A Non-Relativistic Model of Tetraquarks

EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2020

PER LUNDHAMMAR

KTH SKOLAN FÖR TEKNIKVETENSKAP A Non-Relativistic Model of Tetraquarks

Per Lundhammar [email protected] SA114X Degree Project in Engineering Physics, First Level Department of Physics KTH Royal Institute of Technology Supervisor: Tommy Ohlsson

June 1, 2020 Abstract

In this thesis, a non-relativistic model of tetraquark in a diquark-antidiquark con- figuration is investigated. Using a variation of the Cornell potential, the Schrödinger equation is solved numerically, and the four-body problem of the tetraquark is separated into to three two-body problems. The splitting structure is accounted for by a spin-spin interaction term. Several numerical fits are made to different types of meson data to obtain the free parameters of the model, and subsequently, the masses of diquarks and tetraquarks with different constituents are determined. An introduction to the subject of exotic hadrons is presented as well as an overview of the experimental progress concerning tetraquarks. The results obtained in this thesis are then discussed by comparison with other relativistic models and experimental results.

Sammanfattning

I detta arbete undersöks en icke-relativistisk modell av tetrakvarkar i en dikvark- antidikvark-konﬁguration. Genom att använda en variation av Cornellpotentialen löstes Schrödingerekvationen numeriskt och det fyrkroppsproblem som tetrakvarkar utgör de- lades upp i tre tvåkroppsproblem. Modellen tar även hänsyn till systemets spinn-spinn- koppling. Flera numeriska anpassningar gjordes för olika typer av mesondata för att bestämma de fria parametrarna i modellen. Därefter bestämdes massorna av dikvarkar och tetrakvarkar med olika sammansättningar av deras beståndsdelar. En introduktion till exotiska hadroner presenteras samt en översikt av de experimentella framstegen gäl- lande tetrakvarkar. Resultaten diskuteras och jämförs med andra relativistiska modeller och experimentella resultat. Contents

1 Introduction 2

2 Background Material 4 2.1 The Standard Model ...... 4 2.2 Tetraquarks ...... 6 2.3 Symmetries and Conservation Laws ...... 7

3 Investigation 9 3.1 Model ...... 9 3.1.1 Color Structure ...... 11 3.2 Numerical Analysis ...... 12 3.2.1 Data Sets ...... 12 3.3 Results ...... 12 3.3.1 Diquarks ...... 15 3.3.2 Tetraquarks ...... 15 3.3.3 Comparison with Other Works ...... 16 3.4 Discussion ...... 17

4 Summary and Conclusions 21

1 Chapter 1

Introduction

In the history of natural sciences, the exact nature of matter has been extensively discussed and through the ages this concept has been refined and expanded upon. The essential ingredient of matter has been described to be some fundamental building block and the idea that matter consists of a collection of atoms is derived to Democritus in the 5th century B.C. although it was not until the renaissance that the concept was discussed in a physics manner [1]. Since then the concept of the atom has been first disputed in the end of 19th century, confirmed in the beginning of the 20th century and observed to have constituents of negatively charged electrons, positively charged protons, and neutral neutrons some decades after. Yet, the understanding of what is holding these constituents together in the atom was not well understood. The first significant theory tackling this problem was proposed by Yukawa in 1934. This lead to the proposition and later discovery of the π meson (or “pion”). In 1927, Dirac introduced a relativistic framework of quantum mechanics which was supposed to describe free electrons, but to every positive energy solution the equation implied the existence of a negative energy solution suggesting electrons radiating infinite amount of energy. What may have seemed as a disastrous defect later turned out to be described by the discovery of the positron, a positively charged twin of the electron. It turns out that Dirac’s framework of describing free electrons is a universal feature of quantum field theory; namely that for every particle there exists a corresponding antiparticle with same mass but opposite charge [2]. Several other great contributions were made in the years following 1930 such as the discovery and classification of the neutrinos, originally conjectured based on only conservation principles, or symmetry principles. Also, the so-called strange particles and subsequently baryons emerging from cosmic rays which later gave rise to the greatly suc- cessful classification scheme of mesons and baryons called the Eightfold Way. The idea of quarks as the fundamental constituents of matter was introduced in 1964 by Gell-Mann and Zweig independently. They proposed that all hadrons are made out of quarks giving an understanding of the Eightfold Way classification scheme of mesons and baryons [3]. With the introduction of quarks and the Eightfold Way followed many new contributions to the field of particle physics and in the current view all matter is described by the Stan- dard Model, introduced in chapter 2. Even if we can trace the idea of some fundamental constituents of matter back to the ancients the experimental reality is modern.

2 This thesis aims to give an introduction to the field concerning exotic hadrons and in particular tetraquarks as well as applying some variation of existing models in the framework of tetraquark states. An outline of the field of particle physics and in particular Quantum Chromodynamics (QCD), the theory of the strong force, will be presented and some interesting features of the theory connected to tetraquarks will be discussed. Lastly, a more in-depth study is done on a tetraquark system consisting of a diquark and an antidiquark. In chapter 2, the main theory surrounding particle physics called The Standard Model is introduced followed by an overview of the experimental progress of the exotic hadrons and ending with an introduction to symmetries connected to the modeling procedure used in this field. In chapter 3, the model studied in this thesis is laid out and several numerical fits are made to experimental data. The result of this modeling scheme is presented at the end of this chapter. Finally, in chapter 4, a summary and concluding remarks concerning the result obtained in chapter 3 are given.

3 Chapter 2

Background Material

In this chapter, an investigation of the underlying theory describing quarks and composites of quarks is discussed. The Standard Model is introduced as well as an overview of the experimental progress concerning the subject of tetraquarks. Lastly, an introduction to how symmetries contribute to the modeling scheme of quarks is presented.

2.1 The Standard Model

The current view of matter is that it is all made out of three kinds of elementary particles called leptons, quarks, and force mediators. The leptons and quarks are further catego- rized by their diﬀerent quantum numbers. The leptons inhabit charge, electron number Le, muon number Lµ, and tau number Lτ . These are then grouped in pairs as shown in table 2.1, where the symbol for each lepton is given in the ﬁrst column.

Name Charge Le Lµ Lτ Electron e −1 1 0 0 Electron neutrino νe 0 1 0 0 Muon µ −1 0 1 0 Muon neutrino νµ 0 0 1 0 Tau τ −1 0 0 1 Tau neutrino ντ 0 0 0 1 Table 2.1: Lepton classiﬁcation.

