Measurement of the CKM angle γ using B0 → DK *0 with D → K0S π + π − decays

JHEP08(2016)137 Published o SISSA by Sp inge Recei ed:May 10, 2016 Re ised:July 4, 2016 Accep ed:Augus 10, 2016 Published:Augus 24, 2016 Measu emen o he CKM angle γusing B0→DK∗0 wi h D→K0 Sπ+π−decays The LHCb collabo a ion E-mail: [email p o ec ed] Abs ac : A model-dependen ampli ude analysis o he decay B0→D(K0 Sπ+π−)K∗0 is pe o med using p o on-p o on collision da a co esponding o an in eg a ed luminosi y o 3.0 b−1, eco ded a √s= 7 and 8 TeV by he LHCb expe imen . The CP iola ion obse ables x±and y±, sensi i e o he CKM angle γ, a e measu ed o be x−=−0.15 ±0.14 ±0.03 ±0.01, y−= 0.25 ±0.15 ±0.06 ±0.01, x+= 0.05 ±0.24 ±0.04 ±0.01, y+=−0.65 +0.24 −0.23 ±0.08 ±0.01, whe e he i s unce ain ies a e s a is ical, he second sys ema ic and he hi d a ise om he unce ain y on he D→K0 Sπ+π−ampli ude model. These a e he mos p ecise mea- su emen s o hese obse ables. They co espond o γ= (80+21 −22)◦and B0= 0.39 ±0.13, whe e B0is he magni ude o he a io o he supp essed and a ou ed B0→DK+π− decay ampli udes, in a Kπ mass egion o ±50 MeV a ound he K∗(892)0mass and o an absolu e alue o he cosine o he K∗0decay angle la ge han 0.4. Keywo ds: B physics, CKM angle gamma, CP iola ion, Fla o physics, Had on-Had on sca e ing (expe imen s) A Xi eP in : 1605.01082 Open Access, Copy igh CERN, o he bene i o he LHCb Collabo a ion. A icle unded by SCOAP3. doi:10.1007/JHEP08(2016)137 JHEP08(2016)137 Con en s 1 In oduc ion 1 2 The LHCb de ec o 4 3 Candida e selec ion and backg ound sou ces 5 4 E iciency ac oss he phase space 6 5 Analysis s a egy and i esul s 6 5.1 In a ian mass i o B0→DK∗0candida es 7 5.2 CP i 8 6 Sys ema ic unce ain ies 11 7 De e mina ion o he pa ame e s γ, B0and δB017 8 Conclusion 17 The LHCb collabo a ion 25 1 In oduc ion The S anda d Model can be es ed by checking he consis ency o he Cabibbo-Kobayashi- Maskawa (CKM) mechanism [1,2], which desc ibes he mixing be ween weak and mass eigens a es o he qua ks. The CKM phase γcan be exp essed in e ms o he elemen s o he complex uni a y CKM ma ix, as γ≡a g [−VudVub∗/VcdVcb∗]. Since γis also he angle o he uni a i y iangle leas cons ained by di ec measu emen s, i s p ecise de e mina ion is o conside able in e es . I s alue can be measu ed in ee-le el p ocesses such as B±→ DK±and B0→DK∗0, whe e Dis a supe posi ion o he D0and D0 la ou eigens a es, and K∗0is he K∗(892)0meson. Since loop co ec ions o hese p ocesses a e o highe o de , he associa ed heo e ical unce ain y on γis negligible [3]. As such, measu emen s o γin ee-le el decays p o ide a e e ence alue, allowing sea ches o po en ial de ia ions due o physics beyond he S anda d Model in o he p ocesses. The combina ion o measu emen s by he BaBa [4] and Belle [5] collabo a ions gi es γ= (67 ±11)◦[6], whils an a e age alue o LHCb de e mina ions in 2014 ga e γ= 73+9 −10◦[7]. Global i s o all cu en CKM measu emen s by he CKM i e [8,9] and UT i [10] collabo a ions yield indi ec es ima es o γwi h an unce ain y o 2◦. Some o he CKM measu emen s included in hese combina ions can be a ec ed by new physics con ibu ions. – 1 – JHEP08(2016)137 Since he phase di e ence be ween Vub and Vcb depends on γ, he de e mina ion o γin ee-le el decays elies on he in e e ence be ween b→cand b→u ansi ions. The s a egy o using B±→DK±decays o de e mine γ om an ampli ude analysis o D-meson decays o he h ee-body inal s a e K0 Sπ+π−was i s p oposed in e s. [11,12]. The me hod equi es knowledge o he D→K0 Sπ+π−decay ampli ude ac oss he phase space, and in pa icula he a ia ion o i s s ong phase. This may be ob ained ei he by using a model o desc ibe he D-meson decay ampli ude in phase space (model-dependen app oach), o by using measu emen s o he phase beha iou o he ampli ude (model- independen app oach). The model-independen s a egy, used by Belle [13] and LHCb [14, 15], inco