On the UV renormalizability of noncommutative field theories

TL;DR

This paper investigates the UV renormalizability of noncommutative field theories, revealing that proper renormalization requires setting conditions at nonzero external momenta due to UV/IR mixing effects.

Contribution

It demonstrates that noncommutative field theories are renormalizable when renormalization conditions are set at nonzero momenta, using Wilsonian RG and analyzing spontaneous symmetry breaking.

Findings

Renormalization conditions at nonzero momenta ensure renormalizability.

Green's functions exhibit different scaling behaviors based on zero momenta configurations.

Ward identities hold in the broken phase with an infrared regulator.

Abstract

UV/IR mixing is one of the most important features of noncommutative field theories. As a consequence of this coupling of the UV and IR sectors, the configuration of fields at the zero momentum limit in these theories is a very singular configuration. We show that the renormalization conditions set at a particular momentum configuration with a fixed number of zero momenta, renormalizes the Green's functions for any general momenta only when this configuration has same set of zero momenta. Therefore only when renormalization conditions are set at a point where all the external momenta are nonzero, the quantum theory is renormalizable for all values of nonzero momentum. This arises as a result of different scaling behaviors of Green's functions with respect to the UV cutoff ($\Lambda$) for configurations containing different set of zero momenta. We study this in the noncommutative $\phi^4$ theory and analyse similar results for the Gross-Neveu model at one loop level. We next show this general feature using Wilsonian RG of Polchinski in the globally O(N) symmetric scalar theory and prove the renormalizability of the theory to all orders with an infrared cutoff. In the context of spontaneous symmetry breaking (SSB) in noncommutative scalar theory, it is essential to note the different scaling behaviors of Green's functions with respect to $\Lambda$ for different set of zero momenta configurations. We show that in the broken phase of the theory the Ward identities are satisfied to all orders only when one keeps an infrared regulator by shifting to a nonconstant vacuum.