PHYSICAL REVIEW D 73, 072003 (2006)

Final report of the E821 muon anomalous magnetic moment measurement at BNL
G. W. Bennett,2 B. Bousquet,10 H. N. Brown,2 G. Bunce,2 R. M. Carey,1 P. Cushman,10 G. T. Danby,2 P. T. Debevec,8
M. Deile,13 H. Deng,13 W. Deninger,8 S. K. Dhawan,13 V. P. Druzhinin,3 L. Duong,10 E. Efstathiadis,1 F. J. M. Farley,13
G. V. Fedotovich,3 S. Giron,10 F. E. Gray,8 D. Grigoriev,3 M. Grosse-Perdekamp,13 A. Grossmann,7 M. F. Hare,1
D. W. Hertzog,8 X. Huang,1 V. W. Hughes,13,* M. Iwasaki,12 K. Jungmann,6,7 D. Kawall,13 M. Kawamura,12 B. I. Khazin,3J. Kindem,10 F. Krienen,1 I. Kronkvist,10 A. Lam,1 R. Larsen,2 Y. Y. Lee,2 I. Logashenko,1,3 R. McNabb,10,8 W. Meng,2J. Mi,2 J. P. Miller,1 Y. Mizumachi,11 W. M. Morse,2 D. Nikas,2 C. J. G. Onderwater,8,6 Y. Orlov,4 C. S. Özben,2,8J. M. Paley,1 Q. Peng,1 C. C. Polly,8 J. Pretz,13 R. Prigl,2 G. zu Putlitz,7 T. Qian,10 S. I. Redin,3,13 O. Rind,1 B. L. Roberts,1
N. Ryskulov,3 S. Sedykh,8 Y. K. Semertzidis,2 P. Shagin,10 Yu. M. Shatunov,3 E. P. Sichtermann,13 E. Solodov,3
M. Sossong,8 A. Steinmetz,13 L. R. Sulak,1 C. Timmermans,10 A. Trofimov,1 D. Urner,8 P. von Walter,7 D. Warburton,2
D. Winn,5 A. Yamamoto,9 and D. Zimmerman10(Muon g  2 Collaboration)
1

Department of Physics, Boston University, Boston, Massachusetts 02215, USA
2
Brookhaven National Laboratory, Upton, New York 11973, USA
3
Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
4
Newman Laboratory, Cornell University, Ithaca, New York 14853, USA
5
Fairfield University, Fairfield, Connecticut 06430, USA
6
Kernfysisch Versneller Instituut, Rijksuniversiteit Groningen, NL-9747 AA, Groningen, The Netherlands
7
Physikalisches Institut der Universität Heidelberg, 69120 Heidelberg, Germany
8
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
9
KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan10
Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455, USA
11
Science University of Tokyo, Tokyo, 153-8902, Japan
12
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
13
Department of Physics, Yale University, New Haven, Connecticut 06520, USA(Received 26 January 2006; published 7 April 2006)
We present the final report from a series of precision measurements of the muon anomalous magnetic
moment, a  g  2=2. The details of the experimental method, apparatus, data taking, and analysis are
summarized. Data obtained at Brookhaven National Laboratory, using nearly equal samples of positive
and negative muons, were used to deduce a Expt  11659208:05:43:3  1010 , where the statisticaland systematic uncertainties are given, respectively. The combined uncertainty of 0.54 ppm represents a
14-fold improvement compared to previous measurements at CERN. The standard model value for a
includes contributions from virtual QED, weak, and hadronic processes. While the QED processes
account for most of the anomaly, the largest theoretical uncertainty,  0:55 ppm, is associated with firstorder hadronic vacuum polarization. Present standard model evaluations, based on e e hadronic crosssections, lie 2.2 –2.7 standard deviations below the experimental result.
DOI: 10.1103/PhysRevD.73.072003

PACS numbers: 13.40.Em, 12.15.Lk, 14.60.Ef

I. INTRODUCTIONa  12g  2:The muon magnetic moment is related to its intrinsic
spin by the gyromagnetic ratio g :~   g


q ~
S;
2m



(1)

where g  2 is expected for a structureless, spin- 12 particle of mass m and charge q  e. Radiative corrections(RC), which couple the muon spin to virtual fields, introduce an anomalous magnetic moment defined by*Deceased.

1550-7998= 2006=73(7)=072003(41)$23.00(2)

The leading RC is the lowest-order (LO) quantum
electrodynamic process involving the exchange of a virtualphoton, the ‘‘Schwinger term,’’ [1] giving a QED; LO 
=2  1:16  103 . The complete standard model valueof a , currently evaluated to a precision of approximately
0.6 ppm (parts per million), includes this first-order termalong with higher-order QED processes, electroweak
loops, hadronic vacuum polarization, and other higherorder hadronic loops. The measurement of a , carriedout to a similar precision, is the subject of this paper. The
difference between experimental and theoretical values fora is a valuable test of the completeness of the standard

072003-1© 2006 The American Physical Society

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

model. At sub-ppm precision, such a test explores physics
well above the 100 GeV scale for many standard model
extensions.
The muon anomalous magnetic moment was measured
in a series of three experiments at CERN and, most recently in our E821 experiment at Brookhaven National
Laboratory (BNL). In the first CERN measurement [2]muons were injected into a 6-m long straight magnet where
they followed a drifting spiral path, slowly traversing themagnet because of a small gradient introduced in the field.
The muons were stopped in a polarimeter outside themagnet and a measurement of their net spin precession
determined a with an uncertainty of 4300 ppm. The result
agreed with the prediction of QED for a structurelessparticle. The second CERN experiment [3] used a magnetic ring to extend the muon storage time. A primary
proton beam was injected directly onto a target inside thestorage ring where produced pions decayed to muons, asmall fraction of which fell onto stable orbits. The muon
precession frequency was determined by a sinusoidal
modulation in the time distribution of decay positrons,
measured by detectors on the opposite side of the ringfrom the injection point. The result to 270 ppm agreed
with QED only after the theory had been recalculated [4].
The CERN-III experiment [5] used a uniform-field storage
ring and electric quadrupoles to provide vertical containment for the muons having the ‘‘magic’’ momentum of
3:1 GeV=c. At this momentum, the muon spin precession
is not affected by the electric field from the focusing
quadrupoles. Additionally, pions were injected directly
into the ring, which resulted in a higher stored muon
fraction and less background than proton injection. The
CERN-III experiment achieved a precision of 10 ppm foreach muon polarity. CPT symmetry was assumed, and theresults were combined to give a 7.3 ppm measurement,
which agreed with theory. The result served as the first
confirmation of the predicted 60 ppm contribution to afrom hadronic vacuum polarization.

The present BNL experiment follows the general technique pioneered by CERN-III, but features many innovative improvements. A continuous superconducting magnet,
having high field uniformity, is used instead of a lattice of
discrete resistive magnets. A direct current, rather thanpulsed, inflector magnet permits the ring to be filled at
33 ms intervals, the bunch extraction interval from the
AGS. Muons are injected directly into the storage ring,which increases the storage efficiency and reduces theintense hadron-induced background ‘‘flash.’’ A pulsed
kicker places the muons onto stable orbits and centersthem in the storage region. The electrostatic quadrupoles
permit operation at about twice the field gradient of the
CERN experiment. The transverse aperture of the storage
region is circular rather than rectangular, in order to reducethe dependence of the average field seen by a muon on its
trajectory. The magnetic field is mapped using an array of
NMR probes, mounted on a trolley that can be pulled
through the vacuum chamber. Waveform digitizers provide
a time record of energy deposition in calorimeters. The
records are used to determine electron energies and times
and to correct for multiparticle overlap —‘‘pileup.’’ (Note:
In this manuscript, we use electron to represent either the
positron or electron in the generic  ! e decay chain.)
Combining the results of four positive muon runs and a
final run using negative muons, a was determined to a
precision of 0.54 ppm. A summary of the CERN and BNL
measurements is given in Table I. This paper reviews the
BNL E821 results, all reported in Letters [6,8–10] or Brief
Reports [7]. Many of the key experimental components
have been described in separate papers; brief summaries
are given here. The paper begins with the basic principle ofthe experimental method including subsections on theapparatus. The dynamics of muon storage, which shape
the observed decay electron distribution, are discussed
next. Then the data analysis is described and the paper
concludes with a discussion of the theoretical standard
model value for a and its comparison to the final result.

TABLE I. Summary of a results from CERN and BNL, showing the evolution of experimental precision over time. The average is obtained from the 1999, 2000 and 2001 data sets only.
Experiment
CERN ICERN IICERN IIICERN III
BNL
BNL
BNL
BNL
BNL
Average

Years

Polaritya  1010

Precision [ppm]Reference

1961
1962 –1968
1974 –1976
1975–1976
1997
1998
1999
2000
2001



11450000(220000)
11661600(3100)
11659100(110)11659360(120)11659251(150)11659191(59)11659202(15)11659204(9)11659214(9)11659208.0(6.3)

4300
270
10
10
13
5
1.3
0.73
0.72
0.54

[2]
[3]
[5]
[5]
[6]
[7]
[8]
[9]
[10]
[10]











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FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

II. EXPERIMENTAL METHOD
A. Overview
The cyclotron !c and spin precession !s frequencies fora muon moving in the horizontal plane of a magnetic
storage ring are given by:
!
~c

qB~
;
m

!
~s  

gqB~qB~
 1  
:
2m
m

(3)

The anomalous precession frequency !a is determinedfrom the differenceg  2 qB~
qB~
 a
:
!
~a!
~s !
~c  
2
m
m(4)

Because electric quadrupoles are used to provide verticalfocusing in the storage ring, their electric field is seen in themuon rest frame as a motional magnetic field that canaffect the spin precession frequency. In the presence ofboth E~ and B~ fields, and in the case that ~ is perpendicular~ the expression for the anomalous precesto both E~ and B,
sion frequency becomes



 ~
  E~q1!
~a:a B~  a  2
c
m
 1

dPy;  
 1=2n y 1   y cosdyd
n y  y2 3  2y(5) y The coefficient of the ~  E~ term vanishes at the magic
momentum of 3:094 GeV=c, where   29:3. Thus a
can be determined by a precision measurement of !a and
B. At this magic momentum, the electric field is used only
for muon storage and the magnetic field alone determines
the precession frequency. The finite spread in beam momentum and vertical betatron oscillations introduce small(sub-ppm) corrections to the precession frequency.
The longitudinally polarized muons, which are injected
into the storage ring at the magic momentum, have a time1
0.8
0.6

n(y)

n* (y)
0.4

α*(y)

0.2

with (6)and(7)

q 2y  1:e 3  2y(8)

Figure 1(a) shows the quantities n y and  y. Electrons
with y < 0:5 are emitted preferentially along the (negative)
muon spin direction and those with y > 0:5 are more likely
emitted opposite to the spin. Because both n and  arelarger for y > 0:5, decay electrons tend to emerge in the
direction opposite to the muon spin. Like the muon spin,
the angular distribution of the electrons in the muon rest
frame rotates at the angular frequency !a . Figure 1(b)shows the same differential quantities in the boosted laboNormalized number (N) or Asymmetry (A)Relative Number (n) or Asymmetry (A)dilated muon lifetime of 64:4 s. A measurement periodof typically 700 s follows each injection or ‘‘fill.’’ The
net spin precession depends on the integrated field seen bya muon along its trajectory. The magnetic field used in
Eq. (5) refers to an average over muon trajectories during
the course of the experiment. The trajectories of the muonsmust be weighted with the magnetic-field distribution. To
minimize the precision with which the average particle
trajectories must be known, the field should be made asuniform as possible.
Because of parity violation in the weak decay of themuon, a correlation exists between the muon spin anddecay electron direction. This correlation allows the spin
direction to be measured as a function of time. In the rest
frame of the muon—indicated by starred quantities —thedifferential probability for the electron to emerge with anormalized energy y  E =Emax (Emax  52:8 MeV) at anangle  with respect to the muon spin is [11]

A(y)
0

-0.2

1

N
0.8
0.6
2

NA
0.4
0.2
0

A
-0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1
y

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1
y

FIG. 1. Relative number and asymmetry distributions versus electron fractional energy y in the muon rest frame (left panel) and in
the laboratory frame (right panel). The differential figure-of-merit product NA2 in the laboratory frame illustrates the importance of thehigher-energy electrons in reducing the measurement statistical uncertainty.

072003-3

Million events per 149.2 ns

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

representative electron decay time histogram is shown in
Fig. 2.
To determine a , we divide !a by !
~ p , where !
~ p is themeasure of the average magnetic field seen by the muons.
The magnetic field, measured using NMR, is proportional
to the free-proton precession frequency, !p . The muon
anomaly is given by:

10

1

10

10

10

-1

a 

-2

-3

0

20

40

60

80
100
Time modulo 100 µ s

FIG. 2. Distribution of electron counts versus time for the
3:6  109 muon decays in the R01  data-taking period. The
data is wrapped around modulo 100 s.

ratory frame (n ! N,  ! A) (here, Emax  3:1 GeV
and A is the laboratory asymmetry). As discussed later,
the statistical uncertainty on the measurement of !a isinversely proportional to the ensemble-averaged figureof-merit (FOM) NA2 . The differential quantity NA2 ,shown in the Fig. 1(b), illustrates the relative weight byelectron energy to the ensemble average FOM.
Because the stored muons are highly relativistic, thedecay angles observed in the laboratory frame are greatlycompressed into the direction of the muon momenta. The
lab energy of the relativistic electrons is given by
Elab  E  p c cos   E 1  cos :(9)Because the laboratory energy depends strongly on the
decay angle  , setting a laboratory threshold Eth selects
a range of angles in the muon rest frame. Consequently, theintegrated number of electrons above Eth is modulated at
frequency !a with a threshold-dependent asymmetry. The
integrated decay electron distribution in the lab frame has
the form
Nideal t  N0 expt=  1  A cos!a t   ; (10)where N0 , A and are all implicitly dependent on Eth . Fora threshold energy of 1.8 GeV (y  0:58 in Fig. 1(b)], theasymmetry is  0:4 and the average FOM is maximized. A

!a =!
~p
!a


!L  !a !L =!
~ p  !a = !
~p

R
;
R

(11)

where !L is the Larmor precession frequency of the muon.
The ratio R  !a =!
~ p is measured in our experiment andthe muon-to-proton magnetic moment ratio
 !L =!p  3:1833453910

(12)

is determined from muonium hyperfine level structure
measurements [12,13].
The BNL experiment was commissioned in 1997 using
the same pion injection technique employed by the CERN
III experiment. Starting in 1998, muons were injected
directly into the ring, resulting in many more stored muons
with much less background. Data were obtained in typically 3– 4 month annual runs through 2001. In this paper,
we indicate the running periods by the labels R97–R01.
Some facts about each of the runs are included in Table II.
B. Beamline
Production of the muon beam begins with the extractionof a bunch of 24 GeV=c protons from the AGS. The
protons are focused to a 1 mm spot on a 1-interaction
length target, which is designed to withstand the very
high stresses associated with the impact of up to 7 1012 protons per bunch. The target is composed oftwenty-four 150-mm diameter nickel plates, 6.4-mm thickand separated by 1.6 mm. To facilitate cooling, the disks
rotate at approximately 0.83 Hz through a water bath. The
axis of rotation is parallel to the beam.
Nickel is used because, as demonstrated in studies for
the Fermilab antiproton source [14], it can withstand theshock of the instantaneous heating from the interaction of
the fast beam. The longitudinal divisions of the target
reduce the differential heating. The beam strikes the outerradius of the large-diameter disks. The only constraint on
the target transverse size is that a mis-steered proton beam

TABLE II. Running periods, total number of electrons recorded 30 s or more after injection having E > 1:8 GeV. Separate
systematic uncertainties are given for the field (!p ) and precession (!a ) final uncertainties.
Run Period
R97
R98
R99
R00
R01

Polarity

Electrons [millions]

Systematic !p [ppm]

Systematic !a [ppm]Final Relative Precision [ppm]



0.8
84
950
4000
3600

1.4
0.5
0.4
0.24
0.17

2.5
0.8
0.3
0.31
0.21

13
5
1.3
0.73
0.72






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FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

AGS

U−V line
VD3
VD4

V line

D6

Pion Decay Channel

D5

4
D3,D

1,
D

2

Q1
Q2

D

U lineK3−K4

Pion Production Target

Beam Stop

K1−K2

Inflector

g−2 Ring

FIG. 3. Plan view of the pion/muon beamline. The pion decay
channel is 80 m and the ring diameter is 14.1 m.

does not allow production from a part of the target thatwould result in a high flux of pions entering the storage
ring during muon injection running. This region corresponds to the outer edge of the disks. Otherwise, theproduction target transverse size is defined by the beam
size. With the large radius disks, shock damage of thetarget is distributed over the disk circumference as thedisks rotate. Still, it was necessary to replace the target
after each running period, typically following an exposureof 5  1019 protons, or when nickel dust was observed in
the target water cooling basin.
Pions are collected from the primary target at zero angle
and transferred into a secondary pion-muon-decay channel, designed to maximize the flux of polarized muons
while minimizing pion contamination. A schematic representation of the beamline is shown in Fig. 3 and selected
proton beam and pion beamline parameters are given in
Table III. Downstream of the target, pions produced in theforward direction within a solid angle of 32 mrad (horizontal) and 60 mrad (vertical) are collected by quadrupoles
Q1 and Q2. A momentum-dispersed image of the target iscreated at the K1-K2 slits. The momentum dispersion is
0:04%=mm and under typical running conditions the pion

momentum width was p=p 0:5%. The momentumrecombined beam then traverses an 80 m quadrupole magnet FODO (alternating focusing and defocusing) straightsection. A momentum-dispersed image of the target iscreated at the K3-K4 slits. Here the momentum dispersion
is 0:06%=mm. The momentum-recombined beam is thenfocused to allow passage through the hole in the back leg ofthe ring magnet, through the strong vertically focusing
storage-ring fringing field, the inflector, and into the storage volume. For pion injection, the K1-K2 slits select the
momentum of the pion beam and the K3-K4 slits are
effectively not used. For muon injection, the K3-K4 slits
reject most of the pion beam and select particles having thestorage-ring momentum.
The beamline and ring were operated in different modes
for the five running periods. In the R97 period, pions ofcentral momentum 0.5% above the magic momentum were
injected directly into the storage ring. With typically 5 1012 protons on target, 108 particles were injected into themuon storage ring per fill. The muon capture efficiency
was 25 ppm, consistent with Monte Carlo estimates. The
majority of the pions strike obstructions in the ring or
decay to muons that are not stored. This initial particle‘‘flash’’ induces a considerable background in the electron
detectors. All subsequent running periods used direct muon
injection, where muons of the magic momentum were
injected into the ring and then kicked transversely onto
stable trajectories.
While the momentum of the downstream section of thebeamline —after the K3-K4 slits—was always set to themagic momentum of 3:094 GeV=c, the upstream captureand decay sections were adjusted to meet the competing
demands of high muon flux and low pion contamination.
The number of stored muons is maximized when the upstream beamline is tuned 0.5% above the magic momentum. However, this small momentum difference does notprovide adequate pion rejection at the K3-K4 slits.
The muon transmission to the storage ring entrance, thepion survival fraction F past K3-K4 (a figure-of-merit forpion contamination), and the muon polarization, were
calculated using the BTRAF beamline transport program[15]. The results were compared to measurements (SeeTable IV). The muon transmission efficiency is determined

TABLE III. Selected AGS proton beam and secondary pion beamline characteristics.
Proton BeamProtons per AGS cycle
Cycle repetition rate
Proton momentum
Bunches per cycle
Bunch width ( )Bunch spacing
a

