Precise control of a strong X-Y coupling beam transportation for J-PARC muon g-2/EDM experiment

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 (X-Y coupling) and a pulsed kicker system are key to controlling the 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 a five-dimensional phase-space correlation in addition to strong X-Y coupling to control the stored vertical beam size to a level as small as one-third that achievable by X-Y coupling alone.


Introduction
A new measurement of the muon's anomalous magnetic moment a µ = (g − 2)/2 and its electric dipole moment (EDM) is being prepared at the J-PARC muon facility at the MLF, MUSE [1].These physical quantities are suitable probes for exploration beyond the standard model in elementary physics.These parameters were measured experimentally using a difference between the two angular frequencies of the spin procession frequency and orbital cyclotron frequency in a homogeneous magnetic field with no electric field, as Eq. 1.
Here, two dipole moments of the muon are introduced: where c is the speed of light, q the unit charge, and m and g the mass and gyromagnetic ratio of the muon, respectively.If we assume a non-zero η, assuming that ⃗ β • ⃗ B = ⃗ β • ⃗ E = 0, the first term in Eq. 1, which expresses the muon magnetic moment, is orthogonal to the second, which includes the EDM-related term η, which is required to be extremely small from the standard model.The tilt angle of ω to ⃗ B is proportional to the magnitude of the EDM and is of the order of 1 mrad, considering the upper limit from the previous experiment E821 [2] (| ⃗ d µ | = 0.9 × 10 −19 e • cm).To achieve 100-times better sensitivity, the goal is 0.01 mrad.
At J-PARC, a slow muon source and muon LINAC technology have been developed [1] to obtain a low emittance muon beam with a momentum of 300 MeV/c.Thereafter, muons are stored in a 3 T storage volume, and 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 meet this technical challenge, a new beam injection scheme called the three-dimensional spiral injection scheme is 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. 1) and its spiral motion is compressed by the Lorentz force owing to the static radial fringe field (B R ).A vertical kick (pulsed radial magnetic field) is applied to store the beam upon its arrival in the storage region.A small static weak focusing field in the fiducial volume maintains the beam in the storage region.

Vertical Beam Motion Control by Pulsed Kicker
The relation between a single track motion and the time structure of a pulsed kicker current is introduced.The details of the design concept of the kicker coil shape are discussed in [5].

Single Track Motion Controlled by Pulsed Kicker
Figure 2 illustrates the vertical position of a single track as a function of time.The effective magnetic field along the trajectory is also presented.We employed a half-shine-shaped kicker pulse with T K = 120 ns duration time.The effective radial field along a single track is A correlation between the vertical positions and pitch angles along the trajectory is also shown.The role of the vertical pulsed kicker is to guide the trajectory such that both the vertical position and pitch angle are zero when the kick current returns to zero.There is freedom to design the trajectory during the kick period, depending on the kicker coil shape and kicker duration and current provided that the integrated B R L along the trajectory satisfies Eq. 5 [5].

Beam Motion with Expected Beam Phase Space
Considering the expected beam phase space ( ϵ x,y ∼ 0.6 [mm-mrad]) and momentum dispersion ∆p/p ∼ 5 × 10 −4 from the upstream beam line, we can see how the trajectories of the particles differ (Fig. 3).The upstream beam transport line [6] is dedicated to controlling the X-Y coupling in the beam frame at the injection point, as shown in Fig. 4. X-Y coupling is a strong tool for controlling the stored beam distribution to z<10 cm.However, there is no clear hint to distinguish black and red subgroups.We must determine how to control the beam phase space to achieve |z|< 3 cm.
As a trial, we attempted a stronger kicker case, as shown in Fig. 5, to confirm how well the beam distribution was controlled after the kick.Without changing the beam conditions, including the X-Y coupling, or the kicker coil shape, we shortened T K (120 → 85 ns) and increased I 0 (0.9 → 2 kA).
To satisfy the experimental requirements, a smaller vertical beam distribution |z| after the kick is favored.The trajectories in the red lines correspond to |z|<3cm, indicating ideal beam injection and storage.By contrast, the black lines are stored in a weak focusing magnetic field   The left-hand plot in Fig. 6 shows the time slice of the vertical phase space at the end of the kick for the moderate-kicker case.The middle and right-hand plots in Fig. 6 display the integrated B R L distributions.These plots indicate that a stronger kick can control B R L better.This is consistent with the stronger kink shape in Fig. 5 than in Fig. 4.
However, a high voltage of V >80 kV is required for the kicker coils, which may cause severe technical difficulties in the kicker conductor design.Therefore, the strong-kicker case shown in Fig. 5 is not a realistic solution.To maintain V <30 kV, we need to consider how the beam phase space should be controlled from Fig. 3 trajectories.

