Different sources of uncertainty, corresponding to different nuisance parameters, can be treated as fully correlated (100% correlation), anti-correlated (−100%), or independent (0%). The correct assignment of correlations depends on the specific uncertainties in hand. Partially correlated uncertainties are treated by splitting them to fully (un)correlated subcomponents.
The nuisance parameters enter Eq. (9.1) as follows, turning it into a profile likelihood function that depends both on the parameter of interest µ and on the full set of nuisance parameters~θ:
L({ni}|µ,~θ) =
∏
i(µsi(~θ) +bi(~θ))ni
ni! e−(µsi(~θ)+bi(~θ))f(θ0|θ). (9.11)
For a given µ, one can maximize the likelihood function to find the maximum- likelihood estimators~θML(µ)for the nuisance parameters.
When these maximum-likelihood estimators are used to profile the nuisance parame- ters in the test statistic, Eq. (9.2) becomes
˜qµ =−2 ln L({ni}|µ,
~θML(µ)) L({ni}|µML,~θML(µML))
. (9.12)
When the limit is set with this test statistic, full information on the systematic and statistical uncertainties defined by the nuisance parameters is incorporated in the fit. For each nuisance parameter, deviation from the default value is allowed, but associated with a penalty in the fit as described by the pdf.
9.4 Statistical uncertainties
The limited number of events in each bin of the templates introduces statistical uncer- tainties that need to be taken into account. In the following, we discuss two different methods to achieve this: a traditional Barlow-Beeston method and an approximative Barlow-Beeston-lite approach [183] that has recently gained popularity in the CMS analyses.
In the traditional Barlow-Beeston method, each bin of each template is attached a separate nuisance parameter corresponding to the statistical uncertainty of the events in that particular bin. As the population of a bin follows a Poisson distribution, the uncertainty attached is simply the square root of the event count.
140 9. Statistical methods
If the background consists of p processes with separate templates, and each template has n bins, np shape uncertainties are calculated by varying the event yield in each bin of each template up and down. As statistical fluctuations in bins are independent of each other, all these uncertainties are incorporated as np independent nuisance parameters in the likelihood.
As templates can have a large number of bins, this method often introduces a great number of shape uncertainties, and the limit calculation with full uncertainties can become computationally heavy. Therefore an approximative method, known as the Barlow-Beeston-lite approach, is often used instead. In this method, contributions from different background processes are summed into one total event count per bin, and this total count is assigned a single Gaussian-constrained uncertainty, reducing the number of new nuisance parameters to n.
In the Barlow-Beeston-lite approach the ML estimate of each nuisance parameter has an analytic formula that depends only on bi (the total number of background events in template bin i) and ni(the number observed events in bin i). Thus this part of the likelihood can be minimized analytically before proceeding to numerical minimization, which leads to reduced computation time and better fit stability.
PART III
Data Analysis
Chapter 10
Analysis strategy
In Chapters 10–13, the data analysis methods used to search for charged Higgs bosons decaying via H± →τ±ν
τ are presented in detail. The results follow in Chapter 14. This search is the first CMS analysis on this decay channel based on 13 TeV collision data. A preliminary version of the analysis with a partial data set is documented in Ref. [72]. The final analysis and its results are published in Ref. [73].
As discussed in Section 3.2, the production of H± is possible via various processes
and the mass of the H± determines the dominant process. In all production modes,
the final state includes a H±, a top quark, and possibly a b quark. The top quark
decays further into a b quark and a W boson, which in turn can decay hadronically or leptonically.
In this analysis, we concentrate on the hadronic final state of the H± →τ±ν
τ decay channel. In the hadronic final state, also denoted as the τh+ jets final state, both the tau
lepton (from H±) and the W± boson (from a top quark) decay hadronically. As the
total hadronic branching fraction is approximately 2/3 for both τ and W±, almost half
of all H± →τ±ν
τ signal events evolve into the τh+ jets final state. The leptonic final
states where the tau lepton from the H± decay or the W± boson decay leptonically
are targeted by a separate analysis and discussed briefly in Section 10.4.
