2.1 Requirements and design considerations
The FCC-ee lepton collider is designed to provide e+e− collisions with centre-of-mass energies from 88 to 365 GeV. The centre-of-centre-of-mass operating points with most physics interest are around 91 GeV (Z pole), 160 GeV (W± pair-production thresh-old), 240 GeV (ZH production) and 340–365 GeV (tt threshold and above). The machine should accommodate at least two experiments operated simultaneously and deliver peak luminosities above 1 × 1034cm−2s−1per experiment at the tt threshold and the highest ever luminosities at lower energies.
The layout of the FCC-ee collider follows the layout of the FCC-hh hadron collider infrastructure, which has been developed with a view to its integration with the existing CERN accelerator complex as injector facility. As is the case for the hadron collider, beam with adequate quality can be provided by an upgrade of the existing injector complex. Alternatively, a dedicated optimised injector could be built. Care has been taken to ensure easy implementation of transfer lines from the SPS to the future collider tunnel.
2.2 Layout and key parameters 2.2.1 Layout
The design goal is to maximise the luminosity for each energy under the following constraints:
– Apart from ±1.2 km around each interaction point (IP), the machine should follow the layout of the 97.75 km circumference hadron collider [9].
– There should be two interaction points, located in the straight sections at PA and PG as shown in Figure2.1.
– Synchrotron radiation power should be limited to 50 MW/beam at all energies.
Figure2.1 shows the layout of the FCC-ee together with FCC-hh. For FCC-ee, the design principles are
– A double ring collider.
– A horizontal crossing angle of 30 mrad at the IP, with the crab waist collision scheme [11,12].
– The critical energy of the synchrotron radiation of the incoming beams towards the IP is kept below 100 keV at all beam energies.
– A common lattice for all energies, except for a small rearrangement in the RF section for the tt mode. The betatron tune, phase advance in the arc cell, final focus optics and the configuration of the sextupoles are set to the optimum at each energy by changing the strengths of the magnets.
– The length of the free area around the IP (`∗) and the strength of the detector solenoid are kept constant at 2.2 m and 2 T, respectively, for all energies.
– A “tapering” scheme, which scales the strengths of all magnets, apart from the solenoids, according to the local beam energy, taking into account the energy loss due to synchrotron radiation.
– Two RF sections per ring placed in the straight sections at PD and PJ. The RF cavities will be common to e+ and e− in the case of tt.
– A top-up injection scheme to maintain the stored beam current and the luminosity at the highest level throughout the physics run. It is therefore necessary to have a booster synchrotron in the collider tunnel. The integrated luminosity will be
Fig. 2.1. The layouts of FCC-hh (left), FCC-ee (right), and a zoomed view of the trajectories across interaction point PG (right middle). The FCC-ee rings are placed 1 m outside the FCC-hh footprint in the arc. In the arc the e+ and e− rings are horizontally separated by 30 cm. The main booster follows the footprint of the hadron collider. The interaction points are shifted by 10.6 m towards the outside of FCC-hh layout. The beams coming toward the interaction points are straighter than the outgoing ones in order to reduce the synchrotron radiation at the IP.
reduced by more than an order without the top-up, due to ramping (∼1/2), reduction of the beam–beam parameter (∼1/2 − 1/4), lower beam current (∼1/2) at a lower injection energy, loss of stability of the machine (∼1/2), etc.
The FCC-ee inherits two important aspects from the previous generations of e+e− circular colliders. At and above the tt threshold, the FCC-ee will encounter strong synchrotron radiation with the associated rapid damping. This situation is reminiscent of earlier high-energy colliders, especially LEP2. By contrast, at the Z pole, FCC-ee will operate with much less damping, but with a high beam current and a large number of bunches. This mode of operation mode was successfully established by several high-intensity colliders, such as the two B factories and DAΦNE.
There are two reasons for choosing a double-ring collider. Firstly, at low energies, especially at Z, more than 16 000 bunches must be stored to achieve the desired luminosity. This is only possible by avoiding parasitic collisions with a double-ring collider. Secondly, at the highest energy tt, although the optimum number of bunches reduces to ∼30, the double ring scheme is still necessary to allow “tapering” [162].
The local energy of the beam deviates by up to ±1.2% between the entrance and the exit of the RF sections, with the result that the orbit deviation due to the horizontal dispersion in the arc and the associated optical distortion becomes intolerable. The optics may even fall into an unstable region. The tapering scheme restores the ideal orbit and optics almost completely. In the case of a single ring, the tapering scheme cannot be applied to the e+e− beams simultaneously.
