The main part of the beam-delivery line is the final-focus system which is characterized by a very large de-magnification. This means that the linear optics is dominated by the chromatic aberrations in the quadrupole doublets involved, aberrations which should be corrected by a minimal number of sextupoles separated from the quadrupoles by adequate betatron phases. In addition, in order to preserve the cleanness of the colliding beams and to limit backgrounds and losses, the beam halo has to be eliminated by collimation before the interaction region.
Consequently, a so-called conventional main-beam delivery complex is made of three major subsystems which follow one another in the delivery line and are as independent as possible. Following the beam after its extraction from the main linac, the first subsystem is the collimation section made of several collimators separated by adequate betatron-phases and located at positions where the b-amplitudes are large to keep low beam densities at the absorbers. Momentum collimation requires non-zero dispersion and dipole magnets in addition. All these needs lead to a section length which may be large and scales up with the collision energy. The second subsystem is the chromatic correction section which typically contains a minimum of two pairs of non-interleaved sextupoles and dipoles for generating some dispersion. Again this system requires high b-values at the sextupoles and may extend over a long distance which increases with the energy. The last subsystem is the telescope of the final focus, the length of which depends on the de-magnification, the required beam-size at the IP and the synchrotron radiation effect in the last doublet, which is linked to the beam energy and the length of the free space left between the two final doublets. In addition, and to be complete, a betatron-matching section in front of the chromaticity correction subsystem and an insertion for diagnostics might be necessary.
Such a basic beam-delivery line has been studied in detail for the Next Linear Collider (NLC) [2.28], and also for the Japan Linear Collider (JLC-I) [2.29] and TESLA [2.30]. In this context, laws for scaling the length of the different subsystems with the beam energy have been proposed [2.31] . Scaling from 1.5 to 3 TeV with the relative-energy variable Ur =(TeV)/1.5 and assuming a vertical b-function amplitude at the IP of the order of 0.1 mm, the law which has been obtained writes:
where LBD represents the total length (both sides of the IP) of the beam-delivery line and where the parts due to the final focus (FF) and to the collimation system (COLL) are separated.
Starting from this basic design, modifications have been worked out in other studies [2.32]. As a major difference, an asymmetric dispersion has been proposed in order to concentrate the sextupole chromatic effects into the second sextupole of the pair, which tends to reduce chromogeometric aberrations. Another change consists of using non-linear magnetic field elements in the collimation subsystem in order to limit the betatron amplitude there and reduce the chromaticity and the section length. With these modifications, the law proposed [2.33] for the scaling of the length from 0.5 to 3 TeV and written for a different relative-energy variable Ur =E (TeV)/0.5 looks as follows:
where the starting length comes from the 0.5 TeV design of the JLC beam delivery.
At 3 TeV, the first law gives a total length of the order of 21 km and the second of 10 km (value given in Fig. 1.1). More precise evaluation of the space needed can result only from design studies; the present status of the baseline design of the beam delivery is given below, more investigations remaining necessary.
The baseline proposed for the ensemble telescope and chromaticity-correction section, called final focus in Section 126.96.36.199, has a total length of 6.2 km (Fig.2.25). Since the system studied includes the concept of asymmetric dispersion, this should be compared to the result of the second scaling law after ignoring the term linked to the collimation. The two values are remarkably consistent, for this law gives LFF = 6 km.
A further significant reduction of the total final-focus length (both sides added) is only possible if the chromaticity correction is applied locally near the last doublets (the main source of chromaticity). There are essentially two ways which can be envisaged to achieve this. The first one, or short final-focus scheme [2.34] , is based on the generation of some finite dispersion across the final doublet (but of zero amplitude at the IP) and on the addition of sextupoles close to the doublet quadrupoles in order to compensate their chromaticity. The second one, or `ultra-short' final-focus scheme [2.35] , relies on the use of microwave quadrupoles placed near the magnetic quadrupoles of the doublet and playing a role similar to the one of the sextupoles, in the presence of a correlation between the momentum and the position along the bunch. The applicability of these schemes to CLIC at 3 TeV has still to be investigated before they can be adopted; in particular, questions related to the presence of finite dispersion, to flexibility and orthogonal tuning of the parameters in the first case, and to the tight tolerance about the energy variations required in the second.
