The construction cost of different collider elements increases or decreases with the size of the ring. The optimum size is a function of the maximum beam energy. In 1976, B. Richter performed a cost optimisation of circular electron-positron colliders [4]. For a maximum c. m. energy of about 365 GeV (top quark production), he found that a collider diameter of 100 km is close to the optimum. A similar circumference value of about 90 km is obtained when extrapolating from the size and energy of more recent machines (PETRA, TRISTAN and LEP) [5].

Serendipitously, a circumference of 90–100 km is exactly the size required for a 100 TeV hadron collider. Namely, the beam energy of a hadron collider is given by E = e c B ρ , (3) where B is the dipole field, ρ the bending radius. Doubling the field compared with the LHC, and increasing the radius or circumference by a factor 3–4 yields a factor 6–8 increase in proton energy to about 100 TeV in the centre of mass.

In addition, the size of 90–100 km required for both FCC lepton and hadron colliders also matches the local topology of the Lake Geneva basin, where possible tunnel locations are bounded on two sides by the Jura and (Pre-)Alpes, respectively, and where, in addition, the collider should pass around the Salève mountain.

A linear collider still features moderate arcs in its beam delivery system, and also faces the issues of synchrotron radiation emitted in the final quadrupole magnets (Oide effect) and in collision (beamstrahlung), which ultimately limit the achievable beam size and the maximum beam energy of such colliders.

Indeed, some bending magnets are an integral part of the beam delivery systems, e.g., for the collimation of off-energy particles, and for the chromatic correction of the final focus. Synchrotron radiation emitted in these bending magnets can increase the beam size at the interaction point (IP), either directly due to the resulting increase of the horizontal emittance, or due to incomplete chromatic correction for particle energy changes that occur within the system [6]. These effects call for reduced bending as the beam energy is increased. At the same time, at higher energy the incoming geometric beam emittance adiabatically decreases, allowing for stronger sextupole magnets. In consequence, the geometry and the length of the beam delivery system change with beam energy. Two historical examples from the CLIC beam delivery design in Figure 1 illustrate the beam-delivery footprint and length changes that may be required when increasing the collision energy from 500 GeV to 3 TeV. The initial tunnel layout should accommodate and provide space for the high-energy geometry. Even with the modified, optimised geometry synchrotron radiation is by no means negligible. For example, synchrotron radiation in the bending magnets caused a factor of about two loss in luminosity in the 2003 CLIC BDS design at 3 TeV (Figure 1, left picture) [7]; a similar situation was found for the SLC at a beam energy of only 45.6 GeV [8]. Such questions will also need to be addressed for a proposed 3 TeV energy upgrade of the International Linear Collider [9], or for upgrades of linear colliders to even higher energies, based on plasma acceleration.

A second limit set by synchrotron radiation in linear colliders arises in the final quadrupole magnets, where photon emission leads to an energy change, and thereby to a different focal length and increase in the vertical spot size (“Oide effect”) [10], as is illustrated in Figure 2.

The third, and perhaps most important limitation due to synchrotron radiation at linear colliders relates to the one emitted during the collision in the electromagnetic field of the opposite beam, also called “beamstrahlung”. The strength of the beamstrahlung is characterized by the parameter ϒ, defined as [11,12] ϒ ≡ γB/B _c = (2/3)ℏω _c /E _e , with B c = m e 2 c 2 / ( e ℏ ) ≈ 4.4 GT the Schwinger critical field, ℏω _c = (3/2)ℏcγ ³/ρ the critical photon energy as introduced by Sands [13], E _e the electron (or positron) energy before radiation, B the local magnetic field, ρ = e/(pB) the local bending radius, γ the relativistic Lorentz factor corresponding to E _e , p ≈ E _e /c the particle momentum, e the electron charge, and c the speed of light. The average ϒ during the collision of three-dimensional Gaussian bunches is ϒ = 5 r e 2 6 α N b σ z σ x * + σ y * , (4) where α denotes the fine structure constant (α ≈ 1/137), r _e ≈ 2.8 × 10^–15 m the classical electron radius, N _b the bunch populaiton σ _z the rms bunch length, and σ x ( y ) * the rms horizontal (vertical) spot size at the collision point.

