measurement of the ckm angle
TRANSCRIPT
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2020-214LHCb-PAPER-2020-030
April 12, 2021
Measurement of the CKM angle γ
and B0s-B
0s mixing frequency
with B0s → D∓s h
±π±π∓ decays
LHCb collaboration†
Abstract
The CKM angle γ is measured for the first time from mixing-induced CP viola-tion between B0
s→ D∓s K±π±π∓ and B0
s→ D±s K∓π∓π± decays reconstructed in
proton-proton collision data corresponding to an integrated luminosity of 9 fb−1
recorded with the LHCb detector. A time-dependent amplitude analysis is per-formed to extract the CP -violating weak phase γ − 2βs and, subsequently, γ bytaking the B0
s −B0s mixing phase βs as an external input. The measurement yields
γ = (44 ± 12)◦ modulo 180◦, where statistical and systematic uncertainties arecombined. An alternative model-independent measurement, integrating over thefive-dimensional phase space of the decay, yields γ = (44 + 20
− 13)◦ modulo 180◦. More-
over, the B0s −B0
s oscillation frequency is measured from the flavour-specific controlchannel B0
s→ D−s π+π+π− to be ∆ms = (17.757± 0.007 (stat)± 0.008 (syst)) ps−1,
consistent with and more precise than the current world-average value.
Published in JHEP 03 (2021) 137
© 2021 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.
†Authors are listed at the end of this paper.
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1 Introduction
Within the Standard Model of particle physics, the charge-parity (CP ) symmetry betweenquarks and antiquarks is broken by a single complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1]. The unitarity of this matrix leads to the condi-tion VudV
∗ub + VcdV
∗cb + VtdV
∗tb = 0, where Vij are the complex elements of the CKM matrix.
This equation can be visualised as a triangle in the complex plane with angles α, β and γ.It is a key test of the Standard Model to verify this unitarity condition by over-constrainingthe CKM matrix with independent measurements sensitive to various distinct combina-tions of matrix elements. In particular, measurements of γ ≡ arg[−(VudV
∗ub)/(VcdV
∗cb)] in
tree-level decays provide an important benchmark of the Standard Model, to be comparedwith loop-level measurements of γ and other CKM parameters. The world-average value,γ =
(72.1 + 4.1
− 4.5
)◦[2, 3], is dominated by a combination of LHCb measurements obtained
from analyses of beauty meson decays to open-charm final states [4]. Improved directmeasurements are needed to set a conclusive comparison to the indirect predictions fromglobal CKM fits, γ =
(65.7 + 0.9
− 2.7
)◦[5] or γ = (65.8± 2.2)◦ [6].
This paper presents the first measurement of the CKM angle γ with B0s→ D∓s K
±π±π∓
decays.1 The data set is collected with the LHCb experiment in proton-proton (pp)collisions at centre-of-mass energies2 of 7, 8 and 13 TeV, corresponding to an integratedluminosity of 9 fb−1. In these decays, the interference between b→ c and b→ u quark-leveltransitions achieved through B0
s −B0s mixing provides sensitivity to the CP -violating weak
phase γ−2βs [7,8]. The mixing phase, βs, is well constrained from B0s → J/ψK+K− [9,10]
and related decays [11–14] and is taken as an external input. The leading-order Feynmandiagrams for B0
s → D−s K+π+π− and B0
s → D−s K+π+π− decays are shown in Fig. 1. The
amplitudes for both processes are of the same order in the Wolfenstein parameter λ [15],O(λ3), so that interference effects are expected to be large. To account for the strong-phasevariation across the phase space of the decay, a time-dependent amplitude analysis is per-formed. An alternative, model-independent approach analyses the phase-space integrateddecay-time spectrum and is pursued as well; this method is conceptually similar to theanalysis of B0
s→ D∓s K± decays [16,17]. However, a coherence factor needs to be introduced
as an additional hadronic parameter, which dilutes the observable CP asymmetry sinceconstructive and destructive interference effects cancel when integrated over the entirephase space. The topologically similar and flavour-specific decay B0
s→ D−s π+π+π− is used
to calibrate detector-induced effects. This mode is also employed to make a precise mea-surement of the B0
s−B0s mixing frequency, which can be related to one side of the unitarity
triangle. The relative branching fraction of these decay modes was measured by LHCb tobe B(B0
s→ D∓s K±π±π∓)/B(B0
s→ D−s π+π+π−) = (5.2± 0.5(stat)± 0.3(syst)) % [18].
The paper is structured as follows. After introducing the amplitude analysis formalismand the differential decay rates in Sec. 2, the LHCb detector, the event reconstruction andcandidate selection are described in Sec. 3. Section 4 presents the measurement of theB0s mixing frequency, followed by the analysis of the B0
s→ D∓s K±π±π∓ signal channel
in Sec. 5. Experimental and model-dependent systematic uncertainties are evaluated inSec. 6, the results are discussed in Sec. 7, and our conclusions are given in Sec. 8.
1Inclusion of charge-conjugate modes is implied throughout except where explicitly stated.2Natural units with ~ = c = 1 are used throughout the paper.
1
ss s
b c
W+
s
uK+π+π−
D−sB0s
s s
b u
W−
c
sD−s
K+π+π−B0s
Figure 1: Leading-order Feynman diagrams for (left)B0s and (right)B0
s decays to theD−s K+π+π−
final state, where the π+π− subsystem exemplarily hadronises in conjunction with the kaon.
2 Phenomenology of the decay
Assuming tree-level processes are dominant, transitions from B0s and B0
s flavour eigenstatesto the final state f = D−s K
+π+π− are described by decay amplitudes
〈f |B0s 〉 ≡ Ac(x), 〈f |B0
s〉 ≡ r ei(δ−γ) Au(x), (1)
with a relative magnitude r and (constant) strong- and weak-phase differences δ andγ, respectively. A set of five independent kinematic observables (e.g. invariant-masscombinations of the final state particles or helicity angles) fully describes the phase spacex of the decay. The hadronic amplitudes Ac(x) and Au(x), where the superscript c (u)refers to a b→ c (b→ u) quark-level transition, contain the strong-interaction dynamicsand are given by a coherent sum over intermediate-state amplitudes, Ai(x):
Ac(x) =∑i
aci Ai(x), Au(x) =∑i
aui Ai(x). (2)
The complex amplitude coefficients aci and aui need to be determined from data. Sincethe hadronisation process is different for B0
s → f and B0s → f decays, their respective
amplitude coefficients are distinct (aci 6= aui ). To ensure that the parameters r and δ donot depend on the convention employed for the amplitude coefficients, the magnitudesquared of the hadronic amplitudes is normalised to unity when integrated over the phasespace (with four-body phase-space element dΦ4) and the overall strong-phase differencebetween Ac(x) and Au(x) is set to zero, i.e.
∫|Ac(x)|2 dΦ4(x) =
∫|Au(x)|2 dΦ4(x) = 1
and arg(∫
Ac(x)∗Au(x) dΦ4(x))
= 0. Within this convention, the decay fractions andinterference fractions for b→ c (b→ u) transitions are defined as
Fc(u)i ≡
∫ ∣∣∣ac(u)i Ai(x)
∣∣∣2 dΦ4, Ic(u)ij ≡
∫2 Re[a
c(u)i a
c(u)∗j Ai(x)A∗j(x)] dΦ4. (3)
2.1 Amplitude formalism
The isobar model is used to construct the intermediate-state amplitudes Ai(x) [19–21].Within this model, the four-body decay B0
s → h1 h2 h3 h4 proceeds via two isobar states R1
and R2 (typically associated to intermediate resonances), which gives rise to two distinctdecay topologies; quasi-two-body decays B0
s → (R1 → h1 h2) (R2 → h3 h4) or cascade
2
decays B0s → h1 [R1 → h2 (R2 → h3 h4)]. The intermediate-state amplitude for a given
decay channel i can be parameterised as
Ai(x) = BLB0s(x) [BLR1
(x)TR1(x)] [BLR2(x)TR2(x)]Si(x) , (4)
where the form factors BL account for deviations from point-like interactions and thepropagator TR describes the lineshape of resonance R. The angular correlation of thefinal state particles subject to total angular momentum conservation is encoded in thespin factor S. The Blatt–Weisskopf penetration factors [22,23] are used as form factors.They depend on the effective interaction radius rBW, the break-up momentum b and theorbital angular momentum L between the final-state particles. The explicit expressionsfor L = {0, 1, 2} are
B0(b) = 1, B1(b) = 1/√
1 + (b rBW)2 and B2(b) = 1/√
9 + 3 (b rBW)2 + (b rBW)4. (5)
Resonance lineshapes are described by Breit–Wigner propagators,
TR(s) = (m20 − s− i
√sΓ(s))−1, (6)
where s is the square of the centre-of-mass energy. The energy-dependent decay width,Γ(s), is normalised to give the nominal width, Γ0, when evaluated at the nominal mass,m0. For a decay into two stable particles R→ AB, the energy dependence of the decaywidth can be described by
√sΓR→AB(s) = m0 Γ0
m0√s
(b
b0
)2L+1BL(b)2
BL(b0)2, (7)
where b0 is the value of the break-up momentum at the resonance pole [24]. Specialisedlineshape parameterisations are used for the f0(500)0 (Bugg [25]), K∗0 (1430)0 (LASS [26,27])and ρ(770)0 (Gounaris–Sakurai [28]) resonances. The lineshapes for non-resonant statesare set to a constant.
The energy-dependent width for a three-body decay R → ABC is computed byintegrating the squared transition amplitude over the phase space,
ΓR→ABC(s) =1
2√s
∫|AR→ABC |2 dΦ3, (8)
as described in Ref. [29]. For the K1(1270)+ → ρ(770)0K+ cascade decay chain, mixingbetween the ρ(770)0 and ω(782)0 states is included [30], with relative magnitude andphase fixed to the values determined in Ref. [31]. More details are given in Appendix A.The spin factors are constructed in the covariant Zemach (Rarita–Schwinger) tensorformalism [32–36]. The explicit expressions for the decay topologies relevant for thisanalysis are taken from Refs. [29, 37].
2.2 Decay rates
Since B0s mesons can convert into B0
s and vice versa, the flavour eigenstates are anadmixture of the physical mass eigenstates BL and BH ,
|BL〉 = p|B0s 〉+ q|B0
s〉, |BH〉 = p|B0s 〉 − q|B0
s〉, (9)
3
where the complex coefficients are normalised such that |p|2 + |q|2 = 1. The light, |BL〉,and heavy, |BH〉, mass eigenstates have distinct masses, mL and mH , and decay widths, ΓLand ΓH . Their arithmetic means (differences) are denoted as ms and Γs (∆ms = mH−mL
and ∆Γs = ΓL − ΓH). The time evolution of the flavour and mass eigenstates can bedescribed by an effective Schrodinger equation [38,39] resulting in the following differentialdecay rates for initially produced B0
s or B0s mesons [39–41]:
dΓ(B0s → f)
dtdΦ4(x)= |〈f |B0
s (t)〉|2 ∝ e−Γs t
[(|Ac(x)|2 + r2 |Au(x)|2
)cosh
(∆Γs t
2
)+(|Ac(x)|2 − r2 |Au(x)|2
)cos (∆ms t)
− 2 rRe(Ac(x)∗Au(x) ei(δ−(γ−2βs))
)sinh
(∆Γs t
2
)− 2 r Im
(Ac(x)∗Au(x) ei(δ−(γ−2βs))
)sin (∆ms t)
],
dΓ(B0s → f)
dtdΦ4(x)= |〈f |B0
s(t)〉|2 ∝ e−Γs t
[(|Ac(x)|2 + r2 |Au(x)|2
)cosh
(∆Γs t
2
)−(|Ac(x)|2 − r2 |Au(x)|2
)cos (∆ms t)
− 2 rRe(Ac(x)∗Au(x) ei(δ−(γ−2βs))
)sinh
(∆Γs t
2
)+ 2 r Im
(Ac(x)∗Au(x) ei(δ−(γ−2βs))
)sin (∆ms t)
].
(10)
Here, the magnitude of q/p is set to unity (i.e. no CP violation in mixing is as-sumed [3,42,43]) and the phase between q and p can be related to the mixing phase βs,arg(q/p) ≈ −2βs [40, 41]. The decay rates to the CP -conjugate final state f = D+
s K−π−π+
(with phase-space point x ≡ CP x), dΓ(B0s → f) and dΓ(B0
s → f), follow from the ex-pressions for dΓ(B0
s → f) and dΓ(B0s → f) in Eq. (10) by replacing Ac(x) → Ac(x),
Au(x) → Au(x) and −(γ − 2βs) → +(γ − 2βs). This assumes no CP violation in thehadronic decay amplitudes (i.e., aci = aci and aui = aui ), as expected for tree-level-dominateddecays.
It is also instructive to examine the decay rates as functions of the decay time only, bymarginalising the phase space
dΓ(B0s → f)
dt∝
[cosh
(∆Γs t
2
)+ Cf cos (∆ms t)
+A∆Γf sinh
(∆Γs t
2
)− Sf sin (∆ms t)
]e−Γs t,
dΓ(B0s → f)
dt∝
[cosh
(∆Γs t
2
)− Cf cos (∆ms t)
+A∆Γf sinh
(∆Γs t
2
)+ Sf sin (∆ms t)
]e−Γs t.