Beam Phase Space Study
We investigated how a smaller |z| distribution can be realized while maintaining a high voltage at the kicker coil V <30 kV.
One issue to be considered is the nonlinear magnetic field effect through the beam injection channel in the yoke.The left-hand picture in Fig. 7 shows a magnified view of the beam injection channel from above, through the yoke of the storage magnet (OPERA-3D model).The plot on the right shows the magnetic flux distribution along the beam trajectories.These nonlinear magnetic field components, particularly at the channel exit, may affect the beam phase space and cause unclear differences between the red and black distributions.In addition to X-Y coupling, we studied the beam phase space using three parameters: timesliced |r|, θ, and vertical distributions z g that are from six components of each trajectory in the global coordinate system.⃗ r = (x g , y g , z g ), ⃗ p = (p x , p y , p z ) (6) The upper plots in Fig. 8 show the correlations among these three parameters at different positions along the beam injection trajectory.As introduced previously, the red and black distributions are separated at z g ∼ 0.95 m (inside the yoke of the storage magnet); however, not at z g ∼ 1.40 m (entrance point of the channel in the yoke of the storage magnet).It seems that the correlation of time-sliced |r|, θ, and vertical distributions z g is not sufficient to distinguish between the red and black subgroups.

Correlation Finding with Five Phase-Space Parameters through the Channel
Each beam trajectory has six phase-space parameters, but it is fair to consider five independent, because the beam momentum can be considered fixed.
We introduce two other angles: We define an n × 5 matrix M red as where n is the number of trajectories in the red subgroup and r, θ, zg , ψ, and φ denote the mean values of the red subgroup in Fig. 8.We apply singular value decomposition to M red , and obtain the eigenvector of the smallest eigenvalue: ⃗ q = (q r , q θ , q z , q ψ , q ϕ ) (9) We define an N × 5 matrix M tot of all trajectories, and estimate vector ⃗ d = M tot ⃗ q.Here, N is the sum of the black and red subgroup trajectories, and total N components of ⃗ d are residues.We find that ⃗ d indicates the difference between the red and black subgroups, as shown in the lower histograms in Fig. 8.The eigenvector ⃗ q at different time slices indicates a correlation of the five-dimensional phase space.

Conclusion
X-Y coupling can control the stored beam distribution to z<10 cm.In addition to strong X-Y coupling, a five-parameter phase-space correlation should be considered to control the precise vertical beam motion in the storage volume.Design work for a magnetic shield tube at the injection channel, which controls the distribution of ⃗ d to be narrower, is ongoing.An additional multipole magnet at the injection point is also under consideration.

Figure 2 .
Figure 2. Left: Vertical motion of a single track as a function of time.Right: Correlation of vertical position and pitch angle.

Figure 4 .Figure 5 .
Figure 4. X-Y coupling in the beam frame at the injection point, controlled in the upstream beam transport line [6].

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

Figure 7 .
Figure 7. Left: Topside view of channel in the yoke and trajectories (OPERA-3D).Right: Magnetic field along the trajectories.Nonlinear magnetic field effects need to be considered.

Figure 8 .
Figure 8. Upper: |r| − θ − z g correlations with different time slices.It is hard to distinguish between red and black groups.Lower: Residue vector ⃗ d components indicate a hint to control beam as |z|< 3 cm.