Abstract
We study a self-interacting scalar $\varphi^4$ theory on the $d$-dimensional noncommutative torus. We determine, for the particular cases $d=2$ and $d=4$, the counterterms required by one-loop renormalization. We discuss higher loops in two dimensions and two-loop contributions to the self-energy in four dimensions. Our analysis points toward the absence of any problems related to the UV/IR mixing and thus to renormalizability of the theory. However, we find another potentially troubling phenomenon which is a wild behavior of the two-point amplitude as a function of the noncommutativity matrix $\theta$.
Highlights
One of the motivations for considering quantum field theories on noncommutative spaces was the hope that they may be ultraviolet (UV) finite
We demonstrate that the insertion of these diagrams into internal lines of other diagrams does not lead to any divergences, so that there is no UV/IR mixing on T2θ
We saw that in the φ4 theory on T2θ (i) all superficially divergent diagrams can be renormalized by the counterterms that we have proposed, and (ii) the insertion of renormalized superficially divergent diagrams does not lead to any problems with convergence
Summary
One of the motivations for considering quantum field theories on noncommutative spaces was the hope that they may be ultraviolet (UV) finite. If the noncommutativity parameter satisfies the so-called Diophantine condition or is rational, the heat kernel coefficients (and the one-loop counterterms) assume a very simple form if written in terms of a suitably defined trace operation on the algebra of smooth functions on the torus. We shall use this observation to formulate our proposal for a (presumably) renormalizable φ4 theory on NC torus. 3 we consider the two-point functions at one-loop order and analyze their renormalization and variation with θ. The behavior of some double sums is analyzed in Appendices A and B
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