Large Direct CP Violation in B^0 ->pi^+ pi^- and an Enhanced Branching Ratio for B^0 ->pi^0 pi^0

Recent measurements of B^0 ->pi pi decays reveal two features that are in conflict with conventional calculations: the channel B^0 (Bbar^0) ->pi^+ pi^- shows a large direct CP-violating asymmetry, and the channel B^0 (B^0) ->pi^0 pi^0 has an unexpectedly high branching ratio. We show that both features can be understood in terms of strong-interaction mixing of pi pi and D Dbar channels in the isospin-zero state, an effect that is important because of the large experimentally observed ratio Gamma(B^0 / Bbar^0 ->D^+ D^-) / Gamma (B^0 / Bbar^0 ->pi^+ pi^-) approx. 50. Our dynamical model correlates the branching ratios and the CP-violating parameters C and S, for the decays B^0 (Bbar^0) ->pi^+ pi^-, B^0 (Bbar^0) ->pi^0 pi^0, B^0 (Bbar^0) ->D^+ D^- and B^0 (Bbar^0) ->D^0 Dbar^0.

The Belle collaboration has presented new data [1] which support their original evidence [2] for large direct CP violation in the decays B 0 (B 0 ) → π + π − , the asymmetry parameter C (= −A) being measured to be C = −0.58 ± 0.15 ± 0.07. In a related development, both the Babar [3] and Belle [4] collaborations have reported a sizable branching ratio for the decay B 0 (B 0 ) → π 0 π 0 , with an average value Br(B 0 /B 0 → π 0 π 0 ) = (1.9 ± 0.6) × 10 −6 . Both of these observations are unexpectedly large from the standpoint of conventional calculations [5,6,7] based on a short-distance, effective weak Hamiltonian and the assumption of factorization of products of currents in matrix elements for physical hadron states. In this paper, we carry out a calculation based upon the idea [8] of final-state interactions involving the mixing of ππ and DD channels. This dynamics provides a natural, correlated explanation of the new experimental facts, and leads to several further predictions.
To fix notation, we write the three B → ππ amplitudes as A(B 0 → π + π − ) = N (λ u a 1 + λ c a p ) A(B 0 → π 0 π 0 ) = N (λ u a 2 − λ c a p )/ √ 2 (1) A(B − → π − π 0 ) = N λ u (a 1 + a 2 )/ √ 2 Here a 1 , a 2 , a p are, in general, complex numbers and N is a positive normalization factor. The parameters λ u and λ c are CKM factors, defined as λ u = V ub V * ud , λ c = V cb V * cd , with magnitudes |λ u | ∼ = 3.6 × 10 −3 , |λ c | ∼ = 8.8 × 10 −3 and phases given by λ u = |λ u |e −iγ , λ c = −|λ c |, with γ ≈ 60 • [9]. The amplitudes in Eq. (1) are defined so that their absolute square gives the branching ratio, and they satisfy the isospin relation [10] 1 From the results of the models discussed in [5,6,7], the parameters appearing in Eq. (1) have the following rough representation. The constants a 1 , a 2 , a p are approximately real (to within a few degrees), with magnitudes a 1 ≈ 1.0, a 2 ≈ 0.2, a p ≈ −0.1. The normalization factor is N ≈ 0.75; it is here fixed by the empirical branching ratio for B − → π − π 0 . The fact that the parameters a 1 , a 2 , a p are nearly real implies immediately that there is very little direct CP -violating asymmetry between B 0 → π + π − and B 0 → π + π − , as well as in the channels π 0 π 0 and π ± π 0 . Furthermore, the absolute branching ratios following from the above parametrization are as follows (with experimental values given in parentheses): The most striking feature is the strong enhancement of the π 0 π 0 rate compared to this model expectation.
