D0–D0 mixing [329, 330]). The remaining, irreducible, theoryuncertainties are then from the electroweak corrections.

u

u

b

c

W

s

uWt, c, u

b, s, d

u

u

b

c

W

u

s

Figure 17: A B− → D0K− box diagram electroweak correction (left) with a different CKMstructure than the leading weak decay amplitude (right).

Table 5: Ultimate NP scales that can be probed using different observables listed in thefirst column. They are given by saturating the theoretical errors given respectively by 1)δγ/γ = 10−6, 2) optimistically assuming no error on fB, so that the ultimate theoretical erroris only from electroweak corrections, 3) using SM predictions in Ref. [30], 4) optimisticallyassuming perturbative error estimates δβ/β 0.1 % [331], and 5) from bounds for ReC1(ImC1)from UTfitter [262].

Probe ΛNP for (N)MFV NP ΛNP for gen. FV NP

γ from B → DK1) Λ ∼ O(102 TeV) Λ ∼ O(103 TeV)B → τν2) Λ ∼ O(1 TeV) Λ ∼ O(30 TeV)b→ ssd3) Λ ∼ O(1 TeV) Λ ∼ O(103 TeV)

β from B → J/ψK0S

4) Λ ∼ O(50 TeV) Λ ∼ O(200 TeV)K − K mixing5) Λ > 0.4 TeV (6 TeV) Λ > 103(4) TeV

The challenge to determine this uncertainty is that the hadronic elements can nolonger be determined solely from the experiment. Not all electroweak corrections matterthough — the important ones are the corrections that change the CKM structure. Forinstance, vertex corrections and Z exchanges do not affect γ, but corrections from box

56

diagrams carry a different weak phase. The dominant contribution is effectively due to tand b running in the loop. For b→ usc transitions there is a tree level contribution with∼ VubV

∗cs CKM structure, while the box diagram has ∼ (VtbV

∗ts)(VubV

∗cb). Since this has the

same weak phase, it does not introduce a shift in γ. For b→ csu transitions, on the otherhand, the tree level is ∼ VcbV

∗us, while the box diagram ∼ (VtbV

∗ts)(VcbV

∗ub), as illustrated in

Fig. 17. The two contributions have different weak phases, which means that the shift δγis non-zero.

The size of this effect is estimated by integrating over both t and b at the same time.The electroweak corrections in the effective theory are then described by a local operatorwhose matrix elements are easier to estimate. Although the Wilson coefficient of theoperator contains large logarithms, log(mb/mW ), for O(1) estimates, the precision obtainedwithout resummation is sufficient. If one resums log(mb/mW ) then nonlocal contributionsare also generated. As a rough estimate only the local contributions need be kept. Theirreducible theory error on γ is conservatively estimated to be δγ/γ < O(10−6) (mostlikely it is even δγ/γ . O(10−7)).

This limit is far beyond the achievable sensitivity of any foreseeable experiment.Nevertheless, it is interesting to consider what could be learnt in case such small deviationscould be observed. Assuming MFV one can probe ΛNP ∼ 102 TeV, while assuming generalflavour-violating (FV) NP one can probe ΛNP ∼ 103 TeV (where MFV and general FV NPscales are defined as in Ref. [262]). This is by far the most precise potential probe of MFV,as shown in Table 5, due to the small theoretical uncertainty.

Since an experimental precision of δγ/γ ∼ 10−6 is not achievable in the near future,the NP scale reach must be adjusted for more realistic data sets. This is easily done, sincethe scale ΛNP probed goes as the fourth root of the yield. With the LHCb upgrade, anuncertainty of < 1 on γ can be achieved (see Sec. 3.4.6), so that NP scales approachingΛNP ∼ 5(50) TeV can be probed for MFV (general FV) NP.

3.4.3 Current LHCb experimental situation

First results from LHCb in this area include a measurement using B− → DK− with theGLW and ADS final states [6].43 A measurement of the branching ratio of B0

s → D∓s K±

has also been performed [139]. Several other analyses, including studies of GGSZ-typefinal states, are in progress.44

These measurements all share common selection strategies. They benefit greatly fromboosted decision tree algorithms, which combine up to 20 kinematic variables to effectivelysuppress combinatorial backgrounds. Charmless backgrounds are suppressed by exploitingthe large forward boost of the D+

(s) meson through a cut on its flight distance.

43 Results from preliminary GLW-type analyses using B0 → DK∗0 [332] and B− → DK−π+π− [333]have been reported at ICHEP 2012.

44 At CKM 2012, LHCb presented results of a model-independent GGSZ analysis of B− → DK− withD → K0

Sπ+π− and D → K0

SK+K− [7], preliminary results from a ADS-type analysis of B− → DK−

with D → K3π [334], a preliminary determination of γ from combined results using B− → DK−

and B− → Dπ− [335], and preliminary results on the time-dependent CP violation parameters inB0s → D∓s K

± [328].

57

In the GLW/ADS analysis [6] of 1.0 fb−1 of√s = 7 TeV data collected in 2011, the

CP eigenstates D → K+K−, π+π−, and the quasi-flavour-specific D → π−K+ decay areused. The CP asymmetries defined in Eq. (58), and the ratios R± defined in Eq. (61),are measured for both the B → DK signal and the abundant B → Dπ control channel.The latter has limited sensitivity to γ but provides a large control sample from whichprobability density functions are shaped, and can be used to help reduce certain systematicuncertainties. The control channel is also used to measure three ratios of partial widths

RfK/π =

Γ(B− → [f ]DK−) + Γ(B+ → [f ]DK

+)

Γ(B− → [f ]Dπ−) + Γ(B+ → [f ]Dπ+), (70)

where f represents KK, ππ and the favoured Kπ mode. The signal yields are estimatedby a simultaneous fit to 16 independent subsamples, defined by the charges (×2), the Dfinal states (×4), and the DK or Dπ final state (×2). Figure 18 shows the projections ofthe suppressed π±K∓ subsamples. It is crucial to control the cross feed of the abundantB− → Dπ− decays into the signal decays. This is achieved using the two LHCb ring-imaging Cherenkov detectors [336]. The systematic uncertainties are dominated byknowledge of the intrinsic asymmetry of the detector in reconstruction of positive andnegative B meson decays, and by the uncertainty on the particle identification requirements.The results are

RCP+ = 1.007± 0.038± 0.012 ,

ACP+ = 0.145± 0.032± 0.010 ,

R− = 0.0073± 0.0023± 0.0004 ,

R+ = 0.0232± 0.0034± 0.0007 ,

where the first error is statistical and the second systematic; RCP+ is computed fromRCP+ ≈ 〈RKK

K/π, RππK/π〉/RKπ

K/π with an additional 1 % systematic uncertainty assigned to

account for the approximation; ACP+ is computed as ACP+ = 〈AKKK , AππK 〉. From the R±one can also compute

RADS = 0.0152± 0.0020± 0.0004 ,

AADS = −0.52± 0.15± 0.02 ,

asRADS = (R−+R+)/2 andAADS = (R−−R+)/(R−+R+). To summarise, theB± → DK±

ADS mode is observed with ≈ 10σ statistical significance when comparing the maximumlikelihood to that of the null hypothesis. This mode displays evidence (4.0σ) of a largenegative asymmetry, consistent with previous experiments [313–315]. The combinedasymmetry ACP+ is smaller than (but compatible with) previous measurements [337,338],and is 4.5σ significant. The maximum likelihood is compared with that under the nullhypothesis in all three DK final states, diluted by the non-negligible correlated systematicuncertainties. From this, with a total significance of 5.8σ, direct CP violation is observedin B± → DK± decays.

58

)2 cEv

ents

/ ( 5

MeV

/

10

20

30 )2 cEv

ents

/ ( 5

MeV

/

10

20

30

-KD

]-π+π[→-B

LHCb

+KD

]-π+π[→+B

LHCb

5200 5400 56000

100

200

5200 5400 56000

100

200

-πD

]-π+π[→-B

LHCb

)2c) (MeV/±Dh(m5200 5400 5600

)2c) (MeV/±Dh(m5200 5400 5600

+πD

]-π+π[→+B

LHCb

Figure 3: Invariant mass distributions of selected B± ! [!+!!]Dh± candidates. See the caption of Fig. 1for a full description.

)2 cEv

ents

/ ( 5

MeV

/

5

10

15 )2 c

Even

ts /

( 5 M

eV/

5

10

15

-KD

]+K-π[→-B

LHCb

+KD

]-K+π[→+B

LHCb

5200 5400 56000

10

20

30

40

5200 5400 56000

10

20

30

40

-πD

]+K-π[→-B

LHCb

)2c) (MeV/±Dh(m5200 5400 5600

)2c) (MeV/±Dh(m5200 5400 5600

+πD

]-K+π[→+B

LHCb

Figure 4: Invariant mass distributions of selected B± ! [!±K"]Dh± candidates. See the caption ofFig. 1 for a full description. The dashed line here represents the partially reconstructed, but Cabibbofavoured, B0

s ! D0K!!+ and B0s ! D0K+!! decays where the pions are lost. The pollution from

favoured mode cross feed is drawn, but is too small to be seen.

8

Figure 18: Invariant mass distributions of selected B± → [π±K±]Dh± candidate events: (left)

B− candidates, (right) B+ candidates [6]. In the top plots, the track directly from the B vertexpasses a kaon identification requirement and the B candidates are reconstructed assigning thistrack the kaon mass. The remaining events are placed in the bottom row and are reconstructedwith a pion mass hypothesis. The dark (red) curve represents the B → DK± events, the light(green) curve is B → Dπ±. The shaded contribution are partially reconstructed events and thethin line shows the total fit function which also includes a linear combinatoric component. Thebroken line represents the partially reconstructed B0

s → D0K+π− decays where the pion is lost.

The analysis of the B0s → D∓s K

± decay mode [139] is based on a sample correspondingto an integrated luminosity of 0.37 fb−1, collected in 2011 at a centre-of-mass energy of√s = 7 TeV. This decay mode has been observed by the CDF [339] and Belle [340]

collaborations, who measured its branching fraction with an uncertainty around 23 % [189].In addition to B0

s → D∓s K±, the channels B0 → D−π+and B0

s → D−s π+are analysed. They

are characterised by a similar topology and therefore are good control and normalisationchannels. Particle identification criteria are used to separate the CF decays from thesuppressed modes, and to suppress misidentified backgrounds.

The signal yields are obtained from unbinned extended maximum likelihood fits tothe data. The fits include components for the combinatorial background and severalsources of background from b hadron decays. The most important is the misidentifiedB0s → D−s π

+ decay. Its shape is fixed from data using a reweighting procedure [341] whilethe yield is left free to float. A similar procedure is applied to a simulated data sample toextract the shape of the B0 → D−K+ misidentified background. The fit results are shownin Fig. 19.

Correcting the raw signal yields for selection efficiency differences gives

B(B0s → D∓s K

±)

B(B0s → D−s π

+)= 0.0646± 0.0043± 0.0025 , (71)

59

]2

) [MeV/c+

Ks­

Dm(5200 5400 5600 5800

)2

Ev

en

ts /

( 1

4 M

eV

/c

0

20

40

60

80

100

120

140

160

180 LHCb +Ks

­ D→s

0B+

Ks

­ D→

0B

+K­

D→0B+πs

­ D→s

0B(*)+

Ks

(*)­ D→s

00B

ps

(*)­ D→b

0Λ

)+ρ, +π(s

(*)­ D→s

0B

Combinatorial

Figure 19: Mass distribution of the B0s → D∓s K

± candidate events [139]. The stacked backgroundshapes follow the same top-to-bottom order in the legend and in the plot.

where the first uncertainty is statistical and the second is systematic. Using the measuredrelative yield of B0 → D−π+, the known B0 → D−π+ branching fraction [189], and therecent fs/fd measurement [144], the branching fractions

B(B0s → D−s π

+) = (2.95± 0.05± 0.17 +0.18−0.22)× 10−3 , (72)

B(B0s → D∓s K

±) = (1.90± 0.12± 0.13 +0.12−0.14)× 10−4 (73)

are obtained, where the first uncertainty is statistical, the second is the experimentalsystematic uncertainty, and the third is from the fs/fd measurement. Both measurementsare significantly more precise than the previous world averages [189].

3.4.4 Measurements of γ using loop-mediated two-body B decays

CP violation in B0(s) decays plays a fundamental role in testing the consistency of the

CKM paradigm in the SM and in probing virtual effects of heavy new particles.With the advent of the B factories, the Gronau-London (GL) [342] isospin analysis of

B → ππ decays has been a precious source of information on the phase of the CKM matrix.Although the method allows a full determination of the weak phase and of the relevanthadronic parameters, it suffers from discrete ambiguities that limit its constraining power.It is however possible to reduce the impact of discrete ambiguities by adding informationon hadronic parameters [343,344]. In particular, as noted in Refs. [345–347], the hadronicparameters entering the B0 → π+π− and the B0

s → K+K− decays are connected by U-spin,so that experimental knowledge of B0

s → K+K− can improve the extraction of the CKMphase with the GL analysis. Indeed, in Ref. [344], the measurement of B(B0

s → K+K−)was used to obtain an upper bound on one of the hadronic parameters.

LHCb has reported preliminary measurements of the time-dependent CP asymmetriesusing decays to CP eigenstates, namely B0 → π+π− and B0

s → K+K− [348], thereby

60

permitting the use of the U-spin strategy proposed by Fleischer (F) [345–347] to extractthe CKM phase from a combined analysis of B0 → π+π− and the B0

s → K+K− decays.However, as shown explicitly below, this strategy alone suffers from a sizeable dependenceon the breaking of U-spin symmetry. In Ref. [349], the authors propose to perform acombined analysis of the GL modes plus B0

s → K+K− to obtain an optimal determinationof the CKM phase within the SM. They show that this combined strategy has a milderdependence on the magnitude of U-spin breaking, allowing for a more solid estimate of thetheory error. The experimental data used for such a determination of γ are summarised inTable 6.

Table 6: Experimental data on B → ππ and B0s → K+K− decays. The correlation column

refers to that between Sf and Cf measurements. Except for the preliminary results in Ref. [348],all other measurements have been averaged by HFAG [43]. The CP asymmetry of B+ → π+π0

has been reported for completeness, although it has not been used in the analysis. New resultson time-dependent CP violation in B0 → π+π− reported by Belle at CKM2012 [350] are notincluded.

Channel B × 106 Sf (%) Cf (%) Corr. Ref.B0 → π+π− 5.11± 0.22 −65± 7 −38± 6 −0.08 [351–356]B0 → π+π− – −56± 17± 3 −11± 21± 3 0.34 [348]B0 → π0π0 1.91± 0.23 – −43± 24 – [351,355,357]B+ → π+π0 5.48± 0.35 – −2.6± 3.9 – [354,355,358]B0s → K+K− 25.4± 3.7 17± 18± 5 −2± 18± 4 0.1 [348,356,359]

The time-dependent asymmetry for a B meson decay to a CP eigenstate f can bewritten, with the same notation as Eqs. (65) and (66),45 as

ACP (t) =Sf sin(∆mt)− Cf cos(∆mt)

cosh(

∆Γ2t)

+Df sinh(

∆Γ2t) , (74)

where Cf and Sf parametrise direct and mixing-induced CP violation respectively, andthe quantity Df is constrained by the consistency relation

(Cf )2 + (Sf )

2 + (Df )2 = 1 . (75)

The LHCb preliminary results on direct and mixing-induced CP violation parametersin B0 → π+π− and B0

s → K+K− decays [348] are shown in Table 6. The measurementsof Cπ+π− and Sπ+π− are compatible with those from the B factories, whereas CK+K− andSK+K− are measured for the first time and are consistent with zero within the currentuncertainties.

Beyond the SM, NP can affect both the B0(s)–B

0(s) amplitudes and the b→ d(s) penguin

amplitudes. Taking the phase of the mixing amplitudes from other measurements, for

45 In the LHCb preliminary results on B0 → π+π− and B0s → K+K− decays [348] a different notation

has been used: Adirf ≡ −Cf , Amix

f ≡ Sf , A∆Γf ≡ −Df .

61

example from b → ccs decays, one can obtain a constraint on NP in b → s (or b → d)penguins. Alternatively, assuming no NP in the penguin amplitudes, one can obtaina constraint on NP in mixing. The analysis discussed here is based on a simplifiedframework [349], using as input values sin 2β = 0.679± 0.024 [43] and 2βs = (0± 5) [138]obtained from b→ ccs decays. The optimal strategy will be to include the combined GLand Fleischer analysis in a global fit of the CKM matrix plus possible NP contributions.

The GL and Fleischer analyses were formulated with different parametrisations of thedecay amplitudes. In order to use the constraints in a global fit one can write46

A(B0 → π+π−) = C(eiγ − deiθ) , A(B0 → π+π−) = C(e−iγ − deiθ) ,A(B0 → π0π0) = C√

2(TeiθT eiγ + deiθ) , A(B0 → π0π0) = C√

2(TeiθT e−iγ + deiθ) ,

A(B+ → π+π0) = A(B0→π+π−)√2

+A(B0 → π0π0) , A(B− → π−π0) = A(B0→π+π−)√2

+A(B0 → π0π0) ,

A(B0s → K+K−) = C ′ λ

1−λ2/2 (eiγ + 1−λ2

λ2 d′eiθ′) , A(B0

s → K+K−) = C ′ λ1−λ2/2 (e−iγ + 1−λ2

λ2 d′eiθ′) ,

(76)

where the magnitude of VubV∗ud has been reabsorbed in C, and the magnitude of

VcbV∗cd/(VubV

∗ud) has been reabsorbed in d. In the exact U-spin limit, one has C = C ′,

d = d′ and θ = θ′. Isospin breaking in B → ππ has been neglected, since its impact onthe extraction of the weak phase is at the level of 1 [360–363]. The physical observablesentering the analysis are

B(B → f) = F (B)|A(B → f)|2 + |A(B → f)|2

2, (77)

Cf =|A(B → f)|2 − |A(B → f)|2|A(B → f)|2 + |A(B → f)|2 , Sf =

2Im(e−iφM (B)A(B→f)

A(B→f)

)1 +

∣∣∣A(B→f)A(B→f)

∣∣∣2 ,

where φM (B0) = 2β, φM (B0s ) = −2βs in the SM, and F (B0) = 1, F (B+) = τB+/τB0 = 1.08,

F (B0s ) = τB0

s/τB0(m2

B0/m2B0s)√

(M2B0s− 4M2

K+)/(M2B0 − 4M2

π+) = 0.9112.

In the GL approach, one extracts the probability density function (PDF) for the angleα = π − β − γ of the UT from the measurements of B(B → ππ), Sπ+π− , Cπ+π− andCπ0π0 . Using the unitarity of the CKM matrix, it is possible to write the B → ππ decayamplitudes and observables in terms of α instead of γ and β. However, for the purposeof connecting B → ππ to B0

s → KK it is more convenient to use the parametrisation inEq. (76). In this way, α (or, equivalently, γ), is determined up to discrete ambiguities,that correspond however to different values of the hadronic parameters. As discussed indetail in Ref. [344], the shape of the PDF obtained in a Bayesian analysis depends onthe allowed range for the hadronic parameters. For example, using the data in Table 6,solving for C and choosing flat a priori distributions for d ∈ [0, 2], θ ∈ [−π, π], T ∈ [0, 1.5]and θT ∈ [−π, π] the PDF for γ in Fig. 20 is obtained, corresponding to γ = (68± 15)

(γ ∈ [25, 87] at 95 % probability). Using instead the Fleischer method, one can obtain aPDF for γ given a range for the U-spin breaking effects. In this method it was originally

46 Note that the use here of the symbol C to denote a colour-suppressed amplitude is not related to itsuse to denote direct CP violation parameters in time-dependent analyses.

62

suggested to parametrise the U-spin breaking in C ′/C using the result one would obtainin factorisation, namely

rfact =

∣∣∣∣C ′C∣∣∣∣fact

= 1.46± 0.15 , (78)

where the error obtained using light-cone QCD sum rule calculations [364] has beensymmetrised. However, this can only serve as a reference value, since there are non-factorisable contributions to C and C ′ that could affect this estimate. In this analysis, thenon-factorisable U-spin breaking is parametrised as follows

C ′ = rfactrCC , Re(d′eiθ′) = rrRe(deiθ) , Im(d′eiθ

′) = riIm(deiθ) , (79)

with rC , rr and ri uniformly distributed in the range [1− κ, 1 + κ].In Fig. 20 the PDF for γ obtained with the Fleischer method for two different values

of the U-spin breaking parameter κ = 0.1, 0.5 is shown. The method is very precise forsmall amounts of U-spin breaking (κ = 0.1), but becomes clearly worse for κ = 0.5. Thus,a determination of γ from the Fleischer method alone is subject to uncertainty on the sizeof U-spin breaking.

]°[γ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0

0.01

0.02

]°[γ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0

0.02

0.04

]°[γ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0

0.005

0.01

0.015

Figure 20: From left to right: PDF for γ obtained using the GL method as described in the text;PDF for γ obtained using the Fleischer method for κ = 0.1, 0.5 [349]. Here and in the following,dark (light) areas correspond to 68 % (95 %) probability regions.

The result of the combined GL+F analysis is given in Fig. 21, where the PDF for γfor κ = 0.1 and 0.5 is shown. The result of the combined analysis is much more stableagainst the allowed amount of U-spin breaking. In Fig. 21 the 68 % probability region forγ obtained using the combined method as a function of κ is also shown, and compared tothe GL result. The combined method shows a considerable gain in precision even for verylarge values of κ.

