The main characteristics of a linear accelerator are determined by the properties of its rf source (klystrons) and accelerating cavities. For a pulsed ILC-type superconducting linac, one of the currently available rf sources is the Toshiba E3736 Multi-Beam Klystron [3] . This source has the following well-tested specifications: rf frequency―1.3 GHz; peak rf power―10 MW; average power―150 kW; efficiency―65%; pulse length―1.5 ms; repetition rate―10 Hz. If the repetition rate of the Toshiba klystron is increased by a factor of two, while its peak power is reduced by the same factor (thus keeping the average power constant) one obtains the klystron specifications presented in Table 1. For such a klystron, a suitable 20 Hz pulse modulator has to be developed.

A very important parameter that determines, to a large extent, the power conversion efficiency of a klystron is its perveance, defined by

K ≡ I 0 U 3 / 2 (3)

In this expression, I 0 is the beam current and U is the anode voltage. Since there is an upper limit to the applied voltage, low perveance can be only obtained by operating with low currents. For single-beam klystrons, this requirement is not compatible with the need for high output power. With this in mind, multi-beam klystrons (MBK) were originally developed in the 1960s [4] . An MBK is a parallel assembly of low-current (low-perveance) beamlets within a common rf structure, which efficiently generates high output power. Using Equation (3), the output rf power of an MBK can be expressed as

P k = η I 0 U = η K U 5 / 2 (4)

where η is the klystron efficiency and I 0 = N b I b is the total beam current; N b is the number of beamlets and I b is the current carried by each beamlet. For Toshiba’s E3736 MBK, η = 65 % , U = 116   kV and I 0 = 134   A . Hence, K = 3.4 × 10 − 6   A / V 3 / 2 , klystron’s peak power P k = 10   MW and its average power P ¯ k ≡ P k × τ R = 150   kW for τ = 1.5   ms and R = 10   Hz . Since the klystron has six beamlets, I b = 22.3   A .

In order to increase the beam power of the SCL, R could be increased to 20 Hz (see Equation (1)). As already mentioned, the peak power of the klystron would then have to be reduced to 5 MW, so that its average power is still 150 kW for the klystron pulse length τ = 1.5   ms (see Table 1).

The basic properties of a 1.3 GHz superconducting ILC-type cavity are presented, e.g., in [5] . There are two important parameters that characterize rf cavities: the accelerating gradient E acc and the unloaded quality factor Q 0 . The former is a measure of how much the energy of a particle is increased over a given length of the linac (typically expressed in units of MV/m), while the latter specifies how well the cavity can sustain the stored rf power. A higher value of Q 0 implies a lower rate of power loss realtive to the stored energy¹. ILC-type cavities must have a nominal Q 0 greater than 1 × 10 10 (a dimensionless parameter) at E acc = 31.5   MV / m .

Each ILC-type cryomodule for the proposed SCL would contain eight niobium 9-cell cavities and a quadrupole magnet at its centre. Other major components of such a cryomodule are the vacuum vessel, thermal and magnetic shields, cryogenic piping, interconnections, etc. The inactive spaces between cavities or cryomodules (the “packing fraction”) are responsible for a substantial reduction in the average accelerating gradient of the linac. The beam physics and the lattice design of the superconducting L-band linac described in [6] are applicable to the SCL.

The average usable accelerating gradient in ILC-type cavities is E acc = 29.3 ± 5.1   MV / m [7] . Taking into account an estimated linac “packing fraction” of about 70%, the effective accelerating gradient of the SCL is E eff ≈ 20   MV / m . Hence, the total length of a 20 GeV linac is ~1000 m.

¹The Q factor of an rf cavity is defined as Q = 2π´ (energy stored/energy dissipated per cycle). For large values of Q, the Q factor is approximately the number of oscillations required for the energy of a freely oscillating system to fall off to e^−2π, or 0.2%, of its original value.

Since the length of an ILC 9-cell cavity is 1 m, a linac with E b = 20   GeV would require about N cav = ( 20000 MeV ) / ( 29 MeV ) = 690 cavities. The average input rf power per cavity is thus P ¯ cav = P b / N cav ≈ 22   kW , and the corresponding peak power P cav ≡ P ¯ cav / D = 916   kW . Although this value is acceptable for a pulsed linac with D ~ 2 % , it would be prudent to use two rf couplers per cavity. In that case the peak rf power per coupler would be about 460 kW.

