Mass renormalization in a toy model with spontaneously broken symmetry

TL;DR

This paper analyzes mass renormalization in a simplified model with spontaneous symmetry breaking, highlighting a scheme that clarifies the hierarchy problem and can be extended to more complex theories.

Contribution

It introduces a renormalization scheme that effectively handles tadpole contributions and clarifies the hierarchy problem in models with spontaneous symmetry breaking.

Findings

Tadpole contributions are canceled by the renormalization of the vacuum expectation value.

Mass renormalization reintroduces tadpole effects, essential for finiteness.

The scheme generalizes to models with multiple fermions and scalars.

Abstract

We discuss renormalization in a toy model with one fermion field and one real scalar field phi, featuring a spontaneously broken discrete symmetry which forbids a fermion mass term and a phi^3 term in the Lagrangian. We employ a renormalization scheme which uses the MSbar scheme for the Yukawa and quartic scalar couplings and renormalizes the vacuum expectation value of phi by requiring that the one-point function of the shifted field is zero. In this scheme, the tadpole contributions to the fermion and scalar selfenergies are canceled by choice of the renormalization parameter delta_v of the vacuum expectation value. However, delta_v and, therefore, the tadpole contributions reenter the scheme via the mass renormalization of the scalar, in which place they are indispensable for obtaining finiteness. We emphasize that the above renormalization scheme provides a clear formulation of the hierarchy problem and allows a straightforward generalization to an arbitrary number of fermion and scalar fields.