CKM angle γ measurements at LHCb

The CKM angle γ remains the least known parameter of the CKM mixing matrix. The precise measurement of this angle, as a Standard Model benchmark, is a key goal of the LHCb experiment. We present four recent CP violation studies related to the measurement of γ, including amplitude analysis of B± → DK± decays, the ADS/GLW analysis of B± → DK*0 decays and the time-dependent analysis of B± → DK±sK± decays.


Introduction
The CKM angle γ, dened as γ ≡ arg − V ud V * ub V cd V * cb , is the least known CKM parameter. B-factories and the LHCb experiment measured γ with an uncertainty larger than 10 • [1,2,3]. To compare, global ts like CKMtter [4] and UTt [5] obtain an estimation of γ with an error about 2 • . This angle is directly measurable through tree processes, without signicant loop contribution. Hence the extraction of its value is very clean and has a theoretical relative uncertainty lower than 10 −7 [6]. Therefore a precise measurement of γ provides an excellent standard candle to check the consistency of the CKM paradigm in the Standard Model and to probe some new physics. The present paper summarises four measurements of γ performed by the LHCb collaboration and presented at the BEACH 2014 conference.

Time-integrated measurements of γ
Given its denition, γ is approximately the phase dierence between the quark transitions b → cūs and b → ucs and the interference between these two transitions is sensitive to this angle. The interference is obtained by reconstructing the D 0 andD 0 mesons produced in these decays in an identical nal state (Fig. 1). In the case of a three body D meson decay a Dalitz plot analysis can be carried out. This method is called GGSZ [7,8] and two recent LHCb results are reported in section 2.1. In the case of a two body D meson 1 decay a counting analysis is developed following the so called GLW [9,10] or ADS [11,12] methods. A recent ADS/GLW result from LHCb is presented in section 2.2. All of these methods can be applied to the channels B ± → DK ± and B 0 → DK * 0 . Since these B mesons decays are self-tagged, no time-dependent analysis is required. 1 in the following D stands for either a D 0 or aD 0 meson.

Measurements with 3-body D meson decays
The B ± → D(K 0 S h + h − )K ± decay amplitude h stands for either a charged pion or a charged kaon can be written as where A D (AD) is the D 0 (D 0 ) decay amplitude, D represents the D meson phase space, δ B is the strong phase dierence and γ is the weak phase dierence between the D 0 andD 0 channels. The D meson phase space is parameterised by two squared invariant masses, for instance m 2 (K 0 S π + ) and m 2 (K 0 S π − ) on a D meson Dalitz plot. The sensitivity to γ arises from large asymmetries in some region of the Dalitz plot. In order to evaluate γ, the strong phase variation over the Dalitz plot must be known. This can be done in two dierent ways: with a model-dependent (MD) method using BaBar's amplitude model [13], or with a model-independent (MI) method using CLEO-c measurements as inputs [14]. Both methods involve tting the D meson Dalitz plot to extract the polar coordinates (r B ± , δ B , γ). The parameter r B ± is the ratio of the magnitudes of the suppressed and favoured B decay amplitudes. The polar coordinates are not estimated directly from the Dalitz t, but the cartesian coordinates x ± = r B ± cos(δ B ± γ) and y ± = r B ± sin(δ B ± γ).

Model-dependent analysis
This section presents the model-dependent analysis of the B ± → D(K 0 S π + π − )K ± signal, using a sample of proton-proton collision data at a centre-ofmass energy of 7 TeV corresponding to an integrated luminosity of 1 fb −1 . Full details can be found in Ref. [15]. The analysis is carried out in two distinct stages. First a t to the B meson reconstructed invariant mass is performed on the selected B ± → D(K 0 S π + π − )K ± and B ± → D(K 0 S π + π − )π ± candidates. This t determines the signal and background fractions in the data sample. A total yield of 637 B ± → D(K 0 S π + π − )K ± signal events and 8866 events of the B ± → D(K 0 S π + π − )π ± control channel are found. Then a t to the Daliz plot determines the CP violation observables (x ± , y ± ). The signal and background yields and the parameters of the B invariant mass probability distribution function are xed to the values obtained in the rst stage. The model used to described the amplitude of the D 0 → K 0 S π + π − decay over the phase space is the one determined by the BaBar collaboration in Ref. [13]. Fitting simultaneously the distributions in the D 0 → K 0 S π + π − phase space for the B ± → D(K 0 S π + π − )K ± and the B ± → D(K 0 S π + π − )π ± candidates enables to take into account the variation of eciency over the phase space. The B ± → D(K 0 S π + π − )π ± decay is a good proxy to get the eciency variation, since it has a kinematic topology similar to the signal one and CP violation can be neglected in this channel. The resulting values of the cartesian coordinates are: 010 −0.008 ± 0.001, y − = +0.013 ± 0.048 +0. 009 −0.007 ± 0.003, where the rst uncertainty is statistical, the second systematic and the third due to the amplitude model used to describe the D 0 → K 0 S π + π − decay. The leading experimental systematic errors are due to the eciency and background description uncertainties. The constraints obtained on the polar coordinates are r B ± = 0.06 ± 0.04, δ B = (115 +41 −51 ) • and γ = (84 +49 −42 ) • . These results are consistent with those of the LHCb model-independent analysis based on the same data set [16].

