First run II measurement of the $W$ boson mass at the Fermilab Tevatron

We describe a measurement of the $W$ boson mass $m_W$ using $200{\mathrm{pb}}^{-1}$ of $\sqrt{s}=1.96\mathrm{TeV}$ $p\overline{p}$ collision data taken with the CDF II detector. With a sample of 63 964 $W\to e\nu$ candidates and 51 128 $W\to \mu \nu$ candidates, we measure $m_W=[80.413\pm 0.034(\mathrm{stat})\pm 0.034(\mathrm{sys})=80.413\pm 0.048]\mathrm{GeV}/c^2$. This is the single most precise $m_W$ measurement to date. When combined with other measured electroweak parameters, this result further constrains the properties of new unobserved particles coupling to $W$ and $Z$ bosons.

Figure 1

The one-loop contribution to the $W$ boson mass from top and bottom quarks.

Figure 2

Higgs one-loop contributions to the $W$ boson mass.

Figure 3

One-loop squark contributions to the $W$ boson mass.

Figure 4

Leading-order annihilation of a quark and antiquark inside the proton and antiproton, respectively, producing a $W^{+}$ or $Z^0$ boson. The quark (antiquark) has energy $x_p{E}_p$ ($x_{\overline{p}}{E}_{\overline{p}}$), where $E_p$ ($E_{\overline{p}}$) represents the total proton (antiproton) energy. The production occurs at a partonic center-of-mass energy $Q$. The $u\overline{u}\to Z^0$ and $d\overline{u}\to W^{-}$ processes are similar.

Figure 5

A $W$ boson event, with the recoil hadron momentum (${\overrightarrow{u}}_T$) separated into axes parallel ($u_{\Vert }$) and perpendicular ($u_{\perp }$) to the charged lepton.

Figure 6

A cut-away view of a section of the CDF detector. The slice is along the $y$-axis at $x=0\mathrm{cm}$.

Figure 7

End view of the silicon detector. The innermost layer (Layer 00) is attached to the beam pipe, and is surrounded by five concentric layers of silicon wafers (SVX II). The outermost layers are the intermediate silicon layers (ISL), which sit just inside the outer tracking chamber.

Figure 8

End view of a section of a central outer tracker (COT) end plate. The COT consists of eight concentric “superlayers,” separated azimuthally into cells, each containing 12 sense wires and sandwiched by field sheets. The end plates contain precision-machined slots where each cell’s sense wires and field sheets are held under tension. The radius at the center of each superlayer is shown in cm.

Figure 9

The distribution of material upstream of the COT traversed by reconstructed electron trajectories in $W\to e\nu$ data events, in units of radiation lengths. The peaks at $\approx 0.08$, $\approx 0.13$, and $\approx 0.24$ correspond, respectively, to trajectories outside the silicon detector ($|z|>45\mathrm{cm}$), within the silicon detector, and crossing silicon barrels ($|z|\approx 15$, 45 cm). The mean of the distribution is 19%.

Figure 10

The ratio of the photon conversion cross section at a given photon energy to the cross section at $E_{\gamma }=100\mathrm{GeV}$ [55]. We use this function to scale down the cross section obtained from the Bethe-Heitler equation [Eq. (13)] [49].

Figure 11

The ratio of the Compton scattering cross section at a given photon energy to the pair-production cross section at $E_{\gamma }=100\mathrm{GeV}$ [55]. This ratio is applicable for photons traversing silicon.

Figure 12

The mean $p_T$ loss as a function of ${log}_{10}(p_T/\mathrm{GeV})$ for electrons with $p_T>400\mathrm{MeV}$ (solid line) and photons (dashed line) traversing the time-of-flight system and solenoid.

Figure 13

The electron (solid line) and photon (dashed line) $p_T$ leakage into the hadronic calorimeter. Top: The mean $p_T$ leakage as a function of ${log}_{10}(p_T/\mathrm{GeV})$. Bottom: The distributions of $p_T$ leakage for high-$p_T$ ($>10\mathrm{GeV}$) photons and electrons.

Figure 14

The muon identification efficiency as a function of the recoil component in the direction of the muon ($u_{\Vert }$).

Figure 15

The electron track trigger efficiency as a function of track $\eta$, for electrons identified in the calorimeter. The solid line shows the double-Gaussian parametrization of the data.

Figure 16

The electron identification efficiency measured in $Z\to ee$ data as a function of the recoil component in the direction of the electron ($u_{\Vert }$). Background is subtracted using the number of like-charge lepton events observed at a given $u_{\Vert }$. The $E/p<2$ requirement is not included in this efficiency measurement.

Figure 17

The definitions of the local tilt ($\Delta \tau$) and azimuthal shift ($R\Delta \varphi$) alignment corrections applied to each COT cell.