In addition to these six leptons, there are six corresponding antileptons, with all signs reversed. In total that gives twelve leptons. The quarks are attributed charge, and flavour number, D, U, S, C, B, and T representing down, up, strange, charm, bottom, and top flavor, respectively. These are grouped in as is shown in table 2.2, where the symbol for each quark is given in the first column. Each quark is also ascribed one of three colors red, green, and blue. Like for the leptons, there are also corresponding antiquarks with reversed signs for each prescribed number, and the color is changed to its corresponding anticolor. This gives in total 36 possible quarks. There are four fundamental forces or interactions between particles; gravity, electromagnetic interaction, weak interaction, and

4 Name Charge Spin Iz D U S C B T Down d −1/3 1/2 −1/2 1 0 0 0 0 0 Up u 2/3 1/2 1/2 0 1 0 0 0 0 Strange s −1/3 1/2 0 0 0 −1 0 0 0 Charm c 2/3 1/2 0 0 0 0 1 0 0 Bottom b −1/3 1/2 0 0 0 0 0 −1 0 Top t 2/3 1/2 0 0 0 0 0 0 1

Table 2.2: Some properties of the diﬀerent quarks. the strong interaction. The Standard Model is the theory of describing three of these fundamental interactions, namely the electromagnetic, weak, and strong interaction. This is done by considering an interaction as mediated by a diﬀerent kind of particle called intermediate gauge bosons [2].

Gravitation The force responsible for keeping the planets in orbit attracts bodies with a magnitude proportional to their masses. It is well described by Newton’s law of gravitation or Ein- stein’s general relativity but no widely accepted quantum theory of gravity has been found. In particle physics, this force plays a negligible role and is omitted when describing interactions between particles.

The Electromagnetic Interaction The mediating gauge boson of the electromagnetic interaction is the photon and acts between charged particles. The force between charged particles, light- and other electromagnetic radiation are all consequences of this interaction. The theory of describing electromagnetic interaction with the photon as a mediating particle is called quantum electrodynamics (QED) and has its origins in the 1940s.

The Weak Interaction This interaction acts between quarks, and operates by changing the color of the interacting quarks. As the name suggest, it is comparably weaker than the other interactions and is the only interaction able to change quarks into other quarks. The mediators of the weak interaction are called Z and W ± bosons.

The Strong Interaction The strong force is acting between quarks and is described with the gluon as the mediating particle. It is responsible for holding composites of quarks together, as well as the binding of nucleus in the nucleon. The theory of the strong force is called quantum chromodynamics (QCD) and emerged in the 1960s.

Quarks and gluons have not been observed as isolated particles but form composite structures called hadrons. This is a consequence of what is known as color conﬁnement and constraints the stable hadrons to consist of only colorless composites of quarks. A hadron is colorless if the color assigned to its constituent quarks adds up to zero. The combination of red, green, and blue adds up to zero as well as the combination of a color

5 and its corresponding anticolor. The two main types of hadrons are called mesons and baryons. Mesons consist of a quark and a corresponding antiquark forming a colorless hadron. Baryons consist of three quarks such that they form an colorless hadrons. Apart from mesons and baryons, the existence of other types of hadrons consisting of more than three quarks are not excluded in the quark model. As an example, the tetraquarks which consist of two quarks and two antiquarks, pentaquarks consisting of four quarks and one antiquark or glueballs which consist entirely of gluons. The main focus of this thesis is tetraquarks and a short overview of the experimental progress of these particular hadrons will be laid out in following section.

2.2 Tetraquarks

The existence of hadrons consisting of four or more quarks was proposed as early as 1970s, but not until the beginning of the 21st century the first claimed observations were made [4]. Regarding tetraquarks, the first claimed discovery was made in 2003 by the Belle experiment observing a resonance peak at (3872.0 ± 0.6) MeV, it was named X(3872) [5]. Many proposed exotic hadrons only appear in one decay mode, though the X(3872) particle could be seen in several other decays as was discovered by the BaBar and D0 experiments. Later, the ATLAS, CMS, and LHCb experiment were able to contribute to massive amount of data concerning the X(3872) particle and its current mass is determined to be (3871.69 ± 0.17) MeV [6]. It is the most studied exotic particle but its nature is still unknown. It has similar properties as the charmonium state cc¯ and was first believed to be an undiscovered excited charmonium particle but a closer investigation of the decay mode X(3872) → J/ψπ−π+ shows a violation in isospin symmetry that is not conventional for a charmonium particle. Considering the X(3872) particle as an exotic particle, a common explanation is that it has a quark content of two quarks and two antiquarks ccu¯ u¯. Though, how it is bound together is still an open problem. Since the discovery of the X(3872) particle, many new exotic hadron candidates has been discovered with a final state of a pair of heavy quarks and a pair of light antiquarks. These candidates are named by the experimental collaborations as X, Y or Z states, collec- tively referred to as XYZ states [7]. The dynamics of the XYZ systems involves both the short distance and long distance behavior of QCD which make theoretical predictions difficult. Hence many competing phenomenological models currently exist for these states. Many models view the exact nature of the inner structure of the tetraquarks to consist of so-called diquark-antidiquark pairs [8]. A diquark is a bound quark-quark pair, and an antidiquark an antiquark-antiquark pair. They are not by themselves colorless but are proposed in the context of tetraquarks to form colorless combinations. Modeling tetraquarks containing only heavy quarks is easier to study theoretically since several simplifications can be justified. In chapter 3, we will study a non-relativistic model view- ing tetraquarks as consisting of diquarks and antidiquarks interacting much like ordinary quarkonia.

6 2.3 Symmetries and Conservation Laws

Symmetries are an important tool when studying physical systems. This became apparent in 1917 when Noether published her theorem relating symmetries and conservation laws of a system. Noether’s theorem can be stated as follows: For every symmetry of a physical system Nature yields a conservation law; also, every conservation law implies some underlying symmetry. Some examples include conservation of energy which relates to symmetry in time translation, or conservation of angular momentum which can be seen as a consequence of rotational symmetry in space. The symmetries of a system are operators acting on the system, and the set of all symmetries exhibits the same properties as the deﬁning properties of a mathematical group with composition as the group operation. That is, the composition of symmetries is also a symmetry (closure), the action of not changing the system is also a symmetry (identity), to every symmetry there is a symmetry where the composition does not change the system (inverse), and the composition order of three symmetries does not matter (associativity). Thus, the symmetries of a system can be understood through the lens of group theory [2]. A powerful tool used to study group symmetries is what is called representations of groups. A representation of a group is a mapping of the elements of the group onto a set of linear operators. The fact that the group representations live in linear space makes this construction powerful. A representation is said to be reducible if it has an invariant subspace, that is if any vector in the invariant subspace remains in the subspace after the action of any representation. Furthermore, a representation is said to be irreducible if it is not reducible and completely reducible if it can be decomposed into a direct sum of irreducible representations.

