On Evaluation of Nonplanar Diagrams in Noncommutative Field Theory

Abstract

This is a technical work about how to evaluate loop integrals appearing in one loop nonplanar (NP) diagrams in noncommutative (NC) field theory. The conventional wisdom says that, barring the ultraviolet/infrared (UV/IR) mixing problem, NP diagrams whose planar counterparts are UV divergent are rendered finite by NC phases that couple the loop momentum to the external NC momentum \rho^{\mu}=\theta^{\mu\nu}p_{\nu}. We show that this is generally not the case. We find that subtleties arise already on Euclidean spacetime. The situation is even worse in Minkowski spacetime due to its indefinite metric. We compare different prescriptions that may be used to evaluate loop integrals in ordinary theory. They are equivalent in the sense that they always yield identical results. However, in NC theory there is no a priori reason that these prescriptions, except for the defining one built in Feynman propagator, are physically justified. Employing them can lead to ambiguous results. For \rho^2>0, the NC phase can worsen the UV property of loop integrals instead of always improving it in high dimensions. We explain how this surprising phenomenon comes about from the indefinite metric. For \rho^2<0, the NC phase improves the UV property and softens the quadratic UV divergence in ordinary theory to a bounded but indefinite UV oscillation. We employ a cut-off method to quantify the new UV non-regular terms. For \rho^2>0, these terms are generally complex and thus also harm unitarity. As the new terms are not available in the Lagrangian, our result casts doubts on previous demonstrations of one loop renormalizability based exclusively upon analysis of planar diagrams, especially in theories with quadratic divergences.