High precision measurements of the ground state hyperfine
structure interval of muonium and of the muon magnetic
moment
Article (Published Version)
http://sro.sussex.ac.uk
Liu, W, Boshier, M G, Dhawan, S, van Dyck, O, Egan, P, Fei, X, Grosse Perdekamp, M, Hughes,
V W, Janousch, M, Jungmann, K, Kawall, D, Mariam, F G, Pillai, C, Prigl, R, zu Putlitz, G et al.
(1999) High precision measurements of the ground state hyperfine structure interval of muonium
and of the muon magnetic moment. Physical Review Letters, 82 (4). pp. 711-714.
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VOLUME
82, NUMBER 4 PHYSICAL REVIEW LETTERS 25J
ANUARY
1999
High Precision Measurements of the Ground State Hyperfine Structure Interval of Muonium
and of the Muon Magnetic Moment
W. Liu,
1
M.G. Boshier,
1
S. Dhawan,
1
O. van Dyck,
2
P. Egan,
3
X. Fei,
1
M. Grosse Perdekamp,
1
V.W. Hughes,
1
M. Janousch,
1,4
K. Jungmann,
5
D. Kawall,
1
F.G. Mariam,
6
C. Pillai,
2
R. Prigl,
1,6
G. zu Putlitz,
5
I. Reinhard,
5
W. Schwarz,
1,5
P. A. Thompson,
6
and K.A. Woodle
6
1
Department of Physics, Yale University, New Haven, Connecticut 06520-8121
2
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
3
Lawrence Livermore National Laboratory, Livermore, California 94550
4
ETH Zürich, Institute for Particle Physics, CH-5232 Villigen-PSI, Switzerland
5
Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
6
Brookhaven National Laboratory, Upton, New York 11973
(
Received 21 August 1998)
High precision measurements of two Zeeman hyperfine transitions in the ground state of muonium in
a strong magnetic field have been made at LAMPF using microwave magnetic resonance spectroscopy
and a resonance line narrowing technique. These determine the most precise values of the ground
state hyperfine structure interval of muonium Dn 4 463 302 765s53d Hz s12 ppbd, and of the ratio
of magnetic moments m
m
ym
p
3.183 345 13s39ds120 ppbd, representing a factor of 3 improvement.
Values of the mass ratio m
m
ym
e
and the fine structure constant a are derived from these results.
[S0031-9007(98)08281-7]
PACS numbers: 36.10.Dr, 12.20.Fv
Muonium (m
1
e
2
, M) is the hydrogenlike bound state
of a positive muon and an electron. However, unlike for
hydrogen, theoretical predictions of its ground state hyper-
fine structure can be obtained with high precision since the
complications of proton structure are absent. This enables
a most sensitive test of two-body bound state QED to be
made, with the assumption of e 2muniversality. Hy-
perfine transition measurements may also be used to ex-
tract the fundamental constants m
m
ym
p
and m
m
ym
e
.In
addition, using the theory, the fine structure constant a
can be determined and used to test the internal consistency
of QED through comparison with a determined from the
electron anomalous g factor, a
e
.
In a static magnetic field the muonium ground state
1
2
S
1y2
energy levels are described by the Hamiltonian [1]
H hDnI
m
? J 2m
m
B
g
0
m
I
m
?H1m
e
B
g
J
J?H, (1)
where I
m
is the muon spin operator, J is the electron to-
tal angular momentum operator, H is the external static
magnetic field, and the muon (electron) Bohr magneton is
denoted by m
m
B
sm
e
B
d. The gyromagnetic ratios of an elec-
tron bound in muonium, g
J
, and of a muon in muonium,
g
0
m
, differ from the free values, g
e
and g
m
, by binding cor-
rections [2]
g
J
g
e
√
1 2
a
2
3
1
a
2
2
m
e
m
m
1
a
3
4p
!
, (2)
g
0
m
g
m
√
1 2
a
2
3
1
a
2
2
m
e
m
m
!
. (3)
In a strong field the ground state splits into four
substates defined by the magnetic quantum num-
bers sM
J
, M
m
d, and the transitions for s1y2, 1y2d $
s1y2, 21y2d designated n
12
, and s21y2, 21y2d $
s21y2, 1y2d designated n
34
, are observed by a microwave
magnetic resonance technique. The transition frequencies
based on (1) are given by the Breit-Rabi formula [1]
n
12
2
m
m
B
g
0
m
H
h
1
Dn
2
f
s
1 1 x
d
2
p
1 1 x
2
g , (4)
n
34
1
m
m
B
g
0
m
H
h
1
Dn
2
f
s
1 2 x
d
1
p
1 1 x
2
g , (5)
where x sg
J
m
e
B
1 g
0
m
m
m
B
dHyshDnd is proportional to
the magnetic field strength, H. We use the Larmor
relation, 2m
p
H hn
p
, and NMR to determine H in
terms of the free proton precession frequency, n
p
, and the
proton magnetic moment, m
p
. Then using (4), (5), and
measurements of the transition frequencies n
12
and n
34
,
we extract Dn and m
m
ym
p
for positive muons [3].
