ub + V cd V tb = 0, (1) cb + V td V

Transcription

1 REVIEW OF β, α AND γ MEASUREMENTS K. TRABELSI KEK (High Energy Accelerator Research Organization) Institute of Particle and Nuclear Studies 1-1 Oho, Tsukuba-shi, Ibaraki-ken, 35-81, Japan Precision measurements of the angles of the Unitarity Triangle are part of the program to test the Standard Model and the Kobayashi-Maskawa model of CP violation. The most recent results of the B-factories are summarized. 1 Intro In the Standard Model (SM) of particle physics, the weak interaction couplings of quarks are described by the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix 1,2. The CKM matrix, V, must be unitary and the non-zero imaginary part of the CKM matrix is the origin of the CP violation in the SM. The unitarity relationv V = 1 results in a total of nine expressions, that can be written as i=u,c,t V ij V ik = δ jk. Of the off-diagonal expressions (j k), three can be transformed into the other three leaving six relations, in which three complex numbers sum to zero, which therefore can be expressed as triangles in the complex plane. One of these relations, V ud V ub + V cd V cb + V td V tb =, (1) is of particular importance to the B system, being specifically related to flavor changing neutral current b d transitions. The tree terms in Eq. 1 are of the same order and this relation is commonly known as the Unitarity Triangle (UT). Two popular naming conventions for the UT angles exist in the literature: α φ 2 = arg [ V tdvtb ] V ud Vub, β φ 1 = arg In this report the (α,β,γ) set is used. [ V cdvcb V td Vtb ] [, γ φ 3 = arg V udv ub V cd V cb ]. (2) 2 β determination Measurements of CP asymmetries in the proper-time distribution of neutral B decay to a common final CP eigenstate state, f, provide direct information on the angles of the UT. The time-dependent asymmetry: A f ( t) = Γ(B ( t) f) Γ(B ( t) f) Γ(B ( t) f) + Γ(B ( t) f) (3)

2 can be written as A f ( t) = S f sin( m t) C f cos( m t) (4) with S f = 2 I(λ) 1+ λ and C 2 f = 1 λ 2 1+ λ. Here λ = q 2 p A f contains terms related to B B mixing and to the decay amplitude (the eigenstates of the effective Hamiltonian in the B B system are B ± >= p B > ±q B >). The B J/ψK decay is dominated by a single tree-level quark transition b cc s, up to a correction smaller than a fraction of a percent 3. Neglecting effects due to CP violation in the mixing (by taking q/p = 1), S J/ψK = η f sin 2β, where η f is the CP eigenvalue of f, and C J/ψK =. The asymmetries measured in this process and in other decays dominated by b cc s have already provided a precise measurement of sin 2β 4 : A f sin 2β =.67 ±.23. (5) This result combines the measurements of Belle in J/ψK 5 and ψ(2s)k S 6 as well as those of BaBar in J/ψK, ψ(2s)k S, χ c1k S, η ck S and J/ψK 7, which are summarized in Table 1. For these modes, C b cc s =.5 ±.19. It is interesting to notice that the measurement of sin 2β is still dominated by statistics, whereas C b cc s is close to be dominated by systematics (the possible effect of tag side interference on the C measurement). The sin 2β value permits two solutions for β (in [, π]) at (21. ±.9) and (69.. ±.9). Time-dependent angular analysis of B J/ψK and time-dependent Dalitz analyses of B Dh (D K S π+ π,h = π, η, ω) measuring cos 2β > have excluded the second solution at a high confidence level (Fig. 1, right), implying: β = (21. ±.9). (6) Table 1: Results of fitting for CP asymmetries in the charmonium modes. Parameter BaBar Belle J/ψKS.657 ±.36 ± ±.38 stat J/ψKL.694 ±.61 ± ±.57 stat J/ψK.666 ±.31 ± ±.31 ±.17 ψ(2s)ks.897 ±.1 ± ±.9 ±.31 χ c1 KS.614 ±.16 ±.4 η c KS.925 ±.16 ±.57 J/ψK (KS π ).61 ±.239 ±.87 All charmonium.687 ±.28 ± ±.29 ±.18 The b sq q penguin-dominated decays have the same weak phase as the b cc s amplitude up to corrections (at most at the 1% level) from subleading u-quark penguin diagrams leading to an effective angle β eff. Since penguin loop contributions are sensitive to physics beyond the SM, it is important to have an unambiguous estimate of the deviation S S b sq q S b cc s. Various estimates, using different theoretical approaches such as QCDF, pqcd and SCET, find a small value and a positive sign for S for modes such as φk, η K. The various modes measured by Belle and BaBar are consistent with S b cc s (Fig 2). More statistics will be needed for a mode-by-mode study. 3 α determination The direct constraint on α comes from mixing-induced CP-violating measurements, through the combination of the two-body isospin analyses of B ππ and B ρρ, and the Dalitz plot analysis of B ρπ.

