Cusps in K --> 3pi decays: a theoretical framework

Abstract

Based on the analysis of 6.031×10 7 K ± → π 0 π 0 π ± decays, the NA48/2 collaboration has recently determined the S-wave ππ scattering lengths a 0 − a 2 with high precision. In addition, the scattering length a 2 has been independently measured, although less precisely so. The present article discusses in detail one of the theoretical frameworks used in the data analysis.

On the ΣN cusp in the pp → pK+Λ reaction

“…see Ref. [13] for the method to extract the s-wave ππ scattering lengths from K → 3π. Signs of such a cusp in the pp → pK + Λ reaction have been first observed in inclusive measurements with single-arm magnetic spectrometers at Saclay [12] and COSY-HIRES [14].…”

On the three-particle analog of the Lellouch-Lüscher formula

“…A brief review of the essential points of the approach is given in Ref. [32]. The main difference of the approach from the conventional ones consists in the treatment of the relativistic corrections to the internal particle lines.…”

“…In order to rectify the counting rules, one then has to amend the procedure for the calculation of the Feynman integrals -dimensional regularization plus minimal subtraction does not suffice. The modification of the procedure, which is equivalent to the change of the renormalization prescription, is described in detail in [32], and we refer an interested reader to that article.…”

On the role of final-state interactions in Dalitz plot studies

“…[23] for earlier, related formulations.) Again, care has to be taken when performing the dispersive integral, as F (s) shows singular behavior at the pseudothreshold s = (M V − M π ) 2 . Equations ( 8) and ( 9) can be solved iteratively: starting from an arbitrary input function F (s), we can calculate the inhomogeneity F (s) according to Eq.…”

“…Final-state interactions can be of significance for various reasons: if they are strong, they can significantly enhance decay probabilities; they can significantly shape the decay probabilities, most prominently through the occurrence of resonances; besides resonances, also new and non-trivial analytic structures can occur, such as threshold or cusp effects (for the prominent role cusp effects have played recently in studying pion-pion interactions, see Ref. [2] and references therein); and finally, they introduce strong phases or imaginary parts, the existence of which is a prerequisite for the extraction of CP-violating phases in weak decays (see e.g. Ref.…”