Semiclassical analysis of the ballistic weak localization in chaotic billiards: Role of partially time-reversed pairs of paths

The ballistic weak-localization (WL) effect in chaotic billiards is studied using an extended semiclassical approximation. We show that diffraction-induced partially time-reversed pairs of paths are essential to describe the WL effect. By incorporating such pairs, we obtain the WL correction to both the transmission and reflection coefficients, which satisfies the unitarity. This result provides an answer for the puzzle addressed by Baranger et al. [Phys. Rev. Lett. 70 (1993) 3876] that the WL correction arises only in the reflection coefficient but not in the transmission coefficient so long as the usual semiclassical approximation is employed. We also find the full expression of the WL correction in the presence of a weak magnetic field.