Characterisation of Models of New Physics at the LHC

(1)

Master Thesis

Characterisation of Models of New Physics at the LHC

Author: Jakub Salko Supervisor: L. Panizzi Co-supervisor: V. Ellajosyula

Topical reviewer: R. Enberg

(2)

Abstract

The goal of this thesis is the development and application of model-independent analysis techniques to characterise New Physics states of different kinds from their signatures at the LHC and future colliders. In the first part of the thesis, we outline the general steps from the building of the simplified model and identification of the parameter space to the characterisation potential of the current searches with the focus on the characterisation between different scenarios of Beyond the Stan- dard Model physics. We then demonstrate the outlined procedure on two specific and relevant examples: discrimination of exotic decays of vector-like quarks and determination of the spin of a dark matter candidate.

Abstrakt

M˚alet med denna avhandling är utvecklingen och tillämpningen av modellobero- ende analystekniker för att karakterisera nya fysiktillst˚and av olika slag fr˚an deras signaturer vid LHC och framtida kolliderare. I den första delen av avhandlingen redogör vi för de allmänna stegen fr˚an uppbyggnaden av den förenklade modellen och identifiering av parameterrummet till karaktäriseringspotentialen för de aktuella sökningarna med fokus p˚a karaktäriseringen mellan olika scenarier för Beyond the Standard Model-fysik. Vi demonstrerar sedan det beskrivna förfarandet p˚a tv˚a specifika och relevanta exempel: differentiering av exotiska sönderfall av vektorlik- nande kvarkar och bestämning av spinnet hos en kandidat för mörk materia.

(3)

Introduction

Currently, one of our most fundamental theories of nature is the Standard Model (SM) of particle physics. Over the history of its development, it celebrated many remarkable successes but even after several decades since its formulation, it still leaves some bugging questions unanswered. There are several problems within the theoretical framework of the SM, such as the hierarchy problem, naturalness or the stability of the SM vacuum.

Perhaps even more burning problems are the discrepancies in predicted and observed phenomena such as existence of Dark Matter (DM) or non-zero neutrino masses, where the SM tells us these should not be observed.

There is a huge theoretical and experimental effort put into explaining the missing pieces of the SM. Unfortunately, these attempts to probe the physics Beyond the Standard Model (BSM) have gone without a success so far.

One of the problems of doing BSM physics is that there is not one specific way of formulating the new theory. In fact, there are countless ways of extending the SM by introducing new symmetries and particles or even leaving behind the Quantum Field Theory framework of the SM and introducing brand new physical concepts. It is needles to say that physicists all over the world are taking every path imaginable in order to provide a better idea of how the universe works.

Among many of the attractive approaches to describe the New Physics (NP), it is not surprising to find several theoretical scenarios which are equally feasible to explain one thing or another. This poses a challenge for experimentalists who need to be ready for not only providing experimental evidence for new phenomena but also to be able to distinguish among different theoretical scenarios predicting such phenomena.

In this thesis, we aim to provide a simple framework for studying several NP scenarios giving the same (or very similar) observational signatures by largely model-independent means employing only simplified models. We describe how to contrast the theoretical predictions with currently available experimental results, how to estimate the regions of simplified parameter space where discovery could be possible and how to characterize different theoretical scenarios in order to discriminate among them in case of possible future observation.

This paper is organized as follows. In Part I, we layout the general strategy for the phenomenological study. The starting point for the analysis is a simplified Lagrangian and its implementation into a format that can be used by event generators which is described in Sec. 1.1. We proceed with the discussion of limiting the number of free parameters describing the physics of interest in order to identify the relevant parameter space in Sec. 1.2. This parameter space is then scanned over to find exclusion bounds by recasting the results of past experimental searches which is outlined in Sec. 1.3 and Sec. 1.4. The general strategy outline is concluded is Sec. 1.5 by describing a simple χ² test to probe if the potentially observable signal of New Physics could be attributed to one specific theoretical scenario or if the discrimination among the studied models is not realizable within the framework described.

(5)

Further, in Part II, we apply the outlined strategy to two specific examples. Namely, these are the exotic decays of vector-like quarks in Sec. 2 and the discrimination of spin of t-channel dark matter candidates in Sec. 3. Finally, we summarize in Sec. 4.

(6)

Part I

General analysis framework

1 From theory to experiment

In this part of the thesis, we try to provide a simple description of the analysis framework which can be used for a broad range of different phenomenological studies. Some parts of this framework are already well established and known throughout the community.

In such cases, we refer the reader to a dedicated literature rather than providing a full description at each step. We try to keep the discussion in this part as general as possible and postpone the direct application to Part II where we apply the general framework to two different examples.

1.1 Model building

There are two main approaches for how to build a BSM theory. One way is to start with what is already known (the physics of the SM) and finding simple extensions that would explain the missing pieces while keeping the known physics unspoiled. Such a bottom- up approach is very popular for solving problems of the SM with the building blocks already existing within the SM. The Standard Model Effective Field Theory (SMEFT) or the Higgs Effective Field Theory (HEFT) are two great examples of this approach.

For more in-depth review of the effective field theory approach as well as some details on the SMEFT and HEFT, see Ref. [1].

On the other hand, the top-down approach to constructing a BSM theory starts with some kind of fundamental principle or symmetry which goes beyond the SM, builds a full theory around it and then tries to match its predictions to both the SM physics and observable effects of the physics beyond the SM. As an example of a theory built with a top-down approach, any Grand Unified Theory (GUT) [2], Supersymmetric theory (SUSY) [3] or String theory [4] can be used.

