Effective field theory in t-channel single top production at different orders in perturbation theory

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BACHELOR THESIS

EFFECTIVE FIELD THEORY IN

T-CHANNEL SINGLE TOP PRODUCTION

AT DIFFERENT ORDERS IN

PERTURBATION THEORY

by

Jop de Jong

University of Amsterdam, VU Amsterdam

Faculty of Science (FNWI)

Report Bachelor Project Physics and Astronomy , size 15 EC , conducted

between 04 - 01 – 2021 and 09 - 04 – 2021

Supervisor

dhr. prof. dr. M. Vreeswijk

Second examiner

dhr. dr. I. B. van Vulpen

Student number 11072989 Email jopfrisodejong@gmail.com Submitted on 13 - 04 - 2021 Institute Nikhef

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Summary

The standard model predicts the interactions of the known fundamental particles with great accuracy, but we know it is not complete. Therefore it is necessary to look for particles or other physics phenomena beyond the standard model (BSM). While new particles could be produced directly by increasing the energy of the accelerator, the effects they generate can also be looked for by looking at the known interactions with great precision. This is done in a systematic way by extending the standard model Lagrangian with higher dimensional operators, suppressed by a certain energy scale. This technique is called Effective Field Theory (EFT). Using EFT, the space of BSM theories at low energies can be parametrized, and these parameters measured in experiments. This is done by the comparison of data with simulations. These simulations could be done at Leading Order since it is easier, but it could be that the higher order corrections look like contributions of EFT parameters. To test this, in this thesis the EFT parameters are measured from a Next to Leading Order (NLO) simulation using LO predictions. Since NLO describes the data more accurately, this should give an estimate on the biases which can be expected when using LO predictions.

The experiment is done in the context of t-channel single top production at a center of mass energy √s = 13 TeV, for a complex valued co¨efficient of the dimension 6 OtW

operator. Because the single top is produced via the charged current weak interaction, it is polarized which induces angular correlations with its decay products. In previous work it was shown that co¨ordinates can be found such that the distribution of the decay products are sensitive to the OtW operator, which are used here to measure the parameters [1, 2].

The real and imaginary parts of the co¨efficient are measured separately. It was found that there were significant deviations from the true values of the parameters, especially when the EFT parameters were large. In particular the NLO standard model sample measured at Re(CtW) = 0.47+0.16−0.15and Im(CtW)s = 0.32+0.12−0.13using a LO prediction. The results indicate

that using LO predictions could lead to false measurements of EFT parameters.

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Popular summary (Dutch)

De deeltjesfysica houdt zich bezig met alle tot nu toe bekende fundamentele deeltjes, en de interacties tussen deze. De interacties worden beschreven door de sterke kracht die quarks bij elkaar bindt tot protonen en neutronen, de zwakke kracht die zorgt voor radioactief ver-val en electromagnetisme. De meest complete theorie van deze krachten en de dynamiek van de deeltjes zelf heet het standaardmodel. Alle deeltjes van het standaardmodel zijn reeds ontdekt, met als laatste de top quark en het Higgs deeltje. Maar we weten dat het standaardmodel niet compleet is, ten eerste natuurlijk omdat zwaartekracht er niet in zit, en daarnaast kan het bijvoorbeeld ook niet verklaren waarom er meer materie dan antimaterie is in ons heelal.

Daarom wordt er gezocht naar nieuwe fysica die dit kan verklaren, bijvoorbeed door nieuwe deeltjes die te zwaar zijn om op het moment geproduceerd te worden in deeltjesver-snellers. Maar omdat ze te zwaar zijn, betekent niet dat ze geen tekens achterlaten in de huidige reacties. Door de grote hoeveelheid data die verkregen wordt door deeltjesversnellers zoals de LHC op Cern, is het mogelijk om heel precies te kijken naar de interacties die we op het moment kennen, en zo tekenen van nieuwe fysica te kunnen vinden. Om dit op een sytematische manier te doen wordt Effective Field Theory (EFT) gebruikt, waarbij we alle mogelijke nieuwe fysica kunnen beschrijven door slechts enkele parameters. Deze parameters kunnen dan experimenteel vastgesteld worden.

De interactie waar in dit artikel naar wordt gekeken is de productie van een enkele top quark. Dit is het zwaarste elementaire deeltje, een soort zwaardere versie van de up quark die deel uitmaakt van protonen. De enkele top quark wordt geproduceerd door de zwakke kernkracht, en door de studie van de enkele top quark kunnen we deze interactie dus onder-zoeken. Door de grote massa van de top quark is de verwachting dat effecten van nieuwe fysica hier goed zichtbaar zijn. De ATLAS detector op Cern heeft nu naar verwachting ook ongeveer 10 miljoen top quarks gezien, genoeg om te kijken of de productie gebeurt zoals het standaardmodel voorspelt.

In de fysica worden altijd benaderingen gemaakt, het is onmogelijk om de meeste dingen precies uit te rekenen. In de deeltjesfysica is dit een fundamenteel feit, door quantumeffecten springen deeltjes in bestaan, en deze deeltjes kunnen weer andere deeltjes uitstralen, tot in het oneindige. Gelukkig worden de effecten van dit soort secondaire stralingen steeds kleiner hoe meer deeltjes er worden uitgezonden, waardoor we de natuur goed kunnen beschrijven door alleen de eerste paar stralingscorrecties uit te rekenen. Maar bij de berekening van de EFT parameters zijn dit soort kleine correcties juist hetgene dat we zoeken. De vraag die onderzoekt wordt in dit artikel is of de benadering die wordt gedaan kan leiden tot een me-ting van een EFT parameter, terwijl dit alleen komt door de benadering. Dit wordt gedaan door de voorspelling te vergelijken met de voorspelling + de eerste correctie.

Als er een positief resultaat uit deze test komt, zou dit betekenen dat er voorzichtig moet worden omgegaan met de benaderingen die worden gedaan als je naar nieuwe fysica zoekt.

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1 Introduction

The Standard Model (SM) is the most successful theory describing the fundamental particles and the interactions between them. It consists of 12 fermions and their antiparticles, 4 gauge bosons that act as the force carrying particles and the Higgs boson, in the framework of a quantum field theory. All particles of the standard model have currently been experimentally observed, with the most recent ones being the top quark (1995) and the Higgs boson (2013) [3, 4]. It is incredibly successful at describing both the electromagnetic force, as well as the strong and weak nuclear forces.

While it is currently the best theory out there, we know it is not complete. The grav-itational force is not included and it cannot explain the matter/ antimatter asymmetry in the universe, as well as the nature of dark matter and the masses of the neutrinos. For this reason there are multiple theories extending the standard model trying to explain these effects, collectively called Beyond the Standard Model (BSM) theories. These effects can be parametrised in a model independent way as new interactions between standard model parti-cles, using the framework of Effective Field Theory (EFT). With a Standard Model Effective Field Theory (SMEFT), the parameter space describing BSM theories can be systematically explored, and the SMEFT parameters bounded with experiments.

The standard model allows for the calculation of cross sections and the shape of distribu-tions which are measured by the detector. For these calculadistribu-tions perturbation theory is used, where terms are included up to a certain order in a coupling constant. These predictions are always approximate, including higher order terms makes them more precise. It can be useful to simulate only at Leading Order (LO) for time and space reducing purposes, but it needs to be verified that LO predictions do not introduce large biases. To estimate these biases, I will be conducting a measurement of EFT parameters from a LO prediction of a Next to Leading Order (NLO) sample both generated by a Monte Carlo (MC) event generator. Since real data is better described at higher orders, this should give an estimate on biases when using LO predictions to measure SMEFT parameters. The process in which these effects will be studied is the production of a single top quark at a center of mass energy√s = 13 TeV, which is suited for studying BSM effects in the top-electroweak sector of the SM. The top quark is produced in association with a light quark by a proton-proton collision, and decays subsequently to a W boson and a bottom quark. The observables which will be used as probes for the EFT parameters are the angular distributions of the leptons, since these are correlated with the polarization axis of the top, and the momentum distribution of the leptons in the plane transverse to the beam line [5, 6].

