BCJ-relations

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BCJ-relations

Gravity as the square of Yang-Mills Theory

Amsterdam, 07/07/2017

Author

Pieter van der Hoek

10747834

Universiteit van Amsterdam Amsterdam

Report Bachelor Project Physics and Atsronomy

Size 15 EC

Conducted between 03-04-2017 and 07-07-2017 at University of Amsterdam

Commissioned by Nikhef

Supervisor Prof. Dr. E. Laenen Second assessor Prof. Dr. A. Castro

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Abstract

In this thesis it is explained at an undergraduate level how to derive simple expres-sions for gravity amplitudes. This is being done by first calculating and comparing QED amplitudes and QCD amplitudes. After that, color is separated from dynamics in the QCD amplitude and BCJ-relations are then used to find a general expression for tree level amplitudes in Yang-Mills theories. By qualitatively introducing the KLT-relations for gravity, it is found that there might be some connection between Yang-Mills theories and gravity theories. Eventually, we define a formula to calculate gravity amplitudes in a compact way. If this formula holds, it might as well connect gravity to the Standard Model.

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Populairwetenschappelijke samenvatting

In de natuurkunde onderscheiden we op het niveau van elementaire deeltjes vier fundamentele krachten: de elektromagnetische kracht, de zwakke kernkracht, de sterke kernkracht en de zwaartekracht. Van deze krachten zijn de eerste drie verenigbaar in het zogeheten Standaard Model. Theoretici in de natuurkunde zijn er vooralsnog niet in geslaagd zwaartekracht aan dit Standaard Model toe te voegen.

In deze scriptie heb ik gekeken naar amplitudes voor reacties tussen elementaire deeltjes. Deze amplitudes zijn voornamelijk wiskundige objecten, die gekwadrateerd kunnen wor-den, om een maat te geven voor de kans dat een bepaalde interactie verloopt. Amplitudes zijn heel waardevol voor het testen van theorie¨en. Door een proces namelijk heel vaak te laten verlopen kan men zien of de eerder genoemde kanswaarde wordt gevonden. Als een kanswaarde zich verhoudt tot een statistisch gevonden frequentie dat een proces vol-gens een bepaalde interactie verloopt, dan is deze kanswaarde waarschijnlijk op de juiste manier berekend. In dit geval wordt een theorie volgens welke de amplitude was berekend aannemelijker.

Ik heb in deze scriptie eerst amplitudes uit de theorie van de kwantumelektrodynam-ica (QED) en van de kwantumchromodynamkwantumelektrodynam-ica (QCD) behandeld. QED beschrijft de elektromagnetische interacties en QCD beschrijft de interacties van de sterke kernkracht. Nadat ik voor beide theorien een amplitude had berekend voor een proces in iedere the-orie, was te zien dat deze amplitudes veel met elkaar gemeen hadden. Vervolgens heb ik me erop gericht om de QCD-amplitude te herschrijven en zo deze overeenkomst tussen QED en QCD meer naar voren te laten komen. Hiervoor heb ik gebruik gemaakt van de BCJ-relaties. Uiteindelijk kon ik, gebruikmakend van deze BCJ-relaties, een zeer een-voudige uitdrukking vinden voor QCD-amplitudes.

In het laatste deel van deze scriptie heb ik gekeken naar zwaartekrachtsamplitudes. Nu zijn zwaartekrachtsamplitudes normaal gesproken heel moeilijk te berekenen, maar er zijn vereenvoudigingen mogelijk. Gebruikmakend van zogenaamde KLT-relaties kon ik bijvoorbeeld een eenvoudige uitdrukking vinden voor de zwaartekrachtsamplitudes. Uit de gevonden uitdrukking bleek dat ook de zwaartekrachtsamplitudes een onderliggend verband lijken te hebben met de QCD-amplitudes die ik eerder had berekend. Waar dit verband vandaan komt is nog niet bekend en of dit verband in alle zwaartekrachtspro-cessen standhoudt, is niet bewezen. De eerste vooruitzichten laten echter zien dat dit waarschijnlijk wel het geval is. Wellicht geeft deze formule het uiteindelijke inzicht om zwaartekracht te verenigen met het Standaard Model, maar dat zullen theoretici in de komende jaren moeten gaan onderzoeken.

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1 Introduction

This thesis is designed to be understandable for students which are in their second year of their bachelor degree in physics and astronomy. Readers who are on a higher lever of education, might want to skip the first and second chapter. While reading this thesis, the reader is in a way being taken along a journey through time. In chapter 2 we start by calculating QED-amplitudes in the traditional way by using Feynman rules. This was the normal approach in the middle of 20th century. In chapter 3 we do the same for QCD, as this was being done some decades later [1]. In chapter 4 we direct our attention towards an overal generalization of the derived QCD amplitude and find out that this can be done by using BCJ-relations, after their founders, Bern, Carasco and Johansson. This was firstly being done at the beginning of this century [2]. In chapter 5 we concentrate on simplifying gravity calculations, thereby arriving at the KLT-relations, as first being proposed by Kawai, Lewellen and Tye in 1989 [3]. In the final chapter of this thesis we will combine all former information to give one an insight into the questions currently being asked at the very fron-teer of theoretical physics. This thesis yields nothing new, yet it provides an unique way for undergraduates to dive deep into theoretical physics. As for this section, we will first take a look into the meaning of scattering amplitudes. In this way I hope to show the reader why theories like these described at the end of this thesis are of great importance in the field of theoretical and particle physics. The aim of this chapter is thus to give a first insight into the type of research we are doing here and to give those who are new to the field a tool to relate this thesis to their current knowledge. This chapter will be very specifically di-rected towards the subject of this thesis, so I won’t cover the whole story. Therefore, I would like to refer to ref. [1], if one would like a broader introduction into the field of particle physics.

1.1 Different particles

Particle physics is all about the interactions between subatomic particles. A broad group of researchers has been and still is searching for the processes which describe the universe at its smallest. In the decades of research which currently lie behind us, one of the greatest triumphs of physics is the Standard Model, which categories the different types of particles in a very understandable way.

In short there are bosons and fermions. Bosons can be divided into gauge bosons with spin 1, which carry one of the four fundamental forces and the Higgs Boson with spin 0. The electromagnetic force is carried by the photon (γ). The photon couples only to particles with electromagnetic charge, but has no electromagnetic charge on its own. The weak interaction

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is carried by either the W±-bosons or the Z0-boson, where the superscript denotes the elec-tromagnetic charge of the particle. The weak-interaction is the only force that changes the flavor of the particles within a generation via the W±-bosons. It’s also unique in the sense that the masses of its bosons are quite big compared to the other massless gauge bosons. The last gauge boson is the gluon, mostly denoted with just a g, which carries the strong inter-action. This boson couples to all particles with color charge (red (r), blue (b) and green (g), and their anti-charges: r, b and g). There are actually 8 different gluons with the following charges: rr,rg, rb, gr, gg, gb, br, bg and bb. The Higgs Boson is not of interest in this thesis, but this scalar boson assigns mass to other particles.

The fermions, all with spin 1/2, can be further categorized into quarks and leptons, all with spin 1/2. Quarks carry both color and electromagnetic charge and can appear in different flavors, therefore quarks interact with all gauge bosons. Quarks come in three different gener-ations. The first generation is formed out of the up and the down quark with electromagnetic charges +2/3 and -1/3. The second heavier generation is formed out of the charm and the strange quark carrying the same charges, the heavy top and bottom quark form the third generation. These six different quarks are called the six different flavors of the quark. In-cluding color charge, there are thus 18 different quarks. The group of leptons also consist of 6 different flavors, categorized in three different generations. Each generation consist of a negatively charged particle, the electron (e−), muon (µ−) or tau (τ−) and their correspond-ing neutrino’s with no electric charge (νe, νµ and ντ). Leptons do not carry color charge and

therefore won’t interact with gluons.

For every fermion there is a corresponding anti-particle with opposite charge. All negatively charged particles thus have positively charged anti-particles as counterparts and vice-versa. The anti-color charges are named above as belonging to gluon next to the color charges. Quarks and anti-quarks can be combined into hadrons. These hadrons consist out of three particles (baryons) or two particles (mesons). All hadrons have an integer electromagnetic charge.

The particles listed above have all been discovered in experiments and therefore, the Standard Model has proven to work as a description of the microcosmos, however it is not complete. Of particular interest for this thesis is for example the not (yet) discovered hypothetical graviton with predicted spin 2. This particle would couple to all particles with (relativistic) mass, in-cluding itself. On large scales, the properties of the graviton should result in the phenomena as described by the General Theory of Relativity. However, a broadly accepted theory has not been found yet, able to describe both the micro- and macrocosmos. Many more particles have been proposed as extensions to the Standard Model, but these are of no further interest for this thesis [1].

