Measurement of the $W$ boson mass with the D0 detector

We give a detailed description of the measurement of the $W$ boson mass, $M_W$, performed on an integrated luminosity of $4.3{\mathrm{fb}}^{-1}$, which is based on similar techniques as used for our previous measurement done on an independent data set of $1{\mathrm{fb}}^{-1}$ of data. The data were collected using the D0 detector at the Fermilab Tevatron Collider. This data set yields $1.68\times 10^6$ $W\to e\nu$ candidate events. We measure the mass using the transverse mass, electron transverse momentum, and missing transverse energy distributions. The $M_W$ measurements using the transverse mass and the electron transverse momentum distributions are the most precise of these three and are combined to give $M_W=80.367\pm 0.013(\text{stat})\pm 0.022(\text{syst})\mathrm{GeV}=80.367\pm 0.026\mathrm{GeV}$. When combined with our earlier measurement on $1{\mathrm{fb}}^{-1}$ of data, we obtain $M_W=80.375\pm 0.023\mathrm{GeV}$.

Figure 1

The (a) $p_T^e$ and (b) $m_T$ spectra for simulated $W$ bosons without detector resolution effects and $W$ boson transverse momentum $p_T^W=0$ (solid line), with the natural $p_T^W$ spectrum at the Tevatron (shaded area), and with the natural $p_T^W$ distribution and all detector resolution effects included (points). All curves are normalized to unit area.

Figure 2

(a) Definition of $\eta$ and $\xi$ axes for $Z\to ee$ events. (b) Definition of $u_{\parallel }$ and $u_{\perp }$. The variable $u_{\parallel }$ is negative when opposite to the electron direction.

Figure 3

Side view of one quadrant of the D0 detector, not showing the muon subdetector system. The calorimeter segmentation and tower definition are shown in both CC and EC. The lines extending from the center of the calorimeter denote the pseudorapidity (${\eta }_{\text{det}}$) coverage of cells and projected towers. The solenoid and tracking detectors are shown in the inner part of the detector.

Figure 4

Instantaneous luminosity profiles for Run IIa and Run IIb. The instantaneous luminosity is given as a multiple of $36\times 10^{30}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. Since there were 36 $p\overline{p}$ bunch crossings per turn in the Tevatron Collider, the values in the abscissa can be interpreted as the average bunch instantaneous luminosity.

Figure 5

The 13 calorimeter towers defined as the electron reconstruction cone. The cone is centered on the tower with the highest transverse energy. A circle of radius $\mathrm{\Delta }R=0.2$ is shown for comparison.

Figure 6

Overview of the material in front of the CC. This drawing shows a cross-sectional view of the central tracking system in the $x\text{−}z$ plane. Also shown are the locations of the solenoid, the preshower detectors, the luminosity monitor, and the calorimeters.

Figure 7

The average shower energy deposition profile (along the shower axis) of electrons with $E=45\mathrm{GeV}$ simulated using the GFlash parametrization [53]. The depth of each readout section of the central calorimeter is indicated for an electron with (a) normal incidence $\eta =0$ and with (b) non-normal incidence $\eta =1$.

Figure 8

The ratio of data to simulation for the means of the EM layer energy fraction distributions in $Z\to ee$ events for each of the first three EM layers and each of the $15\eta$ categories shown before the correction described in Sec. 5c is applied. Each of the three horizontal lines indicates the result of a fit of a common constant to the 15 data points from a given EM layer.

Figure 9

Fit for $n{X}_0$, the amount of uninstrumented material (in radiation lengths) added to the nominal material in the improved simulation of the D0 detector. The solid and dotted vertical lines show the best fit and 1 standard deviation uncertainties for $n{X}_0$. This fit is performed with the $Z\to ee$ data sample from our $1{\mathrm{fb}}^{-1}$ measurement [28].

Figure 10

Stability check: results of the fit for $n{X}_0$, performed separately for each of the three layers (EM1, EM2, and EM3). The result of the combined fit is also shown for comparison.

Figure 11

The ratio of data to simulation for the means of the EM layer energy fraction distributions in $Z\to ee$ events for each of the first three EM layers and each of the $15\eta$ categories shown after the correction described in Sec. 5c is applied. Each of the three horizontal lines indicates the result of a fit of a common constant to the 15 data points from a given EM layer.