po a es measu emen s om CLEO [16] o he Ddecay s ong phase in bins ac oss he phase space. The p esen pape epo s a new unbinned model-dependen measu emen , ollowing he me hod used by he BaBa [17–19], Belle [20–22] and LHCb [23] collabo a ions in hei analyses o B±→D(∗)K(∗)±decays. This me hod allows he s a is ical powe o he da a o be ully exploi ed. The sensi i i y o γdepends bo h on he yield o he sample analysed and on he magni- ude o he a io Bo he supp essed and a ou ed decay ampli udes in he ele an egion o phase space. Due o colou supp ession, he b anching ac ion B(B0→D0K∗0) = (4.2± 0.6)×10−5is an o de o magni ude smalle han ha o he co esponding cha ged B-meson decay mode, B(B+→D0K+) = (3.70 ±0.17) ×10−4[24]. Howe e , his is pa ially com- pensa ed by an enhancemen in B0, which was measu ed o be B0= 0.240+0.055 −0.048 in B0→ DK∗0decays in which he Dis econs uc ed in wo-body inal s a es [25]; he cha ged de- cays ha e an a e age alue o B= 0.097 ±0.006 [8,9]. Model-dependen and independen de e mina ions o γusing B0→D(K0 Sπ+π−)K∗0decays ha e al eady been pe o med by he BaBa [26] and Belle [27] collabo a ions, espec i ely. The model-independen app oach has also been employed ecen ly by LHCb [28]. Fo hese decays a ime-independen CP analysis is pe o med, as he K∗0is econs uc ed in he sel - agging mode K+π−, whe e he cha ge o he kaon p o ides he la ou o he decaying neu al Bmeson. The K∗0meson is one o se e al possible s a es o he (K+π−) sys em. Le ing X0 s ep esen any such s a e, he B-meson decay ampli ude o DK+π−may be exp essed as a supe posi ion o a ou ed b→cand supp essed b→ucon ibu ions: A(B0→DX0 s)∝ |Ac|A +|Au|ei(δB0−γ)¯ A , A(B0→DX0 s)∝ |Ac|¯ A +|Au|ei(δB0+γ)A ,(1.1) whe e |Ac,u|a e he magni udes o he a ou ed and supp essed B-meson decay ampli- udes, δB0is he s ong phase di e ence be ween hem, and γis he CP- iola ing weak phase. The quan i ies Ac,u and δB0depend on he posi ion in he B0→DK+π−phase space. The ampli udes o he D0and D0mesons decaying in o he common inal s a e , A ≡ HD0and ¯ A ≡ HD0, a e unc ions o he K0 Sπ+π− inal s a e, which can be comple ely speci ied by wo squa ed in a ian masses o pai s o he h ee inal-s a e pa icles, chosen o be m2 +≡m2 K0 Sπ+and m2 −≡m2 K0 Sπ−. The o he squa ed in a ian mass is m2 0≡m2 π+π−. Making he assump ion o no CP iola ion in he D-meson decay, he ampli udes A and ¯ A a e ela ed by ¯ A (m2 +, m2 −) = A (m2 −, m2 +). – 2 – JHEP08(2016)137 The ampli udes in eq. (1.1) gi e ise o dis ibu ions o he o m dΓB0∝ |Ac|2|A |2+|Au|2|¯ A |2+ 2|Ac||Au| RehA? ¯ A ei(δB0−γ)i, dΓB0∝ |Ac|2|¯ A |2+|Au|2|A |2+ 2|Ac||Au| RehA ¯ A? ei(δB0+γ)i, (1.2) which a e unc ions o he posi ion in he B0→DK+π−phase space. In eg a ing only o e he egion φK∗0o he B0→DK+π−phase space in which he K∗0 esonance is dominan , 2 B0≡RφK∗0dφ|Au|2 RφK∗0dφ|Ac|2.(1.3) The unc ional P(A, z, κ) = A 2+|z|2¯ A 2+ 2κRezA?¯ A,(1.4) desc ibes he dis ibu ion wi hin he phase space o he D-meson decay, PB0(m2 −, m2 +)∝ P(A , z−, κ), PB0(m2 −, m2 +)∝ P(¯ A , z+, κ),(1.5) whe e he cohe ence ac o κis a eal cons an (0 ≤κ≤1) [29] measu ed in e . [30], pa ame e ising he ac ion o he egion φK∗0 ha is occupied by he K∗0 esonance, and he complex pa ame e s z±a e z±= B0ei(δB0±γ).(1.6) A di ec de e mina ion o B0,δB0and γcan lead o bias, when B0ge s close o ze o [17]. The Ca esian CP iola ion obse ables, x±=Re(z±) and y±=Im(z±), a e he e o e used ins ead. This pape epo s model-dependen Ca esian measu emen s o z±made using B0→ D(K0 Sπ+π−)K∗0decays selec ed om pp collision da a, co esponding o an in eg a ed lu- minosi y o 3 b−1, eco ded by LHCb a cen e-o -mass ene gies o 7 TeV in 2011 and 8 TeV in 2012. The measu ed alues o z±place cons ain s on he CKM angle γ. Th oughou he pape , inclusion o cha ge conjuga e p ocesses is implied, unless speci ied o he wise. Sec ion 2desc ibes he LHCb de ec o used o eco d he da a, and he me