Chirally Constraining the $\pi \pi$ Interaction in Nuclear Matter

A general prescription for the construction of $\pi \pi$ interaction potentials which preserve scattering length constraints from chiral symmetry when iterated in scattering equations is derived. The prescription involves only minor modifications of typical meson-exchange models, so that coupling constants and cut-off masses in the models are not greatly affected. Calculations of $s$-wave $\pi \pi$ scattering amplitudes in nuclear matter for two models are compared with those for similar models which violate the chiral constraint. While the prescription tends to suppress the accumulation of the near sub-threshold strength of the $\pi \pi$ interaction, an earlier conjecture that amplitudes which satisfy chiral constraints will not exhibit an instability towards $\pi \pi~s-$wave pair condensation appears to be incorrect. At the same time, however, conventional $\pi \pi$ interaction models which fit scattering data well can readily be adjusted to avoid the instability in nuclear matter without recourse to exotic mechanisms.