Single-top quark production as probe for physics beyond the Standard Model using Effective Field Theory

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BACHELOR THESIS

Single-top quark production as probe for physics beyond

the Standard Model using Effective Field Theory

by

Nick Boon, 10504230

University of Amsterdam

Faculty of Science (FNWI)

Report Bachelor Project Physics and Astronomy, size 12 EC,

conducted between 03-04-2017 and 30-06-2017

at

ATLAS group

Supervisor:

Second Assessor:

dr. M. Vreeswijk

dr. I.B. van Vulpen

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Summary

The Standard Model (SM) predicts with great accuracy, but it is still incomplete. It provides no complete ex-planation for the matter-antimatter asymmetry that is observed [12], [13]. In this thesis, Effective Field Theory is applied to t-channel single-top production, going beyond the SM.

The top quark is special, because its mass is close to the ElectroWeak Symmetry Breaking (EWSB)-scale, al-lowing it to weakly decay. The weak interaction only couples left-handed, introducing spin-correlations between the top and its decay products. These correlations can be studied by analyzing specific angular distributions.

Angles are defined that have proven to be sensitive to top polarization in other works [4], [13], [16]. The distribution for each angle is plotted for a sensible range of the coupling constant, calculated from limits of ear-lier results from ATLAS [2]. To make the results more representative of experimental data, an event selection is added that discard events that cannot be measured in a particle detector.

Effective Field Theory allows one to add higher-dimensional operators to the Standard Model. These oper-ators are suppressed by a certain energy scale. EFT implies these higher dimension operoper-ators are not directly

measurable, but only indirectly at lower energies. This energy scale equals 1013_{GeV for dimension-5 operators,}

which is currently unattainable, but is 1 TeV for dimension-6 operators. For this reason, only dimension-6 oper-ators are discussed. In EFT, the Standard Model cross-section changes to a second-order polynomial equation in coupling coefficient c. It is possible to isolate every term from the SM- and ±c-distribution. This allows one to recombine the terms for arbitrary c as is verified here.

In cursory analysis, the CtW-operator had the most dramatic effect by even flipping the sign of the slope of

some distributions compared to the Standard Model. This motivated further analysis into this specific operator. The results before event selection show significant effects, but after event selection only insignificant devi-ations from the Standard Model remain, even with the SM-background and non-statistical errors set to zero.

The data had a luminosity of ∼7.15 fb−1 compared to the 40 fb−1 the LHC produced over last year [19], so

improvements can be made here. Three angles, χt_l, χt_b and χtw, are most promising with relatively large

signifi-cance for tested values of c.

In future research the luminosity should be increased to actual LHC-values and the SM-background should be taken into account to obtain more representative results to guide future experiments.

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Popular summary (dutch)

De huidige elementaire deeltjes in het Standaard Model. Afbeelding van https://en.wikipedia. org/wiki/Elementary_particle.

De deeltjesfysica houdt zich niet alleen bezig met de ele-mentaire deeltjes (hiernaast afgebeeld), maar ook met

de mogelijke interacties tussen al deze deeltjes. In

de loop der jaren is alle kennis bijeengekomen in

het Standaard Model. Hierin kunnen, met enkele

tientallen parameters, alle tot nu toe bekende moge-lijke interacties en elementaire deeltjes worden beschre-ven.

Het Standaard Model verklaart echter niet alle

ob-servaties die worden gedaan. Het biedt

bijvoor-beeld geen verklaring voor de grote aanwezigheid van

materie in vergelijking met antimaterie.

Antimate-rie lijkt in alle opzichten op gewone mateAntimate-rie, maar is in veel eigenschappen precies tegenovergesteld aan

normale materie. Als de materie positief

den is, is het bijbehorende anti-deeltje negatief

gela-den. Bij het samenkomen van materie en

antimate-rie vindt er annihilatie plaats en blijft er slechts energie over.

Het Standaard Model voorspelt hetzelfde gedrag voor materie en antimaterie; slechts bij enkele specifieke reacties is een kleine afwijking mogelijk waarbij CP-schending kan

optreden. Hierbij wordt het behoud van lading (charge) ´en

pariteit (parity), een spiegelsymmetrie dat verschilt tussen materie en antimaterie, geschonden. Bij deze reacties kan materie en antimaterie verschillend reageren. De bekende reacties zijn echter niet voldoende om de volledige

ongelijkheid te verklaren. Om een complete verklaring te vinden zal moeten worden gekeken naar theorie¨en die

voorbij het huidige Standaard Model gaan.

In dit artikel is gekeken naar een theorie genaamd EFT, een afkorting voor Effective Field Theory. Hier

wor-den nieuwe interacties, die eigenlijk pas bij hele hoge energie¨en direct kunnen worden waargenomen, toegepast

op lagere energieën. Het is namelijk mogelijk dat deze interacties tóch op lagere energieën plaatsvinden, maar

dat deze verborgen zitten als alternatieve interactie in waargenomen reacties. Het is bijvoorbeeld mogelijk dat deeltje A via interactie 1 naar deeltje B vervalt, maar via interactie 2 is ook mogelijk met een hele kleine kans. Voor het eindresultaat maakt het niet uit met welke interactie A→B plaatsvindt, maar verdelingen van bepaalde variabelen kunnen wel veranderen. Door deze verdelingen te onderzoeken kan wellicht het bestaan van nieuwe interacties worden bevestigd en zo een verklaring worden gevonden voor het grote verschil in aanwezigheid van materie en antimaterie.

Er wordt specifiek gekeken naar de productie en het daaropvolgende verval van een enkel top-quark, het zwaarste elementaire deeltje dat momenteel bekend is, vanuit de botsing tussen twee protonen. De massa van het top-quark is zo groot, dat het anders vervalt dan alle andere quarks. Hierdoor blijft, in tegenstelling tot de andere quarks, de spin-informatie behouden. Met deze spin-informatie kunnen bepaalde hoekverdelingen worden geanalyseerd. Uit deze distributies blijkt dat het effect van de nieuwe interacties niet significant groot zijn.

