Extracting the unitarity angle gamma in Bs to D0 h0, D0bar h0 Decays

The recently observed color-suppressed B0bar to D0 pi0, D0 eta, D0 eta', Ds+ K-, D0 K0bar, D0 rho0 and D0 omega decay modes all have rates larger than expected. The color-suppressed Bs to D0 phi, D0bar phi modes, which were suggested for the extraction of the unitarity angle gamma in the Gronau-London method, could be larger than the previous estimation by one order of magnitude. Several new theoretical clean modes in Bs decays are suggested for the extraction of gamma. The proposed Bs to D0 h0, D0bar h0 decay modes with h0 = pi0, eta, eta', rho0, omega in addition to h0 = phi are free from penguin contributions. Their decay rates can be estimated from the observed color-suppressed B0bar to D0 h0 rates through SU(3) symmetry. A combined study of these D0 h0, D0bar h0 modes in addition to the D0 phi, D0bar phi modes is useful in the extraction of gamma in the Bs system without involving the Bs-Bsbar mixing. Since the b to u and b to c transitions belong to the same topological diagram, the relative strong phase is likely to be small. In this case, the CP asymmetries are suppressed and the untagged rates are very useful in the gamma extraction.

The extraction of the unitarity angle γ ≡ arg V * ub , where V is the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, is important in completing or testing the Standard Model (SM). Several theoretical clean ways of the weak phase extraction were proposed using interference effects. At B factories, the extraction is performed in the DK system, using the interference effect of B → D 0 K and D 0 K decays in D CP K final states, where D CP are the CP eigenstates of D 0 and D 0 mesons, or to some common f CP K, f CP states [1,2,3,4,5,6].
Similarly, the color-suppressed D CP φ mode was also proposed in the extraction of γ in the B s system [1]. An alternative method made use of the B s -B s mixing was proposed using color-allowed B s → D ± s K ∓ decays with time-dependent tagging [7]. Due to the large rate (10 −4 ) in the color-allowed decays, this scenario has been seriously considered at LHCb [8].
It is well known that in the SM, the ∆m Bs in the B s system is much larger than the one in the B d system. Experimental searches give ∆m Bs > 14.5 ps −1 [9]. The measurement of the time-dependent asymmetry in the B s system is challenging. Furthermore, the deviation of the recently measured sin 2β eff in penguin-dominated modes from the sin 2β (β ≡ arg V * td ) extracted from charmonium modes may hint at New Physics contributions in the b → s transitions [10,11]. In this case, the ∆m Bs can easily be much larger than the SM expectation (see, for example [12]). Therefore, an extraction of γ without relaying on the B s -B s mixing is complementary to the D ± s K ∓ program and is indispensable to the γ program in the B s system.
Although the Gronau-London D CP φ method [1] does not need time-dependent tagging, its usefulness is questioned by the smallness of the color-suppressed decay rate, which is estimated to be as small as 10 −6 [7]. However, color-suppressed B 0 → D ( * )0 π 0 , D 0 η (′) , D 0 ω, D 0 ρ 0 , D + s K − , D 0 K 0 decay modes were observed with branching ratios significantly larger than earlier theoretical expectations based on naive factorization [13].
Similar enhancement in the color-suppressed decay rates in the B s system is expected. In particular, the D 0 φ rate is expected to be larger than the previous estimation. In addition to the Dφ mode, several other theoretical clean modes are suggested in this work. The proposed tree D 0 h 0 , D 0 h 0 decay modes, where h 0 = π 0 , η, η ′ , ρ 0 , ω, in additional to the D 0 φ, D 0 φ modes are useful to extract γ without time-dependent tagging. As we shall see later, the extraction done only with untagged rates can also be useful.
In this study, the γ extraction method is similar to the B d → DK and B s → Dφ method. It will be useful to briefly review the DK method and the present experimental status at B factories. To be specific, the amplitude ratio r B and the strong phase difference δ B for the color-allowed B 0 → D 0 K − and color-suppressed D 0 K − decays, which are governed by different CKM matrices as depicted in Fig. 1, are defined as The weak phase γ is removed from A(B − → D 0 K − ) in the δ B definition. Since the strong phase difference arises from that in the color-suppressed and color-allowed amplitudes, it is expected to be non-vanishing. The r B and δ B parameters are common to the γ determination methods of Gronau-London-Wyler (GLW) [1,2], Atwood-Dunietz-Soni (ADS) [3] and "DK Dalitz plot" [4,5], where one exploits the interference effects of Note that an average of r B = 0.10 ± 0.04 is found by the UT f it group, by combining analyses using all three methods [20]. As the strength of interference is governed by the size of r B , the larger error in the γ value of BaBar reflects the smallness of their r B . Given the present experimental situation that Belle and BaBar have quite different r B values and that the critical role it plays in the γ extraction, it is important to compare with a theoretical or phenomenological prediction of r B . In a recent work, we obtained r B = 0.09 ± 0.02 [18]. The predicted r B agrees with the UT f it extraction [20] and does not differ much from the naive factorization expectation. Furthermore, the r B value prefers the lower value of the BaBar experiment and disfavors the Belle result. A similar r B was found experimentally in the DK * analysis [10,19].
