Probing CP-Violation in Top-Higgs Interactions

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Probing CP-Violation in Top-Higgs Interactions

A Bachelor Thesis submitted at the Institute for Experimental Physics, University of Hamburg

Finn Labe

02.02.2018

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The search for physics beyond the standard model of particle physics that can explain the unsolved mysteries of the universe, such as the matter-antimatter asymmetry, is of high importance in todays research. The interaction of quarks with the Higgs boson is a candidate for such physics, possibly resulting in a new source of CP-violation. This work analyses multiple observables on their sensitivity to the CP coupling in this interaction by performing an analysis using simulated data for tth¯ production in ppcollider events. The data is evaluated on parton level both before and after acceptance cuts are applied. In addition, a detector simulation for the CMS detector at the LHC is executed. A hypothesis test that can evaluate the observables is presented and their usability in a real detector experiment is discussed.

In the final state leptons a promising candidate for observables sensitive to the CP coupling in the process is found.

Die Suche nach Physik jenseits des Standardmodells der Teilchenphysik, welche bisher ungeklärte Fragen des Universums wie etwa die Materie-Antimaterie Asym- metrie erklären kann, ist in der aktuellen Forschung von großem Interesse. Die Wechselwirkung zwischen Quarks und dem Higgs-Boson ist ein Kandidat für solche Physik, welche möglicherweise eine neue Quelle an CP-Verletzung beinhaltet. Die vorliegende Arbeit analysiert verschiedene Observablen auf ihre Sensitivität auf die CP-Kopplung dieser Wechselwirkung, mithilfe eine Analyse von simulierten Daten für dietth¯ Produktion in ppColliderevents. Die Simulationsdaten werden auf Par- tonenniveau sowohl ohne als auch mit Cuts an den Daten untersucht, außerdem wird eine Detektorsimulation für den CMS-Detektor am LHC durchgeführt. Ein Hypothesentest wird genutzt, um die Observablen zu bewerten, und die Nutzbarkeit dieser in einem realen Detektorexperiment wird diskutiert. Es wird gezeigt, dass die Leptonen im Endzustand des Prozesses einen vielversprechenden Kandidaten für Observablen, die sensitiv auf die CP-Kopplung im Prozess sind, darstellen.

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1. Introduction

1.1. Standard Model of particle physics

The standard model of particle physics (SM) classifies all currently known elemental particles and describes the interactions between them.

The elemental particles are grouped into leptons, quarks and bosons. There are six quarks (u, d,c,s,tandb), six leptons (e⁻,µ⁻,τ⁻and the corresponding neutrinosν_e,ν_µandν_τ) and four gauge bosons (g,γ,ZandW). The SM also includes a fifth boson, the scalar Higgs boson. Three forces are known that interact with the elemental particles, the electromagnetic interaction, the weak interaction and the strong interaction. The SM is not including the gravitational force.

The SM is not able to explain every observation that can be made in reality, which motivates the search for physics beyond the Standard Model (BSM).

1.2. Matter-antimatter asymmetry

In particle physics, one differentiates between baryonic matter and antibaryonic matter. For each charged particle, there exists an oppositely charged antiparticle.

As everything today, from the small objects in the everyday life to the stars and planets seen in the sky, is made up almost entirely from matter, something must have happened that created this excess of matter, due to the fact that in the beginning of the universe an equal amount of matter and antimatter has been created. Some kind of process must interact with matter and antimatter differently and thus contribute to the matter-antimatter asymmetry as it can be observed today.

One important condition for the asymmetry (apart from a violation of the baryon number and interactions outside of the thermal equilibrium) is a source of CP-symmetry violation (This condition, as well as the others mentioned, has been proposed by Andrei Sakharov in 1967, see [17]). A CP-violating process has different results when all particles are exchanged with their antiparticles. The required process for the matter-antimatter asymmetry thus has to violate the CP-symmetry to produce more matter than antimatter. In the SM, the electromagnetic and the strong interactions are unaltered by a CP-conjugation. In contrast, the weak interaction does violate the CP-symmetry (and fulfills the requirement of violating the baryon number), but unfortunately this violation is not strong enough to fully explain the existing matter-antimatter asymmetry that can be observed in the universe [18].

Another process resulting in symmetry-breaking is needed. For nowadays research in particle physics, unraveling the secret behind this process is of high importance as it would help in understanding the universe as it is now. This process can be searched, for example, in top- Higgs interactions.

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1.3. Higgs Boson

In July 2012 the Higgs boson, previously predicted by the electroweak symmetry breaking mechanism, was discovered by both ATLAS [8] and CMS [9] collaborations. The Higgs particle, as part of the Higgs mechanism, a theorie proposed in the 1960s, is responsible for the mass of all elemental particles. For those particles, a mass can be experimentally measured, but not explained in a particle model without the Higgs mechanism. The SM includes the Higgs sector, but it took a long time from the prediction until the Higgs boson could finally be observed in an experiment, as particle accelerators and colliders first had to reach high enough energy and luminosity. The Higgs boson was measured with a mass of 125.09±0.24 GeV [16] and matched the properties predicted by the SM of being a neutral boson with spin 0. Since the discovery, the properties of the Higgs particle have been further examined.

1.4. Overview of this work

The goal of this work is to analyze various observables on their ability to detect a CP-violation in atth¯ generation process. BSM models, such as supersymmetry, where the Higgs boson has no definite CP quantum number, result in a Yukawa coupling having two components, one CP- even and one CP-odd. This gives the possibility of having a CP-mixed as well as both a full CP-even and CP-odd coupling.

Using this approach, processes with different behaviours under a CP-transformation can be generated and analyzed with the observables. Note that this is just a method to implement a CP-dependant process to evaluate different observables, but it is not the only possible source of CP-violation that could be detected with those observables.

Out of all fermions, the top quark is expected to have the largest Yukawa coupling to the Higgs boson. Therefore, the associated production of the Higgs particle with a top quark pair (tth) is of interest, as it allows for a direct measurement of the¯ thYukawa coupling and is sensitive to its CP nature. In this work, the dileptonic final state of tth¯ with the Higgs particle decaying throughh→bb¯ is analyzed, which preserves the spin information from thet-quarks in the leptons. The Feynman diagram for this process is shown in f ig.1

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Figure 1: Feynman diagram showing the processes analyzed in this work (taken from [15] with a small change).

