When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to which point can we trust this cut-off as the true validity limit of our theory?
If we do not trust this regulator, there is no way of seeing that the corrections to the Higgs mass are quadratically divergent, right? So how can we know that, in a renormalizable theory, an elementary scalar should have his mass around the highest possible scale in the theory? Moreover, does it even make sense of talking about the cut-off of a theory when it is renormalizable?
Since the notion of a cut-off, the quadratic corrections to the Higgs mass and naturalness are the driving principle to almost every BSM particle physics model I'm pretty puzzled with this.
Any thoughts, hints or literature that could clarify these ideas?