Searching for New Vector Particles at the Lhc

UNIVERSIDAD TÉCNICA FEDERICO SANTA MARÍA DEPARTAMENTO DE FÍSICA

SEARCHING FOR NEW VECTOR PARTICLES AT THE LHC

Tesis de Grado presentada por Sr. Bastián Díaz Sáez

como requisito parcial para optar al grado de Magíster en Ciencias, Mención Física

Profesor Guía Dr. Alfonso R. Zerwekh

Valparaíso - Chile 2013

Valparaíso, Chile. 2013 TÍTULO DE LA TESIS:

SEARCHING FOR NEW VECTOR PARTICLES AT THE LHC

AUTOR:

BASTIÁN JAIME DÍAZ SÁEZ

TRABAJO DE TESIS, presentado en cumplimiento parcial de los requisitos para el Grado de Magister en Ciencias Mención Física de la Universidad Técnica Federico Santa María.

Dr. Alfonso Zerwekh Universidad Técnica Federico Santa María

Dr. Gorazd Cvetic Universidad Técnica Federico Santa María

Dr. Alfredo Vega Universidad de Valparaíso

Valparaíso, Chile. 2013

2 ...dedicada a mis padres Gloria y Jaime.

3 Agradecimientos

Mis más sinceros agradecimientos a mi familia, sobretodo a mi madre, Gloria, por haberme acompañado siempre hasta el día de hoy, y a mi padre Jaime por haberme apoyado incondicionalmente, sobretodo en los primeros años de universidad. Quisiera agradecer también a mi profesor de tesis Alfonso Zerwekh, que gracias a su buena dis- posición y entrega pudo ser posible este trabajo. Sin duda, quiero agradecer a mi polola Camila por todo el amor, paciencia, apoyo y cariño que me ha brindado. Agradezco a Conicyt por el apoyo económico durante el año 2012 y a la UTFSM por el apoyo ﬁnanciero.

4 Resumen

En el Modelo Estandar de Física de Partículas (SM), eventos de dos-jets (dijet) son producidos en colisiones proton-(anti)proton predominantemente de las interacciones de dos partones de la cromodinámica cuántica (QCD). La fragmentación y la hadronización de los partones finales producen jets hadrónicos. El espectro de masa invariante del dijet predicha por QCD cae suave y rápidamente con el incremento de la masa de éste. Muchas extensiones del SM predicen la existencia de nuevas partículas masivas que decaen en dos partones energéticos (quarks, q, or gluons, g), que potencialmente pueden ser observadas como una resonancia en el espectro de masa invariante del dijet. En modelos quirales de color, el grupo de gauge de color de QCD, SU(3)c, resulta del rompimiento espontáneo del grupo de gauge de color quiral SU(3) × SU(3). Cualquier modelo quiral de color predice la existencia de un axigluon, un gluón masivo que se acopla a los quarks de manera axial, que decae a un par qq¯. En este trabajo mostramos algunos modelos quirales, y veremos como algunos de estos modelos se acoplan a las observaciones hechas en CDF, CMS y ATLAS. El corazón de esta tesis se centra en simulaciones en CalcHEP del proceso p+p → A → jet + jet, donde A corresponde a un axigluón universal, con energías centro momentum a 7, 8 y 14 TeV. Estas simulaciones tienen distintos fines. Una es que a través de las comparaciones entre las secciones eficaces dadas por nuestros resultados con los límites superiores al 95% C.L. en la sección eficaz para una resonancia de modelo-independiente dados por el experimento ATLAS a 7 y 8 TeV, excluimos posibles masas del axigluón e imponemos límites en la constante de acoplamiento gA entre los quarks y el axigluón. También, es mostrado como la sección eficaz del axigluón es afectada cuando se considera adicionalmente el efecto del “smearing” de un detector real y cuando consideramos un axigluón ancho (Γ/M & 0.2), el cual éste último podría potencialmente dar cuenta de la FB anomalía Att¯ observada años atras en el Tevatron. Finalmente, tomando en cuenta que el LHC en un futuro cercano colisionará protones a energías sobre los 14 TeV, hacemos predicciones para la sección eficaz de resonancias a este rango de energías. Todas las simulaciones obtenidas de CalcHEP son analizadas con PAW.

5 Abstract

Within the Standard Model (SM) of Particle Physics, two-jet (dijet) events are produced in proton−(anti)proton collisions predominantly from hard quantum chromodynamics (QCD) interactions of two partons. The fragmentation and hadronization of the outgoing partons produce hadronic jets. The dijet mass spectrum predicted by QCD falls smoothly and steeply with increasing dijet mass. Many extensions of the SM predict the existence of new massive particles that decay into two energetic partons (quarks, q, or gluons, g), which can potentially be observed as a resonance in the dijet mass spectrum. In chiral color models, the SU(3) color gauge group of QCD results from the spontaneous breaking of the SU(3) × SU(3) chiral color gauge group. Any model of chiral color predicts the presence of an axigluon, a massive vector gluon which couple to quarks in a pseudo-vector form, that decays to a pair qq¯. In this work we show some chiral models, and we will se how they coupled to the recent seaches performed by CDF, CMS and ATLAS. The hearth of this thesis is focused on simulations in CalcHEP of the process p + p → A → jet + jet, where A correspond to an universal-axigluon, at 7, 8 and 14 TeV center momentum energies. These simulations have different purposes. One of them is, through the comparation among our axigluon cross section simulations to the cross section upper limits 95% C.L. for a resonance model-independent given by ATLAS experiment at 7 and 8 TeV, we exlude posibles axigluon masses and set limits on the coupling constant gA between quarks and axigluon. Also, it is show how the axigluon cross section is affected when is considered both the smearing effect of a real detector in the analisis of the data FB and a broad axigluon (Γ/M & 0.2), which this latter is viable explanation for the Att¯ anomaly observed at the Tevatron. Finally, taking into account that LHC in the near future will operate at energies above 14 TeV, we make predictions on the cross section for resonances at this range of energies. All the simulation obtained from CalcHEP are analized in the frame of PAW.

6 Contents

1. Introduction 9 1.1. Elementary particle physics ...... 9 1.2. New Particles ...... 11 1.3. Natural Units ...... 12

2. Theoretical fundamentals 13 2.1. Elements of QCD ...... 13 2.1.1. The quantum chromodynamics Lagrangian ...... 13 2.1.2. Conﬁnament and asymptotic freedom ...... 14 2.1.3. Jets ...... 15 2.2. Resonances ...... 16 2.2.1. What are they? ...... 16 2.2.2. Breit-Wigner resonance ...... 18 2.2.3. Narrow with approximation ...... 20 2.3. Non-lineal sigma model ...... 21

3. Motivation for New Physics in the Strong Sector 25 3.1. Enlarging the strong sector symmetry group ...... 25 3.2. The Forward-Backward Asymmetry ...... 25

4. Axigluons models 30 4.1. Flavor-universal models ...... 30 4.1.1. Minimal axigluon model ...... 30 4.1.2. Non-minimal model: Four gauge ﬁelds ...... 32 4.2. Flavor-nonuniversal models ...... 34 4.2.1. Frampton Model ...... 34 4.2.2. A simpliﬁed three-site model ...... 36 4.2.3. A Two-Site Model with a New Vector-like Quark ...... 38

5. Searching resonances in the dijet spectrum 41 5.1. Introduction ...... 41 5.2. Parton-parton scattering ...... 41 5.3. CDF II at TEVATRON ...... 42 5.4. ATLAS at LHC ...... 45 5.4.1. The ATLAS experiment ...... 45 5.4.2. Collider Kinematics ...... 47 5.4.3. Trigger ...... 48

7 5.4.4. Kinematical cuts for pp collision at 7 TeV and 8 TeV ...... 48 5.4.5. Model-independent limits on dijet resonance production ...... 49

6. Simulation and Analysis of data 51 6.1. Introduction ...... 51 6.2. The Simulation ...... 52 6.3. Analisis with Paw ...... 56 6.4. Results at 7 and 8 TeV ...... 60 6.5. Simulation at 8 TeV with smearing ...... 64 6.6. Broad Axigluon ...... 65 6.7. Calculus at 14 TeV ...... 66

7. Conclusions 68

A. Data obtained from simulation 70

B. Electron-proton scattering 74 B.1. Introduction ...... 74 B.2. Elastic scattering e− + µ− → e− + µ− ...... 74 B.3. Elastic scattering e− + p → e− + p ...... 78 B.4. Electron-proton inelastic scattering ...... 80 B.5. Bjorken scaling ...... 82

C. σ-lineal model 86 C.1. Symmetries of the model ...... 86 C.2. Spontaneous symmetry breaking ...... 87

D. Anomalies 90

E. Parity 94

8 1. Introduction

1.1. Elementary particle physics

What is matter made of? This is one of the fundamental question of physics. Physicist through the years have been studying the composition of the matter by different tech- niques. Now, accompanied by the technology, we can “look” inside of the matter to a very small lenght scales where the dynamics and the rules of the nature are very different to our experience. We call this tiny world, the subatomic world. In this subatomic world we have found a lot of families of particles, and at this moment, we have found the basic bricks of matter and how they interact. The best known elementary particle is the electron, which, for example, are passing through the wires of your computer, cellphone, etc. This is what we call current. These particles, also, have the role of make fundamental structures of the matter which surround us: the atom. Atoms are structures constitued of three basic particles: electrons, neutrons and protons. The regularities in Mendeleev’s table were a stepping-stone to nuclei and to particles called protons and neutrons (colllec- tively labeled nucleons), which are “glued” together by a strong or nuclear force to form the nuclei. These subsequently bind with electrons through the electromagnetic force to produce the atoms of the chemical elements. There are a lot of different atoms compiled in the famous Periodic Table, which now contains well over 100 chemical elements. Also, in the last century, experiments have shown that the atomic nucleous is composed by another kind of entities: quarks. In Fig. 1 it is shown an schematic representation of the atom and their constituents. We can see that the lenght scale where these particles live is, of course, not visible to the human eye.

Figure 1.1.: The structure and the subsructures of the atom.

With the years, physicist have been ﬁnding a lot of new particles. Furthermore, they have gone classifyng them, and have arrived to distinguish and classifying the basic constituents of matter. Fig. 2, shows the basic building of matter and the particles

9 through they interact. We talk of building of matter because, as far as we know, they are the smaller things ever detected. For one side we have the six quarks u, d, c, s, t and b. They make up particles called hadrons: mesons (particles composed of quark-antiquark pair) and barions (composed by three quarks). The proton, for example, is a kind of baryon, because it is made of two u and one d quarks. Below quarks, there are the so-called leptons, which are e, µ, τ, νe, νµ and ντ . The best known particle of them is the electron, which composed atoms. Also, in Fig. 2 is shown the force carriers, which are the photon (electromagnetic force), gluon (strong force) and the W’s and Z bosons (weak force). These force carriers do interact with matters particles. However, not all the matter particles “feel” all these forces, for example, leptons do not interact through of the gluon, which is the same to say leptons do not feel the strong force. Conversion of neutrons into protons by so-called weak-interaction is responsible to the radioactive β-decay of nuclei. Also, physicist, have found not only these elementary particles, but the anti-particles of them. This is, the same particles but with opposite electric charge. All this elementary particles have a lot of properties which we distinguish them: mass, electric charge, spin, etc..

Figure 1.2.: The basic constituents of matter, the carrier forces, and the recently discovered Higgs boson.

Recently, it was seen at the LHC (Large Hadron Collider) a new particle and all the evidence indicates that it is the Higgs boson. This, is the quanta of the Higgs ﬁeld, which plays a very important role in nature. Through the so called Higgs mechanism, W and Z gauge bosons acquire mass. Fermions, such as the leptons and quarks in the Standard Model, can also acquire mass as a result of their interaction with the Higgs ﬁeld, but not in the same way as the gauge bosons. Particles and their interactions, are described in a mathematical frame called “Standard Model” (SM) of particle physics. This theory -or, more accurately, a collection of related theories, incorporating quantum electrodynamics (QED), the Glashow-Weinberg-Salam

10 theory of electroweak process and quantum chromodynamics (QCD)- has shown to be very succesfull from 1978, predicting a lot of things in total agreement with experimental data. It achieved the status of “orthodoxy”, but no one pretend that SM is the ﬁnal word on the subject. The majority of the matter particles shown in Fig. 2, are not around us and neither compose the matter where we live. These particles must be, for example, created in laboratories at high energy collisions, or are seen in some radiactive process, or some of them are created at the sun. At the LHC two proton beams collide at very high energy and a big amount of the kinetic energy is transformed into new particles through the famous equation E = mc2. After the collision, ﬁnal state particles produce signal in the detector. ATLAS, is one of the big detectors at the LHC, and this thesis is focused on recently data from this detector. The more energy we have to collide protons, more available energy there is to create known and, possiblly, unknown particles.

1.2. New Particles

With the new era of colliders and detectors in particle physics, especially the LHC, we hope to find new physics at the energies near of the TeV scale. There a lot of expectation for the possible existence of new particles that we have never seen. There are many extensions of the SM that predict the existence of new massive particles that decay into two energetic partons (quarks, q or gluons, g), which can eventually be detected as a narrow resonance in the dijet mass spectrum. Such new states may include an excited composite quark q∗, exemplifying quark substructure; an axigluon predicted by chiral color models; a flavour-universal color-octet coloron; a color-octet techni-ρ meson predicted by models of extended technicolor and topcolor-assisted technicolor; Randall- Sundrum (RS) gravitons (G), predicted in the RS model of extra dimensions; scalar diquarks (D), predicted by a grand unified theory based on the E6 gauge; new gauge bosons (W 0 and Z0), predicted by models that propose new gauge symmetries. Experiments at hadron collider have used the dijet mass spectrum to search for new particles beyond the standard model. Whitin the Standard Model (SM) of particle physics, two-jets events are produced in pp¯ or pp predominantly from hard QCD interaction of two partons. We know that the dijets come from a pair of quarks which hadronize to form jets. We study models where dijets are produced by the decay of a new kind of particle or resonance. The simple process that the experiment have searched for this new kind of particles is depicted in Fig. 3.

11 Figure 1.3.: New vector particle as a intermediate state

In a analogy to QED which originates from the spontaneous symmetry breaking (SSB) of the SU(2) × U(1) symmetry group, some people speculate that QCD originates in a chiral color group SU(3) × SU(3), and that the scale of chiral color breaking is similar to the electroweak mass scale. Such theories predict the existence of many new fundamental particles including the so called axigluon. The axigluon is a massive, spin-1, color-octet particle which coupling to quarks is axial-vector. It mass should be of several hundred of GeV and it should be visible as a resonance in the dijet spectrum. In this thesis we focus our attention to this specific hypotetical new particle. We start in section 2 with a review of the basic concepts of QCD, resonances and non-linear σ-model. In section 3 we will see the fundaments which motivate the search for new physics in the dijet mass spectrum. In chapter 4, we will describe repase some specific axigluon models. In this thesis we work with an universal-axigluon coupling. In chapter 5, it will be introduce kinematical concepts that are useful in collider physics, and we show results found at Tevatron and LHC on the dijet mass distribution. Also, it is shown the kinematical constraints given by ATLAS experiments which will be utile to our simulation, and is shown the recents upper limits on the dijet cross section given by the same experiment. In chapter 6, the core of the thesis, it is explained the methodology of the simulation of the process pp → A → jj, how we analyse the data and finally the results that we obtained. Finally, in chapter 7 are exposed the conclusions of this work.

1.3. Natural Units

µ We work in natural units c = ~ = 1. The four vectors are x = (t, ~x), the partial deriva- µ tives are ∂µ = ∂/∂x . The metric in the coordinates (t, x, y, z) is gµν = (+, −, −, −).

12 2. Theoretical fundamentals

2.1. Elements of QCD

2.1.1. The quantum chromodynamics Lagrangian Quantum chromodynamics is a non-abelian quantum ﬁeld theory of the strong interaction among quarks and gluons based on the SU(3) symmetry group. In physical terms the non-abelian behaviour of QCD is that gluons (gauge mediators) are colored and thus they can interact among themselves. The Lagrangian of QCD is

X f µ f 1 a aµν L = q¯ (x)(iγ Dµ − mf ) q (x) − F F (2.1) i ij j 4 µν flavors f f where qi (x) and q¯i (x) are the quark and antiquark spin 1/2 Dirac ﬁelds. i, j = 1, 2, 3 are ﬂavor f indexes, a = 1,...,8 is a color index and mf is the mass of the fermions. The covariant derivate is given by

a a Dµ = ∂µ − igAµ = ∂µ − igt Aµ where ta are the generators of the group SU(3) in the fundamental representation and a b abc c they satisfy the algebra t , t = if t and g is the gauge coupling constant. a The non-abelian gluon field strength tensor Fµv of the gluon field is defined as i F = taF a = [D ,D ] µν µν g µ ν or, equivalently, by

a a a abc b c Fµν = ∂µAν − ∂νAµ + gf AµAν

a abc where Aµ is the gauge ﬁeld and f are the structure functions. This lagrangian was proposed by Fritzsch, Gell-Mann and Leutwyler (1973), Gross and Wilczek (1973,1973b) and Winberg (1973). It has been observed that the strong interactions are inmersed in an internal symmetry called SU(3). This symmetry implies that our Lagrangian should be invariant under the following transformations:

q(x) → S(x)q(x) (2.2)

q¯(x) → S−1(x)¯q(x) (2.3)

13 −1 i −1 Aµ(x) → S(x)Aµ(x)S (x) − [∂µS(x)] S (x) (2.4) g where S(x) corresponds to an element of the fundamental irreducible representation of SU(3) group, and may be written as

S(x) = exp (iαa(x)ta) with αa(x) arbitrary real-valued functions and ta are the generators of the SU(3) group, also called “Gell-mann matrices” which are the generators of the group in the fundamental representation. The form of the Yang-Mills Lagrangian 2.1 can be derived directly from the gauge symmetry in Eq. 2.2, 2.3 and 2.5.

