Quantum scale-invariant models as effective field theories

We address the question of whether the quantum scale-invariant theories introduced in [1] are renormalizable or play the role of effective field theories that are valid below the Planck scale MP . We show that starting from two-loop level the renormalization procedure requires introduction of counter-terms with structures different from those in the initial Lagrangian, making these theories non-renormalizable and therefore non-predictive above MP . Despite non-renormalizability, the attractive features of these theories, associated with the stability of the Higgs mass agains radiative corrections and the smallness of the cosmological constant, remain intact. Ref. [1] introduced a class of models with scale-invariance at the quantum level. Quantum scale invariance (QSI) would forbid all mass parameters in the Lagrangian, including the Higgs mass and cosmological constant, while the spontaneous breaking of QSI introduces all mass scales, including the gravitational constant. Some intriguing phenomenological consequences of these theories were discussed in [2]. They are related to the stability of the Higgs mass agains radiative corrections, to the cosmological inflation, and existence of dark energy. The key point of phenomenological considerations of [2, 1] is related to the existence of exact symmetry and its spontaneous breakdown. Though the consequences of spontaneously broken QSI theories remain valid independently of the character of these models (renormalizable theories versus effective field theories, valid up to the Planck scale MP ∼ 10 19 GeV), it is interesting to clarify whether these models are renormalizable or can only play the role of effective field theories. A simple toy scalar field model considered in [1] is given by the action S = ∫