(11)
Analogous expressions for the CP -conjugate processes can be written by replacing A∆Γf
with A∆Γf
and Sf with −Sf , where the CP coefficients are defined as
4
Cf =1− r2
1 + r2,
A∆Γf = −2 r κ cos (δ − (γ − 2βs))
1 + r2, A∆Γ
f = −2 r κ cos (δ + (γ − 2βs))
1 + r2,
Sf = +2 r κ sin (δ − (γ − 2βs))
1 + r2, Sf = − 2 r κ sin (δ + (γ − 2βs))
1 + r2. (12)
The coherence factor, κ, results from the integration over the interfering amplitudes acrossthe phase space, κ ≡
∫Ac(x)∗Au(x) dΦ4. It is bounded between zero and unity, and
dilutes the sensitivity to the weak phase. For the two-body decay B0s → D∓s K
± thecoherence factor is κ = 1 [17]. Measured values of A∆Γ
f 6= A∆Γf
or Sf 6= −Sf signifytime-dependent CP violation and lead to different mixing asymmetries for decays into thef or f final states. These mixing asymmetries are defined as [7, 8]
Afmix(t) =Nf (t)− Nf (t)
Nf (t) + Nf (t)=Cf cos (∆ms t)− Sf sin (∆ms t)
cosh(
∆Γs t2
)+ A∆Γ
f sinh(
∆Γs t2
) ,Afmix(t) =
Nf (t)−Nf (t)
Nf (t) +Nf (t)=Cf cos (∆ms t) + Sf sin (∆ms t)
cosh(
∆Γs t2
)+ A∆Γ
fsinh
(∆Γs t
2
) , (13)
where Nf (t) (Nf (t)) and Nf (t) (Nf (t)) denote the number of initially produced B0s (B0
s)mesons decaying at proper time t to the final states f and f , respectively. Flavour-specificdecay modes such as B0
s→ D−s π+π+π− have r = 0 and consequently Cf = 1 as well as
A∆Γf = A∆Γ
f= Sf = Sf = 0.
3 Event reconstruction
The LHCb detector [44, 45] is a single-arm forward spectrometer covering thepseudorapidity range 2 < η < 5, designed for the study of particles containing b orc quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector (VELO) surrounding the pp interaction region [46], a large-areasilicon-strip detector located upstream of a dipole magnet with a bending power of about4 Tm, and three stations of silicon-strip detectors and straw drift tubes [47, 48] placeddownstream of the magnet. The polarity of the dipole magnet is reversed periodicallythroughout the data-taking process to control systematic asymmetries. The trackingsystem provides a measurement of the momentum, p, of charged particles with a relativeuncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV. The minimumdistance of a track to a primary vertex (PV), the impact parameter (IP), is measured witha resolution of (15+29/pT)µm, where pT is the component of the momentum transverse tothe beam, in GeV. Different types of charged hadrons are distinguished using informationfrom two ring-imaging Cherenkov detectors [49]. The online event selection is performedby a trigger [50], which consists of a hardware stage, based on information from thecalorimeter and muon systems, followed by a software stage, which applies a full eventreconstruction. At the hardware trigger stage, events are required to have a muon withhigh pT or a hadron, photon or electron with high transverse energy in the calorimeters.For hadrons, the transverse energy threshold is 3.5 GeV. The software trigger requires
5
a two-, three- or four-track secondary vertex with a significant displacement from anyprimary pp interaction vertex. At least one charged particle must have a transversemomentum pT > 1.6 GeV and be inconsistent with originating from a PV. A multivariatealgorithm [51] is used for the identification of secondary vertices consistent with the decayof a b hadron. The data-taking period from 2011 to 2012 with centre-of-mass energies of 7and 8 TeV (2015 to 2018 with centre-of-mass energy of 13 TeV) is referred to as Run 1(Run 2) throughout the paper.
Simulated events are used to study the detector acceptance and specific backgroundcontributions. In the simulation, pp collisions are generated using Pythia [52] witha specific LHCb configuration [53]. Decays of hadrons are described by EvtGen [54],in which final-state radiation is generated using Photos [55]. The simulated signaldecays are generated according to a simplified amplitude model with an additional purephase-space component. The interaction of the generated particles with the detector, andits response, are implemented using the Geant4 toolkit [56] as described in Ref. [57].
3.1 Candidate selection
The selection of B0s→ D∓s K
±π±π∓ and B0s→ D−s π
+π+π− candidates is performed by firstreconstructing D−s → K−K+π−, D−s → K−π−π+ and D−s → π−π+π− candidates fromcharged particle tracks with high momentum and transverse momentum originating from acommon displaced vertex. Particle identification (PID) information is used to assign a kaonor pion hypothesis to the tracks. Candidate D−s mesons with a reconstructed invariantmass within 25 MeV of the known D−s mass [2] are combined with three additional chargedtracks to form a B0
s vertex, which must be displaced from any PV. The PV with respectto which the B0
s candidate has the smallest impact parameter significance is considered asthe production vertex. The reconstructed invariant mass of the B0
s candidate is requiredto be between 5200 MeV and 5700 MeV. The mass resolution is improved by performing akinematic fit [58] where the B0
s candidate is constrained to originate from the PV and thereconstructed D−s mass is constrained to the world-average D−s mass [2]. When deriving thedecay time, t, of the B0
s candidate and the phase-space observables, x, the reconstructedB0s mass is constrained to its known value [2]. The B0
s proper time is required to be largerthan 0.4 ps to suppress most of the prompt combinatorial background. The consideredphase-space region is limited to m(K+π+π−) < 1950 MeV, m(K+π−) < 1200 MeV andm(π+π−) < 1200 MeV since the decay proceeds predominantly through the low-massaxial-vector states K1(1270)+ and K1(1400)+ [18], while the combinatorial background isconcentrated at high K+π+π−, K+π− and π+π− invariant masses. A combination of PIDinformation and kinematic requirements is used to veto charmed meson or baryon decaysreconstructed as D−s candidates due to the misidentification of protons or pions as kaons.
A boosted decision tree (BDT) [59, 60] with gradient boosting is used to suppressbackground from random combinations of charged particles. The multivariate classifieris trained using a background-subtracted B0
s→ D−s π+π+π− data sample as signal proxy,
while B0s→ D∓s K
±π±π∓ candidates with invariant mass greater than 5500 MeV are usedas background proxy. The features used in the BDT are topological variables related tothe vertex separation, such as the impact parameters of the B0
s candidate and final-stateparticles, the flight distance of the D−s candidate with respect to the secondary vertex, aswell as several criteria on the track quality and vertex reconstruction and estimators ofthe isolation of the B0
s candidate from other tracks in the event. The working point of
6
the BDT classifier is chosen to optimise the significance of the B0s→ D∓s K
±π±π∓ signal.
3.2 Data sample composition
Irreducible background contributions to the selected B0s → D−s π
+π+π− andB0s→ D∓s K
±π±π∓ data samples are disentangled from signal decays on a statistical basisby means of an extended maximum likelihood fit to the reconstructed m(D∓s h
±π±π∓)invariant mass, where h is either a pion or a kaon. A Johnson’s SU function [61] is used asprobability density function (PDF) for the signal component. The shape parameters are ini-tially determined from simulation. To account for small differences between simulation anddata, scale factors for the mean and standard deviation of the signal PDF are introduced.These are determined from a fit to the B0
s→ D−s π+π+π− calibration sample and thereafter
fixed when fitting B0s→ D∓s K
±π±π∓ candidates. Background decays of B0 mesons aredescribed by the same PDF shifted by the known mass difference between B0
s and B0
mesons [2]. The combinatorial background is modelled with a second-order polynomialfunction. The shapes for partially reconstructed B0
s → D∗−s π+π+π−, B0s → D∗−s K+π+π−
and B0 → D∗−s K+π+π− decays, where the D∗−s meson decays to D−s γ or D−s π0, are derived
from simulated decays. The same applies to the shape for misidentified B0s→ D−s π
+π+π−
and B0s → D∗−s π+π+π− decays contributing to the B0
s → D∓s K±π±π∓ sample. The
expected yields of these cross-feed background contributions are estimated by determiningthe probability of a pion to pass the PID requirement imposed on the kaon candidate froma control sample of D∗+ → (D0 → K−π+) π+ decays [62]. All other yields are determinedfrom the fit.
Figure 2 displays the invariant mass distributions of B0s → D−s π
+π+π− andB0s→ D∓s K
±π±π∓ candidates with fit projections overlaid. A signal yield of 148 000± 400(7500±100) is obtained for B0
s→ D−s π+π+π− (B0
s→ D∓s K±π±π∓) decays. The results are
used to assign weights to the candidates to statistically subtract the background with thesPlot technique [63]. Here, the m(D∓s h
±π±π∓) invariant mass is used as discriminatingvariable when performing fits to the decay-time and phase-space distributions [64].
) [MeV]−π
+π
+π
−
sD(m
5200 5300 5400 5500 5600 5700
Can
did
ates
/ (
5 M
eV)
5000
10000
15000
20000LHCb
Data
Fit−
π+
π+
π−
sD→s
0B
−π
+π
+π
−
sD→0
B
Comb. bkg.
Part. reco. bkg.
) [MeV]
±
π±π±
K
±
sD(m
5200 5300 5400 5500 5600 5700
Can
did
ates
/ (
5 M
eV)
200
400
600
800
1000
1200
1400 LHCbData
Fit
±
π±π±
K
±
sD→s
0B
±
π±π±
K
±
sD→0
B
Comb. bkg.
Part. reco. bkg.
MisID bkg.
Figure 2: Invariant mass distribution of selected (left) B0s → D−s π
+π+π− and (right)B0s→ D∓s K
±π±π∓ candidates with fit projections overlaid.
7
4 Measurement of the B0s mixing frequency
A likelihood fit to the background-subtracted decay-time spectrum of the B0s→ D−s π
+π+π−
control channel is performed in order to calibrate detector-induced effects and flavour-tagging algorithms as well as to measure the mixing frequency ∆ms.
4.1 Decay-time resolution
Excellent decay-time resolution is essential in order to resolve the fast B0s −B0
s mixing.The global kinematic fit to the decay topology provides an estimate of the decay-timeresolution for each candidate. The per-candidate decay-time uncertainty, δt, is calibratedby reconstructing B0
s candidates from particles originating directly from the PV. Theseprompt B0
s candidates have a known true decay time of zero, such that the width of thedecay-time distribution is a measure of the true resolution, σt. It is determined in equallypopulated slices of δt. A linear calibration function is used to map the per-candidatedecay-time uncertainty to the actual resolution. On average, the resolution amounts to〈σt〉 = 36.6 ± 0.5 fs, where the uncertainty is statistical only. A decay-time bias, µt, ofapproximately −2 fs is observed which results from the applied selection requirements andthe precision of the detector alignment. The implementation of the decay-time bias anddecay-time resolution in the fit are discussed in Sec. 4.3.
4.2 Flavour tagging
Two complementary methods are used to determine the flavour of the B0s candidates
at production. The opposite-side (OS) tagger [65, 66] exploits the fact that b quarksare predominantly produced in quark-antiquark pairs, which leads to a second b hadronalongside the signal B0
s meson. The flavour of the non-signal b hadron is determinedusing the charge of the lepton (µ, e) produced in semileptonic decays, the charge of areconstructed secondary charm hadron, the charge of the kaon from the b → c → sdecay chain, and/or the charge of the inclusive secondary vertex reconstructed from theb hadron decay products. The same-side (SS) tagger [67] determines the flavour of thesignal candidate by identifying the charge of the kaon produced together with the B0
s
meson in the fragmentation process.Each tagging algorithm provides a flavour-tagging decision, d, which takes the value
d = +1 (d = −1) for a candidate tagged as a B0s (B0
s) meson and d = 0 if no decisioncan be made (untagged). The tagging efficiency εtag is defined as the fraction of selectedcandidates with non-zero tag decision. The tagging algorithms also provide an estimate,η, of the probability that the decision is incorrect. The tagging decision and mistagestimate are obtained using flavour specific, self-tagging, decays. Multivariate classifiersare employed combining various inputs such as kinematic variables of tagging particles andof the signal candidate. These are trained on simulated samples of B0
s→ D−s π+ decays
for the SS tagger and on data samples of B+ → J/ψK+ decays for the OS tagger. Themistag estimate does not necessarily represent the true mistag probability, ω, since thealgorithms might perform differently on the signal decay than on the decay modes usedfor the training. Therefore, it is calibrated for each tagger using the flavour-specific decayB0s→ D−s π
+π+π−.
