Renormalization, Decoupling and the Hierarchy Problem

The hierarchy problem is associated with renormalization and decoupling. We can account for the smallness of the scalar mass against loop corrections and its insensitivity to ultraviolet physics through the decoupling of heavy fields. It is essential to correctly identify the observable physical mass as the renormalized one that depends on the external momentum, as opposed to the constant mass. We reconsider the properties of the renormalized loop corrections, which are finite, independent of regularization and admit a well-defined perturbation. By explicit calculation, we show that any loop corrections to the scalar mass-squared are suppressed as $(p^2-m^2)^2/M^2$, where $p,m$ and $M$ are the external momentum, the scalar pole mass and the heavy field mass in the loop, respectively. This is in accordance with the Appelquist-Carazzone decoupling theorem, which we have explicitized and completed for the case of the scalar mass.