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\begin{document}
\vspace*{4cm}
\title{Analysis of $Z \to l^+ l^-$ Polarization at CMS}
\author{ Nhan Viet Tran on behalf of the CMS collaboration }
\address{Department of Physics and Astronomy, Johns Hopkins University, \\
3400 N. Charles St., Baltimore, MD 21218, USA
}
\maketitle\abstracts{
With approximately 35 $pb^{-1}$ of LHC proton-proton collision data collected by CMS we study the Drell-Yan process $q\bar{q} \to Z \to l^+l^-$.
Differential cross sections with respect to the invariant mass, rapidity, and transverse momentum are presented.
The forward-backward asymmetry is measured as a function of the di-lepton invariant mass, and an analysis of the fully differential distribution leads to the measurement of the Weinberg weak-mixing angle.
}
\section{Introduction}
In the process $q\bar{q} \to Z/\gamma^* \to l^+ l^-$, both the vector and axial vector couplings of electroweak bosons to fermions are present.
In the Standard Model (SM), these couplings depend on the electroweak mixing angle, $\theta_W$.
This results in a forward-backward asymmetry, $A_{FB}$, in the number of Drell-Yan lepton pairs.
In addition to being sensitive to the electroweak mixing angle, any deviation of the $A_{FB}$ from the SM can be a sign of new physics from new gauge bosons, supersymmetry, or extra dimensions.
The measurement of the $A_{FB}$ can also improve QCD measurements and constrain Parton Distribution Functions (PDFs).
The $A_{FB}$ and $\sin^2\theta_{W}$ measurements by CDF~\cite{CDF} and D0~\cite{D0} are given for reference.
In addition to the traditional method of measuring $\sin^2\theta_W$ via the asymmetry, we also considering the measurement of the weak mixing angle via a multivariate likelihood analysis.
Full information about the Drell-Yan process is parameterized as a function of the di-lepton rapidity $Y$, the di-lepton invariant mass $m_{ll}$, and the di-lepton decay angle $\theta^\ast_{CS}$ defined in the
Collins-Soper frame~\cite{CS} to reduce the effect of di-lepton transverse momentum.
The angular distribution for the Drell-Yan process is given in a simplified form as:
\be
\frac{d\sigma}{d\cos\theta^\ast_{CS}} = A(1+\cos^2\theta^\ast_{CS}) + B \cos\theta^\ast_{CS}
\ee
Because the LHC is a $pp$ collider, the quark direction is unknown and the definition of the Collins-Soper frame is defined using the boost direction of the di-lepton pair.
This introduces a dilution of the asymmetric term since we can only determine the quark direction on a statistical basis.
The forward-backward asymmetry is defined as
\be
A_{FB} = \frac{N_F - N_B}{N_F + N_B} = \frac{3B}{8A}
\ee
where $N_F$ ($N_B$) is the number of forward (backward) events.
The $A_{FB}$ is measured as a function of the di-lepton invariant mass where typically the slope of this quantity is sensitive to the electroweak mixing angle.
Without applying any corrections, the uncorrected $A_{FB}$ measurement is distorted from the original parton-level asymmetry because of bin-to-bin migration
due to f{}inite resolution of the detector and QED f{}inal state radiation (FSR).
Moreover, the $A_{FB}$ is further distorted by the detector acceptance and by the unknown quark direction at the LHC.
%The multivariate analysis of the Drell-Yan process may allow us to study the elementary couplings of
%fermions to electroweak neutral f{}ields, such as $Z/\gamma^\ast$ in the Standard Model, as well
%as structure functions of the proton.
%However, as a f{}irst illustration of this technique, we take
As an illustration of the multivariate analysis of the Drell-Yan process, we take
the Standard Model description of electroweak interactions and PDFs in the proton as well-established
and allow only the effective electroweak mixing angle $\theta_ {\rm{eff}}$ to be unconstrained,
which is the same for both leptons and light quarks with the current precision of this analysis.
We illustrate this method with analysis in the di-muon channel process.
