New muon g-2 experimentver.3.0: 2008-10-13
M. Iwasaki a,c,1 , N. Sito d , H. Okada d , Y. Matsuda e , S. Wada b ,K. Midorikawa b , D. Tomono a , K. Ishida a , T. Matsuzaki a ,P. Bakule a and who else ??
a RIKEN Nishina Center, Hirosawa, Wako-shi, Saitama 351-0198, Japanb RIKEN, Hirosawa, Wako-shi, Saitama 351-0198, Japanc Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku,

Tokyo 152-8551, Japand KEK, High Energy Accelerator Research Organization, Oho, Tukuba-shi, Ibaraki

305-0801, Japane University of Tokyo, Graduate School of Arts and Science, Komaba, Meguro-ku,

Tokyo 153-8902, Japan

Abstract
New approach toward precision measurement on muon g-2.
Key words: muon, anomalous magnetic moment

1

Can we measure the muon decay without vertical focusing?

This note discusses a new way of using a magnetic storage ring to measure bothanomalous magnetic moment a = (g − 2)/2 and the electric dipole moment dof the muon.
⃗ the
In a muon storage ring which conﬁnes muon by uniform magnetic ﬁeld B,
muon spin precession due to the cyclotron motion is canceled by the muon
⃗ and relativistic mass factor γ =motion by them selves. If a muon of velocity βc2 −1/2(1 − β )is circulating in a horizontal plane due to a vertical magnetic ﬁeld⃗
⃗ it will according to a Lorentz transformation experience
B and electric ﬁeld E,
1

Contact address: advanced meson@riken.jp

Preprint submitted to Elsevier Science

22 October 2009⃗ ∗ and B⃗ ∗ , in its own rest frame. The
both an electric and a magnetic ﬁeld, E⃗ ∗ = γE
⃗ + γcβ⃗ × B,
⃗ andso-called motional electric ﬁeld is represented as E
⃗ ∗ = −γ β⃗ × E/c,⃗
⃗ = β⃗ · B⃗ = 0. The
motional magnetic ﬁeld is B
with β⃗ · E
spin precession vector due to the muon anomalous magnetic moment ⃗ωa in
the Lab. frame can be simpliﬁed [1] as


(

1
e  ⃗
aµ B − aµ − 2 2⃗ωa = −
mµ
γ β

)





⃗
⃗
β⃗ × Eη⃗ + E  ,+ β⃗ × B
c
2
c

(1)

where aµ = (gµ − 2)/2 ≈ 0.00117.
In the previous muon (g−2) experiments, electro-static quadruple ﬁeld is indispensable to conﬁne the muon into the storage ring as a vertical focusing
component. Therefore, so-called magic momentum 3.094GeV/c was applied
to cancel out the second term in Eq. (1), to measure aµ precisely.
However, this electro-static quadruple introduces so called coherent betatronoscillation (CBO) which deforms the decay-positron time spectrum, and cause
muon beam instability resulting muon loss. There is another severe problem forthe precise g − 2 measurement, which is known as “ﬂash” problem. The muonsat the magic momentum can be produced by collecting the pion decay whose
momentum at about 3 GeV/c. This means that it is very diﬃcult to remove
contaminating pions from the muon beam at the magic momentum. Thus,
decay positron detectors records huge background due to the pion originating
reaction / decay. Actually, the detectors were switched on only ∼ 30 µsec after
the muon injection. This background eﬀect cannot be completely removed by
the detector switching technique, resulting a long term (few tens of µsec)
baseline shifts of the detectors.
Is there any other methods to switch oﬀ the second term in Eq. (1)? If one giveup the vertical focusing, namely set E = 0, then one can free from the second
term at any momentum. This setting is only feasible when the emittance ofthe muon beam is extremely small so as the beam circulates in the storage
ring without escaping from the storage region well beyond the decay positronobservation time window deﬁned by the dilated muon life time (γτµ ).

2

Muon beam requirement

If one measures the muon during Nlif e times of its dilated lifetime, then thenumber of muon circulation in the ring Ncycle can be written as2

Nlif e βγcτµNcycle =2πR()

Nlif e τµ eB
=
,
2π mµ

(2)

where the R is the radius of the ring, p, τµ and mµ are the momentum, thelifetime and the mass of the muon, respectively.
The vertical motion of a muon in the ring without the vertical focusing ﬁeld
is simply a linear diversion according to its tilt angle. Thus the requirementthat the muons do not escape from the stored region within the observation
time window can be written by the initial vertical position y0 and the tangentof the tilt angle in the vertical direction y ′ as√∆y02 + (2πRNcycle |y0′ + ∆y0′ |)2 ≪ DHG(3)

where ∆y0 is the deviation of the initial vertical position, y0′ is the centroid of
the distribution of the muon vertical tilt angle, ∆y0′ is the dispersion of thevertical tilt angle, and DHG is the aperture of the muon stored region in thevertical direction (∼ half gap of the storage ring).
Tn

43.8 MeV

106 MeV

211 MeV

317 MeV

γ
√

2

2

3

4

pµ

106

183

299

409

B[T]

R[m]

∆Tcycle

0.25

1.41

41.8 ns

0.5

0.70

20.9 ns

1.0

0.35

10.4 ns

1.0

0.61

14.7 ns

1.5

0.41

9.8 ns3.0

0.20

4.9 ns1.0

1.00

22.2 ns

1.5

0.66

14.8 ns

3.0

0.33

7.4 ns1.0

1.36

29.5 ns

1.5

0.91

19.7 ns

3.0

0.45

9.8 nsNlife

Nlife γτµ′∆ylimit5

15.5 µs15.2 µrad522.0 µs

8.8 µrad5

33.0 µs

5.4 µrad5

43.9 µs

3.9 µrad

Table 1
Several combinations of muon energies and the magnetic ﬁelds of the storage ring tocompute i) time interval of the muon per cycle ∆Tcycle , ii) observation time window′, assuming ﬁve
Nlife γτµ , and iii) upper limit of the vertical angular divergence ∆ylimit
times of the dilated muon life time for the positron observation window (Nlife = 5)
to cover 99.3% of the decay, and the half aperture of the storage ring as DHG = 5cm.

3Therefore, if one can realize ∆y0 ≪ DHG and y0′ ≪ ∆y0′ , then the Eq. (3) canbe simpliﬁed as|∆y0′ | ≪

DHG
1 DHG
=
.
2πRNcycle
βγ Nlif e cτµ(4)Thus, it is essential to minimize the vertical angular divergence as small as
possible to switch oﬀ the vertical focusing ﬁeld. Unlike the conventional beam
optics, the beam emittance ∆y0 · ∆y0′ has second importance.
The timing requirement is another severe condition. The time interval of themuon per cycle can be written as∆Tcycle =

2πR
βc

(

= 2πγ

)

mµ
,
eB

(5)

The time width of the muon bunch should be well below than this time interval
to separate and determine the number of cycle of the muon in the ring without
ambiguity.
Let us tabulate those number described in this section around the muon energy region of interest. Table 3.4 shows the summary of relevant numbers byassuming ﬁve times of the dilated muon life time for the positron observation
window (Nlife = 5) to cover 99.3% of the decay, and the half aperture of thestorage ring as DHG = 5 cm.

3

The muon source

The most promising technique to obtain the extremely good emittance muon
beam at present is the utilization of muonium laser ionization as a muonsource, and re-acceleration of the muon.

3.1 Laser ionization method to generate muon at rest

It is known that the almost at-rest muonium can be produced into vacuum
nearby the muon stopping target rather eﬃciently (∼ few %) using a thintungsten foil. When a positive muon stop in the foil, it forms muonium in the
foil promptly, and the muonium can evaporate from the foil into vacuum. If
one ionizes the muonium by an intense laser irradiation, then one can obtain
positive muons close to at rest. By accelerating those muons, it is demonstrated that one can have sharp muon beam at RIKEN-RAL [5] starting from4the double-pulsed muon beam (300 nsec apart) delivered by ISIS. This slow
muon source has been developed at RIKEN-RAL to open a new era of theµSR, a research method of magnetic property for condensed matter physics
using muon as a micro-probe, for surface property, for micro-target, and forspintronics. The best muon yield at RIKEN-RAL is observed at the tungsten
temperature at 2100K, by shooting the laser beam at about 1 cm apart fromthe tungsten foil, and 600 nsec after the muon injection to the foil.
To know the achievable vertical angular divergence starting from laser ionized
muon, we need to know muonium motion in vacuum. Let us evaluate the initial
kinetic motion of the laser ionized muon. If we assume that the muon havesame kinetic energy of the muonium at 2100K, then the kinetic energy Ts canbe written as

Ts ≈〈px 〉2 + 〈py 〉2 + 〈pz 〉2
3〈ps 〉2=∼ kT = 0.18 eV,
2mµ
2mµ

(6)

where ps is the mean muonium momentum in one direction, k (= 8.617 × 10−5
eV K−1 ) is the Boltzmann constant, and T is the temperature. The muonium
at this kinetic energy will move in the velocity〈vs 〉 ≈

〈ps 〉6 mm
∼ 1.0 × 104 m/s =,mµ600 nsec(7)

which is almost consistent with the experimental condition described above.
In the RIKEN-RAL conﬁguration, operated laser cross section is ± 5 mm
(in height) × ± 1 mm (in width) shooting parallel to the tungsten foil inthe distance of 1 cm. Thus the eﬀective vertical motion of the muonium is
expected to be〈vs.v.eﬀ 〉 ≈ 〈∆θv.eﬀ 〉 · 〈vs 〉 = √5
102 + 52

× 〈vs 〉 = 4.5 × 103 m/s,

(8)

If one assume that the vertical motion of the muonium is same as the transverse motion, then it will move 10 cm within 22 µs (cf. Table 1 of Nlife γτµ =22 µs), which is slightly larger than the reasonable gap size DHG . Therefore,
one need to reduce the vertical motion for the precise measurement.
Is there any alternative muonium source at lower temperature? It is also reported that one can use SiO2 powder or aerogel as target materials. In bothmaterials, an eﬃcient muonium emission is observed even at the room temperature. Thus we obtain the vertical motion of the muonium from SiO2
5

√

2kT
〈vs.v.eﬀ 〉 ≈ c · 〈∆θv.eﬀ 〉 · √= 1.7 × 103 m/s,
2
3mµ c

(9)

which is factor 2.6 smaller than that of the hot tungsten foil so the muon canbe stored in the storage ring fairly well.
Actually, if one can accelerate the laser ionized muon up to pµ without enlarging the vertical momentum, then one can achieve the vertical angulardivergence
√
′

∆y ≈

√

2
·
3

kT mµ
pµ

1
· 〈∆θv.eﬀ 〉 =
βγ

√

√
2
kT
· 〈∆θv.eﬀ 〉.
·√
3
mµ

(10)

Both equations (4) and (10) are inversely proportional to βγ, so that the
muon-beam conﬁnement condition independent to the energy of the muon
re-acceleration. The diﬃculty depends only on the temperature of the muonium, and hence muonium generation from SiO2 is the key for the precise
measurement.
It should be noted that the eﬃciency of the muonium emission from SiO2
powder (or aerogel) drops after few days of operation in vacuum, so the periodical replacement of the muon stopping target is required. It is also true thatthe powder handling as a muon stopping target is quite diﬃcult, because it
is impossible to make a thin ﬁlm in vertical direction so the detailed R&D isneeded. The momentum transfer caused by the laser ionization is in the orderof 10 eV/c, while the momentum of the thermal muonium is in the order of
0.1 ∼ 1 keV/c, so the eﬀect is negligible.

3.2 Achieved yield and polarization of laser ionized muon at RIKEN-RAL

Table 3.4 summarizes the presently available laser ionized muon beam at
RIKEN-RAL. The ionization method has been developed at the RIKEN-RAL
Muon Facility in UK [5]. The 800 MeV proton beam (160 kW beam power
and 50 Hz cycle repetition) of ISIS at RAL hits the graphite target of 10 mmthickness and produce pions. Some of the produced pions are stopped near thetarget surface and the muons from the pion decay emerge from the production
target with energy of 4 MeV (so called “surface muon”). Those muons havetheir spins polarized nearly 100 % along its motion. The muons are collectedwith a solid angle of around 40 msr and are transported by the conventional
beam channel. The muons are stopped and thermalized in a tungsten target (45 µm thick) set in a vacuum and heated up to 2100 K. Some of themuons diﬀuse to the surface and are thermionically emitted to the vacuum as6

muonium atoms with a probability of around 1 %. The thermal muonium isresonantly ionized through a two-photon 1S → 2P → unbound process to gain
free muons with mean kinetic energy of only 0.2 eV. The laser timing is set
about 400 ns after the muon pulse to maximize the overlap of the laser beamwith the cloud of thermal muonium (muon diﬀusion time in tungsten, velocityof thermal muonium) formed in vacuum near the tungsten foil surface. The
laser is irradiated only at 25 Hz (once every other muon pulse) due to theexisting laser speciﬁcation. It should be noted that half of the initial muonpolarization is lost when the muon makes a muonium. The low energy µ+ are
extracted by a electrical ﬁeld and then accelerated to 9.0 keV. The ion transport optics consisting of several electrostatic quadrupoles, a bending magnet
and an electrostatic deﬂector send the muon to a sample chamber, where a
microchannel plate detector as well as a µSR setup was placed to check themuon yield. Finally about 15 µ+ /s are observed at the sample chamber.
We estimate that around 20000 muons are stopped in the tungsten foil per
pulse. Among those about 1 % (200 /pulse) would be emitted as thermal
muonium. Since about half are lost during low energy µ+ transportation (muondecay etc), the ionization eﬃciency by the present laser system is estimated tobe somewhere less than 1 %. This could be limited by the small overlap of thelaser beam and the thermal muonium cloud and also by the still insuﬃcientlaser beam power.
ISIS beam power

160 kW

ISIS proton beam frequency

50 Hz

pulsed laser irradiation frequency

25 Hz

conventional beam chnnel of? msrmuon yield on tungsten target∼ 2 × 104 per pulse

tungsten target temperature
122 nm pulse laser power at present
yield of accelerated slow muonpolarization of slow muon

2100 K∼ 0.5 µW
∼ 20 Hz50 %Table 2Presently available laser ionized muon at RIKEN-RAL

3.3 New laser system for high intensity ionized muon

One of the key technology brake through needed to realize precise measurement is the substantial improvement of the Lyman alpha laser intensity (λ =122 nm) by at least two order of magnitude.
7

The Lyman alpha laser can be generated in the Kr gas cell by using parallel
beam of 212.55 nm and 846 nm lasers.

DFB currentdriven Laser
Diode (LD)LD excited
Nd:GdVO40.1mJ

laser multiAmp system

5th harmonic
1J

fiber MOPA
system212.55 nm
100 mJ

122 nm
50 µJ
Kr gas cell815-855 nm
DFB current100 mJ
LD
exciteddriven Laser
Cr:LISAF
Diode (LD)MOPA
systemFig. 1. Conceptual design diagram for new laser system

To be discussed:− New scheme of the laser systemlaser power improvement of 100 times
laser pulse width 300 ps− asymmetry recovery with 0.2T axial (longitudinal) ﬁeld
− capillary focusing

3.4 Expected yield of laser ionized muon source

4Eﬀective asymmetry Aeﬀ

In the case of the low energy muon below γ ∼ 4, the momentum measurementof the decay positron by the position sensitive device is more eﬀective than
the total energy measurement by the calorimetric measurement. Thus, one
can the trace back and reconstruct kinematics in the center of mass frame(CM ), so let us discuss in CM of the muon for the simplicity. The diﬀerential
probability for the positron at a normalized energy ηcm = Ecm /Emax (Emax =swith respect to the muon spin is [4]52.8 MeV) and at an angle θcm

1dP (ηcm , θcm )s] with=ncm (ηcm )[1 − αcm (ηcm ) cos θcm
dηdΩcm4π
8(11)J-PARC beam power

1 MW

J-PARC proton beam frequency

25 Hz

pulsed laser irradiation frequency

25 Hz

all solenoid beam chnnel of? msrmuon yield on tungsten target∼ 6 × 106 per pulse

SiO2 target temperature

300 K
×2

muonium emission eﬃciency of SiO2

× 100

laser power
capillary focusing

×3

yield of slow muon

∼ 3.6 MHz∼ 1.1 ×1014 / year

polarization of slow muon

100 %Table 3
Expected yield of laser ionized muon at J-PARC MSL
2ncm (ηcm ) = 2ηcm
(3 − 2ηcm ) and αcm (ηcm ) =2ηcm − 1.
3 − 2ηcm(12)

When one accumulate the decay positrons whose energy are greater than Ecm ,then the eﬀective asymmetry can be written as
∫1

Aeﬀ ≈

th
ncm (ηcm )αcm (ηcm )dηcm /Ncm,

(13)ncm (ηcm )dηcm ,

(14)

th
ηcm

where
∫1
th
Ncm
=
th
ηcm

s
by neglecting the asymmetry smearing eﬀect caused by the cos√ θcm terms. In
the same approximation, the ﬁgure-of-merit estimated by Aeﬀ Ncm as shownin Fig. 2.
th∼ 0.7 where
As shown in the ﬁgure, F OM has maximum at around ηcm
eﬀective asymmetry is Aef f ∼ 0.6 and the eﬃciency of the energy window for
the decay positron is N th /N all ∼ 0.5 (N all ≡ 1). The achievable F OM andmean asymmetry is higher than in the case case of the measurement using
higher muon energy at the magic momentum (γ ∼ 30 where Aeﬀ ≈ 0.4 [2]).

9

1.0
th

N =

n(η)dη

FOM

0.8

0.6

0.4

Aeff
0.2

|αcm|
0.0
0.0

0.2

0.4

th
ηcm

0.6

0.8

1.0

Fig. 2. Eﬀective muon ratio N th beyond energy threshold, asymmetry Aef f , andﬁgure-of-merit F OM (arbitrary scale) in the center of mass frame are shown. Maxith ≈ 0.74. For the comparison with the following
mum F OM is realized at around ηcm
ﬁgures, ηcm regions are hatched in diﬀerent colors. Absolute asymmetry corresponding to the threshold energy is also given.
th
Therefore, one should design the experimental setup to cover ηcm> 0.7 withlarge solid angle acceptance to achieve best accuracy.

5

Expression of the muon (g−2) measurement precision

Let us ﬁrst discuss statistical error. The aµ can be calculated as

aµ =

ωµ
aµ
=
,
(1 + aµ ) − aµ
ωL − ωµ(15)

where ωL is the Larmor spin precession rate of the muon. Thus, the precisionof (g−2) can be written as
√
∆aµ
1
∆ωa
∆ωa
2
(
)
=
≈
=
√
eB
aµ
1 − ωa /ωL ωaωaγτµ maµ Aeﬀ ξ Ne
µ
√
1
2
1
=
·
,
·√
e
τµ ( mµ )aµ Aeﬀ ξγB
Ne
10

(16)

where A is the mean asymmetry, ξ is the muon polarization, and Ne is thetotal number of the recorded positron [2], so the accuracy is inversely proportional to γ, the magnetic ﬁeld B of the storage ring, mean asymmetry A, andmuon polarization ξ. If we measure the magnetic ﬁeld in the unit of tesla, theequation can be written as∆aµ11≈ 0.65·√.
aµ
Aeﬀ ξγB [T ]
Ne

(17)

For simplicity, let’s compute number of muons needed to achieve x ppm accuracy of aµ . From Eq. (17), number of required muon Nµ can be writtenas(Nµ ≈ NeN th ∆Ω·
N all Ω)−1(

≈ 4.2 × 10

11

1
xAeﬀ ξγB)2 (N th ∆Ω·
N all Ω

)−1

(18)
,

where ∆Ω/Ω is the acceptance of the decay positron, N all is the number of
positrons without threshold (N all ≡ 1), and B is in tesla.
√
If we assume ξ = 1, γ = 2 at B = 1 T, Aeﬀ = 0.6, N th /N all = 0.5, ∆Ω/Ω =10 %, and all the muon source improvements described in the previous sections
will work ﬁne, then 1 ppm accuracy can be achieved in about one month.
Therefore, conﬁrmation of the previous work might be done in a reasonable
time range, if one can design an eﬃcient experimental setup at such an low
energy.
On the other hand, if one aim at 0.1 ppm, then it will take more than 10 years
to accumulate required statistics, which is not realistic. Therefore, larger muon
γ or higher magnetic ﬁeld B will be required to improve previous experimental
accuracy substantially. From the present limit of the magnetic ﬁeld, it would
be more realistic to accelerate the muon more.

6

Conceptual design of the experimental setup

The momentum of the positron in the laboratory frame pLab for ηcm ≫ me /mµmwith respect to the muon motion can be written asat an angle θcm

pLab =mµm)
ηcm γ (1 + β cos θcm
2

and

11

cos θLab =m
β + cos θcm.
m
1 + β cos θcm(19)From the equation, it is clear that the positron emission angle in the Lab.
frame θLab only depends on the muon emission angle in the CM frame formgiven β. On the other hand, its momentum depends both on θcm
and ηcm , and
it is proportional to ηcm for given θLab .
100

pr (MeV/c)

γ=3

pµ
0

-100

0

100

200

300

pz (MeV/c)

Fig. 3. Positron momentum in the Lab. frame for the muon at γ = 3 for ηcm = 1.
m = 60 ∼ 120 degree (upper hatched
The hatched region is for left hand decay of θcm
mregion) and right hand decay of θcm = −60 ∼ −120 degree (lower hatched region).

As it is discussed in the previous section, higher γ is preferable to achieve thehigher precision. For the higher γ, the muon re-acceleration cost will increase
rapidly, though let us discuss in the case of γ = 3 to realize the higher precision
in a short running period.
Figure 3 shows an example of the positron highest momentum (ηcm = 1)
for the direction with respect to the muon motion at the muon momentum
at γ = 3. As shown in the ﬁgure, transverse momentum is not negligibly
small at this momentum, and the positron momentum largely depends on theemission angle. Thus, to detect high momentum positron component ηcm > 0.7
eﬃciently, it is quite important to know the positron trajectories as a functionof the emission angle.
Figure 4 shows the decay positron trajectories at the highest momentum in
the CM frame (ηcm = 1) in a step of 10 degrees of the frame. The ﬁgure scalesin terms of the magnetic ﬁeld. The diameter of the muon central orbit is 0.66
m at the magnetic ﬁeld of 3 T as it is indicated in the ﬁgure. As shown in theﬁgure, most of the trajectories come inside of the ring except for the forward
and backward decay.
One of the most attractive idea is to place the particle tracking detector at the12mθcm = 180 deg.

left-hand decay

right-hand decay0mθcm = -180 deg.

θcm = -120m

θcm = 120m

-0.2muon central orbit-0.4θcm = -60m

θcm = 60m

-0.6

B = 3Tmθcm = 0 deg.
-0.2m

θcm = 0 deg.
0-0.2

0.2 (m)00.2 (m)Fig. 4. Positron trajectories in the Lab. frame for the muon at γ = 3 for ηcm = 1.
The decay positron trajectories are plotted in a step of 10 degrees for the left-handdecay (left panel) and for the right (right panel). Uniform magnetic ﬁeld of 3T isassumed.
mgreen hatched region so as to have wide angular coverage for θcm. One can use
highly segmented position detector to measure the positron momentum andto reduce the pileup eﬀect, which is one of the major source of the systematic
error of the g-2 measurement.

The positron tracking gives another advantage. A muon rack has ﬁve kinematical degree-of-freedom (DOF ), and a positron introduce another four kinematical DOF at the muon decay. On the other hand, only ﬁve kinematicalparameters of a positron track can be obtained from the tracking, so the parent muon track cannot be deduced only from one positron. However, the muon
central orbit of each muon injection can be deduced rather precisely, becausemthe positrons decays in 60 < |θcm
| < 120 degree crosses muon central orbit
with ﬁnite angle as shown in Fig. 3.
mThere is another good feature. The sign of the θcm
can be easily determinedfrom the tracing back of the trajectory. Thus, one can extract only asymmetry
part of the time spectrum nasym (t) as

nasym (t) =Nlef t (t) − ζNright (t),Nlef t (t) + ζNright (t)(20)

where Nlef t is the number of decay positron in the left hand decay, Nright isthat for right hand decay, and ζ is the relative eﬃciency diﬀerence between
left- and right- hand decay positrons. It is clear that the eﬀect of the muon
decay and muon escaping loss from the ring are canceled out so it is insensitive
to both eﬀect, which could a possible source of the systematic error. Thus the13

time spectrum of the positron asymmetry nasym could be ﬁtted in a simple
formula asnasym (t) = 2Af it cos(ωaf it t + φf it ) ,(21)with the minimum ﬁtting parameters of three, namely Af it is for the asymmetry amplitude, ωaf it is for the muon g − 2 precession, and φf it is for the
initial phase of the muon spin. The eﬃciency diﬀerence between left and rightpositrons ζ should also deﬁned by the data.
On the other hand, the sum of both spectra can be represented as
N0f ittnsum (t) = Nlef t (t) + ζNright (t) = f it (1 − floss (t, T )) exp(− f it ) , (22)τµ
τµwhere floss (t, T ) represent for the escaping loss of the muon from the ring. If
one can evaluate the floss (t, T ) and control its systematic error small enough,then it is also feasible to improve the accuracy of the muon lifetime measurement.

0

right-hand decay

left-hand decay

-0.2

-0.4

-0.6

-0.2

0

-0.2

0.2 (m)00.2 (m)Fig. 5. Positron trajectories in the Lab. frame for the muon at γ = 3. The decay
positron trajectories are plotted in a step of ∆ηcm = 0.1 for the left-hand decay (leftpanel) and for the right (right panel). Uniform magnetic ﬁeld of 3T is assumed.

