Unitarity and radiative decay processes

Abstract

By means of the unitarity condition, we study the contribution of radiative ππ scattering to the matrix elements for radiative-decay processes such as K→2πγ. In particular, the final-state interaction, phases associated with direct emission (as distinguished from bremsstrahlung) amplitudes in radiative decays depend on the importance of the corresponding direct emission amplitudes in radiative scattering. We consider separately the cases ofT-invariance andT-violation, assumingCPT invariance throughout. In the limit ofT-invariance it is found that the phase of a direct emission amplitude corresponding to a final-state di-pion system of angular momentum ι is just the appropriate ππ scattering phase shift δl in the following cases: 1) The corresponding amplitude is forbidden in radiative scattering (e.g. magnetic emission for all ι in spin-zero boson decays). 2) The nonradiative decay is forbidden (e.g. η-decay and K02 decay). For K± decay the nonradiative decay is suppressed relative to direct emission by the |ΔI|=1/2 rule; hence the approximation of a phase δ1 for electric as well as magnetic amplitudes may be reasonable. For the case ofT-violation, the partial-wave amplitudesD1(Open image in new window) for direct emission in radiative K (\(\left( {\bar K} \right)\)) decay into a final di-pion state of angular momentum ι are related byOpen image in new window=±D*1 exp [2iδ1], if condition 1) or 2) above holds. We also consider the case in which these conditions do not hold. It is found thatCP-violating effects such as a difference in rates and spectra for radiative K+ and K− decays, or an interference between KL and Ks amplitudes remain nonvanishing in the latter case.