Equation for the Multi-Pass Beam Breakup Threshold Current for a Single Mode and a 4x4 Recirculation Matrix

This paper describes a novel method for calculating the threshold of the multi-pass beam breakup (BBU) in recirculating linacs. The method is based on a simple idea of the equilibrium between power deposited by the beam in a higher-order mode (HOM) and Ohmic losses in a cavity. The method is used to derive a formula for the threshold current for a single HOM rotated at an arbitrary angle and a 4x4 recirculation matrix. We also use the method to describe HOM voltage behavior above and below the threshold current.