New Results from the Muon g�2 Experiment

E.P. Sichtermann11, G.W. Bennett2, B. Bousquet9, H.N. Brown2,G. Bunce2, R.M. Carey1, P. Cushman9, G.T. Danby2, P.T. Debevec7,

M. Deile11, H. Deng11, W. Deninger7, S.K. Dhawan11, V.P. Druzhinin3,L. Duong9, E. Efstathiadis1, F.J.M. Farley11, G.V. Fedotovich3, S. Giron9,

F.E. Gray7, D. Grigoriev3, M. Grosse-Perdekamp11, A. Grossmann6,M.F. Hare1, D.W. Hertzog7, X. Huang1, V.W. Hughes11, M. Iwasaki10,

K. Jungmann5, D. Kawall11, B.I. Khazin3, J. Kindem9, F. Krienen1,I. Kronkvist9, A. Lam1, R. Larsen2, Y.Y. Lee2, I. Logashenko1�3,

R. McNabb9, W. Meng2, J. Mi2, J.P. Miller1, W.M. Morse2, D. Nikas2,C.J.G. Onderwater7, Y. Orlov4, C.S. Özben2, J.M. Paley1, Q. Peng1,

C.C. Polly7, J. Pretz11, R. Prigl2, G. zu Putlitz6, T. Qian9, S.I. Redin3�11,O. Rind1, B.L. Roberts1, N. Ryskulov3, P. Shagin9, Y.K. Semertzidis2,

Yu.M. Shatunov3, E. Solodov3, M. Sossong7, A. Steinmetz11, L.R. Sulak1,A. Trofimov1, D. Urner7, P. von Walter6, D. Warburton2, and

A. Yamamoto8.(Muon g�2 Collaboration)

1Department of Physics, Boston University, Boston, Massachusetts 022152Brookhaven National Laboratory, Upton, New York 119733Budker Institute of Nuclear Physics, Novosibirsk, Russia

4Newman Laboratory, Cornell University, Ithaca, New York 148535 Kernfysisch Versneller Instituut, Rijksuniversiteit Groningen, NL 9747 AA Groningen, The Netherlands

6 Physikalisches Institut der Universität Heidelberg, 69120 Heidelberg, Germany7 Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801

8 KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan9Department of Physics, University of Minnesota,Minneapolis, Minnesota 55455

10 Tokyo Institute of Technology, Tokyo, Japan11 Department of Physics, Yale University, New Haven, Connecticut 06520

Abstract. The Muon g� 2 collaboration has measured the anomalous magneticg value, a ��g�2��2, of the positive muon with an unprecedented uncertainty of 0.7 parts per million. The resulta�µ �expt� � 11659204�7��5� �10�10, based on data collected in the year 2000 at BrookhavenNational Laboratory, is in good agreement with the preceeding data ona �µ anda�µ . The measurementtests standard model theory, which at the level of the current experimental uncertainty involvesquantum electrodynamics, quantum chromodynamics, and electroweak interaction in a significantway.

INTRODUCTION

The magnetic moment�µ of a particle with chargee, massm, and spin�s is given by

�µ �e

2mcg�s (1)

in which g is the gyromagnetic ratio. For point particles with spin-12, Dirac theory

predictsg � 2.Precision measurements of theg values of leptons and baryons have historically

played an important role in the development of particle theory. Theg value of the pro-ton, for example, is found to differ sizeably from 2, which provides evidence for a richinternal proton (spin) structure. The leptong values deviate from 2 only by about onepart in a thousand, consistent with current evidence that leptons are point particles withspin-12. The anomalous magneticg value of the electron,ae � �ge�2��2, is among themost accurately measured quantities in physics, and is presently known with an uncer-tainty of about four parts per billion (ppb) [1]. The value ofae is described in terms ofstandard model (SM) field interactions, in which it has a leading order contribution ofα��2π� from the so-called Schwinger term. Nearly all of the measured value is con-tributed by QED processes involving virtual electrons, positrons, and photons. Particlesmore massive than the electron contribute only at the level of the present experimentaluncertainty.

The anomalous magneticg value of the muon,aµ , is more sensitive thanae toprocesses involving particles more massive than the electron, typically by a factor�mµ�me�

2 � 4 �104� A series of three experiments at CERN [2, 3, 4] measuredaµ to afinal uncertainty of about 7 parts per million (ppm), which is predominantly of statisticalorigin. The CERN generation of experiments thus tested electron-muon universalityand established the existence of a� 59 ppm hadronic contribution toaµ . Electroweakprocesses are expected to contribute toaµ at the level of 1.3 ppm, as are many speculativeextensions of the SM.

