Material boundaries in Carroll–Field–Jackiw Lorentz-violating electrodynamics

Abstract

This paper investigates certain aspects of the CPT-odd photon sector of the minimal Standard Model Extension (SME) in the presence of a perfectly conducting plate (perfect mirror). The considered sector is described by Carroll–Field–Jackiw (CFJ) electrodynamics, where Lorentz violation is due to the presence of a single background vector denoted as $\left( k_{AF}\right) ^{\mu }$, which we treat perturbatively up to second order. Specifically, we derive the modified propagator for the electromagnetic field due to the presence of the perfect mirror, and we study the corresponding interaction between the mirror and a stationary point-like charge. Our results reveal that when the charge is positioned near the mirror, a spontaneous torque emerges, which is a unique effect of Lorentz symmetry breaking. Additionally, we demonstrate that the image method for this theory is applicable when the background vector has only the component perpendicular to the mirror.

Appendices

Appendices

Appendix 1: Integrals (24)-(26)

In this appendix, we provide some details on the calculation of the integrals (24), (25), and (26). For this task, we employ the well-known $i\varepsilon$-prescription to shift the poles away from the real axis in the complex plane of $p^{3}$. We can write

$$\begin{aligned} \int \frac{\textrm{d}p^{3}}{\left( 2\pi \right) }\frac{e^{ip^{3}(x^{3}-y^{3})}}{p^{2} }=-\int \frac{\textrm{d}p^{3}}{\left( 2\pi \right) }\frac{e^{ip^{3}\left( x^{3} -y^{3}\right) }}{\left( p^{3}\right) ^{2}-\Gamma ^{2}-i\varepsilon }\ , \end{aligned}$$

with the limit $\varepsilon \rightarrow 0$ implicitly assumed. For $x^{3} -y^{3}>0$, we close the integration contour (infinite semicircle) in the upper half-plane, enclosing the simple pole at $\Gamma +i\varepsilon$. For $x^{3}-y^{3}<0$, we choose the integration contour in the lower half-plane, enclosing the simple pole at $-\Gamma -i\varepsilon$. Calculating the residue of the integrand at these poles and adding these contributions using the residue theorem, we obtain the result (24).

For the integrals (25) and (26), the same procedure is followed. The only distinction is that these integrals have double and triple poles, respectively, which impacts the calculation of the residues.

Appendix 2: Propagator

In this appendix, we give some additional technical details of how the propagator in Eq. (37) was obtained. We start by substituting (31) and (34) in (36), what leads to