One thing common to all of the BSM theories is presence of new particles and their interactions with the SM. While each model has its own specific particle content and is distinct by its mass spectrum of new particles and what interactions are allowed among them, there is an inevitable overlap between the particle contents of different theories. It can then be of some importance to find the overlapping pieces and study them independently on what specific theory they come from. This can be achieved by formulating a simplified model, a simple extension of the SM encompassing the pieces of interest while forgetting about the full complicated theory set up. However, it should be clear how the simplified model comes from a specific BSM theory and how the results based on the simplified model translate back to the full theory.

The simplified models are studied from the perspective of a specific observational signature. For example, one might be interested in studying properties of vector-like quarks in processes with final states featuring same-sign leptons or studying properties of dark matter in final states featuring jets. The simplified model should adopt all

(7)

the pieces of new physics that can give rise to processes probed by the observational signature. One might therefore want to add more than the minimal number of additional particles or even non-renormalisable effective interactions in order to properly include all the relevant new physics effects. The predictions of the BSM effects are then contrasted with the pure SM predictions assumed to contribute to the observational signature as a background. Once the effective Lagrangian is formulated, it can be translated into a format readable by event generators and subjected to a phenomenological study.

For this purpose in the following, we make use of the FeynRules 2.3 [5] package for Mathematica. Upon defining all fields, parameters and the Lagrangian itself, the FeynRules package generates the complete set of Feynman rules. Depending on the formulation of the Lagrangian, the resulting Feynman rules are suited for leading order and/or next-to-leading order calculations. It also provides different output formats meant for different computational tools. In this thesis, we make use of two specific formats, namely the Universal FeynRules Output (UFO) format [6] and CalcHep [7]

format.

1.2 Identification of parameter space

Simplified models are not always as simple to investigate as the name would suggest.

Often times, it is easier to define a simplified model describing several mechanisms producing the signature of interest and extract the specific cases by imposing different restrictions on the model. For example, consider a Lagrangian describing one new scalar and one new pseudoscalar particle with different couplings to the SM particles and among themselves. One possible restriction posed on such a model is to consider only the scalar particle having non-zero couplings to the SM and keeping the pseudoscalar state decoupled from any considered interaction. This approach can be very useful in studying contributions from separate cases to the whole picture. On the other hand, studying the unconstrained model benefits from the ability to study not only specific cases one by one but also to explore the possibility of their simultaneous interplay, if desired.

The parameters remaining in the model restrictions constitute a parameter space of interest. These are in general masses of the new particles and the coupling strengths governing their interactions. However, there can be further constraints imposed on such parameter space. For example, if the process under investigation features a decay of one new particle into another, the kinematics of such process require the mass order hierarchy among the new particles. The values of the couplings should be also kept in the perturbative regime since the analysis sketched here relies on the perturbation theory. On the other hand, the collider experiments are in general less sensitive to cases involving cross sections that are too small. For processes which has the cross section proportional to the coupling it is often reasonable to restrict the range for the values of the couplings by a lower bound which yields cross section below the experimental sensitivity.

Further, the Narrow Width Approximation (NWA) can be used to erase the sensitiv-

(8)

the cross sections in the NWA can factor out the coupling such that the cross section scales as some power of the coupling. The NWA is broadly used in phenomenological and experimental study of both the SM and BSM physics. It can be employed when studying processes involving intermediate resonant states. In general, such resonances can be created both on- and off-shell, but under certain assumptions, the difference between treating the full off-shell process as pure on-shell case is negligible. It then comes with the great advantage that the decay of the resonance can be treated independently of its production process and hence the factorisation of the cross section. The assumptions made in order to be able to rely on the NWA are, [8]:

• the ratio of the total width Γ of the resonance and its mass M should satisfy Γ/M 1,

• the decay products of mass m should satisfy m M,

• the center-of-mass energy√

s of the process subjected to the NWA should satisfy

√s M ,

• non-resonant processes should not (significantly) interfere with the resonant ones,

• the resonant propagator should be separable from the matrix element.

Under these assumptions, the NWA is in general expected to yield results with an error of order O(Γ/M ). However, depending on the specific process under consideration, there can be additional corrections which depend on the mass ratio of the decay products and the resonance, m/M , and/or on the velocity of the decay products, β =p1 − m²/M². In the following, we make use of the NWA assuming all the above conditions are met while explicitly assuring only the first condition. We consider our results valid as long as Γ/M . 1% but we expect to be able to draw some conclusions for Γ/M < 10% as well. We note that the second condition, namely m M is not always satisfied, namely in cases when the new resonance decays into another NP state when probing regions of the parameter space where the two new particles are close in mass. In the cases where m → M , the corrections proportional to inverse powers of β can become important.

However, we assume these corrections to be still negligible for processes studied in the following sections. See Ref. [8] for more details on the NWA assumptions and the possible breakdown of the NWA due to not meeting its assumptions.

Finally, the parameter space relevant for collider phenomenology can be constrained by independent observations. Particle physics progresses at rapid pace in all experimental areas, not only on the collider frontier. The results obtained by the complementary observations can therefore be very useful in guiding the efforts of the phenomenological studies at colliders.

1.3 Simulation and recasting strategy

Once the simplified model is implemented in the desired format, it can be used to simulate physical processes of interest. One of the most widely used tools for this purpose

(9)

is MadGraph5 aMC@NLO [9] (in the present work, we make use of MadGraph5 aMC@NLO v.2.7). It provides a framework for both leading order (LO) and next-to-leading order (NLO) calculation of cross sections and simulation of events at the parton level. These can be then matched to parton showers using the automatic interface to Pythia 8.2 [10].

