Single-top Search using Matrix Element, Likelihood Function, and Neural Networks techniques (supporting material)


Common Pseudo-experiment study:

    The question arises to which extent the results of the Matrix Element (ME), the Likelihood Function (LF), and the Neural Networks (NN) techniques are compatible. Given the fact that all three analyses employ common selection (ie common dataset and MC samples), the compatibility question can be answered by means of simulated experiments (``pseudo-experiments''). The discriminants considered are:
  1. NN analysis outputs: Ot , Os , Ost
  2. ME Extended Probability Discriminant: Pst
  3. LF analysis: Lt , Ls
They are essentially used in four analyses (two combined s+t and two separate s- vs t-channel searches), the discriminants of which are:

- s+t analysis NN: Ost
- s+t analysis ME: Pst
- s vs t analysis NN: (Ot, O s)
- s vs t analysis LF: (Lt , Ls)

Every MC event is run through the ME, LF, and NN analyses, and the four (two 1d and two 2d) discriminants listed above are stored. Once this is done and the four discriminants for every event have been recorded, we proceed to form simulated S+B datasets based on the 955/pb luminosity predictions. A pseudo-experiment is constructed by randomly drawing MC events from the signal and background samples according to the 955/pb expected contributions (used as means of Poisson distributions). Once a pseudo-experiment is formed, the four discriminant distributions are constructed, and then fitted as weighted sums of signal and background reference histograms (5-component fit to b-like, c-like, q-like, ttbar-like, and signal-like templates). The same fitting program is used for any and all of the four discriminant distributions, and all fits are found to be unbiased. The systematic uncertainties are taken into account in a correlated way, ie the same variations in JES, ISR/FSR/pdf etc are used for the ME, LF, separate NN, and combined NN fits.

Let S1, S2, S3, and S4 denote the fitted signal cross sections in units of SM cross-sections for the four analyses, respectively, and let e1, e2, e3, e4 denote the uncertainties returned by these fits. In the infinite statistics limit the fits would return S1,S2,S3,S4=1 and e1,e2,e3,e4=0.

The Table below shows the correlations between pairs of the four fit results.



The following two plots show the fits S1, S2, S3, S4 in units of the SM single-top cross section, along with the fit uncertainties e1, e2, e3, e4, without and with the inclusion of the systematic uncertainties. In the "without" case, only the background rates are included as nuisance parameters.

Starting from the fitted cross-sections Si (i=1,..,4) we form a best linear unbiased estimate (BLUE) [1] for the single-top cross-section. We do this by constructing a covariance matrix from the statisitical (and systematic) correlations. We invert this covariance matrix to obtain a weight for each i and combine the four results with these weights to obtain our best cross section estimate. Once we have this, we can calculate a chisquare value for each pseudo-experiment (and for the data). We can then ask the question: in what fraction of pseudo-experiments does the corresponding chisquare exceed the value observed in the data. This fraction will define our compatibility measure.

For the four analyses, this fraction is 0.65%.



[1] L. Lyons, D. Gibaut and P. Clifford, NIM A 270, 110 (1988). L. Lyons, A. Martin and D. Saxon, Phys. Rev. D 41, 3 (1990). A. Valassi, NIM A 500, 391 (2003)