On the quantum field theory of the gravitational interactions
Received: April
interactions
Damiano Anselmi 0 1 2
Open Access 0 1 2
c The Authors. 0 1 2
0 INFN , Sezione di Pisa
1 Largo B. Pontecorvo 3 , 56127 Pisa , Italy
2 Dipartimento di Fisica “Enrico Fermi”, Universita` di Pisa
We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the S matrix is unitary when the cosmological constant vanishes. The model is the simplest of its class. However, infinitely many similar options are allowed, which raises the issue of uniqueness. To deal with this problem, we propose a new quantization prescription, by doubling the unphysical poles of the higher-derivative propagators and turning them into Lee-Wick poles. The Lagrangian of the simplest theory of quantum gravity based on this idea is the linear combination of R, Rμν Rμν , R2 and the cosmological term. Only the graviton propagates in the cutting equations and, when the cosmological constant vanishes, the S matrix is unitary. The theory satisfies the locality of counterterms and is renormalizable by power counting. It is unique in the sense that it is the only one with a dimensionless gauge coupling.
Models of Quantum Gravity; Beyond Standard Model; Renormalization Reg-
1 Introduction 2 3 4
Superrenormalizable quantum gravity
Coupling to matter
The problem of uniqueness
Fake degrees of freedom
Quantum gravity with a dimensionless gauge coupling
The problem of quantum gravity is the compatibility between renormalizability and
unitarity. It is well known that the Hilbert-Einstein action is not renormalizable by power
counting [1–4]. However, if we include the infinitely many counterterms it generates,
multiplied by independent couplings, it is perturbatively unitary [5]. An option to improve
the ultraviolet behavior of the loop integrals is to add quadratic terms with higher
derivatives. It is then possible to build higher-derivative theories of quantum gravity that are
renormalizable with finitely many couplings [6–9]. However, such theories are not unitary,
at least if they are formulated in the usual ways.
Higher-derivative theories must be formulated properly, because they are less trivial
than one would naively expect. For example, if they are defined directly in Minkowski
spacetime, i.e. by integrating the loop energies along the real axis of the complex energy
plane, they generate nonlocal, non-Hermitian divergences when the free propagators have
complex poles [10], which makes them unacceptable from the mathematical point of view.
On the other hand, the Wick rotation from Euclidean space is obstructed when the free
propagators have poles in the first or third quadrants of the complex energy plane. The
obstruction can actually be overcome by a nonanalytic procedure, which leads to a new
formulation [11] of an interesting subclass of higher-derivative theories, the Lee-Wick (LW)
models [12, 13].
Viewed as nonanalytically Wick rotated Euclidean theories, such models are
perturbatively unitary [14]. Moreover, the new formulation is intrinsically equipped with all that
is needed to define the physical amplitudes properly, with no need of ad hoc prescriptions.
The complex energy hyperplane is divided into disjoint regions Ai of analyticity, which can
be connected to one another by a well defined, but nonanalytic procedure. It is necessary
to work in suitable subsets Oi of the regions Ai, in a generic Lorentz frame, and
analytically continue the results from Oi to Ai at the end. Finally, the nonanalytic behaviors of
the physical amplitudes suggest ways that may facilitate the experimental measurements
of the key parameters of the models.
Old formulations of the Lee-Wick models were based on ad hoc prescriptions, the
best known one being the CLOP prescription of ref. [15].1
Often, such approaches are
unambiguous in some loop diagrams, but ambiguous in others, and do not admit a clear
formulation at the Lagrangian level. In ref. [11] it has been shown that they may give
ambiguous results already at one loop.
In this paper, we investigate the main options for quantum gravity that are offered
by the nonanalytic Wick rotation of Euclidean higher-derivative theories, combined with
extra tools that we introduce anew. We begin with the superrenormalizable Lee-Wick
models, which are unitary when the cosmological constant vanishes. We investigate the
simplest representative of this class of models in detail and show that in various cases a
vanishing cosmological constant is consistent with the renormalization group, before and
after the coupling to matter. However, the theories with similar properties are infinitely
many, which raises the issue of uniqueness. A principle of maximum simplicity could be
used to single out the model studied here, but the principle itself would have t (...truncated)