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\begin{document}
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\begin{center}{\Large \textbf{
Status of the Muon ${\mathbf g-2}$ experiment at Fermilab
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A. Driutti \textsuperscript{1,2*}
on behalf of the Muon g-2 collaboration \textsuperscript{**}
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{\bf 1} {Universit\`a di Udine, Udine, Italy}
\\
{\bf 2} {INFN, Sezione di Trieste, Trieste, Italy}
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* anna.driutti@uniud.it \\
** \url{http://muon-g-2.fnal.gov/collaboration.html}
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\today
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{\it Proceedings for the 15th International Workshop on Tau Lepton Physics,}\\
{\it Amsterdam, The Netherlands, 24-28 September 2018} \\
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\section*{Abstract}
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The aim of the Muon $g-2$ Experiment at Fermilab (E989) is to measure the muon anomalous magnetic moment ($a_\mu$) with a relative precision of 140 parts-per-billion (ppb). This precision, which is a factor of four improvement from the current experimental result, will allow for a much more stringent test of the Standard Model.
This paper present the current status of the experimental measurement of $a_\mu$ after the first physics run.
}
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\section{Introduction}
\label{sec:intro}
The muon anomaly is defined as the relative deviation of the muon $g$-factor from 2, $a_\mu=(g_\mu -2)/2$. The $g$-factor is the ratio between the particle magnetic moment and the spin of a particle. In the framework of the Standard Model (SM) of particle physics, for the spin-$1/2$ particles like the muon, $g$ is expected to be 2 at first order, and the higher order contributions from QED, electroweak, hadronic vacuum polarization, and hadronic light-by-light virtual loops lead to $g > 2$. In particular the most recent SM theoretical prediction is $a^{\rm SM}_\mu = (11659182.04\pm3.56) \times 10^{-10}$~\cite{Keshavarzi:2018mgv}, and the contributions of each of the four main components are summarized in table~\ref{tab:1}.
\begin{table}[ht]
\centering
\begin{tabular}{lcr}
\hline
$a\mu$ Contribution & From & Value ($\times 10^{-10}$) \\
\hline
QED & \cite{Aoyama:2012wk,Aoyama:2017uqe} & $11658471.8971 \pm 0.007$ \\
Electroweak & \cite{Gnendiger:2013pva} & $15.36\pm0.10$ \\
Hadronic Vacuum Polarization & \cite{Keshavarzi:2018mgv} & $684.68\pm2.42$ \\
Hadronic light by light & \cite{Nyffeler:2016gnb} & $9.8\pm2.6$ \\
\hline
Total SM ($a^{\rm SM}_\mu$) & & $11659182.04\pm3.56$ \\
\hline
\end{tabular}
\caption{Current predictions for the Standard Model contributions to $a_\mu$.}\label{tab:1}
\end{table}
Since the muon anomaly arises from higher-order contributions from virtual loops, the comparison between theoretical and experimental $a_\mu$ values is a test of the SM: a possible discrepancy could be a hint of new physics.
Experimentally, $a^{\rm E821}_\mu = 11659209.1(5.4)(3.3)\times 10^{-10}$~\cite{Bennett:2006fi} was measured in 2006 by the E821 collaboration at the Brookhaven National Laboratory (BNL), and is about 3.7 standard deviation ($\sigma$) higher than the most recent theoretical prediction $a^{\rm SM}_\mu$.
The ability to measure $a_\mu$ with a relative uncertainty $\delta a_\mu/a_\mu$ of 140 ppb, as aimed at the Fermilab Muon $g-2$ experiment, has the potential to confirm the discrepancy with a 7~$\sigma$ significance.
\section{Experimental Technique}
\label{sec:ET}
The Muon $g-2$ experiment at Fermilab (E989)~\cite{Grange:2015fou} is an improved version of the previous E821 experiment performed at BNL. A longitudinally-polarized muon beam is injected into a storage ring with a uniform magnetic field, $\vec{B}$ and electric field $\vec{E}$ for vertical confinement. The storage ring used by the E821 collaboration was moved from Brookhaven (New York) to Fermilab (Chicago) in the summer 2013. Inside the ring the muons are subject to both the spin precession frequency $\omega_s$ and the cyclotron frequency $\omega_c$: the difference between them is called anomalous precession frequency $\omega_a$. Under the approximation that the muon velocity is perpendicular to the magnetic field ($\vec{\beta} \cdot \vec{B}=0$), the magnetic field is perfectly uniform and, betatron oscillations of the beam are neglected $\omega_a$ that can be expressed as
\begin{equation}
\label{eq:1}
\vec{\omega}_a = \vec{\omega}_s - \vec{\omega}_c = - \frac{e}{m_\mu} \left[ a_\mu \vec{B} - \left( a_\mu - \frac{1}{\gamma^2 -1} \right) \frac{\vec{\beta} \times \vec{E}}{c}\right],
\end{equation}
\noindent where $\vec{\beta}$ the velocity of the muons in units of $c$, and $\gamma = 1/\sqrt{1 - \beta^2}$ is the Lorentz factor. The muon beam enters the storage ring with a forward momentum of $\simeq 3.094$ GeV/$c$, $i.e$, $\gamma \simeq 29.4$, hence $ a_\mu - 1/(\gamma^2-1) \simeq 0$ and Eq.~\ref{eq:1} simplifies to:
\begin{equation}
\label{eq:2}
\omega_a = - \frac{e}{m_\mu} a_\mu B.