In the classical regime ϒ ≪ 1, and for flat Gaussian beams, the number of photons emitted per beam particle during the collision is [14]. n γ ≈ 2.12 α N b r e σ x * + σ y * . (5)

The parameter n _γ is important, since it describes the degradation of the luminosity spectrum. Namely, the emission of beamstrahlung photons changes the energy of the emitting electron or positron, and thereby the energy of its later collision. The fraction of the total luminosity L _tot at the target centre-of-mass energy L ₀ is determined by n _γ as [15]. L 0 L tot = 1 n γ 2 1 − e − n γ 2 , (6)

To illustrate this degradation with an example, for CLIC at 380 GeV 60% of the total luminosity lie within 1% of the target energy, while at 3 TeV this fraction decreases to only 34%. Figure 3 presents the respective luminosity spectra [16]. In this way, at TeV energies, e⁺e⁻ collisions in linear colliders lose their distinct energy precision.

The muon is about 200 times heavier than the electron, which, according to Eq. 1, implies close to 2 × 10⁹ times less radiation at the same energy and bending radius. On the other hand, muon beams have two drawbacks: their production is not trivial, and the muons decay, with a rather short lifetime of only 2.2 μs at rest. In Section 5.2, we will present an innovative approach to the muon collider.

The radiation emission is suppressed at wavelengths larger than λ sh ≈ 2 d 3 / ρ with d signifying the pipe diameter [17]. Therefore, miniature accelerators with extremely small beam pipe on the micron or nanometre scale, combined with a large bending radius ρ could suppress almost all radiation. An extreme case would be the use of bent-crystals, where d becomes comparable to the inter-atom distance in the crystal lattice.

It is noteworthy that classically a uniform time-independent beam does not emit any synchrotron radiation [18,19]. For example, the CERN ISR operated with high-current stationary beams. In the case of such a coasting beam, residual radiation could arise from shot noise or from beam instabilities. The shot noise might be reduced by suitable manipulations—see e.g., [20] — or by stochastic cooling. The shot noise and, therefore, the associated synchrotron radiation can be markedly reduced in case the cooling is so strong as to produce a crystalline beam [21]. Accelerating a “DC” (or near-DC) beam may be accomplished by induction acceleration [22].

As for the bending fields, also for the accelerating systems, superconducting materials have gained widespread use. Superconducting radiofrequency (RF) cavity systems underpin many modern facilities, the latest examples being the European XFEL at DESY Hamburg, the LCLS-II at SLAC, and FRIB in Michigan. Accelerating fields have been increased from a few MV/m to more than 30 MV/m for multicell cavities, and close to twice this value for single cells. Most SC cavities to date have been based on bulk Nb or in Nb-on-Cu cavities. New cavity treatments (nitrogen doping or nitrogen infusion [24]), innovative production methods (chemical vapor deposition [25], high impulse power magnetron sputtering [26]) and new materials, e.g., Nb₃Sn [27], as for the magnets, etc. promise further significant advances in performance, by factors of 2–10 in quality factor Q ₀ and of 2–3 in maximum accelerating gradient. As an example, for Nb₃Sn, the theoretical ultimate “superheating” field [28] corresponds to a maximum accelerating gradient of ∼ 100 MV/m, about twice the corresponding value for Nb, while the latter is not far from the currently achieved peak values of about 50 MV/m for Nb cavities [27].

