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Register a new userFundamental Scale Invariance
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C. Wetterich
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We propose fundamental scale invariance as a new theoretical principle beyond renormalizability. Quantum field theories with fundamental scale invariance admit a scale-free formulation of the functional integral and effective action in terms of scale invariant fields. They correspond to exact scaling solutions of functional renormalization flow equations. Such theories are highly predictive since all relevant parameters for deviations from the exact scaling solution vanish. Realistic particle physics and quantum gravity are compatible with this setting. The non-linear restrictions for scaling solutions can explain properties as an asymptotically vanishing cosmological constant or dynamical dark energy that would seem to need fine tuning of parameters from a perturbative viewpoint. As an example we discuss a pregeometry based on a diffeomorphism invariant Yang-Mills theory. It is a candidate for an ultraviolet completion of quantum gravity with a well behaved graviton propagator at short distances.
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Weyl invariant theories of scalars and gravity can generate all mass scales spontaneously, initiated by a dynamical process of inertial spontaneous symmetry breaking that does not involve a potential. This is dictated by the structure of the Weyl current, $K_mu$, and a cosmological phase during which the universe expands and the Einstein-Hilbert effective action is formed. Maintaining exact Weyl invariance in the renormalised quantum theory is straightforward when renormalisation conditions are referred back to the VEVs of fields in the action of the theory, which implies a conserved Weyl current. We do not require scale invariant regulators. We illustrate the computation of a Weyl invariant Coleman-Weinberg potential.
We show how Einstein-Cartan gravity can accommodate both global scale and local scale (Weyl) invariance. To this end, we construct a wide class of models with nonpropagaing torsion and a nonminimally coupled scalar field. In phenomenological applications the scalar field is associated with the Higgs boson. For global scale invariance, an additional field --- dilaton --- is needed to make the theory phenomenologically viable. In the case of the Weyl symmetry, the dilaton is spurious and the theory reduces to a sub-class of one-field models. In both scenarios of scale invariance, we derive an equivalent metric theory and discuss possible implications for phenomenology.
Scale invariance in the theory of classical mechanics can be induced from the scale invariance of background fields. In this paper we consider the relation between the scale invariance and the constants of particle motion in a self-similar spacetime, only in which the symmetry is well-defined and is generated by a homothetic vector. Relaxing the usual conservation condition by the Hamiltonian constraint in a particle system, we obtain a conservation law holding only on the constraint surface in the phase space. By the conservation law, we characterize constants of motion associated with the scale invariance not only for massless particles but for massive particles and classify the condition for the existence of the constants of motion. Furthermore, we find the explicit form of the constants of motion by solving the conservation equations.
We present a detailed analysis of the construction of $z=2$ and $z eq2$ scale invariant Hov{r}ava-Lifshitz gravity. The construction procedure is based on the realization of Hov{r}ava-Lifshitz gravity as the dynamical Newton-Cartan geometry as well as a non-relativistic tensor calculus in the presence of the scale symmetry. An important consequence of this method is that it provides us the necessary mechanism to distinguish the local scale invariance from the local Schrodinger invariance. Based on this result we discuss the $z=2$ scale invariant Hov{r}ava-Lifshitz gravity and the symmetry enhancement to the full Schrodinger group.
We show that if the $alpha$-attractor model is realized by the spontaneous breaking of the scale symmetry, then the stability and the dynamics of the vector field that gauges the scale symmetry can severely constrain the $alpha$-parameter as $5/6 < alpha < 1$ restricting the inflationary predictions in a very tiny region in the $n_s - r$ plane that are in great agreement with the latest Planck data. Although the different values of $alpha$ do not make a tangible difference for $n_s$ and $r$, they provide radically different scenarios for the post-inflationary dynamics which determines the standard BBN processes and the large scale isotropy of the universe.
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