Measurement of the CKM angle γ from a combination of LHCb results

JHEP12(2016)087 Published for SISSA by Springer Received:November 10, 2016 Accepted:December 9, 2016 Published:December 19, 2016 Measurement of the CKM angle γfrom a combination of LHCb results The LHCb collaboration E-mail: [email protected] Abstract: A combination of measurements sensitive to the CKM angle γfrom LHCb is performed. The inputs are from analyses of time-integrated B+→DK+,B0→DK∗0, B0→DK+π−and B+→DK+π+π−tree-level decays. In addition, results from a time-dependent analysis of B0 s→D∓ sK±decays are included. The combination yields γ= (72.2+6.8 −7.3)◦, where the uncertainty includes systematic effects. The 95.5% confidence level interval is determined to be γ∈[55.9,85.2]◦. A second combination is investigated, also including measurements from B+→Dπ+and B+→Dπ+π−π+decays, which yields compatible results. Keywords: B physics, CKM angle gamma, CP violation, Hadron-Hadron scattering (experiments) ArXiv ePrint: 1611.03076 Open Access, Copyright CERN, for the benefit of the LHCb Collaboration. Article funded by SCOAP3. doi:10.1007/JHEP12(2016)087 JHEP12(2016)087 Contents 1 Introduction 2 2 Inputs from LHCb analyses sensitive to γ6 3 Auxiliary inputs 8 4 Statistical treatment 10 5 Results 11 5.1 DK combination 11 5.2 Dh combination 12 5.3 Coverage of the frequentist method 13 5.4 Interpretation 17 6 Bayesian analysis 18 6.1 DK combination 20 6.2 Dh combination 20 7 Conclusion 20 Appendices 26 A Relationships between parameters and observables 26 B Input observable values and uncertainties 33 C Uncertainty correlations for the input observables 38 D External constraint values and uncertainties 46 E Uncertainty correlations for the external constraints 48 F Fit parameter correlations 48 References 49 – 1 – JHEP12(2016)087 1 Introduction Understanding the origin of the baryon asymmetry of the Universe is one of the key issues of modern physics. Sakharov showed that such an asymmetry can arise if three conditions are fulfilled [1], one of which is the requirement that both charge (C) and charge-parity (CP) symmetries are broken. The latter phenomenon arises in the Standard Model (SM) of particle physics through the complex phase of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix [2,3], although the effect in the SM is not large enough to account for the observed baryon asymmetry in the Universe [4]. Violation of CP symmetry can be studied by measuring the angles of the CKM unitarity triangle [5–7]. The least precisely known of these angles, γ≡arg[−VudV∗ ub/VcdV∗ cb], can be measured using only tree-level processes [8–11]; a method that, assuming new physics is not present in tree-level decays [12], has negligible theoretical uncertainty [13]. Disagreement between such direct measurements of γand the value inferred from global CKM fits, assuming the validity of the SM, would indicate new physics beyond the SM. The value of γcan be determined by exploiting the interference between favoured b→cW (Vcb) and suppressed b→uW (Vub) transition amplitudes using decay channels such as B+→Dh+,B0→DK∗0,B0→DK+π−,B+→Dh+π−π+and B0 s→D∓ sK±[8– 11,14–21], where his a kaon or pion and Drefers to a neutral charm meson that is a mixture of the D0and D0flavour eigenstates. The inclusion of charge conjugate processes is implied throughout, unless otherwise stated. The most precise way to determine γis through a combination of measurements from analyses of many decay modes. Hadronic parameters such as those that describe the ratio (rX B) or strong phase difference (δX B) between the Vcb and Vub transition amplitudes and where Xis a specific final state of a Bmeson decay, are also simultaneously determined. The ratio of the suppressed to favoured Bdecay amplitudes is related to γand the hadronic parameters by Asup/Afav =rX Bei(δX B±γ), where the + (−) sign refers to the decay of a meson containing a b(b). The statistical uncertainty with which γcan be measured is approximately inversely proportional to the value of rX B, which is around 0.1 for B+→DK+decays [22].1In the B+→Dπ+channel, rDπ Bis expected to be of order 0.005 [23] because the favoured amplitude is enhanced by |Vud|/|Vus| while the suppressed amplitude is further reduced by |Vcd|/|Vcs|with respect to B+→DK+ decays. Consequently, the expected sensitivity to γin B+→Dπ+decays is considerably lower than for B+→DK+decays, although the signal yields are higher. For B0→DK∗0 (and also B0 s→D∓ sK±) decays a higher value is expected [24], rDK∗0 B∼rDsK B∼0.3, which compensates for the lower branching fraction [25],2whilst the expected value for rDKππ Bis similar to rDK B. The current world average, using only direct measurements of B→DK-like decays, is γ= (73.2+6.3 −7.0)◦[26]3(or, using different inputs with an alternative statistical approach, γ= (68.3±7.5)◦[27]4). The previous LHCb combination found γ= (73 +9 −10)◦[28]. 1Updated results and plots available at http://www.slac.stanford.edu/xorg/hfag/. 2See also 2015 update. 3Updated results and plots available at: http://ckmfitter.in2p3.fr. 4Updated results and plots available at: http://www.utfit.org/UTfit/. – 2 – JHEP12(2016)087 This paper presents the latest combination of LHCb measurements of tree-level decays that are sensitive to γ. The results supersede those previously reported in refs. [28–31], including more decay channels and updating selected channels to the full Run 1 dataset of pp collisions at √s= 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb−1. Two combinations are performed, one including all inputs from B→DK-like modes (referred to as DK) and one additionally including inputs from B+→Dπ+and B+→Dπ+π−π+ decays (referred to as Dh). The DK combination includes 71 observables depending on 32 parameters, whilst the Dh combination has 89 observables and 38 parameters. The analyses included in the combinations use a variety of methods to measure γ, which are reviewed in ref. [32]. The observables are briefly summarised below; their dependence on γand various hadronic parameters is given in appendix A. The Gronau-London-Wyler (GLW) method [8,9] considers the decays of Dmesons to CP eigenstates, for example the CP-even decays D→K+K−and D→π+π−. The Atwood-Dunietz-Soni (ADS) approach [10,11] extends this to include final states that are not CP eigenstates, for example D0→π−K+, where the interference between the Cabibbo-allowed and doubly Cabibbo-suppressed decay modes in both the Band Ddecays gives rise to large charge asymmetries. This introduces an additional dependence on the Ddecay dynamics through the ratio of suppressed and favoured Ddecay amplitudes, rD, and their phase difference, δD. The GLW/ADS formalism is easily extended to multibody Ddecays [10,11,33] although the multiple interfering amplitudes dilute the sensitivity to γ. For multibody ADS modes this dilution is parameterised in terms of a coherence factor, κD, and for the GLW modes it is parametrised by F+, which describes the fraction of CP-even content in a multibody decay. For multibody Ddecays these parameters are measured independently and used as external constraints in the combination as discussed in section 3. The GLW/ADS observables are constructed from decay-rate ratios, double ratios and charge asymmetries as outlined in the following. For GLW analyses the observables are the charge-averaged rate and the partial-rate asymmetry. The former is defined as RCP = 2 Γ(B−→DCP K−) + Γ(B+→DCP K+) Γ(B−→D0K−) + Γ(B+→D0K+),(1.1) where DCP refers to the final state of a Dmeson decay into a CP eigenstate. Experimentally it is convenient to measure RCP , for a given final state f, by forming a double ratio that is normalised using the rate for a Cabibbo-favoured decay (e.g. D0→K−π+), and the equivalent quantities from the relevant B+→Dπ−decay mode. Defining the ratio of the favoured B+→D0K+and B+→D0π+partial widths, for a given final state f, as Rf K/π =Γ(B−→D[→f]K−) + Γ(B+→D[→¯ f]K+) Γ(B−→D[→f]π−) + Γ(B+→D[→¯ f]π+),(1.2) the double ratios are constructed as RKK CP ≈RKK K/π RKπ K/π , Rππ CP ≈Rππ K/π RKπ K/π , RKKπ0 CP ≈RKKπ0 K/π RKππ0 K/π , Rπππ0 CP ≈Rπππ0 K/π RKππ0 K/π ,etc. (1.3) – 3 – JHEP12(2016)087 These relations are exact when the suppressed B+→Dπ+decay amplitude (b→u) vanishes and the flavour specific rates, given in the denominator of eq. (1.1), are measured using the appropriate flavour-specific Ddecay channel. The GLW partial-rate asymmetry, for a given Dmeson decay into a CP eigenstate f, is defined as ADh,f CP =Γ(B−→DCP h−)−Γ(B+→DCP h+) Γ(B−→DCP h−) + Γ(B+→DCP h+).(1.4) Similarly, observables associated to the ADS modes, for a suppressed D→fdecay, are the charge-averaged rate and the partial-rate asymmetry. For the charge-averaged rate, it is adequate to use a single ratio (normalised to the favoured D→¯ fdecay) because the detection asymmetries cancel out. The charge-averaged rate is defined as RDh, ¯ f ADS =Γ(B−→D[→¯ f]h−) + Γ(B+→D[→f]h+) Γ(B−→D[→f]h−) + Γ(B+→D[→¯ f]h+),(1.5) whilst the partial-rate asymmetry is defined as ADh, ¯ f ADS =Γ(B−→D[→¯ f]h−)−Γ(B+→D[→f]h+) Γ(B−→D[→¯ f]h−) + Γ(B+→D[→f]h+).