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Beam dynamics corrections to the Run-1 measurement of the muon anomalous magnetic moment at Fermilab

T. Albahri et al. (Muon g−2 Collaboration)
Phys. Rev. Accel. Beams 24, 044002 – Published 27 April 2021
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  • INTRODUCTION
  • EXPERIMENTAL DETAILS
  • DETERMINATION OF THE STORED MUON…
  • ELECTRIC FIELD CORRECTION Ce
  • PITCH CORRECTION Cp
  • DYNAMIC EFFECTS OWING TO TIME-CHANGING…
  • MUON LOSS CORRECTION Cml
  • CORRECTION FOR MUON DISTRIBUTION TIME…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    This paper presents the beam dynamics systematic corrections and their uncertainties for the Run-1 dataset of the Fermilab Muon g−2 Experiment. Two corrections to the measured muon precession frequency ωam are associated with well-known effects owing to the use of electrostatic quadrupole (ESQ) vertical focusing in the storage ring. An average vertically oriented motional magnetic field is felt by relativistic muons passing transversely through the radial electric field components created by the ESQ system. The correction depends on the stored momentum distribution and the tunes of the ring, which has relatively weak vertical focusing. Vertical betatron motions imply that the muons do not orbit the ring in a plane exactly orthogonal to the vertical magnetic field direction. A correction is necessary to account for an average pitch angle associated with their trajectories. A third small correction is necessary, because muons that escape the ring during the storage time are slightly biased in initial spin phase compared to the parent distribution. Finally, because two high-voltage resistors in the ESQ network had longer than designed RC time constants, the vertical and horizontal centroids and envelopes of the stored muon beam drifted slightly, but coherently, during each storage ring fill. This led to the discovery of an important phase-acceptance relationship that requires a correction. The sum of the corrections to ωam is 0.50±0.09  ppm; the uncertainty is small compared to the 0.43 ppm statistical precision of ωam.

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    • Received 11 March 2021
    • Accepted 1 April 2021

    DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.044002

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Beam dynamics
    1. Physical Systems
    Accelerators & storage rings
    1. Properties
    Magnetic moment
    Accelerators & Beams

    Viewpoint

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    Muon’s Escalating Challenge to the Standard Model

    Published 7 April 2021

    Measurements of the muon magnetic moment strengthen a previously reported tension with theoretical predictions, ushering in a new era of precision tests of the standard model.

    See more in Physics

    Authors & Affiliations

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    See Also

    Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm

    B. Abi et al. (Muon g−2 Collaboration)
    Phys. Rev. Lett. 126, 141801 (2021)

    Measurement of the anomalous precession frequency of the muon in the Fermilab Muon g−2 Experiment

    T. Albahri et al. (Muon g−2 Collaboration)
    Phys. Rev. D 103, 072002 (2021)

    Magnetic-field measurement and analysis for the Muon g−2 Experiment at Fermilab

    T. Albahri et al. (The Muon g−2 Collaboration)
    Phys. Rev. A 103, 042208 (2021)

    Article Text

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    Issue

    Vol. 24, Iss. 4 — April 2021

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    • Figure 1
      Figure 1

      The solid black trace represents the shape of the kicker magnetic field. The ringing is owing to an unavoidable impedance mismatch. The dashed red trace represents the average injected muon bunch temporal intensity distribution as measured by the T0 detector. The vertical dotted lines are separated by the 149 ns cyclotron period. The overlay between kicker and beam shapes shows that not all entering muons experience the same magnetic deflection. It is clear from the ringing that muons experience multiple kicks, both inward and outward.

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    • Figure 2
      Figure 2

      Plan view of the g−2 storage ring vacuum chamber and instrumentation highlighting the important components discussed in this paper. The beam enters through a field-free corridor provided by the inflector magnet. Three kicker stations operate in concert to deflect the beam onto a stable orbit during the first turn. Four ESQ stations, each consisting of a short and long section, provide vertical containment. The T0 and IBMS detectors monitor the injected beam intensity, time profile, and spatial distribution.

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    • Figure 3
      Figure 3

      Charging profiles for the 30 nominal ESQ plates driven by either one-step (black line) or two-step (dotted red line) power supplies. The two damaged resistors (solid orange and dotted blue lines), connected to the same one- and two-step power supplies, exhibit markedly different charging profiles during the data-fitting period. The vertical dotted black line at time t=0 represents the arrival time of the muons in the storage ring. The vertical dashed red line at 30  μs indicates the time at which the precession data fits begin. The resistors deteriorated slightly for Run-1d and the precession fit start time was delayed to 50  μs to compensate for the longer time charging profiles.

