Measurement of the CKM angle $\gamma$ from a combination of LHCb results

A combination of measurements sensitive to the CKM angle $\gamma$ from LHCb is performed. The inputs are from analyses of time-integrated $B^{+}\rightarrow DK^+$, $B^{0} \rightarrow D K^{*0}$, $B^{0} \rightarrow D K^+ \pi^-$ and $B^{+} \rightarrow D K^+\pi^+\pi^-$ tree-level decays. In addition, results from a time-dependent analysis of $B_{s}^{0} \rightarrow D_{s}^{\mp}K^{\pm}$ decays are included. The combination yields $\gamma = (72.2^{+6.8}_{-7.3})^\circ$, where the uncertainty includes systematic effects. The 95.5% confidence level interval is determined to be $\gamma \in [55.9,85.2]^\circ$. A second combination is investigated, also including measurements from $B^{+} \rightarrow D \pi^+$ and $B^{+} \rightarrow D \pi^+\pi^-\pi^+$ decays, which yields compatible results.


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][6][7]. The least precisely known of these angles, γ ≡ arg[−V ud V * ub /V cd V * cb ], can be measured using only tree-level processes [8][9][10][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 γ and the value inferred from global CKM fits, assuming the validity of the SM, would indicate new physics beyond the SM.
The value of γ can be determined by exploiting the interference between favoured b → cW (V cb ) and suppressed b → uW (V ub ) transition amplitudes using decay channels such as B + → Dh + , B 0 → DK * 0 , B 0 → DK + π − , B + → Dh + π − π + and B 0 s → D ∓ s K ± [8][9][10][11][14][15][16][17][18][19][20][21], where h is a kaon or pion and D refers to a neutral charm meson that is a mixture of the D 0 and D 0 flavour eigenstates. The inclusion of charge conjugate processes is implied throughout, unless otherwise stated. The most precise way to determine γ is through a combination of measurements from analyses of many decay modes. Hadronic parameters such as those that describe the ratio (r X B ) or strong phase difference (δ X B ) between the V cb and V ub transition amplitudes and where X is a specific final state of a B meson decay, are also simultaneously determined. The ratio of the suppressed to favoured B decay amplitudes is related to γ and the hadronic parameters by A sup /A fav = r X B e i(δ X B ±γ) , where the + (−) sign refers to the decay of a meson containing a b (b). The statistical uncertainty with which γ can be measured is approximately inversely proportional to the value of r X B , which is around 0.1 for B + → DK + decays [22]. In the B + → Dπ + channel, r Dπ B is expected to be of order 0.005 [23] because the favoured amplitude is enhanced by |V ud |/|V us | while the suppressed amplitude is further reduced by |V cd |/|V cs | with respect to B + → DK + decays. Consequently, the expected sensitivity to γ in B + → Dπ + decays is considerably lower than for B + → DK + decays, although the signal yields are higher. For B 0 → DK * 0 (and also B 0 s → D ∓ s K ± ) decays a higher value is expected [24], r DK * 0 B ∼ r DsK B ∼ 0.3, which compensates for the lower branching fraction [25], whilst the expected value for r DKππ B is similar to r DK B . The current world average, using only direct measurements of B → DK-like decays, is γ = (73.2 +6.3 −7.0 ) • [26] (or, using different inputs with an alternative statistical approach, γ = (68.3 ± 7.5) • [27]). The previous LHCb combination found γ = (73 +9 −10 ) • [28]. 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][29][30][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 γ and various hadronic parameters is given in Appendix A. The Gronau-London-Wyler (GLW) method [8,9] considers the decays of D mesons 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 D 0 → π − K + , where the interference between the Cabibbo-allowed and doubly Cabibbo-suppressed decay modes in both the B and D decays gives rise to large charge asymmetries. This introduces an additional dependence on the D decay dynamics through the ratio of suppressed and favoured D decay amplitudes, r D , and their phase difference, δ D . The GLW/ADS formalism is easily extended to multibody D decays [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, κ D , and for the GLW modes it is parametrised by F + , which describes the fraction of CP -even content in a multibody decay. For multibody D decays these parameters are measured independently and used as external constraints in the combination as discussed in Sec. 