2.1.2. Confinament and asymptotic freedom There is experimental evidence that quarks are never seen as free particles, that means that they are confined forming bound states by a force that should be strong at long distances. Quarks tends to make bound states, i.e. mesons and barions, and the mediator of this force between quarks are called gluons. Gluons are massless vector bosons spin-1 and they have the property of interact among themselves. If one tries to separate the quarks, color confinement tends to get toghether again both quarks and if the energy is sufficiently high, it will appear quark-antiquark from de vacuum. At high energies, we have a opposite phenomena. The force between quarks and gluons decrease and they are seen almost as a free particles. This behavior is called Asymptotic freedom (Gross and Wilczek (1973), Politzer (1973)). This can be seen in the running QCD coupling

2 2 αs µ 1 αs Q = = Q2 2 Nf Q2 1 + αs (µ ) β2 ln 2 µ b2 ln 2 ΛQCD where 11N − 2N β = c f 2 12π with Nc = 3 the number of colors and Nf the number of quarks ﬂavors. The parameterµ is an arbitrary scale known as the renormalization point and the physical observables not dependo on its value. We can see that at short distances (corresponding to large values µ 2 2 2 2 4-momentum q squared q = −Q with Q a real number), αs Q → 0 as Q → ∞. Thus, quarks and gluons at short distances interact weakly.

14 Figure 2.1.: Running coupling constant as a function of the transfer momentum Q [GeV].

One of the prominent features of QCD at low-energy is color-confinement: Any strongly interacting system at zero temperature and density must be a color singlet at distance scale larger than 1/Λ QCD. As a consequence, isolated free quarks cannot exist in nature (quark confinement). The color confinement of QCD is a theoretical conjecture consistent with experimental facts. To prove it in QCD is still a challenge that has not been met. Suppose, for example, we have a quark-antiquark pair which is in a color singlet state. One may try to separate the quark from the antiquark by pulling them apart. The interaction between the quarks gets stronger as the distance between them gets larger, similar to what happens in a spring. In fact, when a spring is stretched beyond the elastic limit, it breaks to produce two springs. In the case of the quark pair, a new quark-antiquark pair will be created when pulled beyond certain distance. Part of the stretching energy goes into the creation of the new pair, and as a consequence, one cannot have quarks as free particles. The only way we know how to solve QCD in the strong coupling regime is to simulate the theory on a finite spacetime lattice, or in short lattice QCD.

2.1.3. Jets Due to conﬁnement quarks and gluons are not detected directly, we always see them in structures as mesons and barions. In a detector, we detect baryons a kind of “spray” of highly collimated particles, called jet, which are constitued primarly of hadrons, but also photons and leptons. The jet-formation takes the following steps: 1. Branching parton: A parton (i.e. a gluon, quark or antiquark) has the probability of split into two partons, which are emitted in the direction of the initial parton. The

15 probability of split depends on some constants called color factors which depends on the type of involved parton. For example, for gluon → gluon+gluon the color 1 factor is Cgg = 3, for gluon →quark+gluon the color factor is Cqg = 2 , etc. Also, partons are produced close to the direction of the initial partons, which results in a high degree of collimation of the ﬁnal hadrons. 2. Hadronization: When the parton shower has evolved long enough, the energy of the partons is reduced, such that low-momentum transfer occur. In these conditions, the parton interactions become non-perturbative, and the phase of hadronization begins. During the hadronization, partons are combined into color-singlet states, thus forming the hadrons. The QCD (Quantum Chromodynamics) of the hadronization process are not yet fully understood, but are modeled and parame- terized in a number of phenomenological studies, including the Lund string model and in various long-range QCD approximation schemes. In the ﬁgure we show a scheme of the collision of two protons showing the events occurring at the proccess.

Figure 2.2.: Description of the “hard” and “soft” proton-proton scattering.

The resulting events contains particles that originates from the two outgoing partons (plus initial and final-state radiation) and particles that come from the breakup of the proton-proton (i.e., beam-beam remants). We call “underlying event” everything except the two outgoing hard scattered “jets” and receives contributions from the “beam-beam remnants” plus initial and final-state radiation. In the thesis, we foucus our resonance search just in the hard scattering, this is, just in the dijet final state.

2.2. Resonances

2.2.1. What are they? Resonances are extremely short lived particles with a lifetime of the order of 10−23s. From the experimental point of view, a resonance is a peak located around a certain

16 energy found in differential cross sections of scattering experiments. When we performed collisons of two beams of particles, let’s say, particles A and particles B, we can get the differential cross section in function of the energy of the system. Sometimes, we will find that some peaks exist in the plot of differential cross section vs energy of the system, which we identify as a resonance state. Something occurs in the point of the collision that makes the scattering cross section very high at some energies. The first resonance, 4, was discovered by H. Anderson, E. Fermi, E. A. Long, and D. E. Nagle, working at the Chicago Cyclotron in 1952. This particle could be in four diferent states: 4−,40, 4+ and 4++. One of this states appear in the pion-nucleon scattering p + π+ → 4++ → p + π+, where p is a proton and π+ is a pion with +1 electric charge. The width Γ ∼ 110 MeV may me measured directly from the curve, giving the lifetime from the uncertinty principle

−23 τ ∼ ~/Γ ∼ 10 s

This is the characteristic time of the strong interactions, by which resonances decay. Resonances are states with well-deﬁned spin and parity, mass, charge, etc., and are to be treated on an equal footing with the other elementary particles.

Figure 2.3.: Total cross section for π+p (solid line) and π−p (dashed line) scattering.

Other example of resonance is the scattering of pions

π+π− → ρ → π+π−

Rho meson is a short-living hadronic particle that is an isospin triplet whose three states

17 are denoted as ρ+, ρ0 and ρ−. After the pions and kaons, the Rho mesons are the lightest strongly interacting particle with a mass of roughly 770 MeV for all three states. The rho mesons have a very short lifetime and their decay width is about 145 MeV with the peculiar feature that the decay widths are not described by a Breit-Wigner form. The principal decay route of the rho mesons is to a pair of pions with a branching rate of 99.9. Neutral rho mesons can decay to a pair of electrons or muons which occurs with a branching ratio of 5 × 10−5. Other experiment showing resonances, was perform with electron and positrons annihilation

Figure 2.4.: Scattering cross section for the process e−e+ → qq¯ → hadrons. There are a lot of resonances in the ﬁgure.

2.2.2. Breit-Wigner resonance Let’s look at the decay of an unstable particle R → A + B. Assuming an exponential decay for the R particle at t = 0, the wave function is given by

| ψ(t) |2=| ψ(0) |2 exp (−t/τ) Θ(t)

We can see that the width decay (or known as decay rate or resonance width) is given by 1 d | ψ(t) |2 1 Γ = − = | ψ(t) |2 dt τ

The time dependence of the wave function of a free particle in its rest frame is

18 ψ(t) = ψ(0) exp −iEψt = ψ(0) exp (−iE0t) exp −Γt/2 where E0 is the mass of the particle at t = 0 in the rest frame. An unstable particle can be seen to have a complex energy Eψ = E0 − iΓ/2. The Fourier transform of ψ(t) is

1 χ(E) = √ ψ(t) exp (iEt) dt 2π ˆ ψ(0) = √ exp (it (E − (E0 − iΓ/2))) dt 2π ˆ iψ(0) 1 √ ∼ Γ 2π E − E0 + i 2 where we are assuming, in the second term, that the argument of the exponential function oscillates too fast. Then, the probability density of the Fourier transform of the wave function is the energy density | ψ(0) |2 1 | χ(E) |2= √ 2 2 2π (E − E0) + Γ /4 known as Breit-Wigner Resonance distribution. Undetermined lifetime means undetermined energy and the probability of ﬁnding an unstable particle with a given energy is a distribution around E0. The elastic cross section which a resonance is present, can be derive using the partial wave analysis starting with the cross section formula for pure elastic scattering for the lth partial wave

l 4π 2 4π 1 σel = 2 (2l + 1) sin (δl) = 2 (2l + 1) 2 k k 1 + cot δl

We note that the cross section has a maximum when δl = π/2. For a spinless beam and = (E) target, the phase√ can only depend on the invariant mass of the system, i.e. δl δl , where E = s, so the maximum will occur at some energy M, and we can make an expansion

l 4π 1 σel = 2 (2l + 1) 2 k h h d cot δl(M) i i 1 + cot δl(M) + (E − M) + ... dE E=M In the lowest order we have

19 l 4π 1 σel = (2l + 1) 2 k2 h 2(E−M) i 1 + Γ where we have deﬁned 2 d cot δ (M) ≡ − l Γ dE E=M which is the Breit-Wigner resonance formula for a particle with lifetime τ = 1/Γ:

2 l 4π Γ /4 σel = (2l + 1) k2 (E − M)2 + Γ2/4

This is the Breit-Wigner distribution, when the initial and final states of the proccess are the same. Γ is the total witdth of the resonance which is given by the sum of the partial witdths of decay. This expression can be generalize if the initial and final states 2 are differents. The factor Γ in the numerator is replaced by ΓR→iΓR→f where ΓR→i and ΓR→f are the decay rates of the resonance R into the initial and final states, respectively. Furthermore, if the two initial particles have spin sa and sb , and we know their spin state, we simply use the partial width ΓR→i for these spin states. Then, the general form of the Breit-Wigner distribution for produce a resonance in the particular channel i → R → f is 4π (2l + 1) Γ Γ σl (E) = R→i R→f el 2 2 2 k (2sa + 1) (2sb + 1) (E − M) + Γ /4

The Breit-Wigner resonance shape is only valid when de width of the resonance is less than the mass of it: Γ/M < 1.

2.2.3. Narrow with approximation The narrow-width approximation (NWA) is widely applied to predict the probability for resonant scattering processes when the total decay width Γ of the resonant particle is much smaller than its mass M. Based on the Breit-Wigner distribution, the propagator of the resonance was 1 D q2 = q2 − M 2 + iMΓ

To carry out the approximation on the propagator, we have to impose the condition Γ/M 1. Let’s work the propagator with the change of variables x ≡ q2 − M 2 and ε ≡ MΓ, in the limit when ε → 0:

20 1 1 x − iε lim = lim ε→0+ x + iε ε→0+ x + iε x − iε x ε = lim − i lim ε→0+ x2 + ε2 ε→0+ x2 + ε2 = 1/x − iπδ(x)

2 2 Remembering that we have structures of the type D q dq , and returning to the original variables, we have ´ 1 1 − iπδ q2 − M 2 dq2 = P − iπ δ q2 − M 2 dq2 ˆ q2 − M 2 q2 − M 2 ˆ where P (..) represent the principal part of the argument. The ﬁrst integral is equal to cero: Returning to the variable change x ≡ q2 − M 2, the principal part of 1/x is

−ε ∞ 1 P (1/x) ≡ lim + dx ε→0+ ˆ−∞ ˆε x −ε a 1 = lim lim + dx ε→0+ a→∞ ˆ−a ˆε x −ε a = lim lim ln x |−a + ln x |ε ε→0+ a→∞ = lim lim {ln (−ε) − ln (−a) + ln a − ln ε} ε→0+ a→∞ = 0

Then, the propagator in this approximation is π D q2 = δ q2 − M 2 MΓ where π/MΓ is a normalized factor. This aproximation is known as “narrow width approximation” which treats the resonance as a long-lived states as if it were stable. Then, the advantages of this approximation are that it set the intermediate particle to be on-shell and reduces phase-space-dimension which permits calculations relatively simples (loop calculations feasible).

2.3. Non-lineal sigma model

In apendix C it is shown, through the σ-model, the mechanism whereby the spontaneous symmetry breaking (SSB) is achieved. If we are interested in energies lower than the electroweak scale, we have to be able of integrate σ and then obtain and eﬀective theory containing just the pions (see Apendix C). If we integrate σ we are broken the chiral symmetry explicitly. It follows that, however, we can write a eﬀective Lagrangian chiral

21 invariant in terms of pions [15] [20]. The consecuence of this is that Goldstone bosons will have a more complicated transformation. The σ dynamic is removed making its mass to infinity (i.e., λ → ∞). As a consequence the mexican-hat potential gets infinitely steep in the sigma-direction. This confines the dynamic to the circle (also called chiral circle), defined by the minimum of the potential:

2 2 2 σ + ~π = fπ (2.5) As this condition removes one degree of freedom, which close to the ground state we have < σ >= fπ, and we are left with pionic excitations only. In this way, the ﬁelds can be written in terms of the angle Φ~

Φ(x) 2 σ(x) = fπ cos = fπ + O Φ fπ Φ(x) 3 ~π(x) = fπΦˆ sin = Φ(~ x) + O Φ fπ p which to leading order can be identified with the pion field. Here Φ = Φ~ Φ~ y Φˆ = Φ~ /Φ. Using the Kibble parametrization to write the fields

! ~τ · Φ(~ x) U(x) = exp fπ Φ(x) Φ(x) = cos + i~τ · Φˆ sin fπ fπ 1 = (σ + i~τ · ~π) fπ where U respresent a unitary (2 × 2) matrix. From the restriccion 2.5 we have 1 T r U †U = 1 2 To have an invariant Lagrangian chiral symmetry, U must transform as a bidoublet

U → LUR†

a a with L, R = exp −iθL,Rτ and ψL,R → UL,RψL,R. It can be proved that the kinetic energy for the bosons is given by 1 f 2 (∂ σ)2 + (∂ ~π)2 = π T r ∂ U †∂µU 2 µ µ 4 µ

22 For the other hand, the interaction term is 5 Φ(x) Φ(x) −gψ¯ σ − i~π · ~τγ ψ = −gfπψ¯ cos + iγ5~τ · Φˆ sin ψ fπ fπ " !# Φ(~ x) = −gψ¯ fπ exp iγ5~τ · ψ fπ

= −g fπψ¯ΛΛψ where we have deﬁned ! Φ(~ x) Λ ≡ exp iγ5~τ · 2fπ

Redeﬁning the ﬁelds Ψ = Λψ and Ψ¯ = ψ¯Λ, the interaction term will be

¯ −gfπψΛΛψ = −fΨΨ¯ = −MN ΨΨ¯ where it was used the Goldberg-Treiman relation (with ga = 1). The kinetic term for the nucleon is given by

µ † µ † iψγ¯ ∂µψ → iψ¯ΛΛ γ ∂µΛ Λψ † µ † = iΨΛ¯ γ ∂µΛ Ψ µ † = iΨ¯ γ Λ∂µΛ Ψ where we have used the property {γµ, γ5} = 0. Introducing auxiliary fields ! ~τ · Φ(~ x) ξ = exp i 2fπ where U = ξξ. It is easy to probe that Λ can be written in terms of the new auxiliary fields ξ 1 1 Λ = ξ + ξ† + γ5 ξ + ξ† 2 2 such that after some calculus and algebra, we will find that the kinetic energy for the nucleons can be written as

µ † µ µ µ 5 iΨ¯ γ Λ∂µΛ Ψ = iΨ¯ iγ ∂µ + γ Vµ + γ γ Aµ Ψ where i V = ξ†∂ ξ + ξ∂ ξ† µ 2 µ µ

23 i A = ξ†∂ ξ − ξ∂ ξ† µ 2 µ µ As the dynamic of the fields is restringed to the chiral circle 2.5, the potential V (σ, ~π) dissapear. Therefore, the Lagrangian of the σ−no lineal or Weinberg Lagrangian is f L = iΨ¯ iγµ∂ + γµV + γµγ5A − M Ψ + π T r ∂ U †∂µU µ µ µ N 4 µ Note that the this Lagrangian depend non-linearly on the fieldsΦ. It is instructive to expand the Lagrangian to small fluctationsΦ(~ x)/fπ around the ground state

! ~τ · Φ(~ x) Φ~ 2(x) ξ ' 1 + i + 2 2fπ 8fπ ! ~τ · Φ(~ x) Φ~ 2(x) ξ ' 1 − i + 2 2fπ 8fπ

Then, the currents will be

~τ · Φ(~ x) × ∂µΦ(~ x) Vµ ' − 2 4fπ

~τ · ∂µΦ(~ x) Aµ ' − 2fπ such the Lagrangian to this order is

2 µ 1 1 µ 5 L = iΨ(¯ iγ ∂µ − MN ) Ψ + ∂µΦ~ + Ψ¯ γ γ ~τΨ · ∂µΦ~ 2 2fπ 1 ¯ µ ~ ~ − 2 Ψγ ~τΨ · Φ(x) × ∂µΦ(x) 4fπ where Φ~ can be identifed as the pion fields. The realization of λ → ∞ makes that the σ field has dissapeared and the coupling between nucleons and pions has changed to a pseudovector one, involving the pions derivatives (momenta). Because all coupling have the pions momentum, when the momentum is cero, the interactions will do also. If we had expanded the auxiliary fields ξ to superior order, this would have resulted to loops. This formalism is known as “Chiral perturbation theory”.

24 3. Motivation for New Physics in the Strong Sector

3.1. Enlarging the strong sector symmetry group

According to the current paradigm of particle physics, the low-energy world is described by an SU(3)×SU(2)×U(1) gauge theory, spontaneously broken to SU(3)×U(1) near the Fermi scale MF ∼ 250 GeV. There may be other gauge symmetries above this scale, but it is commonly accepted that SU(3) × SU(2) × U(1) is the full gauge group at MF . The hadronic sector is completely described by color SU(3). At low energies QCD appears to work, but at higher energies there might be more to the story. Also, inspired in the chiral electroweak sector form, we can think the following: why not the strong sector group might come from an bigger symmetry group?. It has the "aesthetic" advantage of rendering the Standard Model more similar in its treatment of the two short range forces, strong and weak interactions. These possibilities motivated the introduction of chiral color as an alternative to pure QCD. The minimal extension of the standard model color sector is the chiral color group SU(3)L × SU(3)R, which breaks to diagonal SU(3)C at some scale, leaving standard QCD at low energies. There are many diﬀerent implementations of chiral color, and all require new particles in varying representations of the gauge groups. The most important model-independent prediction of chiral color is a massive color octet of gauge bosons, the axigluon.

