Transverse emittance growth due to rf noise in crab cavities: Theory, measurements, cure, and high luminosity LHC estimates

The High-Luminosity LHC (HL-LHC) upgrade with planned operation from 2029 onward has a goal of achieving a tenfold increase in the integrated number of recorded collisions thanks to a doubling of the intensity per bunch ($2.2\times {10}^{11}$ protons) and a reduction of ${\beta }^{*}$ (the $\beta$ value in the two high luminosity detectors, namely ATLAS and CMS) to 15 cm. Such an increase in recorded collisions would significantly expedite new discoveries and exploration. Crab cavities are an important component of the HL-LHC upgrade and will contribute strongly to achieving an increase in the number of recorded collisions. However, noise injected through the crab cavity radio frequency (rf) system could cause significant transverse emittance growth and limit luminosity lifetime. We presented a theoretical formalism relating transverse emittance growth to rf noise in an earlier work. In this follow-up paper, we summarize measurements in the super-proton synchrotron (SPS) at CERN that validate the theory, we present estimates of the emittance growth rates using state-of-the-art rf and low-level rf (LLRF) technologies, and we set the rf noise specifications to achieve acceptable performance. A novel dedicated feedback system acting through the crab cavities to mitigate emittance growth will be required. In this work, we develop a theoretical formalism to evaluate the performance of such a feedback system in any collider, identify limiting components, present simulation results to validate these studies, and derive key design parameters for an HL-LHC implementation of such a feedback system.

Figure 1

Crab cavity induced bunch rotation [7].

Figure 2

Horizontal and vertical emittance growth rates as a function of the skew quadrupole angle.

Figure 3

Measured and expected emittance growth rates.

Figure 4

rf noise scaling during the HL-LHC cycle.

Figure 5

LHC main accelerating cavity (measured) and crab cavity (estimated) phase noise.

Figure 6

Block diagram including damper and proposed feedback.

Figure 7

Crab cavity phase feedback (mode 0).

Figure 8

Emittance growth reduction factor as a function of ${\alpha }_0$ and tune distribution.

Figure 9

Emittance growth reduction factor as a function of ${\alpha }_1$ and tune distribution.

Figure 10

Emittance growth reduction factor from phase feedback acting on phase noise. The solid line is the analytical formula [Equation (15)]. The dots are simulation results with one cavity ($N_{cc}=1$).

Figure 11

Emittance growth reduction factor from amplitude feedback acting on amplitude noise. The solid line is the analytical formula [Equation (19)]. The dots are simulation results with one cavity ($N_{cc}=1$).

Figure 12

Emittance growth reduction factor with varying mode 0 measurement error to phase noise ratios. $N_{cc}=1$.

Figure 13

Emittance growth reduction factor with varying mode 1 measurement error to amplitude noise ratios. $N_{cc}=1$.

Figure 14

Emittance growth reduction factor with varying mode 0 measurement error to phase noise ratios. HL-LHC case ($N_{cc}=4$).

Figure 15

Emittance growth reduction factor with varying mode 1 measurement error to amplitude noise ratios. HL-LHC case ($N_{cc}=4$).

Figure 16

Bunch length correction factors for two-dimensional Gaussian and pillbox distributions.

Figure 17

Crab cavity amplitude feedback (mode 1).