K2→π++π−and the Question of Bose Statistics for Pions

$K_2$ decay into ${\pi }^{+}+{\pi }^{-}$ is possible in a $\mathrm{CP}$-invariant way if the ${\pi }^{+}{\pi }^{-}$ wave function contains a term antisymmetric in ${\pi }^{+}{\pi }^{-}$ exchange. This possibility requires a small violation of Bose symmetry for identical particles (extended to a particle-antiparticle pair) which, because the evidence for pion Bose statistics has consisted in the absence until recently of $K_2\to 2\pi$, is consistent with present experimental data. Irrespective of any question of Bose statistics, if $\mathrm{PC}$ is conserved, $K_2\to {\pi }^0+{\pi }^0$ is forbidden. The absence or presence of this latter decay will thus discriminate between the theory discussed and theories in which $\mathrm{PC}$ is violated for the $K-\pi$ system either in the dynamics or because of environmental effects. We consider the contribution of an antisymmetric ${\pi }^{+}{\pi }^0$ state to $K_2{\pi }^{+}$ decay, assuming the same statistical mixing $\mathcal{E}$ for the ${\pi }^{+}{\pi }^0$ final state in $K^{+}$ decay as for the ${\pi }^{+}{\pi }^{-}$ state in $K^0$ decay. If $\Delta T=\frac{1}{2}$ amplitudes dominate, this antisymmetric ${\pi }^{+}{\pi }^0$ state contributes negligibly to the ${K^{+}}_{2\pi }$ decay rate, which is therefore still large compared with what might be expected from electromagnetic corrections to an otherwise exact $\Delta T=\frac{1}{2}$ rule. If $\Delta T=\frac{1}{2}$ amplitudes dominate, then $\mathcal{E}=1.5\mathrm{}\times {10}^{-3\mathrm{}}$, but if $\Delta T=\frac{1}{2}, \frac{3}{2}, \mathrm{and} \frac{5}{2}$ amplitudes are comparable, then a value as large as $\mathcal{E}=0.07\mathrm{}$ is possible. The paper contains a short critical review of what is called for in order to test pion Bose statistics in the light of the recent experiment of Christensen et al., and remarks on the theoretical consequences of impure Bose statistics.