14th International Particle Accelerator Conference,Venice, Italy
ISBN: 978-3-95450-231-8

ISSN: 2673-5490

JACoW Publishingdoi: 10.18429/JACoW-IPAC2023-MOPA110

H. Iinuma† , Ibaraki University, Mito, Japan
M. Abe, K. Sasaki, H. Nakayama, T. Mibe1 , KEK, Tsukuba, JapanS. Ogawa, T. Yamanaka, Kyushu University, Fukuoka, Japan
Y. Sato, Niigata University, Niigata, Japan
1 also at Tokyo University, Tokyo, Japan
Abstract
In this study, a strategy for designing a dedicated beam injection and storage scheme for the J-PARC Muon g-2/EDM
experiment is described. To accomplish a three-dimensional
beam injection into the MRI-type compact storage system, transverse beam phase spaces (so-called X-Y coupling)should be coupled appropriately. A pulsed kicker system isa key to control vertical motion inside the storage volume.
Moreover, dedicated beam phase control through the beam
channel of the storage magnet’s yoke is crucial. We introduce five-dimensional phase-space correlation in addition toa strong X-Y coupling, to control stored vertical beam size
as small as a level of one-third.

Figure 1: Outline of three-dimensional spiral injection
scheme [1, 3].

INTRODUCTION
A new measurement of the muon’s anomalous magnetic
moment 𝑎 𝜇 = (𝑔 − 2)/2 and its electric dipole moment(EDM) is in preparation at the J-PARC muon facility at MLF,
MUSE [1]. These physics quantities are suitable probes forexploring beyond the standard model in elementary physics.
Experimentally, we measure them from a difference of two
angular frequencies of spin procession frequency and theorbital cyclotron frequency in a homogeneous magnetic field
but no electric field, as Eq. (1).
𝜂  ® ®𝑞
𝛽×𝐵(1)
𝜔® = − 𝑎 𝜇 𝐵® +𝑚2Here, two dipole moments of the muon are introduced:𝜇®𝜇 = −𝑔𝑞𝑞𝑠®, 𝑑®𝜇 = 𝜂𝑠®,
2𝑚2𝑚𝑐

(2)

and, 𝑐 and 𝑞 are the speed of light and a unit charge, and𝑚 and 𝑔 are the mass and gyromagnetic ratio of the muon,respectively. If we assume non-zero 𝜂 here, but assumingthat 𝛽® · 𝐵® = 𝛽® · 𝐸® = 0, the first term in equation 1, which
express a muon magnetic moment, is orthogonal to the second which includes the EDM related term 𝜂, required to be
extremely small from the standard model. Tilt angle of 𝜔 to𝐵® is proportional to the magnitude of the EDM and is of theorder of 1 mrad, considering the upper limit from the previous experiment E821 [2] (| 𝑑®𝜇 | = 0.9 × 10−19 𝑒 · cm). To
achieve a 100-times better sensitivity, we should be sensitive
to 0.01 mrad.
∗ Work supported by JPS KAKENHI Grant Numbers JP19H00673 andJP20H05625.
† hiromi.iinuma.spin@vc.ibaraki.ac.jp

MOPA110304

In J-PARC, a slow muon source and muon LINAC technology have been developed [1] to have a low emittance muon
beam of momentum of 300 MeV/𝑐. Thereafter, muons are
stored in a 3 T storage volume, the diameter of the orbital
cyclotron motion becomes only 0.66 m. This is the smallest
storage ring for relativistic energy beams in the world. To realize this technical challenge, a new beam injection scheme
called the three-dimensional spiral injection scheme, as
shown in Fig 1, are being developed [3]. The beam enters
the solenoid through a channel in the top iron yoke 110 cm
above the storage volume (refer to Fig. 2) and its spiral motion is compressed by the Fleming force owing to the static
radial field (𝐵 𝑅 ). A vertical kick (pulsed radial magneticfield) is applied to store the beam on arrival to the storage
region. A small static weak-focusing field in the fiducial
volume maintains the beam in the storage region.

Figure 2: Superconducting solenoid magnet and kicker
coils [1, 4].
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MC1.T12: Beam Injection/Extraction and TransportThis is a preprint - the final version is published with IOP

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PRECISE CONTROL OF A STRONG X-Y COUPLING BEAM
TRANSPORTATION FOR J-PARC Muon g-2/EDM EXPERIMENT∗

14th International Particle Accelerator Conference,Venice, Italy
ISBN: 978-3-95450-231-8

ISSN: 2673-5490

JACoW Publishingdoi: 10.18429/JACoW-IPAC2023-MOPA110

Relation of a single track motion and a time structure ofa pulsed kicker current will be introduced. Details of designconcept of kicker coil shape are discussed in Ref. [5].