hods used o p oduce a ealis ic simula ion o he da a. Sec ion 3ou lines he p ocedu e used o selec candida e B0→D(K0 Sπ+π−)K∗0decays, and sec ion 4desc ibes he de e mina ion o he selec ion e iciency ac oss he phase space o he D-meson decay. Sec ion 5de ails he i ing p ocedu e used o de e mine he alues o he Ca esian CP iola ion obse ables and sec ion 6desc ibes he sys ema ic unce ain ies on hese esul s. Sec ion 7p esen s he in e p e a ion o he measu ed Ca esian CP iola ion obse ables in e ms o cen al alues and con idence in e als o B0,δB0and γ, be o e sec ion 8concludes wi h a summa y o he esul s ob ained. – 3 – JHEP08(2016)137 2 The LHCb de ec o The LHCb de ec o [31,32] is a single-a m o wa d spec ome e co e ing he pseudo apidi y ange 2 < η < 5, designed o he s udy o pa icles con aining bo c qua ks. The de ec o includes a high-p ecision acking sys em consis ing o a silicon-s ip e ex de ec o su ounding he pp in e ac ion egion, a la ge-a ea silicon-s ip de ec o loca ed ups eam o a dipole magne o e e sible pola i y wi h a bending powe o abou 4 Tm, and h ee s a ions o silicon-s ip de ec o s and s aw d i ubes placed downs eam o he magne . The acking sys em p o ides a measu emen o he momen um po cha ged pa icles wi h a ela i e unce ain y ha a ies om 0.5% a low momen um o 1.0% a 200 GeV. The minimum dis ance o a ack o a p ima y e ex (PV), he impac pa am- e e (IP), is measu ed wi h a esolu ion o (15 + 29/pT)µm, whe e pTis he componen o he momen um ans e se o he beam, in GeV. Di e en ypes o cha ged had ons a e dis inguished using in o ma ion om wo ing-imaging Che enko de ec o s. Pho ons, elec- ons and had ons a e iden i ied by a calo ime e sys em consis ing o scin illa ing-pad and p eshowe de ec o s, an elec omagne ic calo ime e and a had onic calo ime e . Muons a e iden i ied by a sys em composed o al e na ing laye s o i on and mul iwi e p opo ional chambe s. The igge consis s o a ha dwa e s age, based on in o ma ion om he calo ime e and muon sys ems, ollowed by a so wa e s age, in which all cha ged pa icles wi h pT> 500 (300) MeV a e econs uc ed o 2011 (2012) da a. The so wa e igge equi es a wo-, h ee- o ou - ack seconda y e ex wi h a la ge sum o he ans e se momen um, pT, o he acks and a signi ican displacemen om he p ima y pp in e ac ion e ices. A leas one ack should ha e pT>1.7 GeVand χ2 IP wi h espec o any p ima y in e ac ion g ea e han 16, whe e χ2 IP is de ined as he di e ence in χ2o a gi en PV econs uc ed wi h and wi hou he conside ed ack. A mul i a ia e algo i hm [33] is used o he iden i ica ion o seconda y e ices consis en wi h he decay o a bhad on. In he o line selec ion, igge signals a e associa ed wi h econs uc ed pa icles. Selec ion equi emen s can he e o e be made on he igge selec ion i sel and on whe he he decision was due o he signal candida e, o he pa icles p oduced in he pp collision, o a combina ion o bo h. Decays o K0 S→π+π−a e econs uc ed in wo di e en ca ego ies: he i s in ol ing K0 Smesons ha decay ea ly enough o he daugh e pions o be econs uc ed in he e ex de ec o , and he second con aining K0 S ha decay la e such ha ack segmen s o he pions canno be o med in he e ex de ec o . These ca ego ies a e e e ed o as long and downs eam, espec i ely. The long ca ego y has be e mass, momen um and e ex esolu ion han he downs eam ca ego y. La ge samples o simula ed B0 (s)→D() K∗0decays and a ious backg ound decays a e used in his s udy. In he simula ion, pp collisions a e gene a ed using Py hia [34, 35] wi h a speci ic LHCb con igu a ion [36]. Decays o had onic pa icles a e desc ibed by E Gen [37], in which inal-s a e adia ion is gene a ed using Pho os [38]. The in e ac ion o he gene a ed pa icles wi h he de ec o , and i s esponse, a e implemen ed using he Gean 4 oolki [39,40], as desc ibed in e . [41]. – 4 – JHEP08(2016)137 3 Candida e selec ion and backg ound sou ces In addi ion o he ha