Now, a closer look on a system where the group of symmetries is continuous will be made. This type of groups symmetries is important when studying systems which are invariant under rotations. Let G denote the group and g an element of the group. Assuming that the group elements depend “smoothly” on a parameter α, that is g = g(α). In this context, “smooth” means simply that there exists some sort of notion of closeness in the sense that if two group elements are “close together” so are their respective parameters. We choose the parameterization of the group elements such that g(α)|α=0 = e, where e is the identity element of the group. Thus, assuming that the neighborhood of the identity e, the group elements can be described by N real parameters αk with k = 1,...,N such that g(α)|α=0 = e. Now defining a representation, denoted D(α), mapping each group element g(α) to a corresponding linear operator D(α). This implies that D(α)|α=0 = 1, with 1 being the identity linear operator. Also, Taylor expanding in the neighborhood of the identity, we find that D(dα) = 1 + idαkXk + ... (2.1) depending on the parameter dα. Each term is understood to be summed over according to Einstein’s summation convention. The second term in the Taylor expansion in eq. (2.1) can be identified as ∂ Xk ≡ −i D(α) , (2.2) ∂αk α=0

7 where Xk with k = 1,...,N are called the generators of the group. The i in eq. (2.1) is included so that if the representations are unitary operators, then the generators will be Hermitian operators. Lie showed in the 19th century that these generators can be described in an abstract sense not including representations. The groups of this kind are therefore called Lie groups [9]. A particular Lie group of interest is the so-called special unitary group SU(N) which is the set of unitary matrices with determinant equal to one and having a total of N 2 − 1 number of generators. It plays a role when considering the spin symmetries or the color structure of hadrons. For example, consider the product spin state of two fermions. Labeling each spin state by its dimension, the product state can symbolically be written as 2 ⊗ 2, since each fermion has either spin up or down. This spin system has SU(2) as its symmetry group where the Pauli matrices constitute the generators. This product state can be reduced into following direct sum of irreducible representations 2 ⊗ 2 = 1 ⊕ 3, (2.3) the triplet 3 having total spin of one and the singlet 1 having total spin of zero [10]. As was mentioned in the beginning of this chapter, hadrons are only stable when the colors of their constituent quarks sum up to zero. This formulation is a consequence of a deeper law; namely that every naturally occurring hadron is a color singlet under the group symmetry SU(3). This means that for a hadron to naturally occur the product state of the composite quarks must decompose to an irreducible representation with dimension equal to one. Mesons consist of quarks and antiquarks yielding the product color state 3⊗3¯, since the quarks can either have color red, green or blue. Antiquarks are deﬁned to be the conjugate representation of their corresponding quark representations symbolized by the bar over the dimension number. This product state can now be decomposed to the following irreducible representations: 3 ⊗ 3¯ = 1 ⊕ 8, (2.4) including a color singlet, and thus by the rule above being a naturally occurring hadron. Similarly, baryons consist of three quarks yielding the decomposition 3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10. (2.5) This method of classifying hadrons is known as the Eightfold Way introduced by Gell- Mann in the 1960s. Hadrons consisting of two quarks have the product color state 3 ⊗ 3 which reduces to 3 ⊗ 3 = 6 ⊕ 3¯, (2.6) and similarly, for two antiquarks 3¯ ⊗ 3¯ = 6¯ ⊕ 3. None of these decompose to a color singlet though the product color state 3 ⊗ 3 ⊗ 3¯ ⊗ 3¯ has the following decomposition into irreducible representations 3 ⊗ 3 ⊗ 3¯ ⊗ 3¯ = 1 ⊕ 8 ⊕ 10¯ ⊕ 10 ⊕ 27 (2.7) including a singlet state. In the modeling scheme in chapter 3, we will consider the system of a diquark consisting of two quarks and an antidiquark consisting of two antiquarks in the triplet state yielding a decomposition into a color singlet [11].

8 Chapter 3

Investigation

In this chapter, the model of tetraquarks viewed as diquark-antidiquark systems is presented and the method used to prescribe some tetraquark states quantitative masses is derived. This is preformed by ﬁrstly considering a quark-antiquark system and describing the Hamiltonian of that system with an unperturbed one-gluon exchange (OGE) potential and a perturbation term taking the system’s spin into account. This gives rise to a model with four parameters which are then ﬁtted to meson data. Secondly, the model is expanded to incorporate quark-quark systems which we call diquarks (the antiquark-antiquark systems are called antidiquarks). With the diquark masses determined the previous model of describing quark-antiquark systems are then used to describe the diquark-antidiquark systems, for which we interpret as bound tetraquark states.

3.1 Model

The modeling scheme is outlined as follows: 1. Fitting a quark-antiquark model to meson data obtaining the parameters of the potential; 2. Using that set of parameters to determine the diquark and antidiquark masses by changing the color constant and string tension of the potential; 3. Calculating tetraquark masses by using diquarks as constituents, see figure 3.1. We will begin by considering the interaction between a quark and an antiquark. In quark bound state spectroscopy, a commonly used potential describing the unperturbed contribution is κα V (r) = S + br, (3.1) r called the Cornell potential, where κ is the so-called color factor and is associated with the color structure of the system, αS the structure constant of QCD, and b the string tension. The first term, VV (r) ≡ καS/r, is called the Coulomb term and is associated with the Lorentz vector structure. It arises from the one gluon exchange between the quarks. The second term is associated with the confinement of the system. A non- relativistic approach is legitimate under the condition that the kinetic energy is far less

9 q q 2 2 q1 q2

q1 q1 q1 q2

Figure 3.1: A schematic overview of the modeling procedure. First, considering the quark-antiquark system q1q¯2. Second, extrapolating the model to also describe the quark- quark, or diquark, system q1q2. Third, modeling the tetraquarks in the same way as the quark-antiquark system but with diquarks as consituents.