In this imageA^{ab,i j} represents all different processes which result in the shown output.Fig.2 and f ig.3 show the individual processes contributing to thetth¯ production.

(a) Quark annihilation

(b) gluon annihilation

Figure 2: s-channel Feynman diagrams for thetth¯ production. Two more can be created when the Higgs boson is emitted from the othert-quark [15].

Figure 3: t-channel Feynman diagrams for the tth¯ production. Three more can be created by exchanging the gluon lines in the t-channel diagrams [15].

On tree level, there are ten diagrams of relevance. The tth¯ output that is desired can be produced by quark annihilation as well as by gluon fusion. Fig.2ashows an example for a quark annihilation process, where two quarks (with the same flavour) annihilate into a gluon, which

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then decays into a pair of top quarks. The Higgs boson is emitted from either the top or antitop quark. Processes for gluon fusion are displayed in f ig.2b. There exist two s-channel processes similar to the quark annihilation one. In addition, one can construct t-channel diagrams (see f ig.3). The Higgs boson is emitted from any of the top or antitop quarks which results in six diagrams (three particles that can each emit the Higgs boson and the same three diagrams again with exchanged gluons). The main background for this process is pp→tt¯+jets, especially ttb¯ b¯ is an irreducible background that has to be taken into consideration.

For the investigation of the CP-nature of the top-Yukawa coupling, an effective lagrangian approach, seen ineq.1 was used.

L_t¯_th=−m_t

v (κ_ttt¯ +i¯tγ₅κ˜_tt)h (1) Here v=246 GeV is the SM Higgs vacuum expectation value, κ_t and ˜κ_t represent a scalar and pseudoscalar interaction, respectively.

With this approach different mixtures of full CP-even coupling, for κ_t =1 and ˜κ_t =0, and full CP-odd coupling, forκt=0 and ˜κt =1, can be examined.

2. Data generation

2.1. Event generation

The event generation was done using MadGraph5_aMCatNLO [2] with the HC_NLO_X0_UFO model [3]. MadGraph5_aMCatNLO is a framework that allows the generation of particle interaction processes, as well as individual events for those. The processes shown above have been generated in MadGraph as follows:

> import model HC_NLO_X0_UFO

> generate p p > x0 t t~ [QCD]

> output

First, the needed model to implement the desired approach for the Yukawa coupling is im- ported. Next, the general process is defined. pis any particle that can be found in a proton, so any ofg, u,d, c, s, ¯u, ¯d, ¯cand ¯s,x0 is the Higgs boson candidate. The expression[QCD]tells MadGraph to use quantum chromodynamics for its calculations.

The decay channels of the final particles have also been specified for MadSpin as follows:

> decay t > w+ b, w+ > lept lept

> decay t~ > w- b~, w- > lept lept

> decay x0 > b b~

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Here,lept was defined as any choice out ofe⁻,e⁺, µ⁻, µ⁺,v_e, ˜v_e, v_µ, ˜v_µ, v_τ and ˜v_τ, so any lepton or antilepton except forτ⁻ andτ⁺.

Using these commands, three different combinations forκ_t and ˜κ_t were chosen:

κ_t=1 and ˜κ_t =0 full CP-even case (SM)

κt=0.5 and ˜κt =0.5 equally mixed case

κt=0 and ˜κt =1 full CP-odd case

For each case processes were generated for a top mass of 173.0 GeV, a Higgs mass of 125.0 GeV and 14 TeV center-of-mass energy. To test whether the process generation worked as expected, the obtained cross sections were compared to the ones from the Higgs characteri- zation paper [12] for the 13 TeV LHC intable1.

13 TeV [fb] 14 TeV [fb] increase CP-even (0⁺) 525.1(7)±2.1 % 564.1±0.8 1.074±0.021 CP-mixed (0^±) 374.1(5)±2.5 % 401.3±0.6 1.072±0.025 CP-odd (0⁻) 224.3(3)±3.2 % 245.4±0.4 1.094±0.033

Table 1: Cross sections of three different models used in this analysis, for the 13 TeV LHC (from [12]) and the 14 TeV LHC (from this analysis). The third column shows the quotient of those values.

The obtained cross sections show a small increase in comparison to the values from [12]. An increase of around 16 % would have been expected when increasing the energy from 13 TeV to 14 TeV. The obtained increase is lower than expected, but the same (inside the error ranges) for each model. The lower increase can be most likely explained by the usage of slightly different versions of the detector simulation, different parton density functions and different choices of scales.

For each individual model, 400.000 events were generated. At this point, they exist as Les Houches events which have neither hadronized nor run through a detector simulation. For parton showering and hadronization, Pythia8 [1] with the Monash tune [19] was used. Pythia8 is a program developed for the generation of high-physics events. It is capable of showering Les Houches events generated with MadGraph. Pythia8 can be run inside Delphes [10] which was used for the detector simulation. The FastJet package [7] was used for the jet clustering.

The CMS Phase II detector (see next section for a short overview) was simulated using 200 pile up (the pile up value has to be given for high-luminosity colliders, as it is possible that a single particle bunch crossing produces more than one detector event, which is described with this value).

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2.2. The CMS detector

The Compact Muon Solenoid (CMS) experiment is a particle detector at the LHC at CERN.

The main feature of the CMS is a superconducting solenoid (13 m long, 6 m in diameter) which creates an axial magnetic field of 3.8 T. It is suited up with several particle detection systems.

The most central part is the pixel detector which is surrounded by a strip tracker. In those detectors, charged particle trajectories can be measured in a range of 0≤φ ≤2π in azimuth and |η|<2.5 in pseudorapidity. The electromagnetic calorimeter, which is used to measure the energy of particles with low masses, surrounds the previous parts. It is complemented by the hadron calorimeter which measures the energy of the hadronic components of the decay products. The last radial part of the CMS are the muon chambers, measuring the momentum of, as the name suggests, muons. As the detector is built cylindrical around the beam line, on both ends there is another calorimeter to be able to measure particles with a high pseudorapidity.