3 sin(2β) sin(2φ 1 ) BaBar PRD 79 (29) 729 BaBar χ c K S arxiv: BaBar J/ψ (hadronic) K S PRD 69 (24) 521 Belle J/ψ K PRL 98 (27) 3182 Belle ψ(2s) K S PRD 77 (28) 9113(R) Average FPCP 29 FPCP ±.28 ± ±.52 ±.4 ± ±.42 ± ±.31 ± ±.9 ± ± η β φ 1 β φ 1 = (68.9 ±.9) β φ 1 = (21.1 ±.9) FPCP 29 DISFAVOURED BY J/ψK*, D*D*K S & Dh ρ Figure 1: Averages of sin 2β from the B factories (left). Constraint on the ρ η plane (left). The amplitude for B π + π contains two terms, conventionally denoted tree (T) and penguin (P) amplitudes, involving a weak CP-violating phase γ and a strong CP-conserving phase δ, respectively: A(B π + π ) = T e iγ + P e iδ (7) Expanding to the first order in r = P / T, we can express the S f parameter of Eq. 4 in the case of B π + π as S π + π = sin 2α + 2r cos δ sin(β + α)cos(2α). (8) In the limit of vanishing small penguins S π + π = sin 2α. Additional inputs are required to determine the penguin pollution. The standard method for obtaining α relies on the isospin triangle construction 8 and requires the knowledge of not only the CP-violating parameters, S π + π and C π + π, but also B(π+ π ), B(π + π ), B(π π ) and C(π π ). Results from the two B-factories are consistent. An earlier discrepancy for C π + π between Belle and BaBar seems to get resolved with more statistics (Fig. 3). Combining these measurements for the ππ system, α = ( ) is obtained 9 if one considers the peak in agreement with the SM value. The situation for ρρ channels is more complicated than ππ because of the vector-vector nature of these modes which implies a mixture of CP-even and CP-odd components. The isospin analysis can be applied to each polarization state but the fact that the measured longitudinal polarization is close to unity simplifies considerably the analysis since the CP-even fraction dominates. Here also, results from the two B-factories are available (Table 2) but not always consistent. The BaBar collaboration obtained a 3.1σ for B ρ ρ with a sample of BB pairs 15 and measured B(B ρ ρ ) = (.92 ±.32(stat) ±.14(syst)) 1 6 and a longitudinal fraction f L = (stat) ±.4(syst). They use this signal to measure for the first time the CP-violating parameters of this mode 15. The situation for Belle, with higher statistics ( ) and similar efficiency, is different: no significant signal is seen (Fig. 4) and an upper limit at 9% C.L. is given, B(B ρ ρ ) < In contrast to the situation of the ππ system, the B ρ ρ decay has a much smaller branching fraction than the other ρρ channels. Recently, BaBar 13 updated their analysis of the B + ρ + ρ mode (Fig. 5). The impact on α is larger than one would naively expect from the improvement of the error on the branching fraction and comes primarily from an increase of the measured branching fraction relative to their previous measurement. The B + ρ + ρ branching fraction determines the length of