NP could affect the determination of γ in the combined method by giving (electroweak)penguin contributions a new CP -violating phase. If one assumes that the isospin analysisof the GL channels is still valid, barring order-of-magnitude enhancements of electroweakpenguins in B → ππ, and if one assumes for concreteness that NP enters only b → spenguins, in the framework of a global fit, one can simultaneously determine γ and the

63

]°[γ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0

0.02

0.04

0.06

]°[γ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0

0.01

0.02

0.03

0.04

κ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

[°]

γ

50

60

70

80

90

Figure 21: From left to right: PDF for γ obtained using the combined method for κ = 0.1,0.5; 68 % probability region for γ obtained using the combined method (filled area) or the GLmethod (horizontal lines) as a function of κ [349].

NP contribution to b→ s penguins. For the purpose of illustration, the value of γ fromtree-level processes, γtree = (76± 9) is used as input [262],47 allowing inspection of theposterior for γ and for the NP penguin amplitude. Writing

A(B0s → K+K−) = C ′

λ

1− λ2/2

(e+iγ +

1− λ2

λ2

(d′eiθ

′+ e+iφNPd′NPe

iθ′NP

)), (80)

A(B0s → K+K−) = C ′

λ

1− λ2/2

(e−iγ +

1− λ2

λ2

(d′eiθ

′+ e−iφNPd′NPe

iθ′NP

)),

and taking uniformly distributed d′NP ∈ [0, 2] and φNP, θ′NP ∈ [−π, π] the PDFs shown in

Fig. 22 are obtained for κ = 0.5. This yields γ = (74± 7), and a 95 % probability upperbound on d′NP around 1. Clearly, the bound is stronger for large values of φNP.

]°[NP

φ

0 50 100 150

Pro

ba

bili

ty d

en

sity

0.005

0.01

0.015

0.02

NPd’

0 0.5 1 1.5 2

Pro

ba

bili

ty d

en

sity

0

1

2

3

4

]°[NP’θ

­100 0 100

Pro

ba

bili

ty d

en

sity

0.002

0.0025

0.003

0.0035

0.004

Figure 22: From left to right: PDFs for φNP, d′NP and θ′NP obtained using the combined methodwith κ = 0.5 [349].

47 Note that the value of γ quoted here differs from that obtained from the full CKM fit (given inTable 2) due to the different inputs used.

64

Finally, B0s → KK decays can also be used to extract 2βs in the SM. The optimal

choice in this respect is represented by B0s → K(∗)0K(∗)0 (with B0 → K(∗)0K(∗)0 as U-spin

related control channels to constrain subleading contributions), since in this channel thereis no tree contribution proportional to eiγ [299, 300]. However, the combined analysisdescribed above, in the framework of a global SM fit, can serve for the same purpose. Toillustrate this point, the GL+F analysis is performed, taking as input the SM fit resultγ = (69.7 ± 3.1) [262] and not using the measurement of 2βs from b → ccs decays. Inthis way, 2βs = (3 ± 14) is obtained for κ = 0.5. The analysis can also be performedwithout using the measurement of γ, in this case the result is 2βs = (6 ± 14). Withimproved experimental accuracy, this determination could become competitive with thatfrom b→ ccs decays. Once results of time-dependent analyses of the B0

(s) → K(∗)0K(∗)0

channels are available these may also provide useful constraints.48

To conclude, the usual GL analysis to extract α from B0 → ππ can be supplementedwith the inclusion of the B0

s → K+K− modes, in the framework of a global CKM fit. Themethod optimises the constraining power of these decays and allows the derivation ofconstraints on NP contributions to penguin amplitudes or on the B0

s mixing phase andillustrates these capabilities with a simplified analysis, neglecting correlations with otherSM observables.

3.4.5 Studies of CP violation in multibody charmless b hadron decays

Multibody charmless b hadron decays can be used for a variety of studies of CP violation,including searches for NP and determination of the angle γ. Due to the resonant structurein multibody decays, these can offer additional possibilities to search for both the existenceand features of NP. Model-independent analyses [366, 367] can be performed to firstestablish the presence of a CP violation effect, and then to identify the regions of thephase space in which it is most pronounced.49 To further establish whether any observedCP violation can be accommodated within the Standard Model, amplitude analyses canbe used to quantify the effects associated with resonant contributions to the decay. Anumber of methods have been proposed to determine γ from such processes [370–379], ingeneral requiring input not only from charged B decays, but also from B0 and B0

s decays(to states such as K0

Sh+h′− and π0h+h′−).50 The potential for LHCb to study multibody

charmless Λ0b decays adds further possibilities for novel studies of CP violation effects.

48 The proposal of Ref. [300] has been recently critically reexamined in Ref. [365]. The present analysisshows no particular enhancement of the contribution proportional to eiγ in B0

s → K+K−, in agreementwith the expectation that B0

s → K(∗)0K(∗)0 should be penguin-dominated to a very good accuracy.49 LHCb has presented preliminary results from model-independent searches for CP violation in B± →

π+π−K± and B± → K+K−K± at ICHEP 2012 [368], and in B± → π+π−π± and B± → K+K−π± atCKM 2012 [369].

50 LHCb has presented preliminary branching fraction measurements of B0(s) → K0

Sh+h′− decays at

ICHEP 2012 [380].

65

3.4.6 Prospects of future LHCb measurements

As discussed above, the angle γ can be determined from both tree-dominated and loop-dominated processes. Comparisons of the values obtained provide tests of NP, and soprecision measurements from both methods are needed. Among the tree-dominatedprocesses, in addition to the modes discussed above, any channel that involves theinterference of b → cus and b → ucs transitions is potentially sensitive to γ. Many ofthese modes can be analysed in the upgraded phase of LHCb, including

1. B+ → DK+π−π+ where, similarly to the B → DK mode, the neutral D can bereconstructed either in the two-body (ADS and GLW-like measurement) or multibody(GGSZ-like measurement) final state. The observation of the CF mode in LHCbdata [381] indicates a yield only twice lower than that for the B → DK mode, whichmakes it competitive for the measurement of γ.51 However, two unknown factorsaffect the expected γ sensitivity. First, since this is a multibody decay, the overlapbetween the interfering amplitudes is in general less than 100 %; this is accountedfor by a coherence factor between zero and unity which enters the interference termin Eqs. (54), (55), (59), (60) as an unknown parameter. Second, the value of rBcan be different from that in B → DK and is as yet unmeasured, although it isexpected [310] that it can be larger in this decay than in B → DK.

2. B0 → DK+π−. Although the rate of these decays is smaller that that of B+ → DK+,both interfering amplitudes are colour-suppressed, therefore the expected value ofrB is larger, rB ' 0.3. As a result, the sensitivity to γ should be similar to thatin the B → DK modes.52 Depending on the content of B0 → D0K+π− andB0 → D0K+π− amplitudes, the optimal strategy may involve Dalitz plot analysis ofthe B0 decay [382, 383]. In this case, control of amplitude model uncertainty willbecome essential for a precision measurement; it can be eliminated by studying thedecays B0 → DK+π− with D → K0

Sπ+π− [384].

3. B0s → Dφ. This mode is not self-tagging, but sensitivity to γ can be obtained

from untagged time-integrated measurements using several different neutral D decaymodes [385,386]. The first evidence for the three-body decay B0

s → D0K+K− hasjust been reported by LHCb [387], and investigation of its resonant structure is inprogress.

4. B+c → DD+

s . B+c production in pp collisions is significantly suppressed, however,

in this mode the magnitude of CP violation is expected to be O(100 %): the twointerfering amplitudes are of the same magnitude because the b→ ucs amplitude iscolour allowed, while the b→ cus amplitude is colour suppressed [388–391].

5. Λ0b → DΛ and Λ0

b → DpK−. Measurement of γ from analysis of the Λ0b → DΛ

decay mode was proposed in Ref. [392]. This method allows one to measure γ in a

51 Preliminary results from a GLW-type analysis of this channel was presented at ICHEP 2012 [333].52 Preliminary results from a GLW-type analysis of B0 → DK∗0 were presented at ICHEP 2012 [332].

66

model-independent way by comparing the S- and P -wave amplitudes. However, thismode is problematic to reconstruct at LHCb because of the poorly defined Λ0

b vertex(both particles from its decay are long-lived) and low efficiency of Λ reconstruction.Alternatively, one can consider a similar measurement with the decay Λ0

b → DpK−.A preliminary observation of this mode in early LHCb data has been reported [393].

Table 7 shows the expected sensitivity to γ from tree level decays in the upgradescenario. The LHCb upgrade is the only proposed experiment which will be able to reachsub-degree precision on γ.

Table 7: Estimated precision of γ measurements with 50 fb−1 for various charmed B decaymodes.

Decay mode γ sensitivityB → DK with D → hh′, D → Kπππ 1.3

B → DK with D → K0Sππ 1.9

B → DK with D → 4π 1.7

B0 → DKπ with D → hh′, D → K0Sππ 1.5

B → DKππ with D → hh′ ∼ 3

Time-dependent B0s → D∓s K

± 2.0

Combined ∼ 0.9

Measurement of γ and 2βs by means of the CP -violating observables from loop-mediateddecaysB0 → π+π− andB0

s → K+K− was discussed in Sec. 3.4.4. Extrapolating the currentsensitivity on C and S to the upgrade scenario, when 50 fb−1 of integrated luminosity willbe collected, LHCb will be able to reach a statistical sensitivity σstat(C) ≈ σstat(S) ' 0.008in both B0 → π+π− and B0

s → K+K−. This corresponds to a precision on γ of 1.4, andon 2βs of 0.01 rad, assuming perfect U-spin symmetry.

67

4 Mixing and CP violation in the charm sector

4.1 Introduction

The study of D mesons offers a unique opportunity to access up-type quarks in flavour-changing neutral current (FCNC) processes. It probes scenarios where up-type quarks playa special role, such as supersymmetric models with alignment [394,395]. It offers comple-mentary constraints on possible NP contributions to those arising from the measurementsof FCNC processes of down-type quarks (B or K mesons).

The neutral D system is the latest and last system of neutral mesons where mixingbetween particles and anti-particles has been established. The mixing rate is consistentwith, but at the upper end of, SM expectations [396] and constrains many NP models [397].More precise D0–D0 mixing measurements will provide even stronger constraints. However,the focus has been shifting to CP violation observables, which provide cleaner tests of theSM [398–400]. First evidence for direct CP violation in the charm sector has been reportedby the LHCb collaboration in the study of the difference of the time-integrated asymmetriesof D0 → K+K− and D0 → π+π− decay rates through the parameter ∆ACP [18]. Noevidence of indirect CP violation has yet been found. As discussed in detail below, theseresults on CP violation in the charm sector appear marginally compatible with the SMbut contributions from NP are not excluded.

The mass eigenstates of neutral D mesons, |D1,2〉, with masses m1,2 and widths Γ1,2

can be written as linear combinations of the flavour eigenstates |D1,2〉 = p|D0〉 ± q|D0〉,with complex coefficients p and q which satisfy |p|2 + |q|2 = 1. The average mass andwidth are defined as m ≡ (m1 +m2)/2 and Γ ≡ (Γ1 + Γ2)/2. The D mixing parameters aredefined using the mass and width difference as xD ≡ (m2−m1)/Γ and yD ≡ (Γ2− Γ1)/2Γ.The phase convention of p and q is chosen such that CP |D0〉 = −|D0〉. First evidence formixing of neutral D0 mesons was discovered in 2007 by Belle and BaBar [401,402] and isnow well established [43]: the no-mixing hypothesis is excluded at more than 10σ for theworld average (xD = 0.63 +0.19

−0.20 %, yD = 0.75± 0.12 %).53

It is convenient to group hadronic charm decays into three categories. The CF decays,such as D0 → K−π+, are mediated by tree amplitudes, and therefore no direct CP violationeffects are expected. The same is true for DCS decays, such as D0 → K+π−, even thoughthese are much more rare. The SCS decays, on the other hand, can also have contributionsfrom penguin amplitudes, and therefore direct CP violation is possible, even though thepenguin contributions are expected to be small. Within this classification, it should benoted that some decays to final states containing K0

S mesons, e.g. D0 → K0Sρ

0, have bothCF and DCS contributions which can interfere [404]. Within the SM, however, direct CPviolation effects are still expected to be negligible in these decays.

LHCb is ideally placed to carry out a wide physics programme in the charm sector,thanks to the high production rate of open charm: with a cross-section of 6.10±0.93 mb [3,4],one tenth of LHC interactions produce charm hadrons. Its ring-imaging Cherenkovdetectors provide excellent separation between pions, kaons and protons in the momentum

53 At HCP 2012, LHCb presented the first observation of charm mixing from a single measurement [403].

68

range between 2 and 100 GeV/c, and additional detectors also provide clean identificationof muons and electrons. This allows high purity samples to be obtained both for hadronicand muonic decays. The large boost of the D hadrons produced at LHCb is beneficial fortime-dependent studies. LHCb has the potential to improve the precision on all the keyobservables in the charm sector in the next years.

In the remainder of this section the key observables in the charm sector are described,and the current status and near term prospects of the measurements at LHCb are reviewed.A discussion of the implications of the first LHCb charm physics results follows, motivatingimproved measurements and studies of additional channels. The potential of the LHCbupgrade to make the precise measurements needed to challenge the theory is then described.

4.1.1 Key observables

Currently the most precise individual measurements of mixing parameters are those of therelative effective lifetime difference between D0 and D0 decays to CP eigenstates (Γ andˆΓ) and flavour specific final states (Γ), yCP , which is defined as

yCP =Γ + ˆΓ

2Γ− 1 ≈ ηCP

[(1− 1

8A2m

)yD cosφ− 1

2(Am)xD sinφ

], (81)

where terms below O(10−4) have been ignored [405], ηCP is the CP eigenvalue of the

final state, φ is the CP -violating relative phase between q/p and Af/Af where(—)

Af are thedecay amplitudes, and Am represents a CP violation contribution from mixing (|q/p|±2 ≈1± Am).54 In the limit of CP conservation yCP is equal to the mixing parameter yD. Theresulting world average value for yCP is 0.87± 0.16 % [406]55 and is consistent with thevalue of yD within the current accuracy.

The CP -violating observable AΓ quantifies the difference in decay rates of D0 and D0

to a CP eigenstate and is defined as

AΓ =Γ− ˆΓ

Γ + ˆΓ≈ ηCP

[1

2(Am + Ad)yD cosφ− xD sinφ

], (82)

where terms below O(10−4) have again been ignored [405] and both mixing and directCP contributions are assumed to be small. The parameter Ad describes the contributionfrom direct CP violation (|Af/Af |±2 ≈ 1 ± Ad). The current world average of AΓ is0.02± 0.16 % [43], consistent with the hypothesis of no CP violation. Due to the smallnessof xD and yD, AΓ provides essentially the same information as a full time-dependent CPviolation analysis of D0 → K+K− decays.

An alternative way to search for CP violation in charm mixing is with a time-dependentDalitz plot analysis of D0 and D0 decays to K0

Sπ+π− or K0

SK+K−. Such analyses have

54 Am can be determined from asymmetries in semileptonic charm decays, with the assumption ofvanishing direct CP violation.

55 New results presented by Belle at ICHEP 2012 [407] are not included in this average.

69

been carried out at the B factories [408, 409]. Also in these cases no CP violation wasobserved.

In time-integrated analyses the measured rate asymmetry is

ACP ≡Γ(D0 → f)− Γ(D0 → f)

Γ(D0 → f) + Γ(D0 → f)≈ adir

CP − AΓ〈t〉τ, (83)

where the direct CP asymmetry contribution is defined as

adirCP ≡

|Af |2 − |Af |2|Af |2 + |Af |2

≈ −1

2Ad (84)

and 〈t〉 denotes the average decay time of the observed candidates.A powerful way to reduce experimental systematic uncertainties is to measure the

difference in time-integrated asymmetries in related final states. For the two-body finalstates K+K− and π+π−, this difference is given by

∆ACP ≡ ACP (K+K−)−ACP (π+π−)

≈ ∆adirCP

(1 + yD cosφ

〈t〉τ

)+(aindCP + adir

CPyD cosφ) ∆〈t〉

τ(85)

where the CP -violating phase φ is assumed to be universal [410], ∆a ≡ a(K+K−)−a(π+π−),a ≡ (a(K+K−) + a(π+π−))/2 and the indirect CP asymmetry parameter is defined asaindCP = −(Am/2)yD cosφ + xD sinφ. The ratio ∆〈t〉/τ is equal to zero for the lifetime-

unbiased B factory measurements [411, 412] and is 0.098 ± 0.003 for LHCb [18] and0.25± 0.04 for CDF [413], therefore ∆ACP is largely a measure of direct CP violation.

The current most accurate measurements of ∆ACP are from the LHCb and CDFcollaborations and are (−0.82 ± 0.21 ± 0.11) % [18] and (−0.62 ± 0.21 ± 0.10) % [414],respectively.56 These results show first evidence of CP violation in the charm sector: theworld average is consistent with no CP violation at only 0.006 % C.L. [43].

4.1.2 Status and near-term future of LHCb measurements

LHCb has a broad programme of charm physics, including searches for rare charm decays(see Sec. 2), spectroscopy and measurements of production cross-sections and asymmetries(see Sec. 5). In this section only studies of mixing and CP violation are discussed. Forreviews of the formalism, the reader is referred to Refs. [405,416,417] and the referencestherein, and for an overview of NP implications to Ref. [410].

Mixing and indirect CP violation occur only in neutral mesons. These are probed in anumber of different decay modes, predominantly—but not exclusively—time-dependentratio measurements. In most cases, the same analysis yields measurements of both mixingand CP violation parameters, so these are considered together. By contrast, direct CP

56 At ICHEP 2012, Belle also presented new results on ∆ACP [415], that are consistent with, but lessprecise than, those from LHCb and CDF.

70

violation may occur in decays of both neutral and charged hadrons, and the primarysensitivity to it comes from time-integrated measurements—though it may affect certaintime-dependent asymmetries as well, as discussed in Section 4.7.1.

Several classes of mixing and indirect CP violation measurements are possible at LHCb,particularly:

• Measurements of the ratios of the effective D0 lifetimes in decays to quasi-flavour-specific states (e.g. D0 → K−π+) and CP eigenstates fCP (e.g. D0 → K−K+).These yield yCP . Comparing the lifetime of D0 → fCP and D0 → fCP yields the CPviolation parameter AΓ.

• Measurements of the time-dependence of the ratio of wrong-sign to right-sign hadronicdecays (e.g. D0 → K+π− vs. D0 → K−π+). The ratio depends on y′Dt and(x′2D + y′2D)t2 (see, e.g., Ref. [416]), where

x′D = xD cos δ + yD sin δ ,

y′D = yD cos δ − xD sin δ ,

and δ is the mode-dependent strong phase between the CF and DCS amplitudes.Note that (x′2D + y′2D) = x2

D + y2D ≡ rM . The mixing parameters can be measured

independently for D0 and D0 to constrain indirect CP violation, and the overallasymmetry in wrong-sign decay rates for D0 and D0 gives the direct CP violationparameter Ad.

• Time-dependent Dalitz plot fits to self-conjugate final states (e.g. D0 → K0Sπ−π+).

These combine features of the two methods above, along with simultaneous extractionof the strong phases relative to CP eigenstate final states. Consequently they yieldmeasurements of xD and yD directly. Likewise, the indirect CP violation parameters|q/p| and φ may be extracted, along with the asymmetry in phase and magnitude ofeach contributing amplitude (in a model-dependent analysis).

• Measurements of the ratio of time-integrated rates of wrong-sign to right-signsemileptonic decays (e.g. D0 → D0 → K+l−νl vs. D0 → K−l+νl). These yield rMand Am.

Within LHCb, analyses are planned or in progress for each of these methods. A measure-ment of yCP and AΓ from the 2010 data sample has been published [19]. In addition, apreliminary result on the time-integrated wrong-sign rate in D0 → Kπ from the 2010sample is available [418].57 A summary of what can be achieved with the 2010–2012prompt charm samples is given in Table 8. Note that the observables are generally relatedto several physics parameters, such that the combined constraints are much more powerfulthan individual measurements. After analysing 2.5 fb−1 of data, the mixing parameters xDand yD are expected to be determined at the level of O(10−4), and AΓ to be measured with

57 Results of charm mixing parameters in wrong-sign D0 → K+π− decays have been presented at HCP2012 [403].

71

a similar uncertainty. This will represent a significant improvement in precision comparedto the current world averages, which have uncertainties σxD = 0.19 %, σyD = 0.12 %, andσAΓ

= 0.23 %.

Table 8: Projected statistical uncertainties with 1.0 and 2.5 fb−1 of LHCb data. Yields areextrapolated based on samples used in analyses of 2011 data; sensitivities are projected fromthese yields assuming 1/

√N scaling based on reported yields by LHCb, and using published

input from BaBar, Belle, and CDF. The projected CP -violation sensitivities may vary dependingon the true values of the mixing parameters.

Sample Observable Sensitivity (1.0 fb−1) Sensitivity (2.5 fb−1)Tagged KK yCP 5× 10−4 4× 10−4

Tagged ππ yCP 10× 10−4 7× 10−4

Tagged KK AΓ 5× 10−4 4× 10−4

Tagged ππ AΓ 10× 10−4 7× 10−4

Tagged WS/RS Kπ x′2D 10× 10−5 5× 10−5

Tagged WS/RS Kπ y′D 20× 10−4 10× 10−4

Tagged K0Sππ xD 5× 10−3 3× 10−3

Tagged K0Sππ yD 3× 10−3 2× 10−3

Tagged K0Sππ |q/p| 0.5 0.3

Tagged K0Sππ φ 25 15

For direct CP violation, control of systematic uncertainties associated with productionand efficiency asymmetries is essential. To date, two techniques have been used to mitigatethese effects:

• Measurement of differences in asymmetry between two related final states, such thatsystematic effects largely cancel—for example, ACP (D0 → K−K+) − ACP (D0 →π−π+) [18]. This is simplest with two-body or quasi-two-body decays. This isdiscussed in more detail in Sec. 4.1.3.

• Searching for asymmetries in the distributions of multi-body decays, such thatdifferences in overall normalisation can be neglected and effects related to lab-framekinematics are largely washed out — for example, in the Dalitz plot distribution ofD+ → K−K+π+ [419].