The high level of automation provided by MadGraph5 aMC@NLO leaves only the need to specify the model implemented into a UFO format and the process to be simulated.

One should always take a good care of defining the process of interest via the generate command of MadGraph5 aMC@NLO. Both LO and NLO processes have their own subtleties which need to be handled properly (e.g. jet matching, treatment of narrow resonances at NLO, etc.). Ideally, the process should be defined as going from the initial state directly to the final state, without specifying intermediate particles present in the process. It is often advisable to select the processes involving specific particles or interaction by specifying coupling orders. This way, MadGraph5 aMC@NLO treats all the intermediate particles as off-shell states and correctly treats the finite width effects and spin correlations.

In cases where the specified final state contains unstable particles (e.g. due to time requirements for the full simulation), one can make use of MadSpin [11] which is ca- pable of decaying narrow resonances while approximately keeping spin correlations and finite width effects. Narrow resonances occurring in NLO simulations also need special treatment which can be taken care of by employing MadSTR plug-in¹. Finally, it is often necessary to split the signal process into different subprocesses. In such cases, it is important to carefully treat separate contributions as well as their interferences.

The framework of MadAnalysis5 [12] (we employ v.1.8) can then be used for con- fronting the simulated events with the already existing experimental results. Hadronized events are subjected to the detector simulation which is provided either by Delphes 3 [13] or FastJet [14] packages. The Public Analysis Database (PAD) [15] within the MadAnalysis5 framework contains implementations of many different searches of CMS and ATLAS collaborations at the Large Hadron Collider (LHC) which can be easily recast in the so called Normal Mode of MadAnalysis5. The Expert Mode of MadAnalysis5 [16] then provides an environment to easily implement other searches which are not yet available in PAD.

The full simulation and recasting framework of FeynRules², MadGraph5 aMC@NLO³ and MadAnalysis5⁴ is well established and documented. For more information on all these tools and their use, see the cited documentation or one of many talks and tutorials available online.

1.4 Obtaining current and projected collider bounds

The main results obtained from recasting the past searches within MadAnalysis5 is the efficiency ε_SR of the event selection in different Signal Regions (SRs) defined in the analysis. The fiducial cross section of the signal process in the specific SR can be then

1Available at https://code.launchpad.net/~maddevelopers/mg5amcnlo/MadSTRPlugin

2FeynRules homepage: http://feynrules.irmp.ucl.ac.be/.

3MadGraph5 aMC@NLO homepage: https://cp3.irmp.ucl.ac.be/projects/madgraph/.

4

(10)

defined as

ˆ

σ_SR = ε_SRσ , (1)

where σ is the theoretical cross section of the signal process. In contrast with the theoretical cross section, the fiducial cross section takes into account the detector and analysis effects and is therefore the actual quantity that can be directly measured experimentally.

It is closely connected to the number of signal events, sSR, observed (or expected to be observed) in the given signal region. This number is given as

sSR = L ˆσSR, (2)

where L is the luminosity of the specific experiment. This can be then directly compared to the number of expected background events b_SR± ∆b_SR in each signal region which is given by the experimental search. This comparison can be done in several ways. In the following, we focus on the calculation of Asimov significance and the CLs method.

To calculate the Asimov significance, z, we make use of an analytical formula given by

z =√ 2

(s + b) ln (s + b)(b + ∆b²) b²+ (s + b)∆b²

− b²

∆b²ln

1 + s∆b² b(b + ∆b²)

1/2

, (3)

which can be derived from the Asimov dataset using the method of likelihood ratio [17].

As a result, we can claim that the signal is excluded at the z-sigma level.

The CL_smethod is then based on the numerical evaluation of the likelihood ratio test- statistic [18]. This method has an advantage that it can be used to combine the results from different independent signal regions in a straightforward way. Our implementation of this method assumes the knowledge of expected number of signal events, s_SR± ∆s_SR, and the expected number of background events, bSR± ∆b_SR, together with the number of observed events, nôbs_SR, in each considered SR. It then generates a large number of toy experiments, n_toy, for each SR. The expected number of signal, sêxp_SR, and background, bêxp_SR, events are chosen at random, assuming they are normally distributed with mean and width equal to s_SR± ∆s_SRand b_SR± ∆b_SR, respectively. The negative values of sêxp_SR or bêxp_SR are automatically set to zero. The actual number of background, b^toy_SR, and signal plus background, s^toy_SR + b^toy_SR, events in a toy experiment is then randomly chosen from a Poisson distribution with the parameter bêxp_SR and sêxp_SR + bêxp_SR, respectively. Finally, the generated values are used to evaluate the test statistic for the background only, ln(Qb), and signal plus background, ln(Q_s+b), hypotheses given as

ln(Q_b) =X

SR

b^toy_SR ln s_SR+ bSR

b_SR

− s_SR

, (4)

and

ln(Q_s+b) =X

SR

(s^toy_SR + b^toy_SR) ln s_SR+ b_SR b_SR

− s_SR

. (5)

Similarly, the observed value of the test statistic given as ln(Q_obs) =X

SR

n^obs_SR ln s_SR+ bSR

b_SR

− s_SR

. (6)

(11)

The fraction of toy experiments with ln(Qobs) > ln(Qb), denoted as pb, and the fraction of events with ln(Q_obs) > ln(Q_s+b), denoted as p_s+b, are determined and the final result is given as

CL_s= p_s+b

p_b . (7)

The signal plus background hypothesis is then considered excluded at confidence level CL when

1 − CLs≤ CL . (8)

The two methods just described provide an easy way how to exclude a specific point of the probed parameter space. After scanning over several points of the parameter space, it is possible to use different interpolation methods to find more precise exclusion bounds. Often times cited bound is 2-sigma, or equivalently 95% CL bound.