3.2. The Forward-Backward Asymmetry

Top quark production at hadron colliders is one of the most active fields of current theoretical and experimental studies. Theoretical predictions for the total production cross section are in very good agreement with experimental results both at the Tevatron at 1.96 TeV and the LHC at 7 TeV. In contrast, sizable differences have been observed between theory predictions for the top quark charge asymmetry and measurements by the CDF and the DØ collaborations at the Tevatron. The discrepancy is particularly pronounced for the subsample of tt¯ pairs with large invariant mass, mtt¯ > 450 GeV, where a 3.4σv effect has been claimed [2]. It is important to mention that the top-antitop production cross section σtt¯ and the invariant mass spectrum dσtt¯/dMtt¯ agree well with the SM predictions. Forward-backward asymmetry is a long-winded name for a very simple number: the difference between the proportion of top quarks traveling in the same direction as the

25 proton (i.e., forward hemisphere, where we deﬁne forward as the direction of the incident proton beam) and the proportion traveling in the opposite direction (backward hemisphere). Since this quantity is a diﬀerence, it could be positive or negative. One can present the asymmetry in the form

Nt(p) − Nt(¯p) Afb = Nt(p) + Nt(¯p) where Nt(j) represent the number of top quarks in the direction of particle j. In the Standard Model, a top quark and an anti-top quark are produced as a pair from a single gluon in a way that does not remember the orientation of the original proton and antiproton beams. Top quarks are tagged through their decay t → bW +and can thus be distinguished experimentally from antitop quarks through the sign of the lepton in the semileptonic mode and eventually also through the b tag. Higher-order corrections generate a small asymmetry of about 4%. Previous studies by the DØ and CDF experiments at Fermilab have found that the asymmetry is actually several times larger than FB the Standard Model prediction [2]. Furthermore, Att¯ seems to be increasingly larger for higher-energy top quark pairs, which would be a sign that the effect is due to new interactions at high energy. If the top quark didn’t care about the difference between a FB proton and an antiproton, we would have Att¯ = 0. One possible explanation of the asymmetry is through of a new heavy particle called Axigluon. The Axigluon would be a massive relative of the massless gluon. It would interact with quarks, but not with other key particles, specifically leptons, photons, or the W and Z bosons, which explains why it has not been spotted in previous experiments. However, because it does couple to quarks, it could be observed at proton colliders such as the Tevatron and the LHC. It would appear as an anomalous peak in the distribution of pairs of quarks and antiquarks produced in the collisions. The effect might be visible for all types of quark, or perhaps only for the top quark. Experimentalists at the Tevatron and the LHC have looked for these peaks, but so far none have been observed. 2 In the SM, at O αS , top and antitop quarks have identical angular distributions. The 3 asymmetry arises at O αs order. Only a small asymmetry is expected from interference between ISR and FSR processes, and between LO and box diagrams in qq¯ → tt¯.

26 Figure 3.1.: Origin of the QCD charge asymmetry in hadroproduction of heavy quarks: interference of ﬁnal-state (a) with initial-state (b) gluon bremsstrahlung plus interference of the box (c) with the Born diagram (d).

According to the Standard Model (SM), this asymmetry at next-to leading order (NLO) calculations, this number is

Afb = 0.050 ± 0.013 σ = 7.62 ± 0.31pb

The mesurements at CDF [13] in the laboratory frame are

pp¯ Afb = 0.193 ± 0.065stat ± 0.024syst σ = 7.5 ± 0.31stat ± 0.34syst ± 0.15thpb with 3.2 fb−1 integrated luminosity data. Particularly striking is the mass dependent asymmetry at the parton frame viewed from 2008 at CDF

tt¯ Afb (mtt > 450GeV ) = 0.475 ± 0.114 tt¯ Afb (mtt < 450GeV ) = −0.116±0.153

It shows that most of the asymmetry arises from tt¯ events with high invariant masses and disagree with the SM within 3.4σ, while events with low invariant masses may even have a negative asymmetry but are still compatibles with the SM within 1σ. This rise in the asymmetry at high energy could be interpreted as further evidence of a new physics signal characterized by energies at or above the weak scale. An important question is how this asymmetry is generated: Just models that are responsable for this asymmetry or models where interference between new physics and QCD generates this asymmetry. If just new physics is the responsable for the asymmetry, it implies that at high energies this physics must be the responsable for the creating of tt¯ above 450 GeV. Most new

27 physics models with resonances that may account for this excess fall into two classes: s- channel and t-channel. The s-channel models involve new colored resonances with axial couplings (axigluons). The t-channel models feature a scalar or vector mediator, denoted M, with a flavor violating coupling λ between u or d and t, and can generate a large forward-backward asymmetry through a Rutherford enhancement. Nevertheless, it is difficult to imagine how such ad hoc couplings can be naturally generated. This new particles are constrained by flavor physics and collider searches. Any axigluon that couples to qq¯ is subject to constraints from dijet resonances and dijet contact interactions searches.

Figure 3.2.: s-channel for the presence of an axigluon.

At LHC, FBA vanished trivially because the proton-proton intial states is symmetric, so, charge asymmetry cannot result in a forward-backward asymmetry as at Tevatron. However a charge asymmetry is still visible for suitable deﬁned distributions. Visible eﬀects at LHC:

• Proton-PDF: Quarks in initial state have on average larger momentum than antiquarks.

• Charge Asymmetry transfers boost diﬀerence to top-antitop ﬁnal state.

28 Figure 3.3.: Left: Forward-Backward asymmetry. Right: Central Forward asymmetry

29 4. Axigluons models

In this section we present some axigluon models. We separate two kind of them: Flavor- universal and Flavor-nonuniversal models. All these models are based in a chiral symmetry group SU(3)L × SU(3)R which is broken at some energy by the spontanous symmetry breaking (SSB) and it arrives to the well known group of symmetry SU(3)c. Any chiral color model predicts the existence of an axigluon, a massive vector gluon which coupled to quarks in a axial form, that decays to qq¯. We present only models in the s-channnel. New physics produced in the t-channel could also generate a substantial forward-backward asymmetry. Naively, one might imagine that a t-channel contribution would not rise with energy, but in fact t-channel physics produces a large energy dependence that might ﬁt the CDF results nicely. However, producing tt¯ pairs via a new t-channel process generates non-trivial ﬂavor issues; We will not discuss these models any further here.

4.1. Flavor-universal models

4.1.1. Minimal axigluon model The simplest way to introduce another color octet vector boson is to extend the QCD gauge symmetry to be the group SU(3)1 × SU(3)2, which spontaneously breaks to its subgroup SU(3)c [17]. This SSB can be achieved by introducing a complex scalar ﬁeld Σ which transforms as a (3, 3)¯ under the SU(3)1 × SU(3)2 symmetry with a non-zero < Σ >= 1√3×3 f V.E.V., 6 Σ. The Lagrangian for this minimal model is

2 1 µν 1 µν f µ L = − GLµνG − GRµνG + tr {DµUD U} (4.1) 4 L 4 R 2 where

GLµν = ∂µlν − ∂νlµ − igL [lµ, lν]

GRµν = ∂µrν − ∂νrµ − igR [rµ, rν]

DµU = ∂µU − igLlµU + igRUrµ † † † † DµU = ∂µU − igRrµU + igLU lµ where lµ and rµ are the gauge ﬁelds of SU(3)L and SU(3)R. The third term in the Lagrangian 4.1 is a non-linear sigma model which is used to make explicit the breaking

30 of the chiral symmetry to the well known strong symmetry SU(3)c. Each Lagrangian term is chiral gauge invariant under the following transformations

† i † AL,R → VL,RAL,RVL,R − (∂µVL,R) VL,R gL,R and, the scalar ﬁeld must transform as a bidoublet

† U → VLUVR where ALµ ≡ lµ, ARµ ≡ rµ and VL,R are the gauge elements of SU(3)L and SU(3)R, respectively. In the unitary gauge, U =1, the third term of the lagrangian gives rise to a nondiagonal mass 2 × 2 matrix term for the gauge bosons lµ and rµ. 1 1 3 L = − G Gµν − G Gµν + f 2tr (g r − g l )2 4 Lµν L 4 Rµν R 2 R µ L µ Let’s diagonalize the mass matrix. The mass matrix for this Lagrangian is

1 2 2 1 2 2 f gR − 2 gRgLf M = 1 2 1 2 2 − 2 gRgLf 2 f gL

f 2 2 2 By imposing| M − I2×2λ |= 0, one ﬁnd two eigenvalues: λ1 = 0 and λ2 = 2 gR + gL . With the normalization condition, the autovector for the eigenvalue λ1 is gL gR Gµ = q lµ + rµ 2 2 gL gR + gL and the autovector corresponding to λ2 is 0 gL gR Gµ = q −lµ + rµ 2 2 gL gR + gL

For simplicity, we deﬁne sin α ≡ g/gL and cos α ≡ g/gR, where g is g g g = L R q 2 2 gL + gR

Using the trigonometric functions, we can write in a simply form the transformation of the ﬁelds

Gµ = cos αlµ + sin αrµ

0 Gµ = − sin αlµ + cos αrµ

31 In the case where gL = gR = g (e.g., α = π/4) the eigenvectors reduces to 1 Gµ = √ (lµ + rµ) (4.2) 2

0 1 G = √ (lµ − rµ) (4.3) µ 2 with their respective eigenvalues mG = 0 and mG0 = gf. We recognize Gµ as the gluon 0 state with mass cero and Gµ as the axigluon state with mass gf. The interaction terms of the theory can be written as

1 ¯ µ 1 ¯ µ L = gψlµγ (1 − γ5) ψ + gψrµγ (1 + γ5) ψ (4.4) 2 2 Inverting equations 4.2 and 4.3 and replacing them into the the Lagrangian 4.5, we left with the Lagrangian in terms of the physical states

g µ g µ L = √ ψG¯ µγ ψ + ψA¯ µγ γ5ψ 2 2

In this minimal model, the gluon-quark√ and axigluon-quark interaction terms have the same coupling constant gQCD ≡ g/ 2. This kind of model are called to have an universal- coupling, because all quarks are coupled to the axigluon with the same strangeth. In the general case, where gL 6= gR, the Lagrangian for the coupling between quarks and axigluon appears modulated by trigonometric functions of a mixing angle: λa λa 1 L = g ψ¯ Ga γµψ + g ψ¯ γµ cot α + γ5 Aa ψ QCD 2 µ QCD 2 sin 2α µ where the angle is deﬁned as tan α = gR/gL. Notice that in this case the axigluon interaction has a vector part as well as an axial-vector part and the coupling constant of the axial-vector part is greater than gQCD. Additionally, it is easy to see that in this scenario the axigluon interaction is universal.

4.1.2. Non-minimal model: Four gauge fields Following the same ideas previously depeloped in the minimal model, here we add two more gauge fields: Lµ and Rµ, which transform like gauge fields under SU(3)L and 0 SU(3)R, respectively, with charasteristic coupling constant g . The Lagrangian that describes the gauge sector of the model, including the effective symmetry breaking term, is

1 1 1 1 L = − G Gµν − G Gµν − F F µν − F F µν 4 Lµν L 4 Rµν R 4 Lµν L 4 Rµν R M 2 M 2 f 2 + tr gl − g0r 2 + tr gr − g0R 2 + tr {D UDµU} g02 µ µ g02 µ µ 2 µ

32 where GLµν, GRµν and U are the same ﬁelds than in the previous section, while FLµν and FRµν are deﬁned by

0 FLµν = ∂µLv − ∂vLµ − ig [Lµ,Lν] 0 FRµν = ∂µRv − ∂vRµ − ig [Rµ,Rν]

As in the minimal model, we follow the same steps. In the unitary gauge a nondiagonal mass matrix is explicitly generated. Using the approximation g0 g and writings our result at ﬁrst order in g/g0 we found the eigenvalues of the physical states

mG = 0

mG0 = gf

mG1 = M

mG2 = M

Then, we have four ﬁelds which we recongnize, respectively, as: Gluon, axigluon, and two degenerate heavy states. The new mass scale M is not constrained and we can suppose that it is large enough to prevent the observation of the heavy states. The normalized mass eigenvectors can be written as

1 1 g g Gµ = √ lµ + √ rµ + √ Lµ + √ Rµ (4.5) 2 2 2g0 2g0 2 0 1 1 g mG0 G = − √ lµ + √ rµ − √ 1 − Lµ (4.6) µ 2 2 2g0 M 2 2 g mG0 +√ 1 − Rµ 2g0 M 2 1 g g 1 1 G = √ lµ + √ rµ − √ Lµ − √ Rµ (4.7) µ 2g0 2g0 2 2 2 −1 2 −1 2 g mG0 g mG0 G = −√ 1 − lµ + √ 1 − rµ (4.8) µ 2g0 M 2 2g0 M 2 1 1 +√ Lµ − √ Rµ 2 2

Now, since we have two ﬁelds that transform as gauge ﬁelds for each group (lµ and Lµ for 0 SU(3)L and rµ and Rµ for SU(3)R) any combination of the form g(1 − k)lµ + g kLµ and 0 0 0 0 g(1 − k )rµ + g k Rµ, where k and k are arbitrary constants, can be used to construct covariant derivatives [21]. In this way, the Lagrangian can be written as

1 ¯ µ 1 0 ¯ µ L = g(1 − k)ψlµγ (1 − γ5) ψ + g kψLµγ (1 − γ5) ψ (4.9) 2 2

33 1 1 + g(1 − k0)ψr¯ γµ (1 + γ ) ψ + g0kψR¯ γµ (1 + γ ) ψ 2 µ 5 2 µ 5 For simplicity, we set k = k0. Inverting Eq.4.5, 4.6, 4.7 and 4.8 and replacing these ﬁelds in the Lagrangian 4.9, we left with the Lagrangian in terms of the physical states

2 2 1 µν 1 0 0µν f g 0 0µ g ¯ µ g ¯ 0 µ L = − GµνG − G G + G G + √ ψGµγ ψ + (1 − χ)ψG γ γ5ψ (4.10) 4 4 µν 2 µ 2 2 µ where χ is deﬁned as m2 χ ≡ G0 k M 2

We have ommited in Lagrangian 4.10 the content of the states G1 and G2, because we suppose that they are very heavy and don’t appear in the visible spectrum. Therefore, we have in the Lagrangian the following: The ﬁrst to terms are the kinetic energy of the gluon and the axigluon, the third correspond to the mass of the axigluon, the fourh is the usual term√ of QCD of the coupling between quarks and gluons, where we recognize gQCD ≡ g/ 2. Finally, the last term correspond to the coupling bettween quarks and the axigluon modulated multiplied by a factor (1 − χ), which is arbitrary and it will be useful to set limits on the coupling constant of axigluon-quarks. It is interesting to say, that this model can be derived from an equivalent way in the language of Deconstruction theory [1]. It is possible to construct the same interactions between quarks and axigluon from a four site model of the gauge group SU(3) × SU(3) × SU(3) × SU(3) and with three link scalar ﬁelds (Σ1, U, Σ2). Inspired in this model we worked in the thesis in the part of the simulation and analisis of the data.

4.2. Flavor-nonuniversal models

4.2.1. Frampton Model

The Frampton model [4] is a chiral color model based on gauge group SU(3)A ×SU(3)B × SU(2)L ×U(1)Y with gauge couplings gA, gB, gL and gY respectively, which then spontaneously breaks to its subroup SU(3)c × U(1)E.M. . The spontaneous symmetry breaking of the strong sector can be achivied by introducing a complex scalar field Σ which transform as a (3, 3)¯ under the SU(3)A × SU(3)B symmetry with a non-zero V.E.V. at the TeV scale, hΣik¯i = uδik¯ and it breaks to the diagonal subgroup SU(3)c QCD color group. Here, quarks of opposite chiralities are charged under differents SU(3) gauge groups: qL, tR, bR transform as triplets of SU(3)A and qR, tL, bL transform as triplets of SU(3)2. tt¯ This assingments is motivated by the fact of to have the correct sign of AFB, because it is shown from the tree-level differential cross section qq¯ → tt¯ [5]. The Kinetic term for the link field becomes the mass term for the massive gauge boson

34 h † µ i 2 2 2 2 1 2 T r (DµΣ) (D Σ) ⊃ u (gAAµ − gBBµ) /2 = u g Gµ /2

q 2 2 where g ≡ gA + gB. Diagonalizing the mass matrix of the two gauge bosons Aµ and Bµ, we obtain the massless QCD gluon

Gµ = cos θAµ + sin θBµ and the massive axigluon state

0 Gµ = − sin θAµ + cos θBµ where, as in the minimal model, we define sin θ = gA/g and cos θ = gB/g. The massless 0 field Gµ is the usual QCD gluon while we call the massive octet vector boson Gµ “axigluon”. With the charge assignments given above, we find the vector and axial-vector 0 couplings of Gµ to the SM quarks, re-scaled by the QCD coupling gQCD, to be 1 1 gt = gq = , gt = −gq = V V tan 2θ A A sin 2θ This means that the Lagrangian after the SSB of the strong sector is

2 2 1 1 0 u g L = − G Gµν − G0 G µν + G0 G0µ + g qγG¯ q 4 µν 4 µν 2 µ QCD µ q,t 0 q 0 t µ 0 +gQCDgV qγG¯ µq + gQCDgAqγG¯ µq + gQCDgAqγ¯ Gµq

There are only two parameters in this model and they can be identified as the axigluon mass MG0 and the gauge coupling mixing angle θ. So, the parametric space has only two parameters. Have been done fits to the observed and unfolded parton-level four-bin data from reference, with the results being shown in Fig. 4.1. In the fit, has been neglected tt the NLO SM constributions to AFB since they just include the leading order contribution from this model.

