[Paper Review] Massive fermion model in 3d and higher spin currents

L. Bonora, M. Cvitan+3 more|arXiv (Cornell University)|Feb 23, 2016

Black Holes and Theoretical Physics18 references4 citations

TL;DR

This paper studies the 3D massive fermion model coupled to external sources, computing one-loop effective actions via Feynman diagrams and dimensional regularization. It derives that in the UV limit, the higher spin current correlators yield a generalized Chern-Simons action for spin 3, while in the IR limit, a distinct higher spin action emerges—both results generalize known CS actions and highlight the role of parity breaking via the fermion mass.

ABSTRACT

We analyze the 3d free massive fermion theory coupled to external sources. The presence of a mass explicitly breaks parity invariance. We calculate two- and three-point functions of a gauge current and the energy momentum tensor and, for instance, obtain the well-known result that in the IR limit (but also in the UV one) we reconstruct the relevant CS action. We then couple the model to higher spin currents and explicitly work out the spin 3 case. In the UV limit we obtain an effective action which was proposed many years ago as a possible generalization of spin 3 CS action. In the IR limit we derive a different higher spin action. This analysis can evidently be generalized to higher spins. We also discuss the conservation and properties of the correlators we obtain in the intermediate steps of our derivation.

Motivation & Objective

• To investigate the one-loop effective action of a 3D massive fermion coupled to external gauge, metric, and higher spin fields.
• To determine the structure of two- and three-point correlators of currents, especially gauge current, energy-momentum tensor, and spin-3 current.
• To identify the local UV and IR limits of the effective action and relate them to generalized Chern-Simons actions.
• To examine the role of parity breaking due to the fermion mass and its impact on correlator structure and symmetry restoration.
• To clarify the conservation properties and symmetry anomalies in higher-point correlators under IR and UV limits.

Proposed method

• Uses Feynman diagram techniques and dimensional regularization to compute 2- and 3-point functions of currents in the massive fermion model.
• Evaluates correlators in both UV and IR limits by taking high- and low-energy limits relative to the fermion mass m.
• Applies perturbative cohomology to analyze gauge and diffeomorphism symmetry in the effective action.
• Derives the local part of the effective action from contact terms in the correlators, focusing on odd-parity components.
• Uses generating functions and tensor structures to systematically construct n-point correlators, particularly for spin-3 currents.
• Imposes symmetrization and conservation constraints on three-point functions to ensure consistency with conformal invariance and current conservation.

Experimental results

Research questions

• RQ1 How does the mass term in a 3D fermion theory affect the structure of higher spin current correlators in the UV and IR limits?
• RQ2 Can the UV limit of the effective action for spin-3 currents be identified with a known generalized Chern-Simons action?
• RQ3 Why do odd-parity correlators remain local in both UV and IR limits, while even-parity ones are non-local?
• RQ4 What is the role of the fermion mass in breaking parity and how does this affect the emergence of Chern-Simons-like actions?
• RQ5 How are gauge and diffeomorphism symmetry anomalies introduced in the three-point functions, and how can they be repaired?

Key findings

• In the UV limit, the effective action for spin-3 currents reproduces a generalized Chern-Simons action proposed earlier in the literature, confirming its emergence from a free massive fermion model.
• In the IR limit, the effective action for the energy-momentum tensor yields a local, odd-parity three-point function that corresponds to a distinct higher spin action different from the UV one.
• The odd-parity two- and three-point correlators are conformally invariant in the fixed-point limit, despite arising from a massive theory.
• Even-parity correlators are non-local in both UV and IR limits, while odd-parity ones are local, indicating a fundamental difference in their structure.
• The IR limit of the gauge current and energy-momentum tensor correlators reproduces the standard Chern-Simons action, confirming known results in a new context.
• Symmetry breaking in three-point functions under IR/UV limits necessitates counterterms, indicating that the effective action must be corrected to restore gauge and diffeomorphism invariance.