dwa e and so wa e igge equi emen s, a e a kinema ic i [42] o cons ain he B0candida e o poin owa ds he PV and he Dcandida e o ha e i s nominal mass, he in a ian mass o he K0 Scandida es mus lie wi hin ±14.4 MeV (±19.9 MeV) o he known alue [24] o long (downs eam) ca ego ies. Likewise, a e a kinema ic i o cons ain he B0candida e o poin owa ds he PV and he K0 Scandida e o ha e he K0 Smass, he econs uc ed D-meson candida e mus lie wi hin ±30 MeV o he D0mass. To econs uc he B0mass, a hi d kinema ic i o he whole decay chain is used, cons aining he B0candida e o poin owa ds he PV and he Dand K0 S o ha e hei nominal masses. The χ2o his i is used in he mul i a ia e classi ie desc ibed below. This i imp o es he esolu ion o he m2 ±in a ian masses and ensu es ha he econs uc ed Dcandida es a e cons ained o lie wi hin he kinema ic bounda ies o he phase space. The K∗0candida e mus ha e a mass wi hin ±50 MeV o he wo ld a e age alue and |cos θ∗|>0.4, whe e he decay angle θ∗is de ined in he K∗0 es ame as he angle be ween he momen um o he kaon daugh e o he K∗0, and he di ec ion opposi e o he B0momen um. The c i e ia placed on he K∗0candida e a e iden ical o hose used in he analysis o B0→DK∗0wi h wo-body Ddecays [25]. A mul i a ia e classi ie is hen used o imp o e he signal pu i y. A boos ed decision ee (BDT) [43,44] is ained on simula ed signal e en s and backg ound candida es lying in he high B0mass sideband [5500,6000] MeV in da a. This mass ange pa ially o e laps wi h he ange o he in a ian mass i desc ibed below. To a oid a po en ial i bias, he candida es a e andomly spli in o wo disjoin subsamples, A and B, and wo independen BDTs (BDTA and BDTB) a e ained wi h hem. These classi ie s a e hen applied o he complemen a y samples. The BDTs a e based on 16 disc imina ing a iables: he B0 meson χ2 IP, he sum o he χ2 IP o he K0 Sdaugh e pions, he sum o he χ2 IP o he inal s a e pa icles excep he K0 Sdaugh e s, he B0and Ddecay e ex χ2, he alues o he ligh dis ance signi icance wi h espec o he PV o he B0,Dand K0 Smesons, he D (K0 S) ligh dis ance signi icance wi h espec o he B0(D) decay e ex, he ans e se momen a o he B0,Dand K∗0, he cosine o he angle be ween he momen um di ec ion o he B0and he displacemen ec o om he PV o he B0decay e ex, he decay angle o he K∗0and he χ2o he kinema ic i o he whole decay chain. Since some o he a iables ha e di e en dis ibu ions o long o downs eam candida es, he wo e en ca ego ies ha e sepa a e BDTs, gi ing a o al o ou independen BDTs. The op imal cu alue o each BDT classi ie is chosen om pseudoexpe imen s o minimise he unce ain ies on z±. Pa icle iden i ica ion (PID) equi emen s a e applied o he daugh e s o he K∗0 o selec kaon-pion pai s and educe backg ound coming om B0→Dρ0decays. A speci ic e o is also applied o emo e con ibu ions om B±→DK±decays: B0→DK∗0candi- da es wi h a DK in a ian mass lying in a ±50 MeV window a ound he B±-meson mass a e emo ed. To ejec backg ound om D0→ππππ decays, he decay e ex o each long K0 Scandida e is equi ed o be signi ican ly displaced om he Ddecay e ex along he beam di ec ion. – 5 – JHEP08(2016)137 The decay B0 s→DK∗0has a simila opology o B0→DK∗0, bu exhibi s much less CP iola ion [30], since he decay B0 s→D0K∗0is doubly-Cabbibo supp essed compa ed o B0 s→D0K∗0. These decays a e used as a con ol channel in he in a ian mass i . Back- g ound om pa ially econs uc ed B0 (s)→D∗( ) K∗0decays, whe e D∗s ands o ei he D∗0o D∗0, a e di icul o exclude since hey ha e a opology e y simila o he signal. The D∗0→D0γand D∗0→D0π0decays whe e he pho on o he neu al pion is no econ- s uc ed lead o B0 (s)→D( ) K∗0candida es wi h a lowe in a ian mass han he B0 (s)mass. 4 E iciency ac oss he phase space The a ia ion o he de ec ion e iciency ac oss he phase space is due o de ec o accep- ance, igge and selec ion c i e ia and PID e ec s. To e alua e his a ia ion, a simula ed