35 tt¯ Figure 4.1.: The ﬁt of the minimal two-site axigluon model to the observed AFB at CDF. The region enclosed by the red dashed lines is excluded by the dijet narrow resonance search at CMS with 2.9 pb−1 luminosity. The region between the dark solid yellow lines is excluded by the search for diquark contact interactions from ATLAS with 3.1 pb−1, so all of the preferred parameter space of this model has been eliminated.

Although a broad G0 may survive to the dijet resonance analyses, searches for quark contact interactions via the dijet angular distributions can be used to constrain the model parameter space. We can see in Fig. 4.1 that dijet contact interaction searches make ruled out the posible parameter space for the axigluon. Therefore, minimal axigluon model tt¯ can not adequately explain the AFB asymmetry observed at CDF. Also, if axigluon were broader, it has been shown that the minimal axigluon model neither is posible because the quark contact interactions also put rule out the posible parameter space without a speciﬁc model.

4.2.2. A simpliﬁed three-site model Other way to extend the minimal single axigluon model is to include an additional color- octet gauge boson [5]. This model is based on the three-site model with the gauge group SU(3)1 × SU(3)2 × SU(3)3 plus two sigma ﬁelds Σ1 and Σ2 which transform as (3, 3)¯ under SU(3)1 × SU(3)2 and (3¯, 3) under SU(3)2 × SU(3)3. For simplicity, it is assumed that the gauge coupling between the groups SU(3)1 and SU(3)3 are the identical and are given by h1, with the corresponding coupling of SU(3)2 being h2. Diagonalizing the gauge boson matrix, we determine the three mass eigenstates to be

36 sin θ sin θ Gµ = − √ Gµ + cos θGµ − √ Gµ 2 1 2 2 3 1 1 G0µ = √ Gµ − √ Gµ 2 1 2 3 cos θ cos θ G00µ = √ Gµ + sin θGµ + √ Gµ 2 1 2 2 3 √ where θ is the mixing angle and is deﬁned as tan θ ≡ 2h2/h1 and the physical region is identiﬁed as 0 ≤ θ ≤ 90° . The mass ratio of the two massive gauge bosons is determined to be

M 00 1 G = MG0 cos θ

Assigning the quantum numbers to SM quarks: qL, tR, bR transform as triplets of SU(3)1 0 and qR, tL, bL transform as triplets of SU(3)3, the couplings of Gµ to quarks to be 1 gt = gq = 0, gt = −gq = V V A A sin θ 00 and the corresponding couplings of Gµ to quarks are 1 ht = hq = , ht = −hq = 0 V V tan θ A A Based on those couplings, we can see that G0 behaves as an axigluon (with pure axial- vector couplings) and G00 as a coloron (with pure vector couplings).

37 tt¯ Figure 4.2.: The ﬁt of the simpliﬁed three-site model to the observed AFB at CDF. Regions enclosed by the red dashed lines are excluded by the dijet narrow resonance searches. The region to the left side of the solid dark yellow line is ruled out by the search for dijet contact interactions. The tt¯ production cross section itself does not further constrain the relevant parameter space.

Similar to the minimal two-site model, there are only two parameters in this model: MG0 and θ. The corresponding 90% and 95% C.L. contour plot is shown in Fig. 12. One can see from this ﬁgure that although there exists a large region that is not ruled out by the dijet narrow resonance searches, however the search for contact interactions in dijets indeed excludes almost all of the parameter space.

4.2.3. A Two-Site Model with a New Vector-like Quark

This model also consider a two-site model with the gauge group SU(3)1 × SU(3)2, but now, it is add a new vector-like (under SM gauge group) fermion, ψL,R, which are SU(2)W singlets and have charge 2/3 under U(1)Y [5]. For simplicity, it is only introduce one vector-like fermion which mixes solely with the up quark. Other vector-like fermions may also be present which mix with the charm and top quarks. Under the extended gauge group, we assign qL, qR, tR, bR, ψL to be triplets of SU(3)1 and, (t, b)L and ψR to be triplets of SU(3)2. Because ψR and uR have the same quantum numbers, it is introduced only one new mixing angle in the right-handed quark mixing matrix such that the new f vector-like fermion only mixes with uR in the ﬂavor basis. The transition from the ﬂavor basis to the mass eigenstate is then parametrized as

38 (m) (f) (f) uR = cos αuR + sin αψR (m) (f) (f) ψR = − sin αuR + cos αψR

0 The couplings of the axigluon Gµ to the various quarks in the mass eigenstate basis are found to be 1 1 g(d,s,b,c) = − tan θ, g(d,s,b,c) = 0, gt = , gt = V A V tan 2θ A sin 2θ

sin2 α sin2 α gu = − tan θ + , gu = − V sin 2θ A sin 2θ When α = π/2, the couplings of the up and top quarks become identical to those in the minimal two-site model. From the above equation, one can see that the axial-vector couplings of the up and top quarks are diﬀerent and one now has the freedom to increase t u 2 gA by reducing θ and to simultaneously decease gV,A at the same time by choosing sin α smaller than sin2θ.

39 Figure 4.3.: The ﬁt for the model with one axigluon plus one additional vector-like tt¯ fermion to the observed AFB at CDF. The region above the red dot-dashed line is excluded at 95% C.L. by the dijet narrow resonance search, while the region above the dark yellow solid line is also excluded at 95% C.L. by the search of dijet contact operator interactions. The projected exclusion limit from the dijet narrow resonance search at the 7 TeV LHC with 1 fb−1 is shown by the red dotted line. The black dashed lines designate the regions (above and to the left) where the given percentage of the tt¯ production cross section for mtt¯ > 450 GeV arises from new physics.

u u Although this model gives the advantage of tunning the coupling constants gV and gA through of the new mixing parameter α, from Fig. 4.3, we conclude that in the two-site plus one vector-like fermion model there exist a small region of parameter space which is allowed by all of the constraints.

40 5. Searching resonances in the dijet spectrum

5.1. Introduction

In the SM, two jets (dijet) events in high energy collision are usually described by ap- pliying QCD to the scattering of beam constituents quarks and gluons. We expect to ﬁnd out axigluons as a resonance in the spectrum of the dijet invariant mass deﬁned as q 2 2 mj1j2 = (Ej1 + Ej2) − (~pj1 + ~pj2) , where E and ~p are the jet energy and momentum, respectively. The analysis technique consist of to search a resonance on top of a smooth and rapidly falling spectrum and relied on the mesured mass spectrum mjj distribution to estimate the background level to this new possible signal. Below we show the results on the dijet spectrum obtained by experiments at Tevatron (CDF) and LHC (ATLAS). The heart of the search for dijet resonances is the measurement of the dijet mass distribution and the estimation of the background. Unlike many other searches in high energy physics, the search for dijet resonances is completely dominated by a single background. The observed dijet mass distribution comes from the dominant process in hadronic collisions: 2 → 2 scattering of partons predicted by perturbative QCD. It is worth saying that the measurement of the dijet mass spectrum is also an important test of perturbative QCD (pQCD) predictions. It provides complementary information to the inclusive jet cross section measurements and comparisons of the measurement with pQCD predictions provide constraints on the parton distribution functions (PDFs) of the proton, in particular at high momentum fraction x (& 0.3) where the gluon distribution is not well constrained.

5.2. Parton-parton scattering

The cross section for the short distance process HAHB → F + X, can be described by the parton-parton cross section of the partons i and j with their respective momentum fraction xA and xB in hadrons HA,B

X σ (H H → F + X) = dx dx f A (x ,Q) f B (x ,Q) σ (x , x ) A B ˆ A B i A j B ij→F A B ij

A B where fi and fj are their PDFs for fi = qi, q¯i, G, etc., xA,B are their momentum 2 2 fractions, and Q is the renormalization scale, which can be taken to be, e.g., sˆ or pT . The total momentum of the subprocess ij → F is therefore pF = xApA + xBpB, where

41 the hadron momenta in the CM are pA,B = (EA,B, ±~p) with EA ∼ EB ∼ p ≡| ~p | in the high energy regime. One can show that the invariant mass square of the subprocess is

2 pF =s ˆ ∼ xAxBs ≡ τs

2 2 where s = (pA + pB) ∼ 4p . In the CM of the subsystem F , we deﬁne the rapidity 1 x y = ln A 2 xB

5.3. CDF II at TEVATRON

pp¯ √CDF II detector performed mesurements of the dijet mass spectrum with collision at s=1.96 TeV. The analysis uses data with an integrated luminosity of 1.13 fb−1 collected between February 2002 and February 2006 with the CDF II detector at the Fermilab Tevatron [6]. The dijet mass spectrum predicted by QCD falls smoothly and steeply with increasing dijet mass. For datailed description of the detector see [*!]. The jet energies measured by the calorimeters are affected by instrumental effects such as calorimeter non- uniformity, non-linearity, and energy smearing. CDF collaboration corrects this effects in several steps, but for our purpose is not relevant here. The systematic uncertainties arise mainly from four sources: the jet energy scale, the jet energy resolution, the unfolding correction, and the integrated luminosity. The dominant source is from the absolute jet energy scale. The size of the uncertainty in the cross section varies from 10% at low mjj to +74 % at high mjj. In Fig. 5.1, the upper plot shows the comparision between the measured dijet mass spectrum to the next-to-leading-order perturbative QCD (NLO pQCD) predictions from fastNLO, and the below plot shows the uncertainty of the rate of data and theory for the dijet invariant mass.

42 Figure 5.1.: (a) The measured dijet mass spectrum for both jets to have |y| < 1 compared to the NLO pQCD prediction obtained using the CTEQ6.1 PDFs. (b) The ratio of the data to the NLO pQCD prediction. The experimental systematic uncertainties, theoretical uncertainties from PDF, the ratio of MRST2004/CTEQ6.1, and the dependence on the choice of renormalization and factorization scales are also shown. An additional 6% uncertainty in the determination of the luminosity is not shown.

CDF searched for narrow mass resonances in the measured dijet mass spectrum by fitting the measured spectrum to a smooth functional form and by looking for data points that show significant excess from the fit. To set the theoretical limits on the dijet mass spectrum, CDF took into account differents theoretical models which predicts the existence of new particles which decay into dijets: excited quark q*, RS graviton G* that decays into qq or gg, W 0 and Z0 new bosons. The dijet mass distributions for the q*, RS graviton, W ’ , and Z’ simulations with the mass of 800 GeV/c2 are shown together in Fig. 5.2. (In the reference [16] one can find a detailed explanation about resonance simulations which have been done by differents collaborations and experiments). We are not interested in these kind of particles, but, the form of the resonance is illustrative to understand the form of the resonance of the dijet mass distribution for collisions pp¯. We will see that in the case of scattering pp, the tail of the resonance at low dijet mass change the form and it is seen a considerably increase of the non-resonants events. The shapes of the distributions are mainly determined by the jet energy resolution and QCD radiation which leads to tails on the low mass side. Since the natural width of these

43 particles is substantially smaller than the width from the jet energy resolution, all the dijet mass distributions appear similar. However, the dijet mass resonance distributions are somewhat broader for q* and RS gravitons than for W ’ and Z’ because q* and RS gravitons can decay into the mode containing gluons, unlike W ’ and Z’. Gluons radiate more than quarks and tend to make the resulting dijet mass distributions broader. As a result, the cross section limits obtained based on the q* and RS graviton resonance shapes are about 20% larger than those obtained with the W ´ and Z´ resonance shapes.

Figure 5.2.: Dijet mass distributions for simulated signals of the q∗ , RS graviton, W ’, and Z’ with the mass of 800 GeV/c2.

For simulations of CDF, the kinematical acceptance for each of the leading two jets to have | y |< 1. The obtained 95% Conﬁdence Level (C.L.) limits on the search of new particles is show in the Fig. 5.3.

Figure 5.3.: Observed 95% C.L. upper limits on new particle production cross sections times the branching fraction to dijets obtained with the signal shapes from q*, Axigluon/Coloron and E6 diquark

The mass exlusion regions obtained in the search at Tevatron for the axigluon and ﬂavour-univeral coloron is 260< M <1250 GeV/c2. Therefore, CDF concluded that the measured dijet mass spectrum is found to be consistent with NLO pQCD predictions based on recent PDFs and does not show evidence of a mass resonance from new particle production.

44 5.4. ATLAS at LHC

Figure 5.4.: ATLAS detector at LHC

5.4.1. The ATLAS experiment

As well as Tevatron, ATLAS experiment made mesurements√ of the dijet mass spectrum at LHC, but this time have performed collisions of pp at s =7 and 8 TeV, at an integrated luminosity of 4.8 fb−1 [8] and 13 fb−1 [9]. Also ATLAS has been looking for new particles as an excited composite quark q*, exemplifying quark substructure; an axigluon predicted by chiral color models; a ﬂavour-universal color-octet coloron; or a color-octet techni-ρ meson predicted by models of extended technicolor and topcolor- assisted technicolor. Below we present the data collected by ATLAS which set upper limits on new resonances for a broad specter of masses. The method employed the search for new phenomena (NP) is the same than at CDF, i.e., to look for localised excesses in the dijet mass mjj distribution (often referred to as “bumps” or “resonances”). No resonances associated with new phenomena have been found in previous studies. The two jets emerging from the collision are reconstructed to determine mjj and the scattering angle in the dijet CM. An essential element of the search method is to apply angular kinematic criteria to select large angle scattering events, which retain the highest pT collisions which also emphasise the angular regions where new phenomena are expected to appear. The dominant Quantum Chromodynamics (QCD) interaction in this high- pT scattering regime involves t-channel processes, leading to angular distributions that peak at small scattering angles. We speciﬁed the kinematical criteria in section 5.5.5. A detailed description of the ATLAS detector has been published in [10]. As well as in the analysis at CDF II, ATLAS data is made on the analysis of the dijet invariant mass distribution. The search for new phenomena in the dijet mass distribution

45 reduces to the search for significant local excesses (“bumps” or “resonances”) above the parameterised background, which is fitted by a smooth function, see Fig. 5.5. The dominant sources of systematic uncertainty, in decreasing order of importance, were the absolute jet energy scale (JES), the background fit parameters, the integrated luminosity, and the jet energy resolution (JER). The understanding of the jet energy scale and its uncertainties improved over the course of the 2012 data taking from the collisons at 8 TeV, but no substantial changes were made in the jet calibration procedure.

Figure 5.5.: The reconstructed dijet mass distribution with statistical uncertainties (filled points with error bars) fitted with a smooth functional form (solid line). The bin-by-bin significance of the data-fit difference in Gaussian standard deviations is shown in the lower panel.

Events are required to have a primary collision vertex deﬁned by two or more charged particle tracks. There must be at least two jets in the event. The highest-pT jet is referred to as the “leading” jet, the second-highest-pT as the “subleading” jet; the dijet invariant mass mjj is calculated from these two jets. The relationship between the cross section to the observed detection rates, per unit time, is

dN = Ldσ where N is the number of events observed per unit time and L has the dimensions of an inverse cross-section per unit time. For example, at Fermilab, Run II, has a luminosity of L = 1032cm−2s−1. LHC has worked with L = 1034cm−2s−1. The integrated luminosity

46 is usually quoteed in papers and corresponds to the size of the data set, measured in pb−1. In 2012, the center-of-mass (CM) energy for proton-proton (pp) collisions at the LHC was raised from 7 TeV to 8 TeV. One important consequence of the increase in LHC energy from 7 TeV to 8 TeV is the raising of the kinematic limit by 1 TeV, which translates almost directly into a 1 TeV increase in the upper limit of accessible mjj. However, a much greater increase in the sensitivity to new√ phenomena comes from another eﬀect, the increase in parton luminosity as a function of √sˆ, the energy in the two-parton CM. The parton luminosity√ rises with beam energy for sˆ greater than 1 TeV, and at the highest dijet masses sˆ is roughly equivalent to mjj . The net eﬀect is that a data sample taken at 8 TeV with a given integrated luminosity will have larger sensitivity to NP processes than a 7 TeV data sample of the same integrated luminosity.

5.4.2. Collider Kinematics The coordinate system and nomenclature used to describe the ATLAS detector is the following. The nominal interaction point is defined as the origin of the coordinate system, while the beam direction defines the z-axis and the x-y plane is transverse to the beam direction. The positive x-axis is defined as pointing from the interaction point to the centre of the LHC ring and the positive y-axis is defined as pointing upwards. The side-A of the detector is defined as that with positive z and side-C is that with negative z. The azimuthal angle ϕ is measured with respect to the x-axis, which points toward the center of the LHC ring, and the polar angle θ is the angle from the beam axis. The kinematic criteria used to select events of interest for the current study involve both energy and angle variables. We will work in cilindrical cordinates. The four-momentum of a produced particle in the collider is

µ p = (E, pT cos ϕ, pT sin ϕ, pz)

p 2 2 with E = p + m , the transverse momentum pT = p sin θ, pz = p cos θ, where p ≡| ~p | and θ is the polar angle. The Lorentz-invariant phase space is d3~p dp dp = dp dp z = p dp dϕ z = p dp dϕdy E x y E T T E T T q 2 2 If we deﬁne the transverse mass as mT = m + pT , it is easy to see that our four- momentum can be written as

µ p = (mT cosh y, pT cos ϕ, pT sin ϕ, mT sinh y) where the new variable y is called rapidity and is deﬁned as 1 E + p E + p p y = ln z = ln z = tanh−1 z 2 E−pz mT E

47 The pseudorapidity η is deﬁned as θ 1 1 + cos θ η ≡ − ln tan = ln 2 2 1 − cos θ where 0 ≤ θ ≤ π corresponding to +∞ ≥ η ≥ −∞. We can see that when a particle has m → 0, or v → c (β = 1), the rapidity is equal to pseudorapidity, y → η. The deﬁnition of pseudorapidity is equivalent to

cos θ = tanh η

There is another parameter which is used in the kinematical constraints is 1 y∗ = (y − y ) 2 1 2

5.4.3. Trigger Trigger is a system that uses simple criteria to rapidly decide which events in a particle detector to keep when only a small fraction of the total can be recorded. Trigger systems are necessary due to real-world limitations in data storage capacity and rates. Since experiments are typically searching for "interesting" events (such as decays of rare particles) that occur at a relatively low rate, trigger systems are used to identify the events that should be recorded for later analysis. Current accelerators have event rates greater than 1 MHz and trigger rates that can be below 10 Hz. The ratio of the trigger rate to the event rate is referred to as the selectivity of the trigger. For example, the Large Hadron Collider (LHC) is anticipated to have an event rate of 1 GHz (109 Hz), and the Higgs boson is expected to be produced there at a rate of at least 0.01 Hz. Therefore the minimum selectivity required is 10−11. There are three trigger levels. The ﬁrst is based in electronics on the detector while the other two run primarily on a large computer cluster near the detector. The ﬁrst-level trigger selects about 100,000 events per second. After the third-level trigger has been applied, a few hundred events remain to be stored for further analysis. This amount of data still requires over 100 megabytes of disk space per second – at least a petabyte each year.

