muon lifetimes in the beam. We refer to this spread of decay rate as a
function of beam particle momentum as “differential decay.” The correction is
thus expressed as

$C_{dd}=-\frac{\Delta\omega_{a}}{\omega_{a}}=\frac{1}{\omega_{a}}\frac{d\phi_{0}}{dt}=\frac{1}{\omega_{a}}\frac{d\phi_{0}}{dp}\left(\frac{dp}{dt}\right)_{dd},$
(25)

where $\left(dp/dt\right)_{dd}$ is the temporal variation of the beam-averaged
momentum as muons decay in proportion to their time-dilated lifetimes,
$\gamma(p)\tau_{\mu}$. The evolution of the momentum distribution can be
approximated by

$\left(\frac{dp}{dt}\right)_{dd}\approx\frac{p_{0}}{\gamma_{0}\tau_{\mu}}\sigma_{\delta}^{2},$
(26)

where $\sigma_{\delta}^{2}$ is the variance of the fractional-momentum
distribution.

In addition to the initial $d\phi_{0}/dp$ from the upstream beamline
(described in Sec. V.4.1), there is an additional correlation that develops
from the non-symmetric kicker and longitudinal bunch structure during the
injection process. Because of differential decay, the ensemble average phase
slightly evolves throughout a fill, interpreted as a slight shift in the value
of $\omega_{a}^{m}$ from the precession data fits. On the basis of the orbital
coordinates ${\bf r}=\\{x,x^{\prime},y,y^{\prime},t_{0}\\}$ (see Table 9),

Table 9: Orbital variables ${\bf r}=\\{x,x^{\prime},y,y^{\prime},t_{0}\\}$. All the coordinates are relative to the reference axis at injection.  $r_{i}$ | Definition
---|---
$x,x^{\prime}$ | Spatial and angular offsets in radial phase space
$y,y^{\prime}$ | Spatial and angular offsets in vertical phase space
$t_{0}$ | Time relative to the nominal injection time.

the linear momentum dependence of
$\phi_{0}(x,x^{\prime},y,y^{\prime},t_{0};p)$ is expanded as:

$\displaystyle\frac{d\phi_{0}}{dp}$
$\displaystyle=\frac{\partial{\phi_{0}}}{\partial{x}}\frac{dx}{dp}+\frac{\partial{\phi_{0}}}{\partial{x^{\prime}}}\frac{dx^{\prime}}{dp}+\frac{\partial{\phi_{0}}}{\partial{y}}\frac{dy}{dp}$
$\displaystyle+\frac{\partial{\phi_{0}}}{\partial{y^{\prime}}}\frac{dy^{\prime}}{dp}+\frac{\partial{\phi_{0}}}{\partial{t_{0}}}\frac{dt_{0}}{dp}+\frac{\partial{\phi_{0}}}{\partial{p}}.$
(27)

Beam tracking studies of the stored muons at injection from gm2ringsim
simulations confirm the validity of this equality. From Eqs. (25) and (27), we
divide the $C_{dd}$ correction into three independent contributions based on
their physical origins, namely: the beamline, p-x correlation, and
$p\text{-}t_{0}$ correlation effects.

####  V.4.1 Beamline effect

The direct correlation between the $g\text{-}2$ phase and momentum drives the
beamline effect:

$C_{dd}^{bl}=\frac{1}{\omega_{a}}\frac{\partial\phi_{0}}{\partial
p}\frac{dp}{dt}\approx\frac{\sigma_{\delta}^{2}}{\omega_{a}\gamma_{0}\tau_{\mu}}\frac{\partial\phi_{0}}{\partial\delta}.$
(28)

After four revolutions of the muon beam around the Delivery Ring (DR) at
Fermilab [44], the magnetic field of the bending dipole magnets contribute to
a momentum-dependent angle advance between the muon spin and momentum by
$\Delta\phi\approx 8\pi a_{\mu}\gamma$, which leads to
$|\Delta\phi/\Delta\delta|=8.6\,\mathrm{mrad}/(\%\delta)$ [8]. For Run-1, beam
tracking simulations and direct measurements of the correlation determined
$|\partial\phi_{0}/\partial\delta|$ at beam injection to be $10\pm
1.6\,\mathrm{mrad}/(\%\delta)$; a result in agreement with the DR-only
contribution $|\Delta\phi/\Delta\delta|$.

The first step to calculate $C_{dd}^{bl}$ is to recreate a
$\phi_{0}\text{-}\delta$ distribution of the stored muons at $t=0$ for each
data subset from a bivariate normal distribution. The correlation is defined
from the $\partial\phi_{0}/\partial\delta$ measurements and the momentum
projection is scaled with the corresponding momentum distributions, determined
in the electric-field correction analysis. Then, a Monte Carlo signal with a
simplified five-parameter version of Eq. (LABEL:equation:omegaAfit) is
prepared out of the $\phi_{0}\text{-}\delta$ distribution, where the
differential decay $e^{-\frac{t}{\gamma(t)\tau}}$ transforms the distribution
over time. Finally, we fit the Monte Carlo signal to extract the shift in
$\omega_{a}^{m}$ due to differential decay.

The difference between the results from the steps described above and Eq. (28)
is negligible. The main purpose of the step-by-step procedure is to test the
sensitivity of $C_{dd}^{bl}$ to two possible systematic effects: correlations
of $\gamma$ and $\phi_{0}$ with the muon-momentum dependence of (a) the
asymmetry, $A$, and (b) emitted positrons, $N$, based on the leading-order
Michel spectrum. Because these effects produce systematic uncertainties below
$2\,\mathrm{ppb}$, we assign a conservative upper limit of $3\,\mathrm{ppb}$
to the differential-decay beamline correction. Table 10 summarizes the
evaluation of $C_{dd}^{bl}$ for all the datasets based on the weighted results
of the procedure for each data subset. The larger $\phi_{0}\text{-}\delta$
correlation induced by the cooling wedge increases the beamline effect in
Run-3a and Run-3b.

####  V.4.2 $p\text{-}x$ effect

At the exit of the inflector, the Muon Campus delivers a muon beam where the
only sizable momentum-phase correlation is the one that is measured for the
differential-decay beamline effect (i.e., $\partial\phi_{0}/\partial\delta$).
This specific feature of the injected beam, which tracking simulations
corroborate, is perturbed due to momentum-orbit correlations that develop
during beam injection, where the radial and vertical phase-space coordinates
$x,x^{\prime},y\,\mathrm{and}\,y^{\prime}$ are the “orbit” coordinates in this
context (see Table 9).

The beam injection is optimized to accommodate the radial beam within the
storage ring admittance. The process introduces correlations between the
radial phase coordinates and momentum, $dx^{\prime}/d\delta$ and $dx/d\delta$,
of the stored muons at injection time ($t=0$). The resulting differential-
decay contribution from injection is hence expressed as

$C_{dd}^{p\text{-}x}=\frac{\sigma_{\delta}^{2}}{\omega_{a}\gamma_{0}\tau_{\mu}}\left(\frac{\partial\phi_{0}}{\partial
x}\frac{dx}{d\delta}+\frac{\partial\phi_{0}}{\partial
x^{\prime}}\frac{dx^{\prime}}{d\delta}\right).$ (29)

While the pion beam decays into muons as it is transported down the muon-
production beamline, the angle $\phi$ between each muon’s momentum in the lab
frame and its spin direction depends on the parental pion momentum, $p_{\pi}$,
as

$\sin\left(\phi\right)\approx\frac{2m_{\mu}}{m_{\pi}^{2}-m_{\mu}^{2}}p_{\pi}\sin\theta,$
(30)

where $\theta$ is the angle between the muon momentum and the pion direction
in the lab frame. In our case, as muons are emitted in the lab frame in a
forward cone of semi-angle $\theta_{max}\approx 12.7\,\mathrm{mrad}$, Eq. (30)
is further simplified to

$\sin\left(\phi\right)\approx 78.8x_{0}^{\prime},$ (31)

where $x_{0}^{\prime}$ is the phase-space coordinate of the muon’s trajectory
at birth. Therefore, a nonzero correlation $\partial\phi_{0}/\partial
x_{0}^{\prime}$ exists, which yields nonzero $\phi_{0}\text{-}x$ and
$\phi_{0}\text{-}x^{\prime}$ correlations in Eq. (29) as muons subsequently
execute betatron oscillations and cross bending magnets along the muon-
production beamline. As shown in Eq. (29), these spin-orbit correlations
couple with $dx/d\delta$ and $dx^{\prime}/d\delta$ to alter the original
phase-momentum relationship before injection.

With beam tracking simulations using the BMAD and gm2ringsim injection models
[8], we calculate the beam correlations necessary to determine the
differential-decay $p\text{-}x$ effect. Figure 17 shows the radial coordinate
versus fractional momentum of the stored muons at injection, which is the
dominant momentum-orbit correlation in $C_{dd}^{p\text{-}x}$.

Figure 17: Average radial coordinate $\langle x\rangle$ of the beam
distribution per momentum offset at injection, from a gm2ringsim tracking
simulation of stored muons. In this example, a nominal configuration of the
injection parameters is implemented in the simulation. The $dx/d\delta$
correlations to quantify $C_{dd}^{p\text{-}x}$ are obtained from these
tracking simulation results.

With Eq. (29) and the simulation results, the $p\text{-}x$-effect contribution
to the differential-decay correction for Runs-2/3 is

$C_{dd}^{p\text{-}x}=-5\pm 6\,\mathrm{ppb}.$ (32)

The uncertainty accounts for several simulation configurations in view of
injection parameter configurations within operational ranges (i.e., inflector
current, beam distributions at the inflector exit, and injection kicker
strengths, pulse shapes, and relative timings).

####  V.4.3 $p\text{-}t_{0}\,$effect

A muon’s spin starts to precess as soon as it enters the storage ring. Typical
muon bunches are $120\,\mathrm{ns}$ long; the spin of muons at the head of the
bunch accumulates an additional precession
$\Delta\phi\approx(120\,\mathrm{ns})\omega_{a}$ relative to muons at the tail
while they enter the ring. This longitudinal phase variation across the bunch,
together with the $t_{0}$-dependent momentum acceptance induced by the time
dependence of the injection kicker, produce the momentum-time effect:

$C_{dd}^{p\text{-}t_{0}}=\frac{1}{\omega_{a}}\frac{\partial\phi_{0}}{\partial
t_{0}}\frac{dt_{0}}{dp}\frac{dp}{dt}\approx\frac{\sigma_{\delta}^{2}}{\gamma_{0}\tau_{\mu}}\frac{dt_{0}}{d\delta}.$
(33)

The method to evaluate $C_{dd}^{p\text{-}t_{0}}$ is similar to the procedure
used for the differential-decay beamline effect explained in Sec. V.4.1,
except for the first step where the muon distributions are prepared from the
momentum-time distributions of the electric-field correction analysis; the
time coordinates are transformed to relative spin phase advance via
$\Delta\phi_{0}=\omega_{a}t_{0}$ (Fig. 18 shows one example).

Figure 18: Momentum-phase distribution from the momentum-time distribution for
one bunch in data subset 2C. The gray markers are the averaged relative spin
phases per fractional momentum, exhibiting the correlation that drives
$C_{dd}^{p\mathrm{-}t_{0}}$.

The $C_{dd}^{p\text{-}t_{0}}$ is evaluated at the bunch level because each of
the bunches in a sequence has characteristically different longitudinal
intensity profiles. The results are then combined to obtain the corrections
per data subset, as shown in Fig. 19.

Figure 19: Momentum-time differential decay correction
$C_{dd}^{p\text{-}t_{0}}$ per data subset (black). In gray crosses, correction
predictions where the ratio between $p\text{-}t_{0}$ correlations and kicker
timing offsets relative to beam injection, based on gm2ringsim beam tracking
simulations, is scaled in proportion to the per-data-subset kicker timing
offsets.

The final momentum-time corrections per Run are summarized in Table 10. The
effect in Run-2 and Run-3a is consistent with zero, whereas a more constant
timing offset between the kicker pulse and injection time leads to the non-
zero correction for Run-3b.

To assess the uncertainties in this correction, we prepare 100 momentum-time
distributions, each seeded by different initial conditions in the fitting
method for the electric-field correction. The $C_{dd}^{p\text{-}t_{0}}$
correction is thereafter calculated for each seed, where the standard
deviation for each set of bunches is treated as the uncertainty. The
uncertainties per data subset are the correlated combination of the
uncertainty from each bunch. An additional uncertainty, added in quadrature
with the previously explained errors, is assigned from the RMS of all the
mean-subtracted data subsets to account for the intrinsic ambiguity in the
momentum-time distributions used to calculate the $p\text{-}t_{0}$ effect.

####  V.4.4 Total effect

The total differential decay correction is the combination of the beamline,
$p\text{-}x$, and $p\text{-}t_{0}$ effects:

$C_{dd}=C_{dd}^{bl}+C_{dd}^{p\text{-}x}+C_{dd}^{p\text{-}t_{0}},$ (34)

summarized in Table 10. To first order, these are uncorrelated; their physical
origin is independent of each other. Therefore, the errors of each individual
differential-decay effect are added in quadrature.

Table 10:  Differential decay corrections (ppb) for Run-2, Run-3a and Run-3b. The corresponding uncertainties (ppb) are enclosed in parentheses. $C_{dd}$ | Run-2 | Run-3a | Run-3b | Section
---|---|---|---|---
Beamline | -12(3) | -17(3) | -20(3) | V.4.1
$p\text{-}x$ | -5(6) | -5(6) | -5(6) | V.4.2
$p\text{-}t_{0}$ | 6(15) | 0(16) | 23(17) | V.4.3
Total | -11(16) | -22(17) | -2(18) | V.4

###  V.5 Phase acceptance correction

The detected $g\\!-\\!2$ phase, as measured by the calorimeter detectors,
varies over time as a function of the transverse beam coordinates of the muons
$(x,y)$. The beam transverse distribution changes with time and creates in-
fill variations of the detected phase that could affect the fit model for
$\omega_{a}^{m}$, where the phase is expected to be time-independent. For this
detector-acceptance effect, we introduce the phase acceptance correction,
$C_{pa}$.

