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31. Accelerator Physics of Colliders

31. Accelerator Physics of Colliders
Revised July 2023 by V. Shiltsev (FNAL) and F. Zimmermann (CERN).
This article provides background for the High-Energy Collider Parameter Tables that follow
and some additional information; see in-depth review and a comprehensive list of references in [1];citations below are limited to widely used textbooks and open access seminal papers and reviews.

31.1 Energy and Luminosity
Collisions of two beams of particles
accelerated to high energies E1,2 provide access to center-of√
mass energies (c.m.e.) Ecme ≈ 2 E1 E2 , assuming a typically small or zero crossing angle. Most of
the 31 colliders that have ever reached the operational stage (seven are operational now) used equalmasses and energies of colliding particles, with c.m.e. equal to twice the beam energy Ecme = 2Eb .
Other machines collide beams of unequal energies, such as electron-proton or electron-ion colliders,or asymmetric B-factories, that produce new short-lived particles, whose decays are more easily
detected and analyzed with a Lorentz boost.
In an accelerator, charged particles gain energy from an electric field, which usually varies intime at a high frequency ranging from 100s of kHz to 10s of GHz. With proper phasing to the RF
field over distance l, the energy gain of a particle with charge Ze is proportional to the average
accelerating gradient G, i.e. ∆Eb = ZeGl. In principle, the highest beam accelerating gradientsachieved to date in operational machines or beam test facilities (G ≈ 100 MV/m in 12 GHz
normal-conducting RF cavities and 31.5 MV/m in 1.3 GHz superconducting ones) allow accessing
high energies over reasonably long linear accelerators (linacs), but cost considerations often call forminimization of RF acceleration via repeated use of the same RF system which, in that case, would
boost the energy in small portions ∆Eb = ZeVRF per turn every time a particle passes throughthe total cavity voltage VRF . Such an arrangement can be realized either in the form of storagering circular colliders or also through novel schemes based on, e.g., recirculating linear accelerators(RLAs) with or without energy recovery. Circular colliders are by far the most common; here, themomentum and energy of ultra-relativistic particles are determined by the bending radius insidethe dipole magnets, ρ, and by the average magnetic field B of these magnets:
p = ZeBρ or Eb [GeV] = 0.3Z(Bρ) [Tm] .

(31.1)Such synchrotron condition assures approximately constant radius of the beam orbit during acceleration. Transverse focusing by quadrupole magnets is needed to keep particles inside the rather
limited space provided by the accelerator beam pipe passing through the magnet apertures. The
maximum field of normal-conducting (NC) magnets is about 2 T, due to the saturation of ferromagnetic materials, and, while this is sufficient for lower energy colliders, such as most e+ e− storagerings, it is not adequate for frontier-energy hadron (or muon) beams, because of the implied need forexcessively long accelerator tunnels and prohibitively high total magnet power consumption. The
development of superconducting (SC) magnets that employ high electric current carrying Nb-Ti
wires cooled by liquid helium below 5 K, opened up the way towards higher fields and to hadroncolliders at record energies [2]. For example, the 14 TeV c.m.e. LHC at CERN, uses double-bore
SC magnets with a maximum field of 8.3 T at a temperature of 1.9 K, in a tunnel of C = 26.7 km
circumference (dipole-magnet bending radius ρ = 2800 m). The double-bore design allows acceleration of the same particle type in opposite directions and also the operation with different particle
species (e.g., protons and heavy ions) in the two apertures, while a single bore magnet implies theuse of particles and antiparticles for the collider application. As the production of anti-particles isenergy consuming and therefore limited, this concept opens the door to high-performance hadron
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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31. Accelerator Physics of Colliders

colliders.
The exploration of rare nuclear and high energy particle physics phenomena requires not only
an appropriately high energy, but also a sufficiently large number of detectable reactions. The
number of events of interest Nexp is given by the product of the cross section of the reaction understudy, σexp , and the time integral over the instantaneous luminosity, L:
Nexp = σexp ·Z

L(t)dt.

(31.2)In the Tables, luminosity is stated in the units of cm−2 s−1 . The integral on the right is referred
to as integrated luminosity Lint , and, reflecting the smallness of typical particle-interaction crosssections is often reported in units of inverse femto- or attobarn, e.g., 1 ab−1 =1042 cm−2 . Colliders
usually employ bunched beams of particles with approximately Gaussian distributions, and for twobunches containing N1 and N2 particles colliding head-on with frequency fcoll , a basic expression
for the luminosity is
N1 N2
L = fcollF(31.3)
4πσx∗ σy∗where σx∗ and σy∗ characterize the rms transverse beam sizes in the horizontal and vertical directions
at the interaction point, and F is a factor of order 1, that takes into account inefficient geometricoverlapping of the beams due to a crossing angle and finite bunch length, and dynamic effects, suchas the mutual focusing of the two beam during the collision (see below). Having nb bunches per
beam increases the frequency of collisions fcoll = nb f0 where f0 is either the revolution frequency ofa circular collider or the repetition rate of a linear one. To achieve a high luminosity, one, therefore,
has to maximize the population and number of bunches, either producing these narrowly or focusing
them tightly, and colliding them at high frequencies at dedicated locations, where products of theirreactions can be registered by particle detectors.
Subsequent sections in this report briefly expand on the beam dynamics behind collider design,comment on the realization of collider performance in a selection of today’s facilities, and end withsome remarks on future possibilities.

31.2 Beam Dynamics
Given the enormous and highly concentrated power carried by modern high energy particle
beams, the main concern of beam dynamics in colliders is stability of motion of i) individual
particles in accelerators, ii) single high-intensity beams of many particles moving together, and iii)
colliding beams [3–5].
31.2.1 Single Particle Dynamics
While a reference particle at the nominal energy proceeds along the design trajectory (reference
orbit) mostly determined by transverse magnetic dipole fields, other particles in the bunch are
kept close by through the focusing effect of quadrupole fields. Assume that the reference particle
carries a right-handed Cartesian coordinate system, with the co-moving z-coordinate pointed in
the direction of motion along the reference trajectory, z = s − vt (with v the reference particle
velocity, and t time). The independent variable is the distance s of the reference particle along thistrajectory, rather than time t, and for simplicity this reference path is taken to be planar. The
transverse coordinates are x (horizontal) and y (vertical), where {x, z} defines the plane of thereference trajectory.
Several time scales are involved, and this is reflected in the approximations used in formulating
the equations of motion. All of today’s high-energy colliders are alternating gradient synchrotronsor, respectively, storage rings and the shortest time scale is set by so-called betatron oscillations.
The linearized equations of motion of a particle displaced from the reference trajectory are:31st May, 2024

31. Accelerator Physics of Colliders

3x00 + Kx (s)x = 0 , y 00 + Ky (s)y = 0 , z 0 = −x/ρ(s) ,Ze ∂By1
Ze ∂Bywith Kx ≡+ 2 and Ky ≡ −
p ∂x
ρ
p ∂x

(31.4)

where ρ = p/ZeBy is the radius of curvature due to the field on the reference orbit. The prime
denotes d/ds and the Maxwell equation in vacuum ∇ × B = 0 helps to eliminate Bx (s) using therelation ∂Bx /∂y = ∂By /∂x. In this linear approximation, the vertical magnetic field By (s) in the(x, z)-plane contains only dipole and quadrupole terms, which are treated as static in time, but
s-dependent.
The solutions of the Hill’s equations (31.4) for x and y with a restoring force periodic in s arethose of quasi-harmonic oscillators:sx(s) =p0

2Jx βx cos ψx , x (s) = −2Jx[αx cos ψx + sin ψx ] ,βx(31.5)

where the action Jx is a constant of integration, αx = αx (s) ≡ −(1/2)dβx (s)/ds, and the envelopeof oscillations is modulated by the beta-function βx (s). A solution of the same form describes the
motion in y. The betatron oscillation phase advances according to dψx /ds = 1/βx ; that is, 2πβxalso plays the role of a local wavelength of oscillations along the orbit. An extremely importantparameter for circular machines is the tune, Qx , which is the number of such oscillations per turnabout the closed path:
I
I
ds
1
1
.
(31.6)
Qx =
dψx =
2π
2π
βx (s)
While the integer part of the tune [Qx,y ] generally characterizes the extent of the focusing lattice, itis the fractional part of the tune {Qx,y } that needs to be well defined and controlled by the machine
operators in order to stay away from potentially detrimental resonances, which may occur underconditions of kQx + lQy = m, where k, l, and m are integers. For example, for the LHC a combination of horizontal and vertical tunes — also called the working point — equal to (Qx , Qy )=(64.31,
59.32) has been selected, such that resonances up to the order of |k| + |l| = 10 or 12 are avoided.
These resonances are driven by high order multipole components of the fields in the magnets, orby self-fields of the beam, or by the electromagnetic fields of the opposite bunch. Normally, thenonlinear components are very weak compared to linear ones, nevertheless, when the nonlinear resonance condition is encountered, the amplitudes of particle oscillations could grow over the beam
lifetime, resulting in the escape of the particles to the machine aperture, in the increase of theaverage beam size, or in both; either of these is highly undesirable phenomena. Careful analysis
of nonlinear beam dynamics is instrumental in determining and optimizing the dynamic aperture,
which is defined as the maximum amplitude of a bounded particle motion.
Neglecting for now all nonlinear effects and usually small x − y coupling, and considering only
the linear dynamics, the beta-function is well defined and satisfies the following equation:
2βx βx00 − βx02 + 4βx2 Kx = 4 .