Figure 12

The data/full MC ratios for the means of the EM layer energy fraction distributions in $W\to e\nu$ events for the (a) EM1, (b) EM2, and (c) EM3 layers. The ratio is shown in five electron $\eta$ bins. The thick horizontal lines indicate the average ratio across the central calorimeter, and the shaded (yellow) band represents the systematic and statistical uncertainty in the mean.

Figure 13

The mean data/full MC ratio for the means of the EM layer energy fractions, for electrons from $W\to e\nu$ decays, in each of the three first layers of the EM calorimeter. The innermost error bar (red) indicates the uncertainties from the $W$ boson sample size. The middle error bar (green) indicates the quadrature sum of the uncertainty from the $W$ boson sample size with the one from the $Z$ boson sample size, determined from the uncertainty in the thickness of the added material. Finally, the outermost error bar (blue) represents the quadrature sum of the two previous uncertainties with the one arising from the limited number of full MC events. In all three layers, the ratio is consistent with unity when all uncertainties are considered.

Figure 14

(a) The true energy spectrum for electrons in simulated $W$ boson events that pass the full selection after reconstruction, and (b) the mean ratio of measured minus true energy to the true energy for electrons from $Z\to ee$ events minus the same quantity for electrons from $W\to e\nu$ events as a function of true electron energy.

Figure 15

The mean electron energy versus $\eta$ for electrons from $W$ boson (black solid line) and $Z$ boson (red dashes) events. The thin lines indicate the 1 standard deviation bands of the energy distributions versus $\eta$.

Figure 16

Each line represents the ratio of the mean EM layer energy fractions simulated with zero-bias overlay to the same sample simulated without overlay. It represents the contribution from extra $p\overline{p}$ interactions and noise to the mean EM layer energy fractions, which is determined separately for the Run IIa sample (dotted lines) and the Run IIb sample (continuous lines). The ratio between the continuous to the dashed line is used as a correction factor to the EM layer energy fractions measured in Run IIa when comparing them to the Run IIb fractions (Fig. 17).

Figure 17

Ratio of the means of the EM layer energy fraction distributions in $Z\to ee$ events between the Run IIa analysis and the present Run IIb analysis, separately for each of the four EM layers and each of the 15 standard $\eta$ categories.

Figure 18

Trigger efficiency as a function of $p_T^e$ for the three triggers used. From the total $4.3{\mathrm{fb}}^{-1}$ of integrated luminosity used in this analysis, $1.6{\mathrm{fb}}^{-1}$ was collected with the $p_T=25\mathrm{GeV}$ (v15) threshold trigger, $2.0{\mathrm{fb}}^{-1}$ with the $p_T=25\mathrm{GeV}$ (v16) threshold trigger, and $0.7{\mathrm{fb}}^{-1}$ with the $p_T=27\mathrm{GeV}$ threshold trigger. When more than one unprescaled trigger is available in a data-taking period, we assign the fast MC event to the trigger with the lowest $p_T$ threshold.

Figure 19

Electron identification efficiency for electrons accompanied by FSR determined from full MC as a function of the fraction of the energy carried by the photon. Each pane corresponds to a different $\mathrm{\Delta }R$ region, and the distributions are integrated over $\eta$ and $E^e$.

Figure 20

Track-matching efficiency as a function of $z_V$ and $\eta$ in data. The efficiency is proportional to the area of the boxes.

Figure 21

The $p_T^e$-dependent perturbation over the track-matching efficiency in 11 bins of $\eta$ in full MC. The slopes change because electrons with higher energy are more easily matched to the calorimeter cluster. This efficiency perturbation is normalized at $p_T^e=45\mathrm{GeV}$. The total track-matching efficiency is the product of the efficiencies shown in Figs. 20 and 21.

Figure 22

Average difference between ${\phi }^{\mathrm{EM}}$ and ${\phi }^{\text{trk}}$ in module units as a function of ${\phi }_{\mathrm{mod}}^{\text{trk}}$ extrapolated to EM3.

Figure 23

Dependence of the electron reconstruction efficiency on the extrapolated track ${\phi }_{\mathrm{mod}}$.

Figure 24

The ratio data/fast MC of electron yield in $W\to e\nu$ events as a function of electron $\phi$ after all other efficiencies have been applied to the fast MC. The ratio is used as a final efficiency correction. The maximum efficiency value is normalized to one.