The time-dependent phase $\phi_{\mathrm{pa}}(t)$ is computed by averaging the
measured phase as a function of transverse coordinates ($x$,$y$) that are
obtained from gm2ringsim. The time dependence of the transverse beam
coordinates is extracted from tracker beam profiles $M^{T}(x,y,t)$, which
generates a time-dependent phase by virtue of the correlation between the
phase and the beam transverse distribution. Figure 20 is a transverse map of
$\phi_{\mathrm{pa}}(x,y)$ averaged over the azimuth, obtained by fitting the
asymmetry-weighted histogram used to extract $\omega_{a}^{m}$ (see Sec. IV.1).

Figure 20: Simulated azimuthally averaged phase maps for the asymmetry-
weighted analysis. The coupling between the overall quadratic-like detected
phase acceptance in the vertical direction and the in-fill reduction in
vertical beam width is the most significant effect on $C_{pa}$.

The tracker stations measure the $M^{T}(x,y,t)$ distribution at two locations
around the ring, but the extraction of the measured $\omega_{a}^{m}$ is
performed by calorimeters at 24 azimuthal locations. Therefore, we extrapolate
the $M^{T}(x,y,t)$ profiles around the ring at using gm2ringsim and COSY
INFINITY beam dynamics simulations. Vertical ($y(\varphi,t)$) and radial
($x(\varphi,t)$) muon coordinates at any given azimuthal position $\varphi$
are calculated by scaling the transverse coordinates from tracker measurements
with the mean and width values from simulated beam distributions as

$y(\varphi,t)=y_{\mathrm{trk}}(t)\frac{y^{\mathrm{rms}}(\varphi,t)}{y_{\mathrm{trk}}^{\mathrm{rms}}(t)},$
(35)

for the vertical width, and

$\displaystyle x(\varphi,t)=$
$\displaystyle\frac{x^{\mathrm{rms}}(\varphi,t)}{x^{\mathrm{rms}}_{\mathrm{trk}}(t)}\cdot[x_{trk}(t)-\bar{x}_{\mathrm{trk}}(t)]+\bar{x}(\varphi,t),$
(36)

for the radial motion of the beam, where $(x^{\mathrm{rms}},y^{\mathrm{rms}})$
are the root mean squares of the transverse beam distributions and $\bar{x}$
is the radial distribution average. The quantities from simulated
distributions on the right-hand side in Eq. (36) and Eq. (35) do not have
subscripts, whereas tracker-based values are denoted with the subscript “trk.”
By modifying the $M^{T}(x,y,t)$ distribution using Eq. (36) and Eq. (35), we
obtain the spatial and time distribution of the muons $M^{c}(x,y,t)$ at each
calorimeter location. Combining the simulated maps with the muon
distributions, a time-dependent phase $\phi^{c}_{pa}(t)$ can be computed for
each calorimeter using the following weighted sum:

$\displaystyle\phi_{\mathrm{pa}}^{c}(t)=\mathrm{arctan}$
$\displaystyle\left[\frac{\sum_{ij}M^{c}(x_{i},y_{j},t)\cdot\varepsilon^{c}(x_{i},y_{j})}{\sum_{ij}M^{c}(x_{i},y_{j},t)\cdot\varepsilon^{c}(x_{i},y_{j})}\right.$
(37) $\displaystyle\left.\frac{\cdot
A^{c}(x_{i},y_{j})\cdot\sin[\phi^{c}_{\mathrm{pa}}(x_{i},y_{j})]}{\cdot
A^{c}(x_{i},y_{j})\cdot\cos[\phi^{c}_{\mathrm{pa}}(x_{i},y_{j})]}\right],$

where acceptance, asymmetry and phase maps for a calorimeter “$c$” are
represented by $\varepsilon^{c}$, $A^{c}$ and $\phi_{\mathrm{pa}}$,
respectively.

The calculation of the phase acceptance correction is done by comparing
$\omega_{a}^{m}$ to the fit of the simulated data. A histogram is generated
for each calorimeter and for each parameter of the $\omega_{a}^{m}$ fit,
including the modified ${g-2}$ phase obtained by fitting
$\phi_{\mathrm{pa}}^{c}(t)$. Simulated data (produced using values extracted
from histograms) are fitted with a constant phase. The difference between
$\omega_{a}^{m}$ and the fit result determines $C_{pa}$ for a given
calorimeter.

Figure 21: Calculation of $\phi_{\mathrm{pa}}$ for calorimeter 13 in data
subset 1D (gray) and data subset 2C (blue) using data from the tracker station
at 180∘. The shown fit function is of the form $\phi+\Delta\phi\cdot
e^{{(-t/\tau_{\phi})}}$.

Figure 21 shows the $\phi_{\mathrm{pa}}$ time evolution for a Run-2 data
subset, superimposed with one from Run-1d for comparison. After replacing the
damaged resistors of the ESQ system from Run-1, the variation of the phase is
highly reduced during Run-2/3, and the $C_{\mathrm{pa}}$ is hence smaller. The
central values of the correction are calculated by taking the average of the
results from all calorimeters. The central values are shown in Table 11, where
further improvement on the effect is observable in Run-3 with respect to
Run-2. This outcome is due to the improved stability of the beam motion thanks
to more optimized kicker settings and a better temperature stability of the
main magnet. The evaluations of the statistical and systematic uncertainties
are also reported in Table 11. The statistical uncertainty, which ranges from
2.0 to $7.8\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$, originates from the
limited number of tracks from the $M^{T}(x,y,t)$ collected by tracker
stations. The sources of systematic uncertainty can be divided into three main
groups. The first one stems from imperfect knowledge of the straw trackers’
alignment, resolution, and acceptance, which directly affects the measured
distribution $M^{T}(x,y,t)$. Next are the uncertainties associated with the
estimation of the phase, asymmetry, and acceptance maps in Eq. (37) estimated
using gm2ringsim. Lastly, the calculation utilizes beam dynamics functions
obtained by simulation to extract the calorimeter $M^{c}(x,y,t)$ distribution
from the tracker-based $M^{T}(x,y,t)$. Uncertainties are estimated by
calculating $C_{pa}$ while varying the beta functions and magnetic field
within expected deviations based on the measurements.

Table 11: Values of the phase-acceptance correction $C_{\mathrm{pa}}$ (ppb) and their statistical, systematic, and total uncertainties (ppb) for each of the Run-2/3 datasets. Quantity | Run-2 | Run-3a | Run-3b
---|---|---|---
Correction | -50 | -16 | -13
Statistical Unc. | 9 | 2 | 3
Systematic Unc. |  |  |
Tracker and CBO | 13 | 8 | 7
Phase maps | 13 | 3 | 3
Beam dynamics | 5 | 3 | 2
Total uncertainty | 21 | 9 | 8

###  V.6 Summary

The beam dynamics corrections and their uncertainties for Run-2/3 are listed
in Table 12.

Table 12:  Values and uncertainties of the beam dynamics corrections (ppb) for Run-2/3. Quantity | Correction | Uncertainty
---|---|---
$C_{e}$ | 451 | 32
$C_{p}$ | 170 | 10
$C_{ml}$ | 0 | 3
$C_{dd}$ | -15 | 17
$C_{pa}$ | -27 | 13
Total | 580 | 40

Each individual correction is highly correlated for different datasets, and
therefore, the per-dataset combination of the uncertainties is fully
correlated. To obtain the total beam dynamics correction uncertainty, we add
the uncertainties of all the individual corrections in quadrature because they
are uncorrelated.

A combination of improvements in the experimental setup (listed in Sec. III.2)
and analysis reduced both the beam dynamics correction magnitudes and
uncertainties in Run-2/3 compared to Run-1. The replacement of the ESQ high-
voltage resistors damaged in Run-1 leads to a smaller and more precise
determination of $C_{pa}$. The muon loss correction is negligible thanks to
the significantly reduced mechanical muon loss rates. With the stronger
injection kickers in Run-3b, the more symmetric momentum distribution requires
a lower electric-field correction, whereas the determination of the momentum-
time beam correlations at injection, as well as an independent reconstruction
of the momentum distribution based on the tracker detector data, reduce the
uncertainty of $C_{e}$. While the differential decay correction was not
included in Run-1, the momentum-time correlations analysis for the electric-
field correction allowed us to fully quantify this correction in Run-2/3.

##  VI Magnetic field measurement

In Eq. (2), $\tilde{B}$, the magnetic field averaged over space and time by
the muons, is expressed as the precession frequency of protons in a spherical
water sample at a reference temperature: $\tilde{\omega}^{\prime}_{p}(T_{r})$.
In this notation, the tilde indicates the muon weighting, and the prime
indicates that the proton magnetic moment is shielded in H2O. The reference
temperature is $T_{r}=$34.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$$, the
temperature at which the shielded proton magnetic moment was measured relative
to the bound-state electron in hydrogen [45]. This section describes the
measurements and analyses leading to $\tilde{\omega}_{p}^{\prime}$, which
follows from the general approach of Run-1 [7].

###  VI.1 Magnetic field measurement principle

The muon-weighted magnetic field is derived from time-dependent maps of the
magnetic field in the muon storage region $\omega_{p}^{\prime}(x,y,\phi,t)$.
The maps are derived from measurements by a set of NMR probes in a trolley
that is pulled through the storage ring every two to three days and maps the
full circumference in about 70 minutes. The field is mapped at the 17 NMR-
probe positions ($x$, $y$) ($x=0$ at $r=R_{0}$) and about 9000 azimuthal
positions $\phi$. Corrections for differences of the physical ring
configuration and from magnetic field transients from the kickers and ESQs,
which are not operating during the trolley measurements, are discussed in
section VI.7.

The trolley’s NMR probes, described in [7], contain samples of proton-rich
petroleum jelly (petrolatum). The trolley probes are calibrated to account for
the sample and the different magnetic environment due to magnetic
perturbations from the aluminum shell, the wheels of the trolley, the other
probes, and other trolley components, including the electronics, cables, etc.
A dedicated calibration magnetometer was used to correct each probe to the
frequency that would be measured with a spherical water sample at temperature
$T_{r}$. The details of this calibration procedure are described in sections
VI.2 and VI.3.

The time-dependent trolley maps are parameterized as

$\omega_{p}^{\prime}(x,y,\phi,t)=\sum_{i=1}^{N_{\mathrm{max}}}m_{i}(\phi,t)f_{i}(r,\theta)\,,$
(38)

where

$f_{i}(r,\theta)=\left\\{\begin{array}[]{ll}1&\text{for }i=1,\\\
\left(\frac{r}{r_{0}}\right)^{\frac{i}{2}}\cos\left(\frac{i}{2}\theta\right)&\text{for
even }i>1,\\\
\left(\frac{r}{r_{0}}\right)^{\frac{i-1}{2}}\sin\left(\frac{i-1}{2}\theta\right)&\text{for
odd }i>1.\\\ \end{array}\right.$ (39)

Here $r_{0}=$4.5\text{\,}\mathrm{c}\mathrm{m}$$ is a reference radius,
$x=r\cos(\theta)$, $y=r\sin{(\theta)}$. The $\cos(\theta)$ and $\sin(\theta)$
terms are referred to as normal and skew moments, and $t$ is the time of the
measurement. The moments $m_{i}(\phi,t)$ are determined from fits of the 17
trolley-probe frequencies at the time $t$ when the trolley is at the position
$\phi$. The parameterization in Eq. (38) is motivated by solutions to a 2-D
Laplace equation and is analogous to a 2-D Taylor expansion around
$(x,y)=(0,0)$ with constraints. The 2-D Laplace-equation solution is strictly
valid only if $B$ has no azimuthal dependence; the impact and validation of
this parameterization and the effect of truncating the parameterization at
$N_{\mathrm{max}}$ are discussed in Sec. 22.

The time-dependence of the moments $m_{n}(\phi,t)$ between trolley runs is
estimated by interpolation making use of a set of 378 NMR magnetometers (fixed
probes) mounted on the outside of the vacuum chambers at 72 azimuthal
positions, called stations. Each fixed probe is read out with a rate of
${\sim}$0.5\text{\,}\mathrm{H}\mathrm{z}$$. Each station has either four or
six NMR probes, half above and half below the storage region, and can
interpolate the magnetic field moments up to $i=4$ or $i=5$, respectively. As
a trolley run proceeds, the moments calculated from the fixed probes at the
stations near the trolley are set equal to the corresponding moments
calculated from the trolley probes at that time, which we call “tying”.
Moments up to $n=4,5$ are tracked with the fixed probes by interpolating in
time between two trolley runs, and higher-order moments are interpolated
assuming linear time dependence. The limitation of this interpolation results
in “tracking errors” that are estimated from the difference between the
moments predicted by the fixed probes and the moments actually measured by the
subsequent trolley run. Studies with different intervals between trolley runs
and at different times after the magnet was ramped to the nominal operating
field were used to reduce the tracking errors and uncertainties.

The muon-weighted field is

$\tilde{\omega}_{p}^{\prime}=\frac{\int\omega_{p}^{\prime}(x,y,\phi,t)M(x,y,\phi,t)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}\phi\,\mathrm{d}t}{\int
M(x,y,\phi,t)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}\phi\,\mathrm{d}t},$ (40)

with the muon distribution $M(x,y,\phi,t)$ determined by a combination of
measurements with the trackers and modeling of beam dynamics (Sec. VI.6.1).
Expanding $M(x,y,\phi,t)$ in the basis introduced in Eq. (38), the muon
weighted azimuth- and time-dependent magnetic field is

$\tilde{\omega}_{p}^{\prime}(\phi,t)=\sum_{i}m_{i}(\phi,t)k_{i}(\phi,t)\,,$
(41)

where

$k_{i}(\phi,t)=\frac{\int M(x,y,\phi,t)f_{i}(x,y)dx\,dy}{\int
M(x,y,\phi,t)dx\,dy}\,.$ (42)

The time-dependent azimuthally averaged field is

$\tilde{\omega}_{p}^{\prime}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{\omega}_{p}^{\prime}(\phi,t)\,\mathrm{d}\phi,$
(43)

which is weighted by the number of detected muon decays and time averaged over
few day intervals.

