Measurement of Helicity Fractions and Spin Correlation in Top Quark Pairs Using Reconstructed Lepton+Jets Events
This analysis has been documented in public CDF note 10048 by the University of Michigan authors Dave Mietlicki, Alexei Varganov, and Dan Amidei.
Abstract
In the standard model, top quarks decay weakly before any hadronization processes take effect, enabling top spin information to be transmitted to the top quark decay products. Standard model top pair production produces a characteristic spin correlation which can be modified by new production mechanisms such as Z' bosons or KaluzaKlein gluons. We report on the observation and measurement of the top pair helicity fractions and spin correlation in 4.3 inverse femtobarns of reconstructed lepton+jet data. In the helicity basis, we find the opposite helicity fraction
f_{o} = 0.80 ± 0.25_{stat} ± 0.08_{syst}


and a spin correlation coefficient
κ = 0.60 ± 0.50_{stat} ± 0.16_{ syst}


Introduction
In this measurement we analyze a 4.3/fb dataset in the lepton+jets channel, consisting of a total of 1001 events. The event selection requires one central lepton with large transverse momentum, missing transverse energy of at least 20 GeV, and 4 or more tight jets, at least one of which must be tagged as a b jet. The background is calculated using a combination of Monte Carlo samples and data samples, and predicts a total of 215 ± 48 background events.
Our analysis revolves around the helicity angles of the lepton, the down quark, and the bottom quark which comes from the hadronically decaying top. The helicity angle, defined as the angle between the decay product momentum (in the top rest frame) and the top quark momentum (in the top quark pair rest frame) carries information about the spin of the parent top quark.
We are able to determine the top and top pair rest frames by using a χ^{2} based kinematic fitter where the top mass is contrained to 172.5 GeV.
In order to validate our event selection and background model, we look at the cosines of these three helicity angles in the figures below.
In these figures, our selected data sample is compared to the sum of our background model and a top pair signal sample created using PYTHIA, which does not contain a spin correlation effect.
Signal Templates
In top quark decays, the angular distributions of the top decay products determined by the helicity of the parent top quark via the equation
where the positive sign is for righthanded top quarks and the negative sign refers to
lefthanded top quarks (the signs are reversed for antitop decays). The correlation coefficient Ai varies for each decay product, being equal to +1.0 for the charged lepton or down quark, 0.41 for the bottom quark, and 0.31 for the neutrino or up quark.
We created our templates by modifying the HERWIG source code to implement this
angular distribution for the charged lepton or down type quark, and then allowing the
internal herwig machinery to propagate the appropriate angular distributions to the
other decay products. Using this modified HERWIG, we then created four different simulated Monte Carlo samples, corresponding to the four possible top pair helicity states,
topright antitopleft (RL) , topleft antitopright (LR), topright antitopright (RR), topleft antitopleft (LL).


cos(θ) for lepton or down type quark variable in top quark decays at truth level for the four different top pair helicity states
 cos(θ) for lepton or down type quark variable in anti top quark decays at truth level for the four different top pair helicity states



cos(θ) for bottom quark variable in top quark decays at truth level for the four different top pair helicity states
 cos(θ) for bottom quark variable in anti top decays at truth level for the four different top pair helicity states



cos(θ) for neutrino or up type quark variable in top quark decays at truth level for the four different top pair helicity states
 cos(θ) for neutrino or up type quark variable in anti top quark decays at truth level for the four different top pair helicity states

With the simulated samples prepared for the four different top pair helicity states, the same and opposite helicity templates were created by combining the LR and RL samples to form an opposite helicity sample (OH) and combining the LL and RR
samples to form a same helicity sample (SH). To show the effect of the top quark pair helicity states on the distributions of interest in this analysis, figures below show the variable cos(θlep+)cos(θlep), comparing the distribution at truth level in HERWIG without spin correlations to the same and opposite helicity templates respectively.


Distribution of cosine product variable using truth momenta for unpolarized HERWIG sample and OH template.
 Distribution of cosine product variable using truth momenta for unpolarized HERWIG sample and SH template.

The figures above assume that the down quark can be identified 100% efficienctly, but one of the difficulties of this analysis is that this is not the case. In order to choose the down quark, we use the special precription according to which the jet closest to the b jet in the W rest frame will be the d jet approximately 60% of the time. Figures below again show cos(θlep)cos(θdown) at truth level, comparing HERWIG without spin correlations to our same and opposite helicity templates, but in these figures the down quark is chosen using this prescription. This probabilistic choice reduces the difference between our templates and uncorrelated herwig, but a significant effect is still present.