\end{equation}
\noindent Therefore the muon anomaly can be then be extracted by measuring $\omega_a$ and $B$ precisely.
The anomalous precession frequency, $\omega_a$, is not directly extracted from the muons but from their decayed high-energy positrons. In fact, because of the parity-violating nature of the $\mu^+ \rightarrow e^+ \nu_e \bar{\nu}_{\mu}$ decay, the high-energy positrons are preferentially emitted in the muon-spin direction. The spin directions of the muons can then be inferred from the decay time and the energy of the positrons, which are reconstructed using the information recorded by the calorimeter system.
The magnetic field is mapped using proton nuclear magnetic resonance (NMR) probes. These NMR probes return the magnetic field strength in terms of precession angular frequency of the free proton, $\omega_p$. Then the average magnetic field experienced by the muons, $\bar{\omega}_p$, is measured by integrating all the $\omega_p$ values over the muon storage region.
The muon anomalous magnetic moment can be then obtained from the measurement of $\omega_a$ and $\bar{\omega}_p$ using:
\begin{equation}
\label{eq:3}
a_\mu = \frac{g_e}{2} \frac{m_\mu}{m_e} \frac{\mu_p}{\mu_e} \frac{\omega_a}{\bar{\omega}_p},
\end{equation}
\noindent where $g_e$ is the electron $g$-factor, ${m_\mu}/{m_e}$ and ${\mu_p}/{\mu_e}$ are the ratios between respectively, the muon and electron masses and the proton and electron magnetic moments. The value of these three factors are known with a precision of 22~ppb from other independent measurements~\cite{Mohr:2015ccw}.
\section{Overview of the E989 Experiment}
\label{sec:EO}
The E989 experiment is planning to reduce the uncertainty on the most recent experimental measurement of $a_\mu$ by reducing both the statistical and systematic uncertainty as shown in Table~\ref{tab:2}.
\begin{table}[h!]
\centering
\begin{tabular}{lcc}
\hline
$\delta a_\mu / a_\mu$ & E821 (BNL) & E989 (FNAL)\\
& [ppb] & [ppb]\\
\hline
$\omega_a$ statistic & 480 & 140 \\
$\omega_a$ systematic & 180 & 70 \\
$\bar{\omega}_p$ systematic & 170 & 70 \\
\hline
{\bf Total} & {\bf 540} & {\bf 140} \\
\hline
\end{tabular}
\caption{Summary of the reported and expected relative uncertainties on the muon anomaly measurements.}\label{tab:2}
\end{table}
The main limitation of the BNL experiment was the statistical uncertainty. The Fermilab experiment is planning to acquire approximately 20 times the BNL statistics, so that the statistical uncertainty will be reduced to 100~ppb. This can be achieved thanks to the accelerator complex at Fermilab, which provides about $10^5$ ($>90\%$ polarized) positive-muons per second, with very low contamination of pions ($\sim 10^{-5}$) and no protons. These muons are generated from a 8 GeV beam consisting of $10^{12}$ protons hitting a target with a rate of 12 bunches/second to produce pions. For each bunch, about $1.2 \times 10^8$ 3.11~GeV monoenergetic $\pi^+$ are produced and stored in a 500~m circumference ring called delivery ring. In the delivery ring the pions decay into muons ($\pi^+\rightarrow\mu^+\nu_\mu$)~\cite{Grange:2015fou}.
The 3.094 Gev/$c$ beam of polarized-$\mu^+$ is then injected into the 15-ton and 14-m diameter superferric storage ring through a field-cancelling inflector magnet.
The ring is made up of rectangular vacuum chambers surrounded by a cryosystem and C-shaped dipole magnets and it is kept mechanically and thermally stable. After injection the muons are moved onto the center of the storage ring's orbit, where the field is more uniform, by a kicker system. The beam is vertically contained in the center region by 4 sets of electrostatic quadrupoles plates placed around the ring, and off-momentum muons are removed using collimators.