5.4.4. Kinematical cuts for pp collision at 7 TeV and 8 TeV ATLAS detector uses the following kinemtical cuts on the ﬁnal states to achieve a better searching: √ 1. The kinematic selection criteria for a collision at s = 7 TeV are used to enrich the sample with events in the hard-scattering region of phase space. Events must satisfy | η1,2 |< 2.8,| y∗ |< 0.6 and mjj > 850 GeV. See[8].

48 √ 2. The following kinematic selection criteria for a collision at s = 8 TeV are used to enrich the sample with events in the hard-scattering region of phase space. Events must satisfy | y1,2 |< 2.8,| y∗ |< 0.6 and mjj > 1000 GeV. Also, the combination of kinematics and trigger selection restricts jets in the analysis to have a minimum pT of 150 GeV. See [9]

5.4.5. Model-independent limits on dijet resonance production ATLAS set limits on dijet resonance production using a Gaussian resonance shape hy- pothesis. Limits are set for a collection of hypothetical signals that are assumed to be m m Gaussian-distributed√ in jj with means ( G) ranging from 1.0 TeV to 4.0 TeV, at a CM s collision =7 TeV, with standard deviations√ (σvG) from 7% to 15% of the mean, and from 1.5 TeV to 4.8 TeV, at a CM collision s=8 TeV, with standard deviations (σvG) from 7% to 15% of the mean. These limits are set to compare with NP models. In this thesis, we compare these data to our simulation and analisis for an axigluon. The details of the comparasion are developed in the sections below. The upper limits for 7 and 8 TeV are depicted in Fig. 19 and 20.

Figure 5.6.: The 95% C.L. upper limits at 7 TeV on σv × A for a simple Gaussian resonance decaying to dijets as a function of the mean mass, mG, for three values of σvG/mG, taking into account both statistical and systematic uncertainties.

49 Figure 5.7.: The 95% C.L. upper limits at 8 TeV on σv × A for a simple Gaussian resonance decaying to dijets as a function of the mean mass, mG, for three values of σvG/mG, taking into account both statistical and systematic uncertainties.

50 6. Simulation and Analysis of data

6.1. Introduction

Based on the model-independent limits on dijet resonance production given by ATLAS collaboration at 7 and 8 TeV, we simulate pp collisions at these energies in the presence of an axigluon, and we compare them to the limits given by the ATLAS experiment. The simulation was performed assuming that the hypotetical signal (a resonance) is just come from a pure contribution of the coupling between an universal-axigluon and quarks. Unfortunately, LHC is not the best place to search axigluons because an axigluon resonance can only be produced by quark-antiquark annihilation, but this mechanism is suppressed at the LHC due to the parton distribution functions (PDF’s) show that is too dificult to find anti-quarks with large momentum and hence it is difficult to produce a very massive axigluon. The consequence of this suppression is that a large amount of dijet events originated by an axigluon in the s-channel are in fact produced at low dijet invariant masses. Nevertheless, in actual measurements, this large tail is hidden under the usual QCD background. For the theoretical side, it means that only the resonant part of the cross section must be extracted from any simulation destined to be compared to experimental data. In the 6.2 section, we present the procedure that we used to make the simulation for the process pp considering quarks and axigluon at leading order. The simulation was done by CalcHEP, a package for calculation of Feynman diagrams at parton level event and integration over multi-particle phase space, to simulate the pp collision ocurring at ATLAS. In the simulation, we consider contributions to the scattering cross section from both the s and t-channel. In subsection 6.3, we present the methodology to analysed the data (i.e. N-tuples), obtained with Calchep, with PAW (Physics Analisis Workstation). This was done imposing the respectively kinematical cuts and, with and without, the smearing effect of the data given by the ATLAS collaboration on the N-tuples (Events). In subsection 6.4 we summarize the results of the analysis made in 6.3 in Tables and we compare both the limits obtained by the simulation and the limits on the cross section given by ATLAS experiment. Also, it is shown limits on the coupling constant gAqq¯ = kgQCD, in function of k, from the model with four gauge fields developed in the subsubsection 4.1.2. In subsection 6.5, we study how the effect of including smearing affect the axigluon cross section calculated in subsection 6.4. Also, in subsection 6.6 is show the cross section for a broad axigluon Γ/M = 0.3, which could potentially explain the asymmetry forward-backward. In the next years the LHC will be operating at energies of 14 TeV. Therefore, in the

51 ﬁnal part of this work, we consider the simulations with the same content of particles (axigluons+quarks), but this time, at 14 TeV. Of course, we can not compare our resuts with anything because LHC has not run at those energies. This time, no smearing eﬀect from the detector was considered.

6.2. The Simulation

The simulation at 7 and 8 TeV are equivalent in the procedure, therefore, in this subsection we will explain just the case for 7 TeV. Inspired in the fact that the signal of the new physics (NP) come from just from the coupling between axigluon and quarks, we create a model in CalcHEP [19], called “Modelo de axigluon”, which consider just the color-octet spin-1 vector particle and all quarks. The model consider the same parameters than QCD, except that the coupling between quarks and axigluon is kgQCD, where k is a arbitrary constant set to 1. It does not matter the value of k, because the coupling is universal and, so, all the quarks are coupled to the axigluon with the same intensity. The axigluon width was set to be undefinite such that, at each simulation (i.e., each axigluon mass simulation), CalcHEP do the calculation of it. At the final of the section, we set respective limits for k for each axigluon mass. Once the model is defined in CalcHEP, we are in conditions to perform the simulations. The simulation consist of generate 10.000 events for each axigluon mass for the process pp → A → j1j2. To achieve this, we use batch files, which allows to perform very complicated and tedious calculations. It consists of a series of keywords together with values for those keywords, with each keyword on a separate line. Most of the options available in the interactive session are supported by keywords in the batch file and thus most calculations can be done using the batch interface. In Calchep and in the batch files, antiparticles are denoted with capital: e.g. u es denoted for the up quark, whereas U is for an anti-up quark. In the fallowing table it is shown a batch file example used for a process pp → A → j1j2, at 7 TeV.

52 Model: Modelo de Axigluon Model changed: False Gauge: Unitary Process: p,p->j,j Composite: p=u,d,U,D Composite: j=u,U,d,D,s,S,c,C,b,B pdf1: cteq6l(proton) pdf2: cteq6l(proton) p1: 3500 p2: 3500 Cut parameter: M(j,j) Cut invert: False Cut min: 1000 Cut max: Run parameter: MV Run begin: 2000 Run step size: 100 Run n steps: 10

Note that in the begining we invoke the model “Modelo de axigluon” which is already created in CalcHEP. In all our simulations we used CTEQ6L parton distribution functions (PDF), because ATLAS collaboration makes its predictions using them. We consider that protons are composed by quarks u, d, u¯ and d¯, where the last two comes from the “sea” inside of the protons. We do not consider the other quarks (c, s, b and their antiparticles) due to the low probability of found them in the proton (see PDF’s). Nor top-antitop quarks are consider because they do not exist in stables states as the proton, because they dacay before the hadronization. The final particles could be u, d, s, c, b and their antiparticles. We do not consider top-antitop quark for the reason of hadronization. “j” denote the jets after the process of hadronization. Note that in the initial state there are not gluons, they do not contribute to the process because, gluon-axigluon vertices with an odd number of axigluons, violate parity. In principle, we could consider the process gg → AA, but this process does not give a resonance, and second, it after hadronization, will decay, at least, to four jets, and our analisis is in the dijet mass invariant spectrum. There is specified a cut parameter in the dijet invariant mass, which are only applied to the production processes. For each cut, all four keyphrases have to be present. The Run parameter is the axigluon mass where, for each simulation, is incresing the axigluon mass in 100 GeV: The first simulation (10.000 events) will be with an axigluon mass 1000 GeV, in the second simulation (10.000 events) the axigluon mass will be 2100 GeV, and so on. It is important to say that, after run the batch file, with this proccedure we got not just the N-tuplas with 10.000 events for each axigluon mass, but we obtained the cross section for each subprosses and the total cross section, see Table 1.

53 NN Subprocess Cross section (fb) Number of events 1 u, u →u, u 421.83 1084 2 u, d →d, u 327.58 855 3 u, U →u, U 267.1 711 4 u, U →d, D 159.73 443 5 u, U →s, S 160.03 444 6 u, U →c, C 159.64 443 7 u, U →b, B 159.15 442 8 u, D →D, u 148.54 418 9 d, u →d, u 327.51 855 10 d, d →d, d 143.4 402 11 d, d →d, d 74.135 224 12 d, D →u, U 80.029 242 13 d, D →d, D 147.31 416 14 d, D →s, S 80.049 242 15 d, D →c, C 80.086 242 16 d, D →b, B 79.539 241 17 U, u →u, U 267.07 711 18 U, u →d, D 159.84 443 19 U, u →s, S 159.87 444 20 U, u →c, C 159.71 443 21 U, u →b, B 159.16 442 22 U, d →d, U 74.166 224 23 U, U →U, U 19.567 73 24 U, D →D, U 29.639 105 25 D, u →D, u 148.53 418 26 D, d →u, U 80.024 242 27 D, d →d, D 147.49 416 28 D, d →s, S 80.095 242 29 D, d →c, C 79.947 242 30 D, d →b, B 79.57 241 31 D, U →D, U 29.643 105 32 D, D →D, D 25.282 90 Total 4515.3 10.000

Table 6.1.: Detailed of the subprocess, cross section and events number for the process pp → A → j1j2 at 7 TeV with an axigluon of mass of 2000 GeV.

Also, in the batch ﬁle is set a distribution parameter which gives us the distribution in invariant mass M(j,j):

54 Dist parameter: M(j,j) Dist min: 4000 Dist max: 0 Dist n bins: 3000 Dist title: p,p->j,j Dist x-title: M(j,j)(GeV)

and an Event generator where we speciﬁed how many events will be simulated per axigluon mass:

Number of events (per run step): 10000 Filename: test NTuple: True Cleanup: False

Therefore, after doing all of this steps, we are ready to analyze the results (N-tuplas), i.e, to extract the resonant events number, with their respective kinematical cuts. In the Fig. 6.1 we show a typical distribitution for the invariant mass mjj of the dijet given by Calchep for an axigluon with mass 1400 GeV.

Figure 6.1.: Invariant mass distribution for an axigluon with mass 1400 GeV

The proccedure described here was performed for the process pp → A → jj at:

• 7 TeV with axigluon mass in the interval [1000 , 4000] GeV

55 • 8 TeV with axigluon mass in the interval [1500 , 4800] GeV

The calculus at 8 TeV was done using an not ﬁxed with decay for the axigluon, and it was done for a broad axigluon satisfaying the condition Γ/M = 0.3. Also, it was performed this proccedure for the same process at 14 TeV and it will be explained below.

6.3. Analisis with Paw

As we said in the introduction, the signal must come from the resonants events because they could show a signal in the dijet invariant mass spectrum, in contrast with the corresponding events with low invariant mass which be hidden under QCD background. So, we have to be able to extract the resonant events from the total (10.000 events) in each N-tuple to have an estimation of the cross section of the resonance. Then, to get the resonance cross section we multiply the cross section of the total process pp → A → jj (we saw in the last√ section that it is given by CalcHEP), for an specific axigluon mass and CM energy s, by the acceptance of it, which this last one is defined as the rate of the number of resonant events and the number of total events. PAW works with histograms, so, our data coming from CalcHEP will be stored through PAW on these. The proccedure to substract the number of resonants events of the N- tuple with PAW will be focous on the sector where the resonance appear. Looking at the Fig. 6.2, we note that the resonants events begin to appear near 1000 GeV. Therefore, the first criteria of events selection with PAW will be to extract only the events with invariant mass in the range MV±400 GeV, where MV is the axigluon mass. Fig. 6.2 shows an histogram created with PAW for a resonance mass MV=1400 GeV between the limits that our criteria set. The histogram was made taking into account the kinematical cuts specified for 7 TeV. What is shown in this figure, is a contribution of two parts: resonance events and background events. The key point of this thesis is to be able to extract the resonant events of the histogram shown in Fig. 6.2. The mechansim was to model the resonance part as a Breit-Wigner distribution, while background is modeled by a quadratic polynomial. Breit-Wigner is justified because the width of the resonance is not broad to model with a gaussian function, and is not to much narrow to be modeled with a Dirac delta distribution. The quadratic polynomial is to account for the background.

56 Figure 6.2.: Histrogram

Before the proccedure of substraction the resonants events, we fit the histogram shown in Fig. 6.2 to get the width of the resonance and the values of some parameters that will be utile. We fit the histogram of Fig. 6.2 with the sum of Breit-Wigner+Quadratic Polynomial. To get a well fit (χ2 ∼ 1) in PAW, we have to introduce intial conditions for the parameters of our fit-functions. Breit-Wigner function has three arbitrary parameters: P1, P2 and P3. The quadratic polynomial has three more: P4, P5 and P6. In total, we have six unknown parameters. But, looking the curve can be estimated the initial conditions for each unkown parameter and PAW itself calculate the parameters to get the best fit. In the left side of Fig. 6.3 we see the fit of the histogram showed in 6.2. The paremeter P3 correspond to the resonance witdh.

57 Figure 6.3.: Left: Fit of histogram with Breit-Wigner+quadratic polynomial with χ2=0.1855E+01. Right: Background ﬁt with quadratic polynomial. Note that the scales are diﬀerents in both plots.

The next step is get the number of resonants events. What we did to extract the resonants events, is rest the background (right side of Fig. 6.3) to the right side of Fig. 6.3, and we left just with the number of resonance events. In Fig. 6.4 the resonants events are shown in green and the rest correspond to the background events. We count the events in green and√ we got the resonants events. In this particular case for a axigluon mass of 1400 GeV and s=7 TeV, the number of resonant events was, aproximately, 4297 of the 10.000 total events done by Calchep. Therefore, the acceptance for this process is 0.4297 and the cross section for the resonance is σR = σCalcHEP × Acceptance, which gives σR = 11.04 pb.

58 Figure 6.4.: Resonants events in green. The rest correspond to the background.

In the following we summarize formally the proccedure used above. It was required fortran files to make the kinematical cuts, the invariant mass, and the functions which we used to fit the histograms. The name of these fortran files are minv.f which takes the invariant mass of each event of the N-tuple, atlascuts7.f, which take into account the respectively kinematical cuts for the process at 7 TeV and the invariant mass, bwp2.f is the file containing the sum between Breit-Wigner and quadratic polinomial functions, and back.f which takes into account the background. Also, in addition to Fortran files, PAW has a scripting language that allows to execute interactive commands listed in a file. The proccedure used with PAW explained above to get the resonants events number and the resonance witdh, is detailed here. All the commands used in PAW are stored in a kind of files called Kumac files. Here we show a Kumac file that was done to analize a pp → A → jj N-tupla given by CalcHEP for the process√ , where the resonance mass is 1400 GeV for a collision at the CM of s=7 TeV.

59 N° step Instructions in PAW 1 h/ﬁle 1 test-MV1400-1.nt 2 1d 100 ! 50 1000 1800 3 1d 110 ! 50 1000 1800 4 nt/proj 100 10.minv.f atlascuts7.f>0 5 nt/proj 110 10.minv.f atlascuts7.f>0 6 vect/crea par(6) 7 vec/input par 150000000 1400 200 213.75 -0.175 0.00003 8 h/ﬁt 100 bwp2.f ! 6 par 9 fun1 200 back.f(x) 50 1000 1800 10 h/op/sub 110 200 300 12 crea vr(50) 13 h/get_vec/content 300 vr 14 sigma r=vsum(vr) 15 vec/print r

Table 6.2.: Description of a typical Kumac ﬁle

In the following we expalain each step given in the instructions of Table 2: 1) open the N-tupla called test-MV1400-1.nt. 2) and 3) Makes up two one dimension histograms called “100” and “110” with 50 bins with a range between 1000 and 1800. 4) and 5) Makes up the projection of the N-tupla onto the emptys histograms 100 and 110, considering the files 10.minv.f and atlascuts7.f. The fact to put atlascuts7.f ”>0” is just the condition for the fortran file take into account the instructions that we require inside the atlascuts7.f, otherwise the kinematical cuts are not consider. 6) Creating a six component vector called “par”. 7) Filling the components of par. These components are filling with the initial conditions P1, P2,P3, P4, P5 and P6. 8) Fitting the histogram 100 with the fortran file bwp2.f, considering the initial conditions given in step 7. 9) Making the background called “200” with back.f in an histogram with 50 bins between 1000 and 1800. 10) Substracting histograms: 110 − 200 to generate a new one called “300”. 12) Creating a 50 components vector called “vr”. 13) Getting the content of 300 per bin and filling each component of vr (one-to-one mapping). 14) Summing all the components of vr and storing this result in the variable “r”. 15) Printing r.

6.4. Results at 7 and 8 TeV

With the proccedure described in 6.3 in mind, we performed simulations at 7 and 8 TeV. In Tables 6.3 and 6.4 it is shown the results for each simulation, after impose kinematical cuts, without smearing, given the resonance masses and their simulated cross section σResonance = σCalchep × Acceptance, the upper limits at 95% of Conﬁdence Level (C.L.)