Value
13

5  100.37 Hz24 GeV=c
6 –12
25 ns
33 ms

Pion Beamline
Horizontal emittance
Vertical emittanceInflector horizontal aperture
Inflector vertical aperture
Pions per protona
Muons per pion decayb

Captured by the beamline channel;
Measured at the inflector entranceb072003-5

Value
42 mm-mrad
56 mm-mrad
9 mm
28 mm1050.012

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

TABLE IV. As a function of the ratio of central-pion-to-muon momentum p =p : From left toright, calculated relative muon transmission fraction; measured relative stored muon flux;
calculated and measured pion transmission fraction into the ring; calculated muon polarization;measured asymmetry A using Eq. (10) and Eth  1:5 GeV. The absolute fraction of muons perpion decay is obtained by multiplying column 2 by 0.018.
p =p

N (Calc)N (Meas)F (Calc)F (Meas)P (Calc)A (Meas)

1.005
1.010
1.015
1.017
1.020

1
0.5
0.29
0.25
0.18

1
0.43
0.21
0.17
0.12

0.78
0.29
0.04
0.002
—

0.80
0.30
0.065
0.016
0.009

0.99
0.98
0.96
0.96
0.95

0.22
0.26
0.30
0.30
0.30

by counting the number of high-energy electrons in thecalorimeters in a given time period well after the hadronic
flash has dissipated. The pion survival fraction is determined from beamline Čerenkov measurements (see below). The asymmetry in the observed (g  2) oscillations
is proportional to the beam polarization, as well as to theasymmetry of the weak decay. It is also affected by thedetector acceptance and resolution. In particular,Monte Carlo simulations using Eth  1:5 GeV predictA  0:30  0:01 for a 100% polarized beam. The measured asymmetry, obtained from fits of the data to Eq. (10),is found to be lower than the prediction when p =p is in
the range 1:005–1:010. The dilution is caused by muons
born from pion decays between the target and the pion
momentum selection at K1-K2, which for this small momentum difference, are transported and stored in the ring.
The ratio p =p  1:017 was chosen as the optimal running condition; it features a high asymmetry and storage
fraction, and an acceptably low pion contamination.
Relative particle species fractions immediately downstream of the K3-K4 slit were determined using a tunable
threshold gas Čerenkov counter. Table V lists the particle
fractions for a positive pion beam, measured during the
R98 period. The pion lifetime is 0:58 s, much less thanthe detector gate-on time. The fraction of pions transmitted
to the storage ring falls as p =p is increased, which is
expected from the momentum selection at K3-K4.
However, the increase in the beam positron fraction was
unanticipated. We believe it is due to radiation in the beam
monitoring elements and vacuum windows between the

momentum selection slits at K1-K2 and K3-K4. Positrons
that are stored in the ring lose energy by synchrotron
radiation; they were observed to spiral into the calorimeters during the first 5 s after injection. Protons, which
could not be identified by the Čerenkov detector, were
estimated to be about one third as numerous as the pions.
Stored protons are discussed in Sec. III D. The antiproton
flux for the R01 running period was negligible, typicallysuppressed by a factor of 500 compared to protons in the
R00 running period.
C. Inflector
The muon beam passes through position and intensity
instrumentation, through a thin vacuum window and into a
1 m long hole in the backleg of the storage-ring magnet, inair. After passing through an opening in the outer coilcryostat and additional position instrumentation, it passes
through a thin vacuum window into the 1.7-m long superconducting inflector magnet [16], whose 1.5-T verticalfield (field integral 2:55 T m) cancels the main storageR = 7112 mm from ring center

Outer cryostat

Inflector

77 mm

Beam
channelp =p

e





=

1.000
1.009
1.014
1.017

0.16
0.14
0.17
0.34

0.014
0.05
0.06
0.34

0.83
0.81
0.77
0.32

59
16
13
1

Beam vacuum
chamber

Inflector
cryostat

Superconducting
coils

TABLE V. The relative fraction of e ,  and  versus thepion-to-muon momentum ratio in the R98 period as determinedby Čerenkov measurements after the K3-K4 slits. Protons are
estimated to be 1=3 the pion flux.

Muon storage
region ρ = 45 mm

Partition wall
Passive superconducting
shieldFIG. 4. The inflector/storage-ring geometry. The downstream
end of the inflector is shown, with the beam channel to the left ofthe storage region (larger radius). The ring center is to the right.
Note the limited space between the pole pieces, which has tocontain the inflector and its cryostat along with the beam vacuum
chamber. The current in the inflector flows into the page in the C
shaped arrangement of conductors just to the left of the beam
channel, and out of the page in the conductors that form abackward ‘‘D’’. The superconductor crosses over the beam
channel to connect the two coils.

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FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)10

x’ [mrad]y’ [mrad]

10

5

5

0

0

-5

-5

-10

-40

-20

0

20

-10

40
y [mm]-40 -20

0

20

40

60

80

100 120x [mm]

FIG. 5. The phase space plot for the inflector exit from a beam transport simulation (x is horizontal; y is vertical). Left plot: py =pz 
y0 vs y. Right plot: px =pz  x0 vs x. The inflector center is displaced from the storage-ring central orbit by 77 mm. The ellipses
represent the storage-ring acceptance. After one quarter turn, the distribution in x has rotated through 90 deg and lies below the ring
acceptance. It is then kicked toward more positive x0 , into the ring acceptance.

ring field, permitting the muons to pass largely undeflected
into the storage ring. The current windings feature a unique
double-truncated cosine theta design [17], which minimizes the flux that leaks outside the inflector volume.
The geometry at the inflector exit is shown in Fig. 4. The
inflector axis is approximately tangent to the storage ring,and it is adjustable by 4 mrad. The beam center in theinflector channel exit is 77 mm from the storage-ring
center.
Placing the inflector cryostat in the limited space between the muon storage region and the outer main magnet
coil restricted the inflector aperture size to 18w mm 56h mm, which is significantly smaller than the 90-mm
diameter storage-ring aperture. The small size limits theflux of incoming muons and introduces a mismatch inphase space with respect to the storage ring. Figure 5 shows
the vertical and horizontal muon beam phase space (y, y0
and x, x0 ) as simulated for the exit of the inflector.
Superimposed on the figures are the storage-ring acceptance ellipses. The muons undergo betatron harmonic motion in the storage ring, following elliptical paths about theorigin in phase space.
The precision magnetic field in the storage region isprotected from the small leakage flux from the end of theinflector by means of a passive superconducting shield.
The inflector is cooled down after the storage-ring magnet
has been energized and the main field is stable. The superconducting shield then pins the main field and traps theinflector fringe field as the inflector is energized. The
disturbance of the main storage-ring field by the inflector
fringe field is negligible. However, in 1997 before installing it into the ring, the first inflector required a repair,which could only be made by cutting through the shield.
The resulting fringe field reduced the storage-ring field by
600 ppm over a 1 azimuthal angle, resulting in unacceptable magnetic-field gradients for the NMR trolley probes

closest to the inflector body. The field in this region had to
be mapped by a special procedure following data taking.
This introduced additional uncertainty into the measurement of the average field, 0.20 ppm in the R99 result.
The damaged inflector was replaced before the 2000
running period. In the new inflector, the superconducting
shield was extended further beyond the downstream end,
and the lead geometry was changed to reduce the fringe
field due to the inflector leads. For both the R00 and R01
running periods, the fringe field of the inflector was
negligible.

FIG. 6. A 3D engineering rendition of the E821 muon storage
ring. Muons enter the back of the storage ring through a fieldfree channel at approximately 10 o’clock in the figure. The threekicker modulators at approximately 2 o’clock provide the shortcurrent pulse, which gives the muon bunch a transverse 10 mradkick. The regularly spaced boxes on rails represent the electron
detector systems.

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360 mmThrough bolt

Shim plate

Iron yoke
Upper push−rod

Inner upper coil

slot
Outer coil

1570 mm

Muon beam

Poles
Inner lower coilSpacer Plates

To ring center

544 mm1394 mm

FIG. 7.

Cross sectional view of the C magnet.

D. Muon storage-ring magnet
The muon storage ring [18] is a superferric ‘‘C’’-shaped
magnet, 7.112 m in central orbit radius, and open on theinside to permit the decay electrons to curl inward to thedetectors (Fig. 6). A 5 V power supply drives a 5177 A
current in the three NbTi/Cu superconducting coils.
Feedback to the power supply from the NMR field measurements maintains the field stability to several ppm. The
field is designed to be vertical and uniform at a central
value of 1.4513 T. High-quality steel, having a maximumof 0.08% carbon, is used in the yoke. Low-carbon steel isused for the poles primarily because the fabrication processof continuous cast steel greatly minimizes impurities suchas inclusions of ferritic or other extraneous material and air
bubbles. An air gap between the yoke and the higher
quality pole pieces decouples the field in the storage region
from nonuniformities in the yoke. Steel wedge shims areplaced in the air gap. Eighty low-current surface-correction
coils go around the ring on the pole-piece faces for active
trimming of the field. The opening between the pole faces
is 180 mm and the storage region is 90 mm in diameter. A
vertical cross section of the storage-ring illustrating someof these key features is shown in Fig. 7. Selected storagering parameters are listed in Table VI.
Attaining high field uniformity requires a series of passive shimming adjustments, starting far from and thenproceeding towards the storage region. First the 12 upperand lower-yoke adjustment plates are shimmed by placing

precision spacers between them and the yoke steel, modifying the air gap. Next the 1000 wedge shims in the yoke
pole-piece air gap are adjusted. With a wedge angle of
50 mrad, adjusting the wedge position radially by 1 mmchanges the thickness of iron at the center of the storage
aperture by 50 m. The wedge angle is set to compensate
the quadrupole component, and radial adjustments of thewedge and other changes to the air gap are used to shim thelocal dipole field. The local sextupole field is minimized bychanging the thickness of the 144 edge shims, which sit on
the inner and outer radial edges of the pole faces. Higher
moments, largely uniform around the ring, are reduced byadjusting the 240 surface-correction coils, which run azimuthally for 360 deg along the surface of the pole faces.
They are controlled through 16 programmable current
elements. With adjustments made, the azimuthally averaged magnetic field in the storage volume had a uniformityof ’ 1 ppm during data-taking runs.
The main temporal variation in the magnetic-field uniformity is associated with radial field changes from seasonal and diurnal drift in the iron temperature. Because of
the C magnet geometry, increasing (or decreasing) theoutside yoke temperature can tilt the pole faces together(or apart), creating a radial gradient. The yoke steel wasinsulated prior to the R98 run with 150 mm of fiberglass to
reduce the magnetic-field variation with external temperature changes to a negligible level.
Inflector
24

Q4

Q1
21
5

C
20

C

Tracebackchambers
270 Fiber
19 monitor
Trolleygarage

6

K1
7

K2

18
8
17
9

Q3
1C
2

TABLE VI. Selected muon storage-ring parameters.
Value

K3

16
10

1C
2

1.4513 T
5200 A
7.112 m
90 mm
180 mm
6 MJ

3
4

14

Nominal magnetic field
Nominal current
Equilibrium orbit radius
Muon storage region diameter
Magnet gap
Stored energy

2

Calibration
NMR probe

23

1
22 2C

15

ParameterC

1

180 Fiber
monitor
13

11
12

Q2

1
2C

1C
2

FIG. 8. The (g  2) storage-ring layout. The 24 numbers represent the locations of the calorimeters immediately downstreamof the scalloped vacuum chamber subsections. Inside the vacuum
are four quadrupole sections (Q1–Q4), three kicker plates (K1–
K3) and full-aperture (C) and half-aperture ( 12 C) collimators.
The traceback chambers follow a truncated scalloped vacuum
chamber subsection.

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E. Electric quadrupoles
Electrostatic quadrupoles are used for vertical focussingof the beam in the storage ring. Ideally, the electrodes
should fill as much of the azimuth as possible; but, spaceis required for the kicker magnet and the inflector. For
symmetry reasons, the quadrupoles are divided into four
distinct regions: Q1–Q4 (see Fig. 8). Gaps at 0 and 90 forthe inflector and kicker magnets, along with empty gaps at
180 and 270 provide a four-fold lattice symmetry.
Overall, the electrodes occupy 43% of the total circumference. The four-fold symmetry keeps the variation in thep
beta function small, max =min  1:04, which minimizes
beam ‘‘breathing’’ and improves the muon orbit stability.
The quadrupole voltage should be set as high as possible
to maximize muon storage, subject to an upper limit of
25 kV (negative muons, can be higher for positive
muons) necessary for reliable operation in the vacuum
chamber, which has a typical vacuum of
1:3 
105 Pa. The plates are charged prior to each fill and thevoltage is held constant through the measuring period,
which extends for at least 700 s. At injection, the plates
are charged asymmetrically to shift the beam horizontallyand vertically against a set of centered circular collimators—the scraping procedure. Approximately 5  15 slater, the plate voltages are symmetrized to enable longterm muon storage. The operating voltages, field indices(which are proportional to the quadrupole gradient), theinitial asymmetric scraping time, and the total pulse length
are given in Table VII.
A schematic representation of a cross section of theelectrostatic quadrupole electrodes [19] is shown in
Fig. 9. The four trolley rails are at ground potential. Flat,rather than hyperbolic, electrodes are used because they areeasier to fabricate. With flat electrodes, electric-field multipoles 8; 12; 16;
in addition to the quadrupole are allowed by the four-fold symmetry (see Fig. 9). Of these, the12- and 20-pole components are the largest. The ratio of
the width (47 mm) of the electrode to the distance between
opposite plates (100 mm) is set to minimize the 12-polecomponent. Beam-dynamics simulations indicated thateven the largest of the resulting multipoles, a 2% 20-polecomponent (at the circular edge of the storage region),
would not cause problems with muon losses or beam
TABLE VII. Quadrupole high voltage, field index, scraping
time and the total pulse length for the different running periods.
Run Period HV [kV]nTscraping [s] Pulse Length [s]

R97
R98
R99
R00

24.4
24.4
24.4
24.2

0.137
0.137
0.137
0.136

16
16
16
16

650
800
900
1400

R01

21.7
25.3

0.122
0.142

7
7

700
700

Trolley
rail

FIG. 9. Schematic view of the electrostatic quadrupoles inside
the vacuum chamber. For positive muons, the top and bottom
electrodes are at
 24 kV; the side electrodes are at
24 kV. The NMR trolley rails can be seen between the electrodes in the V  0 planes. The 90 mm diameter storage region
is depicted by the dashed circle.

instabilities at the chosen values of the field indices. The
scalloped vacuum chamber introduces small 6- and 10pole multipoles into the field shape.
The quadrupoles are charged for  1:4 ms of data taking during each fill of the ring. Cycling the quadrupoles
prevents the excessive buildup of electrons around theelectrodes, electrons which are produced by field emission
and gas ionization and subsequently trapped in the electricand magnetic fields near the quadrupoles. Trapping wasparticularly severe during the R01 running period whennegative muons were injected into the ring. The continuous
motion of the electrons —cyclotron motion in the dipole~ and axial
magnetic field, magnetron motion along E~  B,
oscillations along the vertical axis—ionizes the residual
gas and eventually produces a spark, which discharges theplates.
Slight modifications of the magnetron motion were used
to quench the electron trapping. In the original design,
electrons undergoing magnetron motion were trapped inhorizontal orbits around the vertical plates: the naturalsymmetry of the electric field at the ends caused thecirculating electrons to re-enter the quadrupoles and returnto their starting point in approximately 50 100 s, forthe short (long) sections. To prevent the recirculation oftrapped electrons, the leads were adjusted to rotate the fieldsymmetry at the end of the plate by approximately 25 ; theresulting dipole field largely sweeps away the electrons.
Also, field emission was minimized during data-taking
periods by conditioning the quadrupole electrodes and by
returning them to full voltage very slowly after every set of
NMR trolley measurements.
In addition to creating sparks, trapped electrons can also
distort the field created by the quadrupole electrodes. In aquadrupole field with Ey  ky, the vertical oscillationp
frequency !A  ek=me provides a direct measure of

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the actual gradient k. Reducing the number of trappedelectrons permitted the application of a very large dc
voltage to the electrodes and, in turn, a measurement of
the vertical resonance frequency of those that remained.
Thus we were able to set a sensitive limit for the influenceof trapped electrons on the electric-field configuration as
well as a limit on the magnetic fields that they must also
produce.

1

0.8

0.6

F. Pulsed kicker magnet
0.4

0.2

0

-200

-100

0

100

200

300

400

500

600

700

Time [ns]FIG. 10. The trace is a sample kicker current pulse from one of
the three kicker circuits. The periodic pulses provide a schematic
representation of the unmodified muon bunch intensity during
the first few turns. The vertical axis is in arbitrary units.

The kicker consists of three identical sections, eachdriven by a separate PFN; this division keeps the inductance of a single assembly at a reasonable value. In eachcircuit, an initial voltage of 90 kV on the capacitor
results in a current-pulse amplitude of approximately
4200 A. Figure 10 shows the pulse from one of the net2

B [µT]

Direct muon injection requires a pulsed kicker [20] toplace the muon bunch into the phase space acceptance ofthe storage ring. The center of the circular orbit for muons
entering through the inflector channel is offset from that of
the storage ring and, left alone, muons would strike theinflector exit after one revolution and be lost. A kick ofapproximately 10 mrad ( 0:1 T m) applied one quarterof a betatron wavelength downstream of the inflector exit isneeded to place the injected bunch on an orbit concentricwith the ring center. The ideal kick would be applied
during the first revolution of the bunch and then turned
off before the bunch returns on its next pass.
The E821 kicker makes use of two parallel current
sheets with cross-overs at each end so that the current
runs in opposite directions in the two plates. The 80-mm
high kicker plates are 0.75-mm thick aluminum, electronbeam welded to aluminum rails at the top and bottom,
which support the assembly and serve as rails for the 2kg NMR trolley. The entire assembly is 94-mm high and1760-mm long. This plate-rail assembly is supported on
Macor® insulators that are attached to Macor® plates withplastic hardware forming a rigid cage, which is placed
inside of the vacuum chamber. OPERA [21] calculations
indicated that aluminum would minimize the residual
eddy currents following the kicker pulse, and measurements showed that the presence of this aluminum assembly
would have a negligible effect on the storage-ring precision
magnetic field.
The kicker is energized by an LCR pulse-forming network (PFN), which consists of a single capacitor, resistor,
and the loop formed by the kicker plates. The capacitor ischarged by a resonant circuit and the current pulse iscreated when the capacitor is shorted to ground at the firingof a deuterium thyratron. The total inductance of the PFN
is L  1:6 H, which effectively limits both the peakcurrent and the minimum achievable pulse width. The
resistance is R  11:9  and the capacitance is C 
10:1 nF. For the damped LCR circuit, the oscillation frequency is fd  1:08  106 Hz and the decay time is d 
924 ns. The peak current is given by I0  V0 =2fd L,where V0 is the initial voltage on the capacitor. The resulting current pulse has a base width of 400 ns, which islong compared to the 149 ns cyclotron period. Therefore,
the positive kick acts on the muon bunch in the first,
second, and third turns of the storage ring.

0

-2

-4

-6

-8

-10
-8

-6

-4

-2

0

2

4

6

8

10
12
Time [µ s]FIG. 11 (color online). The magnetic field produced mainly byeddy currents on the kicker plates as measured by an opticalpolarimeter with the crystal at x; y  25 mm; 0 position. The
kicker is fired at 95 kV, producing 4200 A at the kicker plates.
The band shows the range of 0:1 ppm on integrated magnetic
field. The large filled circles correspond to simulations withOPERA, which used the measured current pulse as the input.

