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

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, $λ_1$ and $λ_2$, associated to the two noncommutative extensions $ϕ^*\starϕ\starϕ^*\starϕ$ and $ϕ^*\starϕ^*\starϕ\starϕ$ of the interaction term $|ϕ|^4$ on commutative spacetime. It is shown that the symmetric phase is one-loop renormalizable for all $λ_1$ and $λ_2$ compatible with perturbation theory, whereas the broken phase is proved to exist at one loop only if $λ_2=0$, 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 $θ$.