8
4.3 Decay-time fit
The signal PDF describing the B0s→ D−s π
+π+π− proper-time spectrum is based on thetheoretical decay rate in Eq. (11) taking several experimental effects into account [16,17,68,69]
P(t, dOS, dSS, qf |δt, ηOS, ηSS) ∝[(1− qfAD) p(t′, dOS, dSS, qf |δt, ηOS, ηSS) e−Γs t′ ⊗R(t− t′|µt, σt(δt))
]ε(t),
where
p(t′, dOS, dSS, qf |δt, ηOS, ηSS) ≡[(1− AP )h(dOS, ηOS)h(dSS, ηSS)
(cosh
(∆Γs t
′
2
)+ qf cos (∆ms t
′)
)
+(1 + AP )h(qOS, ηOS)h(dSS, ηSS)
(cosh
(∆Γs t
′
2
)− qf cos (∆ms t
′)
)],
h(d, η) ≡ |d|
[1 + d
(1− 2w(η)
)]εtag + 2 (1− |d|) (1− εtag),
h(d, η) ≡ |d|
[1− d
(1− 2 w(η)
)]εtag + 2 (1− |d|) (1− εtag). (14)
This PDF depends on the tagging decisions of the OS and SS taggers, dOS and dSS; onthe final state, qf = +1(−1) for f = D−s π
+π+π−(f = D+s π−π−π+); and is conditional on
the per-event observables [70] δt, ηOS and ηSS, describing the estimated decay-time errorand the estimated mistag rates of the OS and SS taggers, respectively. The parametersof the Gaussian resolution model, R(t− t′|µt, σt(δt)), are fixed to the values determinedfrom the prompt candidate data sample. The decay-time-dependent efficiency, ε(t), ofreconstructing and selecting signal decays is modelled by a B-spline curve [71,72], whosecubic polynomials are uniquely defined by a set of knots. These are placed across theconsidered decay-time range to account for local variations [73]. Six knots are chosensuch that there is an approximately equal amount of data in-between two consecutiveknots. By fixing the decay width Γs = (0.6624± 0.0018) ps−1 and decay width difference∆Γs = (0.090± 0.005) ps−1 to their world-average values [3], the spline coefficients can bedirectly determined in the fit to the data. The correlation ρ(Γs, ∆Γs) = −0.080 is takeninto account when evaluating the systematic uncertainties.
Both taggers are simultaneously calibrated during the fit as described by the functionsh(d, η) and h(d, η). These also take into account a small dependence of the taggingperformance on the initial B0
s flavour by introducing linear calibration functions for initialB0s and B0
s mesons denoted as ω(η) and ω(η), respectively. Similarly, εtag (εtag) denotesthe tagging efficiency for an initial B0
s (B0s) meson. The calibrated responses of the OS
and SS taggers are then explicitly combined in the PDF. Since the tagging algorithms havebeen retuned for the Run 2 data-taking period [74] to account for the changed conditions,separate calibrations for the two data-taking periods are performed.
The production asymmetries between B0s and B0
s mesons at centre-of-mass energies of
9
7 TeV and 8 TeV are taken from an LHCb measurement using B0s→ D−s π
+ decays [75].After correcting for kinematic differences between B0
s → D−s π+ and B0
s → D−s π+π+π−
decays, the effective production asymmetry, AP = (N(B0s)−N(B0
s ))/(N(B0s) +N(B0
s )),for Run 1 data amounts to AP = (−0.1 ± 1.0)%. The production asymmetry at acentre-of-mass energy of 13 TeV is determined in the fit. A detection asymmetry betweenthe final states is caused by the different interaction cross-sections of positively andnegatively charged kaons with the detector material. The detection asymmetry is definedas AD = (ε(f)− ε(f))/(ε(f) + ε(f)), where ε(f) (ε(f)) denotes the detection efficiency offinal state f (f). It is computed by comparing the charge asymmetries in D± → K∓π±π±
and D± → K0Sπ± calibration samples [76], weighted to match the kinematics of the
signal kaon. Only the decay mode D−s → K−π−π+ is a possible source of detectionasymmetry for B0
s→ D−s π+π+π− decays resulting in an average detection asymmetry of
AD = (−0.07±0.15)% for Run 1 and AD = (−0.08±0.21)% for Run 2 data. A sufficientlylarge subsample of the Run 2 data set is used to reconstruct the calibration modes forthis study.
Figure 3 displays the decay-time distribution and the mixing asymmetry forB0s→ D−s π
+π+π− signal candidates. The latter is weighted by the per-candidate timeresolution and flavour-tagging dilution to enhance the visible asymmetry. All features arewell reproduced by the fit projections which are overlaid. The B0
s −B0s mixing frequency
[ps]t
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π+
π+
π−
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−π
+π
+π
−
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0B
) [ps]sm∆/π2 modulo (t
0 0.1 0.2 0.3
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ixA
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0.1−
0
0.1
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π+π+π−
sD→s
0B
Figure 3: Background-subtracted decay-time distribution of (top) all and (bottom left) taggedB0s→ D−s π
+π+π− candidates as well as (bottom right) the dilution-weighted mixing asymmetryfolded into one oscillation period along with the fit projections (solid lines). The decay-timeacceptance (top) is overlaid in an arbitrary scale (dashed line).
10
Table 1: The flavour-tagging performance for only OS-tagged, only SS-tagged and both OS- andSS-tagged B0
s→ D−s π+π+π− signal candidates.
(a) Run 1 data.
εtag[%] 〈ω〉[%] εeff[%]
Only OS 14.74 ± 0.11 39.09 ± 0.80 1.25 ± 0.16Only SS 35.38 ± 0.18 44.26 ± 0.62 1.05 ± 0.18
Both OS-SS 33.04 ± 0.30 37.33 ± 0.73 3.41 ± 0.33
Combined 83.16 ± 0.37 40.59 ± 0.70 5.71 ± 0.40
(b) Run 2 data.
εtag[%] 〈ω〉[%] εeff[%]
Only OS 11.91 ± 0.04 37.33 ± 0.41 1.11 ± 0.05Only SS 40.95 ± 0.08 42.41 ± 0.29 1.81 ± 0.10
Both OS-SS 28.96 ± 0.12 35.51 ± 0.32 3.61 ± 0.13
Combined 81.82 ± 0.15 39.23 ± 0.32 6.52 ± 0.17
is determined to be∆ms = (17.757± 0.007± 0.008) ps−1,
where the first uncertainty is statistical and the second systematic. The systematicstudies are discussed in Sec. 6. Within uncertainties, the measured production asymmetryfor Run 2 data, AP = (0.1 ± 0.5 ± 0.1)%, is consistent with zero. The calibrated per-candidate mistag probabilities, ωi, are used to compute the effective tagging power asεeff = 1
N
∑i(1 − 2ωi)
2, where N is the total number of signal candidates and a valueof ωi = 0.5 is assigned to untagged candidates. Table 1 reports the observed taggingperformance for Run 1 and Run 2 data considering three mutually exclusive categories:tagged by the OS combination algorithm only, tagged by the SS kaon algorithm onlyand tagged by both OS and SS algorithms. While the flavour taggers suffer from thehigher track multiplicity during the Run 2 data-taking period, they profit from the hardermomentum spectrum of the produced bb quark pair. Combined, this results in a netrelative improvement of 14% in effective tagging power.
5 Measurement of the CKM angle γ
This section first describes the phase-space-integrated decay-time analysis of the signalchannel B0
s→ D∓s K±π±π∓ that allows the determination of the CKM angle γ in a model-
independent way. Afterwards, the resonance spectrum in B0s→ D∓s K
±π±π∓ decays isstudied and a full time-dependent amplitude analysis is performed for a model-dependentdetermination of the CKM angle γ.
5.1 Model-independent analysis
The decay-time fit to the B0s→ D∓s K
±π±π∓ candidates uses a signal PDF based on Eq. (11)with modifications accounting for the experimental effects described in Sec. 4.3. The B0
s
production asymmetry for Run 2 data and the B0s −B0
s mixing frequency are fixed to the
11
[ps]t
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.2 p
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LHCb
±
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±
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) [ps]sm∆/π2 modulo (t0 0.1 0.2 0.3
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ixA
0.5−
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0
0.1
0.2
0.3
0.4
0.5LHCb
−π+π+K−
s=Df
+π−π−K+
s=Df
Figure 4: Decay-time distribution of (left) background-subtracted B0s→ D∓s K
±π±π∓ candidatesand (right) dilution-weighted mixing asymmetry along with the model-independent fit projections(lines). The decay-time acceptance (left) is overlaid in an arbitrary scale (dashed line).
Table 2: CP coefficients determined from the phase-space fit to the B0s→ D∓s K
±π±π∓ decay-timedistribution. The uncertainties are statistical and systematic (discussed in Sec. 6).
Fit parameter Value
Cf 0.631 ± 0.096 ± 0.032
A∆Γf −0.334 ± 0.232 ± 0.097
A∆Γf
−0.695 ± 0.215 ± 0.081
Sf −0.424 ± 0.135 ± 0.033
Sf −0.463 ± 0.134 ± 0.031
values obtained from the B0s→ D−s π
+π+π− data sample, whereas the tagging calibrationparameters are allowed to vary within Gaussian constraints taking into account theircorrelation. The kaon detection asymmetry for B0
s→ D∓s K±π±π∓ decays is determined in
a similar manner as for B0s→ D−s π
+π+π− decays and amounts to AD = (−1.02± 0.15)%for Run 1 and AD = (−0.91± 0.22)% for Run 2 data. The decay-time acceptance is alsofixed to the B0
s → D−s π+π+π− result, corrected by a decay-time dependent correction
factor derived from simulation to account for small differences in the selection anddecay kinematics between the decay modes. Otherwise, the fit strategy is identicalto that discussed in the previous section. Figure 4 shows the decay-time distributionand mixing asymmetries together with the fit projections. The mixing asymmetries forD−s K
+π+π− and D+s K
−π−π+ final states are shifted with respect to each other indicatingmixing-induced CP violation, cf. Eq. (13). The CP coefficients Cf , A
∆Γf , A∆Γ
f, Sf and Sf
determined from the fit are reported in Table 2. They are converted to the parameters ofinterest r, κ, δ and γ − 2βs in Sec. 7.
5.2 Time-dependent amplitude analysis
To perform the time-dependent amplitude fit, a signal PDF is employed which replacesthe phase-space integrated decay rate with the full decay rate given in Eq. (10), but is
12
otherwise identical to the PDF used in Sec. 5.1. Variations of the selection efficiency overthe phase space are incorporated by evaluating the likelihood normalisation integrals withthe Monte Carlo (MC) integration technique using fully simulated decays [29,77–79].
The light meson spectrum comprises a large number of resonances potentially contribut-ing to the B0
s→ D∓s K±π±π∓ decay in various decay topologies and angular momentum
configurations. The full list of considered intermediate-state amplitudes can be found inTable B.1. A significant complication arises from the fact that two (quasi-independent)amplitude models need to be developed simultaneously: one amplitude describes decaysvia b→ c (Ac(x)), the other decays via b→ u (Au(x)) quark-level transitions. A modelbuilding procedure is applied to obtain a good description of the observed phase-spacedistribution while keeping the number of included amplitudes as small as possible. This isaccomplished in two stages. The first stage identifies the set of intermediate-state ampli-tudes contributing at a significant level to either decays via b→ c or b→ u quark-leveltransitions or to both. To that end, the time-integrated and flavour-averaged phase-spacedistribution is examined. A single total amplitude, Aeff(x) =
∑i a
effi Ai(x), is sufficient
in this case, which effectively describes the incoherent superposition of the b → c andb→ u amplitudes, |Aeff(x)|2 = |Ac(x)|2 + r2 |Au(x)|2. This significantly simplified fittingprocedure allows the initial inclusion of the whole pool of considered intermediate-stateamplitudes, limiting the model complexity with the LASSO technique [29,80,81]. Thismethod adds a penalty term to the likelihood function,
− 2 logL → −2 logL+ λ∑i
√∫|aeffi Ai(x)|2 dΦ4, (15)
which shrinks the amplitude coefficients towards zero. The optimal value for the LASSOparameter λ, which controls the model complexity, is found by minimising the Bayesianinformation criterion BIC(λ) = −2 logL+ k logNSig [82], where NSig is the signal yieldand k is the number of amplitudes with a decay fraction above the threshold of 0.5%.The amplitudes with a decay fraction above the threshold at the minimum BIC(λ) valueare selected. The second stage of the model selection performs a full time-dependentamplitude fit. The components selected by the first stage are included for both b→ c andb→ u transitions and a LASSO penalty term for each is added to the likelihood function.As the strong interaction is CP symmetric, the subdecay modes of three-body resonancesand their conjugates are constrained to be the same. The final set of b→ c and b→ uamplitudes is henceforth referred to as the baseline model. The LASSO penalty term isonly used to select the model and discarded in the final fit to avoid biasing the parameteruncertainties.
Table B.2 and B.3 list the moduli and phases of the complex amplitude coefficientsobtained by fitting the baseline model to the background-subtracted B0
s→ D∓s K±π±π∓ sig-
nal candidates. The corresponding decay fractions for the b→ c and b→ u amplitudes aregiven in Table 3. The decay-time projection and mixing asymmetries shown in Fig. 5 areconsistent with those of the phase-space integrated fit in Fig. 4. Invariant-mass projectionsare shown in Figs. 6 and 7 indicating that the model provides a reasonable description ofthe data. Decays via b→ c quark-level transitions are found to be dominated by the axial-vector states K1(1270)+ and K1(1400)+. These resonances are produced by the externalweak current (see Fig. 1). The sub-leading contribution comes from the vector resonanceK∗(1410)+. In b→ u quark-level transitions, the excited kaon states are produced by the
13
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0
0.1
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0.5LHCb
−π+π+K−
s=Df
+π−π−K+
s=Df
Figure 5: Decay-time distribution of (left) background-subtracted B0s→ D∓s K
±π±π∓ candidatesand (right) dilution-weighted mixing asymmetry along with the model-dependent fit projections(lines). The decay-time acceptance (left) is overlaid in an arbitrary scale (dashed line).