1.1 LHC and ATLAS

The Large Hadron Collider (LHC) is a particle accelerator located at Cern in Geneva, Switzerland. At the moment it is the most powerful particle accelerator in existence. It first went live colliding protons in 2008 at a center of mass (CM) energy of√s = 7, 8 TeV in its first run. It later received upgrades and operates at√s = 13TeV. At the end of the second run of operation in 2018 the LHC had an integrated luminosity of 139 fb−1 at this energy, which is enough data to perform precision studies of the properties of SM particles [7]. Along the beam axis are 4 main experiments, ALICE, ATLAS, CMS and LHCb. ATLAS is one of the 2 general purpose detectors at the LHC, with the other being the CMS detector. They look for new particles produced at higher energies, as well as trying to measure properties of the known SM particles. Together with CMS it was the detector that found the Higgs in 2012. The collision itself happens at the center of the experiment, resulting in hundreds of

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Figure 1: ATLAS detector with the definition of the co¨ordinate system used. The z-axis points in the counterclockwise direction of the beamline, the x-axis points to the center of the LHC tunnel and the y-axis points upward. The directions of decay products are defined by a polar angle θ in the xz plane and an azimuthal angle φ in the xy plane. Taken from [8] particles. Most of these decay before they reach the detector, so the only particles which are measured are electrons, muons and hadrons. The detector consists of 3 main parts. The inner detector measures the charge, direction and momentum of electrically charged particles produced in the collision. The calorimeter is designed to stop most of the particles coming from a collision to measure their energy. It stops all particles except muons and neutrinos. The muons are measured by the muon spectrometer. Superconductive coils are used to induce a magnetic field for the measurement of momenta of charged particles via the curvature of their trajectories. The missing energy and momentum is then associated to the neutrinos.

An orthonormal co¨ordinate system is defined by the a z-axis pointing in the counter-clockwise direction of the beamline , the x-axis pointing to the center of the LHC tunnel and the y-axis pointing upward. The origin is defined as the center of the detector. The direction of the decay products is parametrized by a polar angle θ in the xz plane and an angle φ in the yx plane. A diagram of these angles is shown in figure 1.

Instead of θ, the pseudorapidity η is often used, which reduces to the rapidity of the particle in the limit where the particles are massless. It is related to the angle θ by

η = −log tanθ 2 . (1)

1.2 In this work

The goal of this work is to estimate the biases on the measurement of EFT parameters when using LO predictions. This is done in the process of t-channel single top production at √s = 13 TeV for the OtW operator. A description of this process and the SMEFT

framework is given in section 2. The differences between LO and NLO distributions are discussed in section 2.4. Predictions are made using Monte Carlo event generators, which is briefly described in section 3 although the samples were not personally generated by the author. Cuts are applied to the data generated by the event generator, both originating from the kinematic acceptance range for the ATLAS detector as well as for the reduction of background events when measuring actual data. All the applied cuts can be found in section 3.3, and were not chosen by the author. The co¨ordinate system convenient for the measurement of the polarization is defined in section 2.3. The method for the measurement

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of parameters is discussed in section 4, which is validated in sections 5.1 and 5.2. Finally the results of the measurement using LO predictions from NLO samples are shown in section 5.3.

2 Theory

2.1 Standard Model

The standard model consists of fermions and bosons with half integer and integer spin respectively. The fermions of the standard model are described by spinors ψ which solve the Dirac equation

(iγµ∂µ− m)ψ = 0 (2)

where the γµ _{satisfy the Dirac algebra {γ}µ_{, γ}ν_{} = 2g}µν_{. The lowest dimension of the}

γµ _{that satisfy this algebra is 4, which describes spin 1/2 particles and their antiparticles.}

Spin does not commute with the Dirac Hamiltonian, instead the observable which is used is the component of spin along the momentum direction of the spinor, which is called the helicity. For spin 1/2 fermions the helicity states have eigenvalues ±~

2 which are called

the (left-) right- and (right-) left-handed part of the (anti-)particles respectively. From the Goldhaber [9] and Wu [10] experiments it became clear that helicity was important in the weak interaction. For this reason the matrix γ5_{is defined as}

γ5= iγ0γ1γ2γ3 (3)

which helps define the projection operators PR/L as

PR/L =

1 2(1 ± γ

5₎ ₍₄₎

which projects onto the right- and lefthanded chiral components of the spin 1/2 dirac spinors. In this work the main process under consideration is the charged current weak interaction, which is mediated by the massive W± bosons. The vertex factor associated with this process is

−igW

√ 2 γ

µ_P

L (5)

where gW is the coupling constant of the weak interaction [11, p.293]. It is clear from

this form that only left handed component of the particle spinors participate in the weak interaction. Such an interaction is called V - A, corresponding to the vector and axial vector transformation properties of the vertex factor. The weak interaction is suppressed by factors of the W-boson mass in the propagator of the W boson, which is the main reason it is weaker then the other forces. The strong and electromagnetic interactions, which are mediated by the massless gluons and photons, are not suppressed by such factors. For this reason higher order corrections to matrix elements are first only calculated for the strong and electromagnetic interactions.

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2.2 Effective field theory

To describe these BSM corrections in a model independent way, we can view the SM as an Ef-fective Field Theory (EFT). With such a Standard Model EfEf-fective Field Theory (SMEFT), effects of higher energy particles or other new physics phenomena can be systematically included by introducing higher dimensional operators to the Lagrangian L of the theory. In this way the effects of new physics can be described as interactions between SM particles. An operator of dimension d is denoted ˆO[d]. Since the SM Lagrangian is of dimension 4, an operator of dimension d is suppressed by a factor Λd−4_{, where Λ is the energy scale of the}

new physics. The SMEFT Lagrangian then becomes

LSM EF T = LSM + X i Ci Λ ˆ O[5]_i +X i Ci Λ2Oˆ [6] i + ... (6)

where the Ci are called Wilson co¨efficients and the sums run over all operators of that

dimension. The framework relies on the fact that the new physics is too energetic to be produced directly, but the effects of the new physics can be found by measuring the prop-erties of known interactions very precisely. Intuitively, the new physics happens at smaller scales than the SM, and the effective operators shrinks these ”small scales” down to 0 size (introducing a contact interaction). The effects of the new effective operators is that new vertices are allowed in the Feynman diagrams of the interaction. A historical example of this is the Fermi description of weak decay, as described in [11, p.296].

Since the operators of dimension 5 violate the conservation of lepton/ baryon number they are not considered [12]. The number of operators to be considered can be further reduced by only including operators which satisfy the symmetries of the SM, and only those which are relevant to the interaction in question. In t-channel single top production to be considered later, 3 operators are left at O(Λ−2) in the SMEFT Lagrangian [13]. They are

O(3)_ϕQ= i1 2y 2 t ϕ †_τI_D µϕ Qγ¯ µτIQ OtW = ytgW Qσ¯ µντIt ˜ϕWµνI O(3)_qQ,rs= ¯qrγµτIqs _¯ QγµτIQ (7)

where the notation used can be found in [5, 14]. Here qrand qsstand for quark doublet

fields of the first and second generation with r, s flavour indices, ytfor the Yukawa coupling

of the top quark, gW for the SM weak coupling constant, ϕ for the Higgs doublet, Wµν for

the W boson field strength and t for the top quark field. Dµ denotes the covariant

deriva-tive and τI are Pauli matrices. OϕQ only changes the strength of the SM interaction, OtW

introduces a tensor coupling σµν _{between the top and W field, and O}

qQ,rs adds a 4 fermion

contact interaction. Since OϕQ only changes the strength of the SM interaction, it can

cap-ture any change in the measured number of events. Therefore only normalized distributions are considered. The other 2 operators can change the shape of the differential distribution of observables [15]. In this report I will consider the OtW operator in particular. For this

operator the real and imaginary part of the Wilson co¨effient are considered seperately and denoted CtW and CitW respectively. This is both because they have different signatures in

observables and because the imaginary part is particularly interesting since it can introduce new Charge (C) Paritry (P)-violating effects to the SM [5].

