On the UV renormalizability of noncommutative field theories

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 ${\varphi }^4$ 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.

29 pages, 8 figures, uses JHEP.cls, references added

References in corpus (3)

• UV/IR mixing and the Goldstone theorem in noncommutative field theory
• Quantum Aspects of GMS Solutions of Noncommutative Field Theory and Large N Limit of Matrix Models
• One Loop Renormalizability of Spontaneously Broken Gauge Theory with a Product of Gauge Groups on Noncommutative Spacetime: the U(1) x U(1) Case

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• Validity of Goldstone Theorem at Two Loops in Noncommutative U(N) Linear Sigma Model
• Aspects of Open-Closed Duality in a Background B-Field
• Aspects of Open-Closed Duality in a Background B-Field II
• Field Theory on a Non-commutative Plane: a Non-Perturbative Study
• Non-commutative Duality: High Spin Fields and $C{P}^1$ Model with Hopf Term
• UV-IR mixing and the quantum consistency of noncommutative gauge theories
• Cornwall-Jackiw-Tomboulis effective potential for canonical noncommutative field theories
• Supersymmetry and Gravity in Noncommutative Field Theories
• Wave Propagation and IR/UV Mixing in Noncommutative Spacetimes
• Effective Field Theory of a Locally Noncommutative Space-Time and Extra Dimensions