In the longer term, the goal is to extract the CP asymmetries for D0 → K+K−

and D0 → π+π− separately, along with those for other decay modes. To achieve this,it will be necessary to determine the production and detector efficiencies from data.Progress has been made in this area, notably in the D+

s production asymmetry mea-surement [420], which involves determination of the pion reconstruction efficiency fromD∗+ → D0π+, D0 → K−π−π+π+ decays in which one of the D0 daughter pions is not used

72

in the reconstruction.58 The detector asymmetries need to be determined as functions ofthe relevant variables, and similarly, the production asymmetries can vary as functionsof transverse momentum and pseudorapidity. Understanding these systematic effectswith the level of precision and granularity needed for CP asymmetry measurements isdifficult and it cannot be assumed that these challenges will be solved in a short timescale. Moreover, production asymmetries can be determined only with the assumption ofvanishing CP asymmetry in a particular (usually CF) control mode. Therefore ultimatelythe resulting measurements of CP asymmetries for individual decay modes are essentially∆ACP measurements relative to CF decays.

A summary of analyses that are in progress or planned with the 2011–2012 data isgiven below:

D0 → K−K+, π−π+: Updates to the 0.6 fb−1 ∆ACP analysis [18] are in progress, usingboth prompt charm and charm from semileptonic B decays (see Sec. 4.1.3).

D+(s) → K0

Sh+, φh+: A ∆ACP -style analysis is possible by comparing asymmetries in a CF

control mode (e.g. D+ → K0Sπ

+) and the associated SCS mode (e.g. D+ → φπ+),taking advantage of the inherent symmetry of the K0

S → π−π+ and φ → K−K+

decays.59 The different kinematic distributions of the tracks (requiring binning orreweighting) and the CP asymmetry in the K0

S decay need to be taken into account.

D+ → π+π−π+, K+K−π+: A search for CP violation in D+ → K+K−π+ with the model-independent (so-called “Miranda”) technique [366] was published with the 2010 datasample [419], comprising 0.04 fb−1. With such small data samples, detector effectsare negligible. However, from studies of control modes such as D+

s → K−K+π+ itis found that this is no longer the case with 1.0 fb−1 of data or more, so an updatewill require careful control of systematic effects. The π+π−π+ final state should bemore tractable, since the π± interaction asymmetry does not depend strongly onmomentum.

D0 → π−π+π−π+, K−K+π−π+: Previous publications have focused mainly on T -oddmoments [422], but there is further information in the distribution of final-stateparticles. A Miranda-style binned analysis or a comparable unbinned method [367]can be used.60

Baryonic decays: LHCb will collect large samples of charmed baryons, enabling novelsearches for CP -violation effects [424]. Triggering presents a challenge, but triggerlines for several Λ+

c decay modes of the form Λh+ or ph−h′+ are already incorporated,allowing large samples to be recorded. In addition to the considerations outlinedabove for D meson decays, the large proton-antiproton interaction asymmetry andthe possibility of polarisation in the initial state must be taken into account.

58 The pion reconstruction efficiency asymmetry has also been used in the determination of the D+

production asymmetry [421].59 A small difference in kinematic distributions can occur in φ→ K−K+ due to crossing resonances.60 Preliminary results on the D0 → π−π+π−π+ decay were presented at ICHEP 2012 [423].

73

4.1.3 Experimental aspects of ∆ACP and related measurements

The raw asymmetry measured for D∗+-tagged D0 decays to a final state f is defined as:

Araw(f) ≡ N(D∗+ → D0(f)π+s ) − N(D∗− → D0(f)π−s )

N(D∗+ → D0(f)π+s ) + N(D∗− → D0(f)π−s )

, (86)

where N(X) refers to the number of reconstructed events of decay X after backgroundsubtraction. This raw asymmetry arises from several sources: the D∗+ production asym-metry AP, the asymmetry in selecting the tagging slow pion AD(π+

s ), the asymmetry inselecting the D0 decay into the final state AD(f), and the CP asymmetry in the decayACP (f).

Consider the general case of a measured rate n±, an efficiency (or other correction) ε±,and the corrected rate N±, where the subscript refers to D0 or D0. Then:

N+

N−=n+/ε+

n−/ε−=n+

n−

ε−ε+

. (87)

Defining a generic asymmetry Ax as

Ax ≡x+ − x−x+ + x−

,

gives the identityx+

x−=

1 + Ax1− Ax

.

Then applying this to Eq. (87),

1 + An1− An

=

(1 + AN1− AN

)(1 + Aε1− Aε

). (88)

Applying the Taylor series expansion to Eq. (88), gives(1 + 2An + 2A2

n + ...)

=(1 + 2AN + 2A2

N + ...) (

1 + 2Aε + 2A2ε + ...

),

and thusAn = AN + Aε + (terms of order A2) . (89)

Generalising this to include multiple asymmetries, the formula used in the publishedanalysis [18] is obtained

Araw(f) = ACP (f) +AP + AD(π+s ) +AD(f) , (90)

which is correct up to terms of second order in the asymmetries. In practise, for D0 → h+h−,the asymmetries are AP ∼ 1 %, AD(π+

s ) ∼ 1–2 %, and AD(f) = 0 by construction. Thus,the second-order correction is O(10−4).61 Further, AD(π+

s ) and AP are the same for

61 Note that the LHCb dipole magnet creates regions of parameter space with large AD(π+s ), particularly

at the left and right edges of the acceptance. These regions are excluded with fiducial cuts.

74

f = K+K− and f = π+π− (leaving aside differences in kinematic distribution, consideredbelow) and so many terms cancel in the difference:62

∆ACP = Araw(K+K−)−Araw(π+π−) ≈ ACP (K+K−)−ACP (π+π−) .

At the present level of precision, with a statistical uncertainty of around 0.2 %, thisapproximation is perfectly adequate. However, when more data is accumulated — andcertainly after the upgrade — it will be necessary to change the analysis to take second-order terms into account. This can be done using the ratio formulation of Eq. (87),i.e.

NKK,+NKK,−

=(nKK,+nKK,−

)(ε−ε+

), Nππ,+

Nππ,−=(nππ,+nππ,−

)(ε−ε+

)⇒ NKK,+/NKK,−

Nππ,+/Nππ,+=

nKK,+/nKK,−nππ,+/nππ,−

The nuisance asymmetries AP and AD(π+s ) cancel between the K+K− and π+π− final

states because these are properties of the D∗+ and of the tagging slow pion, respectively,which do not depend on the decay of the D0 meson. However, an artificial correlationbetween these asymmetries and the decay mode can arise if the asymmetry varies asa function of some variable63 (e.g. the momentum of the D∗+) and the reconstructeddistributions in this variable are different for the K+K− and π+π− final states (e.g. dueto detector acceptance of the daughter tracks). In such a scenario, the two modes wouldpopulate regions with different raw asymmetries and so the nuisance asymmetries wouldnot cancel fully. Two techniques have been used to address this:

• the data can be partitioned into smaller kinematic regions such that within eachregion the raw asymmetries are constant and/or the K+K− and π+π− kinematicdistributions are equal;

• the data can be reweighted such that the K+K− and π+π− kinematic distributionsare equalised.

The first approach was used in the published LHCb result, and the second in the CDFresult [413].

There is another way in which the formalism could be broken: through the presenceof peaking backgrounds which (a) fake the signal, (b) occur at different levels for theK+K− and π+π− final states, and (c) have a different raw asymmetry from the signal.The signal extraction procedure used in the published LHCb analysis is a fit to the massdifference from threshold δm ≡ m((h+h−)D0π+

s ))−m(h+h−)−m(π+). This is vulnerableto a class of background in which a real D∗+ decay occurs and the correct slow pion isfound but the D0 decay is partly misreconstructed, e.g. D0 → K−π+π0 misidentified as

62 Note in particular that if ACP (K+K−) = ACP (π+π−) = 0, the approximation becomes exact at allorders.

63 The discussion is framed in terms of kinematic variables, since there are clear mechanisms that couldcause problems there, but the same logic can be applied to magnet polarity, trigger conditions, etc.

75

Table 9: Summary of absolute systematic uncertainties for ∆ACP .

Source UncertaintyFiducial requirement 0.01 %Peaking background asymmetry 0.04 %Fit procedure 0.08 %Multiple candidates 0.06 %Kinematic binning 0.02 %Total 0.11 %

D0 → K−K+. This typically creates a background which peaks in δm but is broadlydistributed in m(h+h−). Only cases which lie within the narrow m(h+h−) signal windowwill survive. This is more common for the K+K− final state than for π+π−: the energyof a missing particle can be made up by misidentifying a pion as a kaon, but apart fromD0 → π−e+νe there is little that can fake the kinematics of D0 → π+π−. In practise, thecharged hadron identification at LHCb suppresses these background greatly, and their rawasymmetries are not expected to be very different from the signal. In the published LHCbanalysis, the impact of these backgrounds on the asymmetry was estimated by measuringtheir size and asymmetry in the h+h− mass sidebands and computing the effect of such abackground on the signal with a toy Monte Carlo study. The alternative approach wouldbe to use a full 2D fit to m(h+h−) and δm, which would distinguish this class of peakingbackground from the signal by its m(h+h−) distribution.

The three issues discussed above — terms entering at second order in the asymmetries,non-cancellation due to kinematic correlations, and peaking backgrounds — are particularto this analysis and will require some changes to the procedure as larger data samplesbecome available. In addition, there are more generic systematic uncertainties associatedwith the fit procedure and with the handling of events with more than one candidate.These are summarised in Table 9.

4.2 Theory status of mixing and indirect CP violation

4.2.1 Theoretical predictions for ∆ΓD, ∆mD and indirect CP violation in theStandard Model

As discussed in Sec. 4.1, mixing of charmed mesons provides outstanding opportunities tosearch for physics beyond the SM. New flavour-violating interactions at some high-energyscale may, together with the SM interactions, mix the flavour eigenstates giving mixingparameters that differ from their SM expectations. It is known experimentally that D0–D0

mixing proceeds extremely slowly, which in the SM is usually attributed to the absence ofsuper-heavy quarks.

76

Both SM and NP contributions to mass and width differences can be summarised as

xD =1

2MDΓDRe

[2〈D0|H |∆C|=2 |D0〉+ 〈D0| i

∫d4xT

H|∆C|=1w (x)H|∆C|=1

w (0)|D0〉

],

yD =1

2MDΓDIm 〈D0| i

∫d4xT

H|∆C|=1w (x)H|∆C|=1

w (0)|D0〉.

(91)

These formulae serve as the initial point of calculations of the mass and lifetime differences.They include contributions from local (at charm mass scale) ∆C = 2 interactions generatedby the b-quark [425–429] or NP particles and from SM-dominated time-ordered productsof two ∆C = 1 interaction Hamiltonians (see, however, Ref. [430]).

A simple examination of Eq. (91) reveals that the local ∆C = 2 interactions only affectxD, thus one can conclude that it is more likely that xD receives large NP contributions.Hence, it was believed that an experimental observation of xD yD would unambiguouslyreveal NP contributions to charm mixing. This simple signal for NP was found to notbe realised in nature, but it is interesting that the reverse relation, xD < yD with yDexpected to be determined by the SM processes, might nevertheless significantly affect thesensitivity to NP of experimental analyses of D mixing [431]. Also, it is important to pointout that, contrary to the calculations of the SM contribution to mixing, the contributionsof NP models can be calculated relatively unambiguously [397,432,433].

The calculation of the SM contribution to the mixing amplitudes is rather sophisticated.In the SM xD and yD are generated only at second order in flavour SU(3)f breaking,

xD , yD ∼ sin2 θC × [SU(3)f breaking]2 , (92)

where θC is the Cabibbo angle. Therefore, predicting the SM values of xD and yD dependscrucially on estimating the size of SU(3)f breaking [396,434].

There are currently two approaches, neither of which give very reliable results becausemc is in some sense intermediate between heavy and light. The “inclusive” approach isbased on the OPE. In the mc ΛQCD limit, where ΛQCD is a scale characteristic of thestrong interactions, ∆mD and ∆ΓD can be expanded in terms of matrix elements of localoperators [426–429]. Such calculations typically yield xD, yD < 10−3. The use of the OPErelies on local quark-hadron duality (see, for example, Ref. [435]), and on ΛQCD/Ereleased

(with Ereleased ∼ mc) being small enough to allow a truncation of the series. Moreover, acareful reorganisation of the OPE series is needed, as terms with smaller powers of ms arenumerically more important despite being more suppressed by powers of 1/mc [426–429].The numerically dominant contribution is composed of over twenty unknown matrixelements of dimension-12 operators, which are very hard to estimate. As a possibleimprovement of this approach, it would be important to perform lattice calculations ofthose matrix elements, as well as make perturbative QCD (pQCD) corrections to Wilsoncoefficients of those operators.

The “exclusive” approach sums over intermediate hadronic states, which may bemodelled or fit to experimental data [436–441]. Since there are cancellations between

77

states within a given SU(3)f multiplet, one needs to know the contribution of each statewith high precision. However, the D meson is not light enough that its decays aredominated by a few final states. In the absence of sufficiently precise data on many decayrates and on strong phases, one is forced to use some assumptions. While most studiesfind xD, yD < 10−3, Refs. [436–441] obtain xD and yD at the 10−2 level by arguing thatSU(3)f violation is of order unity. Particular care should be taken if experimental dataare used to estimate the mixing parameters, as the large cancellations expected in thecalculation make the final result sensitive to uncertainties in the experimental inputs.It was shown that phase space effects alone provide enough SU(3)f violation to inducexD, yD ∼ 10−2 [434]. Large effects in yD appear for decays close to threshold, where ananalytic expansion in SU(3)f violation is no longer possible; a dispersion relation canthen be used to show that xD would receive contributions of similar order of magnitude.The dispersion calculation suffers from uncertainties associated with unknown (off-shell)q2-dependences of non-leptonic transition amplitudes and thus cannot be regarded asa precision calculation, although it provides a realistic estimate of xD. As a possibleimprovement of this approach, an estimate of SU(3)f breaking in matrix elements shouldbe performed. In addition, a calculation with Vub 6= 0 should also be done, which isimportant to understand the size of CP violation in charm mixing.

Based on the above discussion, it can be seen that it is difficult to find a clear indicationof physics beyond the SM in D0–D0 mixing measurements alone. However, an observationof large CP violation in charm mixing would be a robust signal of NP.

CP violation in D decays and mixing can be searched for by a variety of methods.Most of the techniques that are sensitive to CP violation make use of the decay asymmetryACP (f) [410,417]. For instance, time-dependent decay widths for D → Kπ are sensitiveto CP violation in mixing. In particular, a combined analysis of D → Kπ and D → KKcan yield interesting constraints on CP -violating parameters yCP and AΓ, as discussed inSec. 4.1.1.

With the D0–D0 transition amplitudes defined as follows:

〈D0|H|D0〉 = M12 −i

2Γ12 , 〈D0|H|D0〉 = M∗

12 −i

2Γ∗12 , (93)

then in the limit where direct CP violation is neglected, one can measure [410,417] fourquantities, xD, yD, Am, and φ, which are described by three physical variables,64

x12 =2|M12|

Γ, y12 =

|Γ12|Γ

, φ12 = arg(M12/Γ12) . (94)

This implies that there is a model-independent relation among experimental quantities [417,442],

xDyD

= −1

2

Amtanφ

. (95)

64 Among various possible phase definitions, only φ12, the relative phase between M12 and Γ12, isconvention-independent and so has physical consequences.

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4.2.2 New physics in indirect CP violation

Indirect CP violation in charm mixing and decays is a unique probe for NP, since withinthe SM the relevant processes are described by the physics of the first two generations toan excellent approximation. Hence, observation of CP violation in D0–D0 mixing at a levelhigher than O(10−3) (which is the SM contribution) would constitute an unambiguoussignal of NP.

The commonly used theoretical parameters x12 and φ12 defined in Eq. (94) can beexpressed in terms of xD, yD and |q/p| as:

x212 = x2

D

(1 + |q/p|2)2

4|q/p|2 + y2D

(1− |q/p|2)2

4|q/p|2 ,

sin2 φ12 =(x2

D + y2D)2(1− |q/p|4)2

16x2Dy

2D|q/p|4 + (x2

D + y2D)2(1− |q/p|4)2

.

(96)

The latest fit65 yields the following ranges [43]

xD ∈ [0.24, 0.99] % , yD ∈ [0.51, 0.98] % , |q/p| ∈ [0.59, 1.26] , (97)

all at 95 % C.L. The fit also provides 95 % C.L. ranges also for the theoretical parametersfrom Eq. (94):

x12 ∈ [0.25, 0.99] % , y12 ∈ [0.51, 0.98] % , φ12 ∈ [−8.4, 24.6] . (98)

It should be noted that the experimental precision on the CP violation parameters is morethan two orders of magnitude away from their SM predictions.

It is reasonable to assume that there are no accidental strong cancellations betweenthe SM and the NP contributions to M12 . Useful bounds can thus be obtained by takingthe NP contribution to saturate the upper limits in Eq. (98). The resulting constraintsare presented in the xNP

12 /x12 − φNP12 plane in Fig. 23. One can also translate the data into

model-independent bounds on four-quark operators, as performed e.g. in Refs. [432,433].The generic NP analysis can also be applied to models with MFV, where new contri-

butions to FCNCs originate only from the Yukawa matrices Yu,d . The relevant basis isthen the up mass basis, where Yu is diagonal, so that flavour violation comes from powersof YdY

†d . The leading contribution is to the operator (uαLγµc

αL)2 (α is a colour index), and

it is given in terms of its Wilson coefficient C1 by

C1 ∝ [y2s(V

∗csVus) + (1 + rGMFV)× y2

b (V∗cbVub)]

2. (99)

Here rGMFV parameterises the effect of resummation of higher powers of the Yukawamatrices when these are important, namely in general MFV (GMFV) models [443].

The contribution to x12 in the linear MFV case (rGMFV = 0) is orders of magnitudebelow the current experimental sensitivity, assuming O(1) proportionality coefficient inEq. (99). Yet in the context of GMFV with two Higgs doublets and large tan β, such that

65Not including results presented at ICHEP 2012 or later.

79

Figure 23: Allowed region (shaded) in the xNP12 /x12− sinφNP

12 plane. The red line corresponds toa GMFV prediction (see text for details) with rGMFV ∈ [−3, 3].

yb ∼ 1, observable signals can be obtained, as shown in Fig. 23 for rGMFV in the range[−3, 3]. Note that strictly speaking rGMFV (and thus the resulting signal) is not bounded,but higher absolute values than those considered here are much less likely in realisticmodels. Indeed in the current example rGMFV & 2 is excluded, as shown in the figure.

The available data on D0–D0 mixing can also be used to constrain the parameterspace of specific theories, such as SUSY and warped extra dimensions (WED) [444]. Thishas been done e.g. in Refs. [432,433] or Refs. [445,446] where the interplay between theconstraints from the K and D systems is presented. Here the influence of improving thecurrent bounds is demonstrated.

Figure 24: Bound on the squark mass degeneracy δ12Q , defined in Eq. (100), as a function of

the experimental constraint on CP violation in D0–D0 mixing, parametrised by sinφexp12 . The

alignment angle from the down sector is λ5C (left panel) or λ3

C (right panel). The solid blue linein each panel is for mQ = mg = 1 TeV and the dashed red line is for mQ = mg = 1.5 TeV.

Within a SUSY framework, one can focus on the first two generations of the left-handed

80

squark mass-squared matrix, m2Q , as the source of flavour violation. As an additional

assumption, the framework can be aligned with the down sector, where the constraintsare generically stronger. As in realistic alignment models (see e.g. Refs. [394, 447]), theoff-diagonal element of m2

Q in the down mass basis (which induces s↔ d FCNCs) is takento be small but not zero, with comparable real and imaginary parts. For concreteness,values of either λ5

C or λ3C (with λC being the Cabibbo angle) are examined, where in both

cases the dominant bounds still arise from D0–D0 mixing and not from the K system [446].The constrained parameter is the squark mass degeneracy, defined by

δ12Q ≡

mQ2−mQ1

mQ2+mQ1

. (100)

In order to analyze the effect of improving the experimental constraints on indirect CPviolation in charm (assuming that no such violation is actually observed), for simplicitythe bound on x12 is kept fixed as in Eq. (98), while that on φ12 is varied. This is shown inFig. 24 for the two alignment angles mentioned above and for two points in the SUSYparameter space mQ = mg = 1 and 1.5 TeV, where mQ is the average squark mass andmg is the gluino mass. The right edge of each of the four lines in the plots marks thecurrent situation, where the dominant constraint is from ∆mD . It is evident that after acertain level of improvement, the bound from CP violation becomes the important one,and this happens more quickly for a weaker alignment model (λ3

C) than for λ5C alignment.

The reason is the larger phase in the former case.To conclude, the experimental search for indirect CP violation in charm is one of the

most promising channels for discovering NP or obtaining strong constraints. This is notnegated by the large hadronic uncertainties in the D system, because of the very smallSM short distance contribution to CP violation in D0–D0 mixing.

4.3 The status of calculations of ∆ACP in the Standard Model

As discussed above, the LHCb collaboration has measured a surprisingly large time-integrated CP asymmetry difference [18],

∆ACP ≡ ACP (D0→ K−K+)−ACP (D0→ π−π+) = (−0.82± 0.21± 0.11) %, (101)

which has recently been supported by a result from the CDF collaboration [414].66 Inclusionof the BaBar and Belle measurements of the individual K−K+ and π−π+ time-integratedCP asymmetries [411,412] and the BaBar, Belle, and LHCb measurements of the indirectCP asymmetry AΓ [19, 402, 448] yields the world average for the direct CP asymmetrydifference [43]

∆adirCP ≡ adir

CP (D0→ K−K+)− adirCP (D0→ π−π+) = (−0.67± 0.16) %. (102)

66 New results presented at ICHEP 2012, including a new result from Belle on ∆ACP [415], are notincluded in the averages discussed here.