(1.6) The equivalent charge asymmetry for favoured ADS modes is defined as ADh,f fav =Γ(B−→D[→f]h−)−Γ(B+→D[→¯ f]h+) Γ(B−→D[→f]h−) + Γ(B+→D[→¯ f]h+).(1.7) Some of the input analyses determined two statistically independent observables instead of those in eqs. (1.5) and (1.6), namely the ratio of partial widths for the suppressed and favoured decays of each initial Bflavour, RDh, ¯ f +=Γ(B+→D[→f]h+) Γ(B+→D[→¯ f]h+),(1.8) RDh, ¯ f −=Γ(B−→D[→¯ f]h−) Γ(B−→D[→f]h−).(1.9) It should be noted that eqs. (1.5) and (1.6) are related to eqs. (1.8) and (1.9) by RADS =R++R− 2, AADS =R−−R+ R−+R+ ,(1.10) if the rates of the Cabibbo-favoured decays for B−and B+are identical. Similar to the ADS approach is the Grossman-Ligeti-Soffer (GLS) method [16] that exploits singly Cabibbo-suppressed decays such as D→K0 SK−π+. The GLS observables are defined in analogy to eqs. (1.5)–(1.7). Note that in the GLS method the favoured decay has sensitivity to γbecause the ratio between the suppressed and favoured amplitudes is much larger than in the ADS approach. It is therefore worthwhile to include the favoured GLS decays in the combinations, which is not the case for the favoured ADS channels alone. The Giri-Grossman-Soffer-Zupan (GGSZ) method [14,15] uses self-conjugate multibody Dmeson decay modes like K0 Sπ+π−. Sensitivity to γis obtained by comparing the – 4 – JHEP12(2016)087 distributions of decays in the D→fDalitz plot for opposite-flavour initial-state Band B mesons. The population of candidates in the Dalitz plot depends on four variables, referred to as Cartesian variables which, for a given Bdecay final state X, are defined as xX ±=rX Bcos(δX B±γ),(1.11) yX ±=rX Bsin(δX B±γ).(1.12) These are the preferred observables for GGSZ analyses. The GLW/ADS and GGSZ formalisms can also be extended to multibody Bdecays by including a coherence factor, κB, that accounts for dilution from interference between competing amplitudes. This inclusive approach is used for all multibody and quasi-two-body Bdecays, with the exception of the GLW-Dalitz analysis of B0→DK+π−decays where an amplitude analysis is performed to determine xX ±and yX ±. Here the term quasi-two-body decays refer to a two body resonant decay that contributes to a three body final state (e.g. B0→DK∗(892)0decays in the B0→DK+π−final state). Time-dependent (TD) analyses of B0 s→D∓ sK±are also sensitive to γ[17–19]. Due to the interference between the mixing and decay amplitudes, the CP-sensitive observables, which are the coefficients of the time evolution of B0 s→D∓ sK±decays, have a dependence on (γ−2βs), where βs≡arg(−VtsV∗ tb/VcsV∗ cb). In the SM, to a good approximation, −2βs is equal to the phase φsdetermined from B0 s→J/ψφ and similar decays, and therefore an external constraint on the value of φsprovides sensitivity to γ. The time-dependent decay rates for the initially pure B0 sand B0 sflavour eigenstates are given by dΓB0 s→f(t) dt=1 2|Af|2(1 + |λf|2)e−Γstcosh ∆Γst 2+A∆Γ fsinh ∆Γst 2 +Cfcos (∆mst)−Sfsin (∆mst),(1.13) dΓB0 s→f(t) dt=1 2|Af|2 p q 2 (1 + |λf|2)e−Γstcosh ∆Γst 2+A∆Γ fsinh ∆Γst 2 −Cfcos (∆mst) + Sfsin (∆mst),(1.14) where λf≡(q/p)·(¯ Af/Af) and Af(¯ Af) is the decay amplitude of a B0 s(B0 s) to a final state f. In the convention used, f(¯ f) is the D− sK+(D+ sK−) final state. The parameter ∆msis the oscillation frequency for B0 smesons, Γsis the average B0 sdecay width, and ∆Γsis the decay-width difference between the heavy and light mass eigenstates in the B0 ssystem, which is known to be positive [34] as expected in the SM. The observables sensitive to γare A∆Γ f,Cfand Sf. The complex coefficients pand qrelate the B0 smeson mass eigenstates, |BL,Hi, to the flavour eigenstates, |B0 siand |B0 si, as |BLi=p|B0 si+q|B0 si and |BHi=p|B0 si−q|B0 siwith |p|2+|q|2= 1. Similar equations can be written for the CP-conjugate decays replacing Sfby S¯ f, and A∆Γ fby A∆Γ ¯ f, and, assuming no CP violation in either the decay or mixing amplitudes, C¯ f=−Cf. The relationships between the observables, γand the hadronic parameters are given in appendix A. – 5 – JHEP12(2016)087 The combinations are potentially sensitive to subleading effects from D0–D0mixing [35–37]. These are corrected for where necessary, by taking into account the D0decaytime acceptances of the individual measurements. The size of the correction is inversely proportional to rX Band so is particularly important for the B+→Dπ+(π+π−) modes. For consistency, the correction is also applied in the corresponding B+→DK+(π+π−) modes. The correction for other decay modes would be small and is not applied. There can also be an effect from CP violation in D→h+h−decays [38–41], which is included in the relevant B+→D0h+(π+π−) analyses using the world average values [22], although the latest measurements indicate that the effect is negligible [42]. Final states that include aK0 Smeson are potentially affected by corrections due to CP violation and mixing in the neutral kaon system, parametrised by the non-zero parameter K[43]. The effect is expected to be O(K/rh B), which is negligible for B+→DK+decays since |K| ≈ 0.002 and rDK B≈0.1 [22]. For B+→Dπ+decays this ratio is expected to be O(1) since rDπ B is expected to be around 0.5% [23]. Consequently, the B+→Dπ+decay modes affected, such as those with D→K0 SK∓π±, are not included in the Dh combination. To determine γwith the best possible precision, auxiliary information on some of the hadronic parameters is used in conjunction with observables measured in other LHCb analyses. More information on these quantities can be found in sections 2and 3, with a summary provided in tables 1and 2. Frequentist and Bayesian treatments are both studied. Section 4describes the frequentist treatment with results and coverage studies reported in section 5. Section 6describes the results of a Bayesian analysis. 2 Inputs from LHCb analyses sensitive to γ The LHCb measurements used as inputs in the combinations are summarised in table 1 and described briefly below. The values and uncertainties of the observables are provided in appendix Band the correlations are given in appendix C. The relationships between the observables and the physics parameters are listed in appendix A. All analyses use a data sample corresponding to an integrated luminosity of 3 fb−1, unless otherwise stated. •B+→Dh+,D→h+h−.The GLW/ADS measurement using B+→Dh+, D0→h+h−decays [44] is an update of a previous analysis [53]. The observables are defined in analogy to eqs. (1.3)–(1.7). •B+→Dh+,D→h+π−π+π−.The ADS measurement using the B+→Dh+, D→K±π∓π+π−decay mode [44] is an update of a previous measurement [54]. The quasi-GLW measurement with B+→Dh+,D→π+π−π+π−decays is included in the combination for the first time. The label “quasi” is used because the D→ π+π−π+π−decay is not completely CP -even; the fraction of CP -even content is given by Fππππ as described in section 3. The method for constraining γusing these decays is described in ref. [33], with observables defined in analogy to eqs. (1.3)–(1.7). •B+→Dh+,D→h+h−π0.Inputs from the quasi-GLW/ADS analysis of B+→Dh+,D→h+h−π0decays [45] are new to this combination. The CP-even – 6 – JHEP12(2016)087 Bdecay Ddecay Method Ref. Status since last combination [28] B+→Dh+D→h+h−GLW/ADS [44] Updated to 3 fb−1 B+→Dh+D→h+π−π+π−GLW/ADS [44] Updated to 3 fb−1 B+→Dh+D→h+h−π0GLW/ADS [45] New B+→DK+D→K0 Sh+h−GGSZ [46] As before B+→DK+D→K0 SK−π+GLS [47] As before B+→Dh+π−π+D→h+h−GLW/ADS [48] New B0→DK∗0D→K+π−ADS [49] As before B0→DK+π−D→h+h−GLW-Dalitz [50] New B0→DK∗0D→K0 Sπ+π−GGSZ [51] New B0 s→D∓ sK±D+ s→h+h−π+TD [52] As before Table 1. List of the LHCb measurements used in the combinations. content of the D→K+K−π0(D→π+π−π0) decay mode is given by the parameter FKKπ0(Fπππ0), as described in section 3. The observables are defined in analogy to eqs. (1.3)–(1.7). •B+→DK+,D→K0 Sh+h−.The inputs from the model-independent GGSZ analysis of B+→DK+,D→K0 Sh+h−decays [46] are the same as those used in the previous combination [28]. The variables, defined in analogy to eqs. (1.11)– (1.12), are obtained from a simultaneous fit to the Dalitz plots of D→K0 Sπ+π−and D→K0 SK+K−decays. Inputs from a model-dependent GGSZ analysis of the same decay [55] using data corresponding to 1 fb−1are not included due to the overlap of the datasets. •B+→DK+,D→K0 SK−π+.The inputs from the GLS analysis of B+→DK+, D→K0 SK−π+decays [47] are the same as those included in the last combination [28]. The observables are defined in analogy to eqs. (1.5)–(1.7). The negligible statistical and systematic correlations are not taken into account. •B+→Dh+π−π+,D→h+h−.The inputs from the LHCb GLW/ADS analysis of B+→Dh+π−π+,D0→h+h−decays [48] are included in