###  VI.2 Absolute calibration with a high-purity water probe

Each trolley probe reading is corrected for the field perturbations caused by
the trolley components to the NMR frequency expected from a bare spherical
water sample at $34.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$. This is done
using an H2O absolute calibration probe installed in the $g\\!-\\!2$ storage
ring. The calibration probe for Run-2 and Run-3 was similar to that described
in detail in [16, 7].

Corrections must be applied to the measured calibration probe NMR frequencies
to those expected from a bare spherical water sample at $T_{r}$. Corrections
to the measured calibration frequency are listed in Table 13 and described
below. These corrections were cross-checked with respect to a 3He magnetometer
in a dedicated high uniform $1.45\text{\,}\mathrm{T}$ solenoid and with
simulations. All corrections are expressed as fractions of the measured NMR
frequency, i.e., $\omega^{\text{corr}}=\omega^{\text{meas}}(1+\delta)$, where
$\omega^{\text{corr}}$ is the frequency corrected for the effect $\delta$. For
corrections $\ll$1 (the largest is
$1.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$), the combination of two
corrections is
$(1+\delta_{a})(1+\delta_{b})\approx(1+\delta_{a}+\delta_{b}+\mathcal{O}(\delta^{2}))$;
only the first-order corrections are applied.

Sample-shape correction $\mathbf{\delta^{b}}$ The calibration probe consists
of a cylindrical sample filled with high-purity water. The temperature-
dependent correction to a spherical sample is

$\delta^{b}(T)=\chi(T_{n})(\epsilon-1/3),$ (44)

where $\chi(T_{n})$ is the susceptibility at the temperature of the
calibration probe for calibration of probe $n$, and
$\epsilon=0.4999(0,-0.0003)$ for the finite cylindrical sample, which was
calculated in closed form from [46] and confirmed by numerical simulation
($\epsilon=1/2$ for an infinite cylinder).

The temperature-dependent volume susceptibility is

$\chi_{V}(T)=\chi_{V}($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)\times\left[\frac{\chi_{m}(T)}{\chi_{m}($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}\right]\times\left[\frac{\rho(T)}{\rho($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}\right],$
(45)

where $\chi_{V}($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)=-9.056\times
10^{-6}$ is the value recommended by CODATA [47] with $3\times 10^{-8}$
uncertainty due to additional measurements at unspecified temperatures [48].
We use the ratio of mass susceptibilities from [49]:

$\displaystyle\frac{\chi_{m}(T)}{\chi_{m}($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}$
$\displaystyle=$
$\displaystyle\frac{\chi_{m}(T)}{\chi_{m}($20\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}\frac{\chi_{m}($20\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}{\chi_{m}($22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)}$
$\displaystyle\approx 1$ $\displaystyle+$ $\displaystyle 1.3881\times
10^{-4}(T-$22\text{\,}\mathrm{\SIUnitSymbolCelsius}$)$ (46) $\displaystyle+$
$\displaystyle\mathcal{O}\left((T-$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$)^{2}\right).$

The temperature-dependent density $\rho(T)$ from [50] was used, because that
is what was used in the analysis by [49].

Material effects $\mathbf{{\delta^{s}}}$ The calibration probe consists of the
sample contained in a glass cylinder NMR sample tube, a concentric glass
cylinder holding the NMR coil wires, a concentric aluminum cylinder shell, end
caps, the temperature sensor, tuning capacitors, connectors, and mounting
fixtures.

Due to their finite magnetic susceptibility, each of these components becomes
magnetized by the external $1.45\text{\,}\mathrm{T}$ field, and the resulting
magnetization contributes to the field measured by the probe. The contribution
depends on the orientation (roll and pitch) of the probe with respect to the
vertical magnetic field. The approximate cylindrical symmetry of the probe
construction mitigates these effects, and a combination of direct measurements
of intrinsic-probe effects $\delta^{s}$, and simulations specific to the
configuration in the $g\\!-\\!2$ storage ring are used to determine the
remaining material corrections. Additionally, the high-permeability pole
pieces of the storage-ring magnet act as magnetic mirrors that create images
of the magnetized calibration-probe components, leading to a correction
$\delta^{s,\text{img}}$ that depends on the probe position.

Sample (im)purity $\mathbf{\delta^{P}}$ Potential impurities, in particular,
dissolved paramagnetic O2 and salts, in the water sample could lead to a shift
of the NMR frequency. Degassed ultra-pure (ASTM Type-1) water from several
vendors was used, with no observed variation within an uncertainty of
$2\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. A variety of additional tests were
performed in which the glass water sample tube was rotated, and different
sample tubes were used. No systematic shifts were observed.

Magnetization dependent effects $\mathbf{\delta^{RD}}$ and
$\mathbf{\delta^{d}}$ The sample magnetization $\vec{M}=\chi_{\rm
H_{2}O}\vec{B}$ can lead to two shifts. Radiation damping is the result of the
oscillating current in the NMR coil that rotates the magnetization toward the
external magnetic field. This leads to a time-dependent precession frequency
shift $\delta^{RD}$ that depends on the magnetization along the magnetic
field, the detuning of the NMR coil, and the coupling between the coil and the
precessing spins (filling factor) [51]. A second, shape-dependent frequency
shift is caused by the dipolar field from the precessing protons,
$\delta^{d}$. Both effects are estimated as in Run-1 [7].

Calibration probe temperature dependence $\mathbf{\delta^{T}}$ The
gyromagnetic ratio of protons diamagnetically shielded in a spherical sample
of water was measured at $34.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ [45].
This diamagnetic shielding is temperature-dependent [52]. The correction from
$T^{\text{cp}}_{n}$, the calibration-probe temperature for calibration of
trolley probe $n$, to $T_{r}$, is $\delta_{n}^{T}=(-10.36\pm 0.30)\times
10^{-9}(T_{r}-T^{\text{cp}}_{n})$. The calibration probe temperature was
measured with a platinum resistive temperature device (PT1000 RTD) with an
accuracy of $0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$, and a different
correction per probe was applied to account for the calibration-probe and
trolley temperature during the calibration of each probe as discussed in the
next section.

Corrections dependent on the calibration-probe environment As noted in the
discussion of material effects, the magnetized components of the calibration
probe contribute to the measured magnitude of the magnetic field that depends
on the orientation with respect to $\vec{B}$ and due to magnetic images.
Additional corrections for the calibration configuration vary with the
individual trolley probe being calibrated and are discussed in Sec. VI.3.

#### Calibration-probe cross checks

Work is underway to cross-check the intrinsic corrections applied to the
calibration probe, i.e., corrections not dependent on the environment
($\delta^{b}$, $\delta^{s}$, $\delta^{RD}$, $\delta^{d}$, and $\delta^{P}$),
using 3He magnetometry and a separate H2O probe based on continuous wave (CW)
NMR. The Mark-I 3He absolute magnetometer provided an indirect
$42\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ cross-check on the calibration
probe [53, 16, 7]. A Mark-II 3He probe was designed and constructed with much
smaller intrinsic corrections, and a campaign is underway to directly
calibrate the muon $g-2$ calibration probes for Run-1 and Runs 3-6.
Preliminary analysis confirms agreement with uncertainties less than
$20\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. The calibration probes were also
compared to the CW H2O NMR probe under development for JPARC’s MuSEUM and
g-2/EDM (E34) experiments [54]. Cross-checks with earlier CW prototypes at
$1.4\text{\,}\mathrm{T}$ and $1.7\text{\,}\mathrm{T}$ showed a tension on the
$\sim$$50\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ with a precision around
$15\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. The same cross-check, with newer
probe versions, performed at $3\text{\,}\mathrm{T}$, is in good agreement with
an uncertainty of $10\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. The discrepancy
with the earlier version is not yet understood; additional work is ongoing.

Table 13: Calibration probe intrinsic corrections and uncertainties. Shape corrections are temperature dependent and hence different for each trolley probe. Thus, the range of all probes is given. Description |  | Corr. (ppb) | Unc. (ppb)
---|---|---|---
Shape, susceptibility | $\delta^{b}$ | -1508.7 to -1507.4 | 6.0
Material effects | $\delta^{s}$ | 10.3 | 5.0
Radiation damping | $\delta^{RD}$ | 0 | 3.0
Proton dipolar field | $\delta^{d}$ | 0 | 2.5
Sample purity | $\delta^{P}$ | 0 | 2.0
Subtotal |  |  | 8.9

###  VI.3 Trolley-probe calibration

Trolley-probe calibration provides a set of corrections to the frequencies
$\omega_{n}^{tr}$ measured by each trolley probe

$\omega^{\prime}_{n}=\omega_{n}^{\text{tr}}(1+\delta_{n}^{calib}),$ (47)

where $\omega_{n}^{\prime}$ is the field that would be measured by a spherical
water sample at $T_{r}=$34.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ at the
position of probe $n$. Corrections for the temperature dependence of the
vaseline-filled trolley probes are discussed in Sec. VI.4.

Calibration campaigns before the start of Run-2 and after Run-3 took place in
vacuum in a dedicated region of the storage ring magnet using the calibration
probe described in Sec. VI.2. Magnetic field gradients applied in all three
directions were used to place the effective volumes of the calibration probe
and each trolley probe within $0.5\text{\,}\mathrm{m}\mathrm{m}$ of the same
position, and the magnetic field in the calibration region was carefully
mapped and shimmed.

The calibration correction was determined from a sequence of measurements
swapping the trolley and calibration probe into the calibration position.
During this swapping, the magnetic field was tracked with fixed probes to
mitigate the effect of drifts. Additionally, the Run-2/3 calibration campaigns
and the Run-1 calibration campaign provided data on the stability of the
trolley-probe calibrations over a three-year period.

Uncertainties from the calibration procedure are listed in Table 14. These
include uncertainties due to mis-alignment of the calibration probe and
trolley probe, temperature corrections of the diamagnetic shielding
$\delta^{T}$, the variance between the calibration constants of different
measurement campaigns and analyzer $\delta^{var}$, the difference between the
active volume of the calibration probe and trolley probe $\delta^{av}$, the
trolley and calibration probes magnetic footprint $\delta^{fp}$ and
$\delta^{cp}$ on each other, the frequency extraction $\delta^{f}$ and the
material effects including the magnetic image in the pole pieces
$\delta^{img}$. The per-probe calibration constants with a graphical
representation is given in Table 28 and Fig. 29, in the appendix.

Table 14: Uncertainties from the calibration procedure on the muon-weighted field. The uncertainties for the individual probes are shown in Table 28. The probe individual corrections due to temperature dependence of the diamagnetic shielding range from $-126.3\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ to $-59.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. Description |  | Uncertainty [ppb]
---|---|---
Swapping and misalignment | $\delta^{tr}$ | 1.6
Temperature of diamag. shielding | $\delta^{T}$ | 5.2
Variance | $\delta^{var}$ | 11.0
Active volume | $\delta^{av}$ | 1.7
Footprint trolley | $\delta^{fp}$ | 8.0
Footprint CP | $\delta^{cp}$ | 4.0
Frequency extraction CP | $\delta^{freq(cp)}$ | 1.0
Material and mag. image | $\delta^{img}$ | 9.0
Subtotal |  | 17.8

###  VI.4 Magnetic field maps

In this section, we describe the detailed extraction of the field maps
$\omega_{p}^{\prime\,\text{tr}}(x,y,\phi,t)$ (Eq. (38)). The transverse
positions are fixed by the probe locations, while the trolley position is
radially constrained by the trolley rails. The trolley azimuthal position is
determined by reading the barcodes etched into the bottom of the vacuum
chambers. Encoders that measure the length of the trolley cables are a backup,
however, the encoder precision is inferior compared to the barcode due to
tension variations in the cables. The 17 trolley NMR probes are triggered in
sequence every ${\sim}$30\text{\,}\mathrm{ms}$$, resulting in a
${\sim}$2\text{\,}\mathrm{H}\mathrm{z}$$ sampling rate for each probe. The
corrected frequencies are interpolated to a grid of azimuthal positions
$\phi_{k}(t)$. Different interpolation schemes were tested and agreed within
$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

The multipole coefficients $m_{i}(\phi_{k}(t_{\text{tr}}))$ are determined for
each $\phi_{k}$ by fitting the corrected frequencies to Eq. 39, where
$t_{\text{tr}}$ is the time when the trolley is at $\phi_{k}$. A lower bound
on $N_{\mathrm{max}}$ is derived from azimuthal averaged fit residuals, which
show a transverse dependence if $N_{\mathrm{max}}$ is chosen too small. An
upper bound comes from degeneracies of the multipoles with our trolley probe
configuration. The truncation at $N_{\mathrm{max}}=12$ of the parametrization
in Eq. (41) is used. The difference between using different minimization
algorithms to extract the multipole coefficients is negligible. Representative
field maps $m_{1}(\phi)$ for three different trolley runs are shown in Fig.
22.

Figure 22: The relative (Rel.) dipole $m_{1}$ coefficient as a function of
azimuth for three field maps with respect to its azimuthal average. A) is from
April 8 2019, the beginning of Run-2, B) is from June 20, 2019, the end of
Run-2, and C) from March 11, the end of Run-3. The peak-to-peak amplitudes are
$76\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$,
$108\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, and
$93\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, respectively, with RMSs of
$14.6\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$,
$20.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, and
$15.8\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$.

Corrections and uncertainties to the trolley multipole coefficients are
presented in Table 15 and summarized here.

Trolley motion effects ($\mathbf{\delta^{motion}}$): The trolley motion in a
nonuniform magnetic field generates eddy currents in the conducting
components, most significantly the aluminum shell. We use the Run-1 correction
for $\delta^{\text{motion}}=$-15\pm
18.\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ from Run-1 analysis [7] estimated
from the comparison of standard continuous motion trolley runs with stop-and-
go runs and from the comparison for clockwise and counter-clockwise trolley
runs.

