RG-improvement of the effective action with multiple mass scales

Improving the effective action by the renormalization group (RG) with several mass scales is an important problem in quantum field theories. A method based on the decoupling theorem was proposed in https://arxiv.org/abs/hep-ph/9210229 and systematically improved https://arxiv.org/abs/hep-ph/9809275 to take threshold effects into account. In this paper, we apply the method to the Higgs-Yukawa model, including wave-function renormalizations, and to a model with two real scalar fields $(\phi ,h)$. In the Higgs-Yukawa model, even at one-loop level, Feynman diagrams contain propagators with different mass scales and decoupling scales must be chosen appropriately to absorb threshold corrections. On the other hand, in the two-scalar model, the mass matrix of the scalar fields is a function of their field values $(\phi ,h)$ and the resultant running couplings obey different RGEs on a different point of the field space. By solving the RGEs, we can obtain the RG improved effective action in the whole region of the scalar fields.