Fig.4 shows a slice of the detector showcasing the mentionned parts.

Figure 4: A slice of the CMS detector, showing the discussed detector parts [5].

3. Observables on generation level

In the examined process, the spins of the original top and antitop quarks are correlated. When one of the top quarks now emits a Higgs boson (see f ig. 2 and f ig.3 for reference) the spin correlation changes if the Higgs boson is pseudoscalar. This is not the case for the scalar Higgs boson. Therefore, an evaluation of the spin information in the top quarks is expected to be sensitive on the CP-coupling in the process. In addition, one can expect that the spin information

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A number of different observables were examined on their potential to distinguish between the different cases of thethCP coupling through the induced spin change. In the following two sections those observables are presented.

In this section observables examined for events involving generated partons are shown. This means that no cuts have been applied to the used events and the data used to calculate the individual observables is data from the partons in the event generation. These observables show the expected results if every property of every particle from the process would be known exactly.

Of course, this is not the case for an analysis of real data, but it is still useful to evaluate the different observables. Onlytth¯ signal events are considered.

The used observables are grouped thematically into “basic observables”, “triple product observables” and “angular observables”.

3.1. Basic observables

The expression “basic observables” summarizes all used observables which can be calculated directly from the basic properties of particles in the process. These observables presented here have also been evaluated in [11]. Fig.5ashows the first example of a basic observable. Here, the differential cross section for the tth¯ production is shown as a function of the transverse momentum of the Higgs boson. The errors in this diagram (as for all other observables displayed in this work) are the errors calculated from the event weights using the root functionSumw2().

(a) Higgs boson (b)t-quark

Figure 5: Distribution of the transverse momentum of the Higgs boson and the top quark for the three different models used in this work.

In f ig. 5b, f ig. 6aand f ig. 6b additional distributions for basic observables are displayed.

For each observable, the distribution was calculated for all three models and then normalized to exclude the cross section information and only leave a simple shape comparison. As one can see, there is a strong disparity between the estimated discrimination potential of different distributions. The transversal momentum of the top quark in the process shows no difference

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between the models at all (f ig.5b), whereas the pseudorapidity plot (for the top quark or Higgs boson, see f ig.6) shows a clearly different behavior for the different models. Compared so a SM Higgs boson, the full CP-odd Higgs boson tends to have a lower absolute value ofη (see f ig.6a), thus being produced more centrally. The CP-mixed case is, as expected, in the middle of the other two models. The associated top quark shows an inverted behavior: for the CP-odd case the top quarks are less central than for the SM case (see f ig.6b).

(a) Higgs boson (b)t-quark

Figure 6: Pseudorapitidy plot of the Higgs boson and the top quark.

The observed results can be explained physically. As explained above, the spin correlation changes when emitting a CP-odd Higgs boson, but not when emitting a SM Higgs boson. This change is visible in angular distributions in the process, which gives angular observables (which the pseudorapidity is as it evaluates the angle of the particle in respect to the beam line) the ability to detect the spin change and thus be sensitive on the CP-coupling in thethinteraction.

One can now construct additional distributions by considering the pseudorapidity difference of, for example, the top quark and the antitop quark (see f ig. 7a). This results in two visibly different shapes for the CP-even and CP-odd case and thus in an observable potentially very well suited for discriminating those cases. As discussed further below (see section5.3), it is generally easier to examine decayed particles (ideally final state particles). Thet and ¯t-quarks decay into a pair of b-quarks. The pseudorapidity difference can also be evaluated for those particles, as seen in f ig.7b.

In addition, the pseudorapidity difference can also be evaluated for other participants of the decay process, such as the two leptons (f ig.8) from the final state.

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(a)t-quarks (b)b-quarks

Figure 7: Pseudorapitidy difference between the two t-quarks and the two b-quarks, respectively.

Figure 8: Pseudorapitidy difference between the two leptons.

The last observable from basic properties is calculated as follows:

b₄= (p^z_t∗p^z_t_¯)/(|~p_t| ∗ |~p_t_¯|) (2) The product of p^z_t and p^z_t_¯ is divided by (|~p_t| ∗ |~p_t_¯|) to normalize it. p^z_t is the longitudinal momentum of thet-quark. This results in another observable with clearly different shapes for the different models, as seen in f ig.9. As this observable evaluates the particle momentum in one direction, it is sensitive on the spin correlation of thet-quarks.

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Figure 9: Distribution for observableb₄, seeeq.2.

An interesting aspect of this kind of observable is the forward-backward asymmetry. This value is defined as

A^Y_FB= σ(Y >0)−σ(Y <0)

σ(Y >0) +σ(Y <0) (3) with σ(Y >0) and σ(Y <0) beeing the total cross section withY above and below zero, respectively.Y is the value of any observable for any event. UsingY =b₄, the following values can be obtained:

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^b_FB⁴ 0.2737±0.0024 0.1066±0.0010 −0.1919±0.0020

Table 2: Forward-backward asymmetry with statistical errors

The same way one can see a clear difference between the plots, the asymmetry values obtained show a clear distinction between the three models and especially between the full CP- even and CP-odd case. The given errors are (as for all forward-backward asymmetries in this work) the propagated statistical uncertanties seen in the observable plots.

3.2. Triple product observables

For the next group of observables different triple product terms can be examined. This type of observables was implemented according to [15], see there for a detailed theoretical derivation as well as additional information beyond the shortened explanation below.