To check whether we have suﬃcient energy acceptance by the suggested detector arrangement, the positron trajectories are plotted again by changing ηcm asshown in Fig. 5. It is clear that we have enough energy coverage 0.7 < ηcm < 1
for both left and right hand decay.
To evaluate the solid angle acceptance, let us assume that we have same ver14

tical angular coverage of ±30 degrees. 2 The rough solid angle for both side
can be simply given as∆Ω ≈ 4πs
1 − cos ∆θcm
∼ 4π · 0.07
2(23)

so that we have about 14 % solid angle coverage in total.
s
In Eq. (13), we neglected the asymmetry reduction caused by the cos θcm
terms. The reduction of the eﬀective asymmetry caused by this term can alsos
be given in a same Eq. (1 − cos ∆θcm)/2, which is 7 % asymmetry reduction.
Fortunately, this eﬀect is not very large.

Note that the present conceptual experimental setup design is only applicableto the muon at γ = 3. This is because the design deeply depends on thepositron boost in the Lab. frame. The measurement at lower energy is morediﬃcult for weaker positron boost.

7

Running time to achieve improved precision

In the present conceptual design, 0.1 ppm accuracy can be achieved by Nµ ∼2.4 × 1013 for Aeﬀ = 0.55, ∆Ω/Ω = 14% and ξ = 1 at γ = 3 and B = 3 T.
This statistical precision can be achieved within 0.25 year at J-PARC 3 GeV
PS, if one can improve the muon source as it is described. Thus it is feasible
to achieve this accuracy quite easy at this condition. If one aiming to realize
the accuracy within a year, there is a safety margin of factor four.
The realization of the experimental setup in a high magnetic ﬁeld of 3 T is
quite diﬃcult in several reasons. We need a very fast kicker to inject a muonbunch into a designed muon central orbit. We also need highly segmented
detector system inside of the ring. We could resolve both diﬃculties quite
easily if we reduce the magnetic ﬁeld. However, the required total muon yield
is inversely proportional to B 2 , and the this is the reason why the atgument
based on the high magnetic ﬁeld. On the other hand, if one wish to measuremuon electric dipole component in the same setup, the required electric ﬁeld
The vertical angular coverage beyond ±30 degrees is extremely diﬃcult. However, those positrons carry weaker information of the spin precession caused by aµ ,because the spin precession due to aµ is in horizontal plane. On the other hand, if
muon have ﬁnite electric dipole moment (EDM), then the spin precession due to
EDM incline the spin precession plane out of the horizontal one slightly. For the
dedicative EDM measurement, one need another conﬁguration of the experimental
setup. Because the sensitivity beyond standard model by muon is proportional tomµ /me for EDM and (mµ /me )2 for g-2, so that we neglect EDM eﬀect in this paper.

2

15

to freeze out the muon spin rotation ωa become bigger for higher magnetic
ﬁeld.
The situation of the muon re-acceleration is in similar situation. The reacceleration cost largely depends on the ﬁnal energy γ (or the kinetic energyTµ ). Thus, we need to ﬁnd out the best solution to achieve this accuracy withina reasonable time range. Therefore, the ﬁnal setup needed to be compromised
based on the detailed design works and simulations.

8

The systematic error using laser ionized muon

The major source of the systematic error in the previous work will be substantially reduced in the present setup. There is no CBO by deﬁnition, andfree from electric ﬁeld and its instability. There is no muon loss comes frombeam instability. Remaining muon loss is the vertical escaping, which can be
reduced by the scraper before the muon acceleration. In the re-accelerated
muon beam, there can hardly be no contaminating pions. So there is no background even just after the muon injection to the storage ring, and hence one
can start measure decay positrons right after the muon injection.
The residual sources of the systematic errors also can be canceled by computing left-right asymmetry ratio described as Eq. 21. The pileup eﬀect will also
be reduce if one use highly segmented detector, which is enabled at the lower
momentum because one can avoid calorimetric measurement of EM shower.
Thus almost of all sources of the systematic errors are either vanish or expected
to be negligibly small compared to the previous experiments.
One of the diﬃcult point could be how to improve the magnetic ﬁeld measurement. The present precision would be already good enough, but it would
be better to improve more. From Eq. 24, aµ can also be written asaµ =ωµ /ωpR=,
ωL /ωp − ωµ /ωp
λ−R

(24)

where ωp is the free-proton precession frequency. One can directly measurethe ωp in the same orbit as muon, if one injects polarized proton into the ring
and analyze the spin motion by the polarimeter located outside of the ring
[whose idea is this?]. One can measure the muon-to-proton magnetic moment
ratio λ = ωL /ωp very precisely, measuring muonium hyperﬁne level structureusing high density intense muonium.
16

9

Summary

As described, there are many advantages to achieve better precision comparedto the previous experiment, which can be summarized as:• no pion background at muon injection,• no electric ﬁeld for the muon beam conﬁnement,• free from magic momentum,• free from coherent betatron oscillation,• minimum muon loss due to the beam instability,• highly segmented position sensitive detecters available,• minimum pileup eﬀect expected,• capable of muon orbit determination bunch-by-bunch, and• simple ﬁtting function with only four parameters, Af it , ωaf it , φf it and ζ.
High statistics to achieve the required accuracy, one need R&D is needed for:• intensity improvement of the muonium ionization laser by factor 100,
• target development to generate muonium at room temperature, and• polarization recovery applying axial magnetic ﬁeld by decoupling µ-e spin.
Although, there is no extremely serious diﬃculty in principle as father as we
can expect at the moment.
The most crucial part of the development will be the muon source. If one cansuccessfully develop the muonium source, one can also improve the precisionof the muonium hyperﬁne splitting, which also need for the better g-2 determination, quite easily. If one can operate the muonium generation target well
below room temperature, then one can also improve the accuracy of the muon
lifetime measurement quite easily.
We demonstrated that it is feasible to achieve 0.1 ppm accuracy of muon g-2
within few months at γ = 3 and B = 3 T at J-PARC MLF. The realizationof re-accerelation and the experimental setup at 3T would be expensive anddiﬃcult to realize, so the further optimization will be required.
The realization of high intensity laser ionized muon, it will become a extremely
useful probe for the condensed mater physics. The intensive µSR studies will
be started even at RIKEN-RAL with the current proton beam power of 140
kW. At last, we would like to refer that this is a big step towards the realizationof the muon collider in future.
17

References[1] G.W. Bennett et al., Phys. Rev. D 73 (2006) 072003.
[2] G.W. Bennett et al., Nucl. Instr. Meth. A 579 (2007) 1096 - 1116.
[3] F. J.M. Farley et al., Phys. Rev. Lett. 93 (2004) 052001.
[4] E. J. Konopinski, Annu. Rev. Nucl. Sci. 9 (1959) 99.
[5] P. Bakule et al, Nucl. Instr. Meth. B 266 (2008) 335 - 346.

18

Addendum

Recently submitted g-2 proposal to J-PARC“ An Experimental Proposal on a New Measurement of the Muon

Anomalous Magnetic Moment g−2 and Electric Dipole Moment at
J-PARC ”
submitted on 18/12/2009

An Experimental Proposal on a
New Measurement of the Muon Anomalous Magnetic Momentg − 2 and Electric Dipole Moment at J-PARC
M. Aoki,9 P. Bakule,11 B. Bassalleck,8 G. Beer,22 A. Deshpande,17 S. Eidelman,3
D. E. Fields,8 M. Finger,5 M. Finger Jr.,5 Y. Fujirawa,19, 14 S. Hirota,19, 6 H. Iinuma,6
M. Ikegami,6 K. Ishida,14 M. Iwasaki,14, ∗ T. Kamitani,6 Y. Kamiya,6 S. Komamiya,19
K. Koseki,6 Y. Kuno,9 O. Luchev,13 G. Marshall,21 Y. Matsuda,20 T. Matsuzaki,14
T. Mibe,6 K. Midorikawa,13 S. Mihara,6 J. Murata,16, 14 W.M. Morse,2 R. Muto,6
K. Nagamine,4, 12, 6 T. Naito,6 H. Nakayama,6 M. Naruki,6 H. Nishiguchi,6 M. Nio,15
D. Nomura,6 H. Noumi,10 T. Ogawa,13 T. Ogitsu,6 K. Ohishi,14 K. Oide,6 A. Olin,21, 22
N. Saito,6, 19, † N.F. Saito,13 Y. Sakemi,18 K. Sasaki,6 O. Sasaki,6 A. Sato,9 Y. Semeritzidis,2B. Shwartz,3 K. Tanaka,6 N. Terunuma,6 D. Tomono,14 T.Toshito,7 V. Vrba,1 S. Wada,13
A. Yamamoto,6 K. Yokoya,6 K. Yokoyama,14 M. Yoshida,6 K. Yoshimura,6 and M. Yoshida9
1

Academy of Science of Czech Republic, Institute ofPhysics, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
2
3

Brookhaven National Laboratory, Upton, NY 11973, USA

Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
4

University of California, Department of Physics

and Astronomy,Riverside, CA 92521-0413, USA
5Charles University, Faculty of Mathematics and Physics,
V Holesovickach 2, CZ-18000 Prague 8, Czech Republic
6

KEK, High Energy Accelerator Research Organization,
1-1, Oho, Tsukuba, Ibaraki, 305-0801, Japan

7

National Institute of Radiological Sciences, 4-9-1,Anagawa,Inage-ku,Chiba,263-8555, Japan
8

University of New Mexico, Albuquerque, New Mexico 87131, USA
9

Graduate School / School of Science, Osaka University,

1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
10

Research Center for Nuclear Physics, Osaka University,
10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

11

STFC, Rutherford Appleton Laboratory, ISIS Muon group & RIKEN-RAL Muon
1

Facility, Building R3, UG16, Chilton,OX11 0QX,Oxfordshire,United Kingdom
12
13

RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Advanced Research Institute, RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan
14

RIKEN Nishina Center for Accelerator Based

Science,2-1 Hirosawa, Wako, Saitama, 351-0198, Japan
15

Theoretical Physics Laboratory, Nishina Center, RIKEN, Wako, Japan 351-0198
16

Physics Department, Rikkyo University, 3-34-1

Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan
17

State University of New York, Department of Physics
and Astronomy,Stony Brook, NY 11794-3800,USA

18

Cyclotron and Radioisotope Center, Tohoku University, Sendai, Miyagi 980-8578, Japan
19

Physics Department, University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

20

Graduate School of Arts and Sciences, the University ofTokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
21
22

TRIUMF, Vancouver, BC, V6T 2A3, Canada

University of Victoria, Finnerty Rd., Victoria, Canada(Dated: December 18, 2009)

Abstract
We propose to measure the anomalous magnetic moment of the positive muon aµ down to thelevel of 0.1 ppm with a novel technique utilizing an ultra-cold muon beam accelerated to 300 MeV/c
and a 66 cm diameter precision magnetic storage ring without focusing field. The beam will bealso useful in measuring the electric dipole moment with the improved sensitivity by three ordersof magnitude. The proposed measurement will provide a rigorous test of the Standard Model ofparticle physics as demonstrated by previous experiments. Our measurement will be complimentary
to the previous measurement that achieved 0.54 ppm accuracy with the “magic“ energy of 3.1 GeV
in a 14 m diameter storage ring. This proposed experiment will have very different systematics
from the previous experiment as well as the Fermilab proposal.

∗

Co-contactperson : Masahiko Iwasaki (masa@riken.jp)†

Contactperson : Naohito Saito (naohito.saito@kek.jp)

2

Contents

I. Overview

6

II. Physics of Muon g − 2

20

A. Experimental Situation

21

B. Theoretical Situation

22

C. Principle of the g − 2 Measurement

22

D. New g − 2 Experiment24

E. The Standard Model Prediction and the hadronic term

25

F. Muon g − 2 in the LHC Era

29

III. Surface Muon Production and Transport

31

A. Proton beam from RCS

31

B. Muon Target and Surface Muon Production

31

C. Surface Muon Yield at J-PARC

33

D. Capture solenoid and muon transport

37

E. Summary

40

IV. Ultra-slow Muon Production

41

A. Overview

41

1. Development of the Low-emittance Muon Beam41

2. Slow muon beam line at RIKEN-RAL muon facility

41

B. Muonium Production

47

1. Muonium Production Target Materials

47

2. Proposed muonium production target search at TRIUMF

48

3. Spatial distribution of muonium in vacuum and overlap with laser

49

C. Laser System

50

D. Spin Polarization

54

1. Muon spin in muonium

54

2. Possible recovery of polarization

55

E. Initial Acceleration

56

1. Cylindrical initial acceleration stage

357

2. Primary design of the initial acceleration

58

F. Summary

59

V. Re-acceleration

61

A. Low β linac

62

B. High β section

69

VI. Muon storage ring magnet and injection

72

A. Injection

72

B. Field configuration and muon trajectory75

C. Kicker

80

D. Possible error fields

84

1. Mechanical disturbances84

2. Electromagnetic disturbances

85

E. Field measurement and its absolute calibration

87

1. Measurement methods for the experiment87

2. Present status

90

F. Cryogenic system

91

VII. Measurement of ωa

93

A. Spin-dependent decay from the V−A theory

93

B. g − 2 precession in the magnetic field

94

C. Lorentz boost

97

D. Energy spectra in the lab frame

98

E. Event selection by lab-energy threshold98

1. Relation between Elab , η and θ∗98th
2. Appropriate Elabfor the maximum FOM

99

F. Confirmation of Geant4100

VIII. Detection of Decay Positron110

A. Requirements

110

B. Silicon tracker

111

C. Absorber/calorimeter

112

4

IX. Readout electronics and DAQ

115

X. Systematic uncertainties

117

XI. Electric dipole moment

122

XII. Schedule and cost

126

XIII. Summary

129

References

130

5

I.

OVERVIEW

1. Muon g − 2 and fundamental physics
The anomalous magnetic moment of the muon is directly sensitive to the electromagnetic, strong, and weak forces and has been calculated to a precision of 0.41 parts per
million from known physics, referred to as the Standard Model. Suggested extensionsof the Standard Model (SM) would modify the muon g − 2 prediction. These include
models with supersymmetry and extra gauge and Higgs interactions. In effect, the SM
prediction of the muon anomalous magnetic moment represents a sum of all known
physics, and a definitive disagreement between experiment and theory signals the presence of new physics beyond the SM. Likewise, the agreement between experiment andtheory would strongly constrain or eliminate many possible extensions of our present
theoretical understanding of the basic forces of nature.
TABLE I: Current theoretical evaluation of the muon g − 2 and its experimental result.
11 658 471.810 (0.016) ×10−10

Kinoshita&Nio [1]15.4 (0.2) ×10−10

Czarnecki et al. [3]LO hadronic

689.4 (4.0) ×10−10

HLMNT prelim.[2]NLO hadronic-9.8 (0.1) ×10−10

HLMNT prelim. [2]light-by-light10.5 (2.6) ×10−10

Prades, de Rafael

QED contribution
Electro-Weak contribution
Hadronic contribution& Vainshtein [4]Theory TOTAL

11 659 177.3 (4.8) ×10−10

Experiment

11 659 208.9 (6.3) ×10−10

world average [5]31.6 (7.9) ×10−104.0σ discrepancy

Exp. − Theory

2. Current Status of Muon g − 2
The advance of the spin direction vs. momentum for the muon in the magnetic fieldof a storage ring is directly proportional to the anomalous magnetic moment, aµ =(g−2)/2. The present experimental value for muon g−2 is from the E821 experiment at
BNL, which achieved a sensitivity of 0.54 parts per million [5]. This measured value, aµ= 0.001165 92080 (54) (33), and the calculated theoretical value, aµ = 0.00116591773
6

(48), disagree by more than 3 standard deviations1 : aµ (expt.) - aµ (theory) = 31.6
(7.9) ×10−10 . The E821 experiment used a 3.1 GeV muon beam stored in a 1.5 T
storage ring of 14 m diameter. The muons were confined in the ring using electrostatic
focusing, employing the ”invention” from the previous CERN experiment of selecting
the muon energy so that the focusing field does not contribute at first order to themuon spin precession in the storage ring, referred to as the ”magic” energy. The
measurement of g − 2 requires two separate measurements to great precision: themuon spin precession frequency, and the magnetic field. We note that although thepresent significant disagreement with the Standard Model from the muon g − 2 is fromthe measurement of a single experiment, E821 carried out the measurement using a”double blind” approach, keeping the measurement of the precession frequency andthat of the field separate until the completion of each analysis. It is worth noting thatthe series of measurements from E821, for each running year, each determined using
the double blind approach, agree.
3. Spin Precession Measurement
The muon produced from pion decay is naturally polarized along the momentum
direction as a result of the V − A weak interaction. The spin orientation, or themagnetic moment will follow its momentum direction, when it is stored in the presence~ and E.
~ The anomalous magnetic moment aµ will provide an additionalof static field, B
precession whose precession vector ω~ a is expressed as"µ¶ ~ ~#
β×E
1e~ − aµ −ω~a = −
aµ B
.
2
mµ
γ −1
c

(1)

Since aµ ∼ α/(2π) ∼ 0.00116, a choice of “magic” energy, γ = 29.4 would reduce the~ Therefore, precision
above formula to an extremely simple form, ω~ a = −e/mµ · aµ B.
~ will provide precision determination of aµ .
measurement of ω
~ a in the precision field B
It should be noted that the same simple formula will be obtained also by eliminating~
the E-fieldin the above formula.
4. A New Experiment
We believe that it is valuable to launch a new experiment to measure this fundamental
1

We present a recent evaluation by K. Hagiwara, R.Liao, A.D. Martin, D. Nomura, and T. Teubner,
presented at Phipsi09 [2].

7quantity with a completely new experimental technique, thus providing a completely
independent measurement. Figure 1 displays the expected sensitivity of this experiment. The goal for this new experiment is to reach 0.1 parts per million sensitivity. We
will discuss the value of this measurement, whether or not new physics is discoveredat the newly started LHC. The discovery of supersymmetry at the LHC, for example,
coupled with the muon g − 2 measurement, provides a sensitive measurement of tanβ,
the ratio of the vacuum expectation values of two Higgs doublets. We emphasize,
however, that the precision measurement of a fundamental quantity like the magnetic
moment of a fundamental particle has its own intrinsic value.

FIG. 1: Newly expected “wiggle plot” from this proposed experiment.

5. Ultra-Cold Muon Beam and Ultra-Precision Magnetic Field
We propose to use the very high intensity of the new J-PARC accelerator to produce,in a sequence of steps, a very cold highly polarized high intensity muon beam, at
320 MeV energy. By using ultra-cold muons, with almost no transverse dispersion,
the muons can be stored for the duration of their lifetime without requiring a focusing
field. Therefore, one is not forced to use the 3.1 GeV ”magic” energy used by E821 at
8

BNL, and we have chosen an energy a factor of ten smaller. Using a 3 T magnet, thestorage ring will be 66 cm in diameter, rather than the 14 m diameter of E821. There
are significant advantages to use a much smaller ring, and significant disadvantages
to use much lower energy. All of the experimental techniques will be different fromE821. The field uniformity of the storage ring should be excellent, following technology
developed for MRI magnets. We are planning a solenoid ring with spiral verticalinjection of tightly bunched muons, kicked into orbit with a magnetic kicker. The
cyclotron period will be 7.4 ns vs. 149 ns for E821. Positrons from muon decays will
spiral in the field and be tracked using an array of radial vanes made up of silicondetectors. For E821, the positrons were detected using calorimeters. The systematic
uncertainty issues for the two experiments will be very different.
6. Laser Ionization of Muonium
The cold muons will be produced by the following sequence. The J-PARC 3 GeV
proton beam, running at 25 Hz, will hit the Muon graphite target in the M2 tunnel atMaterial and Life Science Facility (MLF), and produce pions which will be stopped in
the target. Those π + that stop on the surface of the target decay to 4 MeV µ+ , which
will be collected using a large aperture ”capture” solenoid, and transported using asecond solenoid, to a second target. The transported 4 MeV ”surface muons” will stopin the second target, and form µ+ −electron atoms, muonium (Mu). The Mu atoms
behave like hydrogen atoms and diffuse from the target. It is important that thistarget be at room temperature (300 K ∼ 25 meV) so that the Mu atoms drift slowly.
Silicon aerogel is a likely candidate target to maximize the diffusion. A special new
pulsed laser will ionize the muonium atoms within a few cm of the target, producing
very cold, polarized, muons. A small electric field will begin the acceleration of thecold muons. The polarization of the muons from muonium is naturally 50%.
However, we plan to study the use of a magnetic field to align the Muonium spins
and obtain much higher polarization. The muons are then accelerated by two linacs
to reach 320 MeV.
The expected transverse dispersion of this beam is 10−5 . The expected cold muon
intensity is 4 ×104 /pulse, in a 1 cm diameter spot.
7. R& D for the Ultra-Cold Muon Beam
9

The production of the very high intensity, highly polarized, ultra cold muon beam is amajor challenge. Surface muon signals have already been observed from the M2 Muon
target with flux that is equivalent to or better than RAL-ISIS. The room temperature
muonium production target must be selected, with well-understood characteristics ofthe spacial distribution of the muonium atoms drifting downstream of the target. A
silica-aerogel target is promising. This system is to be studied and optimized in anexperiment at TRIUMF this year (proposed). The required Lyman-α laser, needed toionize the muonium atoms, is designed and a CLBO (CsLiB6 O10 ) crystal is being grown
for this laser at RIKEN. This system is to be tested and optimized in an experiment at
Rutherford Laboratory this year (proposed). The laser requires 100 times the powerof previous lasers at this energy. The purpose of the measurements at Rutherford is
to confirm the anticipated flux of ultra-cold muons, where we expect a linear gain influx with laser power. This must be experimentally demonstrated.
We propose to use linac designs following existing linacs at J-PARC. The linacs used toaccelerate the cold muons are essentially a proton-type linac at lower β using a DriftTube Linac (DTL), and an electron-type linac (disk-loaded type with pitch adjusted
for β) at higher β.
8. Polarized Muon Source
Obtaining as high a polarization as possible is important, with the sensitivity of themeasurement proportional to the square of the muon polarization.
The polarization of the surface muon is nearly 100% as it is naturally polarized frompion decay. The muon picks up an unpolarized electron at muonium protudcion target
to form a muonium (hydrogen-like atom which consists of a muon and an electron).
The polarization of muon is kept when the muon spin and electron spin are parallelin the muonium, while the polarization is lost when they are anti-parallel. Since theproposed laser beam can not distinguish those muonim states, the net polarization of
the ultra-slow muon is 50%.
There is a possibility of further improvement in the polarization by either applying
a longitudinal magnetic field of ∼0.3 T or using a narrow band laser to sweep away
depolarized muons. Studies on such a possibility are necessary, and are in progress.

109. Muon Storage
A study of the muon trajectory for the storage solenoid and the vertical spiral injection
is shown in Figure 2. An actual coil configuration is being studied in collaboration
with a private company as displayed in Figure 3. An anti-Helmholz type kicker with150 ns pulse full width (half-sine) and 1.3 Gauss is being studied to store the muons.
The beam of muons will be in a 2 ns long (full width) bunch with ∆p/p = 10−3 . At
the beginning of the g − 2 measurement, we expect to have 4 ×104 muons, 320 MeV,
6.6 µs lifetime, with a transverse dispersion of 10−5 . In order to match the early andlate muon decay acceptance in the detectors we are considering a very small confining
quadrupole field of field index of order 10−5 .

FIG. 2: Preliminary design of the muon injection into the solenoid storage ring. One eighth of theiron yoke (black and green) is displayed without coils and kickers. The injected muon trajectory
is also shown.

10. Detection of Decay Positrons
Radial vanes of silicon detectors will track the decay positrons. A study for thisdetector is shown in Figure 4, which includes a positron track. Positrons will bemeasured over 5 muon lifetimes, 33 µs. At early time, we will have on average ∼7
positron tracks per 7.4 ns turn. If we have a 50 ns detector pulse width, effectively
11

FIG. 3: One of the examples from the current field calculation studies for the muon storage solenoid
is displayed. The iron yoke is not implemented in these field studies, but simulated with the coils.
Details of the coil configuration for the main field is under study.

there will be ∼50 spiral tracks that should be either reconstructed or rejected as being
compromised by pileup. The pileup requirements are determined by the pileup effecton early to late reconstruction differences for the energy and time of the events. The
detectors and signal widths are being designed to deal with the very high rates at
early times in the measurement. Our present scheme is to use 5 cm × 10 cm silicon
strip detectors with 0.2 mm pitch. The 39 ms time gap between fills would be used to
select hit windows in the detector data, zero-suppressing the rest. Each vane would
consist of two detectors covering an area of 20 cm radially × 40 cm vertically, withtwo dimensions of readout. We plan 32 vanes. The total number of strips would be
96K.
11. Statistics and Required Beam TimeAs mentioned above, we have chosen a lower muon energy for the storage ring for the12

FIG. 4: A muon decay event is displayed together with the preliminary version of the detector
design.

precession measurement. The statistical sensitivity of the frequency measurement isestimated as
δωa
1
1
∼ p
,
ωa
γ Ne+ Pµ2 A2(2)where Ne+ , Pµ , and A denote the number of detected positrons from muon decays,
muon polarization and the analyzing power, respectively. The analyzing power would
depend on the energy cut in the analysis. A 0.1 ppm measurement requires a run of
1×107 seconds with a surface muon flux of 4×108 µ+ /sec. This statistical sensitivity
assumes reaching a muon polarization, Pµ of 0.9. For the minimum polarization of 0.5,
the 0.1 ppm statistical sensitivity requires 4×107 seconds. There is no question that,particularly with completely new production and detector schemes, this measurement
13

will require a significant learning curve. A 0.5 ppm measurement (statistical) will be avery significant initial goal, making a strong impact on the physics, since this statisticalsensitivity is identical to the previous measurement, E821. This step requires 1.6×106
seconds (< a month) of running, for a muon polarization of 0.5.
12. Consideration on Systematics
The identification, control, and measurement of the systematic uncertainties are thecruxes of the muon g−2 measurement. These will be quite different for this experiment,
compared with E821. We introduce this discussion by reproducing the E821 systematic uncertainty tables for the spin precession and magnetic field measurements, withremarks and estimates for this experiment at J-PARC. Table II presents the uncertainties and remarks for the spin precession measurement. The estimates and remarks
for this experiment are based on the current design of the experiment, which is stillin an early stage. We attempt to identify the major issues, and have approaches tocontrol and measure these systematic uncertainties.
TABLE II: Systematic uncertainties for ωa . Details are discussed in Sec. X.
Source of uncertainty

E821-R01 (ppm)[5] This experiment (ppm)

Pileup0.08<0.05(goal)

AGS background

< 0.1

0.0

Lost muons

0.09<<0.09

Timing shifts

< 0.1

<<0.1

E-field, pitch

<0.1

<<0.1

Fitting/binning

<0.1

<<0.1

Coherent Betatron Oscillation

0.07

0.00

Gain changes

0.12<<0.1?