than the rest masses of the constituents, which is usually the case considering heavy- quark bound states. We will formulate the Schrödinger equation in the center-of-mass frame where considering spherical coordinates one can factorize the angular and radial parts. Let µ ≡ m1m2/(m1 + m2), where m1 and m2 are the constituent masses of quark 1 and quark 2, respectively. The time independent radial Schrödinger equation can thus be written as 1 d2 2 d l(l + 1) − + − + V (r) ψ(r) = Eψ(r), (3.2) 2µ dr2 r dr r2 with the orbital quantum number l and energy eigenvalue E. By the substitution ψ(r) = r−1ϕ(r), the equation transforms into 1 d2 l(l + 1) − + + V (r) ϕ(r) = Eϕ(r). (3.3) 2µ dr2 r2 Based on the Breit-Fermi Hamiltonian for one-gluon-exchange one can include a spin-spin interaction of the form 2 2πκα V (r) = − ∇2V (r)S · S = − S δ3(r)S · S . (3.4) S 3(2µ)2 V 1 2 3µ2 1 2 In this model, we incorporate this spin-spin interaction in the unperturbed potential V (r) by replacing the Dirac delta function with a smeared Gaussian function, depending on the parameter σ, in the following way 2πκα σ 3 V (r) = − S √ exp −σ2r2 S · S , (3.5) S 3µ2 π 1 2 as is performed in Ref. [12]. Now, eq. (3.3) takes the form d2 − + V (r) ϕ(r) = 2µEϕ(r) (3.6) dr2 eﬀ

10 with the eﬀective potential

l(l + 1) V (r) ≡ 2µ [V (r) + V (r)] + , (3.7) eﬀ S r2 taking into account the spin-spin interaction. Equation (3.6) can be solved numerically for the energy eigenvalue E and the reduced wavefunction ϕ(r). The mass M of the bound quark-antiquark system can now be expressed as

M = m1 + m2 + E. (3.8)

Note that the model consists of ﬁve unknowns, namely the masses m1 and m2 of the constituents, the structure factor αS, the string tension b, and the parameter in the spin-spin interaction σ.

3.1.1 Color Structure The diﬀerence in color structure between the quark-antiquark and quark-quark systems allow us to expand the model of the quark-antiquark system to also be valid considering a quark-quark system by only changing the color factor κ and the string tension b. The SU(3) color symmetry of QCD implies that the combination of a quark and an antiquark in the fundamental color representation can be reduced to |qq¯i : 3 ⊗ 3¯ = 1 ⊕ 8 which gives the resulting color factor for the color singlet as κ = −4/3 for the quark-antiquark system. When combining two quarks in the fundamental color representation, it reduces as follows: |qqi : 3 ⊗ 3 = 3¯ ⊕ 6, that is a color antitriplet 3¯ and a sextet 6. Similarly, by combining two antiquarks, it reduces to a triplet 3 and an antisextet 6¯. Combining the triplet diquark and antitriplet antidiquark yields: |[qq] − [qq ¯ ]i : 3 ⊗ 3¯ = 1 ⊕ 8, thus forming a color singlet for which the Coulomb part of the potential is attractive. The antitriplet state is attractive and has a corresponding color factor of κ = −2/3, while the sextet is repulsive with color factor κ = +1/3. Thus, we only consider diquarks in the antitriplet state.

The eﬀect of going from a quark-antiquark system with color factor κ = −4/3 to a diquark system with color factor κ = −2/3 is equivalent of introducing a factor 1/2 in the Coulomb part of the potential for the quark-antiquark system. It is common to view the factor of 1/2 as a global factor, since it comes from the color structure of the wavefunction, thus also dividing the string tension b by a factor of 2. For further details see Ref. [13]. We apply this when considering diquarks. Given the parameters of the potential, we obtain the mass of the corresponding diquark in a similar manner as when considering the quark-antiquark system only changing the string tension b → b/2 and κ = −2/3, due to the change in colors structure of the system, and thus ﬁnding the energy eigenvalues of the diquark system.

11 3.2 Numerical Analysis

Fitting the four parameters of the model to experimental data is performed by ﬁnding the parameters v ≡ (m, αS, b, σ) that minimizes the function

2 X 2 χ ≡ wi(Mexp,i − Mmodel,i(v)) , (3.9) i where Mexp,i is the experimental mass of the corresponding mass Mmodel,i(v) which are given in the model as a function of v. These terms are then weighted with wi for each mass. Following Ref. [14], we will only consider wi = 1 giving the same statistical signiﬁcance to all states used as input.

3.2.1 Data Sets The model will be numerically fit to five different data sets. First, a data set consisting entirely of charmonium mesons. Second, a data set consisting of bottomonium mesons. Third, the data set consisting of D mesons. Forth, the data set of B mesons. Fifth, a fit to both the charmonium and meson data will be made. The meson consisting of two charm quarks is a good candidate to be fitted to the model, since it has relatively large constituent mass compared to the light quarks, and therefore, a non-relativistic approach can be justified. Bottomonium, as well as charmonium, is a heavy meson and well suited to the restriction of this model. As reference the data set of charmonium mesons is called I, the data set of bottomonium mesons II, the data set consisting of only D mesons III, the data set consisting of only B mesons is called data set IV, and the data set containing both charmonium and bottomonium data V. The data used are presented in table 3.1. However, note that in the left column we use spectroscopic notation where N denotes the principal quantum number, S the total spin, L the orbital quantum number, and J the total angular momentum quantum number.

3.3 Results

In this section, the result of the fitted data sets and subsequently the resulting masses of different diquarks and tetraquarks are presented. The procedure can be divided into three main parts. First, fitting the model to each data set I-V obtaining five sets of parameter values. Second, using the sets of parameter values obtained by fitting data sets I-IV to calculate the masses of different diquarks. In detail, the sets of parameter values obtained by fitting data set I, II, III, and IV are used to calculate the mass of the cc, bb, qc, and qb diquark, respectively, with q being either an up quark or a down quark. Third, using the calculated diquark masses, calculating the masses of different tetraquarks. The set of parameter values used for this calculation is the one obtained by fitting data set V to the model. The number of free parameters when fitting the model to data sets I and II is four, since the masses of the constituent quarks of those data sets are equal. When fitting