One can write the differential cross section of the process as follows:

dσ(pp→t(n_t)¯t(nt¯)H) =κ_t²f₁(p_i∗p_j) +κ˜_t²f₂(p_i∗p_j) +κ_tκ˜_t

15

∑

l=1

g_l(p_i∗p_j)ε_l (4)

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Here, p_i,_j are any of six four-vectors from the process, namely p_t, p_t_¯, q, Q, n_t and n_t_¯. The vectorsqandQare calculated from the initial gluon 4-momentaq₁andq₂as

q= 1

2(q₁−q₂) Q= 1

2(q₁+q₂).

n_tandn_t_¯are the so called spin vectors calculated from thet, ¯t,l⁻andl⁺momenta as follows:

n_t =−p_t

m_t+ m_t

(p_t∗p_l+)p_l⁺ n_t_¯= p_t_¯

m_t− m_t

(pt¯∗p_l−)p_l−

The variableε_l is a triple product (TP) calculated as ε_l=ε_{α β γ δ} p^α_a p^β_b p^γ_c p^δ_d

where the ε_{α β γ δ} is the Levi-Civita symbol with ε₀₁₂₃ =1 and p_a,b,c,d are four of the six vectors listed above.

The differential cross section is, according to eq.4, calculated from three terms, one being linear in κ_t², one linear in ˜κ_t² and one term is dependant onκtκ˜t. The functions f₁ and f₂ as well asg_l depend only on scalar products of two six momenta, and are therefore even under a parity transformation. In constrast, the TPs are odd under a parity transformation, thus it can be concluded that the total cross section is only dependant on f₁ and f₂. In addition to that, forward-backward asymmetries are only expected in the mixed case and not in the full CP-even or CP-odd cases, as for both those cases the term including the TPs, which is the only CP-odd term, disappears.

All triple products containing q will be excluded (asq cannot be expressed in terms of the momenta of final state particles). Also, only observables containing n_t and n_t_¯ are considered, because only observables containing decay products from botht and ¯t will be sensitive to the spin correlation between those two particles. This leaves just three triple products:

ε₁=ε(p_t,p_t_¯,n_t,n_t_¯) (5)

ε₂=ε(Q,p_t_¯,n_t,n_t_¯) (6)

ε₃=ε(Q,p_t,n_t,n_t_¯) (7) As an example, f ig.10 shows the distribution forε₂(the distributions forε₁andε₃look quite similar to the one forε₂and thus are not displayed here, seeappendix Afor the plots). Table3 shows the asymmetry for all three observables.

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Figure 10: Distribution forε₂.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB¹ −0.0017±0.0001 0.0616±0.0007 0.0044±0.0001 A^ε_FB² 0.0035±0.0000 0.0637±0.0006 0.0005±0.0000 A^ε_FB³ 0.0023±0.0001 −0.0593±0.0005 −0.0038±0.0001

Table 3: Forward-backward asymmetry forε₁,ε₂andε₃

Within the errors the asymmetries for the CP-even and CP-odd case are close to 0. The displayed errors are purely statistical, which might explain the remaining slight deviation from 0. For the mixed case, a larger asymmetry is obtained. The asymmetries for the CP-mixed case are close to the same for all three observables (except from a sign change for ε₃). This is the expected behaviour according to [15]. In addition to the asymmetry, the CP-mixed case seems to be more around higher values forε than the SM case, which is even stronger for the CP-odd one.

It is possible to combine the triple products (ε₁, ε₂, ε₃) to try to improve the results. An example is the following triple product:

ε₄=ε₃−ε₂=ε(Q,p_t−p_t_¯,n_t,n_t_¯) (8) Fig. 5a shows the resulting plot for the combined TP. Table 4 shows the corresponding forward-backward asymmetry.

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Figure 11: Distribution forε₄.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB⁴ −0.0027±0.0000 −0.0706±0.0007 −0.0041±0.0001

Table 4: Forward-backward asymmetry forε4. The asymmetry values can be improved in comparison toε₁,ε₂andε₃.

From ε₁, ε₂ and ε₃ one can extract additional triple product terms, not containing the spin vectors, but instead the lepton momenta. The spin vectors include the momenta of t and ¯t, which require a full momentum reconstruction of the process. If this can be avoided, the analysis would be easier for real data. Inserting the spin vector formulas intoε1,ε2andε3results in

ε(p_t,pt¯,n_t,nt¯) = m_t²

(p_t∗p_l+)(pt¯∗p_l−) ε(p_t,pt¯,p_l−,p_l⁺) (9) ε(Q,p_t_¯,n_t,n_t_¯) = m²_t

(p_t∗p_l+)(pt¯∗p_l−) (ε(p_t,p_t_¯,p_l−,p_l⁺) + ε(p_h,p_t_¯,p_l−,p_l+) + (p_t_¯∗p_l+)

m²_t ε(p_h,p_t_¯,p_t,p_l−))

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ε(Q,p_t,n_t,n_t_¯) = m²_t

(p_t∗p_l+)(pt¯∗p_l−) (−ε(p_t,p_t_¯,p_l−,p_l+) + ε(p_h,p_t,p_l−,p_l⁺) + (p_t_¯∗p_l−)

m²_t ε(p_h,pt¯,p_t,p_l⁺))

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From these formulas another three TPs can be obtained:

ε₅=ε(p_t,p_t_¯,p_l−,p_l+) (12)

ε₆=ε(p_h,p_t,p_l−,p_l⁺) (13)

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ε₇=ε(p_h,p_t_¯,p_l−,p_l+) (14) Again, only triple products containing decay products from bothtand ¯tare considered, which leaves only the TPs with bothl⁺andl⁻. These TPs do not contain the spin vectors, but still contain thet-quark 4-momenta. Fig.12 shows the distribution ofε₅ as an example. The forward- backward asymmetry values for all three TPs can be seen intable5.

Figure 12: Distribution forε₅.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB⁵ 0.0020±0.0001 −0.0616±0.0004 −0.0042±0.0000 A^ε_FB⁶ −0.0056±0.0001 0.0217±0.0001 0.0024±0.0001 A^ε_FB⁷ −0.0021±0.0000 −0.0206±0.0000 0.0020±0.0000

Table 5: Forward-backward asymmetry forε₅,ε₆andε₇.

The asymmetry forε₅is very close to the one forε₁. This is expected, aseq.9 shows that the two TPs are related by a positive definite proportionality factor which has no influence on the asymmetry. This is in constrast toε₂andε₃, as the asymmetries for the TPs derived from those are a lot weaker.