For E acc = 30   MV / m , ohmic losses in an ILC 9-cell cavity amount to P c = 100   W in the CW mode of operation, but only P c = ( 100 × D )     W = 2.4     W (plus static loss) in the pulsed mode with a duty factor D = 0.024 . Because of large ohmic losses, which scale with the square of the accelerating gradient, E acc is generally lower in the CW mode of operation than in the pulsed mode.

A typical ~1 GeV proton linear accelerator consists of three main sections:

• Front end, comprising a proton source and a radiofrequency quadrupole accelerator;

• Medium-velocity linac, which accelerates proton beams to ~100 MeV;

• High-velocity linac, which accelerates protons to energies exceeding 1 GeV.

The most complex part of a proton linac is the low-energy (low-b) section, situated between the proton (or ion) source and the first drift-tube-based accelerating section. The continuous beam of protons coming from an electron cyclotron resonance (ECR) source [8] has to be focused, bunched and accelerated in the first rf structure. These three essential functions are nowadays successfully performed by radio frequency quadrupoles (RFQ) [9] . However, the beam has to be shrunk before it can be fed into an RFQ. This is accomplished within a low-energy beam transport (LEBT) section by means of cylindrical magnets (solenoids).

As soon as the beam is bunched---which is essential for further acceleration― it enters a medium-energy beam transport (MEBT) section, where it is collimated and steered from the RFQ into the medium-velocity linac (MVL). The MEBT may also contain a number of buncher cavities. Inside the MVL, the beam is accelerated to about 100 MeV ( β ~ 0.1 to 0.5). The MVL usually contains normal-conducting drift-tube linac (DTL) and cell-coupled drift tube linac (CCDTL) structures. A DTL incorporates accelerating components of increasing length in order to match precisely the increase in beam velocity, while quadrupole magnets provide strong focusing. The main advantage of using CCDTL structures is that they provide longitudinal field stability.

High-velocity linac (HVL) structures accelerate the beam to energies around 1 GeV. They consist either of normal-conducting side-coupled linac (SCL) structures² or superconducting elliptical cavities. The latter offer some advantages over the former, such as higher accelerating gradients and lower operating costs. The superconducting HVL can also feature spoke resonators, characterized by their simplicity, high mechanical stability and compact size [10] .

²The main reason for using these π/2-mode structures is that long chains of coupled cavities are often required for an efficient use of high-power rf sources [10] .

The European Spallation Source (ESS) is an example of a typical ~1 GeV proton linear accelerator [11] (see Figure 4). The transverse beam size along the linac varies in the range 1 - 4 mm, and the bunch length decreases from 1.2 cm to 3 mm towards the end of the linac.

One of the main concerns in the design of a high-power proton linac is to restrict beam losses. A careful beam dynamics study is therefore needed in order to avoid halo formation, a major source of beam loss. Another important issue is the preservation of beam emittance [6] .

High-power proton linear accelerators have a wide range of applications including spallation neutron sources, nuclear waste transmutation, production of radioisotopes for medical use, etc. A number of laboratories worldwide have ex-

pressed interest in building “proton drivers” that would primarily deliver high- intensity neutrino, kaon and muon beams [12] [13] .

The observed transformation of one neutrino flavor into another (“neutrino oscillation”) indicates that these particles have finite masses. Cosmological data suggest that the combined mass of all three neutrino species is a million times smaller than that of the next-lightest particle, the electron [17] . The phenomenon of neutrino oscillations implies not only the existence of neutrino mass, but also of neutrino mixing. That is, the neutrinos of definite flavor are not particles of definite mass (mass eigenstates), but coherent quantum-mechanical superpositions of such states. Converesely, each neutrino of definite mass is a superposition of neutrinos of definite flavor. Neutrino mixing is large, in striking contrast to quark mixing. Whatever the origin of the observed neutrino masses and mixings, it implies a profound modification of the Standard Model.

Mathematically, the phenomenon of neutrino mixing can be expressed as a unitary transformation relating the flavor and mass eigenstates. The neutrino oscillation rate depends, in part, on (1) the difference between neutrino masses and (2) the three parameters in the transformation matrix known as mixing angles. The complex phase factors in the transformation matrix (also called mixing matrix) are associated with the violation of CP symmetry in the lepton sector. The size of the CP violation is determined both by the phases and the mixing angles.

Experiments with high-intensity neutrino beams are designed primarily to explore the mass spectrum of the neutrinos and their properties under the CP symmetry, and thus provide a deeper insight into the nature of these elusive particles and their role in the universe. For instance, if there is experimental evidence for CP violation in neutrino oscillations, it could be used to explain the observed asymmetry between matter and antimatter [18] .

