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Part 2 - Solutions

Solutions

Exercise 2: Data driven background estimate - sideband fit

  1. Use function SideBandFit(1).\ Background scale factor from sideband fit: \(\alpha = 1.11_{-~0.06}^{+~0.07}\)

  2. Propagate scale factor from exercise a) to the mass window.\ SM background in mass window: (width default masswindow = 10.00 GeV):

    unscaled:  \(N_{\rm bgr}\) = 6.42\ scaled:  \(N_{\rm bgr}\) = \(7.10_{-~0.40}^{+~0.42}\)

  3. The solutions can also be found when running SideBandFit(1)\ In a 10 GeV mass window we expect from background (b=6.42/7.10 events: unscaled/scaled), from signal (s=5.96) and in data we observe (d=16) events.

    option 1: assuming no bkg uncertainty and scaling ( \(b = 6.42, \Delta b = 0.00\) ):\                  p_value = 3.12e-02 \(\rightarrow\) 1.86 sigma.

    option 2: āssuming scaled bkg with bkg uncertainty (\(b = 7.10, \Delta b = 0.41\)):\                  p_value = 3.13e-02 \(\rightarrow\) 1.86 sigma.

    In this case with small bkg uncertainty and small number of events, the uncertainty has very little impact. Try to see what happens at larger luminosities.

Exercise 3: Measurement of the production cross-section

We run the fit using 2 GeV bins, i.e. use a rebin factor of 10.

  1. Run MuFit(10,1).

    Best fit: \(\alpha\) = 1.10 , \(\mu\) = 1.28

  2. Run MuFit(10,2).

    We should 'profile' the uncertainty in \(\alpha\). Just in case, we also show the value if we would just look at the slice at the best value of \(\alpha\).\ Result on mu:

                     best alpha approach: \(\mu = 1.29_{-~0.54}^{+~0.66}\)\                  profiled: \(\mu = 1.27_{-~0.54}^{+~0.66}\)

    Not a strange result as the variables are not so much correlated. Let me point out here that in the real Higgs analysis the signal scale factor is strongly correlated with the actual mass as the production cross-section and branching fraction of the Higgs to 4 muons depends on the mass of the Higgs boson.