ar X iv : h ep - p h / 06 04 08 4 v 1 1 0 A pr 2 00 6 HISKP - TH - 06 / 09 Cusps in K → 3 π decays

The pion mass difference generates a pronounced cusp in K → 3π decays. As has recently been pointed out by Cabibbo and Isidori, an accurate measurement of the cusp may allow one to pin down the S-wave ππ scattering lengths to high precision. Here, we present and illustrate an effective field theory framework that allows one to determine the structure of this cusp in a straightforward manner. The strictures imposed by analyticity and unitarity are respected automatically. 1. The S-wave ππ scattering lengths a 0 , a 2 have been predicted with percent level accuracy some time ago [1, 2], and first steps for an experimental verification of this prediction have been performed in Ref. [3, 4]. Recently, it has been pointed out by Cabibbo and Isidori [5, 6] that isospin violating effects generate a pronounced cusp in K → 3π decays whose experimental investigation may allow one to determine the combination a 0 − a 2 with high precision. A first analysis of data based on this proposal has appeared [7]. (The strong impact of the unitarity cusp on π 0 π 0 scattering close to threshold was already mentioned in [8].) In order for this program to be carried through successfully, one needs to determine the structure of the cusp with a precision that matches the experimental accuracy. In view of the large amount of data available [7], this is a considerable task. In the present letter, we present a method which – we believe – has the potential to achieve this goal.