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1.2 Interactions

After defining the elementary particles and their properties as the tools we need in this thesis, we can take a look at how they interact. After all, physics is all about interactions. Predictions about these interactions can later be tested to either confirm or deny a theory. Now, as can be seen from the properties of the different particles, a simple reaction equation with initial and final particles can happen in infinitely many ways. For example:

qq → qq (1)

The particles could just scatter and exchange momenta via one of the gauge bosons quarks interact with, or annihilate completely to form one of the allowed gauge bosons, which will later decay into the same pair as found in the initial state. Furthermore, the bosons could in the meantime also decay into a fermion-antifermion pair which will shortly live to annihilate again some time later to continue as the initial boson. Also, gluons can interact with them-selves and will therefore be able to split up into two (or even three) gluons, to annihilate a little time later to form the same gluon as initially. This later proces can be accounted for using so-called loop diagrams. More possibilities come into view when considering the flavors of the different particles. The quarks and antiquarks themselves could be of all kind of flavors, since its not defined in the reaction equation, which quarks are involved in the proces. Luckily for physicists not all interactions are equally likely to occur. So, in order to make any predictions about the relative likelihood of equation 1 to occur compared to for exam-ple the likelihood of the reaction qq → e−e+, one is able to calculate the so called absolute value of the amplitude squared (|A|2). This value can be used to make physical predictions, such as cross-sections and decay rates. The amplitude itself is composed of the sum of all subamplitudes of the possible interactions via which the final state particles can be created. This thesis is concerned with the calculation of these amplitudes only on tree level, so no loop diagrams are involved in the calculations.

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2 Feynman Calculus in QED

In order to calculate scattering amplitudes, one needs a thorough understanding of both special relativity and quantum mechanics. Only if one is able to combine both fields of physics in a successful way, one will be able to get numerical results or say anything useful about the likelihood of some interactions. Luckily, these problems can be mathematically simplified by making use of Feynman diagrams and letting the so called Feynman rules act on them. Take for example the Feynman diagram to describe reaction 1 via annihilation to a photon and pair production: q q q q γ

In a Feynman diagram the initial state particles in a reaction are on the left hand side of the diagram, whereas the final state particles are on the right hand side of the diagram. Note however, that apart from this, there is no time-ordening in the Feynman diagram. The particles that exist between the initial and final state are therefore often regarded as virtual particles. Feynman rules make sure that the mathematical description of these particles sum over all time orderings. In order to calculate the amplitude corresponding to a certain Feynman diagram, one assigns a particular value to all ingoing, outgoing and internal lines as well as to the different vertices in a diagram. We will first present the Feynman rules for spinless quantum-electrodynamics (QED) from the appropriate Lagrangian. At the end of this section we will take a look at a particular example to show the stunning elegance of this method. Furthermore, I would recommend reading appendix A about the basics of special relativity to understand the notation we use.

2.1 Deriving the QED Feynman rules for scalar particles

In this section we will derive the Feynman rules from the QED Lagrangian for suppressed spin states. For QED we will in this section only consider an electromagnetic interaction between photons and electrons. The Lagrangian we will use in this section is [4]:

L = −1 4F 2 µν+ |Dµφ|2− m2|φ|2 = −1 4F 2 µν− φ ∗ ( + m2)φ − ieAµ[φ∗(∂µφ) − (∂µφ∗)φ] + e2A2µ|φ|2 (2)

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In the last line we used D = Dµφ = ∂µ+ ieAµφ. In equation 2, Fµν2 = FµνFµν is just the

electromagnetic field strength squared in absence of any sources, where we define for the field strength:

Fµν = ∂µAν − ∂νAµ (3)

Here Aµand Aν denote the electromagnetic four-potential .The φ and φ∗ in equation 2 define

the complex scalar field and its complex conjugate. These are fields from electromagnetic sources and for our purposes come from electrons [4].

Now, there are four terms in the Lagrangian from equation 2. First of all, there are terms which are quadratic in the fields. These terms give rise to propagator terms, which are internal lines in the diagrams [5]. For our purposes, we are only interested in the photon propagator (there is also a fermion propagator), for which the Feynman rule in our case turns out to be [6]:

Aµ _{q →} Aν =

−igµν

q2 (4)

Here gµν is just the Minkowski metric and q is the four-momentum of the interchanged photon.

The next terms we see, are cubic and quartic terms were both types of fields are involved. These terms account for vertices in the diagrams. Let us first consider the first term:

ieAµ[φ∗(∂µφ) − (∂µφ∗)φ] (5)

This term couples Aµ to two fermion fields φ and φ∗ with the electromagnetic coupling

constant e. Now, the derivative terms times the scalar fields φ and φ∗ accounts for the fact that that there is a four-momentum ipµ in the vertex terms for every fermion line involved in

the QED-vertex. We eventually find the following term for the QED vertex [4]:

e−₁ e−₂ γ p1 p2 = −ie(p1+ p2)µ (6)

Here, the arrows denote the charge flow, which means as much as: charge × momentum of the particle. In general an arrow to right gives +pi in the vertex term and an arrow to the

left gives −pi.

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following [4]:

= 2ie2gµν (7)

However, since we will only look at lowest order diagrams for e−e−→ e−e−, we will not need the interaction term for this vertex.

Eventually, one should also have a term for all external lines. However, these external lines are in our cases from scalar fields and will just account for an extra factor of 1 times the total amplitude. The kinetic information of their four-momenta is already involved in the 3-vertices [4].

2.2 Calculating the amplitude for e−e−→ e−_e−

Let us start with a particular example: Consider electron scattering (e−e− → e−_e−_).

At lowest order in QED there are two possible ways this interaction can happen, through u-channel scattering and through t-u-channel scattering, consider the following Feynman-diagrams:

e−₁ e−₃ e−₄ e−₂ γ (a) t-channel (A1) e−₁ e−₄ e−₃ e−₂ γ (b) u-channel (A2)

The scattering amplitude A is the sum over the subamplitudes A1 and A2. Using Feynman

rules and the appropriate diagrams, we first calculate iA1 and iA2. Now the total scattering

amplitude through the t-channel can be shown to be [6]:

−iA1 = ie(p1+ p3)µ −igµν q2 ie(p2+ p4)ν = ie(p1+ p3)µ _−ig µν (p4− p2)2 ie(p2+ p4)ν = ie2(p1+ p3)µ(p2+ p4) µ (p4− p2)2 (8)

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For the u-channel scattering, in a similar way, one finds [6]:

−iA2= ie2

(p1+ p4)µ(p2+ p3)µ

(p3− p2)2

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The total amplitude will now be:

−iA = −i(A₁+ A2) = −ie2

−(p1+ p3)µ(p2+ p4) µ (p4− p2)2 − (p1+ p4)µ(p2+ p3) µ (p3− p2)2 = −i(A1+ A2) = −ie2 − p1p2+ p1p4+ p3p2+ p3p4 (p4− p2)2 − p1p2+ p1p3+ p4p2+ p4p3 (p3− p2)2 (10) This is were the Mandelstam variables come in. Rewriting equations 104, 105 and 106 from appendix A.3 and using p2_i = m2_e with i = 1, 2, 3 or 4, one is able to find:

2p1p2= s − p21− p22 p1p2= s 2− m 2 e = p3p4 (11) 2p1p3= −t + p21+ p23 p1p3= − t 2 + m 2 e= p2p4 (12) 2p1p4= −u + p21+ p24 p1p4= − u 2 + m 2 e= p2p3 (13)

Using these definitions, we can now further simplify the scattering amplitude to be:

−iA = −i(A1+ A2) = −ie2 − p1p2+ p1p4+ p3p2+ p3p4 (p4− p2)2 − p1p2+ p1p3+ p4p2+ p4p3 (p3− p2)2 = −ie2 − s 2 − m 2 e−u2 + m 2 e−u2 + m 2 e+ s2− m 2 e (p4− p2)2 − ie2 − s 2 − m2e−2t + m2e−2t+ m2e+s2 − m2e (p3− p2)2 = −ie2 − s − u t − s − t u A = e2 u − s t − t − s u (14)

This expression is quite simple, it is apart from e2 purely kinematic in its information. In the next chapter we will show that it has some resemblance with a typical QCD amplitude, regarding this kinematic information.