Others––Total (ωa )0.21

0.07 (goal)

13. Precision Measurement of the Magnetic Field
Table III presents the uncertainties from E821 for the magnetic field, and estimates
for this experiment and remarks. The main difference for this experiment is that the14

small storage ring size allows construction of a single magnet structure and local nonuniformity should be at a 1 ppm level, compared a local non-uniformity at a ±50 ppm
level for E821 (across the joints of the 36-sector pole structure of the 14 m diameter
magnet). Even so, E821 achieved an uncertainty in the integrated field (the field thatthe muons saw) of 0.17 ppm. The uncertainty for J-PARC should improve on this.
TABLE III: Systematic uncertainties for B.
Source of errors

E821-R01 (ppm)[5] This experiment (ppm)

Absolute calibration of standard probe

0.05

0.05

Calibration of trolley probes (field mea-

0.09

<0.09

Trolley measurement B0

0.05

<0.05

Interpolation using fixed probes

0.07

<0.07

Uncertainty from muon distribution

0.03

<0.03

Inflector fringe field uncertainty

0.00

0.00

Others0.10–

Total syst. error on ωp

0.17<0.07(goal)

surement on the muon trajectory)

14. Cost Estimate
Here we give a cost estimate. This is very preliminary without engineering designs,
but we will give the basis for each number.
• Surface muon transport
The surface muons produced at the production target will be transported to theexperimental hall for muonium production. The first capture solenoid should
be made radiation resistive. Therefore, the cost estimate is based on the current conceptual design which requires 960 m of minerally insulated coil (MIC).
The super conducting curved solenoid is assumed for further transport line to
maximize the transport efficiency. In addition, the area should be shielded very
carefully. Detailed cost estimate requires more detailed facility design, which isthe next step.
• Laser Ionization of Mu
15

The laser for this experiment is being prepared at RIKEN.
• Muon LINAC
The current design assumes significant overlap in design with the existing J-PARC
LINAC. This approach is expected to reduce the cost of the LINAC.
• Ultra-precision magnet and monitor
The current design of the magnet resulted in the stored energy of ∼30 MJ. We
are working on the reduction of the energy. A major cost driver for the magnet
would be person-power to achieve the required precision. Since the magnet is theheart of the experiment, we will work out the details with the in-house experts.
Consequently, the cost would be minimized.
• Detector
We currently assume a silicon detector for the tracking detector for muon decays.
It would be followed by the absorber/calorimeter.
TABLE IV: Preliminary estimate of the cost of this experiment.
Item

Cost (Oku-yen)

Surface Muon Transport

Facility

Ultra-Cold Muon Source
High-power Laser System

3.0

Initial Acceleration System

0.5

Muon LINAC

15Ultra-precision Magnet
Solenoid

10

Field Monitor

1

Detector System
Silicon Tracker

1.5

Readout Electronics

0.5

TOTAL

32 + Facility

15. Schedule
We intend to start the experiment in 2014. There are four major areas of activities: the16

ultra-cold muon source, the ultra-cold muon beam, the ultra-precision field, and thedetector system. Especially the development of the muon source represents a major
challenge. We plan to perform a test experiment at TRIUMF to optimize the muonium
production target in 2010. In parallel, we plan a test experiment at Rutherford witha newly developed high-power laser system.
The Muon LINAC can be constructed in two years, and be commissioned together
with the source.
We have started a conceptual design of the ultra-precision field employing technology
developed for MRI. We expect to have an engineering drawing in a year so that construction would start in about one year. The measurement scheme with NMR probes
is being developed at this time. A test of this technique at 1.5 T is planned, followed
by a test in a 3 T magnet at National Institute of Radiological Science (NIRS). A
reasonably long commissioning time is allocated for the magnet, since it would be oneof the most time-consuming processes to shim the magnet to sub-ppm level.
The physics production run should be divided into a few phases at least; the first stepwould be to reach the precision of the E821, and then further improvements would
follow.
16. Comparison with BNL-E821, Fermilab Proposal, and This Experiment
There is a proposal submitted to Fermilab PAC [6] to continue the measurement withthe g − 2 ring of BNL-E821. We believe such efforts should continue as well as the
new efforts which we are exploring. Expected precisions and experimental parameters
are summarized in Table V.

17

FIG. 5: Preliminary schedule of the efforts towards the experiment.

TABLE V: Comparison of the previous experiment BNL-E821, Fermilab proposal, and this experiment.
BNL-E821
Muon momentum

Fermilab

This Experiment3.09 GeV/c

0.3 GeV/c

29.3

3

Storage field

B = 1.45 T

B = 3.0 T

Focusing field

Electric Quad.

none/very weakγ

# of detected e+

5.0×109

1.8×1011

1.5×1012# of detected e−

3.6×109−−

0.1 ppm

0.1 ppm

Statistical precision 0.46 ppm

18

†

†

†

†

†

In the following sections we describe in more detail each of the above points, including
some that have not been mentioned above. We are excited about the possibility of measuring
such a fundamental physics observable, with a completely new approach that takes fulladvantage of the new J-PARC accelerator. We believe that this measurement will havelasting impacts.
We request support for the development of this challenging experiment.

19

II.

PHYSICS OF MUON g − 2

The anomalous magnetic moment of the muon, aµ has been measured with ever increasing
precision for more than a half-century and served as a solid testing ground for the Standard
Model of particle physics. aµ is defined from the magnetic moment µ~µ and its spin vector ~s
as
g−2
e~
~s.
(3)
,
where
µ~µ = g22mµThe magnetic moment is parallel to the spin, since spin is the only directionality that anaµ =elementary particle can have. The value of the magnetic moment, equivalently the anomalouspart aµ can be calculated very precisely from the standard model, including electro-weakand QCD corrections. Representative diagrams for these contributions are displayed inFig. 6. Therefore, a precision measurement of aµ can test the total sum of all known physics
contributions directly. If any significant deviation is found, it would suggest an existence of
new physics.

FIG. 6: Representative diagrams contributing to aSMµ . From the left to right: first order QED(Schwinger term), lowest order weak, lowest order hadronic contributions.

Differently from the electron magnetic moment, which is measured to 0.28 ppt(parts-pertrillion) and is used for the most precise determination of the fine structure constant, α,
the muon magnetic moment is more sensitive to new physics thanks to its larger mass by afactor of ∼200. In many extensions of the SM, the sensitivity is proportional to m2l so thatthe muon is more sensitive than an electron by a factor (mµ /me )2 ∼ 4 × 104 . In the case of
the tau lepton, possible new physics contributions can be even larger, however, it is difficult
to store in a ring to measure its magnet moment due to its short life of ∼3×10−13 sec.
Therefore, the muon is in a unique position to probe possible new physics contributions
through precision measurements.
20

A.

Experimental Situation

The most recent experiment to measure aµ and dµ , E821[5] at Brookhaven National
Laboratory (BNL), has measured aµ down to 0.54 ppm2 , and constrained dµ down to ≤1.9 × 10−19 e · cm. The obtained value has been compared to the most updated predictionsof the Standard Model, and aµ exhibits significant deviation of ∼4 sigma [2], while the dµ
limit is still required to be improved to be comparable to the limit from the electron withthe lepton universality. Obviously further clarification is required to conclude the discoveryof new physics contribution for aµ and further improvement is required for dµ .

FIG. 7: “The wiggle plot” from the previous experiment, E821 at BNL. The figure is adopted from
Ref.[5].

2

The experiment measured both positive and negative muons separately to 0.70 ppm. Quoted precision of
0.54 ppm represents the combined results assuming CP T invariance.

21

B.

Theoretical Situation

After the publication from E821, there have been many theoretical activities to investigate the implication of the possible new physics to explain the deviation. For example, apossible effect on charged lepton flavor violation (cLFV) is discussed in Ref.[10]. Furthermore, implications on the electric dipole moment are also discussed in Ref.[11]. These showthat studies of flavor structure in the lepton sector is important, especially in the light ofleptogenesis.

C.

Principle of the g − 2 Measurement

The anomalous part aµ has been determined from the precession frequency ω~ a of the~ Sinceanomalous part of the muon magnetic-moment in the ultra-precision magnetic field, B.
the decay positron tends to be emitted in the spin polarization direction of the muon, theorientation of the magnetic moment can be determined by selecting higher energy positrons.
Figure 7 clearly displays oscillation in the number of such positrons and the oscillation
frequency is |~ωa |.
~ and static electric field E~ is expressed as
The ω~ a in the static magnetic field B"µ¶ ~ ~#
β×Ee~ − aµ − 1aµ B,
ω~a = −
2
mµ
γ −1
c

(4)

and in the presence of the electric dipole moment, an additional rotation would be obtainedaswhere η is related to d~µ as

" Ã
!#
~
e η ~ ~ E
β×B+
,
ω
~η =
mµ 2
c

(5)

e~
d~µ = η
~s.
2mµ

(6)

Experiment will see their vector sum, ω~a +ω
~ η . The Lorentz factor γ can be chosen to cancel
the second term in Eq.(4) to a good precision. Previous precision measurements have been
performed based on this concept namely at the magic momentum, pmagic .
γ
1
magic = 29.3
aµ − 2=0→ pmagic = 3.09 GeV/cγ −1This choice was unavoidable in the previous measurements for following the reasons:22(7)

1. muon beam is produced from pion decays in flight and widely spread in the phasespace.
2. therefore, focusing field is required to keep reasonable fraction of the beam in thestorage ring.
3. the focusing has to be provided electrically; if magnetic, the precession due to aµ would
depend on each muon trajectory.
4. precision control of the electric field in addition to the magnetic field would be challenging. With an approximate cancellation of the electric field by choosing magic γ,
the level of required precision would becomes accessible.
This basic concept of the experiment is so beautiful that the experiment is often referred toas a textbook measurement.
The muon beam as a tertiary beam, however, provided some complications in the previous
experiment. The muon beam was contaminated by residual pions which damage the earlytime of the precession measurements in each injection by providing “flash” into the detector
system. Since the measurement was based on the calorimetry of decay electrons/positrons,
pile-ups of signals originated in “flash” was certainly not welcomed. In addition, muonscannot be fully stored in the ring basically due to its large phase space which originates inthe decay kinematics and beam line acceptance. The coherent betatron oscillation of themuon beam originating in the focusing field provided additional systematics related to themuon orbits.
The previous experimental group, E821, has done their very best to reduce these systematic uncertainties. As a result, their measurement was statistically limited. There isan effort to continue this beautiful experiment at Fermilab by moving the existing muon
storage ring from BNL. We believe that the proposed Fermilab experiment[6] should run
to further improve the precision as a continuation of great efforts made on the experiment.
However, we think it is necessary to launch a new experiment to measure this fundamental
quantity with completely new techniques, and hence with different systematics.

23

D.

New g − 2 Experiment

We propose to measure the muon g − 2 without the focusing field by employing ultracold muon beam, where the transverse momentum dispersion, σ(pT ) is significantly smaller
than its longitudinal momentum, pL . It is easy to see that such a beam can circulate in thestorage ring without the focusing field for the duration of the measurements. Eliminationof the electric field would simplify the precession frequency as
ω~a + ω~η = −e h ~ η ³ ~ ~ ´iβ×B .
aµ B −mµ2(8)Since the rotation axes due to aµ and dµ are orthogonal, separation of these signals should
be possible. As will be described in detail, we aim to measure aµ down to 0.1 ppm, andsearch for an EDM with a sensitivity of 5×10−22 e · cm.
Differently from the beam storage for collider experiments, where the beam has to circulate for millions of turns, muon precession measurements requires only several thousands ofturns because of its short life. For example, we consider the case withσ(pT )∼ 10−5 ,pL(9)with the B = 3 T and p = 300 MeV/c, which corresponds to r = 33.3 cm. Five times its
dilated lifetime corresponds to travel length of 8.3 km, or about 4,000 turns. The beam
spread due to the transverse momentum dispersion in Eq.(9) is 83 mm after 4,000 turns,
which can be easily accommodated in the storage field.
Such an ultra-cold beam can be produced from an ultra-cold muon source, where coldmuons are produced from the resonant laser-ionization of muonium (Mu). The kinetic energyof the Mu is ∼25 meV (momentum∼ 2.3 keV/c), when produced at room temperature. If we
could accelerate them to 300 MeV/c without further increase in the transverse momentum,~ term in Eq. (4) does
the condition (9) is satisfied. Since we have no electric field, the β~ × E
not exist so that any momentum can be “magic” momentum.
While higher momentum is helpful in prolonging the life of muons, lower momentum
is beneficial in reducing the size of the experimental apparatus, and hence the cost of theexperiment. However, the most important benefit emerges from the fact that such a small
magnet for a muon precession measurement can be made in one piece enabling us to control
the field very precisely. Thanks especially to the recent advances in the magneto-resonance24

imaging (MRI) technology, the precision monitoring and control of the magnetic field of∼1 m diameter has reached to 1 ppm precision. This precision can be compared to a local
magnetic-field precision of the previous measurement, ∼100 ppm while the field precision
integrated over full azimuth was better than 1 ppm[5].
As will be described later, the statistical precision of the measurements would depend on
the γ factor as
δω
1
∝ .
ω
γ

(10)

The lower value of γ has to be compensated by larger statistical samples, which we expectfrom the high beam power (∼1 MW) envisioned at J-PARC.

FIG. 8: Newly expected “wiggle plot” from this proposed experiment.

E.

The Standard Model Prediction and the hadronic term

We briefly describe the current status of the standard model prediction. Especially we
discuss the hadronic term evaluation, which represents the largest uncertainty in the prediction.
25

One of the largest contributions to ahad,LO
comes from the 2π final state (about 73%)µmaking high-precision measurements of the corresponding cross section mandatory. CMD2 reported their results on the pion form factor Fπ from 370 to 1380 MeV [18] with asystematic error of 0.6-0.8% below 1 GeV, while SND measured Fπ from 390 to 970 MeV
with a systematic error of 1.3% [19]. KLOE studied Fπ using the method of radiative return√or ISR [20, 21] at 590 < s < 970 MeV with a sample of 3 · 106 events and systematic errorof 0.9% [22]. The |Fπ | values from CMD-2 and SND are in good agreement. The KLOE
data are basically consistent with them with a somewhat different energy dependence. The
contributions to aµ from all three experiments agree. First results on the 2π from BaBarshow a slightly larger cross section [23].
Below 1.4 GeV practically all final states have been measured with consistent results by
the CMD-2 and SND groups in Novosibirsk [24]. Above 1 GeV, various final states withmore than two hadrons were studied by BaBar [25] using the ISR method. They measure forthe first time cross sections of a few new channels and also show that some of the previous
results should be reconsidered.
The results of the ahad,LOevaluation based on Ref. [2] are presented in Table II E forµdifferent energy ranges.
TABLE VI: Updated aµhad,LO according to Ref. [2].
√s, GeVahad,LO, 10−10µmπ − 1.43

603.8 ± 3.1

1.43 − 2.0

34.1 ± 1.5

2.0 − 11.1

41.4 ± 0.9J/ψ, ψ 0

7.9 ± 0.2

Υ(1S − 6S)

0.1 ± 0.0

11.1 − ∞2.1 ± 0.0

Total

689.4 ± 3.6exp ± 1.8radis
It can be seen that due to a higher accuracy of e+ e− data the uncertainty of ahad,LOµ
now much smaller than before [12, 17].
26

The most recent estimate of the next-to-leading order (NLO) hadronic term in [2] gives
ahad,NLO= (−9.8 ± 0.1) · 10−10 . The light-by-light hadronic contribution is estimated onlyµtheoretically [27]. Various predictions range between 80 and 136 (in units of ·10−11 ) with anuncertainty reaching 40 · 10−11 .
Using for the light-by-light term the result (10.5 ± 2.6) · 10−10 [4] and adding all hadronic
contributions, we obtain ahad
= (690.1 ± 4.8) · 10−10 . This result agrees with other estiµmations, e.g.,[32] and its accuracy benefits from the new e+ e− data. Adding all theoretical−10
contributions we obtain ath, i.e. 4.0σ below the experimentalµ = (11659177.3 ± 4.8) · 10

value [2]. Similar deviation is reported by Ref. [32].
How real is a very high accuracy of the leading-order hadronic contribution obtained
above? To a large extent it depends on our understanding of the radiative corrections due
to initial-state radiation and vacuum polarization, and even more important effects of final
state radiation. There is also a question of double counting of the hadronic final states in theleading- and higher-order hadronic terms [29], that of the missing states (e.g., final stateswith neutral particles only).
There is still no explanation for the observed discrepancy between the predictions basedon τ lepton and e+ e− data [17]. The new high-statistics measurement of the two-pion spectral
function by Belle [30] leads to about the same ahad,LO
as before despite some inconsistenciesµ
between Belle and ALEPH. On the other hand, a recent comprehensive analysis of thee+ e− data below 1 GeV and those on the 2π decay of the τ lepton performed in Ref. [31]shows that the two data sets can be reconciled if mixing between the ρ, ω, φ mesons is taken
into account in a consistent way. A recent reevaluation of isospin-breaking effects in [32]somewhat decreases the discrepancy, see Fig. 9. According to it, the difference between theτ and e+ e− -based estimates is (6.8 ± 3.4τ +IB ± 2.9ee ) · 10−10 or 1.5σ only.
Finally, the authors of Ref. [33] performed a reestimation of ahad
after BaBar’s ππ dataµ
−10
(CODATA changed µµ /µp ). The new difference between
and increase of aexp
µ by +0.9 · 10

the experimental value and the theoretical prediction becomes (24.6 ± 8.0) · 10−10 or 3.1σ,
see Fig. 9.
What is the future of this study?

It is clear that it will be extremely difficult toby measuring the cross section of
improve significantly the existing accuracy of ahad,LOµe+ e− annihilation to better than 0.3% as required by future determinations of aµ mentioned
above. One can optimistically expect substantial progress from new high-statistics ISR
27

HMNT (06)JN (09)
Davier et al, τ (09)Davier et al, e+e–

w/o BaBar (09)w/ BaBar (09)

HLMNT prelim. (09)experiment
BNL
BNL (new from shift in λ)
170
SM

aµ

× 10

10

180

190

200

210

– 11659000

FIG. 9: Comparison of aµ from theory and experiment [2].

measurements at KLOE, BaBar and Belle together with the more precise determination of
R below 4-5 GeV from CLEO-c and BES-III. Experiments will start soon at VEPP-2000√
now commissioning, which is a VEPP-2M upgrade up to s=2 GeV with Lmax = 1032
cm−2 s−1 [34]. We can estimate that by 2012 the accuracy of ahad,LOwill be improved fromµ
4.0 · 10−10 by a factor of about 2 and the total error of 3.3 · 10−10 will be limited by the LBL
term (2.6 · 10−10 ).
Let us hope that progress of theory will allow a calculation of ahadfrom first principlesµ(QCD, Lattice). One can mention here a new approach in the QCD instanton model [35] or
calculations on the lattice [36].
In conclusion, it should be emphasized once again that BNL success stimulated significant
progress of e+ e− experiments and related theory. Improvement of e+ e− data led to substanuncertainty. For the first time the accuracy of the theoreticaltial decrease of the ahad,LOµprediction is better than that of the experimental measurement. Future experiments as wellandas development of theory should clarify whether the observed difference between aexp
µ
28

ath
µ is real and what consequences for the Standard Model and for possible New Physics [37]it implies.

F.

Muon g − 2 in the LHC Era

In the end of this section, we briefly mention the importance of this experiment in the
LHC and post-LHC era. One could argue that most of the new physics scenarios can bestudied at LHC. Indeed, the most popular interpretation of the deviation from the Standard
Model observed by the previous experiment is supersymmetry, and it can be studied very
well at LHC, if the energy scale turns out to be appropriate. However, important parametersin the model such as µ and tanβ cannot be determined very well. The muon anomalous
magnetic moment aµ is sensitive to these parameters through following expression;aµ (SUSY) ≈ (sgnµ) 13 × 10−10 tanβ(100GeV 2
).
m̃

(11)

The sensitivity to tanβ is compared to that of LHC experiments in Fig. 10.
However, we emphasize that it represents only one of the possible strengths of this measurement. The precision measurement of the fundamental quantities like magnetic andelectric dipole moments of elementary particles has its own value.

29

FIG. 10: Possible future tanβ determination from the measurement of aµ , assuming that the
MSSM reference point SPS1a is realized [6, 9]. The yellow band is from LHC alone which gives
tanβ LHC−fit = 10.0 ± 4.5 [7, 8]. The darker blue band labelled E821 assumes ∆aµ = (295 ±81) × 10−11 , which comes from the present values for aµ and the Standard-Model contribution,
the lighter blue band labelled FNAL corresponds to ∆aµ = 295(34) × 10−11 , which has similarsensitivity projection to the proposed experiment at J-PARC.

30

III.
A.

SURFACE MUON PRODUCTION AND TRANSPORT
Proton beam from RCS

The J-PARC has been constructed in the south part of the Tokai-JAEA (Japan Atomic
Energy Agency) site, and consists of a 181 MeV LINAC (400 MeV in future), as well as
3 GeV and 50 GeV proton synchrotron rings. About 90% of the 3 GeV, 333 µA (1.0 MW)
beam is sent to the Materials and Life Science facility (MLF) for the production of intense
pulsed neutron and muon beams, while the remaining 10% will be sent to the 50 GeV ringfor further acceleration for the kaon and neutrino physics programs [38] The 3 GeV proton
beam is transported through the beam transport line (the 3NBT line) over a distance ofabout 300 m and focused onto the 24 mm diameter muon target with a spot size that is as
small as possible. The parameters of the 3 GeV proton beam are:1. the number of protons will be 8.3 × 1013 / pulse,
2. the average beam power will be 0.6 MW, while the LINAC energy is runing at 181 MeV,
and will reach 1 MW in the future when the LINAC energy is increased up to 400 MeV(expected in 2014),
3. the repetition rate is 25 Hz, so that each pulse is separated by 40 ms. One proton
pulse consists of two bunches; each with a bunchwidth ∼ 100–120 ns, and separated
by 600 ns, and4. the transverse emittance (²) will be 81π mm mrad (beam core) and 324π mm mrad
(maximum halo).

B.

Muon Target and Surface Muon Production

The MLF consists of the neutron and the muon science facilities which utilizes the 3 GeV,
1 MW, 25 Hz proton beam. In order to reduce the total cost of the project through common
use of the utilities, getting rid of the severe beam dump construction associated with highlevel tritium water handling, and by sharing the beam with the neutron facility, a tandemtarget muon facility was adopted, rather than construct a separate building with our own
proton (1 MW) beam dump, as was the case in KEK-MSL.
31

Through discussion with the neutron science group, the total beam loss induced by inserting the muon production targets is designed to be no more than 10%, which allows MUSE
to install 10 mm and 20 mm thick graphite targets on the beamline upstream of the neutron
target, corresponding to a beam loss of 3.5% and 6.5%, respectively. Detailed calculations
on heat, radiation and duct-streaming in the vicinity of the muon target were performed by
NMTC/JAM and MCNP Monte-Carlo codes [40]. In the case of the 20 mm thick graphitetarget, as much as 3.5 kW is deposited into a 25 mm diameter region of the target throughirradiation of the 3 GeV, 1 MW proton beam.
One possible candidate for the muon production target is a rotating carbon target, which
has been developed at PSI and has been working well for more than ten years [39]. In theend, an edge-cooled non-rotating graphite target was adopted, because of its ease of handlingduring maintenance. In this target, graphite is indirectly cooled by the copper frame, whichsurrounds the graphite. For the graphite material, the isotropic graphite, IG-43 was chosen,
which has a thermal conductivity of 139 W/mK, 1.82 g/cm2 density at 300 K, thermal
expansion of 4.8 ppm/K, Young’s modulus of 10.8 GPa, and Poisson’s ratio of 0.28. In thecopper frame, three turns of cooling pipe, a SUS tube with O.D. 12.7 mm and I.D. 10.7 mm,
are embedded through HIP (Hot Isostatic Press). In order to reduce stress, a titanium buffer
layer of 2 mm is placed between the graphite disk and the copper frame. The copper frame,the 20 mm thick graphite, and the titanium buffer layer are bonded by silver brazing in
vacuum.
Calculations of the heat and stress induced by the heat deposit of the 3.5 kW beamthrough ANSYS demonstrate that the edge-cooled graphite target can be used safely as the
1 MW muon production target. Detailed calculations of the neutron irradiation effect on thethermal conductivity and the thermal stress induced by the proton beam will be reported
elsewhere [41].
One graphite target was installed, from which four sets of the secondary lines are extracted
and extended into two experimental halls (toward the west wing, one decay-surface muonchannel and the super omega channel, and towards the east wing one surface muon channel
and possibly one g − 2 dedicated muon channel).
Muons (µ+ /µ− ), together with neutrinos, are a decay product of pions (π + /π − ). Throughthe pp or pn nuclear reactions between the 3 GeV proton beam and the 20 mm thickgraphite target, π + and π − ) are generated. Experimentally, two kinds of muon beams are32

produced and delivered to experiments; one is the Decay muon beam, and the other isthe Surface Muon beam. Decay muons are obtained from π + /π − decaying in-flight in along superconducting solenoid magnet. Its energy is rather higher at up to ∼165 MeV(250 MeV/c). On the other hand, surface muons are obtained when π + decay after stoppingat the surface of the production target. Therefore they have a lower kinetic energy of around
4 MeV (30 MeV/c), and only positive muons are available.

