Further Constraints on theπ0π0s-Wave Amplitude

On the basis of positivity and crossing symmetry, we derive further constraints on the ${\pi }^0{\pi }^0$ $s$ wave. One of these relates the physical region to the region below threshold, where crossing is easy to apply. It is sufficiently strong that none of the models of unitarized $\pi \pi$ partial waves which we have examined satisfies it. We also derive sufficient conditions which indicate that our results cannot be improved very much. Finally, we show how the knowledge of the $s$ wave below threshold can give us useful information about the physical partial waves.