It was pointed out in Ref. [8] that the CP -violating asymmetries and branching ratios in the B → ππ system would be strongly affected by final-state interactions involving the mixing of the ππ and DD channels in the isospin I = 0 state, as a consequence of the large ratio of partial decay widths Γ(B 0 → D + D − )/Γ(B 0 → π + π − ) ≈ 3 14 |V cb | 2 /|V ub | 2 ≈ 26 expected in the Bauer-Stech-Wirbel model [5]. A large ratio has now been confirmed by the Belle measurement [11] of the branching ratio Br(B 0 /B 0 → D + D − ) = 2.5 × 10 −4 , which is about 50 times larger than Br(B 0 /B 0 → π + π − ). This fact gives new urgency to an investigation of ππ ↔ DD mixing as a way of resolving the puzzling observations in B → ππ decays.
The ππ system exists in the states I = 0 or I = 2, while the DD system has I = 0 or I = 1. Mixing can occur between the isospin-zero states By contrast, the I = 2 ππ state and the I = 1 DD state, given by are unaffected by mixing. The physical decay amplitudes of B 0 to the above four states are These physical decay amplitudes are related to the "bare" amplitudes calculated in the absence of final-state interactions, i.e. with no mixing, which we denote byÃ: Here S denotes the strong-interaction S matrix connecting the isospin-zero states |ππ 0 and |DD 0 which can be written generally as 1 where θ is a mixing angle, and δ 1 and δ 2 are the strong-interaction phase shifts for the elastic scattering of ππ and DD systems in the I = 0 state, at √ s = M B . For any choice of these three parameters, the matrix S 1 2 can be calculated numerically, and the set of four equations (7) solved to obtain the physical amplitudes A π + π − , A π 0 π 0 , A D + D − and A D 0 D 0 in terms of the bare amplitudes. The bare amplitudes are identified with those calculated in the factorization model [5,6,7], which we list below where the first two equations are as in Eq. (1), and the factor N ′ is determined from the empirical [11] branching ratio Br 79. In order to show, in a transparent way, how the mixing mechanism gives rise to large direct CP violation in B 0 → π + π − , as well as an enhanced branching ratio for B 0 → π 0 π 0 , we 1 The two-channel S-matrix has been discussed, in particular in [12,13]. The S 1 2 prescription is given in [6,12]. An alternative prescription, using 1 2 [1 + S] in place of S 1 2 , has been discussed by Kamal [14], and was used in Ref. [8].
consider, for illustration, the case where the elastic phases δ 1 and δ 2 in the S matrix (Eq. (8)) are neglected, so that S 1 2 may be written as The amplitudes A π + π − and A π 0 π 0 for B 0 decay are then given by Clearly for θ = 0, the physical amplitudes reduce to the bare amplitudes. Inserting the bare amplitudes from Eq. (9), we can rewrite A π + π − and A π 0 π 0 as linear combinations of λ u and λ c : Note that the isospin relation in Eq. (2) continues to be fulfilled. The important new feature of the amplitudes in Eq. (12) is the appearance of the imaginary term a m in the coefficent of λ c , in striking contrast to the real term a p . The imaginary nature of this dynamical term is an inescapable consequence of S-matrix unitarity, which enforces the factor i in the off-diagonal matrix element in Eq. (10). The term a m , given in Eq. (13), has a magnitude |a m | ≈ 1.39 sin θ, and dominates the term a p cos θ even for a modest mixing angle ∼ 0.1. We will now show that the mixing term a m has profound consequences for direct CP violation in the decays B 0 → π + π − , and for the branching ratio of the channel B 0 → π 0 π 0 .
1 C and S Parameters for B 0 → π + π − and B 0 → π 0 π 0 The C and S parameters derived from the time-dependent asymmetry between B 0 and B 0 decays into π + π − are defined as C +− is the parameter for direct CP violation (i.e. |A(B 0 → π + π − )/A(B 0 → π + π − )| = 1). Using the amplitude A π + π − in Eq. (12), we obtain The asymmetry parameters C +− and S +− calculated from the above expression are plotted as functions of θ in Fig. 1. Good approximate agreement with data is obtained for θ ≈ 0.2 (see Table 1, where we also list C 00 and S 00 ). We note that the amplitudes in Eq. (12) have been derived from the matrix S 1 2 in Eq. (10), which was obtained from (8) by neglecting the phase shifts δ 1 and δ 2 . We have also explored numerically S matrices with non-zero phases, and indicate in Figs. 1 and 2 two examples, obtained with the values δ 1 = ±10 • , δ 1 + δ 2 = −30 • . Table 1 gives numerical values for a few choices of parameters. In all cases, there is a large direct CP violation.