As for ε₁, ε₂ andε₃, one can construct a combined triple product to improve the results, as shown here:

ε8=2ε₅−ε₆+ε7=ε(p_t+pt¯+p_h,p_t−pt¯,p_l⁺,p_l−) (15) Fig.13 shows the plot for this observable andtable6 displays the asymmetry values.

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Figure 13: Distribution forε₈.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB⁸ 0.0003±0.0000 0.0644±0.0005 0.0008±0.0000

Table 6: Forward-backward asymmetry forε₈.

The combined observable yields a better result than the individual TPs, as it was the case for ε₄in comparison toε₁,ε₂andε₃.

Again, as the momentum of the top quark is not easy to obtain from the detector event for real data, it is substituted by the momenta of theband ¯bquarks, respectively. This results in the triple product

ε9=ε(p_b+pb¯+p_h,p_b−pb¯,p_l⁺,p_l−) (16) which finally contains neither the t or ¯t momenta nor the spin vectors. The distribution is shown in f ig.14 and the forward-backward asymmetry intable7.

Figure 14: Distribution forε₉.

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CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB⁹ −0.0007±0.0000 0.0401±0.0002 −0.0033±0.0000

Table 7: Forward-backward asymmetry forε₉.

The substitution of the t and ¯t momenta with the respective b and ¯b momenta lowers the asymmetry in comparison toε₈, but still shows a clear difference between the mixed case and the full CP-even or CP-odd cases.

Last, fromε₄ one can construct another observable. If the definitions of the spin vectors are inserted into the formula forε₄, one derives

ε₄= m_t²

(p_t∗p_l+)(pt¯∗p_l−)ε(Q,p_t−p_t_¯,p_l−,p_l+) + 1

(p_t∗p_l+) ε(Q,p_t,p_l+,p_t_¯)− 1

(p_t_¯∗p_l−)ε(Q,p_t_¯,p_t,p_l−) As a constant factor does not matter when evaluating an asymmetry, by dividing through

m_t²

(pt∗p_l+)(p¯t∗p_l−) the following formula is obtained:

ε(Q,p_t−p_t_¯,p_l−,p_l⁺) + (pt¯∗p_l−)

m_t² ε(Q,p_t,p_l⁺,p_t_¯) − (p_t∗p_l⁺)

m²_t ε(Q,p_t_¯,p_t,p_l−) By replacingp_tandp_t_¯with their visible contributions (p_b+p_l+ andp_b_¯+p_l−) andQwith ˜Q= (p_b+p_l++p_b_¯+p_l−)/2 as well as defining the constantsw₁= (pt¯∗p_l−)/m_t²,w₂= (p_t∗p_l+)/m²_t andc_b_b_¯ = (1−w₁)p_b − (1−w₂)p_b_¯ the following expression for observableε₁₀ is constructed:

ε10=ε(Q,˜ c_bb¯,p_l−,p_l⁺)−w₁ε(Q,˜ p_b,pb¯,p_l⁺) +w₂ε(Q,˜ p_b,pb¯,p_l−) (17) Fig.15 andtable8 show the results for this observable.

Figure 15: Distribution forε₁₀.

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CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB¹⁰ 0.0058±0.0001 −0.0371±0.0003 −0.0011±0.0000

Table 8: Forward-backward asymmetry forε₁₀.

In comparison to ε4, the asymmetry is weaker. Still, the observable shows a decent discrimination between the mixed and the full cases and has the benefit of not containing the top momenta. In comparison to the other observable fulfilling this criteria,ε₉the asymmetry seems to be smaller as well, leavingε₉as the better observable without top momenta for this analysis, in constrast to the results observed in [15].

In general, it can be noted that the TP observables show no discrimination potential between the full CP-even and CP-odd cases through forward-backward asymmetries, as expected. There seems to be a tendency towards higher values for the TPs for a stronger CP-odd coupling, this will be further examined in section 5. The asymmetry shows a clear difference between the full cases and the mixed case, as only the mixed case has an asymmetry. This makes the TP observables especially interesting, as they are sensitive to CP-mixed couplings, which are relevant as the full CP-odd case has been excluded at 99.98 % confidence level [13].

The sensitivity of the observables to this has been analyzed and it has been shown that it is possible to construct observables not depending on the top momenta that are still sensitive to the forward-backward asymmetries.

3.3. Angular observables

3.3.1. Basic angular observables

The last set of observables that have been examined are different angular observables. Those are of interest as they can measure the spin information in the observed particles directly through the altered angular distributions resulting from the change in the spin correlation, as explained above. An example for a simple angular distribution is shown in f ig.16a.

(a)t-quarks (b)b-quarks

Figure 16: Distributions of the cosine of the angle between the two t-quarks and the two b- quarks, respectively.

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Figure 17: Distribution of the cosine of the angle between the two leptons.

The differential cross section is plotted as a function of the cosine of the angle between the twot-quarks. The different shapes of the distribution for the different models are clearly visible, and not only for thet-quarks, but also for theb-quarks (f ig.16b) and the leptons (f ig.17) being the decay products of thet-quarks. Those decay products seem to conserve the spin information from the top quarks.

As for the TP observables, the forward-backward asymmetry is examined for the angular observables. The resulting values are displayed intable9.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^cos_FB^θ(t^,¯^t) −0.0333±0.0003 −0.1215±0.0013 −0.2869±0.0030 A^cos_FB^θ^(b^,^b)^¯ 0.0347±0.0003 −0.0306±0.000 31 −0.1524±0.0015 A^cos_FB^θ^(l⁺^,l⁻⁾ 0.1010±0.0010 0.0043±0.0000 −0.1519±0.0015

Table 9: Forward-backward asymmetry for the angle between the t-quarks, the b-quarks (decaying from thet-quarks) and the two leptons.

The clearly visible asymmetry in the plots is also visible in the asymmetry values. These basic angular observables have also been presented in [11].