the combination for the first time. The observables are defined in analogy to eqs. (1.3)–(1.4), (1.7)–(1.9). The only non-negligible correlations are statistical, ρ(ADKππ, KK CP , ADKππ, ππ CP ) = 0.20 and ρ(ADπππ, KK CP , ADπππ,ππ CP )=0.08. •B0→DK∗0,D→K+π−.The inputs from the ADS analysis of B0→ D0K∗(892)0,D0→K±π∓decays [49] are included as they were in the previous combination [28]. However, the GLW part of this analysis (with D0→K+K−and – 7 – JHEP12(2016)087 D0→π+π−) has been superseded by the Dalitz plot analysis. The ADS observables are defined in analogy to eqs. (1.7)–(1.9). •B0→DK+π−,D→h+h−.Information from the GLW-Dalitz analysis of B0→DK+π−,D0→h+h−decays [50] is added to the combination for the first time. The “Dalitz” label indicates the method used to determine information about CP violation in this mode. The variables, defined in analogy to eqs. (1.11)–(1.12), are determined from a simultaneous Dalitz plot fit to B0→DK+π−with D0→K−π+, D→K+K−and D→π+π−samples, as described in refs. [20,21]. Note that the observables are those associated with the DK∗(892)0amplitudes. Constraints on hadronic parameters are also obtained in this analysis, as described in section 3. •B0→DK∗0,D→K0 Sπ+π−.Inputs from the model-dependent GGSZ analysis of B0→DK∗0(892), D→K0 Sπ+π−decays [51] are included in the combination for the first time. The observables, defined in analogy to eqs. (1.11)–(1.12), are measured by fitting the D→K0 Sπ+π−Dalitz plot using a model developed by the BaBar collaboration [56]. A model-independent GGSZ analysis [57] is also performed by LHCb on the same data sample. Currently, the model-dependent analysis has the best sensitivity to the parameters x±and y±. Therefore the model-dependent results are used in the combination. The numerical results of the combination change insignificantly if the model-independent results are used instead. •B0 s→D∓ sK±.The inputs used from the time-dependent analysis of B0 s→D∓ sK± decays using data corresponding to 1 fb−1[52] are identical to those used in ref. [28]. Note however that a different sign convention is used here, as defined in eqs. (1.13)– (1.14) and appendix A. 3 Auxiliary inputs The external inputs are briefly described below and summarised in table 2. These measurements provide constraints on unknown parameters and result in better precision on γ. The values and uncertainties of the observables are provided in appendix Dand the correlations are given in appendix E. •Input from global fit to charm data. The GLW/ADS measurements need input to constrain the charm system in three areas: the ratio and strong phase difference for D0→K−π+and D0→π−K+decays (rKπ D,δKπ D), charm mixing (xD,yD) and direct CP violation in D0→h+h−decays (Adir KK,Adir ππ), taken from a recent HFAG charm fit [22]. These do not include the latest results on ∆ACP from LHCb [42] but their impact has been checked and found to be negligible. The value of δKπ Dis shifted by 180◦compared to the HFAG result in order to match the phase convention adopted in this paper. The parameter RKπ Dis related to the amplitude ratio rKπ D through RKπ D≡(rKπ D)2. – 8 – JHEP12(2016)087 DK B r 0.08 0.09 0.1 0.11 0.12 1-CL 0 0.2 0.4 0.6 0.8 1 68.3% 95.5% LHCb ]° [ DK B δ 1-CL 0 0.2 0.4 0.6 0.8 1 120 130 140 150 160 68.3% 95.5% LHCb 0 * DK B r 0 0.1 0.2 0.3 0.4 1-CL 0 0.2 0.4 0.6 0.8 1 68.3% 95.5% LHCb ]° [ 0 * DK B δ 1-CL 0 0.2 0.4 0.6 0.8 1 150 200 250 68.3% 95.5% LHCb π D B r 0 0.01 0.02 0.03 0.04 0.05 1-CL 0 0.2 0.4 0.6 0.8 1 68.3% 95.5% LHCb ]° [ π D B δ 1-CL 0 0.2 0.4 0.6 0.8 1 200 250 300 350 68.3% 95.5% LHCb ]° [ γ 1-CL 0 0.2 0.4 0.6 0.8 1 50 60 70 80 90 68.3% 95.5% LHCb Figure 3. 1 −CL curves for the Dh combination obtained with the Plugin method. The 1σand 2σlevels are indicated by the horizontal dotted lines. – 15 – JHEP12(2016)087 ]° [ γ DK B r 50 60 70 80 90 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ ]° [ DK B δ 50 60 70 80 90 120 130 140 150 160 LHCb ]° [ DK B δ DK B r 120 130 140 150 160 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ 0 * DK B r 50 60 70 80 90 0 0.1 0.2 0.3 0.4 LHCb ]° [ γ ]° [ 0 * DK B δ 50 60 70 80 90 150 200 250 LHCb ]° [ 0 * DK B δ 0 * DK B r 150 200 250 0 0.1 0.2 0.3 0.4 LHCb ]° [ γ π D B r 50 60 70 80 90 0 0.01 0.02 0.03 0.04 0.05 LHCb ]° [ γ ]° [ π D B δ 50 60 70 80 90 200 250 300 350 LHCb ]° [ π D B δ π D B r 200 250 300 350 0 0.01 0.02 0.03 0.04 0.05 LHCb Figure 4. Profile likelihood contours from the Dh combination. The contours show the twodimensional 1σand 2σboundaries, corresponding to 68.3% and 95.5% CL, respectively. – 16 – JHEP12(2016)087 Observable Central value 68.3% Interval 95.5% Interval 99.7% Interval γ(◦) 73.5 [70.5,76.8] [56.7,83.4] [40.1,90.8] rDK B0.1017 [0.0970,0.1064] [0.0914,0.1110] [0.0844,0.1163] δDK B(◦) 141.6 [136.6,146.3] [127.2,151.1] [114.6,155.7] rDK∗0 B0.220 [0.173,0.264] [0.121,0.307] [0.000,0.355] δDK∗0 B(◦) 188 [168,211] [148,239] [120,280] rDπ B0.027 [0.0207,0.0318] [0.0020,0.0365] [0.0008,0.0425] δDπ B(◦) 348.3 [343.2,352.9] [220.5,356.4] [192.9,359.8] Table 4. Confidence intervals and central values for the parameters of interest in the frequentist Dh combination. DK B r 0.05 0.1 )σCoverage (1 0.5 0.55 0.6 0.65 0.7 0.75 LHCb Profile likelihood ) methodµ (LUGINP π D B r 0 0.01 0.02 0.03 )σCoverage (1 0.5 0.55 0.6 0.65 0.7 0.75 LHCb Profile likelihood ) methodµ (LUGINP Figure 5. Dependence of the coverage for the one-dimensional Plugin method (blue circles) and the profile likelihood method (red squares) on rDK Bfor the DK combination (left) and on rDπ Bfor the Dh combination (right). The solid horizontal line shows the nominal coverage at 1σof 68.3%. combination (∼0.03) falls in the regime with good coverage, whilst the expected value, and indeed the value of the second minimum (∼0.005), is in the regime in which the coverage starts to deteriorate. No correction for under-coverage is applied to the confidence intervals quoted in tables 3and 4. 5.4 Interpretation Using the nominal DK combination and the simple profile likelihood method some further interpretation of the results is presented in this section. Performing the DK combination with statistical uncertainties only suggests that the systematic contribution to the uncertainty on γis approximately 3◦. Performing the combination without use of the external constraints (described in section 3) roughly doubles the uncertainty on γ, demonstrating the value of including this information. The origin of the sensitivity to γof the various decay modes and analysis methods in the DK combination is demonstrated in figure 6. It can be seen that B+→DK+decays offer the best sensitivity (see figure 6left) and that the GLW/ADS methods offer multiple – 17 – JHEP12(2016)087 η[%] α(profile likelihood) [%] α(Plugin) [%] DK 68.3 65.1±0.7 67.1±0.7 95.5 93.5±0.4 94.3±0.3 99.7 98.7±0.2 98.8±0.2 Dh 68.3 63.0±0.7 64.3±0.7 95.5 90.9±0.4 91.7±0.4 99.7 95.3±0.3 95.6±0.3 Table 5. Measured coverage αof the confidence intervals for γ, determined at the best fit points, for both the one-dimensional Plugin and profile likelihood methods. The nominal coverage is denoted as η. ]° [ γ 1-CL 0 0.2 0.4 0.6 0.8 1 0 50 100 150 68.3% 95.5% LHCb ]° [ γ 1-CL 0 0.2 0.4 0.6 0.8 1 0 50 100 150 68.3% 95.5% LHCb decays 0 s B decays 0 B decays + B Combination GGSZ GLW/ADS Others Combination Figure 6. 1−CL plots, using the profile likelihood method, for DK combinations split by the initial Bmeson flavour (left) and split by analysis method (right). Left: (orange) B0 sinitial state, (yellow) B0initial states, (blue) B+initial states and (green) the full combination. Right: (yellow) GGSZ methods, (orange) GLW/ADS methods, (blue) other methods and (green) the full combination. narrow solutions compared to the single broader solution of the GGSZ method (see figure 6 right). Figures 7and 8further demonstrate the complementarity of the input methods in the (γvs. δX B) and (γvs. rX B) planes, for the B+and B0systems respectively. 