Noncommutative Field Theory and Spontaneous Symmetry Breaking

Abstract

We investigate the noncommutative analogue of the spontaneously broken linear sigma model at the one-loop quantum level. In the commutative case, renormalization of a theory with a spontaneously broken continuous global symmetry depends on cancellations that enable the limited set of counterterms consistent with that symmetry to remove the divergences even after its spontaneous breaking, while preserving the masslessness of the associated Goldstone modes. In the noncommutative case, we find that these cancellations are violated, and the renormalized one-loop correction to the inverse pion propagator explicitly yields a mass shift which depends on the ultraviolet cutoff. Thus, we cannot naively take the ultraviolet cutoff to infinity first, and then take the external momentum to zero to verify Nambu-Goldstone symmetry realization. However, from the Wilsonian perspective where the cutoff is fixed and physical, the zero external momentum limit of the inverse pion propagator still vanishes, and implies the masslessness of the pion fields at one-loop. This is another demonstration of the failure of ultraviolet and infrared limits to commute in noncommutative field theories, and signals the incompatibility of Nambu-Goldstone symmetry realization with the continuum renormalization of these theories.

17 pages, LaTeX, uses feynmp package; analyzes general case using all orderings of star product in quartic vertices; uses symmetrized Feynman vertex rules; revised combinatorics; reference added; conclusions unaltered; diagrams now appear correctly

References in corpus (2)

• Noncommutative Perturbative Dynamics
• Thermal effects in perturbative noncommutative gauge theories

Cited by in corpus (37)

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• Pair Production of Neutral Higgs Bosons through Noncommutative QED Interactions at Linear Colliders
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• Nonuniform symmetry breaking in noncommutative $\lambda {\Phi }^4$ theory
• The Noncommutative Supersymmetric Nonlinear Sigma Model
• UV/IR Mixing for Noncommutative Complex Scalar Field Theory, II (Interaction with Gauge Fields)
• The Higgs Mechanism in Non-commutative Gauge Theories
• The noncommutative U(1) Higgs-Kibble model in the enveloping-algebra formalism and its renormalizability
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• Scalar field theories in a Lorentz-invariant three-dimensional noncommutative space-time
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• The BRS invariance of noncommutative U(N) Yang-Mills theory at the one-loop level
• Spontaneous Symmetry Breaking in Noncommutative Field Theory
• Noncommutative Linear Sigma Models
• The low energy limit of the non-commutative Wess-Zumino model
• On the Operator Product Expansion in Noncommutative Quantum Field Theory
• Noncommutative Thermofield Dynamics
• Validity of Goldstone Theorem at Two Loops in Noncommutative U(N) Linear Sigma Model
• Two-loop effective potential in noncommutative scalar field theory
• Inhomogeneous chiral symmetry breaking in noncommutative four fermion interactions
• High-Temperature Effective Potential of Noncommutative Scalar Field Theory: Reduction of Degree of Freedom by Noncommutativity
• Cornwall-Jackiw-Tomboulis effective potential for canonical noncommutative field theories
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• Color Grosse-Wulkenhaar models: One-loop $\beta$-functions
• Fuzzy Scalar Field Theories: Numerical and Analytical Investigations
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• The Slavnov-Taylor Identities for the 2+1 Dimensional Noncommutative CP${}^{N-1}$ Model
• Effective Potential and Spontaneous Symmetry Breaking in the Noncommutative phi^6 Model
• Noncommutative approach to diagnose degenerate Higgs bosons at 125 GeV