81

The naive penguin-to-tree amplitude ratio is O([VcbVub/VcsVus]αS/π) ∼ 10−4, yielding∆adir

CP < 0.1 %. This has led to extensive speculation in the literature that the measurementof ∆adir

CP is a signal for NP. This is a particularly exciting possibility, given that reasonableNP models can be constructed in which all related flavour-changing neutral current (FCNC)constraints, e.g., from D0–D0 mixing, are satisfied. A summary of NP interpretations isgiven in Sec. 4.4.1. First, a discussion of ∆adir

CP in the SM is given.The naive expectation for the SM penguin-to-tree ratio is based on estimates of

the “short-distance” penguins with b-quarks in the loops. In fact, there is consensusthat a SM explanation for ∆adir

CP would have to proceed via dynamical enhancement ofthe long-distance “penguin contraction” contributions to the penguin amplitudes, i.e.,penguins with s and d quarks inside the “loops”. Research addressing the direct CPasymmetry in the SM has largely fallen into one of two categories: (i) flavour SU(3)f

or U-spin fits to the D decay rates, to check that an enhanced penguin amplitude canbe accommodated [449–455] (this, by itself, would not mean that ∆adir

CP is due to SMdynamics); (ii) rough estimates of the magnitudes of certain contributions to the long-distance penguin contractions [453,456–458], to check if, in fact, it is reasonable that SMdynamics could yield the enhanced penguin amplitudes returned by the SU(3)f or U-spinfits.

The results obtained using the flavour symmetry decompositions can be summarisedas follows. An SU(3)f analysis of the D→ PP decay amplitudes that incorporates CPviolation effects was first carried out about 20 years ago [437,449,459]. Already in thisstudy the possibility of large direct CP asymmetries was anticipated, e.g., as large asthe percent level assuming that the penguins receive a large enhancement akin to the∆I = 1/2 rule in kaon decays. An updated analysis, working to first order in SU(3)f

breaking, has been presented [450], making use of branching ratio measurements for theD→ Kπ, ππ and D0→ K−K+, K0η decay modes. The authors concluded that ∆adir

CP canbe easily reconciled with the measured branching ratios. This was also the conclusion of astudy based on a diagrammatic SU(3)f amplitude decomposition [451], which considered alarger set of D→ PP decay modes. Again, this is only a statement about the possibilityof accommodating the required amplitudes in the flavour decomposition, not about theirrealisation via long distance QCD dynamics. Both studies observe that a SM explanationof ∆adir

CP could be combined with precise measurements of the individual asymmetriesadirCP (D0→ K−K+) and adir

CP (D0→ π−π+) to obtain predictions for adirCP (D0→ π0π0). The

conclusion, based on current data, is that percent level asymmetries for the latter couldbe realised. Ref. [451] also discusses implications for adir

CP (D+→ K+K0).Studies employing U-spin symmetry [452,453] necessarily focus on amplitude fits to

the smaller set of decay modes D0→ K−π+, π−K+, π−π+, K−K+, as the D0 is a U-spinsinglet, while the four final states and the operators mediating these decays in the SM∆C = 1 effective Hamiltonian each consist of a U-spin triplet and a singlet. Working to

82

first order in U-spin breaking, the four decay amplitudes can be written as

A(D0→ K+π−) = VcsV∗ud

(T − 1

2δT), A(D0→ π+K−) = VcdV

∗us

(T + 1

2δT)

A(D0→ π+π−, K+K−) = ∓12

(VcsV

∗us − VcdV ∗ud

)(T ± δS)− VcbV ∗ub

(P ∓ 1

2δP),

(103)

where the U-spin triplet “tree” amplitude T and the singlet “penguin” amplitude P ariseat 0th order in U-spin breaking, and δT , δS and δP are the first order U-spin breakingcorrections, which transform in turn as a triplet, singlet, and singlet under U-spin. Thesinglet amplitude δS accounts for the large rate difference Γ(D0 → K−K+)/Γ(D0 →π−π+) = 2.8 (after accounting for phase space). A ratio δS/T ∼ 0.5 is found in Refs. [452,453], and in the SU(3)f study of Ref. [450] which effectively contains the above U-spindecomposition. Realisation of Eq. (102) requires |P/T | ∼ 3, for O(1) strong phases andadirCP (D0→ K−K+) ∼ −adir

CP (D0→ π−π+), where the last relation becomes an equality inthe U-spin limit. This amounts to an order of magnitude enhancement of the penguinamplitude beyond the naive estimate.

The CP -averaged experimental “sum-rule” relation,

Σsum-rule =|A(D0→ K−K+) / VcsVus|+ |A(D0→ π−π+) / VcdVud||A(D0→ π−K+) / VcdVus|+ |A(D0→ K−π+) / VcsVud|

− 1 = (4.0± 1.6) % ,

(104)together with the observation of small (≈ 15 %) U-spin breaking in A(D0→ π−K+) vs.A(D0→ K−π+), can be interpreted as suggesting that U-spin is a good symmetry inthese decays [453]. Other authors take the large difference between Γ(D0 → K−K+)and Γ(D0→ π−π+) or δS/T ∼ 0.5 as evidence for large U-spin breaking in SCS decays.In Ref. [453], rather than interpreting the amount of U-spin breaking implied by δS bycomparing it to T , as in other works, δS is compared to P . It is observed that whereas∆adir

CP implies that P must be dominated by the sum of the long distance s- and d- quarkpenguin contractions, nominal U-spin breaking would imply that δS must be dominated bytheir difference. A consistent picture emerges in which direct CP asymmetries of order a fewper mille are not surprising given the size of Γ(D0→ K−K+)/Γ(D0→ π−π+). However,as always in the flavour decomposition approach, accommodation need not translate torealisation by QCD dynamics. One consequence of this picture is that adir

CP (D0 → K0SK

0S)

could be as large as ∼ 0.6 % for O(1) strong phases.Finally, the estimates for the long-distance penguin contractions [458,460] are reviewed

to see if the required enhancement can be realised. Ref. [460] employs the one-gluonexchange approximation. The essential ingredients are: (i) 1/Nc counting; (ii) D branchingratio data which shows that certain formally 1/mc power-suppressed amplitudes are ofsame order as their leading (1/mc)

0 counterparts; (iii) translation of this breakdown ofthe 1/mc expansion to the penguin contraction amplitudes, in the approximation of ahard gluon exchange; (iv) use of a partonic quantity as a rough estimator of the hadronicinteractions, e.g., final state interactions, underlying the penguin contraction “loops”. Thisresults in a rough estimate for ∆adir

CP at the few per mille level. The authors of Ref. [460]thus conclude that a SM explanation is plausible, given that their estimate suffers from

83

large uncertainties. In Ref. [458] the penguin contractions are estimated using isospin andinformation from ππ scattering and unitarity. A fit of the CP -conserving contributionsfrom the CP -averaged branching ratios provides information on the isospin amplitudesand the underlying renormalisation group invariant amplitude contributions. Allowing forthree coupled channel contributions to ππ,KK scattering the authors conclude that theobserved asymmetries are marginally compatible with the SM.

To summarise, flavour SU(3) or U-spin fits to the D→ PP data can accommodate theenhanced penguin amplitudes required to reproduce ∆adir

CP . There is consensus that inthis case adir

CP (D0→ π0π0) could lie at the percent level, while adirCP (D+→ K+K0) could

certainly lie at the few per mille level. Under the assumption of nominal SU(3)f breakingin D→ PP decays, the enhancement of the long-distance penguin contractions requiredto realise ∆adir

CP is not surprising, given the large difference between the D0→ K−K+

and D0→ π−π+ decay rates. It would of course be of interest to extend the above CPviolation studies to the SCS D→ V P and D→ V V decay modes. Finally, among theworks which have attempted to estimate directly the magnitudes of the long distancepenguin contractions, there is no consensus on whether they can be enhanced by an orderof magnitude beyond the naive penguin amplitude estimates, as would be required in orderto explain ∆adir

CP . Ultimately this question will have to be answered directly via latticestudies.

In the following section, future prospects are discussed. In subsequent sections, severaldefinitive CP -violating signals for NP in SCS D decays will be discussed.

4.4 ∆ACP in the light of physics beyond the Standard Model

4.4.1 General considerations

Potential NP contributions to ∆ACP can be parametrised in terms of an effective Hamilto-nian valid below the W and top mass scales

Heff−NP|∆C|=1 =

GF√2

∑i

CNP(′)i Q(′)

i , (105)

where the relevant operators Q(′)i are defined in Ref. [461]. Introducing the ratios RNP,i

K,π as

the relevant NP hadronic amplitudes (matrix elements 〈K−K+, π−π+|Q(′)i |D〉) normalised

to the leading CP -conserving SM contributions and writing CNPi = v2

EW/Λ2NP, the relevant

NP scale ΛNP is given by [461]

(10 TeV)2

Λ2NP

=(0.61± 0.17)− 0.12 Im(∆RSM)

Im(∆RNP,i), (106)

where ∆Ri = RiK + Ri

π and RSMK,π parametrise the unknown hadronic amplitude ratios

associated with the CP -violating SM contributions. Comparing this estimate to the muchhigher effective scales probed by CP -violating observables in D mixing and also in thekaon sector, one first needs to verify if such large contributions can still be allowed by

84

other flavour constraints. Within the effective theory approach, this can be estimated viaso-called “weak mixing” of the effective operators. In particular, time-ordered correlatorsof Heff−NP

|∆C|=1 with the SM effective weak Hamiltonian can, at the one weak-loop order, induce

important contributions to CP violation in both D meson mixing and kaon decays (ε′/ε).On the other hand, analogous correlators quadratic in Heff−NP

|∆C|=1 turn out to be either chirallysuppressed and thus negligible, or yield quadratically divergent contributions, which arethus highly sensitive to particular UV completions of the effective theory [461].

4.4.2 Universality of CP violation in flavour-changing decay processes

The strongest bounds can be derived for a particular class of operators, which transformnon-trivially only under the SU(3)Q subgroup of the global SM quark flavour symmetryGF = SU(3)Q×SU(3)U×SU(3)D, respected by the SM gauge interactions. In particular onecan prove that their CP -violating contributions to ∆F = 1 processes (here F genericallyrepresents a flavour quantum number) have to be approximately universal between the upand down sectors [446]. Within the SM one can identify two unique sources of SU(3)Qbreaking given by Au ≡ (YuY

†u )/tr and Ad ≡ (YdY

†d )/tr, where Yq are the Yukawa matrices and

/tr denotes the traceless part. Then in the two generation limit, one can construct a singlesource of CP violation, given by J ≡ i[Au,Ad] [462,463]. The crucial observation is thatJ is invariant under SO(2) rotations between the Au and Ad eigenbases. Introducing now

SU(2)Q breaking NP effective operator contributions of the form QL =[(XL)ij Qiγ

µQj

]Lµ,

where Qi stands for the left-handed quark doublets, i and j are generation indices, XL isa traceless Hermitian flavour matrix and Lµ denotes a flavour singlet current. It followsthat the CP -violating contributions have to be proportional to J and thus invariant underflavour rotations. The universality of CP violation induced by QL can be expressedexplicitly as [446]

Im(XuL)12 = Im(Xd

L)12 ∝ Tr (XL · J) . (107)

The above identity holds to a very good approximation even in the three-generationframework. In the SM, large values of Yb,t induce a SU(3)/SU(2) flavour symmetrybreaking pattern [443] which allows one to decompose XL under the residual SU(2) ina well defined way. Finally, residual SM SU(2)Q breaking is necessarily suppressed bysmall mass ratios mc,s/mt,b, and small CKM mixing angles. The most relevant implicationof Eq. (107) is that it predicts a direct correspondence between SU(3)Q breaking NPcontributions to ∆ACP and ε′/ε [446]. It follows immediately that stringent limits onpossible NP contributions to the latter require SU(3)Q breaking contributions to the formerto be below the per mille level (for ∆RNP,i = O(1)). As a corollary, one can show thatwithin NP scenarios which only break SU(3)Q, existing stringent experimental bounds onnew contributions to CP -violating rare semileptonic kaon decays K0

L→ π0(νν, `+`−) putrobust constraints on CP asymmetries of corresponding rare charm decays D→ π(νν, `+`−).In particular, the SU(3)Q-violating contribution to the CP asymmetry in D → πe+e− hasbeen shown to be less than 2 % [446].

The viability of the remaining 4-quark operators in Heff−NP|∆C|=1 as explanations of the

85

experimental ∆ACP value depends crucially on their flavour and chiral structure (a full listcan be found in Ref. [461]). In particular, operators involving purely right-handed quarksare unconstrained in the effective theory analysis but may be subject to severe constraintsfrom their UV sensitive contributions to D mixing observables. On the other hand, QEDand QCD dipole operators are at present only weakly constrained by nuclear electric dipolemoments (EDMs) and thus present the best candidates to address the ∆ACP puzzle [461].

Finally, note that it was shown that the impact of universality of CP within thealignment framework is to limit the amount of CP violation in D0–D0 mixing to below∼ 20 %, which is interestingly near the current bound. The expected progress in thismeasurement with the LHCb detector is therefore going to start probing this framework.

4.4.3 Explanations of ∆ACP within NP models

Since the announcement of the LHCb result, several prospective explanations of ∆ACPwithin various NP frameworks have appeared. In the following the implications withinsome of the well-motivated NP models are discussed.

In the MSSM, the right size of the QCD dipole operator contributions can be generatedwith non-zero left-right up-type squark mixing contributions (δu12)LR [410, 464,465]. Sucheffects in ∆ACP can be parametrised as [464]

|∆aSUSYCP | ≈ 0.6 %

( |Im(δu12)LR|10−3

)(TeV

m

), (108)

where m denotes a common squark and gluino mass scale. At the same time dangerouscontributions to D mixing observables are chirally suppressed. It turns out howeverthat even the apparently small (δu12)LR value required implies a highly nontrivial flavourstructure of the UV theory; in particular, large trilinear (A) terms and sizeable mixingamong the first two generation squarks (θ12) are required [464].

Im(δu12)LR ≈Im(A)θ12mc

m≈(

Im(A)

3

)(θ12

0.3

)(TeV

m

)0.5× 10−3 . (109)

Similarly, WED models that explain the quark spectrum through flavour anarchy [444,466–468] can naturally give rise to QCD dipole contributions affecting ∆ACP as [469]

|∆aWEDCP | ≈ 0.6 %

(Y5

6

)2(3 TeV

mKK

)2

, (110)

wheremKK is the Kaluza-Klein (KK) scale and Y5 is the five-dimensional Yukawa coupling inappropriate units. Reproducing the experimental value of ∆ACP requires near-maximal 5DYukawa coupling, close to its perturbative bound [470, 471] of 4π/

√NKK ' 7 for NKK = 3

perturbative KK states. In turn, this helps to suppress unrealistic tree-level contributionsto CP violation in D0–D0 mixing [432,433]. This scenario can also be interpreted withinthe framework of partial compositeness in four dimensions, but generic composite models

86

typically require smaller Yukawa couplings to explain ∆ACP and consequently predictsizeable contributions to CP violation in ∆F = 2 processes [472].

On the other hand, in the SM extension with a fourth family of chiral fermions ∆ACPcan be affected by 3× 3 CKM non-unitarity and b′ penguin operators

|∆a4th genCP | ∝ Im

(λb′

λd − λs

). (111)

However, due to the existing stringent constraints on the new CP -violating phases enteringλb′ [426,473], only moderate effects comparable to the SM estimates are allowed [452].

Finally, it is possible to relate ∆ACP to the anomalously large forward-backwardasymmetry in the tt system measured at the Tevatron [474] through a minimal model.Among the single-scalar-mediated mechanisms that can explain the top data, only thet-channel exchange of a colour-singlet weak doublet, with a very special flavour structure, isconsistent with the total and differential tt cross-section, flavour constraints and electroweakprecision measurements [475]. The required flavour structure implies that the scalarunavoidably contributes at tree level to ∆ACP [476]. The relevant electroweak parametersare either directly measured, or fixed by the top-related data, implying that, for a plausiblerange of the hadronic parameters, the scalar-mediated contribution is of the right size.

4.4.4 Shedding light on direct CP violation via D→ V γ decays

The theoretical interpretation of ∆ACP is puzzling: it is above its naive estimate in theSM and it could well be a signal of NP, but it is not large enough to rule out a possibleSM explanation. It is then important to identify possible future experimental tests able todistinguish standard vs. non-standard explanations of ∆ACP . Among the NP explanationsof ∆ACP , the most interesting ones are those based on a new CP -violating phase in the∆C = 1 chromomagnetic operator. A general prediction of this class of models, that couldbe used to test this hypothesis from data, is enhanced direct CP violation (DCPV) inradiative decay modes [477].

1. The first key observation to estimate DCPV asymmetries in radiative decaymodes is the strong link between the ∆C = 1 chromomagnetic operator (Q8 ∼uLσµνT

agsGµνa cR) and the ∆C = 1 electromagnetic-dipole operator (Q7 ∼

uLσµνQueFµνcR). In most explicit new-physics models the short-distance Wilson

coefficients of these two operators (C7,8) are expected to be similar. Moreover, evenassuming that only a non-vanishing C8 is generated at some high scale, the mixingof the two operators from strong interactions implies C7,8 of comparable size at thecharm scale. Thus if ∆ACP is dominated by NP contributions generated by Q8, itcan be inferred that |Im[CNP

7 (mc)]| ≈ |Im[CNP8 (mc)]| = (0.2–0.8)× 10−2.

2. The second important ingredient is the observation that in the Cabibbo-suppressedD→ V γ decays, where V is a light vector meson with uu valence quarks (V = ρ0, ω),Q7 has a sizeable hadronic matrix element. More explicitly, the short-distancecontribution induced by Q7, relative to the total (long-distance) amplitude, is

87

substantially larger with respect to the corresponding relative weight of Q8 inD→ P+P− decays. Estimating the SM long-distance contributions from data, andevaluating the short-distance CP -violating contributions under the hypothesis that∆ACP is dominated by (dipole-type) NP, leads to the following estimate for themaximal direct CP asymmetries in the D→ (ρ, ω)γ modes [477]:

|adirCP (D→ (ρ, ω)γ)|max = 0.04

∣∣∣∣Im[C7(mc)]

0.4× 10−2

∣∣∣∣× [ 10−5

B(D→ (ρ, ω)γ)

]1/2

<∼ 10 % .

(112)The case of the φ resonance, or better the K+K−γ final state with MKK closeto the φ peak, is more involved since the matrix element of Q7 vanishes in thelarge mc limit for a pure ss state. However, a non-negligible CP asymmetry canbe expected also in this case since: 1) the matrix element of Q7 is not expectedto be identically zero because of sizeable O(ΛQCD/mc) corrections; 2) nonresonantcontributions due to (off-shell) ρ and ω exchange can also contribute to the K+K−γfinal state. Taking into account these effects, the following estimates for the maximaldirect CP asymmetries are obtained [477]:

|adirCP (D→ K+K−γ)|max ≈ 2 % , 2mK <

√s < 1.05 GeV ,

|adirCP (D→ K+K−γ)|max ≈ 6 % , 1.05 GeV <

√s < 1.20 GeV .

(113)

In the first bin, close to the φ peak, the leading contribution is due to the φ-exchangeamplitude. The contribution due to the nonresonant amplitudes becomes moresignificant further from the φ peak, where the CP asymmetry can become larger.

3. In order to establish the significance of these results, two important issues have tobe clarified: 1) the size of the CP asymmetries within the SM, 2) the role of thestrong phases.

As far as the SM contribution is concerned, it can first be noticed that short-distancecontributions generated by the operator Q7 are safely negligible. Using the resultin Ref. [478], asymmetries are found to be below the 0.1 % level. The dominantSM contribution is expected from the leading non-leptonic four-quark operators,for which the general arguments discussed in Ref. [461] can be applied. The CPasymmetries can be decomposed as |aSM

CP (f)| ≈ 2ξ Im(RSMf ) ≈ 0.13 % × Im(RSM

f ),where ξ ≡ |VcbVub/VcsVus| ≈ 0.0007 and RSM

f is a ratio of suppressed over leadinghadronic amplitudes, naturally expected to be smaller than one. This decompositionholds both for f = ππ,KK and for f = V γ channels. The SM model explanationsof ∆aCP require RSM

ππ,KK ∼ 3. While this possibility cannot be excluded fromfirst principles, a further enhancement of one order of magnitude in the D→ V γmode is beyond any reasonable explanation in QCD. As a result, an observation of|adirCP (D→ V γ)| >∼ 3 % would be a clear signal of physics beyond the SM, and a clean

indication of new CP -violating dynamics associated to dipole operators.

Having clarified that large values of |adirCP (D→ V γ)| would be a clear footprint of

non-standard dipole operators, it can be asked if potential tight limits on |adirCP (D→ V γ)|

88

could exclude this non-standard framework. Unfortunately, uncertainty on the strongphases does not allow this conclusion to be drawn. Indeed the maximal values for theDCPV asymmetries presented above are obtained in the limit of maximal constructiveinterference of the various strong phases involved. In principle, this problem could beovercome via time-dependent studies of D(D)→ V γ decays or using photon polarisation,accessible via lepton pair conversion in D → V (γ∗ → `+`−); however, these types ofmeasurements are certainly more challenging from the experimental point of view.

4.4.5 Testing for CP -violating new physics in the ∆I = 3/2 amplitudes

It is possible, at least in principle, to distinguish between NP and the SM as the origin of∆ACP . If ∆ACP is due to a chromomagnetic operator, i.e. due to ∆I = 1/2 contributions,one can measure CP violation in radiative D decays, as explained in the previous section.Examples of NP models that can be tested in this way are, e.g., flavour-violating super-symmetric squark-gluino loops that mediate the c→ ug transition [410,464,465]. On theother hand, if ∆ACP is due to ∆I = 3/2 NP one can use isospin symmetry to write sumrules for direct CP asymmetries in D decays [479]. If the sum rules are violated, then NPwould be found. An example of a NP model that can be tested in this way is an additionof a single new scalar field with nontrivial flavour couplings [476].

