Effective Field Theory of pure Gravity and the Renormalization Group

The general structure of the renormalization group equations for the low energy effective field theory formulation of pure gravity is presented. The solution of these equations takes a particular simple form if the mass scale of the effective theory is much smaller than the Planck mass (a possibility compatible with the renormalization of the effective theory). A theory with just one free renormalized parameter is obtained when contributions suppressed by inverse powers of the Planck mass are neglected. E-mail [email protected] E-mail [email protected] The general principles of quantum mechanics and special relativity lead to identify any relativistic quantum theory at low energy as a quantum field theory [1]. This point of view of field theories as effective low energy theories has important consequences on the understanding of general problems in quantum field theory such as renormalization [2]. The finite number of interactions of a renormalizable field theory are just the dominant terms at low energy of an effective field theory with an infinite number of interactions (usually all the terms compatibles with some symmetry principles). On the other hand the present incompatibility of general relativity and quantum mechanics [3] can be translated into the need for restricting the field theoretical formulation of gravitation to energies below the Planck mass. In field theory this incompatibility appears through the non-renormalizability of the theory of a spin 2 massless interacting particle. But in the framework of effective field theories renormalizability is not a consistency requirement and reliable perturbative predictions after renormalization can be made [4]. Then one can consider the effective field theory of an interacting spin 2 massless particle as the low energy field theoretic formulation of pure gravity. This point of view has been recently advocated by J.F.Donoghue [5] who shows how some large distance quantum gravitational effects can be derived within this framework. At high enough energies new interactions and new degrees of freedom will be required in a new theory based on some unknown general principles [3]. Perturbative renormalization in effective field theories can be done along the same lines of the standard renormalization of renormalizable theories. The example of the effective field theory that reproduces the low energy effects of a renormalizable field theory with a very heavy field can be used to illustrate the main ideas [6]. A well defined effective field theory expansion in inverse powers of the heavy mass is obtained if an appropriate renormalization scheme is used in the effective theory (mass independent substraction scheme). The one loop [7] and two loop [8] calculations of ultraviolet divergences in Einstein theory of gravity using dimensional regularization can be understood as the perturbative determination of the counterterms invariant under general covariant transformations with four and six derivatives which are required in the renormalization of the effective low energy theory of pure gravity. The loop expansion corresponds to the expansion in derivatives and