Measurement of the tt Differential Cross Section, dσ/dM_tt, in 2.7 fb^-1 of Data

Alice Bridgeman and Tony Liss

    University of Illinois

Abstract

We present a measurement of the tt differential cross section with respect to the invariant mass of the tt pair, dσ/dM_tt, with 2.7 fb^-1 of CDF II data.  This analysis uses an in-situ calibration of the jet energy scale to significantly reduce the jet energy scale uncertainty.  This results in a large reduction of the total systematic uncertainty with respect to the 1.9 fb^-1 analysis described here.  We use a regularized unfolding technique to correct the reconstructed tt invariant mass back to the true invariant mass.  We compare the resulting unfolded distribution to the Standard Model expectation, as modeled by Pythia with CTEQ5L Parton Distribution Functions, and find no evidence of a discrepancy with the SM.  We set limits on κ/M_Pl in the Randall-Sundrum model by looking for graviton resonances in the tt invariant mass spectrum, where the mass of the first graviton mode is fixed at 600 GeV.  We find that for such resonances κ/M_Pl > 0.16 at the 95% confidence level.

Public Note

Definition

The differential cross section is defined as:

We bin the M_tt distribution in 9 bins of variable widths.  In each bin the cross section is given by the difference between the number of events in that bin and the expected number of background events divided by the product of the acceptance in the bin, the integrated luminosity and the bin width.  In this analysis we always assume a top quark mass of 175 GeV.  The top signal is modeled by Pythia with CTEQ5L parton distribution functions.

Event Selection

We select events in the lepton+jets decay channel of the tt pair by looking for events with exactly one high p_T isolated lepton, large missing transverse energy and at least four jets with large E_T. The composition of this sample was determined in a separate analysis described here.

Jet Energy Scale Calibration

We use the invariant mass of the dijets from the hadronically decaying W in lepton+jets events to determine the jet energy scale.  We create templates of Monte Carlo events with known deviations from the nominal jet energy scale, Δ_JES, in units of the nominal jet energy scale uncertainty, σ_JES.  In these units Δ_JES=0 corresponds to the nominal jet energy scale.  We use an unbinned maximum likelihood fit to compare the data to the templates. 

Template Plots

The invariant mass templates for the tt signal at different values of ΔJES (middle) and the templates compared to the fitted probability distribution functions (right).
The invariant mass distributions for the backgrounds do not vary appreciably with ΔJES (middle).  The probability distribution function (right) is constant with respect to changes in ΔJES.

We test the performance of the fit in Monte Carlo experiments with known values of Δ_JES.  The results are shown in the plots below.

Validation Plots

We apply a small correction to the fitted value of ΔJES based upon output vs. input plot (middle).  The fitted value versus the true value after correction is shown on the right.
The pull mean (middle) and width (right)at various input values of ΔJES.
The uncertainty returned by the fit is about 50% of the nominal uncertainty, σJES.

Unfolding Technique

We reconstruct the invariant mass of the tt system by adding the four-momentum of the leading 4 jets in the event, the lepton and the missing transverse energy (the z-component is set to zero).  This reconstructed invariant mass is not equal to the true (partonic) invariant mass.  In order to find the number of events in each true bin of M_tt, we unfold the background-subtracted data.  We use the SVD unfolding technique described by Hocker and Kartvelishvili here.

The unfolding matrix varies with Δ_JES .  We parameterize each entry of the 9X9 matrix as a quadratic function of Δ_JES. 

Unfolding Plots

The middle plot shows the difference between the reconstructed and true distribution in each of the 9 bins used for the cross section measurement.  The right plot shows the unfolding matrix at ΔJES=0.
The unfolded distribution as compared to the true distribution for a sample reconstructed Mtt distribution.

Validation

We check the unfolding technique by looking at the pull distributions for the unfolded number of events in each bin at various input Δ_JES values.  The results are in the table below.

Acceptance

The dependence of the acceptance on  Δ_JES is well-described as a linear function of Δ_JES.  We parameterize the cross section denominator in each bin (without the bin width) for 2.7 fb^-1 of luminosity as a linear function of Δ_JES is shown below.

Summary of Analysis Procedure

1. Determine Δ_JES by fitting the dijet distribution.

2. Using the unfolding matrix at the value of Δ_JES obtained in step 1, unfold the background-subtracted reconstructed M_tt distribution to obtain the number of events in each bin of the true M_tt distribution.

3. Divide the number of events in each bin of the unfolded distribution from step 2 by the denominator at the value of Δ_JES obtained in step 1 to get the differential cross section.

Results

The important kinematic distributions for this analysis are shown below.  The backgrounds are normalized according to the predictions of the top pair cross section analysis described here.  The tt signal content is normalized to the difference between the observed data and the predicted backgrounds.  All Monte Carlo distributions are shown at the nominal jet energy scale, Δ_JES = 0.

Kinematic Plots

The transverse momentum of the electron or muon.
The missing transverse energy.
The transverse energy of the leading 4 jets in the event.
The HT distribution.  HT is the scalar sum of the transverse energy of the lepton, missing transverse energy and the transverse energy of the jets.
The invariant mass distribution of the dijets, which is fit for the value of ΔJES.
The reconstructed invariant mass of the tt pair.

Systematic Uncertainties

Systematic uncertainties arise due to the Monte Carlo modeling of the signal and backgrounds.  The dominant systematic uncertainty is the parton distribution function used in the Monte Carlo simulation.  The systematic uncertainties in each bin are summarized in the table below.  The total does not include the 6% uncertainty in each bin due to the luminosity measurement.

Δ_JES Fit and Differential Cross Section

The fit to the dijet distribution gives ΔJES = 1.3+/-0.5.
The unfolded Mtt distribution.
The final differential cross section.  The SM uncertainty reflects all systematic uncertainties, except for the luminosity uncertainty in each bin.
The differential cross section in a table format, with all uncertainties.

Consistency with the Standard Model

We establish the consistency of this result with the Standard Model by comparing the value of a test statistic, the Anderson-Darling statistic, that we observe in the data to the distribution in Standard Model-only pseudo-experiments.

In SM-only pseudo-experiments, 28% of pseudo-experiments have a larger (less consistent with the SM) test statistic than that observed in the data.

Limits on New Physics

We set limits on the ratio κ/M_Pl for gravitons which decay to top quarks in the Randall-Sundrum model, where the mass of the first resonance is fixed at 600 GeV.  We model the gravitons using MadEvent plus Pythia.  We use a CLs technique, where the test statistic is the Anderson-Darling statistic, to set limits.

The unfolded Mtt distributions for the Randall-Sundrum gravitons decaying to top quarks, as compared to the SM.
The distribution of the Anderson-Darling statistic in the SM and the Randall-Sundrum model.  The arrow indicates the value measured in the data.
The expected and observed limits.  We exclude κ/MPl > 0.16 at the 95% confidence level.

1.9 fb^-1 Analysis

Last update 20 October 2008.