DCA > $500\leavevmode\nobreak\ \mathrm{\mu m}$. A hit with a small DCA is not
used in the alignment. This is required to prevent a residual discontinuity as
we cross the LR boundary (Equation B.29-Equation B.32).

  * •

$p$-value > 0.005. Only tracks that have a reasonable fit quality are used.

  * •

Hits > 9. Only tracks that have hits in at least three modules are used.

  * •

$\nicefrac{{$p_{z}$}}{{$p$}}$ > 0.93. Tracks that have large curvature are
removed, required by the approximations used in Equation B.40.

####  7.3.7 Iterative alignment

The process of iterative alignment uses the initial module offsets from PEDE
in the subsequent re-tracking. In iteration two onwards, the tracks are formed
through a corrected detector geometry, and only small alignment corrections
are returned in the subsequent PEDE alignment, as shown in Figure 7.20. The
iterative alignment is considered to converge when the returned alignment
results agree with a previous iteration to within the PEDE error (few
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$).

Figure 7.20: The radial alignment results per module in a tracker station in
the simulation are shown. The results of the reconstructed misalignment, from
the four iterations (I1-I4), are shown as the difference between the truth (T)
and a reconstructed misalignment. The reconstructed misalignment from
iterations 3 and 4 overlap, indicating convergence.

###  7.4 Alignment results in simulation

The mean values of the residuals per module before and after the alignment are
shown in Figure 7.21. The alignment results in both stations after three
iterations are shown in Figure 7.22. The improvement in the mean
$p\textnormal{-value}$ and number of reconstructed tracks as a function of
number of iterations is shown in Figure 7.23.

Figure 7.21: The mean values of the residuals per module before and after the
alignment in the simulation. The mean residual is expected to be at $0$ for an
aligned detector.

Figure 7.22: The final simulation alignment results showing that alignment
stability was reached. S12 was not misaligned vertically. The alignment
results from the previous iteration and the current iteration overlap. The
absolute mean reconstructed ($\langle|\mathrm{Reco}|\rangle$) and the absolute
mean truth ($\langle|\mathrm{Truth}|\rangle$) misalignments per module are
indicated. The alignment convergence, with respect to the truth, was
established within $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radially and vertically,
respectively.

The overall input (i.e. truth) misalignment in station 18 was greater; hence,
after the alignment, the improvement in station 18 is more significant than in
station 12. The alignment stability was reached, and the alignment convergence
was established within $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radially and vertically,
respectively, in simulation.

Figure 7.23: The improvement in the mean $p\textnormal{-value}$ and the number
of tracks as a function of the alignment iteration in simulation. The overall
input misalignment in station 18 was greater, hence, after the alignment, the
improvement in station 18 is more significant than in station 12.

###  7.5 Alignment results with data

The mean values of the residuals per module before and after the alignment,
using run 15922, are shown in Figure 7.24. The alignment results in both
stations after three iterations are shown in Figure 7.25. The results were
obtained with $\mathcal{O}(10^{5})$ tracks, and were then used in the track
reconstruction.

Figure 7.24: The mean values of the residuals per module before and after the
alignment in data. The mean residual is expected to be at $0$ for an aligned
detector.

Figure 7.25: The alignment results with data. The absolute mean reconstructed
($\langle|\mathrm{Reco}|\rangle$) misalignment per module is indicated. The
alignment stability was reached, as seen by the alignment results from the
previous iteration and the current iteration completely overlapping.

The improvement in the mean $p\textnormal{-value}$ and number of reconstructed
tracks as a function of number of iterations is shown in Figure 7.26. The
change in the distribution of $p\textnormal{-values}$ and momentum, as well as
in the beam extrapolation are shown in Figure 7.27 and Figure 7.28,
respectively.

Figure 7.26: The improvement in the mean $p\textnormal{-value}$ and the number
of tracks as a function of the alignment iteration. Figure 7.27: The
distribution of track $p\textnormal{-values}$ and momentum in both stations
before and after the alignment. Stations 12 and 18 have 2.0% and 6.4%
improvement in the mean track $p\textnormal{-value}$, respectively, with only
a small change in the mean track momentum. The peak at 2700 MeV is due to
lost-muons (see Section 3.4.6), as the alignment considers all tracks, not
just $e^{+}$ from $\mu^{+}$ decays, that have passed the alignment cuts
defined in Section 7.3.6.

Stations 12 and 18 have a 2.0% and 6.4% improvement in the mean track
$p\textnormal{-value}$, respectively. This implies that after the alignment
the quality of the reconstructed tracks is higher. Moreover, due to the
improved alignment, more tracks are reconstructed from the same hits in
station 12 and 18, by 1.5% and 4.5%, respectively.

Figure 7.28: The vertical and radial distributions of the extrapolated beam
positions in both stations before and after the alignment.

The extrapolated tracks (for $t>$ $30\text{\,}\mathrm{\SIUnitSymbolMicro s}$)
from stations 12 and 18 have a radial shift towards the centre of the ring of
$164\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$538\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively. The extrapolated
tracks have a vertical shift downwards of
$73\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$144\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for stations 12 and 18,
respectively.

####  7.5.1 Alignment monitoring system and database

As seen from Figure 7.26, the most significant alignment corrections come
after the first iteration. Subsequent iterations yield corrections comparable
to the PEDE uncertainty. Hence, an assessment of the alignment stability
across all $g-{2}$ runs is possible, by running the alignment directly on the
reconstructed tracks from all runs without an iterative alignment. The radial
alignment results per module in station 12 over a 60-hour period (Run-1a
dataset) are shown in Figure 7.29.

Figure 7.29: The radial alignment results in the eight modules of station 12
across 60 hours of $g-{2}$ data (Run-1a dataset).

The alignment stability in both stations, radially and vertically, across
Run-1 and Run-2 is shown in Figure 7.30.

Figure 7.30: The alignment stability, radially and vertically, across Run-1
(online run >10000) and Run-2 (online run >20000) in both stations. The
alignment stability is displayed as the difference between the mean alignment
for a given run and the alignment from a fixed reference run (run 15922). S12
displays stable alignment throughout the two Runs. The deviations in modules 4
(M4) and 6 (M6) in S18 are due to physical module swaps [83] before Run-2, and
after run 25086.

The change in the internal alignment of the two modules is seen in Figure
7.30: before the start of Run-2 in module 4, and after run 25086 in module 6.
These modules were replaced [83] with spare modules, since the existing
modules had developed a slightly higher gas leak rate into the storage ring
vacuum than the specification. New alignment constants for the replaced
modules were derived. The alignment constants were written into a PostgreSQL
database, where each set of constants is associated to a given range of runs.

####  7.5.2 Outlook

The internal alignment, radially and vertically, of the two tracker stations
in Run-1 and Run-2 has been successfully determined. Moreover, the alignment
manual [3], to monitor the alignment in the future, has been produced. There
are, however, a few more activities that are planned to further refine and
improve the alignment:

  * •

Scattering target. A placement of a scattering target, as shown in Figure
7.31, in a known position in the $g-{2}$ storage ring will give a precise
location of the starting position of the tracks. This project is being
developed by S. Foster [84].

  * •

Internal alignment with rotations. The current tracking algorithm (see Section
5.3) assumes that the tracker planes are parallel to each other. The use of a
Kalman filter algorithm, currently in development by A. Luca [85], will allow
for rotations of tracker planes, and hence, will improve the alignment.

(a)

(b)

Figure 7.31: The implementation of a tungsten scattering target in the
simulation [84]. (a) Target location inside of the $g-{2}$ ring relative to
the tracker station. (b) Vertical and radial extrapolated beam positions. The
target’s grid structure in seen in the tracker.

####  7.5.3 Alignment impact on the systematic uncertainties

#####  7.5.3.1 Pitch correction ($\omega_{a}$)

The beam extrapolation determines the radial and vertical positions of the
muon beam in the storage ring. The accuracy of this determination is affected
by detector calibrations, such as the alignment. The internal alignment of the
tracking detectors is now established, with the $1\sigma$ uncertainties on the
mean and width of the beam given in Table 7.1. In the table, the combined
uncertainty, in both stations, from Figure 7.28 was taken.

|  $\mathrm{\sigma_{dR}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dR_{\mathrm{width}}}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dV}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dV_{\mathrm{width}}}}$ [$\text{\,}\mathrm{mm}$]
---|---|---|---|---
S12 $\oplus$ S18 | 0.351 | 0.109 | 0.109 | 0.035
Table 7.1: The contribution of the internal alignment to the beam
extrapolation uncertainty in both stations.

However, the alignment is now accounted for, as the correct module positions
are now used in the track reconstruction. For example, in Chapter 6, the
uncertainty on the pitch correction, $\Delta C_{\mathrm{pitch}}$, from an
averaged randomised internal misalignment was estimated to be 1.5 ppb. After
the implementation of the internal alignment, this uncertainty was eliminated.
This has reduced the total contribution to the uncertainty on the pitch
correction from tracking (see Section 3.4.1) from 8.6 ppb to 8.4 ppb.

#####  7.5.3.2 Field convolution ($\omega_{p}$)

The magnetic field is convoluted with the beam profile measured by the
tracking detectors to find the average field experienced by the muons before
decay, as described in Section 5.3.7. As previously discussed, the measurement
of the beam profile from the tracking detectors is affected by the internal
alignment. A study of an impact of translations on the extrapolated beam
profile from the tracking detectors on the field measurements was performed by
Jason Bono and Saskia Charity [71]. Given the scale of translations induced by
the alignment in Figure 7.28, it is possible to estimate the corresponding
uncertainties on $\langle B\rangle$, and hence $\omega_{p}$ (see Equation
3.2). The results for both stations are given in Table 7.2, with the average
vertical and radial uncertainties on $\langle B\rangle$ of 3.8 ppb and 0.7
ppb, respectively. After the implementation of the internal alignment, these
uncertainties were eliminated. This compares to the overall goal of a 70 ppb
uncertainty on the determination of $\omega_{p}$ at the end of the
experiment’s data taking.

S12 radial | S18 radial | S12 vertical | S18 vertical
---|---|---|---
1.5 ppb | 6.1 ppb | 0.4 ppb | 0.9 ppb
Table 7.2: The alignment contribution to the uncertainty on $\langle
B\rangle$.

##  Chapter 8 Detector curvature

###  8.1 Introduction

A radial curvature in the positions of the tracker modules, as shown in Figure
8.1, changes the measurement of the momentum of the tracks, and the
extrapolated beam position. As described in Section 7.3.4, this detector
curvature must be measured and constrained by a method that is independent of
the internal alignment. One such method is described here, along with the
derivations of the tracking systematic uncertainties arising from the
curvature.

Figure 8.1: An illustration of radially misaligned detectors that cause an
erroneous momentum measurement. The positions of the eight tracker modules are
shown with positive and negative curvatures. Positive curvature is defined as
away from the centre of the $g-{2}$ storage ring.

###  8.2 Impact on alignment and tracking

If the internal alignment is established without constraining for the radial
detector curvature, the recovered alignment results will contain an erroneous
curvature. Even for a case of no input misalignment in simulation, the absence
of the correct constraints will introduce this erroneous curvature, as shown
in Figure 8.2.

(a)

(b)

Figure 8.2: An unconstrained detector curvature yields an erroneous result.
(a) Truth misalignment and reconstructed alignment radially. (b) A circle fit
to the reconstructed module positions from (a).

Re-tracking with the modules placed in the suggested position from Figure 8.2
changes the momentum distribution of the tracks, but not their $\chi^{2}$
distribution, as shown in Figure 8.3. This is suggestive of the local minima
problem – Millepede-II minimises the $\chi^{2}$ function, and without
providing more information, this wrong solution could be a valid solution.

(a)

(b)

Figure 8.3: The detector curvature effect on the reconstructed tracks: (a)
change in the mean momentum due to the detector curvature, (b) the $\chi^{2}$
distribution of tracks is unaffected.

Moreover, if a module is misaligned internally, this is manifested as a
deviation from the overall radial detector curvature in that module, as shown
in Figure 8.4.

(a)

(b)

Figure 8.4: An unconstrained detector curvature with an additional internal
misalignment in one module. (a) Truth misalignment and reconstructed alignment
radially. (b) A circle fit to the reconstructed module positions from (a).

However, if the correct constraints from Equation C.4 are used, the radial
detector curvature effect is removed completely as shown in Figure 8.5.

(a)

(b)

Figure 8.5: Constrained radial detector curvature with: (a) no misalignment,
and (b) internal misalignment.