12 2S+1 Meson N LJ Ex. mass [MeV] Ex. error [MeV] 1 ηc(1S) 1 S0 2983.4 0.5 2 J/ψ(1S) 1 S1 3096.900 0.006 3 χc0(1P ) 1 P0 3414.75 0.31 3 χc1(1P ) 1 P1 3510.66 0.07 1 hc(1P ) 1 P1 3525.38 0.11 3 χc2(1P ) 1 P2 3556.20 0.09 1 ηc(2S) 2 S0 3639.2 1.2 3 ψ(2S) 2 S1 3686.097 0.025 3 ψ(3770) 1 D1 3773.13 0.35 3 χc2(2P ) 2 P2 3927.2 2.6 3 ψ(4040) 3 S1 4039 1 3 ψ(4160) 2 D1 4191 5 3 ψ(4415) 4 S1 4421 4 1 ηb(1S) 1 S0 9390.9 2.8 2 Y (1S) 1 S1 9360.30 0.26 1 hb(1P ) 1 P1 9899.3 0.8 3 χb0(1P ) 1 P0 9859.44 0.52 3 Y (2S) 2 S1 10023.26 0.31 3 Y2(1D) 1 D2 10161.1 1.7 3 χb0(2P ) 2 P0 10232.5 0.6 3 Y (3S) 3 S1 10355.2 0.5 3 Y (4S) 4 S1 10579.4 1.2 + 1 D 1 S0 1869.65 0.20 0 1 D 1 S0 1864.83 0.17 + 1 Ds 1 S0 1968.47 0.33 ∗+ 1 D (2010) 1 S0 2010.27 0.17 ∗0 1 D (2007) 1 S0 2006.97 0.19 + 1 B 1 S0 5279.29 0.15 0 1 B 1 S0 5279.61 0.16 0 1 Bs 1 S0 5336.79 0.23 + 1 Bc 1 S0 6275.1 1.0 Table 3.1: Charmonium and bottomonium data from Ref. [15]. The ﬁrst thirteen rows are charmonium particles, the next nine are bottomonium particles, the next ﬁve are D mesons having a quark content of cq¯, and the last four are B mesons having a quark content of bq¯, with q = u, d. the model to data sets III-V, we use the obtained values for the constituent masses of the charm and bottom quarks. Also, when considering data sets III and IV, we use the value 0.323 GeV as the constituent mass of an up quark or a down quark, taken from Ref. [2].

13 Meson I II III IV V −3 −5 ηc(1S) 1.270 · 10 − − − 3.566 · 10 J/ψ(1S) 3.242 · 10−4 − − − 3.930 · 10−4 −2 −3 χc0(1P ) 1.491 · 10 − − − 6.363 · 10 −4 −3 χc1(1P ) 6.862 · 10 − − − 2.605 · 10 −4 −4 hc(1P ) 3.333 · 10 − − − 5.821 · 10 −4 −3 χc2(1P ) 3.742 · 10 − − − 3.804 · 10 −4 −4 ηc(2S) 4.965 · 10 − − − 8.340 · 10 ψ(2S) 4.804 · 10−4 − − − 8.517 · 10−4 ψ(3770) 1.751 · 10−3 − − − 7.731 · 10−5 −3 −4 χc2(2P ) 1.416 · 10 − − − 4.143 · 10 ψ(4040) 3.883 · 10−3 − − − 1.085 · 10−5 ψ(4160) 4.348 · 10−5 − − − 5.200 · 10−3 ψ(4415) 5.760 · 10−4 − − − 3.030 · 10−3 −2 ηb(1S) − 1.012 · 10 − − 0.1905 Y (1S) − 1.942 · 10−3 − − 0.1389 −3 −2 hb(1P ) − 2.638 · 10 − − 2.028 · 10 −10 −2 χb0(1P ) − 1.295 · 10 − − 1.962 · 10 Y (2S) − 6.398 · 10−3 − − 3.569 · 10−2 −3 −3 Y2(1D) − 1.810 · 10 − − 8.899 · 10 −3 χb0(2P ) − 1.472 · 10 − − 0.7631 Y (3S) − 1.013 · 10−3 − − 1.658 · 10−2 Y (4S) − 4.804 · 10−3 − − 2.510 · 10−2 D+ − − 3.382 · 10−2 − − D0 − − 3.650 · 10−2 − − + −2 Ds − − 4.134 · 10 − − D∗+(2010) − − 1.182 · 10−2 − − D∗0(2007) − − 1.254 · 10−2 − − B+ − − − 2.631 · 10−2 − B0 − − − 2.621 · 10−2 − 0 −2 Bs − − − 3.594 · 10 − + −2 Bc − − − 2.302 · 10 − χ2 1.06 · 10−2 2.78 · 10−2 0.193 0.120 0.466

Table 3.2: Resulting value of χ2 for each ﬁt to the data sets I-V with respective pull for each data point.

In practice, we are solving eq. (3.6) in the eigenbasis of the spin operators S, S1 and 1 S2 thus eﬀectively replacing the product S1 ·S2 with 2 (S(S +1)−S1(S1 +1)−S2(S2 +1)), where S, S1, and S2 is the total spin, spin of quark 1, and spin of quark 2, respectively. However, note that this modeling procedure will not be able to split the mass of particles with the same principal, orbital, and spin quantum numbers but diﬀerent total angular momentum quantum numbers. Solving the Schrödinger equation numerically is done by assuming Dirichlet boundary condition in r = 0 and r = r0 when solving eq. (3.6). The

14 parameter value r0 is chosen so that the energy value is independent of r0 up to five significant digits. This approach was inspired by the method described in Ref. [16]. The minimization of eq. (3.9) is then done by discretizing the free parameters and then performing a random search in the discretized parameter space. The restriction 2 2 of the parameters was chosen to 0.10 ≤ αS ≤ 0.60, 0.01 GeV ≤ b ≤ 0.40 GeV , 0.01 ≤ σ ≤ 1.40, 1.20 GeV≤ m ≤ 1.60 GeV for data set I, and 4.00 GeV≤ m ≤ 5.00 GeV for data set II. The resulting value of χ2 and each pull are presented in table 3.2, and the resulting parameters when fitting the model to respective data set yield the parameters in table 3.3.

2 2 Data set m [GeV] αS b [GeV ] σ χ I 1.46 0.50 0.15 1.10 0.0106 II 4.80 0.35 0.14 1.00 0.0278 III − 0.31 0.06 0.01 0.1934 IV − 0.13 0.05 0.01 0.1200 V − 0.52 0.14 1.09 0.4661

Table 3.3: The resulting parameter values when ﬁtting the diﬀerent data sets to the model.

3.3.1 Diquarks Given the values of the parameters of the model, we find the diquark masses by finding the energy eigenvalues changing b → b/2 and κ → −2/3 to compensate for the change in color structure of the quark-quark system. The sets of parameter values obtained when fitting the data sets I, II, III, and IV are used to calculate the mass of the cc, bb, qc, and qb diquark, respectively. The results of these masses are presented in table 3.4.