A Single Track Motion Controlled by Pulsed Kicker
Figure 3 shows vertical position of single track as a function of time. Effective magnetic field along the trajectory
is also shown. We employ a half-shine shaped kicker pulse
with 𝑇𝐾 = 120 ns duration time.
 𝜋
𝐼 (𝑡) =𝐼0 𝑠𝑖𝑛(𝑡 − 𝑡0) , : (𝑡 − 𝑡0 ) < 𝑇𝐾𝑇𝐾
=0, : (𝑡 − 𝑡0 ) > 𝑇𝐾(3)Figure 4: Beam motion with expected phase space withmoderate kicker parameter [5].

Effective radial field along a single track is
𝐵 𝑅 (𝑧, 𝑅, 𝑡)=𝑎𝑡𝑖𝑐(𝑧, 𝑅)
𝐵 𝑠𝑡𝑅(4)𝜋+ 𝐵 𝑅𝑘𝑖𝑐𝑘𝑒𝑟 (𝑧, 𝑅)𝑠𝑖𝑛( (𝑡 − 𝑡0)).
𝑇𝐾
A correlation of vertical potions and pitch angles along thetrajectory is also shown. Role of vertical pulsed kicker is toguide trajectory for both vertical position and pitch angle arezero when kick current back to zero. There is a freedom todesign trajectory during the kick period, depends on kicker
coil shape and kicker duration time and current as long as
integrated 𝐵 𝑅 𝐿 along the trajectory satisfies Eq. (5) [5].
∫𝐵𝐿 =𝐵 𝑅 (𝑧, 𝑅, 𝑡)𝑑𝑙.
(5)

Figure 5: Beam motion with expected phase space withstrong kicker parameter for trials calculation.

comparing Figs. 4 and 5, the shorter 𝑇𝐾 and bigger 𝐼0 havebig advantage to control |𝑧|. This is because stronger kick has
stronger vertical focusing field as clear in kink shape. However, it costs high voltage on the kicker coils of 𝑉 > 80 kV,
which may cause severe technical difficulties for kicker’s
conductor design. Therefore, strong kicker case of Fig. 5 isnot realistic solution. In order to stay with 𝑉 < 30 𝑘𝑉, weneed to consider how beam phase space should be controlledfrom Fig. 4 trajectories.

This is a preprint - the final version is published with IOP

Vertical Phase Space and Integrated 𝐵 𝑅 𝐿

Figure 3: Left:Vertical motion of a single track as a functionof time. Right: Correlation of vertical position and pitch
angle.

Left plots of Figs. 6 and 7 show time slice of vertical phasespace at the end of the kick. Integrated 𝐵 𝑅 𝐿 distributions
are also shown in the right.
From these plots, stronger kick can control 𝐵 𝑅 𝐿 better.
This is very consistent that kink shape in Fig. 5 is stronger
than that of Fig. 4.

Multi Beam Motion Expecting Beam Phase SpaceNow we consider expected beam phase space and see how
trajectories of particle differ as shown in Fig. 4.
Without change kicker coil shape, but shorter 𝑇𝐾 andbigger 𝐼0 case, vertical beam beam distribution after the kickis well controlled.
To meet experimental requirement, smaller vertical beam
distribution |𝑧| after the kick is favored. Trajectories in redlines corresponds to |𝑧| < 3 cm, which indicate ideal bean injection and storage. On the other hand, black lines are stored
inside the weak focusing magnetic field but |𝑧| < 10 cm. By
MOPA: Monday Poster Session: MOPA
MC1.T12: Beam Injection/Extraction and Transport

Figure 6: Time sliced vertical phase space at the end of themoderate kick and 𝐵 𝑅 𝐿 distribution. Red corresponds to|𝑧| < 3 cm. Closed ellipses are from weak focusing field [5].
MOPA110305

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VERTICAL BEAM MOTION CONTROL
BY PULSED KICKER

14th International Particle Accelerator Conference,Venice, Italy
ISBN: 978-3-95450-231-8

ISSN: 2673-5490

JACoW Publishingdoi: 10.18429/JACoW-IPAC2023-MOPA110

Correlation Finding With Five Phase Space Parameters Through the Channel
Figure 7: The same plots as Fig. 6 but for the strong kick.

Each beam trajectory has six phase space parameters, but
it is fair to set five are independent, because we can treat asbeam momentum is fixed.
In addition to Eq. (7), we invite another two angles:

BEAM PHASE SPACE STUDY
Staying with high voltage at kicker coil 𝑉 < 30 kV, how
do we realize smaller |𝑧| distribution? We studied beamphase space by use of three parameters: |𝑟 |, 𝜃 and vertical
distributions 𝑧 𝑔 which are from six components of eachtrajectory in global coordinate system.

𝑟® =|𝑟 |(𝑥 𝑔 , 𝑦 𝑔 , 𝑧 𝑔 ), 𝑝® = ( 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 )√︃𝑟® · 𝑝®
𝑥 𝑔2 + 𝑦 2𝑔 + 𝑧 2𝑔 , 𝑐𝑜𝑠(𝜃) ==|𝑟 || 𝑝|(6)

Figure 8 shows correlations of these three parameters at
different positions along the beam injection trajectory.