6 Bayesian analysis The combinations are also performed using a Bayesian procedure. Probability (or credible) intervals (or regions) are obtained according to a highest posterior density approach. A highest posterior density interval (region) is defined by the property that the minimum density of any point within the interval (region) is equal to, or larger than, the density of any point outside that interval (region). – 18 – JHEP12(2016)087 ]° [ γ ]° [ DK B δ 0 50 100 150 0 50 100 150 LHCb ]° [ γ DK B r 0 50 100 150 0 0.05 0.1 0.15 0.2 LHCb 0 π /hh' π h3→D, + DK→ + B hh 0 S K→D, + DK→ + B ππ / π KK/K→D, + DK→ + B modes + BAll Full LHCb Combination Figure 7. Profile likelihood contours of γvs. δDK B(left) and γvs. rDK B(right) for various DK subcombinations: (blue) B+→DK+,D0→hπππ/hh0π0, (pink) B+→DK+,D0→K0 Shh, (light brown) B+→DK+,D0→KK/Kπ/ππ, (orange) all B+modes and (green) the full combination. Dark and light regions show the intervals containing 68.3% and 95.5% respectively. ]° [ γ ]° [ 0 * DK B δ 0 50 100 150 150 200 250 300 350 LHCb ]° [ γ 0 * DK B r 0 50 100 150 0 0.2 0.4 0.6 0.8 LHCb ππ / π KK/K→D, *0 DK→ 0 B ππ 0 S K→D, *0 DK→ 0 B modes 0 BAll Full LHCb Combination Figure 8. Profile likelihood contours of γvs. δDK∗0 B(left) and γvs. rDK∗0 B(right) for various DK sub-combinations: (brown) B0→DK∗0,D0→KK/Kπ/ππ, (pink) B0→DK∗0,D0→K0 Sππ, (purple) all B0modes and (green) the full combination. Dark and light regions show the intervals containing 68.3% and 95.5% respectively. – 19 – JHEP12(2016)087 Observable Central value 68.3% Interval 95.5% Interval 99.7% Interval γ(◦) 70.3 [62.4,77.4] [52.6,83.5] [42.1,88.4] rDK B0.1012 [0.0954,0.1064] [0.0900,0.1120] [0.0846,0.1171] δDK B(◦) 142.2 [134.7,148.1] [125.3,153.7] [113.2,157.9] rDK∗0 B0.204 [0.149,0.253] [0.073,0.299] [0.000,0.322] δDK∗0 B(◦) 190.3 [165.8,218.4] [139.5,263.4] [117.8,292.4] Table 6. Credible intervals and most probable values for the hadronic parameters determined from the DK Bayesian combination. 6.1 DK combination Uniform prior probability distributions (hereafter referred to as priors) are used for γand the B-meson hadronic parameters in the DK combination, allowing them to vary inside the following ranges: γ∈[0◦,180◦], δDK B∈[−180◦,180◦], rDK B∈[0.06,0.14]. The priors for δDKππ Band δDsK Bare identical to that for δDK B; the range for δDK∗0 Bis [0◦,360◦]. The allowed ranges for rDK∗0 B,rDKππ Band rDsK Bare [0, 0.45], [0, 0.16] and [0, 0.2]. The remaining auxiliary parameters are constrained with Gaussian priors according to the externally measured values and their uncertainties. A range of alternative prior distributions have been found to have negligible impact on the results for γ. The results are shown in table 6 and in figures 9and 10. The Bayesian credible intervals are found to be in good agreement with the frequentist confidence intervals. 6.2 Dh combination For the Dh combination additional uniform priors are introduced: rDπ B∈[0,0.06], δDπ B∈ [180◦,360◦], rDπππ B∈[0,0.13] and δDπππ B∈[0◦,360◦]. All other priors are as described above for the DK combination. The results are given in table 7and shown in figures 11 and 12. Comparison with the frequentist treatment (section 5.2) shows that the 1σintervals and regions differ between the two treatments, but satisfactory agreement is recovered at 2σ. Such differences are not uncommon when comparing confidence and credible intervals or regions with low enough confidence level and probability, in the presence of a highly non-Gaussian likelihood function. 7 Conclusion Observables measured by LHCb that have sensitivity to the CKM angle γ, along with auxiliary information from other experiments, are combined to determine an improved constraint on γ. Combination of all B→DK-like modes results in a best fit value of – 20 – JHEP12(2016)087 DK B r 0.08 0.09 0.1 0.11 0.12 Probability Density 0 0.02 0.04 0.06 LHCb 68.3% 95.5% ]° [ DK B δ 120 130 140 150 160 Probability Density 0 0.01 0.02 0.03 0.04 0.05 0.06 LHCb 68.3% 95.5% 0 * DK B r 0 0.1 0.2 0.3 0.4 Probability Density 0 0.02 0.04 0.06 0.08 LHCb 68.3% 95.5% ]° [ 0 * DK B δ 150 200 250 Probability Density 0 0.005 0.01 0.015 LHCb 68.3% 95.5% ]° [ γ 50 60 70 80 90 Probability Density 0 0.01 0.02 0.03 0.04 0.05 LHCb 68.3% 95.5% Figure 9. Posterior probability density from the Bayesian interpretation for the DK combination. γ= 72.2◦and the confidence intervals γ∈[64.9,79.0]◦at 68.3% CL , γ∈[55.9,85.2]◦at 95.5% CL . A second combination is investigated with additional inputs from B→Dπ-like modes. The frequentist and Bayesian approaches are in agreement at the 2σlevel, giving intervals of γ∈[56.7,83.4]◦and γ∈[52.1,84.6]◦at 95.5% CL, respectively. Taking the best fit value and the 68.3% CL interval of the DK combination γis found to be γ= (72.2+6.8 −7.3)◦, – 21 – JHEP12(2016)087 ]° [ γ 50 60 70 80 90 DK B r 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ 50 60 70 80 90 ]° [ DK B δ 120 130 140 150 160 LHCb ]° [ DK B δ 120 130 140 150 160 170 DK B r 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ 50 60 70 80 90 0 * DK B r 0 0.1 0.2 0.3 0.4 LHCb ]° [ γ 50 60 70 80 90 ]° [ 0 * DK B δ 150 200 250 LHCb ]° [ 0 * DK B δ 150 200 250 0 * DK B r 0 0.1 0.2 0.3 0.4 LHCb Figure 10. Two-dimensional posterior probability regions from the Bayesian interpretation for the DK combination. Light and dark regions show the 68.3% and 95.5% credible intervals respectively. where the uncertainty includes both statistical and systematic effects. A Bayesian interpretation yields similar results, with credible intervals found to be consistent with the corresponding confidence intervals of the frequentist treatment. The result for γis compatible with the world averages [26,27] and the previous LHCb average, γ= (73+9 −10)◦[28]. This combination has a significantly smaller uncertainty than the previous one and replaces it as the most precise determination of γfrom a single experiment to date. Additional inputs to the combinations in the future will add extra sensitivity, this includes use of new decay modes (such as B+→DK∗+), updates of current measurements to the full Run I data sample (such as B0 s→D∓ sK±) and inclusion of the Run II data sample. Exploiting the full LHCb Run II data sample over the coming years is expected to reduce the uncertainty on γto approximately 4◦. – 22 – JHEP12(2016)087 DK B r 0.08 0.09 0.1 0.11 0.12 Probability Density 0 0.02 0.04 0.06 LHCb 68.3% 95.5% ]° [ DK B δ 120 130 140 150 160 Probability Density 0 0.01 0.02 0.03 0.04 0.05 0.06 LHCb 68.3% 95.5% 0 * DK B r 0 0.1 0.2 0.3 0.4 Probability Density 0 0.02 0.04 0.06 0.08 LHCb 68.3% 95.5% ]° [ 0 * DK B δ 150 200 250 Probability Density 0 0.005 0.01 0.015 LHCb 68.3% 95.5% π D B r 0 0.01 0.02 0.03 0.04 0.05 Probability Density 0 0.05 0.1 0.15 0.2 0.25 LHCb 68.3% 95.5% π D B r 0 0.01 0.02 0.03 0.04 0.05 Probability Density 3− 10 2− 10 1− 10 68.3% 95.5% LHCb ]° [ π D B δ 200 250 300 350 Probability Density 0 0.002 0.004 0.006 0.008 0.01 0.012 LHCb 68.3% 95.5% ]° [ γ 50 60 70 80 90 Probability Density 0 0.01 0.02 0.03 0.04 0.05 LHCb 68.3% 95.5% Figure 11. Posterior probability density from the Bayesian interpretation for the Dh combination. The inset for rDπ Bshows the same distribution on a logarithmic scale. – 23 – JHEP12(2016)087 ]° [ γ 50 60 70 80 90 DK B r 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ 50 60 70 80 90 ]° [ DK B δ 120 130 140 150 160 LHCb ]° [ DK B δ 120 130 140 150 160 DK B r 0.08 0.09 0.1 0.11 0.12 LHCb ]° [ γ 50 60 70 80 90 0 * DK B r 0 0.1 0.2 0.3 0.4 LHCb ]° [ γ 50 60 70 80 90 ]° [ 0 * DK B δ 150 200 250 LHCb ]° [ 0 * DK B δ 150 200 250 0 * DK B r 0 0.1 0.2 0.3 0.4 LHCb ]° [ γ 50 60 70 80 90 π D B r 0 0.01 0.02 0.03 0.04 0.05 LHCb ]° [ γ 50 60 70 80 90 ]° [ π D B δ 200 250 300 350 LHCb ]° [ π D B δ 200 250 300 350 π D B r 0 0.01 0.02 0.03 0.04 0.05 LHCb Figure 12. Two-dimensional posterior probability regions from the Bayesian interpretation for the Dh combination. Light and dark regions show the 68.3% and 95.5% credible intervals respectively. – 24 – JHEP12(2016)087 B+→Dh+π−π+,D→h+h−observables. RDKππ CP = 1 + (rDKππ B)2+ 2κDKππ BrDKππ Bcos δDKππ Bcos γ , ADKππ, Kπ fav =2κDKππ BrDKππ BrKπ Dsin(δDKππ B−δKπ D) sin γ 1+(rDKππ B)2(rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B−δKπ D) cos γ ADπππ, Kπ fav =2κDπππ BrDπππ BrKπ Dsin(δDπππ B−δKπ D) sin γ 1+(rDπππ B)2(rKπ D)2+ 2κDπππ BrDπππ BrKπ Dcos(δDπππ B−δKπ D) cos γ ADKππ, KK CP =2κDKππ BrDKππ Bsin δDKππ Bsin γ 1+(rDKππ B)2+ 2κDKππ BrDKππ Bcos δDKππ Bcos γ+Adir KK ADKππ, ππ CP =2κDKππ BrDKππ Bsin δDKππ Bsin γ 1+(rDKππ B)2+ 2κDKππ BrDKππ Bcos δDKππ Bcos γ+Adir ππ ADπππ, KK CP =2κDπππ BrDπππ Bsin δDπππ Bsin γ 1+(rDπππ B)2+ 2κDπππ BrDπππ Bcos δDπππ Bcos γ+Adir KK ADπππ,ππ CP =2κDKππ BrDKππ Bsin δDKππ Bsin γ 1+(rDKππ B)2+ 2κDKππ BrDKππ Bcos δDKππ Bcos γ+Adir ππ RDKππ, Kπ +=(rDKππ B)2+ (rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B+δKπ D+γ) 1+(rDKππ B)2(rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B−δKπ D+γ) RDKππ, Kπ −=(rDKππ B)2+ (rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B+δKπ D−γ) 1+(rDKππ B)2(rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B−δKπ D−γ) RDπππ, Kπ +=(rDπππ B)2+ (rKπ D)2+ 2κDπππ BrDπππ BrKπ Dcos(δDπππ B+δKπ D+γ) 1+(rDπππ B)2(rKπ D)2+ 2κDπππ BrDπππ BrKπ Dcos(δDπππ B−δKπ D+γ) RDπππ, Kπ −=(rDKππ B)2+ (rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B+δKπ D−γ) 1+(rDKππ B)2(rKπ D)2+ 2κDKππ BrDKππ BrKπ Dcos(δDKππ B−δKπ D−γ) – 31 – JHEP12(2016)087 B0→DK∗0,D→K+π−observables. ¯ ADK∗0, Kπ fav = 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dsin(δDK∗0 B+ ∆¯ δDK∗0 B−δKπ D) sin γ 1+(¯ RDK∗0 BrDK∗0 B)2(rKπ D)2+ 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dcos(δDK∗0 B+ ∆¯ δDK∗0 B−δKπ D) cos γ ¯ RDK∗0, Kπ += (¯ RDK∗0 BrDK∗0 B)2+ (rKπ D)2+ 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dcos(δDK∗0 