One of the basic assumptions of CAIN is that the main part (i.e., the part which contributes to the beam field dominantly) of the high energy beams consists of either (almost) right-going or left-going particles. The longitudinal coordinate s is the right-going direction. (The reason s is used instead of z is only historical since ABEL.) The x and y axes are perpendicular to s and (x,y,s) forms a right-handed orthonormal frame. The time coordinate t is always multiplied by the velocity of light.
In contrast to ABEL, CAIN does not use the longitudinal coordinate attached to the beams.