4 sin(2β eff ) sin(2φ e 1 ff ) CKM28 b ccs φ K η K World Average.67 ±.2 Average Average.59 ±.7 K S K S K S Average.74 ±.17 π K Average.57 ±.17 ρ K S ω K S f K S π π K S φ π K S K + K - K Average Average.45 ±.24 Average Average -.52 ±.41 Average Average.82 ± Figure 2: Comparisons of averages in the different b sq q modes. the common base of the isospin triangles for the B and B decays. The increase in the base lengths flattens both triangles making the four possible solutions degenerate (Fig. 6, left). The α constraint from B ρρ is then significantly improved and dominates the final α average (Fig. 6, right). A combined analysis 9 of ππ, ρρ and ρπ system gives α = ( ). 4 γ determination The extraction of γ stems from direct CP-violation measurements in B DK modes. The method employs the interference between b cus and b ucs when the final state f is accessible to both D and D mesons. The theoretical uncertainty is completely negligible as there is no penguin contributions. Various methods have been proposed to exploit this strategy Table 2: B ρρ inputs from the B-factories. Parameter BaBar Belle Reference B(ρ + ρ ) ( 1 6 ) 25.5 ± ± , 11 f L (ρ + ρ ).992 ± ±.3 1, 11 S L (ρ + ρ ).17 ± ±.3 ±.7 1, 12 C L (ρ + ρ ) +.1 ±.15 ±.6.16 ±.21 ±.7 1, 12 B(ρ + ρ ) ( 1 6 ) 23.7 ± 1.4 ± ± , 14 f L (ρ + ρ ).95 ±.15 ± ±.16 ±.21 13, 14 B(ρ ρ ) ( 1 6 ).92 ±.32 ±.14 < 1. 15, 16 f L (ρ ρ ) ±.4 15 S L (ρ ρ ).3 ±.7 ±.2 15 C L (ρ ρ ).2 ±.8 ±

5 BaBar 21 PRD 65, 5152 BaBar 22 PRL 89, Belle 22 PRL 89, 7181 BaBar 23 LP23 preliminary Belle 23 PRD 68, 121 BaBar 24 PRL 95, Belle 24 PRL 93, 2161 Belle 25 PRL 95, 1181 BaBar 26 hep-ex/6716 Belle 26 PRL 98, BaBar 27 PRL 99, 2163 π + π - C CP Winter 29 Winter ±.45 ± ±.25 ± ± ±.19 ± ±.27 ± ±.15 ± ±.15 ± ±.12 ± ±.11 ± ±.8 ± ±.9 ±.2 BaBar ±.8 ±.2 arxiv: Average ± Figure 3: History of the C π + π by the B-factories. using different choice of the final state f: CP eigenstates (GLW method 17 ), doubly Cabibbo suppressed decays (ADS method 18 ), three-body decays as D K S π + π and D K S K + K (GGSZ method 19 ). The feasibility of the γ measurement crucially depends on the size of r B, the ratio of the B decay amplitudes involved (r B = A(B + DK + ) / A(B + DK + ) ). The value of r B is given by the ratio of the CKM matrix elements Vub V cs / Vcb V us and the color suppression factor, and is estimated to be in the range For different D decays, the B system parameters are common, which means than combining different D channels buys more than just adding statistics. It is then not surprising that threebody decays provide the most sensitivity in the extraction of γ. The γ shift due to D D mixing is less than one degree for doubly Cabibbo suppressed decays and much smaller in other cases, and can eventually be included in the γ determination 21. The effect due to CP violation in the neutral D sector is negligible in the Standard Model and at most at the 1 2 order if one considers new physics in the charm sector. 4.1 CP eigenstates (GLW method) and doubly Cabibbo suppressed decays (ADS method) For the GLW method, one considers four observables: two charge-average decay rates for even and odd CP states, normalized by the decay rate into a D flavor state, R CP ±, and two CP asymmetries for even and odd CP states. In order to avoid dependence of R CP ± on errors in D and D CP branching ratio measurements, one uses a definition of R CP ± in terms of ratio of B decay branching ratios into DK and Dπ final states 2. Studies of B + D CP K +, B + D CP K+ and B + D CP K + have been carried out (Fig. 7 (left)), each consisting of a few ten events or more, but no significant difference (R CP+ R CP ) has been yet observed. The ADS method considers a flavor state in Cabibbo-favored D decays, accessible also to doubly Cabibbo-suppressed D decays. So far, no signal has been observed for such modes and only upper bounds on r B are obtained. The most recent result is from Belle 22 for the D Kπ mode (Fig. 7 (right)) and the 9% C.L. upper limit, r B <.19, is derived.