The basic idea behind the ∆I = 3/2 NP tests [479, 480] is that in the SM the CPviolation in SCS D decays arises from penguin amplitudes which are ∆I = 1/2 transitions.On the other hand, ∆I = 3/2 amplitudes are CP -conserving in the SM. Moreover, thereare no ∆I = 5/2 terms in the SM short-distance effective Hamiltonian, and though suchcontributions can be generated by electromagnetic rescattering (as has been discussed inthe context of B → ππ decays [481,482]) they would also be CP conserving. Observingany CP violation effects in ∆I = 3/2 amplitudes would therefore be a clear signal of NP.

In the derivation of the sum rules it is important to pay attention to the potentiallyimportant effects of isospin breaking. Isospin symmetry is broken at O(10−2), which is alsothe size of the interesting CP asymmetries. There are two qualitatively different sources ofisospin breaking: due to electromagnetic interactions, u and d quark masses, which are allCP -conserving effects, and due to electroweak penguin operators that are a CP -violatingsource of isospin breaking. The CP -conserving isospin breaking is easy to cancel in thesum rules. As long as the CP -conserving amplitudes completely cancel in the sum rules,which is the case in Ref. [479], the isospin breaking will only enter suppressed by the smallCP violation amplitude and is therefore negligible. The electroweak penguin operators, onthe other hand, are suppressed by α/αS ∼ O(10−2) compared to the leading CP -violatingbut isospin conserving penguin contractions of the Q1,2 operators, and can thus also besafely neglected.

Among the SCS decays, the D→ ππ, D→ ρπ, D→ ρρ, D→ KKπ, and D+s → K∗π

modes carry enough information to construct tests of ∆I = 3/2 NP. The sum rulesfor D→ ππ decays have the nice feature that the charged decay D+→ π+π0 is purely∆I = 3/2. In the SM therefore

adirCP (D+→ π+π0) = 0 . (114)

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If this CP asymmetry is measured to be nonzero, it would be a clear signal of ∆I = 3/2NP. However, if it is found experimentally to be very small, it is still possible that this isonly because the strong phase between the SM and NP amplitudes is accidentally small.

This possibility can be checked with more data if time-dependent D(t)→ π−π+ andD(t)→ π0π0 measurements become available,67 or if there is additional information onrelative phases from a charm factory running on the ψ(3770). The strategy amountsto measuring the weak phase of the ∆I = 3/2 amplitude A3 via generalised triangleconstructions that also take isospin breaking into account [479]. If

1√2Aπ−π+ + Aπ0π0 − 1√

2Aπ+π− − Aπ0π0 = 3

(A3 − A3

)(115)

is found to be nonzero, this would mean there is CP -violating NP in the ∆I = 3/2amplitude.

The above results apply also to D→ ρρ decays, but for each polarisation amplitudeseparately. The corrections due to finite ρ width can be controlled experimentally in thesame way as in B→ ρρ decays [484]. As long as the polarisations of the ρ resonancesare measured (or if the longitudinal decay modes dominate, as is the case in B→ ρρdecays), the search for ∆I = 3/2 NP could be easier experimentally in D→ ρρ decayssince there are more charged tracks in the final state. The most promising observablewhere polarisation measurement is not needed is ACP (D+→ ρ+ρ0), which if found nonzero(after the correction for the effect of finite ρ decay widths) would signal ∆I = 3/2 NP.

Another experimentally favourable probe is the isospin analysis of the D0→ π+π−π0

Dalitz plot in terms of D→ ρπ decays [485]. There are two combinations of measuredamplitudes that are proportional to ∆I = 3/2 amplitudes

Aρ+π0 + Aρ0π+ = 3√

2A3 ,

Aρ+π− + 2Aρ0π0 + Aρ−π+ = 6A3 .(116)

A measurement of the second sum can be obtained from the D0→ π+π−π0 Dalitz plot. Ifthe related CP asymmetry

|Aρ+π− + 2Aρ0π0 + Aρ−π+ |2 − |Aρ−π+ + 2Aρ0π0 + Aρ+π−|2 = 36(|A3|2 − |A3|2

), (117)

is found to be nonzero, this would mean that the ∆I = 3/2 NP contribution is nonzero.If it is found to vanish, however, it could be due to the strong phase difference beingvanishingly small.

A definitive answer can be provided by another test that is directly sensitive to theweak phase of A3. This test is possible if the time-dependent D(t)→ π+π−π0 Dalitz plot ismeasured. In this case the relative phases between the D0→ ρπ and D0→ ρπ amplitudescan be obtained (alternatively one could use time integrated entangled decays of ψ(3770)

67 Time-dependent D(t)→ π0π0 measurements could in principle be feasible using photon conver-sions [483].

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at the charm factory). The presence of a weak phase in A3 can then be determined fromthe following sum-rule(

Aρ+π− + Aρ−π+ + 2Aρ0π0

)−(Aρ−π− + Aρ+π− + 2Aρ0π0

)= 6(A3 − A3

). (118)

A non-vanishing result for Eq. (118) would provide a definitive proof for ∆I = 3/2 NP. Asimilar sum rule for the CP asymmetries rather than the amplitudes was given in Eq. (117).In that case the time-integrated Dalitz plot suffices to determine the sum rule inputs.

The sum rules involving D→ K(∗)K(∗)π decays are somewhat more complex becausethere are at least three particles in the final state. Nevertheless, it is possible to constructpurely ∆I = 3/2 matrix elements from appropriate sums of decay amplitudes, and thesecan in principle be determined from amplitude analyses of the multibody final states. It isalso possible to search for CP violation in ∆I = 3/2 amplitudes using D+

s → K∗π decays.The sum √

2A(D+s → π0K∗+) + A(D+

s → π+K∗0) = 3A3 , (119)

is ∆I = 3/2 and can be measured from the common Dalitz plot for D+s → K0

Sπ+π0 decay.

Direct CP violation in this sum, i.e.,

|√

2A(D+s → π0K∗+) + A(D+

s → π+K∗0)|2−|√

2A(D−s → π0K∗−) + A(D−s → π−K∗0)|2 6= 0 ,(120)

would necessarily be due to ∆I = 3/2 NP contributions. Additional information on theabsolute value of |A(D+

s → π+K∗0)| can be obtained from the D+s → π+K+π− three-body

decay. Analogous tests using D+s → ρK∗ decays also exist.

4.5 Potential for lattice computations of direct CP violation andmixing in the D0–D0 system

In searches for NP using charmed mesons, it is obviously crucial to determine accurately thesize of SM contributions. In the next few paragraphs the prospects for such a determinationin the future using the methods of lattice QCD are discussed.

Lattice QCD provides a first-principles method for determining the strong-interactioncontributions to weak decay and mixing processes. It has developed into a precision tool,allowing determinations of the light hadron spectrum, decay constants, and matrix elementssuch as BK and BB with percent-level accuracy. For reviews and collections of recentresults, see Refs. [108,486]. The results provide confirmation that QCD indeed describesthe strong interactions in the non-perturbative regime, as well as providing predictionsthat play an important role in searching for new physics by looking for inconsistencies inunitarity triangle analyses.

Results with high precision are, however, only available for processes involving singlehadrons and a single insertion of a weak operator. For the D0 system, the “high-precision”quantities are thus the matrix elements describing the short-distance parts of D0–D0

mixing and the matrix elements of four-fermion operators arising after integrating out NP.

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The methodology for such calculations is in place (and has been applied successfully tothe K and B meson systems), and results are expected to be forthcoming in the next oneto two years.

More challenging, and of course more interesting, are calculations of the decay ampli-tudes to ππ and KK. For kaon physics, this is the present frontier of lattice calculations.One must deal with two technical challenges: (i) the fact that one necessarily works infinite volume so the states are not asymptotic two-particle states and (ii) the need tocalculate Wick contractions (such as the penguin-type contractions) which involve gluonicintermediate states in some channels. The former challenge has been solved in principleby the work of Luscher [487, 488] and Lellouch and Luscher [489] for the K → ππ case,while advances in lattice algorithms and computational power have allowed the numericalaspects of both challenges to be overcome. There are now well controlled results for theK→ (ππ)I=2 amplitude [490] and preliminary results for the K→ (ππ)I=0 amplitude [491].It is likely that results to ∼ 10 % accuracy for all amplitudes will be available in a fewyears. Note that, once a lattice calculation is feasible, it will be of roughly equal difficultyto obtain results for the CP -conserving and CP -violating parts.

To extend these results to the charm case, one must face a further challenge. This isthat, even when one has fixed the strong-interaction quantum numbers of a final state,say to I = S = 0, the strong interactions necessarily bring in multiple final states whenE = mD. For example, ππ and KK states mix with ηη, 4π, 6π, etc. The finite-volumestates that are used by lattice QCD are inevitably mixtures of all these possibilities, andone must learn how, in principle and in practise, to disentangle these states so as to obtainthe desired matrix element. Recently, in Ref. [492], a first step towards developing acomplete method has been taken, in which the problem has been solved in principle forany number of two-particle channels, assuming that the scattering is dominantly S-wave.This is encouraging, and it may be that this method will allow semi-quantitative results forthe amplitudes of interest to be obtained. Turning this method into practise is expectedto take three to five years due to a number of numerical challenges (in particular the needto calculate several energy levels with good accuracy). It is also expected to be possibleto generalise the methodology to include four particle states; several groups are activelyworking on the theoretical issues. It is unclear at this stage, however, what time scale oneshould assign to this endeavour.

Finally, the possibility of calculating long-distance contributions to D0–D0 mixingusing lattice methods should be considered. Here the challenge is that there are twoinsertions of the weak Hamiltonian, with many allowed states propagating between them.Some progress has been made recently on the corresponding problem for kaons [493,494]but the D0 system is much more challenging. The main problem is that, as for the decayamplitudes, there are many strong-interaction channels with E < mD. Further theoreticalwork is needed to develop a practical method.

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4.6 Interplay of ∆ACP with non-flavour observables

4.6.1 Direct CP violation in charm and hadronic electric dipole moments

Models in which the primary source of flavour violation is linked to the breaking of chiralsymmetry (left-right flavour mixing) are natural candidates to explain direct CP violationin SCS D meson decays, via enhanced ∆C = 1 chromomagnetic operators. Interestingly,the chromomagnetic operator generates contributions to D0–D0 mixing and ε′/ε that arealways suppressed by at least the square of the charm Yukawa couplings, thus naturallyexplaining why they have remained undetected.

On the other hand, the dominant constraints are posed by the neutron and nuclearEDMs, which are expected to be close to their experimental bounds. This result is fairlyrobust because the Feynman diagram contributing to quark EDMs has essentially thesame structure as that contributing to the chromomagnetic operator.

In the following the connection between ∆adirCP and hadronic EDMs in concrete NP

scenarios is discussed, following the analyses of Refs. [464,465].

SupersymmetryThe leading SUSY contribution to ∆adir

CP stems from loops involving up-squarks and gluinosand off-diagonal terms in the squark squared-mass matrix in the left-right up sector, theso-called (δu12)LR mass-insertion. As can be seen from Eqs. (108)–(109) and taking intoaccount the large uncertainties involved in the evaluation of the matrix element, it can beconcluded that a supersymmetric theory with left-right up-squark mixing can potentiallyexplain the LHCb result.

Among the hadronic EDMs, the best constraints come from mercury and neutronEDMs. Their current experimental bounds are |dn| < 2.9× 10−26 e cm (90 % C.L.) and|dHg| < 3.1× 10−29 e cm (95 % C.L.). In the mass-insertion approximation one can find

|dn| ≈ 3× 10−26

( |Im (δu11)LR|10−6

)(TeV

m

)e cm . (121)

and therefore it has to be seen whether a concrete SUSY scenario can naturally accountfor the required level of suppression |Im(δu11)LR| . 10−6.

Generalised trilinear termsWhile scenarios in which flavour violation is restricted to the trilinear terms can beenvisaged, it is natural to generalise the structure of Eq. (109) to all squarks and take

(δqij)LR ∼Aθqijmqj

m, q = u, d , (122)

where θqij are generic mixing angles. This pattern can be obtained when the matrices ofthe up and down trilinear coupling constants follow the same hierarchical pattern as thecorresponding Yukawa matrices but they do not respect exact proportionality.

It is found that θqij can all be of order unity not only in the up, but also in the down

sector, thanks to the smallness of the down-type quark masses entering (δdij)LR. The only

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experimental bounds in tension with this scenario are those on |θu,d11 | coming from theneutron EDM.

Split familiesThe severe suppression of (δu21)eff

RL stemming from the charm mass can be partially avoided ina framework with split families, where the first two generations of squarks are substantiallyheavier than t1,2 and bL, the only squarks required to be close to the electroweak scaleby naturalness arguments. In this case the effective couplings relevant to ∆aSUSY

CP can bedecomposed as follows

(δu12)effRL = (δu13)RR (δu33)RL (δu32)LL , (δu12)eff

LR = (δu13)LL (δu33)RL (δu32)RR . (123)

Notice that this scenario takes advantage of the large (δu33)LR ∼ Amt/m which is assumedto be of order one. The following two options can be considered to explain the LHCbresults:

(δu32)LL = O(λ2), (δu13)RR = O(λ2) → (δu12)effRL = O(λ4) = O(10−3) ,

(δu13)LL = O(λ3), (δu32)RR = O(λ) → (δu12)effLR = O(λ4) = O(10−3) .

(124)

Gluino-squark loops yield an EDM (du) and a chromo-EDM (dcu) for the up quark

proportional to d(c)u ∼ Im [(δu13)LL(δu31)RR] and it turns out that

∣∣∆aSUSYCP

∣∣ ≈ 10−3 ×∣∣∣∣ dn3× 10−26

∣∣∣∣ ∣∣∣∣Im (δu32)RR0.2

∣∣∣∣ ∣∣∣∣ 10−3

Im (δu31)RR

∣∣∣∣ . (125)

In conclusion, the EDM bounds require a strong hierarchical structure in the off-diagonalterms of the RR up-squark mass matrix, as happens in models predicting (δuij)RR ∼(mui/muj)/|Vij|.Supersymmetric flavour modelsIn models where the flavour structure of the soft breaking terms is dictated by anapproximate flavour symmetry, (δuLR)12 is generically flavour-suppressed by (mc |Vus| /m),which is of order a few times 10−4. There is however additional dependence on the ratiobetween flavour-diagonal parameters, A/m, and on unknown coefficients of order one,that can provide enhancement by a small factor. In most such models, the selection rulesthat set the flavour structure of the soft breaking terms relate (δuLR)12 to (δdLR)12 and to(δu,dLR)11, which are bounded from above by, respectively, ε′/ε and EDM constraints. Sinceboth ε′/ε and EDMs suffer from hadronic uncertainties, small enhancements due to theflavour-diagonal supersymmetric parameters cannot be ruled out. It is thus possible toaccommodate ∆ACP ∼ 0.006 in supersymmetric models that are non-minimally flavourviolating, but – barring hadronic enhancements in charm decays – it takes a fortuitousaccident to lift the supersymmetric contribution above the permille level [465].

New-physics scenarios with Z-mediated FCNCEffective FCNC couplings of the Z boson to SM quarks can appear in the SM withnon-sequential generations of quarks, models with an extra U(1) symmetry or models with

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extra vector-like doublets and singlets. The effective FCNC Lagrangian can be written as

LZ−FCNCeff = − g

2 cos θWqiγ

µ[(gZL )ij PL + (gZR)ij PR

]qj Zµ + h.c. , (126)

The chromomagnetic operator is generated at the one-loop level, with leading contributionfrom Z–top exchange diagrams leading to∣∣∆aZ−FCNC

CP

∣∣ ≈ 0.6 %

∣∣∣∣∣Im[(gZL )∗ut(g

ZR)ct

]2× 10−4

∣∣∣∣∣ . (127)

The presence of new CP -violating phases in the couplings (gZL,R)ij are also expected togenerate hadronic EDMs. In particular, one can find

|dn| ≈ 3× 10−26

∣∣∣∣∣Im[(gZL )∗ut(g

ZR)ut

]2× 10−7

∣∣∣∣∣ e cm , (128)

and therefore ∆AZ−FCNCCP = O(10−2) only, provided Im(gZR)ut/Im(gZR)ct . 10−3.

In NP scenarios with Z-mediated FCNCs, the most interesting FCNC processes in thetop sector are t→ cZ and t→ uZ, which arise at the tree level. In particular,

B(t→ cZ) ≈ 0.7× 10−2

∣∣∣∣(gZR)tc10−1

∣∣∣∣2 , (129)

which is within the reach of the LHC for the values of (gZR)tc relevant to ∆AZ−FCNCCP .

New-physics scenarios with scalar-mediated FCNCFinally, it is instructive to analyse a new-physics framework with effective FCNC couplingsto SM quarks of a scalar particle h. The effective Lagrangian reads

Lh−FCNCeff = −qi

[(ghL)ij PL + (ghR)ij PR

]qj h+ h.c. (130)

Also in this case the chromomagnetic operator is generated at the one-loop level, with aleading contribution from h–top exchange diagrams. This leads to∣∣∆aFCNC

CP

∣∣ ≈ 0.6 %

∣∣∣∣∣Im[(ghL)∗ut(g

hR)tc]

2× 10−4

∣∣∣∣∣ . (131)

As in all the other frameworks, the most severe constraints are posed by the hadronicEDMs

|dn| ≈ 3× 10−26

∣∣∣∣∣Im[(ghL)∗ut(g

hR)tu

]2× 10−7

∣∣∣∣∣ e cm . (132)

With scalar-mediated FCNCs, the potentially most interesting signals are the rare topdecays t→ ch or t→ uh, if kinematically allowed. In particular,

B(t→ qh) ≈ 0.4× 10−2

∣∣∣∣(ghR)tq

10−1

∣∣∣∣2 , (133)

which could be within the reach of the LHC.

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4.6.2 Interplay of collider physics and a new physics origin for ∆ACPThe first evidence for direct CP violation in SCS D decays may have interesting implicationsfor NP searches around the TeV scale at the LHC. The NP contribution to ∆adir

CP can befully parametrised by a complete set of ∆C = 1 effective operators at the charm scale. Asshown by the authors of Ref. [461] only a few of these operators can accommodate theLHCb result without conflicting with present bounds from D0–D0 mixing and ε′/ε. Inparticular four-fermion operators of the form Oq = (uRγ

µcR)(qRγµqR) with q = u, d, s arepromising since they do not lead to flavour violation in the down-type quark sector. Thecorresponding Wilson coefficients are defined as 1/Λ2

q. Assuming the SM expectation for∆adir

CP is largely subdominant, the LHCb measurement suggests a scale of Λq ' 15 TeV [461].There is an immediate interplay between charm decay and flavour (and CP ) conserving

observables at much higher energies provided Oq arises from a heavy NP state exchangedin the s-channel. Under this mild assumption Oq factorises as the product of two quarkcurrents and the same NP induces D0–D0 mixing and quark compositeness throughthe (uRγµcR)2 and (qRγµqR)2 operators, respectively. Denoting their respective Wilsoncoefficients by Λuc and Λqq, the relation Λq =

√ΛucΛqq is predicted. The D0–D0 mixing

bound on NP implies Λuc & 1200 TeV [433]. Combining this stringent ∆C = 2 boundwith the ∆C = 1 scale suggested by ∆adir

CP thus generically requires Λqq . 200 GeV, whichis a rather low compositeness scale for the light quark flavours.

Quark compositeness can be probed at the LHC through dijet searches. Actuallyfor the up or the down quark the low scale suggested by ∆adir

CP is already excluded bythe Tevatron [495, 496]. On the other hand dijet searches are less sensitive to contactinteractions involving only the strange quark since the latter, being a sea quark, has asuppressed parton distribution function in the proton. The authors of Ref. [497] showedthat a first estimation at the partonic level of the extra dijet production from a (sRγµsR)2

operator with a scale of Λss ∼ 200 GeV is marginally consistent, given the O(1) uncertaintyof the problem, with the bounds from the ATLAS and CMS experiments [498,499].

One concludes that an Os operator induced by a s-channel exchanged NP can accom-modate the ∆adir

CP measurement without conflicting with ∆C = 2, ε′/ε and dijet searches.Furthermore such a NP scenario makes several generic predictions both for charm andhigh-pT physics: 1) most of the CP asymmetry is predicted to be in the K+ K− channel,2) CP violation in D0–D0 mixing should be observed in the near future, and 3) an excessof dijets at the LHC is expected at a level which should be visible in the 2012 data.

4.7 Future potential of LHCb measurements

4.7.1 Requirements on experimental precision

The ultimate goal of mixing and CP violation measurements in the charm sector is to reachthe precision of the SM predictions (or better). In some cases this requires measurementsin several decay modes in order to distinguish enhanced contributions of higher order SMdiagrams from effects caused by new particles.

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Indirect CP violation is constrained by the observable AΓ (see Eq. (82)). The CP -violating parameters in this observable are multiplied by the mixing parameters xD andyD, respectively. Hence, the relative precision on the CP -violating parameters is limited bythe relative precision of the mixing parameters. Therefore, aiming at a relative precisionbelow 10 % and taking into account the current mixing parameter world averages, thetarget precision would be 2–3× 10−4. Indirect CP violation is expected in the SM at theorder of 10−4, and therefore the direct CP violation parameter contributing to AΓ has tobe measured to a precision of 10−3 in order to distinguish the two types of CP violation inAΓ.

