Tejas Rana

Precision Measurement of the Muon Anomalous Magnetic Moment

Status: Draft / Work in Progress — This page summarizes the current state of my master's thesis research as of December 2024. Results are preliminary and subject to revision.

Abstract

The anomalous magnetic moment of the muon, \(a_\mu = (g-2)/2\), provides one of the most stringent tests of the Standard Model of particle physics. The Fermilab Muon g-2 Experiment measures \(a_\mu\) with unprecedented precision by observing the precession of polarized muons in a magnetic storage ring. This thesis presents an analysis of systematic uncertainties in the extraction of the muon precession frequency \(\omega_a\) from calorimeter data, with particular focus on beam dynamics corrections and pile-up effects. Preliminary results from Run 2/3 data are consistent with earlier measurements and maintain the observed tension with Standard Model predictions.

Introduction

The magnetic moment of a charged lepton can be written as \(\vec{\mu} = g \frac{e}{2m} \vec{S}\), where \(g\) is the gyromagnetic ratio. For a point-like Dirac particle, \(g = 2\) exactly. However, quantum loop corrections introduce a small anomaly, \(a = (g-2)/2\), which can be calculated precisely within the Standard Model and measured experimentally.

For the muon, the anomalous magnetic moment receives contributions from quantum electrodynamics (QED), the electroweak sector, and hadronic physics:

$$a_\mu^{\text{SM}} = a_\mu^{\text{QED}} + a_\mu^{\text{EW}} + a_\mu^{\text{had}}$$

The QED contribution is known to five-loop precision, while the electroweak contribution is computed at two-loop level. The dominant uncertainty arises from hadronic contributions, particularly the hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) scattering terms.

Recent experimental results from Fermilab, combined with the final BNL E821 measurement, yield:

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

This shows a discrepancy of approximately \(4.2\sigma\) from the Standard Model prediction (using data-driven HVP), representing one of the most significant hints of physics beyond the Standard Model.

Methods

Experimental Technique

The Fermilab Muon g-2 Experiment stores polarized positive muons (\(\mu^+\)) in a superferric magnetic storage ring with a highly uniform dipole field \(B \approx 1.45\,\text{T}\). The muons undergo cyclotron motion while their spins precess at the anomaly frequency:

$$\vec{\omega}_a = -\frac{e}{m}\left[a_\mu \vec{B} - \left(a_\mu - \frac{1}{\gamma^2 - 1}\right)\frac{\vec{\beta} \times \vec{E}}{c}\right]$$

At the "magic" momentum \(p = 3.094\,\text{GeV}/c\) (corresponding to \(\gamma = 29.3\)), the electric field term vanishes, simplifying the measurement to a direct comparison of \(\omega_a\) and the magnetic field strength.

Data Analysis Framework

Muons decay via \(\mu^+ \to e^+ \nu_e \bar{\nu}_\mu\), and the positron energy spectrum is correlated with the muon spin direction due to parity violation. High-energy positrons are preferentially emitted along the muon spin direction. The time distribution of detected positrons above an energy threshold \(E_{\text{th}}\) follows:

$$N(t) = N_0 \, e^{-t/\gamma\tau_\mu} \left[1 + A \cos(\omega_a t + \phi)\right]$$

where \(\tau_\mu = 2.197\,\mu\text{s}\) is the muon lifetime, \(A \approx 0.4\) is the asymmetry, and \(\phi\) is an initial phase. In practice, the fit function must account for numerous systematic effects:

$$N(t) = N_0 \, e^{-t/\tau} \left[1 + A(t) \cos(\omega_a t + \phi(t))\right] \cdot C_{\text{CBO}}(t) \cdot C_{\text{VW}}(t) \cdot C_{\text{lost}}(t)$$

Here, \(C_{\text{CBO}}\) accounts for coherent betatron oscillations, \(C_{\text{VW}}\) for vertical waist motion, and \(C_{\text{lost}}\) for muon losses from the storage ring.

Systematic Uncertainty Estimation

The extraction of \(\omega_a\) requires careful treatment of systematic effects. The statistical uncertainty on \(\omega_a\) from a maximum likelihood fit is given by the Cramér-Rao bound:

$$\sigma_{\omega_a}^{\text{stat}} \geq \frac{1}{\sqrt{N}} \cdot \frac{1}{A \cdot \gamma\tau_\mu}$$

Systematic uncertainties arise from several sources, including pile-up corrections, gain variations in the calorimeter, and beam dynamics effects. The total systematic uncertainty budget for Run 2/3 analysis is summarized in the results section.