A very easy extension of the bounds at the current experiment is their projection for luminosities at future experiments. Two benchmark luminosities of interest are L_III = 300 fb⁻¹ for the run 3 of LHC and LHL = 3 ab⁻¹ for the High-Luminosity LHC (HL- LHC). The naive projection can be done by evaluating significances and/or CL_s using the future value of the luminosity in Eqn. 2 while the number of background events is rescaled by a factor η = Lfuture/Lpresent. Both LHC run 3 and HL-LHC are expected to reach center-of-mass energy √

s = 14 TeV and, in principle, the projections should be done using the events generated at this future energy. However, the event generation can be a lengthy process when dealing with large scans of multi-dimensional parameter space. It is then still of some value to project the current bounds to future luminosities using events generated at the energy √

s = 13 TeV of current searches. Since the systematic uncertainties in the expected numbers of signal and background events are expected to saturate at high luminosities while the statistical uncertainties scale down with increasing luminosity, the systematics start to dominate the uncertainties at some point. In the absence of the information on how much of current uncertainties are made of statistics and how much of it is systematics, the total uncertainties should be assumed to be dominated by systematics and therefore fixed to a reasonable value (say, 10% or 5%). In contrary, if the full information on the uncertainties is available, the scaling to future luminosity can be done appropriately. Finally, the number of observed events should be replaced by the rescaled number of background events to obtain the expected rather than observed values. The interpolation of the significances/CL_scan be then used again to find the expected 2-sigma (or 95% CL) exclusion bound or to find the expected 5-sigma discovery reach. We note that the numerical precision of both CL_s method described above and the employed interpolation method is not expected to be accurate enough to give a reliable estimate of the 5-sigma discovery reach. For this reason, we use only the interpolation of Asimov significances to find the 5-sigma discovery reach.

1.5 Discrimination analysis

The region of the parameter space between the current 2-sigma bound and the future expected 5-sigma discovery reach is then of special interest since this is the region where

(12)

the discovery would be possible. We note, that after accumulating larger data sample at some future experiment the discovery is still possible even below the current 2-sigma bound. However, it is less and less likely to find an excess over the SM background as the points further below the current 2-sigma bound are probed. For this reason, we consider only the region of the parameter space between the current 2-sigma bound and the projected 5-sigma discovery reach as the domain of our discrimination analysis.

There are two main motivations for such an analysis. In case of a future discovery of the New Physics signal, the actual measured quantity in the experiment would be the fiducial cross section for the signal process⁵. However, the allowed region of the parameter space for one specific signal where the discovery can take place contains several points with the same fiducial cross section (actually iso-lines of constant ˆσ) which, in case of discovery, needs to be narrowed down to one specific point. The other motivation comes when there are two or more different scenarios which contain points of the same fiducial cross sections. In that case, the points from two different parameter spaces need to be compared and distinguished.

The difference between the points of the parameter space can then show up in different kinematic distributions of the final state objects. There are many different statistical methods of how to distinguish between two different distributions. In the following, we employ an easy χ² test. The standard χ² test compares an observed distribution with some theoretical null hypothesis and gives a measure of how similar the two distributions are [19]. However, rather than comparing observed and null distributions, our goal is to test if two “observed” distributions are compatible with each other. For this purpose, we construct a conservative χ² test as described below.

Suppose that we obtain a distribution of a specific kinematic variable observed in a specific signal region coming from two different theoretical scenarios. Construct the histograms of both distributions with the same range and same number of bins, nbins, and normalize them to the expected number of events at a certain luminosity L. We can calculate the measure

¯ χ² =

nbins

X

i=1

(Xi− Y_i)²

max(X_i, Y_i) + ε^bckg_cont(b_SR+ (ε_systb_SR)²), (9) where Xi and Yi, assumed to be Poisson distributed, represent the contents of the bin i of the two histograms, b_SR is the expected number of background events in the specific signal region under consideration, ε_syst= ∆b_SR/b_SR is its relative uncertainty and ε^bckg_cont is the background contamination of the signal region. Here, we pessimistically assume that the background is distributed in the same way as the signal. The ¯χ² variable is expected to follow χ² distribution with ndf = n_bins− n₀ degrees of freedom, where n₀ is the number of bins with zero content in both compared distributions. This holds as long as the expected number of events in the compared histograms is large enough⁶. We

5However, in processes with unstable resonances decaying into visible particles only, one would also observe characteristic resonance peak in the distribution of the invariant mass of the decay products which is much easier to identify as coming from the New Physics signal.

6In the following, we consider large enough to be & 100 such that under an assumption of Poisson statistics, the uncertainty in the number of expected events in the histogram is < 10%

(13)

then calculate the p-value defined as p =

Z ∞

¯ χ²

χ²(x, ndf ) dx , (10)

and we claim that the two distributions can be distinguished at the confidence level CL if p < 1−CL. This p-value can be also converted into a significance using the formula

Z_χ² = Φ⁻¹(1 − p) , (11)

where Φ⁻¹ is the inverse of the cumulative distribution of the standard Gaussian [19].

Equivalently to the CL exclusion, two distributions are considered distinguishable at Z-sigma level if Z_χ2 > Z.

We note that the results of the χ² test as described here are dependent on the binning of the histograms as well as on the histogram range. In order to obtain the best results from the χ² test, one should find suitable values of the binning and the range of the tested histograms.

