A New Gauge-Invariant Regularization Scheme Based on Lorentz-Invariant Noncommutative Quantum Field Theory

The IR/UV mixing in the non-commutative (NC) field theory is investigated in CarlsonCarone-Zobin (CCZ) formalism of Lorentz-invariant NC field theory provided that the fields are ‘independent’ of the ‘internal’ coordinates θ . A new regularization scheme called NC regularization is then proposed, which removes the Lorentz-invariant IR singularity by subtraction from the theory. It requires the usual UV limit Λ → ∞ to be accompanied with the commutative limit a → 0 with Λa fixed, where a is the length parameter in the theory. The new UV limit gives the usual renormalized amplitude of the one-loop self-energy diagram of φ model. It is shown that the new regularization is gauge-invariant, that is, the non-transverse part of the vacuum polarization in QED is automatically transverse in Lorentz-invariant NCQED but the two transverse pieces, one of which is already transverse in QED, possesses Lorentz-invariant IR singularity which should be ‘subtracted off’ at zero external momentum squared. The subtraction leads to the same result as the renormalized one by Pauli-Villars or dimensional regularizations. Other diagrams with three-point vertices which contribute to the photon self-energy in Lorentz-non-invariant NCQED all vanish due to Lorentz invariance under the assumption adopted, while the tadpole diagram gives a finite contribution to the charge renormalization, which vanishes if Λa → 0. Lorentz-invariant NC φ and scalar Yukawa models are also discussed in the one-loop approximation. A comment is made that Lorentz invariance might lead to a decoupling of U(1) from SU(N) in NC U(N) gauge theory.