I=0ππSWave and Broken Scale Invariance

Renner and Staunton have shown that an $I=0\mathrm{}$ $\pi \pi$ $S$ wave with only an $\epsilon$ resonance cannot satisfy the low-energy theorems of broken scale invariance. Recent experimental results indicate that this partial-wave amplitude also has a strongly inelastic $S^{*}\mathrm{}\left(980\right)\mathrm{}$ resonance. We show that if we also include contributions from this resonance in the calculation of Renner and Staunton the low-energy theorems can be satisfied and suggest that the dimension of the operator which breaks both scale and chiral invariance is one. Although this results is not completely conclusive, because of the approximations used, this calculation does illustrate once again that the breaking of scale invariance cannot be ascribed to the $\epsilon$ resonance alone.