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3 Getting used to QCD

In the previous chapter we showed the power of the kinematic Mandelstam variables in simplifying expressions for scattering amplitudes. The purpose of this chapter is to show that we can use these variables to simplify the scattering amplitudes in the much more complicated case of quantum chromodynamics too. In this section we will not suppress helicity states anymore and use the full Feynman apparatus to calculate the tree amplitude for 22 gluon scattering (gg → gg). The difficulty compared to QED lies in the fact that there are extra degrees of freedom, because of the color charge. This is something we cannot ignore in our discussion of calculating amplitudes.

3.1 Color charge

QCD describes the interactions between particles that carry color charge. As could be read in chapter 1, these particles are gluons and quarks. Quarks are able to carry three different color charges: red(r), blue(b) and green(g) and thus come in three different states [1]: |ri =     1 0 0     , |bi =     0 1 0     , |gi =     0 0 1     . (15)

These are just unit basis vectors for a 3-dimensional space. Color charge is a conserved quantity in particle interactions. According to Noether’s theorem this quantity thus belongs to a certain symmetry. In this case, we are looking at SU(3)-symmetry. For more information about these symmetries and how they form a group, one should check the appendix B. The generators (T ) of this symmetry are related to the so called Gell-Mann matrices, denoted by λi, by T =

√

2λi. The Gell-Mann matrices are [7]:

λ1 =     0 1 0 1 0 0 0 0 0     , λ4 =     0 0 1 0 0 0 1 0 0     , λ6 =     0 0 0 0 0 1 0 1 0     , λ2 =     0 −i 0 i 0 0 0 0 0     , λ5 =     0 0 −i 0 0 0 i 0 0     , λ7 =     0 0 0 0 0 −i 0 i 0     , λ3 =     1 0 0 0 −1 0 0 0 0     , λ8= 1 √ 3     1 0 0 0 1 0 0 0 −2     . (16)

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These matrices play a similar role in SU(3)-symmetry as Pauli matrices do in SU(2)-symmetry, which can be checked in appendix B.3. When looking at the commutation relations of the generators, one finds [8]:

[Ta, Tb] =√2ifabcTc (17)

The structure constants (fabc) of the SU(3) group are completely antisymmetric [7]:

fabc= −facb= −fbac (18)

Furthermore, when calculating all 83 possible combinations of the indices, one will find only 9 non-zero structure constants [7]:

f123= 1, f147 = f246= f257= f345 = f516= f637 = 1 2 and f 458_{= f}678₌ √ 3 2 (19)

3.2 Feynman rules for QCD

We now have enough information to read the Feynman rules for QCD. In a similar way as in chapter 2, these can be derived from a Lagrangian. The full Lagrangian for QCD is more complicated than the one we used in the previous chapter [9]:

L = −1 4(F a µν)2− 1 2ξ(∂µA µ₎2_{+ ∂} µcaDµca+ X f Ψif(i /D + mf)Ψif −δ3 4(∂µA a ν− ∂νAaµ)2+ gδ (3g) 1 fabcAbµAcν∂µAaν− g2δ₁(4g) 4 (f abc_Ab µAcν)2 + δ₂(gh)∂µca∂µca− gδ1(gh)f abc_∂ µcaAbµcc +X f Ψif(iδ (qf) 2 ∂ + δ/ (qf) m − gδ(q₁f)A/ata)Ψif. (20)

I am not going to elaborate on each term in the Lagrangian, since that would not be relevant for this thesis. We will first of all ignore the quark-related counter terms on the last line, as well as the quark terms involving Ψ and Ψ, since we are only interested in pure gluon scattering. I am also not going to elaborate on the ghost terms, denoted by the superscript gh. Most important now, are the terms including the superscripts 3g and 4g. These cubic and quartic terms result in a 3-vertex of gluons and and a 4-vertex of gluons [9].

For our purposes, the Feynman rules we need from the Lagrangian are listed in this section. Firstly, one finds for the gluon propagator [8]:

α, µ β, ν

q → = −i

gµνδαβ

q2 (21)

Here we used the Kronecker delta for the first time in this thesis, which states that α = β in this expression for the propagator. Furthermore, the α and β denote the color of the gluon

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involved, whereas the µ and ν are used as the four-vector indices of space-time. Now for the three gluon vertex, with all momenta (pi) of the gluons incoming, one finds [8]:

p3, γ, λ p1, α, µ p2, β, ν = −gsf αβγ_[g µν(p1− p2)λ+ gνλ(p2− p3)µ+ gλµ(p3− p1)ν] (22) For the four gluon vertex, the following term is found with again all momenta incoming [8]:

γ, λ α, µ β, ν δ, ρ = −ig_s2     fαβηfγδη(gµλgνρ− (gµρgνλ) +fαδη_fβγη_(g µνgλρ− (gµλgνρ) +fαγηfδβη(gµρgνλ− (gµνgλρ)     (23)

Finally, each incoming and outgoing gluon, wil get an polarization state [8]: α, µ p → = µ(p) α, µ ← p = ∗ µ(p) (24)

3.3 calculation of the gg → gg amplitude

In this paragraph we calculate the gg → gg amplitude, using the methods described in ref. [8]. Now for the particular proces we are interested in, there are four different processes on lowest order tree level in which this interaction can occur: The s-channel annihilation, the t- and u-scattering process and the four gluon interaction, or in diagrams:

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Using the Feynman rules of the last paragraph, this leads to the following expression for A1 [8]: iA1 = [µ(p1)ν(p2)∗µ0(p₃)∗_ν0(p₄)]g_s2fabcfa 0_b0_c0 [gµν(p1− p2)λ+ gνλ(p2+ q)µ+ gλµ(−q − p1)ν][ −ig_λλ0δcc 0 q2 ] [gµ0ν0(−p1+ p2)λ 0 + gν0λ0(−p2− q)µ 0 + gλ0µ0(q + p1)ν 0 ] (25)

Or written out, using q2 = (p1+ p2)2 = (p3+ p4)2 = s and replacing the terms with

indices by products over the appropriate vectors:

iA1 = [ −ig2 sfabcfa 0_b0_c s ]{−(p1) · (p2) × (p2− p1) 2_×∗_(p 3) · ∗(p4) + (p1+ 2p2) · (p1) × (p2− p1) · (p2) × ∗(p3) · ∗(p4) − (2p1+ p2) · (p2) × (p2− p1) · (p1) × ∗(p3) · ∗(p4) + (p1) · (p2) × (p1+ 2p2) · ∗(p3) × (p2− p1) · ∗(p4) − (p1+ 2p2) · (p1) × (p1+ 2p2) · ∗(p3) × (p2) · ∗(p4) + (2p1+ p2) · (p2) × (p1+ 2p2) · ∗(p3) × (p1) · ∗(p4) − (p₁) · (p2) × ∗(p3) · (2p1+ p2) × (p2− p1) · ∗(p3) + (p1+ 2p2) · (p1) × (2p1+ p2) · ∗(p4) × (p2) · ∗(p3) − (2p₁+ p2) · (p2) × (2p1+ p2) · ∗(p4) × ∗(p3) · (p1) (26)

Now, this calculation seems quite difficult at first, however due to helicity and polirzation there are some constraints. For example the polarization of a lorentz gauge boson is always transverse, so: µ(k)kµ= (k) · k = 0 [4].

Using this condition in the center of mass frame where the initial gluons collide along the z-axis and all gluons travel in the XZ-plane, we can set up the four vectors of the involved gluons and their polarizations [8]:

pµ₁ = (E, 0, 0, E) µ_R/L(p1) = √1₂(0, 1, ±i, 0)

pµ₂ = (E, 0, 0, −E) µ_R/L(p2) = √1₂(0, −1, ±i, 0)

pµ₃ = (E, Esinθ, 0, Ecosθ) µ_R/L∗(p3) = √1₂(0, −cosθ, ±i, sinθ)

pµ₄ = (E, −Esinθ, 0, −Ecosθ) µ_R/L∗(p4) = √1₂(0, cosθ, ±i, −sinθ)

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θ is the angle between p4 and p1 or p3 and p2. Here the capital R/L denotes the specific

helicity state of the particle, which is simply the spin in the direction of the momentum. R means that the spin is directed towards the momentum direction, whereas L means exactly the opposite. Since there are two helicity states per particle. In total 24 = 16 polarizations are possible for this particular reaction. However, angular momentum needs to be conserved,

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which excludes 10 possible polarization states. Furthermore, QCD is invariant under parity [1], so: A(RR → RR) = A(LL → LL) A(RL → RL) = A(LR → LR) A(RL → LR) = A(LR → RL) (28)

Including crossing symmetry and using the Mandelstam variables (s = (p1+ p2)2, t = (p1−

p3)2, u = (p1− p4)2), one finds [8]: A(RR → RR) (s,b)↔(u,b 0₎ −−−−−−−→ A(RL → RL) A(RL → RL) (u,b 0_)↔(t,a0₎ −−−−−−−−→ A(RL → LR) (29)