Model-independent analysis
This section presents the model-independent analysis of the , using a sample of proton-proton collision data at a centre-ofmass energy of 7 and 8 TeV corresponding to a total integrated luminosity of 3 fb −1 . Full details can be found in Ref. [17]. There is a signicant improvement compared to the former results in Ref. [16], thanks to the increased statistics and a better analysis technique. To know the strong phase variation over the D 0 → K 0 S h + h − phase space the measurements made by CLEO-c, in a particular binning scheme, is used [14]. In this way the analysis is a counting experiment in bins of the Dalitz plot. The expected number of D from B + events falling in a particular bin labelled ±i (the ± sign comes from the phase symmetry with respect to the Dalitz diagonal) can be expressed as where c i and s i are the averaged cosine and sine of the strong phase dierence in bin i (CLEO-c inputs), F i is the expected fraction of pure D 0 events in bin i taking into account the eciency prole over the phase space, and h + B is a normalisation factor. The F i parameters are determined from the B 0 → D * ± µ ∓ ν µ control mode. This is an excellent proxy because the sample has a high purity, a high statistics and the D 0 meson is tagged thanks to the slow pion in the D * + → D 0 π + decay. Some corrections are applied from simulated data to account for reconstruction and selection discrepancies between the B 0 → D * ± µ ∓ ν µ and the B ± → DK ± decays. The t is performed in two steps. First the phase space integrated B ± mass t determines the total signal yields (around 2600) and xes the model used in the second step. This last t is made in each Dalitz bin with all the parameters in Eq. (1) xed but the normalisation factor and the (x ± , y ± ) observables. The resulting values of the cartesian coordinates (Fig. 2) are the most precise to date: where the rst uncertainty is statistical, the second systematic and the third due to the experimental knowledge of the (c i , s i ) parameters. Compared to the 1 fb −1 measurement [16], the statistical uncertainty is reduced thanks to the larger data sample, the experimental systematic is reduced by using the new control mode B 0 → D * ± µ ∓ ν µ , and the (c i , s i ) systematic is also improved by the increased LHCb sample size. The results are:

Measurements with 2-body D meson decays
This section presents the ADS/GLW analysis of the B 0 → DK * 0 decays, using a sample of proton-proton collision data at a centre-of-mass energy of 7 and 8 TeV corresponding to a total integrated luminosity of 3 fb −1 . Full details can be found in Ref. [18]. Compared to the B ± → DK ± decays, both Cabibbo favoured and suppressed diagrams are color suppressed, which brings about a higher interference amplitude (r B 0 is larger than r B ± ). Hence a better sensitivity to γ is expected. However this neutral channel is experimentally more challenging. For the GLW modes the D mesons are reconstructed in two CP eigenstate: K + K − and π + π − . For the ADS modes the D mesons are reconstructed in the K + π − and K − π + nal states. From these decays several observables sensitive to γ can be built. For instance in the GLW modes the CP asymmetries are measured. The parameter κ is the coherence factor introduced to account for the eect of the non resonant B 0 → DK + π − contribution in the K * 0 signal region. And in the ADS mode the ratio of suppressed B 0 → D(π + K − )K * 0 to favoured B 0 → D(K + π − )K * 0 partial widths are measured separately for B 0 andB 0 : The parameters r D and δ D are the magnitude ratio and the phase dierence, respectively, between the amplitudes of the D 0 → K + π − and D 0 → K − π + decays. The signicances of the combined B 0 andB 0 signals for the B 0 → D(K + K − )K * 0 , B 0 → D(π + π − )K * 0 and B 0 → D(π + K − )K * 0 decay modes are 8.6 σ, 5.8 σ and 2.9 σ respectively. Once the production and eciency asymmetries are taken into account the results are: where the rst uncertainties are statistical and the second systematic. A ππ d , R + d and R − d are rst measurements and the A KK d result supersedes the former LHCb one [19]. From these measurements the value of r B 0 (proper to the B 0 → DK * 0 channel) is found to be r B 0 = 0.240 +0. 055 −0.048 . This is the most precise measurement to date. luminosity of 1 fb −1 . Full details can be found in Ref. [20]. In addition to the same tree level processes as in the time-integrated analysis (Fig. 1), the eect of the B s mixing occurs. Hence the interference between mixing and decay amplitudes in B 0 The time-dependent decay rates depend on the CP observables: the strong phase dierence and (γ − 2β s ) the weak phase dierence. This analysis uses an independent measurement of φ s [21] and assumes φ s = −2β s to interpret the results in terms of γ. To discriminate the signal and background components a 3D t is performed on the B s and D s masses along with the log-likelihood dierence between the kaon and pion hypothesis for the companion particle (K ± for B 0 s → D ∓ s K ± signal and π + for B 0 s → D − s π + control mode). Then the output of this multivariate t is used for the decay-time t. Two ts are performed: a background subtracted t, called sFit, using the sWeights [22,23] determined by the multivariate t; and a classical t, called cFit (Fig. 3) where all signal and background time distributions are described. The results of these two ts are in excellent agreement: Parameter sFit tted value cFit tted value The rst uncertainties are statistical and the second systematic. The main sources of systematic arise from the trigger-induced time-dependent eciency, Γ s and ∆Γ s . These results can be interpreted as a condence interval γ = (115 +28 −43 ) • at 68% CL (Fig. 4). This is the rst measurement of γ with B 0 s → D ∓ s K ± decays. The latest LHCb results on the CKM angle γ are reported: the model dependent GGSZ analysis of B ± → DK ± decays, the update to the full available data set of the model indepent GGSZ analysis of B ± → DK ± decays, the ADS/GLW analysis of B 0 → DK * 0 decays and the rst γ measurement with B 0 s → D ∓ s K ± decays. Using these results and the corresponding improvements, the next combination of the LHCb γ measurements should yield a signicant reduction of the uncertainty.