C.

Surface Muon Yield at J-PARC

A surface muon beam is produced when positive pions are stopped near the surface inside
the production target and decay to muons. The surface muon intensity can be estimatedfrom the number of pions stopped near the surface. Calculation was done utilizing themeasured data of pion production cross section and tracking of pions with GEANT 3.21.
In order to reduce the CPU time, instead of starting from the proton beam, we generated
the pions according to an empirical differential cross section equation (generator) which approximates the data taken for the 2.9 GeV proton on a beryllium target [42]. The generator
equation was also compared with the recent pion production cross section data taken by
HARP collaboration at CERN [43] in Fig. 11 and it was confirmed that the generator reproduces the HARP data reasonably well within 20% for low energy regions below 100 MeV.
We are concerned mostly with pions with energy below 50 MeV, since the range of a 50 MeV
pion is 10 g/cm2 (4.5 cm) in graphite. Higher energy pions are less likely to be stopped in
the target and would have only a small contribution to surface muon production. Even so,
pions up to 500 MeV were handled for completeness.
We assumed that the pion is produced uniformly along the proton beam path in thetarget, since the proton interaction length is much larger than the target thickness. We
also assumed the proton beam distribution is uniform inside the 24 mm diameter in the
production target.
By integrating the generator equation for various angles and for pion energy up to
500 MeV, the total cross section for the positive pion production is about 90 mb. It is
calculated from this cross section that the total yield of positive pions is about 0.010 per
3 GeV proton passing through the 2 cm (4.5 g/cm2 ) thick graphite target. This number isconverted to 2.0 x 1013 positive pions generated for the 333 µA proton intensity at J-PARC.
33

FIG. 11: Comparison of pion production differential cross section used in the pion generator (solidline) and that of HARP data (symbols and dotted lines).

FIG. 12: Target geometry and proton beam size used for the calculation.

About 2 x 107 pions (1/106 of the above number) were generated in the simulation to
study how many of those pions are stopped near the target surface. The energy and angle ofeach pion was generated following the distribution of our pion generator as described above.
The tracking of the pions, pion interaction in the target, their decay to muons, and thetracking of muons were handled by GEANT 3.21.
Only pions that stopped inside the production target near the surface will contributeto the generation of surface muons, so the number and depth distribution of the stopping

34

pions is the basic information. Figure 13 shows the number of π + stopped in each depth
region of 0.1 mm steps in units of Million/sec. The area is limited to inside 24 mm radius.
For example it can be seen that 11,000 M π + are stopped within 0.2 mm (2 bins in the
figure) from the backward surface. The number of pions stopping in the forward surface is
9,500 M and is about 15% smaller. The figure also shows that the number of pions stoppingnear the surface is about half (although not exactly) of those in the middle plane. This isbecause stops in the middle have contributions from pions produced in closeby forward andbackward regions, while those stopping at the forward (backward) end are caused by pionsthat were produced toward the downstream (upstream) only.

FIG. 13: π + stopping distribution in the production target along the beam path. Only pions thatare stopping inside 24 mm radius around the center are included. Units for number of π + are M/s.

Figure 14 shows the horizontal distribution in the target. It is seen that the main contributions are from inside the 24 mm diameter region which is directly hit by the protonbeam. There are also pions stopping outside, which amounts to 40% of those inside.
The stopped pions decay to muons, some of which are emitted into the vacuum region
35

FIG. 14: Horizontal distribution of pions stopping in the production target. Only pions stopped
within 0.2 mm from the backward surface (black) or the forward surface (red) and within ±5 mm
vertically from the center level are plotted (see the dashed square in Fig. 12). The model issymmetric around the proton beam axis, so a similar distribution is expected for the verticaldirection.

if they are within 1 mm of the target surface. The muon energy when it exits the target islower as the pions decay deeper inside. Concerning the pions that stopped within 0.2 mmfrom the surface, if the muon is emitted in the direction normal to the target surface, its
momentum when it exits the target ranges from 27.3 MeV/c to 29.8 MeV/c. The numberof expected surface muons is calculated from the solid angle and momentum acceptance ofthe beamline. We should take into account that the actual beam line sees the production
target surface at an angle. The effective depth contributing to the surface muon production
shrinks by the cosine of the emission angle, namely by 0.71 and 0.50 for muon beamlines
seeing the target at 45 degrees and at 60 degrees, respectively. The suggested g − 2 beamline
is at 60 degrees backword from the proton beam direction.

36

TABLE VII: Comparison of the expected muon beam intensity for three surface muon beamlines
Beamline

D-lineSuperOmega

H-2 line

Extraction Angle (degrees)

forward 60

backward 45

forward 60

Pion stops per mm thickness (106 /s)

47,50055,00047,500Solid Angle (msr)

45

400

130

Momentum Acceptance (MeV/c)27.5-30

25-30

25-30

Maximum depth for usable pion (mm) 0.2 cos(60 degrees) 0.4 cos(45 degrees) 0.4 cos(60 degrees)
Expected Surface Muon Rate (106 /s) 17

390

100

Let’s compare the estimated production rate with the measured surface muon rate recently measured at the MLF Muon D-line, which sees the production target at 60 degrees
backward from the proton beam. The design value of the D-line solid angle is 45 msr andthe momentum acceptance is around 10 % FWHM. The momentum acceptance corresponds
to covering the surface muons made from pions that are stopped within 0.2 mm from thesurface. The fact that the beamline sees the production target at 60 degrees reduces theavailable thickness to 0.1 mm. Thus we expect from our pion production simulation thatthe surface muon beam rate will be 9,500 M/s × (0.1/0.2) × (0.045/4π) = 17 M/s, A recent
measurement has observed the surface muon rate to be around 15 M/s (the measured rate
is normalized to 1 MW proton intensity). Thus it is reasonable to use the above simulation
result for estimating the surface muon intensity at a new channel.
The surface muon intensities for other beamlines are estimated from the simulation ina similar way. We compare the expected number of surface muons for three beamlines inTable VII. Here, we note that we expect the surface intensity of 108 /s at the H-2 beamline.

D.

Capture solenoid and muon transport

The strategy of transporting surface muons to the muonium production target throughthe H-2 beam line is very similar to that of the super omega beam line. Figure 15 shows theschematic picture of the H-2 beam line. Firstly, the produced surface muons are captured by
the solenoids called ”capture solenoids”. Then they are transported along the H-2 tunnel by
the super conducting transport solenoids, and finally focused on the muonium production

37

target.

FIG. 15: Schematic picture of the H-2 beam line.

The capture solenoids in the current design consist of three types of normal conducting
solenoids. The schematic view of the capture solenoids is shown in Fig. 16. The solid angle
acceptance of the capture solenoids is 130 msr, which is determined by the φ250 mm pillow
seal for the beam port located at 555 mm distance from the production target. The capturesolenoids are located very close to the production target and must have high radiation
resistance, thus they are made of mineral insulation cables (MIC).

FIG. 16: Schematic picture of the capture solenoids. The blue lines represent trajectories of
30 MeV/c muons.

The transport solenoids consist of a 30◦ 5-segment curved section and a 4.5-m-long38

straight section. The diameter of the transport solenoids is 400 mm. The typical strength ofthe magnetic field is around 1 T at the center of the solenoids. In the middle of the transportsolenoids the dipole fields in both horizontal and vertical directions will be implemented toalign the beam position on the center of the downstream focusing coils.
Between the capture solenoids and the transport solenoids a gate-valve pillowseal isplaced, which enables us to detach the transport solenoid for maintenance without spoilingthe vacuum upstream. We propose to use a φ250 mm gate valve, and in that case about
10% of the muons will be lost at the gate valve. The total transport rate of the capture andtransport solenoids for the 30 MeV/c surface muons is around 80% including the focusing
section.
The transported muons have to be focused on the muonium production target. We areconsidering using two large superconducting coils. The inner diameter of the coils is 740 mm,
and the interval between the two coils is 800 mm. Achieved beam size in the current design
is σx ' σy ' 15 mm as shown in Fig. 17. We are now optimizing the design of the focusing
section considering following points.
1. Making the beam size smaller is important to maximize the number of muoniums ina region for the laser ionization.
2. The focus length in the current design is 300 mm. It is too short if we have to place
the vacuum disconnecting device between the focus coil and the muonium production
target.
3. Some magnetic field is present at the muonium production target. If this creates aproblem, we will place a field-canceling coil.
The Super Omega beam line in the muon facility plans to use the similar configurationof the surface-muon transport which consists of capture solenoids and transport curvedsolenoids. The capture solenoids have already been constructed and installed in the target
area. The tests of the transport solenoids have been in progress. There is a synergy indevelopment of the beam line components and optics design since the beam line configuration
is similar to that of the proposed experiment. Therefore, time and cost for the R&D for the
beam transport can significantly be saved in collaboration with the muon facility group at
the MLF.
39

FIG. 17: Horizontal and vertical distributions of the muons at the focus point.
E.

Summary

The muon facility has been constructed and operated in the MLF at J-PARC. The proposed experiment utilizes a beam line in the muon facility. The surface-muon intensity fromthe graphite target is expected to be 108 /s at the H-2 beam line. The muon transport utilizes
capture and transport solenoids to transport the surface muon to the muonium production
target.

40IV.

ULTRA-SLOW MUON PRODUCTION

A.

Overview

1.

Development of the Low-emittance Muon Beam

We plan to create muonium atoms using the 4.1 MeV surface muon beam, in a room
temperature muonium production target. The muonium atom will be ionized by a new laser
system to provide the intense ultra-cold ionized muon source required for the proposed g − 2experiment at J-PARC. This plan is based on work at Rutherford Appleton Laboratory(RAL) and developed jointly by KEK, RIKEN [45, 46]. There, we have utilized the factthat thermal muonium with mean kinetic energy of only 0.2 eV can be produced in vacuumwith an efficiency of several per cent by stopping the muon beam in a hot tungsten foil [44]or in SiO2 powder [47]. Then thermal muonium atoms can be ionized by intense laser radiation, leaving very cold polarized bare muons to be re-accelerated. Recently the conversion
from a 4.1 MeV muon beam to a 10 keV muon beam using this laser ionization method
was achieved at the RIKEN-RAL muon facility, with an efficiency of about 3×10−5 [46].
Although the efficiency achieved is as low as other methods to provide slow muons, and thescheme has an extra complication due to the operation of the laser system, it has huge potential. The ionization efficiency of muonium atoms depends only on the laser intensity andcan theoretically be as high as 100%, which would result in a conversion efficiency to slow
muons of a few per cent. The energy spread as well as the divergence of the re-accelerated
beam is far smaller than other methods of producing cold muons.
In following sections, we describe the slow muon beam line at the RIKEN-RAL muon
facility, then describe plans to improve the efficiency of the scheme to match our requirement.

2.

Slow muon beam line at RIKEN-RAL muon facility

The slow muon setup at RIKEN-RAL muon facility consists of a laser system, an ultrahigh vacuum target chamber (for production of muonium atoms), a beam line (slow muon
beam optics).
a.

Muonium target

Figure 18 shows the layout of the target chamber and the slow

muon beam line. The initial muon beam with momentum of 28 MeV/c enters the target
41

FIG. 18: A experimental layout of the slow muon beam line at RIKEN-RAL muon facility

chamber through a 50 µm thick stainless window and a 20 µm thick tungsten foil, and is
stopped inside a 45 µm thick tungsten foil heated by DC current up to 2100 K. The area ofthe tungsten foil is 40 mm (W) × 35 mm (H). The total thickness of the foils is optimized
to stop the largest fraction of the incident muons near the rear surface of the hot tungsten
foil, which results in efficient emission of muonium atoms.
b.

Laser system and ionization of muonium The binding energy of the electron in

muonium is 13.6 eV. The muonium can be efficiently ionized through two-photon resonant
ionization via its 2P state. This requires using two laser beams of different wavelengths.
Firstly, for the strongly allowed transition from the 1S to 2P state, tunable radiation around122 nm (Lyman-α) is required. Then for the ionization from the 2P state, a wavelength
shorter than 366 nm is required.
Figure 19 shows a schematic diagram for the laser system. The 122 nm radiation is generated through a two-photon resonant sum-difference frequency mixing scheme in a phasematched krypton gas. This conversion scheme requires two laser beams; one at 212.55 nm

42

SLM

Nd:YAG laser

Nd:YAG laser

Continuum
Powerlite 9025

Continuum
Powerlite 9025

532 nmSLM OPO +Ti:Sapphire amp.
Continuum Mirage 800

532 nm2-passTi:sapphire
amplifier532 nm4-pass 850 nm
Ti:sapphire
amplifier
SHG

Nd:YAG laser
Continuum
Powerlite 7025

OPO + OPA
laser
Continuum Mirage 3000

FHG

FHG

532 nm212.55 nm

815-855 nm

Nd:YAG laser
Continuum
Powerlite 9025

355 nmAir-conditioned laser cabin

0 eV
355 nm2P

355 nmKr-Ar
gas c e l l

UHV
3 x10-8 hPa

122.09 nm(Lyman- α)1STuneable
VUV
116-123 nm-13.6 eVMu
atoms

Muonium
RIKEN-RAL Port 3 Experimental area

FIG. 19: A schematic diagram for the laser system used at the RIKEN-RAL muon facility

which is matched to a two-photon resonance in Kr, the other at tunable wavelength from
815 nm to 855 nm. This added complication pays off for the following reasons. Firstly it
gives a high conversion efficiency (∼10−4 ), compared to conventional third harmonic generation in gaseous media. Second, the tunability of the output wavelength allows one toionize not only muonium, but also residual hydrogen and deuterium atoms in the target
chamber. This provides a useful tool for testing the beam line. Lastly, the bandwidth ofthe output can be customized by using a properly selected infra-red laser. In this setup, thebandwidth of the output was set to be about 180 GHz in order to cover the Doppler widthof the thermal muonium from the tungsten foil heated to 2,000 K. For the ionization fromthe 2P state, the 355 nm laser beam generated as the third harmonic of a Nd:YAG laser isused.
The most crucial part of the laser system is an optical parametric oscillator (OPO) laser
43

TABLE VIII: Summary of the laser output parameters at RAL
Wavelength

355 nm212.55 nm

815–855 nm

116–123 nm

Pulse duration

10 ns

4 ns

12 ns

∼4 ns

Pulse energy

380 mJ

10-12 mJ (×2)

12 mJ (×2)

∼0.5 µJ (×2)

Bandwidth

30 GHz<2 GHz180 GHz180 GHzpumped by frequency-doubling Nd:YAG lasers, which generate single longitudinal mode
(SLM) 850 nm wavelength. The output is amplified to high pulse energies in Ti:Sapphire
amplifiers, and then converted to 425 nm and then to 212.5 nm by BBO crystals. Highquality of the 850 nm laser beam results in high quality and intensity of 212.5 nm, which inturn increases the pulse energy of the Lyman-α wavelength. The current output parametersof the laser system are summarized in Table VIII.
The laser system is operating at a repetition rate of 25 Hz, not the 50 Hz of the muon
repetition rate at the RIKEN-RAL muon facility. This limitation is due to strong thermal
effects associated with operating high power solid laser systems. Otherwise, the laser system
has good long term stability. Stable 24 h/day operation can be sustained for up to 20 days,
limited mainly by the lifetime of flashlamps used in Nd:YAG lasers.
The two Lyman-α beams are shaped by focusing optics to have a width of only 1–2 mmin the horizontal plane and about 5–8 mm in the vertical plane, and introduced parallel tothe tungsten target at about 5 mm from the target surface. The 355 nm beam is introduced
from the top at 30 degrees from the Lyman-α beams and is shaped to have a width of 3 mm
by 25 mm. All laser beams are retro-reflected to maximize the ionization yield. Since theretro-reflecting mirrors are placed at a distance of 0.5 m, the width of the ionizing pulse iseffectively broadened from 4 ns to 7 ns. The overlap of the Lyman-α and 355 nm beams
defines an interaction region where muonium atoms are ionized. The volume of the region
is approximately 1 cm3 .
c.

Slow muon beam line

The slow muons generated by the laser ionization are initiallyextracted by a low gradient electric field between the tungsten foil held at 9.0 kV and a meshelectrode S1 held at 8.8 kV, separated by 14 mm. S1 is the first of three elements forming
an SOA immersion lens, which re-accelerates the slow muons to 9.0 keV as well as focuses
the muons to a spot of 2.5 mm (FWHM). The muons are then transported by electrostatic44

FIG. 20: A beam profile of 9 keV muons at the sample position

quadrupoles, a bending magnet, and an electrostatic deflector, to the sample position at theend. A position-sensitive microchannel plate detector (Roentdek DLD40) can be mounted
at the sample position to measure the beam properties.
d.

Beam parameters

Figure 20 shows the beam spot of the re-accelerated 9 keVmuon beam at the sample position. While the FWHM diameter of the incident 4 MeV
muon beam is 35 mm, the FWHM diameter of the re-accelerated 9 keV muon beam is about
10 times smaller. The superior quality of the re-accelerated beam can be also seen in its
time-of-flight (TOF) spectrum shown in Fig. 21.
While the incident muon beam has a double-pulse structure consisting of 80 ns pulses
separated by 320 ns, the Gaussian fit to the TOF spectrum yields FWHM pulse duration of
7.5 ns. Compared to the effective width of ionizing pulse of 7 ns, the energy variation of there-accelerated beam is very small. The main source of the energy variation is the width of
the ionization region, which is of the order of 2 mm (FWHM). The difference in electrostatic
potential over the 14 mm gap between the tungsten foil and S1 is kept at 200 V, whichmeans that the 2 mm width of the ionization region contributes to the standard deviation of
the muon energy by ∼13 eV. Taking into account other factors, such as the uneven distance45

between the foil and S1 due to bending of the heated tungsten, the standard deviation of there-accelerated beam is estimated to be approximately 14 eV. This estimation is consistent
with the measured µSR asymmetry in a 40 nm thick Al layer on SiO2 substrate [46].
With the current laser parameters, we observed ∼15µ+ /s at the sample position, whichcorresponds to an overall conversion efficiency of 3×10−5 . The number of available muonsis far below that required for the g − 2 experiment.
There are three main issues to be addressed to use this scheme for the proposed g − 2experiment. Firstly, we need to increase the intensity of the incident muon beam. Thiscan be achieved by constructing an all-solenoid muon capture and transport beam line at
the MSL facility. Secondly, we need to study the way to produce muonium more efficiently.
Lastly, we need to increase the ionizing laser power for muonium. Our measurements showthat improvement of the yield can be expected from increases of the pulse energy of the
122 nm and 355 nm beams (Fig. 22). Especially it is worth noting that the power of
122 nm (∼1 µJ/pulse) is still below the saturation intensity of 1S–2P transition. In order toaddress this challenge, we propose to build a new laser system, based on a newly developed
Nd:GdVO4 laser.
In following sections, we first describe a new muonium production scheme under consideration, then describe the new laser system we propose to build. Then the muonium
polarization of the re-accelerated muon beam will be discussed, and last, the initial acceleration scheme as a front end to the g − 2 accelerator/storage complex will be discussed.

FIG. 21: A time-of-flight spectrum of 9 keV muon beam detected by a MCP at the sample position

46

FIG. 22: Dependence of the efficiency of slow muon generation on the laser energies
B.

Muonium Production

1.

Muonium Production Target Materials

Since the transverse momentum is an essential factor in the ultra-cold muon beam forthe new muon g − 2 experiment, we will use a room temperature muonium production
target. The average momentum of the muonium at room temperature is 2.5 keV/c. If this is
kept in the transverse direction during acceleration to 320 MeV, the ratio of the transverse
momentum to the beam momentum is 10−5 .
As described above, hot tungsten was used as a muonium production target at RIKENRAL. An alternative would be to use silica powder which is known to produce muonium in
vacuum with thermal energy at room temperature. There have been several measurements
on muonium production from silica powder. The reported efficiency (per incoming muon)
for muonium production appers to be about 3% for 27 MeV/c incident muons. Thus, theefficiency of muonium emission is similar for hot tungsten and silica powder.
The production of muonium in vacuum from silica powder was actively studied in the47

1980s [47–49]. Muonium is formed with about 60% probability inside silica powder particles [50] and is emitted from the particles with 97% probability [51]. The nominal particle
radius is as small as 3.5 nm (Cab-O-Sil EH-5 fumed silica), and can be larger depending on
the grade. Diffusion through the voids between particles at thermal speeds allows migrationof the polarized muonium atoms over distances of the order of 0.3 mm within the powderin the muon’s lifetime. If we form a thin layer target with such powders, the muonium
may escape the surface of the layer with significant probability, of order 20% for a layer ofthickness 0.3 mm [47–49].
There are other advantages in using a room temperature target such as silica powderrather than high temperature targets. Due to the reduced thermal velocity of muonium, theresonant line for muonium excitation has a smaller Doppler broadening and also the spatial
spread of muonium in vacuum is smaller, so we can use the given laser power much more
efficiently.

2.

Proposed muonium production target search at TRIUMF

Although it is a strong requirement from the muon g − 2 experiment to use a muonium
production target at room temperature, the use of silica powder as the production target
has several practical difficulties. Firstly, the powder must be supported by a surface that is
not vertical, while most of the beamlines are designed to extract beam horizontally using anelectric field. The horizontal spread of muonium in vacuum in the tilted target geometry islarger and the difference in electric potential that each ionized muon feels would widen the
beam energy spread. Secondly, we need flatness of the target surface at the mm level so we
can fire the laser beam through the vacuum region as close as possible to the target where
the muonium density in vacuum is the largest. Since powder targets present challenges inboth cases, we hope to find a self-supporting (solid) sample which has a similar or evenbetter yield of muonium.
Silica aerogel is one candidate. Previous measurements on silica aerogel, made 20 years
ago, found a very low emission rate (about 1/4) compared with that from silica powder.
We expect that more uniform aerogel is now available, for example that used for Cerenkov
counters, where the density of the aerogel has been finely controlled to change the refractive
index.
48

We are planning a measurement at TRIUMF for several muonium production targets.
The measurement aims at 1) testing several new materials and 2) studying the muonium
emission model more precisely. We expect to improve our understanding by systematic
measurements on various samples. We propose to image the muonium position with wirechambers. We stop muons in the sample, and detect muonium atoms which emerge fromthe surface of the sample by tracking positrons from muon decay. Since the ratio of thenumber of emitted muonium atoms to the number of muons staying in the sample is rather
small, we need a position resolution better than the typical separation of the muonium fromthe sample. Since the thermal velocity is of the order of 5 mm/µs and the time duration isof the order of the muon lifetime (2.2 µs), we need a position resolution of several mm. For
this purpose, we plan to use wire chamber tracking.

3.

Spatial distribution of muonium in vacuum and overlap with laser

In order to ionize the muonium efficiently with lasers and collect the muons with extraction optics, it is necessary to know the spatial distribution of the muonium and how it
develops with time after muon stopping. We are using a diffusion model which was successful in analyzing the previous experiments at TRIUMF. It assumes the motion of themuonium is diffusion like in the silica layer and it freely moves after emission in vacuumwith a velocity following a Maxwellian distribution at room temperature. It also assumes anemission angle following a cos θ distribution, which results from the maximum depth fromwhich a muonium atom with a given mean free path can exit at a given angle θ from thenormal.
Figure 23 shows the result of the diffusion model. The distribution of the muonium
position in vacuum (distance from the target surface) is plotted at various timings after themuon stopping. This figure can be used to determine the best timing for the laser shots aswell as the region to be covered by the laser beam. Let us assume we shoot the laser at a
distance of at least 1 mm from the target surface and that the laser beam is spread for 5 mmin the z direction, giving the best timing of around 400 nsec from the second beam pulse
and 1000 ns from the first pulse. In this condition, about 25% of all the emitted muonium
will be inside this region.
The two Lyman–α beams used in the RIKEN-RAL experiment had a beam size of 1–49

FIG. 23: Simulated result based on the diffusion model. Each profile shows the muonium distribution in vacuum at variuos timings after muon stopping.

2 mm in the horizontal plane and about 5–8 mm in the vertical plane, thus the two beams
together covered about 2 mm2 . It would be possible to increase the laser area to 5 mm ×
40 mm and still have 10 times more luminosity, because the total power of the new laser isdesigned to be 100 times larger. So it would be possible to ionize most of the muonium in
the irradiated region.

C.