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2.3 Single top quark production

The top quark is the heaviest fundamental particle in the SM, at a mass of 172.76 ± 0.30 GeV/c2_{[16]. Because of it’s short lifetime, the top quark decays before it hadronises. This}

makes the top quark the only possibility to study an isolated quark. Because of its large mass the top quark is also expected to have a large coupling to the Higgs Field. The top quark is also produced in sufficient numbers at the LHC which allows for precision measure-ments. For these reasons the top quark is an excellent probe for new physics. At the LHC the top quark is mostly produced in t¯t pairs, which is a production mode that can probe the strong interaction. The production of single top quarks is a probe into the electroweak sector. An overview of SMEFT in the single top sector can be found in [13]. At the LHC a single top is primarily produced in the t-channel, where it accounts for rougly 70 % of single top production at√s = 13TeV , the others being the s-channel and tW channel [17, 18]. The t-channel production is particularly interesting since the produced top quark always recoils of a measurable light quark, which gives a reference point allowing for the measure-ment of the polarization [1]. The Feynman diagram for this process is shown in figure 2. The associated quark-jet (q’ in figure 2) is called the spectator jet. The effective operator OtW to be considered here can be inserted at every Wtb vertex. In the figure the Feynman

diagrams of both the 4- and 5-flavour schemes are shown, which are slightly different ways to calculate the matrix element. While the b quark in the diagram always arises from gluon splitting, the difference in the 2 schemes lies in the origin of the b quark. In the 5-flavour scheme the pdf contains the g → b¯b splitting, and the b mass is neglected in the matrix element. In the 4-flavour scheme the b quark is not included in the pdf and the b mass enters the matrix element explicitly. While a resummation of terms leads to the description being approximately the same at NLO, at LO the difference will be more pronounced [19]. In this report LO simulations are done in the 4-flavour scheme, and NLO simulations in the 5-flavour scheme. This is because the 4-flavour scheme is more accurate at LO and NLO EFT predictions were only available in the 5-flavour scheme.

Figure 2: Tree level Feynman diagrams for t-channel single top production in 5,4-flavour scheme left, right respectively (taken from [20]).

The first observable which we will consider is the momentum in the transverse plane of the beamline of the lepton from the W decay, since it is easily observed. In figure 3 are shown the distributions for CtW, CitW = 1. Also shown is the reconstructed pT of the top

quark, since it was shown previously to be sensitive to the OtW operator [15]. In the figure

it can be seen that these observables have some sensitivity to the OtW operator.

Because of the V − A nature of the weak interaction, the top quark is produced with a left handed helicity in the t-channel process. Since the top recoils from the spectator

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30 40 50 60 70 80 90 100 110 120 T p 0.8 0.9 1 1.1 1.2

ratio h1/h2

30 40 50 60 70 80 90 100 110 120 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 d p_T σ d σ 1 SM ctWI_c1_ctW_c0 ctWI_c0_ctW_c1 0 20 40 60 80 100 120 140 160 180 200 T p 0.8 0.9 1 1.1 1.2

ratio h1/h2

0 20 40 60 80 100 120 140 160 180 200 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 T d p σ d σ 1 _SM ctWI_c1_ctW_c0 ctWI_c0_ctW_c1

Figure 3: The normalized differential distribution of the transverse momentum of the ob-served leptons (left) and the reconstructed momentum of the top quark (right). The SM in black, in blue CtW = 1 and in red CitW = 1 with all other Wilson co¨efficients 0. It

can be seen that the co¨effient of the OtW operator has some effect on the shape of these

distributions.

quark, this means that the top quark has a high degree of polarization in the direction of the spectator jet. Since the spin of the top is known, polarisation angles θi can be defined

as the angle between a decay product p and the ith basis vector in the top rest frame. The normalized differential cross section can then be parameterized as

1 σ dσ dcosθi = 1 2(1 + αpPicos(θi)) (8)

where αpis the so-called spin analyzing power of the decay product, Pithe ithcomponent

of the polarization vector ~P = h~σi and θi the angle of the decay product with this direction

[6]. For leptonic decay of the W, αl≈ 1. A co¨ordinate system can be constructed to further

analyze the decay vertex as in [1]. The first axis is chosen as the direction of the spectator jet ~pj since the top is polarized in this direction, the second as being orthogonal to the

production plane spanned by ~pj and the direction of the initial quark ~pq, and the third by

requiring the co¨ordinate system to be right handed. That is

ˆ

z = ˆpj, y = ˆˆ pj× ˆpq, x = ˆˆ y × ˆz. (9)

where all vectors are in the top quark rest frame and ˆv = _|v|~v . The differential dis-tributions of the lepton angles relative to this co¨ordinate system are sensitive to the OtW

operator, which is shown in figure 4. It can be seen that the cos θxand cos θy distributions

are separately sensitive to CtW and CitW respectively.

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1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.8 0.9 1 1.1 1.2

ratio h1/h2

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 x θ d cos σ d σ 1 ctWI_c0_ctW_c1 SM ctWI_c1_ctW_c0 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.8 0.9 1 1.1 1.2

ratio h1/h2

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0 0.05 0.1 0.15 0.2 0.25 y θ d cos σ d σ 1 SM ctWI_c1_ctW_c0 ctWI_c0_ctW_c1

Figure 4: In the figure are shown differential distributions of the leptons at LO in the co¨ordinate system defined in equation 9, on the left cos θx, on the right cos θy. The SM is

shown in black, in blue CtW = 1 and in red CtW = i. It can be seen that these observables

are sensitive to the real and imaginary parts of CtW respectively. Because of this they are

considered separately. The error bars are omitted, but they are all on the order of 0.003.

2.4 Differences at different orders

As discussed further in section 3, when using LO predictions the differential cross section is often scaled by the K-factor defined as the ratio between the NLO and LO cross section, so the shape of the distribution can be used to predict the data. This assumes of course that the LO prediction serves as a good predictor for the shape of the distribution. Since in this report the shape of the signal will be used to try to predict EFT parameters, it will be useful to get a general idea of the differences we can expect of the shape at different orders. In figure 5 the LO and NLO pT distributions are shown for the SM. As can be seen

in the figure, the LO simulation underestimates the low pT contributions. Since CtW has a

comparable effect as shown in figure 3, we can expect the measurement of parameters from NLO distributions using an LO sample to be less than accurate.

30 40 50 60 70 80 90 100

T

p

0.8 0.9 1 1.1 1.2

K-factor

30 40 50 60 70 80 90 100 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 d p_T σ d σ 1 LO NLO 0 20 40 60 80 100 120 140 160 180 200

T

p

0.8 0.9 1 1.1 1.2

K-factor

0 20 40 60 80 100 120 140 160 180 200 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 T d p σ d σ 1 LO NLO

Figure 5: On the left is shown the pT distribution of the leptons and on the right the

reconstructed top quark at LO (black) and NLO (blue). At low pT the distribution is

underestimated at LO.

