Neutron Spin RotationWerner, et al., Physical Review Letters 35(1975)1053

Also see textbook Figure 4.4(a)

2 cm

Neutron Beam

Neutron Detectors

Rotation through ω0T=2πRotation through 4π

Muon Spin Rotationand

Measurement of g-2

Brown, et al., Physical Review D 62(2000)091101

Bennett, et al., Physical Review Letters 92(2004)161802

current effect on the total field seen by the muons was less

than 0.1 ppm 20 !s after injection. The time-varying mag-netic field from the eddy currents was calculated with the

program OPERA "4# and was measured in a full-size straightprototype vacuum chamber with the use of the Faraday effect

"5#. Since the muons circulate in 149 ns, they were kickedseveral times before the kicker pulse died out.

About 104 muons were stored in the ring per proton

bunch. With muon injection, the number of detected posi-

trons per hour was increased by an order of magnitude over

the pion-injection method employed previously. Further-

more, the injection related background $flash% in the positrondetectors was reduced by a factor of about 50, since most of

the pions were removed from the beam before entering the

storage ring.

For polarized muons moving in a uniform magnetic field

B! perpendicular to the muon spin direction and to the planeof the orbit and with an electric quadrupole field E! , which isused for vertical focusing "2,15#, the angular frequency dif-ference, &a , between the spin precession frequency &s and

the cyclotron frequency &c , is given by

&! a!"e

m!a!B! "" a!"

1

'2"1 #(! #E! $ . $1%

The dependence of &a on the electric field is eliminated by

storing muons with the ‘‘magic’’ '!29.3, which corre-sponds to a muon momentum p!3.09 GeV/c . Hence mea-surements of &a and of B determine a! . At the magic

gamma, the muon lifetime is ')!64.4 !s and the (g"2)precession period is 4.37 !s. With a field of 1.45 T in ourstorage ring "6#, the central orbit radius is 7.11 m.The magnetic field in Eq. $1% is the average over the muon

distribution. We obtained the equilibrium radius distribution

by determining the distribution of rotation frequencies in the

ring from the time spectra of decay positrons "1#. The distri-bution, reproduced with a tracking code, was found to be 3

mm toward the outside of the central storage region. This

offset was caused by the mode of operating the kicker. The

calculated and measured radial distributions are shown in

Fig. 1.

The magnetic field seen by the muon distribution was

calculated by tracking a sample of muons in software

through the field map measured by NMR, and by averaging

the field values. The resulting average corresponds within

0.02 ppm to the field value taken at the beam center and

averaged over azimuth. We used the latter to account for

variations with time, and to obtain the present result.

Positrons from the in-flight decay !$!e$*e*! were de-

tected with 24 Pb-scintillating fiber calorimeters "7# placedsymmetrically around the inside of the storage ring. Twenty-

one of these detectors were used in the present analysis. The

observed positron time spectrum shown in Fig. 2 was ad-

equately represented by "2,3#

N0$E %e"t/')"1$A$E %cos„&at$+$E %…# . $2%

The normalization constant N0 depends on the energy thresh-

old, E, placed upon the positrons. The $integral% asymmetry

A depends on E and on the beam polarization. The fractional

statistical error on &a is proportional to A"1Ne

"1/2 , where Ne

is the number of decay positrons detected above threshold.

For an energy threshold of 1.8 GeV where NeA2 is maxi-

mum, A was found to be 0.34 on average.

As in Ref. "1#, the photomultiplier tubes were gated offbefore injection. With the reduced flash associated with

muon injection, it was possible to begin counting as soon as

5 !s after injection in the region of the ring 270° aroundfrom the injection point, and 35 !s in the injection region.

FIG. 1. The equilibrium radius distribution calculated using the

tracking code $histogram% and obtained from an analysis of the

beam debunching at early times $points%.

FIG. 2. The positron time spectrum obtained with muon injec-

tion for E%1.8 GeV. These data represent 84 million positrons.

RAPID COMMUNICATIONS

H. N. BROWN et al. PHYSICAL REVIEW D 62 091101$R%

091101-2

where R!! "!a=!p#0:0037072083$26% and " # !!=!p # 3:183 345 39$10% [11]. This new result is in goodagreement with the average of R!& # 0:003 707 204 8$25%[2] as predicted by the CPT theorem. The difference!R # R!! ! R!& # $3:5' 3:4% ( 10!9. The new aver-age is R! # 0:003 707 206 3$20% and

a!$exp% # 11 659 208$6% ( 10!10 $0:5 ppm%; (4)

in which the total uncertainty consists of 5( 10!10

(0.4 ppm) statistical uncertainty and 4( 10!10 (0.3 ppm)systematic uncertainty. The correlation of systematic un-certainties between the data sets has been taken intoaccount. The combined result for the positive muon [3],a!&$exp% # 11 659 203$8% ( 10!10 (0.7 ppm) has a statis-tical uncertainty of 6( 10!10 (0.6 ppm) and a systematicuncertainty of 5( 10!10 (0.4 ppm). It is shown in Fig. 4together with the new result for the negative muon andtheir average.

