Closed Orbit Distortion and the Beam-Beam Interaction Page: 3 of 15
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The analysis presented here follows that of Hirata and Keil , suitably augmented to include a closed
orbit bump at the interaction point (IP). Our presentation is deliberately quite explicit in the hope that this
study will be useful in further analyses or in B-factory-related experiments.
2.1 Simplifying assumptions.
We assume that there exists a closed orbit bump that splits the closed orbits apart vertically by a distance d in
the immediate neighborhood of the IP. For our purposes, it does not matter how this distance is apportioned
between the e+ and the e beams as long as the total separation of the nominal orbits adds up to d. In
the context of this note, this orbit bump is a calculational device that provides a convenient variable to test
the sensitivity of the closed orbit to the beam-beam force. In practice, such a bump would be intentionally
implemented with appropriate magnets or electrostatic beam separators. We assume that this orbit bump
is nominally closed, i.e., that in the absence of the beam-beam force the orbits coincide exactly with the
nominal orbits in the region "outside" the bump. Because of the beam-beam interaction, however, there
is a residual closed orbit distortion everywhere in the ring. The situation is sketched in Fig. 1. The basic
objective in this note is to compute this residual orbit distortion as a function of d and other parameters.
For the purposes of this section we make the following assumptions:
1. The orbit bump is nominally closed, and exists only in the immediate neighborhood of the IP. The
orbits are parallel-displaced by a distance d in the vertical direction only.
2. The bunches are not tilted.
3. All effects of any parasitic crossings are ignored.
4. The beam sizes are independent of d and have their nominal values.
5. The beam-beam interaction is treated in the impulse (thin-lens) approximation.
6. For the purpose of computing the beam-beam kick, the particle distributions are assumed Gaussian.
The analytical calculation presented in this section addresses only the coupled dipole mode of the beams
(rigid-Gaussian approximation). This calculation can be easily extended to the case in which the orbits are
displaced in an arbitrary direction rather than vertically, and in which the beams are tilted in the transverse
plane . We do not consider these generalizations in this note.
We will remove assumptions 4 and 5 in Sec. 4 by resorting to strong-strong multiparticle tracking simu-
lations, in which the beam sizes are determined dynamically and the beam-beam collision is treated in the
thick-lens approximation. Assumptions 3 and 6, however, will remain in force even then. An extension of
these simulations to include parasitic crossing collisions is straightforward and will be presented separately.
The importance of allowing for a self-consistent treatment of non-Gaussian particle distributions has been
recently emphasized ; an extension of our calculation along these lines remains to be investigated.
2.2 One-turn map.
We assume that the two rings are represented by linear maps. The rings intersect at the IP, which we choose
to be the origin for the azimuthal coordinate s for both rings. We imagine observing the beams at every turn
at a point immediately before the IP. The resultant map that relates turn n to turn n + 1 for an individual
particle at this surface of section is written
[x ]+ [AX,+ L (1)
. .+1 .Mya pf + Ad _
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Furman, M.; Chin, Y.; Eden, J.; /LBL, Berkeley; Kozanecki, W.; /DAPNIA, Saclay /SLAC et al. Closed Orbit Distortion and the Beam-Beam Interaction, report, February 23, 2007; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc887516/m1/3/: accessed December 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.