###  8.3 Methodology

This potential radial detector curvature, $a$ (see Equation 7.30), needs to be
measured independently of the internal alignment, analogous to the global
alignment measurements, described in Appendix A. One property that makes this
measurement possible is the momentum-dependence, $p$, of the extrapolated
tracks with respect to the input detector curvature, as shown in Figure 8.6.

(a)

(b)

(c)

Figure 8.6: The high momentum, $p$, tracks have the largest response to the
input detector curvature, $a$, as seen for the positive detector curvature in
(a) and for the negative detector curvature in (b), shown here as a fraction
of the measured momentum over the truth (no curvature) momentum. The overall
mean momentum of the tracks changes depending on the applied curvature as seen
in (c).

A standalone toy-model for this momentum-dependence of the curvature as a
function of momentum is shown in Figure 8.7.

Figure 8.7: The momentum change ($\Delta p$) caused by a change in the
detector curvature, $a$, as a function of momentum.

The previously seen effect is confirmed: larger curvatures have a larger
impact on the track momentum, and high momentum tracks have the largest
response to a given curvature. To further study this effect, laser-survey data
(see Appendix A) of the modules was used to define a reasonable curvature
range of the modules.

The laser-survey alignment has an error of
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ on an individual module position.
This error can be used to fit a nominal (best fit) curve to the module
positions. The curvature parameter ($a$) is extracted directly from a fit to
Equation 7.30, along with the curvature parameters that lie within a 1$\sigma$
band of the nominal curvature. These are obtained using the Mahalanobis method
[86]. For a three parameter fit, there are 27 possible Mahalanobis fits. These
fits are shown in Figure 8.8, and the $a$ parameter for these fits are shown
in Figure 8.9.

Figure 8.8: The 27 Mahalanobis fits to the survey measurements of station 18.
$\Delta R$ is defined as the difference between the assumed module position
and the position from the survey. The range of these fits encompasses $\pm
1\sigma$ around the nominal curvature. Figure 8.9: The 27 parameters from the
Mahalanobis 1$\sigma$ band fits and the associated curvature ($a$) values; $b$
was defined in Equation 7.30 as the rotation parameter.

####  8.3.1 Data-simulation comparison

Equation 7.30 and the estimated curvature parameters from Figure 8.9 were used
to misalign tracker modules in a curved position in the simulation. A
comparison of the various curvatures with data, where this additional
curvature was not applied, was then performed. The assumption is that the
simulated curvature that best matched the data is the corresponding curvature
in data. The comparison criteria was the extrapolated radial beam position, as
shown in Figure 8.10, for a selection of the most closely matched curvatures
in station 18. The extrapolation on data was performed after internal and
external alignments as described in Section 7.5 and Appendix A, respectively.
Moreover, the simulation’s beam distribution was tuned to match the data in
Run-1 (see Section 3.4.4).

Figure 8.10: Extrapolated radial beam position with data in station 18,
compared to a simulation at various levels of detector curvature ($a$), as a
function of track momentum. Negative curvature in data can be ruled out. It is
also clear that the curvature in data is no larger than $+10^{-6}\
\mathrm{mm}^{-1}$.

####  8.3.2 Comparison methods

The results in Figure 8.10 can now be compared using the $\chi^{2}$ method.
This is done by calculating the $\chi^{2}$ per momentum bin for a given
simulation curvature

$\chi^{2}/\mathrm{DoF}=\frac{1}{\mathrm{N_{bins}}}\sum_{i}^{\mathrm{N_{bins}}}\frac{(\mathrm{data_{i}-simulation_{i}}+\delta
r)^{2}}{(\Delta\mathrm{data_{i}})^{2}+(\Delta\mathrm{simulation_{i}})^{2}},$
(8.1)

as the function of $\delta r$, the amount of radial shift (for all momentum
bins) needed in simulation to align the extrapolated curve with data, such
that

$\frac{\langle\chi^{2}\rangle}{\delta r}=0.$ (8.2)

The assumption is that the simulation result with the smallest corresponding
$\chi^{2}/\mathrm{DoF}$ is the best representation of the curvature in the
data. An example scan over $\delta r$ for a curvature of $a=0.35\times
10^{-6}\ \mathrm{mm}^{-1}$ is shown in Figure 8.11.

Figure 8.11: The $\chi^{2}/\mathrm{DoF}$ change for a curvature of
$a=0.35\times 10^{-6}\ \mathrm{mm}^{-1}$ as a function of the radial shift
($\delta r$).

The final results of the $\chi^{2}$ method are summarised in Figure 8.12a.
Alternatively, as a cross-check, a residual method can be used. This method
calculates $\sigma$ for each momentum bin, and extracts the lowest absolute
value of $\sigma/\mathrm{DoF}$ over many curvatures from

$\sigma/\mathrm{DoF}=\frac{1}{\mathrm{N_{bins}}}\sum_{i}^{\mathrm{N_{bins}}}\frac{\mathrm{data_{i}}-\mathrm{simulation_{i}}}{\sqrt{(\Delta\mathrm{data_{i}})^{2}+(\Delta\mathrm{simulation_{i}})^{2}}}.$
(8.3)

The results from the residual method are summarised in Figure 8.12b.

(a)

(b)

Figure 8.12: The results of the two comparison methods. (a) The $\chi^{2}$
method. The most likely curvature in data, as compared to simulation, has the
smallest value of $\chi^{2}/\mathrm{DoF}$: $a=0.5\times 10^{-6}\
\mathrm{mm}^{-1}$ for station 12, and $a=0.45\times 10^{-6}\ \mathrm{mm}^{-1}$
for station 18. (b) The residual method. The most likely curvature in data is
represented by the simulation with the lowest $|\sigma/\mathrm{DoF}|$:
$a=0.5\times 10^{-6}\ \mathrm{mm}^{-1}$ for station 12, and $a=0.4\times
10^{-6}\ \mathrm{mm}^{-1}$ for station 18.

####  8.3.3 Extrapolation with detector curvatures

To examine the effect of these curvatures, $\sim 4\times 10^{5}$ tracks were
re-fitted with a curvature offset and the tracks were extrapolated back to
determine a beam position. This shift of this beam position, both radially and
vertically, from the nominal, is shown in Figure 8.13.

Figure 8.13: The difference in the extrapolated mean and width of the radial
and vertical beam position for six values of curvature and the nominal value
in the simulation. Due to the tracker UV straw geometry (see Section 5.3.2), a
change in the radial position also induces a change in the vertical position.

Finally, a similar scale of curvatures can be applied to data, as shown in
Figure 8.14. The uncertainty coming from the detector curvature estimated by
comparison with simulation can now be directly extracted from these plots.

Figure 8.14: The difference in the extrapolated mean and width of the radial
and vertical beam position for six values of curvature and the nominal value
in the data.

###  8.4 Results and outlook

The beam extrapolation determines the radial and vertical positions of the
muon beam in the storage ring. The accuracy of this determination is affected
by detector effects, such as the radial detector curvature. To estimate the
contribution of the detector curvature to beam extrapolation, the largest of
the values from the two methods in Section 8.3.2 was used, for a conservative
estimate. This value is $a=0.5\times 10^{-6}\ \mathrm{mm}^{-1}$, and the
corresponding beam extrapolation uncertainties, extracted from Figure 8.14,
are given in Table 8.1.

|  $\mathrm{\sigma_{dR}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dR_{\mathrm{width}}}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dV}}$ [$\text{\,}\mathrm{mm}$] |  $\mathrm{\sigma_{dV_{\mathrm{width}}}}$ [$\text{\,}\mathrm{mm}$]
---|---|---|---|---
S12 $\oplus$ S18 | 1.25 | 0.20 | 0.10 | 0.05
Table 8.1: The contribution of the detector curvature ($a$) to the beam
extrapolation uncertainty in both stations, given $a=0.5\times 10^{-6}\
\mathrm{mm}^{-1}$.

However, there is a motivation to improve the detector curvature estimation,
given it is one of the larger tracking systematics (see Section 3.4.1). Some
of the possible improvements are given below:

  * •

Improved tracking and simulation. An improvement to the tracking algorithms
will allow for a more precise limit on the curvature to be placed. Such
improvements include LR assignment, hit selection, and improving track
quality, and are being developed by G. Sweetmore [87], S. Grant [88], and A.
Luca [85], respectively. Moreover, an improved simulation, that is more
representative of real data will also benefit the curvature study.

  * •

Cosmic muons. An alternative estimation of the curvature is possible with
cosmic ray muons during periods when the magnetic field in the ring if off.
However, there are numerous challenges associated with this study, such as
maintaining long periods without the magnet on, developing a precise straight-
track fitting algorithm, and estimating the residual magnetic field.

###  8.5 Impact on the systematic uncertainties

#####  8.5.0.1 Pitch correction ($\omega_{a}$)

Using Equation 3.11, and an upper bound on $a=0.5\times 10^{-6}\
\mathrm{mm}^{-1}$ from Section 8.3.2, with the corresponding
$\mathrm{\sigma_{dV_{\mathrm{width}}}}=0.05$ mm (see Table 8.1), yields
$\Delta C_{\mathrm{pitch}}=0.7$ ppb.

#####  8.5.0.2 Field convolution ($\omega_{p}$)

A study of an impact of translations on the extrapolated beam profile from the
tracking detectors on the field measurements was performed by Jason Bono and
Saskia Charity [71]. Given the scale of translations induced by the curvature
of $a=0.5\times 10^{-6}\ \mathrm{mm}^{-1}$, of 1.25 mm and 0.1 mm radially and
vertically, respectively (see Table 8.1). It is possible to estimate the
corresponding uncertainties on $\langle B\rangle$, and hence $\omega_{p}$ (see
Equation 3.2). The results for both stations are given in Table 8.2, with the
average vertical and radial uncertainties in both stations on $\langle
B\rangle$ of 12.9 ppb and 0.6 ppb, respectively.

S12 radial | S18 radial | S12 vertical | S18 vertical
---|---|---|---
11.6 ppb | 14.2 ppb | 0.6 ppb | 0.6 ppb
Table 8.2: The detector curvature contribution to the uncertainty on $\langle
B\rangle$.

##  Chapter 9 A preliminary track-based $\omega_{a}$ analysis

This chapter presents preliminary results from the first attempt at an
$\omega_{a}$ analysis with data from the tracking detector. This type of
analysis has the advantage of having a different set of systematic
uncertainties (e.g. pile-up, lost-muons), as compared to the calorimeter-based
analysis. The available statistics in the tracker-based analysis is
significantly lower. However, given the previous $\omega_{a}$ measurement’s
discrepancy with the theory of $2.4$ ppm (see Section 2.1) and if the tracking
detector measures the same value as the BNL, then to exclude the theoretical
prediction at $95\%$ CL requires a measurement with a precision of $1.4$ ppm.
In this chapter, an estimate is made of when this threshold will be reached
with the acquisition of more data, based on the preliminary analysis of Run-1
data.

###  9.1 Analysis overview

The analysis work in this chapter follows closely the methodology of the
$\omega_{a}$ analysis using calorimeter data [41, 57, 66]. The first
statistical assessment of Run-1 data, applying time cuts (see Section 3.3.3)
and momentum cuts (see Section D.2), is summarised in Table 9.1. The number of
tracks (above the $p$ threshold) is $\sim 130$ times less than the number of
$e^{+}$ (above the $E$ threshold) recorded by the calorimeters (c.f. Table 3.1
with Table 9.1).

Dataset | All quality | Quality tracks | Quality tracks
---|---|---|---
name | tracks |  ($t>$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$$) |  ($t>$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, $p>1.8$ GeV)
1a | $3.51\times 10^{7}$ | $2.27\times 10^{7}$ | $7.25\times 10^{6}$
1b | $4.83\times 10^{7}$ | $3.12\times 10^{7}$ | $9.93\times 10^{6}$
1c | $7.23\times 10^{7}$ | $4.60\times 10^{7}$ | $1.46\times 10^{7}$
1d | $1.40\times 10^{8}$ | $8.63\times 10^{7}$ | $2.74\times 10^{7}$
Total | $2.96\times 10^{8}$ | $1.86\times 10^{8}$ | $5.92\times 10^{7}$
Table 9.1: For each dataset in Run-1, the total number of quality tracks (see
Section 5.3.6), as well as the number of tracks above certain time and
momentum cuts, are shown. Applying the time cut gives an estimate of the
number of useful tracks for the EDM analysis, while the additional momentum
cut gives an estimate of the number for the $\omega_{a}$ analysis.