(31.7)In a region free of magnetic fields, such as in the neighborhood of a collider interaction point (IP),usually occupied by particle detectors, a symmetric solution of Eq. (31.7) is a parabola:βx (s) = βx∗ +31st May, 2024

s2
,
βx∗

(31.8)

31. Accelerator Physics of Colliders

4

where, in this case, s denotes the longitudinal distance from the IP. The location of the beam waist
usually coincides with the IP and corresponds to the minimum value of the beta-function βx∗ ; theasterisk is used to indicate IP parameters.
Note that individual quadrupole magnet focuses particles in one plane and defocuses in another,see Eq.(31.4), and a standard way to provide focusing in both planes is to employ an alternating
gradient periodic focusing lattice, consisting of a sequence of equally-spaced quadrupoles with amagnetic field gradient equal in magnitude, but alternating in sign (“focusing quadrupole - drift
space - defocusing quadrupole - drift space” – known as a FODO cell). Eq. (31.7) has stable
periodic solutions βx (s), βy (s) in both planes provided that the focal length of the quadrupoles islonger than half the focusing-lens spacing L, i.e., f = p/(eB2 l) > L/2 (where l is the length of aquadrupole magnet, here assumed to be short l  L, and B2 ≡ |∂By /∂x| the quadrupoles’ field
gradient). In that case, the beta-functions have maxima at the√ focusing quadrupoles and√ minima
at the defocusing ones, equal to, for example, βmax,min = (2 ± 2)L in the case of f = L/ 2, whichcorresponds to a betatron phase advance ∆ψx,y = 90◦ per FODO cell.
Expressing the invariant Jx in terms of x, x0 yields
Jx =


1
x2 + (αx x + βx x0 )2
γx x2 + 2αx xx0 + βx x02 =22βx(31.9)

with γx = γx (s) ≡ (1 + αx2 (s))/βx (s). In a periodic system, these Courant-Snyder parameters [6](frequently referred to as Twiss parameters) α(s), β(s), γ(s) are usually defined by the focusing
lattice; in a single pass system such as a linac, the parameters may be selected to match thex-x0 distribution of the input beam. For a given position s in the ring, the transverse particle
motion in {x, x0 ≡ dx/ds} phase space describes an ellipse, the area of which is 2πJx , where
the horizontal action Jx is a constant of motion and independent of s. If the interior of that
ellipse is populated by an ensemble of non-interacting and non-radiating particles, that area, given
the name emittance, is constant over the trajectory as well and would only change with energy.
In a typical case of the particle’s energy change rate being much slower than betatron motion,and considering a Hamiltonian system (i.e., a hadron collider
or a linear collider, either withoutRsignificant synchrotron radiation), the adiabatic invariant px dx is conserved, and given that forsmall angles px = x0 · βγmc2 , it is common practice to consider an energy-independent normalized
emittance that is equal to the product of the emittance and relativistic factor βγ/π and denotedby εn . For a beam with a Gaussian distribution in {x, x0 }, average action value hJx i and standard
deviations σx , and σx0 , the definition of the normalized rms emittance isεnx ≡ βγhJx i = βγ

σx2 (s)σ 20 (s)= βγ x,βx (s)
γx (s)(31.10)with a corresponding expression for the other transverse direction, y. The angular brackets denotean average over the beam distribution. For 1D Gaussian beam, 95% of the particles are contained
within {x, x0 } phase space area of 6πεn /(βγ). Normalized beam emittances are conserved over theacceleration cycle in linear, static focusing lattices Kx,y (s), and consequently, one would expect thesame εn at the hadron (or linear) collider top energy as the one coming from the very initial low
energy particle source. Unfortunately, that is rarely the case as many time-varying or nonlinear
phenomena come into play. In an e− /e+ storage ring, the normalized emittance is not preserved
during acceleration, but at each energy the beam’s equilibrium emittance is determined by the effectof synchrotron radiation as a balance between radiation damping and quantum excitation [7]. Insuch a ring, for a given accelerator optics, the normalized equilibrium emittance increases with thethird power of the beam energy [8].
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531. Accelerator Physics of Colliders

As for the description of a particle’s longitudinal motion, one takes the fractional momentum
deviation ∆p/p from that of the reference particle as the variable conjugate to z. The factors Kx,yand ρ in Hill’s equations (31.4) are dependent on momentum p, leading to a number of effects: first,
the trajectory of off-momentum particles deviates by ∆x(s) = Dx (s)(∆p/p), where the dispersion
function Dx (s) is determined by the magnetic lattice and is usually positive, periodic, and of theorder of ∼ ρ/Q2x . Second, the radius of curvature and orbit path-length C vary with the momentum
and, to first order, are characterized by the momentum compaction factor αc ,
∆C/C
1
αc ≡
=
∆p/p
C

I

Dx (s)ds ,ρ(s)(31.11)

which typically is of order 1/Q2x . Energy deviations also result in changes of machine focusing
lattice properties and variations of the particle tunes, characterized by the chromaticity Q0x,y ≡∆Qx,y /(∆p/p). The natural chromaticity due to momentum dependence of the quadrupole focusing
is negative and large ∼ −Qx,y . Corresponding chromatic tune variations can, therefore, become
unacceptably large even for relatively small energy deviations (∆p/p) ∼ (0.1 − 1) · 10−3 . To assuretransverse particle stability, usually, the chromaticity is partially or fully compensated by additional
sextupole magnets placed at locations of non-zero dispersion.
Radiofrequency electric fields in s direction provide a longitudinal focusing effect, allowing astable increase of particle energy. The frequency fs of such longitudinal synchrotron oscillations is(expressed in units of revolution frequency f0 , to become the synchrotron tune Qs )
fs
Qs ≡
=
f0

s

(αc − 1/γ 2 )hZeVRF sin(φs ),
2πβcp(31.12)where h = fRF /f0 denotes the RF harmonic number, VRF the RF voltage, and φs = cos−1 (∆E/ZeVRF )the synchronous phase, with ∆E the average energy loss per turn (e.g. due to synchrotron radiationand impedance). The synchrotron tune Qs determines the amplitude of longitudinal oscillations
for a particle with an initial momentum offset, e.g., the rms bunch length σz relates to the rms
momentum spread δp/p as: c(αc − 1/γ 2 ) δp
σz =.
(31.13)2πQs f0p
Similarly to the case of transverse oscillations, the area of the longitudinal phase space {∆E, ∆t},or {γβδp/p = (1/β)∆γ, z = βc∆t}, encircled by a moving particle is an adiabatic invariant, andthe corresponding normalized longitudinal emittance εn,L = βγmcσz (δp/p) is a generally conserved
quantity in hadron accelerators and also in linear accelerators. In the case of lepton storage rings,
synchrotron radiation determines the equilibrium relative momentum spread, which grows linearlywith beam energy [7, 8], and the corresponding bunch length follows from Eq. (31.13). In hadronsynchrotrons, the longitudinal emittance sometimes is intentionally blown up during acceleration,
so as to preserve longitudinal beam stability.
Longitudinal oscillations are the slowest of all the periodic processes which take place in theaccelerators. For example, in the LHC, the frequency of synchrotron oscillations at the top energyof 7 TeV is about fs = 23 Hz, the revolution frequency is f0 =11.3 kHz, the frequency of betatronoscillations is about Qx,y frev ' 700 kHz and the RF frequency is fRF = 400.8 MHz (h = 35640).
It should be noted that longitudinal motion is practically absent in linacs. In the absence ofbending dipoles, dispersion Dx (s) is zero and so are the momentum compaction factor αc and thesynchrotron tune Qs . As a result, ultrarelativistic particles in a linac barely change their relative
positions during acceleration despite significant energy spread.
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31. Accelerator Physics of Colliders

Highest-energy circular colliders face a serious impediment in the form of synchrotron radiation(SR) that causes an energy loss per turn of∆ESR =

1 Z 2 e2 β 3 γ 4,
3ε0ρ

(31.14)

