aug1010082023

Also at ]Novosibirsk State University. Also at ]Oak Ridge National Laboratory.
Also at ]The Cockcroft Institute of Accelerator Science and Technology,
Daresbury, United Kingdom. ††thanks: Deceased. Also at ]Università di Trieste,
Trieste, Italy. Also at ]INFN Gruppo Collegato di Udine, Sezione di Trieste,
Udine, Italy. Also at ]Shanghai Key Laboratory for Particle Physics and
Cosmologyalso at ]Key Lab for Particle Physics, Astrophysics and Cosmology
(MOE). Also at ]Università di Pisa, Pisa, Italy. Also at ]Lebedev Physical
Institute and NRNU MEPhI. Also at ]Università di Pisa, Pisa, Italy. ††thanks:
Deceased. Also at ]Istituto Nazionale di Ottica - Consiglio Nazionale delle
Ricerche, Pisa, Italy. Also at ]Istituto Nazionale di Ottica - Consiglio
Nazionale delle Ricerche, Pisa, Italy. Also at ]Istituto Nazionale di Ottica -
Consiglio Nazionale delle Ricerche, Pisa, Italy. Also at ]Università di Pisa,
Pisa, Italy. Now at ]Alliance University, Bangalore, India. Also at ]INFN,
Sezione di Pisa, Pisa, Italy. Now at ]Istinye University, Istanbul, Türkiye.
Also at ]Shanghai Key Laboratory for Particle Physics and Cosmologyalso at
]Key Lab for Particle Physics, Astrophysics and Cosmology (MOE). Also at
]Università di Napoli, Naples, Italy. Also at ]University of Rijeka, Rijeka,
Croatia. Also at ]Research Center for Graph Computing, Zhejiang Lab, Hangzhou,
Zhejiang, China. Also at ]Shenzhen Technology University, Shenzhen, Guangdong,
China. Also at ]Shanghai Key Laboratory for Particle Physics and Cosmologyalso
at ]Key Lab for Particle Physics, Astrophysics and Cosmology (MOE). Also at
]Novosibirsk State University. Also at ]Shanghai Key Laboratory for Particle
Physics and Cosmologyalso at ]Key Lab for Particle Physics, Astrophysics and
Cosmology (MOE). Also at ]Scuola Normale Superiore, Pisa, Italy. Also at
]Università di Napoli, Naples, Italy. Also at ]INFN Gruppo Collegato di Udine,
Sezione di Trieste, Udine, Italy ††thanks: Deceased. Also at ]INFN, Sezione di
Roma Tor Vergata, Rome, Italy. Now at ]Virginia Tech, Blacksburg, Virginia,
USA. Now at ]Alliance University, Bangalore, India. Also at ]INFN Gruppo
Collegato di Udine, Sezione di Trieste, Udine, Italy. Now at ]Wellesley
College, Wellesley, Massachusetts, USA. Also at ]Novosibirsk State University.
Also at ]Università di Roma Tor Vergata, Rome, Italy. Now at ]Institute for
Interdisciplinary Research in Science and Education (ICISE), Quy Nhon, Binh
Dinh, Vietnam. Also at ]INFN, Sezione di Pisa, Pisa, Italy. The Muon
$g\\!-\\!2$ Collaboration

