Drell-Yan lepton pairs are produced in the process pp → γ^*/Z + X, with the subsequent decay of the γ^*/Z into lepton pairs. We present a measurement of the Δσ/ΔP_T differential cross section using 2.1 fb^-1 of CDF Run II data in the e⁺e^- channel. The cross section presented is measured in P_T bins of variable width and is the averge cross section in the bin: Δσ/ΔP_T. The measurements are for dielectrons in a mass range of 66—116 GeV/c², for all boson rapidities and decay electron phase space, and √s = 1.96 TeV.

Introduction
The measurement is an extension of our 2.1 fb^-1 Drell-Yan dielectron process measurements of dσ/dy [Phys. Lett. B692 232 (2010)] and electron angular distribution coefficients [Phys. Rev. Lett. 106, 241801 (2011)]. Analysis details are available on the CDF EWK web page (under W/Z Cross Sections and Asymmetries). The Δσ/ΔP_T measurement uses the exact same dataset and nearly identical analysis methods used in the decay electron angular distribution measurement because the data is understood well and the simulation of the data is tuned well.

The Data

The electron candidates used in this analysis can be in either the CDF central (C) or the forward end plug (P) calorimeters. The central and plug calorimeters cover the range, |η_det|<1.1 and 1.1<|η_det|<3.5 respectively. Each electron is required to have an associated track and pass standard CDF electron selection and fiducial cuts. Three electron-pair topology categories are used with the following kinematic selections.

1. Central-central (CC)
   • E_T > 25/15 GeV (highest/lowest E_T of pair)
   • |η_det|<1.1
2. Central-plug (CP)
   • E_T > 20 GeV
   • C: |η_det|<1.1
   • P: 1.2<|η_det|<2.8
3. Plug-plug (PP)
   • E_T > 25 GeV
   • 1.2<|η_det|<2.8

Data Driven Measurement

The fully corrected measurement of the differential cross section requires corrections for the detector acceptance, A, and electron selection and reconstruction efficiencies, ε. The combined A×ε is calculated with a simulation of the physics and detector that also unfolds detector resolution effects. Because of detector resolution smearing, the correct modeling of the physics and detector are important.

The simulation modeling is data driven. The modeling starts with the Pythia 6.2 simulation of pp → γ^*/Z (→ ee) + X (at tree level) followed by parton showering. PHOTOS 2.0 adds final state QED radiation to charged particle vertices. This is then followed by the full CDF detector simulation. All data efficiencies, both global and η_det dependent, as well as time-dependent, are measured and incorporated into the simulation. Both the physics and detector models are tuned (iteratively) so that the simulated, reconstructed event distributions match the data.

The tuning of Pythia's γ^*/Z dσ/dy and the decay angular distributions are from our previous measurements using the 2.1 fb^-1 data. The simulated, reconstructed event P_T distribution that pass all analysis cuts disagrees with the data, so we tune the underlying Pythia P_T so that it does.

This plot shows the correction to the underlying Pythia PT distribution that flattens out the data-to-simulation PT distribution ratio. The data statistics for PT>100 GeV/c is marginal, so all events are combined into one bin. The uncertainties are from the data-to-simulation PT distribution ratio.

The correction is the measurement of the P_T distribution used for the physics modeling.

The simulation's calorimetric energy scales and resolutions are important for the measurement, and are tuned to the data. The tuning is from our dσ/dy and electron angular decay distribution analyses. The energy scales and resolutions in the simulation are tuned against the data's dielectron mass (M_ee) and electron E_T distributions. As the backgrounds are very small, they are not subtracted out.

This plot shows the tuned data-vs-simulation Mee histograms for the CC (upper left), CP (upper right), and PP (lower left) dielectron topologies. The black crosses are the data and the red are the simulated data. The χ2s over the 100 bins are 107, 123, and 114 for the CC, CP, and PP topologies, respectively.

The corresponding plots for the ET distributions of central and plug electrons of each dielectron topology are on the left. The upper left is CC-central, uppper right is CP-central, lower left is CP-plug, and lower right is PP-plug. The black crosses are the data and the red are the simulated data. The χ2s over the 100 bins (90 for PP) are 117, 100, 87, and 135 for the CC-central, CP-central, CP-plug, and PP-plug, respectively.

The statistical precision of the data tightly constrains calorimetry modeling uncertainties. The systematic shifts in A×ε from global variations in the simulation energy scales and resolutions allowed by the data are small: Under 0.3%. The simulation statistical uncertainties (√N) are larger in all P_T bins.

The systematic uncertainties in each P_T bin from the propagation of efficiency measurement uncertainties are flat and about 1% over 0<P_T<20 GeV/c, and slowly decreasing after that. They are also 100% correlated across all bins.