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

x

θ

cos

0.8 0.9 1 1.1 1.2

K-factor

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0 0.05 0.1 0.15 0.2 0.25 x θ d cos σ d σ 1 LO NLO (a) cos θx 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

y

θ

cos

0.8 0.9 1 1.1 1.2

K-factor

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1 LO NLO (b) cos θy

Figure 6: Comparison of the shape of the LO (black) and NLO (blue) differential distri-butions of the leptons in the co¨ordinate system described by equation 9 for the SM. The K-factor is also shown below.

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

x

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0 0.05 0.1 0.15 0.2 0.25 x θ d cos σ d σ 1 NLO LO

(a) CitW = −10 and CtW = +10

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

x

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0 0.05 0.1 0.15 0.2 0.25 x θ d cos σ d σ 1 NLO LO (b) CitW = −10 and CtW = −10

Figure 7: Comparison of the shape of the LO (black) and NLO (blue) differential angular distribution of the leptons in the co¨ordinate system described by equation 9 for different EFT parameters. The K-factor is also shown below.

The comparison between the shape of the LO and NLO distribtutions of cos θxand cos θy

of the leptons is shown in figure 6. When comparing these with the distributions of figure 4, it can be seen that the LO/NLO differences between the shapes of the distributions can be comparable to effects from the OtW operator.

The shape of the differences at LO and NLO also changes with the values of the EFT parameters. In figure 7 this is shown in the case of CitW = −10 and CtW = ±10 for the

observable cos θx of the leptons. Comparison of figures 7b and 8 shows that the changes

between the different orders also depend on the value of the second parameter, even though it was shown in figure 4 that by themselves, the other operator had little effect (at least at LO). These differences can then be attributed to interference effects between the operators, which are of the O(Λ−4_{) and are more prominent for higher values of C}

tW and CitW.

These differences can also be observed in the cos θy distribution of the leptons. This is

shown in figure 9 for CitW = ±10 and CtW = −10. Although it looks like the difference here

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

x

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0 0.05 0.1 0.15 0.2 0.25 x θ d cos σ d σ 1 LO NLO

Figure 8: Comparison of LO (black) and NLO (blue) of the distribution of cosθx for the

leptons with CitW = 1.57 and CtW = −10. Comparison with figure 7b shows that the

difference between the different orders depends on other EFT parameters. The K-factor is also shown below.

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

y

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1 LO NLO

(a) CitW = −10 and CtW = −10

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

y

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1 LO NLO (b) CitW = +10 and CtW = −10

Figure 9: Comparison of the shape of the LO (black) and NLO (blue) differential angular distribution of the leptons in the co¨ordinate system described by equation 9 for different EFT parameters. The K-factor is also shown below.

is more regular, it is noteworthy that the difference between the different orders is again dependent on the value of the second parameter. This is shown in figure 10.

The differences between LO and NLO of the pT distributions of the leptons are shown

in figure 11. Earlier it was noted that the pT distribution is underestimated by the parton

shower algorithm at low pT. It can be seen that the behaviour here is no longer so simple.

Because of these large differences between the LO and NLO predictions, we can expect the measurement of EFT parameters using LO predictions from NLO distributions to not be very precise. Because the K-factors are also not constant in the EFT parameter space, a simple multiplication by a global scale factor will not solve the problem.

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

y

θ

cos

0.6 0.81 1.2 1.4

K-factor

1 − −0.8−0.6 −0.4−0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1 LO NLO

Figure 10: Comparison of the LO (black) and NLO (blue) cos θydistribution of the leptons

with CitW = −10 and CtW = 0.53. Comparison with figure 9b shows the difference between

the different orders depends on other EFT parameters.

30 40 50 60 70 80 90 100

T

p

0.6 0.81 1.2 1.4

K-factor

30 40 50 60 70 80 90 100 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 d p_T σ d σ 1 LO NLO

(a) CtW = 10 and CitW= −10

30 40 50 60 70 80 90 100

T

p

0.6 0.81 1.2 1.4

K-factor

30 40 50 60 70 80 90 100 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 d p_T σ d σ 1 LO NLO (b) CtW= −10 and CitW= −10 30 40 50 60 70 80 90 100

T

p

0.6 0.81 1.2 1.4

K-factor

30 40 50 60 70 80 90 100 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 d p_T σ d σ 1 NLO LO (c) CtW= −10 and CitW = 10 30 40 50 60 70 80 90 100

T

p

0.6 0.81 1.2 1.4

K-factor

30 40 50 60 70 80 90 100 T p 0 0.05 0.1 0.15 0.2 0.25 0.3 d p_T σ d σ 1 LO NLO (d) CtW = 3.68 and CitW = 10

Figure 11: Differences in the LO (black) and NLO (blue) lepton pT distributions with different

values for CtW and CitW. The K-factor is also shown below the figures. It can be seen that the

shape of the difference between the NLO and LO distributions is dependent on the values of EFT parameters. The behavior is no longer just that the low pT distributions are underestimated as in

figure 5.

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3 Simulation using MC event generators

The modelling of physics at particle accelerators like the LHC is an extremely difficult computational problem. To make the problem more tractable, the event is separated into multiple regimes. First there is the hard scattering event, which consists of both the mod-elling of the internal structure of the protons as well as the approximation of the matrix element. Because the outgoing partons of the high energy process that we want to study carry colour charge, these particles can radiate gluons like electrically charged particles ra-diate photons, and these gluons can also rara-diate (unlike photons), leading to the formation of jets of particles. These jets then form into hadrons by a process known as hadronisation, and most of these decay before they reach the detector, all of which needs to be modelled. Because of the running of the strong coupling constant, at low energy scales we cannot even apply perturbation theory and we need to resort to a phenomenological model. Luckily, all 3 of these regimes are suitable for simulation Monte Carlo (MC) techniques, which uses random numbers to approximate the high dimensional integrals of the matrix elements and the probabilistic nature of particle production and decay. A representation of a generated event is shown in figure 12. Besides the hard scattering event we are interested in, secondary interactions can also occur like additional interactions between the proton remnants. These other interactions are collectively called soft processes, in contrast with the high energy hard processes.

3.1 Hard subprocesses

Since the individual partons in the proton are highly boosted with respect to the center of mass frame of the interaction, these partons are standing still relative to the hard scat-tering event. This allows us to factorize the scatscat-tering event into a scatscat-tering of partons and the momentum distribution of these partons inside the protons. These distributions are described by Parton Distribution Functions (PDFs) which describe the momentum distri-bution of the partons in the proton. These PDFs are dependent on the factorisation scale µF, as well as the renormalization scale µR. The cross section σ of the process can then be

written σ =X a,b Z 1 0 dxadxb Z fh1 a (xa, µF)fbh2(xb, µF)dσab→n(µF, µR) (10) where fh1

a (xa, µF) is the PDF of parton a w.r.t hadron h1, xa the momentum fraction

carried by this parton, and dσab→n the cross section of the process ab → n which can be

calculated using perturbative QCD [21].

Since the internal interactions of the protons involve low momentum transfers, the cor-responding value of the strong coupling constant is high and perturbation theory cannot be applied, although the evolution of the PDF’s with energy scale is known and described by the DGLAP equations [22]. The PDF’s can then be experimentally measured (for example with deep inelastic scattering of electrons) and then evolved to the energy scale of collisions at the LHC. PDF’s are made available by the NNPDF collaboration. PDF’s of protons at the LHC are available at LO, NLO and NNLO accuracy. In this work the NLO PDF set NNPDF30 nlo as 0118 nf 4 is used, which is available from [23].

The NLO PDF is needed for the LO calculations for the consistent use of the 4-flavour scheme, since at LO the PDF does not contain b-quarks.

