Renormalizability of Alternative Theories of Gravity: Differences between Power Counting and Entropy Argument

It is well known that General Relativity cannot be considered under the standard of a perturbatively renormalizable quantum field theory, but asymptotic safety is taken into account as a possibility for the formulation of gravity as a non-perturbative renormalizable theory. Recently, the entropy argument has however stepped into the discussion claiming for a “no-go” to the asymptotic safety argument. In this paper, we present simple counter-examples, considering alternative theories of gravity, to the entropy argument as further indications, among others, on the possible flows in the assumptions on which the latter is based. We considered different theories, namely curvature-based extensions of General Relativity as f(R), f(G), extensions of teleparallel gravity as f(T), and Horava–Lifshitz gravity, working out the explicit spherically symmetric solutions in order to make a comparison between power counting and the entropy argument. Even in these cases, inconsistencies were found.

Vector fields, RG flows and emergent gauge symmetry

We consider the most general perturbatively renormalizable theory of vector fields in four dimensions with a global SU(N) symmetry and massless couplings. The Lagrangian contains 1 quadratic, 2 cubic and 4 quartic couplings. The RG flow among this set of 7 couplings is computed to 1-loop and a rich phase diagram is mapped out; in particular it is shown that a finite number of asymptotically free RG-flows exist corresponding to non-trivial fixed points for the ratios of the couplings. None of these are gauge theories, i.e. possess only global SU(N) invariance but not a local one. We also include the most general ghost couplings, still with global SU(N) invariance, and compute the RG flow to 1-loop for all 9 resulting couplings. Again asymptotically free RG flows exist with non-trivial fixed points for the ratios of couplings. It is shown that Yang-Mills theory emerges at a particular fixed point. The theories at the other fixed points are marginally relevant gauge symmetry violating perturbations of Yang-Mills theory. The large-N limit is also investigated in detail.

Introduction to renormalization theory and renormalization group (RG)

A straightforward construction of a local, relativistic quantum field theory (QFT) leads to ultraviolet (UV) divergences and a QFT has to be regularized by modifying its short-distance or large energy momentum structure (momentum regularization is often used in this work). Since such a modification is somewhat arbitrary, it is necessary to verify that the resulting large-scale predictions are, at least to a large extent, short-distance insensitive. Such a verification relies on the renormalization theory and the corresponding renormalization group (RG). In this chapter, the essential steps of a proof of the perturbative renormalizability of the scalar φ4 QFT in dimension 4 are described. All the basic difficulties of renormalization theory, based on power counting, are already present in this simple example. The elegant presentation of Callan is followed, which makes it possible to prove renormalizability and RG equations (in Callan–Symanzik's (CS) form) simultaneously. The background of the discussion is effective QFT and emergent renormalizable theory. The concept of fine tuning and the issue of triviality are emphasized.

A note on gauge anomaly cancellation in effective field theories

The conditions for the absence of gauge anomalies in effective field theories (EFT) are rivisited. General results from the cohomology of the BRST operator do not prevent potential anomalies arising from the non-renormalizable sector, when the gauge group is not semi-simple, like in the Standard Model EFT (SMEFT). By considering a simple explicit model that mimics the SMEFT properties, we compute the anomaly in the regularized theory, including a complete set of dimension six operators. We show that the dependence of the anomaly on the non-renormalizable part can be removed by adding a local counterterm to the theory. As a result the condition for gauge anomaly cancellation is completely controlled by the charge assignment of the fermion sector, as in the renormalizable theory.

Fermion spectrum and $$g-2$$ anomalies in a low scale 3-3-1 model

We propose a renormalizable theory based on the $$SU(3)_C\times SU(3)_L\times U(1)_X$$ S U ( 3 ) C × S U ( 3 ) L × U ( 1 ) X gauge symmetry, supplemented by the spontaneously broken $$U(1)_{L_g}$$ U ( 1 ) L g global lepton number symmetry and the $$S_3 \times Z_2 $$ S 3 × Z 2 discrete group, which successfully describes the observed SM fermion mass and mixing hierarchy. In our model the top and exotic quarks get tree level masses, whereas the bottom, charm and strange quarks as well as the tau and muon leptons obtain their masses from a tree level Universal seesaw mechanism thanks to their mixing with charged exotic vector like fermions. The masses for the first generation SM charged fermions are generated from a radiative seesaw mechanism at one loop level. The light active neutrino masses are produced from a loop level radiative seesaw mechanism. Our model successfully accommodates the experimental values for electron and muon anomalous magnetic dipole moments.

Solidity without inhomogeneity: perfectly homogeneous, weakly coupled, UV-complete solids

Solid-like behavior at low energies and long distances is usually associated with the spontaneous breaking of spatial translations at microscopic scales, as in the case of a lattice of atoms. We exhibit three quantum field theories that are renormalizable, Poincaré invariant, and weakly coupled, and that admit states that on the one hand are perfectly homogeneous down to arbitrarily short scales, and on the other hand have the same infrared dynamics as isotropic solids. All three examples presented here lead to the same peculiar solid at low energies, featuring very constrained interactions and transverse phonons that always propagate at the speed of light. In particular, they violate the well known $$ {c}_L^2>\frac{4}{3}{c}_T^2 $$ c L 2 > 4 3 c T 2 bound, thus showing that it is possible to have a healthy renormalizable theory that at low energies exhibits a negative bulk modulus (we discuss how the associated instabilities are absent in the presence of suitable boundary conditions). We do not know whether such peculiarities are unavoidable features of large scale solid-like behavior in the absence of short scale inhomogeneities, or whether they simply reflect the limits of our imagination.

Renormalizing gravity: A new insight into an old problem

It is well known that perturbative quantum gravity is nonrenormalizable. The metric or vierbein has generally been used as the variable to quantize in perturbative quantum gravity. In this paper, we show that one can use the spin connection instead, in which case it is possible to obtain a ghost-free renormalizable theory of quantum gravity. Furthermore in this approach, gravitational analogs of particle physics phenomena can be studied. In particular, we study the gravitational Higgs mechanism using spin connection as a gauge field, and show that this provides a mechanism for the effective reduction in the dimensionality of spacetime.

Perturbatively renormalizable quantum gravity

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. In particular, around the Gaussian fixed point, it supports a Hilbert space of renormalizable interactions involving arbitrarily high powers of the gravitational fluctuations. These interactions are characterized by being exponentially suppressed for large field amplitude, perturbative in Newton’s constant but nonperturbative in Planck’s constant. By taking a limit to the boundary of the Hilbert space, diffeomorphism invariance is recovered whilst retaining renormalizability. Thus the so-called conformal factor instability points the way to constructing a perturbatively renormalizable theory of quantum gravity.

Signs and stability in higher-derivative gravity

Perturbatively renormalizable higher-derivative gravity in four space–time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in Lorentzian flat space–time for no-tachyons, fixes the signs. Feynman [Formula: see text] prescription for these signs further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters make the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space–time dimensions. The theory has a transition point arising from renormalization group (RG) equations, where the coefficient of [Formula: see text] diverges without affecting the perturbative quantum field theory (QFT). Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations push the flow of parameters across the transition point. The flow beyond the transition point is analyzed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton’s constant [Formula: see text] becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a nonperturbative treatment of the theory beyond the transition point.