The differential cross section that we measure is Δσ/ΔP_T, or the average cross section in a P_T bin of width ΔP_T. The cross section in a P_T bin is calculated using: Δσ = N / ( L × (A×ε) ), where N is the background subtracted event count, and L is the integrated luminosity. For this measurement, L = 2057 pb^-1, and it has a 5.8% systematic uncertainty. The A×ε calcuated with the tuned simulation ranges from 0.2 at P_T=0 GeV/c to 0.3 at high P_T. As P_T increases, the electrons become more central, their E_T get larger and their acceptance increases. The A×ε includes the measured electron identification, tracking, and other efficiences, but uncertainties from it are kept and presented separately from the (simulation) statistical uncertainties because they are correlated.

The Δσ/ΔPT versus PT is shown on the left. The black data uncertainties plotted for each point include the data statistical uncertainty, the A×ε statistical uncertainty, and and the 1% efficiency measurement uncertainty, all combined in quadrature. No luminosity uncertainties are included. The horizontal "error bars" represent the extent of the bin. The red ResBos calculation is straight out of the box: 041511 (10.15.09cp) using the resummation and yk grids. The ResBos total cross section is 254 pb.

An ASCII table of the measurement is here; the uncertainties are split into the data and simulation statistical (stat) and 100% correlated efficiency systematic (syst) uncertainties. The 5.8% luminosity uncertainty is not included.

The statistical uncertainty of √N from the bin cross section formula is inadequate with the large smearing at low P_T and it is estimated with the simulation. The 1/(A×ε) bin-by-bin unfolding is a two step procedure: A scaling correction then an acceptance correction. The scaling correction is on reconstructed events, and it rescales the events reconstructed in a P_T bin into reconstructed events produced in the bin. The scaling uncertainities from physics and detector modeling mismatches between the simulation and the data has been mitigated by the tuning of both so that the reconstructed simulation data matches the real data. Denote ρ as the scaling factor that takes the number of reconstructed events in a bin, N_r, and rescales to the number number of reconstructed events produced in the bin, N_g. The production and smearing statistical fluctuations are different. A method to accomodate both needs to be specified for an estimate of the overall "statistical" uncertainty.

The simulation is used to estimate this bin uncertainty. Within the context of the simulation, the event smearing across bins can be described by the number of events produced in a bin that stay in a bin, n_g, the events that migrate out, n_o, and events produced in neighboring bins that migrate in, n_i:

    N_g = n_g + n_o
    N_r = n_g + n_i.

The scaling correction prediction for N_g is:

    N_g^(p) = ρ N_r = ρ(n_g + n_i) = n_g + ((ρ-1)n_g + ρn_i).

where the prediction is denoted with a (p) superscript. An interpretation of the last term, ((ρ-1)n_g + ρn_i), is that it is the scaling correction prediction, and it only predicts n_o. Fluctuations of n_g and n_i away from their expectation values introduce a prediction uncertainty on n_o. The simulation event counts of n_g and n_i, when normalized to the data, are used to estimate this prediction uncertainty on n_o. The method to estimate the overall bin uncertainty combines the expected uncertainty on the events produced in the bin (N_g) with the prediction uncertainty of n_o. Denote the statistical variance of a quantity as δ² followed by the variable. For n_o^(p), δ² represents the propagation of uncertainties from its element variances:

    δ²n_o^(p) = (ρ-1)²δ²n_g + ρ²δ²n_i.

The method treats the uncertainty of N_g and the propagation uncertainty on n_o^(p) as unrelated and the two are added in quadature:

    δ²N_g' = δ²N_g + ((ρ-1)²δ²n_g + ρ²δ²n_i),

where δ²N_g' is the estimate of the overall bin uncertainty of the method. With Poisson statistics (δ²n = n)

    δ²N_g' = N_g + ((ρ-1)²n_g + ρ²n_i).

This overall bin uncertainty is specific to the estimation method just described. Work on evaluating and using different methods is ongoing.

The measured cross section of the bin is combined with δ²N_g' in its fractional form, √[δ²N_g']/N_g , for the estimate of the cross section "statistical" uncertainty of a bin used in the cross section plots and tables. The ratio of the data to the ResBos prediction is shown below:

Due to the large inter-bin smearing at low P_T, interpreting the significance of the data-to-prediction comparison there is difficult. At large P_T (above 40 GeV/c), the smearing is predominantly among nearest neighbor P_T bins so the data-to-prediction comparison shown has much less correlations from smearing. The 40 < P_T < 90 GeV/c region is where the the ResBos resummed, asymptotic, and perturbative cross sections are matched.