00footnotetext: $\diamond$ Deceased

# Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm

D. P. Aguillard  University of Michigan, Ann Arbor, Michigan, USA  T. Albahri
University of Liverpool, Liverpool, United Kingdom  D. Allspach  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  A. Anisenkov  [
Budker Institute of Nuclear Physics, Novosibirsk, Russia  K. Badgley  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  S. Baeßler  [
University of Virginia, Charlottesville, Virginia, USA  I. Bailey  [
Lancaster University, Lancaster, United Kingdom  L. Bailey  Department of
Physics and Astronomy, University College London, London, United Kingdom  V.
A. Baranov  Joint Institute for Nuclear Research, Dubna, Russia  E. Barlas-
Yucel  University of Illinois at Urbana-Champaign, Urbana, Illinois, USA  T.
Barrett  Cornell University, Ithaca, New York, USA  E. Barzi  Fermi National
Accelerator Laboratory, Batavia, Illinois, USA  F. Bedeschi  INFN, Sezione di
Pisa, Pisa, Italy  M. Berz  Michigan State University, East Lansing, Michigan,
USA  M. Bhattacharya  Fermi National Accelerator Laboratory, Batavia,
Illinois, USA  H. P. Binney  University of Washington, Seattle, Washington,
USA  P. Bloom  North Central College, Naperville, Illinois, USA  J. Bono
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  E. Bottalico
University of Liverpool, Liverpool, United Kingdom  T. Bowcock  University of
Liverpool, Liverpool, United Kingdom  S. Braun  University of Washington,
Seattle, Washington, USA  M. Bressler  Department of Physics, University of
Massachusetts, Amherst, Massachusetts, USA  G. Cantatore  [  INFN, Sezione di
Trieste, Trieste, Italy  R. M. Carey  Boston University, Boston,
Massachusetts, USA  B. C. K. Casey  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  D. Cauz  [  Università di Udine, Udine, Italy  R.
Chakraborty  University of Kentucky, Lexington, Kentucky, USA  A. Chapelain
Cornell University, Ithaca, New York, USA  S. Chappa  Fermi National
Accelerator Laboratory, Batavia, Illinois, USA  S. Charity  University of
Liverpool, Liverpool, United Kingdom  C. Chen  Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, China  School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai, China  M. Cheng
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA  R. Chislett
Department of Physics and Astronomy, University College London, London, United
Kingdom  Z. Chu  [  [  School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai, China  T. E. Chupp  University of Michigan, Ann Arbor,
Michigan, USA  C. Claessens  University of Washington, Seattle, Washington,
USA  M. E. Convery  Fermi National Accelerator Laboratory, Batavia, Illinois,
USA  S. Corrodi  Argonne National Laboratory, Lemont, Illinois, USA  L.
Cotrozzi  [  INFN, Sezione di Pisa, Pisa, Italy  J. D. Crnkovic  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  S. Dabagov  [  INFN,
Laboratori Nazionali di Frascati, Frascati, Italy  P. T. Debevec  University
of Illinois at Urbana-Champaign, Urbana, Illinois, USA  S. Di Falco  INFN,
Sezione di Pisa, Pisa, Italy  G. Di Sciascio  INFN, Sezione di Roma Tor
Vergata, Rome, Italy  B. Drendel  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  A. Driutti  [  INFN, Sezione di Pisa, Pisa, Italy  V.
N. Duginov  Joint Institute for Nuclear Research, Dubna, Russia  M. Eads
Northern Illinois University, DeKalb, Illinois, USA  A. Edmonds  Boston
University, Boston, Massachusetts, USA  J. Esquivel  Fermi National
Accelerator Laboratory, Batavia, Illinois, USA  M. Farooq  University of
Michigan, Ann Arbor, Michigan, USA  R. Fatemi  University of Kentucky,
Lexington, Kentucky, USA  C. Ferrari  [  INFN, Sezione di Pisa, Pisa, Italy
M. Fertl  Institute of Physics and Cluster of Excellence PRISMA+, Johannes
Gutenberg University Mainz, Mainz, Germany  A. T. Fienberg  University of
Washington, Seattle, Washington, USA  A. Fioretti  [  INFN, Sezione di Pisa,
Pisa, Italy  D. Flay  Department of Physics, University of Massachusetts,
Amherst, Massachusetts, USA  S. B. Foster  Boston University, Boston,
Massachusetts, USA  H. Friedsam  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  N. S. Froemming  Northern Illinois University, DeKalb,
Illinois, USA  C. Gabbanini  [  INFN, Sezione di Pisa, Pisa, Italy  I. Gaines
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  M. D. Galati  [
INFN, Sezione di Pisa, Pisa, Italy  S. Ganguly  Fermi National Accelerator
Laboratory, Batavia, Illinois, USA  A. Garcia  University of Washington,
Seattle, Washington, USA  J. George  [  Department of Physics, University of
Massachusetts, Amherst, Massachusetts, USA  L. K. Gibbons  Cornell University,
Ithaca, New York, USA  A. Gioiosa  [  Università del Molise, Campobasso, Italy
K. L. Giovanetti  Department of Physics and Astronomy, James Madison
University, Harrisonburg, Virginia, USA  P. Girotti  INFN, Sezione di Pisa,
Pisa, Italy  W. Gohn  University of Kentucky, Lexington, Kentucky, USA  L.
Goodenough  Fermi National Accelerator Laboratory, Batavia, Illinois, USA  T.
Gorringe  University of Kentucky, Lexington, Kentucky, USA  J. Grange
University of Michigan, Ann Arbor, Michigan, USA  S. Grant  Argonne National
Laboratory, Lemont, Illinois, USA  Department of Physics and Astronomy,
University College London, London, United Kingdom  F. Gray  Regis University,
Denver, Colorado, USA  S. Haciomeroglu  [  Center for Axion and Precision
Physics (CAPP) / Institute for Basic Science (IBS), Daejeon, Republic of Korea
T. Halewood-Leagas  University of Liverpool, Liverpool, United Kingdom  D.
Hampai  INFN, Laboratori Nazionali di Frascati, Frascati, Italy  F. Han
University of Kentucky, Lexington, Kentucky, USA  J. Hempstead  University of
Washington, Seattle, Washington, USA  D. W. Hertzog  University of Washington,
Seattle, Washington, USA  G. Hesketh  Department of Physics and Astronomy,
University College London, London, United Kingdom  E. Hess  INFN, Sezione di
Pisa, Pisa, Italy  A. Hibbert  University of Liverpool, Liverpool, United
Kingdom  Z. Hodge  University of Washington, Seattle, Washington, USA  K. W.
Hong  University of Virginia, Charlottesville, Virginia, USA  R. Hong
University of Kentucky, Lexington, Kentucky, USA  Argonne National Laboratory,
Lemont, Illinois, USA  T. Hu  Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai, China  School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai, China  Y. Hu  [  [  School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai, China  M. Iacovacci  [
INFN, Sezione di Napoli, Naples, Italy  M. Incagli  INFN, Sezione di Pisa,
Pisa, Italy  P. Kammel  University of Washington, Seattle, Washington, USA  M.
Kargiantoulakis  Fermi National Accelerator Laboratory, Batavia, Illinois, USA
M. Karuza  [  INFN, Sezione di Trieste, Trieste, Italy  J. Kaspar  University
of Washington, Seattle, Washington, USA  D. Kawall  Department of Physics,
University of Massachusetts, Amherst, Massachusetts, USA  L. Kelton
University of Kentucky, Lexington, Kentucky, USA  A. Keshavarzi  Department of
Physics and Astronomy, University of Manchester, Manchester, United Kingdom
D. S. Kessler  Department of Physics, University of Massachusetts, Amherst,
Massachusetts, USA  K. S. Khaw  Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai, China  School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai, China  Z. Khechadoorian  Cornell University,