Helaas is hieruit niet te achterhalen of deze interacties ook meetbaar zijn in experimenten met de LHC, de grote deeltjesversneller bij CERN. In dit artikel is geen achtergrond vanuit het Standaard Model meegenomen en zijn er ook geen meetonzekerheden meegenomen naast de statistische onzekerheid. Wel is de hoeveelheid data een stuk minder dan al bij de LHC is geproduceerd, waardoor statistische onzekerheden groter zijn. Er zijn wel duidelijk distributies welke sterker van het Standaard Model afwijken en dus meer kans hebben om gedetecteerd te worden. Hier kan vervolgonderzoek naar gedaan worden, zodat er mogelijk kan worden aangetoond dat de nieuwe interacties niet enkel in theorie mogelijk zijn, maar dat deze ook werkelijk plaatsvinden.

Mochten deze nieuwe interacties werkelijk bestaan, dan kunnen deze bij hoge energie¨en, zoals kort na de

Big Bang, ervoor zorgen dat er meer materie dan antimaterie is in het heelal. Hiermee is dan een complete verklaring te geven voor de geobserveerde ongelijkheid tussen materie en antimaterie.

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1 Introduction

1.1 Beyond the Standard Model

The Standard Model (SM) proves to be the most accurate description of the elementary particles and their in-teractions. It can predict the existence of an interaction and if so, what its cross-section, a quantity comparable to the probability for the reaction to occur, is with remarkable accuracy. Its latest addition, the Higgs boson, was first published in 1964 [14], 48 years before its detection was announced in 2012 [1]. Some observations are however not yet fully explained by the current Standard Model.

One of the open questions the Standard Model fails to fully answer is the matter-antimatter asymmetry. In a uniform universe, matter and antimatter should have been equally created during the Big Bang. However, current observations show a matter-filled universe. An unknown interaction, perhaps at higher energy scales, that differs between matter and antimatter, might cause the observed asymmetry. The Standard Model pre-dicts the same behavior for matter and antimatter, although later additions to the SM have established that some CP-violation occurs by complex phases in the CKM-matrix. The CP-violation allows for some matter-antimatter asymmetry, but it is not enough to explain the full asymmetry that is observed [12], [13]. A complete explanation might lie Beyond the Standard Model (BSM).

1.2 Approach in this thesis

In this thesis, Effective Field Theory (EFT) will be used to obtain predictions for angular distributions from polarized t-channel single-top production. It has been shown that certain angles are sensitive to top-polarization [13], making these angles an ideal target to investigate EFT-predictions.

This investigation starts by explaining Effective Field theory, introducing an equation that allows for con-structing distributions with an arbitrary coupling coefficient c. This equation will be validated later in section 4. The theory section continues with the Wtb-vertex and polarized top quarks. The polarization-sensitive angles will be defined in this section (2.2). Afterwards, the limits on the coupling coefficient c for a specific operator, which proved to be most sensitive in cursory analysis, is discussed. These limits, together with the event selection discussed in section 3.2, will be important for calculating realistic results.

Results will be shown in histograms, allowing for a comparison between the Standard Model distribution and BSM-distributions with variable coupling c. The significance of any deviations will be shown in the same section.

1.3 Future searches for New Physics

The only particle accelerator capable of attaining the center-of-mass energy of 13 TeV discussed here is the Large Hadron Collider (LHC) at CERN. Because the BSM-effects discussed here have not been measured before, it is expected to require a large amount of events. Both the ATLAS and CMS-detectors can measure single-top

production and provide the needed data. The LHC itself produced 40 fb−1 of data in 2016 and it is expected

to provide another 40 fb−1 of data again this year [19]. This amount of data is larger than the amount used

here. If significant deviations are found here, it might be possible to see these hints of BSM-effects appearing in LHC-data too.

2 Theory

2.1 Effective Field Theory

A promising model-independent theory is the Effective Field Theory (EFT) or Effective Lagrangian Approach (ELA). In EFT the effect of new operators, whose effects are only directly observable at higher energy scales, can be indirectly probed. These operators are added to the Standard Model, suppressed by a certain energy scale Λ. The only requirements for the (effective) operators is that they are invariant under the existing SM

symmetries, that is SU (3)C× SU (2)L× SU (1)Y [21], and should only contain SM degrees of freedom [18]. The

Standard Model is of dimension-4; a dimension-5 operator must therefore be scaled by a single factor of _Λ1. For

higher dimensions this generalizes to

SM +O5 Λ + O6 Λ2 + O7 Λ3 + . . . (1)

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Operator in [10],[21] C_{f Q}(3) O(3)_φq CtW OtW C4f O (1,3) qq CtG OtG

Table 1: Different notations for the used effective operators.

The only dimension-5 operator that can be constructed violates the lepton number [10], giving rise to a

Ma-jorana mass to neutrino’s. Λ would then be of order 1013 _{GeV [8]. This energy scale is not attainable and}

shall therefore not be considered. Only the next-lowest dimension will be considered here, that is, dimension-6. These operators may violate baryon number conservation [11], and be partially responsible for the observed matter-antimatter asymmetry.

There are 80 independent dimension-6 operators, but only four of these, with real coefficients, are relevant

in single top processes [21]. Three of the operators change the Wtb-vertex, while the C4f-operator introduces

4-fermion interactions [9]. The relevant operators, and notations in other works, are shown in table 1.

In cursory analysis, the CtW-operator showed the most dramatic change when compared to the Standard

Model. In some angles the sign of the slope flipped, which means a huge measurable effect. This result

moti-vated the decision to further investigate the CtW-operator. The results of the cursory analysis is shown for all

operators in the Appendix, but no in-depth analysis is performed on operators other than CtW.

EFT modifies the Standard Model by adding new operators of higher dimensions. The modified matrix element can be written in terms of pure SM and BSM contributions

M(gSM, gBSM) = gSM· OSM+ gBSM· OBSM (2)

As the cross-section of a process is always given by the absolute square of the matrix element, the equation

σ ∝ |M|2= g_SM2 · O2

SM+ gSM· gBSM· 2Re(O∗SMOBSM) + g2BSM· O

2 BSM

≡ SM + cX + c2_Y (3)

can be identified, where X ≡ 2Re(O_SM∗ OBSM) and Y ≡ OBSM2 . The gSM is taken as unity and gBSM is

renamed c. The X-function is called the interference term.

