Beam-energy inequality in the beam-beam interaction

Conditions for ''energy transparency,'' unequal-energy beams having the same beam-beam behavior, are derived for round beams from a Hamiltonian model of the beam-beam interaction. These conditions are equal fractional betatron tunes, equal synchrotron tunes, equal beam-beam strength parameters, equal nominal sizes, equal β*'s and equal bunch lengths. With these conditions the only way to compensate for unequal energies is with the number of particles per bunch