here, ε0 is the permittivity of vacuum. For electrons and positrons it is equal to ∆ESR = 88.5
[keV/turn] Eb4 [GeV]/ρ[m] and requires correspondingly high total RF voltage per turn to replenish
the loss. Above a few hundred GeV, the SR energy loss becomes comparable to beam energy∆ESR ∼ Eb , which makes circular e+ e− colliders impractical for c.m.e. above ∼ 500 GeV.
Dynamics of the particle spin and sophisticated methods to maintain beam polarization along
the acceleration chain, from the polarized sources to collisions, dedicated spin matching procedures
to enable self polarization in e+ /e− storage rings and the resonant depolarization method of ultraprecise c.m.e. calibration are described in [9].
31.2.2 High Intensity Beams
Ultimate collider luminosity calls for high beam currents Ib = Zef0 nb N . Three related majordifficulties include growing RF demands to compensate the synchrotron-radiation power loss P =Ib ∆ESR in e+ /e− beams, the advent of so-called coherent (or collective) beam instabilities, andgrowing demands for minimization of radiation due to inevitable particle losses. Many types ofsingle- and multi-bunch instabilities are caused by beam interactions with electromagnetic fields
induced by the beam itself due to the impedance of the vacuum chambers and RF cavities [4], or
caused by unstable clouds of secondary particles, like electrons or ions, which are formed aroundthe circulating beams [10]. These instabilities can develop as quickly as within tens to thousandsof turns and need to be controlled. Mechanisms that are routinely employed to avoid coherent
instabilities include the use of nonlinear magnets to generate sufficient spread of the tunes andtherefore, provide Landau damping, fast beam-based transverse and longitudinal feedback systems,
and electron/ion clearing (either by weak magnetic or electric fields or by modulation of the primarybeam current profile rendering secondaries unstable, or by reducing the yield of secondary electrons
via either a special coating or extensive beam scrubbing of the vacuum chamber walls).
High current beam operation is sensitive to even minuscule fractional intensity losses caused byparticles’ scattering
at a large angle or with a large energy loss, sufficient for either the particleqamplitudes 2Jx,y βx,y (s), or their dispersive position deviations ∆x = Dx (s)(δp/p) to exceed
the available transverse aperture, usually set by collimators (otherwise, by the vacuum chamberand magnet apertures). This can be due to residual vacuum molecules near the beam orbit orCompton scattering off thermal photons, due to Coulomb scattering off other particles within thesame bunch (Touschek effect), or due to collisions with opposite beam particles and fields, such as
inelastic interaction of protons, Bhabha scattering e+ e− → e+ e− , or radiative Bhabha scattering
e+ e− → e+ e− γ (see corresponding chapters in [11]).
Particles can also get lost on the aperture as a result of much slower mechanisms of diffusioncaused either by the above processes with smaller scattering amplitudes, but stochastically repeatedmany times, such as intensity-dependent multiple Coulomb intrabeam scattering [12], by external
noises such as ground motion or magnetic field fluctuations, or via chaotic mechanisms like Arnold
diffusion, modulational diffusion, or resonance streaming in nonlinear fields, enhanced by minor tune
modulations. Diffusion leads to a slow evolution of the beam distribution function and appearanceof highly unwanted large-amplitude tails and beam emittance growth. The only way to counteract
it is to arrange beam cooling (damping of particle oscillations). The cooling requires a reaction
force opposite to particle momentum arranged such that, on average, the corresponding dissipative
particle energy loss is compensated for by external power [13, 14].
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31. Accelerator Physics of Colliders

In the case of electron or positron storage rings, such cooling occurs naturally due to synchrotron
radiation and provides an automatic route to achieve small equilibrium emittances through a balance between radiation damping and excitation of oscillations by random radiation of individual
photons. Fast radiation damping allows top-up injection of new particles without removing existingones, a useful method to maximize the integrated luminosity of circular e+ e− colliders. Synchrotron
radiation damping will also be an important cooling mechanism for future energy-frontier hadroncolliders, like the proposed FCC-hh and SppC (see below). Four other methods of beam cooling
have been developed and successfully employed to attain low emittances, namely electron coolingand stochastic cooling of heavy particles (ions and antiprotons), laser cooling of ion beams, and theionization cooling of muons.
To avoid damage or excessive irradiation of accelerator components so that these remain accessible for maintenance in the tunnel, sophisticated collimation systems are utilized. These systems
usually employ a series of targets or primary collimators which scatter the halo particles, andnumerous absorbers (sometimes as many as a hundred, which intercept particles in dedicated locations) [15, Ch.9.7]. In the highest energy modern and future colliders, extreme total beam energies
nb N Eb ranging from MJs to GJs and impacting surface energy densities reaching many GJ/mm2
pose one of the biggest challenges for high efficiency and robust particle collimation.
31.2.3 High Luminosity Collisions
Eq. (31.3) for luminosity can be recast in terms of normalized transverse emittances Eq. (31.10)
and the beta-functions β ∗ at the IP as:
L = f0 γnb

N2q

4π εnx βx∗ εny βy∗F.

(31.15)Here, equal bunch populations N are assumed in two Gaussian beams with the same emittances.
Naturally, to achieve a high luminosity, one has to maximize the total beam populations nb N
within the lowest possible emittances, and collide the beams at high frequency at locations where
the focusing beam optics provides the lowest possible values of the amplitude functions β ∗ , theso-called low-beta insertion. The latter requires sophisticated systems of strong focusing elements,∗ issometimes occupying quite a significant fraction of the collider’s total length. The lowest βx,ydetermined by the maximum field gradients and apertures in the interaction region (IR) magnets
and the effectiveness of compensation of chromatic and nonlinear aberrations.
The typical geometric reduction factor is F ≈ 1, and it rarely drops below 0.5 for the majority
of colliders, unless this is specifically required by physics processes under study. The reduction
due to the hourglass effect is caused by the increase in transverse beam sizes as one proceeds awayfrom the IP, where β(s) grows parabolically, as in Eq. (31.8). For long round bunches the effect√scales as F ≈ πA exp(A2 )erfc(A) , where A = β ∗ /σz . Nonzero beam crossing angles θc in thehorizontal plane and long bunches (rms bunch length σz ) will reduce the luminosity, too, by a factorF ≈ 1/(1 + Φ2 )1/2 , where the parameter Φ = σz tan(θc /2)/σx∗ is known as the Piwinski angle.
One of the most common limits to producing high luminosity arises from electric and magnetic
forces of the opposite bunch at the IPs, characterized by a dimensionless beam-beam parameter :ξx,y =∗
r0 N βx,y,∗ (σ ∗ + σ ∗ )2πγσx,yxy(31.16)

where r0 = Z 2 e2 /(4πε0 mc2 ) is the classical radius of the colliding particle (with charge Ze and massm). From Eqs. (31.3) or (31.15) and (31.16), one can note that the path to higher luminosity via
higher beam intensity and smaller beam sizes almost automatically calls for a higher beam-beam
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31. Accelerator Physics of Colliders

parameter as L ∝ ξ. Cited in the Tables, the beam-beam parameter is roughly equal to the betatrontune shift experienced by small-amplitude particles – positive in the case of opposite charge beams,
like e+ e− , and negative for same charge beams as in pp collisions. Beam-beam forces can leadto coherent effects, such as unstable beam oscillations or blow-up of one beam’s size while theother beam remains small or even shrinks (flip-flop effect). The tune spread arising from ξ and thenonlinear nature of beam-beam interactions results in strong diffusion along high-order transverse
resonances kQx + lQy = m and, ultimately, in beam size growth and beam losses. Operational
experience indicates that the aforementioned effects are tolerable below certain beam-beam limit ofξx,y ≈ 0.003 − 0.012 in hadron colliders [16], and – due to strong synchrotron radiation damping –an order of magnitude higher one in e+ e− colliders, with maximum ξx,y ≈ 0.03 − 0.12 [17, 18]. The
accessible beam-beam parameter range can also be restricted by coherent beam-beam instabilities.
These various limits translate into a maximum allowed single bunch intensity N and call for an
increase of the number of bunches nb to achieve higher luminosities.
In linear colliders, where each bunch collides only once, with typically much smaller beam sizeand experiencing much stronger forces, the strength of the collision is measured by the ratio of therms bunch length σz to the beam-beam focal length. This ratio, called disruption parameter Dy , isrelated to ξy via Dy = 4πσz ξy /βy∗ . Stronger beam-beam focusing and higher Dy lead to effectively
smaller beam sizes and a resulting luminosity enhancement; but it also makes the collisions moresensitive to small offsets, resulting in a kink instability. Additional beam-beam effects arising in thecollisions at linear colliders are the emission of beamstrahlung (synchrotron radiation in the field of
the opposing beam), along with e+ e− pair creation, and depolarization by various mechanisms.
Beamstrahlung is relevant for both linear colliders, where it may significantly degrade theluminosity spectrum, and for future highest-energy circular colliders, where it may limit the beam
lifetime, and also increases the energy spread and bunch length of the stored beam. For both typesof colliders, the beamstrahlung is mitigated by making the colliding beams as flat as possible atthe interaction point (σx∗  σy∗ ) to lower the beams’ EM fields. The photon energy spectrum of
the beamstrahlung is characterized by the parameter Υ = (2/3)~ωc /Eb [19], with ~ωc denoting thecritical photon energy. The spectrum strongly deviates from the classical synchrotron radiation
spectrum for Υ approaching or exceeding 1.
For hadron colliders, two fundamental luminosity limits are the beam lifetime, determined byburn-off in the collisions due to inelastic pp interaction dN/dt = −Lσin , and the radiation from thecollision debris, which may induce “quenches” (transitions to the normal-conducting state) of thesuperconducting final quadrupole magnets, and, in the long term, affect the equipment lifetime.
Another limit on the achievable integrated luminosity in circular colliders is set by the minimum
or average turnaround time (the time between the beam abort at the end of a physics fill and thestart of the next physics collisions). Achieving practical filling times with many bunches in thecollider requires either fast cycling injector machines and/or the top-up injection operation. The
latter makes the average luminosity of circular electron-positron colliders approximately equal tothe peak luminosity.