The number of IPs is restricted by the current layout choice for the straight sections in the FCC-hh. The straight sections around PD and PJ do not have large caverns for detectors. The intermediate straight sections at PB, PF, PH and PL are placed asymmetrically in the arcs and are not a suitable location for the FCC-ee RF cavities. Therefore, with this twofold ring symmetry, two IPs are the only solution for the FCC-ee. (It has, however, been demonstrated that four IPs would be possible for a ring of four-fold symmetry.) The resulting beam optics [162] have a complete
periodicity of two. The beam lines for e+ and e− have a mirror symmetry with respect to the line connecting the two IPs and the beam optics are identical.
The crab waist scheme [11,163,164] is essential to boost the luminosity by more than four orders of magnitude at Z, compared to previous colliders. This scheme gives a very small beam size at the IP together with a large crossing angle and small emittances, without exciting harmful synchrotron-betatron resonances associated with the crossing angle [164]. This scheme simply needs a pair of static sextupole magnets at both sides of the IP. These sextupoles are incorporated in the local chromatic correction system (LCCS) [162]. The effect of the crab waist is produced by reducing the strengths of some sextupoles in the LCCS, so there is no need for special hardware. The optimum parameters with the crab waist scheme including β∗s, bunch intensity, bunch length, etc., are obtained by the procedures described in the next section. The optimisation takes into account beamstrahlung – synchrotron radiation caused by the coherent EM field of the opposite bunch [164–168] – and various other beam-beam effects.
The layout around the IP including the crossing angle, the strengths of solenoids and beam pipes are common for all energies. The polarity as well as the strengths of final quadrupoles change according to the beam energy and optimum focusing.
2.2.2 Beam parameter optimisation
One of the main factors determining collider performance is the beam-beam inter-action, which at high energies can gain an extra dimension due to beamstrahlung.
FCC-ee will be the first collider where beamstrahlung plays a significant role in the beam dynamics. Only half of the ring with one IP will be discussed in this section, because the other half will behave in the same way due to symmetry. To avoid confusion, the half-ring tunes will be marked by the superscript∗.
The luminosity per IP for flat beams (σx σy) can be written as:
L = γ
2ere ·Itotξy
β∗y · RHG, (2.1)
where Itot is the total beam current which is in this case determined by the synchrotron radiation power limit of 50 MW per beam. Therefore L can only be increased by making the vertical beam–beam parameter ξy larger and β∗y smaller while keeping the hour-glass factor RHG reasonably large. The latter depends only on Li/βy∗ratio, where Li is the length of interaction area which in turn depends on the bunch length σz and Piwinski angle φ:
φ = σz
here θ is the full crossing angle, see Figure 2.2. The beam–beam parameters for θ 6= 0 become [169]:
Fig. 2.2. Sketch of collisions with a large Piwinski angle.
where Np is the number of particles per bunch. Note that ξx ∝ 1/εx (in head-on collision) transforms to ξx∝ βx∗/σ2zwhen φ 1, and ξydependence on σxvanishes.
In the following, the main parameters that need to be optimised are listed:
– The vertical emittance should be as small as possible, but there are two restric-tions: εy≥ 0.002 · εxand εy≥ 1 pm.
– At Z there is some contribution to εy (0.2–0.3 pm) coming from the detector solenoids. It follows that εx should also be minimised, but there is no gain by reducing it below 0.4 nm.
– An important parameter for the luminosity is βy∗, whose minimum value is 0.8 mm and which is limited by the dynamic aperture.
– It is assumed that ξy can be easily controlled by Np, which implies that the number of bunches is adjusted to keep Itot unchanged.
– Finally, it should be noted that βx∗, the RF voltage (which determines the bunch length and the synchrotron tune), and the betatron tunes are relatively free parameters.
Optimisation at the Z pole
Since the FCC-ee is designed for a wide range of beam energies, parameter optimi-sation looks different at various energies. To find the good working points at the lowest energies (44–47 GeV), a scan of betatron tunes was performed in a simplified model: linear lattice, and weak-strong beam-beam simulations (without coherent instabilities). The results are presented in Figure2.3. Since ξx ξy, the footprint looks like a narrow vertical strip, with the bottom edge resting on the working point.
Particles with small vertical betatron amplitudes have maximum tune shifts and are in the upper part of the footprint, so that the strong resonances of Figure2.3, such as Q∗x+ 2Q∗y= n, come closer. Thus the good region is reduced to the red triangular area bounded by the main coupling resonance Q∗x− Q∗y= n, sextupole resonance Q∗x+ 2Q∗y = n, and half-integer resonance 2Q∗x= 1 with its synchrotron satellites.
