Tune modulated beam-beam resonances in the Tevatron

During the 1988-9 Tevatron collider run, six antiproton bunches were colliding head-on with six proton bunches at 12 crossing points symmetrically distributed around the ring. The normalized transverse proton emittance (95% definition) was typically 25 pi mm-mr on both planes and the antiproton transverse emittance was typically 18 pi mm-mr. The working point (unshifted horizontal and vertical tunes) was near 19.41. Given the working point and typical tune spreads of about 0.024, it is unavoidable that the twelfth-order resonance will affect some of the antiprotons. This resonance was certainly felt by the a>2 sigma particles in the 1988-9 run when proton and antiproton emittances were equal. The situation improved when the proton emittance was increased. The question of whether or not the twelfth will be harmless in future collider runs remains, since the beam-beam tune shift parameter will be much higher than it was in the 1988-9 run. This question is answered by applying the analytic theory of tune-modulated beam-beam resonances to the 1988-9 run and to the main injector upgrade scenario. The analytic theory neglects long-range tune shifts and resonances.<>