2.2 The Wtb-vertex and polarized top-quarks

Within the Standard Model, the top quark has the largest mass of all known quarks of 173.1 ± 0.6 GeV and

a decay width of 1.41+0.19_−0.15 GeV [17], although the width is defined 1.5 GeV here. Because the top mass is of

the order of the electroweak symmetry breaking (EWSB) energy of 246 GeV, the top decays before the strong interaction causes hadronization [20]. This means the top quark decays with the weak interaction, almost always to a W-boson and a low-mass quark. This quark is often (0.957 ± 0.034) a bottom quark [17]. The bottom quark hadronizes, but the W-boson decays further into a lepton-neutrino pair. Hadronic decay of the W is also possible, but not further discussed here because it would be much harder to fully reconstruct the event from the jets. In single-top production, the spin of the top is often (∼95%) in the direction of the spectator quark momentum in the top rest frame [16], due to the weak interaction only coupling with left-handed helicity. In figure 1 the relevant diagram is shown where all particles involved are specified. The spectator quark is defined

as q0.

The cross-section of t-channel single-top production, qb → q0t, is predicted to be 136+5.40_−4.57 pb at √s = 13

TeV in the Standard Model [15]. In assuming the top quark always decays to W q, the branching ratio

Γ(W b)/Γ = 0.957 ± 0.034 [17]. The branching ratio for W → e+ν is 10.75 ± 0.13% [6]. The predicted

cross-section for the diagram in figure 1 is thus 13.94+0.728_−0.666pb.

Some angles have shown to be sensitive to top quark polarization in other works [13], [16]. The added dimension-6 operators could change the distributions for these angles because new (indirect) interactions are made possible. This allows one to test for the existence of dimension-6 operators.

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Figure 1: Diagram of the production and semileptonic decay of a single top quark [7]. Not shown are the

protons, which provide the initial quarks, and the jets from q0 and b. The t-channel W is virtual and therefore

not saved in any dataset, allowing to filter events based on a W -count of one. In the diagram, q0 is called the

spectator quark. From the top quark, all four decay products (W , b, l, v) are visible in the diagram, although detection of the neutrino will be difficult in a particle detector. Note that the Wtb-vertex appears twice here, once in the production of a single top and once in the decay of the top. In generating events, it is assumed that the coupling appears in only one of the two Wtb-vertices, but not both in a single event.

Figure 2: Angles between the top quark spin axis and its decay products. The top quark spin axis is often

(∼95%) along the direction of the spectator quark (q0 in figure 1) in the top rest frame [16]. The χt

W-angle is

not shown here, but is defined by the angle between the top spin axis and the W -momentum.

It can be shown that in the SM the angular decay distributions are simply linear for the cosine of the angles defined in [16]. In the general case, where the top quarks may not be fully polarized, the angular decay distribution is given by:

1 ΓT dΓ d(cos χt i) = 1 2 1 + P αicos χ t i (4)

where P is the polarization of the top quark, i the decay product of the top quark and αi the spin analyzing

power for that angle [4], [16]. Expected values for the spin analyzing power for the SM are shown in table 2.

The angles are made in the top rest frame between the spectator quark momentum (≈ ~Stop) and the produced

b, W , l or v. These angles shall be referred to as χt_i with superscript t referring to the top rest frame and

subscript i referring to a specific decay product, see also figure 2.

More angles are identified in [13], but expected distributions are not given. To define these additional angles,

a coordinate system is defined first. The normal axis ˆN is defined by ˆN = ~pq× ~pW where ~pq and ~pW are the

momenta of the spectator quark and the W+_{-boson in the top quark rest frame respectively. The transverse}

axis ˆT is defined by ˆT = ~pW× ˆN with ~pW again in the top quark rest frame. The three angles are then defined by

• θS _{= angle between the momenta of W in the top quark rest frame and the charged lepton in the W rest}

frame.

• θT _{= angle between the momentum of the charged lepton in the W rest frame and the transverse axis.}

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Decay product αi

W 0.403 ± 0.025

b −0.403 ± 0.025

v −0.324 ± 0.040

l 1.000

Table 2: Spin analyzing power for each of the decay products for semileptonic top quark decays in the Standard

Model [16]. Values can be measured by obtaining the cos(χt_i)-distribution for decay product i and measuring

the slope.

2.3 Limits on C

tW

-operator

The CtW-operator has shown the most dramatic effect on the Standard Model distributions. With some angles,

the slope of the distribution flipped sign between the SM result and with the CtW-operator, see figures 24, 28

and 29 in the Appendix. This difference motivated the decision to only further investigate the CtW-operator.

The results of the basic analysis for all operators are shown in the Appendix.

The ATLAS Collaboration has measured the limits of gR/VL, the fraction of the right-handed coupling

factor and the CKM-matrix element VL= Vtbin the effective operator formalism [5]. They found

Re(gR/VL) ∈ [−0.36, 0.10] (5)

where VL= 1 in the SM [2], [5]. The relation between gR and c is given by

gR=

√

2 CtW

v2

λ2 (6)

where v is the vacuum expectation value (VEV) and λ the New Physics energy scale [3]. v equals 246 GeV and

λ is taken to be 1 TeV. The ATLAS result is then rewritten to CtW-operator limits

Re(CtW) = Re(gR) λ2 v2 1 √ 2 = Re(gR) 1 T eV2 0.2462 _{T eV}2 1 √ 2 ≈ 11.865 Re(gR) (7) to obtain limits of Re(CtW) ∈ [−4.2, 1.2] (8)

This result shall be used when plotting distributions of the earlier discussed angles. c is chosen to run from −3.0 to 1.2 in steps of 0.6 in order to stay within reasonable values for c.

3 Approach

3.1 Obtaining events

Although the datasets were provided as-is, a short overview of the used programs and settings is given here.

Events are calculated at LO using MadGraph with √s = 13 TeV. MadSpin is used for top-decay to account

for the spin correlations between the polarized top and its decay products. Pythia 8 is used for the showering

of the produced quarks from b and q0. As mentioned earlier in figure 1, there are two Wtb-vertices in the

event-diagram, but only one Wtb-vertex can exhibit BSM-behavior per event. For every dataset, a combination

of active operator and its strength c, 105_{events are generated. The corresponding luminosity will be calculated}

in section 4.3 and shown in table 4.

Initially, the value of c for each of the effective operators is chosen in such a way that the cX-term has the

largest contribution to the full equation. This is calculated by differentiating cX/(SM + cX + c2Y ) and setting

this to zero. The calculation is not further discussed in this thesis, as these values were also provided. The optimized c-values are shown in table 3.