####  9.1.1 Five-parameter fit

The first step in the analysis was a simple five-parameter fit to the number
of quality tracks as a function of time

$N(t)=N_{0}e^{-t/\tau}[1+A\cos(\omega_{a}t+\phi)],$ (9.1)

where $N_{0}$ is the overall normalisation, $\tau$ is the time-dilated muon
lifetime, $A$ is the asymmetry, and $\phi$ is the phase. The value of
$\omega_{a}$ returned from the fit is blinded in software and hardware (see
Section 4.1.2), so that a value of $R$ (see Equation 4.2) is reported instead.
The result of the fitting procedure is shown in Figure 9.1. Such a plot is
known colloquially as a wiggle plot. The choice of the width of a bin in this
histogram is $149.2$ ns, the cyclotron period. It is important to note that
the start-time of the fit should be at the edge of a bin; otherwise, the fit
quality is negatively affected.

The quality of the above fit can be assessed in terms of the distribution of
the fit pulls (i.e. normalised residual between the fit function in a given
time bin and the data), as shown in Figure 9.2a.

Figure 9.1: Five-parameter fit to data from the Run-1a dataset in both tracker
stations (S12 and S18).

(a)

(b)

Figure 9.2: Five-parameter fit (a) pulls and (b) FFT from the Run-1a dataset.

The non-centred distribution of fit pulls with $\sigma$ significantly larger
than $1$ suggests that the chosen function doesn’t describe the data well.
This can be further verified by transforming the residuals from the time to
the frequency domain, using a fast Fourier transform (FFT) [89]. A large peak
at $0$ MHz is observed, which is likely due to effects from lost-muons or
pile-up. Moreover, a peak at the CBO frequency (see Section 3.4.2) is also
seen, which also couples to the $g-{2}$ frequency, $f_{a}$,

$f_{a}=\frac{2\pi}{\omega_{a}}\approx 0.23\ \mathrm{MHz}.$ (9.2)

The expected frequencies of the peaks in Figure 9.2b depend on the field-
index, $n$ (see Section 3.3.3 and Table 3.1.).

####  9.1.2 Nine-parameter fit

The presence of the large FFT peak at the CBO frequency motivates
incorporating additional terms in the fit function, which describe the impact
of the CBO on the number oscillation, such that

$N(t)=N_{0}e^{-t/\tau}[1+A\cos(\omega_{a}t+\phi)]\cdot C(t),$ (9.3)

where $C(t)$ is the CBO function [90] given by

$C(t)=1.0+e^{-t/\mathrm{T_{CBO}}}\mathrm{A_{CBO}}\cos(\omega_{\mathrm{CBO}}t+\phi_{\mathrm{CBO}}).$
(9.4)

Equation 9.4 is analogous to the original five-parameter function, but with
each term now describing the CBO parameters: lifetime ($\mathrm{T_{CBO}}$),
amplitude ($\mathrm{A_{CBO}}$), angular frequency ($\omega_{\mathrm{CBO}}$),
and phase ($\phi_{\mathrm{CBO}}$). This fit is shown in Figure 9.3a. With the
CBO terms in the fit, the $\frac{\chi^{2}}{\mathrm{DoF}}$ is improved. To
verify that the correct model was used to describe the CBO’s influence on the
oscillation, the FFT analysis was repeated as shown in Figure 9.3b. The CBO
peak is no longer present.

(a)

(b)

Figure 9.3: Nine-parameter (a) fit and (b) FFT from the Run-1a dataset.

####  9.1.3 Ten-parameter fit

The remaining feature in the FFT spectrum in Figure 9.3b that needs to be
accounted for is the peak at $0$ MHz. The physical origin of this feature are
lost-muons (see Section 3.4.6). This can be incorporated by adding a term
accounting for the lost-muons in the fit, such that the fit equation becomes

$N(t)=N_{0}e^{-t/\tau}[1+A\cos(\omega_{a}t+\phi)]\cdot{C(t)}\cdot\Lambda(t),$
(9.5)

where $\Lambda(t)$ is

$\Lambda(t)=1-K_{\mathrm{LM}}\int^{t}_{t_{0}}L(t^{\prime})e^{t^{\prime}/\tau}dt^{\prime},$
(9.6)

the lost-muon term, with an arbitrary fit parameter $K_{\mathrm{LM}}$ (i.e.
normalisation). In this analysis, $\Lambda(t)$ was taken from [66]. It should
be noted, however, that [66] defines the rate of lost-muons for a calorimeter-
based determination of $\omega_{a}$. The different acceptance of the tracking
detectors means it is at best an approximation of the lost-muon rate for a
track-based $\omega_{a}$ analysis. Nevertheless, incorporating the lost-muon
term yielded improved results, as compared to the nine-parameter fit, as shown
in Figure 9.4. The fit $\frac{\chi^{2}}{\mathrm{DoF}}$ can be further improved
by adding systematic effects to the fit (e.g. pile-up), or performing a
dedicated lost-muon analysis in the tracking detector. However, the fit
quality is sufficient for this preliminary analysis to be able to estimate the
statistical precision on the tracker-based determination of $\omega_{a}$.

Figure 9.4: Ten-parameter fit to data from the Run-1a dataset.

With the ten-parameter fit, the slow effect is now diminished as seen in the
improved distribution of fit pulls in Figure 9.5a, and the prominent peak at
$0$ MHz is significantly reduced, as seen in Figure 9.5b.

(a)

(b)

Figure 9.5: Ten-parameter fit (a) pulls and (b) FFT from the Run-1a dataset.

###  9.2 Results

The statistical uncertainties on $R$, obtained from a ten-parameter fit, for
all four Run-1 datasets are summarised in Table 9.2, while the corresponding
plots are contained in Section D.1.

Dataset | 1a | 1b | 1c | 1d
---|---|---|---|---
$\delta R$ (ppm) | $18$ | $15$ | $13$ | $10$
Table 9.2: $\delta R$ from the fit to the tracker data across Run-1 datasets.

The combined statistical uncertainty can be approximated [91] via

$\delta\bar{R}=\frac{1}{\sqrt{\sum_{i}\delta R_{i}^{-2}}}=7\ \mathrm{ppm},$
(9.7)

which assumes a Gaussian distribution of errors and no correlations between
the datasets. To verify this estimation, the statistical uncertainty on $R$ of
1.4 ppm from a fit of $930\times 10^{6}$ calorimeter events [66] in the Run-1a
dataset can be used to predict the expected precision on the tracker data of
$59\times 10^{6}$ events, yielding

$\delta R=\sqrt{\frac{930}{59}}\times 1.4=6\ \mathrm{ppm}.$ (9.8)

The estimated uncertainties in Equations 9.7 and 9.8 compare well, given the
precision on the calorimeter data is derived from a fit with better estimates
of systematic effects.

It is now possible to calculate the number of tracks required to reach a
defined precision on $R$, and hence $\omega_{a}$, given that $59.2\times
10^{6}$ tracks yield a $\sim 7$ ppm uncertainty. Therefore, to reach a
precision on the $\omega_{a}$ measurement of 1.4 ppm, and thus confirm the
BNL-SM discrepancy at $95\%$ CL, should the same value be measured, would
require $1.3\times 10^{9}$ tracks. While to reach a comparable precision of
the final $\omega_{a}$ measurement at the BNL, of 0.54 ppm [1], would require
the experiment to accumulate $8.7\times 10^{9}$ tracks.

###  9.3 Outlook

For the tracker-based determination of $\omega_{a}$ to become competitive,
more tracker data must be acquired, and the tracking efficiency increased.
This efficiency is described in Section 5.3.8, and will be applied to the
already-collected data, yielding at least a factor of four increase in the
number of tracks.

Using Figure 3.17a, a relation can be made between the raw number of $e^{+}$
collected in Run-1 and the number of tracks that were used in this preliminary
$\omega_{a}$ analysis. A projection can than be made, using Figure 3.18a, on
the expected number of tracks that can be used in the determination of
$\omega_{a}$. This projection, shown in Figure 9.6, is based on the assumption
that the increase in the quality data (see Section 3.8.1) will be between
$60\%$ and $80\%$ [92, 93], as compared to Run-1. Given the current
improvements in tracking, a $95\%$ CL limit can be placed by Run-3. With
further improvements, a comparable precision to the one achieved at the BNL
$g-{2}$ (with calorimeter data) can be reached by Run-5 with the Fermilab
$g-{2}$ tracking detectors alone.

Figure 9.6: Projected precision on $\omega_{a}$ with tracker data.

##  Chapter 10 Systematic studies of the EDM

This chapter describes the methodology of the EDM measurement using data from
the tracking detectors. Section 10.2 contains the results of the EDM analysis
in the simulation. The impact of radial and longitudinal magnetic fields on
the EDM and $\omega_{a}$ determination is introduced in Section 10.3, while
the results of their measurements are given in Section 10.4 and Section 10.5,
respectively.

###  10.1 Introduction

The effect of a potential EDM of the muon, which was introduced in Section
2.2, would result in a tilt of the muon spin precession-plane as shown in
Figure 10.1. This tilt changes the observed precession frequency by an
addition of the precession resulting from a muon EDM [6], $\omega_{\eta}$,

$\boldsymbol{\omega_{\eta}}=\eta\frac{e}{2m_{\mu}}\left(\frac{\boldsymbol{E}}{c}+\boldsymbol{\beta}\times\boldsymbol{B}\right).$
(10.1)

Equation 10.1 describes the muon experiencing a torque in the presence of the
electric field produced by the ESQs, as well as the apparent electric field in
the muon rest-frame resulting from the Lorentz boost to the laboratory-frame
magnetic field [15]. $\boldsymbol{\omega_{\eta}}$ is orthogonal to the
vertically oriented $\boldsymbol{\omega_{a}}$, and is pointing radially
inwards towards the centre of the $g-{2}$ storage ring. This results in the
spin precession-plane tilting out from the orbit plane towards the centre of
the storage ring.

(a)

(b)

Figure 10.1: The tilt in the precession-plane of the muon due to an EDM, in
the muon rest-frame. The spin ($\boldsymbol{s}$) of the muon precesses in the
circle that is orthogonal to the vertical magnetic field ($B_{y}$). The
momentum vector ($\beta$) is along the $z$-axis. The centre of the storage
ring is along the $x$-axis in the orientation. (a) A zero EDM produces no
tilt. b) The presence of an EDM tilts the plane by an angle $\delta$. The
precession of $\boldsymbol{s}$ now has a vertical component.

There are two important consequences of a non-zero muon EDM: an increase in
the observed precession frequency, and an introduction of an oscillation of
the average vertical angle of the emitted positrons from the muon decay. The
magnitude of the observed precession frequency is now given by

$\omega_{a\eta}=\sqrt{\omega_{a}^{2}+\omega_{\eta}^{2}}.$ (10.2)

The resultant tilt angle of the spin precession-plane in the muon rest-frame,
$\delta$, is given by

$\delta=\tan^{-1}\left(\frac{\omega_{\eta}}{\omega_{a}}\right)=\tan^{-1}\left(\frac{\eta\beta}{2a_{\mu}}\right).$
(10.3)

As the positrons are preferentially emitted along the direction of muon spin
(see Figure 2.3), the tilt will make the positrons on average downward going
when the spin is pointing towards the centre of the $g-{2}$ storage ring, and
upward going when the spin points away from the centre of the ring. Therefore,
the observed average vertical angle varies with the magnitude of
$\omega_{a\eta}$, and this angle is the observable that is used in the $g-{2}$
experiment to measure the EDM using the tracking detectors.

Finally, it is essential to emphasise that when the muon spin and momentum
vectors are aligned, the maximum $\omega_{a}$ signal is observed. However, the
vertical angle oscillation is at the maximum when the spin is pointing
radially outwards, that is, $90^{\circ}$ out-of-phase with the momentum
vector. This property is used to distinguish between an EDM and other effects,
which are in-phase with $\omega_{a}$, that can also impact the vertical angle
oscillation. This is discussed in Section 10.3, for potential non-zero radial
and longitudinal magnetic field components.

###  10.2 EDM search with the tracking detectors

The goal of the Fermilab $g-{2}$ experiment is to measure $\delta$ to within
$0.4\text{\,}\mathrm{\SIUnitSymbolMicro rad}$, resulting in a sensitivity to
an EDM of $\sim 10^{-21}\leavevmode\nobreak\ e\cdot$cm [14]. The tracking
detectors will realise an EDM measurement through the direct detection of an
oscillation in the average vertical angle of the decay positrons. The tracking
detectors measure this vertical angle, $\theta_{y}$, through the momentum
components of a track

$\theta_{y}=\tan^{-1}\left(\frac{p_{y}}{p_{z}}\right).$ (10.4)

For example, a distribution of measured vertical angles in the Run-1a dataset
are shown in Figure 10.2, with a width of $9.368(2)$ mrad.