(14)

Part II

Applications to specific physical scenarios

We now proceed with applying the general framework sketched in Part I to two specific examples. The first example concerns the exotic decays of vector-like quarks, specifically the discrimination potential between different spin and charge assignments of an intermediate decay products of vector-like quark X_5/3. The second example then works with dark matter candidates of different spins coupling to coloured mediators such that the dark matter pair production can proceed via t-channel process.

2 Exotic decays of vector-like quarks

In this working example, we focus on the phenomenology of a vector-like quark (VLQ) with electric charge +5/3, X_5/3. Vector-like quarks are coloured⁷ particles just like quarks but with their left- and right-handed components belonging to the same represen- tation of the electroweak gauge group (possibly with different left- and right-handed couplings). Their existence is required in several BSM theories such as Composite Higgs [20]

or Little Higgs [21] models as well as models with extra dimensions or in GUTs based on E₆ gauge group [22].

In experimental searches, the X_5/3 is usually assumed to decay into the only possible SM decay channel, namely X_5/3 → W⁺t. In fact, any up-type quark is theoretically allowed to appear in the decay, however, the decay into the third generation quark is in general assumed to dominate. Exploring the possibility of X_5/3 being pair-produced at hadron colliders has a great advantage due to production cross section being governed by pure QCD processes. Simplified models describing the physics of X_5/3 can therefore contain only a few free parameters, which is the most often employed strategy.

However, X_5/3 is always accompanied by additional particles in the BSM theories.

The presence of these NP states opens up a possibility of additional decay channels and, as was shown in Ref. [23], such exotic decays can have comparable or even dominant branching ratios over the only SM channel. In our previous work [24], we focused on the decay process with doubly charged scalar boson, S⁺⁺, in the decay chain of X_5/3 which further decayed into a pair of same-sign leptons. We obtained very stringent bounds on the mass of the VLQ at about M_X = 1.54 TeV. This bound is largely independent of the chirality of the relevant couplings as well as of the mass of the doubly-charged scalar.

The previous work on the exotic decays of X_5/3featuring doubly-charged scalars was highly motivated by the so called 331 model which is based on SU(3)c×SU(3)_L×U(1)_X gauge group. Besides the doubly-charged scalar, the 331 model also features singly- charged scalars, S⁺, or doubly-/singly-charged vector bosons, V⁺⁺/V⁺ which can all further decay into SM leptons. In such cases the decay of X_5/3 into the SM particles

7Charged under SU(3)cgauge group of the SM

(15)

can proceed via decay chains

X_5/3→











S⁺⁺b → l⁺l⁰⁺b V⁺⁺b → l⁺l⁰⁺b S⁺t → l⁺ν_l⁰t V⁺t → l⁺ν_l⁰t

(12)

where l and l⁰ denote two (possibly) different leptons, ν_l denotes l-flavoured neutrino while b and t stand for bottom and top quarks, respectively. In case of a leptonic decay of the top quark present in the decays of the singly-charged bosons, all four decay chains result in a same-sign lepton pair signature accompanied by b-jets. This is the same signature probed by the current searches looking for X_5/3 which focus on the SM decay into W⁺ and top quark.

2.1 Model description

Even though this working example is highly motivated by the 331 model by its particle content and interactions, we construct the simplified Lagrangian without any reference to this model.

Our Lagrangian can be written in the form

L = L_SM+ Lkin+ LX+ LS+ LV , (13) where LSM stands for the full SM Lagrangian, L_kin contains standard kinetic and mass terms for the new vector-like quark X_5/3 and for all four additional bosons. The mass terms define the masses of all new particles as free parameters. In the following, MX

stands for the mass of X_5/3, MBdenotes mass of any of the new bosons while the masses of the specific bosons are denoted as M_S and M_V for scalar and vector case, respectively in case where the charges are irrelevant or by M_S++, M_S+, M_V++ and M_V+, in case where both the spin and the charge of the boson needs to be specified. The interaction parts of the Lagrangian can then be written as

L_X = √gw

2κ^XW_L X¯_5/3W P/ Lt + (L ↔ R) + h.c.

+ κ^XSpp_L X¯_5/3S⁺⁺P_Lb + κ^XSp_L X¯_5/3S⁺P_Lt + (L ↔ R) + h.c.

+ κ^{XV pp}_L X¯_5/3V/⁺⁺PLb + κ^{XV p}_L X¯_5/3V/⁺PLt + (L ↔ R) + h.c. ,

(14)

L_S= X

l,l⁰=e,µ

(g_L^S)^ll⁰¯l^cS⁺⁺PLl⁰+ (L ↔ R) + X

l,l⁰=e,µ

(g^S)^lν^l0¯l S⁻νl⁰+ h.c. , (15)

L_V = X

l,l⁰=e,µ

(g_L^V)^ll⁰¯l^cV/⁺⁺PLl⁰+ (L ↔ R) + X

l,l⁰=e,µ

(g^V)^lν^l0¯l /V⁻ν_l⁰+ h.c. . (16) Here, gw stands for the weak coupling constant, P_L/R = ¹₂(1 ∓ γ⁵) are the usual pro-

c

(16)

conjugate. All coupling strengths κ^X_L/R of X_5/3 to bosons are real parameters while coupling strengths g_L/R^S/V of doubly-charged bosons to leptons and coupling strengths g^S/V of singly-charged bosons to leptons are real 2 × 2 matrices in the two-flavour space.