31.3 Recent High Energy Colliders
In this and the following section, elaboration is made on various issues associated with someof the recently operating colliders, particularly factors which impact integrated luminosity. Only
general references are provided, where further information can be obtained. A more complete list ofrecent colliders and their parameters can be found in the High-Energy Collider Parameters tables.
31.3.1 Tevatron
The first superconducting synchrotron in history, the Tevatron [20] was converted into a protonantiproton collider in 1985. Its 4.4 T dipole magnets employed Nb-Ti superconducting cable op31st May, 2024

931. Accelerator Physics of Colliders√
erating at 4.5 K, requiring what was then the world’s largest cryogenic system. With s up to
1.96 TeV it was the highest energy collider for 25 years and delivered more than 12 fb−1 of theintegrated luminosity to each pp̄ detector experiments (CDF and D0) before being shut off in 2011.
The route to high integrated luminosity in the Tevatron was governed by the antiproton production rate, the turn-around time to produce another store, and the resulting optimization of store
time [21]. The antiproton production complex consisted of three 8 GeV p̄ accelerators (the Accumulator, Debuncher, and Recycler – the latter was the first high energy accelerator built withpermanent magnets), and employed 25 independent stochastic cooling systems and one pioneeringhigh-energy electron cooling set-up to accumulate up to a record high value of 25·1010 p̄ per hour.
Despite severe parasitic long-range interactions of the two beams, each consisting of 36 bunches
placed on helical orbits by two dozen ±150 kV high-voltage (HV) separators, a total beam-beamtune shift parameter of nIP ξ ≈0.025-0.03 was achieved, a record for hadron beams, with nIP = 2
primary collision points. Other notable advances in the accelerator science and technology included
advanced longitudinal beam manipulation techniques of slip-stacking and momentum mining andthe first operational use of electron lenses for beam collimation and for compensation of long-range
beam-beam effects [22]. The Tevatron ultimately achieved luminosities a factor of 430 higher thanthe original design specification.
31.3.2 HERA
The first lepton-proton collider, the 6.4 km long Hadron-Elektron-Ring-Anlage (HERA) at
DESY in Germany [23], operated between 1992 and 2007 and delivered nearly 1 fb−1 of integrated√
luminosity at s of about 320 GeV to the electron-proton collider experiments H1 and ZEUS [24,Ch.10.5]. It was the first facility to employ both applications of superconductivity: 5 T magnets
in the 920 GeV proton ring and SRF accelerating structures to provide about 12 MW of RF power
to compensate for synchrotron radiation losses of 30 GeV lepton beams (positrons or electrons, ina conventional-magnet ring). With proper orbit and optics control, the HERA lepton beam would
naturally become transversely polarized to about 60% (within about 40 minutes) thanks to the
Sokolov-Ternov effect. Special magnets called spin rotators were implemented on either side of thecollider IPs to produce 30–45% longitudinal polarization at the experiments.
31.3.3 LEP
Installed in a tunnel of 26.7 km circumference, LEP [25] was the largest circular e+ e− collider
built so far. LEP was operated from 1989 to 2000 with beam energies ranging from 45.6 to104.5 GeV. The synchrotron radiation loss per turn reached some 3% of beam energy and it was
the total available RF voltage and power, respectively, that determined LEP maximum energy
and luminosity of 1032 cm−2 s−1 . At a beam energy of 98 GeV, LEP operated with a beambeam parameter ξ ≈ 0.083 , i.e., the total beam-beam tune shift for four interaction points wasnIP ξ ≈ 0.33. In the last year of operation, 288 SRF cavities were powered by 36 klystrons withan average power of 0.6 MW each and provided VRF = 3.63 GV. Up to about 60 GeV, LEP used
resonant depolarization to measure the beam energy with 0.001% accuracy.
31.3.4 SLC
Based on an existing 3-km long 2.85 GHz warm RF linac, the SLC [26] was the first and only
linear collider. It was operated from 1987 to 1998 with a constant beam energy of 45.6 GeV, up to
about 80% electron-beam polarization, quasi-flat beams, a final-focus optics with local chromatic
correction based on four interleaved sextupoles and βy∗ ≈ 1 mm. In its last year, SLC achieveda peak luminosity of about 3 × 1030 cm−2 s−1 , roughly half of the design value. The SLC had
a high-efficiency positron source providing 5 × 1012 e+ per s for 120 Hz injection into the linac.
It also employed the BNS damping to suppress the single-bunch beam break up instability, and31st May, 2024

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31. Accelerator Physics of Colliders

also demonstrated an about 2-fold increase of luminosity from disruption enhancement due to themutual focusing of the colliding electron and positron bunches at the interaction point,

31.4 Presently Operating Colliders
31.4.1 LHC
With a beam energy of 6.8 TeV (to be raised to the design value of 7 TeV through consolidation and magnet training), the superconducting Large Hadron Collider [27] presently is the world’s
highest energy collider. In the latest runs, peak luminosities of up to 2.1 × 1034 cm−2 s−1 have beenachieved at 13 TeV cme - more than twice the design value (the current status is best checkedat the Web site [28]). To meet its luminosity goals, the LHC operates with a high beam currentof approximately 0.5 A, leading to stored energies of about 330 MJ per beam. Controlled energy
deposition and component protection are given a high priority and a sophisticated highly efficientsystem of more than 100 beam collimators is employed [29]. At the energy of about 7 TeV per
particle, synchrotron radiation poses a challenge, as the cryogenic system must remove roughly
7 kW due to synchrotron radiation, intercepted with a specially designed beamscreen inside thevacuum chamber, at a temperature of about 5–20 K, to be compared with a temperature of 1.9 K
for the magnet cold bore. The elevated temperature allows for a more energy-efficient removal of
beam-induced heat. The beamscreen also provides an effective cryo-pump for the vacuum system.
When synchrotron-radiation photons hit the beamscreen, they can generate photoelectrons. These
photoelectrons, and also any other electrons generated in the vacuum system, e.g. by residual-gas
ionization, are accelerated in the electric field of the beam and may multiply via secondary-electron
emission, with consequent electron cloud development. To mitigate this issue, the beamscreen isregularly subjected to beam-induced surface conditioning (scrubbing), thereby lowering the secondary emission yield. The two proton beams of 2556 bunches spaced by 25 ns are contained inseparate pipes throughout most of the circumference and are brought together into a single 130
m long beam pipe at the interaction points. To avoid approximately 30 head-on collisions a small
crossing angle of about 0.3 mrad is employed, which reduces the luminosity by about 15%. Still,
the bunches moving in one direction experience multiple long-range encounters with the counterrotating bunches and the resulting perturbations of the particle motion substantially contribute to
the beam lifetime reduction. The dominant source of approximately 8 hour characteristic luminosity decay time is proton burn-off due to inelastic pp interaction with σin ≈ 81 mbarn, corresponding
to pile-up of up to 50 (number of events per individual bunch crossing). In special physics runs witha few bunches and large β ∗ , the LHC achieved a head-on beam-beam tune shift of nIP |ξ| ≈ 0.02
with nIP = 2, about twice as high as in regular operation.
The Tables also show the LHC luminosity performance in Pb-Pb collisions, which for the ATLAS and CMS experiments well exceeded the design value, while for the ALICE experiment, theluminosity is levelled near the Pb-Pb design value of 1027 cm−2 s−1 . The LHC can also provide
Pb-p collisions as it did in 2013 and 2016, and other ion-ion or ion-proton collisions, at different
energies.
In the coming years, an ambitious upgrade program, HL-LHC [30], with the accompanying LHC
Injectors Upgrade [31], has as its target an order-of-magnitude increase in integrated luminosity
through doubling the proton beam current, the utilization of new larger aperture Nb3 Sn superconducting final quadrupoles to allow squeezing the β ∗ to as low as 10 cm, superconducting compact
crab cavities and luminosity leveling also for ATLAS and CMS as its key ingredients.
31.4.2 Electron-Positron Rings
Asymmetric energies of the two beams allow for the enhancement of B-physics research andfor interesting interaction region designs. SuperKEKB operates with 7 GeV electron and 4 GeV
positron beams since 2018 and is aiming for luminosities of 8 × 1035 cm−2 s−1 [32]. By summer
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31. Accelerator Physics of Colliders

2022, a world record luminosity of 4.7 × 1034 cm−2 s−1 has been reached, still at rather low beam
currents. Vertical beam-beam tune shifts of ξy ≈ 0.05 for the 4 GeV positron beam, and ξy ≈ 0.03
for the 7 GeV electron beam have been achieved. These values are still about a factor of two lower
than at the previous KEKB. Since 2020 SuperKEKB operates with a virtual crab-waist collision
scheme, first developed for the FCC-ee design [33]. The original crab-waist scheme, based on
additional sextupole magnets, was earlier implemented at DAΦNE [34]. The general crab-waist
concept combines a large Piwinski angle Φ, and an extremely low βy∗ ( σz ) with the cancellationof the transverse betatron resonances which occur under conditions of kQx + lQy = n, where k, l, n
are integers. The latter is achieved by means of existing or additional electromagnetic sextupoles
with special betatron phase advances to the collision point [35]. The crab-waist collision scheme
has become a design choice for all proposed future e+ e− circular colliders.
Beside SuperKEKB and DAΦNE, three other e+ e− ring colliders currently in operation are√√
VEPP-2000 with s up to 2.0 GeV, BEPC-II with s up to 4.6 GeV and VEPP-4M with maximum
c.m.e. of 12 GeV [1].
31.4.3 RHIC
The Relativistic Heavy Ion Collider [36] employs 3.45 T Nb-Ti superconducting magnets, andcollides combinations of fully-stripped ions such as H-H (p − p), p-Al, p-Au, d-Au, h-Au, Cu-Cu,
Cu-Au, Zr-Zr, Ru-Ru, Au-Au, and U-U over a wide energy range. The high charge per particle (+79
for gold, for instance) makes intra-beam scattering of particles within the bunch a special concern,even for seemingly moderate bunch intensities. In 2012, 3-D stochastic cooling was successfully
implemented in RHIC [37] and is now routinely used. With stochastic cooling, steady increases
in the bunch intensity, and numerous other upgrades, RHIC now operates at 44 times the Au-Au
design average luminosity. Unique among high energy colliders, RHIC heavy ions beams cross the√transition energy γrmt = 1/ αc during acceleration – see Eqs.(31.11, 31.12) – a point where thederivative with respect to momentum of the revolution period is zero. This period of time is kept
as short as allowed by the magnet ramp rate and must be dealt with carefully.
RHIC is also unique in its ability to accelerate and collide polarized proton beams. As proton
beam polarization must be maintained from its low-energy source, successful acceleration throughthe myriad of depolarizing resonance conditions in high energy circular accelerators has taken years
to accomplish [38]. An energy of 255 GeV per proton with 60% final polarization per beam has
been realized. As part of a scheme to compensate the head-on beam-beam effect, electron lenses
operated routinely during the record high beam-beam parameter polarized proton operation at 100
GeV energy in 2015 [39].