60 σ95% given by ATLAS, and the rate of σ95%/σR. ATLAS collaboration, gives three upper limits for 7 and 8 TeV, respectively, for diﬀerents narrow widths resonances: Γ/M with values 7, 10 and 15 %. Also, in Table 6.3 it is shown the σ95% for a narrower resonance of Γ/M = 0.07, while in Table 6.4 it is shown the σ95% for Γ/M = 0.10.

√ s =7 TeV

Mass [GeV] σS[pb] σ95%[pb] σ95%/σR 1000 70.45 0.67 0.009 1100 40.99 0.51 0.012 1200 27.31 0.22 0.008 1300 18.42 0.12 0.006 1400 11.04 0.11 0.009 1500 7.80 0.11 0.014 1600 4.65 0.11 0.023 1700 3.12 0.11 0.035 1800 2.03 0.097 0.047 1900 1.44 0.075 0.052 2000 0.98 0.058 0.059 2100 0.72 0.047 0.065 2200 0.41 0.040 0.097 2300 0.34 0.036 0.105 2400 0.20 0.031 0.155 2500 0.13 0.027 0.207 2600 0.093 0.021 0.225 2700 0.064 0.017 0.265 2800 0.038 0.012 0.315 2900 0.029 0.0075 0.258 3000 0.019 0.0052 0.273 3200 0.0071 0.0032 0.450 3400 0.0034 0.0021 0.617 3600 0.0016 0.0016 1.000 3800 0.00055 0.0012 2.181 4000 0.00017 0.0010 5.882

Table 6.3.: Cross section of the simulation with CalcHEP σS after imposing Kinematical cuts at 7 TeV, no smearing. Also, it is shown the upper limits on the cross section σ95% given by ATLAS.

61 √ s = 8 TeV

Mass [GeV] σS[pb] σ95%[pb] σ95%/σR 1500 11.41 0.14 0.012 1550 10.15 0.12 0.011 1600 8.67 0.11 0.012 1650 6.95 0.097 0.013 1700 5.65 0.085 0.015 1750 4.60 0.068 0.014 1800 4.00 0.055 0.013 1850 3.26 0.041 0.012 1900 2.81 0.031 0.011 1950 2.16 0.026 0.012 2000 2.04 0.024 0.011 2100 1.42 0.022 0.015 2200 0.87 0.021 0.024 2300 0.63 0.019 0.030 2400 0.43 0.017 0.039 2500 0.37 0.015 0.040 2600 0.24 0.013 0.054 2700 0.16 0.011 0.068 2800 0.13 0.0097 0.074 2900 0.082 0.0087 0.106 3000 0.062 0.0075 0.120 3200 0.027 0.0051 0.188 3400 0.016 0.0030 0.187 3600 0.0065 0.0022 0.338 3800 0.0038 0.0017 0.447 4000 0.0019 0.0014 0.736 4200 0.00079 0.0011 1.392 4400 0.00026 0.00082 3.153 4600 0.00012 0.00060 5.000 4800 0.00002 0.00046 23.000

Table 6.4.: Cross section of the simulation with CalcHEP σS after imposing Kinematical cuts at 8 TeV, no smearing. Also, it is shown the upper limits on the cross section σ95% given by ATLAS.

Fig. 6.5 shows the limits on the resonance mass at 7 TeV. It is shown the axigluon cross section done in CalcHEP and PAW, σR, with the three 95% C.L. upper limits predictions by ATLAS for independent-model resonances for Γ/M of 0.07, 0.10 and 0.15.

62 Because upper limits shows that the region above the color curves is excluded by ATLAS experiment, the universal-axigluon could be exist only with masses up to 3600 GeV.

Figure 6.5.: Cross section simulation (black line) in Calchep after the kinematical cuts in PAW vs upper limits given by ATLAS (color lines) at 7 TeV

√ Fig. 6.6 shows the same comparation of Fig. 6.5, but it was done at s=8 TeV. In this ﬁgure we note that the excluded region for the axigluon is more strict than the previous one at 7 TeV, because Axigluon with masses below 4100 GeV are exluded.

Figure 6.6.: Cross section simulation (black line) in CalcHEP after impose kinematical cuts in PAW (no smearing) vs 95% C.L. upper limits given by ATLAS (color lines) at 8 TeV.

Now, let’s invoke the results obtained from the model SU(3) × SU(3) with four gauge ﬁelds developed in section 4.1.2. From diﬀerent experiments we have upper limits on the

63 cross section for resonance masses, therefore, we must impose the following inequality for to keep the axigluon invisible to the dijet observation at ATLAS:

2 (1 − χ) σk=0 ≤ σ95% where σk=0 ≡ σQCD and σ95% is the observed 95% Conﬁdence level (C.L.) cross section. For comodity, we will use k ≡ 1 − χ, then we ﬁnally found the following inequality σ k2 ≤ 95% σk=0

The rate σ95%/σk=0 is shown in Tables 6.3 and 6.4. We solve these inequality for 7 and 8 TeV and Fig. 6.7 shows the limits on the constant k for 7 and 8 TeV for the resonance mass. As we increase the axigluon mass, the limits on the constant k are less stringent. Limits on the constant k at 8 TeV are more stringent than at 7 TeV.

Figure 6.7.: Limits on k in function of the axigluon mass. As we have an inequality on k2, we have upper and lower limits on the value of k. Interior lines correspond to the limits impose by comparasion between our simulation and upper limits given by ATLAS at 8 TeV, whereas the external lines correspond to the the comparation at 7 TeV.

6.5. Simulation at 8 TeV with smearing

For 8 TeV simulation, we also studied the eﬀect due to the resolution of the detector (hadron calorimeter) on our limits. A typical resolution for a hadron calorimeter is

64 ∆E 0.5 = ⊕ 0.03 E pE(GeV )

We smeared our events using this parametrization for the resolution of the calorimeter and we re-extracted the resonant cross section using the same cuts. Of course, due to the smearing procedure we ﬁtted the peak with a gaussian function and not a Breit-Wigner one. Fig. 6.8 shows the results. The continuous line represents our extracted resonant cross section for diferents values of the axigluon mass (no smearing), see Fig. 6.6, the line with dots is the experimental limit set by ATLAS for resonances, and the dotted line represent our simulation in CalcHEP including smearing. Every resonance producing a cross section above the experimental limit is excluded. We see that, in the context of the Minimal Axigluon Model, that the axigluon is excluded if it is lighter than 4100 GeV. The fact of including smearing “relax” the limits on the axigluon mass and now could be exist axigluon with masses near above of 3600 GeV. This limit is comparable with that recent limits obtained by CMS.

Figure 6.8.: Predicted resonant cross sections with (continuous line) and without (dashed line) detector resolution compared to ATLAS 95% C.L. upper limits to narrow resonances decaying into dijet (dots with continuous line). In both cases we used k = 1.

6.6. Broad Axigluon

So far, we have considered an axigluon which decays only into standard quarks with universal coupling. This kind of axigluon is relatively narrow with Γ/M ∼ 0.1. Nevertheless, phenomenologically it is more important the case of a broad axigluon (Γ/M & 0.2) since

65 tt¯ it is a viable explanation for the AFB anomaly observed at the Tevatron. An axigluon may be broad if its width is dominated by a strong decay channel into non-standard particles or even to the top-quark. We tried to remain model-agnostic and we studied a broad axigluon in an illustrative case. Again, we generated events for dijet production due to axigluon exchange but this time we set the value of Γ/M = 0.3. We observed that for axigluon heavier than 3 TeV it was impossible to ﬁt any resonant structure and consequently the axigluon was unobservable by searches in the dijet spectrum. However, for masses in the range 1 TeV ≤ M < 3 TeV, resonant structures were recognizable and the resonant cross sections could be extracted. In Fig. 6.9 it is shown the comparasion between the extracted resonant cross section for the broad axigluon (continuous line) and the one obtained previously for a narrow axigluon (dashed lined). In both cases we used k = 1. We see that the broad axigluon produces larger cross sections. This is mainly due to the fact that, in the broad axigluon case, the axigluon resonance explores lower invariant masses suﬀering less PDF suppression and thus enhancing the integrated cross section.

Figure 6.9.: Resonant cross section for a broad axigluon (Γ/M=0.3, continuous line) compared to the resonant cross section for an axigluon decaying into standard quarks. In both cases we used k = 1.

6.7. Calculus at 14 TeV √ We now move toward our expectations for the next run of the LHC at s = 14 TeV. We performed the same calculus done for 7 and 8 TeV, but this time it was increase the center momentum energy to 14 TeV and we used the same kinematical cuts imposed by ATLAS at 8 TeV. We did not consider here the smearing eﬀects. The numerical results are shown in de Apendix A. Also, considering the maximal value of k for each mass given in Fig.

66 6.7 for the simulation at 8 TeV, we performed the simulation at 14 TeV not only in the minimal Axigluon model, but considering these maximal values for k. In Fig. 6.10 two curves are shown. The ﬁrst one (dots with continuous line) represents the resonant cross section predicted by the Minimal Axigluon Model considering the kinematical cuts used for 8 TeV and no smearing. The second one, (squares with continuous line) represents the resonant cross section obtained considering the maximum value of k allowed by current data for each axigluon mass. Our results tell us that if a (narrow) axigluon exists but has escaped our searches because it is couple to quarks weakly enough, then it would be produced at LHC at 14 TeV with maximum resonant cross section laying somewhere in the range 0.1−1 pb.

Figure 6.10.: Resonant cross section predicted for the 14 TeV LHC using k=1 (dots with continuous line) and the maximum value of k allowed by current data (squares with continuous line).

67 7. Conclusions

This research has the advantage of being in the middle of the theory and experimental data, and thus, it was very enriching from the standpoint to learn from both areas. We could explore the motivations for the existence of massive resonant particles at the TeV scale. Following this, we could study differents axigluon models coming from chiral gauge groups, and how they fit to the available parameter space. However, we centered our attention on an independent-model analysis to compare the results of our simulations to the mesurements on the dijet spectrum given by ATLAS experiment. With the tools of both CalcHEP and PAW, we could set predictions on the cross section for differents axigluon masses at 7, 8 and 14 TeV. With these simulations in hand, we compared them to the mesurements on the dijet spectrum given by ATLAS collaboration and, in this way, we could set limits on the universal-axigluon mass for gAqq¯ = gQCD (k = 1). The first results were for 7 TeV, which ruled out axigluon masses below ∼ 3600 GeV. Also, the analysis at 8 TeV showed that the possibles axigluon masses still are more heavy: M &4000 GeV. However, it was saw that when is including the smearing effect on the data, the axigluon cross section at 8 TeV makes less strict the limits on the axigluon masses: M &3600 GeV. These numbers, of course, are very big these days for the energies which LHC run. However, we hope that with the increase in the energy in the next few years, it will be more accesible to see beyond. We established limits on the universal coupling constant gAqq¯ (actually in the k parameter) based on the phenomenological approximation given by the model with four gauge fields explained in section 4.1.2. (gAqq = kgQCD). These limits on the coupling constant did result to be very close to cero for small masses (1000 - 3000 GeV), which is positive because it makes the axigluon invisible to experiments, meanwhile it increase its value for big axigluon masses (&3000). Furthermore, based on the fact that a broad axigluon (Γ/M = 0.3) could be a possible FB explanation of the Att¯ anomaly at Tevatron, we showed that the possibles resonants structures were in the range 1 TeV≤ M <3 TeV, and they have larger cross section than the axigluons with not narrower widths. We observed that for axigluon heavier than 3 TeV it was impossible to fit any resonant struture and consequently the axigluon was unobservable by searches in the dijet spectrum. With respect to the metodology used to search for resonaces in the dijet spectrum, there are differents ways. CMS made their simulation relaying on the well known Narrow Width Approximation. ATLAS, on the other hand, computed the leading order cross section considering all the possible channels (not only the s-channel) and the integrated the diferential cross section in the neighborhood of the resonance (specically in the interval [0.7M, 1.3M] where M is the axigluon mass). These differents criterias lead to signicantly different limits which originate some level of controversy.

68 However, it is unclear whether the ATLAS and CMS methodology of fitting the data for the background shape and normalization effectively absorbs some of the cross section of these additional processes into the background, setting limits on only the bump like component of the resonances. The same can be said for the remainder of the axigluon tail at masses outside the search window which is ignored by both experiments. Further, both experiments ignore coherence between QCD and axigluons, which may significantly affect the tails of the distribution, and is likely an issue when the tail has a larger cross section than the peak.

69 A. Data obtained from simulation

Here we present the data obtained with Calchep and after the analisis with PAW. In pp → A → jj each of the tables is√ specified the resonance mass at which the process for a specific energy s was done. The number of resonants events (N.E.R.), P3 (≡ Γ) and χ2 are quantities obtained after the analisis with PAW described in section 6.3. For each mass, is given the cross section given by CalcHEP, σC , for the given process at the specified resonance mass and the cross section for just for the resonance σR after extracting the resonance part considering the kinematical cuts. Finally, it is shown the rate Γ/M calculated by our simulation for each resonance mass. In Tables A.1 and A.2 was not considered the smearing effect. In Table A.3 is shown the simulations results at 14 TeV.

70 2 Mass[GeV] N.E.R P3 [GeV] χ σC [pb] σR[pb] Γ/M 1000 5661.00 94.125 0.2794E+01 124.45 70.45 0.094 1100 5090.30 98.092 0.2233E+01 80.531 40.99 0.089 1200 5087.96 105.64 0.9480E+00 53.680 27.31 0.088 1300 5018.73 116.32 0.2181E+01 36.720 18.42 0.089 1400 4297.57 114.62 0.1855E+01 25.700 11.04 0.081 1500 4247.15 131.94 0.1198E+01 18.380 7.80 0.087 1600 3469.36 126.22 0.1675E+01 13.410 4.65 0.078 1700 3133.30 140.05 0.1552E+01 9.961 3.12 0.082 1800 2707.82 140.69 0.9981E+00 7.531 2.03 0.078 1900 2503.24 165.10 0.1266E+01 5.786 1.44 0.086 2000 2169.90 165.07 0.1356E+01 4.515 0.98 0.082 2100 2019.15 188.91 0.1242E+01 3.575 0.72 0.089 2200 1455.70 165.11 0.1364E+01 2.868 0.41 0.075 2300 1492.85 227.27 0.9500E+00 2.330 0.34 0.098 2400 1069.00 195.04 0.1068E+01 1.915 0.20 0.081 2500 8156.84 214.48 0.1021E+01 0.169 0.13 0.085 2600 7517.62 224.15 0.1359E+01 0.125 0.093 0.086 2700 6967.61 237.04 0.1353E+01 0.093 0.064 0.087 2800 5579.21 209.98 0.1966E+01 0.069 0.038 0.074 2900 5634.64 251.58 0.1005E+01 0.053 0.029 0.086 3000 4963.84 268.07 0.1284E+01 0.040 0.019 0.089 3200 5524.65 250.98 0.1044E+01 0.013 0.0071 0.078 3400 4284.97 274.68 0.9034E+00 0.008 0.0034 0.080 3600 8295.46 389.85 0.9327E+00 0.002 0.0016 0.108 3800 6896.59 329.76 0.1168E+01 0.0008 0.00055 0.086 4000 3590.45 299.27 0.1107E+01 0.0005 0.00017 0.074

Table A.1.: Numerical data from the simulation of pp → A → jj at 7 TeV.

71 2 Mass [GeV] N.R.E. P3 [GeV] χ σC [pb] σR[pb] Γ/M 1500 6181.07 116.50 0.1385E+01 18.470 11.41 0.077 1550 8193.12 134.09 0.1419E+01 12.390 10.15 0.086 1600 6559.07 137.56 0.1243E+01 13.230 8.67 0.085 1650 7878.55 146.41 0.1148E+01 8.826 6.95 0.088 1700 5875.77 143.77 0.1088E+01 9.624 5.65 0.084 1750 7249.40 140.97 0.9858E+00 6.358 4.60 0.080 1800 5638.57 153.44 0.1645E+01 7.098 4.00 0.085 1850 7045.93 158.36 0.2077E+01 4.631 3.26 0.085 1900 5308.87 171.59 0.1198E+01 5.307 2.81 0.090 1950 6352.59 152.62 0.1412E+01 3.410 2.16 0.078 2000 5090.09 188.38 0.1333E+01 4.020 2.04 0.094 2100 8430.72 188.41 0.1384E+01 1.689 1.42 0.089 2200 6953.70 179.32 0.1091E+01 1.256 0.87 0.081 2300 6747.42 181.07 0.1490E+01 0.941 0.63 0.078 2400 6157.27 190.24 0.1306E+01 0.710 0.43 0.079 2500 6947.80 243.15 0.1430E+01 0.541 0.37 0.097 2600 5896.44 233.37 0.9601E+00 0.416 0.24 0.089 2700 5222.59 225.66 0.1237E+01 0.323 0.16 0.083 2800 5425.51 262.05 0.1241E+01 0.253 0.13 0.093 2900 4113.26 237.18 0.7435E+00 0.201 0.082 0.081 3000 3872.83 271.40 0.7362E+00 0.161 0.062 0.090 3200 7747.18 241.81 0.1018E+01 0.036 0.027 0.075 3400 7682.39 324.91 0.1064E+01 0.021 0.016 0.095 3600 5075.54 271.15 0.1289E+01 0.013 0.0065 0.075 3800 4789.93 354.35 0.1045E+01 0.008 0.0038 0.093 4000 3822.66 402.80 0.1135E+01 0.005 0.0019 0.100 4200 6911.74 358.87 0.1402E+01 0.001 0.0007 0.085 4400 4866.56 297.94 0.7183E+00 0.0005 0.0002 0.067 4600 4902.39 337.70 0.7334E+00 0.0002 0.0001 0.073 4800 4500.23 350.32 0.9332E+01 0.0001 0.00003 0.072

Table A.2.: Numerical data from the simulation of pp → A → jj at 8 TeV.