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works superimposed on a schematic representation of thetime and width of the muon bunch as it passes the locationof a single kicker section. While a square-wave current
pulse—bracketing the injected bunch and turning off completely before the next revolution —would be ideal, theactual pulse waveform acts both positively and negatively
on the bunch during the first five turns in the ring. The
injection efficiency is estimated to be 3–5%. Even at thismodest efficiency, direct muon injection is a significant
improvement compared to pion injection, not only in storage efficiency, but also because of the reduction of thehadronic flash.
The magnetic field produced by a prototype kicker wasmeasured [20] using a magnetometer based on the Faraday
effect. The main magnetic field, and the transient field
following the kicker pulse, were measured to a few percent.
Excellent agreement was obtained between OPERA andmeasurement for the field at the peak of the current pulse,
and for the residual magnetic field, see Fig. 11. The residual magnetic field 30 s after the main pulse contributes
less than 0.1 ppm to the integrated magnetic field seen by
the muon beam.

chamber. These probes have a cylindrical water
sample oriented with its axis along the tangential
direction [Fig. 12(c)]. They continuously monitor
the field during data taking.
(iv) Seventeen probes mounted inside a trolley that canbe pulled through the storage ring to measure thefield (Fig. 13). The probes on board the trolley are
identical in design and shape to the fixed probes(Fig. 12(c)).
Initially the trolley and fixed probes contained a water
sample. Over the course of the experiment, between run
periods, the water samples in many of the probes were

G. Field measurement instrumentation
Precision measurements of the magnetic field are madeusing the free-induction decay (FID) of nuclear magnetic
resonance (NMR) of protons in water [22]. The various
NMR probes are calibrated with respect to an absolutecalibration probe, which is a spherical sample of water[see Fig. 12(a)]. The functionality of the NMR measurement system and the reliability of the absolute calibration
probe were established in a wide bore superconducting
magnet. The very same equipment was used to calibrate
the field in a muonium microwave experiment at Los
Alamos National Laboratory [12], where the Zeeman effect in the ground state was measured to obtain the muon
magnetic moment. The suite of NMR probes used in E821
includes:(i) A calibration probe with a spherical water sample[Fig. 12(a)], that provides an absolute calibration
between the NMR frequency for a proton in a water
sample to that of a free proton [23]. This calibration
probe is employed at a plunging station located at aregion inside the storage ring, where special emphasis was put on achieving high homogeneity of
the field.
(ii) A plunging probe [Fig. 12(b)], which can be inserted into the vacuum in the plunging station at
positions matching those of the trolley probe array.
The plunging probe is used to transfer calibrationfrom the absolute calibration probe to the trolleyprobes.
(iii) A set of 378 fixed probes placed above and below
the storage-ring volume in the walls of the vacuum

FIG. 12. Schematics of different NMR probes. (a) Absoluteprobe featuring a 10 mm diameter spherical sample of water.
This probe was the same one used in Ref. [12] to determine themuon-to-proton magnetic moment ratio. (b) Plunging probe,
which can be inserted into the vacuum at a specially shimmed
region of the storage ring to transfer the calibration to the trolleyprobes. A spherical water sample is enclosed inside the coil Ls .
(c) The standard probes used in the trolley and as fixed probes.
The resonant circuit is formed by the two coils with inductances
Ls and Lp and a capacitance Cs made by the Al housing and a
metal electrode. A cylindrical water plus CuSO4 sample ofapproximately 10 mm length is used.

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proximately 50 kHz. The typical FID signal decays exponentially and the time between the first and last zerocrossing—the latter defined by when the amplitude has
decayed to about 1=3 of its initial value [22]—is of order
1 ms. The interval is measured with a resolution of 50 ns
and the number of crossings in this interval is counted. The
ratio gives the frequency fFID for a single measurement,
which can be converted to the magnetic field Breal at thelocation of the probe’s active volume through the relationB  BreffBreal  Bref 1  real
 Bref 1  FID : (13)
Bref
fref
The analysis procedure, which is used to determine theaverage magnetic field from the raw NMR data, is discussed in Section IVA.
FIG. 13 (color online). Photograph of the NMR trolley, whichmeasures the magnetic field in the storage ring. The array of 17
NMR probes, which are located inside the trolley housing,82(1) mm behind the front of the trolley. Electronics occupies
the back part of the device. At the location of the probes, the field
perturbation by these materials is less than 2 ppm and isaccounted for by the calibration method. The probe numbers
and placement are given by the schematic.

replaced with petroleum jelly. The jelly has several advantages over water: low evaporation, favorable relaxation
times at room temperature, a proton NMR signal almost
comparable to that from water, and a chemical shift (andthus the NMR frequency) having a negligible temperature
coefficient.
The free-induction decay signals are obtained after
pulsed excitation, using narrow-band duplexing, multiplexing, and filtering electronics [22]. The signals fromall probes are mixed with a standard frequency fref 
61:74 MHz corresponding to a reference magnetic field
Bref . The reference frequency is obtained from a synthesizer, which is phase-locked to the base clock of the
LORAN-C broadcast frequency standard [24], accurate
to 1011 . In a typical probe, the nuclear spins of the water
sample are excited by an rf pulse of 5 W and 10 s length
applied to the resonance circuit. The coil Ls and thecapacitance Cs form a resonant circuit at the NMR frequency with a quality factor of typically 100. The coil Lp
serves to match the impedances of the probe assembly andthe cable. The rf pulse produces a linearly polarized rf field
H~ in coil Ls , orthogonal to the dipole field. The rf pulse
rotates the macroscopic magnetization in the probe by 90.
The NMR signal from the precessing magnetization at thefrequency fNMR is picked up by the coil Ls of the same
resonance circuit and transmitted back through a duplexer
to the input of a low-noise preamplifier. It is then mixed
with fref to obtain the intermediate frequency fFID . We set
fref smaller than fNMR for all probes so that fFID is ap-

H. Detector systems, electronics and data acquisition
1. Electromagnetic calorimeters
Twenty-four electromagnetic calorimeters are placed
symmetrically around the inside of the storage ring, adjacent to the vacuum chamber, which has a scalloped shapeto permit decay electrons to exit the vacuum through a flatface cutout upstream of each calorimeter (see Fig. 8). The
calorimeters are used to measure the decay electron energy
and time of arrival. They are constrained in height by themagnet yoke gap. The width and depth were chosen to
optimize the acceptance for high-energy electrons, andminimize the low-energy electron acceptance. Each calorimeter is 140 mm high by 230 mm wide and has a depth of
13 radiation lengths (150 mm). The 24 calorimeters intercept approximately 65% of the electrons having energy
greater than 1.8 GeV. The acceptance falls with decreasing
electron energy.
Because of the high rate (few MHz) at early times
following injection, fast readout and excellent pulse separation (in time) are necessary characteristics of the design.
They are achieved by using a plastic-scintillator-based
sampling calorimeter read out by photomultiplier tubes(PMTs). The calorimeter (Pb/SciFi) volume is made of
52% lead alloy, 38% scintillating fiber, and 10% epoxy.
The detector provides good light yield and, in the limited
space, adequate shower containment. The 1 mm scintillating fibers are epoxied into grooved metal plates in a nearlyclose-packed geometry. They are oriented radially, terminating on four lightguides that pipe the light to Hamamatsu
R1828 2-inch PMTs (see Fig. 14). The individual PMT
gains and times were carefully balanced because the four
analog signals are added prior to sampling by a waveform
digitizer. The system is described in detail in Ref. [25].
Prior to use in the experiment, each calorimeter wascharacterized and calibrated at the AGS test beam. The
response deviates from a linear form by less than 1% up to
3 GeV, falling short by 2.5% at 4 GeV. The detector fractional energy resolution
is approximately 7.0% at 1.9 GeVp
and scales as 1= E. While the initial absolute calibration

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FIG. 14. Schematic of a Pb/SciFi calorimeter. The inset shows
the close-pack fiber-lead grid. The detector is subdivided into
four quadrants, each viewed by an independent PMT (notshown). A pulsed nitrogen laser provides a time calibration pulse
into each of the four quadrants. The front scintillator hodoscope,with its five-fold segmentation, is visible on the entrance face ofthe calorimeter.

and quadrant balancing were performed at the test beam,
final PMT balancing was established and gain stability
versus time was maintained by using the electron energy
spectra acquired during the data-collection periods. The
PMTs employed transistorized bases to maintain gainstability over the very wide dynamic range of rates andbackground levels encountered during the data acquisition
period [26].
During normal data collection, an intense burst of particles strikes the magnet pole pieces and calorimeters at the
instant of injection. The detectors immediately downstream of the inflector exit are particularly affected.
Because this ‘‘prompt flash’’ would completely saturate
the PMTs, they are gated off prior to injection and turnedback on some tens of microseconds later, depending on thelocation of the calorimeter with respect to the inflector exit.
The output of a PMT was suppressed by a factor of 106 by
exchanging the bias on dynodes 4 and 7. When the dynodes
are reset, the gain returned to better than 99% of its steadystate value in approximately 1 s.
The prompt flash creates an enormous number of neutrons, many of which are thermalized and captured inside
the Pb/SciFi calorimeters. Capture gamma rays can knockout electrons, which traverse the scintillating fibers. The
generated light gives an elevated background pedestal thatdiminishes with the time dependence:  t1:3 for t >5 s. The effective decay time was minimized by doping
detectors in the first half of the ring—where the capture
rate is highest—with a natural boron carbide powder. The
10
B component (20%) has a high thermal-neutron capture
cross section.
To monitor the detectors, a 300 ps uv (  337 nm)
pulse from a nitrogen laser is directed through a splitter
system into an outside radial corner of each of the quad-rants of all calorimeters. The uv pulse is absorbed by asample of scintillating fibers. These fibers promptly emit alight pulse having the same fluorescence spectrum as produced by a passing charged particle. The calibration pulse
propagates through the entire optical and electronic readout system. A reference photodiode and PMT—locatedwell outside the storage ring—also receive a fixed fractionof the light from the laser pulse. The laser was fired every
other fill, in parallel with the beam data, during 20-minuteruns scheduled once per 8-hour shift. Firings alternatedamong four points in time with respect to beam injection.
These points were changed for each laser run, providing a
map of any possible gain or timing changes. Timing shifts
were found to be limited to less than 4 ps from early-to-latetimes, corresponding to an upper limit of 0.02 ppm systematic uncertainty in !a . No overall trends were seen in thelaser-based gain change data. Observed gain changes weregenerally a few tenths of a percent, without a preferredsign. Unfortunately, the scatter in the these calibration
measurements is much greater than is allowed by statistics,
and the mechanism responsible has not been identified.
Consequently, the laser-based gain stability (seeSec. IV B 2) could be established to no better than a few
tenths of a percent. Ultimately, monitoring the endpoints ofthe electron energy spectra provided a better measurementof gain stability.
2. Special detector systems
Hodoscopes consisting of five horizontal bars of scintillator —the front scintillating detectors (FSDs)—are attached to the front face of the calorimeters (Fig. 14). The
FSDs are used to measure the rate of ‘‘lost muons’’ fromthe storage ring and to provide a vertical profile of electrons on the front face of the calorimeters. The FSD signals
are also used as a check on the pulse times reconstructedfrom the calorimeter waveforms. The individual scintillators are 235 mm long (radial), 28 mm high and 10 mmthick. They are coupled adiabatically to 28-mm diameter
Hamamatsu R6427 PMTs, located below the storage-ringmagnet midplane. The PMT bases are gated off at injection, following a scheme similar to that used in the calorimeter PMT bases. (Eventually, over the run periods,
about half of the FSD stations were instrumented with
PMTs.) An FSD signal is recognized by a leading-edge
discriminator and is recorded by a multihit time-to-digitalconverter (MTDC).
An xy hodoscope with 7 mm segmentation is mounted
on the front face of five calorimeters. These positionsensitive detectors (PSDs) have 20 horizontal and 32 vertical scintillator sticks read out by wavelength-shifting
fibers and a Philips multianode phototube (later replaced
by a multichannel DEP hybrid photodiode). The MTDC
recorded event time and custom electronics coded the xy
profile, providing information on albedo and multiplicity
versus time. In some cases, a calorimeter was equipped

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with both PSD and FSD, providing efficiency checks ofeach. The vertical profiles of both FSD and PSD provide
sensitivity to the presence of an electric dipole moment,
which would tilt the precession plane of the muon spin.
Depending on the average AGS intensity for each running period, either a thin scintillator or Čerenkov counter islocated in the beamline, just outside the muon storage ring.
This ‘‘T0 ’’ counter records the arrival time and intensity(time) profile of a muon bunch from the AGS. Injected
pulses not exceeding a set integrated current in T0 are
rejected in the offline analysis because they do not provide
a good reference start time for the fill and because they are
generally associated with bad AGS extraction.
Scintillating-fiber beam-monitors (FBM), which are rotated into the storage region under vacuum, were used toobserve directly the beam motion in the storage ring during
special systematic study runs. Separate FBMs measure thehorizontal and vertical beam distributions. Each FBM is
composed of seven fibers centered within the muon storage
region to 0:5 mm. The 90-mm long fibers are separated
by 13 mm. One end is mated to a clear fiber, which carries
the light out of the vacuum chamber to PMTs mounted on
top of the storage-ring magnet. The PMTs are sampled
continuously for about 10 s using 200 MHz waveform
digitizers. The beam lifetime with the fiber beam monitors
in the storage region is about one half the 64 s muondecay lifetime.
The scalloped vacuum chamber is truncated at one
location and a 360 m-thick mylar window (72 mm
wide by 110 mm high) replaces the aluminum flat exit
face to allow electrons to pass through a minimum of
scattering material. A set of four rectangular drift chambers—the traceback system [27]—is positioned between
this window and calorimeter station #20. Each chamberconsists of three layers of 8-mm diameter straw drift tubes
oriented vertically and radially. An electron that exits thewindow and passes through the four chambers is tracked by
up to 12 vertical and 12 radial measurements. Althoughdecay electrons are not always emitted tangentially, theirradial momenta are sufficiently small so that an extrapolation of the track back to the point of tangency with thecentral muon orbit yields a measurement of the actual

muon-decay position, as well as its vertical decay angle.
The former is helpful in establishing the average positions
and therefore the average field felt by the muons.
3. Waveform digitizers and special electronics
Waveform digitizers (WFDs) were used to collect raw
data from the PMT signals from the calorimeter and fiber
monitors, and to monitor voltages in the electrostatic quadrupoles and fast kicker as a function of time. Each WFD
consists of a motherboard and an application-specificdaughtercard. The motherboard features four independent
8-bit digitizers with 64 kbytes of memory per channel. For
the calorimeter, signals from the four phototubes are
summed on the daughtercard and the resulting signal is
directed into two adjacent channels on the motherbaord,
which sample the waveform on alternate phases of a
200 MHz clock, thus effectively providing a 400 MHz
sampling rate. Data from each clock phase is written into
separate memory buffers every 20 ns. Zero-suppressed
operation, if invoked, is provided by comparators locatedon the daughtercard. For the calorimeter pulses, roughly
16 ns of the baseline are recorded prior to the time of thetrigger and about 64 ns worth of samples are recorded after
the trigger. For the fiber beam monitors and the quadrupole
and kicker voltage readout, four different versions of thedaughtercards were used. Sampling rates varied: 200 MHz
for the fiber monitors and kicker, 2 MHz for the quad
voltage monitor. These WFDs were operated without
zero suppression.
The time words written to memory in each of the twophases have a fixed but arbitrary offset in any datacollection cycle. To resolve the ambiguity, a 150 ns triangular pulse is directed into a fifth analog input on eachdaughtercard immediately prior to each fill. By reconstructing this ‘‘marker pulse,’’ the unknown offset is determined unambiguously, and the data streams can then be
combined in the offline reconstruction program. Typicalcalorimeter pulses, with samples from the two phasesinterleaved, are shown in Fig. 15.
The clock signals for the WFD, MTDC and NMR are allderived from the same frequency synthesizer, which isFIG. 15. Examples of waveform digitizer samples from a calorimeter. The two WFD phases are alternately shaded. The left panel
shows a simple, single pulse, while the right panel has two overlapping pulses separated by 7.5 ns.

072003-14

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

synchronized to the Loran C time standard [24]. There isno correlation between the experiment clock and the AGS
clock that determines the time of injection. This effectively
randomizes the electronic sampling times relative to theinjection times at the 5 ns time scale, greatly reducing
possible systematic errors associated with odd-even effects
in the WFDs.
III. BEAM DYNAMICS
A. Overview
The muon storage ring uses the electrostatic quadrupoles
to obtain weak vertical focussing [28,29]. The field index isgiven by:
R @Ey
n 0
;
vB0 @yy  Ay cosyThe maximum accepted horizontal and vertical angles aredefined by the 45 mm radius of the storage volume, rmax ,givingp
rmax 1  nhmax 
(18)
R0
p

r
n
vmax  max :
R0

(14)

where R0 is the central orbit radius, B0 is the dipole
magnetic field, and v is the muon speed. The coordinate
system is shown in Fig. 16. High voltage of 24 kV
applied to the quadrupole plates gives an n value of
0.137, which is the field index that was used in the R9799 periods. The R00 period used n  0:135 and the R01
running was split into ‘‘low-’’ and ‘‘high-n’’ subperiods
with n  0:122 and 0.142, respectively. The field index
value determines the stored muon beam betatron motion,
which in turn can perturb the electron time distribution, as
discussed in Sec. III C.
For an ideal weak-focusing ring, where the quadrupole
field is constant in time and uniform in azimuth, thep
horizontal pand vertical tunes are given by x  1  n
and y  n: Consider the motion of a particular muon
having momentum p compared to the magic momentum
p0 . Its horizontal and vertical betatron oscillations aredescribed byx  xe  Ax cosx

about the equilibrium orbit (xe ) and the horizontal midplane, respectively, with xe given byp  p0:(17)xe  R0p0 1  n



0x 

0y :

(19)

In general, the muon storage fraction increases forhigher quadrupole voltage. The number of stored muonsis plotted versus high voltage in Fig. 17. As the highvoltage is increased, the vertical phase space increases,
while the horizontal phase space decreases. The operating
quadrupole high voltage is chosen to avoid the beamdynamics resonances, which take the form Lx  My 
N, where L, M, and N are integers. The resonance lines andthe storage-ring working line given by 2x  2y  1 areshown in Fig. 18. Typical working values are x  0:93
and y  0:37 for n  0:137, giving hmax  5:9 mrad andvmax  2:3 mrad.
Numerical calculations, which include the finite extent
of the quadrupole electrodes, were used to select the exact
high-voltage values. Confirmation of resonance-induced
muon losses was made during special runs tuned to the

(15)
(16)

Here  s=R0 , where s is the azimuthal distance aroundthe ring and Ax and Ay are amplitudes of the oscillations

FIG. 16. Coordinate system used to define the beam-dynamics
expressions. An end view (left panel) with the negative muon
beam into the page; the center of the storage ring is to the right,
off the scale of this figure. The magnetic dipole field is oriented
down. A top view (right panel) with the beam travelling in thes direction.

FIG. 17. The normalized number of stored muons (squares)
and the expected dependence (circles) for short runs varying thequadrupole high voltage. Here, KSEC is a relative measure of
the proton intensity at the target as viewed by a secondary
emission counter.

