Self-consistent simulations of beam-beam interaction in future $e^{+}{e}^{-}$ circular colliders including beamstrahlung and longitudinal coupling impedance

For the past generation $e^{+}{e}^{-}$ storage ring colliders, we usually used natural bunch length or its impedance lengthened value in beam-beam simulations instead of considering the impedance directly. In the future colliders, such as FCC-ee and CEPC, the beam-beam interaction becomes essentially three dimensional. In order to increase the luminosity, the future accelerators will collide very intense beams of high energy with low emittances and small beta functions at the collision points exploiting the crab waist collision scheme with a large Piwinski angle. For these extreme parameters several new effects become important for the collider performance such as beamstrahlung, coherent X-Z instability, 3D flip-flop so that the longitudinal beam dynamics should be also treated in a self-consistent manner. In this paper we describe the numerical code for the self-consistent 3D beam-beam simulations including beamstrahlung and the longitudinal beam coupling impedance and study interplay of different effects arising in beam-beam collisions of the future colliders.

Figure 1

Longitudinal distribution (collision off) obtained using elegant and ibb considering the CEPC longitudinal impedance model.

Figure 2

The longitudinal impedance of CEPC.

Figure 3

The horizontal beam size growth rate versus horizontal tune with and without longitudinal coupling impedance (ZL). Beamstrahlung (BS) effect is turned on.

Figure 4

Evolution of RMS energy spread considering longitudinal wakefield (ZL) with and without collision. Beamstrahlung effect is turned on during collision.

Figure 5

The horizontal beam size growth rate versus horizontal tune with and without beamstrahlung effect (BS). The stable equilibrium bunch length and energy spread is same between the two cases, where ${\sigma }_{z,\mathrm{SR}}$ and ${\sigma }_{p,\mathrm{SR}}$ are tuned in latter case.

Figure 6

The horizontal beam size growth rate versus horizontal tune with and without longitudinal coupling impedance (ZL), where beamstrahlung (BS) effect is turned off in both cases. The stable equilibrium bunch length and energy spread is same between the two cases, where ${\sigma }_{z,\mathrm{SR}}$ and ${\sigma }_{p,\mathrm{SR}}$ are tuned in latter case.

Figure 7

Incoherent synchrotron tune with and without longitudinal wake (ZL), where beamstrahlung (BS) effect is turned off. The stable equilibrium bunch length and energy spread is same between the two cases, where ${\sigma }_{z,\mathrm{SR}}$ and ${\sigma }_{p,\mathrm{SR}}$ are tuned in latter case.

Figure 8

Growth rate of horizontal beam size during bootstrapping injection for different tune. The left/right figure is for the case with and without impedance respectively.