Assuming the hadronic energy density has the decomposition ${\Theta }_{00}={\overline{\Theta }}_{00}+\delta +u$ (where ${\overline{\Theta }}_{00}$ is chiral- and scale-invariant, $\delta$ is chiral-invariant but violates scale invariance, and $u$ violates both chiral and scale invariance), we find rigorous bounds on the dimensions of $\delta$ and $u$ in the (actual) SU(2)-symmetric limit and in the SU(3)-symmetric limit. If we further assume that $\delta$ is a $c$ number, we derive a new sum rule which leads to satisfactory bounds on the dimension of $u$ in the SU(2)-symmetric limit but not in the SU(3)-symmetric limit. Saturation of this sum rule with a scalar meson having a mass of 700 MeV leads to a value of 1.9 for the dimension of $u$.

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