The muons for the experiment were derived from the
linear accelerator at the Clinton P. Anderson Meson
Physics Facility (LAMPF) at Los Alamos, which pro-
duced an 800 MeV proton beam in 650 ms pulses at
120 Hz. Interactions of the proton beam with a graphite
target produced many p
1
, whose parity violating decay
at rest near the surface of the target yielded negative he-
licity m
1
. These were transported to the experimental
apparatus by the stopped muon channel (SMC) [4], which
was tuned to accept m
1
of 28 MeVyc which were nearly
100% polarized, had a FWHM Dpyp of 10%, and yielded
an average intensity of a few 310
7
m
1
ys. Impurities in
the beam were reduced by gas barriers and an E 3 H
separator which reduced the ratio e
1
ym
1
in the beam
from about 10 to 0.03 [5].
0031-9007y99y82(4)y711(4)$15.00 © 1999 The American Physical Society 711
VOLUME
82, NUMBER 4 PHYSICAL REVIEW LETTERS 25J
ANUARY
1999
At the end of the SMC, the muons entered the bore of a
large superconducting solenoidal MRI magnet at a field of
1.7 T. Centered in the solenoid was a copper pillbox mi-
crowave cavity containing either 0.8 or 1.5 atm of 99.97%
pure krypton gas, with an O
2
content ,5 ppm (ppm:
parts per 10
6
). Collisions in the gas slowed the muons,
which stopped and formed polarized muonium predomi-
nantly in the s1y2, 21y2d and s21y2, 21y2d states through
the spin-preserving electron capture reaction m
1
1 Kr !
m
1
e
2
1 Kr
1
[1]. With the time constant of the muon
lifetime, t
m
2.197 03s4d ms [6], the muons would decay
weakly via m
1
! e
1
1n
e
1 ¯n
m
, where the momentum
and angle of the decay e
1
are functions of the muon po-
larization. Since high momenta decay positrons are emit-
ted preferentially in the direction of the muon spin, by
driving the n
12
and n
34
transitions with an applied mi-
crowave magnetic field perpendicular to the static field H,
the muon spin could be flipped and the angular distribution
of high momenta positrons changed from predominantly
upstream to downstream with respect to the beam direc-
tion [1]. An aluminum endcap and polyethylene absorber
downstream of the cavity absorbed positrons with energies
below 35 MeV, followed by two scintillators operated in
coincidence to detect the high momenta decay e
1
.
We define the signal, Ssn, Hd sN
on
yN
off
2 1d,
where N
on
is the number of positrons detected downstream
of the microwave cavity with an applied microwave mag-
netic field of frequency n in a magnetic field of strength
H, and N
off
the number with no applied microwave field.
Close to resonance, S becomes large and positive. In
practice, the number of detected positrons was normalized
to the number of incident muons, whose flux and profile
were monitored with a thin plastic scintillator and wire
chamber upstream of the microwave cavity. A schematic
of the apparatus is shown in Fig. 1.
The cavity was designed to be resonant simultaneously
in the TM
110
mode at the n
12
transition frequency of about
1897.5 MHz, and in the TM
210
mode at the n
34
frequency
of roughly 2565.8 MHz. In a typical resonance line scan,
one mode of the cavity was excited for 10 cycles of the
FIG. 1. A schematic of the experimental apparatus.
proton beam and N
on
counts were accumulated, then for
10 cycles the microwave field was turned off and N
off
was
measured. In the next 20 cycles the other transition was
measured.
Resonance curves were obtained using two techniques.
In the magnetic field sweep of the resonance lines at
fixed microwave frequency, data were taken at a particular
field for 10 sec, then the current of a 69 cm bore
normal conducting modulation coil was increased or
decreased. About 120 steps were used to change H by
60.005 T about a central field of 1.7 T, with each step
corresponding to a change in the microwave transition
frequency of about 3.2 kHz. Using small steps reduced
the field measurement error associated with induced
currents. In the microwave frequency sweep at fixed
magnetic field strength, 20 steps of about 20 kHz were
made while quartz tuning bars located inside the krypton
gas target were moved to keep the cavity in tune.
It was important to measure and stabilize all variables
which affect the center and shape of the resonance curves.
