Sign of the induced gravitational constant

We study the restrictions that analyticity and positivity impose on $\psi \left(q^2\right)$, the two-point function constructed with the trace of the energy-momentum tensor of an asymptotically free gauge theory with dynamical symmetry breaking. The Adler-Zee formula for the induced gravitational constant is ${\left(16\mathrm{}\pi G_{\mathrm{ind}}\right)}^{-1\mathrm{}}=\left(\frac{1}{12}\right){\psi }^{\prime }\mathrm{}\left(0\right)\mathrm{}$. We place particular emphasis on the question of the sign of $G_{\mathrm{ind}}$. This sign is related to the distribution and type of zeros of $\psi \left(q^2\right)$. We prove that for the case of asymptotic freedom $\psi$ has at most two zeros. We investigate all possible cases and show that the $G_{\mathrm{ind}}$ is potentially positive in three but each has a different sign of $\psi \mathrm{}\left(0\right)\mathrm{}$, one $\psi \mathrm{}\left(0\right)>0\mathrm{}$, one $\psi \mathrm{}\left(0\right)<0\mathrm{}$, and one $\psi \mathrm{}\left(0\right)=0\mathrm{}$. Thus in the end only one case can survive. We derive convergent sum rules for ${\left(G_{\mathrm{ind}}\right)}^{-1\mathrm{}}$ in each case and determine the sign of $\psi \mathrm{}\left(0\right)\mathrm{}$, and the sign of $\psi (-Q^2)$ as $Q^2\to \infty$. This last sign gives us a check on whether the renormalization procedure is consistent with positivity.