Direct CP violation is not expected to be as large as the current world average of∆ACP in most other decay modes. However, a few large CP violation signatures areexpected in various models, as discussed in the previous sections. Estimations based onflavour-SU(3) and U-spin symmetry lead to expectations of adir

CP (D+→ K+K0) & 0.1% andadirCP (D0 → K0

SK0S ) ∼ 0.6%. Considerations assuming universality of ∆F = 1 transitions

lead to a limit of adirCP (D→ πe+e−) . 2%. Enhanced electromagnetic dipole operators can

lead to adirCP (D→ V γ) of a few %, equivalent to the influence of chromomagnetic dipole

operators on ∆ACP . Additional information can be obtained from time-dependent studiesof D→ V γ decays or from angular analyses of D→ V l+l− decays.

Analyses of ∆I = 3/2 transitions involve asymmetry measurements of several relateddecay modes. Examples are the decays D→ ππ, D→ ρπ, D→ ρρ, D→ KKπ, andD+s → K∗π. The number of final state particles in these decays varies from two to six

(counting the pions from K0S decays) and many of these modes contain neutral pions

in their final state. The precision for modes involving neutral pions or photons will belimited by the ability of the calorimeter to identify these particles in the dense hadronicenvironment. An upgraded calorimeter with smaller Moliere radius would greatly extendthe physics reach in this area.68

In general, a precision of 5×104 or better for asymmetry differences as well as individualasymmetries is needed for measurements of other SCS charm decays. While measurementsof time-integrated raw asymmetries at this level should be well within reach, the challengelies in the control of production and detection asymmetries in order to extract the physicsasymmetries of individual decay modes. This can be achieved by assuming that there isno significant CP violation in CF decay modes.

4.7.2 Prospects of future LHCb measurements

Numbers of events in various channels are projected directly from the numbers reconstructedin the 2011 data set, in most cases. This involves assumptions that the prompt charmcross-section will increase by a factor of 1.8 when doubling the centre-of-mass energyfrom

√s = 7 TeV to

√s = 14 TeV, that the integrated luminosity will increase from 1 fb−1

to 50 fb−1, and that the trigger efficiency for charm will increase by a factor of 2 as thecurrent hardware trigger requirement is effectively removed (or substantially relaxed).Additionally, a factor of 3.5 times greater efficiency in channels with K0

S→ π−π+ daughters

68 Such an upgrade to the calorimeter system is not in the baseline plan for the LHCb upgrade [24, 25].

97

is predicted based on progress made in the trigger software between 2011 and 2012. Thisprimarily results from reconstructing candidates which decay downstream of the vertexdetector. The results of this exercise are summarised in Table 10 for D0 decays and inTable 11 for D+ and D+

s decays.

Table 10: Numbers of D0 and D∗+→ D0π+ signal events observed in the 2011 data in a varietyof channels and those projected for 50 fb−1. These channels can be used for mixing studies,for indirect CP violation studies, and for direct CP violation studies. As discussed in the text,the numbers of events in any one channel can vary from one analysis to another, depending onthe level of cleanliness required. Hence, all numbers should be understood to have an inherentvariation of a factor of 2. To control systematic uncertainties with the very high level of precisionthat will be required by the upgrade, it may be necessary to sacrifice some of the statistics.

Mode 2011 yield 50 fb−1 yield(103 events) (106 events)

Untagged D0→ K−π+ 230 000 40 000D∗+→ D0π+; D0→ K−π+ 40 000 7 000D∗+→ D0π+; D0→ K+π− 130 20D0→ K−K+ 25 000 4 600D0→ π−π+ 6 500 1 200D∗+→ D0π+; D0→ K−K+ 4 300 775D∗+→ D0π+; D0→ π−π+ 1 100 200D∗+→ D0π+; D0→ K0

Sπ−π+ 300 180

D∗+→ D0π+; D0→ K0SK−K+ 45 30

D∗+→ D0π+; D0→ K−π+π−π+ 7 800 1 400D∗+→ D0π+; D0→ K−K+π−π+ 120 20D∗+→ D0π+; D0→ π−π+π−π+ 470 85D∗+→ D0π+; D0→ K−µ+X – 4 000D∗+→ D0π+; D0→ K+µ−X – 0.1

Estimating the physics reach with the projected data sets requires a number ofassumptions. The statistical precision generally improves as 1/

√N . Estimating the

systematic error, and therefore ultimate physics reach, is more of an art. It is often thecase that data can be used to control systematic uncertainties at the level of the statisticalerror, but the extent to which this will be possible cannot be reliably predicted. In somecases controlling systematic uncertainties will require sacrificing some of the statisticsto work with cleaner signals or with signals which populate only parts of the detectorwhere the performance is very well understood. Estimates of sensitivity to CP violation inmixing generally depend on the values of the mixing parameters – the larger the numberof mixed events, the larger the effective statistics contributing to the corresponding CPviolation measurement.

The estimated statistical precisions for parameters of mixing and CP violation inthe D0 system are presented in Table 12. The precision for measuring (x′2D, y

′D) using

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Table 11: Numbers of D+ and D+s signal events observed in the 2011 data in a variety of channels

and those projected for 50 fb−1. These channels can be used for direct CP violation studies. Asdiscussed in the text, the numbers of events in any one channel can vary from one analysis toanother, depending on the level of cleanliness required. To control systematic uncertainties withthe very high level of precision that will be required by the upgrade, it may be necessary tosacrifice some of the statistics.

Mode 2011 yield 50 fb−1 yield(103 events) (106 events)

D+→ K−π+π+ 60 000 11 000D+→ K+π+π− 200 40D+→ K−K+π+ 6 500 1 200D+→ φπ+ 2 800 500D+→ π−π+π+ 3 200 575D+→ K0

Sπ+ 1 500 1 000

D+→ K0SK

+ 525 330D+→ K−K+K+ 60 10

D+s → K−K+π+ 8 900 1 600

D+s → φπ+, (φ→ K−K+) 5 350 1 000

D+s → π−π+π+ 2 000 360

D+s → K−π+π+

D+s → π−K+π+ 555 100

D+s → K−K+K+ 50 10

D+s → K0

SK+ 410 260

D+s → K0

Sπ+ 33 20

the time-dependence of the wrong-sign (WS) to right-sign (RS) Kπ rate comes fromextrapolating the BaBar [401] and Belle [500] sensitivities.69 The precision for measuringrM using the ratio of WS to RS Kµν events assumes the central value to be 2.5× 10−5.The S/B ratio is assumed to be 30 times better than reported by BaBar [501] for theirsimilar Keν analysis. Background can be reduced by a factor of 10 using LHCb’s excellentvertex resolution to remove candidates with decay time less than twice the D0 lifetime – arequirement which only modestly reduces the WS signal as its decay time distributionhas the form dN/dt ∝ t2 e−Γt. In addition, the excellent vertex resolution and thedecay time requirement allow the neutrino momentum, and hence the D∗+ − D0 massdifference to be measured with better resolution than was possible in the e+e− experiments.BaBar demonstrated that using a doubly-tagged sample of semileptonic decay candidatesprovides the same mixing sensitivity as the more traditional singly-tagged sample [502]. Bycombining singly- and doubly-tagged samples, it should be possible to effectively double

69 The LHCb measurements of charm mixing parameters from wrong-sign Kπ decays [403] are consistentwith the estimated sensitivities.

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Table 12: Estimated statistical uncertainties for mixing and CP violation measurements whichcan be made with the projected samples for 50 fb−1 described in Table 10.

Sample Parameter(s) PrecisionWS/RS Kπ (x′2D, y

′D) O[(10−5, 10−4)]

WS/RS Kµν rM O(5× 10−7)WS/RS Kµν |p/q|D O(1%)D∗+→ D0π+; D0→ K−K+, π−π+ ∆ACP 0.015%D∗+→ D0π+; D0→ K−K+ ACP 0.010%D∗+→ D0π+; D0→ π−π+ ACP 0.015%D∗+→ D0π+; D0→ K0

Sπ−π+ (xD, yD) (0.015%, 0.010%)

D∗+→ D0π+; D0→ K−K+, (π−π+) yCP 0.004% (0.008%)D∗+→ D0π+; D0→ K−K+, (π−π+) AΓ 0.004% (0.008%)D∗+→ D0π+; D0→ K−K+π−π+ AT 2.5× 10−4

Table 13: Estimated statistical uncertainties for CP violation measurements which can be madewith the projected D+ samples for 50 fb−1 described in Table 11.

Sample Parameter(s) PrecisionD+→ K0

SK+ phase-space integrated CP violation 10−4

D+→ K−K+π+ phase-space integrated CP violation 5× 10−5

D+→ π−π+π+ phase-space integrated CP violation 8× 10−5

D+→ K−K+π+ CP violation in phases, amplitude model (0.01− 0.10)

D+→ K−K+π+ CP violation in fraction differences, amplitude model (0.01− 0.10)%D+→ π−π+π+ CP violation in phases, amplitude model (0.01− 0.10)

D+→ π−π+π+ CP violation in fraction differences, amplitude model (0.01− 0.10)%D+→ K−K+π+ CP violation in phases, model-independent (0.01− 0.10)

D+→ K−K+π+ CP violation in fraction differences, model-independent (0.01− 0.10)%D+→ π−π+π+ CP violation in phases, model-independent (0.01− 0.10)

D+→ π−π+π+ CP violation in fraction differences, model-independent (0.01− 0.10)%

the statistics.The projected sensitivities for the two-body direct CP violation measurements are

relatively solid: the 2011 ∆ACP measurements provide benchmark samples with fullanalysis cuts including fiducial cuts necessary to control systematic uncertainties formeasuring ∆ACP . The systematic errors for the separate ACP (K−K+) and ACP (π−π+)measurements will be more challenging and may require sacrificing statistical precision.The projections for measuring yCP and AΓ using K−K+ and π−π+ should also be robustas the same samples will be used for these analyses as for the ACP measurements.

The projected precision for measuring (xD, yD) from D0→ K0Sπ−π+ comes from scaling

the Belle [408] and BaBar [503] sensitivities. The statistical precisions could be even better

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as LHCb’s prompt sample will be enhanced at higher decay times where the mixing effectsare larger. By contrast, D0 mesons from semileptonic B decays should be unbiased in thisvariable, providing a useful sample at lower decay times.

The estimated statistical precisions for DCPV in D+ measurements are presented inTable 13. The estimates for the phase-space integrated CP violation rates are scaled by1/√N and are then increased by a factor of two to allow for using tighter cuts to control

systematic uncertainties. The estimates for measuring CP violation in the magnitudesand phases of quasi-two-body amplitudes contributing to three-body final states comefrom scaling the BaBar sensitivities for time-integrated CP violation in D0→ π−π+π0

and D0→ K−K+π0 by 1/√N . The angular moments of the cosine of the helicity angle

of the D decay products reflect the spin and mass structure of the intermediate resonantand nonresonant amplitudes with no explicit model dependence. The difference betweenthe angular moment distributions observed in D0 and D0 decays provides sensitivity toCP violation in the magnitudes (or fractions) and phases of amplitudes about equal tothat of model-dependent fits. The angular moment differences are robust, in the sensethat they are model-independent, but they are less specific compared to the results frommodel-dependent analyses: they indicate only the spins and mass ranges where particleand antiparticle amplitudes differ, but do not identify a specific CP -violating intermediatestate or how much it varies. The sensitivity to CP violation in any contributing amplitudedepends on how much it contributes to the three-body decay, and also on the otheramplitudes with which it interferes. For this reason, ranges of sensitivity are indicatedrather than single values. No sensitivities for CP violation measurements in three-body D+

s

decay channels are estimated explicitly. They can be estimated roughly by extrapolatingfrom the numbers for D+ decays by scaling by 1/

√N . These estimates should be degraded

slightly as the lifetime of the D+ is about twice that of the D+s meson, making it easier to

select clean D+ samples.

4.8 Conclusion

LHCb has proven its capability of performing high-precision charm physics measurements.The experiment is ideally suited for CP violation searches and for measurements ofdecay-time-dependent processes such as mixing.

Finding evidence for a non-zero value of ∆ACP has raised the question of whetheror not this may be interpreted as the first hint of physics beyond the SM at the LHC.Within the SM the central value can only be explained by significantly enhanced penguinamplitudes. This enhancement is conceivable when estimating flavour SU(3) or U-spinbreaking effects from fits to D → PP data. However, attempts at estimating the longdistance penguin contractions directly have not yielded conclusive results to explain theenhancement.

Lattice QCD has the potential of assessing the penguin enhancement directly. However,several challenges arise which make these calculations impossible at the moment. Followingpromising results on K → ππ decays, additional challenges arise in the charm sector asππ and KK states mix with ηη, 4π, 6π and other states. Possible methods have been

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proposed and results may be expected in three to five years time.General considerations on the possibility of interpreting ∆ACP in models beyond the

SM have led to the conclusion that an enhanced chromomagnetic dipole operator is required.These operators can be accommodated in minimal supersymmetric models with non-zeroleft-right up-type squark mixing contributions or, similarly, in warped extra dimensionalmodels. Tests of these interpretations beyond the SM are needed. One promising groupof channels are radiative charm decays where the link between the chromomagnetic andthe electromagnetic dipole operator leads to predictions of enhanced CP asymmetries ofseveral percent. These can be measured to sufficient precision at the LHCb upgrade.

Another complementary test is to search for contributions beyond the SM in ∆I = 3/2amplitudes. This class of amplitudes leads to several isospin relations which can betested in a range of decay modes, e.g. D → ππ, D → ρπ, D → KK, etc. Several ofthese measurements, such as the Dalitz plot analysis of the decay D0 → π+π−π0, can beperformed at LHCb.

Beyond charm physics, the chromomagnetic dipole operators would affect the neutronand nuclear EDMs, which are expected to be close to the current experimental bound.Similarly, rare FCNC top decays are expected to be enhanced, if kinematically allowed.Furthermore, quark compositeness can be related to the ∆ACP measurement and tested indijet searches. Current results favour the NP contribution to be located in the D0 → K−K+

decay as the strange quark compositeness scale is less well constrained. Measurements ofthe individual asymmetries of sufficient precision will be possible at the LHCb upgrade.

The charm mixing parameters have not yet been precisely calculated in the SM. Aninclusive approach based on an operator product expansion relies on the expansion scalebeing small enough to allow convergence and furthermore involves the calculation of alarge number of unknown matrix elements. An exclusive approach sums over intermediatehadronic states and requires very precise branching ratio determinations of these finalstates which are currently not available. Contrary to the SM, contributions beyond theSM can be calculated reliably. With the SM contribution to indirect CP violation being< 10−4, the LHCb upgrade is ideally suited to cover the parameter space available forenhanced asymmetries beyond the SM. Measurements in several complementary modeswill permit the extraction of the underlying theory parameters with high precision.

The LHCb upgrade will allow to constrain CP asymmetries and mixing observablesto a level of precision which, in most of the key modes, cannot be matched by any otherexperiment foreseen on a similar timescale. This level of precision should permit us notonly to discover CP violation in charm decays but also to unambiguously understand itsorigin.

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5 The LHCb upgrade as a general purpose detector

in the forward region

The previous sections have focussed on flavour physics observables that are sensitiveto physics beyond the SM. However, LHCb has excellent potential in a range of otherimportant topics. As discussed in this section, the detector upgrade will further enhancethe capability of LHCb in these areas, so that it can be considered as a general purposedetector in the forward region. LHCb may also be able to make a unique contribution tothe field of heavy ion physics, by studying soft QCD and heavy flavour production in pAcollisions. The first pA run of the LHC will clarify soon the potential of LHCb in this field.

5.1 Quarkonia and multi-parton scattering

The mechanism of heavy quarkonium production is a long-standing problem in QCD. Aneffective field theory, non-relativistic QCD (NRQCD), provides the foundation for much ofthe current theoretical work. According to NRQCD, the production of heavy quarkoniumfactorizes into two steps: a heavy quark-antiquark pair is first created perturbativelyat short distances and subsequently evolves non-perturbatively into quarkonium at longdistances. The NRQCD calculations depend on the colour-singlet (CS) and colour-octet(CO) matrix elements, which account for the probability of a heavy quark-antiquark pairin a particular colour state to evolve into heavy quarkonium. The CS model [504, 505],which provides a leading-order (LO) description of quarkonia production, was first usedto describe experimental data. However, it underestimates the observed cross-section forsingle J/ψ production at high pT at the Tevatron [506]. To resolve this discrepancy theCO mechanism was introduced [507]. The corresponding matrix elements were determinedfrom the large-pT data as the CO cross-section falls more slowly than that for CS. Morerecent higher-order calculations [508–511] close the gap between the CS predictions andthe experimental data [512] reducing the need for large CO contributions.

Traditionally, quarkonia production studies at hadron colliders have focussed on thestudy of J/ψ , ψ(2S) and Υ (nS) decays to dimuon or dielectron pairs [512]. The LHCbprogramme so far has followed this pattern with measurements of many cross-sectionsalready published [513–517]. As an example of the quality of the data, Fig. 25 shows theΥ mass distribution. By the time of the upgrade in 2018, data samples correspondingto several fb−1 will have been collected at

√s = 7, 8 and 14 TeV and the results will be

dominated by systematic uncertainties. Therefore, new probes of quarkonia productionwill be pursued. Two possibilities are detailed here: multiple quarkonia production andquarkonia production via hadronic decay modes. These studies will profit from the higherintegrated luminosity and improved trigger. These modes provide clear signals in thedetector and will be relatively uneffected by the increased pile-up.

As the cross-sections for charmonium production at the LHC are large [513–515,517],the question of multiple production of these states in a single proton-proton collisionnaturally arises. Studies of double hidden charm and hidden and associated open charmproduction have been proposed as probes of the quarkonium production mechanism [518].

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)2) (MeV/c+µ­

µM(

9000 10000 11000

)2

Ca

nd

ida

tes

/ (

25

Me

V/c

1000

2000

3000

4000

5000

6000

= 7 TeVsLHCb

Figure 25: Invariant mass distribution of selected Υ candidates from 25 pb−1 of data collected in2010 [516]. The Υ (1S), Υ (2S) and Υ (3S) states are clearly resolved. The results of a maximumlikelihood fit are superimposed.

In proton-proton collisions contributions from other mechanisms, such as double partonscattering (DPS) [519–521] or the intrinsic charm content of the proton [522], are possible.First studies of both processes have been carried out with the current LHCb data; moredetails can be found in Refs. [296,523].

LO colour singlet calculations for the gg → J/ψJ/ψ process in perturbative QCDexist and give results consistent with the data [524–526]. In the LHCb fiducial region(2 < yJ/ψ < 4.5, pT

J/ψ < 10 GeV/c, where yJ/ψ and pTJ/ψ represent the rapidity and transverse

momentum of the J/ψ , respectively) these calculations predict the J/ψJ/ψ production cross-section to be 4.1± 1.2 nb [526] in agreement with the measured value of 5.1± 1.0 nb [523].Similar calculations exist for the case of double Υ (1S) production. For the case of J/ψ plusΥ (1S) production no leading order diagrams contribute and hence the rate is expected tobe suppressed in Single Parton Scattering (SPS). This leads to an “unnatural” ordering of

the cross-section values: σJ/ψJ/ψgg > σ

Υ (1S)Υ (1S)gg > σ

Υ (1S)J/ψgg .

The DPS contributions to all these double onia production modes can be estimated,neglecting partonic correlations in the proton, as the product of the measured cross-sectionsof the sub-processes involved divided by an effective cross-section [519–521, 527]. Thevalue of the latter is determined from multi-jet events at the Tevatron to be σDPS

eff =14.5 ± 1.7 +1.7

−2.3 mb [528]. At√s = 7 TeV the contribution from this source to the total

cross-section is similar in size to the LO contribution from SPS. For DPS the ordering ofthe cross-section values is: σ

J/ψJ/ψDPS > σ

Υ (1S)J/ψDPS > σ

Υ (1S)Υ (1S)DPS .

The expected cross-sections for a few double quarkonia processes, together with their

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yields, are summarized in Table 14. Measurements of the cross-sections and properties inthese modes will allow the two contributions to be disentangled.

Table 14: Expected cross-sections in the LHCb acceptance and yields for double quarkoniaproduction with 50 fb−1 at

√s = 14 TeV.

Mode σgg [ nb ] Yield [SPS] σDPS [ nb ] Yield [DPS]J/ψ J/ψ 7.2 270 000 11 430 000J/ψ ψ(2S) 3.2 14 000 4.0 19 000ψ(2S) ψ(2S) 0.4 180 0.6 300J/ψ χc0 - - 4.3 200J/ψ χc1 - - 6.6 14 000J/ψ χc2 - - 8.6 11 000J/ψ Υ (1S) 0.0036 360 0.27 20 000J/ψ Υ (2S) 0.0011 90 0.07 5300J/ψ Υ (3S) 0.0005 50 0.035 2000Υ (1S) Υ (1S) 0.014 1100 0.0027 200

As well as probing the production mechanism these studies are sensitive to a potentialfirst observation of tetraquark states [526] and of χb and ηb states decaying in the doubleJ/ψ mode. Based on the cross-sections and branching ratios given in Ref. [529], 500 (1500)fully reconstructed χb0(1P ) (χb2(1P )) are expected with the upgraded detector and thesedecays will be visible at LHCb. In the case of the ηb state, several estimates exist, based onvalues of the branching ratio ηb → J/ψJ/ψ ranging from 10−6 to 10−8 [530], correspondingto yields of 0.02 to 5 events.

The upgraded detector is expected to have excellent hadron identification capabilitiesboth offline and at the trigger level. As discussed in Ref. [531], this allows charmoniumstudies to be performed in hadronic decay modes. A particularly convenient mode is thepp final state. This is accessible for the J/ψ , ηc, χcJ , hc and ψ(2S) mesons. Extrapolatingfrom studies with the current detector large inclusive samples of these decays will becollected. For example around 0.5 million ηc → pp will be collected.

