CP Violation in the Decays B ->K Kbar in the Presence of D Dbar<->K Kbar Mixing

We have recently shown that the large direct CP violation observed in the decay B^0 ->pi^+ pi^-, and the enhanced branching ratio for B^0 ->pi^0 pi^0, can be understood by invoking a small mixing of the pi pi system with the dominant D Dbar channel. We examine here the analogous effect of D Dbar<->K Kbar mixing on the rare decays B^0 ->K^0 K^0bar, B^- ->K^- K^0 and B^0 ->K^+ K^-. We find (a) significant values for the asymmetry parameters C and S in B^0 / B^0bar ->K^0 K^0bar, (b) a possible enhancement of the suppressed mode B^0 ->K^+ K^-, (c) a correlation between the three decay channels following from a triangle relation between amplitudes A_{K^0 K^0bar} - A_{K^+ K^-} = A_{K^- K^0}. The pattern of asymmetries and branching ratios is compared with that derived from the short-distance QCD penguin interaction.

The rare decays B 0 → K 0 K 0 , B − → K − K 0 and B 0 → K + K − have a rather simple description in the conventional approach, based on the short-distance Hamiltonian for b → dss and the assumption of factorization of hadronic matrix elements [1,2,3,4]. The dominant interaction is the QCD penguin transition b → dg * → dss, which leads to the following amplitudes [1] (the absolute square of an amplitude gives the branching ratio): Here λ t = V tb V * td , λ c = V cb V * cd are CKM factors, and ∆P is a term of order (m 2 c − m 2 u )/m 2 b that we discuss below. The normalization N 0 contains the penguin factor αs 12π ln m 2 t m 2 u , as well as decay constants and wave-function overlaps. A typical estimate is N 0 ≈ 0.12, which implies a branching ratio Br(B 0 /B 0 → K 0 K 0 ) ≈ 1.0×10 −6 . The notable feature of Eqs. (1) is the equality of the decay rates of B 0 → K 0 K 0 and B − → K − K 0 , as well as equal CP -violating asymmetries between B 0 → K 0 K 0 and B 0 → K 0 K 0 , and between B − → K − K 0 and B + → K + K 0 . Also notable is the suppression of the decay B 0 → K + K − (which persists even after inclusion of annihilation contributions, the branching ratio being ∼ 10 −8 [4]).
Of particular interest to us is the term ∆P in Eq. (1), which represents the residual contribution of cc and uu intermediate states in the QCD penguin diagrams [5,6]. This complex contribution depends on the effective (time-like) q 2 of the virtual gluon that mediates the B → KK transition. An explicit expression for ∆P (q 2 ) is given in Ref. [1], and suggests that |∆P | can be as large as 0.3 for q 2 eff ≈ 1 3 m 2 b . In the limit in which ∆P is neglected, the asymmetry parameters C and S for the decay B 0 /B 0 → K 0 K 0 , defined as vanish identically: C = S = 0. Non-zero values of C and S are thus associated with effects of cc intermediate states, embedded in the function ∆P (q 2 ). The behaviour of C and S as a function of q 2 is shown in Fig. 1 In the present note, we re-examine the amplitudes for the B → KK channels, from the standpoint of inelastic final-state interactions coupling the KK and DD systems, and suggest an alternative to the term ∆P λ c in Eq. (1). The motivation comes from a recent analysis of the decay channels B → ππ [7], in which we showed that a mixing of the strong-interaction channels ππ and DD in the isospin I = 0 state helps to resolve the puzzling observations of large direct CP violation in B 0 /B 0 → π + π − [8] and an enhanced branching ratio for B 0 /B 0 → π 0 π 0 [9]. There is a natural explanation for these correlated effects in terms of final-state interactions in systems of physical hadrons, which respect the strong-interaction conservation laws. Experiment [10] shows that DD systems are produced with large branching ratio (i.e. large decay amplitude, proportional to λ c ). Production of such systems in a real intermediate state, followed by a small mixing into the final state, results in an important imaginary contribution to the decay amplitude. This gives rise to asymmetries, and can enhance an (initially suppressed) branching ratio. The same physical idea will be implemented here, via the mixing of DD and KK states with the same strong-interaction quantum numbers. Introduction of the DD as an intermediate state in the decay B → KK acts as a long-distance substitute for the cc contribution in the QCD penguin loop parametrized by ∆P (q 2 ). Since the discussion is on the level of physical hadronic channels, no reference is made to the uncertain variable q 2 , which appears in the shortdistance contribution. Here we deal with the contributions to individual exclusive decays, of empirically-established [10], real intermediate states of physical hadrons. We obtain different, correlated results for asymmetries and branching ratios.
Mixing occurs between the following pairs of states, possessing definite strong-interaction quantum numbers.
As a consequence, the physical decay amplitudes of B 0 to these states are [7] A + HereÃ + KK ,Ã + DD ,Ã − KK ,Ã − DD denote the "bare" decay amplitudes, in the absence of final-state interactions. The mixing effects are contained in the rescattering matrices S where θ + and θ − are two phenomenological mixing angles. The physical decay amplitudes of B 0 to the states K 0 K 0 and K + K − can then be written as For the charged decay mode B − → K − K 0 , involving the I = 1 state K − K 0 , mixing with the state D − D 0 , with mixing angle θ − , implies For the bare amplitudes, we use the leading term of the QCD penguin model expressed by Eqs. (1):Ã where N ′ = 1.79 is determined [7] from the measured branching ratio Br(B 0 /B 0 → D + D − ) ∼ = 2.5×10 −4 [10]. With this input, the physical decay amplitudes, for small mixing angles (cos θ ± ≈ 1, sin θ ± ≈ θ ± ) take the form The amplitudes satisfy the triangle relation (for all θ ± ) In what follows, we discuss the observable consequences of the amplitudes in Eqs. (10) and compare them with those obtained from the short-distance penguin model expressed by Eqs. (1).

