Two-beam resistive-wall wake field

In storage-ring colliders, two beams propagating in opposite direction share a common beam pipe over parts or all of the ring circumference. The resistive-wall wake field coupling bunches of these two beams is different from the conventional single-beam wake field. The magnetic force and the longitudinal electric force experienced by a probe bunch invert their sign, while the transverse electric force does not. In addition, the distance between driving and probing bunches is not constant, but the net wake field must be obtained via an integration of the force experienced over the drive-probe distance. We here derive the two-beam resistive-wall wake field for a round beam pipe.