3.3.2. Other angular distributions

Another way of examining the angular distributions (and thus the spin information) in the process is shown in this section. For the second group of angular observables, three angles are defined:

θ₁¹²³: Angle between (123) in lab frame and 1 in (123) rest frame θ₃²³: Angle between (23) in (123) rest frame and 3 in (23) rest frame

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θ₄³: Angle between 3 in (23) rest frame and 4 in 3 rest frame

The numbers 1 to 4 denote different particles from the process. Particles 1, 2 and 3 are any permutation oft, ¯tand h, particle 4 is any decay product of one of those. Different combinations of two of those angles are evaluated using different combinations of sine and cosine according to the following formulas:

trig₁(θ₁¹²³)trig₂(θ₄³) (18)

trig₁(θ₁¹²³)trig₂(θ₃²³) (19)

trig₁(θ₃²³)trig₂(θ₄³) (20) Heretrig_1,2 is either sinor cos. This construction allows for a large amount of options for different observables. To start with, a number of distributions using eq.20, withtrig_1,2=cos, have been examined. ¯thas been used as particle 2,has particle 3. For particle 4, different decay products have been used. This gives the following formula:

cos(θ_h^th^¯ )cos(θ₄^h) (21)

As an example, f ig.18 shows the distribution forl⁺ and ¯bt¯as particle 4.

(a) Particle 4:l⁺ (b) Particle 4: ¯bt¯

Figure 18: Plots for the angular distribution explained above using different particles as particle 4.

Two different shapes are obtained for the different decay products, both showing an angular asymmetry between the models. More figures using other choices (l⁻,b_t,b_h, ¯b_h,W⁺ andW⁻) for particle 4 can be found in appendix A, as they have a similar shape as either of the plots shown in f ig.18. Again, forward-backward asymmetries are evaluated.

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CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^4=l_FB⁺ 0.3607±0.0026 0.3297±0.0026 0.2733±0.0024 A^4=l_FB⁻ 0.1330±0.0011 0.1053±0.0010 0.0678±0.0009 A^4=b_FB ^t 0.1533±0.0013 0.1289±0.0013 0.0844±0.0011 A⁴⁼_FB^b^¯^¯^t 0.3830±0.0029 0.3522±0.0029 0.2923±0.0026 A^4=b_FB ^h 0.1314±0.0010 0.1095±0.0011 0.0672±0.0009 A⁴⁼_FB^b^¯^h 0.3619±0.0026 0.3328±0.0027 0.2749±0.0024 A^4=W_FB ⁺ 0.1892±0.0011 0.1287±0.0008 0.0205±0.0000 A^4=W_FB ⁻ 0.1929±0.0012 0.1278±0.0007 0.0356±0.0002

Table 10: Forward-backward asymmetry foreq.21 using different particles as particle 4.

All choices for particle 4 show a certain sentitivity to the different models in the forward- backward asymmetries. For these angular observables (as well as the observable b₄ shown above) see [14] and [13] for reference.

As stated above, a lot of different combinations can be evaluated. Five examples (apart from ones created usingeq.21) are shown here (see [4] for comparison):

c₁=sin(θ_h^t¯^th)sin(θ_¯^t^¯

bt¯) c₂=sin(θ_h^t^th^¯ )cos(θ_b^t^¯

t) c₃=sin(θ_t^t¯^th)sin(θ_W^h+)

c₄=sin(θ_t_¯^t¯^th)sin(θ_b^h

h) c₅=sin(θ_h^t¯^th)sin(θ_t_¯^t¯^t)

(a)c₁ (b)c₂

Figure 19: Plots of the distributions of observablesc₁andc₂.

Fig.19aand f ig.19bshow two of the observables mentioned (the remaining graphics can be found inappendix A). Note that for observablec₁, in contrast to other angular observables, only

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is also true for observablesc₃,c₄andc₅. The observables seem to only have a weak sensitivity to the CP-coupling.

4. Observables after acceptance cuts and using detector data

The last section presented observables which were evaluated using parton data of all generated events. To now examine the observables under realistic conditions, simulated events were used for generated hadrons, a detector simulation was applied using the simulation framework Delphes [10], and an event selection was performed applying selection cuts. The DAnalysis Framework for ROOT [6] was used for both event selection and the calculation of all observables. As insection3, onlytth¯ signal events are considered.

Data after acceptance cuts has then been analyzed on parton level (but with cuts using detector simulation data) to use the particle properties that could be obtained by a momentum reconstruction in a real analysis. In addition, detector data for the leptons as well as for the jets was evaluated, as this data can be obtained directly from the detector.

4.1. Event selection

After the generation process, 400.000 events for each model are available for the analysis.

Acceptance cuts must be applied to the events to exclude events that could not be identified astth¯ events in a real analysis. The first requirement was the presence of exactly two leptons having a p_T >20 GeV and |η|<2.5 each. The second condition for each event was at least four jets and at least two b-tagged Jets, each having p_T >30 GeV and |η|<2.5. These cuts exclude a percentage of events for each model as seen intable11.

total Events after lepton cuts after jet cuts CP-even (0⁺) 400 000 210 367 195 013 CP-mixed (0^±) 400 000 207 146 191 172

CP-odd (0⁻) 400 000 201 643 184 689

Table 11: Event count after different steps of the event selection

As one can see, the selection leaves a little below half of the data from each model for further analysis.

The same three categories as insection3 are used to group the observables.

4.2. Basic observables

For the basic observables, plots for data with applied cuts have been generated. As one example, f ig.20ashows the difference of the pseudorapidity of the two top quarks.

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(a) with cuts (b) without cuts

Figure 20: Pseudorapitidy difference between the two top quarks for data with and without acceptance cuts, using data from the generated partons.

The plot shows only little difference to the one fromsection3.1 generated without any cuts (shown again in f ig. 20bfor comparison). The only thing visible from the graphics is a little worse statistics (as expected for plotting with less events). Apart from that, it seems that applying acceptance cuts to the data does not interfere with this observable. The same is true for all other basic observables usingh,t, ¯t,bor ¯bproperties. Images can be found inappendix A.

As before, particles are replaced with their decay products. Insection3.1, thet-quarks were substituted with their associated b-quarks. Using detector data, those can now be substituted again, by using the jets that are forming in the detector. From all jets (fullfilling the selection criteria mentionned above), two were matched to the twob-quarks decaying from thet-quarks.

