Left-Right Symmetric Model
± Putting lower bounds on the mass of the heavy, charged WR gauge boson
Melissa Harris 940609-1885 melissa.harris@physics.uu.se
A thesis presented for the degree of MSc: Master in Physics Supervised by Rikard Enberg, Andreas Ekstedt and Johan L¨ofgren
Theoretical High Energy Physics Uppsala University Sweden Abstract
In this project I have studied the left-right symmetric model (LRSM) as a candidate beyond standard model theory of particle physics. The most common version of the theory, called the minimal LRSM, has been studied and tested extensively for several decades. I have therefore modified this minimal LRSM by adapting the scalar sector and computing the mass of the ± charged right-handed gauge bosons WR for this particular scalar sector. I carried out a study of the theory and implemented it into FeynRules, in order to simulate LHC events using ± MadGraph. This allowed computation of the cross-section for the decay WR → tb as a function of the mass of W ±, which was compared with CMS data for the same decay, with R √ proton-proton collisions at a centre of mass energy of s = 13 TeV. The final result was a ± constraint on the mass of WR , with a lower bound of MWR ≥ 3 TeV.
1 Popul¨arvetenskaplig Sammanfattning
Inom partikelfysik kallas den mest v¨alk¨andaoch accepterade teorin f¨orpartikelfysikens stan- dardmodell (SM). Aven¨ om teorin har testats noggrant och i de flesta fall st¨ammer¨overens med experimentella resultat finns det vissa fenomen som den inte kan f¨orklara. P˚agrund av SMs tillkortakommanden finns en gren av fysiken som kallas bortom standardmodellen (BSM). D¨ar utvecklas teorier f¨oratt f¨orb¨attraSM och f¨orklarade fenomen som SM inte kan. Dessa teorier kallas s˚af¨or att de, snarare ¨anatt b¨orjafr˚anb¨orjanoch utveckla en helt ny modell, bygger vidare p˚aSMs framg˚angargenom att l¨aggatill nya delar.
I det h¨arprojektet har jag studerat den v¨anster-h¨ogersymmetriska modellen (LRSM) som kandidatteori f¨orpartikelfysik BSM. Den vanligaste varianten av teorin, kallad minimal LRSM, har testats noggrant under flera decennier. D¨arf¨orhar jag modifierat denna minimala LRSM genom att anpassa den skal¨arasektorn och ber¨aknamassan hos de laddade h¨ogerh¨anta gauge- bosonerna WR i denna specifika skal¨arasektor. Jag genomf¨ordeen studie av teorin och imple- menterade den i FeynRules f¨oratt simulera LHC-h¨andelsermed hj¨alpav MadGraph. Detta gjorde det m¨ojligt att ber¨aknatv¨arsnittet f¨ors¨onderfallet WR → tb som funktion av massan hos WR, vilket j¨amf¨ordesmed data f¨ordetta s¨onderfallfr˚anCMS-detektorn i proton-proton- kollisioner vid masscentrumenergi sqrts = 13 TeV. Slutresulatet ¨aren l¨agregr¨ansf¨ormassan hos WR, MWR = 3 TeV.
2 Contents
1 Introduction 5 1.1 Success and Limitations of the Standard Model ...... 5 1.2 Looking Beyond the Standard Model ...... 5 1.3 Why Study the Left-Right Symmetric Model? ...... 6 1.4 Outline of the Project ...... 7
2 Overview of the Standard Model 8 2.1 Group Structure ...... 8 2.1.1 SU(3) ...... 8 2.1.2 SU(2) ...... 8 2.1.3 U(1) ...... 9 2.2 Chirality ...... 9 2.3 Matter Particle Content ...... 9 2.4 Gauge Transformations ...... 10 2.5 The Higgs Mechanism ...... 11
3 Left-Right Symmetric Model 14 3.1 Gauge Group and Multiplet Structure ...... 14 3.2 Scalar Sector ...... 16 3.3 The Left-Right Symmetric Lagrangian ...... 16 3.3.1 Gauge Field Lagrangian ...... 16 3.3.2 Fermionic Gauge Lagrangian ...... 17 3.3.3 Scalar Lagrangian ...... 17 3.3.4 Yukawa Lagrangian ...... 17 3.3.5 Higgs Potential Lagrangian ...... 18 3.4 Left-Right Symmetry as Parity ...... 19 3.5 Spontaneous Symmetry Breaking ...... 20 3.5.1 Symmetry Breaking: Step 1 ...... 21 3.5.2 Symmetry Breaking: Step 2 ...... 22 3.5.3 Symmetry Breaking: Step 3 ...... 23 3.6 Physical Consequences of Symmetry Breaking ...... 25 3.6.1 Neutral Gauge Boson Masses ...... 25 3.6.2 Charged Gauge Boson Masses ...... 29
4 Running Simulations of the LRSM 31 4.1 FeynRules ...... 31 4.1.1 Gauge Groups ...... 31 4.1.2 Indices ...... 32 4.1.3 Fields ...... 32 4.1.4 Parameters ...... 33
3 4.1.5 Lagrangian ...... 33 4.2 MadGraph ...... 34
5 Comparison of Simulations with CMS Data 35 5.1 Production and Decay of W boson ...... 35 5.2 Cross-section and the Narrow Width Approximation ...... 36 5.3 CMS Search for W Boson ...... 36 5.4 Putting Mass Limits on W Boson ...... 36
6 Conclusion 38 6.1 Future possibilities in this project ...... 38 6.2 Future work beyond this thesis ...... 38
7 References 40
4 1 Introduction
1.1 Success and Limitations of the Standard Model
In the study of particle physics, the most well known and accepted theory to date is the stan- dard model (SM) of particle physics. The SM is a combination of the Glashow-Weinberg-Salam theory of electroweak interactions and the theory of quantum chromodynamics (QCD). The theory associates each known particle with a quantum field and predicts which interactions oc- cur and the probabilities of such interactions. All of the particles occurring in the SM have been experimentally observed, with the final observation being the Higgs boson in 2012 by ATLAS and CMS [1], [2]. The SM has predictive power and is in strong agreement with experimental data.
Despite the success of the SM, the theory has its shortcomings. There are a number of phenom- ena which can’t be explained by the SM [3]. According to the SM, the neutrinos are massless, however experiments have shown that the neutrinos do have masses and mixing occurs between them. Additionally, there is no particle in the SM which can explain the existence of cold dark matter. The asymmetry of matter over anti-matter is yet another feature of the observable universe which can’t be explained by the SM.
There is another reason why physicists are not satisfied with the SM. There are 19 free pa- rameters which describe the theory and determine, for example, the particle masses. These parameters can only be determined by experiment and there is no explanation as to why they have the values they do. There exists a hierarchy in the masses of the three generations of particles which is not explained. It is therefore natural to question the completeness of the SM and look for explanations of how these parameters arise.
1.2 Looking Beyond the Standard Model
Due to the shortcomings of the SM, there is a branch of physics called beyond the standard model (BSM), where theories are developed to improve the SM and explain the phenomena which the SM can’t. These theories are so called, because rather than starting from the very beginning and developing an entirely new model, they build on the success of the SM by adding extensions to the theory. One benefit to this approach is that the SM describes processes which occur at energies which are reachable by current accelerators. Therefore, it is entirely possible that BSM theories can also be probed in particle accelerators. The field of high energy physics is therefore important for both theorists and experimentalists. From the theoretical point of view, a theory can be studied in detail and computer programs can be used to predict observa- tions at accelerators. These predictions can then be compared to data which is collected and processed by the experimentalists.
5 A natural way to extend the SM is to analyse the group structure. The SM is based on the gauge group:
GSM = SU(3)C × SU(2)L × U(1)Y
It is a direct product of the electroweak group SU(2)L × U(1)Y with the QCD group SU(3)C . The electroweak group is broken down via a process called spontaneous symmetry breaking to give the group describing the electromagnetic interactions observed in nature:
SU(2)L × U(1)Y → U(1)EM
This process is caused by having a scalar field with a non-zero vacuum expectation value in- cluded in the theory. It is also the process responsible for the matter particles and gauge bosons acquiring their masses. The energy scale of this electroweak symmetry breaking is de-
fined roughly by the vacuum expectation value of the Higgs field, νH ' 246 GeV [4].
It is predicted that the SM is an effective theory of a more complete theory which has the Planck scale of ∼ 1019GeV as its energy scale. This is known as a Grand Unified Theory (GUT). There then exists a gauge group to describe the GUT at this higher energy, which contains the SM at lower energies:
GGUT ⊃ SU(3)C × SU(2)L × U(1)Y
A number of symmetry breaking steps are then responsible for breaking down the GUT gauge group to the SM gauge group. It is not yet known which group corresponds to GGUT , and searches for a suitable candidate are ongoing [5]. There are two main approaches in this search.
The first is a so-called ”top-down” approach, where the GGUT is hypothesised, then broken down in a number of steps to GSM . The alternative method is a ”bottom-up” approach, beginning with GSM and gradually extending the group. The latter is the approach taken in the left-right symmetric model (LRSM).
