Design of beam optics for the future circular collider $e^{+}{e}^{-}$ collider rings

A beam optics scheme has been designed for the future circular collider-$e^{+}{e}^{-}$ (FCC-ee). The main characteristics of the design are: beam energy 45 to 175 GeV, 100 km circumference with two interaction points (IPs) per ring, horizontal crossing angle of 30 mrad at the IP and the crab-waist scheme [P. Raimondi, D. Shatilov, and M. Zobov, arXiv:physics/0702033; P. Raimondi, M. Zobov, and D. Shatilov, in Proceedings of the 22nd Particle Accelerator Conference, PAC-2007, Albuquerque, NM (IEEE, New York, 2007), p. TUPAN037.] with local chromaticity correction. The crab-waist scheme is implemented within the local chromaticity correction system without additional sextupoles, by reducing the strength of one of the two sextupoles for vertical chromatic correction at each side of the IP. So-called “tapering” of the magnets is applied, which scales all fields of the magnets according to the local beam energy to compensate for the effect of synchrotron radiation (SR) loss along the ring. An asymmetric layout near the interaction region reduces the critical energy of SR photons on the incoming side of the IP to values below 100 keV, while matching the geometry to the beam line of the FCC proton collider (FCC-hh) [A. Chancé et al., Proceedings of IPAC’16, 9–13 May 2016, Busan, Korea, TUPMW020 (2016).] as closely as possible. Sufficient transverse/longitudinal dynamic aperture (DA) has been obtained, including major dynamical effects, to assure an adequate beam lifetime in the presence of beamstrahlung and top-up injection. In particular, a momentum acceptance larger than $\pm 2\%$ has been obtained, which is better than the momentum acceptance of typical collider rings by about a factor of 2. The effects of the detector solenoids including their compensation elements are taken into account as well as synchrotron radiation in all magnets. The optics presented in this paper is a step toward a full conceptual design for the collider. A number of issues have been identified for further study.

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Figure 1

The left figure shows the schematic layout of the FCC-ee collider rings. The two rings are horizontally separated by 0.6 m in the arc, and their center is placed on the center of the FCC-hh hadron rings. The straight sections correspond to the hadron ring, shown in the right figure. Two IPs are located in the straight sections A and G, and the rf sections are located in D and J. There exist intermediate straight sections B, F, H, L in the arcs. Beams cross over in the rf sections. The south IP (G) is shown enlarged in the middle of the left figure. The green line indicates the FCC-hh beam line, along with also the $e^{+}/e^{-}$ booster synchrotron for FCC-ee can be placed.

Figure 2

The beam optics of the arc unit cell of FCC-ee. Upper and lower plots show $\sqrt{{\beta }_{x,y}}$ and dispersions, respectively. Here we show a supercell consisting of 5 $90°/90°$ FODO cells. The horizontal/vertical sextupoles SF1.1/SD1.1 are paired with SF1.2/SD1.2 via $-I$ transformations.

Figure 3

The beam optics of the FCC-ee IR, corresponding to ${\beta }_{x,y}^{*}=(1\mathrm{m},2\mathrm{mm})$. Upper and lower plots show $\sqrt{{\beta }_{x,y}}$ and dispersions, respectively. The beam passes from the left to the right in this figure. The optics is asymmetric to suppress the synchrotron radiation toward the IP. Dipoles are indicated by yellow boxes, and those in region (e) have a critical energy of the SR photon below 100 keV at the $t\overline{t}$. Sextupoles for the LCCS are located at (a–d), and sextupoles at (a,d) play the role of crab sextupoles.

Figure 4

The beam optics of FCC-ee around the IP. Plots show $\sqrt{{\beta }_{x,y}}$ (top), horizontal/vertical dispersions (middle), and the solenoid field (bottom), respectively. QC1/QC2 denote the vertical/horizontal focusing quadrupoles. The distance ${\ell }^{*}$ between the face of QC1 and the IP is 2.2 m. The vertical dispersion is locally confined.

Figure 5

The beam optics of the FCC-ee half ring, corresponding to ${\beta }_{x,y}^{*}=(1\mathrm{m},2\mathrm{mm})$. Upper/lower plots show $\sqrt{{\beta }_{x,y}}$ and horizontal/vertical dispersions, respectively. These plots start and end in the middle of the rf sections, and the IP is located at the center. Sections marked by (a,b) correspond to the intermediate straight sections B, F, H, L in Fig. 1.

Figure 6

Dynamic apertures after an optimization of sextupoles via particle tracking. (a, c): ${\beta }_{x,y}^{*}=(1\mathrm{m},2\mathrm{mm})$, 50 turns at $t\overline{t}$, (b, d): ${\beta }_{x,y}^{*}=(0.5\mathrm{m},1\mathrm{mm})$, 2,650 turns at $Z$. (a, b): $z-x$ plane with $J_y/J_x=0.2\%$ for (a) and 0.5% for (b). (c, d): $x-y$ plane with $\delta =\mathrm{\Delta }ϵ=0$. The aperture was searched both in (a,b) $x$ and $p_x$ or (c,d) $y$ and $p_y$ directions. The initial conditions are chosen according to Eqs. (1), (2). The number of turns is chosen to correspond to about 2 longitudinal damping times at each energy. The blue lines show the DAs required for the beamstrahlung and the top-up injection. Effects in Table 2 are taken into account except for the radiation fluctuation and the beam-beam effect.

Figure 7

Variation of the momentum compaction factor ${\alpha }_p$, momentum spread ${\sigma }_{ϵ}$, length of the dipole magnet ${\ell }_{\mathrm{D}}$, and focusing strengths of dipole, horizontal/vertical focusing quadrupoles $k_{\mathrm{D},\mathrm{QF},\mathrm{QD}}$ ($=B^{\prime }\ell /B\rho$) of $90°/90°$ FODO cell as functions of $J_z$. The horizontal equilibrium emittance is fixed equal to ${ϵ}_x=1.25\mathrm{nm}$ at 175 GeV.

Figure 8

The crab-waist scheme: Two beams cross at the IP with a horizontal crossing angle $2{\theta }_x$. The crab-waist scheme shifts the vertical waist of a beam (solid) in $s$-direction by $\mathrm{\Delta }s$ onto the center of the other beam (dashed).

Figure 9

Poincaré plots in $x-p_x$ (left) and $z-\delta$ (right) planes for a particle starting at $x=10{\sigma }_x$, $p_x=y=p_y=z=\delta =0$, depicted by the red dots. The numbers, 0, 1, 2 are turns. The synchrotron radiation loss in the quadrupoles excites the synchrotron motion as shown in the right plot. The amount of the energy loss in the first turn $\mathrm{\Delta }{p}_1$, and the peak amplitude of the synchrotron motion $\mathrm{\Delta }p$ agrees with the estimation, Eq. (c10).

Figure 10

The dynamic aperture in the $z-x$ plane without(left)/with(right) synchrotron radiation in the quadrupoles at 175 GeV. The parabolas on the left show the amplitude of the synchrotron motion given by Eq. (c10). For a value of $n=\mathrm{\Delta }x/{\sigma }_x$, if an unstable region exists within these parabolas, then the motion larger than $n$ becomes unstable.

Figure 11

The dynamic aperture in the $z-x$ plane at 175 GeV, ${\beta }_{x,y}^{*}=(1\mathrm{m},2\mathrm{mm})$, with synchrotron radiation fluctuation. The lines correspond to the aperture where 75% of 100 samples were stable for 50 turns. The error bars indicate the range of stability between 50% and 100% for 100 samples. The case without fluctuation is plotted in Fig. 6.