Emittance growth suppression with a multibunch feedback in high-energy hadron colliders: Numerical optimization of the gain and bandwidth

A transverse feedback system can effectively mitigate the emittance growth caused by injection oscillations and machine noise in hadron beams. However, as its action on the beam depends on beam position measurements of finite accuracy, it introduces additional noise on its own. The machine noise is in general strongest at low frequencies. Hence, the feedback is less needed at high frequencies. In this paper, two theories for the reduction of the machine noise induced emittance growth rate, with a bunch-by-bunch feedback, have been extended to a multibunch feedback. The extended theories show quantitative agreement with sophisticated macroparticle simulations. The emittance growth caused by the beam position measurement noise is numerically found to be only weakly dependent on the feedback's cutoff frequency, while it is strongly dependent on the single-bunch gain. The ultimate goal of this study is to find the optimal transverse feedback bandwidth and gain, determined by the minimization of the total emittance growth rate. The optimum depends on the ratio between the amplitudes of the beam position measurement error and the machine noise, the power spectrum of the machine noise, the response of the feedback filters, and the magnitude and details of the detuning. For the illustrative case of the Large Hadron Collier during collision in run 2, the optimum is found at the currently lowest possible cutoff frequency of 0.5 MHz, with a single-bunch damping time of approximately 270 turns. Using a chromaticity of 15 units, the minimal emittance growth rate at this cutoff frequency is 72% lower than with a bunch-by-bunch feedback. If the beam position measurement error can be reduced relative to the machine noise, the optimum will shift to larger single-bunch gains, or equivalently shorter single-bunch damping times.