The choice of $\mu^+\mu^-$ , as opposed to $e^+e^-$ , is motivated by the simpler description of
detector and background effects in this f{}irst study; however, we do not expect any
limitation in the method for future application to other f{}inal states.
The formalism is built as an analytic description of the process at leading order where (next-to-)next-to-leading order effects are considered as corrections to the model.
The description of the $pp \to Z/\gamma^\ast \to \mu^+\mu^-$ process is given by a probability distribution function
of the triple differential cross-section in the observables , $P_{pp}(m_{ll},Y,\cos\theta^*_{CS}; \sin^2\theta_ {\rm{eff}})$.
This includes effects from PDFs, the partonic luminosity and the dilution, which requires an analytical parameterization of the PDFs.
Then detector effects such as resolution and FSR, $R(m_{ll})$, and acceptance, $G(m_{ll},Y,\cos\theta^*_{CS})$, are included such that we have the f{}inal description of the signal probability distribution function,
\be
P_{sig}(m_{ll},Y,\cos\theta^*_{CS}; \sin^2\theta_ {\rm{eff}}) = [ P_{pp}(m_{ll},Y,\cos\theta^*_{CS}; \sin^2\theta_ {\rm{eff}}) \times R(m_{ll}) ] \times G(m_{ll},Y,\cos\theta^*_{CS})
\ee
Information about the electroweak mixing angle is contained in the correlated three-dimensional shapes of the observables.
The motivation for the introduction of the method is the improvement in statistical sensitivity;
the increase in sensitivity is approximately 40\% over traditional methods.
Further details of the measurements of both $A_{FB}$ and $\sin^2\theta_ {\rm{eff}}$ can be found in the public CMS results~\cite{PAS}.
\section{Measurement of $A_{FB}$}
\begin{figure}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\center{
\psfig{figure= AFB_uncor_e.ps,height=2.5in}
\psfig{figure= AFB_uncor_mu.ps,height=2.5in}
}
\caption{Uncorrected $A_{FB}$ for the di-electron channel (left) and the di-muon channel (right).
\label{fig:afb}}
\end{figure}
In these proceedings, we present the uncorrected $A_{FB}$ vs. di-lepton mass and compare it to events generated with the POWHEG
Next-to-Leading Order (NLO) generator and with detailed GEANT-based CMS simulation and reconstruction.
Selection of both reconstructed electrons and muons require standard isolation and quality requirements
which are detailed in the inclusive $W$ and $Z$ boson measurements at CMS~\cite{inclusive}.
The muons are required to have a $p_T > 20$~GeV and a pseudorapidity $| \eta | < 2.1$.
The electrons are required to have an $E_T > 20$~GeV after energy scale corrections and $| \eta | < 2.5$ excluding the region from $ 1.442 < |\eta| < 1.560$.
The uncorrected forward-backward asymmetry is given in Fig.~\ref{fig:afb} for the 2010 CMS dataset with an integrated luminosity of $36~{\rm{pb}}^{-1}$.
The expectation from simulation is also given in Fig.~\ref{fig:afb}; and for the given data sample, we observed agreement between the simulation and data.
\section{Measurement of $\sin^2\theta_ {\rm{eff}}$}
For the measurement of $\sin^2\theta_ {\rm{eff}}$, we perform a single parameter f{}it to extract the value of the electroweak mixing angle.
The same isolation and quality requirements on the muons from the $A_{FB}$ measurement are used.
Looser phase space cuts are made in the Collins-Soper frame on the muons to increase sensitivity and accommodate the analytical acceptance, $p_T(CS) < 18$ and $|\eta|(CS) < 2.3$.
In addition, there is a cut on the di-lepton transverse momentum to decrease contributions from next-to-leading order effects, $p_T(l^+l^-) < 25$ GeV.
The f{}it value from simulation $\sin^2\theta_ {\rm{eff}} = 0.2306 \pm 0.0004$ is in good agreement with the generated value $\sin^2\theta_ {\rm{eff}} ({\rm{gen}}) = 0.2311$.
