If I'm doing perturbation theory and it's telling me that I have a correction as big as the largest scale in my problem (cut-off scale), it means I cannot trust the answer. It does not meant the answer is $m_\phi^2 \propto \Lambda^2$.
**The renormalized mass could still be far beyond $\Lambda$**, but the current approach cannot see that.

I disagree with this about a small point, but for now let's assume it is absolutely correct. Then you still have a scalar field that you would like to be massless but your calculation says its mass is of the order of $\Lambda$ or higher. This means that the hierarchy problem is still there and we are only arguing about a detail on how it is formulated.

Now the small point: It is actually very useful to know how the mass scales with the cutoff and there is a lot of information in knowing that $m_\phi^2 \propto \Lambda^2$ as opposed to for example $m_\phi^2 \propto \log\frac{\Lambda^2}{\mu^2}$ or anything else.

The way to think about it is this:
Imagine another "fictitious" cutoff $\Lambda_f$ with $\Lambda_f\ll\Lambda$. Then your previous calculation will give $m_\phi^2 \propto \Lambda_f^2$, but now you are in a region where you can trust perturbation theory! Your calculation says that if you use to different fictitious cutoffs with $\Lambda_{f1}=2\Lambda_{f2}$ then the mass correction for the second theory will be 4 times bigger than the mass correction for the first theory.

Hope this helps!

This post imported from StackExchange Physics at 2014-05-04 11:40 (UCT), posted by SE-user Heterotic