in the fit. The signal is independently considered in each signal region but neglected in the control regions. This background prediction is conservative since any signal contribution in the control regions is attributed to back-ground and thus results in a possible overestimation of the backback-ground in the signal regions. In this analysis this contribution is negligible due to the requirement of leptons in the control regions. This fit configuration is used to extract the 95% CL model independent upper limits on the visible cross section.
Model dependent signal fit: Both control and signal regions are used in the fit. The signal contribution is taken into account as predicted by the tested model in all the regions. The model dependent signal fit configuration is used to interpret the results of this analysis in terms of the different new physics models that are studied.
4.3 Statistical tests
This section describes the general procedure used to search for a new phe-nomena in the context of a frequentist statistical test. If the purpose of the analysis is to discover a new signal process, the null hypothesis, H0, is defined as describing the known SM processes, to be tested against H1, which includes both background as well as the signal model. Instead, if the purpose of the analysis is to set limits on a signal process, the model with signal plus background plays the role ofH0, tested against the background-only hypothesis,H1. In the outcome of such search, the level of agreement of the observed data with a given hypothesisH is quantified by computing the probability, under the assumption of H, of finding data with equal or less incompatibility with the prediction ofH.
According to Equation 4.10, each process is multiplied by a normalization factor,µ. A background-only hypothesis is constructed by fixingµsignal= 0, while a signal+background hypothesis will be defined as havingµsignal>0.
To test an hypothesized value ofµ, the profile likelihood can be defined as the ratio:
λ(µ) = L(µ,~θ)ˆˆ L(ˆµ,~θ)ˆ
, (4.11)
where µ here is the shortcut for µsignal and ~θ ⊃ {µno signal, ~α}. θ~ˆˆ in the numerator denotes the value of ~θthat maximizesL for the specifiedµ(it is a conditional maximum likelihood estimator ofθ, and therefore a function of
µ). The denominator is the maximized (unconditional) likelihood function.
Based on Equation 4.11, the test statisticqµis defined as:
qµ=−2 lnλ(µ). (4.12) Higher values of qµ correspond to increasing compatibility between the data andµ. Thep-value, defined to quantify the level of agreement between the data and the different hypotheses, is defined as:
pµ= Z ∞
tµ,obs
f(qµ|µ0)dqµ, (4.13) where f(qµ|µ0) denotes the PDF of qµ under the assumption of the sig-nal strength µ0. The estimations of f(qµ|µ0) can be done with pseudo-experiments using Monte Carlo methods (Toy MC). These methods are computationally heavy, especially when upper limits are calculated. For this reason, an approximation valid in the large sample limit is normally used to describe the profile likelihood ratio instead (asymptotic approximation).
In the large sample limit, where the asymptotic approximation becomes exact, the PDF ofqµ assuming that the fitted strength parameter ˆµfollows a gaussian of meanµ0 and standard deviationσ is found to be [62]:
f(qµ|µ0) = 1 2√
qµ
√1 2π×
"
exp −1 2
√
qµ+ µ−µ0 σ
2!
+ exp −1 2
√
qµ−µ−µ0 σ
2!#
.
(4.14)
Figure 4.1 illustrates the previous equation, for the particular case of qµ=1 under a signal plus background and a background-only hypotheses, namely µ0 = 1 and µ0 = 0, respectively. In this example, the requirement that the p-value computed from the f(qµ=1|1) PDF is smaller than 0.05, would be enough to exclude the signal model at 95% confidence level (CL).
However, the PDFs for both hypotheses could be similar. These are cases in which the analysis has very low sensitivity and the effect produced by a statistical fluctuation could allow the exclusion of both the null (in this case, the signal plus background) and the alternate (background-only) hypotheses at the same time. In an attempt to address this spurious exclusion, theCLs
method is developed. TheCLssolution bases the test not only on the rejec-tion of the null hypothesis but rather in the p-value of the null hypothesis divided by one minus the p-value of the alternate hypothesis. Following the same illustrative example from Figure 4.1, in which the existence of a given signal model is tested, theCLs+b,CLb andCLscan be defined, respectively,
4.3. STATISTICAL TESTS 61
q1
5 10 15 20 25 30
10-4
10-3
10-2
10-1
1
=1|1) : f(qµ
H0
=1|0) : f(qµ
H1
=1.87e-03 ps+b
=1.22e-01 pb
= 2.13e-03 CLb
CLs+b s = CL
1, obs
q
R. Caminal - PhD Thesis
Figure 4.1: Illustration of the PDF ofqµ=1 under two different hypothesis:
signal plus background (null,µ= 1) and background-only (alternative,µ= 0). TheCLs+b,CLb and CLs are also shown for this particular example.
as:
CLs+b =ps+b
CLb= 1−pb
CLs= CLs+b CLb .
(4.15)
In the work presented in this thesis, theCLsis calculated for each signal model under evaluation. The models for which CLs < 0.05, are excluded at 95% CL. With the CLs method, CLs ≈ CLs+b in the cases where the analysis is sensitive to the signal process under study. Instead, in the cases where the analysis is insensitive, CLb is be small, thus increasing the value of CLs and therefore avoiding the exclusion of the signal model.
Chapter 5
The ATLAS detector at the LHC
The analysis described in this Thesis is performed using proton-proton col-lision data produced in the Large Hadron Collider and detected and re-constructed by the ATLAS detector. This chapter introduces the CERN’s accelerator complex and describes the main aspects of the ATLAS detector at the LHC.
5.1 The Large Hadron Collider
The Large Hadron Collider (LHC) [63] is a circular superconducting parti-cle accelerator installed in a 27 km long underground tunnel (between 45 m and 170 m below the surface) that used to host the Large Electron-Positron (LEP) collider. On the accelerator ring four detectors (ALICE [64], AT-LAS [65], CMS [66] and LHCb [67]) have been built around four different interaction points to reconstruct and study the collisions delivered by the LHC. The LHC is designed to collide protons at a center of mass energy of
√s= 14 TeV.
Since 2010, the LHC has delivered proton-proton (pp) collisions at center of mass energies of 7 TeV and 8 TeV (in 2011 and 2012, respectively), about half of its nominal energy. The LHC has produced also lead ion (Pb-Pb) collisions with a per-nucleon center of mass energy √
sN N = 2.76 TeV and proton-ion (p-Pb) collisions with√
sN N = 5.02 TeV.