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. Indi idual g oups o membe s ha e ecei ed suppo om A H Founda ion (Ge -
many), EPLANET, Ma ie Sk lodowska-Cu ie Ac ions and ERC (Eu opean Union), Conseil
G´en´e al de Hau e-Sa oie, Labex ENIGMASS and OCEVU, R´egion Au e gne (F ance),
RFBR and Yandex LLC (Russia), GVA, Xun aGal and GENCAT (Spain), He chel Smi h
Fund, The Royal Socie y, Royal Commission o he Exhibi ion o 1851 and he Le e hulme
T us (Uni ed Kingdom).
Open Access. This a icle is dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (CC-BY 4.0), which pe mi s any use, dis ibu ion and ep oduc ion in
any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
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