Table 3: Decay fractions of the intermediate-state amplitudes contributing to decays via b→ cand b → u quark-level transitions. The uncertainties are statistical, systematic and due toalternative amplitude models considered.
Decay channel F ci [%] F u
i [%]
B0s → D∓s (K1(1270)± → K∗(892)0 π±) 13.0± 2.4± 2.7± 3.4 4.1± 2.2± 2.9 ± 2.6
B0s → D∓s (K1(1270)± → K± ρ(770)0) 16.0± 1.4± 1.8± 2.1 5.1± 2.2± 3.5 ± 2.0
B0s → D∓s (K1(1270)± → K∗0(1430)0 π±) 3.4± 0.5± 1.0± 0.4 1.1± 0.5± 0.6 ± 0.5
B0s → D∓s (K1(1400)± → K∗(892)0 π±) 63.9± 5.1± 7.4± 13.5 19.3± 5.2± 8.3 ± 7.8
B0s → D∓s (K∗(1410)± → K∗(892)0 π±) 12.8± 0.8± 1.5± 3.2 12.6± 2.0± 2.6 ± 4.1
B0s → D∓s (K∗(1410)± → K± ρ(770)0) 5.6± 0.4± 0.6± 0.7 5.6± 1.0± 1.2 ± 1.8
B0s → D∓s (K(1460)± → K∗(892)0 π±) 11.9± 2.5± 2.9 ± 3.1
B0s → (D∓s π
±)P K∗(892)0 10.2± 1.6± 1.8± 4.5 28.4± 5.6± 6.4 ± 15.3
B0s → (D∓s K
±)P ρ(770)0 0.9± 0.4± 0.5± 1.0
Sum 125.7± 6.4± 6.9± 19.9 88.1± 7.0± 10.0± 20.9
spectator-quark interaction. Here, no clear hierarchy is observed. There are sizeable contri-butions from the axial-vector resonances but also from the pseudoscalar stateK(1460)+ andfrom the quasi-two-body process B0
s → (D∓s π±)P K∗(892)0, where (D∓s π
±)P denotes anon-resonant two-particle system in a P -wave (L = 1) configuration. Interference fractionsof the b→ c and b→ u intermediate-state amplitudes are given in Tables B.6 and B.7. Size-able interference effects between the decay modes B0
s → D∓s (K1(1270)± → K∗(892)0 π±),B0s → D∓s (K1(1400)± → K∗(892)0 π±) and B0
s → (D∓s π±)P K∗(892)0 are observed since
the overlap of their phase-space distributions is significant. A net constructive (destructive)interference effect of all amplitude components of around +26% (−12%) remains for b→ c(b→ u) quark-level transitions when integrated over the phase space.
The mass and width of the K1(1400)+ and K∗(1410)+ resonances are determined fromthe fit to be
mK1(1400) = (1406± 7± 6± 11) MeV, ΓK1(1400) = (195± 11± 12± 16) MeV,
mK∗(1410) = (1433± 10± 23± 8) MeV, ΓK∗(1410) = (402± 24± 47± 22) MeV,
14
in good agreement with the PDG average values [2]. The uncertainties are statistical,systematic and due to alternative models considered as detailed in Sec. 6. The ratio ofthe B0
s → D−s K+π+π− and B0
s → D−s K+π+π− decay amplitudes as well as their strong-
and weak-phase difference are measured to be
r = 0.56± 0.05± 0.04± 0.07,
δ = (−14± 10± 4± 5)◦,
γ − 2βs = ( 42± 10± 4± 5)◦,
where the angles are given modulo 180◦. The coherence factor is computed by numericallyintegrating over the phase space using the baseline model resulting in
κ = 0.72± 0.04 (stat)± 0.06 (syst)± 0.04 (model).
Full PDF
2) |x(cA|
2) |x(uA | 2r
) [GeV]
±
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K(m
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) [GeV]
±
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±
D(m
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)
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300 LHCb
Figure 6: Invariant-mass distribution of background-subtracted B0s→ D∓s K
±π±π∓ candidates(data points) and fit projections (blue solid line). Contributions from b→ c and b→ u decayamplitudes are overlaid.
15
±(1270)1K
±
sD→s
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±(1400)1K
±
sD→s
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±(1410)*
K
±
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±(1460)K
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Others) [GeV]
±
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) [GeV]
±
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ield
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)
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±
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±
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Figure 7: Invariant-mass distribution of background-subtracted B0s→ D∓s K
±π±π∓ candidates(data points) and fit projections (blue solid line). Incoherent contributions from intermediate-statecomponents are overlaid.
6 Systematic uncertainties
The systematic uncertainties on the measured observables are summarised in Table 4 forthe decay-time fits to B0
s→ D−s π+π+π− and B0
s→ D∓s K±π±π∓ decays and in Tables 5
and B.4 for the time-dependent amplitude fit to B0s→ D∓s K
±π±π∓ decays. The varioussources of systematic uncertainties are described in the following.
The overall fit procedure is tested by generating pseudoexperiments from the default fitmodel using the measured values and subsequently fitting them with the same model. Foreach pseudoexperiment and fit parameter, a pull is calculated by dividing the differencebetween the fitted and generated values by the statistical uncertainty. The means of thepull distributions are assigned as systematic uncertainties due to an intrinsic fit bias. Aclosure test using a large sample of fully simulated signal candidates shows a non-significantbias for the determination of ∆ms, which is assigned as a systematic uncertainty.
The statistical subtraction of the residual background relies on the correct descriptionof the reconstructed m(D∓s h
±π±π∓) invariant mass distribution. Alternative parameterisa-
16
Table 4: Systematic uncertainties on the B0s mixing frequency determined from the fit to
B0s→ D−s π
+π+π− signal candidates and on the fit parameters of the phase-space integrated fitto B0
s→ D∓s K±π±π∓ signal candidates in units of the statistical standard deviations.
Systematic ∆ms Cf A∆Γf A∆Γ
fSf Sf
Fit bias (+ closure test) 0.39 0.09 0.00 0.01 0.02 0.11
Background subtraction 0.18 0.17 0.19 0.18 0.11 0.06
Correlations 0.09 0.11 0.32 0.24 0.04 0.06
Acceptance 0.02 0.03 0.19 0.22 0.02 0.02
Resolution 0.16 0.24 0.01 0.03 0.12 0.10
Decay-time bias 1.00 0.06 0.00 0.00 0.10 0.08
Nuisance asymmetries 0.02 0.06 0.06 0.06 0.06 0.05
VELO z scale 0.26
VELO alignment 0.44
∆ms 0.02 0.01 0.01 0.12 0.12
Total 1.21 0.34 0.42 0.38 0.24 0.23
tions are tested for the signal and each background component. The yields of the cross-feedcontributions to the B0
s→ D∓s K±π±π∓ candidates estimated from a combination of simu-
lated data and control modes are fixed to zero or doubled. The standard deviations ofthe obtained differences to the default fit values are assigned as a systematic uncertaintydue to the background subtraction. The background subtraction technique relies on theassumption of independence between the reconstructed m(D∓s h
±π±π∓) invariant mass andthe observables that are used in the final fit. The impact of ignoring the small correlationbetween reconstructed mass and decay time observed for the combinatorial background isdetermined with pseudoexperiments in which the correlation is included at generationand neglected in the fit.
The systematic uncertainties related to the decay-time acceptance as well as due to thelimited experimental knowledge of Γs and ∆Γs are studied simultaneously. Pseudoexperi-ments are generated assuming the default configuration and subsequently fitted under boththis default and an alternative configuration in which the acceptance parameters togetherwith Γs and ∆Γs are randomised within their uncertainties (taking their correlation intoaccount). The bias of the mean of the resulting pull distribution is added in quadratureto the pull width in order to arrive at the final systematic uncertainty.
Systematic effects originating from the calibration of the decay-time uncertainty esti-mate are studied with two alternative parameterisations which either slightly overestimateor underestimate the decay-time resolution. Due to the high correlation between the decay-time resolution and the calibration of the flavour taggers, their systematic uncertaintyis studied simultaneously. As a first step, the decay-time fit to the B0
s → D−s π+π+π−
candidates is repeated using the alternative decay-time error calibration functions. Newtagging calibration parameters are obtained which are then used (together with therespective decay-time error calibration function) for the fits to the B0
s → D∓s K±π±π∓
candidates. The largest deviations of the central values from their default values areassigned as a systematic uncertainty for each fit parameter. A systematic uncertainty dueto the limited knowledge of the decay-time bias which is fixed in the fit is evaluated by
17
Table 5: Systematic uncertainties on the physical observables and resonance parameters deter-mined from the full time-dependent amplitude fit to B0
s→ D∓s K±π±π∓ data in units of the
statistical standard deviations. The systematic uncertainties for the amplitude coefficients aregiven in Table B.4.
Systematic mK1(1400) ΓK1(1400) mK∗(1410) ΓK∗(1410) r δ γ − 2βs
Fit bias 0.00 0.14 0.14 0.42 0.06 0.13 0.13
Background subtraction 0.15 0.28 0.15 0.41 0.15 0.15 0.22
Correlations 0.23 0.27 0.18 0.07 0.13 0.05 0.08
Time acceptance 0.01 0.01 0.00 0.00 0.06 0.04 0.07
Resolution 0.11 0.33 0.09 0.03 0.30 0.27 0.26
Decay-time bias 0.00 0.02 0.01 0.00 0.02 0.12 0.01
Nuisance asymmetries 0.00 0.01 0.00 0.00 0.01 0.07 0.03
∆ms 0.00 0.01 0.00 0.00 0.03 0.06 0.03
Phase-space acceptance 0.17 0.34 0.32 0.21 0.35 0.08 0.06
Acceptance factorisation 0.25 0.41 0.76 0.36 0.49 0.08 0.05
Lineshapes 0.52 0.53 0.43 0.29 0.34 0.14 0.13
Resonances m,Γ 0.07 0.06 0.02 0.02 0.02 0.01 0.01
Form factors 0.51 0.60 2.03 1.74 0.12 0.13 0.07
Amplitude model 1.59 1.50 0.77 0.88 1.60 0.51 0.47
Total 1.80 1.86 2.38 2.10 1.79 0.67 0.63
randomising the value within its uncertainty.The systematic uncertainty from the production and detection asymmetries and ∆ms
(in case of B0s→ D∓s K
±π±π∓ decays) which are fixed in the fit are evaluated by means ofpseudoexperiments, analogously to the procedure performed for the decay-time acceptance.
The precision with which the B0s flight distance can be determined is limited by the
knowledge of the overall length of the VELO detector (VELO z scale) and the position ofthe individual VELO modules (VELO alignment). This VELO-reconstruction uncertaintytranslates into a relative uncertainty on ∆ms of 0.02% [68] with other parameters beingunaffected. In the fit to the B0
s → D∓s K±π±π∓ candidates, the VELO-reconstruction
uncertainty is then implicitly included in the systematic error due to the ∆ms uncertaintydescribed above.
The treatment of the phase-space acceptance relies on simulated data. The integrationerror due to the limited size of the simulated sample used to normalise the signal PDFis below 0.2% and thus negligibly small. To assess the uncertainty due to possible data-simulation differences, alternative phase-space acceptances are derived by varying theselection requirements (for the simulated sample only) on quantities that are expectednot to be well described by the simulation. It is assumed that the phase-space acceptanceis independent of the decay time. This assumption is tested by using only simulatedcandidates within specific decay-time intervals to calculate the MC normalisation integrals.Four approximately equally populated decay-time intervals are chosen and the samplevariance of the fitted values is assigned as a systematic uncertainty.
The lineshape parameterisations for the ρ(770)0 and K∗0 (1430)0 resonances are replaced
18
by a relativistic Breit–Wigner propagator given by Eq. (7) as part of the systematic studies.Moreover, energy-dependent decay widths of three-body resonances are recomputed takingonly the dominant K±π±π∓ decay mode into account. For each alteration, a time-dependent amplitude fit is performed and the standard deviation of the obtained fitresults is assigned as a systematic uncertainty. Systematic uncertainties due to fixedresonance masses and widths are computed with the same procedure used for the otherfixed parameters mentioned above. Similarly, pseudoexperiments are performed in whichthe Blatt–Weisskopf radial parameter rBW (set to 1.5 GeV−1 by default) is varied uniformlywithin the interval [0, 3] GeV−1 to assign a systematic uncertainty due to the form-factormodelling.
Several modifications to the baseline model are tested to assign an additional uncer-tainty due to the choice of amplitude components: all amplitudes selected by the firststage of the model selection are included for both b → c and b → u transitions, theamplitudes with the smallest decay fraction are removed, additional sub-decay modes ofselected three-body resonances are considered, higher orbital angular momentum states areincluded where applicable, the orbital angular momentum state of non-resonant two-bodystates is set to other allowed values, additional cascade and quasi-two-body amplitudes(which were removed by the first stage of the model selection) are considered. In total,twelve amplitude models with similar fit quality as the baseline model are identified. Thefit results for those are summarised in Tables B.8 and B.9. The largest deviations from thebaseline values for r, δ and γ − 2β are 0.19, 10◦ and 12◦ observed for alternative models8, 1 and 11, respectively. The standard deviation of the twelve fit results is taken as themodel uncertainty. No model uncertainty is assigned to the amplitude coefficients sincethey are, by definition, parameters of a given model.
