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Flavor Physics and CP Violation Conference, Victoria BC, 2019
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\title{Measurements of the muon anomalous magnetic moment}
\author{Nam H. Tran\\
  on behalf of the Muon $g-2$ Experiment at Fermilab}
\affiliation{Boston University, Boston, MA, US 02215}
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\begin{abstract}
  The over 3-sigma discrepancy between the latest experimental and theoretical
  values of the anomalous magnetic of the muon has been one of the best
  hints of new physics beyond the Standard Model. Ongoing efforts
  to settle the issue from the experimental side are happening at Fermilab and
  J-PARC. The E989 at Fermilab is using the same key technologies as the last
  experiment at BNL (E821), but with great improvement in accuracy. The first
  physics data has been taken, and data analysis is going to be finalized soon.
  The E34 at J-PARC adopts a novel approach, is
  well advanced in R\&D, and recently received Stage-2 Approval. The
  experimental principles, techniques, and prospects of these measurements are
  presented.
\end{abstract}

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\section{Introduction}
The magnetic dipole moment, $\vec{\mu}$, of an elementary particle can be
expressed in terms of its spin, $\vec{S}$, as:
\begin{equation}
  \vec{\mu} = g\frac{q}{2m}\vec{S},
\end{equation}
where $q$ and $m$ are charge and mass of the particle respectively. The
dimensionless gyromagnetic ratio $g$ for a spin 1/2, structureless particle can
be calculated from Dirac equation to be exactly 2. Measured values of $g$ is
slightly different from 2, this anomaly $a$ is defined as:
\begin{equation}
  a = \frac{g - 2}{2}.
\end{equation}

The Standard Model (SM) provides very precise predictions of the anomaly of the
muon, $a_\mu$, at sub-part-per-million level. Experimentally,
there has been a series of measurements with increasing accuracy at CERN and
Brookhaven National Laboratory (BNL). The last measurement, the E821 at BNL,
was done with an uncertainty of 540 part-per-billion (ppb)~\cite{BNL2006}, and
showed a 3.7 standard deviation from the SM
prediction by Keshavarzi and colleagues~\cite{Alex2018}. This discrepancy
motivates new and better measurements of $a_\mu$, two of such experiments are:
\begin{itemize}
  \item E989 at Fermilab: this is the successor of the E821 experiment, 
    inheriting the muon storage ring, general experimental techniques, as well
    as analysis methods. The E989 will improve the precision by four times
    by collecting 20 times more data, and reducing systematic uncertainties.
  \item E34 at Japan Proton Accelerator Research Complex (J-PARC): a completely
    new experiments with novel approaches, aiming for a measurement with
    comparable accuracy compares to that of the BNL experiment. It will be an
    important cross check of results from E821 and E989.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Principle of measurements}
When a muon is injected perpendicularly into a uniform dipole magnetic field,
$\vec{B}$, it will move in a circle with the cyclotron frequency:
\begin{equation}
  \vec{\omega}_c = -\frac{e}{\gamma m_\mu}\vec{B},
  \label{eq:omega}_c
\end{equation}
where $\gamma$ is the Lorentz factor.

Assuming spin of the muon is also perpendicular to the magnetic field, the spin
will precess with a fixed frequency:
\begin{equation}
  \vec{\omega}_s = -g_\mu \frac{e}{2m_\mu}\vec{B} -
  (1-\gamma)\frac{e}{\gamma m_\mu}\vec{B}.
  \label{eq:omega_s}
\end{equation}

The difference between $\vec{\omega}_c$ and $\vec{\omega}_s$:
% \begin{align}
  % \vec{\omega}_a &= \vec{\omega}_s - \vec{\omega}_c \\
  % &= -\frac{g-2}{2}\frac{e}{m_\mu} \vec{B} \\
  % &= -a_\mu \frac{e}{m_\mu}\vec{B}.
% \end{align}
\begin{equation}
  \vec{\omega}_a = \vec{\omega}_s - \vec{\omega}_c = -a_\mu \frac{e}{m_\mu}\vec{B}.
  \label{eq:omega_a0}
\end{equation}

The magnetic field strength in~\cref{eq:omega_a0} can be measured most
precisely using NMR technique using a sample with high hydrogen content.
Introducing the Larmor precession frequency of the proton in the magnetic
field, $\omega_p$, the proton magnetic moment, $\mu_p$, the electron
$g$-factor, $g_e$, the electron mass, $m_e$, and the electron magnetic moment,
$\mu_e$, the~\cref{eq:omega_a0} can be rearranged into the form:
\begin{equation}
  a_\mu = \frac{g_e}{2} \frac{\omega_a}{\omega_p} \frac{m_\mu}{m_e} \frac{\mu_p}{\mu_e}.
  \label{eq:omega_a1}
\end{equation}

The ratio $\omega_a/\omega_p$ is experimentally measured in the muon $g$-2
experiments, while other quantities $g_e/2$, $m_\mu/m_e$, and $\mu_p/\mu_e$ are
known to 0.26 part-per-trillion, 22 ppb, and 3 ppb,
respectively~\cite{codata}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{The E989 experiment at Fermilab}
%% TODO: this paragraph is from James Tau 2016, need paraphrasing
The Muon $g − 2$ experiment at Fermilab aims to measure the anomalous magnetic
moment of the muon to a precision of 140 ppb, reducing the experimental
uncertainty by a factor of 4 compared to the previous measurement at E821
The measurement technique adopts the storage ring concept used for
E821, with magic-momentum muons stored in a highly uniform 1.45 T magnetic
dipole field. The spin precession frequency is extracted from an analysis of
the modulation of the rate of higher-energy positrons from muon decays,
detected by 24 calorimeters and 3 straw tracking detectors. Compared to the
E821 experiment, muon beam preparation, storage ring internal hardware, field
measuring equipment, and detector and electronics systems are all new or
significantly upgraded.

\subsection{Magnetic field}
\subsection{Muon beam line}
\subsection{Detector systems}
\subsection{Status}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The E34 experiment at J-PARC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}

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\begin{acknowledgments}
This work was supported in part by the US DOE, Fermilab.
\end{acknowledgments}

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% Create the reference section using BibTeX:
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\bibitem{Alex2018} A. Keshavarzi{\em et al.}, Phys.~Rev.~D 97, 114025 (2018).
\bibitem{BNL2006} G.W. Bennet{\em et al.}, Phys.~Rev.~D 73, 072003 (2006).
\bibitem{codata} Mohr, Peter J. and Newell, David B. and Taylor, Barry N., Rev.~Mod.~Phys.~88, 035009 (2016).


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