31.5 Future High Energy Colliders and Prospects
Modern nuclear physics and high energy particle physics face critical questions which requirenext-generation high-energy and high-intensity experiments using hadron-hadron, lepton-lepton,
and lepton-proton colliding-beam facilities. Understanding the structure of the proton and neutron
directly from the dynamics of their quarks and gluons governed by the quantum chromodynamics
calls for new ion-ion and electron-ion colliders. Two types of colliders are generally aspired by the
HEP community [40, 41]: i) Higgs factories with a c.m.e. of 240–250 GeV in e+ e− collisions for
precision studies of the Higgs boson (mH = 125 GeV) and exploration of the Higgs sector in greater
detail, including measurements of Higgs couplings to fermions and vector bosons, self-coupling,rare decays, mass and width, that can also deliver other electroweak precision physics, e.g. on
the Z-pole (91 GeV c.m.e.), at the W -pair threshold (about 160 GeV), and when run as a top
quark factory (365–380 GeV); and ii) colliders with c.m.e. levels significantly beyond those of the
LHC to explore the energy frontier for potential discoveries through direct searches in pp, µµ, ande+ e− interactions. In addition, precision physics at future high-luminosity factories operating at
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1231. Accelerator Physics of Colliders

the τ -charm energy also provides sensitivity to new physics at multi-TeV energies and beyond. A
comprehensive review of the future colliders’ projects, ideas, and R&D activities can be found in
Ref. [1]; Ref. [42] presents comparative analysis of implementation challenges of numerous future
HEP collider proposals. Below we only briefly summarize leading collider proposals for construction
over the next several decades which rely mostly on currently available technologies, such as normalconducting (NC) or SC RF and/or NC or SC magnets, some of them requiring reasonable scopeand duration mission-oriented development programs, as well as advanced schemes based on plasma
acceleration and other innovative ideas. Tentative parameters of some of the colliders discussed, or
mentioned, in this section are summarized in Table 31.1 and Table 31.2.
31.5.1 Ion-Ion and Electron-Ion Colliders
NICA (Nuclotron-based Ion Collider fAcility) is a new accelerator complex under construction
at the Joint Institute for Nuclear Research (JINR, Dubna, Russia) to study properties of hot anddense baryonic matter, strong interactions between quarks and gluons, and spin physics [43]. NICA
will provide a variety of beam species, ranging from protons and polarized deuterons to massive
gold ions. The 500 m circumference SC magnet based collider is designed for average luminosity√
in heavy ion and light ion interactions at sN N =4–11 GeV of 1×1027 cm−2 s−1 for a variety of√
nuclei up to 197 Au79+ , and for polarized proton and deuteron collisions at s=12–27 GeV with
L=(1–10)×1031 cm−2 s−1 . NICA major accelerator challenges include strong intrabeam scatteringand space-charge effects which will be mitigated by extensive use of electron and stochastic coolingsystems.
The recently announced Electron-Ion Collider (EIC) for nuclear physics research will be built
at Brookhaven National Laboratory in the US and arrange collisions between an the reconfigured
RHIC with a 41–275 GeV proton beam and a 5-18 GeV electron beam stored in a new ring (eRHIC)[44]. The EIC physics requirements [45], include highly polarized (Pe,n ∼70%) electron and nucleon
beams (as the precision of measurements of interest scales as LPe2 Pn2 ), a spectrum of ion beams from√deuterons to the heaviest nuclei (U or Pb), variable c.m.e. values from s =20 GeV to 140 GeV, highluminosities of up to 1033−34 cm−2 s−1 , as well as possibilities of having more than one interaction
region. Main accelerator design challenges on the path to the required energy, luminosity, andpolarization, include the development of SRF crab-cavities and advanced SC magnets for interaction
region focusing, energy-recovery linac (ERL) based electron cooling of hadron beams, essential
to attain luminosities two orders of magnitude beyond the predecessor HERA ep collider, andhigh intensity polarized particle sources, augmented by the development of special magnets andoperational techniques to preserve the polarization through the acceleration process to the collisions,
including swap-out injection.
31.5.2 Higgs/Electroweak Factories
Higgs factory proposals generally aim at improving the precision of coupling measurements ofHiggs boson, top quark, W and Z by an order of magnitude or more compared with previous studies.
Two proposals for ∼100 km circumference circular e+ e− colliders have recently gained momentum:
the Future Circular Collider (FCC-ee) at CERN [33] and the Circular Electron-Positron Collider(CEPC) in China [46]. Design philosophy of these machines assumes use of the maximum RF power
available to compensate O(100 MW) synchrotron radiation losses PSR = 2I · ∆ESR and operation
at the beam-beam limit ξy that yields peak luminosity:L=3
16πr02 (me c2 )PSR ξy ρ,βy∗ γ 3(31.17)that scales approximately as 1/Eb3.5 for practical limits on P , ξy and βy∗ . The short beam lifetimeat the high target luminosity, due to radiative Bhabha scattering, requires these machines to be
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31. Accelerator Physics of Colliders

Table 31.1: Tentative parameters of selected future e+ e− high-energy colliders. Parameters
associated with different beam energy scenarios are comma-separated; H and V indicate horizontal
and vertical directions.
FCC-eee+ e−46, 120, 183
90.66
4

CEPC
e+ e−
46, 120, 180
100
2

ILC
e+ e−
125, 250
20.5, 31
1

CLIC
e+ e−
190, 1500
11, 501

17, 0.6, 0.15

15, 0.65, 0.07

0.2, 0.3

0.1, 0.6

140, 5.0, 1.25
3307
≥10, 0, 0
0.025, 0.3, 2.5

115, 5.0, 0.5
3000
5–10, 0, 0
0.025, 0.68, 2.6

Energy spread (rms, 10−3 )1.09, 1.43, 1.92

1.3, 1.7, 2.0

Bunch length σz (rms, mm)

15.5, 4.7, 2.2
H: 9, 13, 40
V: 0.04, 0.04, 0.05
H: 63, 167, 568
V: 0.17, 0.3, 0.6
H: 11, 24, 100
V: 0.07, 0.1, 0.1630
crab waist
26, 5.4, 0.8
97, 88, 134
1.3, 0.9, 2.0
0.01, 0.04, 0.06
400, 400, 800
21, 12, 1611200, 440, 60
1270, 27, 4.9
on Eb (top off)
5.7, 10.6, 20.1
100
n/a
—

8.7, 3.9, 2.9
H: 5.9, 14, 38
V: 0.04, 0.04, 0.11
H: 24, 150, 493
V: 0.12, 0.3, 1.7
H: 13, 33, 104
V: 0.09, 0.1, 0.27
33
crab waist
24, 4.8, 1.3
110, 127, 100
0.6, 1.3, 0.8
0.02, 0.04, 0.05
650
14, 13, 20
11951, 249, 35
804, 16.7, 3.3
on Eb (top off)
17.4, 24.9, 27.6
60
n/a
—

SpeciesBeam energy Eb (GeV)
Circumference or length (km)Interaction regions
Est. integrated luminosity
per experiment (ab−1 /year)
Peak lumi. L/IP (1034 cm−2 s−1 )Rep.rate (Hz, frev for rings)
Polarization (%)
Time between collisions (µs)

IP beam size σ ∗ (rms, µm)
Emittance, εn (rms, µm)
β ∗ at interaction point (cm)Full crossing angle θc (mrad)
Crossing scheme
Piwinski angle Φ = σz θc /(2σx∗ )Beam-beam param. ξy (10−3 )Disruption parameter Dy
Average Upsilon Υ (10−2 )RF frequency fRF (MHz)Particles per bunch N (1010 )Bunches per beam nb
Average beam current Ib (mA)
Injection energy (GeV)
RF gradient G (MV/m)
Total SR power loss (MW)
Total beam power (MW)Key technology

1.4, 1.8
1.5, 6550
80/30 (e− /e+ )80/0 (e− /e+ )0.55
0.0005e− : 1.9, 1.2
3.5
e+ : 1.5, 0.7
0.3
0.07, 0.044
H: 0.52, 0.47
H: 0.15, 0.04
V: 0.008, 0.006
V: 0.003, 0.001
H: 5, 10
H: 0.95, 0.66
V: 0.035, 0.035
V: 0.03, 0.02
H: 1.3, 1.1
H: 0.8, 0.69
V: 0.041, 0.048
V: 0.01, 0.0068
14
20
crab crossing
crab crossing
0
0
n/a
n/a
35, 2513, 8
3, 6
17, 500
1300
11994
2
0.52, 0.37
1312 (pulse) 352, 312 (trains at 50 Hz)
0.021
0.014, 0.009
5.0 (linac)
9.0 (linac)
31.5
72, 100n/an/a5.3, 10.5
5.6, 28high grad. SC RF
two-beam accel.