1.3 Why Study the Left-Right Symmetric Model?
The left-right symmetric model is formed by modifying the electroweak gauge group. A right- handed SU(2)R group is added and the charge on U(1) is modified to a new charge denoted by Y˜ :
GLR = SU(2)L × SU(2)R × U(1)Y˜
This model was first suggested by physicists Jogesh Pati and Abdus Salam, in an attempt to introduce left-right symmetry. The model is attractive to study, as it removes the left-right asymmetry which occurs in the standard model. There is no obvious reason why the left- handed and right-handed particles should obey different physics, and the LRSM takes care of this, leaving the SM as a less symmetric effective theory.
6 Another advantage of the LRSM is its connection to parity. Scientists Goran Senjanovi´cand Rabi Mohapatra developed the LRSM to account for spontaneous parity breaking, which had previously not been achieved [6].
The LRSM introduces new gauge bosons and scalar particles, which phenomenologically have larger masses than the SM particles. This opens up the possibility of detecting them in exper- imental collider physics.
1.4 Outline of the Project
± In this project, I have chosen to focus on the charged heavy gauge bosons WR which are present in the left-right symmetric model. I split my research into three steps: a study of the theory; implementation of the theory into a program to run LHC simulations; comparison of the sim- ulations with actual LHC data.
My thesis is structured as follows. In section 2, I give an overview of the standard model of particle physics, which is useful as most features are re-visited in the LRSM. I describe the main features of the left-right symmetric model in section 3, giving more attention to the spon- taneous symmetry breaking mechanisms and the resulting gauge boson masses. In section 4, I describe the programs FeynRules and MadGraph which I used to create simulations of my model. I compare these simulations to results from the LHC in section 5.
7 2 Overview of the Standard Model
The SM of particle physics is a renormalisable quantum field theory which combines the Glashow-Weinberg-Salam theory of electroweak interactions with QCD. The theory allows not only tree level, but also higher order calculations, where perturbation theory probes quantum effects. The calculations agree with experimental data to a high precision, and to date the SM is the most well known and accepted theory of particle physics [7].
2.1 Group Structure
The SM is defined by the group:
GSM = SU(3)C × SU(2)L × U(1)Y (1)
GSM is a direct product of three Lie groups. SU(3)C describes the theory of QCD, where the subscript C refers to colour. The product SU(2)L × U(1)Y describes the electroweak theory, where L refers to left-handed and Y to hypercharge.
If a field Ψ is charged under one of the groups, it undergoes transformations of the form:
a Ψ → eiaT Ψ
a where the T are generators of the group and a are parameters defining the transformation. Both are labelled with the subscript a which runs from a = 1, 2, ..., n, where n is the number of generators required to define the group. The generators are elements of the corresponding Lie Algebra, which satisfy certain criteria and can be expressed as square matrices. These matrices are not unique and one set of matrices is called a representation. The most common represen- tation is the fundamental representation, but the adjoint representation is also frequently used, and appears in the SM.
2.1.1 SU(3)
Special unitary groups SU(N) with N ≥ 2 are non-abelian, meaning their transformations are non-commutative. The number of generators required to specify an SU(N) Lie group is: n = N 2 − 1. Therefore, for SU(3), the value of a runs from 1 to 8. The 8 generators of SU(3) in fundamental representation are the Gell-Mann matrices, denoted λa. In the SM, only the quarks are colour charged and transform under this group.
2.1.2 SU(2)
The number of generators of SU(2) is 3. In the fundamental representation, the generators are
τa 2 , where the τa are the familiar Pauli-Sigma matrices. The label L of the SM SU(2)L group refers to handedness of particles. Only left-handed particles transform under the SU(2)L group in the SM.
8 2.1.3 U(1)
The unitary group U(1) is an abelian Lie group, which is specified completely by its charge. In the SM, this is the hypercharge Y .
2.2 Chirality
The term chirality refers to whether a particle is left-handed or right-handed. For a massless particle the chirality is equivalent to the helicity, h, which is the dot product between the spin and momentum of the particle. If the particle travels in the same direction as its spin, then h = 1 and the particle is identified as right-handed. Conversely, a left-handed particle travels in the opposite direction to its spin and has h = −1. An anti-particle simply has the opposite sign of helicity to its corresponding particle.
The case for massive particles is more complex, since the dot product between momentum and spin is reference frame dependent, and therefore not a simple Lorentz-invariant property. The chirality of a massive particle is related to an intrinsic property called isospin, analogous to spin. In the SM, left-handed particles are arranged into an isospin doublet, which can be rotated by three SU(2)L transformations. The non-existence of an SU(2)R group means that right-handed particles have a trivial isospin representation, and don’t transform under isospin rotations. A theory such as the SM in which left and right-handed particles are treated differently, is called a chiral theory [8].
2.3 Matter Particle Content
In total there are 12 matter particles in the SM. These consist of 6 leptons and 6 quarks, which are arranged into left-handed doublets and right-handed singlets, each with 3 generations.
The lepton sector consists of: e, µ, τ and the corresponding neutrinos: νe, νµ, ντ . The quark sector consists of three up-type quarks u, c, t and three down-type quarks d, s, b, standing for up, charm, top, down, strange and bottom. The left-handed doublets are then compactly represented by: ! ! νL uL LL = ,QL = eL dL
The right-handed singlets are denoted:
νR, eR uR, dR
The representations of the multiplets with respect to SU(3)C ,SU(2)L,U(1)Y respectively are:
LL = LL(1, 2, −1) QL = QL(3, 2, 1/3)
νR = νR(1, 1, 0) uR = uR(3, 1, 4/3)
eR = eR(1, 1, −2) dR = dR(3, 1, −2/3)
9 The representations sum up how each field transforms under the SM group. For example LL does not transform under SU(3)C , is a doublet under SU(2)L and has a hypercharge value of -1.
2.4 Gauge Transformations
The Lie groups which make up the SM group GSM are gauge groups, meaning they correspond to gauge transformations of the fields. Gauge transformations are continuous, local transfor- mations. If a quantum field theory is required to be gauge invariant, this means that gauge transformations of the fields leave the Lagrangian unchanged. A gauge transformation of a general quantum field Ψ is represented by:
a iaT a Ψ → e Ψ ' (1 + iaT )Ψ (2) where the parameter a = a(x) depends on spacetime. An important consequence of the spacetime dependence is that the derivative of a field is no longer gauge invariant. Therefore, gauge covariant derivatives are required:
Dµ = ∂µ + igAµ (3) where a vector field Aµ and a coupling constant g have been introduced. Provided that this new field transforms as: 1 A → A − ∂ (x) (4) µ µ g µ then the covariant derivative DµΨ remains invariant under gauge transformations. If a field is charged under more than one group, then equation (3) is simply extended. For example, for a
field charged under two groups, we would need to introduce two vector fields Aµ and Bµ, and respective couplings gA and gB. The covariant derivative would then be:
Dµ = ∂µ + igAAµ + igBBµ (5)
The vector fields Aµ which were introduced are known as vector bosons, or gauge bosons. For each gauge group in the SM, one of these fields needs to be introduced to ensure the whole theory is gauge invariant. In addition, each group transformation has an associated coupling constant. The vector bosons can then be written in terms of the generators of the group as: a Aµ = AµaT . Therefore, in the SM we have the following vector bosons and coupling constants:
a SU(3)C : gC ,Gµ a = 1, 2, ..., 8 a SU(2)L : g, Wµ a = 1, 2, 3 00 0 U(1)Y : g ,Bµ
a The vector bosons which mediate the strong interaction are the 8 gluons, Gµ. The vector a bosons responsible for electroweak interactions are the 3 W bosons Wµ and the hypercharge
10 0 boson Bµ. The electroweak gauge bosons are in fact not the physical mass eigenstates observed 1 2 ± 3 0 in nature. Wµ and Wµ mix together to form the familiar charged Wµ . Similarly, Wµ and Bµ 0 mix together to form the neutral Zµ and the photon Aµ. This will be addressed further in the following section.
2.5 The Higgs Mechanism
The Higgs mechanism is the name given to the spontaneous symmetry breaking (SSB) in the SM. This is the process by which matter particles and gauge bosons acquire masses. Prior to SSB, mass terms for leptons, quarks and vector bosons are forbidden, as they are not gauge invariant. This is solved by introducing a scalar field φ with a non-zero vacuum expectation value (vev). This report deals with several symmetry breaking steps involved in the LRSM, therefore it is instructive to give an overview of the simpler case of the SM.
The scalar field introduced into the SM is a complex scalar called the Higgs field, and has the form: ! φ+ φ = , φ = φ (1, 2, 1) φ0
The Higgs field is a colour singlet, an SU(2)L doublet and has a hypercharge value of 1. In the Higgs mechanism, we assign a vev to one or more of the components. In order to assure that the vacuum is electrically neutral, it is only the uncharged field φ0 which takes on a non-zero vev, which can be taken to be real. The overall vev of the Higgs field is then given by: ! 1 0 φv = √ 2 νH The process is called symmetry breaking because the vev of the Higgs field breaks the elec- troweak gauge group down to the electromagnetic gauge group U(1)EM :
SU(2)L × U(1)Y → U(1)EM (6)
This occurs because the vev of the Higgs field isn’t invariant under the whole electroweak group gauge transformation; it is only invariant under U(1) transformations.