The smallness of the ratio r B would demand larger statistics of data for the γ program in the DK ( * ) system. In fact, the smallness of r B is precisely the reason that ADS and DK Dalitz methods are needed in additional to the original GLW method. However, these methods usually bring in additional uncertainties, such as the fourth uncertainties in the extracted γ value quoted above.
We now return to the B s system. By replacing the spectator quark in the previous case, we have B s → Dφ decays replacing the role of B → DK ( * ) decays, as depicted in Fig. 1, in the γ program [1]. Unlike the B case, both B s → D 0 φ and D 0 φ modes are color suppressed decays. Consequently, the corresponding b → u and b → c amplitude ratio is estimated as [9,10], which is several times greater than r B , giving a much prominent interference effect [1]. The B s → D 0 φ decay can be related to other decays by using the topological approach [21], which is closely related to SU(3) symmetry. Indeed the B s → D 0 φ decay is similar to other color-suppressed modes, such as B 0 → D 0 ρ 0 , D 0 ω, as one can see by replacing ss and V us in the second diagram of Fig. 1 by dd and V ud , respectively. These modes were observed with B(B 0 → D 0 ρ 0 ) = (2.9 ± 1.1) × 10 −4 and B(B 0 → D 0 ω) = (2.5±0.6)×10 −4 [9], which are larger than naive factorization expectations.
In addition to the color-suppressed diagram the B 0 → D 0 ρ 0 and D 0 ω amplitudes receive annihilation diagram contributions (similar to the second diagram shown in Fig. 2), but with different relative signs. The measured rates roughly satisfy B(B 0 → D 0 ρ 0 ) ≃ B(B 0 → D 0 ω) and, consequently, imply the sub-dominant role of the annihilation contribution plays in these modes. Assuming SU(3) symmetry and neglecting the annihilation contribution, the B s → D 0 φ rate can be estimated from these decay rates by using 1 where τ B d ,Bs are the lifetime of B d,s mesons with τ Bs /τ Bu ≃ 0.95 [9]. Our estimation of the B s → D 0 φ rate is one order of magnitude larger than the previous one [7]. The Gronau-London method should be useful in the extraction of γ in the B s system.
After realizing the applicability of the Gronau-London method in the B s system, we propose several additional theoretical clean modes adding to the γ program. The tree B s → D 0 h 0 , D 0 h 0 decays with h 0 = π 0 , η, η ′ , ρ 0 , ω, do not contain any penguin contribution.
The B s → D 0 η, D 0 η ′ modes receive contributions from color-suppress tree and W -exchange diagrams as depicted in Fig. 1 and 2, while others are pure weak annihilation modes.
The B s → D 0 h 0 rates can be estimated by using the B 0 → D 0 h 0 rates in the topological amplitude approach [21]. We have 1 In the right-hand-side of the equation, the annihilation amplitude only enters quadratically. Its contribution can be safely neglected. Also note that the B 0 → D 0 ρ 0 (ω) amplitude has an additional factor of 1/ √ 2 due to the ρ 0 (ω) wave function. and where C, C ′ , C ′′ and E, E ′ , E ′′ are (complex) color-suppressed and W -exchange amplitudes, respectively, containing possible final-state-interaction (FSI) effects, and ψ = 39.3 • is the mixing angle of the η and η ′ non-strange and strange contents [22]  with η q = (uū + dd)/ √ 2 and η s = ss. The color suppressed rates are measured to be  [9,13]. These decay rates are much larger than the naive factorization expectations. There are some theoretical efforts in understanding the largeness of these decay modes [14,15,16,18]. Considering, for example, the B 0 → D + s K − decay, in the rescattering approach [18]. Its large rate is feed from the color-allowed D + π − one, through the rescattering process D + (cū)π − (ud) → D + s (cs)K − (sū) with the annihilation (creation) of uū (ss) quark pair in the initial (final) state.
The measured B 0 → D 0 h 0 rates are useful in estimating B s → D 0 h 0 rates. In the SU (3) limit, we have C = C ′ and E = E ′ . For B s → D 0 η, D 0 η ′ modes, we have To further estimate D 0 η and D 0 η ′ rates, we need information on R ≡ E ′ /C ′ . Using the measured color-suppressed B 0 decay rates and Eq. (3), it is straightforward to obtain the best fitted value of E/C = 0.26 e ±i72 • . By assuming which are of the same order as B(B s → D 0 φ).
The pure W -exchange B s → D 0 π 0 decay rate can be estimated in a similar manner as In fact, when take into account the SU(3) breaking effects, the B s → D 0 π 0 decay rate could be larger than the above estimation, since unlike the B 0 → D s K decay no creation of the ss pair is needed in the final state (see Fig. 2).