Figure 10.2: The distribution of measured vertical angles ($\theta_{y}$) of
tracks in the Run-1a dataset in tracking stations 12 and 18.

####  10.2.1 Simulating a large EDM signal

In order to test and develop the analysis techniques, a simulation with a
known value of the EDM was used. The chosen value of the EDM was
$d_{\mu}=5.4\times 10^{-18}\ e\cdot{\mathrm{cm}}$, which corresponds to a
value $\sim 30$ times larger than the EDM limit set at the BNL experiment
[15]. This value was chosen to resolve the EDM signal with a relatively low
number of available tracks from simulation and permitted a cross-check with
previous simulation results [68].

It is possible to analytically predict the observed oscillation amplitude,
$A_{\mathrm{EDM}}$, of $\theta_{y}$

$A_{\mathrm{EDM}}=a_{\mathrm{EDM}}\delta^{\prime},$ (10.5)

where $\delta^{\prime}$ is the precession-plane tilt angle in the lab-frame,
and $a_{\mathrm{EDM}}$ is the asymmetry of $\sim 0.13$ [94], which accounts
for the fact that not all positrons are emitted in the direction of the
polarisation vector. The tilt angle in the muon rest-frame, $\delta$, is
accessible via a Lorentz boost (see Section D.3)

$\delta^{\prime}=\tan^{-1}\left(\frac{\tan(\delta)}{\gamma}\right).$ (10.6)

Finally, substituting $\eta$ from Equation 2.7 into Equation 10.3 allows
$A_{\mathrm{EDM}}$ to be directly expressed in terms of fundamental constants
and the input EDM signal ($d_{\mu}=5.4\times 10^{-18}\ e\cdot{\mathrm{cm}}$)

$A_{\mathrm{EDM}}=a_{\mathrm{EDM}}\tan^{-1}\left(\frac{2d_{\mu}\beta
m_{\mu}c}{a_{\mu}\gamma e\hbar}\right)=0.22\ \mathrm{mrad}.$ (10.7)

####  10.2.2 EDM measurement strategy in the simulation

To observe the oscillation of the vertical angle, and hence measure
$A_{\mathrm{EDM}}$, it is necessary to consider how $\theta_{y}$ changes with
time, as shown Figure 10.3a, where $t$ is the track time in a fill. This time
is then modulated by the $g-2$ period ($\sim$
$4.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$), $T_{g-2}$,

$T_{g-2}=\frac{2\pi}{\omega_{a}}.$ (10.8)

This is analogous to the method used in the BNL EDM analysis [95], which is
known as the period-binned analysis. This modulation method is useful for
looking at periodic effects, such as the vertical angle oscillation. Periodic
effects that do not have the same period as $T_{g-2}$ (e.g. betatron
oscillations mentioned in Section 3.4.2) will generate a flat background when
many periods are considered, while the signal of interest adds constructively.
The time modulation, $t^{\mathrm{mod}}_{g-2}$, is given by

$t^{\mathrm{mod}}_{g-2}=\left(\frac{t}{T_{g-2}}-\mathrm{int}\left[\frac{t}{T_{g-2}}\right]\right)T_{g-2}.$
(10.9)

(a)

(b)

Figure 10.3: (a) $\theta_{y}$ versus time in fill. (b) $\theta_{y}$ versus
modulated time in fill.

$\theta_{y}$ was then profiled (i.e. averaged) into $50$ ns bins, as shown in
Figure 10.4. The distribution of $\theta_{y}$ in each time bin is Gaussian, as
shown in Figure 10.2. The observed oscillation of the averaged vertical angle,
$\langle\theta_{y}\rangle$, is given by

$\langle\theta(t)\rangle=A_{\mathrm{EDM}}\sin(\omega_{a}t)+c,$ (10.10)

where $c$ is the overall offset in $\theta(t)$. Equation 10.10 is the fit
function used in Figure 10.4. The fitting is done using the scipy.optimize
`Python` library. The optimal momentum and time cuts for this analysis are
discussed in Section 10.5.4. The extracted value of $A_{\mathrm{EDM}}$ is
$0.19(2)$ mrad, which agrees well with the prediction from Equation 10.7.

Figure 10.4: Averaged vertical angle oscillation with 4.3M tracks in
simulation from both stations (S12 and S18).

###  10.3 The significance of non-zero magnetic fields

The $g-{2}$ experiment is designed to have only the vertical component of the
magnetic field, with both radial and longitudinal components being zero. In
the experiment, the vertical magnetic field, $B_{y}$, is measured directly by
the NMR probes to an accuracy of 70 ppb, as described in Section 3.5. However,
no such direct measurements presently are possible to determine the radial,
$B_{x}$, or longitudinal, $B_{z}$, fields. Non-zero $B_{x}$ or $B_{z}$ can
tilt the precession-plane of the muon, as shown in Figure 10.5. This becomes
an important systematic effect for both the EDM and $\omega_{a}$ analyses, as
can be seen by adding two aforementioned magnetic field terms to Equation 3.1

$\boldsymbol{\omega_{a}}=a_{\mu}\frac{e}{m_{\mu}}\boldsymbol{B}=a_{\mu}\frac{e}{m_{\mu}}(B_{y}\boldsymbol{\hat{y}}+B_{z}\boldsymbol{\hat{z}}+B_{x}\boldsymbol{\hat{x}}).$
(10.11)

It is important to note, that $B_{x}$ would have the same effect on the
precession-plane as the EDM (c.f. Figure 10.1). It is, therefore, crucial to
estimate $B_{x}$ in order to not bias the results of a search for the muon
EDM. Various strategies exist to estimate $B_{x}$, both direct [96] and
indirect. The approach taken in this thesis is the indirect measurement of
$B_{x}$ using the tracking detectors, as described in Section 10.4.

(a)

(b)

Figure 10.5: The tilt in the precession-plane of the muon due to non-zero
field components, in the muon rest-frame. The spin ($\boldsymbol{s}$) of the
muon precesses in the circle that is orthogonal to the vertical magnetic field
($B_{y}$). The momentum vector ($\beta$) is along the $z$-axis. The centre of
the storage ring is along the $x$-axis in the orientation. (a) The tilt due to
the radial field ($B_{x}$), which is along the $x$-axis. (b) The tilt due to a
non-zero longitudinal field ($B_{z}$), which is along the $z$-axis.

A non-zero $B_{z}$, on the other hand, would tilt the precession-plane in the
direction of the muon momentum vector. This produces an effect that manifests
as an EDM-like signal that is in-phase with the magnetic dipole moment (i.e.
in-phase with $\omega_{a}$) and out-of-phase with the EDM (see Section 10.1).
Therefore, one way to measure $B_{z}$ is to perform a search for the in-phase
signal using data from the tracking detectors, as described in Section 10.5.

###  10.4 Preliminary estimation of $B_{x}$

####  10.4.1 Introduction

In the BNL experiment, the EDM limit is equivalent to a 30 ppm $B_{x}$, while
the determination of $B_{x}$ was estimated with a precision of 10 ppm [15].
Had the experiment had enough statistics, they would have been limited at
$\left|d_{\mu}\right|\sim 4.5\times 10^{-20}\ e\cdot{\mathrm{cm}}$ due to the
uncertainty on $B_{x}$, as demonstrated in Figure 10.6. Therefore, in the
Fermilab $g-{2}$ experiment more precise measurements of $B_{x}$, both direct
and indirect, are motivated to allow for an improved measurement of the EDM.
An accuracy of 0.2 ppm is required to probe $d_{\mu}$ at the
$10^{-21}\leavevmode\nobreak\ e\cdot$cm level.

Figure 10.6: The value of the EDM, $d_{\mu}$, and the corresponding value of
$B_{x}$. For example, to realise the $g-{2}$ experiment’s goal of measuring
$d_{\mu}$ to $10^{-21}\leavevmode\nobreak\ e\cdot$cm, $B_{x}$ must be known to
0.2 ppm. Plot courtesy of J. Price [97].

The aim of this preliminary study was to estimate $B_{x}$ by varying the
electric field produced by the ESQs (see Section 3.3.3) and measuring the
resultant mean vertical beam position, $\langle y\rangle$, using the tracking
detectors, and using the relation [98]

$B_{x}=\frac{nB_{0}\langle y\rangle}{R_{0}},$ (10.12)

where $n$ is the field-index (see Section 3.3.3), $R_{0}$ is the storage ring
radius, and $B_{0}$ is the magnetic field strength. $n$ contains the electric
field gradient of the ESQs, which is directly proportional to the HV applied
to the ESQs – QHV. Increasing the value of QHV should result in the observed
value of $\langle y\rangle$ decreasing, and vice versa. Equations 10.12 and
3.6 allow the estimation [99] of $B_{x}$

$B_{x}\sim(\Delta\langle y\rangle\Delta\mathrm{QHV})$ (10.13)

where $\mathrm{QHV}$ is the variable, and $\langle y\rangle$ is the
observable, and $\Delta\langle y\rangle\Delta\mathrm{QHV}$ can be determined
from tracker data. The objective is to verify the viability of such a
measurement.

####  10.4.2 Results

There were two periods in Run-2 that had a change in QHV: 20 June 2019 and 24
March 2019.

The QHV settings versus time for 20 June are shown in Figure 10.7. To ensure
the measurement of $\langle y\rangle$ is only influenced by the QHV, settings
of other $g-{2}$ components (e.g. kicker voltages) are required to be
constant. The difference in the measured values of $\langle y\rangle$, at a
given QHV setting, between the two tracker stations, as seen in Figure 10.7b,
are expected and their origin discussed in Section 3.4.5.

(a)

(b)

Figure 10.7: 20 June 2019. (a) QHV versus time. b) The QHV from (a) can be
split into three periods of nominal, low, and high QHV values.

$\langle y\rangle$ in tracker station 12 (20 June) is plotted against (QHV)-1
in Figure 10.8a. Using the obtained slope from Figure 10.8a and Equation
10.13, an estimation of $B_{x}$ is possible. The final result, $\langle
B_{x}\rangle=2.1(6)$ ppm, is summarised in Figure 10.8b.

(a)

(b)

Figure 10.8: (a) $\langle y\rangle$ versus QHV in S12 on 20 June 2019. The
slope, $\Delta\langle y\rangle\Delta\mathrm{QHV}$, can be directly extracted
from the plot. (b) The estimated values of $B_{x}$, in ppm, in S12 and S18 on
20 June and 24 March. The weighted mean, $\langle B_{x}\rangle$, and the
$1\sigma$ uncertainty band are shown.

####  10.4.3 Outlook

In this study, the preliminary-estimated value of $B_{x}$ in Run-2 of $2.1(6)$
ppm falls short of the required precision of 0.2 ppm, as outlined in Section
10.4.1. Therefore, more precise measurements of $B_{x}$, both direct and
indirect, are motivated to allow for a more precise determination of the EDM.
An improved method to determine $B_{x}$ was suggested by B. Kiburg [100]. This
method will determine $B_{x}$ by a variation of QHV together with an applied
radial field gradient, $\Delta B_{x}$, from the correction coils (see Section
3.5), that will induce a large, and known, $\Delta B_{x}$

$\langle\Delta y\rangle\sim\frac{1}{\Delta\mathrm{QHV}}\cdot(B_{x}\pm\Delta
B_{x}),$ (10.14)

where $B_{x}$ is the unknown variable in this equation. A direct measurement
of $B_{x}$ is also being developed by M. Fertl [101].

###  10.5 Estimation of $B_{z}$

The presence of a longitudinal magnetic field, $B_{z}$, tilts the precession-
plane of the muon, as shown in Figure 10.5. It is therefore imperative to
accurately and precisely measure $B_{z}$. As discussed in Section 10.3, a non-
zero $B_{z}$ introduces an up-down oscillation of the vertical track angle,
in-phase with $\omega_{a}$. This oscillation is used in this analysis to
determine $B_{z}$. This analysis follows a similar methodology as the EDM
simulation study described in Section 10.2.2. Following the method used in the
BNL experiment [95], two fits are performed. Firstly, the number of quality
tracks is time-modulated according to Equation 10.9, and $\phi$ is determined
using the five-parameter fit of Equation 9.1 but with $\omega_{a}$ fixed at
the value measured by the BNL experiment,
$\omega_{a}=$$1.439\,311\text{\,}\mathrm{MHz}$ [1]. The average vertical angle
oscillation, $\langle\theta_{y}(t)\rangle$, is then fitted to the function

$\langle\theta_{y}(t)\rangle=A_{B_{z}}\cos(\omega_{a}t+\phi)+A_{\mathrm{EDM}}\sin(\omega_{a}t+\phi)+c,$
(10.15)

where $A_{B_{z}}$ is the amplitude due to the longitudinal magnetic field,
$A_{\mathrm{EDM}}$ is the EDM amplitude, and $c$ is an overall offset.

