Outgoing analytic branch of Klein-Gordon step momentum ($q(z)=\sqrt{z-m{c}^2}\sqrt{z+m{c}^2}/(\hslash c)$)

Set $z=E-V_0+i0$ and continue both roots from the upper half-plane. The right-hand wavenumber is positive above the positive-energy threshold, positive imaginary inside the gap, and negative below the negative-energy threshold. Its group velocity in the last region remains rightward because both momentum and kinetic energy are negative. Choosing a positive real root there instead specifies an incoming wavepacket from the right.