Estimation of the Total Cross Section Uncertainty

The total error of the cross section result includes multiple sources of uncertainties. Each source of uncertainty is taken into account by producing, for each MC sample in Tab. 2.3, an equivalent template where the upward or downward variation is applied. The analysis chain is then run again, for each of these variation templates, thus obtaining a different result, whose excursion from the nominal result contains the information about the impact of that specific source of uncertainty. All the sources of systematic uncertainties described in Sec.7.1are treated in this manner. Additionally, also the statistical errors associated to data and to the MC production need to be computed, as well as error associated with the cross section uncertainties. Following a frequentist approach, pseudo-experiments are eventually employed in order extract the total uncertainty and the contributions of each systematic.

Data Statistical Uncertainty

The event yield resulting from running the analysis on the nominal templates is used in order to evaluate the impact of the statistical error associated to the limited number of events collected in data. In practice, this is determined by performing pseudo-experiments where a number is drawn from a Poisson distribution. The expectation value of this distribution

is given by the original number ν_j constituting the event yield of the analysis for the jth MC sample (cf. Eq. 7.8). The sum of the draws then constitutes a pseudo-data template, deviating from the nominal sum. To build the uncertainty distribution, 10,000 pseudo-experiments are performed.

MC Statistical Uncertainty

The impact of the statistical error associated to the limited number of MC events available to the analysis is also evaluated. For each MC samplej, a the number of events in each bin is reset in a pseudo-experiment, drawing a random number according to a Poisson distribution, with an expectation value given by the original event yield νj3. The resulting yields of all processes are added up and used as template for the evaluation of the uncertainty. Again, 10,000 dedicated pseudo-experiments are performed in total to simulate the impact of this uncertainty.

Cross Section Uncertainty

To model the background in the fit, the cross sections reported in Tab. 7.1 are used. The impact of the expected uncertainties on the final result is evaluated, again, by means of pseudo-experiments where the number of expected events in the analysis background is varied. In practice, this is done in three steps. In the first place, a random number x is drawn from a log-normal distribution⁴ with a mean of one and a standard deviation equal to the corresponding (relative) cross section uncertainty δσˆ_j from :

lnN(x;µ_j, σ_j) = 1

x^q2πσ_j² exp



−1 2

lnx−µ_j σ_j

!2

 (7.13)

with

µ_j =−1

2σ_j², (7.14)

σ²_j = ln[(δσˆ_j)²+ 1]. (7.15) In Equations (7.13), (7.14) and (7.15), µ_j and σ_j are the two parameters needed to build the log-normal distribution belonging to the j−th process. Secondly, the expectation value of the nominal total number of events of the j −th process is scaled by the multiplication factorx. Finally, the Poisson statistics is employed again to draw randomly a new number of events N_j^xsec, using the value previously scaled as mean of the distribution. Analogously to

3Since only the event yield is used to extract the signal in this analysis, without involving the comparison of distribution shapes, there is no need to consider the statistical uncertainty associated to the binning.

4A log-normal distribution is chosen to ensure that only positive numbers can be drawn.

the treatment of the data and MC statistics systematics, the sum of all the (independently) shifted templates serves as pseudo-data in the fit. In total, again, 10,000 pseudo-experiments are performed.

Systematic Uncertainties

The limited number of signal events available to the analysis after the event selection does not allow to use the shape of the distribution of the most discriminating observable, p^{W t}_t , which is instead used to select the events by setting an upper threshold on its value. Therefore, since the cross section extraction is based on a simple event count, all the systematics can be treated as simple rate uncertainties. These are taken into account by varying, in each channel m, for each process j, the expected number of events ν_mj according to a quantity

ν_mj^syst =ν_mj ·



1 +

Nsyst

X

i

δ_i·[Θ(δ_i)·_imj++ Θ(−δ_i)·imj−]



 . (7.16)

Here, ν_mj^syst is the expectation value of the total yield, shifted according to the effect of each of the N_syst systematic uncertainties on the acceptance of the process. In practice, this is done by using the nuisance parameter, δ_i, which is drawn at random from a standardized Gaussian distribution (with mean at zero and standard deviation of one). This nuisance parameter is then used in each pseudo-experiment to define the strength and sign of the ith systematic excursion. The quantity Θ(δ_i) is the Heavyside step function. In this framework, it is used to distinguish between the application of relative acceptance uncertainties resulting from the use of upward variation templates from the respective downward ones.

The uncertainty associated to the luminosity measurement, ν_mj is also varied at random according to Eq. (7.16), but in this particular case the efficiency shifts are given by imj±=

±(δL/L).

The ν_mj^syst quantities built in Eq. 7.16 represent now the new expectation values obtained from applying a systematic variation. These quantities are, again, taken as the mean of a Poisson distributions from which the total number of observed events in the current pseudo-experiment, N_mj^syst, is randomly determined. At this point, the contributions from all of the j processes are summed up to produce a new template. This template is then used as pseudo-data, in the sense that is treated like the real pseudo-data, in order to the extract the cross section from the fit described in Sec. 7.2.1. For each source of systematic uncertainty described in Sec. 7.1.1 and 7.1.2, 10,000 pseudo-experiments are performed. In an identical fashion, this procedure is applied to assess the systematic impact of the statistical uncertainty associated to the data and the MC, the one from the cross section uncertainty in the background fit, and the luminosity. In each iteration, the nominal analysis templates are then fitted to the pseudo-data using the likelihood shown in Eq. 7.7.

The distributions of the β_j parameters obtained from each pseudo-experiment are then further exploited to quantify the resulting uncertainty associated to each systematic. The

standard deviation σ_i(βW t-channel) of the distribution of the βW t-channel extracted from the 10,000 fits on the pseudo-data can be used as the estimator of the error associated to the i−th systematic on the measured cross section of the W t-channel.

Finally, the effect of the correlations between the different sources of uncertainty is assessed using 10,000 pseudo-experiments where all sources of uncertainties are combined. A new distribution of βW t-channel is thus obtained, and its standard deviation, which now contains the effect of the correlations of all the systematic error sources, is used as the error on the final cross section result.