To search for CP violation in neutrino oscillations, a 100 kiloton water Cherenkov detector could be built at Okinoshima, located at a distance L 2 ≈ 650   km from KEK. Using the proposed KEK “proton driver”, the detector at Okinoshima and the existing 0.022 Mt Super-Kamiokande detector (situated at a distance L 1 ≈ 300   km from KEK), the neutrino mass hierarchy could be determined either by comparing the ν e appearance probabilities measured at the two vastly different baseline lengths L₁ and L₂, or by measuring at L₁ and L₂ the neutrino energy of the first oscillation maximum. Once the mass hierarchy is determined, the CP-violating phase in the mixing matrix can be measured with a precision of ± 20 ∘ , assuming that 2.5 × 10 21 protons are delivered on target for both ν e and ν ¯ e beams [19] .

The main challenge in the design of a multi-MW neutrino beam facility is to build a proton target that could dissipate large amounts of deposited energy, withstand the strong pressure waves created by short beam pulses, and survive long-term effects of radiation damage. Simulation studies of the pion production and energy deposition in different targets (liquid mercury jet, tungsten powder jet, solid tungsten bars and gallium liquid jet) are presented in [20] . Those studies also provide estimates of the amount of concrete shielding needed to protect the environment from the high radiation generated by each target. A proof-of- principle demonstration of a 4 MW target station comprising a liquid mercury jet inside a 20 T solenoidal magnetic field is described in [21] . Alternatively, one could use a rotating, gas-cooled tungsten target that would require the least amount of development effort, and would also have good thermal and mechanical properties [11] . A 15 MW proton beam could be separated by a series of magnets into four beam lines. Each of the four beams would be focused by a series of quadrupoles and correctors to an assembly consisting of four targets and the same number of magnetic horns (see, e.g., [22] ).

To maximize the discovery potential of a neutrino beam facility, it is important to properly design the magnetic horn that focuses the charged particles produced in the proton target. For proton beam pulses lasting 1 ms, a DC horn has been designed at KEK by Yukihide Kamiya [23] . The toroidal magnetic field of the horn, characterized by B ( r ) = const . , is generated by hollow aluminium conductors containing water. The strength of the magnetic field B = 0.2   T , and its length l = 5   m ; hence, B ⋅ l = 1   T ⋅ m . The radius of the magnet, r, is determined by r = L tan ( θ ) + l tan ( θ / 2 ) , where θ ≈ 0.03 + 0.3 / p is the initial angle a charged pion makes with respect to the proton beam direction, L is the distance from the target to the horn, and p is the pion momentum. For example, if L = 5   m then r ≈ 5   m . The total power generated in the conductors is about 10 MW.

Electron and positron beams, polarized and/or unpolarized, can be used to study the structure of composite particles and the dynamics of strong interactions, as well as to search for new physics beyond the Standard Model. A detailed description of the physics potential of a facility that can provide such beams (e.g., the upgraded CEBAF facility at Jefferson Lab or the proposed KEK superconducting linac) is presented in [15] [16] .

Polarized positrons are created in a conversion target by circularly polarized photons, which themselves are produced when polarized laser light is Compton- backscattered on a high-energy electron beam [24] . Circularly polarized photons can also be produced by bremsstrahlung from polarized electrons [25] . Using polarized electrons and positrons, the nucleon electromagnetic form factors and generalized parton distributions can be determined in a model-independent way [16] .

Among the physics topics discussed in [15] , parity violation in electron-elec- tron (Mø ller) scattering is of particular interest. Mø ller scattering is a purely leptonic process that allows high-precision tests of the Standard Model. At four-momentum transfers much smaller than the mass of the Z boson ( q 2 ≪ m   Z 2 ), the parity-violating asymmetry, A, is dominated by the interference between the electromagnetic and neutral weak amplitudes [26] . By definition,

A ≡ d σ R − d σ L d σ R + d σ L ≈ f Z R − f Z L f γ (5)

In this expression, d σ R ( d σ L ) is the differential cross-section for right-handed (left-handed) electron scattering on an unpolarized target:

d σ R , L ∝ | f γ + f Z R , L | 2 ≈ | f γ | 2 + 2 f γ f Z R , L (6)

where f γ and f Z R , L are the scattering amplitudes with γ and Z exchange, respectively. From the four Feynman diagrams in Figure 6 of [27] , one can readily obtain the Born amplitudes for Mø ller scattering mediated by photons and Z bosons. The weak neutral current amplitudes are functions of the weak mixing (or Wienberg) angle θ w , which relates the weak coupling constants g w and g Z to the electromagnetic coupling constant. As shown in [27] , the polarization asymmetry for polarized electron scattering on an unpolarized target is given by