The process outlined in equation (6) is called partial symmetry breaking, because there is still a remaining U(1) symmetry. The form of the vev can tell us which groups were broken and which remain. Through SSB, a group G with generators tG, can be broken down to a subgroup H, with generators tH . The unbroken generators annihilate the vacuum, meaning v they satisfy tH φ = 0. The broken generators no longer annihilate the vacuum. Therefore, to find the generators of the unbroken group after SSB in the SM, one needs to solve: ! ! ! a b 0 0 √ = c d νH / 2 0
11 With the additional constraint that the resulting generators must be hermitian, there is only one generator that annihilates the vacuum, which corresponds to the electric charge Q of U(1)EM : ! 1 0 Q = 0 0
The unbroken generator must be written as a linear combination of the generators of SU(2)L and U(1)Y . It is then simple to identify the generator of U(1)EM as: τ Y Q = 3 + 1 (7) 2 2
This relates the electric charge of the resulting U(1)EM group to the third generator of SU(2)L and the charge of U(1)Y [9].
Each broken generator corresponds to one degree of freedom which is lost. In the SM, there were in total 4 generators for the direct product SU(2)L × U(1)Y . After SSB we were left with only 1 generator, meaning 3 degrees of freedom were removed. This is accounted for by the massless gauge bosons gaining a longitudinal component, which is what gives them mass. In the SM, this means that 3 gauge bosons acquire a mass via SSB, and one remains ± massless. This corresponds to the massive vector bosons, Wµ , Zµ and the massless photon, Aµ.
Mass terms for the vector bosons are found by evaluating the Higgs field kinetic term in the 2 † µ Lagrangian at its vev. The kinetic term for φ is |Dµφ| = (Dµφ) (D φ), where the gauge covariant derivative of φ is: ~τ 1 D φ = ∂ φ + igW~ · φ + ig00B0 φ µ µ µ 2 2 µ
This can be seen from equation (5), where the gauge fields from SU(2)L have been written as ~ ~τ Y 0 Wµ · 2 and the U(1)Y gauge field as 2 Bµ with Y = 1 for the Higgs field. From this point, I will omit the subscript µ for ease of reading, as during these calculations it is simply a label. Substituting the Higgs vev into the kinetic term gives the result:
2 2 νH 2 1 2 1 2 00 0 32 |D φ| v = g W + iW W − iW + g B − gW (8) µ φ=φ 8 We can see that we now have mass terms for the gauge bosons, but with cross terms between W 3 and B0. W 1 and W 2 don’t have cross terms and are therefore already mass eigenstates, but they don’t correspond to the complex charged fields. By redefining the gauge fields, it is possible to rewrite (8) so that it only contains mass terms for the physical gauge fields.
The vector bosons W 1 and W 2 are combined in the following way: 1 W ± = √ (W 1 ∓ iW 2) (9) 2
12 W ± are the familiar charged vector bosons. The vector bosons W 3 and B0 are combined in the following way:
0 3 A = cos θW B + sin θW W (10) 0 3 Z = − sin θW B + cos θW W
3 0 We see that the fields W and B have been rotated about an angle θW , which is called the Weinberg angle, defined by:
g00 g sin θW = cosθW = (11) pg2 + g002 pg2 + g002
Substituting these results into equation (8) gives the mass terms:
2 2 2 2 002 2 νH g + − νH (g + g ) 2 |D φ| v = W W + Z µ φ=φ 4 8 Since we expect mass terms of the form:
2 2 + − 1 2 2 |D φ| v = M W W + M Z µ φ=φ W 2 Z we see that the masses of Z and W ± are related to the vev of the Higgs field by [10]: 1 M 2 = (g2 + g002)ν2 (12) Z 4 H 1 M 2 = g2ν2 (13) W 4 H
13 3 Left-Right Symmetric Model
In this section I will give an overview of the left-right symmetric model, including the gauge groups defining the theory and how the particles are arranged into multiplets with respect to these groups. As my main focus in this study is the gauge boson sector, I will give more details on the features of the theory which relate to the gauge bosons and their masses. This includes a detailed account of the scalar sector and the corresponding spontaneous symmetry breaking mechanism. This is followed by diagonalisation of the vector bosons and explicit calculation of their masses.
The LRSM has been studied extensively since the 1970s; for one example of an early review see [11]. The most common and popular version of the model is called the minimal LRSM and is defined by its scalar sector. The usual SM Higgs doublet is replaced by a bi-doublet and two complex triplets are introduced into the theory. One of the complex triplets is right-handed and the other is left-handed. The vev of the right-handed triplet is large, resulting in the gauge 0 ± bosons Z and WR having much larger masses than the SM gauge bosons.
In this paper I am exploring an alternative symmetry breaking pattern, which occurs as a result of modifying the scalar sector. This version of the model still includes a complex right- ± 0 handed scalar field giving large masses to WR and Z . However, I have included an additional ± right-handed scalar which contributes only to the WR mass. This leads to a mass hierarchy, ± 0 where the WR are heavier than Z . Many features of my model are the same as the mini- mal LRSM, and therefore strongly resemble the literature. However, the scalar sector I have explored has not been previously published.
3.1 Gauge Group and Multiplet Structure
The gauge group defining the left-right symmetric model (LRSM) is given by:
G = SU(3)C × SU(2)L × SU(2)R × U(1)Y˜ (14) where the U(1) gauge group has a charge Y˜ different to the standard model hypercharge
Y [12]. In what follows, the SU(3)C group is not always explicitly addressed. This is because the modifications from the SM and the symmetry breaking mechanisms apply predominantly to the electroweak subgroup. Therefore, I mostly refer to the left-right symmetric subgroup as:
GLR = SU(2)L × SU(2)R × U(1)Y˜ (15)
Changes need to be made to the fermion multiplets to allow for the addition of a second SU(2) group. As in the SM, the left-handed fermions are arranged in doublets which transform under
SU(2)L, denoted by: ! ! νL uL LL = QL = eL dL
14 There are three generations of both lepton and quark multiplets. In contrast to the SM, the right-handed singlets are grouped together to give corresponding right-handed doublets which transform under SU(2)R. This requires the addition of a right-handed neutrino, which is not present in the SM. Therefore, the right-handed doublets are represented by: ! ! νR uR LR = QR = eR dR
The LRSM group (15) will eventually be broken down to the electromagnetic group U(1)EM , with associated electric charge Q, just as it is in the SM. Therefore, the electric charge can be related to the generators of each group in (15). Modification from the SM relation in (7) gives:
Y˜ Q = T + T + 1 (16) 3L 3R 2 where the T3L(R) are the third generators of SU(2)L(R). The generators of SU(2)L and SU(2)R 1 ˜ are given by Ti = 2 τi, where τi are the familiar Pauli matrices. Y is the generator, or charge, of U(1)Y˜ . The relation in (16) is a natural extension to (7), but will be explicitly proven in subsequent sections.
The matrix Q acts on the LRSM multiplets to give the electric charges of the fields. Since the fermions have known values of Q, we can use equation (16) to find the values of Y˜ . We see ˜ ˜ 1 ˜ that for leptons Y = −1 and for quarks Y = 3 . This is equivalent to assigning: Y = B − L, baryon number minus lepton number.
Therefore, with respect to each of the three groups in (15), the left-handed and right-handed doublets have the following representations.
LL = LL(2, 1, −1) 1 Q = Q (2, 1, ) L L 3
LR = LR(1, 2, −1) 1 Q = Q (1, 2, ) R R 3 The group structure of the LRSM determines which gauge bosons, and corresponding couplings exist:
~ 1 2 3 SU(2)L : gL, WLµ = (WLµ,WLµ,WLµ) (17) ~ 1 2 3 SU(2)R : gR, WRµ = (WRµ,WRµ,WRµ) 0 U(1)B−L : g ,Bµ
As in the SM, the gauge fields denoted above are the unphysical fields. Mixing between them occurs to produce seven physical gauge bosons. Three correspond to the SM massive vector ± bosons, which are now denoted WL and Z. There are an analogous three massive vector bosons ± 0 arising from the SU(2)R group, WR and Z . Finally, there is also the massless photon.
15 3.2 Scalar Sector
In order to break the group (15) down to U(1)EM , the LRSM requires a scalar sector which is more extensive than the SM. A full description of the symmetry breaking mechanism is given in section 3.5. Here I will give a brief overview of the scalar fields included in the model.
There are 3 scalar fields included in this version of the LRSM:
• Φ1 = Φ1(1, 3, 0)
• Φ2 = Φ2(1, 3, 2)
• Φ3 = Φ3(2, 2, 0)
Φ1 and Φ2 are both triplets under SU(2)R. The only field which is charged under U(1)Y˜ is ˜ Φ2, and in section 3.5 I explicitly deduce the value of Y .Φ3 is a bi-doublet under the direct product: SU(2)L × SU(2)R, and has a similar symmetry breaking effect to the SM Higgs field.