Other advanced accelerating concepts can reach much higher gradients. For example, plasma acceleration routinely achieves fields of 100 GV/m, which is 3,000 times higher than the Nb cavities proposed for the International Linear Collider. The accelerating plasma waves can be driven either by a high-energy charged particle beam or by a laser. Comprehensive concepts have been developed for electron-positron colliders based on either beam-driven [29,30] or laser-driven plasma acceleration [31,32]. Beam quality, pulse-to-pulse stability, and energy efficiency of plasma accelerators [33] are critical issues addressed by ongoing R& D programs. High-energy colliders are arguably the most demanding application of plasma acceleration. Possible ultimate limits of plasma acceleration arise from the scattering of beam particles off plasma nuclei and plasma electrons, and from the emission of betatron radiation [34]. Both of these effects might be partially mitigated by accelerating in a hollow plasma channel. For realizing e⁺e⁻ colliders, not only electrons but also positrons must be accelerated in the plasma, while preserving the beams’ transverse and longitudinal emittance. For this purpose, more complex plasma excitation schemes may need to be developed, e.g., [35, 36].

Thanks to their higher electron density, even larger gradients can be generated in crystals. The maximum field is given by [37]. E 0 ≈ m e c ω p e ≈ 100 G e V m n 0 1 0 18 c m − 3 , (7) with ω _p the angular plasma frequency and n ₀ the electron density. With n ₀ ≈ 10²² cm⁻³ to 5 × 10²⁴ cm⁻³ in a crystal, peak gradients of 10–1000 TV/m would be within reach. Accelerating crystal waves could be excited by X-ray lasers [37].

The Gamma factory [38] is based on resonant scattering of laser photons off partially stripped heavy-ion beam in the existing LHC or in the planned FCC-hh. Profiting from two Lorentz boosts, the Gamma factory acts as a high-stability laser-light-frequency converter, with a maximum photon frequency equal to ν _γ,max = 4γ ² ν _laser, where γ is the relativistic Lorentz factor of the partially stripped ion beam. This allows the production of intense bursts of gamma rays with photon energies of up to several 100 MeV.

In particular, the Gamma factory can serve as a powerful source of e⁺ (yielding 10¹⁶–10¹⁷ e⁺/s—five orders of magnitude higher than the state of the art), μ (10¹¹–10¹²/s), π, etc. The positron rate available from the Gamma factory would be sufficient for a LEMMA type muon collider [39,39]. The Gamma factory would also allow for doppler laser cooling of high-energy beams, and, thereby, provide an avenue to a High Luminosity LHC based on laser-cooled isocalar ion beams [41].

The LEMMA scheme for a muon collider is based on the annihilation of positrons with electrons at rest [39]. The cross section for continuum muon pair production e⁺e⁻ → μ ⁺ μ ⁻ has a maximum value of about 1 μb at a centre-of-mass energy of ∼0.230 GeV, which corresponds to a positron beam energy of about 45 GeV, exactly as required for the FCC-ee operating as a TeraZ factory and provided by the FCC-ee full-energy booster [42].

Challenges with the LEMMA-type muon production scheme relate to the emittance preservation of muons and muon-generating positrons upon multiple traversals through a target, and the merging of many separate muon bunchlets, due to production by many separate positron bunches or positron bunch passages.

These challenges may potentially be overcome by [43]:

• Operating the FCC-ee booster with a barrier bucket and induction acceleration, so that all positrons of a cycle are merged into one single superbunch [44], instead of ∼ 10,000 separate bunches.

Since the positron bunch will be mismatched to the nonlinear plasma channel, filamentation and significant transverse emittance growth may result [43].

For a typical initial plasma electron density of n _e = 10²³ m⁻³, and assuming a density enhancement by a factor of 1,000, due to the electron pinch in the positive electric field of the positron beam, the positrons annihilate into muon pairs at a rate of 10^–8 m⁻¹.

As described in the CDR [45], the FCC-ee booster can accelerate 3.5 × 10¹⁴ positrons every 50 s. Using the much more powerful Gamma Factory positron source, with a rate of 10¹⁶–10¹⁷ e⁺ s ⁻¹ [38], and injecting into the booster during one or a few seconds, of order 10¹⁷ e⁺ can be accumulated, at the booster injection energy of ∼20 GeV. The positrons can be captured into a single barrier RF bucket, with a final length of ∼ 5  m, at which the longitudinal density would be about 1,000 times higher than the peak bunch density in the collider ring (without collision), possibly compromising the beam stability.