072003-15

PHYSICAL REVIEW D 73, 072003 (2006)νy2νy = 11=5νy = 2x + 3ν
y=2

3

3ν
x

y=

0.40 ν

ν =3νx + y

y

20000
10000
0
10

1

0.90

2

0.85

−

=1

0.95

1.00

12

14

16

18

20
Time [ µs]FIG. 19 (color online). Intensity at a single detector station
shortly after injection. The rapid modulation repeats at thecyclotron frequency as the muon bunch circles the ring. The
width of the bunch grows with time because of the finite p=p of
the stored muons. The slow variation in the maximum amplitude
is at the (g  2) frequency.

2 ν =νx + y

2νx

3νy

4

x

0.25
0.80

30000

n = 0.148
n = 0.142
n = 0.137
n = 0.126
n = 0.122
n = 0.111
n = 0.100

νy =

=0

3

3νy
ν −

60000

40000

4νx +

0.35 3ν = 1

−2
νy

2ν
y

3ν

−

x+

2ν
x

2ν

0.45

70000

50000

νx = 11
νy = −
4
−
νx
=0
ν
y
2
νx −

=2

0.50

0.30

Counts/5 ns

G. W. BENNETT et al.

νxFIG. 18 (color online). The tune plane showing resonancelines. Three of the n values used to run the experiment, 0.122,
0.137, 0.142, are indicated on the arc of the circle defined asv2x  v2y  1. They do not intersect any of the resonance lines,
contrary to nearby tunes, which are also shown on the arc.

resonances x  3y  2 and 2x  3y  3 (n  0:126
and n  0:148).

distribution of angular frequencies will cause the bunchedbeam to spread out around the ring over time, in a mannerthat depends uniquely on the momentum distribution. In
particular, the time evolution of any finite frequency slice
is readily specified. A given narrow bin of frequencies
contributes linearly to the time spectrum. The total time
spectrum is a sum over many of these frequency components, with amplitudes that can be determined using 2
minimization. The momentum distribution is then determined from the frequency distribution (or equivalently,

B. Fast rotation
The stored muon momentum distribution is determinedby analyzing the debunching of the beam shortly after
injection. This fast-rotation analysis is based on a few
simple ideas. High-momentum muons trace a larger average radius-of-curvature than low-momentum muons.
However, because all muons travel at the same speed(0:9994c, constant to a part in 105 over the aperture), thehigher (lower)-momentum muons have a smaller (larger)
angular frequency. A finite range in angular frequencies
causes the initial bunched beam to spread azimuthally overtime. Figure 19 shows the decay electron rate at one
detector station from 10 to 20 s following injection.
The rapid rate fluctuation reflects the 149 ns cyclotron
period of the bunched beam. The slower modulation iscaused by the (g  2) spin precession. The rapid bunch
structure disappears over time, as the beam uniformly fills
the ring in azimuth.
The initial bunched beam is modeled as an ensemble ofparticles having an unknown frequency distribution and anarrow time spread (rms 25 ns, occupying 60 deg ofthe ring). The model assumes that every time slice of thebeam has the same frequency profile but the time width isleft as a fit parameter, as is the exact injection time. The

FIG. 20. The distribution of equilibrium radii dN=dxe , asdetermined from the fast-rotation analysis. The dashed verticalline is at 7112 mm, the magic radius; the dotted line is at7111 mm. The solid circles are from a debunching model fit to
the data, and the dashed curve is obtained from a modified
Fourier analysis.

072003-16

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

(20)

The result of the fast-rotation analysis from the R00
period is shown in a plot of the beam radius-of-curvature
distribution shown in Fig. 20. The smooth curve is obtained
from a modified Fourier transform analysis. The peak ofthe distribution lies below the nominal magic radius of
7112 mm but the mean is somewhat larger, 7116  1 mm
for R00 and 7115  1 mm for R01. The rms width is about
10 mm.
C. Coherent betatron oscillations

Intensity [arb units]

Special short runs were devoted to observing the muon
beam motion directly using the scintillating fiber beam
monitors (FBM) described in Section II H 2. Figure 21
shows pedestal-subtracted data from the FBM, which sits
180 deg around the ring from the inflector exit. Data
from fibers 2, 4, and 6 (where 4 is the central fiber) are
shown. The left (right) panel shows the fiber intensity
representing the radial (vertical) profile of the beam versus
time shortly after injection. The average size of the signal
across the aperture versus time reveals the longitudinalmotion of the bunched beam at one point in the storage
ring. The rapid oscillation (period  149 ns) in intensity,
which is approximately in phase for every fiber, corresponds to the cyclotron motion of the beam.
The relative size of the signals determines the momentsof the transverse distributions, which oscillate in time.
Oscillations in the width arise from a mismatch between
the inflector and storage-ring apertures. The former is
18 mm and the diameter of the latter is 90 mm. A narrow‘‘waist’’ is therefore imposed on the beam at injection,
which then alternately widens and narrows as the beam
Radial

4000

5000

6000

7000

8000

9000 10000
Time [ns]circulates. Because the deflection provided by the kick issmaller than that required to center the beam in the storage
aperture, the centroid of the beam oscillates. When the
beam is observed at one location (e.g., at a particular
detector station), the width and centroid oscillate at frequencyp
fCBO  fc  fx  fc 1  1  n  476 kHz;the coherent betatron oscillation (CBO) frequency. The
corresponding period is 14.1 turns.
Prominent oscillations are evident in both the mean andwidth of the radial distributions measured with the FBM. In
the left panel of Fig. 21, the traces are stacked from top tobottom corresponding to the intensity of the muons at low,
central, and high radii within the storage aperture. The
vertical arrow illustrates a time when the bunch width isnarrow in radial profile. It reaches a maximum width
approximately 1:1 s later. The breathing frequency corresponds to the horizontal CBO, which for this R01 low-n
period data, is 425 kHz. The fact that the inner and outerradial fiber intensities peak somewhat out of phase indicates that the mean radial position oscillates, again, at thesame CBO frequency.
The interference of multiple frequencies is also apparent
in the vertical profiles shown—with a different time axis
range—in the right panel of Fig. 21. The top-to-bottom
beam intensity is given by the three shown fiber traces,
which are stacked in the same sense. While the dominantfeature is the repetitive cyclotron frequency, a slower,
undulating pattern can also be seen. This pattern —with aperiod roughly indicated by the two vertical dashed lines—
is related to the vertical oscillations of the beam. Recall
that the phase space simulation plots in Fig. 5 show that thebeam initially fills only the central 56 mm of the storage
Intensity [arb units]from the radial distribution) byp  p0R  R0 1  n:
p0
R0

PHYSICAL REVIEW D 73, 072003 (2006)

Vertical

4000

4500

5000

5500

6000

6500 7000
Time [ns]FIG. 21. Digitized waveform from the FBM located 180 from the inflector. The fast-rotation period is observed in both the verticaland the horizontal profiles. Left panel: Radial profile from an inner, middle, and outer fiber trace (stacked top to bottom with an offset).
The width minimum is indicated by the arrow. The outer fibers do not peak at the same time because the radial mean is oscillating at
the same frequency as the width. Right panel: The vertical profile from a top, middle and bottom fiber. The two dashed lines are placed
at approximately 3.3 turns, the period of the vertical betatron oscillation. A vertical waist is apparent halfway between the dashed lines.

072003-17

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

TABLE VIII. Important frequencies and periods in the (g  2)
storage ring for n  0:137.
Physical frequency

Variable Expression Frequency Period

Anomalous precession
Cyclotron
Horizontal betatron
Vertical betatron
Horizontal CBO
Vertical waist

fa
fc
fx
fy
fCBO
fVW

e
2m a Bv
2R0p

1  nf cp
nf c
fc  fx
fc  2fy

0.23 MHz 4:37 s
6.71 MHz 149 ns
6.23 MHz 160 ns
2.48 MHz 402 ns
0.48 MHz 2:10 s
1.74 MHz 0:57 s[cm]

aperture because it is limited by the height of the inflector
channel. However, after one quarter of a vertical betatronoscillation, it fills the full 90 mm. The minimum width at
56 mm is called the vertical waist (VW) (see, for example,
halfway between the dashed lines in the figure, where theouter intensities are p
small).
The vertical width is modulated
at frequency 1  2 nfc  2:04 MHz (low-n R01 data).
The period is 3:3 turns, or 0:49 s.
The frequencies associated with the various beam motions are revealed in Fourier analyses of both the fiber
monitor and muon-decay electron time spectra. However,the lifetime of the oscillations can only be determined by
the latter measurements, which do not affect the beam
directly. The observed CBO lifetime is about
100–140 s. The VW lifetime is much shorter, about
25 s. These lifetimes are determined primarilyp by thetune spread. For the p
CBO,


n=2
1  n. For thexVW, y  n=N n where N is the harmonic of the
CBO frequency. From the FBM data, the VW has CBO
frequency harmonics with relative amplitudes: 0.6, 0.28,
0.08 and 0.04 for N  1, 2, 3 and 4, respectively. The
observed lifetimes imply an effective field index spreadof n ’ 103 , in agreement with computer simulations.
A summary of important frequencies for the case n 
0:137 is given in Table VIII.
The electron traceback system makes a more direct, but
still nondestructive, measurement of the muon phasespace. Figure 22 shows a representative distribution of
the radial mean and width of the muon population from
110 to 150 s after injection during the R99 period. The
fundamental CBO frequency is the dominant feature ofboth plots. The amplitude is damped with an exponential
lifetime consistent with that found in other studies.
D. Muon losses
Small perturbations in the magnetic or electric fields
couple to the horizontal and vertical betatron oscillations.
If the forces arising from these perturbations are periodic,
the amplitude of the oscillatory motion can grow withoutbound until the muon is lost. The losses are minimized bychoosing a field index that does not lie on one of theresonance lines shown in Fig. 18, by minimizing the electric and magnetic-field perturbations, and by ‘‘scraping’’
the beam during the first 7–16 s after injection.
Two perturbations of the field are of particular concern:a nonzero average radial component of the magnetic field
and the component of the main dipole field that varies ascoss=R0 . The latter, the first harmonic component, displaces the orbit horizontally. It was measured with the
NMR system and shimmed at the beginning of each running period to be less than 30 ppm, well below the target
upper limit of 240 ppm. Monthly variations were kept
under 20 ppm. A nonzero radial field displaces the averageorbit vertically. The radial magnetic field was adjusted
using current shims so as to maximize the number of stored
muons. This centered the beam height to 0:5 mm withrespect to the collimators.
Beam scraping, which is discussed in detail in Ref. [19],refers to the application of asymmetric voltages to thequadrupole plates shortly after injection. Scraping eliminates muons having trajectories close to the collimators.
The scraping procedure displaces the beam downward by
about 2 mm and recenters the central orbit in the horizontal plane, moving it 2 mm inward within two of thequadrupole regions and 2 mm outward in the other two
regions. Muons at the extreme edges of the storage-ring

2
1.8
1.6

Width

1.4
1.2
1
0.8

f= ~470 kHz

Mean

0.6
0.4
0.2
0
110

115

120

125

130

135

140

145

150
Time [µ s]FIG. 22 (color online). The radial mean and rms width as determined by the traceback detector system in the R99 period. Note thatthe width includes resolution smearing; the true width is  1:55 cm. Each fit (solid lines) includes a damped exponential and an
oscillatory term having the frequency of the horizontal CBO effect.

072003-18

PHYSICAL REVIEW D 73, 072003 (2006)
Data / Fit

Mean [mm]FINAL REPORT OF THE E821 MUON ANOMALOUS . . .
2

1

1.02
1.015
1.01

0

1.005
1

-1

0.995
-2

0.99

0

10000

20000

30000

40000

0.985

50000
Time [ns]100FIG. 23. Mean vertical position of the beam as measured by
FSD Station 22 during the R01 period. The beam moves vertically with a time constant of 5 s after the end of scraping.

phase space then strike a collimator (see below) and areremoved from the ring. The quadrupole plate voltages arerestored to symmetric values (with a 5 s time constant)
after 7–15 s (see Table VII). The reduced effective aperture typically results in a 10% loss of stored muons, inagreement with analytical calculations and computer tracking simulations. The effect of scraping on the averageheight of the muon distribution as a function of time, as
measured with the FSD counters, is shown in Fig. 23. The
mean beam position for Station 22 in the R01 period starts
a little over 1 mm low and rises as the scraping voltages areturned off.
The collimators, which define the muon storage region,
are made of 3-mm thick copper. They have an inner radiusof 45 mm and an outer radius of 55 mm. Eight locations areavailable for the collimators in the bellows adjoining thevacuum chambers (three more are blocked by the kicker
and one is unused due to proximity to the inflector exit).
The inner half circles of the four collimators opposite theinflector are removed in order to avoid scattering those
low-momentum muons which, because of only a twothirds kick on the first turn, would otherwise strike thecollimators on the first turn and be lost.
The muon-decay fitting function Nt must account formuon losses. A fit to the simple function f  Cet= , fordata binned in the (g  2) period and for times t 
300 s, gives a satisfactory description of the data.
However, an extrapolation of the fit back to early times
reveals that the data lie above the fit as shown in the ratio ofdata to fit in Fig. 24, demonstrating that the fractional lossof muons is greater early in the measurement period.
The relative rate of muon losses can be measured directly using a coincidence of three FSD scintillator hits inthree successive detector stations. A 3.1 GeV muon loses
approximately 100 MeV by ionization in each calorimeter,well below the WFD hardware threshold. A muon-loss
event therefore consists of a threefold FSD coincidence

200

300

400

500
600
Time [ µs]FIG. 24 (color online). A fit to the simple function, f 
Cet= is made for the time period 300–600 s after injection
for a subset of the pileup-subtracted R01 data. The fit is extrapolated back to early times and the ratio data to fit is plotted
for all times. The excess of data compared to the fit at early times
indicates the nonconstant loss rate with time. The bin width is the(g  2) period.

with no accompanying calorimeter signal. After randomcoincidence background subtractions, the time-dependent
loss function is constructed, up to an unknown efficiency
factor, which is typically about 6%.
Beam protons were stored during the positive muon run
periods R97–R00. Like muons, a fraction of these protons
exit the ring because of orbit perturbations. While proton
losses have a negligible effect on the decay positron distribution, they do form a significant background to the lostmuon spectrum, which must be removed. A proton-loss
event is defined as a muon-loss candidate having large
energy in the third calorimeter, presumably from a hadronic shower. The proton-loss signal was studied using dataacquired when the quadrupole voltages were dropped tozero, long after all muons had decayed in a given fill. The
uncertainty in the shape of the muon-loss spectrum isdominated by uncertainty in the proton-loss component—assuming, in addition, that the muon-loss monitor
samples a constant fraction of the actual ring losses versus
time in the fill. The antiproton component for the R01
period with negative muons was negligible.
E. Electric-field and pitch correction
The magnetic field is adjusted so that the magic momentum p0 (see Eq. (5)) is at the center of the aperture. The(g  2) precession !a of muons at this radius is notaffected by the electric field, which is, in any case, zeroat the center of the quadrupoles. However, muons of higher
or lower momentum are subject to a linearly rising fieldand deviate from the magic momentum, so their (g  2)
frequency is reduced. We apply a correction for this effect.

072003-19

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

As the muons oscillate about their equilibrium radii xe ,they experience a mean radial electric field hEr i nB0 =R0 xe . Differentiating Eq. (5) with respect to p,
and averaging over the muon ensemble gives 2 !axe
 22 n1  n
(21)
!a
Ro
where hx2e i is obtained from the fast-rotation analysis.
While the radial distribution uncertainty is based on data,
it is checked against the simulation. Both methods agree on
the mean-square width to a precision of 5%. This uncertainty is added in quadrature with the uncertainty in themean muon radial position with respect to the electrostatic
quadrupoles R  0:5 mm (  0:01 ppm in a ), andwith the uncertainty in the mean vertical position of thebeam 1 mm (  0:02 ppm in a ). As a quantitative
example, the R01 electric-field correction to the precession
frequency for the low-n subperiod is 0:47  0:05 ppm.
The pitch correction [30] accounts for the fact thatvertical betatron motion decreases slightly the magnetic
field felt by the stored muons in their rest frame. When theorbit is inclined at angle to the horizontal, to sufficientaccuracy !a is reduced by the factor 1  12 2 . If m is theangular amplitude of the vertical oscillation, the average22over the ensemble of muons is 1  14 h mi. Here h mi
222
nhy i=R0 , where hy i is the mean-squared vertical spreadof the stored muons, which was measured by the traceback
system, confirming results from the tracking simulation.
The correction to the measured !a due to vertical betatron oscillations is calculated for each n value setting. The
systematic error is estimated as the difference between thecorrection calculated from the simulated and measured
muon distributions (  0:03 ppm) added in quadrature
with the uncertainty in the mean muon radial position
with respect to the electrostatic quadrupoles and the uncertainty in the mean vertical position of the beam. For the
R01 low-n subperiod, the correction is 0:27 
0:04 ppm.
A tracking program, simulating the muon spin precession in the storage ring—including the correct azimuthal
quadrupole electric field, confirmed the validity of theanalytic electric-field and pitch corrections. The equations
given in Ref. [31] were used for the orbit and the spin
motion. The validity of the electric-field and pitchcorrection expressions are verified to 0.01 ppm in a .
IV. DATA ANALYSISThe calculation of a requires the determination of themuon spin precession frequency, !a , and the averagemagnetic field seen by the muons, !
~ p . ‘‘Precession’’ and‘‘field’’ data are recorded separately and analyzed independently by different groups in the collaboration. To prevent
bias, each analyzer adds a secret offset to any frequency
result presented publicly. When all the analyses of a given

type—precession or field—are deemed internally consistent, the individual offsets are replaced by a common offset, to facilitate comparisons between different analyses.
Only when all the analyses of each type are in agreement,
and all systematic investigations finished, are the final
offsets removed. Then a is calculated from the true !aand !
~ p results.
Two independent determinations of !
~ p were made perrunning period. The NMR tools, described in Section II G,
were used to provide the measurements as follows. Every
3– 4 d, the field distribution in the storage volume ismapped by the NMR trolley. The time-dependence of the
field is measured using the fixed-probe NMR system,
which samples and records the magnetic field around thering at approximately 0.1 Hz. Special calibration measurements are made before and after the data-taking periods at
a fixed location inside the ring vacuum, where the field
uniformity is especially good, to transfer the absolutecalibration to the plunging probe and to the trolley probes.
The calibration of trolley probes against the plunging
probe demonstrates the stability of the trolley probe over
the run period.
Four or five independent determinations of !a weremade per running period. The principal data are the waveform digitizer records for each calorimeter station, fromwhich decay electron times and energies are extracted.
Events above an energy threshold are included in electron
time distributions, such as the one shown in Fig. 2. The
precession frequency is extracted from a 2 minimization
fitting procedure, where the fit function is based on
Eq. (10), plus small perturbations from known physicaleffects.
The data-taking was organized in short (  45 min )runs, during which the macroscopic conditions remained
stable. A database was used to mark each run for special
characteristics, such as laser calibration, field index andscraping time, special radial magnetic-field settings andother systematic tests. Approximately 1000 –1300 good
production runs remain per data-taking period after complete commissioning of the apparatus and excluding special runs for systematic studies. The vast majority of runs
took place under ideal operating conditions. However,whole runs were removed if they failed a KolmogorovSmirnov test that was designed to compare each run
against a standard data set; this test was aimed specificallyto search for hardware problems that may have gone unnoticed. One special virtue of the method is that it does notrely on any information associated with the precession
frequency. Runs were also discarded if the power supplies
for the surface-correction coils were not stable. Before
being included in the final data sample, each fill within arun was checked against a set of quality standards: injection bunch intensity and shape must not vary substantially;no AGS extraction misfires; no quadrupole sparks or missing quadrupole traces from the data stream; no kicker