3.3 The Left-Right Symmetric Lagrangian
The LRSM Lagrangian is given by:
L = Lg + Lf + Ls + LY + LHP (18) with subscripts denoting gauge field, fermionic, scalar, Yukawa and Higgs Potential respectively. In the following subsections I will give an overview of each term. Many terms are an obvious extension or modification from the SM Lagrangian, by taking the right-handed doublets as a natural analogy to the left-handed doublets.
3.3.1 Gauge Field Lagrangian
The term Lg describes the kinetic terms for the gauge fields and interactions between them. It is given by: 1 1 1 1 L = − F a F µν − F a F µν − Ga Gµν − B Bµν g 4 Lµν La 4 Rµν Ra 4 µν a 4 µν
The abelian field strength of U(1)Y˜ is:
Bµν = ∂µBν − ∂νBµ
The non-abelian field strengths of SU(2)L, SU(2)R and SU(3)C have the following form:
a a a a b c Fµν = ∂µAν − ∂νAµ − fbcAµAν
a where A = WL,WR and G and fbc are the structure constants of the group.
16 3.3.2 Fermionic Gauge Lagrangian
The fermionic gauge Lagrangian includes kinetic terms for the fermions and interactions be- tween fermion and gauge fields. By putting left- and right-handed fermions on the same footing, it is given by:
¯ µ ¯ µ Lf = ΣΨ=L,Q ψLiγ DµψL + ψRiγ DµψR where the gauge covariant derivatives are given by ∂ + ig ~τ · W~ + ig0 B−L B , for ψ µ L 2 Lµ 2 µ L Dµ = (19) ~τ ~ 0 B−L ∂µ + igR 2 · WRµ + ig 2 Bµ, for ψR.
0 The gauge field corresponding to U(1) transformations is Bµ, with coupling g , and the fields ~ corresponding to left (right) SU(2) transformations are WµL(R) with coupling gL(R)
3.3.3 Scalar Lagrangian
The scalar Lagrangian is formed from the covariant derivatives of the three scalar fields, which give the kinetic terms for the scalars, and interactions between the scalar and gauge fields.
When the scalar vevs are substituted into Ls, the gauge bosons acquire mass terms.
† µ † µ † µ Ls = (DµΦ1) (D Φ1) + (DµΦ2) (D Φ2) + Tr (DµΦ3) (D Φ3)
This Lagrangian will be explained and analysed extensively in sections 3.5 and 3.6.
3.3.4 Yukawa Lagrangian
The Yukawa Lagrangian couples a left-handed fermion to a right-handed fermion and a scalar. If the scalar field has a non-zero vev, then when this vev is substituted into the Yukawa La- grangian, it leads to mass terms for the fermions.
Mass terms for fermions can be of two types; Dirac and Majorana. The most common of the two is the Dirac mass term and in the SM all massive particles have Dirac mass terms. The charged fermions are distinct from their antiparticles and therefore exhibit Dirac mass terms of the form:
¯ ¯ ¯ Lm,Dirac = −mΨΨ = −m ΨLΨR + ΨRΨL (20)
The Yukawa Lagrangian must therefore yield Lm,Dirac after the vevs are substituted in. In our LRSM we have 3 scalar fields to choose from. However, the overall charge of each Lagrangian term must be zero. Therefore, since Y˜ (Ψ)¯ = −Y˜ (Ψ), the scalar field coupled to the fermions ˜ must satisfy Y = 0. The only possible scalar is therefore the bi-doublet field Φ3. The Yukawa Lagrangian leading to fermion Dirac mass terms in the LRSM is:
¯ ψ ¯ ψ ˜ LY (Φ3) = −Σψ=Q,L{ψLiΓijΦ3ψRj + ψLi∆ijΦ3ψRj + h.c}
17 For the leptons, ΓL and ∆L are diagonal matrices of the lepton Yukawa couplings. For the quarks, ΓQ and ∆Q are products of the CKM mixing matrix and the quark Yukawa couplings.
LY (Φ3) results in charged lepton and quark masses just as in the SM, simply by using a bi-doublet as opposed to the SM Higgs doublet. However, a consequence of this is that the neutrinos also gain mass, equal to the corresponding charged lepton. This is counteracted by including Majorana mass terms, which can be formulated to cancel out the neutrino masses, in order to agree with observations.
Majorana mass terms can only be formed from particles which are identical to their anti- particles. This is because Majorana terms mix fermions with their antifermions, which is forbidden by charge conservation and only possible for uncharged fermions. The only fermion for which this is true is the neutrino [13]. Majorana mass terms for the neutrinos are of the form:
c c Lm,Majorana = −mLν¯LνL − mRν¯RνR + h.c (21)
c T Majorana terms require the charge conjugated field: νL = Cν¯L , where the charge conjugation matrix satisfies [14]
T −1 T Cγα C = −γα,C = −C
This means that (21) couples particles to particles, and anti-particles to anti-particles via terms T ˜ such asν ¯LCν¯L . Therefore, the Y charges in the Lagrangian term don’t automatically cancel. Since Y˜ (ν) = −1, we require a scalar field with Y˜ = 2 to ensure the overall term is charge neutral. The only scalar field satisfying this is the charged triplet field Φ2. This field couples only to right-handed neutrinos. The Yukawa Lagrangian giving rise to Majorana mass terms is:
T −1 LY (Φ2) = LRiGRijC iτ2Φ2LRj + h.c
The coupling matrix is denoted GRij. The form of the vev of Φ2 must ensure that no Majorana terms for the charged leptons occur, as they are forbidden.
3.3.5 Higgs Potential Lagrangian
The extended Higgs sector of the LRSM leads to a complicated Higgs potential Lagrangian. This has been analysed for the minimal LRSM in the literature, see for example [15], [16] and [17]. Essentially, the Higgs potential contains all possible Lorentz invariant, gauge invariant, renormalisable combinations of the scalar fields and their derivatives. In my project, I will not be focussing on the Higgs particle spectrum and the Higgs interactions. Therefore, I have not addressed the form of the potential terms, as this would be extensive and wouldn’t contribute to my results.
18 3.4 Left-Right Symmetry as Parity
There is an additional symmetry of the minimal LRSM which makes the theory particularly attractive to study. The minimal LRSM is symmetric under parity transformations, which effectively switch the left- and right-handed fields.
The scalar sector which I have chosen to study is not completely symmetric under these parity transformations, since I have omitted the left-handed complex triplet and added a real right- handed triplet. However, I will briefly outline the concept here for two reasons. It demonstrates that the LRSM has a higher degree of symmetry than the SM, making it an attractive theory to explore. Furthermore, it shows why it is natural and fairly common to take the left- and right-handed coupling constants gL and gR to be equal.
It is known that at the energy scale in nature, parity is not conserved. At the higher energy scale of the LRSM it can be restored, by identifying the LR symmetry as a parity transforma- tion. Consequently, the model is confined a little more. This parity transformation is written as:
~ µ ~ µ WL,R(x) → (µ)WR,L(ˆx)
Bµ(x) → (µ)Bµ(ˆx)
ψ 0 ψL,R(x) → VR,Lγ ψR,L(ˆx)
† Φ3(x) → Φ3(ˆx) where:
xˆ = (x0, −~x), (µ = 0) = 1, (µ = 1, 2, 3) = −1
The parity symmetry is then spontaneously broken, when the LRSM gauge group is broken down to the SM gauge group. This is more theoretically attractive than having to explicitly break parity by treating left- and right- handed multiplets differently. The Lagrangian is only invariant under these transformations if:
ψ † ψ † ψ † ψ † gL = gR =: g (VR ) ΓψVL = Γψ (VR ) ∆ψVL = ∆ψ (22)
The simplest solution is:
ψ † † VL,R = 1 ⇒ Γψ = Γψ, ∆ψ = ∆ψ (23)
An alternative method is to apply a symmetry with respect to charge conjugation, or both parity and charge. For more details on P, C and CP transformations in the minimal LRSM see [18] and [19].
19 3.5 Spontaneous Symmetry Breaking
In this section, I will give a detailed overview of the extended scalar sector of the LRSM model, and how it results in spontaneous symmetry breaking.
There are several possible mechanisms for spontaneously breaking the gauge group of the LRSM down to the observed U(1)EM group. As previously mentioned, the most common method is called the minimal LRSM and the symmetry breaking occurs in two steps. First, the LRSM group is broken down to the SM group, then the SM group is broken down to U(1)EM . This is done using two complex scalar triplets and one scalar bi-doublet. One of the triplets transforms under SU(2)L and the other under SU(2)R. The vev of the left-handed triplet is required to be small for the phenomenological reason that a large vev would contribute significantly to the ± oblique T-parameter, meaning essentially the LRSM Z and WL boson masses can’t differ too much from the observed SM Z and W ± boson masses. Therefore the left-handed triplet vev ± 0 is often taken to be zero. The vev of the right-handed triplet is large, leading to WR and Z ± being heavier than the SM WL and Z. For details of this mechanism, see for example [12], [15] and [20].