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 where D CP refers to the final state of a D meson decay into a CP eigenstate. Experimentally it is convenient to measure R CP , for a given final state f , by forming a double ratio that is normalised using the rate for a Cabibbo-favoured decay (e.g. D 0 → K − π + ), and the equivalent quantities from the relevant B + → Dπ − decay mode. Defining the ratio of the favoured B + → D 0 K + and B + → D 0 π + partial widths, for a given final state f , as the double ratios are constructed as 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), are measured using the appropriate flavour-specific D decay channel. The GLW partial-rate asymmetry, for a given D meson decay into a CP eigenstate f , is defined as Similarly, observables associated to the ADS modes, for a suppressed D → f decay, 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 →f decay) because the detection asymmetries cancel out. The charge-averaged rate is defined as whilst the partial-rate asymmetry is defined as The equivalent charge asymmetry for favoured ADS modes is defined as Some of the input analyses determined two statistically independent observables instead of those in Eqs. (5) and (6), namely the ratio of partial widths for the suppressed and favoured decays of each initial B flavour, It should be noted that Eqs. (5) and (6) are related to Eqs. (8) and (9) by 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 → K 0 S K − π + . The GLS observables are defined in analogy to Eqs. (5)(6)(7). Note that in the GLS method the favoured decay has sensitivity to γ because 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 D meson decay modes like K 0 S π + π − . Sensitivity to γ is obtained by comparing the distributions of decays in the D → f Dalitz plot for opposite-flavour initial-state B and B mesons. The population of candidates in the Dalitz plot depends on four variables, referred to as Cartesian variables which, for a given B decay final state X, are defined as These are the preferred observables for GGSZ analyses. The GLW/ADS and GGSZ formalisms can also be extended to multibody B decays by including a coherence factor, κ B , that accounts for dilution from interference between competing amplitudes. This inclusive approach is used for all multibody and quasi-two-body B decays, with the exception of the GLW-Dalitz analysis of B 0 → DK + π − decays where an amplitude analysis is performed to determine x X ± and y X ± . Here the term quasi-two-body decays refer to a two body resonant decay that contributes to a three body final state (e.g. B 0 → DK * (892) 0 decays in the B 0 → DK + π − final state).
Time-dependent (TD) analyses of B 0 s → D ∓ s K ± are also sensitive to γ [17][18][19]. Due to the interference between the mixing and decay amplitudes, the CP -sensitive observables, which are the coefficients of the time evolution of B 0 In the SM, to a good approximation, −2β s is equal to the phase φ s determined from B 0 s → J/ψ φ and similar decays, and therefore an external constraint on the value of φ s provides sensitivity to γ. The time-dependent decay rates for the initially pure B 0 s and B 0 s flavour eigenstates are given by where The parameter ∆m s is the oscillation frequency for B 0 s mesons, Γ s is the average B 0 s decay width, and ∆Γ s is the decay-width difference between the heavy and light mass eigenstates in the B 0 s system, which is known to be positive [34] as expected in the SM. The observables sensitive to γ are A ∆Γ f , C f and S f . The complex coefficients p and q relate the B 0 s meson mass eigenstates, |B L,H , to the flavour eigenstates, |B 0 s and |B 0 s , as |B L = p|B 0 s + q|B 0 s and |B H = p|B 0 s − q|B 0 s with |p| 2 + |q| 2 = 1. Similar equations can be written for the CP -conjugate decays replacing S f by Sf , and A ∆Γ f by A ∆Γ f , and, assuming no CP violation in either the decay or mixing amplitudes, Cf = −C f . The relationships between the observables, γ and the hadronic parameters are given in Appendix A. 10.
The combinations are potentially sensitive to subleading effects from D 0 -D 0 mixing [35][36][37]. These are corrected for where necessary, by taking into account the D 0 decay-time acceptances of the individual measurements. The size of the correction is inversely proportional to r X B and 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][39][40][41], which is included in the relevant B + → D 0 h + (π + π − ) analyses using the world average values [22], although the latest measurements indicate that the effect is negligible [42]. Final states that include a K 0 S meson 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 /r h B ), which is negligible for B + → DK + decays since | K | ≈ 0.002 and r DK B ≈ 0.1 [22]. For B + → Dπ + decays this ratio is expected to be O(1) since r Dπ B is expected to be around 0.5% [23]. Consequently, the B + → Dπ + decay modes affected, such as those with D → K 0 S K ∓ π ± , are not included in the Dh combination. To determine γ with 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 Secs. 2 and 3, with a summary provided in Tables 1 and 2. Frequentist and Bayesian treatments are both studied. Section 4 describes the frequentist treatment with results and coverage studies reported in Sec. 5. Section 6 describes the results of a Bayesian analysis.