Ithaca, New York, USA  N. V. Khomutov  Joint Institute for Nuclear Research,
Dubna, Russia  B. Kiburg  Fermi National Accelerator Laboratory, Batavia,
Illinois, USA  M. Kiburg  Fermi National Accelerator Laboratory, Batavia,
Illinois, USA  North Central College, Naperville, Illinois, USA  O. Kim
University of Mississippi, University, Mississippi, USA  N. Kinnaird  Boston
University, Boston, Massachusetts, USA  E. Kraegeloh  University of Michigan,
Ann Arbor, Michigan, USA  V. A. Krylov  Joint Institute for Nuclear Research,
Dubna, Russia  N. A. Kuchinskiy  Joint Institute for Nuclear Research, Dubna,
Russia  K. R. Labe  Cornell University, Ithaca, New York, USA  J. LaBounty
University of Washington, Seattle, Washington, USA  M. Lancaster  Department
of Physics and Astronomy, University of Manchester, Manchester, United Kingdom
S. Lee  Center for Axion and Precision Physics (CAPP) / Institute for Basic
Science (IBS), Daejeon, Republic of Korea  B. Li  [  School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai, China  Argonne National
Laboratory, Lemont, Illinois, USA  D. Li  [  School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai, China  L. Li  [  [  School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai, China  I. Logashenko
[  Budker Institute of Nuclear Physics, Novosibirsk, Russia  A. Lorente Campos
University of Kentucky, Lexington, Kentucky, USA  Z. Lu  [  [  School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China  A. Lucà
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  G. Lukicov
Department of Physics and Astronomy, University College London, London, United
Kingdom  A. Lusiani  [  INFN, Sezione di Pisa, Pisa, Italy  A. L. Lyon  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  B. MacCoy  University
of Washington, Seattle, Washington, USA  R. Madrak  Fermi National Accelerator
Laboratory, Batavia, Illinois, USA  K. Makino  Michigan State University, East
Lansing, Michigan, USA  S. Mastroianni  INFN, Sezione di Napoli, Naples, Italy
J. P. Miller  Boston University, Boston, Massachusetts, USA  S. Miozzi  INFN,
Sezione di Roma Tor Vergata, Rome, Italy  B. Mitra  University of Mississippi,
University, Mississippi, USA  J. P. Morgan  Fermi National Accelerator
Laboratory, Batavia, Illinois, USA  W. M. Morse  Brookhaven National
Laboratory, Upton, New York, USA  J. Mott  Fermi National Accelerator
Laboratory, Batavia, Illinois, USA  Boston University, Boston, Massachusetts,
USA  A. Nath  [  INFN, Sezione di Napoli, Naples, Italy  J. K. Ng  Tsung-Dao
Lee Institute, Shanghai Jiao Tong University, Shanghai, China  School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China  H.
Nguyen  Fermi National Accelerator Laboratory, Batavia, Illinois, USA  Y.
Oksuzian  Argonne National Laboratory, Lemont, Illinois, USA  Z. Omarov
Department of Physics, Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, Republic of Korea  Center for Axion and Precision Physics
(CAPP) / Institute for Basic Science (IBS), Daejeon, Republic of Korea  R.
Osofsky  University of Washington, Seattle, Washington, USA  S. Park  Center
for Axion and Precision Physics (CAPP) / Institute for Basic Science (IBS),
Daejeon, Republic of Korea  G. Pauletta  [  Università di Udine, Udine, Italy
G. M. Piacentino  [  Università del Molise, Campobasso, Italy  R. N. Pilato
University of Liverpool, Liverpool, United Kingdom  K. T. Pitts  [  University
of Illinois at Urbana-Champaign, Urbana, Illinois, USA  B. Plaster  University
of Kentucky, Lexington, Kentucky, USA  D. Počanić  University of Virginia,
Charlottesville, Virginia, USA  N. Pohlman  Northern Illinois University,
DeKalb, Illinois, USA  C. C. Polly  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  J. Price  University of Liverpool, Liverpool, United
Kingdom  B. Quinn  University of Mississippi, University, Mississippi, USA  M.
U. H. Qureshi  Institute of Physics and Cluster of Excellence PRISMA+,
Johannes Gutenberg University Mainz, Mainz, Germany  S. Ramachandran  [
Argonne National Laboratory, Lemont, Illinois, USA  E. Ramberg  Fermi National
Accelerator Laboratory, Batavia, Illinois, USA  R. Reimann  Institute of
Physics and Cluster of Excellence PRISMA+, Johannes Gutenberg University
Mainz, Mainz, Germany  B. L. Roberts  Boston University, Boston,
Massachusetts, USA  D. L. Rubin  Cornell University, Ithaca, New York, USA  L.
Santi  [  Università di Udine, Udine, Italy  C. Schlesier  [  University of
Illinois at Urbana-Champaign, Urbana, Illinois, USA  A. Schreckenberger  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  Y. K. Semertzidis
Center for Axion and Precision Physics (CAPP) / Institute for Basic Science
(IBS), Daejeon, Republic of Korea  Department of Physics, Korea Advanced
Institute of Science and Technology (KAIST), Daejeon, Republic of Korea  D.
Shemyakin  [  Budker Institute of Nuclear Physics, Novosibirsk, Russia  M.
Sorbara  [  INFN, Sezione di Roma Tor Vergata, Rome, Italy  J. Stapleton
Fermi National Accelerator Laboratory, Batavia, Illinois, USA  D. Still  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  D. Stöckinger
Institut für Kern- und Teilchenphysik, Technische Universität Dresden,
Dresden, Germany  C. Stoughton  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  D. Stratakis  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  H. E. Swanson  University of Washington, Seattle,
Washington, USA  G. Sweetmore  Department of Physics and Astronomy, University
of Manchester, Manchester, United Kingdom  D. A. Sweigart  Cornell University,
Ithaca, New York, USA  M. J. Syphers  Northern Illinois University, DeKalb,
Illinois, USA  D. A. Tarazona  Cornell University, Ithaca, New York, USA
University of Liverpool, Liverpool, United Kingdom  Michigan State University,
East Lansing, Michigan, USA  T. Teubner  University of Liverpool, Liverpool,
United Kingdom  A. E. Tewsley-Booth  University of Kentucky, Lexington,
Kentucky, USA  University of Michigan, Ann Arbor, Michigan, USA  V. Tishchenko
Brookhaven National Laboratory, Upton, New York, USA  N. H. Tran  [  Boston
University, Boston, Massachusetts, USA  W. Turner  University of Liverpool,
Liverpool, United Kingdom  E. Valetov  Michigan State University, East
Lansing, Michigan, USA  D. Vasilkova  Department of Physics and Astronomy,
University College London, London, United Kingdom  University of Liverpool,
Liverpool, United Kingdom  G. Venanzoni  [  University of Liverpool,
Liverpool, United Kingdom  V. P. Volnykh  Joint Institute for Nuclear
Research, Dubna, Russia  T. Walton  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  A. Weisskopf  Michigan State University, East Lansing,
Michigan, USA  L. Welty-Rieger  Fermi National Accelerator Laboratory,
Batavia, Illinois, USA  P. Winter  Argonne National Laboratory, Lemont,
Illinois, USA  Y. Wu  Argonne National Laboratory, Lemont, Illinois, USA  B.
Yu  University of Mississippi, University, Mississippi, USA  M. Yucel  Fermi
National Accelerator Laboratory, Batavia, Illinois, USA  Y. Zeng  Tsung-Dao
Lee Institute, Shanghai Jiao Tong University, Shanghai, China  School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China  C.
Zhang  University of Liverpool, Liverpool, United Kingdom

