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ATLAS and CMS combined their results recently and it shows that higgs mass is ~125 GeV.

Theoretical physicists try to figure out the "low mass" of the Higgs which causes hierarchy problem at high energy scales. They support that we need to have a model beyond the SM to explain the issue.

I understand the mass correction of higgs getting bigger and bigger when you are getting close to high energy scales. What I do not understand is observables such as amplitude, cross section and physical mass not be effected of any cut-off scale. So, what makes us to think any model beyond SM? Why do we need it?

EDIT: I know Higgs mass has a quadratic diverge contribution and all articles I read so far say that quadratic divergence is much worse than the logarithmic divergence and it should be fine-tuned. I am looking more of a physical and intuitive explanation.

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