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 B and 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.  [46] As before [47] As before [52] As before [44] is an update of a previous analysis [53]. The observables are defined in analogy to Eqs. (3)(4)(5)(6)(7).
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 F ππππ as described in Sec. 3. The method for constraining γ using these decays is described in Ref. [33], with observables defined in analogy to Eqs. (3)(4)(5)(6)(7).
The inputs from the model-independent GGSZ analysis of [46] are the same as those used in the previous combination [28]. The variables, defined in analogy to Eqs. (11)(12), are obtained from a simultaneous fit to the Dalitz plots of D → K 0 S π + π − and D → K 0 S K + K − decays. Inputs from a model-dependent GGSZ analysis of the same decay [55] using data corresponding to 1 fb −1 are not included due to the overlap of the datasets. [47] are the same as those included in the last combination [28]. The observables are defined in analogy to Eqs. (5)(6)(7). The negligible statistical and systematic correlations are not taken into account. [48] are included in the combination for the first time. The observables are defined in analogy to Eqs. • B 0 → DK * 0 , D → K + π − . The inputs from the ADS analysis of B 0 → D 0 K * (892) 0 , D 0 → K ± π ∓ decays [49] are included as they were in the previous combination [28]. However, the GLW part of this analysis (with D 0 → K + K − and D 0 → π + π − ) has been superseded by the Dalitz plot analysis. The ADS observables are defined in analogy to Eqs. (7)(8)(9). [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. (11)(12), are determined from a simultaneous Dalitz plot fit to B 0 → DK + π − with D 0 → 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) 0 amplitudes. Constraints on hadronic parameters are also obtained in this analysis, as described in Sec. 3.
Inputs from the model-dependent GGSZ analysis of B 0 → DK * 0 (892), D → K 0 S π + π − decays [51] are included in the combination for the first time. The observables, defined in analogy to Eqs. (11)(12), are measured by fitting the D → K 0 S π + 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.
The inputs used from the time-dependent analysis of B 0 s → D ∓ s K ± 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. (13)(14) and Appendix A.10.

