Abstract
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 [1] and systematically improved [2] 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 (φ, 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 (φ, 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.
Highlights
JHEP03(2018)165 where V (0)({φa}) is the tree level potential, si and ni represent the spin and the number of degrees of freedom respectively
The wave function renormalization is given by a diagram containing both of the scalar and the fermion fields in the loop, and their mass scales are generically different
We studied renormalization group (RG) improvement of the effective action of a Higgs-Yukawa model and a two real scalar model based on the decoupling approach [2]
Summary
We review the decoupling method of the RG improved effective potential proposed by Casas, Clemente and Quiros [2]3 with a slight generalization to include wave function renormalizations. In this paper we generalize to consider an effective action in order to study the wave function renormalization as well as renormalization of coupling constants. One may choose the heaviest mass Mi({φa}) in the loop diagram as the renormalization scale μ, but such a choice cannot absorb the threshold corrections in the effective coupling constants unless we expand loop integrals with respect to 1/Mi. For example, if two particles with masses, M and m, exchange in a loop diagram, we will have a Feynman parameter integral such as dz log[(zM 2 + (1 − z)m2)/μ2]. If we expanded it with respect to m2/M 2, we would have a simple logarithmic factor log(M 2/μ2) with a single mass scale of the heavy field It would generate diverging Feynman parameter integrals,. The integrations over the Feynman parameter z in G and Gcan be explicitly performed, and we can translate θG and θGinto step functions with the decoupling scales, μG(φ) and μG(φ);
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