sample gene a ed uni o mly o e he D→K0 Sπ+π−phase space is used, a e applying co - ec ions o known di e ences be ween da a and simula ion ha a ise o he ha dwa e igge and PID equi emen s. The igge co ec ions a e de e mined sepa a ely o wo independen e en ca ego ies. In he i s ca ego y, e en s ha e a leas one ene gy deposi in he had onic calo ime e , associa ed wi h he signal decay, which passes he ha dwa e igge . In he second ca ego y, e en s a e igge ed only by pa icles p esen in he es o he e en , excluding he signal decay. The p obabili y ha a gi en ene gy deposi in he had onic calo ime e passes he ha dwa e igge is e alua ed wi h calib a ion samples, which a e p oduced o kaons and pions sepa a ely, and gi e he igge e iciency as a unc ion o he dipole magne pola i y, he ans e se ene gy and he hi posi ion in he calo ime e . The e iciency unc ions ob ained o he wo ca ego ies a e combined acco ding o hei p opo ions in da a. The PID co ec ions a e calcula ed wi h calib a ion samples o D∗+→D0π+,D0→ K−π+decays. A e backg ound sub ac ion, he PID e iciencies o kaon and pion candi- da es a e ob ained as unc ions o momen um and pseudo apidi y. The p oduc o he kaon and pion e iciencies, aking in o accoun hei co ela ion, gi es he o al PID e iciency. The a ious e iciency unc ions a e combined o make wo sepa a e global e iciency unc ions, one o long candida es and one o downs eam candida es, which a e used as inpu s o he i o ob ain he Ca esian obse ables z±. To smoo h ou s a is ical luc ua ions, an in e pola ion wi h a wo-dimensional cubic spline unc ion is pe o med o gi e a con inuous desc ip ion o he e iciency ε(m2 +, m2 −), as shown in igu e 1. 5 Analysis s a egy and i esul s To de e mine he CP obse ables z±de ined in eq. (1.6), an unbinned ex ended maximum likelihood i is pe o med in h ee a iables: he B0candida e econs uc ed in a ian mass mB0and he Dali z a iables m2 +and m2 −. This i is pe o med in wo s eps. Fi s , he signal and backg ound yields and some pa ame e s o he in a ian mass PDFs a e de e mined wi h a i o he econs uc ed B0in a ian mass dis ibu ion, desc ibed in sec ion 5.1. An ampli ude i o e he phase space o he D-meson decay is hen pe o med o measu e z±, using only candida es lying in a ±25 MeV window a ound he i ed B0 – 6 – JHEP08(2016)137 ) 2 (GeV − 2 m 1 2 3 ) 2 (GeV + 2 m 0.5 1 1.5 2 2.5 3 a bi a y uni s 0.4 0.6 0.8 1 Simula ion LHCb ) 2 (GeV − 2 m 1 2 3 ) 2 (GeV + 2 m 0.5 1 1.5 2 2.5 3 a bi a y uni s 0.4 0.6 0.8 1 Simula ion LHCb Figu e 1. Va ia ion o signal e iciency ac oss he phase space o (le ) long and ( igh ) downs eam candida es. mass, and aking he esul s o he in a ian mass i as inpu s, as explained in sec ion 5.2. The c i [45] lib a y has been used o pe o m hese i s. Candida e e en s a e di ided in o ou subsamples, acco ding o K0 S ype (long o downs eam), and whe he he candida e is iden i ied as a B0o B0-meson decay. In he B-candida e in a ian mass i , he B0 and B0samples a e combined, since iden ical dis ibu ions a e expec ed o his a iable, whils in he CP iola ion obse ables i (CP i ) hey a e kep sepa a e. 5.1 In a ian mass i o B0→DK∗0candida es An unbinned ex ended maximum likelihood i o he econs uc ed in a ian mass dis- ibu ions o he B0candida es in he ange [4900,5800] MeV de e mines he signal and backg ound yields. The long and downs eam subsamples a e i ed simul aneously. The o al PDF includes se e al componen s: he B0→DK∗0signal PDF, backg ound PDFs o B0 s→DK∗0decays, combina o ial backg ound, pa ially econs uc ed B0 (s)→D∗( ) K∗0 decays and misiden i ied B0→Dρ0decays, as illus a ed in igu e 2. The i model is simila o ha used in he analysis o B0→DK∗0decays wi h D-meson decays o wo-body inal s a es [25]. The B0→DK∗0and B0 s→DK∗0componen s a e each desc ibed as he sum o wo C ys al Ball unc ions [46] sha ing he same cen al alue, wi h he ela i e yields o he wo unc ions and he