Tune modulated beam-beam resonances in the Tevatron

During the 88--89 Tevatron collider run, 6 antiproton bunches were colliding head-on with 6 proton bunches at 12 crossing points symetrically distributed around the ring. Typical intensities were 7 x 1010 and 2.5 x 1010 particles per bunch for protons and antiprotons respectively. The normalized transverse proton emittance (95% definition) was typically 25 π mm-mr on both planes and the antiproton transverse emittance was typically 18 π 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 12th order resonance will affect some of the antiprotons. This resonance was certainly felt by the α > 2σ particles in the 88--89 run when proton and antiproton emittances were equal. The situation improved when the proton emittance was increased. In this paper, we determine the range of critical resonances by applying the analytic theory of tune-modulated beam-beam resonances to the 88--89 run and to the main injector upgrade scenario. The analytic theory neglects long-range tune shifts and resonances. 3 refs., 4 figs