Consistência nos cálculos perturbativos das anomalias gravitacionais bidimensionais

Abstract

The present study has as its main goal calculating the gravitational anomalies (Einstein and Weyl) that emerge in a theory where Weyl fermions are coupled to a two-dimensional curved space-time. The eventual divergences and/or indeterminations, which are typical of perturbative calculations in quantum field theories, are treated with an alternative approach of universal nature, where regularizations play no relevant role. The perturbative amplitudes are manipulated in a way such that their mathematical content remains intact until the end of the calculations: the basic divergences are not integrated, but organized as standardized objects free from physical content, while the finite part is written in terms of a class of well-behaved functions and carries all the physical content of the amplitude. The Green function investigated, called by us gravitational amplitude, is calculated at 1-loop level and gives the fermionic correction to the graviton propagation. The gravitational amplitude is organized as a sum of sub-amplitudes that can be identified, eventually, with amplitudes that belong to usual gauge theories, e.g, the Schwinger electrodynamics. Besides the organizational character, this systematization allows us to identify the origin of the possible anomalous terms. Through the contraction of the sub-amplitudes with the external momentum and with the metric we can build a set of relations between the amplitudes, which we call relations among Green functions. These are used as consistency constraints that must be satisfied by the calculated sub-amplitudes. The preservation of the symmetry content associated with each sub-amplitude is tested through the verification of theWard identities. We show that it is possible to obtain the usual results to the gravitational anomalies, which are well known from the literature, and come from the the finite part of the amplitude. However, within this frame where the gravitational anomalies emerge, the relations among Green functions associated with the same contractions are violated too, showing a violation on the linearity of the integration operation.