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# First muon acceleration and muon linear accelerator for measuring the muon
anomalous magnetic moment and electric dipole moment

Masashi Otani<EMAIL_ADDRESS>High Energy Accelerator Research
Organization (KEK), Ibaraki 319-1106, Japan

###### Abstract

Muon acceleration using a radio-frequency accelerator was recently
demonstrated for the first time. Measurement of the muon anomalous magnetic
moment and electric dipole moment at Japan Proton Accelerator Research Complex
is the first experiment using accelerated muon beams, and construction will
begin soon. The radio-frequency accelerator used in the experiment and the
first muon acceleration are described in this paper.

C07, C31, G07

##  1 Introduction

A remarkable development of accelerator science from the beginning of the 20th
century to the present has enabled the use of various quantum beams, such as
electron, proton, and ion beams, and has opened the way to greater human
understanding, including the discovery of the Higgs boson and quantum beam
therapy etc. The acceleration of muons using a radio-frequency accelerator was
recently demonstrated for the first time bib:Bae18 , and has opened a new era
of accelerator science using accelerated muon beams.

A muon is an elementary particle similar to an electron, with an electric
charge of $-e$ and a spin of $\frac{1}{2}$, but with a mass 200-times heavier.
After the successful generation of muons using a proton driver half a century
after their discovery during the study of cosmic-rays bib:And37 ; bib:Mic75 ,
muons are now widely used in various scientific fields. In recent years, the
demand for muon acceleration has been increasing in many different areas. For
example, a muon collider, where muons are accelerated to a high energy for
colliding, is one of the future plans of particle physics bib:Man19 ;
bib:Bos18 . In the material and life sciences, one promising application of
muon acceleration is in the construction of a transmission muon microscope
bib:tmm . If the muons can be cooled to thermal temperature and subsequently
re-accelerated, transmission muon microscopes will be realized. Among the
future programs, a new experiment (E34 experiment) in the Materials and Life
Science Experimental Facility (MLF) of the Japan Proton Accelerator Research
Complex (J-PARC) is planning to measure the muon anomalous magnetic moment
($a_{\mu}$) and search for the electric dipole moment (EDM) as a pioneer in
muon acceleration bib:Abe20 .

Although the discovery of the Higgs boson using the Large Hadron Collider
(LHC) has established the Standard Model (SM) as a successful description of
particle interactions, we are still confronted with many problems that can be
solved only through experimental clues. One of the most interesting clues is
the precise measurement of $a_{\mu}$, which has paved the way for
understanding the nature of elementary particles through the quantum effects.
In a series of three experiments at European Organization for Nuclear Research
(CERN) bib:Cha62 ; bib:Bai72 ; bib:Bai79  and an experiment at Brookhaven
National Laboratory (BNL) bib:Ben06 , successive improvements in the accuracy
of the measurements had enabled a deeper understanding of the SM and there is
a large discrepancy between measurements and predictions of the SM. After many
years of scrutiny, challenging calculations and corrections were conducted by
stimulated theorists and still 3.7 sigma deviation remains bib:Aoyama20 . This
discrepancy should be addressed by new measurements. A new experiment at Fermi
National Accelerator Laboratory (FNAL E989 bib:Gra15 ) is currently being
conducted, with the storage ring from the BNL experiment being reused. They
published their first result in early FY2021 bib:abi21 . The result is
consistent to that of the previous BNL experiment and the the tension between
experiment and the SM calculation becomes 4.2 sigma. It strengthens the
importance for confirming the tension with with independent measurements
different from BNL and FNAL.

As described above, continuous studies have continued improving the accuracy
of $a_{\mu}$. In particular, the improvement of the beam has been one of the
driving forces moving the measurements forward. Our ancestors at CERN, BNL,
and FNAL have been struggling with beam-related uncertainties in their
measurements because they use muons obtained directly from a pion decay with a
large emittance. The J-PARC E34 experiment bib:Abe20  aims to measure
$a_{\mu}$ with a precision of 0.1 ppm using an unprecedentedly low-emittance
muon beam realized by the acceleration of thermal muons.

The reminder of this paper is organized as follows. In Section 2, the J-PARC
E34 experiment, particularly the radio-frequency linear accelerator (linac)
dedicated to muons is described. Section 3 describes the first demonstration
of muon acceleration. A summary and outlook are shown in Section 4.

##  2 Muon linac for the J-PARC E34 experiment

In this section, details of the muon linac will be described after successive
descriptions explaining an overview of the J-PARC E34 experiment.

The J-PARC E34 experiment aims to measure $a_{\mu}$ with a precision of 0.1
ppm and search for the EDM with a sensitivity of approximately $10^{-21}$
$e\cdot$cm using a low-emittance muon beam realized by an accelerated thermal
muon beam. The total emittance in the transverse direction is required to be
1.5$\pi$ mm mrad to realize the measurement with a 3 T compact MRI-type magnet
with sufficient injection efficiency. The cyclotron radius with this magnet is
333 mm that is about a factor of 20 smaller than that for the BNL and FNAL
experiments. Because of the high uniformity of the magnetic field in the muon
storage region, the uncertainty due to the field uniformity is much smaller
than in the BNL and FNAL experiments. Figure 1 shows the experimental setup.
The pulsed high-power primary proton beam generates secondary surface muons
produced by $\pi^{+}$ decay near the surface of the production target
bib:Kawamura18 . The produced surface muons are extracted and thermalized to
form muoniums (bound state made up of a positive muon and an electron), which
are then emitted into vacuum region adjacent to the muonium production target
bib:Bak13 ; bib:Beer14 ; bib:Bea20 . The paired electron in the muonium is
knocked out by a laser, and thermal muon (3 keV/c) is generated. After
acceleration to 300 MeV/c, the muon beam has an extremely low emittance such
that it can be injected and stored in a high precision compact storage magnet
bib:Iinuma16 ; bib:Abe18 , where the time dependence of the positrons from the
muon decay is measured for the measurement of the anomalous spin precession
bib:Aoyagi20 ; bib:Kishishita20 .

Figure 1: Overview of the J-PARC E34 experiment.

###  2.1 Overview of muon linac

Because muons have a finite lifetime of approximately 2.2 $\mu$s, they need to
be accelerated faster to avoid decay losses to obtain the necessary
experimental statistics. From this perspective, a linac is one of the best
options for muon acceleration.

Owing to its intermediate mass between that of protons and electrons, the
change in velocity upon acceleration is slower than that of electrons, as
shown in Fig. 2. For this reason, both proton and electron linac technologies
will be used to accelerate the muon to near the speed of light.

Figure 2: Particle velocity as a function of kinetic energy for electron
(black line), muon (red hatched line), and proton (blue dot-dash line)

Accelerators should be designed appropriately to reduce costs. To reduce the
expenses, a spare radio-frequency quadrupole (RFQ) of the J-PARC linac
bib:Kondo13  will be used as a first stage acceleration. It operates with a
resonant frequency of 324 MHz. The L-band high-power klystron developed for
the KEKB linac upgrade bib:Kubosaki11  will be used as the RF power supply for
the acceleration cavity in the high-energy section to achieve a further cost
reduction.

A schematic of the muon linac is shown in Fig. 3. The RFQ bunches and
accelerates the muons to 0.3 MeV after the initial electrostatic acceleration
bib:Kondo15 . After the RFQ, an inter-digital H-mode drift tube linac (IH-DTL)
is employed during the particle velocity $\beta=0.08$ to $0.28$ (4 MeV)
bib:Otani16 . After the muon is accelerated to $\beta=0.28$, a disk-and-washer
(DAW) type coupled cavity linac (CCL) with an operational frequency of 1296
MHz is employed bib:Otani19daw . Because the $\beta$ variation is modest in
the high-$\beta$ region, to realize a sufficiently short distance, the design
emphasis has been shifted to achieving a high accelerating gradient. A disk-
loaded structure (DLS) traveling-wave linac is used when $\beta$ is greater
than $0.7$ (42 MeV) bib:Kondo17 . The details of each acceleration cavity are
described below.

Figure 3: Configuration of the muon linac. In this figure, $W$ is kinetic
energy and $\beta$ is the ratio of the muon velocity to the speed of light.

###  2.2 Initial electrostatic acceleration

In front of the RFQ, there is an electrostatic lens called a Soa lens
bib:Can86 , which accelerates and extracts the thermal muons.