(aug10)

###### Abstract

We present a new measurement of the positive muon magnetic anomaly,
$a_{\mu}\equiv(g_{\mu}-2)/2$, from the Fermilab Muon $g\\!-\\!2$ Experiment
using data collected in 2019 and 2020. We have analyzed more than 4 times the
number of positrons from muon decay than in our previous result from 2018
data. The systematic error is reduced by more than a factor of 2 due to better
running conditions, a more stable beam, and improved knowledge of the magnetic
field weighted by the muon distribution, $\tilde{\omega}^{\prime}_{p}$, and of
the anomalous precession frequency corrected for beam dynamics effects,
$\omega_{a}$. From the ratio $\omega_{a}/\tilde{\omega}^{\prime}_{p}$,
together with precisely determined external parameters, we determine
$a_{\mu}=116\,592\,057(25)\times 10^{-11}$ (0.21 ppm). Combining this result
with our previous result from the 2018 data, we obtain
$a_{\mu}\text{(FNAL)}=116\,592\,055(24)\times 10^{-11}$ (0.20 ppm). The new
experimental world average is $a_{\mu}(\text{exp})=116\,592\,059(22)\times
10^{-11}$ (0.19 ppm), which represents a factor of 2 improvement in precision.

Introduction.—A precise experimental measurement of the muon magnetic anomaly
$a_{\mu}$ provides a stringent test of the Standard Model (SM) as it can be
theoretically predicted with high precision. Any deviation between experiment
and theory may be a sign of physics beyond the SM. We report a new measurement
of $a_{\mu}$ using data collected in 2019 (Run-2) and 2020 (Run-3) by the Muon
$g\\!-\\!2$ Experiment at Fermilab. The data constitute a fourfold increase in
detected positrons compared to our previous measurement (Run-1) [1, 2, 3, 4].
Analysis and run condition improvements also lead to more than a factor of 2
reduction in the systematic uncertainties, surpassing the experiment’s design
goal [5].

Our Run-1 publications describe the principle of the experiment, previous
results, and experimental details [1, 2, 3, 4]. The experiment uses $3.1$
GeV/c polarized muons produced at the Fermilab Muon Campus [6]. Muons are
injected into a $7.112$-m radius storage ring that was moved, and
significantly upgraded, from the BNL experiment [7, 8]. Two key components of
the storage ring are kicker magnets that direct the injected muons onto the
central orbit of the storage ring [9] and electrostatic quadrupoles (ESQs)
that provide vertical focusing of the stored beam [10]. The anomalous spin
precession frequency $\omega_{a}$—the difference between the muon spin
precession frequency and the cyclotron frequency—is measured by recording the
time dependence of the number of high-energy positrons detected in a series of
calorimeters located on the inner radius of the storage ring [11]. The
magnetic field is mapped every few days using a trolley instrumented with
nuclear magnetic resonance (NMR) probes housing petroleum jelly [12]. These
probes are calibrated using a retractable water-based cylindrical probe [13].
This enables the expression of the magnetic field in terms of the precession
frequency of shielded protons in a spherical sample $\omega^{\prime}_{p}$, for
which the relation between precession frequency and magnetic field is
precisely known. After weighting for the muon spatial distribution, the
precession frequency is denoted $\tilde{\omega}^{\prime}_{p}$. Changes in the
field between trolley measurements are tracked using NMR probes embedded in
the vacuum chamber walls above and below the muon storage volume [3].
Dedicated instrumentation is used to measure transient magnetic fields caused
by the pulsing of the kickers and ESQs. The spatial distribution of the muon
beam within the storage ring as a function of time since injection is inferred
from positron trajectories recorded using two tracking detectors [14].