During early analysis, the CtW-operator showed the most drastic change, which motivated to further

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dataset c-value SM 0 C4f -1.4 C_{f Q}(3) 16.8 CtG 35.5 CtW 12.3

Table 3: Values of c for each operator where the contribution of the interference term is at its maximum. The

values are calculated by differentiating cX/(SM + cX + c2_{Y ) and setting this to zero. This calculation is not}

performed here. Note that only one operator is active per dataset. The C4f-operator allows only negative c.

equation 3.

The events are generated according to the diagram shown in figure 1. The particles up until the W-boson are fixed for every event to ensure polarized single-top production and decay. Another constraint only allows

W+ decay to a positron-neutrino pair. Other decay modes will not be considered here.

Not all generated particles are stored in the datasets. Virtual particles, like the t-channel W , and particles far off-shell are not saved. The constituent quarks of the initial protons, with exception of the initial quarks q and b, are also not stored in any dataset.

3.2 Analysis

For analysis, the program ROOT is used. This data analysis framework, made by CERN, can work with large datasets and contains all necessary tools to calculate and present results.

As the generated data is based on truth information, no complex calculations are necessary to identify the particle types in an event. However, some baseline cuts are necessary to ensure correct identification. These

cuts drop < 5% of the events. A single W+, e+, ve, top and at least one bottom quark is required to ensure

correct identification of the specific particles in figure 1. Although the diagram shows two W -bosons, the virtual t-channel W -boson is not saved, leaving only one per event. The bottom quark, at the t → W b-vertex in figure 1, is identified by choosing the closest match to the expected Lorentz vector magnitude. The spectator quark

(q0) is defined as the low-mass (anti-)quark with the highest transverse momentum.

Results from truth information are not realistic for real experiments, therefore a second situation is defined where practical considerations are taken into account. In reality not all events can be perfectly reconstructed; particles may have their momentum along the beam pipe making detection impossible or the particle is not energetic enough for detection. These events will be filtered during event selection. The event selection will only be used in section 5.2; all other results are without event selection.

The event selection uses the following criteria: • M Et ≥ 25 GeV • M tW ≥ 25 GeV • P te+≥ 25 GeV • Etve≥ 25 GeV • P tb ≥ 25 GeV • P tspectator≥ 25 GeV • ηspectator≤ 2

If an event fails to meet any of these criteria, the event is discarded.

As boosting to the top rest frame is necessary, the TLorentzVector-class is used in ROOT. Boosts are always made from the lab frame to the required frame, since boosting twice may result in different results by Wigner rotations [13]. The angles between the (boosted) Lorentz vectors are then calculated from the definitions given

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in section 2.2. Finally, the value of the cosine of the specific angle is added to the histogram with the event weight. After all events, the histograms are fitted using a linear or quadratic function to show the shape of the distribution.

4 Testing the theory

4.1 Approach

In this section the validity of equation 3 is investigated for the CtW-operator. Each of the seven datasets has

∼9.6 · 104_{events after baseline cuts, with c = 0 (SM) and c = {±12.3, ±6.0, ±1.0}. The X and Y -functions can}

be isolated from equation 3 by

X = O(+c) − O(−c)

2c (9)

Y =O(+c) + O(−c) − 2SM

2c2 (10)

where O refers to the specific operator used. Equation 3 can then be verified by reconstructing the c = ±6.0 from the calculated X and Y from c = ±12.3 and vice-versa. This method will also be used to obtain distributions

for arbitrary c in order to estimate whether BSM-effects can be measured with the current limits on the CtW

-operator.

4.2 Effects of top width

The first attempt to validate equation 9 for X resulted in different distributions as shown in figure 3 (left) for

cos(χt_l). However, by equation 3, the X and Y -functions should be independent of c. It was later found that the

event generator calculated different top widths by taking BSM-effects into account. In order to obtain correct X and Y , the SM -part of equation 3 must be equal for the ±c-distributions. In calculating the top width with BSM-effects this resulted in widths of 4.48 GeV for c = +12.3 and 0.58 GeV for c = −12.3. These datasets are defined as loose, because the top width depends on c. Datasets where the top width is fixed to the SM-value are defined as fixed. Fixing the top width value to the SM-expectation of 1.5 GeV for c = ±1.0, c = ±6.0 and c = ±12.3, only minor deviations are found as shown in figure 3 (right). The only exception is c = 1.0, where

the deviations in some bins, especially near cos(χt

l) = 1, are exceeding one standard deviation from the other

two distributions. This is probably caused by subtracting two near-equal distributions, with the only difference being a small BSM addition at c = 1.0.

The next validation reconstructs the c = 1.0 and c = 6.0 distribution from the c = 12.3 dataset, using equations 9 and 10 to isolate X and Y respectively, combining them with equation 3 to obtain the distribution

for arbitrary c. The result for cos(χt

l) is shown in figure in figure 4 (left) for the loose case. Again, large

deviations are visible, which is no surprise considering the calculation for X was already invalid. After fixing the value for the top width, the distribution has deviations mostly within one standard deviation as shown in figure 4 (right). The c = ±1.0 distributions show some larger deviations, due to the uncertainties on X by subtracting two near-equal distributions with only small BSM-effects as shown in figure 3.

4.3 The normalization function

Fixing the top width has no effect on the shape of the distributions, as shown in figure 5 for cos(χt

l). From this

figure it can be concluded that a normalization factor, which depends on the top width, makes the loose width calculation for X invalid. In calculating X from the ±c distributions, the Standard Model was assumed to be independent of c, but with different top widths between +c and −c this is not the case. In order to recover the normalization factor, needed to reconstruct the correct cross section at certain c, a table is created for the

measured cross sections at fixed and loose widths of several values of c for the CtW-operator, as shown in table

4. This table also shows the number of events after baseline cuts, the cross-section and luminosity per dataset.

The SM-result of 13.47 pb agrees with the calculated 13.94+0.728_−0.666 pb in section 2.2.

The normalization function will be referred to as f (c) and is defined by

σ(c, Γtop(c = 0)) σ(c, Γtop(c)) = σ(c)f ixed σ(c)loose = Γtop(c) Γtop(c = 0) ≡ f (c) (11)

where σ is given by equation 3 and Γtop(c) is the top width at c. Γtop(c = 0) is defined as the SM-width of 1.5

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) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 − 0.08 − 0.06 − 0.04 − 0.02 − 0 0.02 X1 loose X6 loose X12.3 loose ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 X1 fixed X6 fixed X12.3 fixed

Figure 3: Comparison of the X-function, extracted using equation 9, for c ∈ {1.0, 6.0, 12.3} for the CtW-operator

with loose (left) and fixed (right) top width. Deviations exceeding the statistical uncertainty are visible in the left graph, but are within statistics in the right graph.

) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0 0.2 0.4 0.6 0.8 1 1.2 SM ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 SM ctW1 fixed -ctW1 fixed ctW6 fixed -ctW6 fixed c = 1.00 fixed c = -1.00 fixed c = 6.00 fixed c = -6.00 fixed

Figure 4: Distribution for cos(χt_l) for the SM and c = ±1.0 with loose (left) and fixed (right) top width. The

data is simulated directly (ctW#) and calculated from c = ±12.3 (c = #). Large deviations are visible in the left graph between the same-colored data. In the right graph, the deviations between the same-colored distributions are mostly within statistical error.

) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.5 1 1.5 2 2.5 3 3.5 SM ctW1 loose ctW1 fixed -ctW1 loose -ctW1 fixed ctW6 loose ctW6 fixed -ctW6 loose -ctW6 fixed ctW12.3 loose ctW12.3 fixed -ctW12.3 loose -ctW12.3 fixed ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SM ctW1 loose ctW1 fixed -ctW1 loose -ctW1 fixed ctW6 loose ctW6 fixed -ctW6 loose -ctW6 fixed ctW12.3 loose ctW12.3 fixed -ctW12.3 loose -ctW12.3 fixed

Figure 5: Unnormalized (left) and normalized (right) distribution of cos(χt

l) for the SM and all CtW-datasets

with fixed and loose top width. Because deviations between same-colored distributions are within statistics after normalization, the difference between the fixed and loose distribution is only a normalization factor.

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Fixed width Loose width c N σ L N σ L Γtop(c) f (c) 12.3 96932 58.47 pb 1.658 fb−1 96578 19.43 pb 4.969 fb−1 4.483 GeV 3.009 6.0 96757 29.34 pb 3.297 fb−1 96436 16.32 pb 5.910 fb−1 2.692 GeV 1.798 1.0 96397 15.16 pb 6.358 fb−1 96347 13.82 pb 6.972 fb−1 1.646 GeV 1.097 SM 96305 13.47 pb 7.147 fb−1 96305 13.47 pb 7.147 fb−1 1.500 GeV 1.000 −1.0 96261 11.72 pb 8.211 fb−1 96353 13.23 pb 7.282 fb−1 1.329 GeV 0.8859 −6.0 95599 8.698 pb 10.99 fb−1 96227 16.60 pb 5.797 fb−1 0.7828 GeV 0.5240 −12.3 96048 16.19 pb 5.934 fb−1 96468 41.41 pb 2.329 fb−1 0.5840 GeV 0.3910

Table 4: Measured cross sections, number of events and luminosity for various coupling strengths with fixed

and loose top width. Also shown is the top width for the specific c, Γtop(c), calculated by MadGraph. The

normalization function value f (c) is calculated by σf ixed/σloose.

c 10 − −5 0 5 10 f(c) 0.5 1 1.5 2 2.5 3 Normalization function c 10 − −5 0 5 10 0.014 − 0.012 − 0.01 − 0.008 − 0.006 − 0.004 − 0.002 − 0 0.002 Relative residuals

Figure 6: Graph showing the normalization function for tested values of the coupling constant c for the CtW

-operator (left) and the relative residuals for the fit (right). The coefficients for the second-order polynomial fit (drawn in red) are given in table 5.

The calculated values of the normalization function are shown in the f (c)-column in table 4. A plot of the normalization function is shown in figure 6. The normalization function is expected to be describable by a

second-order polynomial, because its origin lies in (SM + c BSM )2_{. The coefficients for the fit, described by}

P2

i=0pi· ci, are shown in table 5. The fit shows excellent agreement with the points plotted, indicating its use

for arbitrary c within the [−12.3, 12.3]-range is valid.

It is now possible to calculate X from arbitrary c. Equation 9 is modified by first transforming the loose ±c distributions to the fixed width distribution to obtain

X0=

f (+c)O(+c) − f (−c)O(−c)

2c . (12)

where X0is defined as the X-function for top width Γtop(c = 0) (fixed) and f (c) as the normalization function

shown in figure 6. This is verified by calculating the X0 from the loose datasets and comparing these to the

fixed datasets. The result for cos(χt

l) is shown in figure 7. The left figure is calculated from equation 9 and

shows inconsistent results for the loose distributions. The right figure uses equation 12 and shows that all distributions agree, with exception of the c = ±1.0 distribution. This deviation is probably due to having only a small BSM-effect at low c.

Coefficient Value

p0 1.000

p1 0.106369

p2 0.00461804

Table 5: Coefficients for P2

i=0pi· ci, the fitted normalization function shown in figure 6. The p0 coefficient is

1.000 by definition. This function is used to transform fixed to loose distributions and vice-versa for arbitrary c.

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) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 − 0.05 − 0 0.05 0.1 0.15 X1 loose X1 fixed X6 loose X6 fixed X12.3 loose X12.3 fixed ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 X1 loose X1 fixed X6 loose X6 fixed X12.3 loose X12.3 fixed

Figure 7: Calculating X without scaling of O(c) and O(−c) (left) and X0 with scaling (right) for cos(χtl) using

equations 9 and 12 respectively. The result with scaling shows only minor deviations, with exception of c = 1.0

especially near cos(χt

l) = 1. These deviations originate from subtracting two near-equal distributions, just as

in figure 3. ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.03 − 0.02 − 0.01 − 0 0.01 0.02 Y1 loose Y1 fixed Y6 loose Y6 fixed Y12.3 loose Y12.3 fixed ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.02 − 0.01 − 0 0.01 0.02 0.03 Y1 loose Y1 fixed Y6 loose Y6 fixed Y12.3 loose Y12.3 fixed

Figure 8: Calculating Y without (left) and Y0with (right) transforming the loose ±c distributions by multiplying

with the normalization function f (c) for cos(χt

l). The c = 1.0 result shows large deviations with the c = 6.0

and c = 12.3 result, but this may be explained by the large uncertainties due to having a small BSM-effect at small c.