####  10.5.1 EDM blinding

As seen from Equation 10.15, the EDM amplitude is directly accessible. This is
not desirable when analysing data, as the aim is to measure $A_{B_{z}}$
without revealing $A_{\mathrm{EDM}}$. A software-level blinding (see Section
4.1.2) can be applied to data by injecting an unknown EDM amplitude to
$\theta_{y}$

$\theta_{y}^{\mathrm{Blinded}}=\theta_{y}+A_{\mathrm{EDM}}^{\mathrm{Blinded}}(\mathbf{G},S)\cdot\sin(\omega_{a}t^{\mathrm{mod}}+\phi),$
(10.16)

where $A_{\mathrm{EDM}}^{\mathrm{Blinded}}$ is the unknown injected amplitude,
which is a function of a set of parameters, $\mathbf{G}$, and a blinding
string, $S$. $\mathbf{G}$ is set to produce a value of $A_{\mathrm{EDM}}$
comparable to the EDM limit set at the BNL experiment ($\sim 10^{-19}\
e\cdot{\mathrm{cm}}$), while $S$ allows a unique blinding per dataset.

The blinding procedure can be verified directly using the simulation, by
comparing the fit-parameters before and after the blinding, as shown in Figure
10.9. The unblinded fit parameters ($A_{B_{z}}$ and $c$), as well as
$\frac{\chi^{2}}{\mathrm{DoF}}$, remain unchanged, while $A_{\mathrm{EDM}}$ is
blinded.

(a)

(b)

Figure 10.9: Testing the blinding in the simulation with a large truth $B_{z}$
and no input EDM: a) unblinded, and b) blinded.

In data, a comparison was made between the blinding from Equation 10.16 and a
randomised double blinding:
$2A_{\mathrm{EDM}}^{\mathrm{Blinded}}\times(1+0.25\times\mathrm{rand}[0,1])$.
This procedure allowed an additional unknown signal, without revealing the
original blinding.

####  10.5.2 Fitting in data

The fit of Equation 9.1 to the number of quality tracks is shown in Figure
10.10a. This plot is analogous to the wiggle plot in Figure 9.1, but with the
time modulation of Equation 10.9 applied. The subsequent fit of Equation 10.15
to the vertical angle oscillation, using the value of $\phi$ from Figure
10.10a, is shown in Figure 10.10b, for tracks in a 100 MeV range.

(a)

(b)

Figure 10.10: Fit results in station 12 in the Run-1a dataset for: a) the
number oscillation, and b) $\langle\theta_{y}\rangle$ oscillation.

####  10.5.3 Momentum-binned analysis

The fit in Figure 10.10b is performed in a narrow momentum range (rather than
a simultaneous fit to all data) due to a variation in the overall offset in
the vertical angle ($c$) as a function of momentum, as demonstrated in Figure
10.11a. This is due to the vertical beam motion, discussed in Section 3.4.
Negative $c$ is indicative of the centre of the beam being vertically lower
than the centre of the detector. The global alignment effects were excluded as
a cause, due to the two stations measuring the same variation in $c$, while
having different corrections from the alignment (see Table A.1).

Tracks of lower momentum exhibit an even more significant change in $c$ and
therefore are not considered in this analysis. However, due to the optimal
momentum cuts for this analysis (see Section 10.5.4.3), and the effect of
dilution described in Section 10.5.5, the lowest and highest momentum tracks
do not contribute significantly to the measurement of $A_{B_{z}}$ – regardless
of the observed change in $c$ – and represent only a small population of all
tracks (c.f. Figure 5.14).

Therefore, fits in individual momentum bins of 100 MeV are performed to
measure $A_{B_{z}}$, separately for each station and dataset, in the momentum
range of 1100 MeV to 2300 MeV, as shown in Figure 10.11b for the Run-1d
dataset.

(a)

(b)

Figure 10.11: Momentum-binned analysis fit results in Run-1d dataset. The
tracks are split by momentum in 100 MeV bins, with the bin centre values
displayed on the $x$-axis. (a) The overall offset (parameter $c$) as the
function of momentum. (b) $A_{B_{z}}$ fit parameter in each momentum bin, and
the uncertainty-weighted mean ($\langle A_{B_{z}}\rangle$).

$\langle A_{B_{z}}\rangle$ per dataset is determined as the uncertainty-
weighted mean. The fitting results in the four Run-1 datasets are summarised
in Figure 10.12a, while the equivalent results in 1999 and 2000 at the BNL
[15] experiment are summarised in Figure 10.12b.

(a)

(b)

Figure 10.12: (a) Run-1 $A_{B_{z}}$ fitting results. The mean result in each
dataset is in agreement with the overall Run-1 mean within $1\sigma$. (b) The
equivalent results from 1999 and 2000 measurements at the BNL $g-{2}$
experiment [15].

####  10.5.4 Optimising cuts selection

The results in Figure 10.10 were obtained with a particular choice of cuts:
fit-start and end-times ($30.6\leavevmode\nobreak\
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$<t<500.0$
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$) and momentum ($1800$ MeV $<p<3100$
MeV, for the number oscillation). Moreover, the analysis relied on a choice of
the $g-{2}$ period ($T_{g-2}\approx 4.4\leavevmode\nobreak\
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$$), as well as the phase ($\phi$)
extracted from the fit of Equation 9.1. In this section, these choices will be
motivated, and their stability assessed.

#####  10.5.4.1 Dependence of fit on data subset

In order to assess the robustness of the fit, fits can be performed to
different subsets of the data and the results compared. For example, varying
the fit start-time, where data at a later start-time is a subset of data at an
earlier time. The difference in a given fit parameter for the two datasets has
an allowed difference, $\sigma_{\Delta_{21}}$, where 2 denotes the smaller
dataset and 1 the larger, is given by

$\langle(x_{1i}-x_{2i})^{2}\rangle=\sigma^{2}_{2i}-\sigma^{2}_{1i}\rightarrow\sigma_{\Delta_{21}}=\sqrt{\sigma^{2}_{2i}-\sigma^{2}_{1i}}.$
(10.17)

This equation is valid for any fit function, regardless of the number of
parameters and possible correlations among them [102]. The results for
$A_{B_{z}}$ with different fit start-times are shown in Figure 10.13.

#####  10.5.4.2 Fit start-time

First of all, a careful choice of the fit start-time is necessary – it should
be a multiple of $T_{g-2}$ because data is modulated using Equation 10.9. This
is demonstrated in Figure 10.14.

The second requirement is that the fit start-time must be greater than
$30\text{\,}\mathrm{\SIUnitSymbolMicro s}$ (see Section 3.3.3), and should be
as low as possible to minimise the statistical uncertainty. A fit start-time
of $30.56\text{\,}\mathrm{\SIUnitSymbolMicro s}$ was therefore chosen, with
the variation of $A_{B_{z}}$ at later fit start-times shown in Figure 10.13.

Figure 10.13: The variation of $A_{B_{z}}$ with the the fit start-time. Each
fit start-time in this plot is a multiple of $T_{g-2}$. The red band, given by
Equation 10.17, indicates the allowed set-subset variation.

(a)

(b)

Figure 10.14: Fit residuals using different fit start-times. (a) The fit
start-time is not a multiple of $T_{g-2}$. This results in a large feature at
$0.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$. (b) The residual spectrum is
uniform, with a correct choice of the fit start-time.

#####  10.5.4.3 Momentum cuts

Equation 10.15 is fitted in momentum bins of 100 MeV due to the strong
momentum dependence of $c$. Using the simulation with a truth value of
$B_{z}=1700$ ppm, it was verified that a symmetric cut on momentum is more
appropriate than an asymmetric one (see Section D.2), as shown in Figure
10.15.

(a)

(b)

Figure 10.15: Simulation results with $B_{z}=1700$ ppm. $A_{B_{z}}$ versus
momentum cut: a) asymmetric and b) symmetric. Symmetric cuts give a less
biased determination of $A_{B_{z}}$, and particularly when the momentum
dependent dilution factor (see Section 10.5.5) is accounted for.

Tracks of higher momentum predominantly originate further away from a station,
as seen in Figure 5.16b. Therefore, high momentum tracks with large angles
($\theta_{y}$) miss the tracker station – this can be thought of as a cone
effect. As a consequence, the SD of $\theta_{y}$ ($\sigma_{\theta_{y}}$) is
narrower for higher momentum tracks, as compared with lower momentum tracks,
as measured by the detectors. This is demonstrated in Figure 10.16.

Figure 10.16: The change in $\sigma_{\theta_{y}}$ with $p$ in momentum bins of
100 MeV.

This effect directly leads to a reduction in the uncertainty of the
$A_{B_{z}}$ parameter in the fit at high momentum, as shown in Figure 10.17a
where each momentum bin has the same number of tracks.

However, considering a non-fixed number of tracks in each momentum bin results
in the distribution shown in Figure 10.17b, where past 1600 MeV the $\delta
A_{B_{z}}$ is increasing with a deceasing number of tracks per bin.

(a)

(b)

Figure 10.17: The change in $\delta A_{B_{z}}$ with $p$. (a) Fixed number of
tracks, $N_{\mathrm{bin}}=6.6\times 10^{4}$, in each momentum bin. (b) The
non-fixed number of tracks per bin follows the momentum distribution of Figure
5.14.

The mid-momentum tracks are thus the most sensitive in measuring $A_{B_{z}}$.
A momentum range of 1100 MeV to 2300 MeV is therefore used to fit Equation
10.15, with separate fits performed in 100 MeV momentum bins.

#####  10.5.4.4 Assessment of period and phase

In order to modulate data using Equation 10.9, $T_{g-2}$ must be known a
priori to a reasonable precision. Using the $g-{2}$ frequency
$f_{a}=0.2290735$ MHz, as measured by the BNL experiment [1], yields the value
of $T_{g-2}=$ $4.365\,411\text{\,}\mathrm{\SIUnitSymbolMicro s}$. This choice
can be evaluated by changing $T_{g-2}$ by $-30$ ppm to $+30$ ppm and recording
the change in $A_{B_{z}}$, as shown in Figure 10.18a.

The small variations in Figure 10.18a imply that there is no real systematic
effect there. However, extracting the slope of the fit allows an assessment of
the magnitude of the systematic uncertainty from the choice of $T_{g-2}$, as
described in Section 10.5.4.5, to be made.

The change in $A_{B_{z}}$ due to the variation of $\phi$ is shown in Figure
10.18b.

(a)

(b)

Figure 10.18: The change in $A_{B_{z}}$ due to the variation of: a) $T_{g-2}$
and b) $\phi$.

#####  10.5.4.5 Systematic uncertainty evaluation

Small variations of $A_{B_{z}}$ due to the choices of $T_{g-2}$ and $\phi$
were observed. The systematic uncertainty for each, and the total systematic
uncertainty, on $A_{B_{z}}$ are summarised in Table 10.1. Slopes from Figures
10.18a and 10.18b were used in the estimation, as well as the error from the
fit on $\phi$ in Figure 10.10a. The total systematic uncertainty of
$\sim$0.1\text{\,}\mathrm{\SIUnitSymbolMicro rad}$$ was found to be negligible
compared to the statistical uncertainty of
$2.2\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ (see Figure 10.12a).

Systematic source | Value [$\text{\,}\mathrm{\SIUnitSymbolMicro rad}$]
---|---
$T_{g-2}$ | $0.022$
$\phi$ | $0.085$
Total | $0.088$
Table 10.1: Systematic uncertainties on $A_{B_{z}}$.