Laser System

To ionize the muonium atoms we will apply two photon absorption by muonium inthe ground state: vacuum ultraviolet (VUV) laser light centered on the wavelength of the
Lyman-alpha (Ly-α) line of the muonium atom photoexcites muonium to the 2p state, thenanother UV photon (360 nm or shorter) removes the electron. To generate the Ly-α laser
light we utilize the two-photon resonant four-wave mixing process in gaseous Kr atoms bypumping in 212.55 nm light (ω1 ) to excite it from the bound state to the 4p5 5p state by50

two-photon absorption, then 820 nm light (ω2 , tunable 815 ∼ 850 nm) generates the Ly-α
light by difference frequency generation (Fig. 24). In this method the key issue is how toincrease the power of the Ly-α laser to be sufficient for the target slow muon production.
Generation of high intensity VUV is difficult because of an absence of a non-linear frequency
conversion crystal in wavelength shorter than 200 nm. Therefore historically researchers applied the optical conversion processes in gaseous atoms, such as Kr, Xe, and Hg, to generate
coherent light in VUV region. However the conversion efficiency has always been an issue
in this method. Conversion efficiency was measured to be ∼10−4 to 10−5 with power output
10 µJ/laser pulse at most, even though the Ly-α is such an important wavelength. Here weset our goal to be 100 µJ, 1 ns, 80 GHz spectral width, and 25 ∼ 50 Hz repetition rate of thepulsed Ly-α output by having high-intensity pump lasers (especially ω1 ), and a conversion
efficiency of 10−3 by performing experiments and simulations to find the best conditions forthis wavelength conversion process.
1. ω1 and ω2 laser
The VUV generation in Kr occurs in a range of the so-called small-signal gain where
the conversion efficiency is proportional to P12 P2 , where P1 and P2 are the laser powerfor ω1 and ω2 respectively. To achieve the above-mentioned goals, it is estimated thatboth P1 and P2 need to be approximately 100 mJ/pulse when the pulse duration is
1 ns. A light source of 100 mJ/pulse of 212.55 nm at 50 Hz (when synchronized withthe muon source at RIKEN-RAL) already sounds challenging, but it is technicallyfeasible by utilizing the following laser system:(a) ω1 laser systemi. fundamental for the ω1 laser
To generate high-intensity 212.55 nm laser light we utilize the 5th orderharmonic of 1062.75 nm (fundamental), which is an output of a distributed
feedback laser diode (DFB-LD) that enables oscillation at the single longitudinal mode with narrow linewidth (∼2 GHz). The fine tuning on the central
wavelength of the DFB-LD by changing the LD temperature is very important in this application because the two-photon resonant absorption in Kr
has a well-defined excitation energy. Since the LD has relatively short opticalcavity length, it has less jitter and the temporal shape of the input electric
51current directly appears as the laser output. Therefore it is possible to change
the laser repetition rate and pulse duration just by controlling the electric circuit. To obtain sufficient power for the ω1 laser it is necessary to have enoughpower of the fundamental, which we estimate to be approximately 1 J/pulse
considering the conversion efficiency of the nonlinear crystal. To amplify theseed output from the DFB-LD we consider the multiple amplification stages
as shown on Fig. 25.
ii. Fiber amplifier
The seed output from the DFB-LD is first amplified by the fiber amplifier
to an order of 10 µJ /pulse. For the non-linear wavelength conversion it isimportant to keep the linear polarization and large extinction ratio, and to
avoid the spectral broadening due to the self-phase modulation in the fiber
medium. Currently we have completed the DFB-LD and amplification stages
up to this point with the expected output energy (10 µJ /pulse).
iii. Regenerative and Multi-pass amplifier
The second amplification stage is a regenerative amplifier with Nd:YGAG
crystal as a gain medium, which resembles the traditional Nd:YAG crystal
but has Ga substituting part of the Al atoms, so that the central wavelengthis tuned to the target wavelength (1062.75 nm). Currently this part is underdevelopment. If the amplification turns out to be not sufficient with the regenerative amplifier only, we will consider an additional multi-pass amplifier.
iv. Nd:GdVO4 single pass amplifier
Finally the pulse train is amplified by a single pass amplifier with a
Nd:GdVO4 crystal as the gain medium which our collaborators have recently
developed [52]. Its gain spectrum is peaked at 1062.9 nm with 2 nm bandwidth, so it can amplify our fundamental very close to the center of its
spectrum. If the amplification turns out to be still not sufficient we will add
one more stage for further amplification.
v. Non-linear wavelength conversion
The amplified fundamental pulse (1 J, 1 ns pulse duration) is now converted
to 1 by the nonlinear frequency conversion system as show in Fig. 25. We

52

utilize an LBO crystal for the second harmonic generation and CLBO crystals
for the 4th and 5th order harmonic generation, with a goal of 10% for theconversion efficiency (thus 100 mJ/pulse of 212.55 nm). An important point
here is the use of CLBO crystals, which gives high conversion efficiency,
although a low damage threshold has prevented it from being practical. We
have studied the characteristics of the crystal in detail and found an approach
for long-term stable operation.
(b) ω2 laser system
For the ω2 laser we use the DFB-LD as an oscillator in a similar way as for theω1 laser system, and then amplify the output pulses by the optical parametricamplification (OPA) with an LBO crystal. In this case the pump laser for OPA
is a portion of the second harmonic generated during the ω1 process describedabove (subsection 1(a)v). Our goal here is 100 mJ, 1ns pulse duration of 815 ∼850 nm. This output level should not be so difficult because the technology ismore established in this wavelength region.
2. Ly-α laser light generation using Kr gas
Both the ω1 and ω2 lasers are collimated and incident on a gas cell for the Ly-α laser
light generation. The gas cell contains Kr and Ar gas by a constant fraction for the
wavelength conversion. Here Ar is necessary to satisfy the phase-matching condition,∆k = 0, where k indicates a wave vector and ∆k = kLy−α - (2kω1 + kω2 ). Since Kr
has negative dispersion in the wavelength of Ly-α, it is necessary to have Ar to givea positive dispersion so that the system has a solution for ∆k = 0 and to achieve thebest conversion efficiency. Unfortunately there was no work so far that systematicallystudied the conversion efficiency as a function of the relevant parameters, such as
the partial pressure ratio, absolute pressure, beam intensity, and so on. We needto start by measuring some fundamental optical parameters such as the third-ordersusceptibility of Kr at Ly-α wavelength, therefore it requires a serious R&D to achieve
10−3 conversion efficiency. We have started doing experiments and simulations at alaser laboratory in RIKEN-Wako. We are planning on finishing the Ly-α laser system
during the second half of 2010 and installing it at RIKEN-RAL for the experiment ofslow muon production.
53

Kr

ωLy-α = 2 ω1 - ω2

Kr 4p55p

ω2815~850nm

ω1212.55 nm

ωLy-α
121.5~122.2 nm
Lyman α100 µJ@122 nmω1
212.55 nm
Kr 4p6

FIG. 24: Schematic diagram of four-wave mixing process in Kr

ω1 Fiber Laser / Solid-State Amplifier / Nonlinear Frequency Conversion System
Solid-State Amp. System
Fiber system

Regenerative
AmplifierFiber amp

1J

100 mJ
10 mJ
xN

DFB-LD

10 µJ@25~50 Hz
1062.75 nm

Single-pass
AmplifierMulti-pass
AmplifierFundamental is indicated as ω here

Telescope

Nonlinear Frequency Conversion
Beam
Splitter

100 mJ@212.55 nm (5ω)ω, 4 ω

CLBO

ω

Beam
Splitterω + 4 ω 4ω

2ω

λ/2

CLBO

ω

LBO

2ω + 2 ω2ωω+ωFIG. 25: laser system for generating ω1 , 212.55 nm
D.

Spin Polarization

1.

Muon spin in muonium

Muonium is a bound state of a muon and an electron. Its hyperfine splitting between thespin singlet state and the triplet state has been measured as νHFS (= 2πωHFS ) = 4.463 GHz [53].
54Suppose the incident surface muon is 100% polarized, and the electron it captures is
unpolarized, then half of the muonium ensemble is formed in the state where the spin of
the electron is parallel to that of the muon, denoted as | + +i, and another half of theensemble is in the state with spin anti-parallel, | + −i. Since the | + +i state is a spin-triplet
energy eigenstate, the total spin of muonium and the muon spin are stationary. Therefore,
the muon polarization in the | + +i state is kept. On the other hand, the | + −i state is
not an energy-eigenstate, but a mixture of a spin singlet and a triplet states. The | + −i
state oscillates to the | − +i state in a time scale of 1/(4.463 GHz) = 0.224 ns [54]. The
average thermal velocity of the muonium at room temperature is 8 mm/µs. It takes at leasta hundred ns for muonium to diffuse/drift to the region of the laser beam (about 1–2 mmfrom the target surface). Therefore, by the time muonium is ionized, the | + −i state andthe | − +i state are equally populated, resulting in zero polarization of the muon. Since
the initial ensembles of the | + +i and the | + −i states are equal, the polarization of theultra-slow muon is (100% + 0%)/2 = 50%.

2.

Possible recovery of polarization

Two possibilities of recovery of muon polarization are being considered.
a. Longitudinal magnetic fieldIf the magnetic field is applied in parallel to the initial

muon polarization, total muonium spin is no longer a good quantum number. In the limitof the strong magnetic field, the spin component along the magnetic field becomes a goodquantum number. Thus, the muon spin in the | + −i state will also be “locked”. The
dependence of the muon polarization Pµ with the magnetic field strength B can be written
as [54]1 + 2x2,
2(1 + x2 )(12)(ge µe + gµ µµ )B = 13[T−1 ]B.
2~ωHFS

(13)

Pµ =
with
x=

Figure 26 shows Pµ as a function of B in the region 0 < B < 0.5 T. The muon polarization
increases from 50% to 96% with a longidudinal magnetic field of 0.3 T in the region of themuoninum target and laser ionization.
55

Polarization

Muon polarization vs Longitudinal B-field
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.05

0.1 0.15

0.2

0.25 0.3

0.35 0.4

0.45 0.5
B (Tesla)

FIG. 26: Muon polarization in muonium as a function of the londitudinal magnetic field

The primary challenge of this method is the muon beam transport after the ionization
under the circumstance of the magnetic field. The muon transport in the initial accelerationmust be carefully designed so that transverse kick to the muons from the fringing field of
the spin holding magnet is sufficiently small to preserve the transverse dispersion. We plan
to study feasibility of introducing such magnetic field and beam transport configuration toincrease the muon polarization.
b.

Narrow band laser

The singlet state and triplet states are selectively ionized if thelaser band width is narrower than the hyperfine splitting νHFS =4.463 GHz. The narrow band
laser can be used to sweep out the depolarized muons in the unfavorable singlet state inadvance to the normal ionization scheme. This way, the polarization increases from P =50%
to (100%+0%)/1.5 = 66%, and we lose 1/4th of muons by sweep out. We gain the figure of
merit (∝ N P 2 ) by a factor of 1.33.

E.

Initial Acceleration

In this section, we describe the muon initial acceleration scheme after the muonium laser
ionization, up to the main acceleration using LINACs, as described in section V. A key issue
is to minimize / confine the muon bunch time-width. Because we need to inject the muonbunch into an extremely limited space-time of the acceleration RF field of the LINACs,
56

we need both excellent emittance and a sharp time-width at the same time. The timewidth must be much narrower than the RF frequency, otherwise one loses the control of theoff-phase component.
Another key issue is to avoid introducing additional relative muon motion in a muonbunch at the initial acceleration, because the relative motions viewed by the rest frame of
the bunch is the temperature of the muon bunch. This heating-free acceleration is one of themost critical parts to realize the new g − 2 experiment. We propose to construct the initialacceleration stage keeping cylindrical symmetry to reduce second-order aberration, except
for a higher order aberration compensator such as electro-static sextupoles. For the designof the initial acceleration stage, one can apply the technique developed for the electron
microscope. Together with a conventional initial acceleration scheme, we also propose to
apply a muonium motion-cancellation mechanism using the Doppler shift of 1S-2P transitionenergy by the high-intensity chirped Lyman-α laser, for further temperature reduction of
the muon bunch.
Thus, we need to control muon beam very precisely both longitudinal and radial motion bypulse extraction field and cylindrical initial acceleration stage, respectively, as it is describedin detail as follows.

1.

Cylindrical initial acceleration stage

The simplest and ideal muon acceleration scheme, up to required muon energy for new
muon g−2 measurement, is acceleration of parallel and static electric field, namely CockcroftWalton type, because there is absolutely no possible heat source during the acceleration.
However, it is not realistic at all to accelerate more than few tenth of MeV. Therefore, one
need to combine several acceleration method to reach the required muon energy.
In the low energy region, however, one of the good model is parallel and static electricfield, but the present muon source is not that simple because of rather large source size.
Before the injection to the acceleration field, one need to focus the laser-ionized muon toutilize only homogeneous region of the field, otherwise the system becomes unreasonablyhuge.
The better model is electron microscope instead of the simple Cockcroft-Walton. If all
the initial acceleration components have cylindrical symmetry without magnetic field, the57

equation of the particle tracks close to the beam axis can be represented asd2 x
1 dΦ dx
1 d2 Φ
++x=0
dz 2 2Φ dz dz 4Φ dz 2

(14)

d2 y
1 dΦ dy
1 d2 Φ
++y = 0,
dz 2 2Φ dz dz 4Φ dz 2

(15)

where Φ is the electric potential on axis, Φ = φ(0, 0, z). From the equations, it is clear thatthe tracks are independent of the charge mass ratio e/m, thus the trajectories should besame for electrons and muons. If there is higher order aberration in the initial acceleration
stage, the muon bunch can easily heat-up because uncontrollable relative motions are introduced due to the aberration. The equations have cylindrical symmetry, so the second orderaberration cannot be exist, which simplifies the aberration control. Because there exists an
MV class electron microscope, it is feasible to apply that technology for the muon initialacceleration.

2.

Primary design of the initial acceleration

Figure 27 shows the schematic setup for the initial acceleration scheme. The surface
muon from the production target is transferred to the muonium generation target by a largesolid angle solenoid channel, and accumulated to the central axis by a capillary method [55]using capillary shaped heavy material. The ionization laser will be injected just behind themuonium target to have maximum yield.

primarymuon+V0+Vp(t)+V0verticaldispersion
matching

GND

rebuncherstaticcapillary target
aberration
initialacceleration
focusing
compensator
focusing
field
laser (einzel lens)

final
focusing

FIG. 27: Schematic setup for the initial acceleration.

The extraction field will be generated by applying slightly higher voltage to the muonium
target holder than the bias of the initial focusing element. Double einzel lenses will be used
to form a weakly-focused parallel beam to confine the muon beam diameter. After the58

double einzel lenses, one can place an electrical element to cancel the vertical dispersion
in the case where a pulsed extraction field is applied. The weakly-focused parallel beam isfurther accelerated by the static parallel field, and the higher order aberration is canceled by
an aberration compensator composed by four einzel lenses and two sextupole lenses locatedbehind the static parallel field.
To compress the muon beam bunch in time, one needs a re-buncher to rotate the phaseof the muon bunch, and then apply final focusing elements before injection into the LINAC.
By a static extraction field, longitudinal momentum dispersion depends on the ionization
position along the axis, so one can realize very short time-width muon bunch in the orderof a few psec at few 100 kV acceleration, assuming primary muon is few cm in size. The
pulse width after the re-buncher is dominantly limited by the dispersion of the path length
at the initial focus. Obviously, the weaker focusing design results in shorter time spreadof the muon bunch. On the other hand, one can produce a linear correlation between
the ionization timing and the longitudinal velocity of the cold muons, hence a correlation
between the ionization timing and the longitudinal position dispersion, by using a pulsed
extraction field. If one applies a chirped-laser pulse whose central energy shifted in time,then we can sweep thermal transverse momentum in time according to the Doppler shiftof the muonium, so one can cool the muon below the room temperature. In this case, thetime-width after the re-buncher is limited as ∼ ∆llp /(β0 c), where ∆llp is the longitudinallaser size and β0 c is the central velocity after the initial acceleration, and few 10 psec pulse
width can be realized at few 100 kV acceleration, assuming laser size of about 1 mm. One
needs more detailed design study to finalize the configuration of the final focusing elements
to fulfill the matching condition with LINAC injection.

F.

Summary

The proposed experiment utilizes the ultra-cold muon source which can be produced by
the laser ionization of muoniums. The laser ionization technique has been developed by KEK
and RIKEN, and continuously improved at the RIKEN-RAL muon facility. The efficiency
to produce the ultra-cold muon from the surface muon beam is 3×10−5 at RAL.
A new laser system is currently under development at RIKEN to increase the laser powerby a factor of 100. With the 100 times intense laser, we expect the efficiency to be two59

orders of magnitude higher, i.e. 3×10−3 , which will be confirmed by the test experiments
at RAL and J-PARC. The surface muon intensity at the J-PARC H-2 beam line will be
108 /s (see section IV B). Therefore, a conservative estimate of the intensity of the ultra-slow
muon beam would be 3 ×105 /s. Here, we note that one can expect additional factors ofimprovements in the effficiency from (1) better muonium-production efficiency with newtarget material, and (2) better ionization efficiency at room temperature (as opposed to2,000 K for the hot tangsten target). These improvement factors are to be determined in
the test experiment proposed at TRIUMF in 2010.
The default polarization of the ultra-cold muon is 50%. R&D to improve the polarization
is in progress based on the ideas of introduction of a londitudinal magnetic field, or a narrow
band laser. Initial acceleration will use the weak static electric field, follwed by double einzel
lenses, the static acceleration field, a re-buncher, and a focusing elements. Cooling of themuon beam even below the room temperature would be possible by utilizing a chirpedlaser.

60

V.

RE-ACCELERATIONThe ultra-cold muons created in the muonium target regeon must be reaccelerated toaround 300 MeV/c in momentum (210 MeV kinetic energy) to then be injected into themuon g − 2 storage ring. To suppress muon loss in the course of reacceleration, the muons
should be accelerated in a sufficiently short period of time compared with the muon life
time of 2.2 µs. To realize fast reacceleration, we plan to build a linac dedicated for thispurpose. A linac has an overwhelming advantage over circular accelerators in shortening
the reacceleration time.
Another essential requirement for the reacceleration is the suppression of transverse emittance growth. The ultra-cold muons will have an extremely small transverse momentum ofaround 30 keV/c with an tranverse spacial extent of around 10 mm. Assuming an ideal acceleration to 300 MeV/c, the typical transverse divergence angle of muons becomes 10−5 rad.
These are the requirement for injection into the storage ring. Therefore, the beam at injection should have essentially the same normalized transverse emittance as that at the muonsource. Consequently, the transverse emittance growth in the linac should be suppressed asmuch as possible to meet this requirement.
In the design consideration of this linac, we separate the acceleration into two sections,
namely, a low-β section and a high-β section. In the low-β section, we assume the acceleration of muons from around 10 keV to 43 MeV in kinetic energy. This corresponds toaccelerationg β from 0.01 to 0.7. The muons accelerated in the low-β section are delivered
to the high-β section, where they are further accelerated to 210 MeV.
In the low-β section, the particle β increases rapidly with kinetic energy. Accordingly,
it becomes very important to adopt an adequate accelerating structure matching β. To
the contrary, the β variation in the high-β section is modest. Then, its design emphasis is naturally shifted to achieving a high accelerating gradient to realize sufficiently fast
reacceleration.
In the following two subsections, we review the proposed design for the low-β section andthe high-β section of the linac, respectively.

61

A.

Low β linac

As mentioned above, the particle β changes rapidly in the low-β section as its kinetic
energy increases. Then, a suitable accelerating structure should be chosen for each β range.
This situation is essentially the same as in a usual proton linac. Therefore, we basicallyfollow the conventional configuration for a proton linac in the design of the low-β section.
In addition, we put the emphasis on making the best use of J-PARC experience to reduce
the R&D burden. It should also be noted here that the space-charge effect is not an issue
for the linac even at its lowest energy, because the number of muons assumed per bunch isas small as around 50,000. This is an important difference from a usual proton linac, where
the design emphasis is often put on suppressing the space-charge effect.
We here divide the low-β section into three parts, namely, the front-end part (β < 0.08),
the low-β part (0.08 < β < 0.45), and the medium-β part (0.45 < β < 0.7). For the low-β part,we plan to adopt a DTL (Drift Tube Linac) of Alvarez type with the operating frequencyof 324 MHz. This operation frequency is chosen to be the same as the J-PARC DTL [56].
With this choice, we can use the existing J-PARC klystron for its RF source avoiding an
R&D burden.
As for the medium-β section, we plan to adopt an ACS (Annular Coupled Structure
linac) [57]. The ACS is a variation of a CCL (Coupled Cavity Linac), and adopted for theenergy upgrade of the J-PARC linac. The ACS is advantageous in suppressing the dipole
component of the RF field, having an annular symmetry for the coupling cell [57]. The
operation frequency for the ACS is chosen to be 972 MHz with a three-fold frequency jumpfrom DTL. This frequency is again the same as J-PARC ACS to utilize 972-MHz klystronsdeveloped for it [56].
It should be noted that both the DTL and the ACS are standing-wave structure, and it
is a conventional choice for a proton linac. This is constrasted with the high-β section wherea traveling-wave structure is assumed.
In the linac design, it is important to optimize the particle β where the accelerating
structure is switched from one to another. We here assume the transition β of around0.45 between DTL and ACS. While it is a conventional choice for a proton linac, there
still remains a room for further optimization. We should note here that the transition β isusually determined to optimize the effective shunt impedance. In other words, we will have62

wider flexibility by tolerating inefficient use of RF power. As efficient power consumption
is not such a pressing requirement for our purpose, the choice of the transition β will be
subject to further optimization in the coming design work.
For the front-end part, we plan to adopt a Cockcroft-Walten injector instead of an RFQ(Radio-Frequency Quadrupole linac). In a modern proton linac, an RFQ is usually adopted
for the front-end, where a continuous beam injected from an ion source is gradually bunched
and RF captured for acceleration. This bunching capability greatly improves the transmission efficiency of a proton linac compared with a classical Cockcroft-Walton injector.
However, this bunching capability is not necessarily advantageous for our purpose, because
the muons generated at the muon source already has a bunch structure generated by an
excitation laser. Meanwhile, an RFQ is known to have a substantial nonlinear component of
the RF field due to unavoidable deviation of its vane shape from the ideal one [58]. The nonlinear component can be a source of transverse emittance growth, which should be strictly
suppressed for our purpose. While we need further study on the effect of the nonlinear
component, we here assume a Cockcroft-Walton injector for the front-end with an RFQ asan alternative option.
To adopt a Cockcroft-Walten injector, it is required to have an additional RF cavity
for bunch compression. The bunch length from the muon source is assumed to be around
100 ps. As this bunch length is too long to be accepted by a 324-MHz DTL, we need toperform bunch compression before injecting into the DTL. To this end, we plan to havea buncher cavity between the Cockcroft-Walten injector and the DTL. We here assume asimple single-gap cavity of 324 MHz for this purpose.
Figure 28 shows the proposed configuration of the low-β linac. Plan A shows the present
baseline design with a Cockcroft-Walten injector, and Plan B shows an alternative plan withan RFQ as the front-end. In both plans, we can adopt the same design for DTL and ACS.
In the baseline design, the low-β linac consists of a 0.34-MeV Cockcroft-Walten injector, asingle-gap buncher cavity, a 13.8-MeV DTL, and a 43-MeV ACS. The DTL section consistsof one DTL tank driven by a 324-MHz klystron. Meanwhile, the ACS section consists of six
ACS tanks. Two neighboring ACS tanks are connected with a bridge coupler, and driven
by a 972-MHz klystron. We plan to adopt klystrons of J-PARC type both for 324 MHz
and 972 MHz. The saturation power for the klystrons is 3.0 MW for both 324 MHz and
972 MHz. As for the buncher cavity, we plan to adopt a solid state amplifier as its RF
63

source. The required RF power for the buncher cavity is estimated to be kW level or even
less, and is easily fed by a solid state amplifier.
Table IX and X show the main parameters for the DTL and ACS, respectively. It isreadily seen in these tables that the main parameters are comparable with the J-PARC
design except that the muon DTL and ACS cover a little wider β range. This difference
arises because we omit the SDTL (Separate-type DTL) [59] section in the low-β linac to
reduce the number of transitions. On the other hand, SDTL is adopted in J-PARC to
improve the power efficiency in the β range between DTL and ACS [56].
As shown in these tables, we plan to adopt the synchronous phase of -30 degree to attainsufficient acceptance in the longitudinal phase space. With this choice of the synchronous
phase, the RF field provides the muon beam with longitudinal focusing, but in turn it
defocuses the beam transversely. Accordingly, it is required to provide transverse focusing
to compensate the RF defocusing force. In the DTL, we plan to equip each drift tube witha quadrupole magnet as in a conventional DTL. The transverse focusing is provided withthese DTQ’s (Drift Tube Quadrupole magnets) with a so-called FODO lattice. In the ACS
section, a quadrupole doublet is placed at each inter-tank spacing to provide transverse
focusing. An additional solenoid magnet is also assumed before the DTL injection to matchthe beam size by focusing the beam from the Cockcroft-Walten injector.
For this baseline design, proper beam transport through the low-β section is confirmedwith the Trace3D code [60] as shown in Fig. 29. In this figure, the calculation is performed
from the exit of the Cockcroft-Walten injector to the exit of ACS. We assume a 2.4-m longmatching section between the injector and DTL, and a solenoid magnet and a buncher
cavity are assumed there for transverse focusing and bunch compression, respectively. The
Trace3D code calculates the time-evolution of the phase-space beam ellipses, and hence
beam envelopes, along the linac. The top left figures in Fig. 29 show the phase-space beam
ellipses assumed at the injector exit, and the top right figures those obtained at the ACS
exit. The bottom figure shows the calculated beam envelope along the beam line from theinjector exit (at the left end) to the ACS exit (at the right end). It is seen in this figure thatthe beam can be matched with a rather simple configuration of a solenoid magnet and abuncher cavity. Then, the beam envelope through the low-β linac is sufficiently controlled.
Proper beam acceleration is also confirmed with a preliminary particle simulation. The
simulated particle distribution at the ACS exit is shown in Fig. 30. In this simulation,
64

FIG. 28: Schematic layout of low-β linac.

an initial Gaussian distribution is generated at the DTL entrance with 50,000 simulation
particles. Then, they are tracked through DTL and ACS assuming a realistic RF field
and a hard-edge quadruople magnetic field. For this tracking, the IMPACT code [61] isutilized turning off the space-charge calculation. An RMS emittance growth of around 7%
is observed in the simulation. This growth is believed to be induced by the nonlinear natureof the RF defocusing force. While the shape in the transverse phase space also shows some
distortion in this figure, it is assumed to be at a tolerable level.

