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In this letter we study the self-energy of a point-like charge for the electromagnetic field in a non-minimal Lorentz symmetry breaking scenario in a (n + 1)-dimensional space time. We consider two variations of a model where the Lorentz violation is caused by a background vector dν that appears in a higher derivative interaction. We restrict our attention to the case where dμ is a time-like background vector, namely d2 = dμdμ > 0, and we verify that the classical self-energy is finite for any odd spatial dimension n and diverges for even n. We also make some comments regarding obstacles in the quantization of the proposed model.

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