No statistically significant effect on the results is observed when repeating the analysison subsets of the data, splitting by data-taking period or tagging category (OS-tagged,only SS-tagged or both OS- and SS-tagged).
7 Results
To interpret the parameters determined in the model-independent fit,Pobs ≡ (Cf , A
∆Γf , A∆Γ
f, Sf , Sf ), in terms of the physical observables Λ ≡ (r, κ, δ, γ − 2βs),
the equations for the CP coefficients in terms of these physical variables reported inEq. (12), P (Λ), need to be inverted. This is accomplished by minimising the likelihoodfunction [83]
− 2L(Λ) = −2 exp
(−1
2(P (Λ)− Pobs)
T V −1 (P (Λ)− Pobs)
). (16)
Here, V denotes the experimental (statistical and systematic) covariance matrix of themeasured observables, see Appendix C. Figure 8 displays the confidence levels (CL)for the physical parameters Λ obtained from the model-independent method, wherethe physical boundary of the coherence factor is enforced. The 1 − CL = 68.3% (1σ)confidence intervals are given in Table 6 together with the results of the full time-dependent amplitude analysis. Considering the difference in statistical sensitivity ofthe two methods and that the model-dependent uncertainty only affects the full time-dependent amplitude fit, a good agreement between the measurements is observed. As a
19
cross-check, pseudoexperiments are performed to study the distribution of the test statis-tic Q2 =
∑i(Λ
MIi − ΛMD
i )2/(σstat(ΛMIi )2 − σstat(Λ
MDi )2), where ΛMI
i (ΛMDi ) and σstat(Λ
MIi )
(σstat(ΛMDi )) denote the measured value of the physical observable Λi and its statistical
uncertainty obtained with the model-independent (model-dependent) method. It is foundthat p = 33% of the pseudoexperiments have a larger Q2 value than observed on data,considering only the statistical uncertainty. The p-value increases to p = 49% when theuncertainty due to the amplitude modelling is included.
The measured ratio of the b→ u and b→ c decay amplitudes is qualitatively consistentwith the naive expectation based on the involved CKM elements (r ≈ 0.4). Note that theparameters r, κ and δ are determined in a limited phase-space region (cf. Sec. 3.1) andmight differ when the full phase space is considered.
r
0 0.2 0.4 0.6 0.8 1
CL
−1
0.2
0.4
0.6
0.8
1
68.3%
95.5%
LHCb
κ
0 0.2 0.4 0.6 0.8 1
CL
−1
0.2
0.4
0.6
0.8
1
68.3%
95.5%
LHCb
]° [δ
CL
−1
0.2
0.4
0.6
0.8
1
50− 0 50
68.3%
95.5%
LHCb
]° [s
β2−γ
CL
−1
0.2
0.4
0.6
0.8
1
0 50 100 150
68.3%
95.5%
LHCb
Figure 8: The 1−CL contours for the physical observables r, κ, δ and γ − 2βs obtained with themodel-independent fit.
Table 6: Parameters determined from the model-independent and model-dependent fits tothe B0
s→ D∓s K±π±π∓ signal candidates. The uncertainties are statistical, systematic and (if
applicable) due to alternative amplitude models considered. The angles are given modulo 180◦.
Parameter Model-independent Model-dependent
r 0.47 + 0.08− 0.08
+ 0.02− 0.03 0.56± 0.05± 0.04± 0.07
κ 0.88 + 0.12− 0.19
+ 0.04− 0.07 0.72± 0.04± 0.06± 0.04
δ [◦] −6 + 10− 12
+ 2− 4 −14± 10 ± 4 ± 5
γ − 2βs [◦] 42 + 19− 13
+ 6− 2 42± 10 ± 4 ± 5
20
8 Conclusion
Mixing-induced CP violation in B0s→ D∓s K
±π±π∓ decays is studied for the first timeusing 9 fb−1 of proton-proton collision data recorded by the LHCb detector. A time-dependent amplitude analysis is performed, disentangling the various intermediate-statecomponents contributing via b→ c or b→ u quark-level transitions. The CP -violatingweak phase γ − 2βs is measured and used to determine the CKM angle γ by takingthe mixing phase, −2βs = φs = (−2.35± 1.43)◦ [10], as an external input. This resultsin a measured value of γ = (44 ± 12)◦ modulo 180◦. A model-independent fit to thephase-space integrated decay-time spectrum yields a compatible, but statistically lessprecise, value of γ = (44+ 20
− 13)◦ modulo 180◦. These results correspond to 4.4σ and 4.6σ
evidence for mixing-induced CP violation and agree with the world-average value of theCKM angle γ [2,3] within 2.2σ and 1.4σ for the model-dependent and model-independentmethods, respectively. The B0
s −B0s oscillation frequency is measured from flavour-specific
B0s→ D−s π
+π+π− decays to be ∆ms = (17.757 ± 0.007 ± 0.008) ps−1. This is the mostprecise measurement of this quantity and consistent with the world-average value [2]and theoretical predictions [84]. Both the measurement of the CKM angle γ and of theB0s −B0
s mixing frequency are vital inputs for global fits of the CKM matrix.
Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments for theexcellent performance of the LHC. We thank the technical and administrative staff at theLHCb institutes. We acknowledge support from CERN and from the national agencies:CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3(France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSWand NCN (Poland); MEN/IFA (Romania); MSHE (Russia); MICINN (Spain); SNSFand SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF(USA). We acknowledge the computing resources that are provided by CERN, IN2P3(France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain),GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebtedto the communities behind the multiple open-source software packages on which wedepend. Individual groups or members have received support from AvH Foundation(Germany); EPLANET, Marie Sk lodowska-Curie Actions and ERC (European Union);A*MIDEX, ANR, Labex P2IO and OCEVU, and Region Auvergne-Rhone-Alpes (France);Key Research Program of Frontier Sciences of CAS, CAS PIFI, Thousand Talents Program,and Sci. & Tech. Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia);GVA, XuntaGal and GENCAT (Spain); the Royal Society and the Leverhulme Trust(United Kingdom).
21
Appendices
A Lineshapes
Lineshape parameters for resonances contributing to B0s→ D∓s K
±π±π∓ decays are fixedto the values given in Table A.1. The running-width distributions of the three-bodyresonances are shown in Figure 9.
Table A.1: Parameters of the resonances included in the B0s→ D∓s K
±π±π∓ baseline model.
Resonance m0 [ MeV] Γ0 [ MeV] JP Sourceρ(770)0 775.26± 0.25 149.1± 0.8 1− [2]K∗(892)0 895.55± 0.20 47.3± 0.5 1− [2]K1(1270)+ 1289.81± 1.75 116.11± 3.4 1+ [85]K(1460)+ 1482.4± 15.6 335.6± 10.6 0− [85]K∗0(1430)0 1425± 50 270± 80 0+ [2]
[GeV] s 1 1.5 2
[G
eV
](s
)Γ
0
m /
s
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
+(1270)1K
LHCb
[GeV] s 1 1.5 2
[G
eV
](s
)Γ
0
m /
s
0
0.05
0.1
0.15
0.2
0.25
0.3 LHCb+(1400)1K
[GeV] s 1 1.5 2
[G
eV
](s
)Γ
0
m /
s
0
0.1
0.2
0.3
0.4
0.5
0.6 LHCb+(1410)
*K
[GeV] s 1 1.5 2
[G
eV
](s
)Γ
0
m /
s
0
0.1
0.2
0.3
0.4
0.5
0.6LHCb
+(1460)K
Figure 9: Running width distributions of the three-body resonances included in the baselinemodel for B0
s→ D∓s K±π±π∓ decays: (top left) K1(1270)+, (top right) K1(1400)+, (bottom left)
K∗(1410)+ and (bottom right) K(1460)+.
22
B Amplitude model
The full list of intermediate-state amplitudes considered for the model building procedureis given in Table B.1. In addition to those, the presence of unknown states decaying intoD−s π
+π−, D−s K+π−, D−s π
+ or D−s K+ is investigated. These states are typically fitted
with huge decay widths (effectively indistinguishable from a non-resonant state) and smalldecay fractions. Therefore, such states (if existent in the relevant mass range) are notresolvable with the current data sample.
Table B.1: Intermediate-state components considered for the B0s→ D∓s K
±π±π∓ LASSO modelbuilding procedure. The letters in square brackets and subscripts refer to the relative orbitalangular momentum of the decay products in spectroscopic notation. If no angular momentum isspecified, the lowest angular momentum state compatible with angular momentum conservationand, where appropriate, parity conservation, is used.
(a) Cascade decays
B0s → D∓
s [K1(1270)±[S,D]→ π±K∗(892)0]
B0s → D∓
s [K1(1270)± → π±K∗0 (1430)0]
B0s → D∓
s [K1(1270)± → K± f0(500)0]
B0s → D∓
s [K1(1270)± → K± f0(980)0]
B0s → D∓
s [K1(1270)±[S,D]→ K± ρ(770)0]
B0s → D∓
s [K1(1270)±[S,D]→ K± ρ(1450)0]
B0s → D∓
s [K1(1400)±[S,D]→ π±K∗(892)0]
B0s → D∓
s [K1(1400)±[S,D]→ K± ρ(770)0]
B0s → D∓
s [K(1460)± → K± f0(500)0]
B0s → D∓
s [K(1460)± → K± ρ(770)0]
B0s → D∓
s [K(1460)± → π±K∗(892)0]
B0s → D∓
s [K(1460)± → π±K∗(1430)0]
B0s → D∓
s [K∗(1410)± → π±K∗(892)0]
B0s → D∓
s [K∗(1410)± → K± ρ(770)0]
B0s → D∓
s [K∗2 (1430)± → π±K∗(892)0]
B0s → D∓
s [K∗2 (1430)± → K± ρ(770)0]
B0s → D∓
s [K1(1650)± → π±K∗(892)0]
B0s → D∓
s [K1(1650)± → K± ρ(770)0]
B0s → D∓
s [K∗(1680)± → π±K∗(892)0]
B0s → D∓
s [K∗(1680)± → K± ρ(770)0]
B0s → D∓
s [K2(1770)± → π±K∗(892)0]
B0s → D∓
s [K2(1770)± → K± ρ(770)0]
B0s → D∓
s [K2(1770)± → π±K∗2 (1430)0]
B0s → D∓
s [K2(1770)± → K± f2(1270)0]
B0s → D∓
s [K∗2 (1980)± → π±K∗(892)0]
B0s → D∓
s [K∗2 (1980)± → K± ρ(770)0]
B0s → D∓
s [K∗2 (1980)± → π±K∗
2 (1430)0]
B0s → D∓
s [K∗2 (1980)± → K± f2(1270)]
B0s → K±[Ds1(2536)∓ → D∓
s ρ(770)0]
B0s → K±[Ds1(2536)∓ → D∓
s f0(500)0]
(b) Quasi-two-body decays
B0s → f0(500)0(D∓
s K±)S
B0s → f0(500)0(D∓
s K±)P
B0s → f0(980)0(D∓
s K±)S
B0s → f0(980)0(D∓
s K±)P
B0s → ρ(770)0(D∓
s K±)S
B0s [S, P,D]→ ρ(770)0 (D∓
s K±)P
B0s → f2(1270)0(D∓
s K±)S
B0s → K∗(892)0 (D∓
s π±)S
B0s [S, P,D]→ K∗(892)0 (D∓
s π±)P
B0s → K∗
0 (1430)0 (D∓s π
±)S
B0s → K∗
2 (1430)0 (D∓s π
±)S
B0s → (D∓
s K±)S (π±π∓)S
23
Tables B.2 and B.3 list the moduli and phases of the complex amplitude coefficientsobtained by fitting the baseline model to the background-subtracted B0
s→ D∓s K±π±π∓
signal candidates. The systematic uncertainties are summarised in Table B.4. The fitresults are used to derive the amplitude ratio, ri, and strong-phase difference, δi, for agiven decay channel, defined as
ri = r
∣∣∣∣auiaci∣∣∣∣√∫|Ac(x)|2 dΦ4(x)∫|Au(x)|2 dΦ4(x)
,
δi = δ − arg(aci) + arg(aui )− arg
(∫Ac(x)∗Au(x) dΦ4(x)
).
The values are given in Table B.5. Tables B.6 (b → c amplitudes) and B.7 (b → uamplitudes) list the interference fractions ordered by magnitude, for the baseline model.Figures 10 and 11 show additional fit projections of the baseline model. Here, theacoplanarity angle, φK+π−,D−s π+ , is defined as the angle between the two decay planesformed by the K+π− system and the D−s π
+ system in the B0s rest frame; boosting into
the rest frames of the two-body systems defining these decay planes, the two helicityvariables cos θK+π− and cos θD−s π+ are defined as the cosine of the angle of the K+ or D−smomentum with the B0
s flight direction.The decay fractions for several alternative models are summarised in
Tables B.8 and B.9.