####  10.5.5 Conversion of $A_{B_{z}}$into $B_{z}$

With the optimal cuts justified, and systematic effects on $A_{B_{z}}$
evaluated, it is now possible to convert the observed amplitude into $B_{z}$.
$A_{B_{z}}$ is related to the precession-plane tilt in the lab-frame,
$\delta^{\prime}$, in the same way as $A_{\mathrm{EDM}}$ in Equation 10.5.
However, given that the muon motion is along $z$ (c.f. Figure 10.5), there is
no Lorentz boost in this direction. Therefore, an estimate of $B_{z}$ can be
derived directly from the measured angle in the lab-frame

$\delta^{\prime}=\tan\left(\frac{B_{z}}{B_{y}}\right)\approx\frac{B_{z}}{B_{y}}=\frac{A_{B_{z}}}{d_{B_{z}}(p)},$
(10.18)

for $B_{y}\gg B_{z}$ which is the case. In Equation 10.18, $d_{B_{z}}(p)$ is
the asymmetry-dilution factor, analogous to $a_{\mathrm{EDM}}$ in Equation
10.5. This asymmetry factor was estimated from the simulation, as shown in
Figure 10.19a, where a parabolic function was fitted to the simulation data,
given by

$d_{B_{z}}(p)=ap^{2}+bp+d_{0}.$ (10.19)

(a)

(b)

Figure 10.19: (a) The estimated asymmetry-dilution ($d_{B_{z}}(p)$) from the
simulation, and a fit of Equation 10.19. (b) The covariance matrix of the fit
parameters.

Finally, the uncertainty on $\frac{B_{z}}{B_{y}}$ in momentum bin $i$ is given
by

$\delta\left(\frac{B_{z}}{B_{y}}\right)^{i}=\left|\frac{A^{i}_{B_{z}}}{d_{B_{z}}^{i}}\right|\sqrt{\left(\frac{\delta
A^{i}_{B_{z}}}{A^{i}_{B_{z}}}\right)^{2}+\left(\frac{\delta
d^{i}_{B_{z}}}{d^{i}_{B_{z}}}\right)^{2}},$ (10.20)

where $\delta A^{i}_{B_{z}}$ is the fit error (e.g. Figure 10.10b), and
$\delta d^{i}_{B_{z}}$ is computed using the variance of the parameters in
Figure 10.19b.

Using Equations 10.18 and 10.20, the measured value of $A^{i}_{B_{z}}$ in each
momentum bin in Figure 10.11b, and the value of $d^{i}_{B_{z}}$ from Equation
10.19, $\frac{B_{z}}{B_{y}}$ was estimated (per momentum bin), as shown in
Figure 10.20.

(a)

(b)

Figure 10.20: (a) $B_{z}$ results, in ppm, in each of the momentum bins in the
Run-1d dataset. The uncertainty-weighted mean ($\langle{B_{z}}\rangle$) is
indicated. The large uncertainties at high $p$ is the consequence of the small
$d^{i}_{B_{z}}$ values in Equation 10.20 – their contribution to
$\langle{B_{z}}\rangle$ is minimal. (b) The zoomed version of the plot for
$p<2200$ MeV.

####  10.5.6 Analysis results

The measured values of $B_{z}$ from the four Run-1 datasets are summarised in
Figure 10.21. The results from the individual datasets are compared to the
mean Run-1 result of $\langle{B_{z}}\rangle=-11.8(50.3)$ ppm, and are found to
be in agreement to better than $1\sigma$.

Figure 10.21: $B_{z}$ results, in ppm, in all Run-1 datasets. The mean value
of ${B_{z}}$ was obtained from a weighted mean of the four datasets, its
$1\sigma$ uncertainty band is indicated in green and $2\sigma$ band in purple.

###  10.6 Outlook

The analyses presented in this chapter - measurements of $B_{x}$ and $B_{z}$
using the tracking detectors - were not attempted at the BNL experiment. The
preliminary measurement of $B_{x}$ confirmed that such a technique is viable
using the tracking detectors, and yielded a value of $2.1(6)$ ppm in Run-2.

The measured value of $B_{z}$ in Run-1 is consistent with zero, with a
precision of 50 ppm. Additionally, the measurement is in agreement with the
direct measurement of $B_{z}=0.14(36.9)$ ppm [96] before Run-1.

The precision ($\delta B_{z}=50$ ppm) will be improved with the increase in
the number of reconstructed tracks by adding new data – Run-2 data-
reconstruction is currently ongoing. Moreover, the track-reconstruction with
an improved tracking efficiency is currently ongoing on the UK grid (see
Section 3.8.2). The final precision is likely to reach 5 ppm by Run-5, as
shown in Figure 10.22. This is more precise than the design goal of $<7$ ppm,
which would correspond to the uncertainty on $\omega_{a}$ of $<20$ ppb [103].

Figure 10.22: Projected precision on $B_{z}$ with an increase in the data
volume.

##  Chapter 11 Conclusions

The Fermilab Muon $g-{2}$ experiment will measure the muon magnetic anomaly,
with a precision of 140 ppb, and will search for the muon EDM, with at least a
factor of 10 improvement on the world’s best measurement. The three data-
taking periods, Run-1, Run-2, and Run-3, have been completed, with
preparations for Run-4 currently ongoing. Essential in reducing the systematic
uncertainty on the measurement of $\omega_{a}$ are the straw tracking
detectors, which perform track extrapolation backwards to the muon decay point
and forwards to the calorimeters. Moreover, systematic effects, such as the
vertical pitch, require a correction that is accessible via a measurement of
the vertical width of the beam by the tracking detectors. The beam profile
from the tracking detectors is also convoluted with the magnetic field map to
find the field experienced by the muons at the point of decay.

In order to accurately reconstruct the beam profile, the tracking detectors
must be correctly aligned. The alignment algorithms were validated using
simulation, which converged after three iterations with
$\mathcal{O}$($10^{5}$) tracks to within $3\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ radially and $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ vertically. The
track-based internal alignment was implemented with data from Run-1. The
number of reconstructed tracks has increased by $3\%$ due to the position
calibration from the alignment. Moreover, an improvement in the mean track
$p\textnormal{-value}$ of $4\%$ was achieved. After the alignment procedure,
the uncertainty contribution from the tracker misalignment to the pitch
correction is now negligible.

An alignment manual [3] allowing future alignment determinations, has been
produced. The derived alignment constants were written into a `PostgreSQL`
database, where each set of constants is associated with a given range of
runs. Additionally, the stability of the alignment results was verified using
Run-2 data.

A potential EDM of the muon would increase the observed $\omega_{a}$ signal
and tilt the precession-plane of the muon. The tracking detectors will realise
an EDM measurement through the direct detection of an oscillation in the
average vertical angle of the positron from the muon decay. An observation of
a muon EDM would be evidence of new physics and would provide a new source of
CP violation in the charged lepton sector.

Essential in measuring the EDM, as well as $\omega_{a}$, are accurate and
precise estimations of potential non-zero radial and longitudinal magnetic
fields, which can tilt the precession-plane of the muon. The radial and
longitudinal magnetic fields were estimated using data from the tracking
detector. A preliminary estimation of the radial field resulted in a value of
$2.1(6)$ ppm. An in-depth analysis of the vertical angle oscillation yielded a
value of the longitudinal field consistent with zero: $-11.8(50.3)$ ppm.
Moreover, the first analysis of $\omega_{a}$ with data from the tracking
detector was made. The projected precision will allow an independent cross-
check of the BNL and Fermilab Run-1 determinations of $\omega_{a}$, using the
Run-1 to Run-3 datasets from the tracking detector.

At the time of writing, the experiment-wide effort on the analysis of Run-1
data is nearing completion, with the announcement of the first result expected
in the coming months. This could be a fascinating time for the scientific
community: only time will tell if the measurement by the experiment will be
the harbinger of new physics!

##  Appendix A Global alignment of the tracking detectors

This section on the global alignment briefly summarises the work done by Dr
Horst Friedsam [104], Dr Leah Welty-Rieger [105] and Dr Joe Price [106].

The global alignment established an absolute position of the tracker stations
relative to the rest of the experiment. This procedure consisted of two parts:
physical survey measurements of the modules by the Alignment and Metrology
Department at Fermilab, and the implementation of these measurements in the
software framework.

###  A.1 Survey alignment

The first step in the survey of the tracker compared measurements of a single
tracker module with the design model. Agreement was found within
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, with the survey points used in the
comparison indicated in Figure A.1. The design model was then used to
transform from a point in a local tracker coordinate system to the global
coordinate system. As the tracker modules are mounted into a vacuum chamber,
as shown in Figure 5.2, the global alignment requires the radial and vertical
positions of the vacuum chambers to be determined. The measurements were
performed using the API Laser Tracker as shown in Figure A.2. These
measurements and the design model was then used to estimate the position of
the module flange, the straws themselves, and the carbon fibre post, with the
error on the estimated positions determined to be
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Figure A.1: The survey points used in the global alignment: the centre of the
carbon fibre post and top and bottom of first two and last two straws in a
layer.

(a)

(b)

Figure A.2: The implementation of the survey measurements: a) the API Radian
Laser Tracker uses interferometer to accurately measure the vacuum chamber
position relative to the rest of the experiment, (b) the API I-360 Wireless
Probe was used to survey the individual module flanges.

###  A.2 Global alignment in simulation

The survey measurements described above resulted in four global corrections
per station: a radial translation, a vertical translation, a pitch (radial)
rotation, and a roll (vertical) rotation, as summarised in Table A.1.

| vertical position [$\text{\,}\mathrm{mm}$] | vertical angle [rad] | radial position [$\text{\,}\mathrm{mm}$] | radial angle [rad]
---|---|---|---|---
S12 | $-0.55$ | $+4.11\times 10^{-5}$ | $-1.51$ | $-1.12\times 10^{-3}$
S18 | $-0.61$ | $-4.13\times 10^{-4}$ | $-2.75$ | $-2.36\times 10^{-3}$

Table A.1: The corrections from survey alignment applied in the simulation to
S12 and S18.

A comparison between the extrapolated beam position determined before and
after the implementation of the global alignment is shown in Figure A.3 for
the vertical position. After the alignment, one would not expect the radial or
vertical beam distributions from the two stations to match exactly. This is
the case due to the closed-orbit effect (see Section 3.4.5), as the two
tracker stations see different beams. The summary of the change in the
vertical and radial extrapolated beam position after the global alignment is
presented in Table A.2.

(a)

(b)

Figure A.3: Mean vertical ($\langle y\rangle$) extrapolated beam positions: a)
before global alignment, and (b) after global alignment.

$\Delta\langle$vertical$\rangle$ S12 [$\text{\,}\mathrm{mm}$] |  $\Delta\langle$vertical$\rangle$ S18 [$\text{\,}\mathrm{mm}$] |  $\Delta\langle$radial$\rangle$ S12 [$\text{\,}\mathrm{mm}$] |  $\Delta\langle$radial$\rangle$ S18 [$\text{\,}\mathrm{mm}$]
---|---|---|---
$-0.50\pm 0.06$ | $-1.02\pm 0.06$ | $-0.35\pm 0.06$ | $-0.33\pm 0.06$

Table A.2: The change in the extrapolated beam position after applying the
survey corrections to both stations.

###  A.3 Global alignment uncertainty contribution

The Mahalanobis method [86] was used to estimate the uncertainty on the beam
extrapolation arising from the uncertainty on the global alignment. Nine
Mahalanobis points are derived from the two-parameter straight line-fit. The
results from placing the modules in one of these nine positions and
determining the extrapolated beam positions are shown in Figure A.4.

Figure A.4: The extrapolated vertical beam position (for run 15922) using
parameters from one of the nine Mahalanobis bands. The mean and the width was
estimated from a Gaussian fit to each of the nine distributions.

To estimate the uncertainty on the pitch correction from the global alignment,
the largest vertical beam width of the eight cases in Figure A.4, relative to
the nominal case, was used. This change
($\mathrm{\sigma_{dV_{\mathrm{width}}}}$) corresponds to $0.01$ mm. Using
Equation 3.11 and the estimated $\mathrm{\sigma_{dV_{\mathrm{width}}}}$ leads
to $\Delta C_{\mathrm{pitch}}$ of $0.3$ ppb. This is negligible, compared to
the total tracker pitch correction uncertainty of 8.6 ppb (in Run-1), as
described in Section 3.4.1.

##  Appendix B Derivations of analytical equations for internal alignment

This section provides derivations of the equations used in Chapter 7.