Accelerating the long positron superbunch containing 10¹⁷ e⁺ by 25 GeV, from 20 to 45 GeV, requires a total energy of 0.4 GJ, or, if accelerated over 2 s, about 200 MW of RF power. This translates into an induction acceleration voltage of ∼2 MV per turn, which is three orders of magnitude higher than the induction voltage of the KEK digital accelerator [46], but about 10 times lower than the induction RF voltage produced at the LANL DARHT-II [47], at much higher or lower repetition rate, respectively. On the ramp and at top energy, the full bunch length l _b can conceivably be compressed to the assumed l _b ≈ 5 m, by squeezing the gap of the barrier bucket (which requires substantially more voltage for the barrier RF system)—also see [21,46]. Tentative parameters of the positron superbunch are compiled in Table 1. We assume that the booster ring runs near the coupling resonance so that the emittance is shared between the two transverse planes.

When the accelerated and compressed positron bunch is sent into the plasma channel, we consider that the plasma electron distribution quickly acquires a nearly stationary shape, while any remaining plasma ions are slowly repelled away from the positron beam. In the stationary phase, the electron distribution approaches a shape that mimics the one of the positron beam, with a density n e , stat ≈ N b 2 π l b σ ⊥ 2 , (8) so as to neutralize the electric field. With an average rms size of σ _⊥ ≈ 10 μm, we obtain n e , stat ≈ 1 0 26 m⁻³. Considering a 100 m long plasma channel yields ∼ 1 0 11 μ pairs, with an initial muon energy of ∼22 GeV, and an initial lifetime of 0.5 ms at this energy.

In particular, once the electron distribution is nearly stationary, the longitudinal fields inside the plasma can be neglected. The resulting transverse emittance of the produced muons can be optimized by adjusting positron beam parameters and the optical functions at the entrance to the plasma [43]. In addition, a phase rotation (bunch compression) of the muons may be required, since the initial bunch length ∼ 5  m, of the positrons or resulting muons, will still be too long for collider operation.

Overall, the described scheme, sketched in Figure 5, would produce about 10¹² muon pairs per cycle, with a cycle length of order 3 s. Even at an energy of 50 TeV, the muons would decay with a lifetime of only 1.1 s. This kind of cycle/lifetime ratio of about 3:1 might still be considered acceptable. On the other hand, for collision at a muon beam energy of seven TeV in the existing LHC ring, the muon lifetime would be only 0.15 s, and the scheme would be considerably more challenging.

Radiofrequency (RF) systems are used to keep a charged particle beam bunched, and to feed energy to the beam, be it for purposes of acceleration or to compensate for the energy lost due to synchrotron radiation. In superconducting continuous-wave RF cavities, almost no power is lost to the cavity wall and all RF power entering the cavity can be transferred to the beam highly efficiently. Then, in the overall power budget, the RF power source is the most inefficient element. For RF frequencies above about 400 MHz, and for high power applications, historically klystrons have been the RF power source of choice on particle accelerators.

It is most remarkable that about 80 years after the invention of the klystron by the Varian brothers, a revolution in klystron technology is underway. Using advanced bunching techniques, it is expected that the klystron efficiency can be raised from the present 50–60% level to about 90%, which would translate into a significant energy saving [48]. Prototypes of such novel highly-efficient klystrons are being manufactured both by CERN, in collaboration with industry, for FCC, CLIC and ILC, and, in China, for the CEPC project.

In parallel, the efficiency of alternative RF power sources, such as inductive output tubes or solid-state amplifiers [49], is also being improved.