Figure 4 shows a schematic view of the Soa lens. The Soa lens consists of two
mesh electrodes and three cylindrical electrodes. The first mesh electrode
(target mesh) covers the downstream surface of the silica aerogel target. The
laser ionization region is between the target mesh and second mesh electrode
(S1). The voltage applied to the target mesh and S1 is set to 5.7 and 5.6 keV,
respectively, corresponding to the input energy of the RFQ. The dimensions of
the electrodes are designed to cover the primary surface muons (rms size of 31
mm and 14 mm in the horizontal and vertical directions, respectively
bib:Otani18_surfmu ) and to provide a sufficient extraction efficiency for
ultra slow muons. The voltage applied to other electrodes (S2, S3 and S4) is
determined using the simulation described below such that the phase space of
the beam matched with the design acceptance of the RFQ and a high transmission
efficiency is obtained.

Figure 4: Schematic of the SOA lens. Lengths in the figure are expressed in
millimeters.

Input muon distributions are estimated using a simulation based on
measurements. The simulation for a surface muon beamline (MLF H-line) is
constructed using the g4beamline bib:g4bl ; bib:Kawamura18 . The absolute
number of surface muons is normalized by measurements from another beamline
(MLF D-line bib:Stra10 ) that uses the same muon production target in J-PARC
MLF. The stopping distribution inside the silica aerogel is estimated using
GEANT4 bib:g4 . The simulation for muonium diffusion is developed based on a
three-dimensional random walk. The simulation parameters of the thermal
temperature and the diffusion constant were determined from our measurements
at Canada’s particle accelerator centre (TRIUMF) bib:Beer14 . The muon
trajectories are simulated using the GEANT4 simulation where the electrostaic
field of the Soa lens calculated OPERA bib:opera  is implemented.

Figure 5 shows the transverse phase space distributions at the entrance of the
RFQ. The difference between the horizontal and vertical directions is due to
the difference in the primary surface muon distribution at the entrance of the
silica aerogel. Owing to the spatial distribution of the muoniums in the laser
ionization region, the muons are distributed over time with a full width of
approximately 10 ns. The transmission efficiency in the Soa lens is estimated
to be 72%, including a 17% decay loss. Because the structure of the mesh
electrodes (target mesh and S1) is not implemented in the GEANT4 simulation,
the transmission efficiency of the mesh electrode was estimated to be 78%
based on the products of the aperture ratio of the two meshes. In total, the
efficiency is 56% in the initial electrostatic acceleration.

Figure 5: Phase space distributions at the entrance of the RFQ: (a) Horizontal
divergence angle x’ vs. x, (b) the vertical divergence angle y’ vs. y, (c) y
vs. x, and (d) $\Delta W$ vs. time

###  2.3 RFQ

After the initial acceleration by the Soa lens, the RFQ accelerates the muons
to 340 keV. In addition to the acceleration, the RFQ bunches the muons at a
frequency of 324 MHz.

The principles of the RFQ were first invented in 1969 bib:Kap70_1 ;
bib:Kap70_2  and they proved at Los Alamos National Laboratory (LANL) in 1980
bib:Sto81 . The RFQ consists of a four electrodes excited with quadrupole-
mode. The electrode is modulated longitudinally, generating the axial electric
field. By changing modulation pattern gradually so that the synchronous phase
is changed from $-90$ degree to higher, the beam bunching accomplished.
Because the RFQ supply velocity-independent electric focusing, it has great
advantage in a low-velocity part compared with conventional linacs that used
velocity-dependent magnetic focusing.

Table 1 lists the design parameters of the RFQ for the muon acceleration along
with that for negative hydrogen ion (H-). To accelerate muons using the spare
of the J-PARC RFQ, which was originally developed for H- acceleration, the
intervane voltage must be reduced to the mass ratio to match the particle
velocity. As a result, the required power is reduced to the square of the mass
ratio. The input and output energies are also scaled to the design velocity;
the input and output $\beta$ are 0.01 and 0.08, respectively.

Table 1: Parameters of the RFQ II for the H- and muon acceleration. | H- | muon
---|---|---
Frequency (MHz) | 324
Number of cells | 295
Length (m) | 3.17
Intervane voltage (kV) | 82.9 | 9.3
Power (kW) | 330 | 4.2
Injection energy (keV) | 50 | 5.6
Extraction energy (MeV) | 3 | 0.34

In order to confirm that the RFQ can accelerate muons without any problems,
particle simulations were performed using PARMTEQM bib:parmteqm . Figure 6
shows the phase space distributions at the RFQ exit, and Table 2 summarizes
the input and output beam parameters. The horizontal and vertical normalized
rms emittances at the RFQ exit are 0.30 and 0.17 $\pi$ mm mrad, respectively.
The simulated transmission is 94.7%. Since the transit time through one cell
of the RFQ is half the resonant frequency, the transit time through the entire
RFQ can be calculated as $\frac{1}{324\times 10^{6}}\times\frac{1}{2}\times
295=455$ ns. Ignoring relativistic corrections, the muon lifetime is 2.2
$\mu$s, and the survival rate of muons passing through the RFQ is
$\exp(-0.455/2.2)=0.813$. Therefore, the total transmission is 77.0%.

Figure 6: Phase space distributions at the RFQ exit: (a) Horizontal divergence angle x’ vs. x, (b) vertical divergence angle y’ vs. y, (c) y vs. x, and (d) $\Delta W$ vs. rf phase ($=$ time) Table 2: Input and output beam parameters of the beam dynamics simulation of the RFQ. In this table, $\alpha$ and $\beta$ are Twiss parameters.  | Input | Output
---|---|---
$\alpha_{x}$ | 0.32 | -1.51
$\beta_{x}$ [mm/mrad] | 0.041 | 0.21
$\varepsilon_{x}$ [$\pi$ mm mrad, rms, normalized ] | 0.376 | 0.297
$\alpha_{y}$ | 0.092 | 0.606
$\beta_{y}$ [mm/mrad] | 0.080 | 0.076
$\varepsilon_{y}$ [$\pi$ mm mrad, rms, normalized ] | 0.106 | 0.167
$\alpha_{z}$ | - | 0.17
$\beta_{z}$[deg/MeV] | - | 1360
$\varepsilon_{z}$[$\pi$ MeV deg, rms, normalized ] | - | 0.0381
Time width | 10 ns (full width) | -
Energy spread | 0.00989 keV (rms) | -
Transmission | 94.7%
Transient time | 455 ns
Survival rate | 81.3%
Transmission total | 77.0%

The operation test of the RFQ has already been conducted bib:Otani15_pasj .
The RFQ was powered on by a solid state amplifier at up to 6 kW and a 25 Hz
repetition. Vacuuming is done with an ion pump and reaches $10^{-6}$ Pa.
Figure 7 shows the forward, reflection, and pick-up power in the RFQ when 5 kW
power is applied. There is a small reflection because the coupling was tuned
to be overcoupled originally for a high current H- beam. A few hours of
operation at 5 kW was successfully achieved without any problems, such as an
RF failure from a spark. The background associated with the RF operation was
measured using a micro-channel plate detector connected downstream of the RFQ.
The detector count rates were consistent and negligibly small with and without
the RF operation, within a statistical uncertainty range of 0.1 Hz.

In conclusion, we are ready for a muon acceleration using the RFQ.

Figure 7: Forward, reflection wave, and pick-up power in RFQ with nominal
power of 5 kW

###  2.4 IH-DTL

After the initial acceleration and bunching by the RFQ, the IH-DTL accelerates
the muons to 4.3 MeV.

The IH-DTL has alternating drift tubes up and down through the stem allowing
the TE11 mode to be used for acceleration. The IH-DTL was first proposed in
Japan in 1949 bib:Mor49 . Although there were several efforts for this
inventions bib:Ble56 ; bib:Zei62 ; bib:Pot69 , an IH-DTL was realized after a
quarter century as the heavy ion post-accelerator at the Munich tandem
accelerator bib:Nol79 . Compared to the conventional Arvaretz DTL, a higher
acceleration efficiency is achieved, particularly within the range of
$\beta=0.1$–$0.2$ bib:Rat05 . In recent IH-DTLs, the alternating phase
focusing (APF) method bib:Good53 ; bib:Min99 , which was successfully
implemented in the Hadron therapy machine bib:Iwata06 , enables simultaneous
focusing in the horizontal and vertical directions using only electric fields,
resulting in a higher acceleration efficiency. Owing to the small focusing
strength in the APF method, there is a limit to its use in high current
machines, but it can be applied in the muon linac, which has a small current.