constructed with a full-energy injector ring installed in the same tunnel to top off the electron
and positron currents in the collider rings operating at constant energy. Beamstrahlung introduces
an additional beam lifetime limitation depending on momentum acceptance (so that achieving
sufficient off-momentum dynamic aperture becomes one of the design challenges), as well as somebunch lengthening.
These ambitious, large-scale projects based on well-established technologies are not extendable
to TeV or multi-TeV energies, but they offer several important advantages that include the potential
for much higher luminosities, and, thus, higher precision, the ability to operate multiple experiments
simultaneously, and their ∼100 km circular tunnels that could later house O(100 TeV) hadroncolliders. The high energy efficiency of circular e+ e− colliders is further boosted by advances in RF
power sources, by improved SC cavities, and by innovative low-power magnet systems includingones based on high-temperature superconductors (HTS) at moderate magnetic field.
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31. Accelerator Physics of Colliders

Table 31.2: Tentative parameters of selected future high-energy hadronic and muon colliders.
Parameters associated with different particle species for NICA and EIC, and different beam-energy
scenarios for a muon collider, are comma-separated. Quantities are, where appropriate, r.m.s.;
unless noted otherwise, energies refer to beam energy; H and V indicate horizontal and verticaldirections. Parameters of HL-LHC can be found in the High-Energy Collider Parameters review
tables.

SpeciesBeam energy Eb (TeV)
Circumference C (km)Interaction regions
Est. integr. luminosity
per exp. (ab−1 /year)
Peak luminosity L (1034 cm−2 s−1 )Rep.rate (Hz, frev for rings)
Time between collisions (µs)
Energy spread (rms, 10−3 )Bunch length σz (rms, mm)
IP beam size σ ∗ (H/V rms, µm)

NICA
ion-ion, pp
10−3 ·(4.5/u, 13)
0.503
2

EIC
ep, e-ion
0.01(e),0.275(p)
3.834
1(2)

FCC-hh
pp
40-58
90.66
4

SPPC
pp
62.5
100
2

µµ colliderµ+ µ−0.063, 50.3, 101, 2

10−8,−3 (ii, pp)

0.1

0.2–1.0

0.6

0.001, 2.0

10−7,−2 (ii, pp)
5.9·105
0.077
1.6 (Au)
600
360

1.05
7.8·104
0.009
0.6 (e), 0.7 (p)7 (e), 60 (p)95/8.5
11.3/1.0 (e),9.2/1.6 (p)45/5.6 (e),80/7.2 (p)72/100 (e),12 (p)591
17.2(e), 6.9(p)1160
2500(e),1000 (p)on Eb (e), 25 (p)0.248 (e), 3.80 (p)> 70(e), >70(p)10(e), < 10−6 (p)strong hadroncooling

5–30
3307
0.025
0.1
80
6.7-3.5 (init.)

4.3
3000
0.025
0.1
60
3.0 (init.)

0.008, 20
15, 5
1, 33
0.04, 1
63, 1.575, 0.9

2.2 (init.)

1.2 (init.)

200, 25

110–3050

1.7, 0.15

5–15
400
10
9648
500
≥1000
14-20
0
1.0-4.25
Nb3 Sn/HTS
magnets

15
400/200
4
10082
190
3200
20
0
2.2
HTS
magnets

22, 78805/1300
400, 1801640, 9 (peak)
on Eb
10
0
10−3 , 0.16
muon prod.
& cooling

Emittance εn (H/V rms, mm mrad)1.1

Beta function at IP β ∗ (H/V cm)

60

Beam-beam param. ξ (10−3 H/V)

25
13/39
0.23
22
480
1-3.8
1.8
0(i), >50(p)10−6electron andstoch. cooling

RF frequency fRF (MHz)Particles per bunch N (1010 )Bunches per beam nb
Average beam current Ib (mA)
Injection energy (GeV)
Peak magnetic field B (T)Polarization (%)
SR power loss/beam (MW)Key technology

For more than four decades, efforts have been devoted to developing high-gradient RF technology linear e+ e− colliders in order to overcome the synchrotron radiation limitations of circular e+ e−
machines. The International Linear Collider (ILC), with a c.m.e. of 250 GeV in e+ e− collisions, has√been under consideration for more than two decades and could potentially be upgraded to s=500
GeV and further to 1 TeV. CERN’s Compact Linear Collider (CLIC) design, developed since themid-1980s, also includes possible upgrades, from an initial 380 GeV c.m.e. to ultimately 3 TeV,
which would enable searches for new particles of significantly higher masses.
The primary challenge confronting a high energy, high luminosity single pass collider design isthe beam power requirement, so that measures must be taken to keep the demand within bounds
as illustrated in a transformed Eq.(31.15):
L=

1 Pwall η√
Nγ HD .
8παr0
s σy∗

(31.18)Here, Pwall is the total wall-plug power of the collider, to be converted into beam power Pb =2f0 N Eb with efficiency η, Nγ ≈ 2αr0 N/σx∗ is the number of beamstrahlung photons emitted per
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31. Accelerator Physics of Colliderse± (α ≈ 1/137 denotes the fine-structure constant), and the last factor HD , typically between 1
and 2, represents the enhancement of luminosity due to the pinch effect, the additional focusing
occurring during the collision of oppositely charged bunches. The management of Pwall leads to an
upward push on the bunch population N with an attendant rise in the energy radiated due to theelectromagnetic field of one bunch acting on the particles of the other (beamstrahlung). Keeping
a significant fraction of the luminosity close to the nominal energy represents a design goal, whichis met if Nγ does not exceed a value of about 1. A consequence is the use of flat beams, whereNγ is managed by the beam width, and luminosity adjusted by the beam height, thus the explicit
appearance of the vertical beam size σy∗ .
The ILC [47, 48] is based on 1.3 GHz standing-wave superconducting accelerating structures
operating at 2 K with 31.5 MV/m average gradient, up to 8 nm vertical beam size at the IP, andluminosity comparable to the LHC. Progress toward higher field gradients and Q values continues
to be made, with nitrogen-doping techniques being a recent example [49]. CLIC is based on anovel two-beam acceleration scheme [50]. Here, room-temperature NC copper high-gradient 12
GHz accelerating structures are powered by a high-current 1.9 GeV drive beam to efficiently enableaccelerating gradients of up to 100 MV/m (though optimal G=70 MV/m for the first CLIC stage√
at s=380 GeV, and for this stage an alternative RF power drive option with 12 GHz klystrons
powering is also being considered). To reach their design luminosities, both CLIC and ILC require
unprecedented rates of positron production about 40 times the world record set by the SLC positronsource, and very tight control of imperfections, such as O(10 µm) accuracy of pre-alignment of themain linac and beam delivery system components at the level, suppression of fast vibrations of
the quadrupoles due to ground motion to O(1 nm) level at frequencies above 1 Hz, advanced
beam-based trajectory tuning, and mitigation of the effect of wakefields [51].
Recent developments of the RF technologies show promise of more compact linear e+ e− Higgs/Electroweak factories based on either ∼70 MV/m travelling-wave 1.3 GHz SRF structures [52]or on 70–120 MV/m high-efficiency, distributed RF coupling, normal-conducting cold copper 5.7
GHz accelerating cavities at liquid nitrogen temperature [53].
There are a number of alternative ideas proposed for studies of the Higgs/Electroweak physics,
such as high-energy, high-luminosity e+ e− collider in a 100 km tunnel using ERLs to accelerateparticles to collision energy in 4 to 6 passes and return up to 81% of the energy back into the
SRF cavities on deceleration turns, thus, lowering the required facility power several-fold [54];similar power recovery in one pass can greatly improve efficiency of linear colliders [55–57]; anarrangement of γγ collision through near-IP conversion of high energy electron beams into intense
photon beams by backward Compton scattering off a high-power laser [58,59] or off an FEL photon
pulse [60,61]; µµ Higgs factory with unprecedented 0.004% energy resolution [62], and a high-energy
lepton-hadron collider bringing into collision a 60-GeV electron beam from an ERL with the 7 TeV
protons circulating in the LHC (LHeC) [63]. At lower energies, Super Tau-Charm Factory proposals
aim at the production and precise study of charmonium states and of the tau lepton [64, 65].
31.5.3 Energy Frontier Circular Colliders
Several hadron and lepton colliders have been proposed to extend the energy reach beyond
the LHC. As noted above, ambitious plans have been proposed to upgrade the FCC and CEPC
to hadron colliders – FCC-hh at CERN and Super Proton Proton Collider (SPPC) in China,
respectively – by means of next- or next-next generation SC magnets installed in the arc sections of√
the 100 km rings, so as to enable s of the order 100 TeV or above [66, 67]. Comparable discovery
reach is expected for a circular 10–14 TeV muon collider [68], significantly beyond that of practicale+ e− linear colliders.
The maximum beam energy Eq.(31.1) is directly proportional to the magnetic field and to the
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31. Accelerator Physics of Colliders