Difference in configuration ($\mathbf{\delta^{config}}$): During the trolley
runs, the collimators that radially constrain the stored-muon distribution are
retracted, and the trolley rails are in a different position than when the
muons are stored. The effect of these two configuration changes is estimated
from calculations of the the magnetic field produced by the diamagnetic copper
and paramagnetic aluminum in the respective configurations. The uncertainty of
the Run-1 correction of $\delta^{\text{config}}=$-7\pm
22.\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ 999Note that in Ref. [7],
$-5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ is used for the central value of
the effect from the garage alone. is dominated by a discrepancy in the
calculation and what a local fixed probe measures. The same value is used for
Run-2/3. The effect from the collimators on the azimuthally averaged field is
smaller than $1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

Trolley frequency extraction ($\mathbf{\delta^{freq}}$): Trolley NMR-probe FID
analysis is described in [56]. Briefly, the phase function (phase vs time) for
the free-induction-decay (FID) signals is extracted from in-phase and Hilbert-
transform quadrature signals. The phase functions are fit to polynomials of
varying order from two to six and for a varying time ranging from 0.20 to 0.75
of $T_{2}^{*}$ (the FIDs are not exponential, so in this case, we refer to the
time for the FID amplitude to reach $1/e$ of the maximum). The frequency-
extraction correction $\mathbf{\delta^{freq}}$ on $m_{1}$ is below
$12\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. Potential effects from incorrect
$t_{\text{FID}}=0$ on the $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ level
are shown to be negligible. Temperature changes affect the phase function of
FIDs. This effect on the extracted precession frequencies is included in the
correction below. The uncertainty due to correcting from the
$\approx$25\text{\,}{}^{\circ}\mathrm{C}$$ trolley temperature during field
mapping to around $\approx$33\text{\,}{}^{\circ}\mathrm{C}$$ temperature
during calibration is $5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

The total uncertainty from the frequency extraction, taking the Run-2/3 beam
shapes and correlations between the multipoles into account, is shown in Table
15. In Run-1, this correction had a different meaning because every trolley
NMR position was treated as an independent point with frequency extraction
uncertainty of $10\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$. In fact the NMR
sample active volume is ${\sim}$1.8\text{\,}\mathrm{c}\mathrm{m}$$, while the
measurements are separated by ${\sim}$0.5\text{\,}\mathrm{c}\mathrm{m}$$
leading to oversampling.

Trolley temperature dependence ($\mathbf{\delta^{temp}}$): A dedicated study
in the Argonne National Laboratory magnet facility with two temperature-
controlled probes to track magnet drifts revealed a temperature dependence of
the vaseline frequency of $-0.8\pm
0.2\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}\mathrm{/}\mathrm{C}$. However, a
conservative uncertainty of
$2\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}\mathrm{/}\mathrm{\SIUnitSymbolCelsius}$
is used, since the uncertainty is dominated by the frequency extraction
uncertainties discussed above.

The trolley-probe NMR frequencies are not actively temperature corrected,
rather, we apply a correction and uncertainty $\delta^{\text{tr,temp}}$. The
temperature difference of the trolley probes with respect to the mean
temperature during the calibration
($33.1\text{\,}\mathrm{\SIUnitSymbolCelsius}$) range from
$-8.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$-1.9\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The temperature-dependent
frequency correction is calculated using the temperature dependence of
$-0.8\pm 2.0\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}\mathrm{/}\mathrm{C}$. The
muon weighted corrections for the three datasets are
$-3.6\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$,
$-5.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$, and
$-6.0\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$, respectively. In addition, the
temperature spread during one field map is $1.8\pm
0.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and an uncertainty of
$1\text{\,}{}^{\circ}\mathrm{C}$ on the temperature sensor is used. The
resulting uncertainties for Run-2, Run-3a, and Run-3b are listed in Table 15.

Trolley transverse and azimuthal position ($\mathbf{\delta^{xy}}$,
$\mathbf{\delta^{azi}}$): The trolley position is constrained in the
transverse plane by the rails. A laser tracker was used to estimate rail
distortions before the vacuum chambers were installed. The effect in the
transverse plane $\delta^{\text{xy}}$ is evaluated by taking the Run-2 and
Run-3 beam shapes into account by running one of the analysis chains with and
without incorporating rail distortions. The observed difference of
$11.8\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ (Run-2),
$4.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ (Run-3a), and
$1.8\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ (Run-3b) are used to correct the
other analysis. The corresponding uncertainties are listed in Table 15. For
Run-2/3 the corrections are smaller than for Run-1 due.

The azimuthal trolley position is determined using the barcode except for
small gaps between adjacent vacuum chambers and for barcode errors, where
cable-length encoders are used. A conservative estimate of the azimuthal
position resolution of $2\text{\,}\mathrm{m}\mathrm{m}$ leads to a systematic
uncertainty of
$\delta^{\mathrm{azi}}=$4\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ on the
average dipole field.

Parametrization ($\mathbf{\delta^{param}}$) and Azimuthal averaging
($\mathbf{\delta^{avg}}$): The finite number of measurements and the
parametrization of Eq. (38) lead to additional uncertainty with three
contributions: A. an uncertainty due to the truncation $N_{\text{max}}$ in Eq.
(38), B. uncertainty due to interpolation between the finite number of
azimuthal slices and C. the use of 2D multipole expansion, which is only valid
if there is no azimuthal magnetic field dependence. The uncertainty due to the
choice of $N_{\text{max}}$ is estimated from the residuals of the fits to Eq.
(38) weighted by the azimuthally averaged beam distribution within $\Delta
l=$1\text{\,}\mathrm{mm}10\text{\,}\mathrm{mm}$$ around each probe.

The uncertainty due to the interpolation between these finite azimuthal slices
was determined by interpolating with linear, quadratic, and cubic splines. To
estimate the effect of 2D multipole expansion, the averaged magnetic fields
following the above analysis approach were compared to an analytic azimuthal
average using simulated magnetic fields based on a toroidal 3D multipole-based
field description. The observed differences from such comparisons are
$<$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$.

Table 15: Corrections and uncertainties from the spatial field maps. A single value per line indicates the same value for all datasets. Description |  | Corr. [ppb] | Uncertainty [ppb]
---|---|---|---
|  |  | Run-2 | Run-3a | Run-3b
Motion effects | $\delta^{motion}$ | -15.0 | 18.0
Configuration | $\delta^{config}$ | -7.0 | 22.0
Freq. extraction | $\delta^{freq(tr)}$ | - | 19 | 18 | 16
Temperature | $\delta^{temp}$ | - | 9.2 | 13.8 | 15.2
Transverse pos. | $\delta^{xy}$ | - | 10.0 | 9.9 | 9.0
Azimuthal pos. | $\delta^{azi}$ | - | 4.0
Parameterization | $\delta^{param}$ | - | 3.4 | 6.3 | 7.6
Azi. averaging | $\delta^{avg}$ | - | 0.8 | 1.4 | 1.7
Subtotal |  |  | 37.2 | 38.5 | 38.1

###  VI.5 Magnetic field tracking

The fixed probes track the magnetic field between trolley runs (see Sec. VI.4)
for moments up to $i=5$. For higher-order moments, we use linear interpolation
in time. Fixed-probe tracking entails the following steps: 1) extracting
fixed-probe moments defined in Eq. (38); 2) tying the fixed-probe moments to
the trolley-map moments; 3) parameterizing the moments as a function of
azimuth and time.

####  VI.5.1 Fixed probe moments

Linear combinations of measurements from the four or six fixed probes at each
station provide fixed-probe moments $m_{i}^{\text{fp}}(\phi_{s},t)$ following
the procedure described in [7].

To reduce the effect of probe noise, the $m_{5}^{\text{fp}}(\phi_{s},t)$
moment is first tied to the measured $m_{5}$ from the trolley run pair (see
Sec. VI.5.2) before the change of moment basis.

Fixed probes in three stations close to the inflector experience large
gradients resulting in very short FIDs and increased frequency uncertainty
(noise). Two additional probes with a PEEK housing are installed inside the
vacuum chamber at the position of one of the stations. These additional
measurements verified that linear interpolation of the moments from
neighboring stations gives a better estimate than the determination from the
noisy fixed probe frequencies. Therefore, the multipole moments for these
three stations are linear interpolations from their neighboring stations.

The relative fixed probe frequency extraction is very robust and the
uncertainty from the fixed probe frequency extraction
$\delta^{\text{freq(fp)}}$ is
${\sim}$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$, consistent with Run-1 [7].
Non-linear temperature changes of the yoke and thus the fixed probes are on
the $0.06\text{\,}{}^{\circ}\mathrm{C}$ level, and thus the uncertainty due to
fixed probe temperature is negligible. Linear components are canceled by
tracking between two subsequent field maps.

Fixed probe data are subject to general data quality cuts (Sec. III.1).
Additionally, events with FID amplitudes or FID power more than seven standard
deviations from the probe’s mean amplitude and power are removed.

####  VI.5.2 Tying fixed probe to trolley-map moments

The change of the magnetic field at a fixed-probe station before or after
$t_{s}^{\text{tr}}$, the time the trolley passes the station at $\phi_{s}$
during a trolley run, is

$\displaystyle\Delta
m_{i}^{\text{fp}}(\phi_{s},t)=m_{i}^{\text{fp}}(\phi_{s},t)-m_{i}^{\text{fp}}(\phi_{s},t_{s}^{\text{tr}}),$
(48)

where $m_{i}^{\text{fp}}(\phi_{s},t_{s}^{\text{tr}})$ is the moment measured
using the fixed probes within station $s$ averaged around the time the trolley
passes by that station.

To determine $t_{s}^{\text{tr}}$, we make use of the fact that the material
effects of the trolley and its onboard electronics produce a characteristic
field perturbation (footprint) that is measured by the fixed probes when the
trolley passes. The time of the largest field perturbation sets
$t_{s}^{\text{tr}}$ and the trolley’s azimuthal location sets $\phi_{s}$.
Varying the station positions $\phi_{s}$ by
${\sim}$0.25\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}$$ has an effect less than
$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.

The field perturbation due to the trolley when passing a fixed probe station
is removed from the fixed-probe data and replaced with a linear interpolation
of $m_{i}(\phi_{s},t)^{\text{fp}}$ based on the $30\text{\,}\mathrm{s}$ before
and after $t_{s}^{\text{tr}}$. The effect of the trolley footprint replacement
is tested on data in regions without footprint by comparing the field
estimated by the replacement algorithm and the actual measured data. The
uncertainty is listed in Table 16 and is similar to Run-1, as described in
[7].

####  VI.5.3 Fixed-probe tracking

For azimuth $\phi$ and time $t$ for one or more trolley runs at $t_{k}$ the
fixed-probe tracked moments are

$\displaystyle m_{i}(\phi,t)=$
$\displaystyle\sum_{k}W_{k}(t)\left(m_{i}^{\text{tr}}(\phi,t_{k})+\sum_{s}W_{s}(\phi)\sum_{j}J_{ij}(\phi_{s})\Delta
m_{j}^{\text{fp}}(\phi_{s},t)\right)$ (49)

where $k$ labels the trolley runs, and $W_{k}(t)$ is the weighting of each
trolley run at time $t$. The azimuthal weighting factor $W_{s}(\phi)$
interpolates between stations on either side of $\phi$,
$J_{ij}(\phi_{s})=\frac{\partial m_{i}^{\text{tr}}(\phi_{s})}{\partial
m_{j}^{\text{fp}}(\phi_{s})}$ is the Jacobian that relates small changes of
the fixed probe moments to changes of the trolley moments for station $s$, and
$\Delta m_{i}^{\text{fp}}({\phi_{s}},t)$ is defined in Eq. (48).

Ideally, magnetic field tracking uses two consecutive trolley runs, e.g.
$k=1,2;2,3$ etc.. Occasional unplanned magnet incidents, such as the loss of
magnet power allow tracking only from the trolley run before the incident, in
which case $W_{k}(t)=1$.

Field changes not tracked by the fixed probes lead to errors of the
$m_{i}(\phi,t)$ that is a maximum at the midpoint between the two paired
trolley runs. To quantify this, tracking from a single trolley run is used to
predict the field moments at the later trolley run. The difference between the
predicted and measured field moments for the second trolley run is called the
tracking offset. The tracking offset can be modeled as a random walk process
caused by changes in the magnet shape. For tracking using a pair of
consecutive trolley runs, the random walk becomes a Brownian bridge that uses
a linear interpolation between the first and second trolley run (see Ref. [7]
for details). A single parameter $M$ parametrizes the rate of the process.

The distribution of the azimuthally–averaged tracking offsets can be used to
account for potential correlations between different stations. In order to
reduce the statistical error, The random-walk parameters are determined from
the azimuthally–averaged tracking offsets for all of Run-2/3. We determine
$M=$0.018\text{\,}\mathrm{H}\mathrm{z}\mathrm{/}\sqrt{s}$$ for the $m_{1}$
coefficient. Similar rate of change parameters are determined for each
multipole moment. The resulting uncertainties, taking the muon-weighted
corrections for the different datasets and the correlations between the
different multipole moments into account, are summarized in Table 16. Note
that this uncertainty is statistically independent and hence reduces if
multiple datasets are combined.

We observe that the tracking offset depends on the time after the magnet was
ramped up and shows a characteristic azimuthal dependence that is largest at
magnet yoke boundaries as shown in see Fig. 23. A dedicated measurement was
performed, repeatedly measuring the field with the trolley for
$60\text{\,}\mathrm{h}$ after the magnet was ramped. We use the azimuthally
averaged tracking offset to estimate the bias. We model the effect by an
exponential function with amplitude and time constants as parameters. The
amplitude and time constant may depend on the history of the magnet before the
ramp. Therefore, we determine a correction and uncertainty conservatively; the
result is an initial amplitude of $(100\pm 100)$ ppb and a relaxation time
constant of $12\text{\,}\mathrm{h}$. The correction and uncertainty depend on
the time periods relative to the magnet ramp time in which muon data have been
taken. The resulting correction and uncertainties are listed in Table 16.