This means that we can derive the amplitudes of all polarization states from just the amplitude over one polarization state. Take for example A(RR → RR), using expression 26 for the s-channel amplitude and similar expressions for the other amplitudes, one finds [8]:

iA1(RR → RR) = −ig2fabcfa 0_b0_c cosθ (30) iA2(RR → RR) = ig2fab 0_c fba0c19 + 7cosθ − 11cos 2_{θ + cos}3_θ 4(1 − cosθ) (31) iA3(RR → RR) = ig2faa 0_c fbb0c19 − 7cosθ − 11cos 2_{θ − cos}3_θ 4(1 + cosθ) (32) iA4(RR → RR) = ig2[fabcfa 0_b0_c cosθ +1 4f aa0c_fbb0c_{(3 + 2cosθ − cos}2_θ)] ₍₃₃₎ +1 4f ab0c_fba0c_{(3 − 2cosθ − cos}2_θ) ₍₃₄₎ iA(RR → RR) = −2ig2[faa0cfbb0cs t + f ab0c_fba0cs u](= iA(LL → LL)) (35) Here we again made use of the Mandelstam relations of the appendix A.3, this time for massless particles, yielding:

2p1p2 = s − p21− p22 = s = 2p3p4 s = 4E2 2p1p3 = −t + p21+ p23 = −t = 2p2p4 t = 2E2(1 − cosθ) 2p1p4 = −u + p21+ p24= −u = 2p2p3 u = 2E2(1 + cosθ)

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Now using equations 28 and 29, we find for all possible polarization states: iA(RR → RR) = −2ig2[faa0cfbb0cs t + f ab0_c fba0cs u](= iA(LL → LL)) iA(LR → LR) = 2ig2[faa0cfbb0cu t + f abc_fa0b0cu s](= iA(RL → RL)) iA(LR → RL) = −2ig2[fabcfa0b0ct

s− f

ab0_c

fba0ct

u](= iA(RL → LR))

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This gives for the total amplitude:

iA = iA(RR → RR) + iA(LL → LL) + iA(LR → LR) + iA(RL → RL) + iA(LR → RL) + iA(RL → LR) = 2(iA(RR → RR) + iA(LR → LR) + iA(LR → RL)) = 2(−2ig2[faa0cfbb0cs t + f ab0_c fba0cs u] + 2ig 2_[faa0_c fbb0cu t + f abc_fa0_b0_cu s] − 2ig2[fabcfa0b0ct s+ f ab0c_fba0ct u]) = 4ig2(faa0cfbb0c(u − s) t + f ab0_c fba0c(t − s) u + f abc_fa0_b0_c(u − t) s ) A = 4g2(faa0cfbb0c(u − s) t + f ab0c_fba0c(t − s) u + f abc_fa0b0c(u − t) s ) (37)

Now, let me pause for a moment. This amplitude seems to be very familiar to the electromagnetic one derived in the previous chapter. There is a coupling constant raised to the power 2 and apart from the color factors in expression 37, both amplitudes depend strongly on kinetic variables. Therefore, one can wonder wether it is possible to separate the color factors from the kinetic variables. In the following chapter I will show that this indeed the case.

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4 The duality between kinetmatic and color factors

In this chapter we will direct our attention towards the color factors. As it turns out, there is more to them then we just derived in the previous chapter. At the end of this chapter we will be able to show that more or less the same goes for the kinematic factors. This underlying connection will turn out to be very useful in eventually calculating amplitudes. When trying to derive the relations of this section, it is important to note that legs 3 and 4 are interchanged in typical amplitudes relative to the former chapters. This is done in order to state t = (p1+ p4)2 and u = (p1+ p3)2.

4.1 Separating color from kinematics

In chapter 3 we found an expression for the gg → gg tree amplitude (equation 37). It became clear that the amplitude seems to be a sum over several colour factors times a kinetic part. In this paragraph we will show that it is possible to separate color from kinematics and construct a general sum expression to calculate QCD amplitudes. This starts at the level of Feynman rules. Let us first rewrite Feynman rules for the three gluon vertex[10]:

p3, c, λ

p1, a, µ

p2, b, ν = −gsfabc[gµν(p1− p2)λ+ gνλ(p2− p3)µ+ gλµ(p3− p1)ν]

= gs(ifabc)(i[gµν(p2− p1)λ+ gνλ(p3− p2)µ+ gλµ(p1− p3)ν])

= gs(ifabc)(iV3µνλ(p1, p2, p3)) (38)

In the last line we defined the purely kinetic tensor element V₃µνλ. At first, this does not seem like much progress, but there is an expression which we can use to further generalize equation 38, namely by using equation 17. This equation can be used by multiplying both sides with Td and taking the trace of both sides, giving [11]:

[Ta, Tb] = i√2fabcTc Tr(TdTaTb− Td_Tb_Ta_{) = i}√_2fabc_Tr(Td_Tc₎

ifabc= √1 2Tr(T

a_[Tb_{, T}c_]) ₍₃₉₎

Where we used the cyclic invariance of traces and the following property in the last line:

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When using the antisymmetry of V₃µνλ upon exchanging two legs, we find[10]:

gs(ifabc)(iV3µνλ(p1, p2, p3)) =

gs √ 2[Tr(T a_Tb_Tc_)iVµνλ 3 (p1, p2, p3) − Tr(TaTcTb)iV3µνλ(p1, p2, p3)] = √gs 2[Tr(T a_Tb_Tc_)iVµνλ 3 (p1, p2, p3) + Tr(T a_Tb_Tc_)iVµλν 3 (p1, p3, p2)] = √gs 2 X P(2,3) Tr(Ta1_Ta2_Ta3_)iVµ1µ2µ3 3 (p1, p2, p3) (41)

In the last step the indices are renamed for a more compact expression. We sum over the permutations of legs 2 and 3.

Using the Fierz identity: (ifabe)(ifcde) = 2[Tr(TaTbTcTd)−Tr(TaTbTdTc)−Tr(TbTaTcTd)+ Tr(Tb_Ta_Td_Tc_{)], we can find an expression for the four gluon vertex in a similar way [10]:}

c, λ

a, µ

b, ν

d, ρ

= −ig2s

2 [fabefcde(gµλgνρ− gµρgνλ) +facefbde(gµνgρλ− gµρgνλ)

+fadefbce(gµνgρλ− gµλ_gνρ_)]

= g

2 s

2 [(if

abe_)(ifcde_)(gµλ_gνρ_{− g}µρ_gνλ_{) + (if}ace_)(ifbde_)(gµν_gρλ_{− g}µρ_gνλ₎

+ (ifade)(ifbce)(gµνgρλ− gµλgνρ)] = g 2 s 2 X P(2,3,4) Tr(Ta1_Ta2_Ta3_Ta4_)i(2gµ1µ3_gµ2µ4 _{− g}µ1µ4_gµ2µ3_{− g}µ1µ2_gµ3µ4₎ = g 2 s 2 X P(2,3,4) Tr(Ta1_Ta2_Ta3_Ta4_)iVµ1µ2µ3µ4 4 (42)

In general, using again the Fierz identity to absorb the quartic 4-gluon vertex term into the cubic 3-gluon terms, QCD-amplitudes on tree level can now be written as follows [10]:

A_n(pi, λi, ai) = gsn−2

X

σ∈Sn−1

Tr(Ta1_Taσ(2)_{. . . T}aσ(n)_)A

n(σ(1)λ1, σ(2)λ2, . . . , σ(n)λn) (43)

This expression is a sum over Sn−1 permutations of the n external particles. This means the

first external particle is fixed, since unfixing it would not add anything new to the probability amplitude. pn and λn denote the respective momenta and helicity states of the external

particles. An are the so-called color ordered partial amplitudes. These amplitudes consist

as sum only over planar diagrams (s- and t-channel). They can be created by the following colour ordered Feynman rules for the interaction vertices, where the 4-vertex term is actually

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not needed[10]: p3, λ p1, µ p2, ν = i[gµ1µ2(p1− p2)µ3 + gµ2µ3(p2− p3)µ1 + gµ3µ1(p3− p1)µ2] (44) λ µ ν ρ = i(2gµ1µ3_gµ2µ4 _{− g}µ1µ4_gµ2µ3 _{− g}µ1µ2_gµ3µ4₎ ₍₄₅₎

We now successfully found an expression to separate color and kinetic variables. The color ordered partial amplitudes will be of particular importance in the rest of this thesis. Four our purposes we will from now on suppress helicity and polarization states, yielding [2]:

Atree_n (1, 2, 3, . . . , n) = gn−2_s X

P(2,3,...,n)