Figure 25

The reconstruction and identification efficiency as a function of true $p_T^e$ in full MC and fast MC for electrons in (a) $Z\to ee$ and (b) $W\to e\nu$ events. In $Z\to ee$ events, when the probed electron has high $p_T^e$, the other electron in the event is soft. When the soft electron is not properly identified, the $Z\to ee$ event is not identified either. Thus, we observe a drop in the identification efficiency of $Z\to ee$ events with high $p_T^e$ electrons, but not in $W\to e\nu$ events.

Figure 26

The reconstruction and identification efficiency as a function of ${\eta }_{\text{det}}$ in full MC and fast MC for electrons in (a) $Z\to ee$ and (b) $W\to e\nu$ events.

Figure 27

Electron efficiency correction (data/full MC) as a function of (a) SET and (b) instantaneous luminosity. The contribution of each electron selection requirement to the efficiency correction is shown by the ratios derived from samples in which only the HMatrix (blue line), loose track match (red line), and tight track match (magenta line) criteria are used (see Sec. 4f for the definition of each criterion).

Figure 28

Electron energy correction determined from full MC as a function of fraction of the energy carried by the photon, when the electron is fully identified in the presence of the photon. For values of the energy fraction $X$ where the FSR efficiency is zero (Fig. 19), we do not show the corresponding energy correction.

Figure 29

(a) Result of the $Z$ boson mass fit per ${\eta }_{\text{det}}$ category prior to applying $\eta$-dependent corrections (Table 2). (b) Result of the translation into one relative gain constant per ${\eta }_{\text{det}}$ bin.

Figure 30

Sampling contribution $S_{\mathrm{EM}}/\sqrt{E_0}$ to the electron fractional energy resolution as a function of the electron energy $E_0$ for four different electron incident angles $\eta =0.0$, 0.5, 0.75, and 1.0. The strong energy and angular dependencies are caused by the energy lost in the uninstrumented material before the calorimeter.

Figure 31

Central values and 1 standard deviation contours of the fits for electron energy scale and offset to the collider data. Instantaneous luminosity is given in units of $36\times {10}^{30}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$.

Figure 32

Mean $\mathrm{\Delta }E$ as a function of (a) SET, (b) instantaneous luminosity, (c) ${\eta }_{\text{det}}$, and (d) $u_{\parallel }$ comparing full and fast MC.

Figure 33

(a) Mean $\mathrm{\Delta }{u}_{\parallel }$ as a function of $L$ separately for subsamples with different SET. Instantaneous luminosity $L$ is given in units of $36\times {10}^{30}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. (b) Mean $\mathrm{\Delta }{u}_{\parallel }$ as a function of $u_{\parallel }$ separately for various bins of SET.

Figure 34

(a) The fraction of electron showers that leak outside the reconstruction cone in the CC, and (b) the fraction of the electron transverse momentum that is added to the recoil system for clusters without in-cone FSR photons, both as a function of electron $\eta$.

Figure 35

The distribution of the recoil relative transverse momentum and $\mathrm{\Delta }\phi$ resolutions for full MC (boxes) and fit (contours) for (a) $4.5<p_T^V<5.0\mathrm{GeV}$ and for (b) $18<p_T^V<20\mathrm{GeV}$.

Figure 36

Data and fast MC comparison of the (a) mean and (c) width of the ${\eta }_{\mathrm{imb}}$ for the ten different bins in $p_T^Z$. The $\chi$ value per $p_T^Z$ bin for the (b) mean and (d) width of the ${\eta }_{\mathrm{imb}}$.

Figure 37

Tight track match efficiency as a function of the electron $p_T^e$ measured relative to the loose track match requirement.

Figure 38

Probability of a jet object that passes the loose track match requirement to pass the tight track match requirement.

Figure 39

The (a) $m_T$, (b) $p_T^e$, and (c) ${\cancel{E}}_T$ distributions for the three backgrounds $Z\to ee$ (red lines), multijet (black lines), and $W\to \tau \nu$ (blue lines) with absolute normalization.

Figure 40

(a) The dielectron invariant mass distribution in $Z\to ee$ data compare to the fast MC and (b) the $\chi$ values, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$ for each bin in the distribution. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 41

Distributions of (a) $m_T$, (b) $p_T^e$, and (c) ${\cancel{E}}_T$ for data and fast MC with backgrounds. The $\chi$ values are shown below each distribution, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC and ${\sigma }_i$ is the statistical uncertainty in bin $i$. The fit ranges are indicated by the double-ended horizontal arrows.