The model can be further straightforwardly generalized in several ways. Firstly, the VLQ can be allowed to decay into a pair of doubly-charged boson and any down-type quark or into a singly-charged boson and any up-type quark. However, past searches for X_5/3 are most sensitive to signal with a third generation quark present in the final state and so we focus only on the decays into top and bottom quarks. Secondly, singly-charged scalars can decay into a pair of quarks. Such decays do not result in the same-sign lepton signature in the final state and so we do not treat this case in the present work. Lastly, bosonic decays featuring tau lepton pair or tau lepton and corresponding neutrino could be allowed. This possibility was already explored in Ref. [25]. However, the hadronic decays of tau leptons decrease the efficiency of the selection of signal events containing same-sign lepton pair which further results in lower sensitivity to the signal. We therefore consider decays only into electrons, muons and corresponding neutrinos.

The validity of our model implementation was tested by reproducing the results of our previous work from Ref. [24] as will be discussed in Sec. 2.3.

2.2 Parameter space

The simplified model defined in Sec. 2.1 contains 39 free parameters. These are 5 masses of the new particles, 5 left-handed and 5 right-handed couplings of X_5/3 to bosons, 4 independent components in each left-handed and each right-handed coupling matrix of doubly-charged bosons to leptons and 4 independent components in each coupling matrix of singly-charged bosons to leptons.

In order to reduce the dimensionality of this parameter space, we restrict ourselves to a few specific coupling assignments. Firstly, we focus on the case of only left-handed couplings. We do not expect the coupling chirality to significantly affect the results while working in the NWA. Secondly, we treat 4 cases of different bosons present in the decay chain of X_5/3 separately such that the branching ratio of the VLQ is always equal to 1 (BR(X_5/3 → B q) = 1). In the following, we refer to these four cases as XSPP, XVPP, XSP and XVP for X_5/3 coupling only to S⁺⁺, V⁺⁺, S⁺ and V⁺, respectively. Finally, we restrict ourselves to only one non-zero coupling of any of the bosons to a lepton pair at a time. For the doubly-charged bosons, this restriction completely specifies the final state. We therefore refer to cases of different couplings as ee, µµ and eµ. On the other hand, for the singly-charged bosons, the presence of the same-sign lepton pair in the final state requires the W bosons from the decay of the top quark to decay leptonically, which can result in different final states. In the following, we focus on the cases of singly-charged leptons coupling to a pair of lepton and a neutrino of the same flavour.

We refer to the two distinct cases as eve and mvm for coupling to eν_e pair and µν_µpair, respectively. The case of mixed flavour lν pair is not expected to result in observable differences compared to the two studied case. This is due to the fact that the missing transverse energy, which identifies the presence of the neutrinos in the experiment, is

(17)

blind to the neutrino flavour.

Finally, we make use of the NWA to factor out the cross section for the pair- production process followed by the decays of both VLQs as

σ_{f ull} = σ_pp→X

5/3X¯_5/3BR²(X_5/3→ B q)BR²(B → l l⁰(ν_l⁰)) = σ_pp→X

5/3X¯_5/3. (17) Here, σ_pp→X

5/3X¯5/3 is the cross section for the pair-production of X_5/3 in the proton- proton collisions. This process is driven purely by QCD processes which means that the pair-production cross section is independent of any free parameter of our simplified Lagrangian other than the mass of the VLQ. To assure the validity of the NWA as stated in Sec. 1.2, we set all non-zero couplings such that the width over mass ratio governed by the specific coupling is 1%.

In the end, we ended up with 3 different parameter settings for each doubly-charged boson case (XBPP ee, XBPP µµ, XBPP eµ) and 2 different parameter settings for each singly-charged boson case (XBP eve, XBP mvm). For each parameter setting, the full parameter space of the simplified model then collapses into a 2 dimensional mass plane with the restriction M_X > M_B+ m_q, with m_q being the mass of either bottom or top quark, depending on the charge of the boson B such that the decay of the VLQ is kinematically allowed.

2.3 Collider bounds

In this section, we focus on the collider phenomenology of the pair-production of X_5/3 with subsequent decays as stated in Eqn. 12, to cast the exclusion bounds on the mass plane for different cases of the parameter settings introduced in Sec. 2.2.

2.3.1 Simulation strategy

Since the cross section for the pair-production of X_5/3 can be factored out, it can be evaluated independently of the event generation. Additionally, the QCD corrections are not expected to significantly change the kinematics of the generated events and so the cross section can be evaluated at different order compared to the event generation order which can significantly speed up the whole simulation procedure. In the following, we make use of the cross section evaluated at NLO with next-to-next-to-leading logarithm (NNLL) soft gluon resummation calculated by Hathor [26] using the NNLO MSTW2008 PDF set [27]. It was obtained from Ref. [28].

We then used our implementation of the simplified model in MadGraph5 aMC@NLO to generate 10⁵ LO hard scattering events which were further matched to parton showers and hadronized using the Pythia 8 interface. The simulations were done in 4 Flavour Number Scheme (4FNS, massive b and t quarks and 4 massless quarks) with the PDF sets NNPDF 3.0 [29] (specifically NNPDF30 nlo as 0118 nf 4) provided by the LHAPDF 6 library [30]. The decays of X_5/3and new bosons B where handled as part of the generated process within MadGraph5 aMC@NLO while the decays of top quarks and W bosons were handled by MadSpin.

(18)

Table 1: The number of observed events and the number of background events in different signal regions of the CMS search in the final states with same-sign lepton pairs as reported by the CMS collaboration in Ref. [31].

