Eur. Phys. J. C (2007) 50: 405-422
https://doi.org/10.1140/epjc/s10052-006-0201-7

Regular Article - Theoretical Physics

K→3π final state interactions at NLO in CHPT and Cabibbo’s proposal to measure a₀-a₂

¹ Department of Physics and Astronomy, The University of Glasgow, Glasgow, G12 8QQ, UK
² CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, Campus de Fuente Nueva, 18002, Granada, Spain
³ Departament de Física Teòrica, IFIC, CSIC-Universitat de València, Apt. de Correus 22085, 46071, València, Spain

^* e-mail: Prades@ugr.es

Received: 28 June 2006
Revised: 1 September 2006
Published online: 26 January 2007

Abstract

We present the analytical results for the K→3π final state interaction phases at next-to-leading order (NLO) in CHPT. We also study the recent proposal of Cabibbo to measure the ππ scattering lengths combination a₀-a₂ from the cusp effect in the π⁰π⁰ energy spectrum at threshold for K⁺→π⁰π⁰π⁺ and K_L→π⁰π⁰π⁰, and we give the relevant formulas to describe it at NLO. We estimate the theoretical uncertainty of the a₀-a₂ determination at NLO in our approach and obtain the result that it is not smaller than 5% if added quadratically and 7% if added linearly, for K⁺→π⁰π⁰π⁺. One gets similar theoretical uncertainties if the neutral K_L→π⁰π⁰π⁰ decay data below threshold are used instead. For this decay, there are very large theoretical uncertainties above threshold due to cancellations, and data above threshold cannot be used to get the scattering lengths. We do not include isospin corrections apart from two-pion phase space factors which are physical. We compare our results for the cusp effect with those due to Cabibbo and Isidori.