The magnetic field provided by the solenoid was stable
to 1 part in 10
7
to 10
8
per hour, with a peak to peak in-
homogeneity of #1 ppm over the volume in which muo-
nium was formed. A pulsed NMR magnetometer [7] with
eight probes measured the field outside the cavity sev-
eral times per second in terms of the free induction decay
precession frequency of protons in a zero susceptibility
H
2
O 1 NiCl
2
(, 0.15 Molyl) solution. A radially mov-
able NMR probe determined the relation of the field
outside to the inside of the cavity. Finally, the NMR
frequency of the movable probe was related to the equiva-
lent free proton precession frequency by comparison with
an absolute calibration probe [8] containing a spherical
water sample for which the dominant shielding and sus-
ceptibility corrections, including those from air, have been
made. This allowed the absolute field at each point to
be known to .0.1 ppm. Another important experimental
variable was the microwave power which was actively
stabilized to #7 3 10
24
across a line. Typically 3–
5 watts of microwave power were used, where the Q of
the cavity was 14000 at the n
12
frequency in the TM
110
mode, and 19000 at n
34
in the TM
210
mode. The tem-
perature of the krypton gas target was monitored with
thermocouples and RTD sensors to a precision of 0.2
±
C,
and was stable within 0.1
±
C. The gas pressure was
measured by a Mensor gauge to a precision of ,5 3
10
24
atm. Finally, because the magnetic field was not
perfectly homogeneous, each volume element in the cav-
ity had a slightly different resonance line center. In order
to weight each element correctly, the stopped muons spa-
tial distribution was obtained with a planar wire chamber.
A major improvement of this experiment over the
earlier work [9] is the use of a resonance line narrowing
technique [10]. In the conventional approach, the signal,
S, is obtained by counting positrons throughout the beam-
on time. In terms of the muon lifetime, t
m
, and the square
712
VOLUME
82, NUMBER 4 PHYSICAL REVIEW LETTERS 25J
ANUARY
1999
of the transition matrix element, jb
ij
j
2
, the signal shape is
roughly described by
S ,
2jb
ij
j
2
4p
2
sn
ij
2nd
2
14jb
ij
j
2
1 1yt
2
m
, (6)
where n
ij
sij 12, 34d is the transition frequency and n
is the applied microwave magnetic field frequency. In the
limit of zero microwave power (jb
ij
j
2
! 0) the linewidth
.1yspt
m
d 145 kHz. Observing “old muonium” atoms
which have lived several times longer than t
m
while
interacting coherently with the microwave magnetic field
results in narrower lines with larger signal amplitude.
This was achieved through the use of an E field chopper
[5] which modulated the m
1
beam with a 4 ms beam-
on period followed by a 10 ms beam-off period, during
which the beam was nearly 99% extinguished. Decay
positron counts were accumulated in eleven 0.95 ms
wide windows, with each successive window observing
increasingly old muonium atoms. Lines narrower than the
conventional linewidth by a factor of 3 were observed, as
were significantly larger signal amplitudes. Both features
compensated the reduction in counting rate and reduced
some systematic errors.
The centers of the resonance curves were determined
by fitting a theoretical resonance line shape [10,11] to the
data. The theoretical signal incorporated the measured
magnetic field distribution, the ideal microwave power
distributions, the muon stopping distribution, the solid
angle for detection of an e
1
from m
1
decay, the effects
of the residual unchopped beam, and integrals over the
duration of the muon pulse and volume in which muonium
was formed. In total, about 200 conventional resonance
lines and 1070 old muonium lines were analyzed (see
Fig. 2). The x
2
distributions of the fits indicated good
agreement between the theoretical line shape and the
experimental data. The transition frequencies resulting
from the fits were then transformed to their values in
a magnetic field strength corresponding to a free proton
precession frequency of 72.320 000 MHz. The data were
taken at 0.8 and 1.5 atm, and so were corrected for a
small quadratic pressure shift [9,12] and then extrapolated
linearly to zero pressure to obtain Dn and m
m
ym
p
.
The results obtained from each sweeping method were
consistent, and were combined to yield our final results
n
12
sexpd 1 897 539 800s35d Hz s18 ppbd , (7)
n
34
s
exp
d
2 565 762 965
s
43
d
Hz
s
17 ppb
d
, (8)
Dnsexpd 4 463 302 765s53d Hz s12 ppbd , (9)
m
m
ym
p
3.183 345 13s39ds120 ppbd , (10)
where the linear correlation coefficient of n
12
and n
34
is 20.073 and that of m
m
ym
p
and Dn is 0.11. The
sources of uncertainty in the results, primarily statistical,
are outlined in Table I (ppb: parts per 10
9
).
FIG. 2. Resonance curves obtained by sweeping the magnetic
field using a conventional method, and from different time
windows after muonium production are shown on the left.
Microwave frequency sweep curves are on the right. The solid
curves are fits to the theoretical line shape [10,11].