Table B.2: Moduli and phases of the amplitude coefficients for decays via b → c and b → uquark-level transitions. The uncertainties are statistical and systematic.
Decay channel Ac(x) Au(x)
|aci | arg(aci)[◦] |aui | arg(aui )[
◦]
B0s → D∓s K1(1270)± 0.50± 0.05± 0.05 24± 4± 6 0.51± 0.27± 0.25 −129± 18± 26
B0s → D∓s K1(1400)± 1.0 0.0 1.0 0.0
B0s → D∓s K
∗(1410)± 0.45± 0.05± 0.05 53± 6± 6 0.81± 0.20± 0.15 2± 13± 11
B0s → D∓s K(1460)± 0.78± 0.20± 0.16 −92± 10± 15
B0s → (D∓s π
±)P K∗(892)0 0.40± 0.04± 0.04 −33± 5± 4 1.21± 0.26± 0.22 −7± 8 ± 17
B0s → (D∓s K
±)P ρ(770)0 0.12± 0.03± 0.03 109± 9± 14
Table B.3: Moduli and phases of the amplitude coefficients for cascade decays. The amplitudecoefficients are defined relative to the respective three-body production amplitude coefficients inTable B.2 and are shared among b→ c and b→ u transitions. The uncertainties are statisticaland systematic.
Decay channel |ai| arg(ai)[◦]
K1(1270)± → K± ρ(770)0 1.0 0.0
K1(1270)± → K∗(892)0π± 0.90± 0.09± 0.10 46± 7± 4
K1(1270)± → K∗0(1430)0π± 0.46± 0.05± 0.03 120± 5± 6
K∗(1410)± → K∗(892)0 π± 1.0 0.0
K∗(1410)± → K± ρ(770)0 0.66± 0.05± 0.05 −165± 6± 5
24
Tab
leB
.4:
Syst
emati
cu
nce
rtain
ties
on
the
fit
para
met
ers
of
the
full
tim
e-d
epen
den
tam
pli
tud
efi
ttoB
0 s→D∓ sK±π±π∓
data
inu
nit
sof
the
stati
stic
al
stan
dard
dev
iati
on
s.T
he
diff
eren
tco
ntr
ibu
tion
sare
:1)
fit
bia
s,2)
back
gro
un
dsu
btr
act
ion
,3)
corr
elati
on
of
obse
rvab
les,
4)
tim
eacc
epta
nce
,5)
reso
luti
on
,6)
dec
ay-t
ime
bia
s,7)
nu
isan
ceasy
mm
etri
es,
8)
∆ms,
9)
ph
ase
-sp
ace
acc
epta
nce
,10)
acc
epta
nce
fact
ori
sati
on
,11)
lin
esh
ape
mod
els,
12)
mas
ses
and
wid
ths
of
reso
nan
ces,
13)
form
fact
or.
Par
amet
er1
23
45
67
89
1011
1213
Tot
al
B0 s→
D∓ sK
1(1
270)±|ac i|
0.09
0.32
0.17
0.01
0.36
0.05
0.00
0.02
0.60
0.11
0.31
0.07
0.48
0.99
B0 s→
D∓ sK
1(1
270)±
arg(ac i)
0.10
0.37
0.12
0.03
0.13
0.05
0.05
0.02
0.40
0.31
0.98
0.21
0.58
1.33
B0 s→
D∓ sK
1(1
270)±|au i|
0.17
0.26
0.14
0.04
0.41
0.07
0.01
0.03
0.35
0.18
0.60
0.02
0.26
0.94
B0 s→
D∓ sK
1(1
270)±
arg(au i)
0.12
0.77
0.21
0.08
0.39
0.05
0.11
0.04
0.66
0.55
0.52
0.07
0.54
1.46
K1(1
270)±→
K∗ (
892)
0π±|ai|
0.01
0.32
0.27
0.01
0.31
0.01
0.00
0.01
0.17
0.36
0.55
0.15
0.55
1.03
K1(1
270)±→
K∗ (
892)
0π±
arg(ai)
0.14
0.11
0.11
0.01
0.13
0.02
0.00
0.00
0.19
0.14
0.33
0.07
0.34
0.59
K1(1
270)±→
K∗ 0(1
430)
0π±|ai|
0.10
0.48
0.14
0.00
0.05
0.01
0.00
0.00
0.10
0.25
0.42
0.05
0.17
0.73
K1(1
270)±→
K∗ 0(1
430)
0π±
arg(ai)
0.24
0.33
0.53
0.00
0.14
0.01
0.00
0.00
0.27
0.46
0.16
0.08
0.85
1.22
B0 s→
D∓ sK∗ (
1410
)±|ac i|
0.27
0.35
0.14
0.02
0.29
0.04
0.00
0.01
0.87
0.32
0.09
0.10
0.20
1.10
B0 s→
D∓ sK∗ (
1410
)±ar
g(ac i)
0.32
0.29
0.10
0.01
0.40
0.02
0.00
0.01
0.23
0.21
0.46
0.06
0.64
1.04
B0 s→
D∓ sK∗ (
1410
)±|au i|
0.17
0.16
0.17
0.03
0.25
0.07
0.07
0.03
0.38
0.32
0.26
0.05
0.34
0.77
B0 s→
D∓ sK∗ (
1410
)±ar
g(au i)
0.21
0.42
0.41
0.05
0.13
0.01
0.01
0.01
0.07
0.26
0.21
0.05
0.47
0.87
K∗ (
1410
)±→
K±ρ(7
70)0|ai|
0.58
0.37
0.26
0.00
0.19
0.02
0.10
0.00
0.43
0.45
0.31
0.04
0.06
1.03
K∗ (
1410
)±→
K±ρ(7
70)0
arg(ai)
0.05
0.42
0.24
0.00
0.19
0.04
0.06
0.00
0.58
0.03
0.22
0.05
0.19
0.84
B0 s→
D∓ sK
(146
0)±|au i|
0.02
0.22
0.28
0.05
0.13
0.04
0.01
0.02
0.22
0.18
0.63
0.04
0.19
0.82
B0 s→
D∓ sK
(146
0)±
arg(au i)
0.14
0.51
0.10
0.06
0.30
0.01
0.19
0.02
0.39
0.48
0.11
0.10
1.15
1.46
B0 s→
(D∓ sπ±
) PK∗ (
892)
0|ac i|
0.06
0.37
0.11
0.01
0.13
0.05
0.01
0.02
0.09
0.32
0.58
0.09
0.62
1.00
B0 s→
(D∓ sπ±
) PK∗ (
892)
0ar
g(ac i)
0.03
0.60
0.29
0.03
0.19
0.04
0.02
0.02
0.22
0.10
0.43
0.06
0.29
0.90
B0 s→
(D∓ sπ±
) PK∗ (
892)
0|au i|
0.08
0.26
0.15
0.05
0.20
0.02
0.01
0.02
0.18
0.26
0.62
0.04
0.33
0.86
B0 s→
(D∓ sπ±
) PK∗ (
892)
0ar
g(au i)
0.20
0.75
0.66
0.05
0.54
0.06
0.15
0.04
0.36
0.68
0.31
0.13
1.37
1.98
B0 s→
(D∓ sK±
) Pρ(7
70)0|ac i|
0.11
0.46
0.14
0.01
0.12
0.03
0.00
0.00
0.32
0.05
0.71
0.06
0.37
1.01
B0 s→
(D∓ sK±
) Pρ(7
70)0
arg(ac i)
0.18
0.66
0.66
0.01
0.23
0.01
0.00
0.01
0.20
0.22
1.12
0.06
0.47
1.59
25
Table B.5: Amplitude ratio and strong-phase difference for a given decay channel. The uncer-tainties are statistical and systematic.
Decay channel ri δi[◦]
B0s → D∓s K1(1270)± 0.31± 0.20± 0.16 −109± 22± 28
B0s → D∓s K1(1400)± 0.30± 0.03± 0.02 44± 10± 4
B0s → D∓s K
∗(1410)± 0.54± 0.18± 0.15 −7± 17± 14
B0s → (D∓s π
±)P K∗(892)0 0.92± 0.24± 0.20 69± 15± 18
Table B.6: Interference fractions (ordered by magnitude) of the b → c intermediate-stateamplitudes included in the baseline model. Only the statistical uncertainties are given.
Decay channel i Decay channel j Iij [%]
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → (D∓s π±)P K∗(892)0 −15.4 ± 3.1
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1400)
± → K∗(892)0 π±) −12.2 ± 5.8
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → (D∓s π±)P K∗(892)0 −10.2 ± 1.2
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K± ρ(770)0) 5.8 ± 0.5
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 4.1 ± 0.3
B0s → (D∓
s K±)P ρ(770)0 B0s → D∓
s (K1(1270)± → K± ρ(770)0) 3.1 ± 0.5
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → (D∓s K±)P ρ(770)0 1.3 ± 0.3
B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) B0s → D∓
s (K1(1270)± → K± ρ(770)0) −1.3 ± 0.9
B0s → (D∓
s K±)P ρ(770)0 B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) −0.5 ± 0.3
B0s → (D∓
s π±)P K∗(892)0 B0s → (D∓
s K±)P ρ(770)0 −0.2 ± 0.1
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K± ρ(770)0) −0.1 ± 0.5
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K1(1270)± → K± ρ(770)0) −0.1 ± 0.1
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K∗0 (1430)π±) −0.1 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → (D∓s K±)P ρ(770)0 0.0 ± 0.1
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K∗0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → (D∓
s π±)P K∗(892)0 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → (D∓
s K±)P ρ(770)0 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) B0s → (D∓
s K±)P ρ(770)0 0.0 ± 0.0
26
Table B.7: Interference fractions (ordered by magnitude) of the b → u intermediate-stateamplitudes included in the baseline model. Only the statistical uncertainties are given.
Decay channel i Decay channel j Iij [%]
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → (D∓s π±)P K∗(892)0 9.5 ± 2.6
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1400)
± → K∗(892)0 π±) 9.2 ± 3.2
B0s → D∓
s (K(1460)± → K∗(892)0 π±) B0s → (D∓
s π±)P K∗(892)0 −5.5 ± 0.9
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 4.1 ± 0.8
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → (D∓s π±)P K∗(892)0 −3.7 ± 5.9
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K± ρ(770)0) −1.2 ± 0.6
B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) B0s → D∓
s (K1(1270)± → K± ρ(770)0) −0.4 ± 0.3
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.1 ± 0.3
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → (D∓
s π±)P K∗(892)0 −0.1 ± 0.0
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K± ρ(770)0) 0.0 ± 0.2
B0s → (D∓
s π±)P K∗(892)0 B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K(1460)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K(1460)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K(1460)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K(1460)± → K∗(892)0 π±) B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) B0s → D∓
s (K1(1270)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K∗0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K± ρ(770)0) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K(1460)± → K∗(892)0 π±) B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K1(1270)
± → K∗0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K(1460)± → K∗(892)0 π±) B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) B0
s → D∓s (K∗(1410)± → K∗(892)0 π±) 0.0 ± 0.0
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) B0s → D∓
s (K1(1270)± → K∗
0 (1430)π±) 0.0 ± 0.0
27
) [GeV]
±
π±
Ks
±
D(m
3 4 5
Yie
ld /
(0.0
5 G
eV
)
50
100
150
200
250
300
350 LHCb
) [GeV]±Ks
±
D(m
3 4 5
Yie
ld /
(0.0
5 G
eV
)
50
100
150
200
250
300
350 LHCb
) [GeV]
±
πs
±
D(m
2 3 4 5
Yie
ld /
(0.0
6 G
eV
)
50
100
150
200
250 LHCb
±
π±
Kθcos
1− 0.5− 0 0.5 1
Yie
ld /
(0.0
5)
50
100
150
200
250
300
350 LHCb
±πs
±
Dθcos
0 0.2 0.4 0.6 0.8 1
Yie
ld /
(0.0
3)
100
200
300
400
500
600
700
800LHCb
±πs
±
D,
±
π±K
φ2− 0 2
Yie
ld /
(0.1
6)
50
100
150
200
250
300
350
LHCb
Figure 10: Invariant-mass and angular distributions of background-subtracted B0s→ D∓s K
±π±π∓
candidates (data points) and fit projections (blue solid line). Contributions from b → c andb→ u decay amplitudes are overlaid, colour coded as in Fig. 6.
) [GeV]
±
π±
Ks
±
D(m
3 4 5
Yie
ld /
(0.0
5 G
eV
)
50
100
150
200
250
300
350 LHCb
) [GeV]±Ks
±
D(m
3 4 5
Yie
ld /
(0.0
5 G
eV
)
50
100
150
200
250
300
350 LHCb
) [GeV]
±
πs
±
D(m
2 3 4 5
Yie
ld /
(0.0
6 G
eV
)
50
100
150
200
250 LHCb
±
π±
Kθcos
1− 0.5− 0 0.5 1
Yie
ld /
(0.0
5)
50
100
150
200
250
300
350 LHCb
±πs
±
Dθcos
0 0.2 0.4 0.6 0.8 1
Yie
ld /
(0.0
3)
100
200
300
400
500
600
700
800LHCb
±πs
±
D,
±
π±K
φ2− 0 2
Yie
ld /
(0.1
6)
50
100
150
200
250
300
350
LHCb
Figure 11: Invariant-mass and angular distributions of background-subtracted B0s→ D∓s K
±π±π∓
candidates (data points) and fit projections (blue solid line). Incoherent contributions fromintermediate-state components are overlaid, colour coded as in Fig. 7.
