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.

中文翻译：

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高能强子对撞机中多束反馈的辐射增长抑制：增益和带宽的数值优化

横向反馈系统可以有效地缓解强子束中注入振荡和机器噪声引起的发射率增长。但是，由于其对光束的作用取决于有限精度的光束位置测量，因此它自身会引入额外的噪声。通常，机器噪音在低频时最强。因此，在高频下不需要反馈。在本文中，两个具有逐串反馈的降低机器噪声感应发射率增长率的理论已扩展为多串反馈。扩展的理论显示了与复杂的宏观粒子模拟的定量一致性。从数字上发现，由光束位置测量噪声引起的发射率增长仅微弱地取决于反馈的截止频率，而它强烈依赖于单束增益。这项研究的最终目标是找到最佳的横向反馈带宽和增益，该带宽和增益取决于总发射率增长率的最小化。最佳值取决于光束位置测量误差的幅度与机器噪声之间的比率，机器噪声的功率谱，反馈滤波器的响应以及失谐的幅度和细节。对于运行2碰撞时大型强子煤矿井的说明性情况，最佳选择是在当前最低的0.5 MHz截止频率下进行，单束阻尼时间约为270转。使用15个单位的色度，在此截止频率下的最小发射率增长率比逐束反馈低72％。