Effects of the Space-like Penguin Diagrams on CP Asymmetries in Exclusive B Decays

The space-like penguin contributions were ignored on little ground in many previous studies of nonleptonic B transitions and CP violation. Taking the penguin-dominated channels B− u → K̄0π− and B− u → K0K− for example, we illustrate the non-negligible effects of the space-like penguin diagrams on CP asymmetries. Some qualitative remarks are given on the gluonic penguin picture and direct CP violation in exclusive two-body decays of B mesons. Alexander von Humboldt Research Fellow 1 Within the standard electroweak model, direct CP violation is expected to manifest itself significantly in some exclusive B-meson decays which are dominated by the one-loop gluonic penguin transitions b → qg∗ → q(qg q̄g) (q = d or s and qg = u, d, or s) [1-4]. Non-zero CP asymmetries can arise through the interference between two independent amplitudes that have both different CP violating phases and different CP conserving phases. In the timelike penguin diagram, the necessary strong phases are provided by different loop effects of internal u and c quarks involving real (on-shell) particle rescattering. In contrast, the spacelike penguin diagram can only provide an overall CP conserving phase due to final state hadronization. Thus the time-like penguin amplitudes play the key role in giving rise to CP violation, while the space-like penguin amplitudes only take effects by modifying the dispersive or absorptive parts of the time-like ones. In many previous studies of CP violation in B decays, the contributions from those annihilation-type diagrams were neglected for the argument that they should be formfactor suppressed or helicity unfavoured. This argument becomes questionable, however, for the space-like penguin channels since their amplitudes can be remarkably enhanced by the hadronic matrix elements involving (V − A)(V + A) or (S − P )(S + P ) currents [5]. To date, a phenomenological demonstration of the space-like penguin effects on CP asymmetries in exclusive B decays has been lacking. In this work we shall take the penguin-dominated decay modes B− u → K̄π and KK for example to illustrate the non-negligible contributions of the space-like penguin amplitudes to CP asymmetries. A simple kinematic picture is presented for decays of the type B → PP in order to minimize the uncertainty with the gluonic momentum transfer. We calculate decay amplitudes by using the effective weak Hamiltonian and factorization approximation. Our numerical results show that the space-like penguin diagrams can significantly affect the CP violating signals in exclusive B transitions. It is therefore worthwhile to reexamine some previous works on the penguin-dominated B decays and CP violation, in which the space-like penguin effects were ignored on little ground. The transitions B− u → K̄π and B− u → KK are of great interest for studying direct CP violation [1-4] and extracting the Kobayashi-Maskawa (KM) phase parameters [6]. They have two advantages for our present purpose. First, they occur only through the tree-level annihilation and penguin channels in the quark-diagram scheme [5,7]. The former can be safely neglected in comparison with the space-like penguin contribution. Second, the electroweak penguin effects on these two decay modes are rather small [8], thus one may simply apply the 2 QCD-loop induced Hamiltonian to them. The one-loop gluonic penguin Hamiltonian for B decays is given by [1-5] Heff(∆B = −1) = − GF √ 2 αs 8π [