Renormalization and running couplings in higher derivative theories

The renormalization group has a crucial role in modern physics, however some of its features have not been completely understood yet. While its perturbative realization in 2-derivative theories in d = 4 spacetime has been widely studied, other classes of theories can still hide some subtleties. Higher derivative theories, and in particular quadratic gravity, could furnish a UV completion to general relativity within the framework of quantum field theory. For this reason, a detailed study of the renormalization group of this class of theories is of great importance and is the main object of this thesis. Higher derivative theories suffer from the Ostrogradskij instability at the classical level, which translates into ghost particles in the spectrum at the quantum level, with related problems with unitarity and negative norm states. In recent years many solutions to this pathology have been suggested in order to obtain a well-defined quantum theory and we review some of them. Then, we study the nonperturbative renormalization group of a higher derivative shift-invariant scalar model. In the theory space, we find an interesting region where the renormalization group trajectories flow between the two free Gaussian fixed points corresponding respectively to the 2- and 4-derivative kinetic term. From the perturbative point of view, the fourth power of transferred momentum in the propagator reduces the degree of UV divergence of Feynman diagrams, but at the same time it introduces new off-shell IR divergences. We notice that not all renormalization prescriptions are sensible to this type of infrared effects, potentially leading to running couplings that do not resum all the large logarithms of momenta in scattering amplitudes. We define a “physical” prescription using a momentum subtraction renormalization scheme and we apply it to various higher derivative theories. In particular, we focus on the higher derivative scalar toy model already studied non- perturbatively and on some quantum field theories in curved spacetime. We find that shift invariance seems to protect the universality of one-loop beta functions from IR effects and that quadratic gravity, according to its physical running, has an asymptotically free sector without tachyonic particles, in contradiction with older results which predict asymptotic freedom only in the presence of a scalar tachyon. Finally, we observe that the same type of IR effects can also emerge in 2-derivative theories in d = 2 spacetime. For this reason, we study the renormalization group of the CP (1) non-linear sigma model (NLSM). We observe that in this case the symmetries of the theory seem to protect the running of couplings from IR effects, preserving one-loop universality.