Soft Breaking Naturalness in Majorana Fermion Loop induced Scalar and Majoron Amplitudes

Majorana fermion loops contribute a shift-symmetry violating effective potential to the coupled scalar field. When the Majorana fermion is light, a larger number of symmetry violating insertions let the loop contribution to grow increasingly infrared sensitive to the Majorana mass. With explicit calculations, we calculate the symmetry violating amplitudes, show that the scalar potential’s natural scales serve as nonzero kinematic matching points which remove the superficial sensitivity to the symmetry-violating Majorana mass, continuously restoring the shift symmetry in agreement with 't Hooft naturalness. In the spontaneous broken phase, the scalar radial vacuum expectation value readily ensures the infrared safety of symmetry violating Majorana fermion loops, and all $n$-point Majoron functions develop a leading linear dependence on the symmetry violating parameter in the soft breaking limit, which matches the expansion of the Coleman-Weinberg determinant.