Emittance growth, tune shift, and the bunched-beam bunched-beam interaction

It is pointed out that it is important to evaluate and understand both the emittance growth and nature of the tune shift due to multiple beam-beam crossings of bunched beams of heavy ions. Within the Relativistic Heavy Ion Collider, fully stripped ¹⁹⁷Au ions (charge 79e⁺) will survive up to 10 h in the collider, with six beam crossings per revolution. With this motivation, the authors have developed a fully relativistic theory of both the average emittance growth and the average tune shift for the bunched-beam bunched-beam interaction that is based on a convolution integral over the densities of the two interpenetrating bunches. In order to calculate this integral, the authors chose to work in a frame where one bunch of the collider is stationary and the other is highly relativistic. This frame has the additional advantage that the microscopic heavy ion interaction becomes perpendicular in nature. In this frame the convolution integral acquires many simplifying and physically interesting features