Compactified Quantum Fields. Is There Life beyond the Cut-off Scale?

A consistent definition of high dimensional compactified quantum field theory without breaking the Kaluza–Klein tower is proposed. Possible consequences are discussed for the scalar model. Introduction High dimensional field theoretical models are interesting from two main points of view. First one is the possibility to construct Unified Theories from high dimensional gravities at classical field level, using the Kaluza–Klein mechanism (see [6] for a review). Another one is their connection to string models. It is known, that certain super Yang–Mills and supergravity models are low energy effective models for superstrings, (see e.g. [4]). The high dimensional field theories are non-renormalisable if treated quantically. Compactification reduces the formal dimensionality of the model. The exact compactified model, however, contains in its spectrum not only the low dimensional propagating field(s) but also an infinite number of Kaluza–Klein (KK) massive modes. The natural strategy would be integrating out this modes in order to get the low dimensional effective action for remaining propagating fields. Loop contribution of each KK field taken separately is renormalisable and vanishes due to large mass, but when summing over all the KK tower it diverges. The possible one loop counter-terms appearing after integration KK modes were computed many years ago [8], and are known to lead to non-renormalisability of the effective action. These days phenomenological aspects of the compactified high dimensional models are intensively studied in the framework of GUT building, although the nature of compactified models as QFT in these investigations is not yet clear, [2, 1]. The aim of the present work is finding compactification geometry dependence of the low dimensional effective model. In a previous work [7], we considered compactification of scalar field to a circle. To be fully consequent one has to compare contribution from KK modes with one of high energy modes of low dimensional fields. In the case of S compactification one can see that high energy modes are “stronger” than KK ones (see the note at the end of the next section). Vanishing of non-renormalisable contributions in this case (as well as in any compactifications with an odd number of compact dimensions) seems to be an artefact of computational scheme. Indeed, e.g. divergencies in dimensional regularisation scheme arise as poles of Gamma-function, which is Work supported by RFBR grant # 99-01-00190, INTAS grant # 95-0681, Scientific School support grant # 96-15-06208 and HLP 99-10. 1 2 CORNELIU SOCHICHIU known to produce no poles for negative half-integer values, in spite of divergence of defining integral. Below a more careful definition of QFT compactification will be given. We will show that there exists a possibility of consistent description of a high dimensional compactified QFT model. Namely, one can define such a model when the compactification size is of the order of cut off specific to given low dimensional theory. The plan of the paper is as follows. First we review the possible terms arising in computation of the low dimensional effective action. After that we find renormalisability conditions for KK contribution, and analyse dependence of the low dimensional model on the compactification geometry. 1. Compactified Model Renormalisation Consider D+ p-dimensional model of a field φ described by the classical action,