Diquark E [MeV] M [MeV] cc 220.9 3140.9 bb 71.21 9671.2 qc 232.0 2015.0 qb 218.0 5341.1

Table 3.4: Results for diquark masses, where q represents either an up or a down quark. 2S+1 3 All diquarks are calculated in the ground state N LJ = 1 S1.

3.3.2 Tetraquarks With the diquark masses in table 3.4 known, we use those masses as the constituents of the model used for the quark-antiquark system, together with the set of parameters obtained from ﬁtting data set V to calculate the total mass of diﬀerent composites of diquarks and antidiquarks. The result is presented in table 3.5.

15 2S+1 Tetraquark N LJ E [MeV] M [MeV] 1 1 S0 −162.7 3867.4 3 qcqc 1 S1 −73.46 3956.5 1 2 S0 531.7 4561.7 1 1 S0 189.0 10871.2 3 qbqb 1 S1 193.6 10875.8 1 2 S0 522 11204.6 1 1 S0 −305.7 5976.1 3 cccc 1 S1 −254.0 6027.8 1 2 S0 378.0 6659.8 1 1 S0 −74.78 19267.6 3 bbbb 1 S1 −72.11 19270.3 1 2 S0 277.8 19620.2 Table 3.5: Results for tetraquark masses. The constituent diquarks are from the ground 2S+1 1 state diquarks found in table 3.4. All tetraquarks are calculated in N LJ = 1 S0, 2S+1 3 2S+1 1 N LJ = 1 S1, and N LJ = 2 S0.

3.3.3 Comparison with Other Works Similar models have been proposed in Refs. [13, 17]. In Ref. [13], the authors were using the same model but also taking into account perturbation in spin-orbit and tensor interactions and only considering fully charmed diquarks and tetraquarks (cc and cccc¯ ) and it can thus be compared to the parameter set fitted to data set I. The models are identical for those states where the perturbation energy is zero. The authors of Ref. [17] considered the X(3872) particle under the hypothesis that its constituents are consisting of a diquark qc and an antidiquark qc¯ and could be compared to the parameters found by fitting data set III. They fit the model in order to investigate if Z(4430) could be an exited state of X(3872). Furthermore, we will compare the masses of diquarks and tetraquarks calculated in this model to those presented in Refs. [13, 18–23]. This is displayed in tables 3.6, 3.7, and 3.8.

2 Source M [GeV] αS b [GeV ] σ [GeV] Ref. [13] 1.4622 0.5202 0.1463 1.0831 I 1.46 0.50 0.14 1.10 Ref. [17] − 0.30 0.015 − III − 0.31 0.06

Table 3.6: Model parameters of the Cornell potential (and spin correction) from diﬀerent works.

16 Diquark M [MeV] Ref. [18] Ref. [19] Ref. [13] Ref. [20] qc 2015.0 2250.0 2099 − − qb 5341.1 − 5451 − − cc 3140.9 − − 3133.4 − bb 9671.2 − − − 8670

Table 3.7: Masses of diquarks, in units of MeV, from other works and the diquark mass 2S+1 3 M calculated in this thesis. All diquarks are in the ground state N Lj = 1 S1.

2S+1 Tetraquark N LJ M [MeV] Ref. [21] Ref. [22] Ref. [13] Ref. [23] 1 1 S0 3867.4 3820 − − − 3 qcqc 1 S1 3956.5 − − − − 1 2 S0 4561.7 − − − − 1 1 S0 10871.2 − 10807 − − 3 qbqb 1 S1 10875.8 − 10820 − − 1 2 S0 11204.6 − − − − 1 1 S0 5976.1 − − 5969.4 − 3 cccc 1 S1 6027.8 − − 6020.9 − 1 2 S0 6659.8 − − 6663.3 − 1 1 S0 19267.6 − − − 19306 3 bbbb 1 S1 19270.3 − − − 19329 1 2 S0 19620.2 − − − − Table 3.8: Result from diﬀerent works investigating tetraquarks as diquark-antidiquark systems.

3.4 Discussion

In table 3.2, the pulls and χ2-values from fitting the data sets I-V are presented. Com- paring the values of χ2 among the fits we see that the values of data sets I and II are about an order of magnitude smaller than the ones of data sets III-V. Since we as- sume that all mesons can be treated non-relativistic, the discrepancy in the χ2-value between the data sets I and II and the data sets III and IV could be explained by this assumption. However, one could expect that, when fitting the model to the data set V, the χ2-value would be of the same order of magnitude as the χ2-value when fitting the model to the data sets I and II, since the data set V consists of quarkonia, which is well suited for this model. Comparing the pull values obtained for data set V, we observe that the charmonium mesons fit the model with the parameter values obtained for this fit, significantly better than the bottomonium mesons. The smallest pull from this data set is 1.085 · 10−5, for the ψ(4040) charmonium meson, and the largest pull is 0.7631, for the χb2(2P ) bottomonium meson. This deviation in pull-values is difficult to explain. It could originate from the fitting procedure implemented in this thesis, that it is not suitable to assigning the same parameters for both charmonium and bottomonium mesons, or simply the inclusion of more data points contributing to the value of χ2. The pull-values from the fit to data set I varies from the smallest value of 4.348 · 10−5, for the