The task of the final-focus system is to focus the two main beams to the transverse design spot sizes of 43 nm and 1 nm at the interaction point (IP), where the opposing beams are collided. The momentum bandwidth of the final-focus system should be of the order of 1% in order to accommodate the expected beam energy spread from the linac. Tolerances on magnet position and field stability are another critical issue in the final-focus design. Both optics development and tolerance analysis for the CLIC final-focus made use of the design program FFADA [2.36].
A baseline optics of a 3-TeV final-focus system is shown in Fig. 2.25 . It consists of horizontal and vertical chromatic correction sections (CCX and CCY) followed by a final transformer. The total length is 3100 m per side.
The final transformer, made from two quadrupole doublets, demagnifies the beam by a factor 15 horizontally, and 50 vertically. The gradient of the final quadrupole is taken to be 450 T/m. For comparison, the gradient achieved in an NLC permanent magnet prototype was 500 T/m [2.37] For a large-aperture superconducting quadrupole the gradient would be a 30% increase from present design values of 320 T/m [2.38]. With the assumed quadrupole gradient, the chromaticity of the final doublet is 6900 in the horizontal and 27 000 in the vertical plane, where chromaticity is defined as the relative spot-size increase (added in quadrature) divided by the r.m.s. energy spread. The beta functions at the entrance to the final quadrupole of 15 km horizontally and 88 km vertically correspond to r.m.s. beam sizes of 59mm and 24mm for the nominal CLIC parameters. This translates into a beam stay-clear of 50-140s for a permanent magnet with 3.3-mm bore radius and of 450-1100 s for a superconducting quadrupole with 2.7 cm radial aperture.
Fig. 2.25 : Beta function and dispersion
for the 3-TeV baseline final-focus system,
plotted as a function of the longitudinal position. The interaction point is on the right.
The large chromaticity of the final doublet is compensated in the two chromatic correction sections. Each of these comprises a pair of sextupoles, separated by a - I transformation and placed an integer multiple of p in betatron phase apart from the final doublet. The dispersion function is nonzero only at the second sextupole of each pair. An odd-dispersion optics like this [2.39] has two advantages: (1) it reduces the number of bending magnets and the amount of synchrotron radiation by a factor of 2, and (2) it avoids many of the fifth-order chromogeometric aberrations arising from the chromatic breakdown of the - I between the sextupoles, which limit the momentum bandwidth. Thanks to the - I separation the individual sextupole pairs do not generate third-order geometric aberrations. The second-order dispersion from the CCX is adjusted to cancel the second-order dispersion produced in the CCY. The ratio of dispersion values, or, alternatively, the ratio of bending angles in CCX and CCY is thus constrained.
In the present design the net bending angles for the dipole regions in CCX and CCY are 63 mrad and 230 m rad, respectively. The peak beta functions at the CCY sextupoles are about 1000 km, and the maximum value of the dispersion is 0.1 m.
The achievable luminosity is calculated with the FFADA code, which tracks two random sets of particles through the entire system to the interaction point, using MAD, and then convolves them on a grid. The luminosity was maximized for a 1% flat energy spread by varying the total length, the ratio of CCX and CCY lengths, the bending angles, and the strengths of the last two quadrupoles. Figure 2.26 displays the luminosity of the optimized system as a function of the full-width momentum spread, assuming a flat energy distribution. For the expected energy spread, close to 1%, the luminosity (without pinch) is about 80% of the ideal luminosity that would be attained for a perfectly linear and achromatic optics. Figure 2.27 depicts the dependence of the transverse r.m.s. spot sizes on the energy spread. The vertical spot size is about 30% larger than the ideal linear value, the blow-up being due to synchrotron radiation (SR) in the second-to-last quadrupole magnet Q2. The horizontal spot size for small energy spread is close to the ideal value, but it increases rapidly with increasing energy spread. The strength of Q2 has been adjusted such that for 1% energy spread the horizontal blow-up is similar in magnitude to the vertical one. Hence, the final parameter choice is a trade-off [2.40] between Oide effect [2.41], [2.42] (vertical beam size increase due to synchrotron radiation in the last quadrupoles, favouring a weak second-to-last magnet Q2) and the momentum bandwidth in the horizontal plane (demanding a small horizontal chromaticity and, thus, a strong quadrupole Q2).