We incorporated major instrumental improvements with respect to Run-1.
Resistors in the high-voltage feedthroughs for the ESQ system that were
damaged in Run-1 were replaced before Run-2. This upgrade greatly improved
transverse beam stability. Thermal insulation was added to the storage ring
magnet before Run-2 to remove diurnal temperature variations. Increased
cooling power and improved air circulation in the experimental hall installed
before Run-3 reduced seasonal temperature variations. The magnitude and
reliability of the kicker field were improved between Run-1 and Run-2, and
again within Run-3. Because of these improvements, the data are analyzed in
three sets—Run-2, Run-3a, and Run-3b. A full description of the hardware
upgrades, operating conditions and analysis details will be provided in an in-
depth paper currently in preparation.

The data are blinded by hiding the true value of the calorimeter digitization
clock frequency. This blinding factor is different for Run-2 and Run-3.

We obtain the muon magnetic anomaly from 111We use the shielded proton-to-
electron magnetic moment ratio [16] and the electron $g$-factor [17, 18]
measurement. The CODATA-2018 result is used for the muon-to-electron mass
ratio [19], which is determined from bound-state QED theory and measurements
described in Ref. [20]. The QED factor $\mu_{e}(H)/\mu_{e}$ is computed by
theory with negligible uncertainty [19].

$\displaystyle
a_{\mu}=\frac{\omega_{a}}{\tilde{\omega}^{\prime}_{p}(T_{r})}\frac{\mu^{\prime}_{p}(T_{r})}{\mu_{e}(H)}\frac{{\mu_{e}(H)}}{\mu_{e}}\frac{m_{\mu}}{m_{e}}\frac{g_{e}}{2},$
(1)

where this experiment measures two frequencies to form the ratio
${\mathcal{R}}_{\mu}^{{}^{\prime}}=\omega_{a}/\tilde{\omega}^{\prime}_{p}(T_{r})$,
where $T_{r}=34.7$ ∘C is the temperature at which the shielded proton-to-
electron magnetic moment is measured [16]. The ratio of the measured
frequencies must be corrected for a number of effects, which shift the value
of ${\mathcal{R}}^{\prime}_{\mu}$ by $+$622 ppb in total. We write the ratio
in terms of measured quantities and corrections as

${\mathcal{R}}_{\mu}^{{}^{\prime}}\approx\frac{f_{\rm
clock}{\cdot}\,\omega_{a}^{m}\left(1+C_{e}+C_{p}+C_{pa}+C_{dd}+C_{ml}\right)}{f_{\rm
calib}\,{\cdot}\,\langle\omega^{\prime}_{p}(\vec{r})\times
M(\vec{r})\rangle(1+B_{q}+B_{k})}.$ (2)

The numerator consists of the clock-blinding factor $f_{\text{clock}}$, the
measured precession frequency $\omega_{a}^{m}$, and five corrections $C_{i}$
associated with the spatial and temporal motion of the beam. In the
denominator, we separate $\tilde{\omega}^{\prime}_{p}(T_{r})$ into the
absolute NMR calibration procedure (indicated by $f_{\rm calib}$) and the
magnetic field maps, which are weighted by the muon spatial distribution and
positron count [$\langle\omega^{\prime}_{p}(\vec{r})\times M(\vec{r})\rangle$,
where the average is over all points $\vec{r}$ within the storage region]. We
apply corrections $B_{i}$ to the magnetic field to account for two fast
magnetic transient fields that are synchronized to the muon storage period.
The uncertainties and correction values for the elements of Eq. 2 are shown in
Table 1.

Quantity | Correction | Uncertainty
---|---|---
(ppb) | (ppb)
$\omega_{a}^{m}$ (statistical) | - | 201
$\omega_{a}^{m}$ (systematic) | - | 25
$C_{e}$ | 451 | 32
$C_{p}$ | 170 | 10
$C_{pa}$ | $-27$ | 13
$C_{dd}$ | $-15$ | 17
$C_{ml}$ | 0 | 3
$f_{\text{calib}}\,{\cdot}\,\langle\omega^{\prime}_{p}(\vec{r})\times M(\vec{r})\rangle$ | - | 46
$B_{k}$ | $-21$ | 13
$B_{q}$ | $-21$ | 20
$\mu^{\prime}_{p}(34.7^{\circ})/\mu_{e}$ | - | 11
$m_{\mu}/m_{e}$ | - | 22
$g_{e}/2$ | - | 0
Total systematic for ${\mathcal{R}}^{\prime}_{\mu}$ | - | 70
Total external parameters | - | 25
Total for $a_{\mu}$ | 622 | 215
Table 1: Values and uncertainties of the ${\mathcal{R}}^{\prime}_{\mu}$ terms
in Eq. 2, and uncertainties due to the external parameters in Eq. 1 for
$a_{\mu}$. Positive $C_{i}$ increases $a_{\mu}$; positive $B_{i}$ decreases
$a_{\mu}$ (see Eq. 2). The $\omega_{a}^{m}$ uncertainties are decomposed into
statistical and systematic contributions. All values are computed with full
precision and then rounded to the reported digits.