ail pa ame e s ixed om simula ion. The sepa a ion be ween he cen al alues o he B0→DK∗0and B0 s→DK∗0PDFs is ixed o he known B0-B0 smass di e ence. The a io o he B0→DK∗0and B0 s→DK∗0 yields is cons ained o be he same in bo h he long and downs eam subsamples. The combina o ial backg ound is desc ibed wi h an exponen ial PDF. Pa ially econs uc ed B0 (s)→D∗( ) K∗0decays a e desc ibed wi h non-pa ame ic unc ions ob ained by applying ke nel densi y es ima ion [47] o dis ibu ions o simula ed e en s. These dis ibu ions de- pend on he helici y s a e o he D∗0meson. Due o pa i y conse a ion in D∗0→D0γand D∗0→D0π0decays, wo o he h ee helici y ampli udes ha e he same in a ian mass dis- ibu ion. The B0 s→D∗K∗0PDF is he e o e a linea combina ion o wo non-pa ame ic – 7 – JHEP08(2016)137 m(DK*) (MeV) 5000 5200 5400 5600 5800 Candida es / [18 MeV] 0 20 40 60 80 100 LHCb 0 DK*→ 0 B 0 *K D→ 0 s B Combina o ial 0 D*K*→ 0 B 0 *K D*→ 0 s B 0 ρ D→ 0 B Figu e 2. In a ian mass dis ibu ion o B0→DK∗0long and downs eam candida es. The i esul , including signal and backg ound componen s, is supe imposed (solid blue). The poin s a e da a, and he di e en i componen s a e gi en in he legend. The wo e ical lines ep esen he signal egion in which he CP i is pe o med. unc ions, wi h he ac ion o he longi udinal pola isa ion in he B0 s→D∗K∗0decays unknown and accoun ed o wi h a ee pa ame e in he i . Each o he wo unc ions de- sc ibing he di e en helici y s a es is a weigh ed sum o non-pa ame ic unc ions ob ained om simula ed B0 s→D∗(D0γ)K∗0and B0 s→D∗(D0π0)K∗0decays, aking in o accoun he known D∗0→D0π0and D∗0→D0γb anching ac ions [48] and he app op ia e e i- ciencies. The PDF o B0→D∗K∗0decays is ob ained om ha o B0 s→D∗K∗0decays, by applying a shi co esponding o he known B0-B0 smass di e ence. In he nominal i , he pola isa ion ac ion is assumed o be he same o B0→D∗K∗0and B0 s→D∗K∗0 decays. The e ec o his assump ion is aken in o accoun in he sys ema ic unce ain ies. The B0→Dρ0componen is also desc ibed wi h a non-pa ame ic unc ion ob ained om he simula ion, using a da a-d i en calib a ion o desc ibe he pion-kaon misiden i ica ion e iciency. This componen has a e y low yield and, o imp o e he s abili y o he i , a Gaussian cons ain is applied, equi ing he a io o yields o B0→Dρ0and B0 s→DK∗0 o be consis en wi h i s expec ed alue. The i ed dis ibu ion is shown in igu e 2. The esul ing signal and backg ound yields in a ±25 MeV ange a ound he B0mass a e gi en in able 1. This ange co esponds o he signal egion o e which he CP i is pe o med. 5.2 CP i A simul aneous unbinned maximum likelihood i o he ou subsamples is pe o med o de e mine he CP iola ion obse ables z±. The alue o he cohe ence ac o is ixed o he – 8 – JHEP08(2016)137 To e alua e he sys ema ic unce ain y due o he choice o ampli ude model o D→K0 Sπ+π−, one million B0→DK∗0and one million B0 s→DK∗0decays a e simula ed acco ding o he nominal decay model, wi h he Ca esian obse ables ixed o he nominal i esul . These simula ed decays a e i ed wi h al e na i e models, each o which includes a single modi ica ion wi h espec o he nominal model, as desc ibed in he nex pa ag aph. Each o hese al e na i e models is i s used o i he simula ed B0 s→DK∗0decays o de- e mine alues o he esonance coe icien s o he model. Those coe icien s a e hen ixed in a second i , o he simula ed B0→DK∗0decays, o ob ain z±. The sys ema ic unce - ain ies a e aken o be he signed di e ences in he alues o z± om he nominal esul s. The ollowing changes, labelled (a)-(u), a e applied in he al e na i e models, leading o he unce ain ies shown in able 3: −ππ S-wa e: he F- ec o model is changed o use wo o he solu ions o he K-ma ix ( om a o al o h ee) de e mined om i s o sca e ing da a [53] (a), (b). The slowly a ying pa o he non esonan e m o he P- ec o is emo ed (c). −Kπ S-wa e: he gene alised LASS pa ame isa ion used o desc