UV/IR mixing and the Goldstone theorem in noncommutative field theory

TL;DR

This paper explores UV/IR mixing and noncommutative IR singularities in scalar field theory, demonstrating the conditions for renormalizability, the validity of the Goldstone theorem, and the absence of certain IR singularities in the broken phase.

Contribution

It provides explicit one-loop calculations of IR singularities in noncommutative scalar field theory and clarifies the conditions under which the Goldstone theorem holds.

Findings

Symmetric phase is one-loop renormalizable for all compatible couplings.

Broken phase exists at one loop only if =0, consistent with Ward identities.

Broken phase is free of quadratic noncommutative IR singularities.

Abstract

Noncommutative IR singularities and UV/IR mixing in relation with the Goldstone theorem for complex scalar field theory are investigated. The classical model has two coupling constants, $\lambda_1$ and $\lambda_2$, associated to the two noncommutative extensions $\phi^*\star\phi\star\phi^*\star\phi$ and $\phi^*\star\phi^*\star\phi\star\phi$ of the interaction term $|\phi|^4$ on commutative spacetime. It is shown that the symmetric phase is one-loop renormalizable for all $\lambda_1$ and $\lambda_2$ compatible with perturbation theory, whereas the broken phase is proved to exist at one loop only if $\lambda_2=0$, a condition required by the Ward identities for global U(1) invariance. Explicit expressions for the noncommutative IR singularities in the 1PI Green functions of both phases are given. They show that UV/IR duality does not hold for any of the phases and that the broken phase is free of quadratic noncommutative IR singularities. More remarkably, the pion selfenergy does not have noncommutative IR singularities at all, which proves essential to formulate the Goldstone theorem at one loop for all values of the spacetime noncommutativity parameter $\theta$.