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The cross section dσab→n in equation 10 is proportional to the matrix element Mab→n

squared, which can be calculated as a sum over Feynman diagrams F

Mab→n=

X

i

F_ab→n(i) (11)

with initial state ab and final state n. Since each additional interaction adds another factor of the associated coupling constant α, this expansion is truncated at a particular order. This leads to a Leading Order (LO) prediction called the tree level process, with Next to Leading Order (NLO) and NNLO corrections to this prediction involving additional loop corrections and initial and final state radiation of the tree level process. When the matrix element is calculated at LO, the normalization of the distribution is often badly described and the prediction can only be trusted for the shape of the distribution. In this case the prediction is often multiplied by a single constant called the K-factor so the prediction better matches the experimental data [21]. Unless stated otherwise all matrix elements in this work are generated with MadGraph5 aMC@NLO, available from [24].

3.1.1 Parton showers

As mentioned before, because the outgoing partons of the last section are coloured they cause a ”shower” of particles which are observed in the detector as jets. These jets have the same total energy and momentum as the initial parton. The showering of such partons is a Markov process, and can therefore be simulated as an evolutionary process by choosing to add a parton to the final state one at a time. Such an algorithm is called a parton shower algorithm. The algorithm is formulated as an evolution down from the energy scale of the hard scattering event to the scale ΛQCD where QCD becomes strongly interacting, on the

order of 1 GeV.

Since the parton shower includes in its description the emission of a gluon from one of the initial or final state particles of the tree level process, particular care needs to be taken to avoid double counting when using a NLO or higher prediction of the matrix element. Here that is done by giving certain events in MadGraph5 aMC@NLO negative weights. The parton shower algorithm used is PYTHIA 8, available from [25].

3.1.2 Hadronisation

The nature of the strong interaction is such that the potential energy between 2 colour charged particles actually increases as a function of distance, by approximately 1 GeV/fm [11, p.252]. That means that when the energies of the partons get sufficiently low, all partons will cluster into colourless combinations which we observe as hadrons. This process, known as hadronisation, happens at an energy scale where QCD is strongly interacting and can therefore not be calculated from first principles using perturbation theory. Phenomenological models were therefore developed with many free parameters which could be tuned to fit the data. While these models are based on QCD, they are at the moment not derivable from first principles. Since many of the hadrons produced are unstable, their decay is also modelled until we arrive at particles considered stable on the scale of the detector.

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Figure 12: A pictographical representation of a generated event. The hard scattering event and associated parton shower are depicted in red, the hadronisation of the partons in light green and the decay of these hadrons in dark green. Photon radiation in yellow happens at all stages. A secondary scattering is also shown in purple. Taken from [26].

3.2 Soft subprocesses

The hadrons involved in the hard scattering process are complex bound states of partons and a sea of virtual gluons. These additional partons and gluons which are not involved in the hard scattering event can also interact with each other, leading to additional par-ticles throughout the event which can contribute to any observable. These soft processes, like hadronisation, happen at energy scales where QCD is strongly interacting. They are therefore also modelled using phenomenological models.

3.3 Event selection

There are 2 kinds of cuts applied on the results from the event generator. The first are kinematic cuts based on the acceptance of the ATLAS detector, the second are cuts for reducing the number of background events when measuring real data. All the cuts applied are

• lepton pT > 25 GeV

• for all observed particles |η| < 2.5 • Emiss

T > 35 GeV

• transverse mass of the lepton- Emiss

T system > 60 GeV

• the invariant mass of the top quark is in the range 120.6 − 234.6 GeV • mass of the spectator jet-top quark system > 320 GeV

• the scalar sum of the pT of all observed particles must be larger then 190 GeV

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• A trapezoidal requirement to reject background events ηj< 4ηÈmiss T b+ a ∩ ηj > 4ηÈmiss T b− a ∩ ηj> 0.44ηÈmss T b+ b ∪ ηj< 0.44ηÈmiss T b− b

where a and b are 10 and 2 respectively, and j stands for the spectator jet • mass of the lepton- b-jet system < 153 GeV.

4 Method

For the measurement of EFT parameters from distributions we need a continuous probability density function (pdf) of the data as a function of the parameter to be measured. The predictions which we will use are taken from a MC event generator with a particular set of EFT parameters, but we need to interpolate between these templates to make the pdf continuous in the SMEFT parameterspace. To construct such a pdf from a set of templates a method called morphing is used, which uses the structure of the process to be studied to interpolate between templates. An overview of the technique can be found in [27].

4.1 Morphing

The basic assumption of morphing is the fact that the cross section σ of a process is pro-portional to the matrix element M squared. Since the matrix element can be calculated as a sum of Feynman diagrams, we can write the matrix element as a sum of SM and EFT contributions. To illustrate the technique, consider the problem where a single vertex of the matrix element is modified by an EFT contribution. Then assuming for simplicity that all Wilson co¨efficients are real

σ(CSM, CEF T) ∝ |M|2 = |CSMMSM+ CEF T Λ2 MEF T| 2 = |CSM|2|MSM|2+ 2 1TeV2 Λ2 CEF TCSMRe(M ∗ SM· MEF T) + 1TeV4 Λ4 |CEF T| 2_|M EF T|2 = |CSM|2OSM+ Re(CEF TCSM)Oint+ |CEF T|2OEF T (12) where there is a pure SM term, an interference term between the EFT operator and the SM, and a pure EFT term, and the scale Λ is normalized to 1 TeV. This is a polynomial equation with 3 unknowns (the values Oi), which we can solve if we have 3 distinct solutions.

Suppose we have generated samples with parameters (CSM, CEF T) = (1,0), (1,1) and (0,1).

Then we can write down the system of equations

   σ(1, 0) σ(1, 1) σ(0, 1)   =    1 0 0 1 1 1 0 0 1       OSM Oint OEF T   . (13)

The matrix appearing in equation 13 is called the coupling matrix (CP ). By inverting the coupling matrix we can calculate the Oi’s in terms of the input templates. In this

particular case,

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   OSM Oint OEF T   =    1 0 0 −1 1 −1 0 0 1       σ(1, 0) σ(1, 1) σ(0, 1)   . (14)

This allows for the calculation of the cross section for any (CSM, CEF T) with the help

of equation 12, the weights of the templates can be read of the columns of CP−1. In this example

σ(CSM, CEF T) = (1CSM2 − 1CSMCEF T+ 0CEF T2 )σ(1, 0)

+(0C_SM2 + 1CSMCEF T+ 0CEF T2 )σ(1, 1)

+(0C_SM2 − 1CSMCEF T+ 1CEF T2 )σ(0, 1).

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In general, the morphing cross section can be written as a sum over k templates, where k is the number of terms appearing in the analogue of equation 12. The requirement that CP be invertible translates to the requirement that the parameters of the templates are linearly independent.

4.2 Morphing in single top production

In t-channel production of the single top, an EFT operator can enter the Feynman diagram in 2 places, at the production and decay of the top quark. To apply the morphing principle, the narrow width approximation will be assumed where we will be neglecting the integral over the invariant momentum of the top quark. In that case the total cross section for single top production as in figure 2 can be written as the product of the production and decay cross section σ = σprod· σdec ∝ |M|2 prod· |M| 2 dec. (16)

In the case considered here, the OtW operator can be added both at the production and

decay vertex. The expansion analogous to equation 12 is

σ(CtW, CitW) = |MSM+ CtW Λ2 MtW+ CitW Λ2 MitW| 2 dec· |MSM + CtW Λ2 MtW+ CitW Λ2 MitW| 2 prod =X i KiOi (17) where the sum runs over all possible Ki which are combinations of {1,C_ΛtW2 ,

CitW

Λ2 } with

a total power of 4 (this includes terms of the form _Λ141

2_C2 tW, 1 Λ8C 1 tWC 3

itW). This means

there are 4+3−1₄ = 15 terms. By generating 15 solutions to this equation we can then write down and invert the coupling matrix, allowing us to calculate weights of the input templates and calculate the cross section for arbitrary input parameters.

