Pion-Pion Interaction from Threshold Anomalies inK+Decay

Abstract

It is our purpose to discuss in this note, for this kind of problem, a general method which allows in principle the determination of certain cross sections which are out of reach of direct experimental measurement.The method is based on the observation of threshold anomalies of the kind of a cusp or rounded step in cross sections for reactions with three particles in the final state. 2 ' 3 The necessary requirements are that we must have (at least) two different reactions with three particles in the final state, that one of the particles is common to both the considered reactions, and that the phenomenon is analyzed by keeping the total energy fixed. In processes of this kind it is very important to have a good energy resolution for the incoming beam; a spread of the initial energy can in fact destroy the threshold effect which usually involves a small portion of the final spectrum. In this respect an ideal situation presents itself in the three-particle decays of an unstable particle whose mass is well determined. As an example we will discuss the decay of the K* meson into three pions, for which all the requirements for the appearance of the effect are satisfied*:#+^+ + ^0+7^ (b) 8 H 0 tJberall, Phys. Rev 0 119, 365 (1960). See also D. Yovanovitch, Phys 0 Rev. J17, 1580 (1960), for experimental confirmation. 9 A. Fujii and H. Primakoff, Nuovo cimento 12, 327 (1959); L. Wolfenstein, Nuovo cimento 13, 319 (1959); M 0 Morita and A. Fujii, Phys. Rev. 118, 606 (1960); G. Flammand and K. W. Ford, Phys e Rev. 116, 1591 (1959).If we fix our attention on the ir + meson coming from the decay (b) and we vary its kinetic energy E x starting from its maximum value, we see that at first there can be no corresponding TT + from (a) with the same energy, the decay (a) being there energetically forbidden due to the heavier mass of the charged with respect to the neutral pions. By decreasing E± and therefore by increasing the relative energy E 2° of the (7T°, 7T°) pair, we get to the point E^ = 2 m* -2m° = 9.2 Mev where a (TT", TT + ) pair can be created at rest (in its center-of-mass system) in a (7T°, 7r°) collision. By increasing E 2 b over that value it is by now clear that the decay (a) will also be energetically possible so that£ 2 > or the corresponding E 19 acts essentially as the threshold energy for the reaction 7r° + 7r°-*7r~ + 7r+. At E x a threshold effect of the kind of a cusp or rounded step will be observed in the spectrum of the 7r + coming from decay (b) and the size of such an effect will depend critically on the lowenergy value of the pion-pion charge exchange cross section. Since at threshold the (ir" 9 ir + ) pair is in a relative S wave, from conservation of parity (here applicable) and total angular momentum, it follows that the threshold effect will appear in the relative S wave of the (ir°, TT°) pair. Further, due to the small relative energy of the {if 0 9 ff°) P ai r at the 7r~7r + threshold, the relative waves with l> 0 will be negligible, so that the 7T+ common to both decays will also be in the S wa...

Determination of the S-wave π π scattering lengths from a study of K ±→π ± π 0 π 0 decays

“…The existence of this threshold anomaly had been first predicted in 1961 by Budini and Fonda [5], as a result of the charge exchange scattering process π + π − → π 0 π 0 in K ± → π ± π + π − decay. These authors had also suggested that the study of this k Sezione dell'INFN di Perugia, 06100 Perugia, Italy.…”

“…3) can be interpreted [5,6] as a threshold effect from the decay K ± → π ± π + π − contributing to the K ± → π ± π 0 π 0 amplitude through the charge exchange reaction π + π − → π 0 π 0 . In the formulation by Cabibbo [6] the K ± → π ± π 0 π 0 decay amplitude is described as the sum of two terms:…”

“…More recently, Cabibbo [6] has proposed an interpretation of the cusp-like anomaly along the lines proposed by Budini and Fonda [5], but expressing the K ± → π ± π 0 π 0 decay amplitude in terms of the π + π − → π 0 π 0 amplitude at threshold, a x . In the limit of exact isospin symmetry a x can be written as (a 0 − a 2 )/3, where a 0 and a 2 are the Swave ππ scattering lengths in the isospin I = 0 and I = 2 states, respectively.…”

“…However, samples of K ± → π ± π 0 π 0 decay events available in those years were not sufficient to observe the effect, nor was the M 00 resolution. As a consequence, in the absence of any experimental verification, the article by Budini and Fonda [5] was forgotten.…”

Isospin violation, light quark masses, and all that

“…Refs. [23,24]) since it allows for a precise extraction of the pion-pion scattering lengths [25][26][27]. The cusp effect is driven by the rescattering from two neutral pions through an intermediate charged pion pair and should thus also be visible in other processes like η → 3π 0 (as discussed before) or η → ηπ 0 π 0 .…”