While at present the RF power sources are the dominant contributors to overall grid-to-beam power transmission inefficiency, a few percent additional losses each occur in the electrical network between utility high-voltage interconnect point and RF power source, and in the wave guides and couplers feeding the generated RF power into the accelerating cavities, respectively.

For high fields, superconducting magnets are most efficient, as no energy is lost, and electric power is mostly required for the cryogenic system. For lower fields, up to of order 1 T, permanent magnets are most energy efficient. An example is the Fermilab Recycler Ring [50], which was built almost entirely from permanent magnets. Even adjustable permanent magnets have been designed and built for applications at light sources, colliders, and plasma accelerators [51]. Other ingenious solutions for energy saving can be found, depending on the respective application. For example, for the FCC-ee double-ring collider, twin dipole and quadrupole magnets at low field (of order 0.05 T, for the dipoles) have been designed [52], which promise a significant power reduction compared with the magnets of comparable fields at earlier colliders.

Recovering the energy of the spent beam after one or several collisions is another effective measure to improve overall energy efficiency, if a significant fraction of the overall electric power is stored in the beam, as typically is the case for beams accelerated in superconducting linacs [53].

A comparison of ERL-based colliders proposed half a century ago with several recent concepts is presented in Table 2. The main differences between proposals from the 1970s and today are the collision of flat beams instead of round beams, much smaller (vertical) beam sizes, combined with higher beam current, yielding, on paper, of order ∼10,000 times higher luminosity than the proposals from half a century ago.

Also minimisation of beam loss can improve the energy efficiency of accelerators, such as ERLs. For proposed future higher-energy facilities, machine protection and beam collimation systems become ever more challenging due to their unprecedented beam power or stored energy. For example, the FCC-hh design features a stored beam energy of 8.3 GJ [54], which is more than a factor 20 higher than for the LHC.

Large circular storage rings like the FCC-ee, and even the FCC-hh, can serve as ultimate storage-ring light sources, with diffraction limited emittances down to photon wavelengths of λ min ≈ 4 π ε x . (9)

For FCC-ee the geometric emittance ɛ _x , of the collider or of the full-energy booster, scales as γ ², and the lowest value of ɛ _x ≈ 50 pm is reached at the injection energy of 20 GeV, resulting in λ _min,ee ≈ 650 pm. With a beam current of 1.5 A or higher, this could represent a formidable light source. Conversely, for FCC-hh the normalized proton beam emittance γɛ _x shrinks during proton beam storage at 50 TeV to ∼ 0.2 μ  m [54], corresponding to a geometric emittance of 4 pm, and the associated minimum wavelength is λ _min,ee ≈ 50 pm, still more than an order of magnitude lower than for the FCC-ee. The FCC-hh design beam current is 0.5 A.

The FCC-ee ring emittance could be further reduced by factors of 10–100 through the addiition of damping wigglers, pushing the accessible wavelength into the 10 pm regime.

A more detailed study of synchrotron light produced by such low-emittance FCC-ee beams passing through realistic undulator configurations has been performed recently [55]. For hadron storage rings, their use as a light source was discussed in the past, e.g., for the Superconducting Super Collider (SSC) [56].

In addition, also Free Electron Lasers (FELs) based on ERLs designed for high-energy physics colliders can offer outstanding performance in terms of average brightness, and in their wavelength reach down into the few picometre range [57], e.g., in the case of the LHeC-ERL based FEL, with a beam current of ∼20 mA.

Various approaches have been suggested for using beams in a storage ring for the detection of gravitational waves [58–61] including the construction of special optics with regions of extremely high beta functions that would serve as gravitational wave antennae [61,62]. Exploration of such possibilities continues.

With advanced cooling and manipulation schemes, storage rings might eventually be used as quantum computers [63,63]. Indeed, combining the storage rings of charged particles with the linear ion traps used for quantum computing and mass spectrometry would enable a large leap in the number of ions serving as qubits in the quantum computing. Such an approach holds the promise of significant advances in general quantum calculations and, especially, in simulations of complex quantum systems.