1212.5131 (Patrizia Vitale et al.)

Noncommutative field theories on $R^3_λ$: Toward UV/IR mixing
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Patrizia Vitale, Jean-Christophe Wallet

We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a "foliation" of $\mathbb{R}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to $\mathbb{R}^3_\lambda$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to the square of the angular momentum supplemented by a term representing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. When only the angular momentum part of the kinetic operator is involved, the planar 1-loop contribution is logarithmically divergent while the non-planar contribution has a soft behavior in the external momenta, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the "radial part". We find that the resulting theory is finite to all orders in perturbation expansion.

View original: http://arxiv.org/abs/1212.5131