Figure 23: Top: Tracking offset (inability to track field) as a function of
azimuth (azi.) around a yoke boundary. Different colors indicate different
times after the magnet ramp. Bottom: Amplitude of effect at
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ as a function of time after magnet
ramp. The x show the azimuthally averaged values scaled up by a factor of x10.
A dedicated campaign of back-to-back trolley runs was performed in Run-6 to
study this effect.

A detailed comparison between interpolation analyses from two groups was
performed to identify inconsistencies and bugs in the analysis, while the
individual groups had individual software blinds. Comparisons performed on the
azimuthal averaged field and on a station-by-station basis agree within a few
ppb after relative unblinding. The difference in analysis results due to
different analysis choices is added as additional uncertainty and listed in
Table 16.

Table 16: Corrections and uncertainties (in parenthesis) from magnetic field tracking. A single value per line indicates the same value for all datasets. All values are given in units of ppb. Description | Correction (Uncertainty)
---|---
| Run-2 | Run-3a | Run-3b
Tying |  |  |
Trolley footprint | (7.0)
Fixed probe resolution | (1.0)
Tracking |  |  |
Brownian bridge | (15.4) | (10.7) | (16.0)
Magnet ramp effect | -3.0 (3.0) | -10.0 (10.0) | -3.0 (3.0)
Fixed probe temperature | (0) | (0) | (0)
Analysis choices | (1.8) | (2.5) | (1.5)
Subtotal | (17.3) | (16.5) | (17.8)

The multipole moments averaged over azimuth and weighted by the detected muons
(including DQC) $\langle m_{i}\rangle_{\phi,t}$ are listed in Table 17 for all
three datasets. The lowest order $\langle m_{i}\rangle_{\phi,t}$, the normal
and skew quadrupoles, are shown as a function of time over the full dataset in
Fig. 24.

Table 17: Field multipole moments in ppb (see Eq. (38)) averaged over azimuth and time (including DQC) per dataset. The Run-3 the experiment hall temperature was m ore stable than Run-2 due to a climate-control upgrade.  Multipole | Run-2 | Run-3a | Run-3b
---|---|---|---
$m_{2}/m_{1}$ | 331 | -113 | -14
$m_{3}/m_{1}$ | 611 | -6 | -43
$m_{4}/m_{1}$ | -310 | 23 | 17
$m_{5}/m_{1}$ | 383 | 40 | 35
$m_{6}/m_{1}$ | 94 | -9 | -20
$m_{7}/m_{1}$ | 217 | 127 | 127
$m_{8}/m_{1}$ | -24 | -22 | -21
$m_{9}/m_{1}$ | 23 | 15 | 12
$m_{10}/m_{1}$ | -697 | -725 | -727
$m_{11}/m_{1}$ | -167 | -203 | -215
$m_{12}/m_{1}$ | -1068 | -1056 | -1057

###  VI.6 Muon weighted magnetic field

####  VI.6.1 Muon beam distribution

The muon beam distribution $M(x,y,\phi)$ is reconstructed from measured
tracker profiles combined with beam-dynamics calculations of the azimuthal
dependence of the muon distribution. around the ring. Two trackers provide
well-localized muon beam distributions with an azimuthal sensitivity with an
RMS of $4.9\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$4.8\text{\,}\mathrm{\SIUnitSymbolDegree}$, respectively. Following Eq. (40),
the mapped magnetic field is weighted by the muon distribution to determine
the magnetic field seen by the muons.

Tracker profiles $M^{\text{T}}_{\text{i}}(x,y)$ for the muon-weighted magnetic
field are accumulated in time intervals of
$T_{\text{interval}}=$$2\text{\,}\mathrm{h}3\text{\,}\mathrm{h}$ and corrected
for detector resolution and acceptance. Only positrons with decay times
between the analysis start time
$t_{\text{start}}=$30.2876\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and end
time $t_{\text{end}}=$650.0644\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ enter
the tracker profiles. The time intervals $T_{\text{interval}}$ are chosen to
contain more than $6\text{\times}{10}^{5}\text{\,}$ total tracks, avoid gaps
$>$6\text{\,}\mathrm{h}$$, stay within a trolley-run pair and contain entire
$\omega_{a}$ DAQ runs.

The measured beam profiles at azimuthal locations where the tracker detectors
do not provide beam diagnostics are reconstructed from tracker profiles by
shifting the mean and scaling the transverse widths of the distribution
relative to the tracker station using

$\displaystyle\langle x\rangle(\phi)$
$\displaystyle=x_{\text{COD}}(\phi)+D_{x}(\phi)\langle\delta\rangle,$ (50)
$\displaystyle\langle y\rangle(\phi)$ $\displaystyle=0,$ (51) $\displaystyle
x_{\text{RMS}}(\phi)$
$\displaystyle=\bigg{[}\frac{\beta_{x}(\phi)}{\beta_{x}(\phi_{\text{tkr}})}\left(x_{\text{RMS}}^{2}(\phi_{\text{tkr}})-D_{x}^{2}(\phi_{\text{tkr}})\delta_{\text{RMS}}^{2}\right)$
$\displaystyle+D_{x}^{2}(\phi)\delta_{\text{RMS}}^{2}\bigg{]}^{1/2},$ (52)
$\displaystyle y_{\text{RMS}}(\phi)$
$\displaystyle=\left[\frac{\beta_{y}(\phi)}{\beta_{y}(\phi_{\text{tkr}})}y_{\text{RMS}}^{2}(\phi_{\text{tkr}})\right]^{1/2}.$
(53)

The beam widths $x_{\text{RMS}}$ and $y_{\text{RMS}}$ at the the azimuth of
the tracker stations $\phi_{\text{trk}}$ are extracted from the tracker
profiles $M^{\text{T}}_{\text{i}}(x,y)$. The beta functions $\beta_{x}(\phi)$,
$\beta_{y}(\phi)$, and radial dispersion function $D_{x}(\phi)$ are determined
from the optical lattice calculated with the COSY INFINITY-based model of the
storage ring. The mean and RMS fractional momentum $\langle\delta\rangle$ and
$\delta_{\text{RMS}}$ are extracted from the fast-rotation analysis discussed
in Sec. V.1. The average fractional momentum is
$\sim$$0.07\text{\,}\mathrm{\char 37\relax}$ except for Run-3b, which is lower
($\sim$$0.01\text{\,}\mathrm{\char 37\relax}$) owing to stronger injection
kickers, whereas the RMS of the distribution is
$\sim$$0.1\text{\,}\mathrm{\char 37\relax}$. The field indices are listed in
Table 1.

Closed orbit distortions (COD) shift the ideally circular closed orbit away
from the equilibrium position. Azimuthal variation in the vertical dipole
component of the magnetic field causes a radial COD

$\displaystyle
x_{\text{COD}}(\phi)\approx\frac{R_{0}}{n}\frac{b_{1}(m_{1})}{B_{0}}\cos\left(\phi-\phi_{1}(m_{1})\right),$
(54)

where $R_{0}$ is the nominal radius, $B_{0}$ is the nominal field, $n$ is the
effective field index given in Table 1, and $b_{1}(m_{1})$ and
$\phi_{1}(m_{1})$ are the $N=1$ Fourier amplitude and phase of $m_{1}(\phi)$.
The Fourier components are extracted with an FFT from field maps in each
$T_{\text{interval}}$, and $x_{\text{COD}}$ is calculated for each individual
$T_{\text{interval}}$. The amplitudes of the radial COD range from
$0.6\text{\,}\mathrm{mm}1.5\text{\,}\mathrm{mm}$ and
$0.2\text{\,}\mathrm{mm}0.4\text{\,}\mathrm{mm}$ for Run-2 and Run-3,
respectively.

An azimuthally varying radial magnetic field would cause a vertical COD.
Because the radial field dependence on azimuth is not measured during the
experiment, $y_{\text{COD}}$ is set to zero and considered separately as a
systematic. Misalignments of the electric quadrupole plates also cause radial
and vertical CODs by steering the beam. These are considered separately as a
systematic.

Each tracker station is extrapolated separately, and the reconstructed
distributions from both stations are averaged to get the nominal beam
distribution.

Figure 4 in Sec. III.3.3 illustrates azimuthally averaged muon beam
distributions based on the beam extrapolation around the ring of tracker
measurements.

####  VI.6.2 Muon weighting

Following Eq. (40), the reconstructed muon beam distribution $M(x,y,\phi,t)$
(see Sec. VI.6.1) is projected onto the moments used to describe the magnetic
field for time intervals $T_{\text{interval}}$ and evaluated every
$5\text{\,}\mathrm{\SIUnitSymbolDegree}$ because the azimuthal variation of
the beam moments is small. Since the tracker profiles and thus the beam
moments are only determined every $2\text{\,}\mathrm{h}3\text{\,}\mathrm{h}$,
the field moments $m_{i}(t,\phi)$ are averaged in time, weighted by the number
of muons in the storage region $N_{\mu}(t)$. Eq. (41) is used to calculate the
muon-weighted field per $T_{\text{interval}}$ and azimuthal bin $\phi_{i}$.
Additional averaging over all azimuthal bins and thus implementing Eq. (43)
yields the muon-weighed field per time interval $T_{\text{interval}}$.
Averaging all time intervals within a dataset, weighting by $N_{\mu}(t)$ and
accounting for DQC cuts, yields the muon weighted magnetic field
$\tilde{\omega}_{p}^{\prime}$ per dataset defined in Eq. (40), listed in Table
26 for each dataset.

The improvement in the kick for dataset Run-3b reduces the $k_{2}$ and $k_{5}$
parameters (see Eq. (42)) since the muon distribution is more centered. This
has the effect that weighted moments $m_{i},i>1$ are reduced, and thus
systematic uncertainties that only couple though moments with $m_{i},i>1$ are
reduced as well. The beam multipole projections averaged over azimuth over the
times when muons are stored to extract $\omega_{a}$ ($\langle
k_{i}\rangle_{\phi,t}$) are listed in Table 18 for all three datasets. Figure
24 provides an overview of the muon-weighted field as a function of time.

Table 18: Average beam multipole projections in each dataset, including DQC. Projections are normalized to beam profile intensity and are unitless. Beam Projection | Run-2 | Run-3a | Run-3b
---|---|---|---
$k_{1}$ | 1.000 | 1.000 | 1.000
$k_{2}$ | 0.139 | 0.136 | 0.073
$k_{3}$ | -0.001 | -0.006 | -0.005
$k_{4}$ | 0.001 | -0.001 | 0.000
$k_{5}$ | 0.081 | 0.076 | 0.046
$k_{6}$ | 0.000 | -0.001 | 0.000
$k_{7}$ | -0.001 | -0.001 | -0.006
$k_{8}$ | -0.002 | -0.001 | 0.003
$k_{9}$ | 0.001 | 0.001 | 0.000
$k_{10}$ | -0.004 | -0.003 | 0.001
$k_{11}$ | 0.000 | 0.000 | 0.000
$k_{12}$ | -0.001 | -0.001 | 0.001
Figure 24: The relative muon-weighted magnetic field
($\tilde{\omega}_{p^{\prime}}$) as a function of time for the Run-2 (left
side) and Run-3a and Run-3b (right side). The dipole $m_{1}$ contribution
alone is shown in gray below. On this scale, they barely differ. The lower two
plots show the tracked $m_{2}$ and $m_{3}$ moments.

####  VI.6.3 Systematics

Tracker-specific systematics cause uncertainties in the beam distribution,
which lead to uncertainties in $\tilde{\omega}_{p}^{\prime}$. The relevant
uncertainties for muon weighting are tracker resolution
$\delta^{\text{reso,tkr}}$, acceptance $\delta^{\text{accept,tkr}}$, and
alignment $\delta^{\text{align,x,tkr}}$, $\delta^{\text{align,y,tkr}}$. These
systematics are evaluated by varying each parameter by $1\text{\,}\sigma$,
producing corresponding beam distributions in the usual time intervals
$T_{\text{interval}}$ and evaluating the effect on
$\tilde{\omega}_{p}^{\prime}$ averaged over each dataset. The resulting
uncertainties are listed in Table 19.

The tracker acceptance uncertainty is
$\leq$2\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ from changing the acceptance
function by $\pm 20\%$, and the resolution uncertainty is
$<$1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ by changing the radial and
vertical resolution by $\pm$0.5\text{\,}\mathrm{m}\mathrm{m}$$. Changing the
tracker alignment in $x$ and $y$ by $\pm$0.6\text{\,}\mathrm{m}\mathrm{m}$$
yields uncertainty on the size of $1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$.
The uncertainty due to tracker profile statistics are insignificant.

The muon-weighted field should be calculated for muons that enter the
$\omega_{a}$ determination and thus are seen by the calorimeters. Because the
spatial acceptance from tracker and calorimeters is different, the muon
distribution from the tracker would have to be corrected for calorimeter
acceptance. However, the effect is small and thus is only treated as an
uncertainty.

As discussed above, an azimuthal radial magnetic field variation can
contribute to $y_{\text{COD}}$. Since the radial magnetic field was only
measured in pre-Run-1 while no vacuum chambers were installed, the effect is
estimated by assuming an amplitude of $0.5\text{\,}\mathrm{m}\mathrm{m}$,
which is a factor two larger than the pre-Run-1 measured value, for the $N=1$
COD and the worst case phase.

Misalignments of the electric quadrupole plates cause an $x_{\text{COD}}$ or
$y_{\text{COD}}$ by steering the beam. The expected COD calculations use the
central displacements of the electric quadrupole plates measured in a survey.
Survey uncertainties cause uncertainties in the CODs. These effects were
evaluated using the same method from Run-1 [7], resulting in a correction and
uncertainty listed in Table 19.