The IH-DTL with the APF method for accelerating muons was designed using
several types of simulation software bib:Otani16 . The arrangement of the
drift tubes and acceleration gaps, i.e., the so-called longitudinal beam
dynamics design, should be determined according to the evolution of the beam
velocity, which is determined by the energy gain at each gap. Unlike
conventional accelerating cavities where transverse focusing is performed
conducted using additional elements such as quadrupoles, an APF cavity
requires simultaneous non-independent longitudinal and transverse beam
dynamics designs. This beam dynamics design was performed based on linacsapf
bib:jameson14 , where the beam dynamics is calculated using the so-called
drift-kick-drift approximation. To achieve less emittance growth and a better
transmission, a nonlinear optimization is applied to the synchronous phase
array. The cavity was designed using CST-MW Studio bib:cst  based on the beam
dynamics design. In order to mitigate the peak electric field and obtain a
higher efficiency, the dimensions of the drift tube were changed from
bib:Otani16 . Table 3 summarizes the basic parameters of the IH-DTL cavity.

Table 3: Parameters of the IH-DTL cavity. Because the frequency is tuned to 324 MHz with conductive tuners, the frequency shown here is slightly smaller than 324 MHz. In this table, $E_{surf.}$ is the surface electric field and $E_{kp}$ is the Kilpatrick limit at 324 MHz (17.8 MV/m). Parameters | Values
---|---
Frequency (MHz) | 323.3
Number of cells | 16
Length (m) | 1.32
Synchronous phase [deg.] |  $-44$–$+48$
Unloaded $Q$ | $1.04\times 10^{4}$
Power (kW) | 322
Peak $E_{surf.}$ (MV/m) | 35.3 ($\sim 1.99E_{kp}$)

The input distribution was given from the upstream RFQ simulation. The input
was used for the beam dynamics simulation after simulation of the beam
transport line to match the beam to the IH-DTL. The beam dynamics simulation
is performed using GPT bib:gpt . Figure 8 shows the phase space distributions
at the IH-DTL exit, and Table 4 summarizes the output beam parameters. The
vertical emittance growth is larger than the horizontal one, which is due to
the vertical meandering track caused by the vertical electric field and
horizontal magnetic fields of the first and last cells, which is inevitable in
IH-DTL structures. The transmission is 99.97% and it is sufficiently large.
The beam transit time $t_{\mathrm{trans.}}$ is 25 ns and the muon survival
rate is calculated to be
$\exp(t_{\mathrm{tran.}}/\tau_{\mu}\overline{\gamma})=98.9$%, where the
average Lorentz factor during acceleration is labeled $\overline{\gamma}$ and
$\tau_{\mu}$ is the muon lifetime (2.2 $\mu$s).

Figure 8: Phase space distributions at the IH-DTL exit: (a) Horizontal divergence angle x’ vs. x, (b) vertical divergence angle y’ vs. y, (c) y vs. x, and (d) $\Delta W$ vs. rf phase ($=$ time) Table 4: Output beam parameters of the IH-DTL. In this table, $\alpha$ and $\beta$ are Twiss parameters.  | Output
---|---
$\alpha_{x}$ | -4.3
$\beta_{x}$ [mm/mrad] | 0.38
$\varepsilon_{x}$ [$\pi$ mm mrad, rms, normalized ] | 0.316
$\alpha_{y}$ | -3.4
$\beta_{y}$ [mm/mrad] | 0.33
$\varepsilon_{y}$ [$\pi$ mm mrad, rms, normalized ] | 0.190
$\alpha_{z}$ | -1.0
$\beta_{z}$[deg/MeV] | 174
$\varepsilon_{z}$[$\pi$ MeV deg, rms, normalized ] | 0.0274
Transmission | 99.97%
Transient time | 25 ns
Survival rate | 98.9%
Transmission total | 98.9%

An error study of the beam dynamics was conducted by assuming several possible
cases bib:Otani16pasj . The error in the on-axis electric field owing to a
fabrication error is estimated to be approximately 2% using the CST-MW Studio
assuming a general fabrication error of 100 $\mu$m. The emittance growth due
to the fabrication errors is estimated to be 10%, which still satisfies the
requirement. It also shows that the error field can be controlled by some
movable conductive tuners installed on the cavity side wall.

A prototype of the IH-DTL was fabricated (Fig. 9) to confirm the design and
evaluate the performance bib:Nakazawa19 . The prototype was a three-piece
design with two semi-cylindrical shells attached to the center frame where the
drift tubes are mounted. It has the first five drift tubes of the actual
design, and the total length is approximately 0.5 m. The resonant frequency
and unloaded quality factor ($Q_{0}$) were measured and are consistent with
the CST-MW Studio calculation at 0.1% and 11%, respectively. The on-axis
electric field distribution was measured using the bead-pull method bib:bpm
and agreed with the CST-MW Studio calculation within 3% uncertainty. The
emittance growth owing to the field error is estimated to be 3%, which is
negligible. A high power coupler was fabricated and a high power test will
soon be conducted. Based on the experience with the prototype, we completed a
detailed design of the actual IH-DTL and will soon start production.

Figure 9: Prototype of the IH-DTL: (Left) overall structure, (Right) center
plate, where the five drift tubes are implemented.

###  2.5 DAW-CCL

After the IH-DTL, the DAW-CCL accelerates the muons to 40 MeV.

A DAW-CCL is a cylindrical cavity with conductive washers placed across a
disk-shaped disc. A DAW-CCL was first proposed in early 1970s bib:and72 ;
bib:Mur72  and full scale cavity was demonstrated in late 1970s bib:and76 .
The first DAW-CCL was operated in a proton and H- linac at the Moscow meson
factory bib:Esi88 . In Japan, DAW-CCLs were developed in KEK and Kyoto for
electron bib:Ina86 ; bib:Iwa94 . DAW-CCLs have advantages over other CCLs,
such as a higher shunt impedance and higher coupling between the accelerating
and coupling cells.

The DAW-CCL cavity is designed using several types of softwares bib:Otani19daw
. As a first step, the two-dimensional dimensions are optimized using Poisson
Superfish bib:sf  for a higher acceleration efficiency and lower peak-to-
average value ($E_{\mathrm{max}}/E_{0}$) in satisfying the confluence
condition. After the two-dimensional calculation, the three dimensional
calculations including the washer support are conducted using CST-MW Studio.
The bi-periodic L-support bib:ao00 , in which a washer is fixed by two
supports and a pair of supports are located azimuthally 90 degrees apart from
the adjacent supports, is adopted among several support structures because
perturbation to the acceleration mode can be minimized by adjusting the
support structure. Finally, the dispersion curve is investigated to check
whether an unfavorable mode exists around the operation frequency.

Figure 10 shows the three-dimensional model (left) and the dispersion curve
(right) of the designed cavity for $\beta=0.3$. TM02 and TM01 are accelerating
and coupling mode, respectively. Because of the bi-periodic structure, some
stop bands appear in $\pi/2$. Although the TM11 mode is near the operational
frequency, the cavity is tuned during the optimization process such that the
operational frequencies sit within the stop band at $\pi/2$. Although the
dipole mode passband TE11 crosses the line where the phase velocity matches
the speed of the muons, it is not considered to be a problem because the muon
beam current is negligible and the transverse kick owing to this mode is
estimated to be much smaller than the requirement. Table 5 summarizes the
cavity parameters for $\beta=0.3,0.4,0.5,$ and $0.6$.

Figure 10: (Left) Three-dimensional model of the DAW-CCL. The electric field of the acceleration mode in the two-cell model is shown by the colored arrows. (Right) Dispersion curve with optimized cavity in $\beta=0.3$ calculated using CST-MW Studio. The dotted line represents target frequency (1.3 GHz) and the dashed line corresponds to the phase velocity of the muons. The sup mode is associated with the L-support.  Table 5: Parameters of the DAW-CCL cavity for $\beta=0.6,0.5,0.4,$ and $0.3$. For the simplicity, the resonant frequency is designed to be 1.3 GHz. In this Table, L is the cell length, $\mathrm{f_{a}}$ is the resonant frequency of the accelerating mode, $\mathrm{f_{c}}$ is the resonant frequency of the coupling mode, ZTT is the effective shunt impedance per unit length, $E_{\mathrm{max}}/E_{0}$ is the peak to average axial electric-field, and $\lambda$ is the resonant wavelength.  Parameters | Values
---|---
$\beta$ | 0.6 | 0.5 | 0.4 | 0.3
$\mathrm{L}$ | $\beta\lambda/4$
$\mathrm{f_{a}[GHz]}$ | 1.300 | 1.300 | 1.299 | 1.301
$\mathrm{f_{c}[GHz]}$ | 1.299 | 1.301 | 1.302 | 1.301
$\mathrm{ZTT[M\Omega/m]}$ | 57.8 | 46.3 | 33.8 | 18.0
Transit time factor | 0.84 | 0.85 | 0.84 | 0.81
$E_{\mathrm{max}}/E_{0}$ | 4.4 | 4.8 | 5.1 | 5.0
$\mathrm{Q_{0}}$ | $2.91\times 10^{4}$ | $2.41\times 10^{4}$ | $1.91\times 10^{4}$ | $1.42\times 10^{4}$
Synchronous phase [deg.] | -30