𝑠𝑖𝑛(𝜓) =𝑝𝑧, 𝑐𝑜𝑠(𝜙) = √︃
| 𝑝|𝑝𝑥(7)(8)

And then, we define 𝑛 × 5 matrix: M𝑟 𝑒𝑑 as|𝑟®0 | − 𝑟¯©
­ |𝑟®1 | − 𝑟¯
M𝑟 𝑒𝑑 = ­­ .
­ ..
« |𝑟®𝑛 | − 𝑟¯

𝜃0
𝜃1..
.

𝑧 𝑔0 − 𝑧¯𝑔𝑧 𝑔1 − 𝑧¯𝑔..
.

𝜓0
𝜓1..
.

𝜃𝑛𝑧 𝑔𝑛 − 𝑧¯𝑔

𝜙0ª𝜙1 ®
.. ®®
. ®

𝜓𝑛𝜙𝑛 ¬¯ 𝑧¯𝑔 , 𝜓¯here, 𝑛 is number of red-subgroup’s trajectories. 𝑟,¯ 𝜃,¯and 𝜙 are mean values of red-subgroup in Fig. 8. We apply
singular value decomposition on M𝑟 𝑒𝑑 , and obtain an eigen
vector of the smallest eigen value:𝑞® = (𝑞 𝑟 , 𝑞 𝜃 , 𝑞 𝑧 , 𝑞 𝜓 , 𝑞 𝜙 ).

Figure 8: Zoom up view of channel in the yoke. |𝑟 | − 𝜃 − 𝑧 𝑔correlation at different points are also shown.

.
𝑝𝑥2 + 𝑝𝑦2(9)Now, we define 𝑁 × 5 matrix:M𝑡𝑜𝑡. of all trajectories, and® Total 𝑁 components of 𝑑® areestimate vector 𝑑® = M𝑡𝑜𝑡. 𝑞.
residues and indicate how differ red and black sub-groups
as in lower histograms in Fig. 10.

Red and black distributions, as introduced previous section, are separated at 𝑧 𝑔 ∼ 0.95 m (inside the storage magnet’s yoke), however, not for 𝑧 𝑔 ∼ 1.40 m (entrance pointof the channel in the storage magnet’s yoke). It seems thatthree parameters are not enough to distinguish red and black
sub-groups.

Figure 10: Upper:|𝑟 | − 𝜃 − 𝑧 𝑔 correlations do not help todistinguish red and black groups. Lower:Residue vector 𝑑®
components indicates a way of precise beam phase control.

CONCLUSION AND NEXT

Figure 9: X-Y coupling in the beam frame, at the injection
point, controlled in the upstream beam transport line [6].
MOPA110306

Besides strong X-Y coupling, five-parameter phase spacecorrelation should be consider to control precise verticalbeam motion in the storage volume. Based on this study,
specification of additional multipole magnet at the injection
point is now under considering.
MOPA: Monday Poster Session: MOPA
MC1.T12: Beam Injection/Extraction and TransportThis is a preprint - the final version is published with IOP

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Upstream beam transport line [6] is dedicated to control
so-called X-Y coupling in the beam frame, at the injection
point, as in Fig. 9. However, there is no obvious hint indicating how to control beam phase space to achieve |𝑧| < 3 cm.

ISSN: 2673-5490

REFERENCES[1] M. Abe and J-PARC g-2/EDM Collaboration, “A new approachfor measuring the muon anomalous magnetic moment andelectric dipole moment”, Prog. Theor. Exp. Phys., vol. 2019,p. 053C02, 2019. doi:10.1093/ptep/ptz030[2] G. W. Bennett et al., (Muon (g-2) Collaboration) “Improved
limit on the muon electric dipole moment”, Phys. Rev. D,
vol. 80, p. 052008, 2009.
doi:10.1103/PhysRevD.80.052008[4] M. Abe et al., “Design method and candidate of a magnet
for muon g- 2/EDM precise measurement in a cylindrical homogeneous volume”, Nucl. Instrum. Meth. Phys. Res. Sect. A,
vol. 890, pp. 51–63, 2018.
doi:10.1016/j.nima.2018.01.026[5] H. Iinuma et al., “Design work of pulsed radial kick field for
J-PARC muon g-2/EDM experiment”, in Proc. PASJ2022.
https://www.pasj.jp/web_publish/pasj2022/
proceedings/PDF/TUP0/TUP036.pdf[6] H. Iinuma, H. Nakayama, M. Abe, K. Sasaki, and T. Mibe,“Design of a Strong X-Y Coupling Beam Transport Line for
J-PARC Muon g-2/EDM Experiment”, IEEE Trans. Appl. Supercond., vol. 32, no. 6, pp. 1–5, Sep. 2022.
doi:10.1109/TASC.2022.3161889

This is a preprint - the final version is published with IOP[3] H. Iinuma et al., “Three-dimensional spiral injection scheme
for g-2/EDM experiment at J-PARC”, Nucl. Instrum. Meth.
Phys. Res. Sect. A, vol. 832, pp. 51–62, 2016.
doi:10.1016/j.nima.2016.05.126

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14th International Particle Accelerator Conference,Venice, Italy
ISBN: 978-3-95450-231-8