One part of Distance conjecture states that free theory (Higher spin) are at infinite distance away from any arbitrary point on conformal manifold where the distance is measured with respect to Zamolodchikov metric. For example in Yang Mills theory the marginal operator is the lagrangian itself. So if I do a naive distance estimate like following: $$\mathcal{L} = \frac{1}{g^2}F^2+\frac{dg}{g^3}F^2$$ then the metric is $\langle\frac{dg}{g^3}F^2\frac{dg}{g^3}F^2\rangle$ with the kinematic factors stripped off. Now to first order the metric goes like $\frac{(dg)^2}{g^6}\langle F^2F^2\rangle \approx \frac{(dg)^2}{g^6} g^4$ since $F \sim (\partial A)^2$, each propagator gives $g^2$ and we have two contraction. Therefore $g=0$ is an infinite distance away point as distance goes like $\log g$.
Now doing same bookkeeping for $\phi^4$ theory tells me free theory is at finite distance $$\mathcal{L} = (\partial \phi)^2+\lambda\phi^4$$ The metric turns will be $\langle d\lambda \phi^4d\lambda \phi^4\rangle \sim d\lambda^2$ so $\lambda=0$ is not infinite distance away point.
I can't figure what stupidity I'm doing in my argument.