65

TABLE IX: Main parameters for DTL.
Muon DTL

J-PARC DTL

Number of cells per tank

33

146

Number of tanks

1

3

Operating frequency

324 MHz324 MHz

Cavity type

Alvarez-type

Alvarez-type

Averaged accelerating field E0

2.5 MV/m

2.5 MV/mSynchronous phase φs-30 degree-30 degree

Kinetic energy at the entrance

0.34 MeV

3.0 MeV

Kinetic energy at the exit

13.8 MeV

50.1 MeVβ at the entrance

0.080

0.080

β at the exit

0.462

0.313

Wall loss per tank

0.83 MW

1.1 MW

Total length

8.82 m27.1 mTraversing time

99 ns

66

TABLE X: Main parameters for ACS.
Muon ACS

J-PARC ACS

Number of cells per tank17

17

Number of tanks

642

Operating frequency

972 MHz972 MHz

Cavity type

ACS

ACS

Averaged accelerating field E04.36 MV/m

4.12 MV/m

Synchronous phase φs-30 degree-30 degree

Kinetic energy at the entrance

13.8 MeV

190 MeV

Kinetic energy at the exit

43 MeV

400 MeVβ at the entrance

0.465

0.555

β at the exit

0.703

0.712

Wall loss per tank

0.82 MW0.73 MW

Total length

14.76 m107.1 mTraversing time

80 ns

67

FIG. 29: The beam envelope along the low-β linac calculated with Trace3D.

FIG. 30: The particle distribution at the exit of the low-β linac simulated with IMPACT

68

B.

High β section

The high β section of the muon linear accelerator is designed to re-accelerate from
100 MeV/c to 300 MeV/c. As the basic design of the high β section, a disk-loaded travelingwave accelerating structure, which is similar to an electron linear accelerator, was selected.
The advantage is its high electric field of around 20 MeV/m to shorten the accelerator
length. Figure 31 is the typical accelerating structure of the electron linac. The emittanceand bunch length of the muons are small enough to accelerate using such a disk-loaded
structure, though the available phase space becomes less due to the higher frequency. However we have to evaluate the phase stability and determine the frequency to minimize theemittance growth.

FIG. 31: Disk-loaded accelerating structure.

The particular design item of the disk-loaded structure for the muon acceleration, which isdifferent from the general accelerating structure for the electron accelerator, is the variationof the disk spacing corresponding to the muon velocity.
The design procedure of the phase velocity matched disk-loded structure is as followings,• The disk spacing is generated in turn to pass the center of each cavity at a given RF
phase for a test particle.
69• The test particle is tracked through the electromagnetic field of the cavity which is
calculated utilizing the series expansions and impedance matching method [62] withoutmesh.
• The particles at the end of the low β section are tracked in the cavity chain.
As the initial condition for the accelerating structure, we selected the RF frequency of2856 MHz which is used in many electron accelerators. The optional candidate of the RF
frequency is around 1300 MHz for the wider phase space. Further the constant gradientstructure was adopted, the input power is 8 MW for each accelerating structure and theratio of the power attenuation constant usually described as τ is selected to 0.5. Figure 32
shows the disk spacing and the length of the accelerating structure corresponding to themuon momentum. It consists of only four accelerating structures and the total length isaround 10 m.

FIG. 32: Cell length corresponding to the muon velocity.

The total system based on this desgin of the high β linac is shown in Fig. 33. One 50 MWclass klystron can drive four accelerating structures to accelerate muons from 100 MeV/c to
300 MeV/c.
70

FIG. 33: Schematic diagram of the high β sectino of the muon re-acceleration.

The muons from the end of the low–β section are tracked through these accelerating
structures generated for the center RF phase of 80 degree.
Figure 34 is the longitudinal distribution at the end of the high–β section.
As a result, the 98% of muons are passed in the accelerating phase.

FIG. 34: Longitidinal distribution of the muon at the exit of the high–β Linac.

71

VI.

MUON STORAGE RING MAGNET AND INJECTION

The muon storage ring for this experiment will be a precession field 3 Tesla solenoid withcylindrical iron poles and return yoke. The solenoid is being designed now in collaboration
with a private company to a specification of a < 1 ppm variation of the field locally within thestorage region. The storage region is defined on the mid-plane, with a radius of 33.3±5.0 cm
and height of ±10 cm. An engineering study is shown in Fig. 35. Note that the coils arelocated at 80 cm radius.

FIG. 35: An example of engineering studies by a private company.

A.

Injection

For injection, the muon beam must be injected into the storage ring with minimum
interference to the storage field. In the BNL g − 2 experiment, as displayed in left side ofFig. 36, a device called an inflector was used to inject the muon beam, to avoid deflection by
the fringe field. As the beam crossed the storage region, the beam was kicked horizontally72

by the kicker to move to the central orbit through an angle large enough to prevent thebeam from striking the inflector after one turn. This procedure works only if the radius ofstorage ring is large enough compared to the inflector wall thickness. The limitation comesfrom the required kick-angle within the first turn in the storage ring (within 142.9 ns) to
avoid hitting the inflector itself. In the BNL g − 2 experiment, the radius of storage ringwas 711 cm and the wall thickness was ∼ 1 cm. Therefore the kick angle was ∼ 1.5 mrad.
In our experiment, the radius of the center beam orbit is 33.3 cm. This corresponds to arequired horizontal kick angle of 30 mrad within the first turn in the ring (within 7.4 ns). The
kick angle would be an order of magnitude larger than the kick with any existing technology.
The horizontal kick does not work for our case.
Therefore, we have chosen a spiral injection scheme instead as displayed on right sideof Fig. 37. A solenoidal magnet is suitable for this injection scheme. An unique point of
this scheme is to build in a radial fringe field to replace the role of the inflector. The beam
will enter the solenoid through a hole in the iron return yoke iron, within a pitch angle of55◦ downward. (Note that the BNL E821 used a similar hole in the iron return yoke, which
was easily compensated for by adding additional iron beside the hole.) The definition ofpitch angle θ here is the angle between the muon momentum and horizontal plane (the x − zplane) perpendicular to the solenoid axis (the y-axis),
Vy.
Vx2 + Vz2

θ = sin−1 p(16)

In this way, the beam will be separated from the injecting point by ∼10 cm vertically, so
there would not be no disturbance for the beam trajectory.
The beam momentum is deflected to vertically by a radial magnetic field, which will bebuilt in the solenoid fringe field, which is shown as BR in the right side of Fig. 37. The
magnetic field should be carefully shimmed not only for the vertical field for the muon
storage but also for the radial field so that only small vertical momentum remains when the
spiraling beam reaches the beginning of the storage region.
The injected beam will then spiral through the good field (BR ) storage region and amagnetic kicker will be used to deflect the beam into a stable orbit at the center of thestorage region. The duration of the kick can be for a number of cyclotron (or revolution)
periods, since a longer kicker pulse allows a low kicker voltage and more stability. Multiturn-kick by the longer pulse shape kicker would be easier.
73

FIG. 36: Injection scheme for previous experiment (BNL g − 2) from top view. They have inflector
and horizontal kicker system.

FIG. 37: Schematic representation of spiral injection for our case. A radial fringe field, shown as
BR , deflects the vertical component of the beam momentum to the horizontal component.

74

B.

Field configuration and muon trajectory

Figure 38 introduces the magnet configuration planned for the storage ring. Note that
we will discuss the muon injection scheme in a conceptually designed magnetic field in thissection. More practical design is in progress collaborating with Hitach co. The left picture
depicts a solenoidal conductor (pink), with one-eighth of the iron pole tip (green) and iron
return yoke (black). The right picture depicts calculated muon beam injection trajectorywith this magnet configuration together with one-eighth of the iron pole tip and iron return
yoke.
Figure 39 displays the vertical (along the solenoid axis) and radial components of themagnetic field (By and BR ) along the muon beam trajectory, which is introduced in Fig. 38.
Note that the magnetic field changes smoothly as the muon moves. This is an important
point to find stable trajectory.
Since the magnetic field is symmetric at a given radial position (R =√x2 + z 2 ) anddoes not change dramatically even in the fringe field region, optical parameters of the muon
beam for a stable injection can be obtained. To determine the optimal correlation between
vertical position displacement (∆y) and duration of pitch angle (∆θ), we calculate several
thousands of beam trajectories with random distribution of ∆y and ∆θ by using Opera.
p
We set a proper injection point on the beam center trajectory : R0 = x20 + z02 = 34.6 cm,
y0 = 104.6 cm, with a pitch angle of θ = 23.4◦ . A definition of pitch angle θ is shown in
Fig. 40.
Then we tried several thousands of trajectories in a proper area of (∆y, ∆θ) as shown inblack points in Fig. 41.
Here, ∆Y and ∆φy are written as:∆y = y − y0 in cm,
Vy − Vy0
in rad.
∆θ =
Vy0

(17)

Figure 43 displays several example trajectories shown in green points in Fig. 41, together
with a beam center trajectory.
More practical design as displayed in Fig. 35 is in progress collaborating with a privatecompany.

75

FIG. 38: Top: A conceptual illustration of the storage ring magnet: solenoid coils, one-eighth ofiron pole tip and return yoke. Bottom: Calculated spiral trajectory of a single muon particle with
γµ = 3 by Opera.

76

Magnetic field [T]

0

−0.5
By
−1
BR

−1.5
−2
−2.5
−3
−3.5
0

20

40

60

80

100

Vertical position along the y−axis [cm]FIG. 39: Magnetic field along the muon beam trajectory as function of vertical axis (solenoid axis).
The trajectory is introduced in Fig. 38. Note that the magnetic field changes smoothly as themuon moves.

FIG. 40: A definition of θ is shown.

77∆ θ (rad)

0.03

0.02

0.01

0

−0.01

−0.02

−0.03
−3

−2

−1

0

1

2

3

∆ y (cm)

FIG. 41: Input parameters of (∆y, ∆θ) with regards to initial point of beam center trajectory.
(Parameters for beam center trajectory is (∆y, ∆θ) = (0, 0). ) Parameters of successful injections
are shown in green. The fringe field provides local focusing effect as discussed in Fig. 42.

FIG. 42: A cartoon to give an intuitive idea of local focusing effect.

78x (cm)

40
30
20
10
0
−10
−20
−30
−40
−40

−30

−20

−10

0

10

20

30

40

y (cm)z (cm)

120
100
80
60
40
20
0
−100 −80 −60 −40 −20

0

20

40

60

80

100

x (cm)
FIG. 43: Samples of possible muon beam trajectories in the acceptable acceptance for the kicker
and the storage ring.

79

C.

Kicker

The role of the kicker is to move the muon beam onto a stable orbit. In our case, themuon beam will be guided to the good field region by the solenoidal field only which includes
a small radial field above the good field region, reducing the pitch angle of the incoming
beam to zero degree. The pitch angle of the beam in the good field region is planned tobe the order of a few mrad. Therefore the required vertical kick is a few mrad, which isacceptable strength. We plan to apply the kick for several cyclotron periods along the orbit.
This allows us to apply a lower kicker voltage without strict jitter requirements, and with
better stability.
Figure 44 depicts one of candidate of kicker set-up in the storage ring: Anti-Helmholtz
coil system coil system, shown in red four loops. Figure 45 depicts a zoom on the crosssection of a kicker system. Pairs of coils above and below the storage region have different
radii and are separated 6 cm vertically. The upper and lower coils are separated by 24 cm
and do not interfere with the storage and detector volumes.
The current direction of the pair of upper coils and the pair of lower coils are the same,but the current is opposite between upper and lower coils. Such a coil system generates
mainly a radial field remains around the region of |y| < 10 cm as shown in Fig. 45. This
radial field is cylindrically symmetric in the radial direction and applies a vertical kick to
the muon beam.
Figure 46 displays an example electrical circuit to drive the kicker coils. In this case, thesystem can apply a 3.5 mrad kick vertically with a half-sine shaped pulse with a 150 ns (20cyclotron periods).
Bkick (t) = Bpeak · sin(ωt),

(18)

here,
Bpeak = 1.3 × 10−4 T,
π
ω=
,
Tkick
Tkick = 150 ns.
The simulated current shape as a function of time is displayed in the lower plot in Fig. 47.
During 150 ns, the vertical drift length of the muon beam is about 7 cm.
80

FIG. 44: A kicker system (Anti-Helmholtz coil system) is shown in red lines, in the storage ringmagnet. To give an idea of its size, the expected muon beam trajectory, iron pole tip and returnyoke are also shown.(Solenoid coil is not shown.)

FIG. 45: Cross-section view of the kicker system (zoom up). Pink triangles depicts direction of
the magnetic field.

81

FIG. 46: Example circuit for the kicker. The airs of two ring coil system the storage ring is
indicated as the red dotted box.

FIG. 47: Top: Voltage as function of time. V(1) and V(2) correspond pickup points in the circuit,
Fig. 46. Bottom: Kicker current as function of time. Half-sin shape can be seen as introduced in
Equation 18.

82

Major issues for kicker are the following;• Kicker field flatness as function of radial position,• Residual voltage in the circuit may cause the electric field in the space,• The eddy current in the cryostat wall will be generated.
We are studying to determine the best configuration for the coils and expect to achieve sufficient field flatness. For eddy currents, this causes an error field and needs to be considered
carefully. In the current example circuit, the residual voltage V(2) in Fig. 47 remains for∼ 100 ns after the kicker-pulse. We need to estimate the magnitude of this effect on themagnetic field.

83

D.

Possible error fields

Possible sources of error magnetic field are the following

1.

Mechanical disturbances• Static structural error
Although the coil can be designed such that the field quality in the storage area isin the 10−7 field quality, the construction error can cause the error field. In orderto achieve field quality of 10−7 the construction error must be the order of 10−6 mthat is far beyond the achievable construction precision. The precision of 10−3 m is
the reasonable specification for the 1 m big solenoid. The iron return yoke shouldhave same order of the influences. This indicates that achievable field quality with themain coil is the order 10−4 T range. The structural error can be expressed by radial
or longitudinal deformation modes that have angular spectrum of d = kcm cos(mφ)(or ksm sin(mφ)) as shown Fig. 48, where kxm is the deformation amplitude, and n is
the mode index. Influences to the beam area from each mode differ from one more toanother, generally smaller by ratio of (a/r)m where a is the radius of beam area, r is
the radius of the solenoid, and m is the deformation mode index. The deformation
mode and its influence to the field should be simulated to evaluate the necessary
corrector scheme. Field compensation using magnetic shims may be used for the
initial optimization. The magnet structure is such that the access to the coil area
is limited and that correction using magnetic shim may not be the final tuning. An
appropriate corrector coil scheme must be designed and implemented properly. The
corrector coils may be operated by external power supplies. The current stability of
the corrector should be better than 10−4 .
• Structural error by temperature change
The thermal contraction ratio of iron is about 10−5 /K. In order to achieve 10−6 m
accuracy shown above, the temperature must be controlled by the order of ±0.1 K.
The iron yoke must be encased in the shield room with the very precise temperature
control, i.e. ±0.1 K.

84

FIG. 48: The structural errors expressed by radial or longitudinal deformation modes.

• Mechanical vibration
The mechanical vibration is also a big concern for the field error. The seismic vibration
measured at KEK Tsukuba (Fig. 49) shows the value exceeding 10−6 m. The seismic
vibration at the experiment site should be performed and proper vibration dumping
system should be designed to avoid magnet system mechanical vibration.

2.

Electromagnetic disturbances• Magnetic materials
Although the system maybe designed to avoid to any uncontrollable magnetic materials, some materials such as welded stainless steels maybe inevitable. One must confirm
that those should not produce an error field exceeding the field range correctable by
the corrector coils. The magnetic material close to the beam area may cause the error

85

FIG. 49: The seismic vibration measured at KEK Tsukuba.

field corresponds to the error field introduced by higher order deformation index n,that may not be corrected by the corrector coils. One should design the system very
carefully to avoid magnetic materials close to the beam area.
• Superconductor
a. Magnetization The magnetization of superconductor, which is an inherent property of type III superconductor, is another big concern. The average magnetizationof the superconducting wire can be expressed as M = 4/3µ0 Jc ak/π, where M is themagnetization, µ0 is the permeability of the vacuum, Jc is the current critical current
density, which is the function of the local magnetic field and temperature, a is the diameter of superconducting filaments, and k is the volumetric ratio of superconducting
filaments in the wire. In general Jc can be as large as the order 109 A/m2 , a is an orderof 10−5 m, and k is about the order of 0.1. The magnetization can be estimated as
an order 10−4 T. The influence to the beam area may be further reduced and can becorrected by corrector coils if the error is stable. The issue is that the magnetization86

can be changed due to the flux creep effect. The flux creep is the movement of thepinned flux vortex in the superconductor that generally introduced by statistical thermal excitation of the vortex. The effect decreases the magnetization by a logarithmic
function. A sufficient idle time after magnet excitation may reduce the influences,
however the idle time can be too long to achieve 10−7 stability with a reasonable experiment efficiency. Another way to reduce influence is to reduce temperature aftermagnet excitation. The procedure, however, must be done after all the field optimization is made. The magnet excitation and field optimization procedure should be wellstudied to achieve 10−7 field accuracy and stability.
b. Strand coupling current The strand coupling current is the coupling current between the superconducting filaments. The influences are generally very small and thetime constants are in the range of 10–100 ms. The reasonable idle time after magnet
excitation should avoid the problem.
c. Cable coupling current (if cable is used) In case the cable is used, the couplingcurrent between the cable strands can cause a various kinds of error fields and thosecan be 10−4 T level. The coupling current also includes the various time constants
that can vary from the order of 10 ms to hours. It also causes the change in local fieldin the coil that affects the superconductor magnetization. Those are very complicatedand almost uncontrollable to achieve 10−7 field quality. One should avoid using a cablefor the magnet coil. The coil should be wound from a wire.
d. Persistent current operation Time constants are generally very large and should
not be a problem. There are some coupling with above three effects, however after areasonable idle time after magnet excitation all the effects should be stable such that
operation current change become stable.

E.

Field measurement and its absolute calibration

1.

Measurement methods for the experiment

There are many methods of measuring magnetic field: Hall probe, flux measurementswith pick up coils, nuclear magnetic resonance (NMR), and so on. The optimal choicedepends on the nature of the field to be measured; AC or DC field, gradient or absolute87

field, etc.
In this experiment, a superconducting solenoid having a high uniformity of the central
field is adopted for the muon storage magnet. From the field measurement view point, themeasurement region around the solenoid could be divided into two regions, that is, the beam
injection region and the beam storage region. (We also need to measure/control the tracking
region with moderate precession.)
The beam injection region is the region from the end of the injection beam line to thebeam storage region around the solenoid center, and it includes a stray field region outside
the solenoid. The muon beam passes through the beam injection region only one time,
therefore,the tolerance of the magnetic field error is much larger than that in the beamstorage region. It is enough to measure the magnetic field in the beam injection region withan accuracy of 100 ppm.
There are two field measurement methods satisfying that accuracy; the Hall probe method
and the flux measurement method. Although an NMR method also meets the required
accuracy, below 10 ppm, such a very high accuracy can be reached only in a stable andhomogeneous field, generally 1%/s and 0.1%/cm, so it is not suitable for a field measurement
in a high gradient field like a stray fringe field. Table XI summarizes the advantages anddisadvantages of the Hall probe and flux measurement methods. It is necessary to consider
the cryostat and support structure as well as the accuracy, further design study is required.

From the beam optics study, the beam storage region should be cylindrical shape withan inner diameter of about 280 mm, a outer diameter of 380 mm and a height of 400 mm(no interference with the kicker coils), and the region must have high field homogeneity at a
ppm level locally, and be measured along the muon storage orbit (integrated) to < 0.1 ppm,
with a central field of 3 T. The NMR method is the only solution for measuring with thisprecession.
It is not easy to measure this magnetic field at a sub-ppm level, since there are no magnets
with such a high homogeneity at 3 T which can be used to fine-tune the NMR system. So,
the adjustment of the NMR system at a sub-ppm level of accuracy is planned to progress inparallel with the development of the superconducting solenoid for muon storage.

88

TABLE XI: Summary of advantages and disadvantages of the Hall probe and flux measurement.
MethodAdvantage

Disadvantage

Hall probe

SimpleNon-linear output

Inexpensive

Sensitive to temperature

Commercially available

Long term drift

Fast measurement

Need to compensate Planar Hall effect

Small probe size
Flux measurement

SimpleNeed change in flux

Passive device

Need calibration of geometry

Linear output

Need integrator device

Drift-free device

This strategy has two steps of development of the NMR measurement system. In thefirst step, a commercial NMR system with an accuracy of 1 ppm will be purchased for arough evaluation of the solenoid field at the initial R&D phase. Based on the measurement
results, the solenoid field would be regulated to reach a field homogeneity of 1 ppm in the
beam storage region.
In the second step, an NMR system which can reach a sub-ppm level precision will bepurchased, and would be fine-tuned using the solenoid in collaboration with the company
making the NMR system.
In both steps, a probe positioning system made of nonmagnetic materials should be
developed, which can control the probe position in the solenoid in the cylindrical coordinate
system. The precision of 1 mm should be sufficient for the position control, since thesize of the probe would be 5 mm in diameter. That precision could be easily achievablewith commercial devices, however, materials composing the positioning system should be
carefully selected. It should be placed near the solenoid so that a limited space around thesolenoid could be made effective use, and the accuracy of the position control could be easily
improved.

89

2.

Present status

In the first step of the development of the NMR measurement system, a field distribution
measurement in a 3 T solenoid is planned using an NMR system with 1 ppm accuracy. The
objectives of the measurement are to fine-tune the NMR system and to build the prototype
of positioning system.
Arrangements to use a 3 T solenoid are successfully progressing with the National Institute of Radiological Sciences at Chiba city. The solenoid is designed for MRI medical
applications, and placed horizontally, which is different from the solenoid for the g − 2 experiment, which will be placed vertically. However, this MRI magnet will be useful to check
whether the components of the positioning system work in a high magnetic field and provide
a high accuracy field measurement. Figure 50 shows the overview of the field measurement
system for the horizontal solenoid. Design work is in progress.
3T solenoid
Encoder &Ultrasonic motor

Rotating disk NMR probe
Ceramic bearings
Rotating shaft

Timing belt
Roller &Stepping motor

Moving stage

Guide rail

FIG. 50: Overview of the positioning system for the field distribution measurement in the MRI
solenoid at NIRS.

90

F.

Cryogenic system

A conceptual design proposal of the storage ring cryogenic system is shown in Fig. 51.
The requirements to the cryogenics from the view point of good field quality of the solenoid
are 1) sub-micron mechanical stability, 2) good temperature controllability and stabilityof superconducting coil. There are also additional requirements to the room temperaturesystem to avoid mechanical vibration to the magnet system. The temperature controllabilityof room temperature iron yoke should be guarantee the temperature stability of ±0.1 K.
Since the superconducting solenoid will be made from the wire and operated with a relatively
low current, the system may be cooled with several cryo-coolers. The cooling of the coil,
however, should be made by liquid helium to ensure the good temperature stability. The
temperature control can be made with a pressure control of the helium vessel. The precisepressure control system should be implemented. The cryo-coolers are used for recondensationof helium. Number of corrector coils as well as their operation current may affect to thecryogenics heat load. They should be optimized such that the system can be achieved withcryo-coolers. Since the vibration of the system may cause the devastating effect to thefield quality vibration dumpers should be carefully implemented to achieve vibration level
below 0.1 micron for the solenoid system. The study now being performed for the ILCIRQ(International Linear Collider Interaction Region Quadropole magnet) aiming for the 10 nm
level stability and their R&D results may be used for this system as well. Although this is
not a part of cryogenics the biggest challenge for the system is the control of the iron yoke
temperature. The temperature stability and uniformity of ±0.1 K is currently required witha preliminary design study. The room temperature shield room design must be performed
very carefully to achieve this limit.

91

FIG. 51: A conceptual design proposal of the storage ring cryogenic system.

92

VII.
A.

MEASUREMENT OF ωaSpin-dependent decay from the V−A theory

We begin this section by introducing the spin-dependent µ decay and its analyzing power.
The positive muon decays into a positron, a neutrino and anti-neutrino; µ+ → e+ + ν¯µ + νe .
Parity non-conservation in the weak decay and helicity conservation lead to an asymmetry
in the angular distribution of the decay positrons with respect to the direction of the µ-spin(θS ). Viewing the decay in the muon rest frame, the decay e+ has its maximum momentum
and is 100% polarized when the ν¯µ and νe are emitted to the opposite direction of the e+ , asshown in Fig. 52. Here, e+ and ν¯µ are right-handed, νe is left-handed. The maximum value ofthe positron momentum is approximately half of the mass of µ; (p∗e )max = mµ /2 = 53 MeV/c.
We write for the decay electron momentum relative to its maximum,
η=

p∗e
.
(p∗e )max(19)µ+ rest frame
νµ

µ

+

e+ momentume+ spinνe

FIG. 52: Viewing the decay in the muon rest frame, the decay e+ (right-handed) has its maximum momentum and polarization when ν¯µ (right-handed) and νe (left-handed) are emitted to the
opposite direction to the direction of e+ .