Anomalous precession frequency $\omega_{a}^{m}$.—The time dependence of the
number of positrons from muon decays recorded by calorimeters in a storage
period is given by

$N(t)=N_{0}\eta_{N}(t)e^{-t/\gamma\tau_{\mu}}\\\
\times\left\\{1+A\eta_{A}(t)\cos\left[\omega_{a}^{m}t+\varphi_{0}+\eta_{\phi}(t)\right]\right\\},$
(3)

where $N_{0}$ is the normalization, $\gamma\tau_{\mu}$ is the time-dilated
muon lifetime ($\approx 64.4\,\upmu$s), $A$ is the average weak-decay
asymmetry, and $\varphi_{0}$ is the average phase difference between the muon
momentum and spin directions at the time of muon injection. The normalization,
asymmetry, and phase have time-dependent correction factors, $\eta_{N}$,
$\eta_{A}$, and $\eta_{\phi}$, that account for horizontal $(x)$ and vertical
$(y)$ beam oscillations, including $x$-$y$ coupling.

Nearly all parameters in Eq. 3 have some energy dependence, but it is
particularly strong for $N_{0}$ and $A$. We choose to combine the data in the
statistically optimal way of weighting each positron by its energy-dependent
asymmetry [21].

Seven different analysis groups perform independent extractions of
$\omega_{a}^{m}$ by a $\chi^{2}$ minimization. Each analysis team adds an
independent blind offset to their result in addition to the aforementioned
clock blinding. Two groups perform a new asymmetry-weighted ratio method by
subdividing the data and constructing a ratio that preserves statistical power
whilst reducing sensitivity to slow rate changes [2]. Each fit models the data
well, producing reduced $\chi^{2}$ values consistent with unity. Fourier
transforms of the fit residuals have no unexpected frequencies as shown in
Fig. 1. Scans of fit start and end times, positron energy, and individual
calorimeter stations show variation in $\omega_{a}^{m}$ consistent with
statistical expectations. After unblinding, the analysis groups determine
consistent values for $\omega_{a}^{m}$ and their independently estimated
systematic uncertainties. We combine the six asymmetry-weighted methods
equally for the final central value and verify the result with other less
sensitive methods.

Figure 1: Fourier transform of the residuals from a fit following Eq. 3
excluding $\eta_{N}$, $\eta_{A}$, and $\eta_{\phi}$ (red dashed line), and
from the full fit (black line). The peaks correspond to the missing betatron
frequencies and muon losses. Data are from the Run-3a data set. Inset:
corresponding asymmetry-weighted $e^{+}$ time spectrum (black line) with the
full fit function (red line) overlaid.

The extraction of $\omega_{a}^{m}$ is the only aspect of the result with
significant statistical uncertainty. The number of positrons above 1000 MeV
entering the asymmetry-weighted analyses increased from $15\times 10^{9}$ in
Run-1 to $71\times 10^{9}$ in Run-2/3. This reduces the statistical
uncertainty from 434 ppb to 201 ppb.

The systematic uncertainty on $\omega_{a}^{m}$ is also reduced by a factor
greater than 2 to 25 ppb. The largest reduction comes from our treatment of
pileup, when two positrons enter a calorimeter close in time and are not
separated by reconstruction algorithms. The difference in phase between two
lower-energy positrons and a single higher-energy positron, coupled with a
rate change over the storage period, can bias $\omega_{a}^{m}$. Each
calorimeter comprises a $9\times 6$ array of PbF2 crystals that are read out
independently. Improved clustering of crystal hits in the reconstruction
algorithms reduces the number of unresolved pileup events. In addition, some
groups adopted a method of overlaying waveforms rather than modeling the
reconstruction response to proximate crystal hits. The pileup uncertainty is
reduced from 35 ppb in Run-1 to 7 ppb in Run-2/3.

The other significant reduction is related to transverse beam oscillations.
The repair of the damaged ESQ resistors removes the majority of systematic
effects associated with large changes in the betatron frequencies over a muon
storage period. Additionally, the higher statistical precision allows for
improved empirical modeling of the decoherence envelope, enabling a wider
range of possibilities to be studied. The uncertainty drops from 38 ppb in
Run-1 to 21 ppb but remains the dominant systematic uncertainty for Run-2/3
for $\omega_{a}^{m}$.

Smaller reductions are achieved in the systematic uncertainties from a
residual early-to-late effect and the calorimeter gain correction (see Ref.
[2]), resulting in values of 10 ppb and 5 ppb, respectively.

Beam-dynamics corrections ${C_{i}}$.—Five corrections must be made to convert
the measured frequency $\omega_{a}^{m}$ into the anomalous precession
frequency $\omega_{a}$ in Eq. 1.

The largest correction is due to the electric fields of the ESQs. The effect
on $\omega_{a}$ is minimized by the choice of nominal muon momentum
$3.1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ [10]. The
electric field correction $C_{e}$ is required to account for the momentum
spread of the muon beam.

The muon momentum distribution is determined from the frequency distribution
and debunching rate of the injected beam using calorimeter data. Additionally,
the radial distribution of stored muons over a betatron period is obtained
from tracker data. The debunching analysis takes into account differences in
momentum spread along the injected bunch length that were not included in the
Run-1 analysis. Accounting for this difference and using complementary tracker
information reduces the $C_{e}$ uncertainty from 52 ppb in Run-1 to 32 ppb in
Run-2/3.

A pitch correction $C_{p}$ accounts for the reduction of $\omega_{a}$ caused
by vertical betatron oscillations. We use tracker data to extract the
distribution of vertical betatron amplitudes. The analysis is largely
unchanged from Run-1.

Any temporal change to the muon ensemble-average phase $\varphi_{0}$ in Eq. 3
will bias $\omega_{a}^{m}$. Correlations between the muon decay position and
$\varphi_{0}$ are accounted for through the phase acceptance correction
$C_{pa}$. This correction is evaluated by measuring the transverse beam
distribution throughout the storage period and using simulations to determine
the shifts in average phase at the calorimeters. The size of $C_{pa}$ is
determined by variation in the beam spatial distribution, which is
significantly reduced by replacing the damaged ESQ resistors, and the
associated systematic uncertainty is reduced from 75 ppb to 13 ppb.

