Unified Characterization of HBT Radii in Central and Non-central Collisions

For central collisions, the HBT radii are commonly formulated within the out‑side‑long (osl) coordinate system, and their expressions can be derived from the second‑moment formalism. However, in non‑central collisions, the reaction plane breaks the azimuthal symmetry.

Consequently, the HBT radius formula in osl coordinate system, originally derived for central collisions, is no longer applicable. In this work, we generalize the HBT radius expression for central collisions to the non‑central case by introducing a coordinate system (X，Y，Z) and a rotation matrix. We present a general form of the non‑central HBT radii which are named Ro2，Rs2，Rl 2，Ros2，Rol2，Rsl2, and subsequently simplify these expressions under appropriate symmetry assumptions.