###  B.1 Straight line-fit in 2D

Derivations of Equations 7.18 to 7.21, describing the effect of a
translational misalignment on a line of best fit in 2D is given below, for a
line of the form

$x=mz+c,$ (B.1)

where $x$ is a hit position (i.e. height) along an infinitesimally thin
detector plane, and $z$ is a plane position along the direction of the beam. A
residual, $r_{i}$, on a detector plane is given by

$r=mz_{i}+c-x_{i},$ (B.2)

with the $\chi^{2}$, as a sum of all the residuals in $P$ planes, defined as

$\begin{gathered}\chi^{2}=\sum_{i}^{P}\frac{r_{i}^{2}}{\sigma^{2}}=\frac{1}{\sigma^{2}}\sum_{i}^{P}(mz_{i}+c-x_{i})^{2}=\\\
=\frac{1}{\sigma^{2}}\sum_{i}^{P}(mz_{i}^{2}+c^{2}+x_{i}^{2}+2cx_{i}+2mz_{i}x_{i}-2mz_{i}c),\end{gathered}$
(B.3)

for a constant detector resolution, $\sigma$, and a special case of no
misalignment and no smearing. The aim is to minimise $\chi^{2}$ with respect
to line parameters $m$ and $c$:

$\frac{\partial\chi^{2}}{\partial
m}=\frac{1}{\sigma^{2}}\sum_{i}^{P}(2mz_{i}^{2}+2x_{i}z_{i}-2z_{i}c),$ (B.4)

and

$\frac{\partial\chi^{2}}{\partial
c}=\frac{1}{\sigma^{2}}\sum_{i}^{P}(2c+2x_{i}-2mz_{i}).$ (B.5)

Without the loss of generality, one can define the centre-of-mass of planes to
lie at 0 along $z$, so that $\sum_{i}^{P}z_{i}=0$, which simplifies the above
equations to

$m=\frac{\sum_{i}^{P}z_{i}x_{i}}{\sum_{i}^{P}z_{i}^{2}},$ (B.6)

and

$c=\frac{\sum_{i}^{P}x_{i}}{P}.$ (B.7)

For a realistic case of non-zero characteristic misalignment, $M^{c}_{i}$,
along $x$, and smearing due to the detector resolution, $\delta$, on a plane,
one has a hit position defined by

$x=m_{T}z+c_{T}+\delta_{i}+M^{c}_{i},$ (B.8)

where one can define two variables relating the truth (e.g. $m_{T}$) and
measured line parameters due to the resolution and misalignment effects,
substituting for the measured slope, $m$, from Equation B.6 into Equation B.2

$\delta
m=m_{T}-m=\frac{-\sum_{i}^{P}(\delta_{i}z_{i}+M_{i}^{c}z_{i})}{\sum_{i}^{P}z_{i}^{2}},$
(B.9)

and similarly for the intercept

$\delta c=c_{T}-c=\frac{-\sum_{i}^{P}(\delta_{i}+M_{i}^{c})}{P}.$ (B.10)

Using Equations B.8, B.9 and B.10 the residual in a plane can be defined as

$r_{i}=mz_{i}+c-x_{i}=\delta mz_{i}+\delta c+\delta_{i}+M^{c}_{i}.$ (B.11)

It is now possible to derive an analytical form of the mean residual, $\langle
r_{p}\rangle$ in a detector layer $p$, over $N\rightarrow\infty$ tracks

$\begin{gathered}\langle
r_{p}\rangle=\frac{\sum_{n}^{N}r_{n,p}}{N}=\frac{\sum_{n}^{N}(\delta
m_{n}z_{p}+\delta c_{n}+\delta_{p,n}+M^{c}_{p})}{N}=\\\
=M^{c}_{p}-\frac{\sum_{i}^{P}M_{i}^{c}}{P}-\frac{z_{p}\cdot\sum_{i}^{P}{M^{c}_{i}\cdot
z_{i}}}{\sum_{i}^{P}{z_{i}^{2}}},\end{gathered}$ (B.12)

where $\sum_{n}^{N}\delta_{p,n}\rightarrow 0$ was used. The second term in
Equation B.12 gives a hint of an effect of misalignment in other layers on the
residual in a particular layer $p$. The analytical form of the standard
deviation of the distribution of residuals in a layer $p$, over
$N\rightarrow\infty$ tracks is given by

$\begin{gathered}\sigma^{2}_{r_{p}}=\frac{\sum_{n}^{N}(r_{p,n}-\langle
r_{p}\rangle)^{2}}{N}=\\\ =\frac{1}{N}\left(\delta m_{n}z_{p}+\delta
c_{n}+\delta_{p,n}+\frac{\sum_{i}^{P}M^{c}_{i}}{P}+z_{p}\frac{\sum_{i}^{P}M^{c}_{i}z_{i}}{\sum_{i}^{P}z_{i}}\right)^{2}=\sigma^{2}-\frac{z_{p}^{2}\sigma^{2}}{\sum_{i}^{P}z_{i}^{2}}-\frac{\sigma^{2}}{P}.\end{gathered}$
(B.13)

Finally, an analytical equation for $\chi^{2}$ on a fit to a misaligned
detector for $N\rightarrow\infty$ tracks can be derived

$\begin{gathered}\chi^{2}=\frac{1}{N}\sum_{n}^{N}\frac{\sum_{i}^{P}r_{i}^{2}}{\sigma^{2}}=\frac{1}{N\sigma^{2}}\sum_{n}^{N}\sum_{i}^{P}(\delta
m_{n}z_{i}+\delta c_{n}+\delta_{i,n}+M^{c}_{i})^{2}=\\\
=P-2+\frac{\sum_{i=1}^{P}(M^{c}_{i})^{2}-\frac{\sum_{i=1}^{P}{(M^{c}_{i})^{2}}}{P}-\frac{2\cdot\sum_{i=1}^{P}M^{c}_{i}\cdot
z_{i}}{\sum_{i=1}^{P}(z_{i})^{2}}}{\sum_{i=1}^{P}(\sigma_{i}^{\mathrm{det}})^{2}}.\end{gathered}$
(B.14)

The derived equations for $\langle r_{p}\rangle$, $\sigma^{2}_{r_{p}}$, and
$\chi^{2}$ apply for a 2D case of a straight line-fit to a misaligned detector
planes along $x$, with a given detector resolution on hits. Nevertheless, the
derived equations are quite instructive of the effect of misalignment, and
give correct analytical predictions for a 2D toy-model, as shown in Figures
7.6, LABEL: and 7.7.

###  B.2 Residual derivatives with circle-fit in 2D

In the case of a circle-fit using straight tracks the residual, $r$, is given
by

$r=\mathrm{DCA}(x,m,c)-h=\frac{|c+mz-x|}{\sqrt{m^{2}+1}}-h,$ (B.15)

with two local derivatives,

$\frac{\partial r}{\partial c}=\frac{c+mz-x}{\sqrt{m^{2}+1}|c+mz-x|},$ (B.16)

and

$\frac{\partial r}{\partial
m}=\frac{(m^{2}+1)z(c+mz-x)-m|c+mz-x|^{2}}{(m^{2}+1)^{3/2}|c+mz-x|}.$ (B.17)

The action of an anticlockwise rotation is given by

$\begin{pmatrix}z_{m}^{\prime}\\\ x_{m}^{\prime}\\\
\end{pmatrix}=\begin{pmatrix}\cos\phi&-\sin\phi\\\ \sin\phi&\cos\phi\\\
\end{pmatrix}\begin{pmatrix}z_{m}\\\ x_{m}\\\
\end{pmatrix}=\begin{pmatrix}z_{m}\cos\phi-x_{m}\sin\phi\\\
z_{m}\sin\phi+x_{m}\cos\phi\\\ \end{pmatrix}.$ (B.18)

The transformation back to the global coordinates is given by

$\begin{pmatrix}z\\\ x\\\
\end{pmatrix}=\begin{pmatrix}z_{m}^{\prime}+z^{\mathrm{centre}}\\\
x_{m}^{\prime}+x^{\mathrm{centre}}\\\
\end{pmatrix}=\begin{pmatrix}z_{m}\cos\phi-
x_{m}\sin\phi+z^{\mathrm{centre}}\\\
z_{m}\sin\phi+x_{m}\cos\phi+x^{\mathrm{centre}}\\\ \end{pmatrix}.$ (B.19)

The equation for a residual, under a 2D rotation, can be written using
Equation B.19, and is given by

$\begin{gathered}r=DCA(z(\phi),x(\phi),m,c)-r=\frac{|c+mz(\phi)-x(\phi)|}{\sqrt{m^{2}+1}}-r=\\\
=\frac{|c+mz_{m}\cos\phi-mx_{m}\sin\phi+mz^{\mathrm{centre}}-z_{m}\sin\phi-
x_{m}\cos\phi-x^{\mathrm{centre}}|}{\sqrt{m^{2}+1}}-r.\end{gathered}$ (B.20)

Ultimately, an expression for $\frac{\partial r}{\partial\phi}$ in terms of
global and local parameters is used

$\frac{\partial
r}{\partial\phi}=\frac{\partial\mathrm{DCA}(z(\phi),x(\phi),m,c)}{\partial\phi}.$
(B.21)

This can be obtained using the chain rule

$\frac{\partial
r}{\partial\phi}=\frac{\partial\mathrm{DCA}(\phi)}{\partial\phi}=\frac{\partial\mathrm{DCA}(d)}{\partial
d}\frac{\partial d(\phi)}{\partial\phi},$ (B.22)

where $d$ represents a measurement (e.g. straw displacement in $z$ or $x$).
Using Equation B.22 one can write the equation for the residual derivative
expressed without the use of the truth parameters (i.e. $\phi$)

$\begin{gathered}\frac{\partial r}{\partial\phi}=\frac{\partial r}{\partial
z}\frac{\partial z}{\partial\phi}+\frac{\partial r}{\partial x}\frac{\partial
x}{\partial\phi}=\frac{m(c+mz-x)}{\sqrt{m^{2}+1}\cdot|c+mz-x|}\times(-z_{m}\sin\phi-
x_{m}\cos\phi)\ +\\\
\frac{c+mz-x}{\sqrt{m^{2}+1}\cdot|c+mz-x|}\times(z_{m}\cos\phi-
x_{m}\sin\phi),\end{gathered}$ (B.23)

at this point, a substitution from Equation B.19 can be used to simplify the
expression to

$\begin{gathered}\frac{\partial
r}{\partial\phi}=\frac{m(c+mz-x)}{\sqrt{m^{2}+1}\cdot|c+mz-x|}\times(-x^{\prime}_{m})+\frac{c+mz-x}{\sqrt{m^{2}+1}\cdot|c+mz-x|}\times(z^{\prime}_{m})=\\\
\frac{\partial r}{\partial z}(-x+x^{\mathrm{centre}})+\frac{\partial
r}{\partial x}(z-z^{\mathrm{centre}}),\end{gathered}$ (B.24)

where all inputs come either from measurements or assumption of the ideal
geometry.

###  B.3 Residual derivatives in 3D

The straw wire can be parametrised as

$\boldsymbol{r}_{W}(s)=\begin{pmatrix}x\\\ y\\\ z\\\
\end{pmatrix}=\begin{pmatrix}x_{W}+m_{W}s\\\ s\\\ z_{W}\\\
\end{pmatrix}=\boldsymbol{u}s+\boldsymbol{r}_{W}^{0}=\begin{pmatrix}m_{W}\\\
1\\\ 0\\\ \end{pmatrix}s+\begin{pmatrix}x_{W}\\\ 0\\\ z_{W}\\\ \end{pmatrix},$
(B.25)

as the position on the straw is constant with $z$, and the slope in $x$ is due
to the stereo angle on the straw wire, $\theta_{W}$, of $\sim 7.5^{\circ}$
with $m_{W}=\tan(\theta_{W})$. A geometrical approach can be taken to express
the requirements on the DCA from Equation 7.27 as

$\mathrm{DCA}=\frac{|\boldsymbol{u}\times\boldsymbol{v}\cdot(\boldsymbol{r}^{0}_{W}-\boldsymbol{r}^{0}_{T})|}{|\boldsymbol{u}\times\boldsymbol{v}|}=|(\boldsymbol{r}^{0}_{W}-\boldsymbol{r}^{0}_{T})\cdot\boldsymbol{n}|,$
(B.26)

which projects the distance between the intercept of the lines, to the normal,
$\boldsymbol{n}$, between the two lines.