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 D and 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 , taken from a recent HFAG charm fit [22]. These do not include the latest results on ∆A CP from LHCb [42] but their impact has been checked and found to be negligible. The value of δ Kπ D is shifted by 180 • compared to the HFAG result in order to match the phase convention adopted in this paper. The parameter R Kπ D is related to the amplitude ratio r Kπ • Input for D 0 → K ± π ∓ π 0 and D 0 → K ± π ∓ π + π − decays. The ADS measurements with D 0 → K ± π ∓ π 0 and D 0 → K ± π ∓ π + π − decays require knowledge of the hadronic parameters describing the D decays. These are the ratio, strong phase difference and coherence factors of the two decays: Recently an analysis of D 0 → K ± π ∓ π + π − decays has been performed by LHCb [63] that is sensitive to r K3π D , δ K3π D and κ K3π D . Furthermore, an updated measurement has been performed using CLEO-c data, and the results have been combined with those from LHCb [58] to yield constraints and correlations of the six parameters. These are included as Gaussian constraints in this combination, in line with the treatment of the other auxiliary inputs.
• CP content of D → h + h − π 0 and D → π + π − π + π − decays. For both the three-body D → h + h − π 0 and four-body D → π + π − π + π − quasi-GLW measurements the fractional CP -even content of the decays, F KKπ 0 , F πππ 0 and F ππππ , are used as inputs. These parameters were measured by the CLEO collaboration [59]. The uncertainty for the CP -even content of D → π + π − π + π − decays is increased from ±0.028 to ±0.032 to account for the non-uniform acceptance of the LHCb detector following the recommendation in Ref. [44]. For the D → h + h − π 0 decay the LHCb efficiency is sufficiently uniform to avoid the need to increase the F + uncertainty for these modes.
Constraints from Ref. [60] on all three are included, along with an additional constraint on the branching fraction ratio R K S Kπ D from Ref. [61]. The results corresponding to a limited region of the Dalitz plot, dominated by the K * (892) + resonance, are used here. The quantity .
The linear correlation coefficient between • Constraints on the B 0 → DK * 0 hadronic parameters. The quasi-two-body B 0 → DK * 0 ADS and model-dependent GGSZ measurements need input on the coherence factor κ DK * 0 B and the parametersR DK * 0 , which relate the hadronic parameters of the quasi-two-body B 0 → DK * 0 ADS and GGSZ measurements (barred symbols) to those of the B 0 → DK + π − amplitude analysis (unbarred symbols). The resulting values are taken from the LHCb GLW-Dalitz analysis described in Ref. [50]. These are taken to be uncorrelated with each other and with the x DK * 0 ± , y DK * 0 ± parameters that are determined from the same analysis.
• Constraint on φ s . The time-dependent measurement of B 0 s → D ∓ s K ± determines the quantity γ − 2β s . In order to interpret this as a measurement of γ, the weak phase −2β s ≡ φ s is constrained to the value measured by LHCb in B 0 s → J/ψ hh decays [62]. It has been checked that using the world average instead has a negligible impact on the results.

Statistical treatment
The baseline results of the combinations are presented using a frequentist treatment, starting from a likelihood function built from the product of the probability density functions (PDFs), f i , of experimental observables A i , where A obs i are the measured values of the observables from an input analysis i, and α is the set of parameters. For each of the inputs it is assumed that the observables follow a Gaussian distribution where V i is the experimental covariance matrix, which includes statistical and systematic uncertainties and their correlations. Correlations in the systematic uncertainties between the statistically independent input measurements are assumed to be zero. A χ 2 -function is defined as χ 2 ( α) = −2 ln L( α). The best-fit point is given by the global minimum of the χ 2 -function, χ 2 ( α min ). To evaluate the confidence level (CL) for a given value of a parameter, e.g. γ = γ 0 in the following, the value of the χ 2 -function at the new minimum is considered, χ 2 ( α min (γ 0 )). The associated profile likelihood function for the parameters is L( α min (γ 0 )). Then a test statistic is defined as ∆χ 2 = χ 2 ( α min ) − χ 2 ( α min ). The p-value, or 1 − CL, is calculated by means of a Monte Carlo procedure, described in Ref. [64] and briefly recapitulated here. For each value of γ 0 the test statistic ∆χ 2 is calculated, and a set of pseudoexperiments, A j , is generated according to Eq. (17) with parameters α set to the values at α min . A new value of the test statistic, ∆χ 2 , is calculated for each pseudoexperiment by replacing A obs → A j and minimising with respect to α, once with γ as a free parameter, and once with γ fixed to γ 0 . The value of 1 − CL is then defined as the fraction of pseudoexperiments for which ∆χ 2 < ∆χ 2 . This method is sometimes referred to as the "μ", or the Plugin method. Its coverage cannot be guaranteed [64] for the full parameter space, but can be verified for the best-fit point. The reason is that for each value of γ 0 , the nuisance parameters, i.e. the components of α other than the parameter of interest, are set to their best-fit values for this point, as opposed to computing an n-dimensional confidence region, which is computationally impractical. The coverage of the frequentist combinations is discussed in Sec. 5.3.

Results
Results for the DK combination are presented in Sec. 5.1 and for the Dh combination in Sec. 5.2. The coverage of the frequentist method is discussed in Sec. 5.3 whilst an interpretation of the results is provided in Sec. 5.4. The rate equations from which the observables are determined are invariant under the simultaneous transformation is the strong phase for each B → DX decay considered. Only the solution most consistent with the determination of γ from the global CKM fit [26,27] is shown.