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    • Figure 4
      Figure 4

      Positron counts as a function of time as seen by all the calorimeters combined for the Run-1d dataset for the time ranges: (top) 4–5 and (bottom) 4–14  μs with respect to the beam injection. The time binning period is 1 ns. The amplitude modulation in the bottom panel is from the muon spin rotation frequency ωam.

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    • Figure 5
      Figure 5

      The frequency distribution obtained with a cosine transform of the fast rotation signal from the Run-1d dataset. The background is a consequence of the missing data before the start time of 4  μs. A cardinal sine (sinc) background fit has been applied to the black points and is subtracted from the whole frequency range as a correction.

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    • Figure 6
      Figure 6

      Top: the radial (closed-orbit) distribution extracted by both the Fourier method and the debunching analysis for the Run-1d dataset. Bottom: the radial distribution for the four Run-1 datasets as determined by the Fourier method. In both plots, the equilibrium radius is defined such that a magic-momentum muon is at 0 mm.

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    • Figure 7
      Figure 7

      (a) The vertical distribution of muons from a subset of Run-1a after applying tracker resolution and acceptance corrections. (b) The fitted distribution of vertical oscillation amplitudes, before (dashed red line) and after (solid blue line) the azimuthal averaging and calorimeter acceptance corrections described in the text.

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    • Figure 8
      Figure 8

      Additional electrostatic potential at r=5  cm versus time for the ESQ powered through damaged resistors. The skew dipole (dashed blue line) and normal quadrupole (solid black line) terms are shown relative to the nominal case and correspond to a 2D Taylor expansion based on data collected after Run-1 using an HV probe. The vertical dashed red line at 30  μs after injection indicates the nominal start of the precession fits. Scraping ends at 7  μs after injection, visible at the abrupt kinks in the curves. Nominal ESQ plates rise with a nominal RC time constant of τ≈5  μs, but these ESQ plates show prolonged and asymmetric time constants which lead to focusing and vertical steering errors during the data-taking period.

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    • Figure 9
      Figure 9

      The calculated βy (top), βx (middle), and dispersion Dx (bottom) functions versus azimuth in the ring at times 30 and 500  μs after injection for the Run-1a dataset. The calculation includes the effect of the damaged resistors used in the Q1L ESQ (hatched region) and the measured magnetic field distortions versus azimuth. The vertical shaded regions correspond to the locations in azimuth of the short and long ESQ sections.

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    • Figure 10
      Figure 10

      Tracker measurements of ωCBO in different time slices during the muon fill from the 270° tracker station. (a) and (b) show the Run-1a and Run-1d datasets, respectively. The fit function and parameters are noted in the figure, together with their uncertainties. The difference between the two datasets is attributed to worsening performance of the damaged ESQ resistors. The time dependence of the frequency is included in the fits of the positron data from the calorimeters to accurately incorporate the CBO acceptance dependence.

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    • Figure 11
      Figure 11

      The 180° tracker station determination of the vertical mean (a) and width (b) versus time in fill for Run-1d. The rapid oscillations owing to vertical and horizontal betatron motion have been randomized out to reveal the underlying time dependence of the mean and width (see also Sec. 8a).

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    • Figure 12
      Figure 12

      The change in the vertical mean over the time period from 40 to 300  μs in a fill during Run-1d as measured by the 24 calorimeter stations. The calorimeter data are corrected for acceptance. The blue line corresponds to the cosy model of the vertical closed-orbit evolution during the fill and its prediction of the vertical mean change around the ring. The red squares are from the gm2ringsim muon tracking simulation. This implementation of cosy does not utilize beam tracking, so the error band is purely systematic, while the gm2ringsim errors are statistical. Both simulations have been anchored to data from the tracking station at 270° and have been shifted azimuthally to align with the maximum acceptance for each calorimeter. The azimuthal structure predicted by the simulations due to the damaged resistors is clearly visible in the data.

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    • Figure 13
      Figure 13

      The intensity distribution of where muons first strike the r0=45  mm radius collimators (black circle). For any particular muon, this will occur when its horizontal and vertical betatron oscillations conspire such that the transverse displacement is at r0 and, at the same time, it is at the azimuthal location in the storage ring of a collimator. Some muons, which will eventually be lost, can survive hundreds to thousands of turns before this condition is met.

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    • Figure 14
      Figure 14

      The integrated fractional muon loss versus time in fill from 30  μs following muon injection. The four curves are from the different run groups. The smaller loss fraction curves are from the n=0.120 datasets (1b and 1c) and the greater loss fractions are from the n=0.108 datasets (1a and 1d). The absolute scale is determined from the Kloss parameter following final precession frequency fits. The uncertainty bands on the curves come from two different precession frequency analyses and whether a small empirical slow correction term to ensure stability of Kloss versus energy is included.

