Luminosity reduction caused by phase modulations at the HL-LHC crab cavities

The design of the High-Luminosity Large Hadron Collider (HL-LHC) requires two pairs of crab cavities to be installed either side of Interaction Points (IPs) 1 (ATLAS) and 5 (CMS) to compensate for the geometric reduction in luminosity due to the beam crossing angle at the IP. The HL-LHC beam current is a factor of two larger than the LHC design value. The existing RF system has insufficient power to use the existing low level RF (LLRF) scheme for HL-LHC and therefore a new scheme is proposed which results in an irregular bunch pattern in the ring; here in referred to as a phase modulation. In this paper we study the effect of this phase modulation on the crab cavity scheme and the resulting impact on peak luminosity. We have developed an analytical model to calculate the luminosity and its dependence on the related beam and RF parameters. We compare this model to tracking simulations in PYTRACK and show a good agreement between the model and simulations. In the case of a coherent phase error between the counter-rotating bunch trains, having the maximum expected time shift of 100 ps (0.25 radians at the RF frequency), the reduction of analytical peak luminosity is found to be 1.89% when the crabbing voltage is 6.8 MV. For incoherent phase errors, the luminosity reduction for a 100 ps phase error is 5.67%; however the expected incoherent phase error is significantly less than 100 ps. These reductions are not foreseen as an issue when the crabbing scheme is used for luminosity levelling during physics experiments.