Preliminary Results

Precession Frequency Extraction

The five-parameter fit to the positron time spectrum yields the anomaly frequency with high precision. After applying all corrections, the blinded frequency ratio \(R = \omega_a / \tilde{\omega}_p\) (where \(\tilde{\omega}_p\) is the measured proton NMR frequency) is determined. The unblinding procedure will reveal the final result upon collaboration approval.

Sample plot showing simulated signal oscillation overlaid with an exponential decay envelope. The oscillation frequency corresponds to the anomaly precession frequency omega_a.
Figure 1: Simulated positron count rate as a function of time, showing the characteristic oscillation at the anomaly frequency \(\omega_a\) modulated by the muon decay exponential. The blue curve represents a five-parameter fit to the data. This plot was generated from Monte Carlo simulation data using the official collaboration fitting framework; actual data remain blinded.

Systematic Uncertainty Analysis

My contribution focuses on the estimation of systematic uncertainties from beam dynamics effects. The coherent betatron oscillation (CBO) frequency is extracted independently from tracker data and used to constrain the fit model:

$$C_{\text{CBO}}(t) = 1 + A_{\text{CBO}} \, e^{-t/\tau_{\text{CBO}}} \cos(\omega_{\text{CBO}} t + \phi_{\text{CBO}})$$

The CBO frequency \(\omega_{\text{CBO}} \approx 2\pi \times 0.37\,\text{MHz}\) is determined by the storage ring focusing system and must be precisely measured to avoid biasing the extracted \(\omega_a\).

Histogram showing the energy spectrum of detected positrons with statistical error bars. The spectrum shows the Michel edge near 52 MeV and the energy threshold used for analysis.
Figure 2: Energy spectrum of detected positrons from muon decay, showing the Michel spectrum endpoint near 52 MeV. Events above the threshold \(E_{\text{th}} \approx 1.8\,\text{GeV}\) (lab frame) are used in the \(\omega_a\) analysis. Error bars represent statistical uncertainties; the solid line shows the expected spectrum from simulation.

Comparison with Theory

The Standard Model prediction for \(a_\mu\) involves contributions from multiple sectors. The leading-order hadronic vacuum polarization can be written as a dispersion integral:

$$a_\mu^{\text{HVP,LO}} = \frac{1}{4\pi^3} \int_{m_\pi^2}^{\infty} ds \, K(s) \, \sigma(e^+e^- \to \text{hadrons})$$

where \(K(s)\) is a known kernel function and the cross-section \(\sigma\) is determined from experimental data. Recent lattice QCD calculations have provided an independent determination of \(a_\mu^{\text{HVP}}\), with results that reduce the tension with experiment but are in some tension with the data-driven approach.

The electroweak contribution involves loop diagrams with \(W\), \(Z\), and Higgs bosons. At leading order:

$$a_\mu^{\text{EW}} = \frac{G_F m_\mu^2}{8\sqrt{2}\pi^2} \left[\frac{5}{3} + \frac{1}{3}(1 - 4\sin^2\theta_W)^2 + \mathcal{O}\left(\frac{m_\mu^2}{M_W^2}\right)\right]$$

where \(G_F\) is the Fermi constant and \(\theta_W\) is the weak mixing angle.

Discussion

The preliminary results from this analysis are consistent with previously published Fermilab measurements. The systematic uncertainty from beam dynamics corrections has been reduced compared to Run 1 analysis through improved modeling of the CBO decoherence and the use of tracker-constrained fits.

Key findings from the systematic studies include:

  • The CBO systematic uncertainty is reduced by approximately 30% using the tracker-constrained approach
  • Pile-up corrections are well-controlled at the current statistical precision
  • Gain stability in the calorimeters meets specifications throughout the data-taking period

If the current tension between experiment and theory persists at the \(5\sigma\) level with the complete Fermilab dataset, it would constitute strong evidence for physics beyond the Standard Model. Possible new physics explanations include supersymmetric contributions, dark photon mixing, or other weakly coupled new particles.

Code & Data Availability

Analysis code, fitting scripts, and plotting tools developed for this thesis are available on GitHub:

github.com/tejasrana973

The repository includes a README.md with setup instructions and documentation. Note that official collaboration data files are not publicly available due to collaboration policy; the repository contains only analysis code and scripts for processing public simulation samples.

Selected References

  • Muon g-2 Collaboration, "Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm," Phys. Rev. Lett. 131, 161802 (2023).
  • T. Aoyama et al., "The anomalous magnetic moment of the muon in the Standard Model," Phys. Rept. 887, 1-166 (2020).
  • Budapest-Marseille-Wuppertal Collaboration, "Hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon from lattice QCD," Nature 593, 51-55 (2021).