28
Table B.8: Decay fractions in percent for several alternative amplitude models (Alt. 1 - Alt. 6).Resonance parameters and the observables r, κ, δ, γ − 2βs are also given.
Alt.1 Alt.2 Alt.3 Alt.4 Alt.5 Alt.6
b→ c
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) 12.2 12.0 13.9 11.5 14.7 12.8
B0s → D∓
s (K1(1270)±[D]→ K∗(892)0 π±) 0.4
B0s → D∓
s (K1(1270)± → K± ρ(770)0) 16.2 14.6 21.7 15.6 18.6 16.5
B0s → D∓
s (K1(1270)± → K± ρ(1450)0) 0.3
B0s → D∓
s (K1(1270)± → K∗0 (1430)0 π±) 3.9 3.5 3.7 3.6 3.0 3.6
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) 43.1 61.1 59.8 57.7 69.5 60.6
B0s → D∓
s (K1(1400)± → K± ρ(770)0) 0.5
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) 12.9 14.0 13.6 13.8 24.2 13.5
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 4.9 5.8 4.9 5.3 3.6 5.2
B0s → D∓
s (K(1460)± → K∗(892)0 π±) 1.7 1.3
B0s → D∓
s (K(1460)± → K± ρ(770)0) 0.2
B0s → D∓
s (K(1460)± → K± f0(500)0) 0.6
B0s → D∓
s (K∗(1680)± → K∗(892)0 π±) 5.5
B0s → D∓
s (K∗(1680)± → K± ρ(770)0) 1.9
B0s → D∓
s (K2(1770)→ K∗(892)0 π±) 0.4
B0s → (D∓
s π±)P K∗(892)0 8.8 9.3 9.5 7.7 10.2 9.2
B0s → (D∓
s K±)P ρ(770)0 1.3 0.7 0.1 0.6 0.2 0.5
b→ u
Sum 105.1 121.7 127.6 118.0 151.5 122.1
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) 4.5 2.4 1.1 4.9 2.0 3.4
B0s → D∓
s (K1(1270)±[D]→ K∗(892)0 π±) 0.1
B0s → D∓
s (K1(1270)± → K± ρ(770)0) 6.0 2.9 1.6 6.6 2.6 4.4
B0s → D∓
s (K1(1270)± → K± ρ(1450)0) 0.1
B0s → D∓
s (K1(1270)± → K∗0 (1430)0 π±) 1.4 0.7 0.3 1.5 0.4 1.0
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) 23.3 18.9 26.2 10.3 28.2 23.0
B0s → D∓
s (K1(1400)± → K± ρ(770)0) 0.2
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) 14.8 13.7 12.7 7.2 2.4 11.5
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 5.6 5.7 4.6 2.8 0.4 4.4
B0s → D∓
s (K(1460)± → K∗(892)0 π±) 12.9 10.8 12.6 8.1 12.1 11.9
B0s → D∓
s (K(1460)± → K± ρ(770)0) 1.0
B0s → D∓
s (K(1460)± → K± f0(500)0) 3.7
B0s → D∓
s (K∗(1680)± → K∗(892)0 π±) 5.6
B0s → D∓
s (K∗(1680)± → K± ρ(770)0) 2.0
B0s → D∓
s (K2(1770)→ K∗(892)0 π±) 0.9
B0s → (D∓
s π±)P K∗(892)0 32.5 33.4 31.3 38.3 27.4 30.7
Sum 103.9 88.7 90.7 84.4 83.0 91.4
mK1(1400) [MeV] 1401 1406 1404 1401 1397 1408
ΓK1(1400) [MeV] 189 193 192 196 199 193
mK∗(1410) [MeV] 1434 1437 1436 1442 1419 1434
ΓK∗(1410) [MeV] 391 400 386 382 453 395
r 0.50 0.55 0.59 0.48 0.54 0.58
κ 0.64 0.75 0.74 0.77 0.71 0.75
δ [◦] −4 −15 −14 −24 −19 −18
γ − 2βs [◦] 40 44 42 50 52 51
29
Table B.9: Decay fractions in percent for several alternative amplitude models (Alt. 7 - Alt. 12).Resonance parameters and the observables r, κ, δ, γ − 2βs are also given.
Alt.7 Alt.8 Alt.9 Alt.10 Alt.11 Alt.12
b→ c
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) 13.3 12.8 15.5 24.0 11.2 13.3
B0s → D∓
s (K1(1270)± → K± ρ(770)0) 17.5 17.5 19.1 14.9 17.7 19.9
B0s → D∓
s (K1(1270)± → K∗0 (1430)0 π±) 3.3 4.0 4.3 2.8 3.2 3.7
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) 66.0 93.1 55.5 77.9 44.7 63.5
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) 12.5 16.5 13.3 13.7 15.1 12.7
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 5.4 5.9 6.4 4.7 5.8 5.1
B0s → (D∓
s π±)S K
∗(892)0 5.1
B0s → (D∓
s π±)P K∗(892)0 11.4 19.8 8.5 5.5 10.8
B0s [P ]→ (D∓
s π±)P K∗(892)0 1.7
B0s [D]→ (D∓
s π±)P K∗(892)0 4.7
B0s → (D∓
s K±)S f0(500)0 1.8
B0s → (D∓
s K±)S f0(980)0 1.5
B0s → (D∓
s K±)S f2(1270)0 0.1
B0s → (D∓
s K±)S ρ(770)0 0.3
B0s → (D∓
s K±)P ρ(770)0 1.6 3.5 1.3 0.5
B0s [P ]→ (D∓
s K±)P ρ(770)0 0.2
B0s [D]→ (D∓
s K±)P ρ(770)0 0.9
b→ u
Sum 129.6 177.5 127.2 144.4 107.1 129.0
B0s → D∓
s (K1(1270)± → K∗(892)0 π±) 3.3 5.2 1.1 10.6 2.5 2.5
B0s → D∓
s (K1(1270)± → K± ρ(770)0) 4.4 7.2 1.4 6.6 3.9 3.7
B0s → D∓
s (K1(1270)± → K∗0 (1430)0 π±) 0.8 1.6 0.3 1.2 0.7 0.7
B0s → D∓
s (K1(1400)± → K∗(892)0 π±) 19.8 35.8 27.8 7.7 17.1 22.4
B0s → D∓
s (K∗(1410)± → K∗(892)0 π±) 14.2 9.8 9.9 11.7 4.0 14.5
B0s → D∓
s (K∗(1410)± → K± ρ(770)0) 6.1 3.5 4.8 4.0 1.5 5.8
B0s → D∓
s (K(1460)± → K∗(892)0 π±) 11.7 2.6 13.2 8.8 13.6 12.3
B0s → (D∓
s π±)S K
∗(892)0 22.3
B0s → (D∓
s π±)P K∗(892)0 25.8 68.5 33.6 43.1 29.3
B0s [P ]→ (D∓
s π±)P K∗(892)0 5.8
B0s [D]→ (D∓
s π±)P K∗(892)0 16.3
B0s → (D∓
s K±)S f0(500)0 0.7
B0s → (D∓
s K±)S f0(980)0 0.6
B0s → (D∓
s K±)S f2(1270)0 0.0
B0s → (D∓
s K±)S ρ(770)0 0.4
B0s → (D∓
s K±)P ρ(770)0 4.4
B0s [P ]→ (D∓
s K±)P ρ(770)0 0.3
B0s [D]→ (D∓
s K±)P ρ(770)0 1.1
Sum 86.5 156.3 97.9 73.0 87.8 91.0
mK1(1400) [MeV] 1405 1398 1404 1365 1406 1406
ΓK1(1400) [MeV] 193 247 188 203 184 190
mK∗(1410) [MeV] 1430 1443 1447 1440 1427 1434
ΓK∗(1410) [MeV] 406 432 419 373 406 399
r 0.57 0.75 0.58 0.45 0.54 0.57
κ 0.73 0.81 0.73 0.70 0.75 0.73
δ [◦] −16 −18 −12 −14 −19 −14
γ − 2βs [◦] 41 45 44 52 53 41
30
C Interpretation of the CP coefficients
The statistical and systematic correlation matrices of the CP coefficients obtained fromthe phase-space integrated fit are given in Table C.1 and Table C.2.
Figure 12 visualises the measured CP coefficients in the complex plane, whereλf ≡ (q/p) (
∫Au(x) dΦ4/
∫Ac(x) dΦ4). The points determined by (−A∆Γ
f , Sf) and
(−A∆Γf, Sf ) are proportional to r κ e±δ−(γ−2βs), whilst an additional constraint on r arises
from Cf .
Table C.1: Statistical correlation of the CP coefficients.
Cf A∆Γf AΓ∆
fSf Sf
Cf 1.000 0.135 0.168 0.040 0.042
A∆Γf 1.000 0.504 −0.082 −0.052
A∆Γf
1.000 −0.036 −0.113
Sf 1.000 0.014
Sf 1.000
Table C.2: Systematic correlation of the CP coefficients.
Cf A∆Γf AΓ∆
fSf Sf
Cf 1.000 0.039 0.042 −0.136 −0.006
A∆Γf 1.000 0.208 −0.031 −0.023
A∆Γf
1.000 −0.021 −0.035
Sf 1.000 −0.278
Sf 1.000
1.5− 1− 0.5− 0 0.5 1 )
2|fλ|+/(1fλ2 Re
1−
0.5−
0
0.5
1 )2 | fλ|
+/(
1f
λ2 Im
2fC−1
)fS, Γ∆
fA−(
)f
S, Γ∆
fA−(
)sβ2−γ(−
Combination
LHCb
Figure 12: Visualisation of how the CP coefficients contribute towards the overall constraint onthe weak phase, γ − 2βs. The difference between the phase of (−A∆Γ
f , Sf ) and (−A∆Γf, Sf ) is
proportional to the strong phase δ, which is close to 0◦ and thus not indicated in the figure.
