While falling vertically downwards under gravity, a body breaks...

When a body falls vertically downwards under gravity, the only external force acting on the system is the gravitational force, which acts in the vertically downward direction.

There is no external force acting on the body in the horizontal direction. According to the principle of conservation of linear momentum, if the net external force acting on a system in a given direction is zero, the velocity of the centre of mass in that direction remains constant.

Since the initial horizontal velocity of the body is zero, the horizontal velocity of the centre of mass remains zero even after the body breaks into two parts. Therefore, the centre of mass of the two parts taken together does not shift horizontally.