DK combination
The DK combination consists of 71 observables and 32 parameters. The goodness of fit computed from the χ 2 value at the best fit point given the number of degrees of freedom is p = 91.5%. The equivalent value calculated from the fraction of pseudoexperiments, generated from the best fit point, which have a χ 2 larger than that found in the data is p = (90.5 ± 0.2)%. Table 3 summarises the resulting central values and confidence intervals that are obtained from five separate one-dimensional Plugin scans for the parameters: . These are shown in Fig. 1. Due to computational constraints the two-dimensional contours, shown in Fig. 2, are obtained via the profile likelihood method in which the value of the test statistic itself (∆χ 2 ) is used. Except for the coverage, as described in Sec. 5.3, this is verified to be a good approximation of the Plugin method. The parameter correlations obtained from the profile likelihood method are given in Appendix F.1.

Dh combination
The Dh combination includes observables measured from B + → Dπ + and B + → Dπ + π − π + decays, in addition to those measured in the DK combination, for a total of 89 observables and 38 parameters. The goodness of fit calculated from the χ 2 is p = 72.9% and calculated from the pseudoexperiments is p = (71.4 ± 0.3)%. Table 4 gives the results of the one-dimensional Plugin scans for γ, r Dπ . The scans are shown in Fig. 3. Two solutions are found, corresponding to r Dπ B values of 0.027 and 0.0045 for the favoured and secondary solutions, respectively. Figure 3 shows that the secondary solution is suppressed by slightly more than 1σ. Consequently, the 1σ interval for γ is very narrow because the uncertainty scales inversely with the central value of r Dπ B . As with the DK combination, the two-dimensional scans are performed using the profile likelihood method and are shown in Fig. 4. The two solutions and the non-Gaussian contours are clearly visible. The parameter correlations obtained from the profile likelihood method for both solutions are given in Appendix F.2. The coverage for the Dh analysis is examined in Sec. 5.3, where it is found that the coverage is slightly low and then starts to degrade when the true value of r Dπ B is less than 0.01, reaching a minimum around 0.006, before the behaviour of the DK combination is recovered at very low values. Recently, attempts have been made to estimate the value of r Dπ B using the known branching fractions of B 0 → D 0 K 0 and B 0 → D 0 π 0 decays and SU(3) symmetry [23], predicting a value of r Dπ B = 0.0053 ± 0.0007, consistent with the secondary solution observed in the data. Using this as an additional external input in the Dh combination gives γ = (71.8 +7.2 −8.6 ) • , which shows that when r Dπ B is small the uncertainties on γ are dominated by the B → DK inputs. This behaviour is similarly reflected by the 95.5% and 99.7% confidence intervals for the Dh combination when no external constraint on r Dπ B is used. The goodness of fit calculated from the χ 2 is p = 70.5% and calculated from pseudoexperiments is p = (69.7 ± 0.6)%.
Given the poor expected additional sensitivity from the B → Dπ-like modes, coupled

Coverage of the frequentist method
The coverage of the Plugin (and the profile likelihood) method is tested by generating pseudoexperiments and evaluating the fraction for which the p-value is less than that obtained for the data. In general, the coverage depends on the point in parameter space.    Following the procedure described in Ref. [30], the coverage of the profile likelihood and one-dimensional Plugin method intervals are tested. The coverage is determined for each method using the same pseudoexperiments; consequently their uncertainties are correlated.
The results for the best fit points are shown in Table 5. Figure 5 shows the coverage of the 1σ intervals as determined from pseudoexperiments for the DK (Dh) combination as a function of the value of r DK B (r Dπ B ) used to generate the pseudoexperiments. It can be seen that the coverage for the DK combination degrades as the true value of r DK B gets smaller. This behaviour has previously been observed by the CKMfitter group [26]. The fitted value found in this combination, r DK B ≈ 0.1, is well within the regime of accurate coverage. The dependence of the coverage for the Dh combination on r Dπ B shows similar behaviour, where the coverage begins to degrade when the true value reaches r Dπ B < 0.01, worsening until the true value of r Dπ B becomes so small that the Dπ modes offer no sensitivity and the behaviour seen in the DK combination is recovered. The fitted value of r Dπ B in the Dh 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 3 and 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 γ is approximately 3 • . Performing the combination without use of the external constraints (described in Sec. 3) roughly doubles the uncertainty on γ, demonstrating the value of including this information.
The origin of the sensitivity to γ of the various decay modes and analysis methods in the DK combination is demonstrated in Fig. 6. It can be seen that B + → DK + decays offer the best sensitivity (see Fig. 6 left) and that the GLW/ADS methods offer multiple narrow solutions compared to the single broader solution of the GGSZ method (see Fig. 6 right). Figures 7 and 8 further demonstrate the complementarity of the input methods in the (γ vs. δ X B ) and (γ vs. r X B ) planes, for the B + and B 0 systems respectively.