The same method is used for the Y -function. The original equation, equation 10, is rewritten by first transforming the distributions from loose to fixed by multiplying with the normalization function, to obtain

Y0=

f (+c)O(+c) + f (−c)O(−c) − 2SM

2c2 (13)

where define Y0is defined as the Y -function with width Γtop(c = 0). The comparison between results of equation

10 and equation 13 is shown in in figure 8 for cos(χt

l). The left graph for Y -function shows large deviations

from the other distributions. The right graph for Y0 shows agreement between c = 12.3 and c = 6.0, but not

for c = 1.0. This can again be explained by the large uncertainties of this distribution, originating from the small BSM-effects at small c. It is best to reconstruct from the c where the interference term (with X) is largest

in order to have the lowest statistical uncertainties. Even so, at small c the c2Y -term is not that large; any

inaccuracies in Y will therefore mostly disappear.

4.4 Calculating known distributions from other distributions

When the SM, X0, Y0-distributions for an angle and the angle-independent f (c) are known, it is possible to

calculate the distribution of a specific angle for arbitrary c by

σ(c) = 1

f (c) SM + cX0+ c

2_Y

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)

l t

χ

cos(

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(pb)

σ

0.2 0.4 0.6 0.8 1 1.2 _{ctW1 loose} -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose

Figure 9: Distributions comparing the loose c = ±1.0 and c = ±6.0 to the calculated distributions from the loose c = ±12.3 dataset. Equation 14 is used to obtain the calculated distributions. Results for the other angles are shown in figure 30 in the Appendix.

where f (c) is defined by the equation shown in table 5. Definitions for X0and Y0 are given in equations 12 and

13 respectively.

The results from using equation 14 for cos(χt_l) with c = ±1.0 and c = ±6.0 is shown in figure 9. The results

for the other angles are shown in the Appendix in figure 30. The difference between the loose and calculated distributions, with the exception of some bins, are within a standard deviation.

4.5 Error calculation

Errors are automatically propagated by ROOT, but are only correct if the variables are uncorrelated. A manual error calculation is needed for correlated variables. In previous calculations, two situations arise where corre-lated variables are used.

The first is when calculating a distribution at arbitrary c from SM, X0 and Y0 as in last section. The full

equation for any c from the initial datasets is given by

σc= 1 f (c) SM + cX0+ c 2_Y 0 = 1 f (c) SM + cf (+c 0_)O(+c0_{) − f (−c}0_)O(−c0₎ 2c0 + c 2f (+c0)O(+c0) + f (−c0)O(−c0) − 2SM 2c02 (15)

where c0 is the coupling constant at which the ±CtW-distribution was generated. For the distributions here,

this equals 12.3. In this equation, the same distributions are used twice but ROOT cannot know about this correlation, so the error calculation is done manually.

The error on a function f is calculated by

δ2_f =X i ∂f ∂i 2 δ2_i (16)

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where the derivative is taken to every variable i in f with an error. The error on the calculated cross-section is then given by

δc= 1 f (c) s 1 − c 2 c02 2 δ2 SM + f (+c0)2 _c 2c0 + c2 2c02 2 δ2 O(+c0₎+ f (−c0)2 − c 2c0 + c 2c02 2 δ2 O(−c0₎. (17)

The second case is when calculating the significance of the deviations with the Standard Model. The

significance is calculated by S = σc− SM pδ2 SM+ δ2c = 1 f (c) SM + cX0+ c 2_Y 0 − SM pδ2 SM+ δc2 = 1 f (c) SM + cf (+c0)O(+c0)−f (−c_2c0 0)O(−c0)+ c 2 f (+c0)O(+c0)+f (−c0)O(−c0)−2SM 2c02 − SM pδ2 SM+ δ2c (18)

Following equation 16, the error on the significance is given by

δS= s 1 δ2 SM+ δ2c s 1 − c 2 c02 ₁ f (c)− 1 2 δ2 SM + f (+c0₎2 f (c)2 _c 2c0 + c2 2c02 2 δ2 O(+c0₎+ f (−c0₎2 f (c)2 −c 2c0 + c2 2c02 2 δ2 O(−c0₎ (19) This updated error calculation is incorporated in all previously shown distributions where necessary.

5 Results

In this section only results for the CtW-operator for arbitrary c is shown. For the other operators, normalized

plots for all angles with optimized c as shown in figure 3 are shown in figures 24-29 in the Appendix.

The graphs shown in this section are calculated from the loose c = ±12.3 distributions using equation 15. The results before event selection, but after the baseline cuts, are shown first. After event selection the results should be more realistic and are shown in section 5.2.

5.1 All events

The distributions for the cross-section and corresponding significance of the deviations from the SM for all angles, whose definitions are given in section 2.2, are shown in figures 10-16. This is before event selection and with arbitrary c, ranging from −3.0 to 1.2 in steps of 0.6. The solid lines show a second-order polynomial fit to the data, which only helps to see the shape of each distribution.

Note that only errors arising from statistics are taken into account; the systematic error, detector effects and the SM-background are all set to zero. This increases the significance since deviations are calculated only against the SM-signal with smaller errors. A deviation is defined significant if the significance is larger than 5σ.

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) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.4 0.6 0.8 1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 2 4 6 8 10 12 14 16 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 10: Cross-section (left) and significance (right) distributions for cos(χt

l) before event selection. Only

c ≤ 1.80 has significant deviations for negative cos(χt

l). Barely significant deviations are also visible for c = −3.00

at cos(χt l) ≈ 1. ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 1 2 3 4 5 6 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

b) before event selection. Significant

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) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 2 4 6 8 10 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

v) before event selection. Significant

deviations are visible for c ≤ −2.40 at cos(χt

w) ≤ −0.60 and cos(χtw) ≥ 0.45. c = −1.80 also shows significant

deviations at cos(χt w) ≤ −0.80 and cos(χtw) ≥ 0.90. ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.3 0.4 0.5 0.6 0.7 0.8 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 1 2 3 4 5 6 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 13: Cross-section (left) and significance (right) distributions for cos(χt_w) before event selection. Significant

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) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.45 0.5 0.55 0.6 0.65 0.7 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 14: Cross-section (left) and significance (right) distributions for cos(θN_{) before event selection. No}

significant deviations are visible.

) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.3 0.4 0.5 0.6 0.7 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 1 2 3 4 5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 15: Cross-section (left) and significance (right) distributions for cos(θS_{) before event selection. The only}

significant deviation visible is for c ≤ −2.40 at cos(θS_{) ≈ 1.}

) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 2 4 6 8 10 12 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 16: Cross-section (left) and significance (right) distributions for cos(θT_{) before event selection. Significant}

deviations are visible for c ≤ −2.40 at cos(θT_{) ≤ −0.6, c = −1.80 at cos(θ}T_{) ≤ −0.9 and c = −3.00 at}

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5.2 After event selection

In this section results are shown after event selection. The event selection tries to obtain data more represen-tative of a real experiment by discarding event that cannot be measured due to practical considerations. The conditions for the event selection are listed in section 3.2. The cross-section distributions and the significance of the deviations from the SM are shown in figures 17-23. Again, the SM-background, systematic error and (other) detector effects are set to zero. The fit is no longer shown as distributions are no longer described by second-order polynomials.

After event selection the number of event samples is reduced considerably. This increases the statistical uncertainty per bin, making it already much harder to obtain significant deviations in the data. The shapes are also different from the truth distributions. This has a dramatic effect on the deviations from the SM for all angles. ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.5 1 1.5 2 2.5 3 3.5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

l) after event selection. The

event selection lowered the cross-section everywhere, but it also changed the shape of the distribution. Before event selection the cross-section distribution was almost linear, but the event selection had large effects on

cos(χt

l) > 0.2 as the distribution shows a large decline. The significant deviations from the truth distribution is

gone. ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.5 1 1.5 2 2.5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

b) after event selection. The event

selection had large effects on the region cos(χt

b) < 0 and the last bin, where the cross-section has dropped to

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) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.5 1 1.5 2 2.5 3 3.5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

v) after event selection. The

significant deviations from the truth distributions are gone.

) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.5 1 1.5 2 2.5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

w) after event selection. No

significant deviations are visible.

) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.2 0.4 0.6 0.8 1 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 21: Cross-section (left) and significance (right) distributions for cos(θN_{) after event selection. The}

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) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.2 0.4 0.6 0.8 1 1.2 1.4 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 22: Cross-section (left) and significance (right) distributions for cos(θS_{) after event selection.} _The

deviations are far from significant.

) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 SM c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ) σ Significance ( 0.5 1 1.5 2 2.5 c = 0.60 loose c = 1.20 loose c = -0.60 loose c = -1.20 loose c = -1.80 loose c = -2.40 loose c = -3.00 loose

Figure 23: Cross-section (left) and significance (right) distributions for cos(θT_{) after event selection. No}

signif-icant deviations are visible.

6 Conclusion and discussion

It can be concluded that the CtW-operator shows a remarkable dependency on the top width. This conclusion

was drawn from figure 5, which showed that the difference between a distribution with the top width fixed to the SM-expectation and a distribution with BSM top width is just a normalization factor. Without accounting for these differences between ±c-distributions, the EFT approach is invalid due to the difference in normal-ization. This is immediately a problem when calculating the interference term X, where both the +c and −c distributions are needed with the same SM -term.

Using the normalization function, defined in equation 11, equation 15 is used to obtain the distribution for the same angle for arbitrary c. To calculate this distribution, only the SM and specific ±c-distributions are needed for an angle. Because the normalization function is defined by the cross-sections, it is angle-independent; it only depends on the operator and c.

With the results from fitting the normalization function with a second-order polynomial, an equation is obtained for this function for arbitrary c. The coefficients are shown in table 5. The full calculation, equation 15, has been validated by calculating the loose c = ±1.0 and c = ±6.0-distributions from the loose c =

±12.3-distributions for all seven angles. Only minor deviations were found in figures 9 (cos(χt_l) and 30 (others).

In the previous section, it was shown that deviations from the Standard Model are significant before event selection, exceeding 10σ in some cases, but insignificant after event selection. This is with systematic errors

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and background set to zero. In a real experiment systematic errors are always introduced and the SM-background is never perfectly eliminated after event selection, so it would be invalid to conclude much about the results of an actual experiment based on current results. It is however possible to identify the most promis-ing angles to guide future experiments.

After the event selection, the uncertainties grow larger and deviations are no longer significant. The χt

l,

χt

b and χ

t

w angles are however most promising with relatively large significance compared to other angles for

c = {−3.00, −2.40, 1.20}. The χt

v also shows large significance, but only for c = −3.00 and is difficult to

re-construct considering the angle is taken with the neutrino momentum. It is therefore suggested to prioritize

researching the χt

l, χtb and χtw angles.

The effect of the CtW-operator may be possible to measure, but probably only with a large number of events.

In this thesis the data had a luminosity of < 10 fb−1 while the LHC produced 40 fb−1 of data last year and

expects to produce another 40 fb−1of data this year [19]. It may be worthwhile to redo this analysis at higher

luminosity to better determine if the deviations are significant.

Another result shown is the validity of equation 3, which may help in fitting real data. It remains a question whether an experiment can isolate the signal enough from the background; only the signal itself was compared

here. Also important is whether the physical value of c for CtW is far enough from zero, the Standard Model,

to even measure the indirect effects using a particle detector with the current resolution at√s = 13 TeV. A

study in which background and estimated systematic errors are accounted for may prove useful before real data is analyzed. It can guide experiments to perhaps measure significant deviations from the Standard Model.

After a study at higher luminosity and accounting for SM-background, analysis on real data, perhaps on LHC’s Run-II and beyond, might uncover some additional clues for CP-violation. If such clues are found, it may fully explain the abundance of matter over antimatter in the universe.

7 Acknowledgements

Thanks to Marc de Beurs for providing the necessary datasets, optimized values of c and his help during this project, including proofreading earlier versions of this thesis. Also thanks to Marcel Vreeswijk for helping with some theory questions and proofreading.

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[2] G. Aad, B. Abbott, J. Abdallah, O. Abdinov, R. Aben, M. Abolins, O.S. AbouZeid, H. Abramowicz, H. Abreu, R. Abreu, et al. Search for anomalous couplings in the Wtb vertex from the measurement of double differential angular decay rates of single top quarks produced in the t-channel with the ATLAS detector. Journal of High Energy Physics, 2016(4):1–46, 2016, hep-ph/1510.03764.