Using the track parametrisation from Equation 7.8 the DCA in 3D is given by

$\openup
3.0pt\begin{gathered}DCA(x_{T},y_{T},m_{x},m_{y},x_{W},z_{W},m_{W})=\frac{|\begin{pmatrix}1\\\
-m_{W}\\\ m_{W}m_{y}-m_{x}\end{pmatrix}\cdot\begin{pmatrix}x_{W}-x_{T}\\\
-y_{T}\\\ z_{W}\end{pmatrix}|}{|\begin{pmatrix}1\\\ -m_{W}\\\
m_{W}m_{y}-m_{x}\end{pmatrix}|}=\\\
=\frac{|m_{y}m_{W}z_{W}-m_{x}z_{W}+m_{W}y_{T}+x_{W}-x_{T}|}{\sqrt{(m_{W}m_{y}-m_{x})^{2}+(m_{W})^{2}+1}},\end{gathered}$
(B.27)

and the residual is simply

$r=\mathrm{DCA(x_{T},y_{T},m_{x},m_{y},x_{W},z_{W},m_{W})}-h.$ (B.28)

The four local derivatives $\frac{\partial r}{\partial x_{T}}$,
$\frac{\partial r}{\partial y_{T}}$, $\frac{\partial r}{\partial m_{x}}$, and
$\frac{\partial r}{\partial m_{y}}$ are then given by:
For the $x$-intercept

$\frac{\partial r}{\partial
x_{T}}=\frac{m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}}{-|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|\sqrt{\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1}},$
(B.29)

for the $y$-intercept

$\begin{gathered}\frac{\partial r}{\partial
y_{T}}=\frac{m_{W}(m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W})}{|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|\sqrt{\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1}}.\end{gathered}$
(B.30)

The slope (defined in Equation 7.10)

$\begin{gathered}\frac{\partial r}{\partial
m_{x}}=\frac{\left(m_{W}m_{y}-m_{x}\right)|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|}{\left(\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1\right){}^{3/2}}-\\\
-\frac{z_{W}(m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W})}{|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|\sqrt{\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1}},\end{gathered}$
(B.31)

and

$\begin{gathered}\frac{\partial r}{\partial
m_{y}}=\frac{m_{W}z_{W}(m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W})}{|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|\sqrt{\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1}}-\\\
-\frac{m_{W}\left(m_{W}m_{y}-m_{x}\right)|m_{W}m_{y}z_{W}+m_{W}y_{T}-m_{x}z_{W}-x_{T}+x_{W}|}{\left(\left(m_{W}m_{y}-m_{x}\right){}^{2}+m_{W}^{2}+1\right){}^{3/2}}.\end{gathered}$
(B.32)

And the two global derivatives for radial and vertical translations are

$\frac{\partial r}{\partial x_{W}}=-\frac{\partial r}{\partial x_{T}},$ (B.33)

and

$\frac{\partial r}{\partial y_{W}}=\frac{\partial r}{\partial y_{T}}.$ (B.34)

The 3D rotation matrix is given by

$\scriptstyle\boldsymbol{R}=\begin{pmatrix}\cos\phi\cos\psi&\cos\theta\sin\psi+\sin\theta\sin\phi\cos\psi&\sin\theta\sin\psi-\cos\theta\sin\phi\cos\psi\\\
-\cos\phi\sin\psi&\cos\theta\cos\psi-\sin\theta\sin\phi\sin\psi&\sin\theta\cos\psi+\cos\theta\sin\phi\cos\psi\\\
\sin\phi&-\sin\theta\cos\phi&\cos\theta\cos\phi\\\ \end{pmatrix},$ (B.35)

where $\theta$, $\phi$ and $\psi$ are the Euler angles. There are three
possible rotations (see Figure 7.3) around the centre of the tracker module. A
rotation around the $y$-axis is the same as considered previously in Equation
B.23, but needs to be extended to a 3D geometry. With the constraint that a
point along the straw will have the same vertical height ($y$) before and
after the above rotation with the derivatives for the anticlockwise rotation
$\phi$ along the $y$-axis given by

$\begin{pmatrix}x\\\ y\\\
z\end{pmatrix}=\begin{pmatrix}x_{m}^{\prime}+x^{\mathrm{centre}}\\\
y_{m}+y^{\mathrm{centre}}\\\ z_{m}^{\prime}+z^{\mathrm{centre}}\\\
\end{pmatrix}=\begin{pmatrix}z_{m}\sin\phi+x_{m}\cos\phi+x^{\mathrm{centre}}\\\
y_{m}+y^{\mathrm{centre}}\\\ z_{m}\cos\phi-
x_{m}\sin\phi+z^{\mathrm{centre}}\\\ \end{pmatrix}.$ (B.36)

The global derivatives for the anticlockwise rotation $\phi$ along the
$y$-axis is

$\frac{\partial r}{\partial\phi}=\frac{\partial r}{\partial
z_{W}}(-x_{W}+x^{\mathrm{centre}})+\frac{\partial r}{\partial
x_{W}}(z_{W}-z^{\mathrm{centre}}).$ (B.37)

and similarly for the rotation along the $z$-axis

$\frac{\partial r}{\partial\psi}=\frac{\partial r}{\partial
x_{W}}(-y_{W}+y_{\mathrm{centre}})+\frac{\partial r}{\partial
y_{W}}(x_{W}-x_{\mathrm{centre}}),$ (B.38)

and $x$-axis

$\frac{\partial r}{\partial\theta}=\frac{\partial r}{\partial
y_{W}}(-z_{W}+z_{\mathrm{centre}})+\frac{\partial r}{\partial
z_{W}}(y_{W}-y_{\mathrm{centre}}).$ (B.39)

The fifth local derivative in the presence of a magnetic field is given by

$\frac{\partial r}{\partial\kappa}=\frac{\partial
r}{\partial\left(\frac{1}{p_{z}}\right)}=\frac{\partial r}{\partial
m_{x}}\frac{\partial m_{x}}{\partial\kappa}+\frac{\partial r}{\partial
m_{y}}\frac{\partial m_{y}}{\partial\kappa}=\frac{\partial r}{\partial
m_{x}}p_{x}+\frac{\partial r}{\partial m_{y}}p_{y}.$ (B.40)

The vector containing derivatives of the residuals with respect to the fitted
track parameters is given by

$\frac{\partial r}{\partial\boldsymbol{b}}=\begin{pmatrix}\frac{\partial
r}{\partial x_{T}}\\\ \frac{\partial r}{\partial y_{T}}\\\ \frac{\partial
r}{\partial m_{x}}\\\ \frac{\partial r}{\partial m_{y}}\\\ \frac{\partial
r}{\partial\kappa}\end{pmatrix}.$ (B.41)

##  Appendix C Control and steering of PEDE

This section summarises the steering and constraint options used in PEDE.

The PEDE steering file is given by

⬇

1* g-2 Tracker Alignment: PEDE Steering File

2ConstraintFile.txt ! Constraints text file

3Cfiles ! Following bin files are Cfiles

4Data.bin ! Binary data file

5method inversion 10 0.01

The constraint equation is given by

$c=\sum_{i}^{N}l_{i}\cdot f_{l},$ (C.1)

where $c$ is is the constraint value, $l$ is the global parameter label, $f$
is the parameter factor, and the summation is taken over the parameters
contributing to the constraint. The constraints given in Section 7.3.4 can be
written as the inputs to PEDE as follows:

  1. 1.

Constraint on the overall radial translation

$0=\sum_{i=1}^{N}l_{i}\cdot 1=1211\cdot 1+1221\cdot 1+...1281\cdot 1,$ (C.2)

where, for example, label $1281$ corresponds to station number 12, module
number 8 and parameter number 1 (i.e. radial shift). The factor of $1$ simply
means equal weighting for all parameters.

Equation C.2 is written in the PEDE format as follows

⬇

1! Radial translational fixing (no overall movement):

2Constraint 0

31211 1

41221 1

51231 1

61241 1

71251 1

81261 1

91271 1

101281 1

  2. 2.

Constraint on the overall vertical rotation

$0=\sum_{i=1}^{N}l_{i}\cdot(N+1-2\cdot i)=1212\cdot 7+1222\cdot
5+...1282\cdot-7,$ (C.3)

⬇

1! Vertical rotational constraint

2! around the centre of the station:

3Constraint 0

41212 7

51222 5

61232 3

71242 1

81252 -1

91262 -3

101272 -5

111282 -7

  3. 3.

Constraint on the radial bowing effect

$0=\sum_{i}^{N}l_{i}\cdot(N+1-2\cdot i)^{2}=1211\cdot 49+1221\cdot
25+...1281\cdot 49,$ (C.4)

⬇

1! Bowing effect: parabloic radial term

2! describing the track curvature

3! around the centre of the station:

4Constraint 0

51211 49

61221 25

71231 9

81241 1

91251 1

101261 9

111271 25

121281 49

##  Appendix D Ancillary analysis plots and derivations

This section contains accompanying plots and derivations to Chapters 9 and 10.

###  D.1 Ancillary wiggle plots

Wiggle plots and ten-parameter fits across the four datasets: Run-1a (Figure
D.1), Run-1b (Figure D.2), Run-1c (Figure D.3), and Run-1d (Figure D.4).

Figure D.1: Ten-parameter fit in the Run-1a dataset.

Figure D.2: Ten-parameter fit in the Run-1b dataset. Figure D.3: Ten-parameter
fit in the Run-1c dataset. Figure D.4: Ten-parameter fit in the Run-1d
dataset.

###  D.2 Momentum cuts used in the wiggle plots

Imposing a momentum cut, $p_{\mathrm{min}}$, on data in Figure 9.1 is
motivated by the muon decay asymmetry ($A$) and the number of tracks ($N$)
used in the fit, such that the optimal value of $p_{\mathrm{min}}$ corresponds
to the quantity $NA^{2}$ reaching the maximum [102]. That is, the statistical
uncertainty on $\omega_{a}$ (or $R$) is minimised when $NA^{2}$ is maximised.

If the momentum fraction, $y$, is defined to be

$y=\frac{p_{\mathrm{min}}}{p_{\mathrm{max}}},$ (D.1)

then, in the lab-frame, both $N$ and $A$ can be expressed [11] as

$N(y)\propto(y-1)^{2}(-y^{2}+y+3),$ (D.2)

and

$A(y)=\frac{y(2y+1)}{-y^{2}+y+3}.$ (D.3)

The above expressions are plotted in Figure D.5a. Analytically, the optimal
asymmetric momentum cut ($p_{\mathrm{min}}$) occurs at $y\sim 0.6$ or
$p_{\mathrm{min}}\sim 1800$ MeV. This was empirically verified using the
Run1-a dataset, with the optimal momentum cut determined in Figure D.5b.

(a)

(b)

Figure D.5: (a) The analytical change in $NA^{2}$ terms as the function of $y$
in the lab-frame, not including the effect of the detector acceptance. (b) The
measured change in $NA^{2}$ with $p_{\mathrm{min}}$, as determined from the
Run1-a dataset.

###  D.3 Lorentz boost of the tilt angle

The tilt angle of the muon spin precession-plane is reduced by the
relativistic effect of length contraction [94]. This is demonstrated in Figure
D.6.

Figure D.6: The muon rest-frame and the lab-frame (i.e. the detector-frame)
are shown. The coordinate components of the lab-frame are primed (′) for
clarity.

The tilt angle, in the muon rest-frame, is given by

$\tan(\delta)=\frac{\Delta z}{\Delta y},$ (D.4)

which, in the lab-frame, is equivalent to

$\Delta z^{\prime}=\frac{\Delta z}{\gamma},$ (D.5)

and

$\Delta y^{\prime}=\Delta y.$ (D.6)

These allow for the tilt angle in the lab-frame to be expressed as

$\tan(\delta^{\prime})=\frac{\Delta z^{\prime}}{\Delta
y^{\prime}}=\frac{\Delta z}{\gamma\Delta y}.$ (D.7)

Therefore,

$\delta^{\prime}=\tan^{-1}\left(\frac{\tan(\delta)}{\gamma}\right).$ (D.8)

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  *[SM]: Standard Model
  *[BNL]: Brookhaven National Laboratory
  *[ppb]: parts-per-billion
  *[EDM]: electric dipole moment
  *[CP]: charge parity
  *[DAQ]: data acquisition
  *[ESQs]: electrostatic quadrupoles
  *[NMR]: nuclear magnetic resonance
  *[CBO]: coherent betatron oscillation
  *[VW]: vertical waist
  *[ppm]: parts-per-million
  *[SiPM]: silicon photomultiplier
  *[SAM]: sequential access via metadata
  *[OSG]: Open Science Grid
  *[POMS]: Production Operations Management Service
  *[FE]: front-end
  *[ASDQ]: amplified shaper discriminator charge
  *[TDC]: time-to-digital converter
  *[HV]: high voltage
  *[BE]: back-end
  *[FLOBBER]: front-end low voltage optical box to back-end readout
  *[LV]: low voltage
  *[SC]: slow control
  *[DCA]: distance of closest approach
  *[LR]: left-right
  *[LSR]: least squares regression
  *[DoF]: degrees of freedom
  *[SD]: standard deviation
  *[FFT]: fast Fourier transform