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# Frequently Asked Questions

### Accelerator

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.

The beam-beam parameter is measuring the linear part of the deflection (for small offsets wrt center of the bunch). For example for the vertical plane, the beam-beam parameter is given by:

The beam-beam parameter can be expressed in terms of beam parameters. It depends on:

- the bunch population ~ N,
- the inverse of the energy ~ 1/gamma,
- the betatron function at the IP ~ beta*,
- the inverse of the beam size (sigma) products ~1/(sigma
_{x}*(sigma_{x}+sigma_{y})) for horizontal plane x, ~1/(sigma_{y}*(sigma_{x}+sigma_{y})) for vertical plane y.

For flat beams (sigma_{x} >> sigma_{y}) the vertical beam-beam tune parameter can be approximated by:

Because the beam-beam force is non-linear, it also introduces a spread of the tunes (--> beam-beam tune shift).

**In a simple model:** as long as the beam-beam force is below a critical value (the beam-beam limit), nothing happens. Above the limit the beam sizes start to blow up such a to maintain the beam-beam parameter equal to or smaller than the beam-beam limit value. Large tails develop, the lifetime goes down due to the tails. In that regime (at or above beam-beam limit) the luminosity no longer scales with N^{2}, but only with N (due to the increase in beam size). Indeed one can express the luminosity as:

When the beam-beam parameter is constant, the luminosity ~N !

In general one tries to operate the machine just below or at the beam-beam limit.

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/E^{3}. 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):

Since the beam size is given by (z=x,y, epsilon is the emittance):

The beam size grows roughly like s between IP and first quadrupole.

When the betatron function b* at the IP (beta measures the beam envelope and is defined by the optics) is large compared to the bunch length, to a very good approximation the beam size is constant along the length of the colliding bunch (case of LEP beta* 5 cm for bunch length of 1 cm, and current LHC beta* of 60 cm for bunch length of 9 cm). For FCC-ee bunch length (2 mm) and beta* (1 mm) become comparable, as a consequence the beam size varies along the colliding bunch. Since head and tail have larger sizes, the luminosity is reduced as compared to the case of constant beam size. The H factor measures this loss of luminosity.

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.

Wigglers are employed to increase the damping and lengthen bunches (for example at injection energy) in order to stabilize the beams. They can also be employed to control the emittance (mainly the horizontal one) of the beam by providing a controlled emittance increase. This may be useful in the presence of beam-beam effects to control the blow and therefore the beam-beam parameter of the beams.

A more sublte way to use the wiggler is by using for example a field B0 in two outer dipoles and a field 2B0 in the inner magnet. The total length of the 2 magnets with field B0 and of the magnet with field 2B0 is the same. In that case there is no net deflection as in any other wiggler. With such an asymmetric arrangement it is possible to speed up the build-up of polarization. The price to pay is increased energy spread (can be a problem) and a lower asymptotic polarization level.

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 10^{12} 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).

At LEP, the energy calibration was performed after the collision periods, and the measurement had to be extrapolated back to the runnning conditions, with the corresponding systematic uncertainty of 2 MeV on the beam energy. At the FCC-ee, the number of bunches is large enough (more than 10,000 at the Z pole!) to select of few pilot bunches - say 100 bunches - which would not collide, and on which resonant depolarization could be applied while the other bunches are colliding. The systematic uncertainty related to the extrapolation would therefore disappear, and each of the 100 measurements would have a precision better than 0.1 MeV on the beam energy.