Figure 42

Variations in $M_W$ determined from fits to the $m_T$ spectrum as the fit range is changed. (a) Impact of varying the lower edge of the $m_T$ fit range, and (b) the impact of varying the upper edge. For each of the variations the differences between the result from the varied range and the result from the nominal range are shown. The uncertainties represent the statistical uncertainties of the varied range fits.

Figure 43

The measured $M_W/M_Z$, separately for the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables and in four bins of instantaneous luminosity, in units of $36\times 10^{30}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. The error bars for each observable represent the statistical uncertainty due to the limited size of the $W$ boson sample. The shaded (yellow) bands indicate the contribution from the $Z$ boson statistics, which is fully correlated for the three observables. The three vertical lines with hashed bands indicate the results from the three observables for the full data sample. When systematic uncertainties are considered, the measured $M_W/M_Z$ values are consistent.

Figure 44

The measured ratio $M_W/M_Z$, separately for the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables and for four data-taking periods. The uncertainties for each observable represent the combined statistical uncertainty due to limited $W$ statistics and $Z$ statistics. The three vertical lines with hashed uncertainties indicate the results from the three observables for the full data sample.

Figure 45

Measured $M_W$ from the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables, separately for five different regions in electron $|{\eta }_{\text{det}}|$. The error bars for each observable represent the statistical uncertainty of the $W$ boson sample. The three vertical lines with hashed bands indicate the results from the three observables for the full data sample. When systematic uncertainties are considered, the measured $M_W$ values are consistent.

Figure 46

The measured $M_W$ from the $m_T$, $p_T^e$ and ${\cancel{E}}_T$ observables, separately for positive and negative $u_{\parallel }$. The three vertical lines with hashed bands indicate the results from the three observables for the full data sample.

Figure 47

The measured ratio $M_W/M_Z$, separately for the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables and for four ${\phi }_{\mathrm{mod}}$ selection variations. The numbers in parentheses indicate which fraction of the CC EM module around its center is included in the electron fiducial region. The three vertical lines with hashed bands indicate the results from the three observables for the full data sample.

Figure 48

The measured ratio $M_W/M_Z$, separately for the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables and for two $u_T$ variations. The three vertical lines with hashed bands indicate the results from the three observables with the nominal $u_T$ requirement.

Figure 49

The measured ratio $M_W/M_Z$, separately for the $m_T$, $p_T^e$, and ${\cancel{E}}_T$ observables and for eight bins in recoil $\phi$.

Figure 50

The D0 Run II measurement of $M_W$ shown with the world-average mass of the top quark $m_t$ [82] at 68% C.L. by area. The new world average for $M_W$ [81] is also shown. The thin blue band is the prediction of $M_W$ in the Standard Model given by Eq. (1), assuming $M_H=125.7\pm 0.4\mathrm{GeV}\mathrm{.}$

Figure 51

(a) Comparison between data and fast MC for the $p_T^e$ distribution of the electrons from $Z\to ee$, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 52

(a) Comparison between data and fast MC for the $p_T^Z$ distribution of $Z\to ee$ events, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 53

(a) Comparison between data and fast MC for the $u_T$ distribution of $Z\to ee$ events, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 54

(a) Comparison between data and fast MC for the $u_{\parallel }$ distribution of electrons in $Z\to ee$ events, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 55

(a) Comparison between data and fast MC for the $u_{\perp }$ distribution of electrons in $Z\to ee$ events, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 56

(a) Comparison between data and fast MC for the ${\cancel{E}}_T$ distribution of in $Z\to ee$ events, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 57

(a) Comparison between data and fast MC for the $u_T$ distribution in $W\to e\nu$ data, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 58

(a) Comparison between data and fast MC for the $u_{\parallel }$ distribution in $W\to e\nu$ data, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 59

(a) Comparison between data and fast MC for the $u_{\perp }$ distribution in $W\to e\nu$ data, and (b) $\chi$ value per bin, where ${\chi }_i=\mathrm{\Delta }{N}_i/{\sigma }_i$. $\mathrm{\Delta }{N}_i$ is the difference between the number of events for data and fast MC, and ${\sigma }_i$ is the statistical uncertainty in bin $i$.

Figure 60

Comparison between data and fast MC for the (a) $\eta$ distribution of electrons from $W\to e\nu$, and (b) ${\eta }_{\det }$ distribution of electrons from $W\to e\nu$.

Figure 61

Comparison between data and fast MC for the (a) instantaneous luminosity distribution of $W\to e\nu$ events, and (b) SET distribution of $W\to e\nu$ events.