The present muong� 2 experiment at Brookhaven National Laboratory (BNL) hasdeterminedaµ� of the positive muon with an uncertainty of 0.7 ppm from data collectedin the year 2000, and aims for a similar uncertainty onaµ� of the negative muon frommeasurements in 2001. The continuation of the experiment by a single running period,if funded, should bring the design goal precision of 0.4 ppm within reach.

EXPERIMENT

The concept of the experiment at BNL is the same as that of the last of the CERNexperiments [2, 3, 4] and involves the study of the orbital and spin motions of polarizedmuons in a magnetic storage ring.

The present experiment (Figure 1) is situated at the Alternating Gradient Synchrotron(AGS), which in the year 2000 delivered up to 60� 1012 protons in twelve 50 ns(FWHM) bunches over its 3 s cycle. The 24 GeV protons from the AGS were directedonto a rotating, water-cooled nickel target. Pions with energies of 3.1 GeV emitted from

FIGURE 1. Top view of theg� 2 apparatus. The beam of longitudinally polarized muons enters thesuperferric storage ring magnet through a superconducting inflector magnet located at 9 o’clock andcirculates clockwise after being placed onto stored orbit with three pulsed kickers modules in the 12o’clock region. Twenty-four lead scintillating-fiber calorimeters on the inner, open side of the C-shapedring magnet are used to measure muon decay positrons. The central platform supports the power suppliesfor the four electrostatic quadrupoles and the kicker modules.

the target were captured into a 72 m straight section of focusing-defocusing magneticquadrupoles, which transported the parent beam and naturally polarized muons fromforward pion decays. At the end of the straight section, the beam was momentum-selected and injected into the 14.2 m diameter storage ring magnet [5] through a field-free inflector [6] region in the magnet yoke. A pulsed magnetic kicker [7] located atapproximately one quarter turn from the inflector region produced a 10 mrad deflectionwhich placed the muons onto stored orbits. Pulsed electrostatic quadrupoles [8] providedvertical focusing. The magnetic dipole field of about 1.45 T was measured with an NMRsystem [9] relative to the free proton NMR frequencyωp over most of the 9 cm diametercircular storage aperture. Twenty-four electromagnetic calorimeters [10] read out by400 MHz custom waveform digitizers (WFD) were used on the open, inner side of the C-shaped ring magnet to measure muon decay positrons. The decay violates parity, whichleads to a relation between the muon spin direction and the positron energy spectrumin the laboratory frame. For positrons above an energy thresholdE, the muon-decaytime-spectrum

N�t� � N0�E�exp

��tγτ

��1�A�E�sin�ωat �φ�E��� � (2)

in whichN0 is a normalization,γτ � 64µs is the dilated muon lifetime,A an asymmetryfactor, φ a phase, andωa the angular difference frequency of muon spin precessionand momentum rotation. In our measurements, the NMR and WFD clocks were phase-locked to the same LORAN-C [11] frequency signal.

The muon anomalous magneticg value is evaluated from the ratio of the measuredfrequencies,R � ωa�ωp, according to:

aµ �R

λ �R� (3)

in whichλ � µµ�µp is the ratio the muon and proton magnetic moments. The value withsmallest stated uncertainty,λ � µµ�µp � 3�18334539�10� [12], results from measure-ments of the microwave spectrum of ground state muonium [13] and theory [14, 15].

Important improvements made since our preceeding measurement [16] include: theoperation of the AGS with 12 beam bunches, which contributed to a 4-fold increasein the data collected; a new superconducting inflector magnet, which greatly improvedthe homogeneity of the magnetic field in the muon storage region; a sweeper magnet inthe beamline, which reduced AGS background; additional muon loss detectors, whichimproved the study of time dependence; and further refined analyses, in particular ofcoherent betatron oscillations.

DATA ANALYSIS

The analysis ofaµ follows, naturally, the separation of the measurement in the fre-quenciesωp andωa. Both frequencies were analyzed independently by several groupswithin the collaboration. The magnetic field frequencies measured during the runningperiod were weighted by the distribution of analyzed muons, both in time and over thestorage region. The frequency fitted from the positron time spectra was corrected by+0.76(3) ppm for the net contribution to the muon spin precession and momentum ro-tation caused by vertical beam oscillations and, for muons withγ �� 29�3, by horizontalelectric fields [17]. The values ofR � ωa�ωp andaµ were evaluated only after each ofthe frequency analyses had been finalized; at no earlier stage were the absolute values ofboth frequencies,ωp andωa, known to any of the collaborators.