Phase is also correlated with muon momentum owing to the momentum-dependent
phase advance in upstream beamline components [4]. A differential decay
correction $C_{dd}$ is required since the higher-momentum muons have a longer
boosted lifetime than lower-momentum muons. Three separate contributions to
the $C_{dd}$ correction yield a $-15$ ppb correction with 17 ppb uncertainty.
This correction was not applied to the Run-1 analysis.

Muons lost during a storage period can also lead to a change in the muon
momentum distribution. This effect has also been greatly reduced by replacing
the ESQ resistors. The correction factor $C_{ml}$ is evaluated as $0\pm 3$ ppb
compared to a correction in Run-1 of $-11\pm 5$ ppb.

Muon-weighted magnetic field $f_{\rm
calib}\,{\cdot}\,\langle\omega^{\prime}_{p}\times M\rangle$.—The increased
temperature stability in Run-2 and Run-3 due to thermal magnet insulation and
improved hall temperature stability results in a significantly more stable
magnetic field (RMS of $2\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ for Run-2
and $0.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ for Run-3). Additional
systematic measurements of the temperature dependence of the petroleum-jelly-
based NMR probes used in the trolley have reduced the systematic uncertainty
from trolley temperature changes to 9–15 ppb, depending on the data set.

The calibration procedure is improved for Run-2/3 compared to Run-1. Not only
are two calibrations performed, one for each run, but the process is also
optimized, resulting in reduced uncertainties. Small differences between the
sample volume in the calibration and the trolley probes are now corrected. In
addition, correction terms for the calibration probe are determined more
precisely. The overall systematic uncertainty from calibration is below 20
ppb.

As in Run-1, the magnetic field is parametrized in a multipole expansion in
transverse planes. In the current analysis, the number of terms used has
increases from $9$ to $12$, improving the fit quality. The dominant
uncertainties for the spatial field maps—each approximately
$20\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ in magnitude—arise from NMR
frequency extraction [22], the motion effects of the trolley, and the
estimated perturbation by the mechanism used to retract the trolley from the
storage region.

The systematic uncertainty of tracking the field in time using the fixed probe
data between two field maps is estimated by a Brownian bridge model tuned to
the observed mismatch from propagating one map to another. Because of the
larger number of field maps ($69$ in Run-2/3, compared to $14$ in Run-1), the
uncertainty from the field tracking is reduced to 10–16 ppb depending on the
data set. We discovered and corrected a tracking bias as a function of time
after the last magnet ramp-up (3–10 ppb).

The muon weighting follows the same approach used in Run-1. The more uniform
field reduces the uncertainties by around a factor of 2 to 7–13 ppb. The beam
distribution and azimuthally averaged magnetic field from Run-3b are shown in
Fig. 2.

Magnetic field transients $B_{i}$.—Transient magnetic fields synchronized with
beam injection are caused by the pulsing of ESQs and eddy currents in the
kickers. Both effects require corrections to the muon-weighted magnetic field
and are improved significantly compared to Run-1 by additional measurements.

In Run-1, the correction from the magnetic field transient due to vibrations
caused by ESQ pulsing, $B_{q}$, was only measured at a limited number of
locations around the ring. Using the same vacuum-sealed petroleum-jelly-based
NMR probe, but now on a nonconductive movable device, we mapped the transient
fields in the storage region between the ESQ plates azimuthally. This mapping,
in combination with improved methodology and repeated measurements over time,
leads to a reduction of the formerly dominant systematic effect by more than a
factor of 4 to $20\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

The effect of kicker-induced eddy currents $B_{k}$ was measured with the same
fiber magnetometer based on Faraday rotation in terbium gallium garnet
crystals used in Run-1 [3]. An improved setup, mainly to further reduce
vibrations, and more extensive measurements, reduces the uncertainty by around
a factor of $3$ to $13\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

Figure 2: Azimuthally averaged magnetic field contours overlaid on the time-
and azimuthally averaged muon distribution for the Run-3b data set. The field
is more uniform, and the increased kicker strength moves the beam closer to
the center than in Run-1.

Consistency checks.—In addition to the three data subsets described here, the
data are further subdivided based on a number of monitored experimental
parameters to examine possible correlations. These parameters include ring
temperature, magnet current, vacuum pressure, day/night, time since magnet
ramp-up and variables associated with the beam motion. We find no
statistically significant correlations between our results and any of these
parameters.

Calculation of $a_{\mu}$.—Table 2 contains the values of $\omega_{a}$ and
$\tilde{\omega}^{\prime}_{p}$, including all correction terms in Eq. 2, for
the three data subsets and their ratios ${\mathcal{R}}^{\prime}_{\mu}$. The
statistical uncertainty dominates in each subset, and as such, the
${\mathcal{R}}^{\prime}_{\mu}$ values are largely uncorrelated. Nearly all
systematic uncertainties that enter into ${\mathcal{R}}^{\prime}_{\mu}$ are
fully correlated across the subsets. Over the course of this analysis, three
small errors in the Run-1 analysis were identified 222We have updated the
Run-1 measurement with three corrections. The E-field correction $C_{e}$ was
inadvertently calculated using the blinded clock frequency. We have updated
this to use the unblinded clock frequency. This changes the Run-1 result by
$+19$ ppb. The phase acceptance correction $C_{pa}$ for Run-1b was
inadvertently swapped with the $C_{pa}$ for Run-1c. We have applied these
correctly now. This changes the Run-1 result by $+6$ ppb. A correction was
applied to the temperature dependence of the magnetic susceptibility of a
spherical water probe. The applied correction did not include an additional
term that accounts for the temperature dependence of the density of water. We
have now included this term. This changes the Run-1 result by $+2$ ppb. These
corrections are all positive and sum to a $+28$ ppb correction to the Run-1
result.. The total shift in the previous result due to these errors is $+28$
ppb, which has been applied to the value reported in this Letter.