SR ee µµ eµ

n^obs_SR 10 12 26

b_SR 10.9±1.9 11.2±2 23.2±3.7

We then used the Expert Mode of MadAnalysis5 together with Delphes 3 to recast a recent CMS search for X_5/3 from Ref. [31]. This search was looking at both same- sign dilepton and single-lepton final states in the dataset corresponding to an integrated luminosity of 35.9 fb⁻¹ at the center-of-mass energy √

s = 13 TeV. We made use of our implementation of the CMS search in the same-sign dilepton final state which is described in Ref. [24]. There are 3 signal regions defined in the analysis. They all share the same set of cuts and differ only by the flavour of the signal lepton pair, namely ee, µµ and eµ. The signal pair is selected as the same-sign pair that maximizes the scalar transverse momentum (p_T) sum with some further requirements on the lepton isolation and transverse momenta. Several cuts are placed on the invariant masses of any lepton pair present in the event to suppress different backgrounds. Further, cuts on the number of different reconstructed objects and their combined p_T sum are also placed on each event. The number of background events and the number of observed events in different signal regions reported by the CMS collaboration are summarized in Tab. 1. For further details on the analysis, see Ref. [31] and for details of our implementation an validation, see Ref. [24].

This simulation and recasting procedure was used to scan over the mass plane where MX > MB+ mq. The mass of the VLQ was chosen from the range between 600 to 2000 GeV in all different boson cases. The mass of the singly-charged boson was chosen from the range between 200 to 1000 GeV while the doubly-charged bosons had their masses in the range between 600 to 2000 GeV. The differences in the range for singly-/doubly- charged bosons were due to lower expected bounds in the singly-charged case. Both masses were scanned over with a mass step of 200 GeV. To probe the region of small mass splitting between the VLQ and the boson masses, we also simulated events for the mass setting M_X− M_B− m_q= 10 GeV.

2.3.2 Current and projected bounds

The results of our recasting were the selection efficiencies which were then used for the calculation of signal significances in each SR separately as described in Sec. 1.4. The cross sections used for calculation of the fiducial cross sections were those obtained by Hathor at NLO+NNLL order. The CLs method, as described in Sec. 1.4 was then used to combine the results from all three signal regions. The uncertainty in the number of signal events necessary for the CLs calculation was estimated to be 10% based on the systematic uncertainties of the generated datasets as well as on the uncertainties coming from the PDF sets used for the event generation.

(19)

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 0

1 2 3 4 5 6 7 8

Significance

= 800 GeV

S++

M

= 13 TeV) s

-1 ( 35.9 fb

SR = ee Gaussian fit

bound σ 2

= 800 GeV

S++

M

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 0

1 2 3 4 5 6 7 8

Significance

= 800 GeV

S++

M

= 13 TeV) s

-1 ( 35.9 fb

µ µ SR =

Gaussian fit bound σ 2

= 800 GeV

S++

M

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 0

1 2 3 4 5 6

Significance

= 800 GeV

S++

M

= 13 TeV) s

-1 ( 35.9 fb

µ SR = e Gaussian fit

bound σ 2

= 800 GeV

S++

M

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 0

0.2 0.4 0.6 0.8 1

sCL = 800 GeV++

MS

= 13 TeV) s

-1 ( 35.9 fb

Combined SRs Normal CDF fit 95% CL bound = 800 GeV

S++

M

Figure 1: The fitted interpolation functions for the case of X5/3 coupling to S⁺⁺ with the mass of the doubly charged boson MS= 800 GeV.

To obtain the current 2-sigma bounds on the mass plane in each parameter setting, the significances in different signal regions were interpolated by a Gaussian function in each constant MX slice of the plane while the 95% CLs bounds were extracted by interpolating the CLs values by a Normal Cumulative Distribution Function (Normal CDF). The validity of using these functions for interpolation can be seen on Fig. 1 where we show the interpolated functions as fitted on the obtained significances and CLs for the case of X_5/3 coupling to S⁺⁺ in the slice of the mass plane where MS = 800 GeV.

The bound at the kinematic limit was obtained by the same interpolation procedure on the small mass splitting points MX− M_B− m_q= 10 GeV along the kinematic boundary.

This bound in the small mass splitting region is considered to be only approximate due to possible breakdown of the NWA.

Using the same datasets of events generated at √

s = 13 TeV, we further proceeded with projecting the current bounds for the future luminosities and finding 5-sigma discovery reaches as described in Sec. 1.4. The future bounds were obtained from the projected significances in each signal region as well as from the projected CL_s values.

The number of observed events in this case was set to the rescaled number of background events in each signal region. The uncertainty of the number of signal events was again set

(20)

to 10%. However, under an assumption of the 10% uncertainty in the future estimate of the number of background events, we found that the projected 5-sigma discovery reach at the luminosity L_HL barely exceeded the current 2-sigma bound on the mass plane which limited our discrimination analysis. However, under an optimistic assumption of the 5% uncertainty in the number of background events, we found a large enough region suitable for the discrimination analysis and so we use this estimate of the uncertainty⁸. In case of the luminosity LIII, the projected discovery reach did not exceed the current bound on the mass plane and so in the following, we report only on the projection results for the case of L_HL. Due to the numerical precision of the CL_s calculation as well as the precisions of the interpolation, we used only the Gaussian interpolation on the significances to derive the 5-sigma discovery reach in different signal regions.

The current bounds together with the projected bounds and discovery reach at L_HL for all four boson scenarios are shown in Figs. 2 - 7.