The probability for a positron, whose momentum range is η → η + dη, to be emitted into93

a solid angle dΩ at an angle θS is given by the V−A theory[63, 64];dM(η, θS ) = n(η)[1 + a(η)Pµ cos θS ]dηdΩ,n(η) = 2η 2 (3 − 2η),
2η − 1a(η) =.
3 − 2η

(20)~ and pˆe+ and Pµ is the polarization of the muon.
θS is the angle between S
A two-dimensional plot of dM(η, θS ) is displayed in the top plot in Fig. 53. n(η) anda(η) are displayed in the bottom plot in Fig. 53.
The decay event introduced in Fig. 52 corresponds to the right top point in Fig. 53(cos θS = 1 and η = 1).

B.

g − 2 precession in the magnetic field~ = (0, 0, B), the muon moves on the plane
In the presence of a static magnetic field, B~ Here, we assume that both the electric field and the electric dipole
perpendicular to B.
moment are zero.
The cyclotron frequency ω~ c and spin frequency ω~ S are expressed as~qB,m µ γµ
~
~
gq B
qB
ω
~S = −− (1 − γµ ).
2mµm µ γµω~c = −(21)

The anomalous precession frequency ω~ a is determined from the difference:~qB
ω~a = ω~S − ω~c = −
mµ

µ

g−2
2

¶
.

(22)

~ |= 3 T, we obtain cyclotron period
For example we input γµ = 3 (pµ = 300 MeV/c) and | B
Tc = 2πωc = 7.41 ns and anomalous period 2πωa = 2111.7 ns.
~µ ) rotates with the
The µ-decay is described in the muon rest frame, where the µ-spin (Sfrequency of ω~a~µ = (cos(ωa t), sin(ωa t), 0).
S(23)

We introduce the definition of angles in the muon rest frame. Figure 54 (left) displays
the system of the muon rest frame. The direction of the µ beam is along the x-axis, the94cosθS

1

400

0.8

350

0.6
300

0.4
0.2

250

0

200

-0.2

150

-0.4
100
-0.6
50

-0.8
-1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

n(η), a(η)

η
2.5

η

th

2

n(η)

1.5

1
0.5

a(η)

0

−0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

FIG. 53: Top: Probability of positron emission as a function of η and θS for Pµ = 1. Decay
positrons tend to be emitted to the muon spin direction (cos θS ∼ 1) with their highest energy(η ∼ 1). Interestingly, note that cos θS distribution is flat at η = 0.5.
Bottom: Analyzing power a(η) and event fraction n(η) as functions of η, the positron energy
fraction in the muon rest frame. ηth is the optimal threshold to measure the spin precession.

direction of the magnetic field is along the z-axis. The spin vector is displayed as red arrow−→~ = (0, 0, B).
and precesses as Sµ =(cos(ωa t), sin(ωa t), 0) in the static magnetic field B
In Fig. 54 (right), we adopt spherical polar coordinates (1, θ∗ , φ∗ ) in the muon rest frame.
−→
The polar angle is θ∗ . φ∗ is zero when p∗e is in the x − y plane (the muon spin precession
plane). The unit vector of the decay positron momentum is displayed as a blue arrow andis then expressed as−→∗ −→pe /| p∗e | = (cos θ∗ , sin θ∗ cos φ∗ , sin θ∗ sin φ∗ ).
95

(24)

θS is the angle between S~µ and P~e∗ /|P~e∗ |. cos θs is obtained as,p~∗cos θS = S~µ · e∗
|p~e |= sin θ∗ cos φ∗ sin(ωa t) + cos θ∗ cos(ωa t).

(25)

Note that cos θS behaves as a trigonometric function of (ωa t). There are 4 types of decays
we could detect; forward, backward, left, and right as displayed in Fig. 55.
Since S~µ is rotating in the x − y plane, the decay positron spectrum for any direction
behaves as a trigonometric function of (ωa t):µ¶Z Zt
Ne (t) = N0 · exp −n(η){1 + a(η)Pµ cos θS }dηdΩ.
γµ τ

(26)

Finally, the time spectra of four types of acceptance combinations for φ∗ and cos θ∗ arelisted in Table XII.
TABLE XII:R

cos θ∗ d(cos θ∗ ) of several integral regions for forward-decay.

decay typeθ∗φ∗forward0 ≤ θ∗ ≤ π/3

0 ≤ φ∗ ≤ 2π

backward

2π/3 ≤ θ∗ ≤ π

0 ≤ φ∗ ≤ 2π

left

π/4 ≤ θ∗ ≤ 3π/4 −π/3 ≤ φ∗ ≤ π/3

right

π/4 ≤ θ∗ ≤ 3π/4 2π/3 ≤ φ∗ ≤ 4π/3

In the case where we detect decay positrons which are emitted in the direction of the~ and the x-axis (=direction of p̂e+ ).
x-axis only, p̂e+ = (1, 0, 0), θS is the angle between S
Therefore, θS is simply written as
θS = ωa t.
The time spectrum of decay positrons in this case is expressed as¶Zµtn(η){1 + a(η)Pµ cos(ωa t)}dη.
Ne (t) = N0 · exp −
γµ τ

(27)

(28)

In order to extract ωa , we should detect positrons which have the highest asymmetry:a(η) = 1 at η = 1. However, we lose statistics of decay positrons if we are limited only
within a narrow η-bin. We can determine an optimal threshold to measure ωa shown as ηthin Fig. 53 (bottom). However, applying this threshold requires reconstructing the decay inthe muon rest frame, which cannot be done due to an uncertainty in the positron momentum
direction in the muon rest frame. We consider the decay, instead, in the laboratory frame.
96

C.

Lorentz boost

The muon rest frame is rotating in the lab frame, as displayed in Fig. 56. Decay positrons
in the Lab-frame are boosted as follows:

 


∗
Ee
γ γµ βµ
Ee

= µ

,
pe cos θγµ β µ γµp∗e cos θ∗

(29)

and
pe sin θ = p∗e sin θ∗ .

(30)Here, p~∗e = p∗e (cos θ∗ , sin θ∗ cos φ∗ , sin θ∗ sin φ∗ ) and θ∗ and θ∗ are in the spherical polar coordinate system in the muon rest frame. φ∗ is an angle on the plane perpendicular to thex-axis, therefore there is no change between the frames,
φ = φ∗ .

(31)

The left plot in Fig. 57 displays the decay positron energy spectrum in the muon rest
frame. η is normalized by its maximum energy. The right plot displays the energy spectrumof decay positrons in the Lab frame, with a µ-beam momentum of pµ = 300 MeV/c.
Figure 58 displays the Elab vs. cos θlab scatter plot for all positrons generated by Geant4,
Pµ = 1. θlab is the cone angle with respect to the XLab -axis in the rotation frame asshown in Fig. 56. The positrons are boosted towards the direction of the muon momentum(cos θlab ≥ 0.96 or θlab ≤ 0.3 [rad]).
As we introduced, we have spin-dependent spectra by selecting events of η > 0.75 in acertain solid cone angle as in Table XII. In the next section, we will discuss energy spectraof such solid angles.

97

D.

Energy spectra in the lab frame

Figure 59 displays energy spectra in the lab frame for forward, backward, left and rightdecays as seen in the muon rest frame in Fig. 55. The thin black line is the histogram forall decays. The thick black line corresponds to η > 0.75, when η is the positron energy
fraction in the muon rest frame. The pink and purple filled areas are forward and backward
decays. The green and orange thick lines are left and right decays. (Table XII defines theacceptances.)
The emission angle in the muon rest frame determines whether the positrons in the labframe are strongly boosted or not, and cause energy dependence in the lab frame, withforward decays highly boosted, with Elab > 180 MeV.
Figure 60 shows the positron time spectra for 65 M simulated events for a set of positronenergy ranges in the lab frame. This corresponds to Eq. 26. Integral interval for η and theta∗
in the µ frame are related to the lab frame energy Elab interval as discussed in Eq. 33. The
spin precession ”wiggle” is evident for Elab ≥ 175 MeV, and is also visible at low positronenergy.

E.

Event selection by lab-energy thresholdWe can select forward-decay by applying a proper energy threshold in the lab frame.
We do not need to know of the information in the muon rest frame. In this section, wethwill discuss the effective analyzing power as a function of energy threshold (Elab) in the labframe [64]. The first goal of this section is to introduce an effective analyzing power for
forward-decay and the acceptance in the muon rest frame. We can then optimize the energy
threshold, using a figure of merit, for the measurement of ωa . we choose appropriate energy
threshold.

1.

Relation between Elab , η and θ∗th
To transform Elabin the lab frame into a boundary condition in the muon rest frame, we

set the mass of the positron me = 0, βµ = 1 and express the energy or momentum in units98of (p∗e )max (i.e. 1/2mµ ). Equations 29 in this particular case become;Elabcos θ = ηγµ (1 + cos θ∗ ),
mµ /2
Elab
sin θ = η sin θ∗ ,
mµ /2

(32)

θ is the angle of the positron and µ directions in the lab frame. Elab is given by
Elab
=
mµ /2

q
η 2 sin2 θ∗ + η 2 γµ2 (1 + cos θ∗ )2 .

(33)

Taking γµ >> 1 and (Elab /η)2 >> 1, we obtaincos θ∗ =

Elab
− 1.
ηγµ

(34)

From this equation, we relate Elab to θ∗ and η.
The upper scatter plot in Fig. 61 displays correlation between cos θ∗ and Elab . These
events are all generated by Geant4. The boundary of the scatter plot corresponds to
Eq. 34 at η = 1, which is shown in the black solid line in the middle and bottom plots. The
blue and red dotted lines correspond to η = 0.5 and η = 0.75. The filled area in green in
the middle plot is forward-decay in the muon rest frame. The filled area in blue, edged withpink in the bottom plot represents forward-decay selected by Elab . These plots suggest themost of the areas of the two types of selected forward-decays are common.
th
In the next subsection, we will discuss how to find a proper energy threshold Elabat givenγµ .

2.

th for the maximum FOM
Appropriate Elabth
We describe the relative number of events above an energy threshold of Elab ≥ Elabas

C th . From Eq. 26 integration over Ω and η:
R
R
R ∗
∗
n(η)dη
d(cos
θ
)
dφ
R
R
.
C th = R∗dη d(cos θ ) dφ∗th
The analyzing power for Elab ≥ Elab:RRRn(η)a(η){cos θ∗ cos(ωa · t) + sin θ∗ cos φ∗ sin(ωa · t)}dηd(cos θ∗ )dφ∗
th
R
R
R
C ·A=
.
dη d(cos θ∗ ) dφ∗

99

(35)

(36)

The two integrals are not independent in the lab frame but are related to Elab as shownin Eq. 34. By using Eq. 34, two equations become
·
¸
5b b3
b4
th
C = 1−
+ +
,
6
8
48
¸
·
b4
b b3
th
− +
C ·A =
,
6
8
24

(37)

th
th
is unit of mµ /2. The figure of merit for the measurement of ωa is/γµ and Elabb = Elab

C th · A2 . Figure 62 displays C th and A and FOM ∝ as a function of b.
th
In this way, we can select an appropriate E th to have the maximum FOM at any γµ . Elabis 200 MeV for γµ = 3. And we have C th = 0.13 and A = 0.46. This laboratory energy
threshold selects forward positron decays in the muon rest frame, and give the optimum
figure of merit for measuring ωaF.

Confirmation of Geant4

Finally, we estimate C th and A by Geant4 applying Eeth = 200 MeV with a condition ofγµ = 3 and B = 3 T. Event fraction in red area in Fig. 63 is 13 %. This is consistent withexpected value from 62. Time spectrum generated by Geant4 confirms A by applying fit
function,
F (t) = F · exp(−tγµ τ) [1 + A cos(ωa t)] ,(38)

with three free parameters of F , A and ωa .
Extracted anomalous period 2π/ωa from time spectrum is 2112.1 ns, and consistent withexpected value 2111.7 ns within the statistical uncertainty of the simulated events.

100

p+

µ rest frame

|pe+|

Y

Z, B

e

H

Sµ

X, βµ

O

µ rest frame

Z, B
H

P

Y

p+
φ* = YOH

e

|p +|
e

θ* = XOP
X, βµO

FIG. 54: Top: The direction of the the beam (along the x-axis) and the direction of the magnetic
field (along the y-axis) in the muon rest frame are defined. The spin vector, displayed as a red~µ = (cos(ωa · t), sin(ωa · t), 0). p~e is the momentum of the positron. θS is thearrow, precesses as S~µ and P~e∗ /|P~e∗ |. Bottom: We adopt the spherical polar coordinates (1, θ∗ , φ∗ ) with
angle between S
the polar angle θ∗ and φ∗ = 0 on the plane of the muon spin precession (the x-y plane). The unit
vector of the decay positron is P~e∗ / | P~e∗ | =(cos θ∗ , sin θ∗ cos φ∗ , sin θ∗ sin φ∗ ) as displayed as a blue
~ and thearrow. θ∗ is the angle between the blue arrow and the x-axis. φ∗ is the angle between OH
y-axis. Here H is the projection of the blue arrow in the y − z plane.

101µ rest frame

Left
Y

Sµ
p+
e

Backwardp+

Forward
p+

e

e

p+

X, βµ

e

Z, B
Right

FIG. 55: Examples of forward, backward, left and right-decays are shown.

102

Rotation frame
X Rot, βµµ orbit
B

ZRot
Y Rot

Y Lab

X Lab
ZLab
Lab frame

FIG. 56: This figure explains the rotation frame. XLab , YLab and ZLab denote the lab frame. In
the presence of a static magnetic field along the ZLab -axis, a µ particle with non-zero βµ undergoes
cyclotron motion in the XLab − YLab plane. The origin of the rotation coordinate system is the µdecay point. We set the XRot -axis to be the direction of the µ beam (βµ ), and the ZRot -axis isparallel to the ZLab -axis.

103counts

25000
η=
20000

pe*
(pe*)max(pe*)max = 53 MeV/c
15000

10000

5000

count

0
0

0.2

0.4

0.6

0.8

1

η

7000
6000
5000
4000
3000
2000
1000
0
0

50

100

150

200

250

300

350

400
MeV

FIG. 57: Energy spectra simulation generated by Geant4. Top: Emission energy spectrum of
decay positrons in the muon rest frame, normalized by its maximum energy (p∗e )max . Bottom:
Emission energy spectrum of decay positrons in the lab frame. We set the µ-beam momentum to
300 MeV/c for this simulation.

104

cosθlab

1

2000

0.8

1800

Forward positrons

0.6

1600

0.4

1400

0.2

1200

0

1000

−0.2

800

−0.4

600

−0.6

400

−0.8

200

−1
0

50

100

150

200

250

300

350
MeV

0

FIG. 58: Elab vs. cos θlab scatter plot for all positrons generated by Geant4, Pµ = 1. Positrons
with Elab ≥ 200 MeV are emitted in the forward direction (θlab ≤ 0.3 [rad]) with respect to the µcountdirection in the muon rest frame.

7000
all
η > 0.75
ForwardBackward
Left
Right

6000
5000
4000
3000
2000
1000
0
0

50

100

150

200

250

300

350

400
MeV

FIG. 59: Energy spectra of the decay positrons from µ (pµ = 300 MeV/c) in the lab frame. Grayline is same as right plot of Fig. 57. The thin black line is the histogram for all decays. The thickblack line corresponds to η > 0.75. The pink and purple filled areas are forward and backward
decays. The green and orange thick lines are left and right decays. Although we select events
η > 0.75 in the muon rest frame, the emission angle affects the positron energies in the lab frame.

105

25−50 MeV

Count/T cyc

Count/T cyc

Count/T cyc

0−25 MeV

50−75 MeV

102

10 2

102

10

10

10

75−100 MeV

1010000 20000 30000 40000 50000
time [nsec]

100−125 MeVCount/T cyc

1010000 20000 30000 40000 50000
time [nsec]

Count/T cyc

Count/T cyc

10

125−150 MeV

102

10 2

102

10

10

10

2

10

10

Count/T cyc

10

175−200 MeV
2

10

10

10

10000 20000 30000 40000 50000
time [nsec]

225−250 MeV

Count/T cyc

Count/T cyc

150−175 MeV

1
010000 20000 30000 40000 50000
time [nsec]

2

10

10

10000 20000 30000 40000 50000
time [nsec]

250−275 MeV

10 2

102

10

10

10

10000 20000 30000 40000 50000
time [nsec]

1010000 20000 30000 40000 50000
time [nsec]

10000 20000 30000 40000 50000
time [nsec]

275−300 MeV

102

1
0

10000 20000 30000 40000 50000
time [nsec]

200−225 MeVCount/T cyc

10

1
0

10000 20000 30000 40000 50000
time [nsec]

Count/T cyc

Count/T cyc

10

10000 20000 30000 40000 50000
time [nsec]

1010000 20000 30000 40000 50000
time [nsec]

FIG. 60: Positron time spectra generated by Geant4 for 65M events for several energies, with
Pmu = 1. Spectra of Elab < 100 MeV (Elab > 175 MeV) correspond to backward-decay (forwarddecay) in the muon rest frame.

106cosθ*

1
1800

0.8

1600

0.6

1400

0.4
0.2

1200

0

1000

−0.2

800

−0.4

600

−0.6

400

−0.8

200

cosθ*

−1
0

50

100

150

200

250

300

350
MeV

0

1
η=0.5

0.8
0.6
0.4

η=0.75

0.2
0
−0.2
−0.4

η=1

−0.6
−0.8

cosθ*

−1
0

50

100

150

200

250

300
350
Elab (MeV)

200

250

300
350
Elab (MeV)

1
η=0.5

0.8
0.6
0.4

η=0.75

0.2
0
−0.2
−0.4

η=1

−0.6
−0.8
−1
0

50

100

150

FIG. 61: Up: Elab vs. cos θ∗ scatter for all positrons generated by Geant4. The boundary comesfrom Eq. 34 at η = 1.
Middle and bottom: Correlation between cos θ∗ and Elab introduced in Eq. 34. The black solidline corresponds to η = 1. blue and red dotted lines correspond to η = 0.5 and η = 0.75. The filledarea in green (middle plot) is the selected area by parameters in the mu rest frame. The filled areain blue, edged with pink for is for the lab-threshold.

107

1.4

Lab frame

1.2
1

Cth
A

0.8
0.6
0.4

CthA2 (∝ FOM)

0.2
00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b=Eth
/γ
lab µ
FIG. 62: At b = 1.27, FOM is maximum and effective analyzing power is 0.46.

count

7000
6000
5000
4000
3000
2000
1000
0
0

50

100

150

200

250

300

350

400
MeV

FIG. 63: Energy spectrum of decayed positrons. Above Ee > 200 MeV region, which is filled inred, is 13%. This is consistent with C th = 0.13 from Fig. 62.

108

Count/Tcyc

103

χ2/ndf = 5282.2/4941
2π/ ω a=2112.05 [ns]∆ ωa/ ωa = 2.39e−04
A = 0.4593+/−0.0020
3.3M events

102

10

10

10000 20000 30000 40000 50000 60000 70000
time [nsec]

FIG. 64: Time spectrum of decayed positrons of Ee ≥ 200 MeV. From fitting results, amplitude is
0.46. This is consistent with A in Fig. 62.

109

VIII.

DETECTION OF DECAY POSITRON

Detectors to measure the decay positrons from the muon beam are discussed in thissection.

A.

Requirements

After injection into the muon storage region, the positive muons circulate in a plane
parallel to the magnetic field with a radius of 33 cm. They decay into a positron and twoneutrinos with a time-dilated lifetime of 6.6 µs. The anomalous muon precession period is
2.2 µs, which is about 300 times the cyclotron period (7.4 ns), i.e. the muon spin rotates
360 degrees in every 300 turns in the storage region. Because of the parity violation in the
weak decay of the muon, there is a correlation between the muon spin and the directionof the positron momentum. The correlation (analyzing power) has an energy and angulardependence.
In the muon center of mass frame, the positrons tend to decay towards the direction of
the muon spin at the maximum positron energy of Emax =52.8 MeV (the Michel edge). The
analyzing power becomes smaller in the mid-energy range of E/Emax ∼ 0.5, then becomes
negative in the lowest energy range of E/Emax ∼ 0, where the direction of the positron
tends to be opposite of the the muon spin direction. This correlation allows one to measure
the time dependence of the muon spin direction by detecting positrons and their energy as
a function of time.
This correlation remains in the laboratory frame. From Fig. 62, it is clear that higher
analysing power is obtained with the higher energy threshold, while the number of positrons
decreases. The statistical uncertainty is proportional to the figure-of-merit (FOM) C th A2 .
The FOM is maximum at around E th = 190 MeV. The maximum positron energy in thelaboratory frame is 310 MeV. Thus the positron detector should be efficient, especially todetect positrons in the energy region above ∼190 MeV.
Due to the Lorentz boost, the positron direction is confined to a narrow cone parallelto the parent muon direction. The positrons curl into the inner part of the muon storage
region where the magnetic field is as strong as that in the muon storage region (3 T). The
positron momentum component parallel to the magnetic field causes it to drift vertically.

110

The detector system needs to large enough to accept positrons which curl to the inside of themuon storage region with a vertical drift. We have set the volume for the detector system
as a tube inside the muon storage area with 40 cm in height and 27 cm in radius.
The time sequence of the g − 2 measurement is shown in Fig. 65. The muon beam spill
comes every 40 ms schyncronously with the J-PARC RF signal. The proposed rate of themuon beam is 106 /s = 40000 muons per spill. About 10 percent of the decay positrons, 4000
positrons per spill, survive the minimum energy cut mentioned above. We anticipate thatthe contamination in the detector from positrons with other energies would be of the sameorder as the signal positrons. Therefore, about 10000 positrons enter the detector system
per fill. The instantaneous rate of the whole detector system will be 10000 muons/6.6µs= 1.6 GHz at injection. This becomes 10 MHz after 33 µs. The individual detector ratewould be this rate times the number of detector hits per track divided by the numberof channels. The individual detectors must keep their performance (resolution, efficiency,
response function) stable as a function of time in order not to introduce any undesired bias
in the timing distribution which in turn will be a source of systematic uncertainty.
The detectors and their readout electronics must operate under a 3 T magnetic field. Anyelectric field from the detector and readout electronics should be sufficiently small so thatthe electric field in the muon storage region is negligible for the measurement (a few mV/m
for a ppb systematic uncertainty).
We propose to use an array of radial vanes made up of silicon detectors, radially surrounded by fine-segmented timing counters with, an active absorber material (EM calorimeters) on the inner side of the tracking detectors. The conceptual layout of the detector
system is shown in Fig. 66.

B.

Silicon tracker

The instantaneous rate will be as high as 1.6 GHz in the detection volume. High granularity is desired to suppress systematic uncertainty from pile-up. Tracking the positrons withposition sensitive detectors is beneficial in a sense that effects of pile-up can progressively
be suppresesd by requiring hit-to-hit correlations in the track reconstruction. The existanceof the strong magnetic field in the detection volume provides for momentum analysis.
Silicon strip detectors are considerd as the tracking detector since these detectors are111compact, highly segmented, have stable gain, and do not introduce high electric field in thedetection volume. The sensitive area of the silicon detector, we plan to use, is a double-sided
AC-coupled rectangular silicon sensor with vertical and stereo strips with 200 µm pitch. The
size of the sensor is 10 cm wide and 20 cm high. A total of 64 units of silicon strip planes
will be radially placed in the detection volume to efficiently detect the circular tracks of thepositrons. 32 units are used to form a set of vanes in the upper half of the detection volume,
while another 32 units are for the lower half. There are 1500 strips per unit, and 96k strips
in total.
At injection, there will be on average 12 tracks in the first cycrotron period (7.4 ns), andthe instantaneous rate would be about 1 MHz per strip. About 1 ns timing resolution isrequired to disentangle overlapped positron tracks. Figure 67 shows example decay positron
events for three positron energies. It can be seen that at the higher positron energy, thetrack path-length in the detection volume becomes longer, and thus more hits register in the
tracker. On the other hand, acceptance becomes less and less at lower energies. At energies
below 50 MeV, the tracker is completely insensitive to these unwanted positrons, i.e. theradius of curl up is so small that it can not be seen by any vane. Note that we are notsensitive to positrons at the highest energy (E>280 MeV) since the radius of the positron
track is similar to that of muons.

C.

Absorber/calorimeter

The inner tube of the detection volume with a radius of about 10 cm will be filledwith heavy material to block a range of lower energy positron tracks. This is to prevent
unwanted low-energy positron tracks from spiraling and continuing to keep hit the tracking
detector, resulting in an increase of data volume. The absorber material and configuration
will be chosen so that a calorimetoric measurement of positron energy will be possible. The
calorimetric information can be used to crosscheck the calibration of the track momentum
scale using a lower intensity muon beam. The track-integrated information, such as thetotal energy deposit and the energy-weighted position, will be useful in monitoring themuon beam intensity and position during the g − 2 measurement with full beam intensity.
Note that the instantaneous rate of the order of a GHz at injection would not allow retrivalof useful calorimeter information for individual track during the g − 2 measurement, with a112

FIG. 65: Time sequence of the g − 2 precession measurement

FIG. 66: Layout of the positoron detector systemforeseenable calorimeter segmentation.