31
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LHCb collaboration
R. Aaij31, C. Abellan Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H. Afsharnia9,C.A. Aidala84, S. Aiola25, Z. Ajaltouni9, S. Akar64, J. Albrecht14, F. Alessio47, M. Alexander58,A. Alfonso Albero44, Z. Aliouche61, G. Alkhazov37, P. Alvarez Cartelle47, S. Amato2,Y. Amhis11, L. An21, L. Anderlini21, A. Andreianov37, M. Andreotti20, F. Archilli16,A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B. Audurier11,S. Bachmann16, M. Bachmayer48, J.J. Back55, S. Baker60, P. Baladron Rodriguez45,V. Balagura11, W. Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11, W. Barter60,M. Bartolini23,i, F. Baryshnikov80, J.M. Basels13, G. Bassi28, B. Batsukh67, A. Battig14,A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A. Beiter67, V. Belavin41, S. Belin26,V. Bellee48, K. Belous43, I. Belov39, I. Belyaev38, G. Bencivenni22, E. Ben-Haim12,A. Berezhnoy39, R. Bernet49, D. Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12,A. Bertolin27, C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, M. Birch60, F.C.R. Bishop54,A. Bizzeti21,s, M. Bjørn62, M.P. Blago47, T. Blake55, F. Blanc48, S. Blusk67, D. Bobulska58,J.A. Boelhauve14, O. Boente Garcia45, T. Boettcher63, A. Boldyrev81, A. Bondar42,N. Bondar37, S. Borghi61, M. Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48,T.J.V. Bowcock59, A. Boyer47, C. Bozzi20, M.J. Bradley60, S. Braun65, A. Brea Rodriguez45,M. Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D. Brundu26, A. Buonaura49, C. Burr47,A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W. Byczynski47, S. Cadeddu26,H. Cai72, R. Calabrese20,g, L. Calefice14,12, L. Calero Diaz22, S. Cali22, R. Calladine52,M. Calvi24,j , M. Calvo Gomez83, P. Camargo Magalhaes53, A. Camboni44, P. Campana22,D.H. Campora Perez47, A.F. Campoverde Quezada5, S. Capelli24,j , L. Capriotti19,e,A. Carbone19,e, G. Carboni29, R. Cardinale23,i, A. Cardini26, I. Carli6, P. Carniti24,j ,L. Carus13, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M. Cattaneo47, G. Cavallero47,S. Celani48, J. Cerasoli10, A.J. Chadwick59, M.G. Chapman53, M. Charles12, Ph. Charpentier47,G. Chatzikonstantinidis52, C.A. Chavez Barajas59, M. Chefdeville8, C. Chen3, S. Chen26,A. Chernov33, S.-G. Chitic47, V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37,V. Chulikov37, P. Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, P.E.L. Clarke57,M. Clemencic47, H.V. Cliff54, J. Closier47, J.L. Cobbledick61, V. Coco47, J.A.B. Coelho11,J. Cogan10, E. Cogneras9, L. Cojocariu36, P. Collins47, T. Colombo47, L. Congedo18,d,A. Contu26, N. Cooke52, G. Coombs58, G. Corti47, C.M. Costa Sobral55, B. Couturier47,D.C. Craik63, J. Crkovska66, M. Cruz Torres1, R. Currie57, C.L. Da Silva66, E. Dall’Occo14,J. Dalseno45, C. D’Ambrosio47, A. Danilina38, P. d’Argent47, A. Davis61,O. De Aguiar Francisco61, K. De Bruyn77, S. De Capua61, M. De Cian48, J.M. De Miranda1,L. De Paula2, M. De Serio18,d, D. De Simone49, P. De Simone22, J.A. de Vries78, C.T. Dean66,W. Dean84, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P. Dembinski14, A. Dendek34,V. Denysenko49, D. Derkach81, O. Deschamps9, F. Desse11, F. Dettori26,f , B. Dey72,P. Di Nezza22, S. Didenko80, L. Dieste Maronas45, H. Dijkstra47, V. Dobishuk51,A.M. Donohoe17, F. Dordei26, A.C. dos Reis1, L. Douglas58, A. Dovbnya50, A.G. Downes8,K. Dreimanis59, M.W. Dudek33, L. Dufour47, V. Duk76, P. Durante47, J.M. Durham66,D. Dutta61, M. Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede68,V. Egorychev38, S. Eidelman42,v, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S. Ely67, A. Ene36,E. Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47, A. Falabella19, J. Fan3, Y. Fan5,B. Fang72, N. Farley52, S. Farry59, D. Fazzini24,j , P. Fedin38, M. Feo47, P. Fernandez Declara47,A. Fernandez Prieto45, J.M. Fernandez-tenllado Arribas44, F. Ferrari19,e, L. Ferreira Lopes48,F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49, M. Ferro-Luzzi47, S. Filippov40,R.A. Fini18, M. Fiorini20,g, M. Firlej34, K.M. Fischer62, C. Fitzpatrick61, T. Fiutowski34,F. Fleuret11,b, M. Fontana12, F. Fontanelli23,i, R. Forty47, V. Franco Lima59,
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M. Franco Sevilla65, M. Frank47, E. Franzoso20, G. Frau16, C. Frei47, D.A. Friday58, J. Fu25,Q. Fuehring14, W. Funk47, E. Gabriel31, T. Gaintseva41, A. Gallas Torreira45, D. Galli19,e,S. Gambetta57,47, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4, M. Garau26,L.M. Garcia Martin55, P. Garcia Moreno44, J. Garcıa Pardinas49, B. Garcia Plana45,F.A. Garcia Rosales11, L. Garrido44, C. Gaspar47, R.E. Geertsema31, D. Gerick16, L.L. Gerken14,E. Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph. Ghez8, V. Gibson54,M. Giovannetti22,k, A. Gioventu45, P. Gironella Gironell44, L. Giubega36, C. Giugliano20,47,g,K. Gizdov57, E.L. Gkougkousis47, V.V. Gligorov12, C. Gobel69, E. Golobardes83, D. Golubkov38,A. Golutvin60,80, A. Gomes1,a, S. Gomez Fernandez44, F. Goncalves Abrantes69, M. Goncerz33,G. Gong3, P. Gorbounov38, I.V. Gorelov39, C. Gotti24,j , E. Govorkova47, J.P. Grabowski16,R. Graciani Diaz44, T. Grammatico12, L.A. Granado Cardoso47, E. Grauges44, E. Graverini48,G. Graziani21, A. Grecu36, L.M. Greeven31, P. Griffith20, L. Grillo61, S. Gromov80,B.R. Gruberg Cazon62, C. Gu3, M. Guarise20, P. A. Gunther16, E. Gushchin40, A. Guth13,Y. Guz43,47, T. Gys47, T. Hadavizadeh68, G. Haefeli48, C. Haen47, J. Haimberger47,S.C. Haines54, T. Halewood-leagas59, P.M. Hamilton65, Q. Han7, X. Han16, T.H. Hancock62,S. Hansmann-Menzemer16, N. Harnew62, T. Harrison59, C. Hasse47, M. Hatch47, J. He5,M. Hecker60, K. Heijhoff31, K. Heinicke14, A.M. Hennequin47, K. Hennessy59, L. Henry25,46,J. Heuel13, A. Hicheur2, D. Hill62, M. Hilton61, S.E. Hollitt14, J. Hu16, J. Hu71, W. Hu7,W. Huang5, X. Huang72, W. Hulsbergen31, R.J. Hunter55, M. Hushchyn81, D. Hutchcroft59,D. Hynds31, P. Ibis14, M. Idzik34, D. Ilin37, P. Ilten64, A. Inglessi37, A. Ishteev80, K. Ivshin37,R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K. Jashal46, A. Jawahery65, V. Jevtic14,M. Jezabek33, F. Jiang3, M. John62, D. Johnson47, C.R. Jones54, T.P. Jones55, B. Jost47,N. Jurik47, S. Kandybei50, Y. Kang3, M. Karacson47, M. Karpov81, N. Kazeev81, M. Kecke16,F. Keizer54,47, M. Kenzie55, T. Ketel32, B. Khanji14, A. Kharisova82, S. Kholodenko43,K.E. Kim67, T. Kirn13, V.S. Kirsebom48, O. Kitouni63, S. Klaver31, K. Klimaszewski35,S. Koliiev51, A. Kondybayeva80, A. Konoplyannikov38, P. Kopciewicz34, R. Kopecna16,P. Koppenburg31, M. Korolev39, I. Kostiuk31,51, O. Kot51, S. Kotriakhova37,30, P. Kravchenko37,L. Kravchuk40, R.D. Krawczyk47, M. Kreps55, F. Kress60, S. Kretzschmar13, P. Krokovny42,v,W. Krupa34, W. Krzemien35, W. Kucewicz33,l, M. Kucharczyk33, V. Kudryavtsev42,v,H.S. Kuindersma31, G.J. Kunde66, T. Kvaratskheliya38, D. Lacarrere47, G. Lafferty61, A. Lai26,A. Lampis26, D. Lancierini49, J.J. Lane61, R. Lane53, G. Lanfranchi22, C. Langenbruch13,J. Langer14, O. Lantwin49,80, T. Latham55, F. Lazzari28,t, R. Le Gac10, S.H. Lee84, R. Lefevre9,A. Leflat39, S. Legotin80, O. Leroy10, T. Lesiak33, B. Leverington16, H. Li71, L. Li62, P. Li16,X. Li66, Y. Li6, Y. Li6, Z. Li67, X. Liang67, T. Lin60, R. Lindner47, V. Lisovskyi14,R. Litvinov26, G. Liu71, H. Liu5, S. Liu6, X. Liu3, A. Loi26, J. Lomba Castro45, I. Longstaff58,J.H. Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,m, S. Luchuk40,M. Lucio Martinez31, V. Lukashenko31, Y. Luo3, A. Lupato61, E. Luppi20,g, O. Lupton55,A. Lusiani28,r, X. Lyu5, L. Ma6, S. Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48,P. Mackowiak14, S. Maddrell-Mander53, O. Madejczyk34, L.R. Madhan Mohan53, O. Maev37,A. Maevskiy81, D. Maisuzenko37, M.W. Majewski34, J.J. Malczewski33, S. Malde62,B. Malecki47, A. Malinin79, T. Maltsev42,v, H. Malygina16, G. Manca26,f , G. Mancinelli10,R. Manera Escalero44, D. Manuzzi19,e, D. Marangotto25,o, J. Maratas9,u, J.F. Marchand8,U. Marconi19, S. Mariani21,47,h, C. Marin Benito11, M. Marinangeli48, P. Marino48, J. Marks16,P.J. Marshall59, G. Martellotti30, L. Martinazzoli47,j , M. Martinelli24,j , D. Martinez Santos45,F. Martinez Vidal46, A. Massafferri1, M. Materok13, R. Matev47, A. Mathad49, Z. Mathe47,V. Matiunin38, C. Matteuzzi24, K.R. Mattioli84, A. Mauri31, E. Maurice11,b, J. Mauricio44,M. Mazurek35, M. McCann60, L. Mcconnell17, T.H. Mcgrath61, A. McNab61, R. McNulty17,J.V. Mead59, B. Meadows64, C. Meaux10, G. Meier14, N. Meinert75, D. Melnychuk35,S. Meloni24,j , M. Merk31,78, A. Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A. Milanes73,E. Millard55, M. Milovanovic47, M.-N. Minard8, L. Minzoni20,g, S.E. Mitchell57, B. Mitreska61,
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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, China4School of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing,China5University of Chinese Academy of Sciences, Beijing, China6Institute Of High Energy Physics (IHEP), Beijing, China7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France9Universite Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France11Universite Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France12LPNHE, Sorbonne Universite, Paris Diderot Sorbonne Paris Cite, CNRS/IN2P3, Paris, France13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany14Fakultat Physik, Technische Universitat Dortmund, Dortmund, Germany15Max-Planck-Institut fur Kernphysik (MPIK), Heidelberg, Germany16Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg, Heidelberg, Germany17School of Physics, University College Dublin, Dublin, Ireland18INFN Sezione di Bari, Bari, Italy19INFN Sezione di Bologna, Bologna, Italy20INFN Sezione di Ferrara, Ferrara, Italy21INFN Sezione di Firenze, Firenze, Italy22INFN Laboratori Nazionali di Frascati, Frascati, Italy23INFN Sezione di Genova, Genova, Italy24INFN Sezione di Milano-Bicocca, Milano, Italy25INFN Sezione di Milano, Milano, Italy26INFN Sezione di Cagliari, Monserrato, Italy27Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy28INFN Sezione di Pisa, Pisa, Italy29INFN Sezione di Roma Tor Vergata, Roma, Italy30INFN Sezione di Roma La Sapienza, Roma, Italy31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands32Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam,Netherlands33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland34AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science,Krakow, Poland35National Center for Nuclear Research (NCBJ), Warsaw, Poland36Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania
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37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI), Gatchina, Russia38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute (ITEP NRC KI), Moscow,Russia39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS), Moscow, Russia41Yandex School of Data Analysis, Moscow, Russia42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI), Protvino, Russia,Protvino, Russia44ICCUB, Universitat de Barcelona, Barcelona, Spain45Instituto Galego de Fısica de Altas Enerxıas (IGFAE), Universidade de Santiago de Compostela,Santiago de Compostela, Spain46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain47European Organization for Nuclear Research (CERN), Geneva, Switzerland48Institute of Physics, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland49Physik-Institut, Universitat Zurich, Zurich, Switzerland50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine51Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine52University of Birmingham, Birmingham, United Kingdom53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom55Department of Physics, University of Warwick, Coventry, United Kingdom56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom58School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom60Imperial College London, London, United Kingdom61Department of Physics and Astronomy, University of Manchester, Manchester, United Kingdom62Department of Physics, University of Oxford, Oxford, United Kingdom63Massachusetts Institute of Technology, Cambridge, MA, United States64University of Cincinnati, Cincinnati, OH, United States65University of Maryland, College Park, MD, United States66Los Alamos National Laboratory (LANL), Los Alamos, United States67Syracuse University, Syracuse, NY, United States68School of Physics and Astronomy, Monash University, Melbourne, Australia, associated to 55
69Pontifıcia Universidade Catolica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2
70Physics and Micro Electronic College, Hunan University, Changsha City, China, associated to 7
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South ChinaNormal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to 12
74Universitat Bonn - Helmholtz-Institut fur Strahlen und Kernphysik, Bonn, Germany, associated to 16
75Institut fur Physik, Universitat Rostock, Rostock, Germany, associated to 16
76INFN Sezione di Perugia, Perugia, Italy, associated to 20
77Van Swinderen Institute, University of Groningen, Groningen, Netherlands, associated to 31
78Universiteit Maastricht, Maastricht, Netherlands, associated to 31
79National Research Centre Kurchatov Institute, Moscow, Russia, associated to 38
80National University of Science and Technology “MISIS”, Moscow, Russia, associated to 38
81National Research University Higher School of Economics, Moscow, Russia, associated to 41
82National Research Tomsk Polytechnic University, Tomsk, Russia, associated to 38
83DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 44
84University of Michigan, Ann Arbor, United States, associated to 67
aUniversidade Federal do Triangulo Mineiro (UFTM), Uberaba-MG, BrazilbLaboratoire Leprince-Ringuet, Palaiseau, FrancecP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, RussiadUniversita di Bari, Bari, Italy
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eUniversita di Bologna, Bologna, ItalyfUniversita di Cagliari, Cagliari, ItalygUniversita di Ferrara, Ferrara, ItalyhUniversita di Firenze, Firenze, ItalyiUniversita di Genova, Genova, ItalyjUniversita di Milano Bicocca, Milano, ItalykUniversita di Roma Tor Vergata, Roma, ItalylAGH - University of Science and Technology, Faculty of Computer Science, Electronics andTelecommunications, Krakow, PolandmUniversita di Padova, Padova, ItalynUniversita di Pisa, Pisa, ItalyoUniversita degli Studi di Milano, Milano, ItalypUniversita di Urbino, Urbino, ItalyqUniversita della Basilicata, Potenza, ItalyrScuola Normale Superiore, Pisa, ItalysUniversita di Modena e Reggio Emilia, Modena, ItalytUniversita di Siena, Siena, ItalyuMSU - Iligan Institute of Technology (MSU-IIT), Iligan, PhilippinesvNovosibirsk State University, Novosibirsk, Russia
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