Noncommutative Field Theory and Spontaneous Symmetry Breaking

TL;DR

This paper studies how noncommutative field theories affect spontaneous symmetry breaking and Goldstone bosons, revealing that UV divergences and symmetry realization behave differently than in commutative theories, especially at the quantum level.

Contribution

It demonstrates the failure of standard renormalization cancellations in noncommutative models, showing the incompatibility of Nambu-Goldstone symmetry with continuum renormalization.

Findings

Noncommutative corrections lead to pion mass shifts dependent on UV cutoff.

Zero external momentum limit still yields massless Goldstone modes in Wilsonian perspective.

UV and IR limits do not commute in noncommutative field theories.

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.