Hadronic decays of heavy bottomonium have received less attention in the litera-ture [530]. The high mass implies a large phase space for many decay modes, butconsequently the branching ratio for each individual mode is reduced. In Ref. [530] it isestimated that the ηb → D∗D branching fraction is 10−5 and the ηb → DDπ rate maybe a factor of ten higher. Though no specific studies have been performed, based on thestudies of double open charm production given in Ref. [296] it is plausible that an ηb signalwill be detected in this mode with the upgraded detector.

5.2 Exotic meson spectroscopy

The spectroscopy of bound states formed by heavy quark-antiquark pairs (c or b quarks),has been extensively studied from both theoretical and experimental points of view since

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the discovery of the J/ψ state in 1974 [532,533] and the discovery of the Υ (1S) state in1977 [534]. Until recently, all experimentally observed charmonium (cc) and bottomonium(bb) states matched well with expectations.

However, in 2003, a new and unexpected charmonium state was observed by theBelle experiment [535] and then confirmed independently by the BaBar [536], CDF [537]and D0 [538] experiments. This new particle, referred to as the X(3872), was observedin B → X(3872)K decays, in the decay mode X(3872) → J/ψπ+π− and has a massindistinguishable (within uncertainties) from the D∗0D0 threshold [512]. Several of theX(3872) parameters are unknown (such as its spin) or have large uncertainties, but thisstate does not match any predicted charmonium state [512]. The discovery of the X(3872)has led to a resurgence of interest in exotic spectroscopy and subsequently many newstates have been claimed. For example: the Y family, Y (4260), Y (4320) and Y (4660), ofspin parity 1−, or the puzzling charged Z family, Z(4050)+, Z(4250)+ and Z(4430)+, so farobserved only by the Belle experiment [539–541], and not confirmed by BaBar [542,543].The nature of these states has drawn much theoretical attention and many models have beenproposed. One possible explanation is that they are bound molecular states of open charmmesons [544]. Another is that these are tetraquarks [545] states formed of four quarks (e.g.c, c, one light quark and one light anti-quark). Other interpretations have been postulatedsuch as quark-gluon hybrid [545] or hadrocharmonium models [546], but experimentaldata are not yet able to conclude definitely. For reviews, see Refs. [512,544,546–550].

The bottomonium system should exhibit similar exotic states to the charmoniumcase. The Belle experiment recently reported the observation of exotic bottomoniumcharged particles Zb(10610)+ and Zb(10650)+ in the decays Zb → Υ (nS)π+ and Zb →hb(nP )π+ [551]. Evidence for a neutral isopartner has also been reported [552].70 Thesestates appear similar to, but narrower than, the Z(4430)+ observed in the charmoniumcase. In addition, neutral states analogous to the X(3872) and the Y states are expectedin the bottomonium system.

Studies of the X(3872) have already been performed with the current detector [554].The 50 fb−1 of integrated luminosity collected with the upgraded detector will contain overone million X(3872)→ J/ψ ππ candidates, by far the largest sample ever collected andallow study of this meson with high precision. A significant fraction of the X(3872) samplewill originate from the decays of B mesons (the remainder being promptly produced)allowing the quantum numbers and other properties to be determined. With such a largesample the missing 3D2 state of the charmonium system [555] will be also be observedand studied with high precision.

Another study being pursued with the current detector is to clarify the status of theZ(4430)+ state. If confirmed, the Z(4430)+ will be copiously produced at

√s = 14 TeV

and the larger data set will allow detailed study of its properties in different B decaymodes, thus setting the basis for all future searches for exotic charged states.

Similar to the charmonium-like states, exotic bottomonium states will mainly besearched for in the Υ (nS)π+π− channel, with Υ (nS)→ µ+µ−. The excellent resolution

70 At ICHEP 2012, Belle reported observations of the Zb states decaying to BB(∗) [553].

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observed in the Υ (nS) analysis [516] allows efficient separation of the three states, whichis crucial in searching for exotic bottomonium states in these channels.

All these studies, and searches for other exotica such as pentaquarks will profit fromthe increased integrated luminosity.

5.3 Precision measurements of b- and c-hadron properties

A major focus of activity with the current LHCb detector is the study of the properties ofbeauty and charm hadrons. This is a wide ranging field including studies of propertiessuch as mass and lifetime, observation of excited b hadrons and the measurements ofbranching ratios. These studies provide important input to pQCD models. Three topicsare considered here: b decays to charmonia, B+

c , and b-baryon decays.One important field being studied with the current detector is exclusive b decays to

charmonia. Studies of these modes are important to improve understanding of the shapeof the momentum spectrum of J/ψ produced in b hadron decays, as measured by the Bfactories [556,557]. To explain the observed excess at low momentum, new contributions tothe total b→ J/ψX rate are needed. Several sources have been proposed in the literature:intrinsic charm [558], baryonium formation [559] and as yet unobserved exotic states [560].One of the first proposed explanations for the excess was a contribution from an intrinsiccharm component to the b-hadron wave-function [558]. This would lead to an enhancementof b-hadron decays to J/ψ in association with open charm. The B-factories have set limitson such decays at the level of 10−5 [189], which considerably restricts, but does not exclude,contributions from intrinsic charm models. The branching ratios of these decays havebeen estimated in pQCD [561]. In the case of B0 → J/ψD0 the branching ratio has beenestimated to be 7×10−7. If this value is correct, several hundred fully reconstructed eventswill be collected with the upgraded detector. Similar decay modes are possible for B0

s andB+c mesons though no limits (or predictions) exist.

Another possibility to explain the shape of the J/ψ spectrum is contributions fromexotic strange baryonia formed in decays such as B+ → J/ψΛ0p. This decay has beenobserved by BaBar [562], with a branching ratio of (1.18 ± 0.31) × 10−5. The relateddecay B0 → J/ψpp is unobserved, with an upper limit on the branching ratio of 8.3× 10−7

at 90 % confidence level [563]. At present, these decays are experimentally challengingdue to the low Q-values involved. The larger data samples available at the time of theupgrade, together with improved proton identification at low momentum, may lead totheir observation.

Compared to the case of B0 and B+, the B0s sector is less well explored both experi-

mentally and theoretically. Decays such as B0s → J/ψK∗0K

∗0and B0

s → J/ψφρ shouldbe observable with the present detector. With the upgraded apparatus, the decay modesB0s → J/ψK0

SK0S and B0

s → J/ψφφ will also become accessible. The latter channel isinteresting as the low Q-value will allow a precision determination of the B0

s mass.As the lowest bound state of two heavy quarks b and c, the B+

c meson forms aunique flavoured, weakly decaying quarkonium system. Studies of the properties of B+

c

mesons such as the mass, lifetime and two-body non-leptonic decay modes are being

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performed with the current detector. As an example, Fig. 26 shows the signals observed forB+c → J/ψπ+ and B+

c → J/ψ 3π+. The large data set collected with the upgraded detectorwill allow these studies to be pursued with higher precision together with first studies ofCP and triple-product asymmetries in the B+

c system. In Table 15 the expected yields ofselected decay modes are estimated extrapolating from the yields of B+

c → J/ψπ+ andB+c → J/ψ3π+ observed with the current detector. As well as studies of the branching

5800 6000 6200 6400 6600 6800

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+πψJ/

Figure 26: Invariant mass distribution of (top) B+c → J/ψ3π+ and (bottom) B+

c → J/ψπ+

candidates using 0.8 fb−1 of integrated luminosity collected in 2011 [564]. The results of maximumlikelihood fits are superimposed.

ratios and searches for NP, these modes will allow precision measurements of the B+c

mass and lifetime to be made. Based on ongoing studies with the current detector, astatistical precision of 0.1 MeV/c2 on the mass will be achieved. The uncertainty on themass will most likely be dominated by systematic errors related to the momentum scale.Precision of 10−4 on this variable would translate to an uncertainty of 0.3 MeV/c2 on themass. Measurements of the B+

c lifetime using the J/ψπ+ decay are ongoing. Extrapolatingthese results to 50 fb−1, a statistical precision of 0.004 ps will be achieved.

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Table 15: Branching ratios and expected yields for selected B+c decays to final states containing a

J/ψ or ψ(2S) meson. The branching ratios for the J/ψ modes are taken from Ref. [565], with theadditional constraint of the ratio of the B+

c → J/ψ3π+ to B+c → J/ψπ+ reported in Ref. [564].

The ψ(2S) mode branching ratios are estimated assuming that they are 0.5 of the J/ψ values, asobserved in many modes (see for example Ref. [566]). Only dimuon modes are considered forthe J/ψ and ψ(2S), and only the K+K−π+ (K+π−π+) modes are considered for the D+

s (D+)modes. The B+

c → K+K∗0 numbers are taken from Ref. [567].

Mode Branching ratio Expected yield [50 fb−1]B+c → J/ψπ+ 2× 10−3 52 000

B+c → J/ψ3π+ 5× 10−3 17 000

B+c → J/ψK+ (1–2)× 10−4 3000–4000

B+c → J/ψK+

1 3× 10−5 1000B+c → ψ(2S)π+ 1× 10−3 3000

B+c → ψ(2S)3π+ 2.5× 10−3 1000

B+c → J/ψD+

s (2–3)× 10−3 1400–1900B+c → J/ψD+ (5–13)× 10−4 8–100

B+c → K+K∗0 10−6 500

The large B+c data set will open possibilities for many other studies. Decay modes of

the B+c meson to a B0

s or B0 meson together with a pion or kaon will also be accessible.Studies of the B+

c → B0sπ

+ decay have been started with the data collected in 2011 wherea handful of events are expected. As discussed in Ref. [565], semileptonic B+

c decays to B0s

can be used to provide a clean tagged decay source for CP violation studies. Finally, signalsof the currently unexplored excited B+

c meson states are expected to be observed [568–571].As discussed in Ref. [567] observation of the B∗+c decay is extremely challenging due to thesoft photon produced in the decay to to the ground state. The prospects for observationof the first P-wave multiplet decays decaying radiatively to the ground state are morepromising.

Large samples of b baryons decaying to final states containing charmonia will also becollected. Precision measurements of the properties of the already known states will bepossible. For example, extrapolating the preliminary studies with 0.3 fb−1 discussed inRef. [572], 10 000 Ξb → J/ψΞ and 2000 Ωb → J/ψΩ events will be collected. This willallow the Ξb (Ωb) mass to be measured to a precision of 0.1 MeV/c2 (0.5 MeV/c2). Preciseb-baryon lifetime measurements, that will allow tests of the heavy quark expansion [146,573, 574], should also be possible. Studies of excited b baryons, for example determinationof the quantum numbers of the Λ∗b baryons that have recently been observed by LHCb(Fig. 27) [575], will also be made.

Baryonic states containing two heavy quarks will also be observable. The lightest ofthese, the Ξcc isodoublet, have an estimated cross-section of O(102) nb [576,577] and soshould be visible with 5 fb−1 collected with the current detector. However, the statistics maybe marginal for follow-on analyses: measurements of the lifetime and ratios of branching

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)2c) (MeV/

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Figure 27: Invariant mass spectrum of Λ0bπ

+π− [575]. The points with error bars are the data,the solid line is the result of a fit to this distribution, and the dashed line is the fitted backgroundcontribution.

fractions, searches for excited states, and so forth. They will certainly be insufficient forangular analyses aimed at confirming the quark model predictions for the spin-parityof these states. These studies will require the statistics and improved triggering of theLHCb upgrade. Heavier states such as the Ωcc, Ξbc, and Ξbb have still smaller productioncross-sections [577]. First studies towards Ξbc detection are in progress. These indicatethat at best a handful of events can be expected in 5 fb−1, but that this state should beobservable with the upgrade.

5.4 Measurements with electroweak gauge bosons

Two of the most important quantities in the LHC electroweak physics programme arethe sine of the effective electroweak mixing angle for leptons, sin2 θlept

eff , and the mass ofthe W -boson, mW . Thanks to its unique forward coverage, an upgraded LHCb can makeimportant contributions to this programme. The forward coverage of LHCb also allowsa probe of electroweak boson production in a different regime from that of ATLAS andCMS, and the range of accessible physics topics is not limited to electroweak bosons. Forexample, tt production proceeds predominantly by gluon-gluon fusion in the central region,but has a significant contribution from quark-antiquark annihilation in the forward region,giving a similar production regime to that studied at the Tevatron.

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5.4.1 sin2 θlepteff

The value of sin2 θlepteff can be extracted from AFB, the forward-backward asymmetry of

leptons produced in Z decays. The raw value of AFB measured in dimuon final statesat the LHC is about five times larger than at an e+e− collider, due to the initial statecouplings, and so, in principle, it can be measured with a better relative precision, givenequal amounts of data. The measurement however requires knowledge of the direction ofthe quark and antiquark that created the Z boson, and any uncertainty in this quantityresults in a dilution of the observed value of AFB. This dilution is very significant in thecentral region, as there is an approximately equal probability for each proton to contain thequark or anti-quark that is involved in the creation of the Z, leading to an ambiguity in thedefinition of the axis required in the measurement. However, the more forward the Z bosonis produced, the more likely it is that it follows the quark direction; for rapidities y > 3,the Z follows the quark direction in around 95% of the cases. Furthermore, in the forwardregion, the partonic collisions that produce the Z are nearly always between u-valence andu-sea quark or d-valence and d-sea quark. The ss contribution, with a less well-knownparton density function, is smaller than in the central region. Consequently, the forwardregion is the optimum environment in which to measure AFB at the LHC. Preliminarystudies [578] have shown that with a 50 fb−1 data sample collected by the LHCb upgrade,AFB could be measured with a statistical precision of around 0.0004. This would give astatistical uncertainty on sin2 θlept

eff of better than 0.0001, which is a significant improvementin precision on the current world average value. It is also worth remarking that the two mostprecise values entering this world average at present, the forward-backward bb asymmetrymeasured at LEP (sin2 θlept

eff = 0.23221± 0.00029), and the left-right asymmetries measured

at SLD with polarised beams (sin2 θlepteff = 0.23098± 0.00026), are over 3σ discrepant with

each other [579]. LHCb will be able to bring clarity to this unsatisfactory situation.More work is needed to identify the important systematic uncertainties on the AFB

measurement. One source of error is the uncertainty in the parton density functions. Withcurrent knowledge this contribution would lead to an uncertainty of almost double thestatistical precision estimate above, but this will reduce when the differential cross-sectionmeasurements from the LHC of the W and Z bosons, and those of Drell-Yan dimuonproduction at lower masses, are included in the global fits to the parton density functions.LHCb has already embarked on this measurement programme. Figure 28 (left) shows theZ → µ+µ− peak obtained with 37 pb−1 of data [580]. Figure 28 (right) shows the measuredasymmetry between W+ and W− production as a function of lepton pseudorapidity. Thismeasurement is already approaching the accuracy of the theoretical uncertainties. The Wand Z measurements described in Ref. [580] are being used to constrain parton densityfunctions by some groups [581]. A preliminary measurement of lower mass Drell-Yanproduction [582] will extend these constraints to lower Q2 (masses above 5 GeV/c2 arecurrently considered) and Bjorken x.

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Figure 28: LHCb Z and W production results from 37 pb−1 at√s = 7 TeV [580]. Left:

Z → µ+µ− peak. Right: W+ −W− production asymmetry, where the bands correspond to theexperimental uncertainties (only indicated within the LHCb acceptance), and the data pointsgive predictions for various different parton density function sets. Note that the kinematic rangeof the ATLAS and CMS experiments only extends up to lepton pseudorapidities of 2.5.

5.4.2 mW

Decreasing the uncertainty on mW from its present error of 15 MeV/c2 is one of the mostchallenging tasks for the LHC (it may also be reduced further at the Tevatron). Althoughno studies have yet been made of determining mW with LHCb itself, it is evident thatthe experiment can give important input to the measurements being made at ATLASand CMS [583]. A significant and potentially limiting external uncertainty on mW willagain come from the knowledge of the parton density functions. These are less constrainedin the kinematic range accessible to LHCb, so that precise measurements of W+, W−,Z and Drell-Yan production in this region can be used to improve the global picture.Improved determinations of the shapes of the differential cross-sections are particularlyimportant. One specific area of concern arises from the knowledge of the heavy quarks inthe proton. Around 20–30 % of W production in the central region is expected to involves and c quarks, making the understanding of this component very important for the mW

measurement. LHCb can make a unique contribution to improving the knowledge of theheavy-quark parton density functions by exploiting its vertexing and particle identificationcapabilities to tag the relatively low-pT final-state quarks produced in processes such asgs → Wc, gc → Zc, gb → Zb, gc → γc and gb → γb. These processes provide directprobes of the strange, charm and bottom partons, and can be probed at high and lowvalues of Bjorken x inside the LHCb acceptance.

5.4.3 tt production

Understanding the nature of top production, and in particular the asymmetry in tt eventsreported by Fermilab [584–588], is of prime concern. As for the measurement of sin2 θlept

eff ,identifying the forward direction of events is crucial. The LHCb acceptance for identifying

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both leptons from tt decays is far smaller than that of ATLAS and CMS (typically 2 %rather than 70 %, according to PYTHIA generator level studies). However, the higher qqproduction fraction and better determined direction in the LHCb forward acceptancecombine to suggest that competitive measurements can be achieved. With the integratedluminosity offered by the upgrade, statistical precision will no longer be an issue, andLHCb measurements of the tt asymmetry will offer a competitive and complementary testof Tevatron observations [589].

5.5 Searches for exotic particles with displaced vertices

Different theoretical paradigms have been proposed to solve the so-called “hierarchyproblem”, the most discussed being SUSY. There are, however, many other ideas includingvarious models involving extra dimensions, Technicolour and little Higgs models. Theseideas approach the hierarchy problem from the direction of strong dynamics [590].

A growing subset of models features new massive long-lived particles with a macroscopicdistance of flight. They can be produced by the decay of a single-produced resonance,such as a Higgs boson or a Z ′ [591, 592], from the decay chain of SUSY particles [593], orby a hadronisation-type mechanism in models where the long-lived particle is a boundstate of quarks from a new confining gauge group, as discussed in Ref. [591]. In thelast case, the multiplicity of long-lived particles in an event can be large, while only onelong-lived particle is expected to be produced in other models. The decay modes may alsovary depending on the nature of the particle, from several jets in the final state [592] toseveral leptons [594] or lepton plus jets [595]. A comprehensive review of the experimentalsignatures is given in Ref. [596].

The common feature amongst these models is the presence of vertices displaced fromthe interaction region. Such signatures are well suited to LHCb, and in particular to theupgraded experiment, which will be able to select events with displaced vertices at theearliest trigger level.

As an example, consider the hidden valley (HV) model already discussed in Ref. [24].In this model the hidden sector, or v-sector, contains two new heavy quarks: U and C.Strassler and Zurek [593] suggest that an exotic Higgs boson could decay with a significantbranching fraction to a pair of π0

v particles, where the π0v is the ’neutral’ member of

the isotriplet of v-isospin 1 hadrons formed by U and C quarks. The π0v can decay in

SM particles and if the mass of the spinless π0v is below the ZZ threshold it will decay

dominantly into bb pairs due to helicity conservation. Here the π0v widths are determined

by their lifetime which could be very long, resulting in narrow states. The final state wouldconsist of four b-jets, each pair being produced from a displaced vertex corresponding tothe π0

v decay as illustrated in Fig. 29. If these decays exist, the lower limit on the Higgsmass set by LEP would be misleading, as it assumes the prompt decay of the Higgs to bbto be dominant.

The potential of LHCb to search for such exotic Higgs decays at√s = 14 TeV has

been discussed in Ref. [24], and is briefly summarised here. The benchmark model usesmH = 120 GeV/c2, mπ0

v= 35 GeV/c2 and τπ0

v= 10 ps. By combining vertex and jet

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Figure 29: Decay of a Higgs via a scalar field φ into two π0v particles, with π0

v charge equal tozero, which subsequently decay into bb jets. (Adapted from Ref. [593].)

reconstruction, the capacity to reconstruct this final state is shown using full simulationof the detector, assuming 0.4 interactions per crossing. Backgrounds to this signal fromother processes, such as the production of two pairs of bb quarks, have been consideredand found to be negligible.

During 2010 and 2011 data taking, an inclusive displaced vertex trigger has beenintroduced in the second level of the software trigger. Preliminary studies [597] havedemonstrated that for an output rate below 1 % of the overall trigger bandwidth, theefficiency of the whole trigger chain on events with two offline reconstructible π0

v verticeswith a minimum mass of 6 GeV and good vertex quality is of the order of 80%. Thisstrategy has been tested up to on average two visible interactions per crossing which iswhat is expected for the upgraded experiment.

Flight distance (mm)0 1 2 3 4 5 6 7 8 9 10

Nor

mal

ized

dis

trib

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n

-410

-310

-210

-110

1 LHCb PreliminaryData

bMC: bMC: HV10MC: BV48

Figure 30: Left: Distribution in x and z, for |y| < 1 mm, of the reconstructed vertices. Thevisible structures reflect the geometry of the vertex detector, with the pairs of silicon sensorsappearing as pairs of vertical bands and the corrugated (“RF”) foil as the two wave shapes. Thegreen shaded region represents the fiducial vacuum volume in which candidates are accepted.Right: Flight distance of offline reconstructed vertices in events outside the matter region. Dataare compatible with bb background. The black points are for data in 36 pb−1 [597], the red lineis a full simulation of bb production and the green dashed line is a full simulation of the HVbenchmark channel. The blue dashed line shows a simulation of a model with baryon numberviolating neutralino couplings.

114

The analysis of the trigger output showed that once vertices arising from hadronicinteractions with material are rejected, the dominant background is compatible with bhadron decay vertices as shown in Fig. 30. Those b hadron vertices are reconstructed withlarge masses because of the presence of fake or cloned tracks. With the present detector,it is difficult to keep the trigger rate down for single candidate events without using tightcuts on the mass and the displacement of the candidates. In the previous model, thetrigger efficiency for events with a single long-lived particle reconstructible in LHCb isonly about 20 %. This efficiency is expected to decrease for models where the mass of thelong-lived particle is smaller. In addition, the number of events with at least one π0

v statein the acceptance is three times higher than the number of events with two π0

v particles.Improving the single candidate efficiency would increase sensitivity to this model. It wouldalso give a better coverage for the models where only one long-lived particle is produced.

