Symmetry Protection and the Hierarchy Problem: From Lattice Models to Goofy

Abstract

The hierarchy problem asks why the Higgs mass remains so far below the highenergy scales at which new physics is expected, despite quantum corrections that tend to drive it upward. It can be reformulated as a question of symmetry because a parameter is stable under radiative corrections when setting it to zero enlarges the symmetry of the theory. We examine this principle of symmetry protection in two complementary settings. First, in a two-dimensional Ising model, extending an existing inverse-Ising blocking procedure to Z2-odd operators, we show numerically that such operators are generated under coarse-graining only when the symmetry is explicitly broken. We then revisit one-loop mass corrections in quantum field theory, contrasting the fermion self-energy, whose mass correction vanishes in the chiral limit, with the scalar self-energy, which has no analogous protection. This identifies the hierarchy problem as the absence of a symmetry forbidding a generic scalar mass operator. Finally, we review the goofy symmetry, a transformation acting non-trivially on kinetic terms that can forbid the Higgs mass while remaining compatible with the Standard Model.