ring circumference, hence,future hadron colliders like FCC-hh and SppC rely on the development of
the technology of 14–16 T Nb3 Sn dipole magnets [69] or dipole magnets based on high-temperature
superconductors (HTS) with a field of up to 18-20 T, including iron-based HTS [70]. Though higher
fields are possible with HTS, more cost-effective might be hybrid magnet designs incorporating NbTi, Nb3 Sn, and an inner layer of HTS and providing fields of about 20 T. Another important
technology is the cryogenic beam vacuum system, which has to cope with unusually high levels ofsynchrotron radiation (up to 15 MW in total, for FCC-hh) in a cold environment. The beam-screenintercepting the radiation inside the cold bore of the magnets should operate at or above 50 K —significantly higher temperature than in the LHC.
Design luminosities of these hadron colliders O(1035 cm−2 s−1 ) will result in a pile-up of events
per crossing O(500) (from up to 50 in LHC) and fast intensity drop due to burn-off. Significant
radiation damping of beam emittances will naturally level luminosity evolution, though the total
beam-beam tune shift nIP ξ ∼ (0.01 − 0.03) might need a special control as it will increase during
the store [71].
Future hadron colliders are characterized by record high stored beam energy, e.g. 6–9 GJ in
FCC-hh, rendering machine protection a paramount concern. A very challenging multi-stage collimation system is needed to avoid local beam loss spikes near cold magnets, which would induce
magnet quenches. The primary and secondary collimators of the LHC are based on carbon-carbon
composite material. For the future hadron colliders, ever stronger materials are being developedand examined, which also feature higher conductivity and, hence, lower impedance. More advanced
options include the use of short bent crystals as primary collimators [72] and the deployment ofhollow electron-beam lenses as non-destructible collimators [73].
It is noteworthy that machines like FCC-hh or SPPC can additionally be used for ion-ion/ionproton collisions; their high-energy proton beams can also be collided with high-intensity O(60
GeV) electrons from an ERL resulting in c.m.e. of 3.5 TeV.
The lifetime of the muon, γτ0 where τ0 =2.2µs, is sufficient to allow fast acceleration to highenergy before most, or all, of the muons decay, and storage for some 300B turns in a ring withan average bending magnets field of B (in units of Tesla). The muon to electron mass ratio of
210 implies removal of the synchrotron radiation barrier and possibility of a muon collider facility
scale to a level compatible with on-site placement at existing accelerator laboratories. High-energy
muon colliders, as presently conceived, are predicted to be more compact, more power-efficient andsignificantly less expensive than the equivalent energy-frontier hadron or e+ e− machines [74], anda neutrino factory could potentially be realized in the course of their construction [75]. The Higgs
production cross-section in the s-channel is enhanced by a factor of (mµ /me )2 compared to that ine+ e− collisions.
The average luminosity of a muon collider,hLi = f0 γ 2cτ0 nb N 2
N r0 γ cτ0 F
F = BPb
2C 4πεn β ∗4πεn β ∗ 8πe

,(31.19)scales with B, the total beam power Pb , and the beam brightness – the third factor above isnothing but the muon beam-beam tune shift Eq. (31.16). There is an obvious incentive to have all
the particles in just one bunch per beam. The beta-function at the two IPs β ∗ ≈ σz scales as 1/γ
within certain range of energies, giving overall scaling hLi ∝ γ 2 with other limiting parameters fixed.
The main challenges to luminosity achievement with decaying particles are related to production
and fast cooling and acceleration of O(1012 ) muons per bunch without emittance degradation.
A multi-TeV c.m.e. high luminosity O(1034 cm−2 s−1 ) muon collider would consist of [76] : (i)a high power proton driver (e.g., 8 GeV 2-4 MW H− SRF linac), (ii) pre-target accumulation
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31. Accelerator Physics of Colliders

and compressor rings, in which high intensity 1-3 ns long proton bunches are formed, (iii) a liquidmercury or other high-power target for converting the proton beam into a tertiary muon beam withenergy of about 200 MeV, (iv) a multi-stage ionization cooling section that reduces transverse andlongitudinal emittances by several orders of magnitude and creates a low emittance beam, similarly
to that recently demonstrated [77], (v) a multistage acceleration system, possibly employing either
rapid cycling synchrotrons or RLAs to accelerate muons in a modest number of turns up to thefinal energy using superconducting RF technology, and, finally, (vi) a 3–14 km diameter collider
ring, where counter-propagating muon beams are stored and collide over the roughly 3000 turns
corresponding to the muon lifetime.
The intense neutrino flux originating from the multi-TeV muon beams decaying in the collider
poses another challenge — the need to minimize the environmental impact. The collider complex
is usually located underground and when the produced neutrinos emerge at the surface, a small
fraction interacts with the rock (and other material) and produces ionizing radiation dose scalingas γ 3 . The impact of this neutrino-induced radiation can be mitigated, for example, by continually
adjusting the orbits of the beams to spread them out on a wider area, by deeper collider tunnels
or with a further reduced emittance of the muon beam so that the required luminosity could beobtained using a substantially smaller number of muons.
31.5.4 Plasma Acceleration and Other Advanced Concepts
Since about the mid-1950s, it has been understood that collective plasma-based accelerators
promise extremely large accelerating gradients, approximately three orders of magnitude greaterthan ∼100 MV/m obtained in conventional breakdown limited RF structures [78]. Ionized plasmas
can sustain electron plasma density waves with electric fields in excess of E0 = cme ωp /e or
E0 ≈ 96 [V/m]qn0 [cm−3 ],(31.20)pwhere n0 denotes the ambient electron number density and ωp = e2 n0 /(me ε0 ) is the electron
plasma frequency [79].
Such gradients can be effectively excited by either powerful external pulses of laser
p light or byelectron bunches if they are shorter than the plasma wavelength λp = c/ωp ≈ 1 mm× 1015 cm−3 /n0 ,or by longer beams of protons if their charge density is modulated with the period of λp . In thepast decade, we have seen impressive progress in the plasma wakefield acceleration of high-quality
beams. Laser-driven electron energy gain of about 8 GeV over 20 cm of plasma with density
3×1017 cm−3 has been demonstrated at the BELLA facility at the Lawrence Berkeley National
Laboratory (LBNL) [80]. Short electron bunches were used to boost the energy of externallyinjected electron bunches by 9 GeV over 1.3 m of ∼1017 cm−3 plasma at the FACET facility in
SLAC [81]. The AWAKE experiment at CERN used self-modulating long 450 GeV proton bunches
to accelerate electrons to 2 GeV over 10 m of 1015 cm−3 plasma [82].
Whether plasma acceleration will find application in an HEP facility is not yet clear, given
the necessity of staging and phase-locking acceleration in multiple plasma chambers [83]. Anothercritical issue is the power efficiency η for a collider based on plasma acceleration, whose luminosity
would still be described by Eq. (31.18). Positrons can be accelerated as well but, contrary toelectrons, plasma-wakefields will naturally defocus positively charged particles. Such fundamental
deficiency can be addressed, e.g., in more complicated plasma acceleration schemes for e+ [84];or by acceleration of electrons in both linacs with conversion into a γγ collider at the IP. In
addition, many novel approaches have been proposed to extend the energy or physics reach of
future particle colliders, reduce their cost, and improve their luminosity or energy efficiency. Those
include: i) dielectric wakefield accelerators in which resonant dielectric accelerating structures are
fed by ultra-short RF pulses of wakefields driven by either collinear or preceding high charge
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electron bunches and withstand 270 MV/m operational accelerating gradients [85]; ii) dielectriclaser accelerators – micron-size dielectric accelerating structures driven by a laser and supportingO(1 GV/m) accelerating fields [86]; iii) compact linear muon crystal colliders with ultimate energies
O(1–10 PeV) [87] based on 1–10 TeV/m wakefield acceleration of muons (instead of electrons or
hadrons) channeling between the planes in crystals or inside carbon nanotubes (CNT) with charge
carrier density ∼1020−22 cm−3 [88]; iv) the Gamma Factory concept [89], where frequent bursts of
gamma rays are generated by repeatedly colliding a partially stripped heavy-ion beam circulatingin the LHC, or in a future higher-energy hadron storage ring like the FCC-hh, with a conventionallaser pulse, profiting from two Lorentz boosts. Active R&D programs are presently underway
worldwide to determine the suitability of novel technologies for use in future high-energy colliders.
References[1] V. Shiltsev and F. Zimmermann, Reviews of Modern Physics 93, 1, 015006 (2021).
[2] A. Tollestrup and E. Todesco, in “Reviews Of Accelerator Science And Technology: Volume
1,” 185–210, World Scientific (2008).
[3] D. A. Edwards and M. J. Syphers, An introduction to the physics of high energy accelerators,
John Wiley & Sons, New York, NY (2008).
[4] A. W. Chao, Physics of collective beam instabilities in high energy accelerators, Wiley, New
York, NY (1993), URL https://cds.cern.ch/record/246480.
[5] S.-Y. Lee, Accelerator physics, World Scientific, Singapore (2018).
[6] E. D. Courant and H. S. Snyder, Annals of physics 3, 1, 1 (1958).
[7] H. Wiedemann, Particle accelerator physics, Springer Nature (2015), URL https://library.
oapen.org/handle/20.500.12657/23641.
[8] M. Sands, Conf. Proc. C 6906161, 257 (1969).
[9] S.-Y. Lee, Spin dynamics and snakes in synchrotrons, World Scientific Publishing Company(1997).
[10] F. Zimmermann, Physical Review Special Topics-Accelerators and Beams 7, 12, 124801 (2004).
[11] A. W. Chao et al., editors, Handbook of accelerator physics and engineering, World Scientific,
Hackensack, USA (2013), ISBN 9789814415842, URL http://www.worldscientific.com/
worldscibooks/10.1142/8543.
[12] A. Piwinski, J. D. Bjorken and S. K. Mtingwa, Physical Review Accelerators and Beams 21,
11, 114801 (2018).
[13] V. Parkhomchuk and A. Skrinsky, Reviews of Accelerator Science and Technology 1, 01, 237(2008).
[14] M. Minty and F. Zimmermann, Measurement and control of charged particle beams, Springer
Nature (2003), URL https://library.oapen.org/handle/20.500.12657/23642.
[15] S. Myers, Particle Physics Reference Library: Volume 3: Accelerators and Colliders, volume 3,
Springer Nature (2020), URL https://library.oapen.org/handle/20.500.12657/39571.
[16] V. Shiltsev et al., Physical Review Special Topics-Accelerators and Beams 8, 10, 101001 (2005).
[17] J. T. Seeman, Observations of the beam-beam interaction, Springer (1986).
[18] R. Assmann, M. Lamont and S. Myers, Nucl. Phys. B Proc. Suppl. 109, 17 (2002).
[19] K. Yokoya and P. Chen, Lect. Notes Phys. 400, 415 (1992).
[20] H. T. Edwards, Annual Review of Nuclear and Particle Science 35, 1, 605 (1985).
[21] V. Lebedev and V. Shiltsev, Accelerator physics at the Tevatron collider, Springer (2014).
31st May, 2024

