Planar phenomena in $CPT$-even chiral QED with Lorentz violation

Quantum gravity models as string theory, loop quantum gravity, and noncommutative field theories allow for the violation of Lorentz symmetry in the Planck scale. In order to examine the implications of such violation in planar models, we present here a $CPT$-even effective model, in the Standard Model extension framework, for Maxwell-Chern-Simons quantum electrodynamics $({\mathrm{QED}}_3)$, with a Lorentz violation tensor parameter coupled to the photon field. The magnetoelectric effect induced by the space-time anisotropy in the model is discussed. We then derive the exact classical photon propagator. The electrostatic interaction is investigated and the long-range classical potential, corrected to second order in the Lorentz-violating tensor parameter, is obtained as well as its radiative correction in the isotropic violation scenario, derived at one-loop approximation from the vacuum polarization tensor. Also, the contribution of the space-time anisotropy to the magnetic moment of the fermion trough the vertex diagram is evaluated on shell and in the forward scattering approximation, in the absence of the topological Chern-Simons term.

Figure 1

Angular parametrization of the relevant planar vectors.

Figure 2

Contribution to the electrostatic potential $\frac{A_{\mathrm{cl}}^0}{(e/8\pi )u^0{v}^0}$ vs $m_{\mathrm{cs}}r$, in the isotropic case with $m_{\mathrm{cs}}\ne 0$.

Figure 3

Circularly symmetric contribution to the electrostatic potential $\frac{A_{\mathrm{cl}}^0}{(e/8\pi )|\overrightarrow{u}||\overrightarrow{v}|}$ vs $m_{\mathrm{cs}}r$, in the anisotropic collinear case with $m_{\mathrm{cs}}\ne 0$.

Figure 4

Contribution to the electrostatic potential $\frac{A_{\mathrm{cl}}^0}{(e/2\pi )|\overrightarrow{u}||\overrightarrow{v}|}$ ($z$ axis) vs $m_{\mathrm{cs}}r$ ($x$ axis) vs $\alpha$ ($y$ axis), in the anisotropic case with $m_{\mathrm{cs}}\ne 0$. (a) Collinear $\overrightarrow{u}$ and $\overrightarrow{v}$; (b) perpendicular $\overrightarrow{u}$ and $\overrightarrow{v}$.

Figure 5

Contribution to the electrostatic potential $\frac{A_{\mathrm{cl}}^0}{(e/4\pi )|\overrightarrow{u}||\overrightarrow{v}|}$ ($z$ axis) vs $\beta$ ($x$ axis) vs $\alpha$ ($y$ axis), in the anisotropic case with $m_{\mathrm{cs}}=0$.

Figure 6

Quantum correction to the electrostatic potential $\frac{A_{\mathrm{q}}^0}{e^3/(48{\pi }^2{m}_o)}$ vs $m_{\mathrm{cs}}r$, with $m_{\mathrm{cs}}/m_o=0.001$.

Figure 7

Quantum correction to the electrostatic potential $\frac{A_{\mathrm{q}}^0}{e^3/(32{\pi }^2)}$ vs $m_{\mathrm{cs}}r$, with $m_{\mathrm{cs}}/m_o=0.001$, in the isotropic case.