T r(Ta1_Ta2_Ta3_{. . . T}an_)Atree

n (1, 2, 3, . . . , n) (46)

4.2 Color ordered partial amplitudes

In this subsection we will solely focus ourselves on the color ordered partial amplitudes. As it turns out, there are some important features which we can use in the following chapters. Let me first of all begin with the fact that the colour ordered partial amplitudes have a few interesting properties [10]. There is first of all the cyclic property:

Atree_n (1, 2, 3, . . . , n) = Atree_n (2, 3, . . . , 1) (47)

This expression was actually used to fix the first leg in equation 43. Next, there is also a reflection property:

Atree_n (1, 2, 3, . . . , n) = (−1)nAtree_n (n, . . . , 3, 2, 1) (48)

We find a ”photon-decoupling” identity, where the sum runs over all cyclic permutations of legs 2, 3, . . . , n:

X

σ∈cyclic

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And finally we find the Kleiss-Kuijf relations:

Atree_n (1, {α}, n, {β}) = (−1)nβ X {σ}i∈OP ({α},{βT})

Atree_n (1, {σ}i, n) (50)

In this last expression the sum runs over all ordered permutations (OP) of {α} and {βT}. That is, within these latter subgroups the order of the legs stays the same. {βT}, means that the order of {β} is reversed.

Using these relations we can obtain some important features for the four legged color ordered partial amplitudes. From equation 46 we know that the following amplitudes are relevant:

Atree₄ (1, 2, 3, 4) Atree₄ (1, 3, 2, 4) Atree₄ (1, 4, 3, 2) Atree₄ (1, 2, 4, 3) Atree₄ (1, 3, 4, 2) Atree₄ (1, 4, 2, 3) (51)

These amplitudes are not all independent, one can for example use reflection and cyclic properties, to show that:

Atree₄ (1, 2, 3, 4) = Atree₄ (1, 4, 3, 2) (52) Atree₄ (1, 3, 4, 2) = Atree₄ (1, 2, 4, 3) (53) Atree₄ (1, 4, 2, 3) = Atree₄ (1, 3, 2, 4) (54)

So, we are now left with three independent amplitudes, which can be combined into the the photon-decoupling identity:

Atree₄ (1, 2, 3, 4) + Atree₄ (1, 3, 4, 2) + Atree₄ (1, 4, 2, 3) = 0 (55) (= Atree₄ (1, 3, 2, 4) + Atree₄ (1, 2, 4, 3) + Atree₄ (1, 4, 3, 2))

Eventually, only two out of six amplitudes need to be used to calculate all the others. Note that this fact could also be found, by using the Kleiss-Kuijf relations:

Atree₄ (1, 2, 4, 3) = −Atree₄ (1, 2, 3, 4) − Atree₄ (1, 3, 2, 4) (56)

In general, using Kleis-Kuijf relations for equation 46 it is shown that one only needs (n − 2)! amplitudes, to calculate all the others [2].

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4.3 Obtaining a kinematic expression

Let us now take a closer look at equation 55. Since the three amplitudes here are equal to zero, we can state that they show some resemblance to the vanishing sum of the kinetic Mandelstam variables from appendix A.3. Here the Mandelstam variables are defined as: s = (p1+ p2)2, t = (p1+ p4)2 and u = (p1+ p3)2 with pi all outgoing massless momenta, in

particular:

Atree₄ (1, 2, 3, 4) + Atree₄ (1, 3, 4, 2) + Atree₄ (1, 4, 2, 3) = 0 = (s + t + u)χ (57) We used the fact that the sum of the Mandelstam variables equals zero in the ultra-relativistic case where particles do not have any mass, which is the case for gluons. χ is a shared function, which includes all polarizations and four-momenta. This is a very important step in the derivation of the BCJ relations. From equation 57 two things become clear. The first thing, is that the amplitudes should be proportional to each other. The next thing might not be directly clear, but a closer look at the amplitude Atree₄ (1, 2, 3, 4), shows that this amplitude is the sum over a s-channel and t-channel diagram, which are treated the same way[10]:

1 2 3 4 (a) s-channel (∝ 1_s) 1 4 3 2 (b) t-channel (∝1_t)

Therefore, we can state that Atree₄ (1, 2, 3, 4) is proportional to u = −(s + t). We can find similar relations for the other two amplitudes, thus yielding [2]:

Atree₄ (1, 2, 3, 4) = uχ, Atree₄ (1, 3, 4, 2) = tχ, Atree₄ (1, 4, 2, 3) = sχ (58)

When multiplying the above equations with either s,t or u on both sides of the equations, we find the following expressions:

tAtree₄ (1, 2, 3, 4) = uAtree₄ (1, 3, 4, 2), sAtree₄ (1, 2, 3, 4) = uAtree₄ (1, 4, 2, 3), (59)

tAtree₄ (1, 4, 2, 3) = sAtree₄ (1, 3, 4, 2) (60)

These are actually the simplest examples of BCJ-relations. Please note that the number of independent amplitudes involved is now equal to one, or more formally (n − 3)!. This might

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not seem like much for now, but the improvement relative to only using Kleiss-Kuijf relations, which was responsible for a factor (n − 2)!, will be huge for higher n [2].

Now, expanding the color ordered amplitudes in terms of diagrams as before, we find the following relations: Atree₄ (1, 2, 3, 4) = ns s + nt t (61) Atree₄ (1, 3, 4, 2) = −nu u − ns s (62) Atree₄ (1, 4, 3, 2) = −nt t + ns s (63)

Here the propagator terms are responsible for the poles, the rest of the amplitude is combined into the numerators ns, ntor nu[10]. These relations can be combined with those in equation

60, to obtain the kinematic numerator identity, for example: Atree₄ (1, 2, 3, 4) =ns s + nt t uAtree₄ (1, 3, 4, 2) t = ns s + nt t u(−nu u − ns s) t = ns s + nt t −nsu s − nst s = nt− nu −ns(u + t) s = nt− nu nss s = nt− nu ns= nt− nu ←→ nu = ns− nt←→ ns− nt− nu = 0 (64)

This expression is the kinetic equivalent of the Jacobi identity. And can be used to further simplify the calculation of scattering amplitudes [2].

4.4 Using the Jacobi identity

We now direct our attention towards the color factors, using equation 46 and 52, 53 and 54, together with the cyclic invariance of traces, we define three color factors. Each color factor belonging to a color ordered amplitude [11]:

c1Atree4 (1, 2, 3, 4) = (Tr(Ta1Ta2Ta3Ta4) + Tr(Ta1Ta4Ta3Ta2))Atree4 (1, 2, 3, 4)

= (Tr(Ta1_Ta2_Ta3_Ta4_{) + Tr(T}a4_Ta3_Ta2_Ta1_))Atree

4 (1, 2, 3, 4) (65)

4 (1, 2, 3, 4) (66)

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This can be further examined by using equations 61, 62 and 63:

Atree_n (1, 2, 3, 4) = g2(c1Atree4 (1, 2, 3, 4) + c2Atree4 (1, 3, 4, 2) + c3Atree4 (1, 4, 2, 3))

= g2(c1( ns s + nt t ) + c2(− nu u − ns s ) + c3(− nt t + ns s )) = g2((c1− c2) ns s + (c1− c3) nt t + (c3− c2) nu u ) (68)

Before we can go any further, we have to take a look at the color generators [11]:

Tr(Ta1_Ta2_Tb_{)T r(T}b_Ta3_Ta4_{) = T}a1 ij T a2 jkT b kiTlmb Tmna3 Tnla4 = Ta1 ij T a2 jk(δkmδil− 1 Nc )δkiδlmTmna3Tnla4 = Ta1 ij T a2 jkT a3 knT a4 ni − 1 Nc )Ta1 ij T a2 ji T a3 lnT a4 nl = Tr(Ta1_Ta2_Ta3_Ta4_{) −} 1 Nc Tr(Ta1_Ta2_)Tr(Ta3_Ta4₎ = Tr(Ta1_Ta2_Ta3_Ta4_{) −} 1 Nc δabδcd Tr(Ta1_Ta2_Ta3_Ta4_{) = Tr(T}a1_Ta2_Tb_)Tr(Tb_Ta3_Ta4_{) +} 1 Nc δabδcd (69)

