Indirect measurement of ${\sin }^2{\theta }_W$ (or $M_W$) using ${\mu }^{+}{\mu }^{-}$ pairs from ${\gamma }^{*}/Z$ bosons produced in $p\overline{p}$ collisions at a center-of-momentum energy of 1.96 TeV

Drell-Yan lepton pairs are produced in the process $p\overline{p}\to {\mu }^{+}{\mu }^{-}+X$ through an intermediate ${\gamma }^{*}/Z$boson. The forward-backward asymmetry in the polar-angle distribution of the ${\mu }^{-}$ as a function of the invariant mass of the ${\mu }^{+}{\mu }^{-}$ pair is used to obtain the effective leptonic determination ${\sin }^2{\theta }_{\text{eff}}^{\text{lept}}$ of the electroweak-mixing parameter ${\sin }^2{\theta }_W$, from which the value of ${\sin }^2{\theta }_W$ is derived assuming the standard model. The measurement sample, recorded by the Collider Detector at Fermilab (CDF), corresponds to $9.2{\mathrm{fb}}^{-1}$ of integrated luminosity from $p\overline{p}$ collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDF Run II data set. The value of ${\sin }^2{\theta }_{\text{eff}}^{\text{lept}}$ is found to be $0.2315\pm 0.0010$, where statistical and systematic uncertainties are combined in quadrature. When interpreted within the context of the standard model using the on-shell renormalization scheme, where ${\sin }^2{\theta }_W=1-M_W^2/M_Z^2$, the measurement yields ${\sin }^2{\theta }_W=0.2233\pm 0.0009$, or equivalently a $W$-boson mass of $80.365\pm 0.047\mathrm{GeV}/c^2$. The value of the $W$-boson mass is in agreement with previous determinations in electron-positron collisions and at the Tevatron collider.

Figure 1

CS coordinate axes $(x,z)$ in the lepton-pair rest frame, along with the laboratory $z$ axis ($z_{\text{lab}}$). The three axes are in the plane formed by the proton (${\overrightarrow{P}}_{\mathrm{A}}$) and antiproton (${\overrightarrow{P}}_{\mathrm{B}}$) momentum vectors within the rest frame. Relative to the laboratory $z$ axis, the transverse component of $-({\overrightarrow{P}}_{\mathrm{A}}+{\overrightarrow{P}}_{\mathrm{B}})$ is the same as the transverse-momentum vector of the lepton pair in the laboratory (${\overrightarrow{P}}_{\mathrm{T}}$).

Figure 2

Muon-pair invariant mass distributions. The upper set of crosses is the background-subtracted data, the middle set of crosses is the EWK background, and the lower set of crosses is the QCD background (same-charge muon pairs). The EWK background is derived from simulation.

Figure 3

Typical dependence of $A_{\text{fb}}$ as a function of the lepton-pair invariant mass. The curve is an analytic calculation. The vertical line is at $M=M_Z$.

Figure 4

Distribution of $C^{+}$ for same-side (SS) pairs in the central $\eta$ region of $(-0.3,0)$. The crosses are the data, and the solid histogram is the generator-level distribution normalized to the data.

Figure 5

Distribution of $C^{+}$ for OS pairs in the central $\eta$ region of $(-0.3,0)$. The crosses are the data, and the solid histogram is the generator-level distribution normalized to the data.

Figure 6

Calibrated muon-pair invariant mass distributions. The crosses are the background-subtracted data and the solid histogram is from the simulation. The comparison of the simulation with the data yields a ${\chi }^2$ of 219 for 200 bins.

Figure 7

Calibrated muon-pair invariant mass distributions over an extended mass range. The crosses are the background-subtracted data and the solid histogram is from the simulation. The comparison of the simulation with the data yields a ${\chi }^2$ of 518 for 400 bins.

Figure 8

Calibrated $P_{\mathrm{T}}$ distribution for the muon with the larger $P_{\mathrm{T}}$. The crosses are the background-subtracted data and the solid histogram is from the simulation.

Figure 9

Calibrated $P_{\mathrm{T}}$ distribution for the muon with the smaller $P_{\mathrm{T}}$. The crosses are the background-subtracted data and the solid histogram is from the simulation.

Figure 10

Adjusted $\cos \vartheta$ distribution in the Collins-Soper frame. The crosses are the background-subtracted data and the solid histogram is from the simulation.

Figure 11

Observed $\phi$ distribution in the Collins-Soper frame. The crosses are the background-subtracted data and the solid histogram is from the simulation.

Figure 12

Raw $A_{\text{fb}}$ measurement in bins of the muon-pair invariant mass. Only statistical uncertainties are shown. The pythia $|y|<1$ asymmetry curve does not include the effect of QED FSR.

Figure 13

Muon-pair $y$ distribution. The crosses are the background-subtracted data and the histogram is the simulation. The measurement is resticted to the region $|y|<1$. The upper curve is the (arbitrarily normalized) shape of the underlying rapidity distribution from pythia.

Figure 14

Event-weighting bias for each of the muon-pair invariant mass bins. The bias is estimated with the simulation, and the uncertainties represent the full precision of the simulation.

Figure 15

Fully corrected $A_{\text{fb}}$. The measurement uncertainties are uncorrelated bin-by-bin unfolding estimates. The vertical line is $M=M_Z$. The pythia calculation uses ${\sin }^2{\theta }_{\text{eff}}^{\text{lept}}=0.232$. The EBA-based resbos calculation uses ${\sin }^2{\theta }_W=0.2233$ $({\sin }^2{\theta }_{\text{eff}}^{\text{lept}}=0.2315)$.

Figure 16

Comparison of the $A_{\text{fb}}$ measurement with the resbos NLO templates. The triangles are the scan points, and the solid curve is the fit of those points to a generic ${\chi }^2$ functional form.

Figure 17

Comparison of experimental measurements of ${\sin }^2{\theta }_{\text{eff}}^{\text{lept}}$. “$Z$ pole” represents the LEP-1 and SLD standard model analysis of $Z$-pole measurements and “light quarks” represents the LEP-1 and SLD results from the light-quark asymmetries; “D0 $ee$ $5{\mathrm{fb}}^{-1}$” represents the D0 $A_{\text{fb}}(M)$ analysis; “CMS $\mu \mu$ $1{\mathrm{fb}}^{-1}$” represents the CMS analysis; “CDF $ee$ $2{\mathrm{fb}}^{-1}$” represents the $A_4$ analysis; and “CDF $\mu \mu$ $9{\mathrm{fb}}^{-1}$” represents this analysis. The horizontal bars represent total uncertainties.

Figure 18

Comparison of experimental determinations of the $W$-boson mass. “TeV and LEP-2” represents direct measurements of the $W$-boson mass; “LEP-1 and SLD (${\mathrm{m}}_{\mathrm{t}}$)” represents the standard model analysis of $Z$-pole measurements; “NuTeV” represents the indirect measurement derived from neutrino scattering at Fermilab; “CDF $ee$ $2{\mathrm{fb}}^{-1}$” represents the $A_4$ analysis; and “CDF $\mu \mu$ $9{\mathrm{fb}}^{-1}$” represents this analysis. The horizontal bars represent total uncertainties.