Hierarchy problem and dimension-six effective operators

Without any mechanism to protect its mass, the self-energy of the Higgs boson diverges quadratically, leading to the hierarchy or fine-tuning problem. One bottom-up solution is to postulate some yet-to-be-discovered symmetry which forces the sum of the quadratic divergences to be zero, or almost negligible; this is known as the Veltman condition. Even if one assumes the existence of some new physics at a high scale, the fine-tuning problem is not eradicated, although it is softer than what it would have been with a Planck scale momentum cut-off. We study such divergences in an effective theory framework, and construct the Veltman condition with dimension-six operators. We show that there are two classes of diagrams, the one-loop and the two-loop ones, that contribute to quadratic divergences, but the contribution of the latter is suppressed by a loop factor of $1/16\pi^2$. There are only six dimension-six operators that contribute to the one-loop category, and the Wilson coefficients of these operators play an important role towards softening the fine-tuning problem. We find the parameter space for the Wilson coefficients that satisfies the extended Veltman condition, and also discuss why one need not bother about the $d>6$ operators. The parameter space is consistent with the theoretical and experimental bounds of the Wilson coefficients, and should act as a guide to the model builders.