19

31. Accelerator Physics of Colliders[22] S. D. Holmes and V. D. Shiltsev, Annual Review of Nuclear and Particle Science 63, 435(2013).
[23] G. Voss and B. Wiik, Annual Review of Nuclear and Particle Science 44, 1, 413 (1994).
[24] S. Myers and H. Schopper, Landolt Börnstein 21 (2013).
[25] D. Brandt et al., Reports on Progress in Physics 63, 6, 939 (2000).
[26] N. Phinney, eConf C00082, MO102 (2000), [arXiv:physics/0010008].
[27] L. Evans, Annual Review of Nuclear and Particle Science 61, 435 (2011).
[28] CERN, The Large Hadron Collider web site: http://lhc.web.cern.ch/lhc/.
[29] G. Valentino et al., Physical Review Special Topics-Accelerators and Beams 15, 5, 051002(2012).
[30] G. Apollinari et al., Technical Report CERN Yellow Report: Monograph 2017-007-M (2017).
[31] H. Damerau et al., LHC Injectors Upgrade, Technical Design Report (2014), URL https://cds.cern.ch/record/1976692.
[32] Y. Ohnishi et al., Progress of Theoretical and Experimental Physics 2013(3), 3 (2013).
[33] M. Benedikt et al., The European Physical Journal Special Topics 228, 2, 261 (2019).
[34] M. Zobov et al., Physical Review Letters 104, 17, 174801 (2010).
[35] P. Raimondi, D. N. Shatilov and M. Zobov (2007), [arXiv:physics/0702033].
[36] M. Harrison, S. Peggs and T. Roser, Annual Review of Nuclear and Particle Science 52, 1,
425 (2002).
[37] M. Blaskiewicz, J. M. Brennan and K. Mernick, Phys. Rev. Lett. 105, 094801 (2010).
[38] M. Bai et al., Physical Review Letters 96, 17, 174801 (2006).
[39] W. Fischer et al., Phys. Rev. Lett. 115, 26, 264801 (2015).
[40] 2020 Update of the European Strategy for Particle Physics by the European
StrategyGroup,
https://home.cern/sites/home.web.cern.ch/files/2020-06/
2020UpdateEuropeanStrategy.pdf (2020), accessed: June 20, 2020.
[41] J. N. Butler et al., arXiv preprint arXiv:2301.06581 (2023).
[42] T. Roser et al., Journal of Instrumentation 18, 05, P05018 (2023).
[43] V. Kekelidze et al., The European Physical Journal A 52, 8, 211 (2016).
[44] E. Beebe-Wang (ed.) et al., Technical Report BNL Formal Report BNL-211943-2019-FORE,
Brookhaven (2019).
[45] A. Accardi et al., The European Physical Journal A 52, 9, 268 (2016).
[46] CEPC Study Group (2018), [arXiv:1809.00285].
[47] C. Adolphsen et al., Technical Report arXiv:1306.6328. CERN-ATS-2013-037. IHEP-ACILC-2013-001. ILC-REPORT-2013-040. INFN-13-04-LNF. JINR-E9-2013-35. JLAB-R-201301. LLNL-TR-635539. comments: See also http://www.linearcollider.org/ILC/TDR . The fulllist of signatories is inside the Report, URL http://cds.cern.ch/record/1601969.
[48] L. Evans and S. Michizono (Linear Collider) (2017), [arXiv:1711.00568].
[49] A. Grassellino et al., Supercond. Sci. Technol. 26, 102001 (2013), [arXiv:1306.0288].
[50] T. K. Charles et al. (CLICdp, CLIC), CERN Yellow Rep. Monogr. 1802, 1 (2018),[arXiv:1812.06018].
[51] T. O. Raubenheimer, Physical Review Special Topics-Accelerators and Beams 3, 12, 121002(2000).
31st May, 2024

20

31. Accelerator Physics of Colliders[52] S. Belomestnykh et al. (2022), [arXiv:2203.08211].
[53] C. Vernieri et al., Journal of Instrumentation 18, 07, P07053 (2023).
[54] V. N. Litvinenko, T. Roser and M. Chamizo-Llatas, Physics Letters B 804, 135394(2020), ISSN 0370-2693, URL http://www.sciencedirect.com/science/article/pii/
S0370269320301982.
[55] M. Tigner, Il Nuovo Cimento (1955-1965) 37, 3, 1228 (1965).
[56] U. Amaldi, Physics Letters B 61, 3, 313 (1976).
[57] V. I. Telnov, Journal of Instrumentation 16, 12, P12025 (2021).
[58] V. Telnov, Journal of Instrumentation 9, 09, C09020 (2014).
[59] S. A. Bogacz et al. (2012), [arXiv:1208.2827].
[60] C. Adolphsen et al. (2022), [arXiv:2207.02095].
[61] T. Barklow et al., Journal of Instrumentation 18, 07, P07028 (2023).
[62] Y. Alexahin, E. Gianfelice-Wendt and V. Kapin, Journal of Instrumentation 13, 11, P11002(2018).
[63] P. Agostini et al., Journal of Physics G: Nuclear and Particle Physics 48, 11, 110501 (2021).
[64] P. Piminov, Physics of Particles and Nuclei Letters 15, 7, 732 (2018).
[65] J. Lan et al., Journal of Instrumentation 16, 07, T07001 (2021).
[66] M. Benedikt et al., The European Physical Journal Special Topics 228, 4, 755 (2019), ISSN
1951-6401, URL https://doi.org/10.1140/epjst/e2019-900087-0.
[67] CEPC Study Group (2018), [arXiv:1809.00285].
[68] H. Al Ali et al., Reports on Progress in Physics 85, 8, 084201 (2022).
[69] D. Schoerling and A. V. Zlobin, Nb3 Sn Accelerator Magnets: Designs, Technologies and Performance, Springer Nature (2019).
[70] E. Kong et al., International Journal of Modern Physics A 34, 13n14, 1940003 (2019).
[71] M. Benedikt, D. Schulte and F. Zimmermann, Phys. Rev. ST Accel. Beams 18, 101002 (2015).
[72] W. Scandale and A. Taratin, Physics Reports 815, 1 (2019).
[73] V. Shiltsev, Journal of Instrumentation 16, 03, P03039 (2021).
[74] K. R. Long et al., Nature Physics 17, 3, 289 (2021).
[75] S. Geer, Annual Review of Nuclear and Particle Science 59, 347 (2009).
[76] C. Ankenbrandt et al., Physical Review Special Topics-Accelerators and Beams 2, 8, 081001(1999).
[77] M. Bogomilov et al., Nature 578, 53 (2020).
[78] V. Veksler, The Soviet Journal of Atomic Energy 2, 5, 525 (1957).
[79] T. Tajima and J. M. Dawson, Physical Review Letters 43, 4, 267 (1979).
[80] A. Gonsalves et al., Physical Review Letters 122, 8, 084801 (2019).
[81] M. Litos et al., Plasma Physics and Controlled Fusion 58, 3, 034017 (2016).
[82] E. Adli et al., Nature 561, 7723, 363 (2018).
[83] D. Schulte, Reviews of Accelerator Science and Technology 9, 209 (2016).
[84] Z.Y. Xu et al., Phys. Rev. Accel. Beams 23, 091301 (2020).
[85] C. Jing, Reviews of Accelerator Science and Technology 9, 127 (2016).
31st May, 2024

21

31. Accelerator Physics of Colliders[86] E. Peralta et al., Nature 503, 7474, 91 (2013).
[87] V. D. Shiltsev, Physics-Uspekhi 55, 10, 965 (2012).
[88] T. Tajima and M. Cavenago, Physical Review Letters 59, 13, 1440 (1987).
[89] D. Budker et al., Annalen der Physik 532, 8, 2000204 (2020).

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