072003-20

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

misfirings; no detector electronics malfunctions; no memory overflows in the onboard electronics; and, no missing
marker pulses.
~p
A. Determination of !p and !
Two independent analyses of the magnetic field weremade for each of the data-taking periods. The methodology
was similar for all analyses. Consistent results were obtained for all run periods. The description below focuses on
one of the analyses for the R00 data-taking period.
1. Absolute calibration probe
The determination of a requires a measurement of thefree-proton precession frequency, !p  2fp , in the storage ring magnetic field. However, the free-induction decay(FID) signals observed with the trolley NMR probes were
not those from free protons, but protons in water with a
CuSO4 additive that shortened the relaxation time of the
FID signal. In addition the field is different from that inwhich the muons were precessing, being perturbed by themagnetization of the materials in the probe and trolley
construction. To determine !p required an absolute calibration probe to relate the trolley probe FID precession
frequencies to that of a free proton in an unperturbed field.
The absolute calibration probe [32] measured the precession frequency of protons in a spherical water sample. It
is constructed to tight mechanical tolerances, using materials of low magnetic susceptibility. The perturbation of theprobe materials on the field at the spherical water sample
was minimized, measured, and corrected for. This allowed
the storage ring magnetic field to be expressed in terms of
the precession frequency of protons in a spherical water
sample !p sph; H2 O; T at temperature T.
A precession frequency measured with the absolutecalibration probe was then related to the spin precession
frequency of a free proton through [33]:!p sph; H2 O; T  1  H2 O; T !p free:(22)

Here H2 O; T accounts for the internal diamagnetic
shielding of a proton in a water molecule, determinedfrom [23]:H2 O; 307:85 K  1 

gp H2 O; 307:85 K gJ H
gp H
gJ H

gp H
gp free 25:79014  106 :The ratio of the g factor of the proton in water to that of theelectron in ground-state atomic hydrogen was measured to
10 ppb [23]. The ratio of the electron to proton g factors inhydrogen is known to 9 ppb [34], and the bound-statecorrections relating the g factor of a free proton to one inhydrogen were calculated in [35,36]. We also corrected forthe measured temperature dependence of the shielding
constant, d H2 O; T=dT  10:3630  109 =K [37].
The overall accuracy of the free-proton precession frequency determined with the absolute calibration probe wasestimated to be 0.05 ppm. This same probe was used in themuonium experiment in which  =p has been determined [12]. Since the magnets were different for the two
experiments, the perturbations of the absolute calibration
probe materials on the field at the spherical water sample
were different. The change in field is of the order of a few
ppb, and is caused by magnetic images of the probe in theg  2 magnet pole pieces.
A cross-check on the accuracy of the calibration probe
comes from measurements made in the muonium experiment where the ground-state hyperfine interval HFS was
measured to 12 ppb, a result that is independent of errors in
the calibration probe. Because the theoretical prediction
for the hyperfine interval, HFS;th , depends on  =p as
a parameter, by equating HFS and HFS;th , we candetermine  =p to 30 ppb [12,33,38]. The value extracted agrees with  =p , determined to 120 ppb fromthe measurement of two Zeeman hyperfine transitions inmuonium in a strong magnetic field [12]. The latter measurement of the magnetic moment ratio is strongly dependent on the magnetic field measured using the absolutecalibration probe, and it is only weakly dependent on
HFS;exp . If we assume the theoretical value is correct,
the consistency of the two determinations of  =p suggests the absolute calibration probe is accurate to better
than 120 ppb.
2. Trolley probe and plunging probe relative calibration
Before and after each data-taking period, the absoluteprobe was used to calibrate the plunging probe, the trolley
center probe, and several of the other trolley probes. The
procedure was carried out at carefully shimmed locationsin the storage ring having good field homogeneity.
Measurements from the 17 trolley probes, taken at suchan azimuthal location, were then compared to those fromthe plunging probe and hence with respect to each other.
The errors which arise in the comparison are caused bothby the uncertainty in the relative positioning of the trolleyprobe and the plunging probe, and on the local field inhomogeneity. The position of the trolley probes is fixedwith respect to the frame that holds them and to the railsystem on which the trolley rides. The vertical and radial
positions of the trolley probes with respect to the plunging
probe are determined by applying a sextupole field andcomparing the change of field measured by the two. The
error caused by the relative position uncertainty is minimized by shimming the calibration field to be as homogeneous as possible. In the vertical and radial directions, thefield inhomogeneity is less than 0:02 ppm=mm, as shownin Fig. 25, and the full multipole components at the calibration position are given in Table IX, along with the072003-21

PHYSICAL REVIEW D 73, 072003 (2006)
vertical distance [cm]vertical distance [cm]G. W. BENNETT et al.
4
0.5

3

−1.0

2
1

−0.5

0

0

0

−1
−2

−1.0
−1.5
−2.0

0.5

−0.5

−4
−4

−3

−2

−1

0

2

0.5

−1.0
−0.5
−0.5

1
0

0
0

0

−1

−0.5

−2

1.0
1.5
2.0
2.5
3.0

−3

3

2.0
1.5
1.0
0.5

4

−1.0
−1.0
−1.5
−2.0

−3

0.5
0.5
1.0

−4

1
2
3
4
radial distance [cm]

−4

−3

−2

−1

0

1
2
3
4
radial distance [cm]FIG. 25. Homogeneity of the field at the (a) calibration position and (b) in the azimuthal average for one trolley run during the R00
period. The contour lines correspond to 0.5 ppm field differences between adjacent pairs.

multipole content of the full magnetic field averaged overazimuth. For the estimated 1 mm position uncertainty, theuncertainty on the relative calibration is less than 0.02 ppm.
Field variation along the azimuthal direction also leads
to calibration uncertainties because the precise azimuthal
location of the trolley probe’s active volume is not known,a priori, to better than a few mm. Azimuthal field gradientswere created at the calibration location by powering correcting coils on the surface of nearby magnet poles. The
gradients were measured with the trolley positioned atvarious locations, typically 10 mm apart in azimuth. The
trolley measurements and the shift in field measured withthe plunging probe determine the azimuthal positioning of
the active volume within the trolley probes with respect tothe plunging probe. The corresponding contribution to therelative calibration of the trolley probes amounts to
0.03 ppm.
The calibration of the NMR probes may vary with themeasured NMR frequency, since the frequency is determined by counting zero crossings in a signal with a decaying baseline. Other factors, such as the temperature andpower supply voltage, may have an effect as well. The
effects were studied and an uncertainty contribution of
0.05 ppm in the field measurement was derived.

The absolute calibration of the trolley probes was madewith the storage ring at atmospheric pressure while themeasurements used in the analysis were made with the ringunder vacuum. The paramagnetism of O2 in the air-filledtrolley creates a small shift in the measured field, which
depends on the positions of the probes within the trolley.
The size of the shift was estimated by taking a tube of thesame size and shape as the trolley, containing an NMR
probe, and filling the tube alternately with pure O2 and N2 .
The resulting correction is 0.037 ppm.
3. Measurement of azimuthal field average with trolleyprobes
The trolley is pulled by a cable around the inside of thestorage ring to measure the magnetic-field integral in thesame volume where the muons are stored. Figure 26 shows
the field measured by the center trolley probe for a typicalmeasurement in the R01 period. The trolley is pulled
clockwise or counterclockwise through the storage ringin about two hours, and the magnetic field is measured atabout 6000 locations. Trolley measurements were madetwo to 3 times per week at varying times during the day ornight. The azimuthal trolley positions were determinedfrom the perturbations to the fixed-probe readings when

TABLE IX. Multipoles at the outer edge of the storage volume (radius  45 mm). The lefthand set are for the azimuthal position where the plunging and calibration probes are inserted.
The right-hand set are the multipoles obtained by averaging over azimuth for a representative
trolley run during the R00 period.
Multipole[ppm]

Calibration Position
NormalSkew

Azimuthal Averaged
Normal
Skew

Quadrupole
Sextupole
Octupole
Decupole

0:71
1:24
0:03
0.27

0.24
0:53
0:10
0.82

1:04
0:29
1.06
0.40

072003-22

0.29
1:06
0:15
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FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

FIG. 26 (color online). The NMR frequency measured with thecenter trolley probe relative to a 61.74 MHz reference versus theazimuthal position in the storage ring for one of the measurements with the field trolley during the R01 period. The continuous vertical lines mark the boundaries of the 12 yoke pieces ofthe storage ring. The dashed vertical lines indicate the boundaries of the pole pieces. The inset focuses in on the measurements
over an interval of two degrees. The point-to-point scatter in themeasurements is seen to be small.

the trolley passes, and from the readings of a pair ofpotentiometers whose resistances were set by the drums
that wind and unwind the trolley cables. In the final R01
running period, the use of an optical encoder system further improved the trolley position measurement.
The trolley probe measurements are averaged over azimuth and a multipole expansion including terms up to 4thorder for a two-dimensional field without an azimuthalcomponent is made followingBr;   B0 

4  nXr
n1 r0an cosn  bn sinn : (23)Here x and y are the radial and vertical directions and thecorresponding cylindrical coordinates are r;  with r  0
at the center of the storage region and   0 defined topoint radially outward. The multipoles ai (normal) and bi(skew) are normalized at r0  45 mm; the contour plot
obtained from one trolley run is shown in Fig. 25(b), whichcorresponds to the multipole content in the right-hand listof Table IX.
The trolley measures the magnetic field in the muon
storage region up to a radius of 35 mm. The magnetic fieldbeyond 35 mm radius is obtained by extrapolating themeasured moments up to and including the decupoles.
Data obtained in 1998 with a special shimming trolley—
having 25 NMR probes positioned to measure the field up
to 45 mm radius—determined the field in the outer region
and the higher multipoles. The shimming trolley ran along
the pole pieces and could only be used with the vacuum
chamber removed. The multipole expansions of the shimming trolley data at six azimuthal positions, both at thepole edge and at the pole center, all give less than 8 ppm at
45 mm for multipoles higher than the decupoles. By as-

suming an 8 ppm multipole at 45 mm and by multiplying
its radial dependence by the falling muon distribution, amaximum uncertainty of 0.03 ppm is implied on the average field seen by the muons, a negligibly small error.
The overall position uncertainty of the trolley, combinedwith the field inhomogeneity, contribute to the error on the
average field B0 . The short-range jitter in azimuth is notcorrelated to the field inhomogeneity and its effect, averaged out over many trolley runs, is estimated to be smaller
than 0.005 ppm. By contrast, the long-range position error
can be a systematic deviation. Combining this uncertainty
with the field inhomogeneity gives a 0.10 ppm uncertainty
in the worst case for runs prior to 2001. Improved longitudinal position measurements reduced this uncertainty to
0.05 ppm for 2001. Both the normal and skew quadrupoles
vary from 0:5 to 0:5 ppm=mm over the azimuth. The
vertical and radial position of the trolley, over most parts of
the ring are constrained within 0:5 mm with respect tothe center of the storage region. The transverse position
uncertainty should not be correlated with the multipoles
and hence there is no effect on average field due to verticaland horizontal position uncertainties. There are some gaps,
about 100 mm in total, where the position uncertainty
could be as large as 3 mm and where the field inhomogeneity could be as large as 1:3 ppm=mm. The contribution
to the average field uncertainty from such an extreme caseis 0.01 ppm, which is negligible. Reverse direction trolleyruns are also used in the analysis, which are important toconfirm the forward direction runs.
The NMR measurements
provide only absolute fieldq
2
2~
magnitudes, jBj  Bx  By  B2z , where Bx , By and Bz
are the radial, vertical and longitudinal components, re~ is used, rather than By tospectively. The quantity jBj
calculate a . This introduces an errorq
22~  jBy j  B2x  B2y  B2z  jBy j ’ Bx  Bz ;jBj
2jBy jfor Bx  By ;(24)

Bz  By :The radial field Bx is measured [39] in some locations inthe storage ring; By and Bz are both estimated using~
Maxwell’s equations. The difference between By and jBj
is less than 0.01 ppm.
4. Field tracking with the fixed probes
The fixed probes are used to track the magnetic field
during data taking —the time between the direct measurements made by the NMR trolley. Of the 378 fixed probes,
which are placed in grooves in the walls of the vacuum
chamber, about half are useful for the analysis. Some
probes are simply too noisy. Other probes, located inregions where the magnetic-field gradients are large,
have very short FID times and therefore make measurements of limited precision. Still other probes failed during072003-23

PHYSICAL REVIEW D 73, 072003 (2006)
B0,trolley -B0,fp [ppm]

G. W. BENNETT et al.
26.4
26.2
26
25.8
25.6
25.4
25.2
25
24.8
24.6
20

30

40

50

60

70
80
90
day from Feb. 1 2001

FIG. 27. The difference between the average field from the trolley measurements and the one determined using a subset of the fixedprobes versus time. The solid vertical lines indicate times when the difference is expected to change because the magnet power iscycled. The vertical dashed line shows when the inflector was powered on.

B [ppm]the data-taking periods because of mechanical and cabledefects. The probes used in the analysis were given aweight related to the region over which they are sensitive.
For example, the field measurements of a fixed probe at thejunction of magnet pole pieces are sensitive to a more
restricted region compared to a probe located at the centerof the poles. Therefore, a higher weight in calculating thefield average from the fixed probes is used for probes near
the pole centers than to probes near the pole edges. The
values for the weights, about 0.3 and 0.7, were chosen so as
to optimize the agreement between the field average determined with the trolley probes and that determined withthe fixed probes. The weights were constant for all datataking periods.
The differences between the average dipole field fromthe trolley measurements and the weighted average fieldfrom the fixed probes, made during the trolley measurements of the R01 running period, are shown in Fig. 27.
850

840

830

820

810

800
0

200

400

600

800

1000

1200

1400
Run

FIG. 28 (color online). The average magnetic field versus run
number (relative to the start of good data) for the R00 and R01
periods. The relatively flat distribution is from the R01 period,
where the feedback system kept the field stable at the few ppm
level or better. In R00, large deviations are seen owing to atemporary fault in the feedback system.

Because of magnet ramping or changes in the surface coilcurrents, the comparison between pairs of adjacent points
has meaning only when trolley measurements were madeduring the same power cycle (the power cycles are indicated by the vertical divisions). The rms distribution of
those differences is 0.10 ppm. The average magnetic field
for the R00 and R01 periods is shown in Fig. 28. The
relatively flat distribution is from the R01 period, where
the feedback system kept the field stable at the few ppm
level or better. The large deviations near run 600 of the R00
data-taking period were caused by a fault in the feedback
system, but even then, the average field was measured with
sufficient accuracy.
5. Average of the field over muon distribution and time
The stray magnetic field from the pulsed AGS magnets
and from the eddy currents caused by the kicker magnet
has an effect on the storage ring magnetic field.
Fortunately, the field measured by the fixed NMR probes
is a time average, which is largely insensitive to suchtransient effects. The field from the AGS magnets is determined by plotting the fixed NMR probe readings obtained during the 2  105 AGS cycles modulo the AGS
period of 3 s. Its effect is less than 0.01 ppm. The effect on
the magnetic field in the storage region from the kicker
magnet eddy currents was measured in a special setup of
the prototype kicker magnet [20], as this transient magnetic
field is largely shielded from the fixed NMR probes by thealuminum vacuum chamber walls. At 30 s after injection, when the !a fits typically begin, the influence of straymagnetic fields on the average magnetic field seen by themuons is 0.02 ppm [20].
A multipole expansion is used to express the average
field felt by the circulating muons. The azimuthally averaged magnetic field is written

072003-24

Br;  1
Xn1

rn cn cosn  sn sinn

(25)

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

where the coefficients cn and sn are the normal and skew
moments, respectively. The muon distribution moments In ,Jn are defined:
Z r0 Z 2
Mr; rdrd
I0 
0

In 
Jn 

Z r0 Z 2
0

rn Mr;  cosnrdrd

0

Z r0 Z 200

rn Mr;  sinnrdrd:0

The average field seen by the muons is then given by11 X
c I  sn Jn :
B  c0 
I0 n1 n n
For the circular storage-ring aperture, only the first few
multipoles of the muon distribution are significant. Of
those, the normal moments (In ) tend to be more important,
except for the R00 period, when the muon beam wasapproximately 2 mm above the midplane for much of theearly data taking. The radial mean I1 is deduced from thefast-rotation analysis. The higher moments require theinstantaneous radial distributions measured by the traceback system, the most important of which is the radial rms
width  15 mm, consistent with the results of the fulltracking simulation.
Combining the muon distribution and the field moments
yields only very small corrections to the lowest-orderterms. In the R00 running period, the beam radial centroid
was 2:3  1:0 mm larger than the nominal aperture centerand the normal quadrupole moment of the magnetic field at
the aperture of the muon storage region was 0.06 ppm of
1.45 T. Specifically, Br r;   fr cos and By r;  fr sin,wherefr  0:06 ppm 1:45 Tr=45 mm. Therefore, the average magnetic field seenby the muons is 0:06 ppm  2:3 mm=45 mm 
0:003 ppm higher than the field at the aperture center.
Because the beam position is not known on a run by run
basis, the rms of the normal quadrupole distribution is used
to estimate the field error: 0.022 ppm. In the verticaldirection, the skew quadrupole from all trolley runs is0:11 ppm with an rms of 0.27 ppm. However, evenwhen the beam was 2 mm high during R00, the correction
was negligible. The associated field error was taken to be
0:27 ppm  2 mm=45 mm  0:012 ppm.
TABLE X. Weighting of the magnetic field Bx; y from a
representative NMR trolley run during the R00 period over themuon distribution Mx; y for the sextupole through decupole
components.
MultipoleMx; yBx; y [ppm]Product [ppb]Sextupole
Octupole
Decupole

0.02
0:003
0.005

0:53
0.10
0.82

11
0:3
4.0

The weighting of the sextupole and higher normal
magnetic-field multipoles over the simulated muon distribution is shown in Table X. The results were checked withfull particle tracking and the complete magnetic-field map
Bx; y; s. All the sextupole and higher skew multipoles ofthe muon distribution are less than 103 . No correction wasmade for moments higher than the quadrupole. Instead,those values were incorporated into the systematic error.
The total systematic error in weighting the magnetic fieldover the muon distribution is 0:03 ppm. To compute theaverage field seen by the muons, the average magnetic fieldis calculated for each run, and then averaged over all runs
with a weight corresponding to the number of decay electrons in each run.
6. Results and systematic errors
The weighted average field and systematic uncertainties
for the three running periods used in the final combined
result are shown in Table XI. A defect in the shield of thesuperconductiong inflector resulted in imperfect shieldingof the inflector field for the R99 period. Over an azimuthal
angle of 1 degree, the residual fringe field lowered themagnetic field by about 600 (3000) ppm at the center(edge) of the storage aperture. This made a separate measurement of the field necessary in this region and led to an
additional uncertainty of 0.2 ppm. The inflector was replaced between the R99 and R00 periods.
B. Analysis of !a
The anomalous muon spin precession at frequency !aleads to a corresponding modulation in the number ofelectrons striking the detectors. A general form of theelectron time spectrum for events having energy E isNE; t  N E; tet= 1  AE; t cos!a t  E; t ;(26)
where explicit energy and time dependencies for the normalization, asymmetry and phase are included. The replacement E ! Eth is a convenient simplification of
Eq. (26) when the spectra include all events above energy
threshold Eth . An example of a histogram of the decay
electron data from the R01 period was presented in Fig. 2.
The anomalous spin precession frequency !a , having period a  2=!a  4:365 s, is the prominent feature.
The asymmetry AEth ; t is  0:4, as the threshold energy
in this histogram is 1.8 GeV. The time-dilated muon lifetime is   64:4 s.
The only kinematic cut applied in the !a analysis is theenergy of a given event; it is therefore important to establish the dependence of the fit parameters on energy. For
example, NE (integrated over all decay times, t) is a
gradually falling distribution in the ideal case, having an
endpoint at 3.1 GeV (see Fig. 1b). Because of the size ofthe detectors and their placement around the storage ring,

072003-25

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)
TABLE XI. Systematic errors for the magnetic field for the different run periods.