However, in this project I will take a slightly different approach. I have explored a scalar ± 0 sector which results in WR being heavier than Z . It will be seen that by achieving this, the LRSM can be linked to the SM with a U(1) extension. One example of such a U(1) extended SM can be seen in the research carried out in [21], from the department in which I conducted my research.
Essentially, I have broken the LRSM group down to the SM group in two steps instead of one. To achieve this, the triplet under SU(2)L is omitted and a real triplet under SU(2)R is introduced. The symmetry breaking steps occur as follows:
1 SU(2)L × SU(2)R × U(1)B−L → SU(2)L × U(1)R × U(1)B−L
As stated in [21] and [22], redefining the gauge fields and scaling the corresponding couplings, allows one to identify:
U(1)R × U(1)B−L ∼ U(1)Y × U(1)Z
This means that after the first symmetry breaking, we have the SM group with a U(1) extension. The additional U(1) group is then broken at the intermediate energy scale to give:
2 SU(2)L × U(1)R × U(1)B−L → SU(2)L × U(1)Y
The effective action of these two steps is breaking the LRSM down to the SM:
SU(2)L × SU(2)R × U(1)Y˜ → SU(2)L × U(1)Y
20 Following this, the final symmetry breaking step results in the following, as in the SM:
3 SU(2)L × U(1)Y → U(1)EM
I will now outline each step in more detail, explaining the extended scalar sector.
3.5.1 Symmetry Breaking: Step 1
In this first step we need to introduce a scalar field which will break down
SU(2)R → U(1)R
This requires a real scalar field in the adjoint representation of SU(2)R, with a trivial repre- sentation of SU(2)L and U(1)Y˜ . I have denoted this field Φ1, and it has the form:
φ1 2 Φ1 = φ , Φ1 = Φ1(1, 3, 0) φ3
1 2 3 The three fields φ , φ , φ are all real. If we were to break down SU(2)R entirely in this step, it a would result in all 3 vector bosons WµR obtaining mass. As I am focussing on the case where ± 0 the charged WµR are heavier than the neutral Zµ boson, this first stage of symmetry breaking, ± which occurs at a higher energy than the subsequent stages, must give mass to only the WµR.
We can therefore deduce some properties of Φ1 by analysing its covariant derivative.
In the adjoint representation of SU(2), the generators are 3 × 3 matrices defined by the totally antisymmetric Levi-Civita tensor:
abc abc TA = −i (24)
Therefore, the covariant derivative:
a a c c b DµΦ1 = ∂µΦ1 + ig(T )abW Φ1 (25) is equivalently written as:
a a ~ ~ a DµΦ1 = ∂µΦ1 − g(W × Φ1) (26)
1 2 After assigning Φ1 a non-zero vev, we require that WµR and WµR obtain masses. Therefore, the cross product tells us that it is the third component of Φ1 which must have the non-zero vev: 0 Φv = 0 (27) 1 √ ν1/ 2
As outlined in [9], it is possible to check the partial symmetry breaking using the following knowledge. For a group G with generators tg which is broken down to a subgroup H with
21 v generators th, the generators of the unbroken group satisfy: thΦ = 0, for some vacuum state v v Φ . Therefore, for our ground state Φ1, it is easy to check that only the third generator of 3 v SU(2)R satisfies: TAΦ1 = 0. This is equivalent to observing that the ground state is no longer invariant under the full SU(2) transformations, but retains invariance under rotations around v the z-axis. This means that Φ1 is invariant under U(1) transformations, which we will denote
U(1)R. This is a useful method to check that the scalar field is responsible for the correct symmetry breaking.
3.5.2 Symmetry Breaking: Step 2
We now require a scalar field Φ2 which will break down the remaining U(1)R group. For this, we introduce a complex scalar field in the adjoint representation of SU(2)R, which also transforms under U(1)Y˜ . This field in fact breaks the entire SU(2)R group, in addition to U(1)Y˜ . At the higher energy scale of Φ1 it is partially broken, then effectively re-broken at the lower energy scale of Φ2, but this time completely. This means that overall Φ2 is breaking:
SU(2)R × U(1)Y˜ → U(1)Y
This scalar field is given by: δ1 ˜ Φ2 = δ2 , Φ2 = Φ2(1, 3, Y ) δ3 where the value of Y˜ will be deduced subsequently. In this case, all three δa are complex scalar fields. It is common to multiply Φ2 by the generators of SU(2) in the fundamental representation to put it into the form: ! 1 δ3 δ1 − iδ2 Φ2 = 2 δ1 + iδ2 −δ3
As Φ2 is in the adjoint representation of SU(2), elements of the Lie algebra act on it as a commutation relation. This means that the electric charge of each complex field can be calculated by:
1 Y˜ QΦ = [ τ , Φ ] + Φ (28) 2 2 3 2 2 2 Computing this gives:
˜ ˜ ! Y δ3 ( Y + 1)(δ1 − iδ2) QΦ = 2 2 2 Y˜ 1 2 Y˜ 3 ( 2 − 1)(δ + iδ ) 2 δ In order for the vacuum to be electrically neutral, the field which takes on a non-zero vev must have an electric charge of zero. This means either Y˜ = 0 and δ3 breaks the symmetry, or Y˜ = ±2 and δ1 ± iδ2 breaks the symmetry. The solution Y˜ = 0 does not completely break the
22 SU(2)R symmetry as required. This is shown by letting each generator act on the vacuum state and observing that the third generator still annihilates the vacuum, i.e isn’t broken. Therefore, we must choose Y˜ = ±2. It was shown in section 3.3.4 that in order to form Majorana mass terms for the right-handed neutrinos we need a right-handed scalar field with Y˜ = 2. This confirms that assigning Φ2 a B − L value of 2 successfully breaks the symmetry. With this charge assignment, it is convenient to reassign the fields as: √ ! δ+/ 2 δ++ Φ2 = √ δ0 −δ+/ 2 It is then simple to identify:
δ1 = δ++ + δ0 δ2 = i(δ++ − δ0) √ δ3 = 2δ+
The vev of Φ2 is therefore: ! v 0 0 Φ2 = √ (29) ν2/ 2 0 where ν2 can be complex valued. In the adjoint representation it is written as: √ ν / 2 2 √ Φv = (30) 2 −iν2/ 2 0
Again, we can check that the remaining symmetry group is U(1)Y by looking for a generator v of U(1)Y with charge Y, such that: Y Φ2 = 0. The most general matrix satisfying this is: Y˜ Y = T R + (31) 3 2
3.5.3 Symmetry Breaking: Step 3
Finally, we require a third scalar field to perform the same symmetry breaking step which occurs in the SM:
SU(2)L × U(1)Y → U(1)EM
This scalar, Φ3, transforms under the direct product SU(2)L × SU(2)R. It is therefore a 2 × 2 matrix of fields which is formed from two SM-like doublets denoted ϕ1 and ϕ2 in the following way [15]:
∗ Φ3 = [ϕ1, ϕ2] (32)
Each of the fields ϕi has the form:
0 ! ϕi ϕi = − (33) ϕi
23 Therefore, overall Φ3 is: ! ϕ0 ϕ+ Φ = 1 2 (34) 3 − 0∗ ϕ1 −ϕ2
The matrix = iσ2 is formed from the second Pauli matrix. This scalar field is in the fundamen- tal representation of SU(2)L and the anti-fundamental representation of SU(2)R. This means that ϕ1 and ϕ2 are both SU(2)L doublets, and the two rows in (32) form SU(2)R doublets [23].
When a field Φαβ has a representation of a direct product of SU(2) groups, the first group acts on the index α and the second group acts on the index β [9]. Putting the bi-doublet in this form means that:
˜ ∗ ∗ Φ3 = Φ3 = [ϕ2, ϕ1]
This effectively exchanges the two doublets inside the bi-doublet.
At this stage, we have already broken SU(2)R × U(1)Y˜ → U(1)Y , so we can check how Φ3 3 Y˜ transforms under the remaining U(1)Y gauge transformations. From the relation: Y = TR + 2 , ˜ 3 and with Y = 0, we are left with Y = TR. Since Φ3 is in the anti-fundamental representation of SU(2)R [24], it therefore transforms as:
† iτ3/2 Φ3 → Φ3UR,UR = e (35)
Substituting in the third Pauli matrix then leads to: ! e−i/2 0 −iτ3/2 Φ3 → Φ3e = Φ3 (36) 0 e+i/2
Each component of Φ3 then transforms as:
0 −i/2 0 ϕ1 → e ϕ1 (37) − −i/2 − ϕ1 → e ϕ1 0 +i/2 0 ϕ2 → e ϕ2 + +i/2 + ϕ2 → e ϕ2
Therefore, the bi-doublet splits into two individual doublets, ϕ1 and ϕ2, which transform under 1 U(1)Y rotations with charges Y = ∓ 2 respectively.