The initial beta functions, at the entrance of the CCX, are about 1 m in both planes. Since the typical beta functions at the end of the linac are 18-65 m, an upstream beta-matching section will be required.
Fig. 2.26 : Relative luminosity loss as
a function of the full-width energy spread for a flat distribution.
The luminosity was calculated by tracking two random distributions of 5000 particles through the final focus to the interaction point (IP)
and there convolving them on a grid, not including beam-beam focusing forces. The ideal reference luminosity is L 0 = 4.6 ¥ 10 34 cm -2 s -1 .
Fig. 2.27 : Relative r.m.s. spot sizes
as a function of the full-width energy spread for a flat distribution.
The ideal linear spot sizes are s x = 43 nm and s y = 1.0 nm.
Figure 2.28 displays jitter and drift tolerances for the horizontal and vertical magnet positions. The jitter tolerances apply to pulse-to-pulse time scales. The tightest jitter tolerance is 0.2 nm for the last quadrupole. The drift tolerances refer to a time scale of several minutes and are of the order of 100 nm.
In Fig. 2.29, we present the tolerances on the magnet pitch angle and on the relative field stability. For the final quadrupole, the pitch jitter tolerance is 0.1 nrad. Typical field stability tolerances are 10-5, a value close to specifications for the LHC power supplies [2.43] .
Finally, it should be noted that a large crossing angle of 20 mrad (total) is required to suppress the multibunch kink instability by parasitic collisions around the IP and to provide sufficient space for the spent beam and collision debris, in particular for the opposite-charge pairs [2.44] . This means that crab-crossing cavities will have to be used to avoid a reduction of luminosity. However, the tolerance on the relative phase of the cavities is approximately 0.06° at 30 GHz (for a few per cent loss of L ).
Fig. 2.28 :Displacement sensitivities for 2% luminosity loss, calculated with the FFADA code. The full bars represent pulse-to-pulse `jitter' tolerances, due to both the induced orbit motion and the spot-size increase at the interaction point. This jitter can be corrected within a few pulses using a fast orbit feedback. The tightest jitter tolerances are about 3 nm (x) and 0.2 nm (y). The open bars are `drift' tolerances referring to increases in the IP beam size only. Since the beam size tuning will be performed only every couple of minutes, the drift tolerances must be met over a longer time-scale. Drift tolerances are of the order of 100 nm.
Fig. 2.29 : Sensitivities to pitch angle (left) and relative field changes (right), calculated with the FFADA code. Again, the full and open bars represent jitter and drift tolerances, respectively. The tightest pitch angle jitter tolerance is 0.1 nrad for the final quadrupole. Field stability tolerances are about 10 -6 .
The collimation system should serve two different purposes: (1) remove the beam halo which otherwise would cause unacceptable background in the particle-physics detector, and (2) protect the downstream systems against the impact of a missteered beam. Point (1) is achieved if the collimator shadows the final doublet apertures on the incoming side. To this end, the collimation aperture should be about 25 s horizontally and 80 s vertically [2.45] , in case the final quadrupole is a permanent magnet, or 400 s and 1000 s for a superconducting final quadrupole with a large bore. The apertures on the outgoing side need not be shadowed by the collimation, since the incoming halo is tiny compared with the wide-angle debris coming from the collision point [2.46] .
If a first beam-halo collimation is performed prior to injection into the main linac, the halo at the entrance to the final focus, due to all known scattering sources, is estimated to be of the order of only 10 3 or 10 4 particles per bunch [2.47] and[2.48] . Thus there may not be a need for a dedicated separate collimation section. Instead the collimators might be installed in the final focus itself, since for every few 10 4 scraped particles a single muon is generated, and the detector should be able to cope with several hundred muons passing it per bunch train. Collimation in the final focus will considerably shorten the overall system length, and, in addition, it will profit from the naturally large final-focus beta functions.
