Amplitude dependent closest tune approach

Recent observations in the LHC point to the existence of an amplitude dependent closest tune approach. However this dynamical behavior and its underlying mechanism remain unknown. This effect is highly relevant for the LHC as an unexpected closest tune approach varying with amplitude modifies the frequency content of the beam and, hence, the Landau damping. Furthermore the single particle stability would also be affected by this effect as it would modify how particles with varying amplitudes approach and cross resonances. We present analytic derivations that lead to a mechanism generating an amplitude dependent closest tune approach.

Figure 1

LHC beam-beam tune footprint up to 6 beam sigma (in blue). Resonance lines up to 10th order are shown [4]. A hypothetical $\mathrm{\Delta }{Q}_{\min }=8\times {10}^{-3}$ stopband is drawn in light red to illustrate the reduction of the available resonance-free area.

Figure 2

Tunes from particle tracking for different initial tunes and for increasing amplitudes of the vertical kicks. The color code represents the sum of the horizontal and vertical actions. The black diagonal line indicates the resonance $Q_x=Q_y$.

Figure 3

Tunes from tracking with linear closest tune approach of 0.015 and nominal octupoles. The vertical kicks ranged from 0.5 mm to 4.5 mm while the horizontal were kept at 0.5 mm. The light red area delimits the linear closest tune approach of 0.015.

Figure 4

Tunes from tracking with linear closest tune approach of 0.015 and $h_{1111}=0$ by powering MOF with opposite polarity than MOD. The vertical kicks range from 0.5 mm to 4.5 mm while the horizontal were kept at 0.5 mm. The light red area delimits the linear coupling stopband of 0.015.

Figure 5

Tunes from tracking with linear closest tune approach of 0.015 and nominal octupoles. The kicks in the horizontal plane ranged from 0.5 mm to 4.5 mm while the kicks in the vertical plane were kept at 0.5 mm. The light red area indicates the linear stopband of 0.015 for all cases. The two black lines show the resonances $Q_x=Q_y$ and $Q_y=1/3$ respectively.

Figure 6

Comparison of tracking simulations with incremental vertical kicks, for optics configurations with $Q_x=64.28$, $Q_y=59.31$ and $Q_x=62.28$, $Q_y=62.31$. In both cases $|C^{-}|=0.01$ and Landau octupoles are applied with a strength of $-3{\mathrm{m}}^{-4}$.