72 2 Mass N.E.R. P3 χ σc [pb] 1500 7664.94 124.30 0.2037E+01 94.640 1600 8032.42 141.73 0.1646E+01 71.890 1700 7564.22 137.57 0.1519E+01 55.280 1800 7575.63 153.02 0.1137E+00 42.980 1900 7473.38 162.03 0.1583E+01 33.750 2000 9048.11 171.21 0.1512E+01 17.510 2100 8469.41 161.83 0.1393E+01 13.930 2200 8554.09 173.43 0.1002E+01 11.120 2300 7968.19 167.45 0.1421E+01 8.924 2400 8373.56 195.68 0.1626E+01 7.195 2500 7778.68 198.84 0.1893E+01 5.828 2600 7950.61 216.74 0.1416E+01 4.742 2700 7616.45 214.28 0.2080E+01 3.877 2800 7482.32 226.75 0.1876E+01 3.183 2900 7083.32 219.67 0.1436E+01 2.624 3000 7529.32 221.01 0.1334E+01 1.917 3200 7588.19 257.26 0.1529E+01 1.319 3400 7128.01 281.83 0.1116E+01 0.919 3600 7604.40 292.61 0.1095E+01 0.564 3800 6779.21 287.05 0.1584E+01 0.399 4000 6597.33 305.90 0.1406E+01 0.286 4200 8864.15 323.76 0.1401E+01 0.128 4400 9089.76 374.00 0.1214E+01 0.096 4600 8775.57 344.70 0.1392E+01 0.064 4800 8600.56 356.56 0.1245E+01 0.043

Table A.3.: Numerical data from the simulation of pp → A → jj at 14 TeV.

73 B. Electron-proton scattering

B.1. Introduction

The scattering experiment electron-proton is the best known experiment to explore the internal structure of the proton. A powerfull technique for exploring the internal structure of a target is to bombard it with a beam of high energy electrons and to observe the angular distribution and energy of scattered electrons. We can apply this results to electron-quark scattering when we are probing the proton structure. Suppose that we want to probe some charge distribution with electrons. To achiveve this, we mesure the scattered electron distribution corresponding to this charge distribution, and by this, we could compare the known cross section of a puntual charge versus the new cross section. dσ dσ =| F (q) |2 dΩ charge distribution dΩ puntual where q is the momentum transfer between the incident electron and the target,q = ki − kf . In this way, we try to discover the target structure through the function F (q) known as form factor. The normalization of it is F (0) = 1. In the more complex analysis showed below for the elastic and inleastic electron-proton scattering, the form factor is replaced by what are called “structures functions”. In this apendix, we show, not in detail at all, elastic and inelastic electron-proton scattering. If you want to have a more detail description of the process, see [11] and [12].

B.2. Elastic scattering e− + µ− → e− + µ−

We understand “elastic” as a process which the initial particles are exactly the same as the ﬁnal. The transition amplitude for the process e− + µ− → e− + µ− at leading order is given by

1 T = −i jµ (x) − jmuon(x)d4x fi ˆ electron q2 µ 1 = −i u¯(k0)γµu(k) − u¯(p0)γ u(p) (2π)4 δ(4) k + p − k0 − p0 q2 µ

The scattering or invariant amplitude is related to the transition amplitude by

74 4 (4) 0 0 Tfi = −i (2π) δ k + p − k − p M

So, the invariant amplitude for this process from the Feynman rules is 1 M = −e2u¯(k0)γµu(k) u¯(p0)γ u(p) q2 µ

We define the transfer momentum as q = k − k0. To obtain the (unpolarized) cross section, we have to take the square of the modulus of M and then carry out the spin sums. To allow for scattering in all possible spin configurations, we average over the spins of the incoming particles and we sum over the spins of the particles in the final state.

2 1 X 2 | M | ≡ | M | (2s − + 1) 2s − + 1 e µ all spin states where se− and sµ− are the spins of the incoming particles. We can separate the sum over the electron and muon spins writings as e4 | M |2 = LµνLmuon q4 e µν where

1 X ∗ Lµν ≡ u¯(k0)γµu(k) u¯(k0)γνu(k) e 2 (e spins)

muon and with a similar expression for Lµν .

Figure B.1.: Electron muon scattering

To calculate the sum of the electron tensor, we use the fact that inside the brackets we have just complex numbers for which the complex and the hermitian conjugate are the same:

u¯(k0)γνu(k)∗ = u¯(k0)γνu(k)† = u¯(k)γνu(k0)

75 Using this, we write the electron tensor in components (the sum of the spinor index is ommited)

1 X X (s) (s0) Lµν = u¯ (k0)γµ u (k)¯u(s)(k)γν u (k0) e 2 α αβ β γ γδ δ s0 s

Due that in the elctron tensor we have the multiplication of the matrix elements, we can 0 (s ) 0 move the uδ (k ) to the left, this is

1 X (s0) X (s) Lµν = u (k0)¯u (k0)γµ u (k)¯u(s)(k)γν e 2 δ α αβ β γ γδ s0 s

Then, we note that has appeared two completness relation for the spinors inside of the electron tensor, so using the completnees relations

X (s) (s) µ uβ (k)¯uγ (k) = (k γµ + m)βγ s

X (s0) 0 0 0µ uδ (k )¯uα(k ) = k γµ + m δα s0 we get the following form of the tensor 1 Lµν = k0 + m γµ (k + m) γν e 2 δα αβ βγ γδ P Noting that the deﬁnition of the trace of a matrix is deﬁned as T rA = i Aii, we write µν Le as the trace of the product of four 4 × 4 matrices, 1 Lµν = T r k0 + m γµ (k + m) γν e 2 To evaluate the trace, we have to use the trace theorems. After the using the trace theorems, we found that the electron tensor is

µν 0µ ν 0ν µ 0 2 µν Le = 2 k k + k k − k · k − m g (B.1)

With an equivalent proccedure for the muon, we found

muon 0 0 0 2 Lµν = 2 pµpν + pνpµ − p · p − M gµν (B.2) where M is the muon mass. Multipling both tensors, B.1 and B.2, we get the scattering amiplitude for the spin-average process e− + µ− → e− + µ−

4 8e 0 0 0 0 2 0 2 0 2 2 | M |2 = k · p (k · p) + k · p k · p − m p · p − M k · k + 2m M (B.3) q4

76 In the relativistic limit, we can neglect the electron and muon mass, given 8e4 | M |2 = k0 · p0 (k · p) + k0 · p k · p0 (k − k0)4

Moreover, in this limit, the Mandelstam variables (Lorentz invariants) become

s ≡ (k + p)2 ' 2k · p ' 2k0 · p0 t ≡ k − k02 ' −2k · k0 ' −2p · p0 u ≡ k − p02 ' −2k · p0 ' −2k0 · p

Thus, at high energies, the unpolarized amplitude scattering in function of the Mandel- stam variables is s2 + u2 | M |2 = 2e4 t2 Now, let’s consider the proccess in the laboratory frame, that is, the frame where the initial µ is at rest. We can then directly apply these results to the electron-quark scattering when probing the structure of the proton. Now, considering expression B.3 and neglecting the electron mass, we have 8e4 | M |2 = k0 · p0 (k · p) + k0 · p k · p0 − M 2k0 · k q4

Considering the q variable instead of p0 through the relation p0 = k − k0 + p, and taking into account that k2 = k02 ' 0 and q2 ' −2k · k0, the ampllitude can be written as 8e4 1 1 | M |2 = − q2 k · p − k0 · p + 2 k0 · p (k · p) + M 2q2 q4 2 2

Evaluating the amplitude in the laboratory frame, this is, when the muon is at rest, p = M,~0 , we found

8e4 θ q2 θ | M |2 = 2M 2EE0 cos2 − sin2 (B.4) q4 2 2M 2 2 The diﬀerential cross section is given by

2 3 0 3 0 1 | M | d k d p (4) 0 0 dσ = 2 0 0 δ p + k − p − k (B.5) (2E) (2M) 4π 2E 2p0 Inserting B.4 in B.5, we obtain

dσ (2αE0)2 θ q2 θ q2 = cos2 − sin2 δ ν + dE0dΩ q4 2 2M 2 2 2M

77 0 2 0 2 θ Performing the integration in E , replacing q = −4EE sin 2 , we arrive to the ﬁnal expresion for the cross section scattering for the proccess e− + µ− → e− + µ− in the laboratory frame ! dσ α2 E0 θ q2 θ = cos2 − sin2 dE0dΩ 2 4 θ E 2 2M 2 2 4E sin 2

This expresion will be very utile in the study the internal structure of the proton.

B.3. Elastic scattering e− + p → e− + p

If the proton were a point Dirac particle, we could just copy our analysis of electron- muon scattering, with M now the mass of the proton. But the proton is not just a simply point charge. Let’s represent the interaction between de photon and the proton, at leading order, by a circle which take into account our ignorance at the vertex. See Fig. B.2.

Figure B.2.: Electron proton elastic scattering

First, suppose that the proton is a charge point particle with a Dirac magnetic moment e/2M. With this in mind, we suppose that our proton it behaves as a muon, and in the diﬀerential cross section for the process e− +µ → e− +µ , we just replace the muon mass by the proton mass ! dσ α2 E0 θ q2 θ = cos2 − sin2 dΩ 2 4 θ E 2 2M 2 2 4E sin 2 where α is the ﬁne structure constant, θ is the angle between the incident electron and the scattered electron in the laboratory reference system and q2 is the momentum E0 transference in the process. The factor E is

78 E0 1 = E 2E 2 θ 1 + M sin 2 which arises from the recoil of the target. The important point is that the electron vertex and the photon propagator are unchanged, then the scattering amplitude of the process is

4 2 ge µν <| M | >= L Wµν (B.6) q4 e where the Wµν function represent the photon-proton vertex. We do not know the tensor Wµν, but we know that it has to depend on p2, p4 and q. But, q = p4 − p2, therefore we have just two independent variables. We will use q and p2. Using p2 ≡ p, there are many posibilities to construct this two-rank tensor: K K K W proton = −K g + 2 p p + 4 q q + 5 (p q + p q ) µν 1 µν M 2 µ ν M 2 µ ν M 2 µ ν ν µ where K0s are functions of the the only one scalar variable, q2. We have introduced in each term a M 2 so all K0s will have the same dimensions. Using the property

µ q Wµν = 0 we found that the functions K0s are not independent: M 2 1 K = K and K = K 4 q2 1 5 2 2

Using this, we found that our hadronic tensor can be written as q q K 1 1 W proton = K −g + µ ν + 2 p + q p + q µν 1 µν q2 M 2 µ 2 µ ν 2 ν

A fundamental problem for any theory of proton structure is to determine these to functions K1and K2. proton Then, using this construction of Wµν in the scattering amplitude B.6 and using that the diﬀerential cross section dσ E0 = <| M |2> dΩ 8πME we found the diﬀerential cross section for elastic scattering of electron-proton dσ α2 E0 θ θ = 2K sin2 + K cos2 dE0dΩ 2 2 4 θ E 1 2 2 2 4M E sin 2

79 where E E0 = 1 + (2E/M) sin2 (θ/2) This formula is known as “Rosenbluth formula” and it was derived in 1950. We see that the scattered energy electron is not an independent variable, but it depends on the initial electron energy and the scattered angle. The form factors can be determined in the experiment and they have been conﬁrmed to agree very well with the theoretical predictions.

B.4. Electron-proton inelastic scattering

One can “see” more inside in the proton by increasing the −q2 of the photon to give a better spatial resolution. This can be done simply by requiring a large energy loss of the bombarding electron. However, because of the larger transfer energy, the proton will often break up and so we have the situation presented in Fig. B.3.

Figure B.3.: Lowest order for inelastic electron-proton scattering.

We can analize the problem in two parts: the leptonic part and the hadronic part. The leptonic part must continue unchanged. However, the hadronic sector is different because, now, we can have multiples final states, and so, now we can not have just a final state as u¯(p0). In this way, the cross section must be

g4 <| M |2>= e Le W µν(X) q4 µν

µν where Lµν represent the leptonic tensor, which is unchanged, and W respresent the hadronic part which parametrize our ignorance about the current form in the other propagator extreme. W µν represent the subprocess γ + p → X, and this tensor must depend on q = p1 − p3, p ≡ p2 and the various outgoing momenta p4, p5,...,pn. µν Due that Lµν is a symmetric tensor, W must be a symmetric tensor too. The only µν variables which W can depend is p1, p3 and q, but, as p3 = p1 + q we have just two

80 independents variables: p1 ≡ p and q, because tha ﬁnal momenta p4, p5, etc., have been integrated. Based in all before, the tensor structure is given by W W W W µν = −W gµν + 2 pµpν + 4 qµqν + 5 (pµqν + pνqµ) 1 M 2 M 2 M 2

2 where Wi are functions depending on the only one scalar variable of the problem: q . Due µν µν to the conservation of the electromagnetic current requires that qµW = qµW = 0. Imposing this condition, we found that not all the functions Fi are independent: p · q W = − W 5 q2 2

p · q 2 M 2 W = W + W 4 q2 2 q2 1

Therefore, the hadronic tensor can be written in the form qµqν W p · q p · q W µν = W −gµν + + 2 pµ − qµ pν − qν 1 q2 M 2 q2 q2 where W1 and W2 are called structure functions. Evaluating the differential cross section with our new construction of W µν we got dσ α2 θ θ = W cos2 + 2W sin dE0dΩ 2 4 θ 2 2 1 2 4E sin 2 where we have made electron mass tend to cero. This is the scattering differential cross section for the inelastic process between the proton and the electron. We must to have in mind that these new structure functions, W1 and W2, depends on two Lorentz-invariant independent variables: q2 and x ≡ −q2/2p · q. We have to remember that in the elastic case, the form factors, K1,2, depended just of one variable, q2 and in this case the variable x is fixed (x = 1). Then, the elastic case 2 2 is a particular case of the inelastic one just taking the constraint ptot = M . It can be shown that the relationship between the structure functions of the inelastic process and elastic process are related by

2 K1,2 q W q2, x = − δ(x − 1) 1,2 2Mq2

The most naïve model at low energies which treat the proton as a point charge set the form factors as

2 2 K1 = −q ; K2 = (2M)

This aproximation is not bad at low energies and the photon does not see the proton

81 structure.

B.5. Bjorken scaling

Bjorken predicted that, at very high energies, the dependence of the inelastic structure functions on q2 fades away, and they become functions of x alone. More precicely, he proposed that

2 MW1(q , x) → F1(x) −q2 W (q2, x) → F (x) 2Mx 2 2

This regime is called “deep inelastic scattering”, where −q2 and q · p = M(E − E0) are both larges, but the ratio 2x = −q2/q · p is not. Experimentally, Bjorken scaling sets in 2 2 2 for −q & 1 GeV and q · p & 3.5 GeV . This behaviour is known as “scaling”, and it was conﬁrmed by SLAC in the early seventies. Scaling is a consequence of the fact that proton is made of point-like constituents. The Callan-Gross relation stated that the Bjorken’s scaling functions are related by

2xF1(x) = F2(x) and this relation reflects the fact that the partons inside of the protons carry spin 1/2. In DIS, the Bjorken scaling and Callan-Gross relation are the first evidence of the existence of quarks. Therefore, we have three stages in the scattering between electron and proton: 1. Low energies: The intermediate photon “sees” the whole proton as a point charge; Mott scattering. 2. High energies: The photon “sees” a single quark - This gives Bjorken scaling and Callan-Gross relation. 3. Intermediate energy: The photon “sees” the proton and its complex structure; this is the hardest case to analyse. In case 2 we can treat the scattering using the old electron-muon scattering. Then, the important point for to make the parton model is to think the proton as a system with many point particles, which are known as partons. Using what we have got in the last section for the interaction of a photon with a point-like particle, we set the structure functions for scattering off a quark of flavor i are

2 i Qi W1 = 2 δ (xi − 1) 2mi

2m Q2 W i = − i i δ (x − 1) 2 q2 i

82 2 where Qi is the quark charge of ﬂavor i, mi is the quark mass and xi = −q /2q ·pi, where pi is the quark momentum. We deﬁne

pi = zip (B.7) where zi is the fraction of the total momentum carried by quark i. It follows that the quark mass and the proton mass are related by

mi = ziM

If zi is a variable, then mi is a variable too. Relation B.7 implies that x xi = zi and hence Q2 2x2M W i = i δ(x − z ); W i = − Q2δ (x − z ) 1 2M i 2 q2 i i

Finally, let’s deﬁne the probability fi(zi) of ﬁnd the ith quark with momentum fraction zi. Integrating over zi, and summing over all the quarks in the proton, we conclude that

X 1 Q2 1 X W = i δ (x − z ) f (z )dz = Q2f (x) 1 ˆ 2M i i i i 2M i i i 0 i X 1 2x2M 2 2M X W = − Q2δ (x − z ) f (z )dz = − x2 Q2f (x) 2 ˆ q2 i i i i i q2 i i i 0 i thus

1 X MW = Q2f (x) ≡ F (x) 1 2 i i 1 i q2 X − W = x Q2f (x) ≡ F (x) 2Mx 2 i i 2 i conﬁrming the Bjorken scaling law. Comparing these two last equations, we obtain

F1(x) = 2xF2(x) which is the Callan-Gross relation. Therefore, instead of have two unknown functions in the scattering cross section, 2 W1,2 q , x , we have now just only one unknown function of one variable to contend with F1(x). Then, the diﬀerential scattering cross section in terms of this one unknown function F1(x) is

83 dσ F (x) α 2 2EE0 θ = 1 1 + cos2 dE0dΩ 2M E sin (θ/2) (E − E0)2 2 where 1 X F (x) = Q2f (x) 1 2 i i i

Have been measured the parton distribution functions in some experiments at diﬀerents Q2. In Fig. B.4 we show some pdf’s for some valence quarks, gluon and charm sea quark.