Discussions of the direct CP -violating parameter C +− are often based on an amplitude for B 0 → π + π − written in the form The parametrization in Eq. (1), based on the models [5,6,7], gives |P ππ /T ππ | = 0.24, and arg(P ππ /T ππ ) = 0. The small phase of the "penguin-to-tree" ratio P ππ /T ππ is a generic feature of these models, and is responsible for the prediction C +− ≈ 0, which is now contradicted by data [1]. In our approach, the role of P ππ /T ππ is played by the ratio "P/T " = − |λ c |(a p cos θ + a m ) For a typical value θ = 0.2, this ratio has the modulus |"P/T "| ≈ 0.77, and a phase arg("P/T ") ≈ −70 • . The difference is a consequence of the term a m in Eq. (19), which reflects the physical final-state interaction of the ππ system, as implemented in our model through ππ ↔ DD mixing.
2 Branching Ratio for B 0 → π 0 π 0 and B 0 → π + π − The branching ratios (averaged over B 0 and B 0 ) may be calculated in our model by taking the absolute square of the B 0 decay amplitudes in Eq. (12), and the corresponding amplitudes for B 0 decay. The results are shown in Fig. 2. It is remarkable that the empirical branching ratio for B 0 /B 0 → π 0 π 0 is accurately reproduced, using the same value θ ≈ 0.2 which accounts for the asymmetry parameter C +− . We also note that the branching ratio B 0 /B 0 → π + π − remains close to its bare value, and can be lowered slightly with the introduction of phases δ 1 and δ 2 . Numerical results for Br(B 0 /B 0 → π 0 π 0 ) and Br(B 0 /B 0 → π + π − ) are listed in Table 1.
3 Branching Ratio for B 0 → D 0 D 0 Since our model treats the ππ and DD states with I = 0 as a coupled system, it also produces predictions for branching ratios and asymmetry parameters in B 0 → D + D − and B 0 → D 0 D 0 . The amplitudes after mixing are Of particular interest is the branching ratio for B 0 /B 0 → D 0 D 0 , since it vanishes at the level of the bare amplitude (Ã D 0 D 0 = 0), and is induced by mixing with the ππ system. For θ = 0.2, ignoring the phases δ 1 , δ 2 , our model predicts To conclude, we have demonstrated a mechanism of final-state interactions among physical hadrons in B 0 → ππ decays which predicts a large direct CP -violating parameter C +− . The same mechanism enhances the theoretical prediction for the branching ratio of B 0 /B 0 → π 0 π 0 to the experimentally observed level. Predictions are made for the C and S parameters of B 0 (B 0 ) → π 0 π 0 decays, and for the branching ratio of B 0 /B 0 → D 0 D 0 . The model makes essential use of the large empirical ratio Γ(B 0 /B 0 → D + D − )/Γ(B 0 /B 0 → π + π − ) ≈ 50. Its success in the present context leads to the expectation that sizable direct CP violation could be observed in other charmless B decays, in which an amplitude of order λ u receives a dynamical contribution proportional to λ c , through mixing with a channel possessing a large branching ratio. The resulting amplitude contains two pieces which are comparable in magnitude and have different weak-interaction and strong-interaction phases. We have treated earlier [15] the charged-particle decays B ± → ηπ ± (and B ± → η ′ π ± ), which are influenced by mixing with the channel B ± → η c π ± , and have predicted significant direct CP violation. Evidence for a sizable violation in B ± → ηπ ± has indeed been reported in one experiment [16], the first ever seen in a charged-particle decay.