As for the previous two steps, we will choose the electrically neutral fields to obtain vevs. These vevs are denoted ν3 and ν4 and can be complex valued. The overall vev of Φ3 is therefore: √ ! v 2ν3 0 Φ3 = √ (38) 0 2ν4
24 √ Note: the factors of 2 are simply a matter of convention, so I chose the vevs which made the following calculations neater. Individually, we have: √ ! ! v 2ν3 v 0 ϕ1 = , ϕ2 = √ (39) 0 2ν4
These vevs both break down the residual: SU(2)Y × U(1)Y to U(1)EM with electric charge Q. This is easily checked, by applying the charge matrix Q to each doublet, using the charges from (32) and seeing that it annihilates the vev, and satisfies:
Y˜ Q = T L + Y = T L + T R + (40) 3 3 3 2
3.6 Physical Consequences of Symmetry Breaking
One consequence of SSB is that the vector bosons acquire masses. In this project, I am fo- cussing on the extended gauge boson sector of the LRSM, so I will go through this process in detail. I will discuss the neutral and charged gauge bosons separately.
The Ls part of the LRSM Lagrangian is responsible for giving mass to the vector bosons. This is because the covariant derivatives couple the scalar fields to the gauge bosons. Evaluating Ls at the vevs gives the mass relations. I will do this for each term separately in:
† µ † µ † µ Ls = (DµΦ1) (D Φ1) + (DµΦ2) (D Φ2) + Tr (DµΦ3) (D Φ3) (41) where each covariant derivative is given by:
a a a DµΦ1 = ∂µΦ1 + igT WRµΦ1
Y˜ D Φ = ∂ Φ + igT aW a Φ + ig0 B Φ µ 2 µ 2 Rµ 2 2 µ 2 τ τ D Φ = ∂ Φ + ig · W~ Φ − Φ · W~ µ 3 µ 3 2 Lµ 3 3 2 Rµ
The covariant derivatives of Φ1 and Φ2 take on the usual form. The form of DµΦ3 ensures that it transforms as:
† DµΦ3 → ULDµΦ3UR (42)
The fundamental representation of SU(2)L transformations act to the left as usual. The anti- † µ fundamental representation of SU(2)R act to the right. This ensures that Tr (DµΦ3) (D Φ3) remains gauge invariant.
3.6.1 Neutral Gauge Boson Masses
3 3 We begin with the three unphysical, massless bosons: WL,WR,B which mix during the sym- metry breaking process to give the three physical vector bosons: A, Z, Z0. The A and Z are
25 the known SM bosons, and Z0 is the additional boson, with a mass much higher than the SM Z. This is a requirement for phenomenological reasons, to account for the fact that this heavy gauge boson hasn’t been experimentally observed.
After evaluating Ls at the vev, we expect the gauge boson mixing to produce mass terms of the form:
1 2 2 1 2 2 1 2 02 M A + M Z + M 0 Z 2 A 2 Z 2 Z During the spontaneous symmetry breaking, the gauge bosons mix in stages, so it is intuitive to look at the mixing which takes place in each step separately.
Since Φ1 is responsible for the first symmetry breaking step, and only contributes to the charged vector boson masses, we can begin to calculate the neutral gauge boson mixing by looking at the symmetry breaking caused by Φ2:
U(1)R × U(1)B−L → U(1)Y
Evaluating the covariant derivative squared of Φ2 at the vev gives the following mass terms for the neutral gauge bosons:
† µ 2 2 3 3µ 0 2 µ 0 3µ (DµΦ2) (D Φ2)neutral = ν2 (g WRµWR + (g ) BµB − 2gg BµWR ) (43)
3 This induces mixing between the WRµ and Bµ gauge bosons, given by the cross term between them. By rotating the fields it is possible to find the mass eigenstates, which I denote B0 and 03 WR . This is done with a rotation matrix, R, in the following way: ! ! ! B B0 cos θ0 sin θ0 U = ,V = U = RVR = (44) 3 03 0 0 WR WR − sin θ cos θ The mass terms therefore satisfy: 1 1 U T M 2U = V T M¯ 2V RT M 2R = M¯ 2 (45) 2 2 1 2 1 ¯ 2 where 2 M is the mass matrix resulting from (43), and 2 M is a diagonal mass matrix, giv- 0 03 ing the masses of the physical gauge bosons B and WR . It is possible to computationally diagonalise M 2 directly without going through this process, however, this way we can find the mass eigenstates as simple rotations of the unphysical fields, for some angle θ0. Computing the 03 0 calculation in (45) results in the boson WR gaining mass, and the boson B remaining massless. 0 Therefore, B corresponds to the remaining U(1)Y rotations, and is the same vector boson that exists in the SM, see section 2.4.
The mass eigenstates at this stage are then:
0 0 0 03 B = cos θ B + sin θ WR (46) 3 0 0 0 03 WR = − sin θ B + cos θ WR
26 Since rotation matrices satisfy: R−1 = RT , these equations are easily inverted. By rotating the fields in this way, we ensure that they are properly normalised, meaning they satisfy:
2 3 2 ! 0 2 03 2 (B) + (WR) = (B ) + (WR ) (47)
Furthermore, by ensuring that the matrix M¯ 2 is diagonal, we obtain the following result:
g0 cos θ0 = −g sin θ0 (48)
From equation (48) it follows from trigonometry that we can express θ0 in terms of the couplings as: −g0 g sin θ0 = cos θ0 = (49) pg02 + g2 pg02 + g2
Next, we look at the mixing induced by Φ3. This is responsible for the breaking of the elec- troweak group:
SU(2)L × U(1)Y → U(1)EM
The mass terms of the neutral gauge bosons are again given by the covariant derivative evaluated v at the vev Φ3: g2 Tr(D Φ )†(DµΦ ) = ν02W 3W 3 + ν02W 3 W 3 − 2ν02W 3W 3 (50) µ 3 3 neutral 2 L L R R L R
02 2 2 where ν = ν3 + ν4 .
3 At this stage, we can substitute in the relation for WR from (46) to get a mass matrix for 0 3 03 the fields B ,WL,WR . We then repeat the calculations in (45) to achieve a diagonal mass matrix for A, Z, Z0. This case is simply the same as the SM, so we already know that the form of the rotation matrix comes from:
0 B = cos θW A − sin θW Z (51) 3 WL = sin θW A + cos θW Z
3 3 where θW is the usual Weinberg angle. Ensuring again that the mass matrix for A, WL,WR is diagonal leads to the relation: sin θ − sin θ0 = W (52) cos θW By substituting in the expression for B0 from (46) into expression (51), the entire rotation from both steps is then:
0 0 0 B cos θW cos θ − sin θW cos θ sin θ A 3 = (53) WL sin θW cos θW 0 Z 3 0 0 0 0 WR − cos θW sin θ sin θW sin θ cos θ Z
27 In a similar study by [23], it was stated that at this stage there still exists mixing between the Z and Z0 gauge bosons, which must be rotated away. However, this mixing is sufficiently small that it can be approximated by a rotion of the form: ! ! ! Z 1, α Z → Z0 −α, 1 Z0 The angle α satisfies α 1 such that cos α ∼ 1 and sin α ∼ α. The same process of diagonali- 0 sation was used to calculate the angle α. However, since α depends on the parameters ν2, ν , 0 θ and θW , it is more complex to solve and I won’t give the full solution here.
The simplest way to calculate the masses of A, Z and Z0 is by computationally diagonalis- ing the overall mass matrix for the vector of initial fields, which is:
g2ν02 −g2ν02 0 B M 2 = 2 02 2 02 2 2 0 2 for basis 3 (54) −g ν g ν + 2g |ν2| −2gg |ν2| WL 0 2 0 2 2 3 0 −2gg |ν2| 2(g ) |ν2| WR The mass terms for the three mass eigenstates are then:
2 MA = 0 (55) q 2 2 2 02 2 2 02 4 04 2 02 2 2 02 4 4 04 MZ = g ν2 + g ν2 + g ν − ν2 g + 2g g ν2 (ν2 − ν ) + g (ν2 + ν ) q 2 2 2 02 2 2 02 4 04 2 02 2 2 02 4 4 04 MZ0 = g ν2 + g ν2 + g ν + ν2 g + 2g g ν2 (ν2 − ν ) + g (ν2 + ν )
There is one problem remaining, which is that we have introduced a new angle θ0, in addition 0 to the unknown U(1)B−L coupling g . It is more useful to express everything in terms of known SM values, which have been tested. We have already related the angle θ0 to the known Wein- 0 berg angle θW in equation (52). We can also relate the LRSM coupling g to the known SM couplings g and g00. See section 2.4 for a reminder of the notations.
There is a simple way to relate the different couplings to each other, which has been outlined in [22]. As explained previously, after breaking down the symmetry U(1)R × U(1)B−L → U(1)Y , we have identified the gauge boson B0 as the boson corresponding to the hypercharge group.