The beam dynamics is designed using PARMILA bib:parmila  and TRACE3D bib:t3d
based on the designed cavity performance. For ease of fabrication, a constant
cell length ($L=\beta_{s}\lambda/2$) in a tank with multiple cells is
designed. The design velocity $\beta_{s}$ and number of cells for each tank
are determined based on the design of the beam dynamics. The inter-tank
spacing is set to 4.5$\beta\lambda$ considering the feasibility of a magnet
installation. The average acceleration field is determined to be 5.6 MV/m
according to the Kilpatrick limit bib:Kil57  and $E_{\mathrm{max}}/E_{0}$; the
bravery factor is set to 0.9 in maximum. The number of cells in a tank is
determined by the limitation of the quadrupole strength owing to transverse
instabilities. There may be instabilities or resonances when the zero current
phase advance ($\sigma_{0}$) is greater than 90 degrees bib:Reiser94 .
Although the muon beam intensity is expected to be much smaller than that in
the region of such instabilities, the number of cells is chosen to match this
criterion. The maximum $\sigma_{0}$ is 83 degrees at the first tank when the
number of cells is ten, which determines the number of cells for all tanks.
Table 6 shows the basic parameters of each DAW tank. Because the shunt
impedance of the DAW cell increases as a function of $\beta$, the power
required for the tank decreases with a higher energy. The total power required
is 4.5 MW. The phase slippage is the greatest in the first tank, ranging from
$-14$ to $+10$ degrees.

Table 6: Parameters of each DAW tank. tank | $N_{\mathrm{cells}}$ | $\beta_{s}$ | length [m] | Energy [MeV] | Power [MW]
---|---|---|---|---|---
1 | 10 | 0.29 | 0.34 | 5.6 | 0.39
2 | $\uparrow$ | 0.33 | 0.38 | 7.1 | 0.35
3 | $\uparrow$ | 0.37 | 0.42 | 8.8 | 0.33
4 | $\uparrow$ | 0.40 | 0.46 | 10.7 | 0.31
5 | $\uparrow$ | 0.43 | 0.50 | 12.7 | 0.30
6 | $\uparrow$ | 0.47 | 0.54 | 14.9 | 0.29
7 | $\uparrow$ | 0.50 | 0.57 | 17.2 | 0.29
8 | $\uparrow$ | 0.52 | 0.61 | 19.7 | 0.29
9 | $\uparrow$ | 0.55 | 0.64 | 22.3 | 0.28
10 | $\uparrow$ | 0.58 | 0.67 | 25.0 | 0.28
11 | $\uparrow$ | 0.60 | 0.69 | 27.9 | 0.28
12 | $\uparrow$ | 0.62 | 0.72 | 30.8 | 0.28
13 | $\uparrow$ | 0.64 | 0.74 | 33.8 | 0.28
14 | $\uparrow$ | 0.66 | 0.77 | 37.0 | 0.28
15 | $\uparrow$ | 0.68 | 0.79 | 40.2 | 0.28

The input distribution was given from the upstream IH-DTL simulation and used
for the beam dynamics simulation after the calculation for the beam transport
line to match the beam to the DAW-CCL. Figure 11 shows the phase space
distributions at the DAW exit, and Table 7 summarizes the output beam
parameters. The total length is 16.3 m with 15 modules, which corresponds to
138$\beta\lambda$, and the beam transit time $t_{\mathrm{trans.}}$ is
calculated as $\frac{138\beta\lambda}{\beta c}=106.0$ ns. The survival
probability is calculated as
$\exp(t_{\mathrm{tran.}}/\tau_{\mu}\overline{\gamma})=96.1$%, where
$\overline{\gamma}=(\gamma_{\mathrm{in}}+\gamma_{\mathrm{out}})/2=1.212$. The
emittance growth is estimated to be less than a few percent, and the output
emittance is 0.32 $\pi$ and 0.21 $\pi$ mm mrad for the horizontal and vertical
directions, respectively. The effect of errors on the emittance owing to a
misalignment of the quadrupole magnet, power errors in the cavity, and other
factors is evaluated to be less than 5% bib:takeuchi19 .

Figure 11: Phase space distributions at the DAW-CCL exit: (a) Horizontal divergence angle x’ vs. x, (b) vertical divergence angle y’ vs. y, (c) y vs. x, and (d) $\Delta W$ vs. rf phase ($=$ time) Table 7: Output beam parameters of the DAW-CCL. In this table, $\alpha$ and $\beta$ are Twiss parameters.  | Output
---|---
$\alpha_{x}$ | 4.1
$\beta_{x}$ [mm/mrad] | 0.23
$\varepsilon_{x}$ [$\pi$ mm mrad, rms, normalized ] | 0.332
$\alpha_{y}$ | -6.9
$\beta_{y}$ [mm/mrad] | 0.36
$\varepsilon_{y}$ [$\pi$ mm mrad, rms, normalized ] | 0.201
$\alpha_{z}$ | -0.002
$\beta_{z}$[deg/MeV] | 24
$\varepsilon_{z}$[$\pi$ MeV deg, rms, normalized ] | 0.108
Transmission | 99.80%
Transient time | 106 ns
Survival rate | 96.1%
Transmission total | 95.9%

A cold model of the first DAW cells was fabricated to confirm the design. The
cold model is made of aluminum. Figure 12 shows a mechanical drawing (left)
and photograph of the assembly (right). The resonant frequency was measured
and consistent with the calculation using CST-MW Studio at 0.4%. The on-axis
electric field distribution was measured and the variation in the fields per
cell was observed, which is considered to be due to the assembly process of
the cavity. Based on the measurement results in the cold model, the actual
DAW-CCL has been designed and the first tanks will be soon fabricated.

Figure 12: Prototype of the DAW-CCL: (Left) mechanical three-dimensional
drawing, (Right) assembly of the prototype. The end plate with the antenna
ports is attached for the low power measurements.

###  2.6 DLS

Finally, muons are accelerated to 212 MeV using the DLS.

DLS is classified as a radio-frequency linac for electron. After the first
invention of the radio-frequency linac by Wideröe, Sloan and Lawrence
bib:Wid28 ; bib:Law31 , Hansen studied an electromagnetic field with radio-
frequency resonator for accelerating electron bib:Hansen38 . The acceleration
of electron using a traveling wave accelerator was demonstrated in late 1940s
bib:Gin48 . Recent electron accelerators using room-temperature cavities have
been based on the results of the Mark III linac bib:Cho55  and SLAC. Unlike
linear accelerators for electrons, which quickly reach the speed of light,
muon linacs, which slowly approach the speed of light, require a gradual
change in the length of the cell.

The geometrical parameters of the DLS cell are designed using Poisson
Superfish. Poisson Superfish generates the standing-wave-mode electric fields
of each cell with open-open and short-short boundary conditions and the
electric field of the traveling wave is represented by superposing these two
fields with a phase difference of $\pi/2$. The L-band structure was adopted to
make the acceptance sufficiently large for the input muon beam and the
conventional $2\pi/3$ acceleration mode is adopted. The synchronous phase is
set to $-10$ degrees to ensure sufficient longitudinal acceptance, and the
average acceleration field ($E_{0}$) is assumed to be 20 MV/m. Figure 13 shows
the parameters of the DLS cells. In this study, a constant impedance design
was adopted for simplicity. The calculated fields are implemented in the beam
dynamics simulation conducted using the GPT. Figure 14 shows the phase space
distributions at the DLS exit, and Table 8 summarizes the output beam
parameters. Almost no emittance growth is observed. The transmission through
the DLS section is 99.9%, and the loss owing to the muon decay is estimated to
be 1%.