Interest in these results stems from their use in making
precision tests of QED and for extracting fundamental
constants. To this end, we combine (9) and (10) with
the results of the last precision experiment [9] to obtain
Dnsexpd 4 463 302 776s51d Hz s11 ppbd , (11)
m
m
ym
p
3.183 345 24s37ds120 ppbd . (12)
A precise value for the ratio m
m
ym
e
can be obtained
from (12) through the relation
m
m
m
e
√
g
m
2
!√
m
p
m
m
!√
m
e
B
m
p
!
206.768 277s24ds120 ppbd , (13)
where we have used m
m
ym
p
from (12), g
m
2
°
1 1 a
m
¢
where a
m
1 165 923s8.5d 3 10
29
[13], and m
p
ym
e
B
1.521 032 202s15d 3 10
23
[14].
To test QED requires a theoretical prediction for Dn,
which is described in leading order by the Fermi formula
DnsFermid
16
3
a
2
cR
`
m
e
m
m
"
1 1
m
e
m
m
#
23
. (14)
Corrections to this formula have been calculated over
the last few decades, with the most recent theoretical
work described in [15]. In modern terms, the ground
state hyperfine splitting is decomposed into the nonre-
coil and recoil contributions of QED diagrams, as well as
weak and hadronic contributions. The hyperfine splitting
comes mainly from the nonrecoil term, which includes the
Fermi term in leading order. The recoil term correction
713
VOLUME
82, NUMBER 4 PHYSICAL REVIEW LETTERS 25J
ANUARY
1999
TABLE I. The sources of uncertainty are outlined in this
table. Run dependent errors are added in quadrature before data
taken under different running conditions are combined. Errors
common to all data sets are then added in quadrature.
dDn dDn
dsm
m
ym
p
d
Run dependent uncertainties [Hz] [ppb] [ppb]
Sweep mode H n H n H n
Statistical error 89 60 20 13 191 129
Kr density fluctuations 2 2 0.4 0.4 0 0
Drift of Kr density calibration 22 11 4.9 2.5 0 0
Muon stopping distribution 8 5 1.8 1.2 19 17
Magnetic field distribution 0 0 0 0 67 54
Microwave power uncertainty 5 9 1.1 2.0 11 20
Subtotal 92 62 21 14 204 142
Uncertainty in combined
results 51 12 117
dDn dDn
dsm
m
ym
p
d
Run independent uncertainties [Hz] [ppb] [ppb]
Apparatus effect on H field 0 0 30
Absolute calibration of H field 0 0 21
Calibration of Kr density 11 2.5 0
Hydrogen contamination 10 2.2 0
Quadratic pressure shift 8.5 1.9 11
Bloch-Siegert term
and nonres. states [19] 2.8 0.6 0
Total common systematic
errors 17 3.9 38
contributes about 800 kHz (180 ppm), while the effects of
hadronic vacuum polarization and the weak contribution
from Z
0
exchange are at the level of 0.250 kHz (56 ppb)
and 20.065 kHz (15 ppb), respectively.
Using a
21
137.035 999 58s52d (3.8 ppb) [16,17],
R
`
10 973 731.568 639s91d m
21
[18], and m
m
ym
e
from (13), a value of Dn can be calculated from the latest
theoretical formulation [16]
Dnstheoryd 4 463 302 563s510ds34ds#100d Hz , (15)
where the errors, in order, are those from m
m
ym
e
, a,
and an estimate of the uncertainty from uncalculated
terms. Differing estimates of the latter have been made
[15]. The fractional difference fDnsexpd 2Dnstheorydgy
Dnsexpd 48 6 120 ppb represents one of the most
sensitive tests of the validity of bound state QED.
Alternatively, one may regard the mass ratio m
m
ym
e
as
a parameter in Dnstheoryd which can be determined by
the experimental result Dnsexpd. This yields the ratios
m
m
ym
e
206.768 267 0
s
55
d
(27 ppb), and m
m
ym
p
3.183 345 396
s
94
d
(30 ppb), where we have used a and
R
`
as in (15). Similarly, we can extract a value for
the fine structure constant a, using m
m
ym
e
from (13),
yielding a
21
137.035 996 3
s
80
d
(58 ppb).
Significant future improvement in determining m
m
ym
p
and Dn would probably require measuring a third transi-
tion and/or a new reference standard for the magnetome-
ter, and the use of a more intense pulsed muon source.
The latter may be achieved at the Japan Hadron Facility
or at the front end of a muon collider.
The authors are pleased to acknowledge the help and
expertise of E. Hoffman, J. Ivie, and the staff of AOT
Division, and the support of P. Barnes, C. Hoffman, and
G. Garvey at LANL. We also thank S.G. Karshenboim
and T. Kinoshita for valuable discussions. Research was
supported in part by the U.S. DOE, BMBF (Germany), the
Alexander von Humbolt Foundation and a NATO grant.
Data analysis was supported in part by the Cornell CTC
and by the NERSC Center.
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