Distribution of cosine product variable using truth momenta for unpolarized HERWIG sample and OH template.
 Distribution of cosine product variable using truth momenta for unpolarized HERWIG sample and SH template.

Measurement Method
With the same and opposite helicity templates established, we can use them in performing our fit. Our fitting method is a binned likelihood fit to the data, using three separate templates  the same helicity template, the opposite helicity template, and the background template.
Figures below show the composition and validation of the background templates. Distribution on the left shows the various components that go into the background template, and their relative sizes, for the variable cos(θlep)cos(θdown). The largest component of our background model consists of W + heavy flavor jet events. We also validate our background model by considering the antitag sample, to verify that our our background shape is properly modeled. Figure on the right side shows the distribution for the product cos(θlep)cos(θdown) in the antitag sample, compared to our background model summed with the small expected contribution from top pair signal events.


Distribution of cos(θlep)cos(θdown) variable for the various components of our background template
 Distribution of cos(θlep)cos(θdown) variable in antitag data sample compared to the sum of our background model and a signal model

We consider two separate helicity angle bilinears in our fit, cos(θlep)cos(θdown) and cos(θlep)cos(θbot). Two 1dimensional likelihood fits could be performed using these two variables, but pseudoexperiments show that there is a significant gain in sensitivity when the two variables are combined into a single 2dimensional fit, so this is the chosen method for our measurement. When performing the fit, the background normalization is Gaussianconstrained to the predicted value, but the same helicity fraction fs and opposite helicity fraction fo are allow to float freely. We do not require that fs and fo be constrained to physical values between 0 and 1, but we do require fs + fo = 1. The fitting method has been checked for bias using pseudoexperiemnts with various input falues for fo and no bias was found.
Systematic Uncertainties
There are a number of systematic effects that contribute to our uncertainty which need to be taken into account. These include uncertainties in the background size and shape, uncertainties in the exact detector response, and uncertainties in the underlying structure of the colliding particles. Each of these uncertainties is handled its own unique way, but all follow the same general procedure. We start with a template consisting of a nominal background and signal model, and then replace either the background or signal model with a model where the appropriate systematic effect has been varied. Our fit is then performed using this new template, and the result compared to the nominal result in order to determine the systematic uncertainty. In all cases, except for the "Bias Around Null" systematic uncertainty, the signal models used had a true opposite helicity fraction of 0.70. The "Bias Around Null" uncertainty results from a small deviation from the expected fit result of 0.50 for fo when using a signal sample composed of top pair events where spin correlation effects are not included. It is believed that this deviation is a statistical fluctuation, but we conservatively include it as a systematic uncertainty.

Summary of systematic uncertainties

Result
With our fitting procedure established and all systematics uncertainties calculated, the final result of our 2dimensional fit of cos(θlep)cos(θdown) vs. cos(θlep)cos(θbot) in data corresponding to an integrated luminosity of 4.3/fb returns an oppsite helicity fraction of
f_{o} = 0.80 ± 0.25_{stat} ± 0.08_{syst}


Converting this to the spin correlation coefficient using κ = 2⋅fo1 yields
κ = 0.60 ± 0.50_{stat} ± 0.16_{ syst}


Figures below show the 1dimensional distributions for cos(θlep)cos(θdown) and cos(θlep)cos(θbot) respectively, where our data is compared to the sum of the background model, same helicity model, and opposite helicity model, with the normalizations determined by the result of our fit for fo.


Distribution of the cos(θlep)cos(θdown) variable
in data compared to the sum of our background model,
the SH template, and the OH template,
where the OH fraction in signal is fo = 0.80
 Distribution of the cos(θlep)cos(θbot) variable
in data compared to the sum of our background model,
the SH template, and the OH template,
where the OH fraction in signal is fo = 0.80

The figure below shows the same 1dimensional distributions for cos(θlep)cos(θdown) as above with a slightly different presentation.

Distribution of the cos(θlep)cos(θdown) variable
in data compared to the sum of our background model,
the SH template, and the OH template,
where the measured value of the spin correlation coefficient is 0.60

These measurements became public CDF result on January 14, 2010.