The momentum deviation $\delta$ used in the beam reconstruction procedure in
Eq. (50) and Eq. (52) slightly differ from different analyzing teams in Sec.
IV. The related systematic uncertainty is determined by varying
$\langle\delta\rangle$ and $\delta_{\text{RMS}}$ by $\pm 0.0001$.

A changing muon distribution over time in a fill can be caused by magnetic
field transient effects from the electric quadrupoles and kicker eddy
currents. Tracker profiles are reconstructed for different times in a fill.
Studies show that the related uncertainties are negligible in Run-2/3.

Table 19: Corrections and uncertainties (in parenthesis) due to spatial muon weighting of the magnetic field.  Description | Correction (Uncertainty) (ppb)
---|---
| Run-2 | Run-3a | Run-3b
Detector effects |  |  |
Tracker acceptance | (2.1) | (1.1) | (0.1)
Tracker resolution | (0.1) | (0.1) | (0.1)
Tracker y-alignment | (10.7) | (0.6) | (0.4)
Tracker x-alignment | (4.5) | (1.3) | (0.3)
Calorimeter acceptance | (1.0) | (0.2) | (0.2)
Closed Orbit Distortion |  |  |
and azimuthal effects |  |  |
yCOD (radial B) | (1.8) | (3.7) | (2.9)
xCOD (quad misalig.) | +1.3 (5.9) | +2.7 (6.7) | +2.5 (6.3)
yCOD (quad misalig.) | -0.9 (0.1) | -0.5 (0.2) | -0.3 (0.2)
Mean momentum offset | (0.2) | (0) | (0)
Subtotal | (13.4) | (7.9) | (6.9)

###  VI.7 Transient magnetic fields

The fixed probe system measures the magnetic field at intervals of
$1.2\text{\,}\mathrm{s}1.4\text{\,}\mathrm{s}$ asynchronous to beam injection.
Thus, any time-dependent, $\text{\,}\mathrm{\SIUnitSymbolMicro s}$-timescale
magnetic field transient that is synchronized with beam injection is not
accounted for in $\tilde{\omega}_{p}^{\prime}$. In addition, the skin-depth
effect in the aluminum of the vacuum chambers reduces the effects on high-
frequency magnetic field transients. Transient magnetic fields synchronized
with beam injection are caused by eddy currents in the kicker and time-varying
fields caused by the pulsing of ESQs. Both effects lead to corrections on the
muon-weighted magnetic field and are improved compared to Run-1 by additional
measurements. Additional transient effects related to magnetic fields in the
booster are $<$7\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$ as determined for
the Run-1 analysis [7].

####  VI.7.1 Transient magnetic fields from kickers

The magnetic field kick of $22\text{\,}\mathrm{m}\mathrm{T}$ to store muons on
the stable orbit is a fast transient field
(${\sim}$150\text{\,}\mathrm{n}\mathrm{s}$$) that introduces eddy currents in
the region of the kicker magnets that lasts longer than the initial kick. NMR
magnetometers are too slow to measure the effect on the magnetic field. The
transient magnetic field has been measured with two magnetometers based on
Faraday rotation using terbium gallium garnet (TGG) crystals [7]. For Run 2/3,
additional measurements with improved setups have been performed using the
same magnetometers.

One of the magnetometers utilizes fibers to guide the light from the laser
source, which is housed in the center of the storage ring magnet, to the 3D
printed magnetometer where the laser light is polarized and sent through two
14.5-mm-long TGG crystals. A polarization-sensitive splitter divides the laser
beam into two returning fibers. The two beam intensities are measured by PIN
diodes; the polarization is reconstructed from the difference. This
differential readout scheme reduced the sensitivity on laser instabilities.
The magnetometer base consists of a glass block with small Sorbothane legs,
lowering the magnetometer’s center of mass and reducing mechanical vibrations.

The measurements in Run-1 [7] were limited by noise picked up from mechanical
vibrations of the kicker cage through the magnetometer and the fibers
themselves. To reduce the noise in the measurements, a PEEK bridge was
machined with Sorbothane legs that allow the magnetometer to be anchored to
the vacuum chamber instead of the cage that holds the kicker plates. In
addition, the returning fibers are routed on top of silicon bands that dampen
out potential vibrations.

Two measurement campaigns in summer 2021 and summer 2022 have been performed.
To calibrate the magnetometer, the magnetic field of the main magnet was
ramped at a constant rate from $0\text{\,}\mathrm{T}$ to
$1.4513\text{\,}\mathrm{T}$ or vice versa. The calibration constants change
from ramp to ramp due to temperature changes affecting the Verdet constant of
the TGG crystal and small tilt angles changing the effective length of the
crystal.

Since the laser was operated in constant current mode, the calibration factor
changed over time, which was tracked by measuring the
$12\text{\,}\mathrm{\SIUnitSymbolMicro T}$ magnetic field transient from
charging the kicker plates prior to the kick.

Figure 25: Magnetic field transient induced by kicker magnets measured by the
optical fiber magnetometer in summer 2021 and summer 2022.

The measured transient field is shown in Fig. 25 for two measurement campaigns
one year apart. The average of the two campaigns is used to estimate the
effect of the measured field perturbations. The effect on $\omega_{a}$ is
estimated by integrating the effect of the transient over the muon lifetime. A
five-parameter fit is used to estimate the overall correction. The corrections
are estimated based on measurements in the first of the three kickers with
upgraded kicker cables and operated at nominal kicker setting of
$53.1\text{\,}\mathrm{k}\mathrm{V}$ as present during Run-3b. The results from
this measurement are scaled to the other kickers, which operate at slightly
different operation voltages ($53.1\text{\,}\mathrm{k}\mathrm{V}$,
$53.0\text{\,}\mathrm{k}\mathrm{V}$, and $55.0\text{\,}\mathrm{k}\mathrm{V}$),
and to the conditions in Run-2 and Run-3a, during which the kickers were
operated at lower voltages ($47.7\text{\,}\mathrm{k}\mathrm{V}$,
$47.1\text{\,}\mathrm{k}\mathrm{V}$, and $47.1\text{\,}\mathrm{k}\mathrm{V}$).
Azimuthally, the kicker transient is treated as uniform within the regions
occupied by the kicker plates. The steep fall-off at the edges was modeled and
confirmed by measurements outside the kicker plates, resulting in a
suppression for the azimuthal average of $0.085$. Overall, this results in
corrections to $\tilde{\omega}^{\prime}_{p}$ of
$-21.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ and
$-22.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ for Run-2/3a and Run-3b,
respectively. The associated uncertainties are summarized in Table 20 and
described briefly below.

The effect of residual vibrations in the measured signal is estimated by
comparing results with the main magnet powered and not powered. The origin of
the perturbations with a time scale of about $1\text{\,}\mathrm{m}\mathrm{s}$
and amplitude of a few $0.1\text{\,}\mathrm{\SIUnitSymbolMicro T}$ remains
ambiguous. The measurement data cannot distinguish between an actual change in
the total magnetic field and mechanical vibrations of the fibers or the
crystal. This ambiguity contributes to the leading systematic uncertainty on
the transient measurement. The observed differences between the two campaigns
is not fully understood and might indicate local variations of the effect.
This ambiguity is accounted for by assigning the observed difference as a
“transient variance” uncertainty. Further contributions to the uncertainty
come from the azimuthal and transverse modeling, as well as from the above-
mentioned calibration procedure and baseline determination. Like the total
effect, the uncertainties are scaled to the different run conditions in the
Run-2 and Run-3a datasets. The scaling and potential differences in pulse
shapes due to using different cables lead to additional uncertainties for
these datasets.

Table 20: Uncertainties to $\tilde{\omega}^{\prime}_{p}$ due to transient magnetic fields from eddy currents in the kicker system. The uncertainties from the two campaigns in 2021 and 2022 are combined for the Run-3b dataset. The values are scaled for the Run-2 and Run-3a datasets accounting for the different run conditions. Description | Uncertainty (ppb)
---|---
| 2021 | 2022 | Run-3b | Run-2/3a
Vibration ambiguity | 8.3 | 12.8 | 10.5 | 9.9
Transient variance |  |  | 4.2 | 3.9
Azimuthal | 3.1 | 4.7 | 3.9 | 3.7
Transverse | 4.4 | 6.8 | 5.6 | 5.3
Calibration | 0.3 | 0.2 | 0.3 | 0.3
Baseline | 2.5 | 0.2 | 1.3 | 1.2
Scaling |  |  |  | 1.7
Pulse shape diff. |  |  |  | 4.2
Subtotal |  |  | 13.3 | 13.3

####  VI.7.2 Transient magnetic fields from ESQs

The beam-synchronous pulsing of the ESQ plates causes time-dependent magnetic
field changes on the $\text{\,}\mathrm{\SIUnitSymbolMicro s}$-timescale. These
fast synchronous changes are not captured by the field maps nor tracked by the
fixed probe system. Besides the asynchronous operations of the fixed probes
with respect to beam injection times, skin depth effects in the aluminum walls
of the vacuum chambers suppress field transients on that time scale. In-situ
measurements are required. While the exact mechanism creating this magnetic
field transition is not fully understood, the effect is associated with the
ESQ plates’ and support structure’s mechanical vibrations. The injection of
muons and associated pulsing of the ESQ plates every $10\text{\,}\mathrm{ms}$
for $8$ bunches drives an oscillation around
$100\text{\,}\mathrm{H}\mathrm{z}$, close to the system’s intrinsic
frequencies around $50\text{\,}\mathrm{H}\mathrm{z}$. The bottom plot in Fig.
26 shows an example of this effect as a function of time at one fixed
location. A second train of eight bunches is injected after
$266.7\text{\,}\mathrm{ms}$, a gap long enough for the vibration to mostly
ring down. This pattern repeats every $1.4\text{\,}\mathrm{s}$ or
$1.2\text{\,}\mathrm{s}$. Since this field changes during the time muons are
stored are not reflected in the direct measurement of
$\tilde{\omega}^{\prime}_{p}$, this transient results in a correction term
$B_{Q}$.

Figure 26: Top) The transient magnetic field from the vibration caused by the
ESQ pulsing for all times as a function of azimuth in the storage ring.
Bottom) The transient magnetic field as a function of time at one specific
location ($-17\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}$). The times during
which muons are stored are highlighted by gray bands. The shown field
transients are scaled up to the ESQ operation voltage.

In Run-1, the transient fields from ESQs were measured in a dedicated
measurement campaign with a set of trolley NMR probes sealed inside plastic
tubes for vacuum compatibility, held in place in the center of the storage
volume on static legs sitting on the trolley rails.

The ESQs span $43.3\text{\,}\mathrm{\char 37\relax}$ of the ring and are
grouped into four stations, each consisting of a short and a long section. The
azimuthal dependence was mapped coarsely for one such section. Significant
differences in the oscillation pattern were observed as a function of azimuth.
The long sections were approximated with two short ones. Due to the static
nature of the used probes, only one measurement per section was feasible for
most sections. The total shift of the magnetic field during the times the
muons are stored averaged around the ring was determined from these spatially
sparse measurements, leading to the dominant systematic uncertainty of the
Run-1 result [5].

In dedicated measurement campaigns, the identical sealed NMR probes were
mounted on a frame that can be moved around the ring using the trolley
infrastructure. The NMR probes were pulsed and read out in the same scheme
used in Run-1 through a dedicated multiplexer of the fixed probe systems, now
through the $\sim$$50\text{\,}\mathrm{m}$ long trolley cable. This scheme
allows mapping of the effect with finer resolution, significantly improving
the precision. In the summer of 2020, a quarter of the ring was mapped, and in
summer 2021, the full ring was mapped. The top plot in Fig. 26 shows the
transients for all times as a function of azimuth around the ring. The
measurements were performed at a reduced ESQ voltage of
$14\text{\,}\mathrm{k}\mathrm{V}$. The confirmed voltage-squared dependence
was used to scale the measurement to the nominal ESQ operations voltage of
$18.2\text{\,}\mathrm{k}\mathrm{V}$.

The effect of the magnetic field perturbations on $\omega_{a}$ in a particular
fill at a particular azimuthal position is estimated by a linear fit of the
magnetic field transient over the muon storage time of around
$700\text{\,}\mathrm{\SIUnitSymbolMicro s}$ of this fill. The effect
accumulates over the muon lifetime in the storage ring [7]. The azimuthally
resolved effect from the different measurement positions is averaged around
the ring, accounting for the different azimuthal spacings between the
measurements. Segments outside vacuum chambers containing ESQs and where no
time-dependent field perturbations are observed don’t contribute. Table 21
shows the total correction
$B_{Q}$=$-21.0(19.5)\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$ due to transient
magnetic fields from the ESQ and lists the corresponding uncertainties, which
are discussed in more detail below.

The frequency extraction from NMR FID signals requires a minimal length of
more than $\sim$$0.5\text{\,}\mathrm{ms}$ for the required resolution. The
time scale of the observed transient changes the field within an FID. Hence,
magnetic field perturbations from outside the fit window of the transient
effect leak into the frequency. Alternatively, the phase function from
multiple FIDs with different delays with respect to the muon injection time
can be combined and fitted directly in the relevant time window. The NMR
probes have a 0.5-mm-thick aluminum shell, and the corresponding skin depth
suppresses higher-frequency components. This effect was evaluated in a
dedicated measurement. The transient caused by the ESQ was mapped partially
one year after Run-3 and around the full ring the year afterward. In addition,
starting mid-Run-3, periodic measurements at static positions were taken. The
different measurements over time are in good agreement. In addition, the fixed
probe system is used to monitor the effect of the transient from outside of
the vacuum chambers parasitically during data taking.