Source of errors

R99 [ppm]

R00 [ppm]

R01 [ppm]

Absolute calibration of standard probe
Calibration of trolley probes
Trolley measurements of B0
Interpolation with fixed probes
Uncertainty from muon distribution
Inflector fringe field uncertainty
Othersa
Total systematic error on !p
Muon-averaged field [Hz]: !
~ p =2

0.05
0.20
0.10
0.15
0.12
0.20
0.15
0.4
61791256

0.05
0.15
0.10
0.10
0.03
—
0.10
0.24
61791595

0.05
0.09
0.05
0.07
0.03
—
0.10
0.17
61791400aHigher multipoles, trolley temperature and its power supply voltage response, and eddy currents
from the kicker.

having the same modulation frequency, !a . However, thetime dependence of E is an important concern. For
example, an energy-scale change as a function of timeafter injection is particularly dangerous because Eth is set
at an energy where d =dE is relatively large (see inset toFig. 29).
Considerable effort was invested in developing a physically motivated functional form that describes the electron
decay spectrum. For example, the coherent betatron oscillation described in Sec. III C modulates N, A and with anamplitude and decay time that must be determined fromthe data. Once a suitable functional form is developed, theremaining task is to show that the frequencies extracted
g-2 phase

their acceptance is a strong function of energy. At 1.8 GeV,
it is approximately 65% and the acceptance drops withdecreasing energy because low-energy electrons, having
small bending radii, can curl between detector stations. At
higher energies, the acceptance rises slowly until, close to
3.1 GeV, the bending radii are nearly equal to the storagering radius. These electrons strike the outer radial edge of acalorimeter, where they are poorly measured because ofincomplete shower containment. Obstacles inside the storage aperture that are in the path of decay electrons (quadrupole and kicker plates, the vacuum chamber wall) can
initiate partial showering, with the result that a higherenergy electron appears in the detectors as a lower-energy
event. The observed energy spectrum is shown in the inset
to Fig. 32. In the ideal case, the asymmetry A is also amonotonically increasing function of energy, starting at
0:2 for E  0 and rising to 1 at 3.1 GeV. It crosses 0
at approximately 1.1 GeV. Because of upstream showeringand finite energy resolution, the measured asymmetry differs from expectation.
The phase E represents the average muon spin angle
as a function of energy at t  0. It is maximally correlated
to !a in the fitting procedure. In the ideal case, the energy
dependence is trivial. The phase angle is constant for E
above the nominal threshold. The actual shape is explainedby two geometrical considerations, both of which dependon energy. The time assigned to a decay electron corresponds to the arrival at the detector, not to the decay. Highenergy electrons are assigned times later after the decay
than low-energy electrons because their average path
length to the detector is longer. In addition, the detector
acceptance is slightly greater when the electron decay
angle is inward—toward the center of the storage ring—compared to when it is outward, and the magnitude of thedifference scales with energy. A plot of phase versus
energy, based on a tracking and detector simulation, isshown in Fig. 29. The energy dependences of N, A anddo not, in principle, affect the anomalous precession frequency !a . Any superposition of decay electrons having atime variation given by Eq. (26), will be of the same form,

1.5

1

1.55

0.5

1.54
0

-0.5

1.3

1.4

1.5

1.6

1.7

1.8

-1

-1.5
0.5

1

1.5

2

2.5

3

GeV

FIG. 29. Relative (g  2) phase versus energy from a tracking
simulation. There is an abrupt shift by  radians at approximately 1.05 GeV. Inset: an expanded view of the phase plot from
1.3 to 1.8 GeV. The required gain stability with time is proportional to the slope of the curve at the energy threshold, as well as
the differential asymmetry, AEth .

072003-26

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

from the resulting fits are immune to a large number ofpossible systematic errors.
The !a analyses all rely on nonlinear 2 minimization
using a fitting function Ft; ,~ having a range of freeparameters ~ that depends on the specific fitting anddata-preparation strategy of the analyzer. The reduced
chi-squared, 2 =dof (dof  degree of freedom), is used
to judge the goodness of fit with the condition thatp
2 =dof  1 with a variance of 2=dof , a property of the
2 distribution. For the typically 4000 degrees of freedom
in the fits described here, an acceptable reduced chisquared is 1  0:022, which is minimized by varying thefree parameters. In the limit of a large number of events ineach bin (e.g., Nk > 30) the variance becomes Gaussian.
The fit range in the different analyses is always truncatedwell before the number of counts in the last bin is fewer
than this limit. In the sections that follow, we describe thedata preparation and fitting procedures.
1. Data preparation and pulse-fitting procedure  2:5 ns imax  2=tan1 xp ;

(27)

where
xp 

Simax  Simax 1;
Simax  Simax 1

and imax is the index of the maximum sample and Si is thevalue of the ith sample. Here xp is a measure of where the−2Marker Pulse

(2

(t=

n

n

ADC Baseline Offset

O

ct
io

s

80 ns

rn

je

Background Pedestal

Ca

lo

Tu

In

Decay Electrons

70

0µ

s

ADC Counts

0)

7

µs

)

Figure 30 depicts a schematic fill as recorded by the
waveform digitizer for one calorimeter station near the
inflector. Prior to injection, the calorimeters are gated off
and a triangular-shaped marker pulse is conveyed to each
WFD to establish correct phasing of the two independent
200 MHz digitizers. The calorimeters are gated on 2 
27 s after the muon bunch is injected. A time-dependentpedestal remains from the prompt hadronic flash. For
detectors immediately downstream of the inflector, thepedestal can be well above the WFD trigger threshold fortens of microseconds, thus initiating a continuous digitization of the analog signals. The slowly decaying pedestal
gives rise to an effective time change of the hardware
threshold. True electron pulses ride on top of this background and must be identified and characterized as E; tpairs. The function of the pulse-finding algorithms is to
identify such events within the roughly 700 s measuring
interval. This task is particulary challenging because theinitial instantaneous rate of a few MHz implies a pileup

rate above 1%. The rate falls by a factor of 104 over themeasurement interval. Because background and electron
pileup can introduce a time dependence in the averagephase, they can also bias the measurement of !a .
The data are reconstructed by two different programs,
g2Off and G2Too. Their pulse-processing algorithms areconceptually similar but the programs share no code. The
two algorithms were developed concurrently and the results obtained from the two were compared. A Monte Carlo
waveform generator was used to check that each was
immune to the difficulties outlined above. The fraction ofpulses that are found by only one of the two algorithms issmall. Slightly different fill-acceptance criteria, together
with intrinsic differences in the time and energy resolutionof each method, lead to a 1–2% difference in the reconstructed events. This difference is accounted for in thecomparison of !a results because it affects the statisticalfluctuations.
The pulse-processing algorithm uses a standard pulse
shape—determined independently for each calorimeter —as a template to extract the time and energy of the electron
signals from the raw waveform samples. These shapes arelargely independent of electron energy, which is proportional to the signal amplitude. The algorithm finds thisamplitude, along with the time of the true maximum andthe pedestal.
The average pulse shapes are built from sample pulses
taken at times long after injection, when the pedestal isconstant, and background and pileup are small. The sample
pulses are aligned with respect to the time of the true
maximum, tm . A pseudotime is defined as

Threshold
Time

FIG. 30. Schematic representation of a WFD record obtained from a single calorimeter in one fill of the storage ring. The dark grayflat band represents an internal baseline added in hardware to prevent underflows. The open triangle marker pulse is inserted externallyto establish the phase relation between the digitizers. The light gray decaying pedestal corresponds to a flash-induced background in
the calorimeters. The narrow spikes, see expansion, represent the individual electron pulses of interest.

072003-27

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

samples lie with respect to tm . If the maximum sample
corresponds to the amplitude of the pulse, xp is nearly 1,
because the pulse is fairly symmetric near its peak. Whenthe maximum sample moves earlier in time with respect totm , then xp decreases, whereas xp increases if it moves
later. Note that xp is a ratio of signal differences, in which
the pedestal and scale cancel.
The true time of the pulse is a complicated function of
the pseudotime. However, for any reasonable pulse shape,t is a monotonically increasing function. Since the
distribution pt of true times is uniform, modulo 5 ns,
the measured distribution of pseudotimes is used to create amapping between pseudotime and true timeRp0 d0t  5 ns R5 0ns
00
0 p d

(28)

0.35
0.3

(29)

j2pulsesis minimized. Here fi t is the average pulse shape andvariables tj , Aj and P are fit parameters, representing thetimes and amplitudes of each pulse and the pedestal. If thequality of the fit is inadequate, a new model that includes
additional pulses is tested. For the typical digitizationinterval—having a single pulse —only a subset of 15
2.5 ns ADC samples centered on the assumed pulse isincluded in the fit. In practice, only the times tj need to
be included directly in the fit. The optimal Aj and P may be
calculated analytically.
2. Energy calibration and stability
The calorimeter energy scale is determined for eachdetector from its measured energy spectrum; a typical
spectrum is shown in Fig. 32. The indicated portion of
the falling edge, in which the number of counts declines
from 80% to 20% of its peak value, is fit by a straight line.
The x-intercept of this line, the endpoint, corresponds toapproximately 3.1 GeV.
The absolute energy scale needs to be known only
approximately. However, it is critical to measure variationsin the energy scale over the fill period. A time-changing
gain—where the relevant time scale is tens of microseconds following the fit start time —can come from true
hardware drifts in the amplification system and from software reconstruction bias. Both effects are associated withthe rate of events. To measure energy-scale stability, the of electrons above threshold as a funcmean energy Ettion of time is constructed, suitably corrected for pileup(see next section) and binned in full (g  2) cycles to avoid
the natural dependence of the average energy on the muon
spin orientation. The gain stability function Gt is related
to the average energy through
x10

0.4

2
X 
X
Si  P 
Aj fi tj i2samples

Events/100 MeV

Pulse amplitude [a.u.]

to fix tm for the pulse within the 5 ns time bin. The time bin
is taken to be 5 ns rather than 2.5 ns because the pulse
shapes are slightly different as viewed by the two flash
ADCs of a given input channel. The pedestal is determinedfrom the average signal far from the maximum.
Once the map between the pseudotime and true time has
been established, the average pulse shape is readily determined. The pedestal-subtracted ADC samples of every
pulse, time-aligned so that the true maximum sits at 0,
are added to a finely binned time histogram. That histogram represents, by construction, the average pulse shape.
As an example, the pulse shapes from the two independent
WFD phases of one calorimeter are shown in Fig. 31. The
shapes and amplitudes are slightly different, as noted
above.
The main pulse-processing routine fits each digitizationinterval using the template average pulse shape for eachdetector. First, the number of candidate pulses is determined by noting local peaks. The expression

D

4

3500

Fitting
range

3000

0.8
0.6

2500
0.4

0.25
2000

0.2

1

0.2
0

0.15

1500

0.1

1000

0.05

0.5

1

1.5

2

2.5

3

3.5

500

0
-10

0

10

20

30

0

40
Time [ns]FIG. 31. Average pulse shapes for detector station 1. WFD
phase 0 (dotted) has the more pronounced oscillatory behavior
compared to phase 1 (solid).

1.5

2

2.5

3

3.5

4
4.5
Energy [GeV]

FIG. 32. Typical measured calorimeter energy distribution
with an endpoint fit superimposed. The inset shows the fullreconstructed energy spectrum from 0.3 to 3.5 GeV.

072003-28

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)(30)

where f is a threshold-dependent constant, approximately
equal to 2 for Eth  1:8 GeV and t  1 refers to late in thefill. With the exception of a few detectors, Gt varies fromunity by at most 0.2% for the fitting period. Typically, theelectron time spectra are constructed applying Gt as acorrection to each electron’s energy.
To determine the sensitivity of !a on Gt, electron time
spectra are reconstructed where the applied Gt is varied
by multiplying the optimal Gt by the multipliers mG 
1, 0, 1 and 2. These spectra are then fit for !a and theslope !a = mG is determined. Typically, the quoted result
from an individual analysis is based on mG  1 and thegain systematic uncertainty is based on the change in !a ifGt is not applied. Because of differences in running
conditions and fit start times, the gain systematic uncertainty differs by running period. The most complete studiesof the gain systematic uncertainty were carried out for the
R00 and R01 analyses, giving 0.13 and 0.12 ppm,
respectively.
3. Correction for multiparticle pileup
Two low-energy electrons arriving close together in time
are interpreted as one equivalent high-energy electron.
Because low-energy electrons have a shorter drift time
and their acceptance peaks at a different orientation of
the muon spin, the (g  2) phase carried by these ‘‘pseudoelectrons’’ is different from a true high-energy electron.
The phase difference, by itself, would not be of concern if
the fraction of pileup pulses were constant over the measurement period. But, the pileup rate is proportional toe2t= , which has a time-constant half that of themuon lifetime. The pileup fraction therefore changes intime, which can pull the (g  2) phase and thus bias !a .
The general treatment of pileup in the analyses is to correct
for pileup, then fit for !a .
Because the full raw waveforms are stored, most multiparticle pileup events are recognized by the fitting algorithms. When two pulses coexist on the same digitizationinterval and are separated by more than 5 ns, the pulsefinding algorithms can identify the individual pulses withnearly 100% reliability. The right panel of Fig. 15 shows
two pulses separated by 7.5 ns. The fitting algorithms
distinguish these pulses and correctly assign their times
and energies.
Truly overlapped pulses cannot be separated, but they
are accounted for on average by subtracting an artificiallyconstructed pileup contribution from the spectra. Consider
a digitization interval having a ‘‘trigger’’ pulse and a‘‘shadow’’ pulse that trails the trigger pulse by a fixed
time offset. The probability to find a pulse at time t is
assumed to be the same as finding a similar pulse at timet  for  Tc , the cyclotron period. The trigger pulse

energy must exceed the  1 GeV hardware trigger threshold, but the shadow pulse can have any energy above thefitting minimum of approximately 0.25 GeV. The twodimensional spectra ST E; t and SS E; t are obtained,
where T and S represent trigger and shadow pulses, respectively. A ‘‘double’’ pulse having an energy equal to thesum of the trigger and shadow is created at a time determined by the average of the two pulses, weighted by theirrespective energies. Such events create the doubles distribution DE; t. Finally, the pileup distribution isPE; t  DE; t  ST E; t  SS E; t:(31)

The spectrum of counts can now be corrected for pileup
by subtracting this distribution. In doing so, care must be
taken to properly evaluate the errors because some of theevents used to construct the pileup spectrum are part of theuncorrected spectrum already, depending on the energy
cut. To sufficient precision, the uncertainty for a bin in
the pileup-subtracted histogram at time t is described by
2 t  1  k expt= Ft, where Ft is the func
tional form describing the pileup-free data and k is determined from the data itself.
A pileup-free spectrum is created from normalized difference spectra. Figure 33 illustrates the uncorrected energy distribution (upper curve), which extends well beyond
the 3.1 GeV natural maximum because of pileup events.
The artificially constructed pileup energy distribution is
also shown (lower curve). It matches the raw distribution at
the highest energies, confirming that pileup has been correctly constructed from the raw spectrum. Below 2.5 GeV,
the constructed pileup spectrum is ‘‘negative’’ because twolower-energy electrons are removed and misinterpreted asa single higher-energy electron; in this figure, the absolutevalue is plotted.
Counts per 100 MeVEt  E1
Gt  G1
f
;

E1
G1

109

108

7

10

106

105

1.5

2

2.5

3

3.5

4

4.5

5

5.5
6
Energy [GeV]

FIG. 33. Uncorrected energy spectrum, including pileup events(dotted, top curve) and artificial pileup reconstruction spectrum(solid, bottom curve). The energy distributions smoothly coincide above about 3.5 GeV, where only true pileup events can
exist. Note, below 2.5 GeV, the pileup spectrum is negative. The
absolute value is plotted.

072003-29

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

4. Elimination of fast rotation
As described in Section III B, muons are injected into
the storage ring in approximately Gaussian bunches withrms widths of 25 ns. The momentum spread causes debunching with a time constant of approximately 20 s andthe leading and trailing edges begin to overlap 5 s after
injection. Approximately 30 s after injection —a typicalfit start time —the underlying microstructure remains, appearing as a rapid modulation of the electron decay spectrum for a given detector. This fast-rotation signal isfiltered from the decay spectra by adding a random fractionof the cyclotron period Tc to the reference time T0 thatmarks the arrival of the bunch at the entrance to the storage
ring, a procedure that reduces the fast-rotation modulation
by a factor of about 500. Furthermore, if the calorimetersignals are aligned in time according to their azimuthal
location and their decay spectra are combined, the fastrotation structure is reduced by an additional factor of 10.
In addition to the slow modulation caused by the (g  2)
precession, the actual rate in a detector station varies significantly over a cyclotron period, from early times until
the bunch structure has disappeared. The corresponding
modulation of the pileup rate is handled automatically by
the shadow pulse subtraction scheme.

(f g−2 )

0.018

0.002

1

1.5

fVO

0.004

fVW

0.006

fVW − fg−2

0.008

2 f CBO

0.01

fCBO + f g−2

0.012

fCBO − fg−2

0.014

fVW + f g−2

fCBO

0.016

muon losses, gain variations

Fourier Amplitude

The pileup systematic uncertainty falls into three categories: efficiency, phase, and unseen pileup. The pileup
efficiency is established by creating a pileup event spectrum and adding it to the raw spectrum. A pileup multiplier,
mpu , is used to construct modified electron time distributions with varying pileup fractions. These spectra are fit todetermine !a = mpu . Equality of the electron energy
spectra early (high rate) and late (low rate) in the fillindicates that pileup is corrected; an uncertainty of 8%on this correction is assigned. The systematic uncertainty
on !a from pileup subtraction efficiency is 0.036 ppm. The
pileup phase reflects the error due to the uncertainty in thephase of the constructed pileup spectrum. Simulations
determine the limits of the phase difference, and the amplitude of pileup subtraction, combined with the phasedifference, yields an uncertainty in !a of 0.038 ppm.
Finally, a 0.026 ppm uncertainty is assigned to the effectof those very-low-energy pulses, unnoticed by the pulsefinding algorithm, which are not included in the constructed pileup spectra. The combined pileup uncertainty
on !a is 0.08 ppm, where the efficiency and phase uncertainties are correlated and add linearly and the unseen
pileup uncertainty is combined in quadrature.

0.02

0
0

0.5

2

2.5

3

Frequency [MHz]FIG. 34. The Fourier spectrum obtained from residuals from a
fit based on the five-parameter, ideal muon decay and spin
precession expression. The horizontal coherent betatron oscillation (CBO) frequency at 466 kHz, its first harmonic, and thedifference frequency between CBO and the (g  2) frequency
are strong peaks. The vertical waist (VW) and CBO verticaloscillation (VO) produce smaller, but still significant, effects athigh frequencies. The low-frequency rise stems from muon loss
and gain distortions of the underlying decay exponential.

