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\begin{document}
\Title{$W$ Boson Mass Measurement from CDF}
\bigskip\bigskip
%+\addtocontents{toc}{{\it D. Reggiano}}
%+\label{ReggianoStart}
\begin{raggedright}
{\it Oliver Stelzer-Chilton\index{Stelzer-Chilton, O.}\\
TRIUMF\\
4004 Wesbrook Mall\\
V6T 2A3 Vancouver, BC, CANADA}
\bigskip\bigskip
\end{raggedright}
\section{Introduction}
The $W$ boson mass receives self-energy corrections due to
vacuum fluctuations involving virtual particles. Thus the $W$ boson
mass probes the particle spectrum in nature, including particles that have yet
to be observed directly. The $W$ boson mass can be calculated at tree level using
the precise measurements of the $Z$ boson mass, the Fermi coupling $G_F$ and the
electromagnetic coupling $\alpha_{em}$. In order to extract information
on new particles, we need to account for the radiative corrections to $M_W$.
With the discovery of a 'Higgs like' particle at the LHC \cite{lhc_higgs}, the measured $M_W$
can be used as a consistency check when compared with the predicted $W$ boson mass in the Standard Model
(including radiative corrections due to the Higgs boson loop).
At the Tevatron, $W$ bosons are mainly produced by valance quark-antiquark annihilation, with initial
state gluon radiation generating a typical transverse boost. The transverse momentum ($p_T^l$) distribution
of the decay lepton has a characteristic Jacobian edge whose location is sensitive to the $W$ boson mass.
The neutrino transverse momentum ($p_T^{\nu}$) can be inferred by imposing $p_T$
balance in the event. The transverse mass, defined as $m_T=\sqrt{2 p_T^l p_T^{\nu}(1-cos[\phi^l-\phi^{\nu}])}$, includes both
measurable quantities in the $W$ decay.
We use the $m_T$, $p_T^l$ and $p_T^{\nu}$ distributions to extract $M_W$. These distributions do not
lend themselves to analytic parameterizations, which leads us to use a Monte Carlo simulation to predict
their shape as a function of $M_W$.
\section{Momentum and Energy Scale Calibration}
The key aspect of the measurement is the calibration of the lepton momentum, which is measured in a cylindrical drift chamber called the Central Outer Tracker (COT).
The electron energy is measured using the central electromagnetic (EM) calorimeter and its angle measurement
is provided by the COT trajectory.
The momentum scale is set by measuring the $J/\Psi$ and $\Upsilon(1S)$
masses using the dimuon mass peaks. The $J/\Psi$ sample spans a range of muon $p_T$, which allows us
to tune our ionization energy loss model such that the measured mass is
independent of muon $p_T$. We obtain consistent
calibrations from the $J/\Psi$, $\Upsilon(1S)$ mass fits shown in Fig. \ref{scales} (left).
\begin{figure}[ht]
\centerline{\epsfxsize=3.3in\epsfbox{pscale.eps}\epsfxsize=3.3in\epsfbox{we_eop_fit.eps}}
\caption{Left: Momentum scale summary: $\Delta p/p$ vs $1/p_T$ for $J/\Psi$, $\Upsilon(1S)$ and $Z$ boson samples. The dotted
line represents the independent uncertainty between J/$\Psi$ and $\Upsilon(1S)$.
Right: Energy scale calibration
using $E/p$ from $W\rightarrow e\nu$ events. \label{scales}}
\end{figure}
The momentum scale extracted from the $Z\rightarrow\mu\mu$ mass fit, shown in the same figure,
is consistent, albeit with a larger, statistics-dominated uncertainty.
Given the tracker momentum calibration, we fit the peak of the
$E/p$ distribution of the signal electrons in the $W\rightarrow e\nu$ sample (Fig.~\ref{scales} right)
in order to calibrate the energy measurement of the electromagnetic (EM) calorimeter. The energy scale
is adjusted such that the fit to the peak returns unity.
The model for radiative energy loss is constrained, by comparing the number of events in
the radiative tail of the $E/p$ distribution. The calorimeter
energy calibration is performed in bins of electron $E_T$ to constrain the calorimeter non-linearity.
The calibration yields a $Z\rightarrow ee$ mass measurement of $M_Z = 91230\pm30_{stat}$ MeV/$c^2$, in
good agreement with the world average ($91187.6\pm2.1$ MeV/$c^2$ \cite{lepwmass}); we obtain the most precise calorimeter calibration by combining the
results from the $E/p$ method and the $Z\rightarrow ee$ mass measurement.
\section{Hadronic Recoil Calibration}
The recoil against the $W$ or $Z$ boson is computed as the vector sum of transverse energy over all calorimeter towers, where the towers
associated with the leptons are explicitly removed from the calculation. The response of the calorimeter
to the recoil is described by a response function which scales the true recoil magnitude to simulate the
measured magnitude. The hadronic resolution receives contributions from ISR jets and the underlying event. The
latter is independent of the boson $p_T$ and modeled using minimum bias data.
The recoil parameterizations are tuned on the mean and $rms$ of the
$p_T$-imbalance in $Z\rightarrow ll$ events as a function of boson $p_T$.
\section{Event Generation and Backgrounds}
We generate $W$ and $Z$ events with {\sc resbos} \cite{resbos}, which captures the QCD physics and
models the $W$ $p_T$ spectrum.
The {\sc resbos} parametrization of the
non-pertubative form factor is tuned on the dilepton $p_T$ distribution in the $Z$ boson
sample. Photons radiated off the final-state leptons (FSR) are generated according to
{\sc Photos} \cite{photos} and checked with {\sc HORACE} \cite{horace}. We use the CTEQ6.6 \cite{cteq} set of parton distribution
functions (PDFs) at NLO and evaluate their uncertainties on the $W$
boson mass and verify that the MSTW2008 \cite{mstw} PDFs give consistent results.