Figure 18

The local tilt (top) and azimuthal shift (bottom) alignment corrections applied to each cell of the inner superlayer of the west end plate. Not shown are a global 0.0021 tilt correction and a small global rotation and shift of the COT that only redefines the axes.

Figure 19

The net wire shift in $y$ as a function of $z$ from gravitational sag only (solid line), including electrostatic effects (dashed line), and including data-based corrections from Eq. (22) (dotted line). The shift is shown at $\varphi =\pi$ and $R=130\mathrm{cm}$.

Figure 20

The difference between $e^{+}$ and $e^{-}E/p$ as a function of $\cot \theta$ (top) and $\varphi$ (bottom) before (solid triangles) and after (open diamonds) track-level corrections.

Figure 21

The $J/\psi \to \mu \mu$ data (circles) and tuned simulation (histogram) distributions of $p_T^{\mu \mu }$ (top) and $\sum _{}^{}q/p_T^{\mu }$ (bottom). The $\sum _{}^{}q/p_T^{\mu }$ is equal to the sum of the track curvatures of muons from a $J/\psi$ decay, divided by $2.11593\times {10}^{-3}$.

Figure 22

The fractional momentum correction for data as a function of the mean inverse momentum of the muons from $J/\psi$ decays. In a linear fit, the intercept corresponds to the scale correction relevant for $W$ and $Z$ boson decays, and the slope corresponds to the remaining unmodeled ionization energy loss after material tuning. The uncertainties are statistical only.

Figure 23

The $m_{\mu \mu }$ fits (histograms) to data (circles) with $⟨p_T^{-1}⟩=(0.15,0.2){\mathrm{GeV}}^{-1}$ (top) and $⟨p_T^{-1}⟩=(0.25,0.3){\mathrm{GeV}}^{-1}$ (bottom). The best fits to the $m_{\mu \mu }=(3.08\pm 0.12)\mathrm{GeV}$ (top) and $m_{\mu \mu }=(3.08\pm 0.11)\mathrm{GeV}$ (bottom) regions correspond to momentum scale corrections of $(-1.54\pm 0.09)\times {10}^{-3}$ (top) and $(-1.65\pm 0.04)\times {10}^{-3}$ (bottom). The arrows indicate the fit regions and the uncertainties are statistical only. The fit ${\chi }^2$ can be improved by adjusting the final-state radiation model, and this effect is incorporated into the systematic uncertainty (Sec. 5b4).

Figure 24

The $p_T^{\mu \mu }$ (top) and $p_T^{\mu }$ (bottom) distributions from ${\rm Y}\to \mu \mu$ decays for the data (circles) and simulation (histogram).

Figure 25

The momentum scale correction $\Delta p/p$ derived from binned likelihood fits to the $m_{\mu \mu }$ data distribution (circles) in the region dominated by ${\rm Y}\to \mu \mu$ decays. The small difference between fits using tracks with (top) and without (bottom) a beam constraint is incorporated into the systematic uncertainty. The arrows indicate the fit region and the uncertainties are statistical only.

Figure 26

The $m_Z$ fit to data (circles) in the $83\mathrm{GeV}<m_{\mu \mu }<99\mathrm{GeV}$ region (denoted by arrows). The uncertainty is statistical only.

Figure 27

The maximum likelihood fit (histogram) for the multiplicative correction factor to the number of tracker radiation lengths using the high-$E/p$ region ($1.19\le E/p<1.85$) in $W\to e\nu$ data (circles).

Figure 28

The energy scale as a function of measured electron calorimeter $p_T$ for $W$ (top) and $Z$ (bottom) boson decays. The fast simulation incorporates a per-particle nonlinear response correction [Eq. (17)]. Combining the individual linear fits to $\xi$, shown in the figure, results in no energy dependence of the energy scale.

Figure 29

The energy scale calibration using the peak $E/p$ region ($0.93\le E/p<1.11$, denoted by arrows) in $W\to e\nu$ data (circles).

Figure 30

The $m_Z$ fit (histogram) to data (circles) in the $75\mathrm{GeV}<m_{ee}<99\mathrm{GeV}$ region (denoted by arrows), using reconstructed track information only. The uncertainty is statistical only.

Figure 31

The $m_Z$ fit to data (circles) in the $81\mathrm{GeV}<m_{ee}<101\mathrm{GeV}$ region (denoted by arrows). The uncertainty is statistical only.

Figure 32

The maximal energy correction factors applied to each tower as a function of $|\eta |$. The decreasing curve corresponds to towers closest to the beam line, and the increasing curve corresponds to towers on the opposite side. The correction factors reduce the $\varphi$ variation in acceptance.

