The Lee-Wick-Chern-Simons pseudo-quantum electrodynamics

The Lee-Wick pseudo-quantum electrodynamics in the presence of a Chern-Simons term is studied in this paper. The paper starts with a non-local lagrangian density that sets the pseudo-Lee-Wick electrodynamics defined on a $1+2$ space-time added to a non-local Chern-Simons topological term. Thus, we obtain the Lee-Wick-Chern-Simons pseudo-electrodynamics as a most complete gauge invariant model that provides a light mass associated with the Chern-Simons parameter, and also includes a Lee-Wick heavy mass. We investigate classical aspects as the potential energy for the interaction of static charges through the gauge propagator. The causality of theory is discussed through the retarded Green function in the coordinate space. The gauge field of the Lee-Wick-Chern-Simons pseudo-electrodynamics is minimally coupled to the fermions sector that includes new degree of freedoms, as a Lee-Wick heavy fermion partner of the electron. The perturbative approach for the theory is presented via effective action in which we obtain the Ward identities. We study the quantum corrections at one loop, as the electron self-energy, the vacuum polarization, and the $3$-vertex. We show that the Lee-Wick mass has a fundamental role in these results, where it works like a natural regulator of the ultraviolet divergences. The $g-2$ factor for the electron is obtained as function of the LW mass, and of the CS parameter. Through the optical theorem, the Lee-Wick-Chern-Simons pseudo-electrodynamics is unitary at the tree level.

💡 Research Summary

The paper presents a novel gauge theory in (2+1)‑dimensional spacetime that combines a non‑local Lee‑Wick (LW) extension of electrodynamics with a Chern‑Simons (CS) topological term. Starting from the four‑dimensional LW Lagrangian, the authors introduce a non‑local operator
(N(\Box)=2(\Box+M^{2})/\sqrt{\Box(\Box+M^{2})})
which encodes the heavy LW mass (M). By imposing that external currents are confined to a plane, they perform a dimensional reduction to obtain a 2+1‑dimensional pseudo‑Lee‑Wick electrodynamics (PLW‑ED) described by
(\mathcal L_{PLW}= -\frac14 F_{\mu\nu}N(\Box)F^{\mu\nu} - j_{\mu}A^{\mu}).

A non‑local CS contribution,
(\mathcal L_{CS}= \frac{\theta}{2},\epsilon^{\mu\nu\rho}A_{\mu}N(\Box)\partial_{\nu}A_{\rho}),
is then added. The parameter (\theta) carries the topological (light) mass characteristic of CS theory. The full Lagrangian (\mathcal L_{PLWCS}= \mathcal L_{PLW}+ \mathcal L_{CS} - j_{\mu}A^{\mu}) remains gauge‑invariant despite the presence of higher‑derivative operators.

From the action the field equation reads
(N(\Box)\partial_{\mu}F^{\mu\nu}+ \theta N(\Box)\tilde F^{\nu}=j^{\nu})
with (\tilde F^{\nu}= \frac12\epsilon^{\nu\alpha\beta}F_{\alpha\beta}). After fixing a covariant gauge (parameter (\xi)), the kinetic operator (O_{\mu\nu}) is decomposed using the transverse projector (\theta_{\mu\nu}), the longitudinal projector (\omega_{\mu\nu}) and the antisymmetric operator (S_{\mu\nu}= -\epsilon_{\mu\nu\rho}\partial^{\rho}). The exact inverse yields the gauge propagator in momentum space:

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