All the measurements are point estimates, and the values in between the
measurement points are unknown, resulting in uncertainty in the azimuthal
averaging. In addition, the mapping was performed in the center of the storage
volume. The radial dependence of the transient was measured on the diagonal
along the ESQ $0\text{\,}\mathrm{V}$-line at one location. A flat dependence
was found up to $2\text{\,}\mathrm{cm}$, where most of the muon beam is
located, and variations up to $25\text{\,}\mathrm{\char 37\relax}$ were
observed at a radius of $4\text{\,}\mathrm{cm}$, at the edge of the storage
volume. As mentioned above, the ESQ can only be operated consistently at
$14\text{\,}\mathrm{k}\mathrm{V}$ with the mapper device present.
Perturbations of the electric field from the mapping device itself might
modify the local forces on the ESQ plates and change the mechanical
oscillation of the system. Other sources for uncertainties are fill-by-fill
intensity variations not accounted for the averaging between the 16 fills and
small changes in the time structures in the second eight bunches between
running conditions and the measurements.

Table 21: Correction and associated uncertainties to $\tilde{\omega}^{\prime}_{p}$ due to transient magnetic fields caused by the pulsing of the ESQ system.  Description | Correction (ppb) | Uncertainty (ppb)
---|---|---
frequency extraction |  | 5
skin depth |  | 2
stability over time |  | 8
azimuthal averaging |  | 11
transverse dependence |  | 5.3
measurement apparatus |  | 10.5
fill-by-fill variations |  | 2
second bunch train |  | 5
Subtotal | -21.0 | 19.5

###  VI.8 Summary and differences with respect to Run-1

The dataset averaged $\tilde{\omega}_{p}^{\prime}$ are listed in Table 26. All
non-negligible uncertainties are summarized in Table 22. For uncertainties
that have been determined on a probe-by-probe basis, the uncertainties are
translated to multipole moments and further to $\tilde{\omega}_{p}^{\prime}$
taking the correlation between moments and the spatial and temporal muon
distribution into account. Uncertainties are highly correlated and thus
treated as fully correlated, except the Brownian bridge-based tracking
uncertainty, which is random in nature and reduced by combining datasets.
Calibration constants and corrections are taken into account in the final
$\tilde{\omega}_{p}^{\prime}$ and are not listed individually. The total
uncertainty on the muon-weighted magnetic field, including corrections from
magnetic field transients, is
$\leq$52\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$, a factor of ${\sim}2$
improvement compared to the Run-1 analysis [7]. The main reason is the
improved understanding of the electrostatic quadrupole transient due to
additional measurements. Overall, the current uncertainty budget is well below
the systematic uncertainty goal from the technical design report of
$<$70\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$.

Table 22: Summary of uncertainties on $\tilde{\omega}_{p}^{\prime}$ for each step in the analysis. A detailed breakdown of each contribution is given in the corresponding section. A single value per line indicates the same value for all datasets. All contributions are assumed to be fully correlated, except the Brownian bridge uncertainty in the Tracking section, which is treated as statistical uncertainty. Description | Uncertainty (ppb) | Section
---|---|---
| Run-2 | Run-3a | Run-3b |
Calibration probe | 8.9 | VI.2
Trolley calibration | 17.8 | VI.3
Spacial Field Maps | 37.2 | 38.5 | 38.1 | VI.4
Tracking | 17.3 | 16.5 | 17.8 | VI.5
Muon Weighting | 13.4 | 7.9 | 6.9 | VI.6
Transient Booster | 7 | VI.7
Transient Kicker | 13.3 | VI.7.1
Transient ESQ | 19.5 | VI.7.2
Sub total uncorrelated | 15.4 | 10.7 | 16.0 |
Sub total correlated | 51.3 | 52.0 | 50.6 |

The major differences in the Run-2/3 analysis of $\tilde{\omega}^{\prime}_{p}$
with respect to the Run-1 analysis are listed below:

  * •

In Run-1, the transverse multipole expansion was truncated at
$N_{\mathrm{max}}=9$, for Run-2/3, $N_{\mathrm{max}}=12$ was used.

  * •

In the frequency extraction of the trolley FIDs, in Run-2/3, slightly earlier
times in the phase function fits were used compared to Run-1.

  * •

While in Run-1 only one of the barcode readers was used to determine the
azimuthal position, in Run-2 and Run-3 the second barcode reader is used as a
cross-check, increasing reliability. This has the advantage that measurements
in the small gaps between adjacent vacuum chamber positions can still be
reconstructed even though one of the barcode readers fails. In addition,
better timing alignment of the barcode and encoder systems is possible due to
additional timing information in the raw data of both systems. These two
developments led to improved reliability of the position determination.

  * •

For Run-2/3, the trolley calibration procedures were improved with respect to
Run-1. The improvements include the following: 1) moving the trolley further
from the calibration position during measurements with the calibration probe;
2) revised corrections to the calibration-probe mounting configuration; 3)
inclusion of improved magnetic image measurements described in Sec. VI.2; 4)
Corrections for second-order gradients near the calibration position due to
the different effective sample volumes of the trolley probe and calibration
probe.

  * •

A ground loop issue that was present in Run-1 was removed between Run-1 and
Run-2.

  * •

Higher-order multipole moments are smaller in Run-2/3 than in Run-1. They were
shimmed out better after Run-1 due to the availability of trolley calibration
constants. This reduces the uncertainty from the rail misalignments, as well
as from muon weighting.

  * •

The temperature dependence of the trolley NMR probes was measured more
precisely for Run-2/3. It was evaluated as $-0.8\pm
2.0\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}\mathrm{/}\mathrm{\SIUnitSymbolCelsius}$.
In Run-1, a temperature dependence of $-0\pm
5.\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}\mathrm{/}\mathrm{\SIUnitSymbolCelsius}$
was used.

  * •

The rate of change parameter $M$ used for the uncertainty evaluation of the
field tracking with a random walk or Brownian bridge model was evaluated in
Run-1 station-by-station, manually including observed correlations. This
approach was chosen due to the statistics of field periods. In Run-2/3, $M$ is
evaluated directly from azimuthal averages, which intrinsically includes
correlations.

  * •

Additional measurements with a dedicated magnetometer with significantly
reduced vibrations lowered the uncertainty on the measurements of transient
magnetic fields from the kickers.

  * •

An extensive azimuthal mapping of the transient magnetic field from the ESQ
system reduced the corresponding uncertainty significantly.

##  VII Overall $\mathbf{\omega_{a}/\tilde{\omega}^{\prime}_{p}}$ consistency
checks

The $R^{\prime}_{\mu}$ ratio values have been investigated for any
inconsistencies and unexpected correlations to external parameters. These
external parameters are representative of the conditions that the experiment
Run-2/3 data had been collected in. Eight external parameters had been
identified for these checks, namely, average temperature of the muon storage
ring, average vacuum pressure of the muon storage ring, magnet current,
inflector current, time of data collection since last magnet ramp up, time of
data collection (day or night), amplitude of CBO and $k_{loss}$.

###  VII.1 Methodology

In order to perform these checks the data were split into five slices based on
the external parameter values, for each of the three Run sets. The
$\omega_{a}$ and $\omega_{p}$ values with their respective uncertainties are
subsequently extracted from each of the fifteen data slices. These in turn are
used to calculate the $R^{\prime}_{\mu}$ ratio and its uncertainty for each of
the data slices. It should be noted that for this study the beam dynamics and
magnetic field transient corrections are assumed to be constant within the
Run-2, Run-3a, and Run-3b datasets.

For the purposes of these tests, we perform a $\chi^{2}$ minimization on the
calculated $R^{\prime}_{\mu}$ ratios and their uncertainties in order to
evaluate the overall optimal error weighted $R^{\prime}_{\mu}$ ratio value for
each external variable studied. Thereafter, the p-value for the sliced
$R^{\prime}_{\mu}$ ratios against the optimal $R^{\prime}_{\mu}$ ratio is
extracted.

Furthermore, the sliced $R^{\prime}_{\mu}$ ratio values are plotted against
the external parameter values for each of the slices and fitted against a
constant. The pull histograms for these plots are then evaluated for any
skewness in order to identify dependencies on the external parameters at hand.

###  VII.2 Results

The p-values for all the different external parameter cross-checks performed
using the methodology described above are summarised in Table 23. In the
Run-2, Run-3a, and Run-3b overall consistency study, none of the sliced
$R^{\prime}_{\mu}$ ratio values show any direct dependency on the eight
investigated external parameters, with p-values within nominal ranges.
Moreover, the pull histograms for each of the external parameter slicing fits
show a Gaussian distribution of the data centered around $0.0\pm 0.2$.

Table 23: $R^{\prime}_{\mu}$ ratio vs. external parameter value with optimal $R^{\prime}_{\mu}$ ratio fit p-values, for combined Run-2, -3a and -3b slicings. External variable | p-value
---|---
Average ring temperature | 0.43
Inflector current | 0.75
Magnet current | 0.13
Time since magnet ramp up | 0.91
Day/Night split | 0.70
Average vacuum pressure | 0.75
Amplitude of CBO | 0.77
$k_{loss}$ | 0.93

The magnet current slicing has a relatively small p-value due to a pull from
the slices containing data from runs 2F, 2H, and 3N. Detailed analysis cross-
checks have been made for datasets 2F, 2H and 3N, by $\omega_{a}^{m}$ and
$\tilde{\omega}^{\prime}_{p}$ analyzers. In these cross-checks no
extraordinary anomaly was discovered by the analyzers, consequently, the
datasets remain valid datasets with statistical fluctuation.

Additionally, a slicing over different datasets was also performed in order to
examine the consistency of the extracted $R^{\prime}_{\mu}$ ratio values over
different datasets and time. The results for this data splitting can be
visualized in Fig. 27. There are no observed inconsistencies for the
$R^{\prime}_{\mu}$ ratio values extracted for different datasets.

Figure 27: $R^{\prime}_{\mu}(T_{r})$ versus data subset. The fit line has a
$\chi^{2}$/ndf$=19.31/19$ with a p-value of $44\text{\,}\mathrm{\char
37\relax}$.

##  VIII Calculation of $\mathbf{a_{\mu}}$

Following Eq. 4, for each dataset, the measured $\omega_{a}^{m}$ is corrected
by adding the beam dynamics corrections, and the ratio
$R^{\prime}_{\mu}(T_{r})=\omega_{a}/\tilde{\omega}^{\prime}_{p}(T_{r})$ is
computed. All uncertainty contributions to $\omega_{a}^{m}$, to the beam
dynamics corrections and to $\tilde{\omega}^{\prime}_{p}(T_{r})$, are
propagated to $R^{\prime}_{\mu}(T_{r})$.

Table 24: Values and uncertainties of the $\mathcal{R}_{\mu}^{\prime}$ terms in Eq. (4) and uncertainties due to the external parameters in Eq. (56) for $a_{\mu}$. Positive $C_{i}$ increase $a_{\mu}$; positive $B_{i}$ decrease $a_{\mu}$. The $\omega_{a}^{m}$ uncertainties are decomposed into statistical and systematic contributions. Quantity | Correction | Uncertainty | Section
---|---|---|---
| (ppb) | (ppb) |
$\omega_{a}^{m}$ statistical | - | 201 | IV.11
$\omega_{a}^{m}$ systematic | - | 25 | IV.10
$C_{e}$ | 451 | 32 | V.1
$C_{p}$ | 170 | 10 | V.2
$C_{ml}$ | 0 | 3 | V.3
$C_{dd}$ | -15 | 17 | V.4
$C_{pa}$ | -27 | 13 | V.5
$\langle\omega_{p}^{\prime}\times M\rangle$ | - | 46 | VI.8
$B_{K}$ | -21 | 13 | VI.7.1
$B_{Q}$ | -21 | 20 | VI.7.2
$\mu_{p}^{\prime}($34.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$)/\mu_{e}$ | - | 11 |  [45] [57]
$m_{\mu}/m_{e}$ | - | 22 | [58]
$g_{e}/2$ | - | 0 | [59]
Total systematic | - | 70 |
Total external parameters | - | 25 |
Totals | 622 | 215 |

Uncertainty contributions that are assumed to be fully correlated between
different Run-2/3 datasets and also between different measurements by the
Fermilab Muon $g\\!-\\!2$ (E989) collaboration are tracked separately from the
statistical uncertainties and the other uncertainty contributions that can be
considered uncorrelated: the magnetic field uncorrelated uncertainty. The
correlation matrix between the ratios is reported in Table 25. The three
$R^{\prime}_{\mu}(T_{r})$ values are found to be statistically consistent and
are fit to obtain the measured $R^{\prime}_{\mu}(T_{r})$ for the Run-2/3
sample. The fit $\chi^{2}$ probability is about 20%. The results are
summarized in Table 26.

Table 25: Correlation matrix of the Run-2/3 datasets measurements of $R^{\prime}_{\mu}(T_{r})$. $R^{\prime}_{\mu}(T_{r})$ | Run-2 | Run-3a | Run-3b
---|---|---|---
Run-2 | 1.00 | 0.05 | 0.03
Run-3a | 0.05 | 1.00 | 0.03
Run-3b | 0.03 | 0.03 | 1.00
Table 26: Run-2/3 datasets measurements of $\omega_{a}$, $\tilde{\omega}^{\prime}_{p}(T_{r})$, and their ratios $R^{\prime}_{\mu}(T_{r})$ multiplied by 1000. Dataset |  $\omega_{a}/2\pi$ (Hz) |  $\tilde{\omega}^{\prime}_{p}(T_{r})/2\pi$ (Hz) | $R^{\prime}_{\mu}(T_{r})\times 1000$
---|---|---|---
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)

Over the course of this analysis, three small errors in the Run-1 analysis [5]
were identified. The total shift in the previous result due to these errors is
$28\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$, resulting in
$R^{\prime}_{\mu}(T_{r})_{\text{Run-1}}=0.0037073004(16)(6)$. The measured
$R^{\prime}_{\mu}(T_{r})_{\text{Run-2/3}}=0.00370730088(75)(26)$ is combined
with the Run-1 result [5], assuming that the systematic uncertainties are
fully correlated, to obtain the Fermilab experimental measurement,
$R^{\prime}_{\mu}(T_{r})_{\text{Run-1/2/3}}=0.00370730082(68)(31)$. This value
is combined with the BNL measurement of $R_{\mu}$ for free protons in vacuum
[2], $R_{\mu}=0.0037072063(20)$, after converting it using the measured
diamagnetic shielding correction $\sigma_{p^{\prime}}(T_{r})$ [45]:

$\displaystyle
R^{\prime}_{\mu}(T_{r})=\frac{R_{\mu}}{1-\sigma_{p^{\prime}}(T_{r})}=0.0037073019(20)~{}.$
(55)

We compared the systematic uncertainties for the BNL and FNAL measurements
and, due to the significant changes in the beam characteristics and detectors
between the experiments, concluded that those uncertainties were largely
uncorrelated between the two experiments. The resulting experimental average
is $R^{\prime}_{\mu}(T_{r})_{\text{Exp}}=0.00370730095(70)$.