Figure B.4.: Parton distribution functions

Other way to understand of how the parton distribution functions appear, is to think the following. In DIS the interaction time of the photon with quarks τγ ∼ 1/Q is much less than the time of the QCD interaction. As a result, the structure functions are proportional to the density of partons with fraction x of the nucleon momentum, weighted with the squared charge. Also it was proven by experiment that at values of Q2 of a few GeV2, in the scaling region, about half of the nucleon momentum, given by the momentum sum rule: " # 1 X f (x) + g(x) xdx = 1 ˆ i 0 i

84 is carried by neutral partons (gluons).

85 C. σ-lineal model

C.1. Symmetries of the model

The origin of the term "sigma model" for a field theory where the scalar values are on a manifold is from Gell-Mann and Levy’s 1960 paper "The Axial Vector Current in β- Decay" which introduced two models: σ-model and σ-nonlinear model. In this apendix we will see just the first one. For more references, see [15] and [20]. The σ − lineal (Schwinger 1958; Polkinghorne 1958; Gell-Mann and Levy 1960) is a effective model with chiral symmetry SU(2)L × SU(2)R with the following fields: an 1 2 3 isoscalar field σ, a pion isotriplet π = π , π , π and an nucleon isodoublet ψ = (p, n) whose Lagrangian is

¯ µ 1 2 1 2 ¯ 5 L = iψγ ∂µψ + (∂µσ) + (∂µ~π) − gψ σ − i~π · ~τγ ψ (C.1) 2 2 2 2 µ λ 2 + σ2 + ~π2 − σ2 + ~π2 2 4 where g is a interaction constant between fermions and bosons, µ2 is a constant parameter undefined yet and λ > 0. The three first terms correspond to the kinetic energy of the fields, the fourth is an interaction term and the last one is a potential made for the isoscalar and the isotriplet. Note that it can not exist a mass term for the fermions mψψ¯ because it breaks the chiral symmetry. There are two ways equivalents to show the chiral invariance of the Lagrangian C.1: i) Showing its invariance under SU(2)L × SU(2)R and ii) showing its invariance under SU(2)V × SU(2)A. i) To see explicitly that the Lagrangian is chiral invariant under SU(2)L × SU(2)R, let’s define a matrix M of 2 × 2 σ1 + i~π · ~τ M = 2×2√ 2

After some algebra, the Lagrangian in terms of this new matrix M is

1 h i √ L = iψγ¯ µ∂ ψ + T r ∂ M †∂µM − g 2 ψ¯ Mψ + ψ¯ M †ψ µ 2 µ R L L R µ2 λ2 − T r[MM] − (T r[MM])2 2 4

86 This Lagrangian is invariant under chiral symmetry whenever both the fermionic ﬁelds transform under the fundamental representation of the symmetry group ψL → exp i~α · T~L ψL

ψR → exp i~α · T~R ψR and M transform under the bifundamental representation M → exp i~α · T~R M exp i~α · T~L ii) The other form is to replace directly the transformation of the ﬁelds under the direct product of SU(2)V × SU(2)A. This is, under SU(2)V

σ → σ0 = σ ~π → π + ~α × ~π

ψ → ψ − i~α · T~V ψ and, under axial transformations SU(2)A

σ → σ − ~π · ~α ~π → ~π + ~ασ

ψ → ψ − i~α · T~Aγ5ψ

C.2. Spontaneous symmetry breaking

The potential ﬁeld of the σ- model is

2 2 µ λ 2 V (σ, ~π) = − σ2 + ~π2 + σ2 + ~π2 2 4 with λ > 0. The sign of µ2 we will analize below because it will be utile to study the extreme conditions of the potential. The ﬁrst derivatives of the potential respect the ﬁelds are

∂V = σ λ σ2 + ~π2 − µ2 ∂σ ∂V 2 2 2 = πi λ σ + ~π − µ ∂πi

87 2 2 2 The extreme condition is then σ = πi = 0 y σ + ~π = µ /λ, for i = 1, 2, 3. From the second derivatives we obtain

∂2V = λ σ2 + ~π2 − µ2 + 2λσ2 ∂σ2 ∂V 2 2 2 2 = λ σ + ~π − µ + 2λπi ∂πi Let’s see the maximum and minimum of the potential:

1. In the case when σ = πi = 0, i = 1, 2, 3, we have

2 2 ∂ V ∂ V 2 2 = 2 = −µ ∂σ ∂πi

If µ2 > 0, we have a maximum of potential. This condition is necesary to make perturbation theory around the minimum of potential. If µ2 < 0, we have a minimum of potential. However, this case makes and absolute minimum and is very bored. In this case mesons have mass.

2. In the case when ﬁelds satisfy the condition

2 2 2 λ σ + ~π − µ = 0 (C.2)

also, we have two posibilities. The ﬁrst one is when µ2 > 0 ∂2V = −µ2 + 2λσ2 ∂σ2 2 ∂ V 2 2 2 = −µ + 2λπi ∂πi

To know the sign of the seconds derivatives, we have to know the value of σ2 y de 2 2 2 πi . Suppose the simply case when< σ >= µ /λ y < πi >= 0. For this choice, the potential has a minimum because ∂2V = −µ2 + 2λ < σ2 >= µ2 > 0 ∂σ2 In this manner, we note that for each point belonging to the hypersphere, given by the condition C.2, is a minimum of the potential and all points are degenerate. This form of potential is known as “mexican hat”. Finally, in the case µ2 < 0, we will have just one potential maximum, therefore, it does not interest to us because it is an unstable point.

The breaking symmetry is maniﬁest when the potential takes the values < σ >= ν ≡ q µ2 λ y < πi >= 0. The point σ = 0 and | ~π |= ν is not an acceptable point because

88 it violate parity. Making perturbation theory, we have σ = ν + δσ and ~π = ~0 + ~π. Replacing this perturbations on the potential, appear a mass term for the sigma field 2 2 mσ = 2µ 6= 0. There are not mass terms for the pions, thus they correspond to the three Goldstone bosons. In term of group theory, from the chiral symmetry we had six generators, which finally three are broken and due to Goldstone theorem, appear three Goldston bosons. This makes three field directions light, and these modes are the three pions, and one field direction heavy, the "sigma". It was a predicted particle, and it was identified with the σv(600) broad resonance, except that this resonance is very strange and was delisted, and is too broad to be a real sigma. Also, because the σ field aquire a expectation value ν, nucleons (fermions) aquire mass too

MN = gπfπ

An equivalent form of see the breaking symmetry is necesary to fermion aquire mass, is paying attention to the mass term

¯ m ψRψL + h.c.

This term, under chiral symmetry transform as ~τ ~τ m ψ¯ exp iβ~ · exp −iβ~ · ψ + h.c. ' mψ¯ 1 + i β~ − β~ + ... ψ R R 2 L 2 L R L R L

~ ~ In this way, the mass term for the fermions is possible only when βL = βR. Therefore, the symmetry is broken from a chiral group symmetry to a vectorial group symmetry

SU(2)L × SU(2)R → SU(2)V

89 D. Anomalies

In QFT, anomalies are proccess which not conserve the symmetry of the rest of the theory. This is equivalent to say that the Ward identity is no longer satisfaced. Ward identities form the basic equations for a consistently quantized theory. The ﬁrst chiral anomaly was enconuntered in an analysis of π0 → γ +γ decay by Steinberger in 1949 and Schwinger in 1951. It was ignored for some time but resurfaced in the work of Sutherland in 1966 who showed that application of current algebra and PCAC to π0 decay gave a very small rate, in contradiction with experiment. It was Adler, and Bell and Jackiw, who later realised important role of the axial anomaly in the Sutherland paradox, and, in fact, the anomaly is sometimes called the Adler-Bell-Jackiw or ABJ anomaly. The sutherland paradox disappears when anomaly cancellation condition is applied. To have an idea of where come from the Ward identities, let’s see the following illustrative example: Imagine a proccess, as in QED, which we have X → Y + γ, where X is an initial state, Y is the ﬁnal state and γ is a photon emitted in the proccess. We can decompose the scattering amplitude in the following form

∗ µ M (X → Y + γ) = εµ × M where M µ comprises the rest of the diagram - the vertices, the internal lines and the external lines of the electron and positron. The net Mμ factor depends on the momenta of all the particles and on the spin states of the electrons and positrons, but it does not ∗ depend on the outgoing photon’s polarization — that dependence is carried by the εμ factor. As Lorentz vectors, the factors M µ are ⊥ to the photon momentum kµ:

µ kµ × M (k, ...) = 0 (D.1) These relations are known as Ward identities. We can see that, due that in QED we µ µ have the current conservation, ∂µJ = 0, which in momentum space is kµJ = 0. For the other hand, the current J µ is inside of the amplitude M µ(k, ...), then we found that the relation D.1 is accomplished. Ward Identities provides the gauge invariance of the scattering amplitude despite the gauge-dependence of the photonic polarization vector εµ(k, λ). Ward identities are essential to proving the renormalizability of a gauge theory. We see that the photonic polarization vector appear in the expresion for the electromagnetic potential

d3k 1 X A (x) = exp (−ikx) ε (k, λ)ˆa + exp (+ikx) ε∗ (k, λ)ˆa† µ ˆ 3 2w µ k,λ µ k,λ (2π) λ=±1

90 Also, we know for the invariance of gauge transformation of QED, that the EM potential satisface Aˆµ(x)[in one gauge] − Aˆµ(x)[in another gauge] = ∂µ some Λ(ˆ x) we must have εµ(k, λ)[in one gauge] − εµ(k, λ)[in another gauge] = kµ some Λ(ˆ x)

µ Consecuently, the gauge invariance of the scattering amplitude M = εµ(k, λ) × M is µ only when we have the Ward identity satisfaced kµ × M (k, ...) = 0. µ ¯ µ In the case of quiral theories, we have currents JAxial ∼ ψγ γ5ψ, where they do not conserve. An expansion of the axial vertex is shown in the Fig. D.1. What is found in the calculus, is that the last graph in the expansion, the fermion triangle, fails to satisfy the axial Ward identities, giving rise to the so-called axial, or chiral, or triangle anomaly [14]. The only way of saving renormalizability is to ensure that the total contribution of the triangle graphs is zero, so the anomalies cancel. This is a condition on the fermion content of the theory

Figure D.1.: Expansion of the axial vertex

Let’s have a global look to the analisis of the anomalies in the case of a theory. Fig. D.2 shows the contributions to the fermion amplitude which is given by

Tkλµ (p1, p2) = Skλµ (p1, p2) + Sλkµ (p1, p2)

91 where, using the Feynman rules, the amplitud for the left diagram in the Fig. D.2 (the right-side diagram is obtained from the ﬁrst by interchanging k ↔ λ and p1 ↔ p2),

4 3 d k i i i Skλµ (p1, p2) = − (−i) T r γk γkγk γk ˆ (2π)4 k − p − m k + p − m k − m

Figure D.2.: Contributions to the fermion amplitude

where has been ignored the coupling constants. After a detailed analisis using the amplitud Skλµ (p1, p2), this is, constructing the relations that will give us the information of the Ward identities for the triangle graphs, we obtain the following equations

0 p1Tkλµ = 0 λ 0 p2 Tkλµ = 0 1 (p + p )µ T 0 = ε pµpν 1 2 kλµ 2π kλµν 2 1 where 1 T 0 = S (p , p ) + S (p , p ) + ε (p − p )ν kλµ kλµ 1 2 λkµ 1 2 2π kλµν 1 2 0 The amplitude of the triangle graph has been redeﬁne Tkλµ → Tkλµ. Therefore, what we can see is the following. The vector Ward identities are now satisfaced, whereas the axial Ward identity cantains an anomaly, which no method of regularization can avoid. The fact that the axial Ward identitie contains an anomaly indicates that the axial current 5 Jµ is not conserved, i.e., in the massless case, there should be an additional term in the 5 expression for Jµ:

e2 ∂ J µ = 2mJ + F F˜µν µ Axial 5 8π2 µν

92 µν where F˜ is the dual of Fµν and their product is deﬁned as 1 F F˜µν = ε (∂µAν − ∂νAµ)(∂ρAσ − ∂σAρ) µν 2 µνρσ Chiral color models which we are dealing (see section 4) require the existence of extra fermions to cancel anomalies. We will assume that it is always possible to set the extra fermions arbitrarily heavy.

93 E. Parity

Parity is an operation that involves a transformation that changes the algebraic sign of the coordinate of an object: ψ(x, y, z) → ψ(−x, −y, −z). Objects as vectors, axialvectors, scalars, etc., depending in the space that they are inmerse, transform differently under this operation. For example, in three dimensions, we say that vectors have negative parity because all their components change sign, and therefore, after the operation we left with the negative vector: P~aˆ = Pˆ (ax, ay, az) = (−ax, −ay, −az) = −~a. To this vectors we called them “vectors” or “polar vectors”. As opposed, the cross product has positive parity because P~cˆ = Pˆ(~a × ~b) = (−~a) × −~b = ~a × ~b = ~c. These kind of objects are called axial or pseudo-vectors. Notice that the cross product of a polar vector and a pseudovector will be a polar vector. In physics, the momentum of a particle in three dimensions is a polar vector, meanwhile the angular momentum, and therefore, the magnetic field is a pseudovector or an axialvector. In addition, scalars have positive parity and pseudoscalars have negative parity. As the dot product of two polar vectors does not change the sign, we call to the result “scalar”, meanwhile the dot product between a polar vector and a pseudovector (or the triple porduct a · (b × c)) does change sign, and so, the result is a pseudoscalar. The terminology extends very simple to special µ 0 relativity: a = a ,~a is called a pseudovector if its spatial components consitute a pseudovector P~a = ~a; p is a pseudoscalar if it goes to into minus itself under spatial inversions P (p) = −p. In quantum field theory, each state is an eigenstate of the parity operator. The parity 2 group forms the Abelian group Z2 and consist of two elements: P and I = P . As we saw in the first part, the parity operator have two eigenvalues: ±1. In particle physics, the operator P should reverse the particle momentum without flipping its spin. This is an inspiration from quantum mechanics in the contexts of bound states. Let’s think in a bound state having a wave function which is symmetric under ~x → −~x, we say that the state has even parity, while if it is antisymmetric under ~x → −~x, the state has odd parity. The L = 0 bound states, for example, have odd parity; the J = 0 transform as a pseudoscalar, while the three J = 1 states transform as the spatial components of a vector. Consider the π− decay. We know that the pion has spin 0, so the muon and the antineutrino spins must be oppositely aligned. Mesurements of the µ− show always that it is right-handed, which implies that the antineutrino must be right-handed too. By the same token, in π+ the antimuon is always left-handed, and this indicates that the neutrino is left-handed. It is not difficult to see that these processes do not respect parity because the mirror image does not exist in nature. In experiments only have been seen left-handed neutrinos and ritght handed antineutrinos. For contrast, if we consider the dacay of the neutral pion π0 → γ + γ. In this case, the photons must to have the same

94 helicity. But this is an electromagnetic proccess, and this kind of proccess respect parity, and thus, on the average, we get just as many right-handed photon pairs as left-handed pairs. Therefore, because neutrinos do not have charge, they interact only through the weak force, and so, weak interactions do not respect parity. It is interesting to think if neutrinos have mass (although it would be pretty small), right-handed neutrinos must exist because the fact of having mass implies that they travel at lower speeds than light. In that case, the question is: where are they?. Spin-parity of some commonly known particles:

State Spin Parity Particle Pseudoscalar (0−) 0 - π, K + Scalar (0 ) 0 + a0, Higgs Vector (1−) 1 - γ, ρ, w, φ, ψ,Y + Pseudovector (axialvector) (1 ) 1 + a1

Table E.1.: Clasiﬁcation of some particles according to their spin-parity.

In QFT we are dealing with currents and they have to be consistent with the Lorentz covariance. Currents have the form ψ¯ (4 × 4) ψ which have deﬁnite properties under Lorentz transformations, where the (4 × 4) matrix is a product of γ- matrices. Dirac spinors transform under a parity transformation as

ψ → ψ0 = γ0ψ

It follows that

† ψψ¯ 0 = ψ0† γ0ψ0 = ψ† γ0 γ0γ0ψ0 = ψ†γ0ψ = ψψ¯ so ψψ¯ is invariant under P : It means that we will consider the object ψψ¯ as a scalar. Also, one can show that ψγ¯ 5ψ under a parity transformation behaves as a pseudoscalar

0 ψγ¯ 5ψ → ψγ¯ 5ψ = − ψγ¯ 5ψ

The object ψγ¯ µψ, under a P transformation, behaves as a four-vector because one can show, in the same way than before, that

0 ψγ¯ 0ψ = ψγ¯ 0ψ 0 ψγ¯ iψ = − ψγ¯ iψ with i = 1, 2, 3. In the following we show the posibles bilinear constructed. In Table E.2 it is shown some bilinears structures and their respective parities.

95 ψψ¯ scalar one component + under P ψγ¯ 5ψ pseudoscalar one component - under P ψγ¯ µψ vector four components Space compts.: - under P ψγ¯ µγ5ψ pseudovector four components Space compts.: + under P ψσ¯ µνψ antisymmetric tensor six components

Table E.2.: Some possibles bilinear covariants

In strong interactions parity is conserved. However, in this thesis we are dealing with the hypotetical particle called “axigluon”. It appears in the Lagrangian coupled to quarks in the form

µ ¯ µ 5 AµJAxial = ψAµγ γ ψ (E.1)

Taking into account that the axigluon ﬁeld Aµ is equivalent, in the Lorentz sense, to the Gluon, axigluon is a vector particle. So, the term E.1 is parity-violating because we have µ the product of a vector particle Aµ and a pseudovector current JAxial. Therfore, the fact of include color-octet vector bosons which couple to quarks in an axial way, means that there would be a parity-violating in the strong sector as well as in the weak sector. I think that there is no a fundamental principle to demand parity conservation at the TeV scale, but, experiments will say the ﬁnal word about this topic.

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