The f{}it result is shown in Fig.~\ref{fig:mc}.
In addition, we run 400 toy experiments and f{}ind the expected statistical error should be 0.0078.
The pull distributions are found to be in good agreement with a unit Gaussian centered at zero.
\begin{figure}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\center{
\psfig{figure= ZDYPLot_CmsMCG-v2_60_120_PAS.ps,height=3.5in}
}
\caption{Projections of the f{}it model on simulation for the observables (a) rapidity, (b) $\cos\theta^\ast_{CS}$, and (c)/(d) di-lepton mass.
\label{fig:mc}}
\end{figure}
The leading systematics come from the alignment and resolution model, the FSR modeling, and the PDF uncertainties.
We also consider contributions from the LO model and initial state radiation, the f{}it model, and QCD background.
The total systematic uncertainty is 0.0036 though these estimates are conservative and sometimes statistically limited.
The total systematic uncertainty is less than the statistical errors.
With the CMS 2010 data sample of $40~{\rm{pb}}^{-1}$, we make a measurement of the weak mixing angle.
The f{}it value was kept blinded until evaluating all systematics and the f{}inal f{}it result is
\be
\sin^2\theta_ {\rm{eff}} = 0.2287 \pm 0.0077 ({\rm{stat.}}) \pm 0.0036 ({\rm{sys.}})
\ee
A f{}inal cross-check of the goodness-of-fit is found to be in agreement with simulation.
\begin{figure}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\center{
\psfig{figure= ZDYPLot_CmsDataG_v3_60_120_PAS.ps,height=3.5in}
}
\caption{Projections of the f{}it model on CMS 2010 data for the observables (a) rapidity, (b) $\cos\theta^\ast_{CS}$, and (c)/(d) di-lepton mass.
\label{fig:dat}}
\end{figure}
\section{Summary}
We have presented the measurement of the forward-backward asymmetry for the Drell-Yan process in $pp$ collision at $\sqrt{s}=7$ TeV.
We have also presented the measurement of the effective electroweak mixing angle based on a multivariate
likelihood f{}it which results in a value of $\sin^2\theta_{\rm{eff}} = 0.2287 \pm 0.0077\pm 0.0036$.
We f{}ind both the $A_{FB}$ distributions and the $\sin^2\theta_{\rm{eff}}$ measurement to be consistent with the Standard Model
predictions within uncertainties.
%\pagebreak
\section*{References}
\begin{thebibliography}{99}
\bibitem{CDF}
CDF Collaboration, ÒMeasurement of the forward-backward charge asymmetry of electron-positron pairs in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeVÓ, Phys. Rev. D 71 (Mar, 2005) 052002. doi:10.1103/PhysRevD.71.052002.
\bibitem{D0}
The D0 Collaboration Collaboration, ÒMeasurement of the Forward-Backward Charge Asymmetry and Extraction of $\sin^2\theta^ {\rm{eff}}_W$ in $p\bar{p} \to Z\gamma^\ast \to e^+e^- + X$ Events produced
at $\sqrt{s}$ = 1.96 TeVÓ, Phys. Rev. Lett. 101 (Nov, 2008) 191801. doi:10.1103/PhysRevLett.101.191801.
\bibitem{CS} J. Collins and D. Soper, ÒAngular Distribution of Dileptons in High-Energy Hadron
CollisionsÓ, Phys. Rev. D 16 (1977) 2219. doi:10.1103/PhysRevD.16.2219.
\bibitem{PAS} CMS Collaboration, "Measurement of Forward-Backward Asymmetry of Lepton Pairs and the Weak-mixing angle at CMS", CMS PAS EWK-10-011 (2011).
\bibitem{inclusive} CMS Collaboration, "Measurement of the W and Z inclusive production cross sections at $\sqrt{s}=7$ TeV with the CMS experiment at the LHC", CMS PAS EWK-10-005 (2011).
\end{thebibliography}
\end{document}
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