Figure 13: Calculated cell parameters of the DLS section. $D$ is the cell length calculated by $\beta\lambda/3$ and $w$ is the kinetic energy of the muons. Figure 14: Phase space distributions at the DLS exit: (a) Horizontal divergence angle x’ vs. x, (b) vertical divergence angle y’ vs. y, (c) y vs. x, and (d) $\Delta W$ vs. rf phase ($=$ time) Table 8: Output beam parameters of the DLS. In this table, $\alpha$ and $\beta$ are Twiss parameters.  | Output
---|---
$\alpha_{x}$ | -1.9
$\beta_{x}$ [mm/mrad] | 0.86
$\varepsilon_{x}$ [$\pi$ mm mrad, rms, normalized ] | 0.331
$\alpha_{y}$ | -3.3
$\beta_{y}$ [mm/mrad] | 1.5
$\varepsilon_{y}$ [$\pi$ mm mrad, rms, normalized ] | 0.211
$\alpha_{z}$ | 0.07
$\beta_{z}$[deg/MeV] | 150
$\varepsilon_{z}$[$\pi$ MeV deg, rms, normalized ] | 1.94
Transmission | 99.9%
Transient time | 30 ns
Survival rate | 99%
Transmission total | 99%

###  2.7 Summary of the muon linac design

Figure 15 shows the emittance evolution in the entire muon linac. There is no
significant growth of the beam emittance and the output emittance is
comparable to the required total emittance of 1.5$\pi$ mm mrad. Table 9
summarizes the transmission, decay loss, and emittance at each section. The
intensity of the low emittance muon beam at the linac exit is estimated to be
$1.2\times 10^{4}$ muons per pulse with a repetition of 25-Hz. Table 10 shows
the breakdown of the estimated transmission efficiency for all the
experimental components and the statistical uncertainty of the $a_{\mu}$
measurement is estimated to be 450 ppb for $2.2\times 10^{7}$ s of data taking
bib:Abe20 . The statistical precision is comparable to that of the BNL
experiment, and we are able to examine the $a_{\mu}$ anomaly with a completely
different scheme.

Figure 15: Emittance evolution from the RFQ entrance to the linac exit.  Table 9: Summary of the particle simulations through the muon linac: Transmission and decay loss in each section, and the emittance at the exit of each section. The transmission in SOA includes that of the mesh electrodes.  | Soa | RFQ | IH-DTL | DAW-CCL | DLS
---|---|---|---|---|---
Transmission [%] | 56 | 95 | 100 | 100 | 100
Decay loss [%] | 17 | 19 | 1 | 4 | 1
$\varepsilon_{x}$ [$\pi$ mm mrad, rms, normalized] | 0.38 | 0.30 | 0.32 | 0.32 | 0.33
$\varepsilon_{y}$ [$\pi$ mm mrad, rms, normalized] | 0.11 | 0.17 | 0.20 | 0.21 | 0.21
Table 10: Breakdown of the estimated transmission efficiency. For the $a_{\mu}$ measurement, $e^{+}$ energy window (12%), detector acceptance (100%), and reconstruction efficiency (12%) should be considered. Details are described in bib:Abe20 .  | H-line |  | muonium
---
emission
| laser
---
ionization
acceleration | injection | kicker
Transmission [%] | 16 | 0.34 | 73 | 40 | 84 | 93

Because the systematic uncertainty is estimated to be less than 70 ppb and the
measurement is statistically limited, further improvement of the beam
intensity directly impacts the sensitivity of the physics. The goal in the
next phase in the J-PARC E34 experiment is to measure $a_{\mu}$ with an
accuracy of 100 ppb, comparable to the goal of the FNAL experiment. As shown
in Table 10, primary improvement should be in the muonium emission efficiency.
The next one is the H-line and the acceleration efficiency. According to the
results of previous muonium measurements bib:Bak13 ; bib:Beer14 ; bib:Bea20 ,
the arrangement of the silica aerogel target is being investigated to enhance
the overlap between the effective area of the target and the laser ionization
region. In the muon linac, the most effective way for the improvement is a
reduction of the decay loss in the low-$\beta$ section. For this purpose,
several efforts have been devoted: a new RFQ dedicated to the muons
bib:Kondo15 , an L-band RFQ combined with CC-DTL bib:kondo20 ;
bib:kondo18linac , and a new scheme of acceleration for the low-energy muons
bib:otani20 .

Further, development is carried out for spin tracking bib:yasuda20 . Because a
spin precession in the muon storage ring starts with the initial spin state
determined by the muon linac exit, the initial state should be understood to
examine the potential systematic uncertainties of the physics measurement. For
this purpose, a spin tracking simulation is being developed. In addition, a
spin-rotator based on the Wien-filter type is designed to enable a spin-
flipping analysis bib:yasuda20jparc , which is usually used in many types of
spin experiments.

##  3 Demonstration of muon acceleration

As described in Section 2, a low emittance muon beam is realized through
thermal muon production and acceleration using the linac dedicated to the
muons. The production of thermal muons has been developed for over a quarter
of a century bib:Mills86 ; bib:Chu88 ; bib:Nagamine95  and has matured
sufficiently and can be used in experiments. Muon acceleration, however, was
an unproven technology. Therefore, a muon acceleration should have been
demonstrated prior to the construction of the actual linac. In addition, beam
monitors are needed to diagnose the muon beam with an unprecedented beam size.

To demonstrate muon acceleration prior to the construction of the experimental
setup that included the thermal muon source system, a faster and simpler
system for slow muons was needed. The scheme of muon cooling using a simple
metal degrader and acceleration using an RFQ proposed at Los Alamos National
Laboratory bib:Miyadera07  are suitable for this purpose. We basically
followed this method, but to further reduce the emittance and separate
accelerated muons from background high energy muons that pass through the RFQ
without being accelerated, we decided to use negative muonium ions
($\mu^{+}e^{-}e^{-}$; $\mathrm{Mu^{-}}$) produced through an electron capture
process bib:Kua89 . Prior to the demonstration of muon acceleration by
$\mathrm{Mu^{-}}$, an experiment dedicated to a $\mathrm{Mu^{-}}$ measurement
was carried out to evaluate the expected amount of accelerating
$\mathrm{Mu^{-}}$ signal.

This chapter describes the $\mathrm{Mu^{-}}$ measurement followed by a
demonstration of muon acceleration along with a description of the development
of the beam monitor.

###  3.1 Development of $\mathrm{Mu^{-}}$ source

As mentioned above, the $\mathrm{Mu^{-}}$ production process can be used to
cool a muon beam down to 1 keV by simply irradiating positive muons
($\mu^{+}$’s) onto a thin metal film such as aluminum. However, since the
first observation during the 1980s bib:Kua89 ; bib:Har86 , there have been no
measurements or no proof that a significant intensity can be obtained for an
acceleration test. Therefore, we decided to conduct the $\mathrm{Mu^{-}}$
measurement prior to the muon acceleration test.

In addition to the estimation of the accelerated muon signal, the method for
identifying the background caused by positrons from muon decay (decay-
positron) was also important. This is because the conversion efficiency from
the primary $\mu^{+}$ to $\mathrm{Mu^{-}}$ is expected to be less than
$10^{-4}$, which results in a large amount of decay-positrons derived from
muon decays at the target. Therefore, the decay-positron background and low
energy muon (LE-$\mu$) measurements were carried out bib:Otani19mcp  using a
micro-channel plate (MCP) detector (Hamamatsu photonics, F1217-01
bib:hamamatsu ) in advance of the $\mathrm{Mu^{-}}$ measurements. In the
decay-positron measurement, the decay-positrons from a muon beam target were
identified by a series of triple scintillation detectors installed in front of
the MCP detector. In the LE-$\mu$ measurement, the 7-keV $\mu^{+}$s are
measured using the same setup as the $\mathrm{Mu^{-}}$ measurement described
below. Figure 16 shows the pulse height distribution of the decay-positron
(red triangle) and LE-$\mu$ (blue box). The difference between the decay-
positrons and LE-$\mu$ in the pulse height distributions was evident, which
can be explained by the single- and multi- channel amplification model
described in Section 4 of bib:Otani19mcp . The results show that event
selection using the pulse height can reject the decay-positron more than 75%
while maintaining the muon efficiency at 90%.

Figure 16: Pulse height distribution for decay-positrons (red triangle) and
LE-$\mu$ (blue box). Red dashed line and blue dotted line show the pulse
height distribution with multi- and single-channel excitation models,
respectively, as discussed in bib:Otani19mcp . This figure is cited from
bib:Otani19mcp .