Figure 33

The average $p_T$ measured by the electromagnetic (top) and hadronic (bottom) calorimeters in the vicinity of the electron shower in $W$ boson decays. The differences $\Delta \varphi$ and $\Delta \eta$ are signed such that positive differences correspond to towers closest to the electron position at shower maximum. The central seven towers inside the box are removed from the recoil measurement. Statistical uncertainties on the values outside the box are $\mathcal{O}(1\mathrm{MeV})$.

Figure 34

The average $p_T$ measured by the electromagnetic (top) and hadronic (bottom) calorimeters in the vicinity of the muon in $W$ boson decays. The differences $\Delta \varphi$ and $\Delta \eta$ are signed such that positive differences correspond to towers closest to the muon position at the shower maximum detector. The central three towers inside the box are removed from the recoil measurement. Statistical uncertainties on the values outside the box are $\mathcal{O}(1\mathrm{MeV})$.

Figure 35

The scales applied to the transverse momentum in the three removed muon towers in the simulation (solid lines), as functions of $u_{\Vert }$ (top), $u_{\perp }$ (middle), and $\eta$ (bottom). The circles show the scales measured using towers separated in $\varphi$ from the muon in $W\to \mu \nu$ data. The scaling functions for removed electron towers have similar shapes.

Figure 36

The definitions of the $\eta$ and $\xi$ axes in $Z$ boson events. The quark and gluon radiation from the boson production points predominantly in the $-\eta$ direction.

Figure 37

The mean $-{\overrightarrow{u}}_T^{\mathrm{meas}}·{\widehat{p}}_T^{\mu \mu }/p_T^{\mu \mu }$, which approximates the recoil response $R$, as a function of dimuon $p_T$ for $m_{\mu \mu }$ in the $Z$ boson mass region. The simulation (lines) uses parameters fit from the electron and muon $p_{\eta }^{ll}+u_{\eta }$ distributions, and models the data (circles) well.

Figure 38

The simulation (lines) and data (circles) distributions of ($p_{\eta }^{ll}+u_{\eta }$) for $Z$ boson decays to muons (top) and electrons (bottom). The combined ${\chi }^2$ is minimized in the fit for the recoil detector response parameters.

Figure 39

The $\sum _i{p}_T^i$ distribution for minimum bias data (solid line) and a single $p\overline{p}$ collision (dashed line), as derived from the minimum bias distribution.

Figure 40

The angle between the measured recoil and the direction opposite $p_T^{ll}$, for simulation (histogram) and data (circles) events where the $Z$ boson decays to muons (top) or electrons (bottom). The ${\chi }^2$ from the $Z\to \mu \mu$ sample is minimized in the fit to the recoil angular resolution. The corresponding uncertainty on $m_W$ is negligible.

Figure 41

The simulation (lines) and data (circles) $p_{\eta }^{ll}+u_{\eta }$ (top) and $p_{\xi }^{ll}+u_{\xi }$ (bottom) resolutions for $Z$ boson decays to muons and electrons. The sum of the four ${\chi }^2$ values is minimized in the fit for the recoil resolution parameters $N_{W,Z}$ and $s_{\mathrm{hard}}$. Since there are four distributions and two fit parameters, each distribution contributes half a degree of freedom to the fit.

Figure 42

The simulation (solid line) and data (circles) $u_{\Vert }$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation uses parameters fit from $Z$ boson data, and the uncertainty on the simulation is due to the statistical uncertainty on these parameters. The data mean ($\mu$) and RMS ($\sigma$) are well modeled by the simulation.

Figure 43

The simulation (solid line) and data (circles) $u_{\perp }$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation uses parameters fit from $Z$ boson data, and the uncertainty on the simulation is due to the statistical uncertainty on these parameters. The data mean ($\mu$) and RMS ($\sigma$) are well modeled by the simulation.

Figure 44

The simulation (solid line) and data (circles) $u_T$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation uses parameters fit from $Z$ boson data, and the uncertainty on the simulation is due to the statistical uncertainty on these parameters. The data mean ($\mu$) and RMS ($\sigma$) are well modeled by the simulation.

Figure 45

The ${\mathrm{p̸}}_T$ distribution of the $W\to e\nu$ candidate sample (triangles) and prediction (solid line), with the ${\mathrm{p̸}}_T$ and $m_T$ selection cuts removed. We fit for the normalization of the hadronic-jet background (dotted line) after fixing the normalization of the $W\to e\nu$ distribution (dashed line) in the peak. Not shown are the $Z/{\gamma }^{*}\to ee$ and $W\to \tau \nu$ backgrounds, whose relative normalizations are fixed from the simulation.

Figure 46

The parametrizations of the backgrounds to the $W\to e\nu$ data sample. The backgrounds to the $m_T$ (top), $p_T$ (middle), and ${\mathrm{p̸}}_T$ (bottom) distributions are included in the $m_W$ fits.

