Electron sample with 250fb-1

Fit results for 10x10 bins:
r_bb=0.95 +/- 0.056 (5.93%)
r_cc=1.3 +/- 0.55 (41.82%)
r_gg=1.2 +/- 0.52 (44.51%)

Errors from the toy study using chi2 ignoring less than 7:
r_bb 0.057 +/- 1.7*10-3
r_cc 0.49 +/- 6.4*10-2
r_gg 0.52 +/- 4.9*10-2
The error on the error is taken from the sigma of a gaussian fit on a plot of the errors, with the error taken from the mean (if you see what a mean - no pun intended).

Errors from the toy study of a toy set to its source using the likelihood function:
r_bb 0.038 +/- 5.8*10-4
r_cc 0.36 +/- 1.4*10-2
r_gg 0.38 +/- 6.9*10-3


The numbers mostly match Roberval's well, although the likelihood r_cc and r_gg are a little lower here.

Plots, same as yesterday:
r_bb
r_cc
r_gg

Pulls:
chi2
likelihood
The chi2 is off slightly. I wasn't going to look into it too much unless anyone thinks it's a cause for concern?

Also, the fit results:
chi2

Electron sample with 500fb-1

I'm running the 250fb-1 numbers at the moment, here are the results for 500fb-1.

The "Toy study using limited templates" is the chi2 fit ignoring less than 7 entries using 5000 tests. I plotted the error, fitted a gaussian to it and that gives the value and the error bars.
The "Toy study error using infinite statistic templates" is the likelihood fit of a toy sample to its source. Again 5000 tests, with a gaussian fitted to the errors.
The "Error from fully simulated data" is the error from the chi2 fit ignoring less than 7 entries.
r_bb
r_cc
r_gg

The fully simulated data matches the toy study for limited statistics and the values for 10x10 bins match Roberval's 500fb-1 numbers quite well. When I post the results for 250fb-1 I'll add the numbers at 10x10 bins too.

The pulls for both of the toy studies are below. A slight low bias for the chi2 but other than good.

Likelihood
Chi2

Results scaled to 250 fb^-1 - muon channel

By simply scaling the data histograms, the central values did not change and the errors are larger, as expected

Likelihood fit

• r_bb = 1.013 ± 0.038
• r_cc = 0.81 ± 0.46
• r_gg = 0.98 ± 0.45

Chi2 fit

• r_bb = 1.012 ± 0.044
• r_cc = 0.87 ± 0.54
• r_gg = 0.93 ± 0.51

The contribution from the MC to the errors in the parameters should be the same as in the 500fb^-1 case because on the MC side nothing changed. The pull distributions for pseudo-experiments with MC only give contributions of 46% in the estimated error arising from the MC finiteness, yielding exactly the same errors as before.

• r_bb = 1.012 ± 0.039 (data) ± 0.020 (MC)
• r_cc = 0.84 ± 0.48 (data) ± 0.25 (MC)
• r_gg = 0.95 ± 0.45 (data) ± 0.23 (MC)

The statistical errors from the data in the chi2 fit are very similar to the ones from the likelihood.

It seems that with 250fb^1 of luminosity, even combining the different channels, the measurement of the branching ratio of the Higgs boson will be quite poor.

Actions from Meeting

It was agreed that the method was now in place, and the muon numbers look good for 500 fb^-1.

The actions required to complete our contribution to the LOI:

• Re-run muon numbers for 250 fb^-1 (Roberval)
• Re-make figure with flavour-likelihood distributions with axes labels swapped and all text enlarged (Hajrah).
• Run final electron numbers for 250 and 500 fb^-1 (Mark).
• Final edit and combination of numbers (Joel).

Results for muon channel - Update

Fit with Poisson Likelihood

• r_bb = 1.013 ± 0.027
• r_cc = 0.81 ± 0.33
• r_gg = 0.98 ± 0.32

With the likelihood method contributions from a finite Monte Carlo sample are not considered.

Generating pseudo-experiments of the data assuming the data is poisson distributed yields pull distributions of gaussian shape with mean O(10^-2) and rms ~1.

Fit with Chi2

• r_bb = 1.012 ± 0.033
• r_cc = 0.84 ± 0.41
• r_gg = 0.95 ± 0.39

In this case, statistical fluctuations arising from the finite MC sample are taken into account. The poisson to gaussian approximation is only valid if the number of events in the bin is larger than 5.  We then considered in the fit only bins with at least 7 entries in the data.

To check that this method is valid and consistent with the likelihood method, pseudo-experiments of the data and the MC were generated and the pull distributions of the fit parameters were calculated considering that the "true" value of the parameters are the ones obtained in the likelihood.

The mean (rms) of the pull distributions are 1.0 (-0.08) , 0.99 (-0.08) and 1.0 (-0.16) for r_bb, r_cc and r_gg, respectively. Notice that r_gg is slightly biased.

In order to estimate the contribution of the finite MC sample to the error of the fit, the pull distributions of the fit parameters were obtained using pseudo-experiments generated for the MC only. The pull distributions are then gaussians with width less than one. That is the fraction of the fit error that is purely due to the MC statistical fluctuations. We obtained widths of 0.6 for the three parameters. Splitting the error contributions from the data and from the MC we obtained:

• r_bb = 1.012 ± 0.027 (data) ± 0.020 (MC)
• r_cc = 0.84 ± 0.34 (data) ± 0.25 (MC)
• r_gg = 0.95 ± 0.31 (data) ± 0.23 (MC)

Ignoring small bins

The pulls from the toy study:
r_bb
r_cc
r_gg

Results from the full data:
r_bb
r_cc
r_gg

Likelihood cut variables

Some plots for likelihood variables I'm only just getting around to putting up:

Jet energy difference
recoil mass
Z candidate cos theta
thrust cos theta