$$\begin{aligned} {\bar{D}}_{\mu \nu }\left( x,y\right)= & {} \frac{i}{2}\int \textrm{d}^{3}\omega _{\parallel } \ \int \textrm{d}^{3}z_{\parallel }\int \frac{\textrm{d}^{3}p_{\parallel }}{\left( 2\pi \right) ^{3}}\int \frac{\textrm{d}^{3}k_{\parallel }}{\left( 2\pi \right) ^{3}}\int \frac{\textrm{d}^{3}q_{\parallel }}{\left( 2\pi \right) ^{3}} \ e^{-ip_{\parallel }\cdot \left( \omega _{\parallel }-x_{\parallel }\right) } \ e^{-ik_{\parallel }\cdot \left( z_{\parallel }-y_{\parallel }\right) } \ e^{-iq_{\parallel }\cdot \left( \omega _{\parallel }-z_{\parallel }\right) }\nonumber \\{} & {} \times \Biggl \{\epsilon _{3 \ \ \mu }^{\ \alpha \gamma } \ p_{\parallel \gamma }\Biggl [-1+\frac{1}{2}\Biggl (\frac{3\left[ i\Gamma \mid x^{3}-a\mid - 1\right] }{p_{\parallel }^{2}}+\mid x^{3} -a\mid ^{2}\Biggr )\nonumber \\{} & {} \times \Biggl (\left[ \left( k_{AF}\right) ^{3}\right] ^{2}+\frac{\left[ \left( k_{AF}\right) _{\parallel }\cdot p_{\parallel }\right] ^{2}}{p_{\parallel }^{2}}\Biggr ) +\frac{\left[ i\Gamma \mid x^{3}-a\mid - 1\right] }{p_{\parallel }^{2}}\Biggl (i\left( k_{AF}\right) ^{3}\left( x^{3}-a\right) \left[ \left( k_{AF}\right) _{\parallel }\cdot p_{\parallel }\right] -2\left( k_{AF}^{2}\right) _{\parallel }\Biggr )\Biggr ]\nonumber \\{} & {} +\frac{\left[ i\Gamma \mid x^{3}-a\mid - 1\right] }{p_{\parallel }^{2}} \Biggl [2\left( k_{AF}\right) _{\mu }\epsilon _{3}^{\ \alpha \gamma \rho }p_{\parallel \gamma }\left( k_{AF}\right) _{\parallel \rho }+i\eta _{\mu 3}\left( \left( k_{AF}\right) _{\parallel }^{\alpha }p_{\parallel }^{2}-p_{\parallel }^{\alpha }\left[ \left( k_{AF}\right) _{\parallel }\cdot p_{\parallel }\right] \right) \nonumber \\{} & {} +i\left( k_{AF}\right) ^{3}\left( \eta _{\parallel \mu }^{\alpha }p_{\parallel }^{2}-p_{\parallel }^{\alpha }p_{\parallel \mu }\right) \Biggr ] +\Biggl [\left( k_{AF}\right) _{\parallel }^{\alpha }p_{\parallel \mu }-\eta _{\parallel \mu }^{\alpha }\left[ \left( k_{AF}\right) _{\parallel }\cdot p_{\parallel }\right] \Biggr ]\left( x^{3}-a\right) \Biggr \}\nonumber \\{} & {} \times \Biggl \{\epsilon _{3 \ \ \nu }^{\ \theta \lambda } \ k_{\parallel \lambda }\Biggl [-1+\frac{1}{2}\Biggl (\frac{3\left[ i\Gamma '\mid y^{3}-a\mid - 1\right] }{k_{\parallel }^{2}}+\mid y^{3} -a\mid ^{2}\Biggr )\nonumber \\{} & {} \times \Biggl (\left[ \left( k_{AF}\right) ^{3}\right] ^{2}+\frac{\left[ \left( k_{AF}\right) _{\parallel }\cdot k_{\parallel }\right] ^{2}}{k_{\parallel }^{2}}\Biggr ) +\frac{\left[ i\Gamma '\mid y^{3}-a\mid - 1\right] }{k_{\parallel }^{2}}\Biggl (i\left( k_{AF}\right) ^{3}\left( y^{3}-a\right) \left[ \left( k_{AF}\right) _{\parallel }\cdot k_{\parallel }\right] -2\left( k_{AF}^{2}\right) _{\parallel }\Biggr )\Biggr ]\nonumber \\{} & {} +\frac{\left[ i\Gamma '\mid y^{3}-a\mid - 1\right] }{k_{\parallel }^{2}} \Biggl [2\left( k_{AF}\right) _{\nu }\epsilon _{3}^{\ \theta \lambda \phi }k_{\parallel \lambda }\left( k_{AF}\right) _{\parallel \phi }+i\eta _{\nu 3}\left( \left( k_{AF}\right) _{\parallel }^{\theta }k_{\parallel }^{2}-k_{\parallel }^{\theta }\left[ \left( k_{AF}\right) _{\parallel }\cdot k_{\parallel }\right] \right) \nonumber \\{} & {} +i\left( k_{AF}\right) ^{3}\left( \eta _{\parallel \nu }^{\theta }k_{\parallel }^{2}-k_{\parallel }^{\theta }k_{\parallel \nu }\right) \Biggr ] +\Biggl [\left( k_{AF}\right) _{\parallel }^{\theta }k_{\parallel \nu }-\eta _{\parallel \nu }^{\theta }\left[ \left( k_{AF}\right) _{\parallel }\cdot k_{\parallel }\right] \Biggr ]\left( y^{3}-a\right) \Biggr \}\nonumber \\{} & {} \times \Biggl \{\Biggl [1+\frac{\left[ \left( k_{AF}\right) _{\parallel }\cdot q_{\parallel }\right] ^{2}}{2 q_{\parallel }^{4}}-\frac{\left[ \left( k_{AF}\right) ^{3}\right] ^{2}}{2 q_{\parallel }^{2}}\Biggr ]\eta _{\parallel \alpha \theta }-\frac{\left[ \left( k_{AF}\right) ^{3}\right] ^{2}}{q_{\parallel }^{4}}q_{\parallel \alpha }q_{\parallel \theta } \nonumber \\{} & {} +2\frac{\left( k_{AF}\right) _{\parallel \alpha }\left( k_{AF}\right) _{\parallel \theta }}{q_{\parallel }^{2}} -2\frac{\left[ \left( k_{AF}\right) _{\parallel }\cdot q_{\parallel }\right] }{q_{\parallel }^{4}}\left[ q_{\parallel \alpha }\left( k_{AF}\right) _{\parallel \theta }+\left( k_{AF}\right) _{\parallel \alpha }q_{\parallel \theta }\right] \nonumber \\{} & {} +\frac{i}{q_{\parallel }^{2}}\left( k_{AF}\right) ^{3}\epsilon _{\alpha \theta \tau 3}q_{\parallel }^{\tau }\Biggr \}\frac{e^{i\Gamma \mid x^{3}-a\mid } \ e^{i\Gamma '\mid y^{3}-a\mid }}{\Gamma \ \Gamma ' \ \Gamma ''} \ , \nonumber \end{aligned}$$

where $\Gamma =\sqrt{p_{\parallel }^{2}}$, $\Gamma '=\sqrt{k_{\parallel }^{2}}$ and $\Gamma ''=\sqrt{q_{\parallel }^{2}}$.

Using the fact that

$$\begin{aligned} \int \frac{\textrm{d}^{3}\omega _{\parallel }}{\left( 2\pi \right) ^{3}} \ e^{-i\left( p_{\parallel }+q_{\parallel }\right) \cdot \omega _{\parallel }}=\delta ^{3}\left( p_{\parallel }+q_{\parallel }\right) \ , \ \ \int \frac{\textrm{d}^{3}z_{\parallel }}{\left( 2\pi \right) ^{3}} \ e^{-i\left( k_{\parallel }-q_{\parallel }\right) \cdot z_{\parallel }}=\delta ^{3}\left( k_{\parallel }-q_{\parallel }\right) \ , \end{aligned}$$

we perform the integrals $\textrm{d}^{3}k_{\parallel }$ and $\textrm{d}^{3}p_{\parallel }$. After, we carry out some straightforward manipulations obtaining the propagator as in Eq. (37).