6 Events/ 1 MeV 7 6 (a) E (GeV) Events/ 2 MeV 6 5 (b) M bc (GeV/c 2 ) Events/ 5 MeV 9 8 (c) M 1 (ππ) (GeV/c 2 ) Figure 4: Projections of the four-dimensional fit onto (a) E, (b) M bc, (c) M 1(π + π ), for candidates satisfying (except for the variable plotted) the criteria E [.5,.5] GeV, M bc [5.27, 5.29] GeV/c 2, and M 1,2(π + π ) [.626,.926] GeV/c 2. The fit result is shown as the thick solid curve; the solid shaded region represents the B ρ ρ signal component. The dotted, dot-dashed and dashed curves represent, respectively, the cumulative background components from continuum processes, b c decays, and charmless B backgrounds. Events / (.1 GeV/c 2 ) (a) 4 Data ρ + ρ 2 ρ + f BB Bkgs qq m ES (GeV/c ) Events / (.2 GeV/c 2 ) 4 2 (b) mπ+ π --(GeV/c ) Figure 5: Projections of the fit (solid curve) onto the (a) m ES and (b) m π + π variables. A requirement on the likelihood ratio that retains 38% of the signal,.1% of the continuum background, and 1.3% of the BB background has been applied. The peak in the BB background at m π + π.78 GeV/c2 is from B + ρ + ω events with ω π + π. 4.2 Three-body decays (GGSZ method) The Belle collaboration uses a data sample that consists of B B pairs 23. The decay chains B + DK +, B + D K + with D Dπ are selected for the analysis. Analysis by the BaBar collaboration 24 is based on B B pairs. The reconstructed final states are B + DK +, B + D K + with two D channels: D Dπ and D Dγ and B + DK + with K + K S π+. The neutral D meson is reconstructed in the K S π+ π final state in all cases for BaBar and Belle collaborations and BaBar also used K S K+ K final state for the DK + and D K + cases. Figure 8 shows the results of the separate B + and B data fits for B DK mode in the x y plane for the BaBar and Belle collaborations. Confidence intervals were then calculated using a frequentist technique. The central values for the parameters γ, r and δ for the combined fit (using the (x ±,y ± ) obtained for all modes) with their one standard deviation intervals are presented in Tab. 3 for the BaBar and Belle analysis. The uncertainties in the model used to parametrize the D K S π+ π decay amplitude

7 C K M f i t t e r Winter9 (prel.) C K M f i t t e r Winter9 (prel.) B ρπ(wa) B ρρ(wa) B ππ(wa) COMBINED.8 1 CL.6.4 CKM fit no α meas. in fit α (deg) Figure 6: (Left) Isospin triangle situation for the B ρρ system. (Right) Confidence level as a function of the angle α extracted from an isospin analysis of the world averages for B ππ, B ρρ and B ρπ. The shaded region is the combination, also shown is the prediction of the CKM fit without including the direct measurements. lead to an associated systematic error in the fit result. These uncertainties arise from the fact that there is not unique choice for the set of quasi-2-body channels in the decay, as well as the various possible parametrizations of certain components, such as the non-resonant amplitude. To evaluate this uncertainty several alternative models have been used to fit the data. Table 3: Results of the combination of B + DK +, B + D K +, and B + DK + modes for BaBar and Belle analyses. The first error is statistical, the second is systematic and the third one is the model error. Parameter BaBar Belle γ (76 ± 22 ± 5 ± 5) ( ± 4 ± 9) r B (DK).86 ±.35 ±.1 ± ±.4 ±.1 ±.5 δ B (DK) (118 ± 63 ± 19 ± 36) ( ± 4 ± 23) r B (D K).135 ±.51 ±.11 ±.5.21 ±.8 ±.2 ±.5 δ B (D K) ( 62 ± 59 ± 18 ± 1) ( ± 4 ± 23) κr B (DK ) ±.37 ±.21 δ B (DK ) ( ± 17 ± 5) Despite similar statistical errors being obtained for (x ±,y ± ) in both experiments, the resulting γ error is much smaller in Belle s analysis. Since the uncertainty on γ scales roughly as 1/r B, the difference is explained by noticing that the BaBar (x ±,y ± ) measurements favor values of r B smaller than the Belle results. 5 Conclusions In this work we have presented a review of the most recent results on UT angles at the B Factories from Belle and BaBar experiments. An error of 5% is obtained for β and α, whereas Dalitz analyses allow to get the first significant measurement of γ.