In the upgraded detector, the track fake rate in the vertex detector is expected to bebelow one percent [25], compared to 6 % in the present detector. Other upgrades to thetracking detectors will also help to reduce the fake rate. Moreover the use of an improveddescription for the complex RF foil shape will give a better control on the backgroundarising from hadronic interactions. It will enable the use of the true shape of the RFfoil, rather than the loose fiducial volume cut used at present, which depending on theconsidered lifetime, rejects 10–30 % of the long-lived particles. Those improvements wouldallow to decrease the thresholds on the single candidates trigger and therefore increase thereach of such searches.

As discussed in Ref. [24] the coupling of vertex information to jet reconstruction willallow to reduce the physical backgrounds. Studies are on-going on this matter. Assuminga Higgs production cross-section at

√s = 14 TeV of 50 pb, an integrated luminosity of

50 fb−1 and a geometric efficiency of 10 %, 250 000 Higgs bosons will be produced inLHCb. If H0 → π0

vπ0v is a dominant decay mode, then LHCb will be in an excellent

position to observe this signal, taking advantage of the software trigger’s ability to selecthigh-multiplicity events with good efficiency.

5.6 Central exclusive production

Central exclusive production (CEP) processes provide a promising and novel way to studyQCD and the nature of new particles, from low mass glueball candidates up to the Higgsboson itself. The CEP of an object X in a pp collider may be written as follows

pp→ p + X + p ,

where the ‘+’ signs denote the presence of large rapidity gaps. At high energies thet-channel exchanges giving rise to these processes can only be zero-charge colour singlets.Known exchanges include the photon and the pomeron. Another possibility, allowed inQCD, but not yet observed, is the odderon, a negative C-parity partner to the pomeronwith at least three gluons. The most attractive aspect of CEP reactions is that they offera very clean environment in which to measure the nature and quantum numbers of thecentrally produced state X.

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Central exclusive γγ [598], dijet [599,600] and χc [601] production has been observedat the Tevatron. LHCb has presented preliminary results on candidate dimuon eventscompatible with CEP [602]. Figure 31 shows the invariant mass of CEP χc candidates.These are events in which only a J/ψ → µ+µ− decay and a γ candidate are reconstructed,with no other activity (inconsistent with noise) seen elsewhere in the detector. Importantobservables in CEP are the relative production rates of χc0, χc1 and χc2. As is evidentfrom Fig. 31, the invariant mass resolution of LHCb is sufficient for this measurement.

Figure 31: Preliminary LHCb results on central exclusive χc production [602]. The J/ψγ invariantmass in data is compared to the expectation of the SuperCHIC Monte Carlo generator [603],which has been normalised to the observed number of events. The relative proportions of χc0,χc1 and χc2 are 12 %, 36 % and 52 % respectively.

Although not part of the baseline for the LHCb upgrade, additional instrumentation isbeing considered which could improve the potential of LHCb to study CEP processes. Forexample, the inclusion of forward shower counters (FSCs) on both sides of the interactionpoint, as proposed in Ref. [604], would be able to detect showers from very forward particlesinteracting in the beam pipe and surrounding material. The absence of a shower wouldindicate a rapidity gap and be helpful in increasing the purity of a CEP sample. Moreambitiously, the deployment of semi-conductor detectors very close to the beam, withinRoman pots, several hundred meters away from the interaction point, as proposed for otherLHC experiments [605] would also be beneficial for LHCb. The ability to measure thedirections of the deflected protons in the CEP interaction provides invaluable informationin determining the quantum numbers of the centrally produced state.

Several important physics goals have been identified for the LHCb CEP programme:

• Accumulation and characterisation of large samples of exclusive cc and bb events.A full measurement programme of these ‘standard candles’ will be essential tounderstand better the QCD mechanism of CEP [606], and may provide vital input ifCEP is used for studies of Higgs and other new particles [607].

• Searches for structure in the mass spectra of decay states such as K+K−, 2π+2π−,K+K−π+π− and pp. A particular interest of this study would be the hunt forglueballs, which are a key prediction of QCD.

116

• Observation and study of exotic particles in CEP processes. For example, a detailedstudy of the CEP process pp → p + X(3872) + p would provide a valuable newtool to aid understanding of this state. This and other states could be searched forin, for example, decays containing DD, which if observed would shed light onto thenature of the parent particle [606].

There are several reasons which make LHCb a suitable detector for realising these goals,particularly with the upgraded experiment:

• Even when running at a luminosity of 1033 cm−2 s−1 LHCb will have low pileup com-pared to ATLAS and CMS. This will be advantageous in triggering and reconstructinglow mass CEP states.

• The higher integrated luminosity that will be collected by the upgraded detector willallow studies to be performed on states that are inaccessible with only a few fb−1.This is true, for example, of central exclusive χb production, which is expected to bea factor of ∼ 1000 less than that of χc mesons [606].

• The particle identification capabilities of the LHCb ring-imaging Cherenkov detectorsystem allow centrally produced states to be cleanly separated into decays involvingpions, kaons and protons.

• The low pT acceptance of LHCb, and high bandwidth trigger, will allow samples ofrelatively low mass states to be collected and analysed.

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6 Summary

As described in the previous sections, LHCb has produced world-leading results across itsphysics programme, using the 1.0 fb−1 data sample of

√s = 7 TeV pp collisions collected

in 2011. The inclusion of the data collected at√s = 8 TeV during 2012 will enable further

improvements in precision in many key flavour physics observables. However, an upgradeto the detector is needed to remove the bottleneck in the trigger chain that currentlyprevents even larger increases in the collected data sample. The upgraded detector withtrigger fully implemented in software is to be installed during the 2018 long shutdownof the LHC, and will allow a total data set of 50 fb−1 to be collected. With such a datasample, LHCb will not only reach unprecedented precision for a wide range of flavourphysics observables, but the flexible trigger will allow it to exploit fully the potential of aforward physics experiment at a hadron collider.

In this section, some highlights of the LHCb physics output so far, and their implicationson the theoretical landscape, are summarised. The sensitivity of the upgraded detectorto key observables is then given, before a concluding statement on the importance of theLHCb upgrade to the global particle physics programme.

6.1 Highlights of LHCb measurements and their implications

6.1.1 Rare decays

Among rare decays, the LHCb limit on the rate of the decay B0s → µ+µ− [13] places

stringent limits on NP models that enhance the branching fraction. The measurement

B(B0s → µ+µ−) < 4.5× 10−9 (95 % confidence level) , (134)

can be compared to the SM prediction B(B0s → µ+µ−)SM = (3.1 ± 0.2) × 10−9 [115].71

This result puts severe constraints — far beyond the ATLAS and CMS search limits — onsupersymmetric models with large values of tan β, i.e. of the ratio of vacuum expectationvalues of the Higgs doublets (see, for example, Refs. [115,128,161]).

The measurement of the forward-backward asymmetry in B0 → K∗0µ+µ− [15] has to beviewed as the start of a programme towards a full angular analysis of these decays. The fullanalysis will allow determination of numerous NP-sensitive observables (see, for example,Refs. [52,53]). The measurements that will be obtained from such an analysis, as well assimilar studies of related channels, such as B0

s → φµ+µ− [68], allow model-independentconstraints on NP, manifested as limits on the operators of the effective Hamiltonian (see,for example, Refs. [41, 42]). Indeed, the first results already impose important constraints.Studies of radiative decays such as B0

s → φγ [16, 17] provide additional information sincethey allow to measure the polarisation of the emitted photon, and are therefore especiallysensitive to models that predict new right-handed currents. Similarly, studies of observables

71 It should be noted that the measured value is the time-integrated branching fraction, and the SMprediction should be increased by around 10 % to allow a direct comparison [135].

118

such as isospin asymmetries [76] are important since they allow to pin down in whichoperators the NP effects occur.

Several new opportunities with rare decays at LHCb are becoming apparent. Theobservation of B+ → π+µ+µ− [85], the rarest B decay yet discovered, enables a newapproach to measure the ratio of CKM matrix elements |Vtd/Vts|. Decays to final statescontaining same-sign leptons [196] allow searches for Majorana neutrinos complementary tothose based on neutrinoless double beta decay. LHCb can also reach competitive sensitivityfor some lepton flavour violating decays such as τ+ → µ+µ−µ+ [190].

6.1.2 CP violation in the B sector

Measurements of the neutral B meson mixing parameters provide an excellent methodto search for NP effects, due to the low theoretical uncertainties associated to severalobservables. The LHCb measurements of the CP -violating phase, φs, and the widthdifference, ∆Γs, in the B0

s system [10,138,214,224] significantly reduce the phase space forNP:

φs = −0.002± 0.083± 0.027 rad , ∆Γs = 0.116± 0.018 (stat)± 0.006 (syst) ps−1 . (135)

However deviations from the SM predictions [118,216] are still possible. Effects of O(0.1)are typical of some well-motivated NP models that survive the present ATLAS and CMSbounds (such as in Ref. [36]). The experimental uncertainty on φs is still a factor of 40larger than that on the prediction, therefore improved measurements are needed to reachthe level of sensitivity demanded by theory. It should also be noted that compared tothe CP -violating phase in the B0 system (2β), φs is much more precisely predicted, andtherefore presents stronger opportunities for NP searches.

In addition, to understand the origin of the anomalous dimuon asymmetry seenby D0 [158], improved measurements of semileptonic asymmetries in both B0

s and B0

systems are needed. LHCb has just released its first results on the B0s asymmetry [240],

demonstrating the potential to search for NP effects with more precise measurements.Moreover, a constraint on, or a measurement of, the rate of the decay B0

s → τ+τ− isimportant to provide knowledge of possible NP contributions to Γ12 (see, for example,Refs. [152,154]).

Among the B0 mixing parameters, improved measurements of both φd (i.e., sin 2β)and ∆Γd are needed. Reducing the uncertainty on the former will help to improve theglobal fits to the CKM matrix [244,258], and may clarify the current situation regardingthe tension between various inputs to the fits (see, for example, Ref. [259]). Anothercrucial observable is the angle γ, which, when measured in the tree-dominated B → DKprocesses, provides a benchmark measurement of CP violation. The first measurementsfrom LHCb already help to improve the uncertainty on γ [6, 7]: further improvements areboth anticipated and needed.

Comparisons of values of γ from loop-dominated processes with the SM benchmark fromtree-dominated processes provide important ways to search for new sources of CP violation.In particular, the study of B0

s → K+K− and B0 → π+π− decays [348], which are related by

119

U-spin, allows a powerful test of the consistency of the observables with the SM [347,349].Similarly, the U-spin partners B0

s → K∗0K∗0 [295] and B0 → K∗0K∗0 are among thegolden channels to search for NP contributions in b → sqq penguin amplitudes [300].Another important channel in this respect is B0

s → φφ [296], for which the CP -violatingobservables are predicted with low theoretical uncertainty in the SM. Studies of CPviolation in multibody b hadron decays [368,369] offer additional possibilities to search forboth the existence and features of NP.

6.1.3 Charm mixing and CP violation

In the charm sector, the evidence for CP violation in the observable ∆ACP has prompteda large amount of theoretical work. The measurement

∆ACP = ACP (K+K−)−ACP (π+π−) = (−0.82± 0.21± 0.11) % , (136)

is different from zero by 3.5 standard deviations [18]. While ACP represents a time-integrated CP asymmetry, ∆ACP originates predominantly from direct CP violation. Theemergent consensus is that while an asymmetry of the order of 1 % is rather unlikely inthe SM, it cannot be ruled out that QCD effects cause enhancements of that size. Furthermeasurements are needed in order to establish if NP effects are present in the charm sector.Among the anticipated results are updates of the ∆ACP measurement as well as of theindividual CP asymmetries in D0 → K+K− and D0 → π+π−. It is of great interest tolook for direct CP violation in decays to other final states, and in decays of other charmedhadrons (D+, D+

s and Λ+c ).

The SM predictions are somewhat cleaner for indirect CP violation effects, and thereforeit is also essential to search for CP violation in charm mixing. New results from time-dependent analyses of D0 → K+K− [19] and D0 → K0

Sπ+π− will improve the current

knowledge, and additional channels will also be important with high statistics.Several authors have noted correlations between CP violation in charm and various

other observables (for example, Refs. [461,476]). These correlations appear in, and differbetween, certain theoretical models, and can therefore be used to help identify the originof the effects. Observables of interest in this context include those that can be measuredat high-pT experiments, such as tt asymmetries, as well as rare charm decays. Among thelatter, it has been noted that CP asymmetries are possible in radiative decays such asD0 → φγ [477], and that searches for decays involving dimuons, such as D0 → µ+µ− [177]and D+ → π+µ+µ− are well motivated.

6.1.4 Measurements exploiting the unique kinematic acceptance of LHCb

The unique kinematic region covered by the LHCb acceptance enables measurements thatcannot be performed at other experiments, and that will continue to be important inthe upgrade era. These include probes of QCD both in production, such as studies ofmulti-parton scattering [523,608], and in decay, such as studies of exotic hadrons like theX(3872) [554] and the putative Z(4430)+ state. Conventional hadrons can also be studied

120

with high precision: one important goal will be to establish the existence of doubly heavybaryons. Central exclusive production of conventional and exotic hadrons can also bestudied; the sensitivity of the upgraded experiment will be significantly enhanced due tothe software trigger.

Measurements of production rates and asymmetries of electroweak gauge bosons inthe LHCb acceptance are important to constrain parton density functions [580]. Withhigh statistics, LHCb will be well placed to make a precision measurement of the sine ofthe effective electroweak mixing angle for leptons, sin2 θlept

eff , from the forward-backwardasymmetry of leptons produced in the Z → µ+µ− decay. Improved knowledge of partondensity functions, as can be obtained from studies of production of gauge bosons inassociation with jets [609], will help to reduce limiting uncertainties on the measurement ofthe W boson. These studies are also an important step towards a top physics programmeat LHCb, which will become possible once the LHC energy approaches the nominal 14 TeV.

The importance of having a detector in the forward region can be illustrated with therecent discovery by ATLAS and CMS of a new particle that may be the Higgs boson. It isnow essential to determine if this particle has the couplings to bosons, leptons and quarksexpected in the SM. In particular, at the observed mass the highest branching ratio isexpected to be for H → bb — however this is a difficult channel for ATLAS and CMS dueto the large SM background. LHCb with its excellent b-hadron sensitivity will be able tosearch for such decays. The forward geometry of LHCb is also advantageous to observenew long-lived particles that are predicted in certain NP models, including some withextended Higgs sectors. Although limits can be set with the current detector [597], thisis an area that benefits significantly from the flexible software trigger of the upgradedexperiment. Models with extended Higgs sectors also produce characteristic signals inflavour physics observables, which emphasises the need for the LHCb upgrade as part ofthe full exploitation of the LHC.

6.2 Sensitivity of the upgraded LHCb experiment to key observ-ables

As mentioned in Sec. 1, the LHCb upgrade is necessary to progress beyond the limitationsimposed by the current hardware trigger that, due to its maximum output rate of 1 MHz,restricts the instantaneous luminosity at which data can most effectively be collected.To overcome this, the upgraded detector will be read out at the maximum LHC bunch-crossing frequency of 40 MHz so that the trigger can be fully implemented in software.The upgraded detector will be installed during the long shutdown of the LHC plannedfor 2018. A detailed description of the upgraded LHCb experiment can be found in theLetter of Intent (LoI) [24], complemented by the recent framework technical design report(FTDR) [25], which sets out the timeline and costing for the project. A summary hasbeen prepared for the European Strategy Preparatory Group [610].

The sensitivity to various flavour observables is summarised in Table 16, which is takenfrom the FTDR [25]. This is an updated version of a similar summary that appears asTable 2.1 in the LoI [24]. The measurements considered include CP -violating observables,

121

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)1.

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)/B(B

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)–

∼10

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122

rare decays and fundamental parameters of the CKM unitarity triangle. More details aboutthese observables are given below. The current precision, either from LHCb measurementsor averaging groups [43, 244, 258], is given and compared to the estimated sensitivity withthe upgrade. As an intermediate step, the estimated precision that can be achieved priorto the upgrade is also given for each observable. For this, a total integrated luminosityof 1.0 (1.5, 4.0) fb−1 at pp centre-of-mass collision energy

√s = 7 (8, 13) TeV recorded in

2011 (2012, 2015–17) is assumed. Another assumption is that the current efficiency of themuon hardware trigger can be maintained at higher

√s, but that higher thresholds will

be necessary for other triggers, reducing the efficiency for the relevant channels by a factorof 2 at

√s = 14 TeV.

In LHCb measurements to date, the CP -violating phase in B0s mixing, measured in

both J/ψ φ and J/ψ f0(980) final states, has been denoted φs. In the upgrade era it willbe necessary to remove some of the assumptions that have been made in the analyses todate, related to possible penguin amplitude contributions, and therefore the observables inb→ ccs transitions are denoted by 2βs = −φs, while in b→ qqs (q = u, d, s) transitions thenotation 2βeff

s is used. This parallels the established notation used in the B0 system (theα, β, γ convention for the CKM unitarity triangle angles is used). The penguin contributionsare expected to be small, and therefore a theory uncertainty on 2βs (B0

s → J/ψ φ) ∼ 0.003is quoted, comparable to the theory uncertainty on 2β (B0 → J/ψ K0

S ). However, largereffects cannot be ruled out at present. Data-driven methods to determine the penguinamplitudes are also possible [238,269,276]: at present these given much larger estimatesof the uncertainty, but improvement can be anticipated with increasing data samples.The flavour-specific asymmetry in the B0

s system, assl in Table 16, probes CP violation inmixing. The “sl” subscript is used because the measurement uses semileptonic decays.

Sensitivity to the emitted photon polarisation is encoded in the effective lifetime, τ eff

of B0s → φγ decays, together with the effective CP -violation parameter 2βeff

s . Two of themost interesting of the full set of angular observables in B0 → K∗0µ+µ− decays [61], areS3, which is related to the transverse polarisation asymmetry [62], and the zero-crossingpoint (s0) of the forward-backward asymmetry. As discussed above, isospin asymmetries,denoted AI , are also of great interest.

In the charm sector, it is important to improve the precision of ∆ACP , described above,and related measurements of direct CP violation. One of the key observables related toindirect CP violation is the difference in inverse effective lifetimes of D0 → K+K− andD0 → K+K− decays, AΓ.

The extrapolations in Table 16 assume the central values of the current measurements,or the SM where no measurement is available. While the sensitivities given includestatistical uncertainties only, preliminary studies of systematic effects suggest that thesewill not affect the conclusions significantly, except in the most precise measurements,such as those of assl, AΓ and ∆ACP . Branching fraction measurements of B0

s mesonsrequire knowledge of the ratio of fragmentation fractions fs/fd for normalisation [144].The uncertainty on this quantity is limited by knowledge of the branching fraction ofD+s → K+K−π+, and improved measurements of this quantity will be necessary to avoid

a limiting uncertainty on, for example, B(B0s → µ+µ−). The determination of 2βs from

123

B0s → J/ψ φ provides an example of how systematic uncertainties can be controlled for

measurements at the LHCb upgrade. In the most recent measurement [138], the largestsource of systematic uncertainty arises due to the constraint of no direct CP violation thatis imposed in the fit. With larger statistics, this constraint can be removed, eliminatingthis source of uncertainty. Other sources, such as the background description and angularacceptance, are already at the 0.01 rad level, and can be reduced with more detailedstudies.

Experiments at upgraded e+e− B factories and elsewhere will study flavour-physicsobservables in a similar timeframe to the LHCb upgrade. However, the LHCb sample sizesin most exclusive B and D final states will be far larger than those that will be collectedelsewhere, and the LHCb upgrade will have no serious competition in its study of B0

s

decays, b-baryon decays, mixing and CP violation. Similarly the yields in charmed-particledecays to final states consisting of only charged tracks cannot be matched by any otherexperiment. On the other hand, the e+e− environment is advantageous for inclusive studiesand for measurements of decay modes including multiple neutral particles [611–615], andtherefore enables complementary measurements to those that will be made with theupgraded LHCb experiment.

6.3 Importance of the LHCb upgrade

The study of deviations from the SM in quark flavour physics provides key informationabout any extension of the SM. It is already known that the NP needed to stabilize theelectroweak sector must have a non-generic flavour structure in order to be compatiblewith the tight constraints of flavour-changing processes, even if the precise form of thisstructure is still unknown. Hopefully, ATLAS and CMS will detect new particles belongingto these models, but the couplings of the theory and, in particular, its flavour structure,cannot be determined only using high-pT data.

Therefore, the LHCb upgrade will play a vital role in any scenario. It allows theexploration of NP phase space that a priori cannot be studied by high energy searches.Future plans for full exploitation of the LHC should be consistent with a co-extensiveLHCb programme.

124

Acknowledgements

The LHCb collaboration expresses its gratitude to its colleagues in the CERN acceleratordepartments for the excellent performance of the LHC. LHCb thanks the technical andadministrative staff at the LHCb institutes, and acknowledges support from CERNand 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute”(Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NASUkraine (Ukraine); STFC (United Kingdom); NSF (USA). LHCb also acknowledges thesupport received from the ERC under FP7. The Tier1 computing centres are supportedby IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (TheNetherlands), PIC (Spain), GridPP (United Kingdom). LHCb is thankful for the computingresources put at its disposal by Yandex LLC (Russia), as well as to the communitiesbehind the multiple open source software packages that are depended upon.

The work of A. Datta was supported by the National Science Foundation under GrantNo. NSF PHY-1068052. D.M. Straub was supported by the EU ITN “Unification in theLHC Era”, contract PITN-GA-2009-237920 (UNILHC). We thank D. Gorbunov for usefulcomments.

125

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