The $\mathrm{Mu^{-}}$ measurements were conducted in a series of three
experiments at the J-PARC MLF (2015A0324 bib:Otani19mcp , 2016A0067
bib:rkita21 , and 2018B0007 bib:Otani19mum ). Figure 17 shows a typical setup
for the experiments. The $\mu^{+}$’s were injected into a $\mathrm{Mu^{-}}$
production target after passing through a steel window. The $\mathrm{Mu^{-}}$
generated in the target was accelerated at up to 20 keV by the SOA
electrostatic lens. Then, the $\mathrm{Mu^{-}}$ was transported to the
detector location through a series of electrostatic quadrupole (EQ1-4), an
electrostatic deflector (ED), and a bending magnet (BM). The energy acceptance
of the beamline is estimated to be 1.4% by the GEANT4 simulation. The MCP
detector was employed to measure the time of flight (TOF) from the
$\mathrm{Mu^{-}}$ production target. The electrical signal from the MCP
detector was amplified using a fast-filter amplifier (ORTEC 579) and digitized
using CAEN V1720. The waveform within an interval of 10 $\mu$sec around each
25-Hz beam pulse was recorded for analysis. A pulse higher than the noise
level was regraded as a signal pulse. The leading edge of the signal pulse was
defined as the signal timing. The pulse height is defined by the maximum
height within the signal window of 40 ns. The $\mu^{+}$ arrival time at the
$\mathrm{Mu^{-}}$ production target was measured with a set of scintillating
counters located at the side of the $\mathrm{Mu^{-}}$ production target.

Figure 17: Schematic of the typical experimental setup for the
$\mathrm{Mu^{-}}$ measurements. This figure is cited from bib:Otani19mum .

In the 2015A0324 experiment, the LE-$\mu$ measurement was conducted to
commission the experimental setup and compare the MCP signal to the decay-
positron signal as described above. The aluminum (Al) thin foil is adopted as
the target and LE-$\mu$ is extracted instead of $\mathrm{Mu^{-}}$ by flipping
the beamline polarity and transported to the MCP detector. The LE-$\mu$ signal
was used to tune the applied voltage of the quadrupole and other experimental
setups. In addition to the tuning using the LE-$\mu$, negative hydrogen ions
generated by ultraviolet light bib:nakazawa19  were used for offline
commissioning of the experimental setup.

The $\mathrm{Mu^{-}}$s were observed during the 2016A0067 experiment, and
high-statistics $\mathrm{Mu^{-}}$ data were obtained in the 2018B0007
experiment using three types of production targets: an Al foil, a C12A7
electride foil bib:hosono03 ; bib:hosono17 , and a steel foil. The electrical
signals of the MCP detector were recorded and analyzed in a way similar to the
2015A0324 experiment. Figure 18 (A) shows a pulse height versus the TOF of the
observed signal. The background events are considered to be decay-positrons
from muons stopped in the experimental apparatus, based on the results of the
analysis of time constants. The decay-positron events can be eliminated
efficiently by the pulse height selection, as described above. After pulse
height selection, the TOF distribution is obtained as shown in Fig. 18 (B).
The two peaks at approximately $-300$ ns and $300$ ns are due to prompt
positrons carried through the surface muon beamline with the same momentum as
$\mu^{+}$’s. Because the prompt positron is faster than $\mu^{+}$, the prompt
positron arrives earlier than $\mu^{+}$. The blue and red curves show the
fitting results assuming the remaining decay-positron background (blue) and
the $\mathrm{Mu^{-}}$ signals (red), respectively. The background is
consistent with the exponential decay curve with the muon decay constant
($\tau_{\mu}=2.2$ $\mu$sec). The $\mathrm{Mu^{-}}$ peak width is consistent
with that of the primary $\mu^{+}$ beam. The time interval of the
$\mathrm{Mu^{-}}$ peaks is consistent with that of the primary proton beam
pulses. The $\mathrm{Mu^{-}}$ TOF is consistent within a few percentage points
with the expectation estimated through the GEANT4 simulation where the initial
energy of $\mathrm{Mu^{-}}$ is assumed to be 0.2 keV.

Figure 18: (A) Pulse height vs. TOF for observed signal. (B) TOF distribution
after pulse height selection. The solid red line and blue hatched line shows
the fitting results assuming the $\mathrm{Mu^{-}}$ event and the decay-
positron background. This figure is cited from bib:Otani19mum .

Based on the measured $\mathrm{Mu^{-}}$ intensity during the 2016A0067
experiment, The expected intensity of the accelerated $\mathrm{Mu^{-}}$ signal
was estimated to be $6.8\pm 1.2$ (stat.)${}^{+0.4}_{-0.0}$ (syst.)/s
bib:rkita21 , where (stat.) represents the statistical error assigned by the
measured $\mathrm{Mu^{-}}$ intensity in the experiment and (syst.) represents
the systematic error. The effect of a misalignment of the $\mathrm{Mu^{-}}$
transport beamline is taken into account as a systematic error. From these
results, it can be concluded that several hundred events of accelerated
$\mathrm{Mu^{-}}$ signals can be expected with a beam time of a few days, and
that the background of the decay-positrons can be sufficiently suppressed by a
pulse height discrimination, such that a sufficient amount of the accelerated
$\mathrm{Mu^{-}}$’s can be observed with a sufficient signal-to-noise ratio.

###  3.2 First muon acceleration

A demonstration of the muon acceleration was conducted in J-PARC MLF over a
6-day period starting on October 24, 2017 (2017A0263) bib:Bae18 .

Figure 19 shows a schematic drawing of the experimental setup. The $\mu^{+}$’s
were incident on the Al foil target used in the $\mathrm{Mu^{-}}$
measurements. The $\mu^{+}$’s were decelerated through the target, and some of
the $\mu^{+}$’s become Mu-’s at the downstream surface of the Al target. Using
the same Soa lens as used in the $\mathrm{Mu^{-}}$measurements, the generated
$\mathrm{Mu^{-}}$’s were accelerated to 5.6 keV and focused on the entrance of
the RFQ. In this experiment, the prototype RFQ bib:kondo06  was used. The
length of this RFQ is equivalent to two-thirds of the RFQ in the muon linac,
and this RFQ was originally designed using KEKRFQ bib:Ueno90  to accelerate
negative hydrogen ions at up to 810 keV. To use this RFQ for a muon
acceleration, the intervane voltage should be normalized to the muon mass, and
the input velocity $\beta$ should be the same as that of H-. Table 11
summarizes the parameters of the prototype RFQ. The $\mathrm{Mu^{-}}$’s are
accelerated to 89 keV and then transported to the MCP detector through the
diagnostic beamline consisting of two quadrupole magnets (QM1 and QM2) and a
bending magnet (BM). Because the expected number of $\mathrm{Mu^{-}}$ signals
is only a few hundred, and it is impossible to obtain the correct field
setting using the $\mathrm{Mu^{-}}$ signals themselves, the diagnostic
beamline was commissioned using negative hydrogen ions generated by
ultraviolet light bib:nakazawa19 , and the field setting of the magnets was
verified prior to the experiment. The electrical signals of the MCP detector
are recorded in a similar way as the experiments of the $\mathrm{Mu^{-}}$
measurements.

Figure 19: Schematic drawing of the experimental setup for the demonstration of muon acceleration. In this figure, TMP is turbo molecular pump.  Table 11: Parameters of the prototype RFQ for the H- and muon acceleration. | H- | muon
---|---|---
Frequency (MHz) | 324
Number of cells | 297
Length (m) | 1.97
Intervane voltage (kV) | 81 | 9.1
Power (kW) | 180 | 2.3
Injection energy (keV) | 50 | 5.6
Extraction energy (keV) | 810 | 89

At the start of the experiment, the polarity of the beamline was set to
transport positively charged particles. In this positive-charge configuration,
muons that passed through the target and were decelerated to 89 keV, which
corresponds to the acceleration energy, were measured. The observed TOF is
consistent with calculations based on the distance between the target and the
MCP detector and the velocity of the 89-keV muon. This confirms the beam
diagnostic system.

After measurement of 89-keV $\mu^{+}$’s, the polarities of the magnets were
flipped to a negative-charge configuration. Figure 20 shows the TOF spectrum
with and without the RFQ operation after the pulse-height cut was applied.
With the RFQ operation, a clear peak was observed at $830\pm 11$ ns. The time
required to reach the RFQ entrance from the target while being accelerated by
the Soa lens was estimated as 307 ns using the GEANT4 simulation. The number
of cells of the prototype RFQ is 297, and thus it takes $\frac{297}{2\times
324\times 10^{6}}=458$ ns to fully accelerate the particles through the 324
MHz field. The length of the diagnostic beamline is 0.91 m, and thus the
transit time of the 89-keV Mu- is 72 ns. The total flight time of the
accelerated Mu- from the target to the MCP detector was calculated as
$307+458+72=837$ ns, which is consistent with the measurement. The hatched
histogram in Fig. 20 represents the simulated TOF spectrum of the accelerated
Mu-. The number of simulation events was normalized to the number of incident
muons of this data set. The 46 ns rms width of the TOF spectrum is consistent
with that from the timing distribution of the primary $\mu^{+}$ at the Al
target.