17 −2 ψ(4160) meson, to 1.491 · 10 , for the χc0(1P ) meson. This is a considerably smaller variation in the pull-values than the ones obtained from the fit to data set II with the −10 −2 smallest value of 1.295·10 , for χb0(1P ), to 1.012·10 , for ηb(1S). Thus, the parameter values obtained from the fit to data set I yields a better fit to the data compared to the parameter values obtained from the fit to data set II, although the value of χ2 is about the same for both fits. Considering the pull-values from the fit to data set III, the smallest value is 1.182 · 10−2, for the D∗+(2010) meson, and the largest value is 4.134 · 10−2, for the D+ meson, which do not deviate as much as the pull-values from the fit to data sets I and II. The same approximate variation is shown by the pull-values from the fit −2 + to data set IV, for which the smallest value is 2.302 · 10 , for the Bc meson, and the −2 0 largest value is 3.594 · 10 , for the Bs meson. Since the model is not able to split the masses for data sets III and IV, and since the experimental masses for these data sets only deviate with about 0.2 GeV-0.3 GeV, it is expected that the resulting pulls are of the same order of magnitude. The model will find a mass for which the pull-values are at most 0.04-0.09, which is what we observe in table 3.2 for the pull-values of data sets III and IV. Overall, the data set I fits the model better than all other data sets, and the data set V fits the model worst. When fitting the model to data set I consisting of cc¯ mesons, the resulting parameter values are similar to those found in Ref. [13]. The modelling procedure is similar, although the authors of Ref. [13] are also taking into account other energy contributions, namely spin-orbit and tensor interactions. Using the parameter values from Ref. [13], the resulting masses of the fully charmed diquarks and tetraquarks were found to be identical. The difference of the parameter sets could most likely be attributed to accuracy in the minimization of eq. (3.9). A comparison between the parameter sets is presented in table 3.6. Fitting the model to data set II consisting only of the b¯b meson, the constituent mass m of the bottom quark is similar to the value 4.73 GeV presented in Ref. [2] though the rest of the parameters deviate from the parameters obtained when fitting the model to the data set I. One could expect that the resulting parameters, apart from the constituent masses, obtained when fitting the model to data sets I and II would be similar since both are modeling quarkonia interacting with an OGE potential. The parameters obtained when fitting the data sets I and V shared some similarities, although the χ2 value is larger with an order of magnitude when fitting the model to data set V than when fitting the model to data set I. This could be a consequence of introducing more data points in the data set V. Since the model is unable to split the masses with same principal, orbital, and spin quantum numbers but different angular momentum quantum numbers the effect of introducing more data points could contribute to a larger value of χ2. Finding the parameter sets for the cq¯ and bq¯ systems is performed under the assumption that the model is valid for heavy-light meson systems. This is questionable since a non-relativistic approach is suitable considering only heavy mesons with small kinetic energy, thus the values obtained from fitting the model to data sets III and IV are hard to justify. Also, since all masses of data sets III and IV are assumed to be in the ground state, the mass obtained from the model is the same for a given parameter set for each meson in those data sets, thus being unable to split the masses of those data sets. This could explain the large values of χ2 for those cases. Comparing the resulting masses for the diquark qc with Refs. [18,19], we observe that

18 it is smaller than that obtained in Refs. [18, 19], see table 3.7. The authors of Ref. [18] considered a more detailed approach with a relativistic model of diquarks taking the kinetic energy as well as a hyper-fine interactions consisting of spin-orbit and tensor interactions into consideration. The authors of Ref. [19] studied the mass of diquarks by the means of the so-called Schwinger–Dyson and Bethe–Salpeter equations. The model takes into account the kinetic energy as well as splitting in the spin-spin, spin-orbit, and tensor interactions. As mentioned earlier a non-relativistic approach to the heavy-light meson system is difficult to justify, since relativistic effects play a significant role in those kinds of systems. In table 3.7, we see that the mass of the qc and qb diquarks are around 100 MeV smaller than those found in Refs. [18,19] which could be explained to originate from relativistic effects. Now, comparing the result of the resulting mass of the cc diquark, we observe a larger conformity with the mass obtained in Ref. [13]. Since the modeling approach and data fitting procedure are similar, the similarity of the cc mass found in Ref. [13] and this thesis is not surprising. When using the parameter set obtained by the authors of Ref. [13], found in table 3.6, the result is actually identical. The model is also well suited to analyze systems of heavy mesons because the contribution of relativistic effects should be small when the main energy comes from the OGE potential. The resulting mass of the bb diquark is bigger than that found in Ref. [20]. The authors of Ref. [20] generate the color-antitriplet diquarks using the so-called QCD Laplace sum rule. The discrepancy between these values is difficult to explain, but considering that the modeling procedure of this thesis and that done in Ref. [23] are very different, it could be a consequence of the simulation method used in Ref. [23] and/or the fitting method used in this thesis. The resulting mass of the qcqc¯ tetraquark is about 47.4 MeV smaller than the one obtained in Ref. [21], see table 3.8. The authors of Ref. [21] took into consideration, apart from spin-spin interaction between the quarks and the diquarks, the spin-orbit and orbit-orbit interactions. Since the constituent diquarks building up the tetraquark studied in Ref. [21] are qq and cc¯ , and from this thesis qc and qc¯ , the difference in the resulting tetraquark mass could be attributed to this disparity in the modeling procedure. Comparing the resulting mass of the qbqb¯ tetraquark with the one found in Ref. [22], we see that the masses obtained in this thesis are about 60 MeV larger than those obtained in Ref. [22]. The method used in Ref. [22] is based on calculating the diquark-gluon vertex in terms of the diquark wave functions in a relativistic approach. Since the model considered in this thesis is under the assumption that the qc and qb can be treated non- relativisticly, the difference in the resulting masses of the qcqc¯ and qbqb¯ compared to the corresponding mass found in Refs. [21, 22] could be a consequence of this assumption. The resulting mass of the cccc¯ is in good agreement with the masses obtained in Ref. [13]. Also, when using the parameters m, αS, b, and σ presented in Ref. [13], see table 3.6, the result is identical. The difference in the resulting masses of the cccc¯ is thus a consequence of the different sets of parameters used in the OGE potential. The resulting mass of the bbbb¯ is about 40 MeV smaller compared to the mass obtained in Ref. [23]. The authors of Ref. [23] used a non-relativistic approach but adopted the variation principle to solve the Schrödinger equation. Also, the potential used in Ref. [23] uses a different confiding part of the potential, which could explain the differences in the results obtained. On the experimental side, the tetraquark cccc¯ is of special interest, since the possibil-

19 ity of pair production of J/ψ mesons can imply the existence of fully-charmed tetraquark states. This have been studied by the LHCb [24, 25], CMS [26], and ATLAS [27] collaborations as well as the Belle collaboration [28]. In Ref. [24], one can see that the diﬀerential production cross-section lies in the range between 6 GeV and 8 GeV and most of the predictions for the cccc¯ tetraquark yield values around 6 GeV. Since the masses of charmonium lie well below this value (3.0 GeV-4.5 GeV) the cccc¯ tetraquark is a special object of study in the sector of exotic hadrons. The results obtained in this thesis suggest that the mass of the fully-charmed tetraquark could be 5976.1 MeV in its ground state.