Note that our estimation of the B s → D 0 π 0 rate is similar to a recent one [23], while our predicted B s → D 0 η, D 0 η rates are smaller than theirs by a factor of 20. This is because, the CKM factor V ud instead of V us was used in [23] for the B s → D 0 η (′) amplitudes.
The extraction of γ in B s → D 0 h 0 modes can be preformed by employing the GLW [1,2] method. It should be clear that other methods, such as ADS [3] and DK Dalitz [4,5] can also be used. However, as r Bs is several times greater than r B , the GLW method should be more favorable in reducing additional uncertainties. By the standard construction, we where D CP ± are defined as (D 0 ± D 0 )/ √ 2, a, b are real numbers with suitable phase convention and δ is the strong phase difference. All four unknowns γ, a, b, δ can be obtained by measuring the four tagged B s → D CP ± h 0 and B s → D CP ± h 0 decay rates. It is useful to where r Bs ≃ R b ≃ 0.4. It should be noted that the measurement of the asymmetry A ± requires tagging, while the measurement of R ± is untagged. In [24], weak annihilation modes of B s → D ± π ∓ having rate similar to B(B s → D 0 π 0 , D 0 π 0 ) were proposed for extracting γ.
However, contrary to our case, time-dependent tagged rates are necessary [24].
As a result of the same topological amplitudes for b → u and b → c transitions, the strong phase difference δ is likely to be small. In this case, a large r Bs value does not necessary lead to a large CP -asymmetry A ± , but it is still very useful in producing the interference effects in the D CP ± h 0 rates. For illustration, using δ = 0, r Bs = 0.4 and γ = 60 • , we obtain 2 Note that an additional negative sign in the last equation is due to the CP quantum number of h 0 and a (−) L factor, where L is the orbital angular momentum.
The measurements of R ± provide γ and r Bs values. The vanishing strong phase approximation is useful in extracting or constraining γ using less data. It can be verified by measuring A ± , when more data is available. Since the b → u and b → c amplitudes are of similar size, the direct CP asymmetry will be very sensitive to the strong phase difference. In fact, similar arguments also apply to B 0 → D 0 K 0 , D 0 K 0 decays. The measurement of direct CP violation in B 0 → D CP K 0 decays, will provide the information of the usefulness of the vanishing strong phase approximation.
It is interesting to give the δ = 0 argument in the rescattering picture. For example, as in the B 0 → D + s K − case, the B s → D 0 π 0 (D 0 π 0 ) rate is mainly feed from the color-allowed D + s K − (D − s K + ) one, through the rescattering D + s (cs)K − (sū) → D 0 (cū)π 0 (uū) [D − s (cs)K + (su) → D 0 (cu)π 0 (ūu)] with the annihilation and creation of ss and uū quark pair in the initial and final states, respectively [18]. The tree-allowed D ± s K ∓ amplitudes do not have any strong phase difference, while the D + s (cs)K − (sū) → D 0 (cū)π 0 (uū) and D − s (cs)K + (su) → D 0 (cu)π 0 (ūu) annihilation rescattering amplitudes are related by charge conjugation, which is respected by strong interactions. Consequently, the strong phase difference in B s → D 0 π 0 and D 0 π 0 amplitudes should be small. The above consideration also applies to other modes, including those with C ′ , C ′′ , as long as they are long distant dominated (as hinted by the B 0 → D 0 h 0 data). For the case of D CP V , the amplitudes C ′ and C ′′ , E ′ and E ′′ can be different in signs [25], but we do not expect a large strong phase difference.
In conclusion, we point out that the large enhancement in color-suppress decay rates observed in B decays suggest similar enhancement in the color-suppress B s decay rates. The GLW method in extracting γ using B s → D 0 φ, D 0 φ is not limited to the color suppressed decay modes as previously believed. We also suggest several new theoretical clean modes in the extraction of γ in B s decays. These modes are color-suppressed B s → D 0 h 0 , D 0 h 0 decays, with h 0 = π 0 , η, η ′ , ρ 0 , ω, in addition to the h 0 = φ case. They are free of penguin contributions. The extraction of γ can be performed as in the D CP φ case. These D 0 h 0 rates are of order 10 −6 ∼ 10 −5 . A combined analysis could be useful in reducing the statistical uncertainties in the γ extraction. No information on the B s -B s mixing is required. While the mixing is sensitive to New Physics, the γ extraction in this case is expected to be insensitive to NP and does not require a ∆m Bs value as predicted by the standard model. It can be considered as a complementary to the D ± s K ∓ method. The r Bs value is expected to be R b ≃ 0.4, while the strong phase difference between b → u and b → c amplitudes, both are of the same topological types, are likely to be small. In this case, the CP asymmetries are suppressed and the untagged measurements will provide very useful information in the extraction of γ.