C and S Parameters for
The asymmetry parameters C and S for direct and indirect CP violation respectively, calculated from the amplitudes in Eqs. (10), are shown in Fig. 3.
, the short-distance model predicts C ≈ 0.2, S ≈ −0.2. While the two trajectories intersect, there is a large domain in which the predictions of the two models are quite different. Since A K + K − arises from the final-state interaction for θ + = θ − , we have C K + K − ∼ = 0 and S K + K − ∼ = − sin 2β ∼ = −0.7.

Effects of a Small Elastic Phase in the Rescattering Matrix
The matrices S 1 2 ± that connect the bare and the physical amplitudes in Eq. (4) can have a more general form than that given in Eq. (5), if the S matrix is allowed to have non-zero elastic phases in the KK and DD channels [7]. For small phases, the net effect is to introduce a further phase factor e iδ on the second term in the B 0 → K 0 K 0 amplitude in Eq. (10). The effect of such a modification on the trajectories in the C − S plane is shown in Fig. 5, for two choices δ = ±20 • . The effect on the average branching ratios is exhibited in Fig. 6.

Comments on other Models for B → KK
Predictions for B → KK decays appear in various analyses dealing with B → P P transitions, where P denotes a pseudoscalar meson. In Ref. [15], an SU (3) symmetry is assumed, and decay amplitudes are written as linear combinations of phenomenological parameters associated with different quark topologies. These parameters are estimated from a fit to all B → P P channels. The model gives equal rates for K 0 K 0 and K − K 0 , and the K + K − rate is strongly suppressed, exactly as in the QCD penguin amplitudes given in Eqs. (1). A similar result occurs in the QCD factorization approach of Ref. [4], which estimates an asymmetry parameter for K 0 K 0 of C ≈ 0.16. In Ref. [16], the C and S parameters for B 0 /B 0 → K 0 K 0 are estimated to be C ≈ 0.12, S ≈ −0.16, similar to Ref. [1], using a model like that in Eqs. (1). In Ref. [17], data on B → P P decays are fitted using elastic scattering effects. The fits have very large elastic scattering phases. Then, they give a branching ratio for the K + K − channel of up to 0.6× 10 −6 , a large asymmetry parameter C for the K 0 K 0 channel, and also a C K + K − of the same sign. Smith [18] has carried out an analysis of B → P P decays which discusses SU (3) elastic scattering, and separately, effects of P P ↔ DD mixing using a parametrization different from ours. In the limit of retaining only the inelastic mixing effect, his results for B 0 /B 0 → K 0 K 0 resemble ours, but the branching ratio B 0 /B 0 → K + K − vanishes. In our approach, an enhanced rate for K + K − occurs as long as the mixing angles θ + and θ − for states of different isopin, Gparity, are different. Finally, there are papers discussing "charming penguins", which attempt to parameterize charm-related, long-distance aspects of the penguin diagram, but make no reference to strong-interaction eigenstates of physical hadrons in real intermediate states. Results may be found in Ref. [19] and the literature cited therein.
In conclusion, we note that current experiments are steadily approaching the level of sensitivity at which the decays B → KK should soon be measurable. One awaits with interest the pattern of branching ratios and asymmetries that these measurements will reveal. Our results here show that this pattern can elucidate further important aspects of the underlying dynamics. The unexpected, recent oberservations of enhanced rates, for π 0 π 0 [9], π 0 ρ 0 [20,21], and ηω [22] are striking consequences of this dynamics [7,23].