Abstract
We consider the noncommutative space \( \mathbb{R}_{\lambda}^3 \), 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 \( \mathbb{R}_{\lambda}^3 \). 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 dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, 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.
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Vitale, P., Wallet, JC. Noncommutative field theories on \( \mathbb{R}_{\lambda}^3 \): towards UV/IR mixing freedom. J. High Energ. Phys. 2013, 115 (2013). https://doi.org/10.1007/JHEP04(2013)115
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DOI: https://doi.org/10.1007/JHEP04(2013)115