The muon magnetic anomaly is computed from

$a_{\mu}=R^{\prime}_{\mu}(T_{r})\frac{\mu^{\prime}_{p}(T_{r})}{\mu_{e}(H)}\frac{\mu_{e}(H)}{\mu_{e}}\frac{m_{\mu}}{m_{e}}\frac{g_{e}}{2}~{}.$
(56)

Here $\mu^{\prime}_{p}(T_{r})/\mu_{e}(H)$ is the ratio of the magnetic moment
of the proton in a spherical water sample at 34.7
$\mathrm{\SIUnitSymbolCelsius}$ and the magnetic moment of the electron in a
hydrogen atom [45] ($10.5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$).
${\mu_{e}(H)}/{\mu_{e}}$ is the ratio of the magnetic moment of the electron
in a hydrogen atom and the magnetic moment of the free electron in vacuum,
obtained with a theory QED calculation [57], whose precision is limited to
$100\text{\,}\mathrm{p}\mathrm{p}\mathrm{t}$ by the number of reported digits.
${m_{\mu}}/{m_{e}}$ is the ratio of the muon and electron masses
($22\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$), taken from the CODATA 2018 fit
[58], primarily driven by the LAMPF 1999 measurements of muonium hyperfine
splitting [60]. ${g_{e}}$ is the electron gyromagnetic factor, computed from
the electron anomaly $a_{e}=(g{-}2)/2$ world average [59]
($100\text{\,}\mathrm{p}\mathrm{p}\mathrm{t}$), dominated by [1].

The measured muon magnetic anomaly for this measurement, this measurement
combined with our Run-1 result, and the combined BNL and FNAL results are

$\displaystyle a_{\mu}^{\text{FNAL Run-2/3}}$ $\displaystyle=$ $\displaystyle
116\,592\,057(25)\times
10^{-11}~{}($0.21\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$),$ $\displaystyle
a_{\mu}^{\text{FNAL Run-1/2/3}}$ $\displaystyle=$ $\displaystyle
116\,592\,055(24)\times
10^{-11}~{}($0.20\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$),$ $\displaystyle
a_{\mu}^{\text{Exp}}$ $\displaystyle=$ $\displaystyle 116\,592\,059(22)\times
10^{-11}~{}($0.19\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$).$

These are displayed in Fig. 28. Values of $R^{\prime}_{\mu}(T_{r})$ and
$a_{\mu}$ with extra digits to facilitate further calculations without loss of
precision due to rounding are provided in the supplement material.

Figure 28: From top to bottom: experimental values of $a_{\mu}$ from BNL E821,
the FNAL 2021 measurement (FNAL Run-1), this measurement (FNAL Run-2/3), the
FNAL combined measurement (FNAL Run-1 + 2/3), and the combined experimental
average (Exp. average). The inner tick marks indicate the statistical
contribution to the total uncertainties.

##  IX Comparison to Theory

In recent years, all aspects of the SM theory prediction $a_{\mu}^{\rm SM}$
have been scrutinized and refined with continued theoretical and computational
efforts. These were summarized by the $g\\!-\\!2$ Theory Initiative [10],
using results from Refs. [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80]. While the QED and electroweak contributions are
widely considered non-controversial, the SM prediction of the muon $g\\!-\\!2$
is limited by our knowledge of the vacuum fluctuations involving strongly
interacting particles, comprising effects called hadronic vacuum polarization
(HVP) and hadronic light-by-light scattering. The latter is currently known at
a level of precision comparable to $a_{\mu}^{\text{Exp}}$, and it is the
leading HVP contribution to the muon magnetic anomaly, denoted by
$a_{\mu}^{\text{HLO}}$, that gives the dominant uncertainty to the SM
prediction. These effects cannot be computed at low-energy scales due to the
non-perturbative nature of QCD at large distances. It is possible to overcome
this problem by means of a dispersion relation technique involving
experimental data on the cross-section of electron-positron annihilation into
hadrons, $e^{+}e^{-}\to\text{hadrons}$. In the last 20 years, the worldwide
efforts of experiments working on $e^{+}e^{-}\to\text{hadrons}$ data in the
energy range below a few GeV have achieved the remarkable uncertainty of 0.6%
on $a_{\mu}^{\text{HLO}}$ [81, 10]. In addition, in the last few years, there
has been significant progress on the first-principles calculation of
$a_{\mu}^{\text{HLO}}$ using lattice QCD which, however, was not yet as
precise as the data-driven dispersive approach compiled in [10]. In 2021, the
BMW collaboration published the first lattice calculation of
$a_{\mu}^{\text{HLO}}$ with sub-percent precision [9]. This result would move
$a_{\mu}^{\rm SM}$ towards $a_{\mu}^{\rm Exp}$ and is compatible with the “no
new physics” scenario but discrepant with the dispersive approach. While the
evaluation of the whole $a_{\mu}^{\text{HLO}}$ from the other lattice groups
is in progress, excellent agreement between the different lattice groups is
found for the so-called intermediate window observable [82, 83, 84, 85, 86].
The evaluation of this intermediate window observable shows a $4$ standard
deviation discrepancy between the lattice and the data-driven computation. On
the $e^{+}e^{-}\to\text{hadrons}$ side, in addition to the known discrepancy
between KLOE [87, 88, 89, 90] and BaBar [91, 92], the recent CMD-3 [93, 94]
result has shown a discrepancy with all previous measurements used in [10].
The origin of this discrepancy is currently unknown and efforts are in
progress to clarify the situation [95]. In view of this situation, a firm
comparison with the theory cannot be established at the moment.

##  X Conclusion

We have reported a measurement of the muon magnetic anomaly to
$0.20\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ precision, based on the first
three years of data. This measurement represents the most precise
determination of this quantity. The statistical and systematic errors have
been reduced by a factor of two with respect to our first measurement [5], due
to greater than four times more data and improved running conditions, analysis
procedures, dedicated measurements, and systematic studies. This measurement
is still statistically limited and the analysis of the remaining data from
three additional years of data is expected to result in an improved
statistical precision by another factor of approximately two.

###### Acknowledgements.

We thank the Fermilab management and staff for their strong support of this
experiment, as well as the tremendous support from our university and national
laboratory engineers, technicians, and workshops. 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, NP, and ASCR (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 (Grant No. 12211540001,
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, No.
734303 and European Union STRONG 2020 project under grant agreement No. 824093
and the Leverhulme Trust, LIP-2021-01.

##  Appendix A Correlations between $\mathbf{\omega_{a}^{m}}$ analyses

Table 27 lists the correlations coefficients between the 19 different
$\omega_{a}$ analyses. The largest allowed statistical differences are between
the event-based analyses and the energy-based analyses. Smaller allowed
statistical differences are between analyses that employ either a common
construction approach or a common histogramming method. The correlation
coefficients do not account for additional allowed systematic differences
between analysis methods.

Table 27: Table of correlation coefficients based on the allowed statistical differences between the 19 different $\omega_{a}$ analysis approaches. They include the different reconstruction procedures and different histogramming methods. They assume a 100% correlation of systematic uncertainties between analysis approaches. | C_T | E_T | I_T | S_T | W_T | B_A | C_A | E_A | I_A | S_A | W_A | B_RT | E_RT | I_RT | B_RA | E_RA | K_Q | KR_RQ
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
B_T | 0.967 | 0.999 | 0.967 | 0.999 | 1.000 | 0.900 | 0.871 | 0.884 | 0.867 | 0.884 | 0.884 | 0.993 | 0.995 | 0.963 | 0.895 | 0.904 | 0.765 | 0.824
C_T |  | 0.967 | 1.000 | 0.965 | 0.967 | 0.891 | 0.900 | 0.875 | 0.896 | 0.874 | 0.875 | 0.961 | 0.963 | 0.996 | 0.887 | 0.895 | 0.756 | 0.815
E_T |  |  | 0.967 | 0.999 | 0.999 | 0.913 | 0.885 | 0.898 | 0.880 | 0.898 | 0.897 | 0.993 | 0.996 | 0.963 | 0.909 | 0.918 | 0.753 | 0.811
I_T |  |  |  | 0.965 | 0.967 | 0.897 | 0.906 | 0.881 | 0.902 | 0.880 | 0.880 | 0.961 | 0.963 | 0.996 | 0.892 | 0.901 | 0.751 | 0.809
S_T |  |  |  |  | 0.999 | 0.915 | 0.886 | 0.900 | 0.882 | 0.902 | 0.899 | 0.992 | 0.995 | 0.961 | 0.911 | 0.920 | 0.751 | 0.809
W_T |  |  |  |  |  | 0.915 | 0.887 | 0.899 | 0.882 | 0.899 | 0.899 | 0.993 | 0.995 | 0.963 | 0.911 | 0.919 | 0.752 | 0.810
B_A |  |  |  |  |  |  | 0.994 | 1.000 | 0.994 | 0.999 | 1.000 | 0.890 | 0.886 | 0.887 | 0.991 | 0.994 | 0.688 | 0.740
C_A |  |  |  |  |  |  |  | 0.994 | 1.000 | 0.993 | 0.994 | 0.862 | 0.857 | 0.896 | 0.986 | 0.988 | 0.681 | 0.732
E_A |  |  |  |  |  |  |  |  | 0.994 | 0.999 | 1.000 | 0.875 | 0.871 | 0.871 | 0.991 | 0.994 | 0.676 | 0.727
I_A |  |  |  |  |  |  |  |  |  | 0.993 | 0.994 | 0.858 | 0.853 | 0.892 | 0.986 | 0.988 | 0.678 | 0.729
S_A |  |  |  |  |  |  |  |  |  |  | 0.999 | 0.875 | 0.871 | 0.870 | 0.990 | 0.993 | 0.677 | 0.728
W_A |  |  |  |  |  |  |  |  |  |  |  | 0.875 | 0.870 | 0.871 | 0.991 | 0.994 | 0.676 | 0.727
B_RT |  |  |  |  |  |  |  |  |  |  |  |  | 0.994 | 0.962 | 0.902 | 0.907 | 0.758 | 0.825
E_RT |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.967 | 0.895 | 0.901 | 0.767 | 0.837
I_RT |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.895 | 0.901 | 0.750 | 0.819
B_RA |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.994 | 0.682 | 0.743
E_RA |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.689 | 0.754
K_Q |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 0.994

##  Appendix B Trolley calibration constants

The trolley calibration constants, including their contributions, are listed
in Table 28. A graphic comparison is shown in Fig. 29. In addition to the
Run-2/3 average, the values and the differences from the dedicated Run-2 and
Run-3 calibration campaigns are shown, in combination with predictions from
COMSOL simulations based on a simplified trolley geometry that only takes into
account the trolley shell but not the interior details.

Table 28: Overview of trolley probe calibration constants $\delta^{\text{calib}}$ and individual contributions for run-2/3. All values are given in ppb. | $\delta^{\text{fp,tr}}$ | $\delta^{\text{fp,cp}}$ | $\delta^{\text{av}}$ | $\delta^{s,\text{img}}$ | $\delta_{n}^{\text{calib}}$
---|---|---|---|---|---
Probe | value | uncertainty | value | uncertainty | value | uncertainty | value | uncertainty | value
1 | 14.3 | 8 | 4.0 | 4.0 | 4.9 | 1.9 | -17.2 | 8.9 | 1469.0
2 | 4.0 | 4.0 | -0.2 | 2.4 | -17.8 | 8.9 | 1336.9
3 | 3.7 | 3.7 | 1.8 | 2.9 | -17.2 | 8.9 | 1523.6
4 | 4.0 | 4.0 | 2.8 | 4.0 | -17.8 | 8.9 | 1358.3
5 | 4.9 | 4.9 | -1.0 | 3.1 | -17.2 | 8.9 | 1514.4
6 | 3.6 | 3.6 | 9.4 | 4.4 | -20.2 | 9.1 | 1734.5
7 | 3.2 | 3.2 | -9.5 | 4.7 | -19.4 | 8.9 | 1903.0
8 | 3.2 | 3.2 | -2.8 | 2.9 | -17.8 | 8.9 | 1195.8
9 | 3.1 | 3.1 | 7.9 | 4.0 | -17.2 | 8.9 | 1367.2
10 | 3.1 | 3.1 | 8.6 | 3.4 | -17.8 | 8.9 | 421.1
11 | 3.1 | 3.1 | 19.7 | 9.1 | -19.4 | 8.9 | 2878.3
12 | 3.7 | 3.7 | 40.9 | 8.1 | -20.2 | 9.1 | 1787.1
13 | 4.4 | 4.4 | -4.4 | 4.4 | -19.4 | 8.9 | 1993.8
14 | 5.7 | 5.7 | 1.5 | 6.1 | -17.8 | 8.9 | 1263.9
15 | 6.5 | 6.5 | -15.2 | 6.5 | -17.2 | 8.9 | 1193.0
16 | 5.5 | 5.5 | -1.0 | 4.4 | -17.8 | 8.9 | 337.2
17 | 4.2 | 4.2 | 4.9 | 8.3 | -19.4 | 8.9 | 2738.5
Figure 29: Top: Trolley calibration constants per trolley probe for Run-2
(blue) and Run-3 (orange) and the combination (black). Predictions from COMSOL
simulations (gray) with simplified geometry, only considers the trolley shell,
show qualitative consistency. Bottom: The difference of Run-2 and Run-3
calibration constants with respect to the combined value that are used for
this analysis.

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