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In this paper, we consider an electrodynamics of higher derivatives coupledto a Lorentz-violating background tensor. Specifically, we are interested in adimension-five term of the CPT-odd sector of the nonminimal Standard-ModelExtension. By a particular choice of the operator $\hat{k}_{AF}$, we obtain ahigher-derivative version of the Carroll-Field-Jackiw (CFJ) term,$\frac{1}{2}\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}D_{\kappa}\squareF_{\mu\nu}$, with a Lorentz-violating background vector $D_{\kappa}$. Thismodification is subject to being investigated. We calculate the propagator ofthe theory and from its poles, we analyze the dispersion relations of theisotropic and anisotropic sectors. We verify that classical causality is validfor all parameter choices, but that unitarity of the theory is generally notassured. The latter is found to break down for certain configurations of thebackground field and momentum. In an analog way, we also study a dimension-fiveanisotropic higher-derivative CFJ term, which is written as$\epsilon^{\kappa\lambda\mu\nu}A_{\lambda}T_{\kappa}(T\cdot\partial)^{2}F_{\mu\nu}$and is directly linked to the photon sector of Myers-Pospelov theory. Withinthe second model, purely timelike and spacelike $T_{\kappa}$ are considered.For the timelike choice, one mode is causal, whereas the other is noncausal.Unitarity is conserved, in general, as long as the energy stays real - even forthe noncausal mode. For the spacelike scenario, causality is violated when thepropagation direction lies within certain regimes. However, there areparticular configurations preserving unitarity and strong numerical indicationsexist that unitarity is guaranteed for all purely spacelike configurations. Theresults improve our understanding of nonminimal CPT-odd extensions of theelectromagnetic sector.

Maxwell electrodynamics modified by a CPT-odd dimension-five higher-derivative term