All other higher-order coupling resonances are suppressed by the crab waist and, therefore, not visible. As seen from the plot, the range of permissible Qx for large ξy is bounded on the right by 0.57–0.58.
At the Z pole and the WW threshold, the main problems associated with the beam interaction come from the two new phenomena found in beam-beam simulations: coherent synchrotron-betatron (x−z) instability [170–172] and 3D flip-flop [172], the latter occurring only in the presence of beamstrahlung. Both instabilities are bound with the horizontal synchro-betatron resonances, satellites of half-integer. In any case, it is necessary to move away from low-order resonances, so Q∗x is chosen close to the upper limit (thus Qx,y move further away from the integer, which facilitates tuning of linear optics). Another requirement is that ξx must be substantially less than the distance between neighbouring satellites, which is equal to the synchrotron tune Q∗s. In other words, it is necessary to reduce the ratio ξx/Q∗s.
Fig. 2.3. Luminosity at Z as a function of betatron tunes. The colour scale from zero (blue) to 2.3 × 1036cm−2s−1 (red). The white narrow rectangle above (0.57, 0.61) shows the footprint due to the beam-beam interaction. A few synchrotron-betatron resonance lines Q∗x− mQ∗s = n/2 are seen.
The first step is to reduce βx∗. However, because of the absence of local horizontal chromaticity correction in the interaction region, attempts to make βx∗too small lead to a decrease in the energy acceptance. β∗x can be reduced to 15 cm at Z, but this is not enough to suppress the instabilities. The next step is to reduce ξxfor a given βx∗, whilst trying to keep ξy unchanged. This can only be done by increasing σz. The most efficient way is to increase the momentum compaction factor αp, because not only does ξx decrease (due to larger σz) but also Q∗s grows. In addition, larger αp raises the threshold of microwave instability to an acceptable level. The only drawback of this approach is that the horizontal emittance εxgrows with the power of 3/2 with respect to αp. For the luminosity, εx is not so important by itself, but εy should be small and it is normally proportional to εx. However, the horizontal emittance at Z with small αp and FODO arc cells with 90◦/90◦ phase advances is small – less than 90 pm. Therefore, even a threefold increase still allows achieving the design vertical emittance εy = 1 pm. Thus, the FCC-ee features a lattice where doubling of αp is achieved by reducing the phase advance per FODO cell in the arcs to 60◦/60◦, see Section2.4.1.
Turning to the dependence on RF voltage: σz∝ 1/√
VRF, Q∗s∝√
VRF. The requirement to keep ξyunchanged means that Np/σz is held constant. Therefore, if
VRFis lowered, ξxdecreases inversely with σz(and not with the square of the inverse bunch length, as it might have seemed at first glance). As a result, ξx/Q∗s does not change, but by lowering Q∗s the order of synchro-betatron resonances located in the vicinity of working point is increased. For this reason VRF is made small and one can find betatron tunes where neither instability manifests itself. For example, the working point is located between high order synchrotron-betatron resonances 2Q∗x− 10Q∗s= 1 and 2Q∗x− 12Q∗s= 1.
At low energies beamstrahlung leads to a significant increase in the energy spread and, correspondingly, the bunch lengthening. If Npis large enough to achieve high ξy, then σzbecomes several times larger; in this case it scales as σz∝pNp. Accordingly, ξy and luminosity also grow ∝ pNp while ξx remains constant. This means that increasing Np does not reach the instability threshold, but only increase the energy spread. In general, Npcan be limited by several factors: ξy, beam lifetime (depends on the energy spread and energy acceptance), and the impedances. The result is close to all these limits, which corresponds to a full exploitation of the available margins.
Optimisation at the WW threshold
As the energy increases to ∼80 GeV, σxgrows due to the synchrotron radiation and the bunch lengthening due to the beamstrahlung decreases, therefore the Piwinski angle drops. In addition, the damping decrement grows with γ3. All this leads to an increase in the instability threshold. For example, at W± it is already possible to work in a lattice with small momentum compaction. However, there is one more important requirement: in order to obtain a resonant depolarisation, which is neces-sary for the energy calibration, the synchrotron tune Q∗s must be larger than 0.025 (see Sect. 2.7). To achieve such a Q∗s value, the momentum compaction has to be increased. Therefore, the same 60◦/60◦lattice was chosen as for Z. Furthermore, the RF voltage must be increased to 750 MV, so the only window for a good working point can be found between 2Q∗x− 4Q∗s= 1 and 2Q∗x− 6Q∗s= 1. In order that insta-bilities do not arise near these resonances, βx∗≤ 20 cm is required. Here it should be noted that with increasing energy, obtaining small beta functions becomes more difficult as this leads to a reduction in the dynamic aperture and momentum accep-tance. Consequently, βy∗was increased to 1 mm. To obtain the higher VRF required, the single-cell cavities used at Z will be replaced by multi-cell ones, whose capacity to damp the higher order modes (HOM) is limited. An important consequence is that the number of bunches should not be smaller than 2000 and, therefore, the luminosity at W± is limited by this factor.
