Local correction of quadrupole errors at LHC interaction regions using action and phase jump analysis on turn-by-turn beam position data

This article shows that the effect of all quadrupole errors present in an interaction region with low $\beta *$ can be modeled by an equivalent magnetic kick, which can be estimated from action and phase jumps found on beam position data. This equivalent kick is used to find the strengths that certain normal and skew quadrupoles located on the IR must have to make an effective correction in that region. Additionally, averaging techniques to reduce noise on beam position data, which allows precise estimates of equivalent kicks, are presented and mathematically justified. The complete procedure is tested with simulated data obtained from madx and 2015-LHC experimental data. The analyses performed in the experimental data indicate that the strengths of the IR skew quadrupole correctors and normal quadrupole correctors can be estimated within a 10% uncertainty. Finally, the effect of IR corrections in the ${\beta }^{*}$ is studied, and a correction scheme that returns this parameter to its designed value is proposed.

Figure 1

Action and phase analysis of 2015-LHC beam trajectories. Each point of the action and phase plots is calculated with measurements of two adjacent Beam Position Monitors (BPMs) (see [6]) that have been preprocessed with averaging techniques, as explained in Sec. 5. The figure also contains a simplified representation of the LHC lattice with short bars corresponding to dipole magnets and long bars corresponding to quadrupole magnets. The effect that local corrections have on the action and phase plots is also illustrated.

Figure 2

Max trajectory whose horizontal and vertical components at $s_e$ are positive and negative respectively. The other three possible combinations produce a total of four kinds of max trajectories.

Figure 3

Three average trajectories with different values of $\mathrm{\Omega }$. The larger the allowed phase range (determined by $\mathrm{\Omega }$), the larger is $N$ the number of trajectories involved in the construction of the average trajectory. All three average trajectories are in phase, and since the first one is a max trajectory, the other two are also max trajectories.

Figure 4

Action plots of the average trajectories shown in Fig. 3. A significant reduction in fluctuations (noise) can be seen as more turns are used to build the average trajectory.

Figure 5

$\beta$ beating due to quadrupole errors introduced in the LHC lattice in IR1 (black), and the effect of adding the corrections of Table 1 to the same lattice (red).

Figure 6

This first turn trajectory shows how betatron oscillations are fed into the $y$ plane due to the skew quadrupole errors placed in IR1 (black). Corrections of Table 1 reduce significantly the induced trajectory in the $y$ plane (red), which indicates a good local compensation of linear coupling at IR1.

Figure 7

Comparison between the $\beta$ beating generated by magnetic errors in IR1, and the corresponding $\beta$ beating after IR and triplet-by-triplet corrections are applied.

Figure 8

Phase plots before corrections (black) and after IR (red) and triplet-by-triplet (blue) corrections are applied. The jumps between the arcs disappear when IR corrections are applied, but a large step can still be seen between the arcs and the IR. After applying triplet-by-triplet corrections, the plot becomes completely flat.

Figure 9

Simulation of the waist shift of the ${\beta }^{*}$ due to magnetic errors and corrections. IR corrections increase the waist shift of the ${\beta }^{*}$, while triplet-by-triplet corrections eliminate this shift by returning the beta functions to their nominal values. Similar plots are found in the $y$ plane.