Theory of bunched beam transverse Schottky signals with stationary rf system and high-order chromaticity

The observation of transverse Schottky spectra provides insight into intrabunch particle dynamics and serves in numerous facilities to derive important beam and machine parameters, such as the betatron tune, first-order chromaticity, and transverse emittance. The original theory confirming the validity of these observations has been, however, developed by taking strong assumptions on the particle motion in both the longitudinal and transverse planes. In this work, we loosen these assumptions, by proving that the relationships linking the spectra with transverse emittance and betatron tune remain universally unchanged regardless of the shape of the rf waveform and chromaticity up to any order, while for the first-order chromaticity, we derive compact formulas for error estimates. The derived theory is supported by macroparticle simulations and applied to typical beam and machine parameters in the Large Hadron Collider.