20 Chapter 4

Summary and Conclusions

The field of exotic hadrons is ever growing and many models have been proposed to predict the masses of those exotic hadrons that have been observed as well as those that have potential to be observed in the future. Considering tetraquarks, and in particular fully-heavy tetraquarks cccc¯ and bbbb¯ , many new propositions of the inner structure of a diquark and an antidiquark pair have been proposed [6]. The observation of the charmed baryon Ξcc could also imply the existence of the fully-charmed tetraquark [23]. In this thesis, we have considered a variation of a model originally suggested to describe the fully-charmed tetraquark cccc¯ in a diquark-antidiquark system using a non-relativistic approach. The modeling procedure consists of first fitting the parameters of Cornell potential added with a smeared Gaussian spin-spin correction to a total of five different data sets consisting of experimental masses of mesons. The potential is often used to describe heavy-meson systems but was assumed to be valid for heavy-light meson systems. The data sets considered were thirteen charmonium mesons, nine bottomonium mesons, the combination of the thirteen charmonium and nine bottomonium mesons, five D mesons and four B mesons. Second, the model was extrapolated to a system consisting of two quarks, called a diquark. This has been performed by considering the difference in the color structure between the quark-antiquark system and diquark system, resulting in a relation between the potentials of the two systems. Using the parameters found by fitting the different data sets, potentials for the diquark systems qc, qb, cc, and bb have been found. Subsequently, the respective diquark mass has been calculated. Lastly, the qcqc¯ , qbqb¯ , cccc¯ , and bbbb¯ tetraquark masses have been calculated by considering the diquark-antidiquark system in the same manner as was performed for the meson systems, using the parameters of the potential found by fitting the data set consisting of thirteen charmonium and nine bottomonium mesons.

Double cc¯ production can be a good start to search for fully-charmed tetraquarks, and as seen in Ref. [24], the diﬀerential production cross-section for J/ψ pairs is between 6 GeV and 8 GeV. The resulting mass obtained in this thesis suggests that the mass of 2S+1 1 the cccc¯ tetraquark is 5975.1 MeV in the state N LJ = 1 S0, and has a mass spectrum as shown in table 3.5.

21 Acknowledgment

I would like to thank Dr. Franz F. Schoeberl for providing me with the Mathematica notebook used in Ref. [16]. It inspired the code used in this work when solving the Schrödinger equation. I would also like to thank my supervisor Prof. Tommy Ohlsson for his useful and constructive recommendations, and valuable support throughout this project.

22 Bibliography

[1] R.J. Blin-Stoyle. Nuclear and particle physics. Chapman & Hall, 1991. [2] D. Griﬃths. Introduction to elementary particles. Wiley-VCH Verlag GmbH, 2008. [3] G. Fraser. The New Physics for the Twenty-First Century. Cambridge University Press, 2006. [4] R.L. Jaﬀe. Exotica. Phys. Rept., 409:1–45, 2005. [5] S.K. Choi et al. Observation of a narrow charmoniumlike state in exclusive B± → K±π+π−J/ψ decays. Phys. Rev. Lett., 91:262001, 2003. [6] G. Cowan and T. Gershon. Tetraquarks and pentaquarks. IOP Publishing, 2018. [7] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. Shen, C. E. Thomas, A. Vairo, and C. Yuan. The XYZ states: experimental and theoretical status and perspectives. arXiv e-prints, 1907.07583, 2019. [8] A. Ali, L. Maiani, and A. D. Polosa. Multiquark Hadrons. Cambridge University Press, 2019. [9] H. Georgi. Lie algebras in particle physics. Front. Phys., 54:1–320, 1999. [10] H.F. Jones. Groups, representations and physics. Institute of Physics Publishing, 1990. [11] S. Sternberg. Group theory and physics. Cambridge University Press, 1994. [12] S. Godfrey and N. Isgur. Mesons in a Relativized Quark Model with Chromody- namics. Phys. Rev. D, 32:189–231, 1985. [13] V.R. Debastiani and F.S. Navarra. A non-relativistic model for the [cc][¯cc¯] tetraquark. Chin. Phys. C, 43:013105, 2019. [14] M. De Sanctis and P. Quintero. Charmonium spectrum with a generalized Fermi- Breit equation. Eur. Phys. J. A, 46:213–221, 2010. [15] M. Tanabashi et al. Review of Particle Physics. Phys. Rev. D, 98:030001, 2018. [16] Wolfgang Lucha and Franz F. Schoberl. Solving the Schrödinger equation for bound states with Mathematica 3.0. Int. J. Mod. Phys. C, 10:607–620, 1999.

23 [17] V.R. Debastiani and F.S. Navarra. Charm tetraquarks in a non-relativistic quark model. J. Phys. Conf. Ser., 630:012047, 2015.

[18] M. N. Anwar, J. Ferretti, and E. Santopinto. Spectroscopy of the hidden-charm [qc][¯qc¯] and [sc][¯sc¯] tetraquarks in the relativized diquark model. Phys. Rev. D, 98:094015, 2018.

[19] M.A. Bedolla. Tetraquark Insights from Quark Models and Schwinger–Dyson Equa- tions. Few Body Syst., 60:24, 2019.

[20] S. Esau, A. Palameta, R.T. Kleiv, D. Harnett, and T.G. Steele. Axial Vector cc and bb Diquark Masses from QCD Laplace Sum-Rules. Phys. Rev. D, 100:074025, 2019.

[21] X. Yan, B. Zhong, and R. Zhu. Doubly charmed tetraquarks in a diquark– antidiquark model. Int. J. Mod. Phys. A, 33:1850096, 2018.

[22] D. Ebert, R.N. Faustov, and V.O. Galkin. Relativistic model of hidden bottom tetraquarks. Mod. Phys. Lett. A, 24:567–573, 2009.

[23] M. Liu, Q. Lü, X. Zhong, and Q. Zhao. All-heavy tetraquarks. Phys. Rev. D, 100:016006, 2019. √ [24] R. Aaij et al. Observation of J/ψ pair production in pp collisions at s = 7 TeV. Phys. Lett. B, 707:52–59, 2012.

[25] R.√ Aaij et al. Measurement of the J/ψ pair production cross-section in pp collisions at s = 13 TeV. JHEP, 06:047, 2017.

[26] V.√ Khachatryan et al. Measurement of Prompt J/ψ Pair Production in pp Collisions at s = 7 TeV. JHEP, 09:094, 2014.

[27] M. Aaboud et al.√ Measurement of the prompt J/ ψ pair production cross-section in pp collisions at s = 8 TeV with the ATLAS detector. Eur. Phys. J. C, 77:76, 2017. √ [28] K. Abe et al. Observation of double c anti-c production in e+ e− annihilation at s approximately 10.6 GeV. Phys. Rev. Lett., 89:142001, 2002.

24 www.kth.se