Diffusion due to Beam-Beam Interaction and Fluctuating Fields in Hadron Colliders.

Emittance growth due to the non-linear beam-beam interaction is a major concern at all hadron colliders. The tunes (i.e. the rotation numbers of the transverse betatron oscillations) are always chosen to avoid low order resonances and the overlap of high order resonances does not create chaotic regions large enough to lead to significant amplitude growth. The dynamics are qualitatively different however when the time-dependence, both deterministic and stochastic, of parameters is included. The effects of deterministic tune modulation are well studied and removing modulation lines from the betatron spectrum reduces particle loss from the tails of the beam [1]. Random fluctuations in the tune, closed orbit and beam size are also present in accelerators. Qualitative arguments [2] and numerical simulations [3] have shown that tune fluctuations lead to emittance growth especially for tunes close to a resonance . Our aim is to provide a quantitative theory to explain these observations. We consider collisions of a proton beam with an opposing beam composed of either leptons as at HERA or hadrons as at the Fermilab Tevatron and the proposed LHC at CERN. For one dimensional motion, the beam-beam potential seen by a proton, assuming a Gaussian charge distribution of the opposing beam, is given by U(x) = C ∫ ∞