To match a jet to a quark, the following formula is used:

∆R= q

∆φ²+∆η² (22)

Here, ∆φ is the difference between φ of any jet and φ of any b-quark. Similar, ∆η is the difference between the η of the jet and theb-quark. The value∆Ris minimized, the jet quark combination which gives the minimal value is matched. In a real analysis, as both the top quarks and the Higgs boson decay into bottom quarks in the examined process, it is not easy to chose the b-quarks decaying from the top quarks (and thus find the corresponding jets) as the information about theb-quarks parent particles is not available. In addition, even the differentiation between b and ¯b-jets is a problem that has to be solved in a real analysis (for example using jet charge measurement). In this analysis, the information about the parent particles of b and b-quarks are taken from the simulation to allocate the right¯ b-jets to their parent top/antitop quark or Higgs bosons.

For the pseudorapidity difference between the jets, one recieves the distribution shown in f ig.21.

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Figure 21: Pseudorapidity difference between the jets decaying from the b-quarks originating from thet-quarks.

The jets seem to conserve the information from their respective particles and thus are also a candidate for analysis.

Detector data can not only be used to replace quarks with jets, but also for the lepton observables. Fig.22a and f ig. 22b show the pseudorapidity difference between the leptons for generation and detector data (both after acceptance cuts have been applied, which were calculated using detector data).

(a) Generation data (b) Detector data

Figure 22: Plots for the pseudorapidity difference between the two leptons for parton data from generation and for detector data.

As one can see, the information included in the parton level observable from the generation carries over to the detector data very well, making this observable a very promising candidate for detecting CP-violation in tth-events. In contrast to the jets, the correct leptons are easy to¯ identify and their charge can be easely measured. No momentum reconstruction is required.

This makes the lepton observables a lot better than the jet observables for a real analysis.

The last basic observable isb₄, which was also evaluated after acceptance cuts.

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Figure 23: Observableb₄after acceptance cuts.

As previously done, the forward-backward asymmetry can be calculated for observable b₄. The values for data with applied cuts are shown intable12.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^b_FB⁴ 0.2472±0.0033 0.0901±0.0013 −0.2225±0.0038

Table 12: Forward-backward asymmetry using parton data after acceptance cuts.

The cuts do not have a too strong effect on the asymmetries which still clearly distinguish the different cases.

4.3. Triple product observables

As for the basic observables, also the TPs have been calculated using data after acceptance cuts.

As the images do not change much, they are only displayed inappendix A.

For each of the observables, the forward-backward asymmetry has been calculated and is displayed intable13. These values use only information from the generated partons, but events have been selected using detector data (as explained above).

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CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB¹ −0.0050±0.0003 0.0553±0.0010 0.0138±0.000 35 A^ε_FB² 0.0037±0.0002 0.0599±0.0007 0.0078±0.0002 A^ε_FB³ 0.0108±0.0003 −0.0564±0.0007 −0.0046±0.0001 A^ε_FB⁴ 0.0012±0.0000 −0.0619±0.0008 −0.0109±0.0003 A^ε_FB⁵ 0.0052±0.0001 −0.0552±0.0005 −0.0132±0.0002 A^ε_FB⁶ −0.0060±0.0000 0.0284±0.0001 0.0049±0.0003 A^ε_FB⁷ −0.0050±0.0002 −0.0229±0.0001 −0.0003±0.0001 A^ε_FB⁸ −0.0073±0.0001 0.0676±0.0008 0.0062±0.0002 A^ε_FB⁹ −0.0089±0.0001 0.0400±0.0001 −0.0008±0.0000 A^ε_FB¹⁰ 0.0077±0.0001 −0.0318±0.0003 −0.0059±0.0001

Table 13: Forward-backward asymmetry after acceptance cuts using parton data for all particles.

As a next step, the detector values for the leptons replace the values from the generation data. This gives additional forward-backward asymmetry results as shown in table 14. All other momenta, as they would require a momentum reconstruction, are still obtained from the generated partons.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^ε_FB¹ −0.0050±0.0003 0.0554±0.0010 0.0102±0.0003 A^ε_FB² 0.0013±0.0001 0.0554±0.0006 0.0098±0.0003 A^ε_FB³ 0.0103±0.0003 −0.0526±0.0007 −0.0006±0.0000 A^ε_FB⁴ 0.0015±0.0000 −0.0599±0.0008 −0.0089±0.0003 A^ε_FB⁵ 0.0050±0.0001 −0.0555±0.0006 −0.0100±0.0001 A^ε_FB⁶ −0.0035±0.0000 0.0275±0.0002 0.0022±0.0001 A^ε_FB⁷ −0.0062±0.0002 −0.0194±0.0000 0.0026±0.0000 A^ε_FB⁸ −0.0080±0.0002 0.0630±0.0008 0.0059±0.0003 A^ε_FB⁹ −0.0073±0.0000 0.0395±0.0001 0.0009±0.0001 A^ε_FB¹⁰ 0.0041±0.0001 −0.0325±0.0004 −0.0013±0.0001

Table 14: Forward-backward asymmetry after acceptance cuts, using detector data for the leptons.

The values show the same behaviour as the other observables. For data after acceptance cuts the asymmetries are a little weaker than for uncut data. The same is true when replacing the lepton momenta from generation with the momenta obtained from detector data. The asymmetries change a little, but the observables still seem to be sensitive on the CP-coupling.

4.4. Angular observables

4.4.1. Basic angular observables

For the basic angular observables evaluating the angles between the t or b-quarks, as for the basic observables, most plots look very similar to the ones made using generator data without acceptance cuts except for, as above, a little worse statistics after cuts have been applied.

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Angular distributions can be examined for detector data as well, using the two jets decaying from theb-quarks (which come from thet-quark decay) or the two leptons. Fig.24 shows the comparison between generation data and detector data (both after acceptance cuts using detector data) for the angle between the leptons, f ig.25 shows the angular distribution of the jets (again, despite the jets being visible in the detector parton data is still needed to correctly match the jets, as explained insection4.2).

