The Hierarchy Problem and Beyond

Abstract

Using standard field theory tools, one renormalizes the SM Lagrangian in order to account for quantum (i.e., loop) corrections. Indeed, this procedure is successfully carried out in the case of the extrapolation of “the effective” SU(3)_C × U(1)_q theory to the weak scale and its embedding in the SU(3)_c × SU(2)_L × U(l)_Y SM, as was described by Altarelli [3]. (See also Fig. 6.1 below.) While most corrections are at most logarithmically divergent and hence correspond a well-understood logarithmic renormalization (and finite shifts) of parameters, there is an important exception. The model now contains the Higgs field whose lightness is not protected by either gauge or chiral symmetries. The leading corrections to the Higgs two-point function (and hence, to its mass parameter in (1.1)) depend quadratically on the ultraviolet cut-off scale. Adopting the view that the SM itself is only an effective low-energy theory, the renormalization procedure is again understood as extrapolating the model towards the more fundamental ultraviolet scale. The quadratic dependence on this scale, however, is forcing the theory to reside at the ultraviolet scale itself, hence leading to an apparent paradox with crucial implications. It amounts to a failure to consistently accommodate fundamental scalar fields within the framework of the SM, undermining the notion of an ultraviolet independent low-energy theory.