A four-dimensional mapping model of colliding proton beams, with cylindrically symmetric charge distribution, is investigated in the weak-strong approximation for unequal "tunes" Q1 ≠ Q2. It is typically observed that the maximum deviation of the weak beam from its ideal path attains after some 103 beam crossings a "terminal blow-up" value, which persists up to 105 crossings and beyond. We first obtain some simple solutions of the mapping equations and show that it is possible to minimize this "terminal blow-up" of the weak beam by selecting "tune ratios" σ ≡ Q2 Q1 such that these simple solutions exist and are stable under small perturbations. The stability condition of the origin also proves to be very useful in determining σ ranges of minimal "terminal blow-up". On the other hand, at some other σ values, a dramatic increase of beam blow-up is observed. Introducing Action-Angle variables, we explain this effect in terms of the lowest order resonances of an effective Hamiltonian, and calculate the location and energy threshold for the overlap of major resonances, at parameter values of interest to present day machines.
ASJC Scopus subject areas
- Nuclear and High Energy Physics