B+ ∆¯ δDK∗0 B+δKπ D+γ) 1+(¯ RDK∗0 BrDK∗0 B)2(rKπ D)2+ 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dcos(δDK∗0 B+ ∆¯ δDK∗0 B−δKπ D+γ) ¯ RDK∗0, Kπ −= (¯ RDK∗0 BrDK∗0 B)2+ (rKπ D)2+ 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dcos(δDK∗0 B+ ∆¯ δDK∗0 B+δKπ D−γ) 1+(¯ RDK∗0 BrDK∗0 B)2(rKπ D)2+ 2κDK∗0 B¯ RDK∗0 BrDK∗0 BrKπ Dcos(δDK∗0 B+ ∆¯ δDK∗0 B−δKπ D−γ) B0→DK+π−,D→h+h−observables. xDK∗0 −=rDK∗0 Bcos(δDK∗0 B−γ) yDK∗0 −=rDK∗0 Bsin(δDK∗0 B−γ) xDK∗0 +=rDK∗0 Bcos(δDK∗0 B+γ) yDK∗0 +=rDK∗0 Bsin(δDK∗0 B+γ) B0→DK∗0,D→K0 Sπ+π−observables. ¯xDK∗0 −=¯ RDK∗0 BrDK∗0 Bcos(δDK∗0 B+ ∆¯ δDK∗0 B−γ) ¯yDK∗0 −=¯ RDK∗0 BrDK∗0 Bsin(δDK∗0 B+ ∆¯ δDK∗0 B−γ) ¯xDK∗0 +=¯ RDK∗0 BrDK∗0 Bcos(δDK∗0 B+ ∆¯ δDK∗0 B+γ) ¯yDK∗0 +=¯ RDK∗0 BrDK∗0 Bsin(δDK∗0 B+ ∆¯ δDK∗0 B+γ) – 32 – JHEP12(2016)087 B0 s→D∓ sK±observables. Cf=1−(rDsK B)2 1+(rDsK B)2, A∆Γ f=2rDsK Bcos(δDsK B−(γ+φs)) 1+(rDsK B)2 A∆Γ ¯ f=2rDsK Bcos(δDsK B+ (γ+φs)) 1+(rDsK B)2 Sf=2rDsK Bsin(δDsK B−(γ+φs)) 1+(rDsK B)2 S¯ f=2rDsK Bsin(δDsK B+ (γ+φs)) 1+(rDsK B)2 B Input observable values and uncertainties The input observable values and their statistical and systematic uncertainties are listed below. The observables labelled Dπ are only used in the Dh combination. B+→Dh+,D→h+h−analysis. The values and uncertainties are taken from ref. [44]. The observables are defined in analogy to eqs. (1.3)–(1.7), and the measured values are ADK,πK ADS =−0.403 ±0.056 ±0.011 , ADπ,πK ADS = 0.100 ±0.031 ±0.009 , ADK,KK CP = 0.087 ±0.020 ±0.008 , ADK,ππ CP = 0.128 ±0.037 ±0.012 , ADπ,KK CP =−0.0145 ±0.0050 ±0.0017 , ADπ,ππ CP = 0.0043 ±0.0086 ±0.0031 , ADK,Kπ fav =−0.0194 ±0.0072 ±0.0060 , RDK,πK ADS = 0.0188 ±0.0011 ±0.0010 , RDπ,πK ADS = 0.0036 ±0.0001 ±0.0001 , RKK CP = 0.968 ±0.022 ±0.021 ±0.010 , Rππ CP = 1.002 ±0.040 ±0.026 ±0.010 , RKπ K/π = 0.0779 ±0.0006 ±0.0019 , where the first uncertainties are statistical and the second are systematic. For RKK CP and Rππ CP the third uncertainties arise from the assumption that rDπ B= 0 as discussed in ref. [44] – 33 – JHEP12(2016)087 and subsequently applies only for the DK combination. Their statistical and systematic correlations are given in tables 9and 10. The relationships between observables and parameters are given in appendix A. B+→Dh+,D→h+π−π+π−analysis. The values and uncertainties are taken from ref. [44]. The observables are defined in analogy to eqs. (1.3)–(1.7), and the measured values are ADK,πKππ ADS =−0.313 ±0.102 ±0.038 , ADπ,πKππ ADS = 0.023 ±0.048 ±0.005 , ADK,ππππ CP = 0.100 ±0.034 ±0.018 , ADπ,ππππ CP =−0.0041 ±0.0079 ±0.0024 , ADK,πKππ fav =−0.000 ±0.012 ±0.002 , RDK,πKππ ADS = 0.0140 ±0.0015 ±0.0006 , RDπ,πKππ ADS = 0.0038 ±0.0002 ±0.0001 , Rππππ CP = 0.975 ±0.037 ±0.019 ±0.005 , RKπππ K/π = 0.0793 ±0.0010 ±0.0018 , where the first uncertainty is statistical and the second is systematic. The third uncertainty for Rππππ CP is again from the assumption that rDπ B= 0 and subsequently applies only for the DK combination. Their statistical and systematic correlations are given in tables 11 and 12. The relationships between observables and parameters are given in appendix A. B+→Dh+,D→h+h−π0analysis. The values and uncertainties are taken from ref. [45]. The observables are defined in analogy to eqs. (1.3)–(1.7), and the measured values are ADK,πKπ0 ADS =−0.20 ±0.27 ±0.04 , ADπ,πKπ0 ADS = 0.44 ±0.19 ±0.01 , ADK,KKπ0 CP = 0.30 ±0.20 ±0.02 , ADK,πππ0 CP = 0.054 ±0.091 ±0.011 , ADπ,KKπ0 CP =−0.030 ±0.040 ±0.005 , ADπ,πππ0 CP =−0.016 ±0.020 ±0.004 , ADK,Kππ0 fav = 0.010 ±0.026 ±0.005 , RDK,πKπ0 ADS = 0.014 ±0.005 ±0.002 , RDπ,πKπ0 ADS = 0.00235 ±0.00049 ±0.00006 , RKKπ0 CP = 0.95 ±0.22 ±0.05 , Rπππ0 CP = 0.98 ±0.11 ±0.05 , – 34 – JHEP12(2016)087 where the first uncertainty is statistical and the second is systematic. The statistical and systematic correlations are given in tables 13 and 14. The relationships between observables and parameters are given in appendix A. B+→DK+,D→K0 Sh+h−analysis. The values and uncertainties are taken from ref. [46]. The results are xDK −= 0.025 ±0.025 ±0.010 ±0.005 , yDK −= 0.075 ±0.029 ±0.005 ±0.014 , xDK +=−0.077 ±0.024 ±0.010 ±0.004 , yDK +=−0.022 ±0.025 ±0.004 ±0.010 , where the first uncertainty is statistical, the second is systematic, and the third is an external uncertainty due to the information on the strong phase variation across the D→ K0 Sh+h−phase space. Correlations between the statistical and systematic uncertainties are given in tables 15 and 16. The relationships between observables and parameters are given in appendix A. B+→Dh+,D→K0 SK−π+analysis. The values and uncertainties are taken from ref. [47]. The observables are defined in analogy to eqs. (1.5)–(1.7) and are RDK,KSKπ ADS = 3.855 ±0.961 ±0.060 , ADK,KSKπ fav = 0.026 ±0.109 ±0.029 , ADK,KSKπ ADS = 0.336 ±0.208 ±0.026 , where the first uncertainty is statistical and the second is systematic. The statistical and systematic correlations are found to be negligible and not included. The relationships between observables and parameters are given in appendix A. B+→Dh+π−π+,D→h+h−analysis. The values and uncertainties are taken from ref. [48]. The observables are defined in analogy to eqs. (1.3), (1.4), (1.7)–(1.9) and – 35 – JHEP12(2016)087 are RDKππ CP = 1.040 ±0.064 ADKππ, Kπ fav = 0.013 ±0.019 ±0.013 , ADπππ, Kπ fav =−0.002 ±0.003 ±0.011 , ADKππ, KK CP =−0.045 ±0.064 ±0.011 , ADKππ, ππ CP =−0.054 ±0.101 ±0.011 , ADπππ, KK CP =−0.019 ±0.011 ±0.010 , ADπππ, ππ CP =−0.013 ±0.016 ±0.010 , RDKππ, Kπ += 0.0107 ±0.0060 ±0.0011 , RDKππ, Kπ −= 0.0053 ±0.0045 ±0.0006 , RDπππ, Kπ += 0.0043 ±0.0005 ±0.0002 , RDπππ, Kπ −= 0.0042 ±0.0005 ±0.0002 , where the first uncertainty is statistical and the second is systematic. For RDKππ CP , the single uncertainty includes both statistical and systematic contributions. The only nonnegligible correlations are the statistical correlations, ρ(ADKππ, KK CP , ADKππ, ππ CP ) = 0.20 and ρ(ADπππ, KK CP , ADπππ,ππ CP ) = 0.08. The relationships between observables and parameters are given in appendix A. B0→DK∗0,D→K+π−analysis. The values and uncertainties are taken from ref. [49]. The ADS observables are defined in analogy to eqs. (1.7)–(1.9) and are ¯ ADK∗0, Kπ fav =−0.03 ±0.04 ±0.02 , ¯ RDK∗0, Kπ += 0.06 ±0.03 ±0.01 , ¯ RDK∗0, Kπ −= 0.06 ±0.03 ±0.01 , where the first uncertainty is statistical and the second is systematic. The statistical correlations are given in table 17, and the systematic correlations in table 18. The relationships between observables and parameters are given in appendix A. B0→DK+π−,D→h+h−analysis. The values and uncertainties are taken from ref. [50]. The results are xDK∗0 −=−0.02 ±0.13 ±0.14 , yDK∗0 −=−0.35 ±0.26 ±0.41 , xDK∗0 += 0.04 ±0.16 ±0.11 , yDK∗0 +=−0.47 ±0.28 ±0.22 , where the first uncertainty is statistical and the second is systematic. The correlations are given in tables 19 and 20. The relationships between observables and parameters are given in appendix A. – 36 – JHEP12(2016)087 B0→DK∗0,D→K0 Sπ+π−analysis. The values and uncertainties are taken from ref. [51]. The results are ¯xDK∗0 −=−0.15 ±0.14 ±0.03 ±0.01 , ¯yDK∗0 −= 0.25 ±0.15 ±0.06 ±0.01 , ¯xDK∗0 += 0.05 ±0.24 ±0.04 ±0.01 , ¯yDK∗0 +=−0.65 ±0.24 ±0.08 ±0.01 , where the first uncertainty is statistical, the second is systematic and the third is from the Dalitz plot fit model. The correlations are given in table 21. The relationships between observables and parameters are given in appendix A. B0 s→D∓ sK±analysis. The values and uncertainties are taken from ref. [52] (with a change in the sign convention, see appendix Afor the explicit definition). The results are Cf= 0.53 ±0.25 ±0.04 , A∆Γ f=−0.37 ±0.42 ±0.20 , A∆Γ ¯ f=−0.20 ±0.41 ±0.20 , Sf=−1.09 ±0.33 ±0.08 , S¯ f= 0.36 ±0.34 ±0.08 , where the first uncertainty is statistical and the second is systematic. The statistical correlations are given in table 22, and the systematic correlations in table 23. The relationships between observables and parameters are given in appendix A. – 37 – JHEP12(2016)087 C Uncertainty correlations for the input observables The correlation matrices of the statistical and systematic uncertainties are given below. The observables labelled Dπ are only used in the Dh combination. ADK,πK ADS ADπ,πK ADS ADK,KK CP ADK,ππ CP ADπ,KK CP ADπ,ππ CP ADK,Kπ fav RDK,πK ADS RDπ,πK ADS RKK CP Rππ CP RKπ K/π ADK,πK ADS 1−0.047 0.002 0.001 0.009 0.005 0.008 0.102 −0.003 0 0 0 ADπ,πK ADS −0.047 1 0.004 0.003 0.017 0.010 0.014 0.015 −0.043 0 0 0 ADK,KK CP 0.002 0.004 1 0.004 −0.007 0.016 0.024 0 0 −0.014 0 −0.001 ADK,ππ CP 0.001 0.003 0.004 1 0.016 −0.036 0.014 −0.001 0 0 −0.038 −0.002 ADπ,KK CP 0.009 0.017 −0.007 0.016 1 0.064 0.092 0 0 −0.001 0 −0.001 ADπ,ππ CP 0.005 0.010 0.016 −0.036 0.064 1 0.053 0 0 0 −0.003 0 ADK,Kπ fav 0.008 0.014 0.024 0.014 0.092 0.053 1 0 0 0 0 0 RDK,πK ADS 0.102 0.015 0 −0.001 0 0 0 1 −0.022 0.040 0.025 −0.114 RDπ,πK ADS −0.003 −0.043 0 0 0 0 0 −0.022 1 −0.005 −0.003 0.011 RKK CP 0 0 −0.014 0 −0.001 0 0 0.040 −0.005 1 0.060 −0.317 Rππ CP 0 0 0 −0.038 0 −0.003 0 0.025 −0.003 0.060 1 −0.176 RKπ K/π 0 0 −0.001 −0.002 −0.001 0 0 −0.114 0.011 −0.317 −0.176 1 Table 9. Correlation matrix of the statistical uncertainties for the B+→Dh+,D0→h+h−observables [44]. – 38 – JHEP12(2016)087 ADK,πK ADS ADπ,πK ADS ADK,KK CP ADK,ππ CP ADπ,KK CP ADπ,ππ CP ADK,Kπ fav RDK,πK ADS RDπ,πK ADS RKK CP Rππ CP RKπ K/π ADK,πK ADS 1 0.36 −0.06 0.27 0.30 0.04 0.09 0.78 −0.43 −0.04 0.23 −0.14 ADπ,πK ADS 0.36 1 −0.03 0.31 0.22 −0.06 −0.55 0.59 −0.47 −0.01 0.12 −0.04 ADK,KK CP −0.06 −0.03 1 −0.02 −0.80 0.09 0.09 −0.10 −0.06 0.03 −0.28 0.07 ADK,ππ CP 0.27 0.31 −0.02 1 0.19 −0.42 −0.01 0.35 −0.28 −0.22 0.11 −0.07 ADπ,KK CP 0.30 0.22 −0.80 0.19 1 0.11 0.09 0.37 −0.21 −0.16 0.20 −0.13 ADπ,ππ CP 0.04 −0.06 0.09 −0.42 0.11 1 0.30 −0.03 0.06 −0.08 −0.03 −0.09 ADK,Kπ fav 0.09 −0.55 0.09 −0.01 0.09 0.30 1 −0.11 0.05 −0.01 −0.02 −0.02 RDK,πK ADS 0.78 0.59 −0.10 0.35 0.37 −0.03 −0.11 1 −0.57 −0.14 0.33 −0.22 RDπ,πK ADS −0.43 −0.47 −0.06 −0.28 −0.21 0.06 0.05 −0.57 1 0.19 0.10 0.02 RKK CP −0.04 −0.01 0.03 −0.22 −0.16 −0.08 −0.01 −0.14 0.19 1 0.17 0.21 Rππ CP 0.23 0.12 −0.28 0.11 0.20 −0.03 −0.02 0.33 0.10 0.17 1 −0.11 RKπ K/π −0.14 −0.04 0.07 −0.07 −0.13 −0.09 −0.02 −0.22 0.02 0.21 −0.11 1 Table 10. Correlation matrix of the systematic uncertainties for the B+→Dh+,D0→h+h−observables [44]. – 39 – JHEP12(2016)087 ADK,πKππ ADS ADπ,πKππ ADS ADK,ππππ CP ADπ,ππππ CP ADK,πKππ fav RDK,πKππ ADS RDπ,πKππ ADS Rππππ CP RKπππ K/π ADK,πKππ ADS 1−0.062 0.002 0.009 0.006 0.082 −0.004 0.002 0.003 ADπ,πKππ ADS −0.062 1 0.005 0.020 0.017 0.013 −0.022 0 0 ADK,ππππ CP 0.002 0.005 1 −0.020 0.024 −0.001 0 −0.018 −0.002 ADπ,ππππ CP 0.009 0.020 −0.020 1 0.097 0 0 −0.004 0 ADK,πKππ fav 0.006 0.017 0.024 0.097 1 0 0 0 0.001 RDK,πKππ ADS 0.082 0.013 −0.001 0 0 1 −0.046 0.041 −0.099 RDπ,πKππ ADS −0.004 −0.022 0 0 0 −0.046 1 −0.004 0.012 Rππππ CP 0.002 0 −0.018 −0.004 0 0.041 −0.004 1 −0.308 RKπππ K/π 0.003 0 −0.002 0 0.001 −0.099 0.012 −0.308 1 Table 11. Correlation matrix of the statistical uncertainties for the B+→Dh+,D0→h±π∓π+π−observables [44]. – 40 – JHEP12(2016)087 Input for D→K0 SK−π+parameters. The following constraints from ref. [60] are used: RKSKπ D= 0.356 ±0.034 ±0.007 , δKSKπ D=−0.29 ±0.32 rad, κKSKπ D= 0.94 ±0.16 . In addition the following contraint from ref. [61] is used RKSKπ D= 0.370 ±0.003 ±0.012 . The correlation between δKSKπ Dand κKSKπ Dis determined from the experimental likelihood to be ρ(δKSKπ D, κKSKπ D) = −0.60. Constraints on the B0→DK∗0hadronic parameters. The values and uncertainties are taken from ref. [50]. The values used are κDK∗0 B= 0.958 ±0.008 ±0.024, ¯ RDK∗0 B= 1.020 ±0.020 ±0.060, ∆¯ δDK∗0 B= 0.020 ±0.025 ±0.110 rad, where the first uncertainty is statistical and the second systematic. These are taken to be uncorrelated. Constraint on φs.The value used is taken from ref. [62] as φs=−0.010 ±0.039 rad . – 47 – JHEP12(2016)087 E Uncertainty correlations for the external constraints xDyDδKπ DRKπ DAdir ππ Adir KK xD1−0.361 −0.332 0.234 0.117 0.146 yD−0.361 1 0.941 0.234 −0.180 −0.221 δKπ D−0.332 0.941 1 0.439 −0.200 −0.237 RKπ D0.234 0.234 0.439 1 −0.078 −0.067 Adir ππ 0.117 −0.180 −0.200 −0.078 1 0.726 Adir KK 0.146 −0.221 −0.237 −0.067 0.726 1 Table 24. Correlations of the HFAG charm parameters (CHARM 2015, “Fit 3”, CP violation allowed) [22]. κK3π DδK3π DκK2π DδK2π DrK3π DrK2π D κK3π D1−0.67 0.04 −0.05 −0.48 −0.04 δK3π D−0.67 1 0.02 0.15 0.12 0.08 κK2π D0.04 0.02 1 0.23 −0.04 −0.04 δK2π D−0.05 0.15 0.23 1 −0.02 0.36 rK3π D−0.48 0.12 −0.04 −0.02 1 −0.03 rK2π D−0.04 0.08 −0.04 0.36 −0.03 1 Table 25. Correlations of the D0→K±π∓π+π−and D0→K±π∓π0parameters from CLEO and LHCb [58]. F Fit parameter correlations DK combination. γ rDK BδDK BrDK∗0 BδDK∗0 B γ1 0.54 0.44 0.21 -0.15 rDK B0.54 1 0.39 0.11 -0.08 δDK B0.44 0.39 1 0.08 -0.05 rDK∗0 B0.21 0.11 0.08 1 -0.13 δDK∗0 B-0.15 -0.08 -0.05 -0.13 1 Table 26. Fit parameter correlations for the DK combination. The fit results are given in table 3. – 48 – JHEP12(2016)087 Dh combination. γ rDK BδDK BrDK∗0 BδDK∗0 BrDπ BδDπ B γ1 0.19 0.23 0.10 -0.07 -0.59 -0.22 rDK B0.19 1 0.23 0.02 0 -0.20 0.02 δDK B0.23 0.23 1 0.02 0 -0.09 0.42 rDK∗0 B0.10 0.02 0.02 1 -0.10 -0.06 -0.03 δDK∗0 B-0.07 0 0 -0.10 1 0.04 0.03 rDπ B-0.59 -0.20 -0.09 -0.06 0.04 1 0.45 δDπ B-0.22 0.02 0.42 -0.03 0.03 0.45 1 Table 27. Fit parameter correlations for the Dh combination solution 1. The fit results are given in table 4. γ rDK BδDK BrDK∗0 BδDK∗0 BrDπ BδDπ B γ1 0.52 0.51 0.22 -0.16 -0.12 0.01 rDK B0.52 1 0.41 0.11 -0.08 0.03 0.10 δDK B0.51 0.41 1 0.11 -0.06 -0.19 -0.01 rDK∗0 B0.22 0.11 0.11 1 -0.13 -0.02 0 δDK∗0 B-0.16 -0.08 -0.06 -0.13 1 0.01 0 rDπ B-0.12 0.03 -0.19 -0.02 0.01 1 0.83 δDπ B0.01 0.10 -0.01 0 0 0.83 1 Table 28. Fit parameter correlations for the Dh combination solution 2. The fit results are given in table 4. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] A.D. Sakharov, Violation of CP invariance, c asymmetry and baryon asymmetry of the universe,Pisma Zh. Eksp. Teor. Fiz. 5(1967) 32 [JETP Lett. 5(1967) 24] [Sov. Phys. Usp. 34 (1991) 392] [Usp. Fiz. Nauk 161 (1991) 61] [INSPIRE]. [2] N. Cabibbo, Unitary symmetry and leptonic decays,Phys. Rev. Lett. 10 (1963) 531 [INSPIRE]. [3] M. Kobayashi and T. Maskawa, CP violation in the renormalizable theory of weak interaction,Prog. Theor. Phys. 49 (1973) 652 [INSPIRE]. [4] M.B. Gavela, P. Hern´andez, J. Orloff and O. Pene, Standard Model CP-violation and baryon asymmetry,Mod. Phys. Lett. A 9 (1994) 795 [hep-ph/9312215] [INSPIRE]. – 49 – JHEP12(2016)087 [5] L. Wolfenstein, Parametrization of the Kobayashi-Maskawa matrix,Phys. Rev. Lett. 51 (1983) 1945 [INSPIRE]. [6] L.-L. Chau and W.-Y. Keung, Comments on the parametrization of the Kobayashi-Maskawa matrix,Phys. Rev. Lett. 53 (1984) 1802 [INSPIRE]. [7] A.J. Buras, M.E. Lautenbacher and G. Ostermaier, Waiting for the top quark mass, K+→π+ν¯ν,B0 s-¯ B0 smixing and CP asymmetries in Bdecays,Phys. Rev. D 50 (1994) 3433 [hep-ph/9403384] [INSPIRE]. [8] M. Gronau and D. Wyler, On determining a weak phase from CP asymmetries in charged B decays,Phys. Lett. B 265 (1991) 172 [INSPIRE]. [9] M. Gronau and D. London, How to determine all the angles of the unitarity triangle from B0 d→DKsand B0 s→Dφ,Phys. Lett. B 253 (1991) 483 [INSPIRE]. [10] D. Atwood, I. Dunietz and A. Soni, Enhanced CP-violation with B→KD0¯ D0modes and extraction of the CKM angle γ,Phys. Rev. Lett. 78 (1997) 3257 [hep-ph/9612433] [INSPIRE]. [11] D. Atwood, I. Dunietz and A. Soni, Improved methods for observing CP-violation in B±→KD and measuring the CKM phase γ,Phys. Rev. D 63 (2001) 036005 [hep-ph/0008090] [INSPIRE]. [12] J. Brod, A. Lenz, G. Tetlalmatzi-Xolocotzi and M. Wiebusch, New physics effects in tree-level decays and the precision in the determination of the quark mixing angle γ,Phys. Rev. D 92 (2015) 033002 [arXiv:1412.1446] [INSPIRE]. [13] J. Brod and J. Zupan, The ultimate theoretical error on γfrom B→DK decays,JHEP 01 (2014) 051 [arXiv:1308.5663] [INSPIRE]. [14] A. Bondar, Proceedings of BINP special analysis meeting on Dalitz analysis, unpublished, 24–26 September 2002. [15] A. Giri, Y. Grossman, A. Soffer and J. Zupan, Determining γusing B±→DK±with multibody Ddecays,Phys. Rev. D 68 (2003) 054018 [hep-ph/0303187] [INSPIRE]. [16] Y. Grossman, Z. Ligeti and A. Soffer, Measuring γin B±→K±(KK∗)Ddecays,Phys. Rev. D 67 (2003) 071301 [hep-ph/0210433] [INSPIRE]. [17] R. Fleischer, New strategies to obtain insights into CP-violation through Bs→D± sK∓, D∗± sK∓,. . . and Bd→D±π∓,D∗±π∓,. . . decays,Nucl. Phys. B 671 (2003) 459 [hep-ph/0304027] [INSPIRE]. [18] R. Aleksan, I. Dunietz and B. Kayser, Determining the CP-violating phase γ,Z. Phys. C 54 (1992) 653 [INSPIRE]. [19] I. Dunietz and R.G. Sachs, Asymmetry between inclusive charmed and anticharmed modes in B0,¯ B0decay as a measure of CP violation,Phys. Rev. D 37 (1988) 3186 [Erratum ibid. D 39 (1989) 3515] [INSPIRE]. [20] T. Gershon, On the measurement of the unitarity triangle angle γfrom B0→DK∗0decays, Phys. Rev. D 79 (2009) 051301 [arXiv:0810.2706] [INSPIRE]. [21] T. Gershon and M. Williams, Prospects for the measurement of the unitarity triangle angle γ from B0→DK+π−decays,Phys. Rev. D 80 (2009) 092002 [arXiv:0909.1495] [INSPIRE]. [22] Heavy Flavor Averaging Group (HFAG), Y. Amhis et al., Averages of b-hadron, c-hadron and τ-lepton properties as of summer 2014, arXiv:1412.7515 [INSPIRE]. – 50 – JHEP12(2016)087 [23] M. Kenzie, M. Martinelli and N. Tuning, Estimating rDπ Bas an input to the determination of the CKM angle γ,Phys. Rev. D 94 (2016) 054021 [arXiv:1606.09129] [INSPIRE]. [24] K. De Bruyn, R. Fleischer, R. Knegjens, M. Merk, M. Schiller and N. Tuning, Exploring Bs→D(∗)± sK∓decays in the presence of a sizable width difference ∆Γs,Nucl. Phys. B 868 (2013) 351 [arXiv:1208.6463] [INSPIRE]. [25] Particle