Coupling running through the Looking-Glass of dimensional Reduction

Abstract

The dimensional reduction, in a form of transition from four to two dimensions, was used in the 90s in a context of HE Regge scattering. Recently, it got a new impetus in quantum gravity where it opens the way to renormalizability and finite short-distance behavior. We consider a QFT model $g{\phi }^4$ with running coupling defined in both the two domains of different dimensionality; the $\gbar(Q^2)\,$ evolutions being duly conjugated at the reduction scale $Q\sim M.$ Beyond this scale, in the deep UV 2-dim region, the running coupling does not increase any more. Instead, it {\it slightly decreases} and tends to a finite value $\gbar_2(\infty) \,< \, \gbar_2(M^2)\,$ from above. As a result, the global evolution picture looks quite peculiar and can propose a base for the modified scenario of gauge couplings behavior with UV fixed points provided by dimensional reduction instead of leptoquarks.

8 pages, 4 figures,Version to match the one which (besides the Appendix) will appear in "Particles and Nuclei (PEPAN), Letters", v.7, No 6(162) 2010 pp 625-631. Slightly edited, one more reference and related numerical estimate added

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• Riemannian $(1+\mathbf{d})$-Dim Space-Time Manifolds with Nonstandard Topology which Admit Dimensional Reduction to Any Lower Dimension and Transformation of the Klein-Gordon Equation to the $1$-Dim Schrödinger Like Equation
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