Each power of a Wilson co¨efficient in this expansion suppresses the associated operator by a factor of Λ−2. Since we are only considering the Λ−2 expansion of the EFT Lagrangian, we can expect EFT operators of dimension 8 to interfere with the terms of power 8 in equation 17. The argument could therefore be made that these terms should not be considered in the

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1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.1 0.11 0.12 0.13 0.14 0.15 0.16 x θ d cos σ d σ 1 morphing_ctWI_c0_ctW_c1 ctWI_c0_ctW_c1 chi2_morphing = 0.692138 (a) cos θx 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 y θ d cos σ d σ 1 morphing_ctWI_c0_ctW_c1 ctWI_c0_ctW_c1 chi2_morphing = 1.011201 (b) cos θy

Figure 13: Morphing examples of the normalized differential cross sections of the cosine of the angles θx, θy defined in equation 9. The morphing function allows us to predict the

distribution of a sample with arbitrary EFT parameters using 15 templates. In this case the morphing is done to a sample with CtW = 1 and CitW = 0.

expansion of equation 17, since these will be effected by unknown higher order operators. In this report however, none of the terms in equation 17 will be neglected.

Examples of the output of the morphing are shown in figure 13. It can be seen that the agreement between the morphing function and the target sample is within the margin of error.

4.2.1 Error calculation morphing

The weights of the input templates calculated in the last section are completely determined by the parameters of the input samples, which are known exactly. This means that the error on the morphing function is completely determined by the statistical errors on the input samples. Since each morphing function is essentially a weighted sum over the input templates, the morphing prediction can be written as

σ(CtW, CitW) =

X

k

wk(CtW, CitW)σ(k) (18)

where the sum runs over all input templates, σ(k)denotes template k and wkthe weight

of template k. The total error on the morphing sample can be calculated by Guassian error propagation, which leads to the formula

δi= v u u t 15 X k=1 w2 k(CtW, CitW)(δ (k) i )2 (19)

where i stands for the bin number and δ_i(k) for the error on bin i of template k. A heat plot of the average error on an 8 bin morphing of the setup used with LO templates is shown in figure 14. The precision of the morphing function is highly sensitive to the parameters of the input templates. Interestingly, it can happen that the morphing function is more precise than any one of the input samples in regions where the input samples overlap.

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10 − −5 0 5 10 tW C 10 − 5 − 0 5 10 itW C 0 0.002 0.004 0.006 0.008 0.01 0.012

Heat plot of the average error on an 8 bin morphing function

Figure 14: The absolute error averaged over the bins of an eight bin morphing function in the CtW, CitW plane with LO input samples. Since the input samples all have approximately

the same number of events, this shows the variable error due to the interpolation. The input parameters to the morphing function are shown as the red triangles. In regions where input parameters overlap it can happen that the morphing function is more precise than any one of the input samples.

4.3 Measurement of parameters

For the measurement of the parameter the likelihood ratio test is used. The likelihood L of a particular experimental outcome is defined as

L(µ) = P (N |µ) (20)

where P (N |µ) denotes the probability density function (pdf) at the outcome of the measurement N given the parameter µ. In the case of multiple independent experiments, the pdf at the outcomes (N1, N2, ..., Nk) is the product of the individual pdfs. The best fit

value of the parameter is found by maximizing the likelihood function by varying µ. This value of µ is usually written as ˆµ. The likelihood ratio is defined as the ratio of L(µ) with L(ˆµ). For practical reasons, the negative log likelihood Λ

Λ = −2logL(µ)

L(ˆµ) (21)

is used, so the likelihoods in the case of multiple experiments can just be added up and Λ can be minimized instead of maximized.

Wilks’ theorem states that Λ has a χ2 _{distribution with dim(µ) degrees of freedom, which}

allows the definition of confidence intervals. In the case of 1 parameter the N sigma confi-dence interval are the values µ for which Λ(µ) < N2_{. This was the reason for the factor of}

2 in equation 21.

In the particular case here, the approximation is made that the first N − 1 bins of both

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the pdf and the MC sample are independent normally distributed random variables. The likelihood of bin i is then

Li = ₁ σi √ 2πExp ∆2 i 2σ2 i (22)

where ∆i is the difference between the expected and measured value and σ2i = σ2pdf,i+

σ_{M C,i}2 is the quadratic sum of the error on the pdf and of the MC sample. The likelihood

of the entire differential distribution is then

−2logL = −2log Nbins−1 Y i=1 Li = −2 Nbins−1 X i=1 log(Li) = − Nbins−1 X i=1 2log(σi) + ∆i σi 2 + C (23)

where ∆i is the difference in bin i, σi as in eq. 22, C a constant and Nbins the number

of bins of the distribution. Since the distributions are normalized, one of the bins can be calculated from the rest and so is not an independent experiment. Therefore the sum in eq 23 runs over all bins except the last one.

Since the precision of the morphing function can increase in regions where multiple input samples overlap, the log(σ) term in equation 23 can cause the likelihood function to be maximized at different values of the parameter when applying the method to a sample in the morphing input (where ∆i = 0 for the correct parameter). This term is therefore

disregarded. The minus log likelihood that will be used in the calculation is

−2logL = Nbins−1 X i=1 ∆i σi 2 . (24)

The likelihood ratio Λ is then found by shifting the minus log likelihood distribution to 0. While the measurement of parameters using the likelihood ratio test can be done with any number of parameters, here 1 parameter will be varied at a time. In the case where more than 1 parameter is nonzero, the other one will be set to the value used to generate the sample.

5 Results

To test the method outlined in the last sections a closure test is done by fitting to the input, this should match exactly since the morphing prediction corresponds exactly to the distri-bution to be fitted. Then the LO templates are used to measure the parameter values from LO samples, and the same for NLO templates. Finally there are the results for measuring parameters from NLO samples with LO templates.

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5.1 Closure test

In figure 15 is shown an example of a plot from the closure test, in which it is visible that the morphing prediction, fit result and distribution to be fitted exactly overlap. All results of the fits in the closure test and the 1 σ error bars are shown in figure 16. As visible there all measured parameters match. The variable error bars can be attributed to the variable precision of the morphing expression. This shows that the fit using the morphing templates works as expected. 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.05 0.1 0.15 0.2 0.25 0.3 y θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 0.000000 morphing Λ (a) cos θx 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 0.000000 morphing Λ (b) cos θy

Figure 15: Example plots of the closure test with LO templates. The fit (black), morphing of the correct parameters (red) and the sample (blue) are all the same.