ibe he K∗ 0(1430)± esonance, is eplaced by a ela i is ic B ei -Wigne p opaga o wi h pa ame e s aken om e . [54] (d). −ππ P-wa e: he Gouna is-Saku ai p opaga o is eplaced by a ela i is ic B ei - Wigne p opaga o [19,49] (e). −Kπ P-wa e: he mass and wid h o he K∗(1680)− esonance a e a ied by hei unce ain ies om e . [50] ( )−(i). −ππ D-wa e: he mass and wid h o he 2(1270) esonance a e a ied by hei unce - ain ies om e . [24] (j)−(m). −Kπ D-wa e: he mass and wid h o he K∗ 2(1430)± esonance a e a ied by hei unce ain ies om e . [55] (n)−(q). −The adius o he Bla -Weisskop cen i ugal ba ie ac o s, BW, is changed om 1.5 GeV−1 o 0.0 GeV−1( ) and 3.0 GeV−1(s). −Two u he esonances, K∗(1410)0and ρ(1450), pa ame ised wi h ela i is ic B ei - Wigne p opaga o s, a e included in he model [19,49] ( ). −The Zemach o malism used o he angula dis ibu ion o he decay p oduc s is eplaced by he helici y o malism [19,49] (u). I esul s in o al sys ema ic unce ain ies a ising om he choice o ampli ude model o δx−= 8 ×10−3, δy−= 7 ×10−3, δx+= 10 ×10−3, δy+= 5 ×10−3. The di e en sys ema ic unce ain ies a e combined, assuming ha hey a e indepen- den o ob ain he o al expe imen al unce ain ies. Depending on he (x±, y±) pa ame e s, he leading sys ema ic unce ain ies a ise om he in a ian mass i , he desc ip ion o he non-Dbackg ound and he i biases. A la ge da a sample is expec ed o educe all h ee o – 15 – JHEP08(2016)137 Desc ip ion δx−δy−δx+δy+ (a) K-ma ix 1s solu ion −2 0.921 (b) K-ma ix 2nd solu ion 0.3 0.3 0.0−0.5 (c) Remo e slowly a ying −0.7 0.2 0.5 0.6 pa in P- ec o (d) Gene alised LASS 2 3 −1 3 → ela i is ic B ei -Wigne (e) Gouna is-Saku ai 0.7 0.0−0.1 0.8 → ela i is ic B ei -Wigne ( ) K∗(1680) m+δm −0.0 0.6 0.1 0.5 (g) m−δm −0.2−0.5 0.2−0.9 (h) Γ + δΓ−0.2 0.2 0.0−0.2 (i) Γ −δΓ 0.2−0.1 0.5−0.2 (j) 2(1270) m+δm −0.1 0.0 0.3−0.2 (k) m−δm −0.0 0.1 0.2−0.2 (l) Γ + δΓ−0.0 0.0 0.2−0.2 (m) Γ −δΓ−0.1 0.0 0.2−0.2 (n) K∗ 2(1430) m+δm 0.3 0.2 0.2−0.2 (o) m−δm −0.4−0.2 0.3−0.1 (p) Γ + δΓ−0.2 0.2 0.1−0.2 (q) Γ −δΓ 0.1−0.1 0.3−0.2 ( ) BW = 0.0 GeV−1−2 0.7−1−0.3 (s) BW = 3.0 GeV−14−242 ( ) Add K∗(1410) and ρ(1450) −0.2−0.2 0.3−0.3 (u) Helici y o malism −6 6 −8 2 To al model ela ed 8 7 10 5 Table 3. Model ela ed sys ema ic unce ain ies o each al e na i e model, in uni s o (10−3). The ela i e signs indica e ull co ela ion o an i-co ela ion. hese unce ain ies. Whils no in insically s a is ical in na u e, he sys ema ic unce ain y due o he desc ip ion o he non-Dbackg ound is p esen ly e alua ed using a conse a- i e app oach due o lack o s a is ics. The o al sys ema ic unce ain ies, including he model- ela ed unce ain ies, a e signi ican ly smalle han he s a is ical unce ain ies. – 16 – JHEP08(2016)137 7 De e mina ion o he pa ame e s γ, B0and δB0 To de e mine he physics pa ame e s B0,δB0and γ om he i ed Ca esian obse ables z±, he ela ions x±= B0cos(δB0±γ), y±= B0sin(δB0±γ),(7.1) mus be in e ed. This is done using he GammaCombo package, o iginally de eloped o he equen is combina ion o γmeasu emen s by he LHCb collabo a ion [7,56]. A global likelihood unc ion is buil , which gi es he p obabili y o obse ing a se o z± alues gi en he ue alues ( B0, δB0, γ), L(x−, y−, x+, y+| B0, δB0, γ).(7.2) All s a is ical and sys ema ic unce ain ies on z±a e accoun ed o , as well as he s a is ical co ela ion be ween z±. Since he p ecision o he measu emen is s a is ics domina ed, co - ela ions be ween he sys ema ic unce ain ies a e igno ed. Cen al alues o ( B0, δB0, γ) a e ob ained by pe o ming a scan o hese pa ame e s, o ind he alues ha maximise L(xobs −, yobs −, xobs +, yobs +| B0, δB0, γ), whe e zobs ±a e he measu ed alues o he Ca esian obse ables. Associa ed con idence in e als may be ob ained ei he om a simple p o ile- likelihood me hod, o using he Feldman-Cousins app oach [57] combined wi h a “plugin” me hod [58]. Con idence le el cu es o ( B0, δB0, γ) ob ained using he la e me hod a e shown in igu es 6,7and 8. The measu ed alues o z±a e ound o co espond o γ=80+21 −22◦, B0= 0.39 ±0.13, δB0=197+24 −20◦. In insic o he me hod used in his analysis [12], he e is a wo- old ambigui y in he solu- ion; he S anda d Model solu ion (0 < γ < 180)◦is chosen. Two-dimensional con idence le el cu es ob ained using he p o ile-likelihood me hod a e shown in igu es 9and 10. 