The possibility of increasing Q∗sfurther to 0.0375 at W± was also considered, in accordance with the desire to improve the conditions for resonant depolarisation. In this case Q∗xfalls between low order resonances 2Q∗x− 2Q∗s= 1 and 2Q∗x− 4Q∗s= 1.
To avoid coherent instabilities it is necessary to reduce βx∗to 15 cm. The momentum acceptance drops accordingly and, as a consequence, luminosity decreases. On the other hand, the number of bunches for this option is larger (2500), though they are shorter. This option is not worse for HOM and the luminosity is about the same as for 2000 bunches with Q∗s= 0.025. However, obtaining Q∗s= 0.0375 would require twice the RF voltage VRF and, thereby, a revised RF staging scenario. Therefore, the current baseline is Q∗s= 0.025.
Optimisation at the ZH cross-section maximum
Polarisation is not an issue at a beam energy of 120 GeV (ZH production) and the optimum parameters are selected as follows:
1. The 90◦/90◦ lattice, which provides naturally smaller emittances.
2. The RF voltage is made as small as possible, but adjusted so that the RF accep-tance (bucket height) is larger than the energy accepaccep-tance due to dynamic aper-ture, resulting in Q∗s≈ 0.018.
3. Q∗xis selected in the range of 0.56–0.58 with the condition that Q∗x≈ 0.5 + Q∗s· (m+0.5) in order to be separated from the low-order synchro-betatron resonances and Q∗y= Q∗x+ (0.03–0.04).
4. A βx∗ at which the coherent instabilities disappear is then sought; in this case, 30 cm is enough.
5. With the given εx and β∗x, the length of interaction area Li≈ 0.9 mm, and this defines the optimum β∗y. However, obtaining small βy∗ at higher energies is more difficult, so 1 mm was chosen.
6. The lattice optimisation was performed for the selected β∗ in order to maximise the dynamic aperture and energy acceptance.
7. A fine scan of betatron tunes was performed to choose the working point more precisely.
8. Then quasi-strong-strong beam-beam simulations were performed with an asym-metry of 3% in the bunch currents (3% is determined by the required beam lifetime and the injection cycle time). At energies W±, ZH, and tt, single high-energy beamstrahlung photons become important and impose a limit on Np. The bunch population is scanned, while the restriction is the lifetime of the weak (less populated) bunch. The maximum Npand luminosity are determined in this way.
Optimisation at the tt threshold and above
At the tt production (beam energy from 170 to 182.5 GeV) the coherent instabilities are suppressed by very strong damping, but another problem becomes dominant:
the lifetime limitation by single high-energy beamstrahlung photons [168]. Thus, in contrast to low energies, βx∗ should be increased in order to make σx larger and thereby weaken the beamstrahlung. With increased σx, Li≈ 1.8 mm is obtained, and βy∗ should be about the same (or slightly smaller). It should be noted that an increase in εxis not profitable since a small εy is needed for high luminosity, so the 90◦/90◦ lattice is used.
2.3 Design challenges and approaches
Based on combinations of existing technologies for e+e− circular colliders developed through the last half century, the FCC-ee will achieve the best ever luminosities at each energy. Although some components need final touches to their design or proto-typing in the phase after the CDR, the fundamental feasibility of their construction has already been proven in other colliders and storage rings.
2.3.1 Synchrotron radiation
The synchrotron radiation (SR) is a key feature for any e+e−storage ring. It is worth comparing the characteristics of FCC-ee with those of LEP2, the highest energy e+e− ring ever operated and PEP-II high energy ring, one of the e+e− colliders with the highest beam current (see Tab.2.2).
While the total radiation power is higher than that of LEP2 by a factor of 4, the critical energy and the energy loss per arc length are only 20% and 10% higher,
respectively. The power dissipation per arc length is less than 1/4 of that at PEP-II.
The level of synchrotron radiation can therefore be handled by existing technology.
Another aspect of the SR is the radiation towards the detector at the IP. This
Another aspect of the SR is the radiation towards the detector at the IP. This