A Born = m e E G F Q w e 2 π α F ( θ ) (7)

where m e is the mass of the electron, E is the incident beam energy, G F is the Fermi coupling constant characterizing the strength of the weak interaction, α is the fine structure constant, and F ( θ ) is a function of the scattering angle in the center-of-mass frame. The weak charge of the electron, Q w e = 1 − 4 s i n 2 θ w , is proportional to the product of the electron’s vector and axial-vector couplings to the Z boson.

Since the value of s i n 2 θ w is close to 1/4, there is an enhanced sensitivity of A to small changes in the Weinberg angle. The value of θ w varies as a function of

the four-momentum transfer, q, at which it is measured. This variation, or “running”, is a key prediction of the Standard Model. The one-loop electroweak radiative corrections to A Born (calculated once the renormalized parameters in (7) are properly defined) reduce its Born value by ~40%. This effect can be attributed to an increase of s i n 2 θ w ( q 2 ) by 3% as the four-momentum transfer “runs” from q 2 = m   Z 2 to q 2 ≈ 0 [28] . The precision with which s i n 2 θ w can be measured in the MOLLER experiment [29] (where a 11 GeV longitudinally polarized electron beam scatters on atomic electrons in a liquid hydrogen target) is shown in Figure 5.2 of [15] .

Many of the electroweak meaurements obtained so far may be combined to provide a global test of consistency with the SM. Since the Higgs-boson mass affects the values of electroweak observables through radiative corrections, it is of fundamental importance to test the agreement between the directly measured value of m H and that inferred from the measurements of electroweak parameters m W , m top and s i n 2 θ w (see Figure 5.2 in [15] ). High-precision electroweak measurements, therefore, represent a natural complement to direct studies of the Higgs sector.

Apart from providing a comprehensive test of the SM, precision measurements of weak neutral current interactions at q 2 ≪ m   Z   2 also allow indirect access to new physics phenomena beyond the TeV energy scale. For instance, such measurements can be used to look for hypothetical Z’ bosons, 4-fermion contact interactions, or very weekly coupled low-mass “dark bosons” [30] .

CP violation was introduced in the SM by increasing the number of quark and lepton families to at least three (M. Kobayashi and T. Maskawa, 1973). This idea became very attractive with the subsequent discovery (in 1977) of the bottom quark, which forms, together with the top quark (discovered in 1995), a third family of quarks. It is a remarkable property of the Kobayashi-Maskawa model that quark mixing and CP violation are intimately related [31] .

A deeper insight into CP violation is expected to be gained from precision measurements of rare kaon decays such as K L 0 → π 0 ν ν ¯ and K + → π + ν ν ¯ . Both decays are theoretically ‘clean’ because hadronic transition amplitudes are matrix elements of quark currents between mesonic states, which can be extracted from the leading semileptonic decays using isospin symmetry. Since photons do not couple to neutrinos, K → π ν ν ¯ decays are entirely due to second-order weak processes determined by Z-penguin and W-box diagrams [31] .

The process K L 0 → π 0 ν ν ¯ proceeds almost entirely through direct CP violation, and is completely determined by “short-distance” one-loop diagrams with top quark exchange. The Standard Model predicts its branching ratio to be [32]

B ( K L 0 → π 0 ν ν ¯ ) = ( 2.43 ± 0.39 ) × 10 − 11   theoretical   value   (8)

This decay is an important source of information on higher-order effects in electroweak interactions, and thus can serve as a probe of new physics (see [33] and references therein).

The decay K + → π + ν ν ¯ receives both CP-conserving and CP-violating contributions. It has theoretical uncertainties that are somewhat larger than those in the process K L 0 → π 0 ν ν ¯ . Since both decays involve one-loop Feynman diagrams with top quark exchange, they can yield valuable measurements of the CKM matrix elements | V t d | and | V t s | . The quantity Im V t s * V t d , which can be obtained from K L 0 → π 0 ν ν ¯ alone, plays a central role in the phenomenology of CP violation in K decays; this quantity is related to the Jarlskog parameter, the invariant measure of CP violation in the Standard model [31] [34] .