For our purposes Nc = 3 which is nothing more than the number of possible basisvectors

within a certain symmetry. Now, inserting this result into equation 68, we see that the factors _N1

cδ

ab_δcd _{perfectly cancel each other. Furthermore, remembering equation 39, we can}

now find an expression for (c1− c2):

c1− c2= Tr(Ta1Ta2Ta3Ta4) + Tr(Ta4Ta3Ta2Ta1) − Tr(Ta2_Ta1_Ta3_Ta4_{) − Tr(T}a1_Ta2_Ta4_Ta3₎ = Tr(Ta1_Ta2_Tb_{)T r(T}b_Ta3_Ta4_{) + Tr(T}a4_Ta3_Tb_)Tr(Tb_Ta2_Ta1₎ − Tr(Ta2_Ta1_Tb_)Tr(Tb_Ta3_Ta4_{) − Tr(T}a1_Ta2_Tb_)Tr(Tb_Ta4_Ta3₎ = Tr([Ta1_{, T}a2_]Tb_)Tr([Tb_{, T}a3_]Ta4₎ = −2fa1a2b_fba3a4 _{= c} s

Looking at this expression, we define [2]: ˜

fabc= Tr([Ta, Tb]Tc) = i √

2fabc (70)

When taking a good look at this last definition, we see that this exactly the vertex one needs to multiply the color ordered 3-vertex (equation 44) with to obtain the original QCD Feynman rule for the 3-vertex (equation 41) again. In general, we could also have found cs by simply

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vertex with equation 70 and assigning a δab to every internal line. In particular, one finds in in one way or another:

cs= c1− c2= ˜fa1a2bf˜ba3a4,

ct= c1− c3= ˜fa2a3bf˜ba4a1,

cu= c3− c2= ˜fa4a2bf˜ba3a1

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Now, these three color factors have a Jacobi identity on their own, which can most easily be seen by the composition of cs,ct and cu out of c1,c2 and c3:

˜

fa4a2b_f˜ba3a1 _{= ˜}_fa1a2b_f˜ba3a4 _{− ˜}_fa2a3b_f˜ba4a1

cu = cs− ct↔ nu= ns− nt

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Were in the last line, we see the striking resemblance between color and kinematics. We now have enough information to finish the calculation in equation 68:

Atree_n (1, 2, 3, 4) = g2(nscs s + ntct t + nucu u ) (73)

When rearranging this formula by interchanging legs 3 and 4 again and using equation 70 to obtain the color factors used in the previous chapter, one will find exactly the same expression as in equation 37, apart from a factor 2. This rearrangement can only been done, because of the invariant nature of the Jacobi identity under permutations, which follows from the standard expression for the Jacobi identity:

[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 (74)

Now using, [A, B] = −[B, A], we can define [11]:

[[A, B], C] = [A, [B, C]] − [B, [A, C]]

[[A, B], C] = [A, [C, B]] − [C, [A, B]] = −[A, [B, C]] + [[A, B], C] [[B, A], C] = [B, [A, C]] − [A, [B, C]]

[[B, C], A] = [B, [C, A]] − [C, [B, A]] = −[B, [A, C]] − [[A, B], C] [[C, A], B] = [C, [A, B]] − [A, [C, B]]

[[C, B], A] = [C, [B, A]] − [B, [C, A]] = [[A, B], C] + [B, [A, C]]

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Using this invariance, one will thus have derived exactly the same amplitude as in the pre-vious chapter, apart from a factor 2 because of helicity and polarization states, which were suppressed in this chapter. Note that by doing this, one will find the specific values of ns,

nt and nu, which could also be derived by using equations 61, 62 and 63 and the photon

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In this chapter we were able to calculate the amplitude for the gg → gg-proces, by using a lot of, yet logical steps. For now the big improvement regarding the efficiency of calculating amplitudes, might not yet be clear. In the following chapters, however, we will show some generalizations and also some very interesting applications of this theory.

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5 Intermezzo: Gravity

As I mentioned in the first chapter of this book, up to now, physicists have not been able yet to fully include gravity into the Standard Model. In this chapter we will try to give the reader some insight into why this is the case and we will give a qualitative approach to the tricks currently being used to effectively calculate gravity amplitudes.

5.1 Problems of the gravity Lagrangian

From general relativity, we can get a very compact, typical expression for the gravity Lagrangian density [12]:

L_EH = 2 κ2

√

−gR (76)

This Lagrangian density is called the Einstein-Hilbert Lagrangian, where κ2 is related to Newtons constant by 32π2GN and R is the curvature scalar. In chapter 2 and 3 we used that

Feynman rules for vertices follow from the cubic and quartic terms. The QED Lagrangian accounted for the 3-vertex between two particles and a photon, whereas a 3 gluon-vertex and a 4 gluon-vertex arose in QCD. Now, the gravity Lagrangian does not have any constraints and thus accounts for an infinite set of vertices. Expressed in diagrams, this can be seen as follows:

Possible gravity vertices: , , , , . . .

Now, this is a real problem. In chapter 1 it was described that the amplitude A gives a measure of probability of a certain interaction to occur when squared. Therefore, the sum of all possible subamplitudes, should be equal to 1. In order tho make this happen, the amplitudes get a normalization constant. An infinite number of possible self-interactions amplitudes, yields an infinite amount of ways a reaction can occur in. Therefore, gravity is non-normalizable in 4 dimensions and thus does not account for any physical quantities when being looked at from a Standard Model viewpoint.[10]

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extremely lengthy and difficult to calculate with[12]: G3µα,νβ,σγ(p1, p2, p3) = iκ sym[− 1 2P3(p1· p2ηµαηνβησγ) − 1 2P6(p1νp1βηµαησγ) + 1 2P3(p1· p2ηµνηαβ− P6(p1· p2ηµαηνσηβγ) + 2P3(p1νp1γηµαηβσ) − P3(p1βp2µηανησγ) + P3(p1σp2γηµνηαβ+ P6(p1σp1γηµνηαβ) + 2P6(p1νp2γηβµηασ) + 2P3(p1νp2µηβσηγα− 2P3(p1ν· p2µηανηβσηγµ)] (77) Here sym stands for the symmetrization of the indices µ ↔ α, ν ↔ β and σ ↔ γ. P3 and P6

stand for the symmetrization over the three external legs. These symmetrizations give three or six terms. Furthermore, now gµν = ηµν+ κhµν, where gµν is just the Minkowski metric.

In total, the above expression accounts for approximately 100 terms [12].

5.2 Gravity amplitudes simplified

After the last paragraph, we could definitely state that gravity calculations are extremely difficult to perform. Luckily, we can simplify the calculations by using on-shell methods [2]. Now, I am not going to explain how these methods exactly work, since that would be beyond the scope of this thesis. I will only explain them qualitatively.

In chapter 3 we already saw that not all possible Feynman diagrams relate to physical quan-tities, for example A(RR → RL) = 0, because of conservation of angular momentum. Now using on-shell methods these quantities can be avoided. On-shell methods rely on the fact that all information is ”on-shell”, so at the legs of the vertices. The power of these methods relies on the fact that the ”shell” could be over both external and internal lines, thus being able to ”cut” every diagram into diagrams of 3-vertices. Therefore, using on-shell methods, we do not need the infinite possible vertices anymore, which LEH gave us. Furthermore, we

can now describe the 3-vertex of gravity in a much more compact form 77 [10]:

G3µα,νβ,σγ(p1, p2, p3) = iκ(ηµν(p1− p2)ρ+ Pcyclic) × (ηαβ(p1− p2)γ+ Pcyclic) (78)

We can now compare this result to the on-shell result of the 3-gluon vertex, from equation (22):

−gfabc(ηµν(p1− p2)ρ+ Pcyclic) (79)

In the previous chapter we showed that we could strip the color factors of the QCD-amplitude. The same sort of convention can be used for gravity, in general [2]:

Mtree_n = (κ 2)

n−2_Mtree

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Now the general expression for M_ntree yields:

M_ntree(1, 2, . . . , n) = i(−1)n+1Atree_n (1, 2, . . . , n) X

perms

f (i1, . . . , ij)f (l1, . . . , lj0)

× ˜Atree_n (i1, . . . , ij, 1, n − 1, l1, . . . , lj0, n)] + P(2, . . . , n − 2)

(81)

This is a sum over all permutations {i1, . . . , ij} ∈ P{2, . . . , [n/2]} and {l1, . . . , lj0} ∈ P{([n/2]+

1), . . . , n − 2}, where j = [n/2] − 1 and j0= [n/2] − 2. Furthermore +P(2, . . . , n − 2) stands for the sum over the preceding expression for all permutations of the terms between the brackets. The functions f and f are given by[2]:

f (i1, . . . , ij) = s1,ij j−1 Y m=1 s1,im+ j X k=m+1 g(im, ik) ! (82) f (l1, . . . , lj0) = s_l 1,n−1 j0 Y m=2 slm,n−1+ m−1 X k=1 g(lk, lm) ! (83)

