Longitudinal diffusion as inflicted by arbitrary band-width random-modulated currents in feeders of detuned cavities

The diffusion coefficient for a bunched p-beam in a synchrotron is presented with two technically imposed items included into the scheme. (1) The role of basic external noise is attributed to random envelopes i^(a,φ)(t) carried by two modulated time-quadrature RF-currents. These represent amplitude, (a) or (small) phase, (φ) noises of a forward current in cavity feeders. (Commonly, the (a, φ)-noise voltages v^(a,φ)(t) at the accelerating gaps are taken as the basic ones). Phase shifts between spectral components v(ω) vs. i(ω) are inevitable due to both, the transfer function from feeder current to the detuned-cavity gap voltage, and the phasor diagram of the gap voltages maintained under beam loading. None of the `pure´ noises v^{(a/)(t) or v}sup (φ/)(t) is observable practically. Only their mixture is. (2) The possibility of an arbitrary ratio of the noise correlation time to the revolution period (i.e., the effect of noise spectrum to stretch over many revolution frequency harmonics) is incorporated, which is important to treat the noise-induced diffusion in the large rings (UNK, LHC, SSC)