Fourier transform of the residuals of such a fit to the R01
data. While the fit removes !a from the residuals, strong
peaks at the horizontal CBO frequency, its first harmonic,
and at the sum and difference between the CBO and !afrequencies are evident. Additionally, small peaks associated with the vertical CBO (VO) and the vertical waist(VW) are seen at higher frequencies. The low-frequency
rise is ascribed to distortions to the exponential envelope
from muon loss and gain changes. These physical terms
motivate development of a multiparameter fitting function.
A general form, which includes all known and relevant
physical perturbations, and assumes an energy threshold
Eth , can be writtenNt 

N0 t=et Vt Bt Ct
1  At cos!a t  t

(32)

with
t  1  Aloss

5. Multiparameter fitting
The electron decay spectra prepared as described above,
fit with the naive five-parameter function in Eq. (10),results in a very poor 2 =dof. Fourier analysis of theresiduals reveals identifiable oscillatory features and slow
changes to the overall spectral shape. Figure 34 shows the072003-30

Zt0

Lt0 et = dt0

(33)

0

Vt  1  et=VW AVW cos!VW t 
Bt  1  Abr et=br

VW 

(34)
(35)

Muon Loss Events

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

10

10

10

PHYSICAL REVIEW D 73, 072003 (2006)

4

3

2

10
500

1000

1500

2000

2500

x10
3000
Time [ns]2FIG. 35. Muon-loss rate vs time from the R00 period. Three
consecutive and coincident FSD station signals form the muonloss signal. The loss function Lt is proportional to this raw dataplot.

Ct  1  et=cbo A1 cos!cbo t 

1

(36)

At  A1  et=cbo A2 cos!cbo t 

2 

(37)

t=cbo A cos!
0e3
cbo t 

3 :

(38)

t 

While the additional terms are necessary to obtain an
acceptable 2 , they are not strongly correlated to !a .
Equation (33) describes t, which is derived fromLt, which in turn is derived from the muon-loss monitor
data, as illustrated in Fig. 35. Muon loss introduces aslowly changing modification to the normal exponential
decay. To determine the absolute rate of muon losses, theacceptance of the detection system must be established by
the Monte Carlo simulation. With an estimated acceptanceof a few percent, results from the fits indicate an approximate fractional loss rate of 103 per lifetime.
Equation (34) defines Vt, which accounts for an acceptance modulation owing to vertical betatron oscillationsof the stored muon beam—the vertical
waist—havingp
angular frequency !VW  1  2 n!c . The vertical
waist dephases with a characteristic decoherence time
VW  25 s. The frequency and decoherence time aredetermined in dedicated early-start-time fits on isolated
detectors —when the vertical waist is large—and they
are found to be consistent with expectations from beamdynamics calculations. The phase is the only free parameter in the physics fits. The term Vt is important in fits thatstart before 30 s.
During scraping, which is completed 7–15 s after
injection, the muon beam is displaced both vertically andhorizontally. As the quadrupole voltages return to theirnominal storage values, the beam returns to the aperture’s
center. The relaxation time of 5  1 s is observed directly in the mean position of decay electrons on the FSD(see Fig. 23). The term Bt modifies the normalization
constant N0 and accounts for the corresponding change inacceptance. As was the case for Vt, Bt is important only
for fits which start at very early times.
The horizontal CBO describes the modulation of thenormalization, phase and asymmetry: N, A and . The
modulation of N and A results from the changing acceptance for detecting decay electrons as the beam oscillates
radially. The modulation in results from changes in thedrift time. The amplitudes and phases in Eqs. (36)–(38) are
difficult to predict and are therefore determined from thefit. Nonetheless, fit results for these parameters are consistent with Monte Carlo predictions, including variations
by detector. The damping of the CBO is adequately described by an exponential having the common decay time
constant cbo in the range 90–130 s (dependent on quadrupole voltage). The horizontal CBO frequency and lifetime are independently determined using the FSD
detectors. As the muons move in and out in the horizontal
plane of the storage ring, the average path length for the
decay electrons to strike the detectors grows shorter andlonger. As a result, the rms vertical width on the face of thedetectors oscillates at the horizontal CBO frequency with
an amplitude that decays with characteristic time cbo . The
CBO frequency and lifetime, measured using the FSD
detectors, are consistent with those obtained from themultiparameter fits to the calorimeter data.
No CBO terms were used in the R98 fitting function.
The small CBO modulation was largely hidden, given the
low statistical power of the data set and the fact that the
final histograms were formed from sums of the individual
detector time spectra. The effect of the CBO would cancel
exactly in the sum if the acceptance around the ring were
uniform. However, kicker and, to a lesser degree, quadrupole electrodes —upstream of particular detector stations—lower the acceptance of those detectors, breaking
the symmetry. Still, the CBO effect is reduced by an orderof magnitude in the sum. However, fits to the much larger
R99 data set gave a poor 2 =dof before accounting for
CBO modulation of the normalization, which was incorporated by multiplying Nideal by the function
2

2

C0 t  1  Acbo et =cbo cos!cbo t 

cbo :(39)

With this modification, the 2 =dof was acceptable (alternatively, Eq. (36) was also used and, in subsequent analyses, became the standard method to account for CBO). The
fit value Acbo ’ 0:01 was in good agreement with thatfound from the full detector simulation. The conservative
systematic error of 0:05 ppm was assigned, based onchanges to the !a frequency when varying the CBO fit
parameters over wide ranges.
After the R00 running was concluded, it was discoveredthat the CBO frequency was unusually close to twice the(g  2) frequency; that is, !cbo  !a  !a . Analytic andsimulation studies indicate that in this case, the fit value of

072003-31∆ωa [arb. units]

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

0.4

large ranges and comparing the !a fit values from fits tothe single detector data and all detector data.
Table XII shows a representative correlation matrix froma fit to the low-n data set in the R01 period. The Vt andBt terms are omitted because they are negligible for thefit start time of 31:8 s in this fit. Note that none of theterms—apart from the (g  2) phase—correlates strongly
with !a .

0.2

6. The ratio-fitting method

1

1999
2000

0.8

2001
0.6

0
400

420

440

460

480
500
CBO Frequency [kHz]FIG. 36. The relative pull (!) versus the CBO modulation
frequency if not addressed by the fitting function. A typical fullvertical scale is several ppm; the actual scale depends on thespecifics of the fit and the data set used. The R00 data wereacquired under run conditions in which !a was very sensitive to
CBO. This sensitivity was minimized in the R01 period where
low- and high-n subperiods, each having CBO frequencies well
below or above twice the (g  2) frequency, were employed.

!a is particularly sensitive to the CBO modulation.
Figure 36 shows the relative pull (!) versus the CBO
modulation frequency if not addressed by the fitting function. As evident in the figure, the R00 data were acquired
under run conditions in which !a was very sensitive to
CBO. Accordingly, the systematic uncertainty for thisperiod is larger than the R01 period, which featured lowand high-n subperiods, each having CBO frequencies well
below or above twice the (g  2) frequency. The CBO
systematic uncertainties were conservatively estimated tobe 0:21 ppm and 0:07 ppm for the R00 and R01 periods, based on varying the fixed CBO fit parameters over

TABLE XII. Correlation matrix covpi pj =
N0

A



!a

!cbo

i

An alternative analysis of !a first randomly sorts theelectron time spectra into four equal subsets, labeled n1 tthrough n4 t. Each is similar to the standard Nt distribution. The four subsets are then recombined in the ratiort 

n1 t  12 T  n2 t  12 T  n3 t  n4 t
n1 t  12 T  n2 t  12 T  n3 t  n4 t

(40)

to produce a spectrum as shown in Fig. 37. Here, T 
2=!a is a (very good) estimate of the spin precession
period. In preparing the ratio histograms, gain corrections
are applied to the data and pileup and fast rotation are
removed. The ratio histograms can then be fit well by athree-parameter function, which accounts for the amplitude, frequency and phase of the spin precession:


1 T 2
r3 t  A cos!a t   :(41)
16 
The final term arises from the exponential decay whenshifting the subsets forward or backward in time by T=2.
Equation (41) describes well the ratio histograms formed
when data from all detectors are combined. In the R01
analysis, it was used successfully to fit the detectorcombined data from both n-value data sets.
However, Eq. (41) is inadequate in fitting individual
detector data subsets, or those of a given field index, as

j from a fit to the low-n data from R01 starting at 31:8 s.

cbo

A1

1

A2

2

A3

3

Aloss

1.000 0:040 0.816 0:010 0:007 0.017 0:022 0.030 0:025 0:023 0.028 0:013 0:071 0.982
0:040 1.000 0:025 0:008 0:004 0.006
0.011 0:014 0:010 0.066
0.038 0:068 0.016 0:037
0.816 0:025 1.000 0:007 0:005 0.010 0:013 0.018 0:015 0:014 0.018 0:010 0:044 0.873
0:010 0:008 0:007 1.000 0:834 0.018 0:025 0.035 0:024 0.054 0:082 0.004
0.068 0:009
0.007
0.004
0.005 0:834 1:000 0:013 0.018 0:026 0.017 0:040 0.058 0:002 0:050 0.007
!a
0.006
0.010
0.018 0:013 1.000 0:004 0.007 0:319 0.002 0:049 0:002 0:016 0.015!cbo 0.017
0.132
0.005
0.016 0:007 0:020
cbo 0:022 0.011 0:013 0:025 0:018 0:004 1.000 0:903 0.003
0.030 0:014 0.018
0.035 0:026 0.007 0:903 1.000 0:006 0:134 0:007 0:010 0.008
0.028
A10:025 0:010 0:015 0:024 0:017 0:319 0.003 0:006 1.000 0:003 0.046 0:001 0.014 0:023
1
0.132 0:134 0:003 1.000 0:006 0.009
0.023 0:021
A2 0:023 0.066 0:014 0.054 0:040 0.002
0.028
0.038
0.018
0:0820:058
0:0490.005
0:0070.046
0:0061.000
0:007
0.005
0.026
2
A3 0:013 0:068 0:010 0.004 0:002 0:002 0.016 0:010 0:001 0.009 0:007 1.000 0:015 0:013
0:071 0.016 0:044 0.068 0:050 0:016 0:007 0.008
0.014
0.023
0.005 0:015 1.000 0:065
3
0.982 0:037 0.873 0:009 0:007 0.015 0:020 0.028 0:023 0:021 0.026 0:013 0:065 1.000
Aloss

N0
A


072003-32

FINAL REPORT OF THE E821 MUON ANOMALOUS . . .

PHYSICAL REVIEW D 73, 072003 (2006)

uncertainty by 10%. The systematic sensitivities are different but of comparable size. This method is equivalent tobinning and analyzing the data in discrete energy bins in
the limit of an infinite number of bins.
8. Internal and mutual consistency

FIG. 37 (color online). Ratio spectrum from the R01 period.
The top band begins 27 s after injection. For display purpose,each successive 100 s band is displaced downward.

was done in an independent analysis of the R01 data. The
ratio function must be expanded to incorporate the CBO
modulation of the normalization, asymmetry and phase, as
described for the multiparameter fitting. Therefore,
Eq. (41) is replaced by the more general nine-parameter
functionr9 t 

2f0 t  f t  f t
2f0 t  f t  f t

(43)

and
f t  Ct0 eT=2 

1  At0  cos!a t  t0 (44)with Ct, At and t defined in Eqs. (36)–(38), and witht0  t  T=2. This expression is sufficient to obtain goodfits to individual data subsets.
7. The asymmetry-weighting method
In the standard multiparameter or ratio fitting describedabove, the statistical error on the frequency !a is
1
2
!a



N hAi2  2
;
2

(45)

where hAi is the average of the asymmetry of the N events
having energy greater than Eth . In the asymmetryweighting method, each decay is weighted by the asymmetry, the weight being implied by the nominal AE
relationship. The resulting statistical error
1
2
!a



N hA2 i 2
2

bins

(42)

where
f0 t  Ct 1  At cos!a t  t

Before the individual offsets assigned to the analyses
can be replaced by a single common offset, each analysis
must demonstrate a high level of internal consistency.
Then, before the common offset is removed, the resultsof the different analyses are examined for mutual
consistency.
Internal consistency demands:(i) a reduced 2 equal to unity within the expected
statistical spread;(ii) no pronounced structures in the fit residuals, both in
the time and frequency domains;(iii) that the fit results must be independent of any
specific subset of data, within expected statisticaland systematic errors.
The optimal set of parameters is obtained by minimizing
2 , defined asX2Ni  Fti ; 
~ 2 2i :
(47)

(46)

is optimized. The difference between Eqs. (45) and (46) isthe asymmetry term, hA2 i vs hAi2 , which reduces the

The error i must account for pileup subtraction. In
some analyses it was calculated during the pileup correction procedure, as a combination of the contents of the bin
and the amount subtracted from that bin. In others, it wasapproximated as 1  ket= Fti ; .
~ The latter method
was inspired by the observation that without pileup subtraction, 2i  Fti ; ,~ that is the best estimate of the error
squared at any point is given by the fit value itself. The
factor (1  ket= ) reflects a small correction because of
the pileup subtraction.
The quality of the fits is tested by splitting the data into
subsets, especially those that enhance or reduce particularsystematic uncertainties. In estimating the impact of statistical fluctuations in the comparison of subset fits, thedata overlap and change in analyzing power (notably theasymmetry) has to be taken into account.
Data subset comparisons include:(i) Start time: As the fit start times increase, the effectsof short time-constant ( < ) perturbations —Vt, Bt, fast rotation, pileup, gain changes—
disappear. Therefore the stability of results versus
start time is one of the most sensitive measures of agood fitting function.
(ii) Run conditions: Fits are carried out on individual
runs or sets of runs having special experimental
conditions. Examples include variations in n values, quadrupole scraping voltage or timing, andradial magnetic field. Data from each of the 12bunches in an AGS cycle were considered separately and compared.

072003-33

G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

229.0745

The expected spread in fitting results for data sets thatpartially overlap depends on the size of the data samples, aswell as on the difference in analyzing power—the asymmetry and the phase of the (g  2) modulation —givings


A
1
2
2
cos 1  2   1 :(48)
diff 
2 1 2A2Here, the larger data set is indicated by subscript 1 and thesmaller, fully-contained subset, by subscript 2. Data setshaving different energy thresholds will have a different
analyzing power. In the case that the analyzing powers
are the same (A1 ’ A2 and 1 ’ 2 ), Eq. (48) reduces toq
2  2;
(49)
’
diff
2
1
a result that is applicable to nearly all of the consistency
tests, and is the basis of the envelope shown in Fig. 38.
Results from the individual analyses were required to bestatistically compatible. When comparing different analyses of the same nominal data set, a correct assessment of
the expected deviations must include a careful analysis of
the data overlap. Each analysis included the same acceptedruns but the data overlap differed at the few percent level,
as explained in Sec. IV B 1, because of the individual
techniques employed. For example, some of the moreaggressive analyses used very early start times or included
a broader energy range of accepted electrons.
Table XIII gives the final, offset-removed, relative precession frequency results for the R99, R00 and R01 running periods based on the 14 analyses. The parameter !a isa /2 [kHz]a /2 [kHz](iii) Electron energy: Data are sorted into discrete energy bins (typically 200 MeV wide) or into setshaving different low-energy thresholds. Fits tothese data sets are naturally quite sensitive to thedetails of energy scale and pileup corrections.
(iv) Detector: Fits are carried out on data from individual detectors, on detector groups, and on the sum of
all detectors. This division is particularly sensitive
to a correct treatment of CBO and of the early-timebackground, which are highly dependent on position around the ring.
Figure 38 is a representative set of important fit subset
tests for one of the R00 multiparameter analyses.
Individual fits all had acceptable reduced 2 . The top left
panel shows the stability of the precession frequency versus start time of the fit. The sideways ‘‘parabolic’’ band
defines the expected range of successive results, given theinitial fit value and the steady diminution of the data set.
The outer envelope represents the statistical uncertainty on
any individual fit. The top right panel shows the !a resultsfor data binned in 200 MeV energy bands, for a fixed start
time. This distribution is flat, as expected. The bottom left
panel shows the result as a function of detector station
around the ring [40]. Data were divided into subsets indexed by both detector number and energy band. The
consistency of the !a values can be verified by calculating
the mean of the ensemble and plotting the deviation of eachindividual fit, normalized by the respective statistical error.
The result, shown in the bottom right panel, is a Gaussianof width 1, as expected.

2/dof = 6.2/8

229.076

229.0744
229.0743

229.075

229.0742
229.074
229.0741
229.073

229.074
229.0739

229.072
229.0738
229.071229.0737
70

80

90

100

2

/dof = 23.9/21

229.076

1.4

110
120
130
Fit start time [ s]229.075

229.074

Number of occurrences60

a /2 [kHz]50

229.073

1.6

1.8

2

2.2

2.4
2.6
2.8
3
3.2
Observed positron energy [GeV]

Mean: 0.024
Width: 0.994

45

0.071
0.050

40
35
30
25
20
15

229.072105229.071
2

4

6

8

10

12

14

16

18

20
22
24
Detector number

0
-6

-4

-2

0
2
4
6
(i)
(i) -fromNo. standard deviationsmean
a < a>/

FIG. 38. From a multiparameter fitting of the R00 period data. Precession frequency: (a) versus start time of the fit; (b) versus
detector; (c) versus energy bin; (d) The difference between the precession frequency from each of the 198 individual fits of energy anddetector subsets of the data compared to the mean. Unit width is expected for a statistical distribution. In a), the sideways parabolic
band defines the expected range of successive results while the outer envelope represents the statistical uncertainty on any individual
fit.

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PHYSICAL REVIEW D 73, 072003 (2006)

TABLE XIII. Comparison of the fitted relative precession frequency R from each of theanalyses. The units are ppm. The uncertainties are statistical and systematic, respectively. Notethat the variation in the magnetic field from year to year has not been corrected for. The labels
A–E represent individual, independent analysis efforts for any given running period.
Analysis
A
B
C
D
E

R99

R00

R01

119:60  1:23  0:08
119:33  1:28  0:19
119:38  1:24  0:22
119:67  1:28  0:17
119:55  1:24  0:22

113:97  0:70  0:26
113:74  0:63  0:34
113:57  0:64  0:36
113:83  0:64  0:35

108:63  0:63  0:24
107:98  0:69  0:28
108:36  0:69  0:22
108:31  0:71  0:23
107:96  0:72  0:19

encoded in all the fitting programs as
1  R  R  106!a  2 0:2291 MHz(50)

where R is the actual free parameter and R is the secretoffset for the given running period. In this convenient form,
the precession results listed in Table XIII are in ppm. The R
values cannot be compared across running periods without
further computation because the mean magnetic fieldchanged from year to year.
The individual a results for the R99, R00, and R01
running periods are shown in Fig. 39. The results within aperiod are highly correlated. The analysis methods arecoarsely distinguished by raw-event production, g2Off(g2) or G2Too (G2) and by the multiparameter (MP) or
ratio-method (R) of fitting. The G2-MP result from the R01
period used the asymmetry-weighting method and the G2MP result from the R00 period was based on an energybinned method (an indirect asymmetry weighting).

with appropriate combinations of uncertainties taken into
account for the different analyses for each period.
For each running period, the final quoted result is obtained from a statistical average of the individual resultsfrom each analysis. This method of combination is robust,
given that each analysis is individually believed to becorrect. Each analysis included a complete and independent evaluation of all fit-related systematic uncertainties.
The methods, being different, feature somewhat different
sensitivities to individual uncertainties. For example,
g2Off and G2Too-based productions have pulse-finding
algorithm
differencesandpileupsensitivities.
Multiparameter versus ratio fitting implies different sensitivity to slow terms such as muon loss. The CBO sensitivity
depends on whether the analysis sums over all detectors ortreats them individually. Gain changes depend on the
pileup subtraction treatment and the start time selected.
The published values take into account these differences in
the assignment of a systematic uncertainty for the final
result.