And for the function g:

g(i, j) =    (si,j if i > j 0 else (84)

Where si,j = sij = (ki+ kj)2 [2]. Apart from the fact that we already see the color ordered

amplitudes (Atree_n ) coming into the equation, equation 81 does not seem like a particular easy to cope with expression at first. However, when looking at the simplest cases, i.e. M₄tree, M₅tree and M₆tree, one finds [2]:

M₄tree(1, 2, 3, 4) = −is12Atree4 (1, 2, 3, 4) ˜Atree4 (1, 2, 4, 3) (85)

M₅tree(1, 2, 3, 4, 5) = is12s34Atree5 (1, 2, 3, 4, 5) ˜Atree5 (2, 1, 4, 3, 5)

+ is13s24Atree5 (1, 3, 2, 4, 5) ˜Atree5 (3, 1, 4, 2, 5) (86)

M₆tree(1, 2, 3, 4, 5, 6) = −is12s34Atree6 (1, 2, 3, 4, 5, 6)[s35A˜tree6 (2, 1, 5, 3, 4, 6)

+ s34s35) ˜Atree6 (2, 1, 5, 4, 3, 6)] + P(2, 3, 4) (87)

These equations are the KLT-relations, for which equation 81 is the general expression. Com-pared to traditional way of calculating amplitudes, by using Feynman diagrams and using Feynman rules directly out of the Lagrangian in equation 76, these equations are incredibly simple and easy to use. The equations have been proved to hold in the low-energy (or ”in-frared”) limit of string theory which is just field theory.

The KLT-relations show a remarkable connection between gravity theory and Yang-Mils the-ory, which we will exploit in the next chapter. Some might even say that this connection

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screams for a universal origin of both theories. The relations themselves might be a hint towards a unification theory. However, since the relation between the two theories has not been proven to hold in the whole energy spectrum, more research is required to make such a statement.

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6 Implications of BCJ on gravity amplitudes

In this final chapter we will see the particular importance of BCJ-relations in calculating gravity amplitudes. Let us start by recalling expression 73 at the end of chapter 4:

Atree n (1, 2, 3, 4) = g2( nscs s + ntct t + nucu u ) This expression can be further generalized as shown in [2]:

Atree n (1, 2, . . . , n) = gn−2 X i nici (Πjp2_j)i (88)

In this latter expression ni and ci are equivalent to the previous kinetic and color factors.

Furthermore, this equations is for all Yang-Mills theory amplitudes, for which QCD is just one of them.

6.1 The four legged amplitude

Let us first take a look at the KLT-relations we expressed in the previous chapter. We start with the simplest case the four-legged amplitude. This can be done by calculating equation 85, using two not necessarily the same gauge-theories:

M₄tree(1, 2, 3, 4) = −is12Atree4 (1, 2, 3, 4) ˜Atree4 (1, 2, 4, 3)

= −isAtree₄ (1, 2, 3, 4) Ãtree₄ (1, 3, 4, 2) =ns s + nt t −n˜u u − ˜ ns s = −is −nsn˜u su − ntñu tu − nsñs s2 − ntn˜s st = is ñu(−tns− snt) + ñs(− tu sns− unt) stu = −is _−t˜_n uns− sntñu− ñsnstu_s − untn˜s stu = −is _−t˜_n u(nu+ nt) − sñunt− untn˜s− ñsnstu_s stu = −is (−t − s)ñunt− tñunu− untn˜s− ñsns tu s stu = −is (−t − s)ñunt− unt( ñu+ ñt) − tñunu− ñsns tu s stu = −is (−t − s)ñunt− untn˜u− untn˜t− tñunu− ñsns tu s stu = −is _−un tn˜t− tñunu− ñsnstu_s stu = i(n˜tnt t + ˜ nunu u + ˜ nsns s ) (89)

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In this calculation, we used a great part of the apparatus derived in chapter 4. In the second line we used the reflective and cyclic invariance of four-legged color ordered partial amplitudes, so that we could use equations 61 and 62 in the next line. Throughout the rest of the calculation we made use of the kinematic Jacobi identities (equation 64) and the vanishing sum of Mandelstam variables (equation 108).

In particular, for pure gravity one will find [2]:

−iMtree 4 (1, 2, 3, 4) = n2_t t + n2_u u + n2_s s (90)

By ”pure”, we mean that the only interacting particles are gravitons, when other particles are involved, one should use equation 89.

6.2 Generalization for gravity amplitudes

From equation 89 one could get the idea that gravity amplitudes can be calculated by using just the multiplication of two Yang-Mills theories. Using equation 88, we can define for two such theories:

1 gn−2A tree n (1, 2, . . . , n) = X i nici (Πjp2_j)i 1 gn−2A˜ tree n (1, 2, . . . , n) = X i ˜ nici (Πjp2_j)i (91)

Comparing these amplitudes to equations 73 and 89, one would now thus expect [2]:

−iM₄tree(1, 2, 3, . . . , n) =X

i

nin˜i

(Πjp2j)i

(92)

Here the sum runs over the same diagrams as in equation 91. In this case, gravity is the double copy of Yang-Mills theory and if equation 92 turns out to be consistent with existing theories, it provides a powerful tool into connecting gravity theory to Yang-Mills theory. The number of both independent Atree_n and independent ˜Atree_n in the KLT-relations (equation 81), turns out to be (n − 3)!, which is an indication that equation 92 might indeed describe the KLT relations in an appropriate way [2].

6.3 Summary, conclusions and further discussion

In this thesis we have used different ways of calculating certain scattering amplitudes. We started the traditional way, by using typical Feynman rules derived from the Lagrangian for QED. Using kinematic variables we found a very simple expression for the typical electron scattering amplitude. In the next chapter we used the same traditional way to do a slightly more difficult calculation to find an expression for the gg → gg process. In the following

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chapter we proved that the same expression could have been found by using BCJ-relations after separating color from kinematics. We also saw how Kleiss-Kuiff relations brought the number of independent amplitudes down to (n-2)!, after which we saw that BCJ-relations could in fact account for (n-3)! independent amplitudes. In chapter 5 we took a closer look at gravitation amplitudes to find the KLT-relations. These relations, combined with BCJ-relations, resulted in a very useful master formula (equation 92), which seems to hold for all KLT-relations, therefore probably correctly describing gravity interactions in the infrared limit of string theory.

Now further examination of the BCJ-relations in combination with KLT-relations needs to be done to see how both gauge theories and gravitation are related to each other. In this thesis we showed that there is certainly some kind of relationship between the two, but it is far from clear where this originates from. In order to construct a new theory which is complete, one should also bring the loop-level diagrams into the calculation. The first researches have proven to be very promising, since BCJ-relations and the kinematic numerator identity (equation 64) in particular, turn out to be very useful in simplifying loop level amplitudes too. In some calculations the normalization problem of gravity, qualitatively described in chapter 5, even seems to be solved, due to the interference of the different subamplitudes. However, in order to state anything definite, one should first have a formal proof that equation 92 indeed holds for all KLT-relations. Furthermore, different parts of the energy spectrum of string theory should also be investigated [2].

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A

Relativistic kinematics

Often, particle interactions involve high energies for particles with small masses, it is therefore no surprise that some key-concepts of special relativity occur throughout this thesis. In this appendix a few basic concepts about special relativity are summarized, furthermore some notation used throughout this thesis is explained, as well as the Mandelstam variables.

A.1 Key-concepts of special relativity

Special relativity describes how physical observables are described in different inertial frames. We can take for example takes two inertial frames, Σ and Σ0, where one moves with a constant speed v away from the other. If we now use Einsteins postulation that the speed of light c is invariant, that is, the same in each inertial frame, we see that the spacetime-interval is an invariant quantity under transformations between the two frames:

c2t2− x2− y2− z2 = c2t02− x02− y02− z02 (93)

This can be proved by imagining a light pulse which is emitted in the overlapping origin of the two inertial frames and then measuring the covered distance and the time passed in both inertial frames. This requirement can only be met if the coordinates are related by the Lorentz transformations. In particular, for some constant speed between the inertial frames on the z-axis, we can write [1]:

       t0 x0 y0 z0        =        γ 0 0 −γβ 0 1 0 0 0 0 1 0 −γβ 0 0 γ               t x y z        (94)

Where we used β = v/c and γ = (1 − β2)−12. In this thesis, I use natural units, thus setting

c = 1.

Equation 94 can be written more compactly: x0µ = Λµνxν, where we used Einstein’s

sum-mation convention to sum over repeated indices. Here, we actually defined the contravariant four-vector: xµ= (t, x, y, z).