The frequency ωp

The analysis of the magnetic field data starts with the calibration of the 17 NMRprobes in the field trolley using dedicated measurements taken during and at the endof the data collection period. In these calibration measurements, the field in the storageregion was tuned to very good homogeneity at two specific calibration locations. Thefield was then measured with the NMR probes mounted in the trolley shell, as well aswith a single probe plunged into the storage vacuum and positioned to measure the fieldvalues in the corresponding locations. Drifts of the field during the calibration measure-ments were determined by remeasuring the field with the trolley after the measurementswith the plunging probe were completed, and in addition by interpolation of the readingsfrom nearby NMR probes in the outer top and bottom walls of the vacuum chamber. Thedifference of the trolley and plunging probe readings forms a calibration of the trolleyprobes with respect to the plunging probe, and hence with respect to each other. The

TABLE 1. Systematic uncertainties for theωp analysis. Theuncertainty "Others" groups uncertainties caused by higher mul-tipoles, the trolley frequency, temperature, and voltage response,eddy currents from the kickers, and time-varying stray fields.

Source of errors Size [ppm]

Absolute calibration of standard probe 0.05Calibration of trolley probe 0.15Trolley measurements ofB0 0.10Interpolation with fixed probes 0.10Uncertainty from muon distribution 0.03Others 0.10

Total systematic error onωp 0.24

plunging probe, as well as a subset of the trolley probes, were calibrated with respectto a standard probe [18] at the end of the running period in a similar sequence of mea-surements in the storage region, which was opened to air for that purpose. The leadinguncertainties in the calibration procedure result from the residual inhomogeneity of thefield at the calibration locations, and from position uncertainties in the active volumesof the NMR probes. These uncertainties were evaluated from measurements in whichthe trolley shell was purposely displaced and known field gradients were applied usingthe so-called surface and dipole correction coils of the ring magnet. The size of theseuncertainties is estimated to be 0.15 ppm, as listed in Table 1. The uncertainty in the ab-solute calibration of the standard probe amounts to 0.05 ppm [18]. The dependencies ofthe trolley NMR readings on the supply voltage and on other parameters were measuredto be small in the range of operation. An uncertainty of 0.10 ppm ("Others" in Table 1) isassigned, which includes also the measured effects from the transient kicker field causedby eddy currents and from AGS stray fields.

The magnetic field inside the storage region was measured 22 times with the fieldtrolley during the data collection from January to March 2000. Figure 2a shows the fieldvalue measured in the storage ring with the center trolley probe versus the azimuthalangle. The field is seen to be uniform to within about�50ppm of its average value overthe full azimuthal range, including the region near 350Æ where the inflector magnet islocated. Non-linearities in the determination of the trolley position during the measure-ments — from the measured cable lengths and from perturbations on the readings fromfixed probes as the trolley passes — are estimated to affect the azimuthal average of thefield at the level of 0.10 ppm. Fig. 2b shows a 2-dimensional multipole expansion of theazimuthally averaged readings from the trolley probes,

By �∞

∑n�0

Cnrn cos�nφ��∞

∑n�0

Dnrn sin�nφ�� (4)

Bx �∞

∑n�0

Cnrn sin�nφ��∞

∑n�0

Dnrn cos�nφ�� (5)

where the coefficientsCn andDn are the normal and skew multipoles, andr andφ denotethe polar coordinates in the storage region. The multipole expansion was truncated in the

azimuth [degree]0 50 100 150 200 250 300 350

B [

pp

m]

700

720

740

760

780

800

820

840

860

880

900

920 Multipoles [ppm]

normal skew

Quad 0.24 0.29

Sext -0.53 -1.06

Octu -0.10 -0.15

Decu 0.82 0.54radial distance [cm]

-4 -3 -2 -1 0 1 2 3 4

vert

ical

dis

tan

ce [

cm]

-4

-3

-2

-1

0

1

2

3

4

-1.5

-1.0

-1.0

-0.5

-0.5

000

0.5

0.5

1.0

1.01.5

FIGURE 2. The NMR frequency measured with the center trolley probe relative to a 61.74 MHzreference versus the azimuthal position in the storage ring (left), and (right) a 2-dimensional multipoleexpansion of theazimuthal average of the field measured with 15 trolley probes with respect to the centralfield value of 1.451 275 T. The multipole amplitudes are given at the storage ring aperture, which has a4.5 cm radius as indicated by the circle.

analysis after the decupoles (n � 4). Measurements with probes extending to larger radiishow that the neglect of higher multipoles is at most 0.03 ppm in terms of the averagefield encountered by the stored muons, in agreement with magnet design calculations.The field averaged over azimuth is seen to be uniform to within 1.5 ppm of its value.