Backgrounds passing the event selection have different kinematic distributions from the $W$ signal
and are included in the template fit according to their normalizations.
\section{Results and Conclusions}
The fits to the three kinematic distributions $m_T$, $p_T^l$ and $p_T^{\nu}$ in the electron and muon channels give
the $W$ boson mass results shown in Table \ref{fits}.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|cc}
{} &{} &{}\\[-1.5ex]
Distribution & Fitted $M_W$ [e-channel] (MeV/$c^2$) & Fitted $M_W$ [$\mu$-channel] (MeV/$c^2$) \\[1ex]
\hline
{} &{} &{} \\[-1.5ex]
$m_T$ &80408$\pm$19$_{stat}$$\pm$18$_{syst}$ & 80379$\pm$16$_{stat}$$\pm$16$_{syst}$\\[1ex]
$p_T^l$ &80393$\pm$21$_{stat}$$\pm$19$_{syst}$ & 80348$\pm$18$_{stat}$$\pm$18$_{syst}$\\[1ex]
$p_T^{\nu}$ &80431$\pm$25$_{stat}$$\pm$22$_{syst}$ & 80406$\pm$22$_{stat}$$\pm$20$_{syst}$\\[1ex]
\end{tabular}
\caption{Fit results from the distributions used to extract M$_W$ with uncertainties.}
\label{fits}
\end{center}
\end{table}
\begin{figure}[h]
\centerline{\epsfxsize=3.3in\epsfbox{wm_mt_fit.eps}\epsfxsize=3.0in\epsfbox{gfitter.eps}}
\caption{Left: Transverse mass fit in the muon decay channel.
Right: $\Delta\chi^2$ vs $M_W$ from the Standard Model fit is shown in the blue band \cite{gfitter}, the world average
measured $M_W$ is represented by the red point. \label{results}}
\end{figure}
The transverse mass distribution for the $W\rightarrow \mu\nu$ channel is shown in Fig. \ref{results} (left).
We combine the six $W$ boson mass fits including all correlations to obtain
$M_W$=80387$\pm$12(stat)$\pm$15(syst) MeV/$c^2$.
The uncertainties for the combined result on $M_W$ are summarized in Table \ref{uncertainty}.
\begin{table}[h!]
\caption{Uncertainties for the combined result on $M_W$.}\label{uncertainty}
\begin{center}
\begin{tabular}{l|r}
{} &{} \\[-1.5ex]
Source & Uncertainty (MeV/$c^2$) \\[1ex]
\hline
{} &{} \\[-1.5ex]
Lepton Energy Scale and Resolution& 7\\[1ex]
Recoil Energy Scale and Resolution& 6\\[1ex]
Backgrounds & 3\\[1ex]
$p_T(W)$ Model & 5\\[1ex]
Parton Distributions & 10\\[1ex]
QED radiation & 4\\[1ex]
\hline
W-boson statistics & 12\\[1ex]
\hline
Total Uncertainty & 19 \\[1ex]
\end{tabular}
\end{center}
\end{table}
With a total uncertainty of
19 MeV/$c^2$, this measurement is the most precise measurement to date.
The new world average becomes $M_W$=80385$\pm$15 MeV/$c^2$ \cite{tevwmass}, which is in good agreement with
the Standard Model prediction of $M_W$=80359$\pm$11 MeV/$c^2$ \cite{gfitter}. This is illustrated in Fig. \ref{results} (right), which shows
the $\Delta\chi^2$ vs $M_W$ from
the Standard Model fit as the blue (grey) band, including (excluding) the 'Higgs like' discovery at the LHC \cite{lhc_higgs} at a mass near $\sim$126 GeV/$c^2$.
The world average measured $M_W$ is represented by the red point.
The updated world average $W$ boson mass impacts
the global precision electroweak fits for the Higgs boson mass $M_H$=94$^{+29}_{-24}$ GeV/$c^2$ \cite{lepwmass} which is also in good agreement
with the discovery at the LHC \cite{lhc_higgs}.
Sensitivity to beyond the Standard Model physics
contributions to $M_W$ requires an improved direct measurement of $M_W$, as well as improvements in the
theoretical prediction of the $W$ boson mass. An improved $W$ boson mass measurement can be achieved by using the full Tevatron
datasets and on the longer term, making precise measurements using LHC data. The theoretical predictions are currently limited by uncertainties on $\alpha_{em}$,
the top quark mass and higher order calculations.
\newline\newline
I would like to thank my colleagues from the CDF collaboration
in particular the $W$ boson mass group for their hard work on
this important analysis.
\newpage
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\bibitem{lhc_higgs}
ATLAS Collaboration and CMS Collaboration, Phys. Lett. B 716 (2012) 1-29.
\bibitem{lepwmass} LEP Collaborations and LEP Electroweak Working Group,
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\bibitem{resbos} C. Balazs {\it et. al.}, Phys. Rev. D 56 (1997) 5558; G. Ladinsky {\it et. al.}, Phys. Rev. D 50 (1994) 4239;
F. Landry {\it et. al.}, Phys. Rev. D 67 (2003) 073016.
\bibitem{photos} P. Golonka and Z. Was, Eur. Phys. J. C 45 (2006) 97.
\bibitem{horace} C.M. Carloni Calame {\it et. al.}, J. High Energy Phys. 0710 (2007) 109.
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\bibitem{mstw} A. D. Martin {\it et. al.}, Eur. Phys. J. C 63 (2009) 189.
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%F. A. Mesmer, Proc. Wien. Acad. Sci. {\bf 13}, 1564, 1593 (1762).
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\end{document}