From these experimental results, it is concluded that the observed TOF peak is
due to the Mu-’s accelerated by the RFQ to 89 keV. The event rate was
estimated as $(5\pm 1)\times 10^{-4}$/s by subtracting the decay-positron
events estimated from the timing region outside the signal range. This is
consistent with the expectation based on the $\mathrm{Mu^{-}}$ measurements.
The intensity of the accelerated $\mathrm{Mu^{-}}$ is limited by the low
conversion efficiency of $\mu^{+}$ to $\mathrm{Mu^{-}}$.

Figure 20: TOF spectra of the negative-charge configuration with RF on and
off. The clear peak of the RF on the spectrum at 830 ns corresponds to the
accelerated Mu-’s. The error bars are statistical. A simulated TOF spectrum of
the accelerated Mu-’s is also plotted. This figure is cited from bib:Bae18 .

###  3.3 Development of beam monitors

After the first demonstration of the muon acceleration described above, the
beam monitors for the accelerated muons were tested using the accelerated
$\mathrm{Mu^{-}}$’s. To diagnose the beam in both the transverse and
longitudinal directions, a beam profile monitor (BPM) and a bunch width
monitor (BWM) are necessary.

The BPM has been developed based on a MCP assembly (Hamamatsu F2225-21P) and a
charge-coupled device (CCD) camera. The avalanche electrons generated by the
MCP are injected into the phosphor screen, from which the light output is
focused onto the CCD camera. The spatial resolution was estimated as 0.3 mm
using an ultraviolet light and surface muons bib:kim18 . During the 2016B0214
experiment, the detector in the setup for the $\mathrm{Mu^{-}}$ measurement
shown in Fig. 17 was replaced with the BPM and the performance against
LE-$\mu$ was demonstrated bib:rkita17 . During the 2017B0006 experiment, the
detector in the setup shown in Fig. 19 is replaced with the BPM and the beam
profiles of the accelerated $\mathrm{Mu^{-}}$’s were measured. The
Kolmogorov–Smirnov test was performed and the measurement and expectation were
consistent within a statistical error (P-value is 21% and 38% for the
horizontal and vertical direction, respectively) bib:Otani18muaccbpm .

The BWM employed a MCP assembly (Hamamatsu F1217-11G) containing two stages of
chevron-type MCPs. The signal-processing electronics of constant-fraction
discriminators bib:Les75 , which were adopted from the technology of the other
experiment bib:Inami14 , measured the timing of the MCP signal. It detects
single muon with a high temporal resolution, accumulates muon events, and
reconstructs the beam bunch by taking the difference from the RF reference
time of the muon linac. A test bench was developed to evaluate the temporal
resolution of the BWM using a picosecond pulse laser (Hamamatsu PLP10-040)
bib:yotsuzuka19 . Recent results have evaluated the resolution of the BWM to
be 40 ps, which corresponds to 1% of the acceleration frequency of 324 MHz and
satisfies the requirement. During the 2018A0222 experiment, the detector in
the setup shown in Fig. 19 is replaced with the BWM to measure the bunch width
of the accelerated $\mathrm{Mu^{-}}$’s. In addition, a buncher cavity
developed as a prototype for the muon linac bib:Otani19buncher  was installed
between QM2 and BM to focus the beam in the longitudinal direction. The bunch
width was successfully measured to be $0.54\pm 0.11$ ns, which is consistent
with the simulation bib:sue20 .

From these measurements, the beam monitors are ready to diagnose the
accelerated muon beam, particular for low energy part. Because these monitors
are based on an MCP and the detection efficiency of an MCP is expected to
decrease as the muon energy increases, to conduct sufficient beam
commissioning, it is necessary to develop a beam monitor for the high-energy
part.

##  4 Summary and outlook

The design of the muon linac for the J-PARC E34 experiment and the first muon
acceleration were presented. The experiment for demonstrating the acceleration
of the thermal muons is scheduled in 2022 at a new muon experiment area (S2)
in J-PARC MLF. The experiment aims to study the laser ionization of the
muonium via the 1S-2S excitation with 244-nm laser bib:Ced21  and to
accelerate the thermal muons using the RFQ. The muon beam emittance before and
after the acceleration will be measured using the beam monitors described in
Section 3.3. After the experiment at S2, the muon acceleration using the RFQ
and IH-DTL at the H-line will be conducted.

We are entering a new era in which the accelerated muons are available. One
can naturally imagine an imaging technique with accelerated muons, allowing
for a better resolution with less time than cosmic-ray muons. It is natural to
think of a collider using an accelerated muon beam. A type of collider in
which muons collide with other particles has been actively discussed bib:Men20
; bib:Kin21 . If a cooling method of negative muons with an efficiency
comparable to that of the muonium laser ionization is realized, a muon
collider will be promised.

The J-PARC E34 experiment will soon be realized as a flagship to new horizon
for a better understanding of nature.

## Acknowledgment

We express our appreciation to the many manufacturing companies involved in
this project, especially to Toshiba Co., who fabricated the prototype RFQ,
Toyama Co.,Ltd., who fabricated the actual RFQ, and TIME Co., Ltd., who
fabricated the buncher cavity and the prototype IH-DTL. This work is supported
by JSPS KAKENHI (Grant Numbers , 25800164, 15H03666, 15H05742, 16H03987,
16J07784, 18H03707, 18J22129, 19J21763, 20J21440, 20H05625, 21K18630,
21H05088, 22H00141) , JST FOREST Program (Grant Number JPMJFR212O) and the
natural science grant of the Mitsubishi Foundation. This work is also
supported by the Korean National Research Foundation grants
NRF-2015H1A2A1030275, NRF-2015K2A2A4000092, and NRF-2017R1A2B3007018; the
Russian Foundation for Basic Research grant RFBR 17-52-50064; and the Russian
Science Foundation grant RNF 17-12-01036. This experiment at the Materials and
Life Science Experimental Facility of the J-PARC was performed under user
programs (Proposal No. 2015A0324, 2016A0067, 2016B0214, 2017A0263, 2017B0006,
2018A0222, and 2018B0007).