By measuring the branching ratios of both K → π ν ν ¯ decay modes, the unitarity triangle of the CKM matrix can be completely determined (see Figure 5), provided the matrix element V c b and the top quark mass are known [34] . Of particular interest is the unitarity triangle parameter s i n 2 β , which can also be determined from the decay B d → Ψ K s . Both determinations of this parameter have to coincide if the Standard Model is valid [33] .

The current branching ratio measurement of the charged decay mode,

B ( K + → π + ν ν ¯ ) = ( 17.3 − 10.5 + 11.5 ) × 10 − 11   measured   value   (9)

is based on the seven candidate events observed by the experiment E787/E949 at Brookhaven [35] . This result is consistent with ( 7.8 ± 0.80 ) × 10 − 11 , the value predicted by the SM [32] .

As shown in Figure 11 of [33] , the kaon yield rises rapidly as a function of the incident proton momentum. From the figure one infers that the minimum energy of the proton beam should be about 20 GeV, for otherwise the kaon yield would be severely reduced. At the proposed KEK facility, the total proton beam energy would be ( 2.5 + 20 ) GeV. A 70 GeV proton synchrotron could be installed, at a later stage, inside the Tristan ring in order to increase the proton beam energy -albeit at the cost of a considerably lower beam power. For a given kaon yield, the required beam power would be lowest at E_b = 30 - 100 GeV [33] .

A charged elementary fermion has a magnetic dipole moment μ = g s ( q / 2 m ) s aligned with its spin s. The proportionality constant g s is the Landé g-factor, q is the charge of the particle and m is its mass. Dirac’s theory of the electron predicts that g s = 2 . For the electron (e), muon ( μ ) and tau lepton ( τ ), this prediction differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment; the anomaly is defined by a ≡ ( g s − 2 ) / 2 ~ 10 − 3 . In the Standard Model, three distinct classes of Feynman diagrams contribute to the value of the anomaly for each lepton species: (1) the dominant QED terms that contain only leptons and photons; (2) terms that involve hadrons; and (3) electroweak terms containing the Higgs, W and Z bosons. The muon anomaly a μ is about ( m μ / m e ) 2 ~ 43   000 times more sensitive to the existence of yet unknown heavy particles than the electron anomaly a e . The value of a μ ( a e ) is sensitive to new physics at the scale of a few hundred GeV (MeV) [36] .

³A much higher level of polarization can be obtained by using a magnetic field to align the particle spins [37] .

The current experimental uncertainty on a μ is ± 0.54 ppm. In a novel g-2 experiment [37] , the main aim of which is to reduce this uncertainty to ± 0.1 ppm, 3 GeV protons impinge on a graphite target and produce pions that are stopped in the target. Some of the positive pions are brought to rest near the surface of the target, where they decay into positive muons with momenta p μ = 30   MeV / c and 100% spin polarization. The muons are collected using a large-aperture solenoid and transported to a silica-aerogel target in which they form muonium (electron― μ + ) atoms. As the atoms slowly diffuse from the target, they are ionized by a pulsed laser to produce 50% polarized muons with very low momenta³. Those “ultra-slow” muons ( 4 × 10 4 / pulse ) are then accelerated to p μ = 300   MeV / c by two linacs, and injected into a magnetic storage ring that contains a 3 T solenoid with a diameter of 66 cm. After injection, the muons circulate orthogonal to the magnetic field B. An orbiting muon decays within 6.6 ms into a positron, a neutrino and an antineutrino: μ + → e + + ν e + ν ¯ μ . The experiment is also designed to measure the electric dipole moment d μ = η ( q ℏ / 2 m ) s of the muon (see below).

The highest-energy positrons, preferentially emitted parallel to the muon spin direction in the μ + rest frame, are Lorentz-boosted to become the highest-energy positrons in the lab frame. Hence, the angular distribution of those positrons has its maximum in the direction of the muon spin [17] . By measuring the energy and time distributions of positrons one can determine the average spin direction. The time spectrum will show the muon lifetime modulated by the spin precession frequency. The relative precession of the spin with respect to the direction of the particle velocity u is given by

ω a + ω η ∝ a μ B − ( η / 2 ) ( β × B ) (10)

where ω a and ω η arise from a μ and d μ , respectively, and β ≡ u / c . Since the rotation axes due to a μ and d μ are orthogonal, the corresponding signals can be separated [37] . In the case of μ μ , the anomalous precession period is 2.2 ms, about 300 times the cyclotron period. Assuming that muons are 100% polarized, 1.5 × 10 12 positrons have to be detected for a measurement precision of 0.1 ppm [37] .