Quantum Kaluza–klein Compactifiation

Kaluza–Klein compactification in quantum field theory is analysed from the perturbation theory viewpoint. Renormalisation group analysis for compactification size dependence of the coupling constant is proposed. Introduction Development of the string theory [1] increased interest to field theoretical and quantum mechanical models in high dimensions (D ≥ 5). The consistency of string theory need it to be formulated in either D = 10 for supersymmetric strings or D = 26 for bosonic ones, while “everyday” physics is four-dimensional. From the other hand most interesting field theoretical models can be consistently formulated at dimensions not exceeding four. Compactification of extra dimensions existent in string theory to have four-dimensional physics at the low energies conciliates these facts. It would be interesting to explain this mechanism dynamically, at least in some approximation at low energies. Much optimism is inspired by the progress in understanding non-perturbative strings and and in special ADS/CFT correspondence [2, 3, 4]. In the present work, however, we address a different question: what is the effect of the compactification on the level of quantum field theory? It is known that certain quantum field theories/gravities can serve as low energy effective actions for string theories. Since the string description must not enter in contradiction with low energy field theoretical one, one can limit oneself to study of the last. As an important example can serve ten-dimensional IIA string model whose low energy field theory is IIA supergravity. As it is believed string theory for large couplings (gstring) results in an eleven-dimensional model (M-theory). Its low energy effective field theory model is D = 11 supergravity. Therefore, stringy corrections for large gstring in IIA supergravity must led to the eleven-dimensional theory (see [5] for a recent review). It is, however, easer to see instead the transition from higher dimensional model to lower one. In classical physics, compactification of a (D + p)-dimensional model to D dimensions is given by “confining” some p spacial dimensions of original space-time to form a compact manifold. From the D-dimensional point of view the spectrum of the compactified model consists of a light field which corresponds to constant or zero mode on the compact directions and an infinite number of massive fields corresponding to non-constant in the compact direction modes or Kaluza–Klein (KK) modes. Masses of KK-modes This work was completed with the support of RBRF grant 96-01-0551, Scientific School Support grant 96-15-96208, and INTAS grant 96-370. 1 2 CORNELIU SOCHICHIU are proportional to inverse compactification size M = R−1 (i.e. the typical size of the compact dimensions). In the limit of strong compactification KK-modes acquire large masses and do not propagate, and decouple at the classical level. In quantum description, however, “virtual” KK-particles can contribute even for energies less than their masses, becoming significant as they are approached. In the limit of zero compactification size one expects to have all the KK-modes decoupled also in quantum theory since their masses become infinite. In fact, the fields not only do not simply decouple in this limit but they may also produce additional divergences. Also, extra divergences appear even for finite compactification sizes due to infinite number of fields. These divergences are natural reflection of the fact that usually similar models in higher dimensions are more divergent. As one can see these divergences can be eliminated in the framework of the standard renormalisation procedure and one is left with renormalised physical parameters which depend on the size of the compact space. In actual paper we are going to consider a simple (toy) model to illustrate above ideas. We will consider D+1-dimensional φ-model compactified (on a circle) to a D-dimensions. The structure of the paper is as follows. In the next section we consider compactified D + 1-dimensional φ model, and in the second one its D-dimensional one-loop effective action. We also analyse the renormalisation group dependence of the effective D-dimensional coupling on the compactification mass M . In the Appendix we give some properties of ζ and Γ-functions used in the body of the paper and describe the computation of the effective action. 1. Compactified φ model Let us consider D+ 1 dimensional φ model described by the following classical action