Since the τh+ jets final state looks similar regardless of the H± mass, the search can
cover a wide mH± range from 80 GeV up to 3 TeV. Examples of possible production
processes, with the following decays that produce the τh+ jets final state, are shown as
tree-level Feynman diagrams in Figure 10.1. The first diagram shows the production of a light H± in a top quark decay, whereas the second shows the heavy H± production
in association with a top quark (in the five-flavor scheme). In both cases, the final state contains the visible decay products of the hadronic tau lepton decay (τh), neutrinos
144 10. Analysis strategy 1 g g g 1 q 1 q� 1 W− 1 b t 1 1 b ντ 11 ντ 11 τh 11 τ+ H+ t 1 g b 1 b 1 q 1 q� 1 W+ 1 b t ντ 11 ντ 11 τh 11 τ− H−
Figure 10.1: Tree-level Feynman diagrams for the production of a light H± in a top quark
decay (left) and of a heavy H± in association with a top quark in the five-flavor
scheme (right). In both cases, the decays producing a hadronic final state are shown. The H± decays to a tau lepton and a neutrino, and the tau lepton decays
hadronically (τh). The top quark decays into a b quark (producing a b jet) and a W± boson that also decays hadronically (producing light quark jets) .
from the H± and τ decays, quarks from the W boson decay, and one or several b
quarks.
The detector fingerprint of the fully hadronic final state can be easily inferred from these diagrams: regardless of the H± production mode, we expect the final state to
contain a τh, missing transverse momentum (pmissT ) due to neutrinos, and a number of
jets including one or several b jets. No isolated electrons or muons are expected to be present in the targeted final state. The event selection criteria designed to extract this type of event topology are described in Chapter 11.
The physics objects are reconstructed using the methods described in Chapter 8, and used in the event selection. The exact identification criteria for different objects are described when the event selection is detailed in Chapter 11.
The special feature of the τh + jets final state is that most of the pmissT in the event
originates from the neutrinos produced in the H± decay (and the subsequent tau
lepton decay). Thus the τh candidate and the pmissT can be used to reconstruct the
transverse mass (mT) distribution of the tau-neutrino system, defined as
mT(~pT(τh),~pTmiss) =
q
2pT(τh)pmissT (1−cos ∆φ(~pT(τh),~pTmiss)), (10.1)
where~pT(τh)is the transverse momentum of the reconstructed τhand ∆φ denotes a
10.1. Data 145
The selected signal region is contaminated by background events that contain a jet misidentified as a τh(jet→τh events), most of which are QCD multijet events. This
background contribution is estimated from a control region with a data-driven method presented in Chapter 12.
The top quark production processes in the SM (tt and single top quark production) also produce a significant irreducible background with a final state that is almost identical to the H± signal events. For example, changing the H± to a W± in Figure 10.1
describes one possible final state of tt production. These processes, as well as the SM processes with W or Z bosons (W+jets, Z/γ∗, WW, WZ, ZZ) produce events with
genuine tau leptons that decay hadronically. The background from all these processes is referred to as the genuine-tau background. A data-driven method for estimating this background, known as tau embedding, is presented in Chapter 12. As this method is still considered experimental, it is not used for the final results, where instead the genuine-tau background is estimated from simulation. To suppress the genuine-tau background, the tau lepton helicity can be used to discriminate between tau leptons originating from H± and W± bosons, as discussed in Section 2.5.2.
Finally, a small background contribution arises from events with electrons or muons misidentified as τh(e/µ→τh). This contribution is estimated from simulation. Also
the background estimations from simulation are presented in Chapter 12.
10.1 Data
The analysis is based on the proton–proton collisions recorded by the CMS experiment during the year 2016. The center-of-mass energy of the collisions was 13 TeV. The amount of collision events delivered by the LHC in 2016 corresponds to an integrated luminosity of 41.07 fb−1. The CMS detector successfully recorded 37.82 fb−1 (92%)
out of it. Finally, 35.92 fb−1(95%) of the recorded data were certified as high-quality
data suitable for physics analyses, including this analysis. The pileup distribution in the data is shown in Figure 10.2. For the data collected in 2016, an average of approximately 23 interactions per bunch crossing was measured [109].
In 2015, the CMS detector recorded 3.80 fb−1 of data [109]. Since the additional
statistical power obtained by including these data is negligible, while they would require separate calibration, optimization and background measurement efforts, the 2015 data are not included in this analysis.
146 10. Analysis strategy
0 20 40 60 80 100
Mean number of interactions per crossing
0 50 100 150 200 250 300 R e c o rd e d L u m in o s it y ( p b ¡ 1/0 .1 0 ) <¹> = 23 ¾pp in=69:2 mb 0 50 100 150 200 250 300
CMS Average Pileup, pp, 2016, ps = 13 TeV
Figure 10.2: The pileup distribution in the pp collision data recorded by the CMS experiment
in 2016. [109]
Following the usual CMS procedure, in order to avoid any subconscious biases this analysis was designed, optimized, validated and reviewed by the Higgs Physics Analysis Group of the CMS Collaboration while keeping the data blinded in the signal region. Only after the analysis methods were fixed and initially approved, the data were unblinded and the final results as presented in Chapter 14 were extracted.