The measurements with the trolley relate the readings of 370 NMR fixed probes in theouter top and bottom walls of the storage vacuum chamber to the field values in the beamregion. The fixed NMR probes are used to interpolate the field when the field trolley is’parked’ in the storage vacuum just outside the beam region, and muons circulate in thestorage ring. Since the relationship between the field value in the storage region and thefixed probe readings may change during the course of the data collection period, thefield mappings with the trolley were repeated typically two to three times per week, andwhenever ramping of the magnet or a change in settings required such. The uncertaintyassociated with the interpolation of the magnetic field between trolley measurements isestimated from the spread of the difference between the dipole moments evaluated fromthe fixed probe measurements and from the trolley probe measurements in periods ofconstant magnet settings and powering. It is found to be 0.10 ppm.

Since the field is highly uniform, the field integral encountered by the (analyzed)muons is rather insensitive to the exact location of the beam. As in earlier works [16,19], the radial equilibrium beam position was determined from the debunching of thebeam following injection and the vertical position from the distribution of counts inscintillation counters mounted on the front faces of the positron calorimeters. Theposition uncertainty amounts to 1 – 2 mm,which contributes 0.03 ppm uncertainty tothe field integral.

The result for field frequencyωp weighted by the muon distribution is found to be,

ωp��2π� � 61791595�15�Hz (0.2 ppm)� (6)

where the uncertainty has a leading contribution from the calibration of the trolleyprobes and is thus predominantly systematic. A second analysis of the field has beenperformed using additional calibration data, a different selection of fixed NMR probes,

4 Billion Positrons with E > 2 GeV

10 2

10 3

10 4

10 5

10 6

10 7

0 10 20 30 40 50 60 70 80 90 100

45-100 µs

100-200 µs

200-300 µs

300-400 µs

400-500 µs

500-600 µs

600-700 µs

700-800 µs

800-850 µs

Time µs

Num

ber

of P

osit

rons

/149

ns

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Frequency [MHz]

Fo

uri

er A

mp

litu

de

[a.u

.]

f cbo,

h

f cbo,

h +

f g-2

f cbo,

h -

f g-2

muo

n lo

sses

, gai

n va

riatio

ns

2 f cb

o, h

f g-2

FIGURE 3. The time spectrum for 4� 109 positrons with energies greater than 2 GeV collected fromJanuary to March 2000, after corrections for pile-up and for the bunched time structure of the injectedbeam (left) were made, and (right) the Fourier transform of the time spectrum, in which muon decay andspin precession (cf. Eq. 2) has been suppressed to emphasize other effects.

and a different method to relate the trolley and fixed probe readings. The results fromthese analyses are found to agree to within a fraction of the total uncertainty onωp.

The frequency ωa

The event sample available for analysis from data collection in the year 2000 amountsto about 4� 109 positrons reconstructed with energies greater than 2 GeV and timesbetween 50µs and 600µs following the injection of a beam bunch. Figure 3a showsthe time spectrum corrected for the the bunched time structure of the beam and foroverlapping calorimeter pulses, so called pile-up [16].

The leading characteristics of the time spectrum are those of muon decay and spinprecession (cf. Eq. 2). Additional effects exist, as seen from the Fourier spectrumin Figure 3b, and require careful consideration in the analysis. These effects includedetector gain and time instability, muon losses, and oscillations of the beam as a whole,so-called coherent betatron oscillations (CBO).

Numerically most relevant to the determination ofωa are CBO in the horizontal plane.CBO are caused by injecting the beam through the relatively narrow 18�w��57�h�mm2

aperture of the 1.7 m long inflector channel into the 90 mm diameter aperture of thestorage region, and have been observed directly with fiber harp monitors plunged into thebeam region for this purpose. The CBO frequency is determined by the focusing indexof the storage ring, and is numerically close to twice the frequencyωa for the quadrupolesettings employed in most measurements so far. Since the calorimeter acceptances varywith the radial muon decay position in the storage ring and with the momentum of thedecay positron, the time and energy spectra of the observed positrions are modulated

TABLE 2. Systematic uncertainties for theωa analysis. Theuncertainty "Others" groups uncertainties caused by AGS back-ground, timing shifts, vertical oscillations and radial electricfields, and beam debunching/randomization.