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).      Table 6 and in Figs. 9 and 10. The Bayesian credible intervals are found to be in good agreement with the frequentist confidence intervals.  Figure 9: Posterior probability density from the Bayesian interpretation for the DK combination. The results are given in Table 7 and shown in Figs. 11 and 12. Comparison with the frequentist treatment (Sec. 5.2) shows that the 1σ intervals 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.     Taking the best fit value and the 68.3% CL interval of the DK combination γ is found to be

Dh combination
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 γ is 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 γ from 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 B 0 s → D ∓ s K ± ) 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 γ to approximately 4 • .

Appendices A Relationships between parameters and observables
The equations given in this section reflect the relationship between the experimental observables and the parameters of interest. For simplicity, the equations are given in the absence of D 0 -D 0 mixing. In order to include the small (< 0.5 • ) effects from D 0 -D 0 mixing, the equations should be modified following the recommendation in Ref. [37], making use of the D 0 decay time acceptance coefficients, M xy , given in Table 8.

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.
The values and uncertainties are taken from Ref. [44]. The observables are defined in analogy to Eqs.
The values and uncertainties are taken from Ref. [44]. The observables are defined in analogy to Eqs. (3-7), and the measured values are The values and uncertainties are taken from Ref. [45]. The observables are defined in analogy to Eqs. (3)(4)(5)(6)(7), and the measured values are

B.4 B
The values and uncertainties are taken from Ref. [46]. The results are 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 → K 0 S h + 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.4.
The values and uncertainties are taken from Ref. [47]. The observables are defined in analogy to Eqs. (5)(6)(7) and are 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.5.
The values and uncertainties are taken from Ref. [48]. The observables are defined in analogy to Eqs. (3,4,(7)(8)(9) and are 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.7.
The values and uncertainties are taken from Ref. [50]. The results are 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.8.
The values and uncertainties are taken from Ref. [51]. The results arē x DK * 0 − = −0.15 ± 0.14 ± 0.03 ± 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.9.
The values and uncertainties are taken from Ref. [52] (with a change in the sign convention, see Appendix A.10 for the explicit definition). The results are 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.10.

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. Table 9: Correlation matrix of the statistical uncertainties for the B + → Dh + , D 0 → h + h − observables [44].   Table 11: Correlation matrix of the statistical uncertainties for the B + → Dh + , D 0 → h ± π ∓ π + π − observables [44].   Table 13: Correlation matrix of the statistical uncertainties for the B + → Dh + , D 0 → h ± h ∓ π 0 observables [52].        Table 21: Correlation matrix of the statistical uncertainties for the B 0 → DK * 0 , D → K 0 S π + π − observables [51].x   The values and uncertainties are taken from Ref. [22]. The observables are Here the value of δ Kπ D has been shifted by π to comply with the phase convention used in the combination. The correlations of the charm parameters are given in Table 24.
D.2 Input for D 0 → K ± π ∓ π + π − and D 0 → K ± π ∓ π 0 decays The values and uncertainties are taken from Ref. [58]. The values used are The correlation matrix is given in Table 25.
D.3 CP content of D → h + h − π 0 and D → π + π − π + π − decays The values and uncertainties are taken from Ref. [59]. The values used are where the first uncertainty is statistical and the second systematic. These are taken to be uncorrelated.