In fact, the invariant spacetime interval can now be found, by multiplying the contravariant four-vector by the covariant four-vector: xµ= (t, −x, −y, −z):

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Now, it does not take much intuition to see that the contravariant and covariant four-vectors are closely related to each other. Using the Minkowski metric gµν,

gµν =        1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1        , (96) we find in general: aµbµ= aµbµ= gµνaµbν (97)

This scalar product is Lorentz invariant. So, if there is a quantity which can be expressed as a scalar product of two four-vectors, this quantity will always be invariant under Lorentz transformations. This feature we will use in appendix A.3.

Now, the four-derivative,

∂µ= ∂ ∂xµ = ∂ ∂t, ∂ ∂x, ∂ ∂y, ∂ ∂z , (98)

can be shown to transform as a covariant vector. Multiplying it with its contravariant coun-terpart, ∂µ= ∂ ∂xµ = ∂ ∂t, − ∂ ∂x, − ∂ ∂y, − ∂ ∂z , (99)

we find the so-called d’Alembertian, denoted by . This is the four-vector equivalent of the Laplacian [1]. = ∂µ∂µ= ∂2 ∂t2 − ∂2 ∂x2 − ∂2 ∂y2 − ∂2 ∂z2 (100) A.2 Four-momenta

In special relativity, energy (E) and momentum (p) are not Lorentz invariant quantities. However, they are still conserved in the same inertial frame. They can be found by the following definitions:

E = γm and p = γmv (101)

Now for velocity v = dx_dt, it can thus be shown that we can define a four-momentum as a contravariant four-vector: pµ = (E, px, py, pz). In the same inertial frame, energy and

momentum are conserved quantities, so the four-momentum is also conserved. Writing a scalar product of four-momenta, we find the following Lorentz-invariant quantity:

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When looking at the rest frame of some particle, this quantity can be shown to be just equal to its mass squared, m2. Four-momenta of multiple particles pµ_i can be added to find new four-momenta. pµ= n X i=1 pµ_i (103)

When writing the scalar product of this four-momentum with itself, we find again a mass squared, which is the invariant squared mass of a system [1].

A.3 Mandelstam variables

Now, as being said in the previous subsection there are conserved quantities, such as energy and momentum and there are invariant quantities, such as the invariant mass of a particle. In order to do calculations in different inertial frames for both initial and the final states in particle processes, one needs variables which are both conserved and invariant under Lorentz transformations.

Three such variables are the Mandelstam variables, s, t and u. These variables are designed to describe the energy exchange between the initial and final state particles in a 2 → 2 particle proces. On lowest order this proces include s-channel (annihilation), t-channel scattering and u-channel scattering between two identical particles. The corresponding Feynman tree-diagrams are [1]: 1 2 3 4 q (a) s-channel 1 3 4 2 q (b) t-channel 1 4 3 2 q (c) u-channel

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The Mandelstam variables [1], s = (p1+ p2)2= (p3+ p4)2 = p2₁+ p2₂+ 2p1p2= p23+ p24+ 2p3p4, (104) t = (p1− p3)2= (p2− p4)2 = p2₁+ p2₃− 2p1p3= p22+ p24− 2p2p4,, (105) u = (p1− p4)2= (p2− p3)2 = p2₁+ p2₄− 2p1p4= p22+ p23− 2p2p3, (106)

are the same as the four-momentum squared q2 of the exchanged boson in the proces of a particular channel. In equation 104, 105 and 106, the pi’s are assigned to the four-vectors

of the particles, or legs, corresponding to the subscript number. The Mandelstam variables are invariant, because they are just the squares of particular four-vectors, as explained in appendix A.1. Due to this invariance of the Mandelstam variables, one can easily use the center of mass frame in a particle problem, where calculations are in general much easier. If one needs to know the value of one of the four-vectors in the labsystem, only 1 transformation is needed after a calculation through the Mandelstam variables.

An important property of Mandelstam variables used in this thesis, can be found, when we again consider the invariant masses for a particular scalar product of four-vectors. Further-more, using four-momentum conservation in a particular 2 → 2 particle proces, we can define p1= p3+ p4− p2. Let us now see what happens when we add up the Mandelstam variables:

s + t + u = (p1+ p2)2+ (p1− p3)2+ (p1− p4)2 = 3p2₁+ p2₂+ p2₃+ p2₄+ 2p1p2− 2p1p3− 2p1p4 = 3m2₁+ m2₂+ m2₃+ m2₄+ 2p1(p2− p3− p4) = 3m2₁+ m2₂+ m₃2+ m2₄+ 2p1(−p1) = 3m2₁+ m2₂+ m2₃+ m2₄− 2m2₁ = m2₁+ m2₂+ m2₃+ m2₄ (107)

In the ultra-relativistic limit where all external particles are massless, equation 107, will show the vanishing sum of Mandelstam variables:

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B

Group theory

As final part of this thesis, I will introduce some group theory. This appendix can be used to see where color factors actually originate from, as well as why symmetries play such a big role in particle physics. I also included the particular example of the SU(2)-group to show how group theory works. All information in this section comes from ref. [5], if not stated otherwise.

Mathematically, a group G is a set of elements {g, h, k, . . . }, where each element k ∈ G can be found by a multiplication of g and h which are both elements of the group G too. This multiplication is not required to be an ordinary multiplication. In general, it is an operation to get k from h and g. This group multiplication is required to be associative (g · (h · k) = (g · h) · k) and will sometimes be commutative (g · h = h · g). In the latter case a group G is called Abelian. Furthermore, both the identity 1 and the inverse elements g−1 of every g ∈ G are in a set of elements of G.

A group has a representation D(G), if the set of matrices D(G) preserves the multiplication structure of the elements g ∈ G, for example:

D(g · h) = D(g)D(h) ∈ D(G) (109)

For every D(g), D(h) ∈ D(G) If we can define a group H ∈ G, this group H is called a subgroup of G.

B.1 Symmetries

Now, let us take a look at symmetry transformations. A symmetry transformation is defined by a transformation that leaves the dynamics of a physical system invariant. So in general, the action functional S, which, being the integral over the Lagrangian, actually defines these dynamics, will be invariant under symmetry transformations. Thus, for symmetry transformations in {g, h, k, . . . }, we can simply write:

S(φg) = S(φh) = S(φk) = S(φ) (110)

Where φ is just some field.

From this definition it can be deduced that symmetry transformation will constitute a group. After all, if we let k = g · h as required for groups, we can see that k has to be a symmetry transformation on its own:

S(φ)−→ S(φk _k) = S(φ)−−→ S(φg·h _g·h)

= S(φ)−→ S(φh _h)−→ S(φg _h·g) = S(φ)−→ S(φ)h −→ S(φ)g

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When all possible symmetry transformations are included in a set {g, h, k, . . . }, we call this group complete. It can easily be checked that indeed the identity 1 and the inverse elements g−1 are included in a complete set.

Now symmetry transformations can either be discrete or continuous. If the symmetries are discrete, they will usually constitute a group with a finite number of elements, defined as a finite group. If the symmetries are continuous and depending on one or more parameters, we call the number of parameters the dimension of the group. A group is called a Lie group, if the dependence on this parameters is analytical.

B.2 Lie algebra

Let us now take a closer look at these Lie groups. If a group depends on just one parameter, we can define its elements as g(ξ). Assuming that the elements can take the form of linear transformations, we can express these elements as matrices. Now using the analytic property of the elements in the Lie group and the fact that the identity g(0) = 1 is part of the group, we can write elements close to this identity as:

g(ξ) = 1 + ξt + O(ξ2) (112)

In this expression t is a matrix that generates the needed transformation. Now using that the elements g(ξ) are analytic, there is always a canonical parametrization possible: g(ξ1)g(ξ2) =

g(ξ1+ ξ2). This can be exploited to write g(ξ) as a series of infinitesimally small steps away

from the identity element:

g(ξ) = [g(ξ/n)]n= lim n→∞ 1 + ξ nt n = exp{ξt} (113)

Now, this can be generalized for n different parameters, so that we obtain:

g(ξ1, . . . , ξn) = exp{ξata} (114)

In this expression (a = 1, . . . , n). Using te fact that two elements of a group constitute another element, we can use this expression to write:

g(ξ1, . . . , ξn) · g(ζ1, . . . , ζn) = exp{ξata} · exp n ζbtb o (115) = exp{ηctc} (116)

This requirement can only be met if the generators close under commutation:

BCJ-relations

BCJ-relations

Pieter van der Hoek

10747834

Contents

1

Introduction

2

Feynman Calculus in QED

3

Getting used to QCD

4

The duality between kinetmatic and color factors

5

Intermezzo: Gravity

6

Implications of BCJ on gravity amplitudes

A

Relativistic kinematics

B

Group theory