Source of errors Size [ppm]

Coherent betatron oscillations 0.21Pileup 0.13Gain changes 0.13Lost muons 0.10Binning and fitting procedure 0.06Others 0.06

Total systematic error onωa 0.31

with the CBO frequency. These modulations affect the normalizationN0, the asymmetryA, and the phaseφ in Eq. 2 at the level of 1%, 0.1%, and 1 mrad at beam injection. Whennot accounted for in the function fitted to the data, the modulations of the asymmetry andphase with a frequencyωcbo�h � 2�ωa may manifest themselves as artificial shifts ofup to 4 ppm in the frequency valuesωa determined from individual calorimeter spectra.The circular symmetry of the experiment design results in a strong cancellation of suchshifts in the joined calorimeter spectrum.

Several approaches have been pursued in the analysis ofωa. In one approach, the timespectra from individual positron calorimeters was fitted in narrow energy intervals usinga fit function as in Eq. 2 extended by the number, asymmetry, and phase modulations.Other approaches made use of the cancellation in the joined calorimeter spectra andeither fitted for the residual of the leading effects, or accounted for their neglect in acontribution to the systematic uncertainty. The results are found to agree, onωa to withinthe expected 0.5 ppm statistical variation resulting from the slightly different selectionand treatment of the data in the respective analyses. The combined result is found to be,

ωa��2π� � 22907411�14��7�Hz (0.7 ppm)� (7)

in which the first uncertainty is statistical and the second systematic. The combinedsystematic uncertainty is broken down by source in Table 2.

RESULTS AND DISCUSSION

The value ofaµ was evaluated after the analyses ofωp andωa had been finalized,

aµ� � 11659204�7��5� � 10�10 (0.7 ppm)� (8)

where the first uncertainty is statistical and the second systematic. This new result is ingood agreement with the previous measurements [4, 16, 19, 20] and drives the presentworld average,

aµ�exp� � 11659203�8� � 10�10 (0.7 ppm)� (9)

- 1

1659

000

10 1

0× µa

Measurements Standard Model

160

180

200

220

160

180

200

220

BNL’98

BNL’99

BNL’00 WorldAverage

, DEHZ’02τ

, DEHZ’02-e+e

References: BNL’98 PRL 86 2227

BNL’99 PR 62D 091101

BNL’00 PRL 89 101804

(had;1) from hep-ph/0208177µ , DEHZ’02 aτ

(had;1) from hep-ph/0208177µ , DEHZ’02 a-e+e

FIGURE 4. Recent measuremens ofaµ and standard model evaluations using the evaluations inRef. [21] of the lowest order contribution from hadronic vacuum polarization.

in which the uncertainty accounts for known correlations between the systematic uncer-tainties in the measurements. Figure 4 shows our recent measurements ofaµ�, togetherwith two SM evaluations discussed below.

In the SM, the value ofaµ receives contributions from QED, hadronic, andelectroweak processes,aµ�SM� � aµ�QED� � aµ�had� � aµ�weak�. The QED andweak contributions can, unlike the hadronic contribution, be evaluated perturbatively,aµ�QED� � 11658470�57�29� � 10�10 [22] and aµ�weak� � 15�1�4� � 10�10 [23].The hadronic contribution is, in lowest order, related by dispersion theory to the hadronproduction cross sections measured ine�e� collisions and, under additional assump-tions, to hadronicτ -decay. Clearly, the hadronic contribution has a long history ofvalues as new data appeared and analyses were refined.

Shortly before the SPIN-2002 conference, Davier and co-workers released a new anddetailed evaluation [21], which now incorporates the high precisione�e� data [24]in the region of theρ resonance from CMD-2 at Novosibirsk, more accuratee�e�

measurements [25, 26] in the 2–5 GeV energy region from BES in Beijing, preliminaryresults from the final ALEPH analysis [27] of hadronicτ -decay at LEP1, as wellas additional CLEO data [28, 29]. The authors note discrepancies between thee�e�

and τ data at the present levels of precision, and obtain separate predictions for thecontribution toaµ�SM� from lowest order hadronic vacuum polarization,aµ�had�1� �685�7� � 10�10 from e�e� data andaµ�had�1� � 702�6� � 10�10 from τ data. Higherorder contributions include higher order hadronic vacuum polarization [30, 31] andhadronic light-by-light scattering [32, 33, 34, 35, 36].

Open questions concern the SM value ofaµ , in particular the hadronic contribution,and the experimental value ofaµ� at sub-ppm precision. The former should benefit from

further theoretical scrutiny and from radiative-return measurements ate�e� factories.We are currently analyzing a sample of about 3� 109 decay electrons fromaµ� datacollected in the year 2001. Stay tuned!

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Department of Energy, the U.S. NationalScience Foundation, the U.S. National Computational Science Alliance, the GermanBundesminister für Bildung und Forschung, the Russian Ministry of Science, and theU.S.-Japan Agreement in High Energy Physics.

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