(a) Generation data (b) Detector data

Figure 24: Plots for the angle between the two leptons for parton data from generation and for detector data, both after applied acceptance cuts.

Figure 25: Plot for the angle between the two jets decaying from the b-quarks (from the t- quarks).

Again, neither the acceptance cuts nor the usage of detector data have a strong negative influence on the results regarding the leptons and also the angle between the jets shows a similar shape as the one between thet orb-quarks and thus shows a sensitivity to the CP-coupling in

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4.4.2. Other angular observables

For all other angular observables that have not been discussed for data after acceptance cuts and using detector data (section3.3.2), the changes are similar to the ones shown above.

Applying acceptance cuts has no strong effects on the distributions except for, as above, the worse statistics. The same is true for using detector data for the leptons instead of generation data, which leaves the observables mostly unchanged. For this reason, images are not shown here (but can be found inappendix A).

Table 15 shows the asymmetries for the other angular observables using generation data after acceptance cuts.Table16 shows the asymmetry values using detector data for the leptons instead (for all distributions that use lepton momenta). Again: all these obserbles use acceptance cuts made according to detector data.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^4=l_FB⁺ 0.3795±0.0042 0.3495±0.0043 0.2910±0.0040 A^4=l_FB⁻ 0.1251±0.0017 0.1054±0.0017 0.0700±0.0016 A^4=b_FB ^t 0.1600±0.0021 0.1441±0.0023 0.1005±0.0021 A⁴⁼_FB^b^¯^¯^t 0.3954±0.0046 0.3620±0.0046 0.3017±0.0043 A^4=b_FB ^h 0.1299±0.0016 0.1120±0.0016 0.0804±0.0017 A⁴⁼_FB^b^¯^h 0.3759±0.0041 0.3484±0.0043 0.2884±0.0040 A^4=W_FB ⁺ 0.2000±0.0017 0.1418±0.0012 0.0471±0.0003 A^4=W_FB ⁻ 0.2056±0.0019 0.1456±0.0013 0.0587±0.0007

Table 15: Forward-backward asymmetry foreq.21 using different particles as particle 4. These values were calculated using events after acceptance cuts.

CP-even (0⁺) CP-mixed (0^±) CP-odd (0⁻) A^4=l_FB⁺ 0.3770±0.0043 0.3460±0.0017 0.2899±0.0041 A^4=l_FB⁻ 0.1232±0.0017 0.1016±0.0017 0.0650±0.0015

Table 16: Forward-backward asymmetry foreq.21 using different particles as particle 4. These values were calculated using events after acceptance cuts and detector data for the leptons.

The asymmetries for data after acceptance cuts do not change significantly in comparison to the ones for data without cuts. The values for the individual observables still show a difference between the CP-even, CP-mixed and CP-odd models.

This does not change when using detector data for the lepton momenta, the observables still seem sensitive to the CP-coupling in the process.

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5. Evaluation of observables

The last two chapters gave an overview on the observables used in this work and showed results in the form of plots and values for the forward-backward asymmetries. In this chapter, another method of evaluating and comparing all observables is presented. It only consideres the full CP-even and CP-odd case, as for most observables these cases are the most different. This comparison can give a general impression on how useful an observable is, as it can discard observables that are not able to even distinguish the two extreme cases.

5.1. Log likelihood comparison

To put a number onto the discrimination potential of the observables, a log likelihood comparison is used. For an event having the value x for an observable, the likelihood L_idescribes the probability that the event has been generated (either by simulation or a real process in a detector) with the modeli. For a normalized distribution, as it is the case for all observables in this work, this is given by the height of the corresponding bin.

To compare two models, the ratio ofL₀⁺ andL₀−is used, so in case of normalized histograms, the ratio of the bin heights. To simplify the handling with this value for further use below, the natural logarithm is used, as well as an additional constant, resulting in the finally used value

L=−2ln(L₀+

L₀−) (23)

To compare two distributions with this value, toy datasets are generated. A toy dataset contains a certain number of eventsN, for which the likelihood of being in modelaorbis calculated. Then, the product of all individual likelihood values for a model is taken asL₀+ (orL₀−) to obtain the likelihood values for the dataset. For a simpler calculation, the natural logarithm allows to just sum up about the individual values forL, as seen ineq.24.

L_avg=−2ln(∏^N_n=1L₀+(n)

∏^N_n=1L₀−(n)) =

N n=1

∑

−2ln(L₀⁺(n) L₀−(n)) =

N n=1

∑

L(n) (24)

For each event of a toy dataset, a random x-value for the observable must be chosen. This value is not chosen uniformly over the range of the observable, but is instead chosen to match the distribution of the observable for a certain model. As an example, this means that for the p_T(t)observable (see f ig.27a), more events are generated having low values for x than events having a higher value.

This process is executed for both compared models. A number of toy datasets, each matching the distribution of the respective model, are created and L_avg is calculated for each of them.

Plotting these values results in two Gaussian looking distributions, one for each model. Those distributions are then fitted with a Gaussian ansatz having the free parametersµ_i(mean) andσ_i (standard derivation). Fig.26 shows an example for the p_T(t)observable using 20 events per

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Figure 26: Example for a log likelihood comparison plot, showing a distribution for each model, which were fitted with a Gaussian ansatz.

The further away the curves are from each other the better the separation power of the observable. This separation powerσ can be calculated using

σ= (µ₀⁺−µ₀⁻) q

σ₀²++σ₀²−

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It is important to note that in this work no background has been taken into consideration while evaluating the observables. Therefore, the obtained results are not absolute, as for a regular hypothesis test the background has to be used as well. This would, for all observables, weaken the distinguishing potential (but not by the same amount for all observables) and require a lot more data to reach the same values forσ. To use the test as explained above, a perfect reduction of all backgrounds would be needed, leaving only signal events (as it is the assumption in this analysis which uses no background data).

5.2. Evaluation results

For each observable listed above a log likelihood comparison has been executed with the same values as in the explanation insection5.1, so 1000 toy datasets with 20 events per toy. The following figures show the generated distributions with the fitted Gaussian functions for a selection of those observables.

Probing CP-Violation in Top-Higgs Interactions