9. Systematic errors in !a
10. Consideration of a muon EDM
Before presenting the final result, we remark on thepossible effect on the precession frequency from a nonzero
muon electric dipole moment (EDM). Equation (11),

G2 R

g2 R

G2 MP

R01
g2 MPbg2 MPaG2 R

g2 MPbg2 MPaR00
G2 R

G2 MPa

g2 R

1

R99
g2 MPa

2

G2 MP

3

g2 MPb∆ a [ppm]

The systematic uncertainties have been described in therelevant sections. Table XIV lists the numerical values,

TABLE XIV. Systematic errors for !a in the R99, R00 and
R01 data periods.
R99 (ppm)

R00 (ppm)

R01 (ppm)

PileupAGS background
Lost muons
Timing shifts
E-field and pitch
Fitting/binning
CBO
Gain changes

0.13
0.10
0.10
0.10
0.08
0.07
0.05
0.02

0.13
0.01
0.10
0.02
0.03
0.06
0.21
0.13

0.08

0.07
0.12

Total for !a

0.3

0.31

0.21

syst !a0-1-2

FIG. 39. Results for the anomalous magnetic moment from the
14 different !a analyses performed in the R99, R00 and R01
running periods. The analysis methods are coarsely distinguished by raw-event production, g2Off or G2Too (G2), andby the multiparameter (MP) or ratio-method (R) of fitting. The
results in a given period are highly correlated: the event samples
largely overlap.

a

a

0.09
a
a
a

In R01, the AGS background, timing shifts, E field and verticaloscillations, beam debunching/randomization, binning and fitting procedure together equaled 0.11 ppm.

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G. W. BENNETT et al.

PHYSICAL REVIEW D 73, 072003 (2006)

which gives the dependence of a on the measured difference frequency (!a ), tacitly assumes that the muon EDM
is zero (d  0). A nonzero EDM requires a modificationof Eq. (5):~ EDM  !
~a 
!
~ !
~a !

q ~
~
  B;2m(51)

where  is a unitless constant proportional to the EDM:q ~S:
d~  
2mc(52)

The interaction of the EDM with the motional electricfield (in the muon rest frame) induces a radial component
in the spin precession vector, which is otherwise purely
vertical. The spin precession plane is tilted radially by
angle :d:
2a 1:1  1016 e cm

(53)

Independent of its sign, an EDM increases the measured
precession frequency,p
!meas  !a 1  2 :(54)
A precession plane tilt causes an oscillation at !meas in thevertical direction of the decay electrons and thus in themean vertical position at the calorimeters. The verticaloscillation reaches extrema when the spin points inward
or outward radially, 90 out of phase with the usual (g  2)
oscillation.
The vertical electron hit distribution on the FSDs was
examined for oscillations having this phase relation.
Additionally, the traceback wire chambers were used tolook for oscillations in the vertical decay angle of theelectrons. Both methods find null results. The final analysisof the EDM studies from our experiment is being completed and we expect to achieve a limit of a few times
1019 e cm. The details will be described in a separate
paper [41]. However, to set the scale of the potential effect,a nonzero EDM at 2:0  1019 e cm would cause a systematic increase to a by 1.6 ppm. A more sensitive limit isobtained by invoking muon-electron universality and thelinear scaling relation expected for many standard model
extensions: d  m =me de . With the current electron
EDM limit, de < 1:6  1027 e cm, linear scaling implies
a muon upper limit below 3:2  1025 e cm. An EDM ofthis magnitude is too small to affect the !a measured inthis experiment. Therefore, we adopt the assumption thatthe measured anomalous precession frequency alone determines a .
C. Final a result
The final a result is obtained by combining the individual a results from all running periods, which are listed

FIG. 40. Results for the E821 individual measurements of aby running year, together with the final average.

in Table I. All E821 results are plotted in Fig. 40 together
with the final average.
Recall that the relation
a 

R
R

~ p is theis used to determine a , where the ratio R  !a =!
experimental measurement: the anomalous precession frequency divided by the event-weighted magnetic field. The
muon-to-proton magnetic moment ratio   =p 
3:1833453910 is obtained independently [12]. The appropriate comparison of results from E821 is made in termsof R. The precession frequencies, radial electric field andpitch corrections, average magnetic field and R ratios are
given in Table XV for the R99, R00 and R01 periods. Total
uncertainties are given for each quantity. While themagnetic-field strength differed slightly from year toyear, and consequently the precession frequency changed,
the agreement in R between periods is excellent.
Correlations in certain systematic uncertainties exist
across running periods. These include the use of a common
absolute calibration probe, perturbations to the storagering field from kicker eddy currents, uncertainty in thelost muon population phase, and the E/pitch correction.
Their combined uncertainty is less than 0.15 ppm. Other
systematic uncertainties are uncorrelated, as are the statistical uncertainties. The errors quoted on combined results
reflect a slight increase compared to a direct weighted error
because of correlations across periods.
The two positive muon values for R can be combined
and compared to the negative muon result:R   0:003707204726

(55)

R   0:003707208326;(56)

giving R  R  R  3:6  3:7  109 , whichis in good agreement with the expectation from CPT
invariance. Assuming CPT invariance, we obtain the average value

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PHYSICAL REVIEW D 73, 072003 (2006)

TABLE XV. Individual !a and !
~ p results, and the ratio for the three high-statistics running
periods. Column 3 gives the relative electric field and pitch corrections, which have been applied
to the !a values quoted in column 2. The total uncertainties for each quantity are given. The
error on the average takes into account correlations between the interperiod systematic
uncertainties.
Period!a =2 [Hz]E/pitch [ppm]!
~ p =2 [Hz]R  ! a =!
~p

R99 ( )R00 ( )
R01 ( )229072.8(3)229074.11(16)229073.59(16)0:818
0:763
0:776

61791256(25)61791595(15)61791400(11)0.0037072041(51)0.0037072050(25)0.0037072083(26)Average———

0.0037072063(20)R  E821  0:003707206420;(57)

loop term [44]. The total QED value isa QED  11658471:9580:0020:1150:085  1010

giving the anomalous magnetic momenta Expt  11659208:06:3  1010 0:54 ppm: (58)(60)

The total uncertainty includes a 0.46 ppm statistical uncertainty and a 0.28 ppm systematic uncertainty, combined inquadrature.

where the first two uncertainties are from the 4 and 5
terms, respectively, and the third is from the uncertainty on
. The value and uncertainty on  are obtained from atominterferometry [44].
The electroweak contribution from one and two loops is

V. THE STANDARD MODEL VALUE OF THE
ANOMALYa Weak  15:40:10:2  1010

Three classes of radiative processes contribute to a :QED loops containing photons and leptons e; ; ; hadronic loops (Had) containing hadrons in vacuum polarization loops; and weak (Weak) loops involving the W, Z, andHiggs bosons. The standard model anomaly is represented
by the expressiona SM  a QED  a Had  a Weak:(59)

While the QED and electroweak contributions are wellunderstood, the hadronic terms remain the subject of intensive study. Recent reviews of the complete standard
model calculation are given by Davier and Marciano [42]and by Passera [43]. Their accounting of the individual
contributions is summarized below; however, we include
only those results that have been published in refereed
journals.
The QED contributions to a are calculated throughfour loops and an estimate is made of the leading five-(61)

where the first error comes from two-loop electroweak
hadronic effects in the quark triangle diagrams and thesecond comes from the uncertainty on the Higgs mass
[42,45].
Establishing an accurate and precise value for the hadronic contributions to a is the source of much theoreticaland experimental work worldwide. The lowest-order hadronic vacuum polarization loop (Had; LO), shown in
Fig. 41(a), contributes approximately 60 ppm to a . This
diagram can be evaluated using the dispersion relation
shown pictorially in Fig. 41(a) and 41(b), which connects
the bare cross section for electroproduction of hadrons to
the hadronic vacuum polarization contribution to a :m 2 Z 1 ds
KsRs;(62)a Had; LO 
23
4m2 swhere

FIG. 41. (a) The lowest-order ‘‘cut’’ hadronic vacuum polarization diagram and (b) the electroproduction of hadrons, which isrelated to (a) through a dispersion relation. (c) The hadronic light-by-light contribution.

072003-37

G. W. BENNETT et al.

Rs PHYSICAL REVIEW D 73, 072003 (2006) 
tot e e ! hadrons
 
  ;
tot e e !   (63)

and Ks is a kinematic factor. The measured cross section
ratio Rs is the critical input to the evaluation. The s2dependence of the kernel weights preferentially the valuesof Rs at low energies (e.g., near the  resonance) and,
consequently, the low-energy region dominates the determination of a Had; LO. The higher-energy region is lesscritical [46].
The most precise data entering the dispersion relation at
low energies are from the CMD2 experiment at
Novosibirsk [47]. The CMD2 experiment measures Rs
by sweeping the center-of-mass collision energy of thepositron and electron beams in the VEPP-II ring. An alternate way of obtaining Rs is exploited by the KLOE
collaboration at Frascati [48]. They operate at a fixede e collision energy corresponding to resonant
production. The hadronic cross sections at lower energies are
obtained from events having an initial-state radiated photon, which reduces the actual center-of-mass collisionenergy.
The contribution to the dispersion integral from threshold to the  mass can also be derived from hadronic 
decays by invoking the conserved vector current hypothesis and by making necessary isospin corrections. For example, the decay rate for  !  0  can be related tothe e e annihilation cross section into   [46,49–
51]. Because the  data only contain an isovector component, the isoscalar piece present in e e annihilation has tobe put in ‘‘by hand’’ to evaluate a Had; LO. The -dataapproach is attractive because a large body of highprecision data exists from the LEP experiments and from
CLEO. Unfortunately, there are significant inconsistencies
between these data and those obtained in direct e e
TABLE XVI. Contributions to the standard model value fora . The error in ppm refers to the full value of a . The errors
listed for (Had; LO) are the quadrature of those from the data andfrom radiative corrections. The higher-order hadronic contribution does not include the hadronic light-by-light term, which islisted separately. In computing the total, the higher-order hadronic is taken from the same reference as the lowest-order
hadronic.
Contribution

Value [1010 ]

Error

QED
Had; LO
Had; LO
Had; HO
Had; HO
Had; LBL
Weak

11658471.958
696.3
692.4
10:0
9:8
12
15.4

0.143
7.2
6.4
0.6
0.1
3.5
0.22

0.012
0.62
0.55
0.05
0.01
0.3
0.02

[44]
[46]
[52]
[46]
[52]
[42]
[42]

Total

11659185.711659182.0

8.0
7.3

0.69
0.62

[46]
[52]

annihilation [46]. For example, the  branching ratios
predicted from the e e data do not agree, nor do theshapes of F obtained from either the e e or  data.
Because of this inconsistency, we compared only to thedirect e e annihilation data using the two recently published analyses [46,52] for the a Had; LO contribution(see Table XVI). Use of the  data leads to a higher
dispersion integral.
Higher-order hadronic contributions fall into twoclasses. The first class represents modifications ofFig. 41(a) with an additional vacuum polarization loop
(hadronic or leptonic), or with a photon loop along withthe hadronic vacuum polarization loop. These contributions, (Had; HO), can be calculated from a dispersion
relation with a different kernel function and experimental
data. Using the kernel function of Krause [53], the evaluations reported in Refs. [46,52] find 10:0  0:6 
1010 and 9:8  0:1  1010 , respectively, which arein good agreement and of sufficient precision compared tothe experimental uncertainty on a .
The hadronic light-by-light (Had; LbL) contribution
shown in Fig. 41(c) must be calculated using a theoreticalmodel. Its evaluation has been the focus of considerable
theoretical activity [54 –59]. In recent work, Melnikov andVainshtein (MV) report 13:62:5  1010 (0.22 ppm) [59],roughly 50% larger than that obtained in earlier efforts byothers [54 –57]. This value is found by ignoring several
small negative contributions. For this reason, in their review [42], Davier and Marciano assign 123:5  1010 for
this contribution, the central value being an alternate result
reported by MV [59]. The conservative uncertainty in the
DM review expands the range to include the earlier results.
We use the DM recommendation in our summary.
The standard model theoretical summary is given in
Table XVI. Two results are presented, representing thetwo slightly different e e -based evaluations of theleading-order hadronic vacuum polarization contribution.
The theoretical expectation should be compared to ourexperimental result (Eq. (58)):a Expt  11659208:06:3  1010 0:54 ppm:The differencea Expt-SM  22:4  10 to 26:1  9:4  1010 ;Error [ppm] Reference(64)
has a significance of 2.2 –2.7 standard deviations. Use ofthe -data gives a smaller discrepancy.
To show the sensitivity of the measured muon (g  2)
value to the electroweak gauge bosons, the electroweak
contribution given in Eq. (61) is subtracted from the standard model values in Table XVI. The resulting differencewith theory isa  38 to 41  10  1010 ;

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PHYSICAL REVIEW D 73, 072003 (2006)

or a 3.7– 4.3 standard deviation discrepancy when theelectroweak contribution is left out.
The standard model value of the muon’s magnetic anomaly is entirely the result of radiative corrections fromintermediate states formed from a wide range of known
particles. It is also sensitive to speculative effects beyond
the standard model such as additional gauge bosons, muonor gauge boson substructure or the existence of extra
dimensions. Its value is potentially quite sensitive to thepresence of as yet undiscovered particles associated withmany generic manifestations of supersymmetry [42,60].
VI. DISCUSSION AND CONCLUSIONS
Experiment E821 at Brookhaven is formally complete.
All of the data are analyzed for a and these independent
muon anomalous magnetic moment evaluations have beenreported [8–10]. Our combined result—based on nearly
equal samples of positive and negative muons—represents
a 14-fold improvement on the CERN-III experiment [5] of
the mid-1970’s. Our final measurement uncertainty is
0.54 ppm, corresponding to 0.46 ppm statistical and0.28 ppm systematic uncertainties, respectively. The magnetic field and muon spin precession systematics are combined. The experimental promise and progress since theproposal submission in 1984 motivated a significant theoretical effort to accurately predict the standard model
expectation for the muon anomaly. The theoretical uncertainty, now 0:62–0:69 ppm, has been reduced by more thanan order of magnitude over the same period. The difference
between the measured and theoretical values, 22  26 
1010 , lies within the expected range for many standard
model extensions.
Our (g  2) Collaboration published individual resultsfrom the yearly running periods and these were comparedto the most up-to-date theory expectations. Improvementsin the theoretical calculations, new input data for the
hadronic vacuum polarization analysis, and corrections to
the theory changed the standard model central value appreciably over this time period. For example, the theoretical values quoted in Table XVI are higher by 24  1010than the theory value quoted for our R99 result [8]. The
dominant theoretical uncertainty is associated with theleading-order hadronic vacuum polarization. Further
work is in progress at Novosibirsk, and additional datafrom both the CMD2 and SND experiments there will be
published in the near future. The B factories at SLAC and
KEK are also using initial-state radiation to measure Rs,
and results should be forthcoming. Continued theoreticalmodelling of the hadronic light-by-light contribution canalso be expected, and initial studies using the lattice havebegun [61]. We are confident that the precision on thestandard model value will be improved, enabling a moresensitive comparison to experiment.
Because our measurement precision was ultimately limited by statistics, the question naturally arises whether thecurrent technique can be extended using a more intense
muon source. We have studied this question and haveoutlined a plan [62] that can reduce the present uncertainty
on a by a factor of 2.5 (or more), to a relative precision of0:2 ppm. The effort requires straight-forward improvements in the magnetic-field uniformity and mapping system and a five-fold increase in the muon production andstorage rate. An important feature of the design is the useof a ‘‘backward’’ decay beam to eliminate the hadronicinduced flash. Other improvements are associated withincreasing the muon transmission fraction, optimizing the
kicker efficiency, and replacing the detectors with segmented calorimeters having independent readout
digitizers.
Historically, precision tests of the standard model have
led to both discoveries and refinements in the predictive
power of the theory. The series of CERN and BNL muon(g  2) experiments —spanning more than 40 yr—has
methodically progressed such that the muon anomaly is
now measured to sub-ppm precision. Over the same time,
the standard model theoretical development has progressed, with QED loops evaluated through fourth orderand estimated through fifth, weak loops through second
order order, and hadronic loops through second order.
Many standard model extensions—SUSY is just one example —suggest leading-order loops that will affect a atthe 1 ppm (or slightly smaller) range. The present sensitivity of the muon anomaly test of the standard model is0:9 ppm, with roughly equal contributions from theory
and experiment. Theory uncertainty improvement can be
expected from new experimental input for the hadronic
contribution and from new calculational approaches for the
hadronic light-by-light term. We have described an approach to improve the experiment uncertainty on the anomaly to 0:2 ppm. Thus, we may expect a significantly
improved sensitivity for the anomaly test in the future. In
the era of the LHC and direct searches for specific standard
model extensions, precision measurements, such as that of
the muon anomaly, represent a continually improving sum
rule of known physics and provide independent insight into
physics at high energies and short-distance scales.
ACKNOWLEDGMENTS
Many people have made important contributions towards the success of E821 over the past 20 years when
this experiment was proposed, built and completed. We
thank N. P. Samios, R. Palmer, R. K. Adair, T. L. Trueman,
M. Schwartz, T. B. W. Kirk, P. D. Bond, M. Murtagh*, S.
Aronson, D. I. Lowenstein, P. Pile and the staff of the BNL
AGS for the strong support they have given; H.
Hirabayashi and S. Kurokawa for their support of themagnet technology provided by KEK; J. M. Bailey, J. R.
Blackburn, W. Earle, M. A. Green, E. Hazen, K. Ishida, J.
Jackson, L. Jia, D. Loomba, W. P. Lysenko, J. Ouyang,
W. B. Sampson, R. T. Sanders, and K. Woodle for their

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PHYSICAL REVIEW D 73, 072003 (2006)

contributions to parts of the preparation and running of theexperiment; J. Cullen*, C. Pai, C. Pearson, I. Polk; L.
Snydstrup, J. Benante, D. von Lintig and S. Kochis, whoplayed major roles in the design and construction of theexperiment. We gratefully acknowledge M. Tanaka* forhis development of the AGS extraction system required forour experiment. We thank U. Haeberlen for his contributions to the development of the NMR magnetometer system. We thank our colleagues J. Bijnens, A. Czarnecki, M.
Davier, E. de Rafael, S. I. Eidelman, A. Höcker, F.
Jegerlehner, T. Kinoshita and W. Marciano for very useful
discussions on the theory. We gratefully acknowledge
L. M. Barkov for establishing the collaboration between
the Budker Institute and E821, both at BNL and on the Rs

measurements at BINP. W. W. Williams* (U. Michigan)
made important contributions in the early phases of thiswork and in the development of the collaboration. This
work was supported in part by the U.S. Department ofEnergy, the U.S. National Science Foundation, the German
Bundesminister für Bildung und Forschung, the Russian
Ministry of Science, the US-Japan Agreement in HighEnergy Physics, the NATO Office of Scientific Research,and the U.S. Partial support to individuals from The
Alexander von Humboldt Foundation (Deile, Prigl andSteinmetz), General Electric (Gray), The Scientific andResearch Council of Turkey (Özben) and the John Simon
Guggenheim Foundation (Hertzog) is acknowledged.

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