The missing transverse momentum,~pTmiss, is defined as the negative vector sum of the ~pT of all reconstructed PF candidates [165]. This~pTmissdefinition incorporates all the
physics objects (muons, electrons, photons, τhcandidates and jets), reconstructed from
the PF candidates and identified as described in the previous sections, but also the unclustered energy, defined as the energy of all the PF candidates not clustered into any physics object. Therefore the estimation of~pTmissis affected by spatial and energy resolutions of all PF candidates, and in addition to genuine momentum imbalance, it can be altered by mismeasurement and detector artifacts.
The~pTmiss is calibrated by propagating the effect of the jet energy corrections described in Section 8.2.3 into it. Corrections to the energy scales of other physics objects are negligible compared to the jet energy corrections, so they are ignored. The Type-I corrected~pTmiss is defined as:
~pTmiss = ~pTmiss,uncorrected−
∑
jets
(~pcorrectedT − ~puncorrectedT ), (8.4)
where ~pcorrectedT (~puncorrectedT ) is a jet pT before (after) the jet energy corrections. To
suppress the effect of pileup jets, only the jets with the corrected pTabove 15 GeV are
included in the sum. As the correction is designed for quark and gluon jets, the jets corresponding to electromagnetic showers from electrons and photons are removed by excluding jets that have>90% of the jet energy deposited in the ECAL. For the same reason, jets containing global muons or standalone muons are excluded from the sum.
Chapter 9
Statistical methods
The statistical analysis aims to draw conclusions about the observed data, under some theoretical hypothesis. If there is no significant excess corresponding to the presence of a signal, how large signals can be excluded based on the observed data? Or if there is an excess in the data, how likely is it to originate from the signal modeled by a given signal model?
In both cases, the parameter of interest in the analysis is the amount of signal, represented by the signal strength modifier µ. The signal strength modifier µ is defined as a parameter that varies the signal yield, thus representing different signal hypotheses. If s (b) is the expected event yield for signal (background) events, the expected total yield is µs+b. The standard workflow of hypothesis testing is followed: First, the null hypothesis is defined and a suitable test statistic is constructed. The observed value of the test statistic is calculated from data, and conclusions are drawn by comparing it to the expected distribution of the test statistic. In this chapter, these steps of the hypothesis testing workflow and the relevant concepts are presented, as applied in the analysis presented in this thesis.
In traditional "cut-and-count" experiments the test statistic was defined simply based on the expected and observed event yields, obtained after online and offline selections. Modern computing techniques enable a more powerful "shape analysis" approach, where a summary statistic is calculated from the selected events and used to derive the test statistic. The summary statistic can be any distribution that discriminates between the background and signal events, such as a reconstructed mass distribution or output of an MVA classifier. In the analysis presented in this thesis, the transverse mass as defined in Chapter 10 is used. We refer to these distributions of the summary statistic as templates.
132 9. Statistical methods
In a shape analysis, the test statistic incorporates both the expected event yield µsi+bi and the observed yield niin each bin of the summary statistic. The normalization of the signal in the templates (si) can be based on a specific theoretical model, or it can be arbitrary, as it is only an initial value for the fit to data. If the production cross section (σ) and the branching fraction to the final state in case (B) are known from the SM (or from a specific BSM model), the signal can be normalized accordingly and the signal strength modifier represents deviation from the theory expectation. In case of more generic searches, such as the H± search presented in this thesis, it is more
convenient to normalize the signal templates to an arbitrary initial value, such as σ =1 pb,B =100%.
In the frequentist paradigm, a probability is defined in an objective way as the relative frequency of an event in the limit of a large number of trials. On the other hand, in the Bayesian paradigm a probability is interpreted as a subjective degree of belief, thus changing as new information is obtained. In the statistical interpretation of most high-energy physics experiments, including this analysis, the frequentist approach is followed. The reason is that we aim to interpret each individual search result in an objective way and independent of previous results. In Bayesian approaches the conclusions would be dependent on prior probability distributions affected by subjective judgments and previous results.
9.1 Exclusion of signal
When no clear excess is observed in data compared to the background expectation, the goal of the statistical analysis is to set a limit on µ, which can be interpreted as a limit on σB by taking into account the initial normalization of the signal template. Here we present a method for signal exclusion, which is agreed upon and used by the ATLAS and CMS experiments and documented in Ref. [177].
The probability to observe ni events in a template bin i when the expected yield is µsi+biis given by the Poisson probability distribution. The combined probability for all bins is
L({ni}|µ) =
∏
i
(µsi+bi)ni
9.1. Exclusion of signal 133
Here{ni}denotes to the ensemble of bins, with nievents in bin i. This value of µ that maximizes this likelihood function for the observed data niis the maximum-likelihood value µML.