Contribution of sigma meson pole to K L - K S mass difference ∗

The hypothesis of σ meson pole dominance in the |∆I| = 1 2 K → ππ amplitudes is tested qualitatively by using the KL-KS mass difference. Dominance of σ-meson pole contribution in the amplitudes for the KS → ππ decays was first proposed as the origin of the well-known |∆I| = 1 2 rule in these decays [1], and recently revived in connection with the direct CP violation in the K → ππ decays [2]. If it is the case, however, the matrix elements, 〈σ|Hw|K〉, should survive and give a significant contribution to the KL-KS mass difference, ∆mK , where Hw is the strangeness changing (|∆S| = 1) effective weak Hamiltonian. Dynamical contributions of various hadron states to hadronic processes in which pion(s) take part can be estimated by using a hard pion technique (with PCAC) in the infinite momentum frame (IMF) [3]. For later convenience, we review briefly it below. As an example, we consider a decay, B(p) → π1(q)π2(p), in the IMF, i.e., p → ∞, and assume that its amplitude M(B → π1π2) can be approximately evaluated at a slightly unphysical point, q → 0, i.e., q → 0 but (p · q) is finite: M(B → π1π2) ≃ lim p→∞, q→0 M(B → π1π2). (1) In this approximation, the σ → ππ amplitude is described in terms of the asymptotic matrix element, 〈π|Aπ−|σ〉, (matrix element of Aπ− taken between π and σ with infinite momentum) as M(σ → ππ) ≃ √ 2 mσ −mπ fπ )