SM ctWI_c2p63158_ctW_c3p68421 ctWI_c-4p73684_ctW_c1p57895 ctWI_c0p526316_ctW_c-6p84211 ctWI_c6p8421_ctW_c-3p68421 ctWI_c6p84211_ctW_c3p84211 ctWI_c-5p78947_ctW_c-5p78947 ctWI_c-5p78947_ctW_c5p78947 ctWI_c-10_ctW_c0p526316 ctWI_c1p57895_ctW_c-10 ctWI_c10_ctW_c3p68421 ctWI_c10_ctW_c-4p73684 ctWI_c4p73684_ctW_c10 ctWI_c-4p73684_ctW_c-10 ctWI_c10_ctW_c-10 ctWI_c-10_ctW_c10 ctWI_c-10_ctW_c-10 1.5 − 1 − 0.5− 0 0.5 1 1.5 measured - true tW C xθ

compared to true value using cos

tW

Measurement of C

xθ

tW Measurement of C SM ctWI_c2p63158_ctW_c3p68421 ctWI_c-4p73684_ctW_c1p57895 ctWI_c0p526316_ctW_c-6p84211 ctWI_c6p8421_ctW_c-3p68421 ctWI_c6p84211_ctW_c3p84211 ctWI_c-5p78947_ctW_c-5p78947 ctWI_c-5p78947_ctW_c5p78947 ctWI_c-10_ctW_c0p526316 ctWI_c1p57895_ctW_c-10 ctWI_c10_ctW_c3p68421 ctWI_c10_ctW_c-4p73684 ctWI_c4p73684_ctW_c10 ctWI_c-4p73684_ctW_c-10 ctWI_c10_ctW_c-10 ctWI_c-10_ctW_c10 ctWI_c-10_ctW_c-10 1.5 − 1 − 0.5− 0 0.5 1 1.5 measured - true itW C yθ

itW

measurement of C

yθ

itW

measurement of C

Figure 16: Results of the closure test by measuring parameters from LO samples also used as templates in the LO morphing function. On the left, the measurement of CtW using the

cos θx distribution of the leptons. On the right the measurement of CitW using the cos θy

distribution of the leptons. Shown is the difference between the measured value with the true value. The 1 σ confidence interval is shown in blue.

5.2 Fitting to the same order

To further prove that this method can be used to measure parameters from differential dis-tributions the parameters are measured from samples using templates of the same order. In

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what follows all parameters of the pseudodata are not used in the templates of the morphing function.

First are the results at LO. In figure 17 is shown an example of the result of the fit to-gether with the likelihood scan for the calculation of the confidence interval. In figure 18 are the differences of the measured parameters and the true parameters shown using cosθxand

cosθy of the leptons for the measurement of CtW and CitW respectively. The measurement

of CtW with the cos θxdistributions seems to be in good agreement, with 2 out of 7 outside

the 1 σ interval, the measurement of CitW with the cos θydistributions is consistently high,

but within limits. This can be explained by a statistical deviations in the templates used in the morphing function, in particular the SM sample, since it is the biggest contributor to the morphing expression close to the SM. The results of the measurement of CtW and CitW

using the pT distributions of the leptons is shown in figure 19. Here the agreement for CtW

is not regular since the 2 samples with nonzero CtW are outliers. For CitW the agreement is

better, only the large outlier with CitW = 3, CtW = 0 measures as minus it’s actual value.

There is another local minimum at CitW ≈ 3.5. The deviation with the true value was

measured at 2.35 σ. It is unclear what the cause is of this large deviation. This shows the pT of the leptons have some sensitivity to these EFT parameters. It can be seen by the size

of the error bars that the pT distribution is more sensitive to CtW than CitW. It is also clear

that both CtW and CitW are more accurately measured by the angular distributions of the

leptons.

The results for the measurement of CtW and CitW using the distributions of cosθx and

cosθyof the leptons respectively at NLO are shown in figure 20. The larger uncertainties here

originate from the greater statistical errors on the pseudodata used. Since the agreement here is good this shows that the method used returns the correct value of the EFT parameter when the prediction is of the same order.

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult

= 1.937339 morphing

Λ

(a) Fit and morphing result.

0.2 − 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 tW C 0 5 10 15 20 25 30 35 Λ ctWI_c0_ctW_c1 ctW = 0.78 - 0.15 + 0.16 at 1 #s confidence ctWI_c0_ctW_c1

(b) Result of likelihood scan over the CtW

param-eter.

Figure 17: Example of result of the measurement of the CtW parameter using LO templates

from a LO sample with parameters that are not in the input. Left: the fit (black), the morphing result (red) in comparison with the sample (blue). Right: the likelihood scan over a range of CtW values. The value measured was CtW = 0.76+0.16−0.15 in comparison with the

correct value of CtW = 1.

5.3 LO to NLO

As mentioned in the introduction, LO predictions could be used to measure EFT param-eters from data since they are easier to generate, but this could also lead to biases in the measurements. Since real data is more like NLO than LO, it is interesting to measure EFT parameters from NLO distributions using LO templates, since this models the biases

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ctWI_c-0p33_ctW_c0 ctWI_c0p66_ctW_c0 ctWI_c1_ctW_c0 ctWI_c0_ctW_c1 ctWI_c-1p5_ctW_c0 ctWI_c3_ctW_c0 ctWI_c0p3_ctW_c3 1 − 0.8− 0.6− 0.4− 0.2− 0 0.2 0.4 0.6 0.8 1 measured - true tW C xθ

tW

Measurement of C

xθ

tW Measurement of C ctWI_c-0p33_ctW_c0 ctWI_c0p66_ctW_c0 ctWI_c1_ctW_c0 ctWI_c0_ctW_c1 ctWI_c-1p5_ctW_c0 ctWI_c3_ctW_c0 ctWI_c0p3_ctW_c3 1 − 0.8− 0.6− 0.4− 0.2− 0 0.2 0.4 0.6 0.8 1 measured - true itW C yθ

itW

measurement of C

yθ

itW

measurement of C

Figure 18: Results of the measurements with LO templates and LO pseudodata not used in the morphing prediction. On the left, the measurement of CtW using the cos θxdistribution

of the leptons. On the right the measurement of CitW using the cos θy distribution of the

leptons. Shown is the difference between the measured value with the true value. The 1 σ confidence interval is shown in blue.

which occur when using LO templates to measure EFT parameters from real data. First the parameters were measured from an NLO SM sample with LO templates. The results are shown in figure 21. The measured parameters from the fit are CtW = 0.47+0.16_−0.15 and

CitW = 0.32+0.12−0.13. This represents a 2.5 sigma deviation in both cases.

In figures 22 and 23 some of the fit results for cos θxand cos θy are shown together with

the morphing function for the correct parameters. From these figures it can be seen that the fit has a better agreement with the sample than the morphing prediction, and also that the morphing prediction can have large deviations from the sample.

In figure 24 are shown results of the fit with LO templates multiple NLO samples, from left to right in increasing distance from the SM. The large confidence intervals on some of the samples originate from the samples lying on the edge of the allowed parameterspace. Since CtW and CitWare both expected to be below 1, the samples closest to the SM are reproduced

separately in figure 25. It can clearly be seen that the deviation from the LO prediction depends on the distance from the SM, and close to the SM there is reasonable agreement. Since the larger parameters are used in the calculation however, using LO predictions will lead to biases.

For the measurements of parameters from the pT distributions of the leptons large shifts

were observed. The results for a fit to the NLO SM sample are shown in figure 26. The measured parameters were CtW = 0.84+0.31−0.27and CitW = 0.04+0.70−0.66. Notable is that

appar-ently the NLO corrections are interpreted as a nonzero EFT parameter for CtW, but not

for CitW. That the NLO corrections can be observed as a CtW contribution is also shown

in figure 27.