The collimators survive the impact of one entire bunch train if the following condition is fulfilled: sUTS > aE/Cp dE /dm , where sUTS is the ultimate tensile strength, a the linear thermal expansion coefficient, Cp the heat capacity, E the elastic modulus, and d E /d m the energy loss per gram of material. This condition can be rewritten as sx sy > a E /( sUTS Cp ) dE /dx kb Nb/(2p ) where dE /dx is the loss of energy per unit of length. Assuming a copper collimator, with E = 120 GPa, a =1.7 ¥ 10-5 K-1, Cp = 0.385 Jg-1 K-1, dE/ (r dx) = 1.44 MeV cm2g-1 (r =material density), and sUTS = 300 MPa, we find (sx sy)1/2 > 200mm, or bx,y >1000 km. Materials with a smaller product a E would be better suited for collimation. We are studying the possible use of carbon composites. For many conceivable failure modes, the emittance of a missteered beam will be significantly blown up in the linac, and a smaller b-function would suffice.
As a back-up option we may investigate a nonlinear collimation system à la KEK [2.49] or TESLA [2.48]. Also such a nonlinear collimation could be integrated into the final focus, e.g., utilizing the sextupoles in the two chromatic correction sections.
In addition, we are considering the installation of high-impedance structures for passive machine protection. In this scheme a missteered off-centre bunch will excite a wakefield in the high-impedance structure that spoils the bunch emittance and also deflects subsequent bunches. Both effects would increase the area of beam impact on a downstream collimator, thereby ensuring its survival.
The task of the exit line is to remove the spent beam and debris from the interaction region, simultaneously ensuring acceptable background in the detector and not causing any vibrations of the final-doublet quadrupoles.
Design of the beam exit line is a major challenge, as the outgoing beam has an energy spread of 100%, and there are copious beamstrahlung photons, carrying a third of the initial beam power, and almost as many coherent pairs as incoming particles [2.50] . Low-energetic pairs of both charge signs might spiral around the solenoidal field and hit apertures between incoming and outgoing quadrupole magnets. The impact of only a small fraction of the incoming beam power near the final quadrupole can induce elastic waves with amplitudes well in excess of the 0.2 nm vertical jitter tolerance [2.51] . Neutron generation in the vicinity of the detector is a further concern. Large apertures should be chosen in order to guide as much of the collision debris as possible away from the IP. The ideal quadrupole magnet might be a double-quadrupole with an open centre, similar in its layout to the magnets built for the LHC cleaning insertions [2.52].
Because of the small spot size at the interaction point, the colliding bunches produce strong electromagnetic fields focusing each other. While this enhances luminosity, the beam particles travel on curved trajectories, emitting beamstrahlung, which is comparable to synchrotron radiation. In the machines with centre-of-mass energies of up to 1 TeV the critical energy of this radiation is below the beam energy, but this is not the case at ECM = 3 TeV.
The beam-beam effects were simulated using GUINEA-PIG [2.53] . The beamstrahlung has a total power of about P = 4.6 MW, but it is emitted into small cones in the forward direction so it does not cause direct background. However, protection of the magnets in the spent beam line is an issue which has not yet been investigated. Also secondary particles -- especially neutrons -- are of concern.
The beam-beam interaction will also cause background via coherent and incoherent pair creation. In the coherent process, a hard beamstrahlung photon turns into an electron-positron pair in the strong field of the oncoming bunch. About 68¥ 108 pairs are produced per bunch crossing [2.54] ; a number comparable to the number of beam particles. Thus these particles influence the beam-beam interaction significantly. Initially they have small angles but they can be deflected by the beams. The detector can not be extended to very small angles in order to stay out of the flux of these particles. At larger angles it can be protected from most of their secondaries with the help of a mask. However, secondary neutrons produced inside of the mask potentially lead to significant background [2.55] . In order not to produce the neutrons in the first place it is therefore necessary to allow an exit hole with an opening angle of about 10 mrad [2.50] . In this case the energy deposited per bunch crossing in the final quadrupoles is comparable to the value expected for TESLA. The number of particles produced via the incoherent process is much smaller than the number of particles in the bunch (about 4.5¥ 105), but they can have significant inherent angles. They can enter the detector causing background -- especially in the vertex detector, which integrates over a full train. The longitudinal magnetic field of the main detector solenoid helps to reduce the number of hits. For Bz = 4 T a minimum radius of 30 mm for the inner layer of the detector seems possible, with a hit density of less than one charged hit per mm2 and bunch train. The background processes can be used as a fast signal for tuning the luminosity [2.56] .
The number of hadronic background events per bunch crossing is high, about four events with a centre-of-mass energy above 5 GeV [2.54] . Their effect on the physics needs to be investigated.