# How is the hierarchy problem consistent with the decoupling theorem?

One the one hand we have the hierarchy problem in it's various forms, in my understanding in it's most serious form one could state it as the observation that if there is a heavy mass scale M in addition to the weak scale m, then when we run the higgs mass (say) mh in the wilsonian renormalization group from some large scale Lambda down to the physical weak scale we will generically find that mh is of order M unless the initial conditions of the renormalization group were very finely tuned such that mh(Lambda) and M cancel very precisely.

On the other hand we have the decoupling theorem, which also can be stated in various ways but most practically for here we can say that the beta function computed at low energies should be independent of high energy scales.

Naively, I would think that you could only have one or the other of these statements. On the one hand the hierarchy problem tells us the Higgs mass computed at low energies is sensitive to high energy physics, on the other hand the decoupling theorem tells us that no low energy observable should be sensitive to high energy physics.

How do we reconcile these? Does the decoupling theorem not apply to the higgs?

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