Negative superluminal velocity and violation of Kramers-Kronig relations in "causal" optical setups

We investigate nonanalyticities (e.g., zeros and poles) of refractive index $n(\omega)$ and group index $n_g(\omega)$ in different optical setups. We first demonstrate that: while a Lorentzian dielectric has no nonanalyticity in the upper half of the complex frequency plane (CFP), its group index -- which governs the pulse-center propagation -- violates the Kramers-Kronig relations (KKRs). Thus, we classify the nonanalyticities as in the (a) first-order (refractive index or reflection) and (b) second-order (group index or group delay). The latter contains the derivative of the former. Then, we study a possible connection between the negative superluminal velocities and the presence of nonanalyticities in the upper half of the CFP. We show that presence of nonanalyticities in the upper half of the CFP for (a) the first-order response and (b) the second-order response are accompanied by the appearance of negative (a) phase velocity and (b) group velocity, respectively. We also distinguish between two kinds of superluminosity, $v>c$ and $v<0$, where we show that the second one ($v<0$) appears with the violation of KKRs.