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9 Appendix

9.1 Normalized distributions for all operators

Fit results Entries 96305 Mean 0.3174 Std Dev 0.4841 Slope 0.4729 ± 0.0022 Start 0.4997 ± 0.0016 ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ) l t χ

Distribution for cos(

Fit results Entries 96305 Mean −0.1108 Std Dev 0.5641 Slope −0.1674 ± 0.0027 Start 0.4997 ± 0.0016 ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.35 0.4 0.45 0.5 0.55 0.6 0.65 ) v t χ

Fit results Entries 96305 Mean 0.1286 Std Dev 0.5601 Slope 0.1958 ± 0.0027 Start 0.4996 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 ) w t χ

Fit results Entries 96305 Mean −0.1868 Std Dev 0.5693 Slope −0.2575 ± 0.0028 Start 0.496 ± 0.002 ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) b t χ

Fit results Entries 96305 Mean −0.1233 Std Dev 0.5007 a −0.3793 ± 0.0046 b −0.1851 ± 0.0024 c 0.6262 ± 0.0025 ) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ) S θ

Fit results Entries 96305 Mean 0.2426 Std Dev 0.5561 a 0.1968 ± 0.0052 b 0.3641 ± 0.0029 c 0.4341 ± 0.0022 ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ) T θ

Fit results Entries 96305 Mean −0.001085 Std Dev 0.605 a 0.1838 ± 0.0056 b −0.001902 ± 0.002913 c 0.4386 ± 0.0023 ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 ) N θ

Figure 24: Normalized distributions for each of the discussed angles in the Standard Model. The second-order

function fit is given by y = ax2_{+ bx + c.}

Fit results Entries 95725 Mean 0.1283 Std Dev 0.5613 Slope 0.1951 ± 0.0027 Start 0.4995 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 ) w t χ

Fit results Entries 95725 Mean −0.1274 Std Dev 0.5661 Slope −0.19 ± 0.00 Start 0.4994 ± 0.0016 ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 ) b t χ

Fit results Entries 95725 Mean 0.0007733 Std Dev 0.604 a 0.1769 ± 0.0056 b 0.001127 ± 0.002917 c 0.4409 ± 0.0023 ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 ) N θ

Figure 25: Normalized distributions for each of the discussed angles with effective operator C4f. The

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Fit results Entries 94979 Mean 0.1319 Std Dev 0.5615 Slope 0.1991 ± 0.0027 Start 0.4996 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 ) w t χ

Fit results Entries 94979 Mean −0.006206 Std Dev 0.6054 a 0.186 ± 0.006 b −0.009128 ± 0.002933 c 0.4377 ± 0.0023 ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.45 0.5 0.55 0.6 0.65 ) N θ

Figure 26: Normalized distributions for each of the discussed angles with effective operator C_{f Q}(3). The values

shown are before event selection. The second-order function fit is given by y = ax2_{+ bx + c.}

Fit results Entries 96295 Mean 0.1252 Std Dev 0.5614 Slope 0.19 ± 0.00 Start 0.4997 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 ) w t χ

Figure 27: Normalized distributions for each of the discussed angles with effective operator CtG. The

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Fit results Entries 96578 Mean 0.1828 Std Dev 0.5497 Slope 0.2724 ± 0.0027 Start 0.4998 ± 0.0016 ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 ) l t χ

Fit results Entries 96578 Mean −0.1723 Std Dev 0.5473 Slope −0.261 ± 0.003 Start 0.4996 ± 0.0016 ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 ) v t χ

Fit results Entries 96578 Mean −0.0496 Std Dev 0.5675 Slope −0.0714 ± 0.0027 Start 0.4977 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.4 0.45 0.5 0.55 0.6 ) w t χ

Fit results Entries 96578 Mean −0.001347 Std Dev 0.5839 Slope −0.003004 ± 0.002810 Start 0.4981 ± 0.0016 ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 ) b t χ

Fit results Entries 96578 Mean 0.1888 Std Dev 0.5638 a 0.1102 ± 0.0053 b 0.2829 ± 0.0029 c 0.463 ± 0.002 ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) T θ

Fit results Entries 96578 Mean −0.00463 Std Dev 0.5941 a 0.1101 ± 0.0055 b −0.006577 ± 0.002863 c 0.463 ± 0.002 ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 ) N θ

Figure 28: Normalized distributions for each of the discussed angles with effective operator CtW. Note that the

slope of cos(χtw) has changed sign compared to the SM. This dramatic change was one of the main reasons to

further investigate the CtW-operator. The second-order function fit is given by y = ax2+ bx + c.

Fit results Entries 96468 Mean −0.2279 Std Dev 0.5297 Slope −0.3439 ± 0.0026 Start 0.4998 ± 0.0016 ) l t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) l t χ

Fit results Entries 96468 Mean 0.1187 Std Dev 0.5616 Slope 0.1803 ± 0.0027 Start 0.4997 ± 0.0016 ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 ) v t χ

Fit results Entries 96468 Mean −0.05386 Std Dev 0.5645 Slope −0.07519 ± 0.00271 Start 0.4962 ± 0.0016 ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.35 0.4 0.45 0.5 0.55 0.6 ) w t χ

Fit results Entries 96468 Mean 0.007214 Std Dev 0.5727 Slope 0.00701 ± 0.00275 Start 0.4981 ± 0.0016 ) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.35 0.4 0.45 0.5 0.55 ) b t χ

Fit results Entries 96468 Mean −0.1887 Std Dev 0.5606 a 0.09085 ± 0.00529 b −0.2829 ± 0.0029 c 0.4695 ± 0.0023 ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6 0.7 0.8 ) T θ

Figure 29: Normalized distributions for each of the discussed angles with effective operator -CtW. Note that

the slope of cos(χt

l), cos(χ

t

v) and cos(χtw) has changed sign compared to the SM. This dramatic change was one

of the main reasons to decide to further investigate the CtW-operator. The second-order function fit is given by

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9.2 Calculating loose C

tW

(±1.0) and C

tW

(±6.0) distributions from loose C

tW

(±12.3)

) b t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.4 0.6 0.8 1 1.2 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) v t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.4 0.5 0.6 0.7 0.8 0.9 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) w t χ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) N θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.5 0.6 0.7 0.8 0.9 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) S θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose ) T θ cos( 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (pb) σ 0.4 0.6 0.8 1 1.2 ctW1 loose -ctW1 loose ctW6 loose -ctW6 loose c = 1.00 loose c = -1.00 loose c = 6.00 loose c = -6.00 loose

Figure 30: Calculating the loose distributions for the other six angles. The calculated distributions are made

from the loose CtW (12.3)-distributions to the loose CtW (1.0) and CtW (6.0)-distributions. The cos(χtl) result

is shown in figure 9. The calculated distributions show generally only minor deviations, with the exception of a few bins.

Single-top quark production as probe for physics beyond the Standard Model using Effective Field Theory

BACHELOR THESIS