Figure 47

The $t_0$ distributions for $W\to \mu \nu$ candidates (solid line), $Z\to \mu \mu$ candidates (dashed line), and identified cosmic rays (dotted line). For comparison, the distribution from $Z$ candidates has been normalized to that of the $W$ candidates, while the cosmic distribution has been normalized to 0.05% of the $W$ boson sample.

Figure 48

The parametrizations of the backgrounds to the $W\to \mu \nu$ data sample. The backgrounds to the $m_T$ (top), $p_T$ (middle), and ${\mathrm{p̸}}_T$ (bottom) distributions are included in the $m_W$ fits. Not shown are the small hadronic-jet and cosmic-ray background distributions.

Figure 49

The simulation (solid line) and data (circles) $p_T^Z$ distributions for $Z$ boson decays to $\mu \mu$ (top) and $ee$ (bottom). The distributions are used to fit for the nonperturbative parameter $g_2$, which determines the most probable value of $p_T^Z$. Since there are two distributions for one fit parameter, each distribution contributes half of a degree of freedom. The mean ($\mu$) and RMS ($\sigma$) are consistent between data and simulation.

Figure 50

The cube root of the fraction of electron (solid line) or muon (dashed line) momentum contained in the radiated photon (top), and the square root of the angle $\Delta R$ (bottom) between the radiated photon and the electron (solid line) or muon (dashed line). The first two bins in the $y^{1/3}$ distribution correspond to photons of energy $E_{\gamma }\le 5\mathrm{MeV}$ for leptons with energy $E_l=40\mathrm{GeV}$.

Figure 51

The simulation (solid line) and data (points) $m_T$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation corresponds to the best-fit $m_W$, determined using events between the two arrows. The uncertainty is statistical only. The large ${\chi }^2$ for the electron fit is due to individual bin fluctuations (Fig. 55) and does not bias the fit result, as evidenced by the small change in the fit $m_W$ when the fit window is varied (Fig. 58).

Figure 52

The simulation (solid line) and data (points) $p_T$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation corresponds to the best-fit $m_W$, determined using events between the two arrows. The uncertainty is statistical only.

Figure 53

The simulation (solid line) and data (points) ${\mathrm{p̸}}_T$ distributions for $W$ boson decays to $\mu \nu$ (top) and $e\nu$ (bottom). The simulation corresponds to the best-fit $m_W$, determined using events between the two arrows. The uncertainty is statistical only.

Figure 54

The negative log of the likelihood ratio $\mathcal{L}/{\mathcal{L}}_0$, where ${\mathcal{L}}_0$ is the maximum likelihood, as a function of $m_W$ for the $m_T$ (solid line), $p_T$ (dashed line), and ${\mathrm{p̸}}_T$ (dotted line) fits in the electron (top) and muon (bottom) channels.

Figure 55

The difference between the data and simulation, divided by the statistical uncertainty on the prediction, for the $m_T$ distributions in the muon (top) and electron (bottom) channels. The arrows indicate the fit region.

Figure 56

The difference between the data and simulation, divided by the statistical uncertainty on the prediction, for the $p_T$ distributions in the muon (top) and electron (bottom) channels. The arrows indicate the fit region.

Figure 57

The difference between the data and simulation, divided by the statistical uncertainty on the data points, for the ${\mathrm{p̸}}_T$ distributions in the muon (top) and electron (bottom) channels. The arrows indicate the fit region.

Figure 58

The shifts in measured $m_W$ for variations in the lower (top) and upper (bottom) edges of the $m_T$ fit range. The electron channel is denoted by open triangles and the muon channel by solid circles. The error bars indicate the expected statistical variations from simulation pseudoexperiments. The dashed (dotted) lines indicate the statistical uncertainty on the $m_W$ fit using the default fit range in the electron (muon) channel.

Figure 59

The shifts in measured $m_W$ for variations in the lower (top) and upper (bottom) edges of the $p_T$ fit range. The electron channel is denoted by open triangles and the muon channel by solid circles. The error bars indicate the expected statistical variations from simulation pseudoexperiments. The dashed (dotted) lines indicate the statistical uncertainty on the $m_W$ fit using the default fit range in the electron (muon) channel.

Figure 60

The shifts in measured $m_W$ for variations in the lower (top) and upper (bottom) edges of the ${\mathrm{p̸}}_T$ fit range. The electron channel is denoted by open triangles and the muon channel by solid circles. The error bars indicate the expected statistical variations from simulation pseudoexperiments. The dashed (dotted) lines indicate the statistical uncertainty on the $m_W$ fit using the default fit range in the electron (muon) channel.