The results for multiple samples are shown in figure 28. It can be seen that the differences using this distribution can be close to the allowed parameter range, even when the distri-bution is close to the SM. This is because the distridistri-butions match nowhere, so the minus

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ctWI_c-0p33_ctW_c0 ctWI_c0p66_ctW_c0 ctWI_c1_ctW_c0 ctWI_c0_ctW_c1 ctWI_c-1p5_ctW_c0 ctWI_c3_ctW_c0 ctWI_c0p3_ctW_c3 2 − 1.5− 1 − 0.5− 0 0.5 1 1.5 2 measured - true tW C T

compared to true value using lep p

tW

Measurement of C

T

tW Measurement of C (a) Measurement of CtW. ctWI_c-0p33_ctW_c0 ctWI_c0p66_ctW_c0 ctWI_c1_ctW_c0 ctWI_c0_ctW_c1 ctWI_c-1p5_ctW_c0 ctWI_c3_ctW_c0 ctWI_c0p3_ctW_c3 6

− ₄− C₂−itW measured - true₀ ₂ ₄ ₆

T

itW

measurement of C

T

itW

measurement of C

(b) Measurement of CitW

Figure 19: Results of the measurements with LO templates and LO pseudodata not used in the morphing prediction using the pT distribution of the leptons. On the left, the

measure-ment of CtW, on the right, the measurement of CitW. The 1 σ confidence interval is shown

in blue.

log likelihood is minimized by maximizing the error on the morphing function. This means of course that the LO prediction does not describe the shape correctly for some parameter values. For the measurement of CtW at least, it seems there is a trend to underestimating

the parameter.

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ictW-c1-ctW-c1 ictW-c1-ctW-c0 ictW-c-1-ctW-c-1 ictW-c10-ctW-c10 ictW-c1-ctW-c-1 ictW-c-1-ctW-c0 ictW-c0-ctW-c-1 ictW-c0-ctW-c1 5 − ₄− ₃− ₂− 1 − measured - true₀ ₁ ₂ ₃ ₄ ₅ tW C xθ

tW

Measurement of C

xθ

tW Measurement of C ictW-c1-ctW-c1 ictW-c1-ctW-c0 ictW-c-1-ctW-c-1 ictW-c10-ctW-c10 ictW-c1-ctW-c-1 ictW-c-1-ctW-c0 ictW-c0-ctW-c-1 ictW-c0-ctW-c1 5 − ₄− ₃− ₂− 1 − measured - true₀ ₁ ₂ ₃ ₄ ₅ itW C yθ

itW

measurement of C

yθ

itW

measurement of C

Figure 20: Results of the measurements with NLO templates and NLO pseudodata not used in the morphing prediction using the pT distribution of the leptons. On the left, the

measurement of CtW, on the right, the measurement of CitW. The 1 σ confidence interval

is shown in blue. 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 7.546573 morphing Λ 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.05 0.1 0.15 0.2 0.25 0.3 y θ d cos σ d σ 1

= 6.014671 morphing

Λ

Figure 21: The result of the fit (black) using LO templates to an NLO SM sample (blue). The (LO) morphing expression of the SM is shown in red. The cos θx distribution of the

leptons on the left is used to measure CtW and the cos θy distribution on the right for the

measurement of CitW. The parameters measured are CtW = 0.47+0.16_−0.15and CitW = 0.32+0.12_−0.13.

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1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

= 45.278867 morphing

Λ

(a) CtW = −6.84 and CitW = 0.52

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 100.081425 morphing Λ (b) CtW= −10 and CitW= 1.57 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 0.005578 morphing Λ (c) CtW = 3.68 and CitW= 2.63 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 x θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 40.763192 morphing Λ (d) CtW = −3.68 and CitW = 6.84

Figure 22: Plots of the (LO) morphing result with the correct parameters (red), the result of the fit (black) and the NLO sample (blue) for cos θx distributions of the leptons with different values

for CtW, CitW. The minus log likelihood Λ of the correct parameters is also shown in the legend.

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 y θ d cos σ d σ 1

= 31.526631 morphing

Λ

(a) CtW = −6.84 and CitW = 0.52

1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 65.664224 morphing Λ (b) CtW= −10 and CitW= 1.57 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 9.415129 morphing Λ (c) CtW = 3.68 and CitW= 2.63 1 − −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 y θ cos 0.05 0.1 0.15 0.2 0.25 0.3 0.35 y θ d cos σ d σ 1

morphing to correct parameters validation fitresult = 18.607155 morphing Λ (d) CtW = −3.68 and CitW = 6.84

Figure 23: Plots of the (LO) morphing result with the correct parameters (red), the result of the fit (black) and the NLO sample (blue) for cos θy distributions of the leptons with different values

for CtW, CitW. The minus log likelihood Λ of the correct parameters is also shown in the legend.

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SM ictW_c1_ctW_c0 ictW_c0_ctW_c1 ictW_c-1_ctW_c0 ictW_c1_ctW_c-1 ictW_c-1_ctW_c-1 ictW_c1_ctW_c1 ictW_c-0p3_ctW_c-3 ictW_c2p63158_ctW_c3p68421 ictW_c-4p73684_ctW_c1p57895 ictW_c0p526316_ctW_c-6p84211 ictW_c6p84211_ctW_c-3p68421 ictW_c-5p78947_ctW_c-5p78947 ictW_c-10_ctW_c0p526316 ictW_c1p57895_ctW_c-10 ictW_c10_ctW_c3p68421 ictW_c-4p73684_ctW_c-10 ictW_c4p73684_ctW_c10 ictW_c10_ctW_c-4p73684 ictW_c-10_ctW_c-10 ictW_c10_ctW_c-10 ictW_c-10_ctW_c10 15 − ₁₀− 5 − ₀ ₅ 10 15 measured - true tW C xθ

tW

Measurement of C

xθ

tW Measurement of C SM ictW_c1_ctW_c0 ictW_c0_ctW_c1 ictW_c-1_ctW_c0 ictW_c1_ctW_c-1 ictW_c-1_ctW_c-1 ictW_c1_ctW_c1 ictW_c-0p3_ctW_c-3 ictW_c2p63158_ctW_c3p68421 ictW_c-4p73684_ctW_c1p57895 ictW_c0p526316_ctW_c-6p84211 ictW_c6p84211_ctW_c-3p68421 ictW_c-5p78947_ctW_c-5p78947 ictW_c-10_ctW_c0p526316 ictW_c1p57895_ctW_c-10 ictW_c10_ctW_c3p68421 ictW_c-4p73684_ctW_c-10 ictW_c4p73684_ctW_c10 ictW_c10_ctW_c-4p73684 ictW_c-10_ctW_c-10 ictW_c10_ctW_c-10 ictW_c-10_ctW_c10 8

− ₆− ₄−CitW₂− measured - true₀ ₂ ₄ ₆ ₈

yθ

itW

measurement of C

yθ

itW

measurement of C

Figure 24: Here is shown the difference between the measured parameter and the true value using LO templates from a NLO sample. On the left, the measurement of CtW using the

cos θx distribution of the leptons. On the right the measurement of CitW using the cos θy

distribution of the leptons. The samples are arranged in increasing distance from the SM from top to bottom. The 1 σ confidence interval is shown in blue.

SM ictW_c1_ctW_c0 ictW_c0_ctW_c1 ictW_c-1_ctW_c0 ictW_c-1_ctW_c-1 ictW_c1_ctW_c-1 ictW_c1_ctW_c1 ictW_c-0p3_ctW_c-3 2 − 1.5− 1 − 0.5− 0 0.5 1 1.5 2 measured - true tW C xθ

tW

Measurement of C

xθ

tW Measurement of C SM ictW_c1_ctW_c0 ictW_c0_ctW_c1 ictW_c-1_ctW_c0 ictW_c-1_ctW_c-1 ictW_c1_ctW_c-1 ictW_c1_ctW_c1 ictW_c-0p3_ctW_c-3 1.5 − 1 − 0.5− 0 0.5 1 1.5 measured - true itW C yθ

itW

measurement of C

yθ

itW

measurement of C

Figure 25: Here the difference is shown between the measured parameter and the true value using LO templates for NLO samples close to the SM. On the left the measurement of CtW

using the cos θxdistribution of the leptons, and on the right the measurement of CitW using

the cos θy distribution of the leptons. The 1 σ confidence interval is shown in blue.

Effective field theory in t-channel single top production at different orders in perturbation theory