Modeling of nonlinear effects due to head-on beam-beam interactions

Abstract

The beam-beam interaction is one of the most severe limitations on the performance of circular colliders, as it is an unavoidable strong nonlinear effect. As one aspires for greater luminosity in future colliders, one will simultaneously achieve stronger beam-beam interactions. We study the limitations caused by strong incoherent head-on beam-beam interactions, using a new code (CABIN) that calculates on a graphics processing unit (GPU), allowing for a detailed description of the long-term particle trajectories in 6D phase space. The evolution of the beam emittance and beam intensity has been monitored to study the impact quantitatively, while frequency map analysis has been performed to understand the impact qualitatively. Results from CABIN have shown good quantitative agreement with dedicated experiments in the Large Hadron Collider (LHC). For large beam-beam tune shifts, alternatives to the LHC tunes have been found to improve the beam quality. Schemes devised to cancel beam-beam driven resonances, by use of specific intermediate phase advances between the interaction points, work very well with zero crossing angle, and the accuracy required is achievable. Due to lack of symmetry, these schemes have an almost negligible impact with a significant crossing angle. The hourglass effect has been found to reduce the detrimental effects caused by the chromaticity and vice versa. The optimal level of the hourglass effect has been achieved when β Ã ¼ 1.5σ s. The ultimate parameters of the Future Circular Hadron Collider (FCC-hh) seem within reach, in absence of residual odd resonances.

Long-term evolution of Landau damping in the presence of transverse noise, feedback, and detuning

“…This is the diffusion equation with a constant diffusivity D 0 =2. This is equal to what one expects from an incoherent noise source with the same variance σ 2 k [18]. If the initial distribution in ðx; p x Þ is a Gaussian with average action hJiðt ¼ 0Þ ¼ 1, it will remain a Gaussian as…”

“…Such a process can be modeled by the Fokker-Planck equation in Eq. (18). The distribution then evolves toward a step where ΔQ ¼ 0, as seen in Fig.…”

“…If we would have assumed constant tunes during the feedback and decoherence process of a single kick, the D in Eq. (18) would have been inside the inner derivative. That partial differential equation (PDE) would have corresponded to a stochastic process where the probabilities of reducing and increasing the action by Δ from an initial action J 0 were equal, or φðΔ þ Δ; J 0 Þ ¼ φðΔ − Δ; J 0 Þ, whereΔ ¼ k 2 L 2 =2 is assumed small.…”

“…II B we studied the Fokker-Planck equation, and found that the problem at hand could be written as Eq. (18) in 1D and Eq. (20) in 2D.…”

Commissioning and the Initial Operation of the LHC

“…This resulted in a peak luminosity of over twice design and was in fact limited there by the cryogenic cooling capacity of the inner triplets. This peak performance was accompanied by impressive availability and a low level of premature dumps following a concerted program of measures outlined in more detail below (5).…”