Transverse emittance growth due to rf noise in the high-luminosity LHC crab cavities

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The high-luminosity LHC (HiLumi LHC) upgrade with planned operation from 2025 onward has a goal of achieving a tenfold increase in the number of recorded collisions thanks to a doubling of the intensity per bunch (2.2e11 protons) and a reduction of ${\ensuremath{\beta}}^{*}$ to 15 cm. Such an increase would significantly expedite new discoveries and exploration. To avoid detrimental effects from long-range beam-beam interactions, the half crossing angle must be increased to 295 microrad. Without bunch crabbing, this large crossing angle and small transverse beam size would result in a luminosity reduction factor of 0.3 (Piwinski angle). Therefore, crab cavities are an important component of the LHC upgrade, and will contribute strongly to achieving an increase in the number of recorded collisions. The proposed crab cavities are electromagnetic devices with a resonance in the radio frequency (rf) region of the spectrum (400.789 MHz). They cause a kick perpendicular to the direction of motion (transverse kick) to restore an effective head-on collision between the particle beams, thereby restoring the geometric factor to 0.8 [K. Oide and K. Yokoya, Phys. Rev. A 40, 315 (1989).]. Noise injected through the rf/low level rf (llrf) system could cause significant transverse emittance growth and limit luminosity lifetime. In this work, a theoretical relationship between the phase and amplitude rf noise spectrum and the transverse emittance growth rate is derived, for a hadron machine assuming zero synchrotron radiation damping and broadband rf noise, excluding infinitely narrow spectral lines. This derivation is for a single beam. Both amplitude and phase noise are investigated. The potential improvement in the presence of the transverse damper is also investigated.