The weighted-average value of the Run-2/3 data is
${\mathcal{R}}^{\prime}_{\mu}(\text{Run-2/3})$ = 0.00370730088(75)(26), where
the first error is statistical and the second is systematic. This value is in
excellent agreement with the adjusted Run-1 value
${\mathcal{R}}^{\prime}_{\mu}(\text{Run-1})$ = 0.0037073004(16)(6). Assuming
that the systematic errors are fully correlated between
${\mathcal{R}}^{\prime}_{\mu}(\text{Run-2/3})$ and
${\mathcal{R}}^{\prime}_{\mu}(\text{Run-1})$, we obtain the combined value of
${\mathcal{R}}^{\prime}_{\mu}(\text{Run-1/2/3})$ = 0.00370730082(68)(31).

From Eq. 1, we arrive at a new determination of the muon anomaly,

$a_{\mu}({\rm FNAL})=116\,592\,055(24)\times
10^{-11}~{}~{}~{}(\text{0.20\,ppm}),$

where the statistical, systematic, and external parameter uncertainties from
Table 1 are combined in quadrature. The combined (BNL and FNAL) experimental
(exp) average becomes

$a_{\mu}(\text{exp})=116\,592\,059(22)\times
10^{-11}~{}~{}~{}(0.19\,\text{ppm}).$

The results are displayed in Fig. 3.

A comprehensive prediction for the SM value of the muon magnetic anomaly was
compiled most recently by the Muon $g\\!-\\!2$ Theory Initiative in 2020 [24],
using results from Refs. [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44]. The leading-order hadronic contribution, known as
hadronic vacuum polarization (HVP), was taken from $e^{+}e^{-}\rightarrow$
hadrons cross-section measurements performed by multiple experiments. However,
a recent lattice calculation of HVP by the BMW Collaboration [45] shows
significant tension with the $e^{+}e^{-}$ data. In addition, a new preliminary
measurement of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ cross section from
the CMD-3 experiment [46] disagrees significantly with all other $e^{+}e^{-}$
data. There are ongoing efforts to clarify the current theoretical situation
[47]. While a comparison between the Fermilab result from Run-1/2/3 presented
here, $a_{\mu}({\rm FNAL})$, and the 2020 prediction yields a discrepancy of
$5.0\sigma$, an updated prediction considering all available data will likely
yield a smaller and less significant discrepancy.

Run |  $\omega_{a}/2\pi$ [Hz] |  $\tilde{\omega}^{\prime}_{p}/2\pi$ [Hz] | ${\mathcal{R}}^{\prime}_{\mu}\times 1000$
---|---|---|---
Run-1 |  |  | 3.7073004(17)
Run-2 | 229077.408(79) | 61790875.0(3.3) | 3.7073016(13)
Run-3a | 229077.591(68) | 61790957.5(3.3) | 3.7072996(11)
Run-3b | 229077.81(11) | 61790962.3(3.3) | 3.7073029(18)
Run-2/3 |  | 3.70730088(79)
Run-1/2/3 | 3.70730082(75)
Table 2: Measurements of $\omega_{a}$, $\tilde{\omega}^{\prime}_{p}$, and
their ratios ${\mathcal{R}}^{\prime}_{\mu}$ multiplied by 1000. The Run-1
value has been updated from [1] as described in the text.

In summary, we report a measurement of the muon magnetic anomaly to $0.20$ ppm
precision using our first three years of data. This is the most precise
determination of this quantity, and it improves on our previous result by more
than a factor of 2. Analysis of the remaining data from three additional years
of data collection is underway and is expected to lead to another factor of 2
improvement in statistical precision.

Figure 3: Experimental values of $a_{\mu}$ from BNL E821 [8], our Run-1 result
[1], this measurement, the combined Fermilab result, and the new experimental
average. The inner tick marks indicate the statistical contribution to the
total uncertainties.

We thank the Fermilab management and staff for their strong support of this
experiment, as well as our university and national laboratory engineers,
technicians, and workshops for their tremendous support. Greg Bock and Joe
Lykken set the blinding clock and diligently monitored its stability.

The Muon $g\\!-\\!2$ Experiment was performed at the Fermi National
Accelerator Laboratory, a U.S. Department of Energy, Office of Science, HEP
User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA),
acting under Contract No. DE-AC02-07CH11359. Additional support for the
experiment was provided by the Department of Energy offices of HEP and NP
(USA), the National Science Foundation (USA), the Istituto Nazionale di Fisica
Nucleare (Italy), the Science and Technology Facilities Council (UK), the
Royal Society (UK), the National Natural Science Foundation of China (Grants
No. 11975153 and No. 12075151), MSIP, NRF, and IBS-R017-D1 (Republic of
Korea), the German Research Foundation (DFG) through the Cluster of Excellence
PRISMA+ (EXC 2118/1, Project ID 39083149), the European Union Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie Grant
Agreements No. 101006726 and No. 734303, the European Union STRONG 2020
project under Grant Agreement No. 824093, and the Leverhulme Trust,
LIP-2021-01.

00footnotetext: $\diamond$ Deceased

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