We therefore know how DµΦ3 should look: ~τ D Φ = ∂ Φ + ig · W~ Φ + ig00YB0Φ (56) µ 3 µ 3 2 L 3 3
The hypercharge of Φ3 was calculated in section 3.5.3. Comparing equation (56) to the original covariant derivative in equation (42) gives the following relation:
g sin θ0 = −g00 (57)
Combining this with our previous relation for sin θ0 in equation (49) gives the following: g00 g0 = (58) q 00 g 2 1 − ( g )
28 0 Using this relation, it can be seen that in the limit that ν2 ν , the mass of the SM Z boson is approximately in accordance with the SM mass:
2 02 2 002 MZ ' ν (g + g )
02 0 1 This is provided that we identify ν with the SM Higgs vev in the following way: ν = 2 νH . The mass of the additional heavy neutral gauge boson, Z0, is then approximately:
4 2 2 2g ν2 02 2 002 M 0 ' + ν (g − g ) Z g2 − g002
2 0 The contribution from ν2 is what gives Z a much larger mass then Z.
3.6.2 Charged Gauge Boson Masses
We can now evaluate LS at the vacuum expectation values to find the mass terms for the ± ± charged gauge bosons, WL and WR . The results are very similar to the minimal LRSM, for which there is extensive literature. It is therefore not necessary to perform the same procedure as for the neutral gauge bosons. Here, I will outline the results, taken from [12].
v v v Substituting Φ1,Φ2 and Φ3 into LS gives the following mass terms.
† µ 2 2 + − (DµΦ1) (D Φ1) = g ν1 WR WR (59) † µ 2 2 + µ− (DµΦ2) (D Φ2)charged = g ν2 WRµWR † µ 2 02 + − 02 + − ∗ + − ∗ + − Tr (DµΦ3) (D Φ3) charged = g ν WL WL + ν WR WR − 2ν3 ν4WL WR − 2ν3ν4 WR WL
02 2 2 where again ν = ν3 + ν4 . The charged gauge bosons take on the usual form: 1 W ± = √ (W 1 ∓ W 2) (60) 2 By identifying:
(W 1)2 + (W 2)2 = 2W +W − we see that our mass terms for the charged bosons are of the form: ! ! W − ν02 −2ν∗ν + + M 2 L ,M 2 = g2 3 4 WL WR − ∗ 2 2 02 WR −2ν3ν4 ν1 + ν2 + ν
± 2 2 2 In the first two steps, no mixing occurs, and the WR simply acquire masses of g (ν1 + ν2 ). In the third step, the left and right handed bosons mix. This mixing can be described by the rotation: ! ! ! W + cos ζ − sin ζeiλ W + 1 = L (61) + −iλ + W2 sin ζe cos ζ WR
29 This leads to the results:
2 ∗ ! iλ ν3ν4 2|ν3ν4| M1 e = − , ζ ' 2 2 |ν3ν4| ν3 + ν4 M2
The L-R mixing angle of the charged gauge bosons has been shown to be small [25], meaning ± ± W1 are essentially the standard model WL , with mass:
M 2 = g2ν02 (62) WL
I have followed this approach, assuming the L-R mixing is negligible, which is very common in LRSM literature, see for example [26], [27] and [28].
By comparing M 2 to the SM case in equation (13) it is simple to confirm that ν0 can be WL related to the Higgs vev νH , by the relation: 1 ν0 = ν (63) 2 H The mass of the heavy charged gauge boson is then:
M 2 = g2(ν2 + ν2 + ν02) (64) WR 1 2
30 4 Running Simulations of the LRSM
The second stage of this project has been to implement the features of the LRSM which I studied into the program FeynRules, which can calculate the tree-level Feynman diagrams for all possible interactions. This was then interfaced with the program MadGraph, which generates events at the LHC and calculates the cross-sections.
4.1 FeynRules
FeynRules is a program written in Mathematica, which allows a model of particle physics to be implemented by specifying certain aspects of the theory. These include the gauge groups, quantum fields and their represenations, parameters and the Lagrangian. The program then cal- culates the underlying Feynman rules of each vertex. FeynRules is used commonly in the field of particle physics, predominantly in searches for new physics. It is vital for theorists to make predictions of these new physics models, which can be compared to experimental data. In order for cross-sections to be determined, first the Feynman diagrams and rules must be calculated. It is common that thousands of diagrams need to be calculated, which is where computational methods become crucial. FeynRules makes it possible to calculate the Feynman diagrams quickly and without error, making it a useful tool in new physics searches [29], [30].
The program operates on an algorithm which uses the canonical quantisation formalism to derive Feynman rules for both renormalisable and effective theories. The theory is written into a model file, then run through a Mathematica notebook with the FeynRules package loaded. FeynRules contains a number of predefined notations, simplifying the process of writing the model file. There are also defined functions allowing the user to run checks on the model. These include checking hermiticity, normalisation and diagonal mass terms.
A FeynRules program file for the SM has been written by the following authors: Claude Duhr, ETH; Neil Christensen, Michigan State University; Benjamin Fuks, IPHC Strasbourg/University of Strasbourg. In addition to this, a number of BSM files have been written, including super- symmetric and dark matter theories. In this project, I took the SM file as a starting point, and modified it in order to create a basic LRSM file.
In the following I will outline the necessary parts of the model file which I used to create my LRSM program.
4.1.1 Gauge Groups
The group structure of each gauge group in the LRSM needs to be specified.
GLR = SU(3)C × SU(2)L × SU(2)R × U(1)Y˜
31 As an example, the group SU(2)L is input into the model file as: SU2L == { Abelian -> False, CouplingConstant -> gw, GaugeBoson -> WLi, StructureConstant -> Eps, Representations -> {Ta,SU2D}, Definitions -> {Ta[a , b , c ]->PauliSigma[a, b, c]/2, FSU2L[i , j , k ]:> I Eps[i, j, k]} }
The group is defined as abelian or non-abelian and has a corresponding coupling constant and gauge bosons. The structure constant is also defined, for the case of SU(2)L using the pre- defined antisymmetric Levi-Civita tensor. Finally, the group can be given a representation, here the fundamental representation, called Ta with indices specified by SU2D. This is defined using the Mathematica function ”PauliSigma”. The adjoint representation is also specified and named FSU2L.
In addition, the groups U(1)Y˜ , SU(2)R and SU(3)C were input in a similar fashion.
4.1.2 Indices The model is dependent on the group and multiplet structure of the theory, and it is therefore necessary to specify all indices. Again, using SU(2)L as an example, the index SU2D was used, where the letter D refers to doublet. Fields which transform under SU(2)L are arranged into doublets, and SU2D = 1 (2) refers to the top (bottom) field in the doublet. An additional index 1,2,3 1,2,3 is needed to describe the three gauge bosons WL , or similarly the three generators τ . This can be achieved by defining the index SU2W, taking on values 1,2 and 3. In FeynRules, this is achieved in the following way.
IndexRange[Index[SU2W]] = Unfold[Range[3]] IndexRange[Index[SU2D]] = Unfold[Range[2]]
Indices for SU(2)R are defined in the same way, denoted SU2DR and SU2WR. It is necessary to also define indices for the generation, colour and gluons.
4.1.3 Fields The implementation of quantum fields into FeynRules is done in two steps; the physical mass eigenstates and the unphysical states. As an example, I will outline how this was achieved for the gauge bosons, since this was my main focus for the project.
The gauge boson mass eigenstates are simple to define, by giving them a name, mass and width. It is also necessary to specify if the particle is self-conjugate, i.e if it is charged or uncharged. The unphysical gauge bosons are then defined in terms of the mass eigenstates. In i order to do this, I used my results from section 3.6. As an example, the three vector fields WR
32 are input in the following way:
V [13] == { ClassName -> WRi Unphysical -> True SelfConjugate -> True Indices -> {Index[SU2WR]} FlavorIndex -> SU2WR Definitions -> {WRi[mu ,1]->(WRbar[mu]+WR[mu])/ Sqrt[2], WRi[mu ,2]->(WRbar[mu]-WR[mu])/ I*Sqrt[2], WRi[mu ,3]->-sp*cw A[mu] + (sp*sw - aa*cp) Z[mu]} }
In FeynRules notation, V[13] specifies that we have a vector field. Each type of field is given a number to identify it. Each of the fields was then defined with respect to the physical fields, + − using the notation WR = WR , W Rbar = WR . The mixings are denoted using the notation: 0 0 0 sp = sinθ , cp = cos θ , sw=sinθW , cw = cosθW . The small Z − Z mixing is denoted aa.
In addition to the vector fields, the fermion and scalar fields are included in the model file in a similar way.
4.1.4 Parameters
All the parameters needed to specify the LRSM need to be supplied to the FeynRules model file. There are two types of parameter: external and internal. The external parameters have fixed values, so are often chosen to be parameters that are experimentally measured to a high precision and accuracy. The internal parameters can then be written in terms of the external, and also other internal parameters.
In my project, the masses of the gauge bosons are of particular interest. They depend on the vevs of Φ1 and Φ2 and these dependencies are given in equations (55), (64) and (62). I inverted these equations in order to have the gauge boson masses as fixed variables, and include them as external parameters.
4.1.5 Lagrangian
The final ingredient required to run FeynRules is the Lagrangian, which was specified in section 3.3. FeynRules contains a number of pre-defined functions which make this task much simpler. This includes functions which automatically calculate the covariant derivative and the field strength of a field.