8 D CP K R CP+ D* CP K R CP+ D CP K R CP- D* CP K R CP- D CP K* R CP+ D CP K* R CP- R CP Averages BaBar 1.6 ±.1 ±.5 Belle 1.13 ±.16 ±.8 CKM28 CDF 1.3 ±.24 ±.12 Average 1.1 ±.9 BaBar 1.3 ±.1 ±.5 CKM28 Belle 1.17 ±.14 ±.14 Average 1.6 ±.1 BaBar 1.31 ±.13 ±.3 CKM28 CKM28 Belle 1.41 ±.25 ±.6 Average 1.33 ±.12 BaBar 1.9 ±.12 ±.4 Belle 1.15 ±.31 ±.12 Average 1.1 ±.12 BaBar 2.17 ±.35 ±.9 CKM28 Average 2.17 ±.36 BaBar 1.3 ±.27 ±.13 Average 1.3 ± CKM28 CKM28 Events / 12.5 MeV 25 (a) E E (GeV).2.3 Figure 7: (Left) Compilation of the R CP ± results. (Right) E distribution for B D supk for the Belle analysis. The signal component is shown by red thicker dashed curve. y D Dalitz K ± x ± vs y ± ICHEP 28 y D * Dalitz K± x ± vs y ± ICHEP BaBar B + Belle B + BaBar B - Belle B - Averages BaBar B + Belle B + BaBar B - Belle B - Averages Contours give -2 (ln L) = χ 2 = 1, corresponding to 6.7% CL for 2 dof x Contours give -2 (ln L) = χ 2 = 1, corresponding to 6.7% CL for 2 dof x Figure 8: Results of signal fits with free parameters x ± = r cos θ ± and y ± = r sin θ ± for B ± DK ± (left) and B ± D K ± (right) from the BaBar and Belle latest publications. The contours indicate one standard deviation. References 1. N.Cabibbo, Phys. Rev. Lett. 51, 531 (1963). 2. M.Kobayashi and T.Maskawa, Prog. Theor. Phys. 49, 652 (1973). 3. M.Gronau, Phys. Rev. Lett. 63, 1451 (1989), H.Boos, T.Mannel and J.Reuter, Phys. Rev. D 7, 366 (24). M.Ciuchini, M.Pierini and L.Silvestrini, Phys. Rev. Lett. 95, (25). 4. Heavy Flavor Averaging Group: 5. Belle Collaboration, K.-F.Chen et al., Phys. Rev. Lett. 98, 3182 (27). 6. Belle Collaboration, H.Sahoo et al., Phys. Rev. D 77, 9113 (28). 7. BaBar Collaboration, B.Aubert et al., Phys. Rev. D 79, 729 (29). 8. M.Gronau and D.London, Phys. Rev. Lett. 65, 3381 (199). 9. CKMfitter group, J.Charles et al., Eur. Phys. Jour. C 41, 1. (25), updated in ckmfitter.in2p3.fr.

9 1. BaBar Collaboration, B.Aubert et al., Phys. Rev. D 76, 527 (27). 11. Belle Collaboration, A.Somov et al., Phys. Rev. Lett. 96, (26). 12. Belle Collaboration, A.Somov et al., Phys. Rev. D 76, 1114 (27). 13. BaBar Collaboration, arxiv: Belle Collaboration, J.Zhang et al., Phys. Rev. Lett. 91, (23). 15. BaBar Collaboration, B.Aubert et al., Phys. Rev. D 78, 7114 (28). 16. Belle Collaboration, C.-C.Chiang et al., Phys. Rev. D 78, (28). 17. M.Gronau and D.London, Phys. Lett. B 253, 483 (1991); M.Gronau and D.Wyler, Phys. Lett. B 265, 172 (1991); M.Gronau, Phys. Rev. D 58, 3731 (1998). 18. D.Atwood, I.Dunietz and A.Soni, Phys. Rev. Lett. 78, 3257 (1997). 19. A.Giri, Y.Grossman, A.Soffer and J.Zupan, Phys. Rev. D 68, 5418 (23). 2. M. Gronau, Phys. Lett. B 557, 198 (23). 21. Y.Grossman, A.Soffer and J.Zupan, Phys. Rev. D 72, 3151 (25). 22. Y.Horii, K.Trabelsi, H.Yamamoto et al., Phys. Rev. D 78, 7191 (28). 23. Belle Collaboration, arxiv: BaBar Collaboration, B.Aubert et al., Phys. Rev. D 78, 3423 (28). 25. G.J.Feldman and R.D.Cousins, Phys. Rev. D 57, 3873 (1998).