The magnetic field inside the storage ring has 1.45~T of magnitude and is as uniform as possible with a target of azimuthal variations not exceeding $\pm50$~ppm point-to-point, and $< 1$~ppm when considering the transverse variations across the storage aperture after averaging around the azimuth.
Uniformity of the field is achieved using various magnetic components placed around the muon region to locally affect $\vec{B}$. In particular, the higher average field magnetic moments along the ring are removed using active surface coils, the dipole field is coarse tuned using iron pole pieces and iron top hats, while localized nonuniformities are cured by gluing iron foils on the pole pieces surface, and quadrupole and sextuple asymmetries are reduced with iron wedges and edge shims. The position of these calibration tools is shown in Figure~\ref{fig:1}.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{magnet}
\caption{Schematic drawn of the $g-2$ storage ring cross-section. The muon beam is schematically depicted as a circle with a plus ($\oplus$) and travels around the ring towards the reader.}
\label{fig:1}
\end{center}
\end{figure}
The E989 systematic uncertainty budget for the $\bar{\omega}_p$ measurement is $70$~ppb. During data-taking fixed NMR probes located above and below the muon region as shown in Figure~\ref{fig:1} are used to determine $\omega_p$. Periodically, the field between the fixed probes is mapped using a trolley with 17 NMR probes that travels like the muons, around ring. To obtain a precise map of the magnetic field, three tracker stations (each made of $\sim 3000$ straws divided into 8 modules) are used to extrapolate from the decayed positrons trajectory the beam profile of the stored muons. In addition, the $xy$ profile of the beam is periodically monitored using a set of scintillating fibers hodoscopes.
To reduce the systematic uncertainty on the $\omega_a$ measurement to $70$~ppb, the E989 experiment has been equipped with 24 new highly segmented electromagnetic calorimeters. Each calorimeter is made of 54 lead fluoride crystals read out by arrays of silicon-photomultiplier (SiPMs). The calorimeters are located around the inner part of the ring to detect the positrons from the decay of the stored muons.
In E821, the main sources of the systematic uncertainty for the $\omega_a$ measurement were related to the gain stability of the calorimeters and the pile-up identification and correction. In E989, an elaborate laser calibration system is used to monitor and ensure the gain stability at the sub-per-mil level~\cite{Anastasi:2016luh}. The systematic uncertainty due to pile-up events, $i.e.$ events where multiple positrons hit a calorimeter at the same time and are reconstructed as a higher-energy single positron, is reduced thanks to the better time and spatial separation allowed by the fast response and the finer segmentation of the calorimeters.
\section{E989 First Physics Run}
\label{sec:FFR}
The E989 first physics run, Run~1, took place between late March and early July 2018. During Run~1, the storage ring magnet was operational for about 95\% of the time and about $\sim 500 e^+$ per muon bunch were recorded. The final raw dataset (no quality cuts applied) contains about twice the statistics accumulated at BNL and the magnetic field was mapped with more than 30 trolley runs.
A blinding factor was applied to the reference clock frequency, and in figures~\ref{fig:2a} and~\ref{fig:2b} the preliminary blind measurements of $\omega_a$ and $\omega_p$ performed with a small part of the Run~1 dataset are shown. Completion of the full Run~1 analysis is expected for the second half of 2019.
\begin{figure}[ht!!!!]
\captionsetup{labelsep=period, format=plain, justification=justified}
\centering
\vspace{0.0cm}
\subfloat[]{\includegraphics[width=0.5\textwidth]{Omega_a.png} \label{fig:2a}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{Omega_p.png} \label{fig:2b}}
\caption{(a) Preliminary positron time spectrum (with no data quality cuts applied). (b) The magnetic field averaged over azimuth and the dipole component as a function of azimuth around the ring. \label{fig:2}}
\end{figure}
The rate of muons stored was about 50\% of the TDR design~\cite{Grange:2015fou}. During the summer shutdown of 2018, upgrades to the accelerator complex and to the experiment muon kicker and electric quadrupole systems are taking place to increase the number of stored muon during Run~2, which will start in late 2018.
\section{Conclusion}
The goal of the muon $g-2$ experiment at Fermilab (E989) is to perform a test of the SM of particle physics measuring the anomalous magnetic moment of the muon with a precision of 140~ppb.
The E989 successfully completed its first physics run, during which a dataset exceeding the statistics of the previous experiment was acquired. In 2019, the E989 collaboration expects to complete the analysis of the Run 1 data and to collect a significantly larger data set.
\section*{Acknowledgements}
This work was supported in part by the US DOE, Fermilab (Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359), the Istituto Nazionale di Fisica Nucleare (Italy), the European Union's Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreements No. 690835 (MUSE).
% FIRST OPTION - write your entries here directly, following the example below, including Author(s), Title, Journal Ref. with year in parentheses at the end, followed by the DOI number.
\begin{thebibliography}{99}
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