## References

  * (1) S. Bae et al., Phys. Rev. AB 21, 050101 (2018).
  * (2) S.H. Neddermeyer, C.D. Anderson, Phys. Rev. 51, 884 (1937).
  * (3) E. G. Michaelis, IEEE Trans. Nucl. Sci.Vol. NS-23, No. 3, 1385 (1975).
  * (4) M. Boscolo, J. P. Delahaye and M. Palmer, Rev. Acc. Sci. Tech., Vol 10, No. 01, 189-214 (2019)
  * (5) M. Boscolo et al., Phys. Rev. AB 21, 061005 (2018).
  * (6) https://slowmuon.kek.jp/index_e.html
  * (7) M. Abe et al., Prog. Theor. Exp. Phys. 2019, 053C02 (2019).
  * (8) G. Charpak, F. J. M. Farley, R. L. Garwin, T. Muller, J. C.Sens, and A. Zichichi, Phys. Lett. 1, 16 (1962).
  * (9) J. Bailey et al., Nuovo Cimento A 9, 369 (1972).
  * (10) J. Baileyet al., Nucl. Phys. B 150, 1 (1979).
  * (11) Bennett G. W. et al. (Muon $g-2$ Collaboration), Phys. Rev. D 73, 072003 (2006).
  * (12) T. Aoyama et al., Phys. Rept. 887, 1 (2020).
  * (13) J. Grangeet al. (Muon $g-2$ Collaboration), arXiv:physics.ins-det/1501.06858
  * (14) B. Abi et al. (Muon $g-2$ Collaboration), Phys. Rev. Lett. 126, 141801 (2021).
  * (15) D. Stratakis et al., Phys. Rev. AB 22, 011001 (2019).
  * (16) N. Kawamura et al., Prog. Theor. Exp. Phys. 2018, 113G01 (2018).
  * (17) Bakule P. et al. , Prog. Theor. Exp. Phys. 2013, 103C01 (2013).
  * (18) G.A. Beer et al., Prog. Theor. Exp. Phys. 2014, 091C01 (2014).
  * (19) J Beare et al., Prog. Theor. Exp. Phys. 2020, 123C01 (2020).
  * (20) H. Iinuma et al., Nucl. Instrum. Meth. A 832, 51 (2016).
  * (21) M. Abe et al., Nucl. Instrum. Meth. A 890, 51 (2018).
  * (22) T. Aoyagi et al., JINST 15, P04027 (2020).
  * (23) T. Kishishita et al., IEEE TNS, 67, 2089 (2020)
  * (24) Y. Kondo et al., Phys. Rev. AB 16, 040102 (2013).
  * (25) M. Kubosaki et al.,, Proceedings of PASJ 2011, 1366 (2011).
  * (26) Y. Kondo et al., Proceedings of IPAC 2015, 3801 (2015).
  * (27) M. Otani et al., Phys. Rev. AB 19, 040101 (2016).
  * (28) M. Otani et al., J. Phys. :Conf. Ser. 1350, 012097 (2019).
  * (29) Y. Kondo et al., J. Phys. :Conf. Ser. 875, 012054 (2017).
  * (30) K. F. Canter, P. H. Lippel, W. S. Crane, and A. P. Mills Jr., in “Positron studies of solids, surfaces and atoms” (World Scientiﬁc, Singapore, 1986) p. 199.
  * (31) M. Otani et al., J. Phys. :Conf. Ser. 1067, 052018 (2018).
  * (32) G4beamline, http://public.muonsinc.com/Projects/G4beamline.aspx
  * (33) P. Strasser et al., Journal of Physics: Conference Series 225, 012050 (2010).
  * (34) GEANT4, http://geant4.cern.ch/
  * (35) OPERA3D, Vector Fields Limited, Oxford, England., https://operafea.com/
  * (36) K. R. Crandall et al., LA-UR-96-1836 (2005).
  * (37) I.M. Kapchinskiy and V.A. Tepliakov, Prib. Tekh. Eksp 2, 19-22 (1970).
  * (38) I.M. Kapchinskiy and V.A. Tepliakov, Prib. Tekh. Eksp 4, 17-19 (1970).
  * (39) J.E. Stovall, K.R. Crandall, R.W. Hamm, IEEE Trans. Nucl. Sci. NS-28 p.1508 (1981)
  * (40) M. Otani et al., Proceedings of PASJ 2015, 56 (2015)
  * (41) H. Morinaga, Phys. Soc. Meeting, Osaka (1949).
  * (42) J.P. Blewett, Proceedings of Symposium du CERN sur les Accelerateurs de Haute Enegie et la Physique des Mesons $\pi$ v.1, 162 (1956).
  * (43) P.M. Zeidlitz and V.A. Yamnitskii, J. Nucl. Energy, Part C4, 121 (1962).
  * (44) J. Pottier, IEEE Trans. Nucl. Sci. 16/3, 377 (1969).
  * (45) E. Nolte et al., Nucl. Instr. and Meth. 158, 311 (1979).
  * (46) U. Ratzinger, CERN Yellow Report, 2005-003, 351 (2005).
  * (47) Good, M.L., Phys. Rev. 92, 538 (1953).
  * (48) S. Minaev and U. Ratzinger, Proceedings of PAC Conf 1999., 3555 (1999).
  * (49) Y. Iwata et al., Nucl. Instr. Meth. A569, 685 (2006).
  * (50) R. A. Jameson, arXiv:physics.acc-ph/1404.5176
  * (51) CST Studio Suite, Computer Simulation Technology (CST). https://www.cst.com/products/CSTMWS
  * (52) M. Otani et al., Proceedings of the PASJ 2016, 858-862 (2016).
  * (53) General Particle Tracer, Pulsar Physics. http://www.pulsar.nl/gpt/
  * (54) Y. Nakazawa et al., J. Phys. :Conf. Ser. 1350, 012054 (2019).
  * (55) Peter A. Mcintosh, Proceedings of 4th European Particle Accelerator Conference, 1283 (1994).
  * (56) V. G. Andrev et al.,Proceedings of Linear Acel. Conf 1972, 114 (1972)
  * (57) B.P. Murin et al., Proceedings of Proton Linac Conf 1972, LA-5115, p.387 (1972).
  * (58) V.G.Andreev et al., Proceedings of Linac Conf. 1976, AECL-5677 p.269 (1976).
  * (59) S. K. Esin et al., Proceedings of Linear Accel. Conf 1988. 657 (1988)
  * (60) S. Inagaki, Nucl. Instrm. and Meth. in Phys. Res. A 251, 417-436 (1986).
  * (61) Y. Iwashita, Nucl. Instrm. and Meth. in Phys. Res. A 348, 15-33 (1994).
  * (62) J.H. Billen and L.M. Young, Los Alamos Report, LA-UR-96-1834 (1996).
  * (63) H. Ao et al., Jpn. J. Appl. Phys. 39, 651 (2000).
  * (64) http://www.laacg.lanl.gov
  * (65) K.R. Crandall and D.P Rustoi, Los Alamos Report, LA-UR-97-886 (1997).
  * (66) W.D. Kilpatrick, Rev. Sci. Instr. 28, 824 (1957).
  * (67) Reiser M “Theory and Design of Charged Particle Beams”1994 JOHN WILEY & SONS, INC.
  * (68) Y. Takeuchi et al., to be published in proceedings of J-PARC symposium.
  * (69) R. Wideröe, Arch. Electrotech., 21, 387 (1928).
  * (70) D.H. Sloan and E.O. Lawrence, Phys. Rev. 38, 2021-2032 (1931).
  * (71) H. Hansen, J. App. Phys. 9, 654 (1938)
  * (72) E.L. Ginzton, W.W. Hansen and W.R. Kennedy, Rev. Sci. Instrum, 19, 89 (1948)
  * (73) M. Chodorow et al., Rev. Sci. Instrum. 26, 134 (1955).
  * (74) Y. Kondo, M. Otani, Proceedings of PASJ2020, 218 (2020) (in Japanese).
  * (75) Y. Kondo, T. Morishita, J. Tamura, M. Otani, Proceedings of LINAC 2018, 794 (2018).
  * (76) M. Otani, Y. Kondo, Proceedings of PASJ2020, 202 (2020) (in Japanese).
  * (77) H. Yasuda et al., Proceedings of PASJ2020, 173 (2020) (in Japanese).
  * (78) H. Yauda et al., to be published in proceedings of J-PARC symposium.
  * (79) A. P. Mills, Jr., J. Imazato, S. Saitoh, A. Uedono, Y. Kawashima, and K. Nagamine Phys. Rev. Lett. 56, 1463 (1986).
  * (80) Steven Chu, A. P. Mills, Jr., A. G. Yodh, K. Nagamine, Y. Miyake, and T. Kuga Phys. Rev. Lett. 60, 101 (1988).
  * (81) K. Nagamine et al., Phys. Rev. Lett. 74, 4811 (1995)
  * (82) H. Miyadera et al., in proceedings of PAC 2007, 3032 (2007).
  * (83) Y. Kuang et al., Phys. Rev. A39, 6109 (1989)
  * (84) D.R. Harshman, et al., Phys. Rev. Lett. 56, 2850 (1986).
  * (85) M. Otani et al., Nucl. Inst. Meth. Phys. Res. Sec. 946, 162693 (2019).
  * (86) http://www.hamamatsu.com/
  * (87) R. Kitamura et al., to be published in Phys. Rev. AB.
  * (88) M. Otani et al., J. Phys. :Conf. Ser. 1350, 012067 (2019).
  * (89) Y. Nakazawa et al., Nucl. Inst. Meth. Phys. Res. Sec. 937, 164 (2019).
  * (90) H. Hosono et al., Science, 301, 626 (2003).
  * (91) H. Hosono et al., Proc. Natl. Acad. Sci. USA 114(2), 233 (2017).
  * (92) Y. Kondo, K. Hasegawa, and A. Ueno, Proceedings of LINAC 2006, 749 (2006).
  * (93) A. Ueno and Y. Yamazaki, Proceedings of Linac Conf. 1990, 329 (1990).
  * (94) B. Kim et al., Nucl. Inst. Meth. Phys. Res. Sec. A. 899, 22 (2018).
  * (95) R Kitamura et al., J. Phys. :Conf. Ser. 875, 012055 (2017).
  * (96) M. Otani et al., J. Phys. :Conf. Ser. 1067, 052012 (2018).
  * (97) B. Leskovar and C. C. Lo, Nucl. Instrum. Meth. Phys. Res., Sect. A123, 145 (1975).
  * (98) K. Inami (Belle-II PID Group), Nucl. Instrum. Meth. Phys. Res., Sect. A766, 5 (2014).
  * (99) M. Yotsuzuka et al., Proceedings of IPAC2019, 2571 (2019).
  * (100) M. Otani et al., Nucl. Inst. Meth. Phys. Res. Sec. A 946, 162693 (2019).
  * (101) Y. Sue, M. Yotsuzuka et al., Phys. Rev. Accel. Beams 23, 022804 (2020).
  * (102) Ce Zhang et al., JPS Conf. Proc. 33, 011125 (2021).
  * (103) M. Lu et al., arXiv:physics.hep-ph/2010.15114
  * (104) K. Cheung, Z. S. Wang., arXiv:physics.hep-ph/2101.10476