Abstract
We study the renormalization group flow in weak power counting (WPC) renormalizable theories. The latter are theories which, after being formulated in terms of certain variables, display only a finite number of independent divergent amplitudes order by order in the loop expansion. Using as a toolbox the well-known SU(2) non linear sigma model, we prove that for such theories a renormalization group equation holds that does not violate the WPC condition: that is, the sliding of the scale $\mu$ for physical amplitudes can be reabsorbed by a suitable set of finite counterterms arising at the loop order prescribed by the WPC itself. We explore in some detail the consequences of this result; in particular, we prove that it holds in the framework of a recently introduced beyond the Standard Model scenario in which one considers non-linear St\"uckelberg-like symmetry breaking contributions to the fermion and gauge boson mass generation mechanism.
Highlights
(UV) divergences by a redefinition of the tree-level parameters
Using as a toolbox the well-known SU(2) non-linear sigma model, we prove that for such theories a renormalization group equation holds that does not violate the weak power-counting (WPC) condition; that is, the sliding of the scale μ for physical amplitudes can be reabsorbed by a suitable set of finite counterterms arising at the loop order prescribed by the WPC itself
Imagine that a change in the scale μ of the radiative corrections is inequivalent to a rescaling of the coefficients of the ancestor amplitudes counterterms, at the given loop order prescribed by the WPC: this would imply that the renormalization group (RG) flow mixes up the hierarchy of UV divergences encoded in the WPC, making it impossible to slide the scale μ between different energies
Summary
Within the aforementioned Weinberg–Gomis approach to renormalization, one can interpret the WPC as a condition dictating at which loop order the coefficient of a particular monomial in the ancestor variables and their derivatives becomes non-vanishing While it has been proven in [6] that the WPC allows for the definition of a symmetric (i.e. compatible with the symmetries of the theory) subtraction scheme, the question has remained open of whether a renormalization group (RG) equation compatible with the WPC exists. As we will prove in this paper, this is not the case, as the RG equation of ancestor amplitudes turns out to be compatible with the WPC This is an important result which acquires particular relevance in the context of the nonlinearly realized electroweak theory introduced in [7,8], in which the WPC has been used as a model-building principle.
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