The significances used to obtain Fig. 2 and Fig. 3 in each signal region are calculated with the assumption of only one non-zero coupling of the doubly-charged boson to a lepton pair which directly corresponds to the signal region. The CL_svalues then combine the results of the three independent SRs. It can be seen from these two figures, that the bounds on the mass plane together with the discovery reaches are very similar between the two spin scenarios. They are also only weakly sensitive to the boson mass. The dependence on the boson mass comes into play only in the region near the kinematic limit where the results need to be considered with caution due to possible breakdown of the NWA.

The most constraining current bound on the mass plane comes from the CLsinterpo- lation and it is found at about M_X = 1.54 TeV independently of the boson mass for both boson spins. This result for the doubly-charged scalar boson case is in agreement with our previous result in Ref. [24], which validates our model implementation as well as our simulation strategy. The bounds coming from the dielectron and dimuon signal regions are about 50 GeV weaker compared to the CL_s results. The projected bounds in the ee and µµ signal regions at LHL luminosity are then pushed to about MX = 1.85 TeV with the projected CL_s bounds found at about M_X = 1.94 TeV. The projected 5-sigma discovery reach in the ee and µµ signal regions is found at about 1.65 TeV. Current and projected bounds together with the discovery reaches obtained in the eµ signal region are in general about 100 GeV weaker than those coming from the dielectron or dimuon signal regions. These results leave us enough space between the current bound and the discovery reach for the discrimination analysis.

The figures for the singly-charged boson scenarios feature the significance results only from two signal regions. This is due to the fact that the choice of the coupling of the boson to a lνl pair does not fix the signal region sensitive to the resulting final state since the second charged lepton in the final state comes from a decay of W boson.

For each singly-charged boson case, we therefore report on results from two different

8We note again, that the future discovery is possible even below the current 2-sigma bound. We resort to this optimistic assumption to provide the discrimination analysis over an extended discrimination region as a proof of concept.

(21)

coupling assignments in two respective figures. The results for the singly-charged scalar boson are shown in Fig. 4 and Fig. 5 while the results for the singly-charged vector boson are shown in Fig. 6 and Fig. 7.

Current bounds coming from the two coupling assignments for a specific boson as well as the bounds from the two different boson cases are all very similar. It can be easily seen that the bounds on the mass plane for the singly-charged bosons are considerably lower that the bounds on the doubly-charged bosons. This is an expected result since the decay chain of the VLQ in the singly-charged boson cases feature W bosons which can decay hadronically and therefore yield the desired final state with lower efficiency while some of the kinematical information is lost to the missing transverse energy due to neutrinos present in the decay chains. It can be also seen that the bounds for the singly-charged bosons are more dependent on the boson mass compared to the doubly- charged cases. As a general trend, the bounds shift towards smaller values of the mass of the VLQ with the increasing boson mass.

The most constraining bounds in the singly-charged boson cases come from the ee and µµ signal region for the bosons coupling to eν_e and µν_µ lepton pair, respectively.

The obtained bound starts at about MX = 1.12 TeV at the boson mass MB = 200 GeV and it slowly decreases to a value of M_X ∼ 900 GeV at the kinematic limit (note that the specific value of the bound at the kinematic limit should be considered with caution). This result is in agreement with the bounds obtained in Ref. [25] for the case of X_5/3 → S⁺t → τ⁺ν t. The bounds coming from the eµ SR as well the CL_s bounds are found to be very similar and weaker by about 100 GeV compared to ee and µµ signal regions. The projected bounds in the ee and µµ signal regions are then pushed to about M_X = 1.49 TeV at M_B= 200 GeV with the CL_sbounds being only slightly weaker. On the other hand, the bounds in the eµ signal region remain to be about 100 GeV lower compared to ee and µµ signal regions. Finally, the discovery reach in the ee and µµ signal regions is at about 1.3 TeV with the eµ signal region yielding the discovery reach of about 1.17 TeV at M_B = 200 GeV.

(22)

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 600

800 1000 1200 1400 1600 1800 2000 2200

[GeV]++SM

b = 0 - m

++

- MS MX

= 13 TeV s

Significance SR = ee

) = 100 % e+

-> e+

BR(S++

= 5 % εsyst

Projected

bound @ 35.9/fb σ

Current 2

discovery reach @ 3/ab σ

5

bound @ 3/ab σ

2

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 600

800 1000 1200 1400 1600 1800 2000 2200

[GeV]++SM

b = 0 - m

++

- MS MX

= 13 TeV s

Significance µ µ SR =

) = 100 % µ+

µ+

->

BR(S++

= 5 % εsyst

Projected

bound @ 35.9/fb σ

Current 2

5

bound @ 3/ab σ

2

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 600

800 1000 1200 1400 1600 1800 2000 2200

[GeV]++SM

b = 0 - m

++

- MS MX

= 13 TeV s

Significance µ SR = e

) = 100 % µ+

-> e+

BR(S++

= 5 % εsyst

Projected

bound @ 35.9/fb σ

Current 2

5

bound @ 3/ab σ

2

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

[GeV]

MX 600

800 1000 1200 1400 1600 1800 2000 2200

[GeV]++SM

b = 0 - m

++

- MS MX

= 13 TeV s

Combined SRs CLs bound @ 35.9/fb

σ Current 2

bound @ 3/ab σ

2

= 5 % εsyst

Projected

Figure 2: Mass plane bounds for the XSPP case (X5/3coupling to S⁺⁺) in different signal regions. The current 2-sigma bound is depicted as a solid green line. Also shown in the figure are the projected 5-sigma discovery reach (solid red line) and the projected 2-sigma bound (solid black line) at the future luminosity LHL. The uncertainty in the number of background events used to obtain the projections was fixed to 5%

of the expected number of background events.

Characterisation of Models of New Physics at the LHC

Master Thesis