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    • Figure 15
      Figure 15

      Phase-momentum correlation from an end-to-end simulation (blue band) and from a data-driven approach (black). The simulation gives the result at the entrance to the storage ring. The three data points are obtained by fits to muon precession frequency data at nominal, reduced, and increased central magnetic field values. The phase reported for these data represented muons stored and fit during the measuring period. The phase dependence on momentum from the data is −10.0±1.6  mrad/%Δp/p0.

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    • Figure 16
      Figure 16

      Four of the eight stored momentum distributions Fi(x) obtained from adjusting the DR high- and low-momentum collimators. The area under each curve is normalized for each DR setting to show the fractional intensity of stored muons with respect to the nominal distribution. The solid gray line is an illustrative muon loss probability function l(x,t) from a model fit to all eight distributions for a time early in the fill. In this time window, the loss probability is greater at low momentum than at high momentum.

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    • Figure 17
      Figure 17

      The muon loss function L(t), normalized to beam intensity, for the lowest (blue) and highest (red) momentum stored muon distributions; see corresponding momentum distributions in Fig. 16. Asymmetry in the loss rates is seen early in the fill.

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    • Figure 18
      Figure 18

      The expected phase shift versus time in fill of the remaining, stored muon population. The uncertainty bands arise from the use of three different l(x) functions.

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    • Figure 19
      Figure 19

      An azimuthally averaged muon spatial distribution for Run-1a as measured by the trackers. The (hatched) collimator defines the 45-mm-radius transverse storage aperture.

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    • Figure 20
      Figure 20

      The azimuthally averaged phase maps for the asymmetry-weighted analysis.

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    • Figure 21
      Figure 21

      Phase versus vertical (a) and horizontal (b) decay coordinate. The Gaussian profiles show (at an exaggerated scale) how muon distributions might evolve from early (dotted red line) to late (dashed blue line) times in the fill. The largest effect is from the reduction in the vertical width (a), where the phase changes to the distribution add coherently. The mean shift (not shown) is smaller, because the increase on one side is balanced on the opposite side. Conversely, the larger phase shift in the radial projection is from the mean motion (b), while the width change (not shown) is largely canceled.

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    • Figure 22
      Figure 22

      Calculation of φpac(t) for calorimeter 19 in Run-1d using data from the tracker station at 180° and Eq. (26). The fit start time for the extraction of ωam for this dataset is 50  μs after injection in order to mitigate the effect of φpac(t).

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    • Figure 23
      Figure 23

      A 2D representation of the equipotentials and electric field in the ESQ region. The circle at radius 45 mm represents the storage ring aperture defined by the collimators.

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    • Figure 24
      Figure 24

      The electric field along the x axis in the midplane (y=0). The purple points are computed from the multipole expansion using Eq. (a11). The black curve uses the values from the field map.

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    • Figure 25
      Figure 25

      Top: contribution to ωa due to the electric field as computed by spin tracking (integration of the Thomas-BMT equation) and integration along trajectories [Eq. (b3)] for closed orbits at varying radial offsets. The vertical closed orbit is zero, and there is no vertical betatron motion. A representative, measured radial distribution is superimposed for reference. Bottom: contribution to ωa due to the vertical oscillation computed with spin tracking and integration along the trajectory [Eq. (b4)] for muons with magic momentum and zero radial betatron amplitude. Note that the formula that assumes linearity overestimates the contribution at large amplitudes.

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    • Figure 26
      Figure 26

      gm2ringsim simulation of detected positron energy as a function of time, with muon decay disabled.

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    • Figure 27
      Figure 27

      The average calorimeter response function showing the deposited energy versus the truth energy, integrated over the decay volume.

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    • Figure 28
      Figure 28

      The asymmetry-weighted relative acceptance maps for all calorimeters. The differences in acceptance derive from the materials just upstream of each station; see Fig. 29.

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    • Figure 29
      Figure 29

      The asymmetry-weighted vertical projections of the phase maps by calorimeter. The calorimeters are ordered sequentially in azimuth with respect to the injection location. The differences between stations is caused by material differences that impact the transmission of positron decays enroute to detectors. They also impact the acceptance (see Fig. 28) and the asymmetry maps. These indices represent: Q, behind ESQ plates; K, behind kicker plates; T, behind tracker stations; IV, in the constrained inflector vacuum chamber; and F, free space in front.

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