The standard model (SM) prediction for a! consists ofQED, hadronic, and weak contributions. The uncertaintyon the standard model value is dominated by the uncer-tainty on the lowest-order hadronic vacuum polarization.This contribution can be determined directly from theannihilation of e&e! to hadrons through a dispersionintegral [12]. The indirect determination using datafrom hadronic # decays, the conserved vector currenthypothesis, plus the appropriate isospin corrections,could in principle improve the precision of a!$had%.However, discrepancies between the # and the e&e! re-sults exist [13,14]. The two data sets do not give consistentresults for the pion form factor. Using e&e! annihilationdata, the corresponding theoretical value is a!$SM% #11 659 181$8% ( 10!10 (0.7 ppm). The value deducedfrom # decay is larger by 15( 10!10 and has a stateduncertainty of 7 ( 10!10 (0.7 ppm). The difference be-tween the experimental determination of a! and thestandard model theory using the e&e! or # data for thecalculation of the hadronic vacuum polarization is 2:7$and 1:4$, respectively.

This is the final analysis of the anomalous magneticmoment from experiment E821 at the Brookhaven

Alternating Gradient Synchrotron. We aim to substan-tially improve our result in a new measurement andlook forward to continued efforts to improve the theo-retical evaluation.

We thank T. Kirk, D. I. Lowenstein, P. Pile, and thestaff of the BNL Alternating Gradient Synchrotron forthe strong support they have given this experiment. Thiswork was supported in part by the U.S. Department ofEnergy, the U.S. National Science Foundation, the U.S.National Computational Science Alliance, the GermanBundesminister fur Bildung und Forschung, the RussianMinistry of Science, and the U.S.-Japan Agreement inHigh Energy Physics.

*Deceased.[1] V.W. Hughes and T. Kinoshita, Comments Nucl. Part.

Phys. 14, 341 (1985).[2] Muon (g! 2) Collaboration, H. N. Brown et al., Phys.

Rev. D 62, 091101 (2000); Muon (g! 2) Collaboration,H. N. Brown et al., Phys. Rev. Lett. 86, 2227 (2001).

[3] Muon (g! 2) Collaboration, G.W. Bennett et al., Phys.Rev. Lett. 89, 101804 (2002).

[4] A. Yamamoto et al., Nucl. Instrum. Methods Phys. Res.,Sect. A 491, 23 (2002); G. T. Danby et al., Nucl. Instrum.Methods Phys. Res., Sect. A 457, 151 (2001); S. I. Redinet al., Nucl. Instrum. Methods Phys. Res., Sect. A 473,260 (2001); R. Prigl et al., Nucl. Instrum. Methods Phys.Res., Sect. A 374, 118 (1996).

[5] X. Fei, V.W. Hughes, and R. Prigl, Nucl. Instrum.Methods Phys. Res., Sect. A 394, 349 (1997).

[6] J. Ouyang et al., Nucl. Instrum. Methods Phys. Res., Sect.A 374, 215 (1996); S. A. Sedykh et al., Nucl. Instrum.Methods Phys. Res., Sect. A 455, 346 (2000).

[7] E. Efstathiadis et al., Nucl. Instrum. Methods Phys. Res.,Sect. A 496, 8 (2003).

[8] Y. K. Semertzidis et al., Nucl. Instrum. Methods Phys.Res., Sect. A 503, 458 (2003).

[9] J. Bailey et al., Nucl. Phys. B150, 1 (1979).

- 11

6590

0010

10

! µa

150

160

170

180

190

200

210

220

230

+µ

-µ

Avg.

]-e+

[e

]"[

Experiment Theory

FIG. 4 (color online). Measurements of a! by E821 with theSM predictions (see text for discussion). Uncertainties indi-cated on the measurements are total uncertainties.

TABLE II. Systematic uncertainties for the combined !aanalysis.

Source of errors Size (ppm)

Coherent betatron oscillations 0.07Pileup 0.08Gain changes 0.12Lost muons 0.09Othersa 0.11Total systematic error on !a 0.21

aAlternating gradient synchrotron background, timing shifts, Efield and vertical oscillations, beam debunching/randomiza-tion, binning, and fitting procedure.

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