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, ${\lambda }_1$ and ${\lambda }_2$, associated to the two noncommutative extensions ${\varphi }^{\ast }\star \varphi \star {\varphi }^{\ast }\star \varphi$ and ${\varphi }^{\ast }\star {\varphi }^{\ast }\star \varphi \star \varphi$ of the interaction term $|\varphi {|}^4$ 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 $\theta$.

30 pages, 7 eps figures. V2: references added

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Cited by in corpus (18)

• Vacuum configurations for renormalizable non-commutative scalar models
• Phase diagram and dispersion relation of the non-commutative λϕ^{4} model in d=3
• Nonuniform symmetry breaking in noncommutative $\lambda {\Phi }^4$ theory
• Noncommutative geometry, gauge theory and renormalization
• The noncommutative U(1) Higgs-Kibble model in the enveloping-algebra formalism and its renormalizability
• Trouble with space-like noncommutative field theory
• On the UV renormalizability of noncommutative field theories
• Non--Commutative Field Theories beyond Perturbation Theory
• Position-dependent noncommutative products: classical construction and field theory
• Spontaneous Symmetry Breaking in Noncommutative Field Theory
• IR Dynamics from UV Divergences: UV/IR Mixing, NCFT, and the Hierarchy Problem
• Canonical Quantization of Noncommutative Field Theory
• Validity of Goldstone Theorem at Two Loops in Noncommutative U(N) Linear Sigma Model
• Renormalization on noncommutative torus
• Cornwall-Jackiw-Tomboulis effective potential for canonical noncommutative field theories
• The Hierarchy Problem: From the Fundamentals to the Frontiers
• Supersymmetry and Gravity in Noncommutative Field Theories
• Effective Potential and Spontaneous Symmetry Breaking in the Noncommutative phi^6 Model