Final State Interactions in Kaon Decays Revisited

We examine effects of final state interactions (FSI) in kaon decays using a new scheme without the Omnés function, and show that while ∆I = 1/2 rule is impossible to be explained in terms of FSI alone, ′/ ∼ 16.5 × 10−4 is obtained through FSI, if the calculation is concerned only with the CP-violating parts. The ππ contribution to the mL −mS mass difference is shown to be about 14%, if the sizes of the CP conserving amplitudes |A0| and |A2| are correctly given. @ @ @ @ @ It seems that the established theory to explain simultaneously both of the ∆I = 1/2 rule and the large ′/ ratio in kaon decays has not yet been obtained [1, 2, 3]. In order to take a step toward resolving the issues much efforts have been made on the evaluation of hadronic matrix elements of quark operators [4, 5, 6]. Among them it is advocated that taking account of final state interactions (FSI) is crucial to enhance or suppress the decay amplitudes besides giving the strong phases [7, 8, 9, 10]. The final state interactions (FSI) in the kaon decays are usually described in terms of the Omnés function [11] as seen in Refs.[12, 7, 8, 9]. The FSI effect written in terms of the Omnés function is, however, involved with some ambiguities, inherent in dispersion integrals, coming from the dependence on a subtraction, a cutoff and a multiplicative arbitrary polynomial function [13, 14, 2]. We propose, therefore, a new scheme in this note, which makes the decay amplitude satisfy the final state interaction theorem without any dispersion relations. In order to evaluate our scheme of FSI, we briefly see FSI in production processes of S-wave dipion states such as γ + γ → ππ, φ → γππ, cascade decays of heavy quarkonia and others. Let us consider a production amplitude of an S-wave dipion state with a mass √ s, called an f -state: It is composed of a direct production term of the f -state given by tree diagrams, which we call the Born term and denote by Bf , and an indirect production term, in which at first an i-state, denoted by Bi, is directly produced and then transformed into the final f -state through the s-channel loop and the S-wave scattering amplitude Tif . The loop integral should be properly regularized, and it may, then, depend on the renormalization scale parameter or a cutoff parameter. We can write the production amplitude, Ff (s) as Ff (s) = Bf (s) + ∑ i Gi(s) · Tif (s), (1) where Gi(s) is the loop integral composed of the meson propagators in the i-th channel, whose imaginary part (discontinuity along the physical cut) is given as ImGi(s) = −Bi(s)ρi(s)θ(s− si), (2) where ρi = pi/(8π √ s) with pi (si) being the CM momentum (threshold energy squared) of the i-th channel. We can easily see that the above production amplitudes satisfy the final state interaction theorem, ImFf = − ∑ i F ∗ i ρiθ(s− si)Tif (3) ∗E-mail: [email protected]