The beam-beam parameter (often also called beam-beam tune shift) characterizes the strength of the beam-beam force in a given plane on the particles of the opposing beam. The figues below show the deflection by the beam-beam force for the case of LEP (flat beams). Note the difference between x and y due to the fact that the horizontal beam size is much larger than the vertical size.

Longitudinal damping time measures the time to damp a longitudinal plane (energy offset - longitudinal coordinate along the bunch) oscillation of a particle. For a centered orbit and a FODO lattice it is 2 as fast as the damping of transverse oscillations. The damping time scales with beam energy as 1/E³. The transverse damping times are given in the FCC-ee parameter table.

Around an IP the transverse beam size grows as the distance to the IP (focus) increases. The betatron function evolves in the drift to the first quadrupole as (where s is the distance from the IP) - here example for the vertical plane (for hor. plane replace y by x):

In general a wiggle is a sequence of small dipole magnets with alternating magnetic field (typically +-+-+- etc). The simplest wigglers have a sequence of N alternating + and - poles.

The synchrotron radiation (SR) induced by the wiggler adds to the total SR energy loss of the ring.

It is the time to ramp the energy of the machine. For FCC-ee it is not relevant since FCC-ee operates at constant energy and the beam are topped-up continously from a booster ring. The booster ring has to deliver on average around 10¹² e⁺ and e^- per second. The booster is cycling its energy between its injection energy (12-40 GeV - to be defined) and the operating energy of FCC-ee. The cycle time is roughly 5 seconds - details to be worked out.

Here is an extract of a lecture I gave in the 2014 Frascati Spring school, which explains the principle of energy calibration with resonant depolarization at LEP in some details (as well as the effect of the moon on the measurement).

This question can not really be answered with e+e- machines such SR is intrisic to such beams. By definition one has to design the machine with the SR, which increases proportionally to E⁴ and 1/R, where E is the energy and R the radius.  A larger machine will have less SR and require less RF voltage and power.

It is the same thing. Beamstrahlung is the bremstrahlung emitted by a particle in the electromagnetic field of the counter-rotating beam.

The energy spread for electron beams is due to the fluctuations of the synchrontron radiation (SR).

DIfferent elements contribute to the energy spread:

• The SR in the arc section (unavoidable),
• The SR in wiggler magnets,
• The beamstrahlung at the IP due to the emission of SR in the strong EM field of the opposing beam.

The beamstrahlung lifetime is mitigated by:

• increasing the bunch length (provided it does not lower the luminosity through the hourglass effect),
• increasing the horizontal beam size (but not too much since it lowers the luminosity),
• reducing the single bunch current.