Data Group, K.A. Olive et al., Review of particle physics,Chin. Phys. C 38 (2014) 090001 [INSPIRE]. [26] CKMfitter Group, J. Charles et al., CP violation and the CKM matrix: assessing the impact of the asymmetric Bfactories,Eur. Phys. J. C 41 (2005) 1 [hep-ph/0406184] [INSPIRE]. [27] UTfit collaboration, M. Bona et al., The 2004 UTfit collaboration report on the status of the unitarity triangle in the Standard Model,JHEP 07 (2005) 028 [hep-ph/0501199] [INSPIRE]. [28] LHCb collaboration, Improved constraints on γ: CKM2014 update,LHCb-CONF-2014-004, CERN, Geneva Switzerland (2014). [29] LHCb collaboration, Improved constraints on γfrom B±→DK±decays including first results on 2012 data,LHCb-CONF-2013-006, CERN, Geneva Switzerland (2013). [30] LHCb collaboration, Measurement of the CKM angle γfrom a combination of B±→Dh± analyses,Phys. Lett. B 726 (2013) 151 [arXiv:1305.2050] [INSPIRE]. [31] LHCb collaboration, Measurement of the CKM angle γfrom a combination of B→DK analyses,LHCb-CONF-2016-001, CERN, Geneva Switzerland (2016). [32] T. Gershon and V.V. Gligorov, CP violation in the Bsystem,arXiv:1607.06746 [INSPIRE]. [33] M. Nayak et al., First determination of the CP content of D→π+π−π0and D→K+K−π0, Phys. Lett. B 740 (2015) 1 [arXiv:1410.3964] [INSPIRE]. [34] LHCb collaboration, Determination of the sign of the decay width difference in the Bs system,Phys. Rev. Lett. 108 (2012) 241801 [arXiv:1202.4717] [INSPIRE]. [35] C.C. Meca and J.P. Silva, Detecting new physics contributions to the D0-¯ D0mixing through their effects on Bdecays,Phys. Rev. Lett. 81 (1998) 1377 [hep-ph/9807320] [INSPIRE]. [36] J.P. Silva and A. Soffer, Impact of D0-¯ D0mixing on the experimental determination of γ, Phys. Rev. D 61 (2000) 112001 [hep-ph/9912242] [INSPIRE]. [37] M. Rama, Effect of D-¯ Dmixing in the extraction of gamma with B−→D0K−and B−→D0π−decays,Phys. Rev. D 89 (2014) 014021 [arXiv:1307.4384] [INSPIRE]. [38] W. Wang, CP violation effects on the measurement of the Cabibbo-Kobayashi-Maskawa angle γfrom B→DK,Phys. Rev. Lett. 110 (2013) 061802 [arXiv:1211.4539] [INSPIRE]. [39] M. Martone and J. Zupan, B±→DK±with direct CP-violation in charm,Phys. Rev. D 87 (2013) 034005 [arXiv:1212.0165] [INSPIRE]. [40] B. Bhattacharya, D. London, M. Gronau and J.L. Rosner, Shift in weak phase γdue to CP asymmetries in Ddecays to two pseudoscalar mesons,Phys. Rev. D 87 (2013) 074002 [arXiv:1301.5631] [INSPIRE]. [41] A. Bondar, A. Dolgov, A. Poluektov and V. Vorobiev, Effect of direct CP-violation in charm on γextraction from B→DK±, D →K0 Sπ+π−Dalitz plot analysis,Eur. Phys. J. C 73 (2013) 2476 [arXiv:1303.6305] [INSPIRE]. – 51 – JHEP12(2016)087 [42] LHCb collaboration, Measurement of the difference of time-integrated CP asymmetries in D0→K−K+and D0→π−π+decays,Phys. Rev. Lett. 116 (2016) 191601 [arXiv:1602.03160] [INSPIRE]. [43] Y. Grossman and M. Savastio, Effects of K0-¯ K0mixing on determining γfrom B±→DK±, JHEP 03 (2014) 008 [arXiv:1311.3575] [INSPIRE]. [44] LHCb collaboration, Measurement of CP observables in B±→DK±and B±→Dπ±with twoand four-body Ddecays,Phys. Lett. B 760 (2016) 117 [arXiv:1603.08993] [INSPIRE]. [45] LHCb collaboration, A study of CP violation in B∓→Dh∓(h=K, π)with the modes D→K∓π±π0,D→π+π−π0and D→K+K−π0,Phys. Rev. D 91 (2015) 112014 [arXiv:1504.05442] [INSPIRE]. [46] LHCb collaboration, Measurement of the CKM angle γusing B±→DK±with D→K0 Sπ+π−,K0 SK+K−decays,JHEP 10 (2014) 097 [arXiv:1408.2748] [INSPIRE]. [47] LHCb collaboration, A study of CP-violation in B±→DK±and B±→Dπ±decays with D→K0 SK±π∓final states,Phys. Lett. B 733 (2014) 36 [arXiv:1402.2982] [INSPIRE]. [48] LHCb collaboration, Study of B−→DK−π+π−and B−→Dπ−π+π−decays and determination of the CKM angle γ,Phys. Rev. D 92 (2015) 112005 [arXiv:1505.07044] [INSPIRE]. [49] LHCb collaboration, Measurement of CP-violation parameters in B0→DK∗0decays,Phys. Rev. D 90 (2014) 112002 [Erratum ibid. D 94 (2016) 079902] [arXiv:1407.8136] [INSPIRE]. [50] LHCb collaboration, Constraints on the unitarity triangle angle γfrom Dalitz plot analysis of B0→DK+π−decays,Phys. Rev. D 93 (2016) 112018 [arXiv:1602.03455] [INSPIRE]. [51] LHCb collaboration, Measurement of the CKM angle γusing B0→DK∗0with D→K0 Sπ+π−decays,JHEP 08 (2016) 137 [arXiv:1605.01082] [INSPIRE]. [52] LHCb collaboration, Measurement of CP asymmetry in B0 s→D∓ sK±decays,JHEP 11 (2014) 060 [arXiv:1407.6127] [INSPIRE]. [53] LHCb collaboration, Observation of CP-violation in B±→DK±decays,Phys. Lett. B 712 (2012) 203 [Erratum ibid. B 713 (2012) 351] [arXiv:1203.3662] [INSPIRE]. [54] LHCb collaboration, Observation of the suppressed AdS modes B±→[π±K∓π+π−]DK± and B±→[π±K∓π+π−]Dπ±,Phys. Lett. B 723 (2013) 44 [arXiv:1303.4646] [INSPIRE]. [55] LHCb collaboration, Measurement of CP violation and constraints on the CKM angle γin B±→DK±with D→K0 Sπ+π−decays,Nucl. Phys. B 888 (2014) 169 [arXiv:1407.6211] [INSPIRE]. [56] BaBar collaboration, P. del Amo Sanchez et al., Measurement of D0-¯ D0mixing parameters using D0→K0 Sπ+π−and D0→K0 SK+K−decays,Phys. Rev. Lett. 105 (2010) 081803 [arXiv:1004.5053] [INSPIRE]. [57] LHCb collaboration, Model-independent measurement of the CKM angle γusing B0→DK∗0decays with D→K0 Sπ+π−and K0 SK+K−,JHEP 06 (2016) 131 [arXiv:1604.01525] [INSPIRE]. [58] T. Evans, S. Harnew, J. Libby, S. Malde, J. Rademacker and G. Wilkinson, Improved determination of the D→K−π+π+π−coherence factor and associated hadronic parameters from a combination of e+e−→ψ(3770) →c¯cand pp →c¯cX data,Phys. Lett. B 757 (2016) 520 [Erratum ibid. Bsubmitted] [arXiv:1602.07430] [INSPIRE]. – 52 – JHEP12(2016)087 [59] S. Malde et al., First determination of the CP content of D→π+π−π+π−and updated determination of the CP contents of D→π+π−π0and D→K+K−π0,Phys. Lett. B 747 (2015) 9 [arXiv:1504.05878] [INSPIRE]. [60] CLEO collaboration, J. Insler et al., Studies of the decays D0→K0 SK−π+and D0→K0 SK+π−,Phys. Rev. D 85 (2012) 092016 [Erratum ibid. D 94 (2016) 099905] [arXiv:1203.3804] [INSPIRE]. [61] LHCb collaboration, Studies of the resonance structure in D0→K0 SK±π∓decays,Phys. Rev. D 93 (2016) 052018 [arXiv:1509.06628] [INSPIRE]. [62] LHCb collaboration, Precision measurement of CP violation in B0 s→J/ψK+K−decays, Phys. Rev. Lett. 114 (2015) 041801 [arXiv:1411.3104] [INSPIRE]. [63] LHCb collaboration, First observation of D0-¯ D0oscillations in D0→K+π−π+π−decays and measurement of the associated coherence parameters,Phys. Rev. Lett. 116 (2016) 241801 [arXiv:1602.07224] [INSPIRE]. [64] B. Sen, M. Walker and M. Woodroofe, On the unified method with nuisance parameters, Statist. Sinica 19 (2009) 301 [INSPIRE]. – 53 – JHEP12(2016)087 The LHCb collaboration R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10, F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An41, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M. Artuso61, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S. Bachmann12, J.J. Back50, A. Badalov38, C. Baesso62, S. Baker55, W. Baldini17, R.J. Barlow56, C. Barschel40, S. Barsuk7, W. Barter40, M. Baszczyk27, V. Batozskaya29, B. Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i, K. Belous37, I. Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, J. Benton48, A. Berezhnoy33, R. Bernet42, A. Bertolin23, F. Betti15, M.-O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42, S. Bifani47, P. Billoir8, T. Bird56, A. Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41, J. Blouw11,†, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N. Bondar31,40, W. Bonivento16, A. Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12, M. Britsch12, T. Britton61, J. Brodzicka56, E. Buchanan48, C. Burr56, A. Bursche2, J. Buytaert40, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M. Calvo Gomez38,m, A. Camboni38, P. Campana19, D. Campora Perez40, D.H. Campora Perez40, L. Capriotti56, A. Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21,i, L. Castillo Garcia41, M. Cattaneo40, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M. Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S. Chen56, S.-F. Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X. Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J. Closier40, V. Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f , L. Cojocariu30, G. Collazuol23,o, P. Collins40, A. Comerma-Montells12, A. Contu40, A. Cook48, G. Coombs40, S. Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz Torres62, S. Cunliffe55, R. Currie55, C. D’Ambrosio40, F. Da Cunha Marinho2, E. Dall’Occo43, J. Dalseno48, P.N.Y. David43, A. Davis59, O. De Aguiar Francisco2, K. De Bruyn6, S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d, P. De Simone19, C.-T. Dean53, D. Decamp4, M. Deckenhoff10, L. Del Buono8, M. Demmer10, D. Derkach35, O. Deschamps5, F. Dettori40, B. Dey22, A. Di Canto40, H. Dijkstra40, F. Dordei40, M. Dorigo41, A. Dosil Su´arez39, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany56, K. Dungs40, P. Durante40, R. Dzhelyadin37, A. Dziurda40, A. Dzyuba31, N. D´el´eage4, S. Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S. Eidelman36,w, S. Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, Ch. Elsasser42, S. Ely61, S. Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S. Farry54, R. Fay54, D. Fazzini21,i, D. Ferguson52, V. Fernandez Albor39, A. Fernandez Prieto39, F. Ferrari15,40, F. Ferreira Rodrigues1, M. Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M. Fiore17,g, M. Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, K. Fohl40, M. Fontana16,40, F. Fontanelli20,h, D.C. Forshaw61, R. Forty40, V. Franco Lima54, M. Frank40, C. Frei40, J. Fu22,q, E. Furfaro25,j, C. F¨arber40, A. Gallas Torreira39, D. Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2, P. Gandini57, Y. Gao3, L.M. Garcia Martin68, J. Garc´ıa Pardi˜nas39, J. Garra Tico49, L. Garrido38, P.J. Garsed49, D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gian`ı41, V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V. Gligorov8, D. Golubkov32, A. Golutvin55,40, A. Gomes1,a, I.V. Gorelov33, C. Gotti21,i, M. Grabalosa G´andara5, R. Graciani Diaz38, L.A. Granado Cardoso40, E. Graug´es38, E. Graverini42, G. Graziani18, A. Grecu30, P. Griffith47, L. Grillo21,40,i, B.R. Gruberg Cazon57, – 54 – JHEP12(2016)087 O. Gr¨unberg66, E. Gushchin34, Yu. Guz37, T. Gys40, C. G¨obel62, T. Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, S. Hall55, B. Hamilton60, X. Han12, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, J. Harrison56, M. Hatch40, J. He63, T. Head41, A. Heister9, K. Hennessy54, P. Henrard5, L. Henry8, J.A. Hernando Morata39, E. van Herwijnen40, M. Heß66, A. Hicheur2, D. Hill57, C. Hombach56, H. Hopchev41, W. Hulsbergen43, T. Humair55, M. Hushchyn35, N. Hussain57, D. Hutchcroft54, M. Idzik28, P. Ilten58, R. Jacobsson40, A. Jaeger12, J. Jalocha57, E. Jans43, A. Jawahery60, F. Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N. Jurik61, S. Kandybei45, W. Kanso6, M. Karacson40, J.M. Kariuki48, S. Karodia53, M. Kecke12, M. Kelsey61, I.R. Kenyon47, M. Kenzie49, T. Ketel44, E. Khairullin35, B. Khanji21,40,i, C. Khurewathanakul41, T. Kirn9, S. Klaver56, K. Klimaszewski29, S. Koliiev46, M. Kolpin12, I. Komarov41, R.F. Koopman44, P. Koppenburg43, A. Kosmyntseva32, A. Kozachuk33, M. Kozeiha5, L. Kravchuk34, K. Kreplin12, M. Kreps50, P. Krokovny36,w, F. Kruse10, W. Krzemien29, W. Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K. Kuonen41, K. Kurek29, T. Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, D. Lambert52, G. Lanfranchi19, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, J. van Leerdam43, J.-P. Lees4, A. Leflat33,40, J. Lefran¸cois7, R. Lef`evre5, F. Lemaitre40, E. Lemos Cid39, O. Leroy6, T. Lesiak27, B. Leverington12, Y. Li7, T. Likhomanenko35,67, R. Lindner40, C. Linn40, F. Lionetto42, B. Liu16, X. Liu3, D. Loh50, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39, H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton57, A. Lusiani24, X. Lyu63, F. Machefert7, F. Maciuc30, O. Maev31, K. Maguire56, S. Malde57, A. Malinin67, T. Maltsev36, G. Manca7, G. Mancinelli6, P. Manning61, J. Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito38, P. Marino24,t, J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez Vidal68, D. Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A. Mauri42, B. Maurin41, A. Mazurov47, M. McCann55, J. McCarthy47, A. McNab56, R. McNulty13, B. Meadows59, F. Meier10, M. Meissner12, D. Melnychuk29, M. Merk43, A. Merli22,q, E. Michielin23, D.A. Milanes65, M.-N. Minard4, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez62, I.A. Monroy65, S. Monteil5, M. Morandin23, P. Morawski28, A. Mord`a6, M.J. Morello24,t, J. Moron28, A.B. Morris52, R. Mountain61, F. Muheim52, M. Mulder43, M. Mussini15, D. M¨uller56, J. M¨uller10, K. M¨uller42, V. M¨uller10, P. Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52, N. Neri22, S. Neubert12, N. Neufeld40, M. Neuner12, A.D. Nguyen41, C. Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Novoselov37, D.P. O’Hanlon50, A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R. Oldeman49, C.J.G. Onderwater69, J.M. Otalora Goicochea2, A. Otto40, P. Owen42, A. Oyanguren68, P.R. Pais41, A. Palano14,d, F. Palombo22,q, M. Palutan19, J. Panman40, A. Papanestis51, M. Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18, A. Pastore14,d, G.D. Patel54, M. Patel55, C. Patrignani15,e, A. Pearce56,51, A. Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5, L. Pescatore47, K. Petridis48, A. Petrolini20,h, A. Petrov67, M. Petruzzo22,q, E. Picatoste Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer52, M. Plo Casasus39, T. Poikela40, F. Polci8, A. Poluektov50,36, I. Polyakov61, E. Polycarpo2, G.J. Pomery48, A. Popov37, D. Popov11,40, B. Popovici30, S. Poslavskii37, C. Potterat2, E. Price48, J.D. Price54, J. Prisciandaro39, A. Pritchard54, C. Prouve48, V. Pugatch46, A. Puig Navarro41, G. Punzi24,p, W. Qian57, R. Quagliani7,48, B. Rachwal27, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S. Rangel2, I. Raniuk45, G. Raven44, F. Redi55, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz68, V. Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, V. Rives Molina38, P. Robbe7,40, A.B. Rodrigues1, E. Rodrigues59, J.A. Rodriguez Lopez65, P. Rodriguez Perez56,†, A. Rogozhnikov35, S. Roiser40, A. Rollings57, V. Romanovskiy37, – 55 – JHEP12(2016)087 A. Romero Vidal39, J.W. Ronayne13, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls68, J.J. Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f , V. Salustino Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes39, R. Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, A. Sarti19,k, C. Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S. Schael9, M. Schellenberg10, M. Schiller40, H. Schindler40, M. Schlupp10, M. Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40, K. Schubert10, M. Schubiger41, M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A. Semennikov32, A. Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T. Shears54, L. Shekhtman36,w, V. Shevchenko67, A. Shires10, B.G. Siddi17,40, R. Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49, N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M. Smith55, H. Snoek43, M.D. Sokoloff59, F.J.P. Soler53, B. Souza De Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O. Stenyakin37, S. Stevenson57, S. Stoica30, S. Stone61, B. Storaci42, S. Stracka24,p, M. Straticiuc30, U. Straumann42, L. Sun59, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, M. Szczekowski29, T. Szumlak28, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F. Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli40, S. Topp-Joergensen57, F. Toriello61, E. Tournefier4, S. Tourneur41, K. Trabelsi41, M. Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N. Tuning43, A. Ukleja29, A. Ustyuzhanin35, U. Uwer12, C. Vacca16,f , V. Vagnoni15,40, A. Valassi40, S. Valat40, G. Valenti15, A. Vallier7, R. Vazquez Gomez19, P. Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M. Veltri18,r, G. Veneziano41, A. Venkateswaran61, M. Vernet5, M. Vesterinen12, B. Viaud7, D. Vieira1, M. Vieites Diaz39, X. Vilasis-Cardona38,m, V. Volkov33, A. Vollhardt42, B. Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß66, J.A. de Vries43, C. V´azquez Sierra39, R. Waldi66, C. Wallace50, R. Wallace13, J. Walsh24, J. Wang61, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42, M. Whitehead40, J. Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams40, M.P. Williams47, M. Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60, J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K. Wraight53, S. Wright49, K. Wyllie40, Y. Xie64, Z. Xing61, Z. Xu41, Z. Yang3, H. Yin64, J. Yu64, X. Yuan36,w, O. Yushchenko37, K.A. Zarebski47, M. Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y. Zhang63, A. Zhelezov12, Y. Zheng63, A. Zhokhov32, X. Zhu3, V. Zhukov9and S. Zucchelli15 1Centro Brasileiro de Pesquisas F´ısicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Universit´e Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Universit´e, Universit´e Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France 6CPPM, Aix-Marseille Universit´e, CNRS/IN2P3, Marseille, France 7LAL, Universit´e Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Universit´e Pierre et Marie Curie, Universit´e Paris Diderot, CNRS/IN2P3, Paris, France 9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany 10 Fakult¨at Physik, Technische Universit¨at Dortmund, Dortmund, Germany 11 Max-Planck-Institut f¨ur Kernphysik (MPIK), Heidelberg, Germany 12 Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany 13 School of Physics, University College Dublin, Dublin, Ireland 14 Sezione INFN di Bari, Bari, Italy 15 Sezione INFN di Bologna, Bologna, Italy 16 Sezione INFN di Cagliari, Cagliari, Italy 17 Sezione INFN di Ferrara, Ferrara, Italy – 56 –