113

FIG. 67: Example positoron trajectries in the detector system at three different energies of
positrons. The green circle is the muon beam orbit. The red trajectory is the trace of the positron
track. The white tracks are photons.

114

IX.

READOUT ELECTRONICS AND DAQ

The global timing structure of the g − 2 measurement is shown in Fig. 65. The beam
spill, which corresponds to the injection of the muon beam to the storage ring, arrives inevery 40 ms (25Hz). After the beam is moved to the good field region, the detector systemstarts the measurement of the decay positron. The measurement will continue for five life
times, i.e. 33 µs. The expected positron rate immediately after injection is 1600 MHz, andthe rate reduces to 10 MHz after 33 µs. The total number of decay positrons per spill is 10k,
which is estimated assuming 10 percent of the decay positrons after the energy cuts (4,000) plus 6,000 as background.
The very high instantaneous event rate requires fine granularity and timing resolution
for the detectors and front-end electronics in order to reduce the probability of signal pileup. The short duration (33 µs) of the measurement allows us to continuously take data of
the signal timing and/or amplitude from the detectors and to store the data in memories
on front-end electronics. The scheme can eliminate dead time and unexpected structure oftiming profile, i.e. the dead time introduces a rate-dependent inefficiency in the read out
which will skew the timing distribution. The long interval (40 ms) between the measurement
and the subsequent spill enables us to send the data after their proper processing to a backend DAQ system.
Figure 68 shows a conceptual block diagram of the front-end board. Signals from thetracker detectors, assuming a silicon strip detector, are processed by Amplifier Shaper Discriminator (ASD) circuits. The binary signal outputs by the ASD are then routed to Timeto-Digital Converter (TDC) circuits, where the arrival times of leading edges are stored ina buffer memory (Spill Buffer). The TDC measurement starts at receiving the “START”
signal and stops at the end of the preset period, which should be shorter than the Buffer
size. The resolution must be of order 1 ns and a dynamic range longer than 33 µs is required
for the TDC and the buffer memory. When the measurement stops, the front-end processor circuit starts data processing. The processor reads raw data from the buffer memory
and performs zero-suppression and re-formatting for the data compaction. A set of spill
data fragments, which includes a fragment header and trailer, is prepared and sent to theback-end DAQ system using a set of a high speed serializer and an optical transmitter (TX)
via an optical fiber. The fragment header and trailer contains Board Number Identifier
115

FIG. 68: Block diagram of the front-end board.

(BNID), Spill Number Identifier (SNID), status of the board, the data size, etc. and Cyclic
Redundancy Check (CRC). The real time control signals, e.g. Clock, Start, Reset and Test
Pulse Trigger, are provided from a set of an optical receiver and a de-serializer (RX). The
bidirectional links (a set of TX and RX) are also used for system configuration, control andmonitoring of a front-end board. The baud rate per the optical link will be higher than
2 Gbps. The foreseen data fragment size per spill is less than 100 kB per link, when each
100mm x 200mm double-sided silicon strip detector (1,500 channels strips) is covered by one
front-end board. The ASD and the TDC will be ASIC’s. Several ASIC’s are on the market
and have some of the desired functionality, but none could have all. We are ready to undertake a preliminary design effort. The back-end DAQ system collects the data fragmentsfrom front-end boards and builds a complete data set for each spill. The PC farm, which
consists of parallel PC’s, receives spill data sets via a Gigabit Ethernet network switch anddoes online analysis.

116

X.

SYSTEMATIC UNCERTAINTIESWe discuss here systematic issues and estimate systematic uncertainties, comparing th
the experience of the BNL E821 experiment.
The measurement consists of two separate precision measurements ; the magnetic field
as seen by the stored muons, and the frequency of the muon spin precession, as observed in
the high energy decay positrons.
For the measurement of B, the field, the main issues are to monitor the field in the storage
ring, vacuum, to map the field frequency, and to calibrate these probes with great precision.
The uncertainties, in these steps are greatly improved if the field is more uniform. The
expectation is for a 1 ppm locally flat field for the solenoid storage ring in this experimentvs. ± 50 ppm local variation for E821. Therefore, most issues for B should be significantly
improved for this experiment.
For the measurement of the spin precession frequency, this experiment will use high granularity tracking to overcome much higher instantaneous intensity than E821. All systematic
uncertainties for ωa derive from differences in detection early in the measurement time compared to late. The high granularity in this experiment leads to a concern that the detector
elements may not see the same proportion of positrons early to late. We must estimate
this uncertainty. Handling of pileup is crucial. this experiment will have the advantage ofno significant flash in in the detectors at injection, which was due to incoming pion alongwith the muon for E821. Several systematic uncertainty issues for E821 were related to the
flash – detector had to be turned off or down at injection, thresholds had to be set higher
accounting for the flash.
There are many differences of the proposed measurement and E821:• The storage ring magnet for this experiment will be much smaller diameter than E821,
and is expected to have a much more locally uniform field.
• The 1.5 T field will be used in this experiment.
• The instantaneous rate in the positron detection in this experiment will be over 100×greater than for E821, before consideration of detector granularity.
• This experiment will primarily use tracking to measure the positrons, while E821 use
calorimeters.
117• The granularity of the detectors will be 96k channels/ 6 hits/ track = 16k, while thegranularity for E821 was 24.
• The detector will be in the storage ring vacuum, while the E821 calorimeters wereoutside the vacuum, separated by a scalloped vacuum chamber wall.
• The detectors will be much closer to the muon decay point and reconstruction to the
decay point using the timed location of the narrow muon bunch may be possible.
The development of the design and its systematic uncertainties will be an ongoing process
throughout the design, measurement, and analysis stages. As it was for E821. At this time
we have used the published systematic uncertainty tables from E821 as a starting point todevelop table for this experiment. These follow, with estimates for J-PARC where possibleat this time, and with remarks.

118

119

< 0.1

0.09

Lost muons

0.08

AGS background

Pileup<<0.090.0<0.05(goal)

small focusing field, should avoid losses entirely

this experiment should reduce resonance loss. Also, if we use a verytion of resonance loss needs to be studied. Much reduced emittance inthere will be no material near the stored beam in this experiment; ques-background of this typethere is no stored beam in this experiment during data-taking, so no

not affect the time.

for the tracking would result in reduced efficiency and probably would

should be less for the same level of pileup. Also, pileup in this experimentmuch closer to the time of the muon decay, so that this uncertainty

observed. For this experiment, the time of the positron track will beapproach used in E821 has this uncertainty for the amount of pileup

to pileup was below threshold, it was not sampled and the correctionof the positron at the calorimeter). When a positron energy contributing

according to the wrong energy (the time of the muon decay vs. the timefrom not identifying two positrons in calorimeter and assigning the time

contribution to below 0.05 ppm. Uncertainty in ωa for E821 comes

start with 133x pileup E821, detector design requires reducing the pileup

Source of uncertainty E821-R01 (ppm) J-PARC (ppm) Remarks for this experiment

TABLE XIII: Systematic uncertainties for ωa are listed for the E821 experiment Run 01 (R01) and for this experiment.

120

<0.1

<0.1

E-field, pitch

Fitting/binning<<0.1?

0.12

–
0.21

Gain changes

OthersTotal (ωa )0.07 (goal)

–

0.00

Coherent Betatron Oscillation 0.07

<<0.1

<<0.1

Any other categories for this experiment are being searched for.

will not be shut down between fills.

injection. There will be no flash in this experiment, and the detector

were from turning off the photomultiplier tubes during the flash fromvature vs. time in store, also being studied. For E821 the gain changes

Tracking efficiency vs. rate issue is being studied. Possible bias on cur-ing with n=3 × 10−5 , fCBO = 490 us, 74 muon lifetimes)

not present in this experiment for small (or no) focusing field (for focus-

and how this might pull the frequency

For E821 this is an inclusive category for unaccounted for fitting issues

Fewer g − 2 cycles per lifetime in this experiment–need to discuss this.

rection if a small magnetic focusing field is used.

Very small pitch correction (tiny transverse divergence), no E-field cor-be turned off between fills. Should be much smaller in this experiment.

experiment, there will be no pions, no flash, and the detector will not

the flash of pions entering the ring with the muons, in E821. For thisThis came from the photomultiplier response to turning the PMs on after< 0.1

Timing shifts<<0.1

E821-R01 (ppm) J-PARC (ppm) Remarks for this experiment

Source of uncertainty

TABLE XIII: (continued)

TABLE XIV: Systematic uncertainties for B.
Source of errors

E821-R01 (ppm) J-PARC (ppm) Remarks for this experiment

Absolute calibration of standard 0.05

0.05

probe

Issue is how spherical is theabsolute probe–same issue forthis experiment

Calibration of trolley probes 0.09

<0.09

Much better field uniformity(field measurement on the muon

expected for this experiment,

trajectory)

so that transferring calibration from absolute probe totrolley probes should haveless uncertainty from position
errors.

Trolley measurement B0

0.05

<0.05

Much better field uniformity
in this experimentInterpolation using fixed probes 0.07

<0.07

Should be smaller in thisexperiment due to expected
uniformity and stability ofsmaller magnet

Uncertainty

frommuon 0.03<0.03

distribution

Should be small with tracking
and bunched beam

Inflector fringe field uncertainty 0.00

0.00

no inflector

Othersa–

Need to consider these, and

0.10

others?
Higher multipoles (should be negligible for solenoid for this experiment)
Trolley temperature
Trolley power supply voltage response
Eddy currents from kicker
Total systematic error on ωp

0.17<0.07(goal)

a

121

XI.

ELECTRIC DIPOLE MOMENTAs the anomalous magnetic moment of the muon, aµ has been measured for more thana half-century, the electric dipole moment of muon, dµ has been similarly searched for longyears. While aµ serves as a solid test ground for the Standard Model of the particle physics,
non-zero dµ value immediately means CP violation in the lepton sector. It is defined ase~d~µ = η
~s.
2mµ

(39)

dµ is parallel to the spin, ~s, as well as the magnetic moment ~µµ does. In the presence of the~ and E,~ the Hamiltonian of the system can be written as
static field B~ − d~µ · E.
~
H = −µ~µ · B(40)~ is odd under P and T transformations. Therefore nonzero value
The second term, −d~µ · Eof the dµ would violate CP , if CP T theorem holds.
The most recent measurement of dµ (as well as aµ ), E821[5] at Brookhaven National
Laboratory (BNL), has constrained dµ down to ≤ 1.9 × 10−19 e · cm. The dµ limit is, however,still required to be improved to be comparable to the limit from the electron with the lepton
universality. Obviously further clarification is required to conclude the indication of the newphysics for aµ and further improvement is required for the dµ .
In the presence of the electric dipole moment, a rotation would be obtained as" Ã!#~e η ~ ~ E
β×B+
.
ω
~η =
mµ 2
c

(41)

~ and static electric field E.
~
in the presence of the static magnetic field B
This EDM rotation is an additional rotation to the muon spin precession:
"
" Ã
!#
µ
¶ ~ ~#
~
1
β
×
E
e
η
E
e
~ − aµ −~+aµ B+
β~ × B. (42)
ω~ all = ω~a + ω~η = −mµγ2 − 1cmµ 2c
Eliminating the terms which relate to the electric field, Eq. 42 would simplifies the precession frequency asω~ all = ω~a + ω~η = −e h ~ η ³ ~ ~ ´iaµ B −
β×B .
mµ
2

(43)

Since the rotation axises due to aµ and dµ are orthogonal each other, separation of thesesignal should be possible. If not, we should rotate muon spin axis from the momentum
122direction to parallel to the magnetic field. In this way, the precession measurement will
become blind to the aµ . In this case, the time spectrum of positron is written as:µ¶t
Ne (t) = N0 · exp −{1 + APµ sin(ωη · t) · cos(ωa · t)}.
γµ τ

(44)

Because ωη is very very smaller than ωa , APµ sin(ωη · t) team can be treat as effective
amplitude, and should growth as a function of time.
In this case, the initial muon spin direction is parallel to the magnetic field. We should
not see spin precession of g − 2, if ωη is precisely zero. This means that we have a sensitivity
to the non-zero dµ . Upper plot in Fig. 69 displays a time spectrum of positron of dµ = 10−19 .
The statistics is the half amount to reach the 0.1 ppm for aµ . Lower plot in Fig. 69 displays
a ratio of time spectra for spin up and spin zero.
Figure 70 displays the same but dµ = 10−22 case.

123decay e + counts per Tc

N(spin ↑)=0.75E+12

1010

109

108

dµ =1.0e−19ωη = 2556.4 rad/s

107

Tη = 2.5 ms

0

5

10

15

20

25

×10−6

30

Spin up/zero ratio per 10 × Tc

[s]

1.05
1.04
Ratio =

1.03

N(spin ↑)N(spin 0)

1.02
1.01
1
0.99
0.98
0.97
0.96
0.950

5

10

15

20

25

×10−6

30
[s]

FIG. 69: Top: Time spectrum of positron of dµ = 10−19 for Pµ = 1 (Initial spin direction is parallelto the magnetic field). Bottom: The Ratio of two time spectrum of positron for spin up and spin
zero.

124

decay e + counts per Tc

N(spin ↑)=0.75E+12

1010

109

108

dµ =1.0e−22ωη = 2.4 rad/s

107

Tη = 2457.0 ms

0

5

10

15

20

25

×10−6

30

Spin up/zero ratio per 10 × Tc

[s]

1.001

1.00081.0006

Ratio =

1.0004N(spin ↑)N(spin 0)

1.0002
1

0.99980.99960.99940.99920.9990

5

10

15

20

25

×10−6

30
[s]

FIG. 70: Top: Time spectrum of positron of dµ = 10−22 for Pµ = 1 (Initial spin direction is parallelto the magnetic field). Bottom: The Ratio of two time spectrum of positron for spin up and spin
zero.

125

XII.

SCHEDULE AND COST

We intend to start the experiment in 2014 as shown in Fig. 71. There are four major areasof activities: the ultra-cold muon Source, the ultra-cold muon beam, the ultra-precision field,
and the detector system. Especially the development of the muon source represents a major
challenge. We plan to perform a test experiment at TRIUMF to optimize the muonium
production target in 2010. In parallel, we plan a test experiment at Rutherford with a
newly developed high-power laser system.
The Muon LINAC can be constructed in two years, and be commissioned together withthe source.
We have started a conceptual design of the ultra-precision field employing technology
developed for MRI. We expect to have an engineering drawing in a year so that construction
would start in about one year. The measurement scheme with NMR probes is being developed at this time. A test of this technique at 1.5 T is planned, followed by a test in a 3 T
magnet at National Institute of Radiological Science (NIRS). A reasonably long commissioning time is allocated for the magnet, since it would be one of the most time-consuming
process to shim the magnet to sub-ppm level.
The physics production run should be divided into a few phases at least; the first stepwould be to reach similar precision to the E821, then further improvements would follow.
A cost estimate is given in Table XV. This is very preliminary without engineering
designs, but we will give the basis for each number.
• Surface muon transport
The surface muons produced at the production target will be transported to experimental hall for muonium production. The first capture solenoid should be maderadiation resistive. Therefore, the cost estimate is based on the current conceptual
design which requires 960 m of minerally insulated coil (MIC). The super conducting
curved solenoid is assumed for further transport line to maximize the transport efficiency. In addition, the area should be shielded very carefully. Detailed cost estimate
requires more detailed facility design, which is the next step.
• Laser Ionization of Mu
The laser for this experiment is being prepared at RIKEN.
126

FIG. 71: Preliminary schedule of the efforts towards the experiment.

• Muon LINAC
The current design assumes significant overlap in design with the existing J-PARC
LINAC. This approach is expected to reduce the cost of the LINAC.
• Ultra-precision magnet and monitor
The current design of the magnet resulted in the stored energy of ∼30 MJ. We areworking on the reduction of the energy. A major cost driver for the magnet would beperson-power to achieve the required precision. Since the magnet is the heart of theexperiment, we will work out the details with the in-house experts. Consequently, thecost would be minimized.
• Detector
We currently assume silicon detector for the tracking detector for muon decays. It
would be followed by the absorber/calorimeter.

127

TABLE XV: Preliminary estimate of the cost of this experiment.
Item

Cost (Oku-yen)

Surface Muon Transport

Facility

Ultra-Cold Muon Source
High-power Laser System

3.0

Initial Acceleration System

0.5

Muon LINAC

15Ultra-precision Magnet
Solenoid

10

Field Monitor

1

Detector System
Silicon Tracker

1.5

Readout Electronics

0.5

TOTAL

32 + Facility

128

XIII.

SUMMARY

We propose to measure the anomalous magnetic moment of the positive muon aµ down to
the level of 0.1 ppm with a novel technique utilizing an ultra-cold muon beam accelerated to
320 MeV and a 66 cm diameter precision magnetic storage ring without focusing field. The
beam will be also useful in measuring the electric dipole moment beyond current precision.
The proposed measurement will provide a rigorous test of the Standard Model of particle
physics as demonstrated by previous experiments. Our measurement will be complimentary
to the previous measurement which was done at 3.1 GeV, the “magic“ energy approach witha 14 m diameter storage ring. This proposed experiment will have very different systematics
from the previous experiment.
There are many challenges in realizing the experiment. Those include• increasing the surface muon flux,• optimizing the muonium target,• realizing the high-power Lyman-α laser,• developing the scheme to maximize the muon polarization,• finishing the actual design of the re-acceleration system,• monitoring the very low intensity muon beam,• realizing the ultra-precision magnetic field with zero electric field, and• developing the high-rate tracking detctor system.
Although we have touched the most of the areas mentioned above, and we have developed
the time-line of these R&D activities, we are almost sure that we will have new challenges
on the way to realizing the experiment. It is the nature of this kind of precision physics,
which requires continuing efforts.

We conclude this proposal by requesting the lab’s deep involvement into thisnew experiment, which has full of new challenges!

129[1] T.Kinoshita and M.Nio, Phys. Rev. D73, 013003 (2006); T.Aoyama et al, Phys. Rev. Lett.
99, 110406 (2007); T.Kinoshita and M.Nio, Phys. Rev. D70, 113001 (2004); T.Kinoshita,
Nucl. Phys. B144, 206 (2005) (Proc. Supp.); T.Kinoshita and M.Nio, Phys. Rev. D73, 053007(2006); A.L. Kataev, arXiv:hep-ph/0602098; M. Passera, J. Phys. G31, 75 (2005).
[2] K. Hagiwara, R. Liao, A.D. Martin, D. Nomura, and T. Teubner, Preliminary results shownat Inernational Workshop, Phipsi09.
[3] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys.Rev.D67 (2003) 073006. Erratumibid.D73 (2006) 119901.
[4] J. Prades, E. de Rafael, and A. Vainshtein, e-Print: arXiv:0901.0306 [hep-ph][5] G.W. Bennett et al. Muon G-2 Collaboration, Phys. Rev. D73 (2006) 072003.
[6] Proposal submitted to Fermilab; The New (g - 2) Experiment: A Proposal to Measure theMuon Anomalous Magnetic Moment to ±0.14 ppm Precision, New (g − 2) Collaboration.
Contactpersons: D. Hertzog, and B. Roberts.
[7] R. Lafaye, T. Plehn, M. Raunch and D. Zerwas, arXiv:0709.3985 [hep-ph].
[8] M. Alexander, S. Kreiss, R. Lafaye, T. Plehn, M. Rauch, and D. Zerwas, Chapter 9 in M. M.
Nojiri et al., arXiv:0802.3672 [hep-ph].
[9] Hertzog, Miller, de Rafael, Roberts and Stöckinger, hep-ph/0705.4617v1.
[10] J. Hisano and K. Tobe, Phys.Lett.B510 (2001) 197-204.
[11] J.P. Archambault, A. Czarnecki, and M. Pospelov. Phys. Rev. D70 (2004) 073006.
[12] S. Eidelman and F. Jegerlehner, Z. Phys. C 67 (1995) 585.
[13] S. Eidelman, Proceedings of ICHEP-06, World Scientific, Vol.1, p.547, 2006.
[14] K. Hagiwara et al., Phys. Lett. B 649 (2007) 173.
[15] F. Jegerlehner, Nucl. Phys. Proc. Suppl. 181-182 (2008) 26.
[16] M. Davier et al., Eur. Phys. J. C 27 (2003) 497.
[17] M. Davier et al., Eur. Phys. J. C 31 (2003) 503.
[18] R.R. Akhmetshin et al., Phys. Lett. B 648 (2007) 28.
[19] M.N.Achasov et al., JETP 103 (2006) 380.
[20] S. Binner, H.J. Kühn, K. Melnikov, Phys. Lett. B 459 (1999) 279.
[21] M. Benayoun et al., Mod. Phys. Lett. A 14 (1999) 2605.

130

[22] F. Ambrosino et al., Phys. Lett. B 670 (2009) 285.
[23] B. Aubert et al., arXiv:0908.3589.
[24] F.V. Ignatov, Nucl. Phys. Proc. Suppl. 181-182 (2008) 101.
[25] B.Aubert et al., Phys. Rev. D 77 (2008) 092002.
[26] K. Hagiwara et al., Phys. Rev. D 69 (2004) 093003.
[27] F. Jegerlehner and A. Nyffeler, Phys. Rept. 477 (2009) 1.
[28] J.F. Troconiz and F.J. Yndurain, Phys. Rev. D 71 (2005) 073008.
[29] Yu.M. Bystritsky et al., JETP Lett. 83 (2006) 51.
[30] M. Fujikawa et al., Phys. Rev. D 78, 072006 (2008).
[31] M. Benayoun et al., Eur. Phys. J. C 55 (2008) 199.
[32] M. Davier et al., arXiv:0906.5443.
[33] M. Davier et al., arXiv:0908.4300.
[34] S.I. Eidelman and S.I. Serednyakov, Nucl. Phys. Proc. Suppl. 131 (2004) 19.
[35] A.E. Dorokhov, Nucl. Phys. A 790 (2007) 481.
[36] T. Blum and S. Chowdhury, Nucl. Phys. Proc. Suppl. 189 (2009) 251.
[37] D. Stöckinger, Nucl. Phys. Proc. Suppl. 181-182 (2008) 32.
[38] S. Nagamiya, T. Nagae, Y. Ooyama, Y. Miyake, H. Takano, J.R. Helliwell, J.C. Peng, Energy
Review 19, 12(1999) 4-23
[39] G.Heidenreich, P. Baumann, A. Geissler, A. Strinning and W. Wagner, PSI Scientific ReportVI (1998) 16[40] N. Kawamura, S. Makimura, P. Strasser, A. Koda, H. Fujimori, K. Nishiyama, Y. Miyake,and MLF Muon Section, Nucl. Instr. and Meth. in Phys. Res. A 600(2009) 114-116[41] S. Makimura, Y. Miyake, N. Kawamura, P. Strasser, A. Koda, K. Shimomura, H. Fujimori,K. Nishiyama, M. Kato, K. Nakahara, R. Kadono, Y. Kobayashi, J. Sagawa, T. Nakamura,
M. Kaneko, H. Ozaki, H. Okamura, T. Suzuki, K. Fujimoto, K. Kira, Nucl. Instr. and Meth.
in Phys. Res. A:600 (2009) 146-149[42] P. A. Piroue and A. J. S. Smith, Phys. Rev. 148 (1966) 1315.
[43] HARP Collaboration, Eur.Phys.J.C53 (2008) 177-204.
[44] A.P. Mills Jr.et al., Phys. Rev. Lett. 56, 1463 (1986)[45] K. Nagamine, et al., Phys. Rev. Lett., 74, 4811 (1995)[46] P. Bakule, et al., Nucl. Instrum. Meth. B., 266, 335 (2008)131

[47] G.A. Beer, et al. Phys. Rev. Lett., 57, 671 (1986)[48] A. C. Janissen et al., Phys. Rev. A 42 (1990) 161.
[49] K. A. Woodle et al., Z. Phys. D 9 (1988) 59.
[50] R. F. Kiefl et al.,Hyp. Int. 6 (1979) 185.
[51] G. M. Marshall et al., Phys. lett. 65A (1978) 351.
[52] T. Ogawa, Y. Urata, S. Wada, K. Onodera, H. Machida, H. Sagae, M. Higuchi and K. Kodaira,Opt. Lett. 28, (1994) 2333
[53] W. Liu et al., Phys. Rev. Lett., 82 (1999), 711[54] V. W. Hughes, C. S. Wu, “Muon Physics volume III”, Academic press, New York (1975).
[55] T. M. Kojima, D. Tomono et al., JPSJ 76, 093501 (2007)[56] Y. Yamazaki ed., “Accelerator Technical DesignReport for J-PARC”, KEK-Report2002-13.
[57] T. Kageyama, et. al., “Development of Annular Coupled Structure”, Procs. of LINAC’94,
Tsukuba, 1994, p. 248 (1994).
[58] K. R. Crandall et. al.,“RFQ design codes”, LA-UR-96-1836.
[59] T. Kato, “Proposal of a separated-type proton drift tube linac for a medium-energy structure”,
KEK Report 92-10 (1992).
[60] K. R. Crandall, D. P. Rusthoi, “TRACE-3D Documentation”, Los Alamos National Laboratory Report, LA-UR-97-886, 1997.
[61] J. Qiang, R. D. Ryne, S. Habib, V. Decyk, J. Comput. Phys. 163, p. 434 (2000).
[62] Masao Nakamura, Japan Journal of Applied Physics, Vol.7, No.3, March, 1968.
[63] E. J. Konopinski,Annu. Rev. Nucl. Sci. 1959.9: 99-158.
[64] F. Combley and E. Picasso, Physics Report (Section C of Physics Letters)

132