8 Conclusion An ampli ude analysis o B0→DK∗0decays, employing a model desc ip ion o he D→ K0 Sπ+π−decay, has been pe o med using da a co esponding o an in eg a ed luminosi y o 3 b−1, eco ded by LHCb a a cen e-o -mass ene gy o 7 TeV in 2011 and 8 TeV in 2012. The measu ed alues o he CP iola ion obse ables x±= B0cos (δB0±γ) and y±= B0sin (δB0±γ) a e x−=−0.15 ±0.14 ±0.03 ±0.01, y−= 0.25 ±0.15 ±0.06 ±0.01, x+= 0.05 ±0.24 ±0.04 ±0.01, y+=−0.65 +0.24 −0.23 ±0.08 ±0.01, – 17 – JHEP08(2016)137 ]°[γ CL−1 0 0.2 0.4 0.6 0.8 1 50 100 150 22− +21 80 68.3% 95.5% LHCb Figu e 6. Con idence le el cu e on γ, ob ained using he “plugin” me hod [58]. 0 B 0.2 0.4 0.6 0.8 CL−1 0 0.2 0.4 0.6 0.8 1 0.13− +0.13 0.39 68.3% 95.5% LHCb Figu e 7. Con idence le el cu e on B0, ob ained using he “plugin” me hod [58]. whe e he i s unce ain ies a e s a is ical, he second a e sys ema ic and he hi d a e due o he choice o ampli ude model used o desc ibe he D→K0 Sπ+π−decay. These a e he mos p ecise measu emen s o hese obse ables ela ed o he neu al channel B0→DK∗0. They place cons ain s on he magni ude o he a io o he in e e ing B- meson decay ampli udes, he s ong phase di e ence be ween hem and he CKM angle γ, – 18 – JHEP08(2016)137 ]°[ 0 B δ CL−1 0 0.2 0.4 0.6 0.8 1 100 200 300 20− +24 197 68.3% 95.5% LHCb Figu e 8. Con idence le el cu e on δB0, ob ained using he “plugin” me hod [58]. Only he δB0solu ion co esponding o 0 < γ < 180◦is highligh ed; he o he maximum is due o he (δB0, γ)→(δB0+π, γ +π) ambigui y. ]° [γ 0 B 0 50 100 150 0 0.2 0.4 0.6 0.8 1 LHCb con ou s hold 68%, 95% CL Figu e 9. Two-dimensional con idence le el cu es in he (γ, B0) plane, ob ained using he p o ile- likelihood me hod. – 19 – JHEP08(2016)137 ]° [γ ]° [ 0 B δ 0 50 100 150 150 200 250 300 350 LHCb con ou s hold 68%, 95% CL Figu e 10. Two-dimensional con idence le el cu es in he (γ, δB0) plane, ob ained using he p o ile-likelihood me hod. gi ing he alues γ=80+21 −22◦, B0= 0.39 ±0.13, δB0=197+24 −20◦. He e, B0and δB0a e de ined o a Kπ mass egion o ±50 MeV a ound he K∗(892)0 mass and o an absolu e alue o he cosine o he K∗0decay angle g ea e han 0.4. These esul s a e consis en wi h, and ha e lowe o al unce ain ies han hose epo ed in e . [28], whe e a model independen analysis me hod is used. The wo esul s a e based on he same da a se and canno be combined. The consis ency shows ha a he cu en le el o s a is ical p ecision he assump ions used o ob ain he p esen esul a e jus i ied. Acknowledgmen s We exp ess ou g a i ude o ou colleagues in he CERN accele a o depa men s o he excellen pe o mance o he LHC. We hank he echnical and adminis a i e s a a he LHCb ins i u es. We acknowledge suppo om CERN and om he na ional agencies: CAPES, CNPq, FAPERJ and FINEP (B azil); NSFC (China); CNRS/IN2P3 (F ance); BMBF, DFG and MPG (Ge many); INFN (I aly); FOM and NWO (The Ne he - lands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Swi ze land); NASU (Uk aine); STFC (Uni ed King- dom); NSF (U.S.A.). We acknowledge he compu ing esou ces ha a e p o ided by CERN, IN2P3 (F ance), KIT and DESY (Ge many), INFN (I aly), SURF (The Ne he lands), PIC – 20 – JHEP08(2016)137 (Spain), G idPP (Uni ed Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Swi ze - land), IFIN-HH (Romania), CBPF (B azil), PL-GRID (Poland) and OSC (U.S.A.). We a e indeb ed o he communi ies behind he mul iple open sou ce so wa e packages on which we depend. 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