33 4.2 MadGraph
After producing a FeynRules model and running the program to calculate the Feynman di- agrams, the results can be interfaced with a second program called MadGraph, which is an LHC event simulation software written in Python. This is done by generating a Universal FeynRules Output, or UFO. The UFO contains the necessary files to run LHC simulations of the model in MadGraph [31].
MadGraph allows the user to load any model, and generate an event, such as a decay or a 2 → n scattering. The program then recomputes the Feynman diagrams and generates the code needed to compute the matrix elements for the event. MadGraph includes a package called MadEvent, which then calculates the cross-section for the event.
Within MadGraph, it is possible to vary external parameters which were defined previously in FeynRules. By running MadGraph a number of times over a range of parameter values, the cross section can then be found as a function of this parameter. In this case, the mass of ± WR was varied and the cross section for its production and decay calculated with MadGraph. This is outlined in more detail in section 5.
34 5 Comparison of Simulations with CMS Data
The final step in my project is to test the LRSM model which I studied and implemented into FeynRules. This can be done by calculating the cross-section for a particular interaction and comparing it with experimental data from the LHC for the same interaction.
Each year the Particle Data Group (PDG) publish reviews and tables, which include sum- maries of the mass, decay channels and decay widths of particles. In addition to updates for the known particles, the PDG also include searches for BSM particles. The non-SM heavy, 0 ± charged particles are often denoted W , which correspond to the right-handed WR in the LRSM. I used the most recent PDG review to look up searches for W 0 through various decay channels. The review provided references to the CMS publications, which give plots of the cross-section for the decay of W 0 as a function of its mass. This was the experimental data which I compared with data from my simulations to test my model.
5.1 Production and Decay of W boson
± There are several possible decay channels of WR . Neglecting potential vertices involving the extended scalar sector, the main decays which can occur are:
± ± • WR → hW
± ± • WR → W Z
± • WR → `ν`
± • WR → N`
± • WR → qq¯
In the list above, h corresponds to the SM Higgs boson, W ± and Z to the SM gauge bosons and N to right-handed neutrinos.
+ ¯ − ¯ The interaction which I have chosen to study is WR → tb and the conjugate decay WR → bt. + The tree-level diagram for the production of WR from proton-proton collisions and its decay into a top and an anti-bottom quark is given in Figure 1.
0 + ¯ Figure 1: Leading order diagram of production of W = WR and its decay to tb [32]
35 5.2 Cross-section and the Narrow Width Approximation
± The quantity being measured at CMS is the cross section: σ(pp → WR → tb). This can be simplified, using the narrow width approximation (NWA), to rewrite:
± ± ± σ(pp → WR → tb) ' σ(pp → WR ) × BR(WR → tb) (65)
This relation is only true in certain cases. The scattering energy must be much larger than ± the mass of the WR , which in turn must be much larger than the mass of the decay products: √ ± s MWR Mt,b. The total width of the WR must be much smaller than its mass. Pro- vided these conditions are met, the NWA assumes that the WR± is produced as an asymptotic state, so there is no interference before and after production, meaning its decay can be treated separately from its production [33].
± ± For W it is assumed phenomenologically that MW Mt,b, since the large mass of W is R R √ R the reason it has not been experimentally observed to date. The requirement that s MW √ R means that at the centre of mass energy currently reached at the LHC of s = 13 TeV, we can ± realistically only apply the NWA in searches for WR with a mass on the order of several TeV. For a particle with a mass on the TeV scale, the requirement that the mass be much greater than its total width will be satisfied. Therefore, it is appropriate in this study to use equation ± (65) to calculate the cross-section for the decay of WR .
5.3 CMS Search for W Boson
I am using a study published by the CMS Collaboration using data collected by the CMS experiment in 2016 [34]. The study looks for heavy resonances decaying to a top and bottom √ quark. The proton-proton collisions occur at a center of mass energy s = 13 TeV and data was collected with an integrated luminosity of 35.9 fb−1. The paper looks for final states of lepton+jets, meaning the SM W boson in Figure 1 decays to a lepton and neutrino and the + ¯ ¯ jets originate from the two bottom quarks. The overall decay is then WR → tb → bb`ν`.
In the CMS study, the 95% confidence level (CL) upper limit on the cross section of W 0 was calculated. A plot was included of both the observed and expected value of σ(pp → ± ± ± WR ) × BR(WR → tb) as a function of the mass of WR . I used a program called EasyNData to extract the data points from the CMS plot [35]. For a detailed description of the signal and background modelling and event selecton, see the paper [34].
5.4 Putting Mass Limits on W Boson
In order to compare my model to the CMS data, I then calculated the product σ(pp → ± ± ± WR ) × BR(WR → tb) using the simulations. With FeynRules I calculated BR(WR → tb) as a function of MWR . Using the FeynRules UFO, I then ran the model within MadGraph ± to calculate σ(pp → WR ), also as a function of MWR . Although versions of both FeynRules
36 and MadGraph now allow next to leading order (NLO) calculations, this requires a more in depth model file in addition to longer running time. An alternative method is to simply multiply the cross-section by 1.3 to account for the NLO effects, which is the approach that I have taken.
After acquiring both experimental data and the results from my simulations, I plotted both using gnuplot. In Figure 2, I show both the CMS results and the results from my simula- tions. I have also included the ± 1 and ± 2 standard deviation uncertainties on the expected cross-section, taken from the CMS paper.
Figure 2: Cross section for pp → W 0 multiplied by branching ratio of W 0 → tb as a function of the mass of the 0 0 ± W . In this case, the W is a right-handed WR gauge boson. The black lines show expected and observed 95% CL upper limit from CMS data. The green and yellow bands show the ± 1 and ± 2 standard deviations on the expected limit, respectively. The red line shows the results from my simulations.
By comparing the simulation results with the CMS data, I can put a lower bound on the mass of a right-handed heavy gauge boson based on my version of the LRSM. This is done by ob- serving from Figure 2 where the cross-section from simulations (red line) meets the observed cross-section from CMS data (solid black line). The mass at which this occurs is the lower limit for MWR , and we can exclude a right-handed charged gauge boson with a lower mass than this. ± Using this approach, Figure 2 shows that the experimental data excludes a WR boson with a mass lower than approximately 3 TeV.
This result is in agreement with similar searches for a heavy W 0 gauge boson published by the PDG [36]. Typically, CMS data puts lower bounds on the W 0 mass between 2-3 TeV.
37 6 Conclusion
In this master thesis project, I successfully carried out a study of the LRSM. This included an overview of the theory, implementation into a model and comparison of my theoretical simulations with real LHC data. This allowed me to put a lower limit on the mass of the ± charged right-handed gauge boson WR in the region of 3 TeV.
6.1 Future possibilities in this project
± Further limits on the mass of WR could be found by carrying out a similar analysis to that ± in section 5. In this project I only looked at the decay of WR to a top and a bottom quark. However, any of the decays in section 5.1 could be tested and compared to CMS data, to back up the results obtained in this study. In particular, several papers have been published by the ± ± CMS Collaboration, searching for WR through the decay WR → `ν`.
Furthermore, the Z0 boson decays could be tested in a similar way, to put limits on the mass of the Z0 boson. This could then be compared to results from studies of the SM with a U(1) extension, which result in a neutral heavy gauge boson. The initial motivation for this project came from the paper [21], which looked at such a U(1) extended model and put mass limits on 0 Z . Therefore, it would be possible to compare their results with constraints on MZ0 calculated from my study of the LRSM.
Another area for future study is the scalar sector of the LRSM. The scalar sector of the minimal LRSM has been studied, however the scalar sector introduced in this thesis has not. In order to explore this, the entire scalar Lagrangian would need to be implemented into the FeynRules model file. This could then be used to compare with CMS data for non-SM Higgs searches.
In section 3.4 I stated how left-right symmetry can be viewed as a parity transformation in the minimal LRSM provided that the left and right-handed gauge couplings are taken to be equal. However, with the alternative scalar introduced in this paper, it is not possible to implement this left-right symmetry as a parity transformation. Therefore, it is not necessary to assume that gL and gR are equal, as I have done. Therefore, the right-handed gauge coupling gR could be varied to test the implications this has on the model.
6.2 Future work beyond this thesis
As mentioned in section 1.2, the ultimate goal in BSM physics is to find a GUT which can be broken down in successive symmetry breaking steps to the SM. The LRSM is a ”bottom-up” approach, where the SM gauge group is extended with a right-handed SU(2)R group and the hypercharge Y is modified to B − L. A natural next step is to look for a way to extend the LRSM group, and explore the symmetry breaking at this higher energy scale, where the LRSM
38 becomes an effective theory.
One limitation in searches for new particles is the maximum centre of mass energy that can be √ reached at accelerators. The current LHC value of s = 13 TeV sets an upper limit of ∼ TeV on the mass of particles which can be produced. Therefore, BSM theories with particles more massive than this can’t be tested until the centre of mass energy is increased. This means that future developments of particle accelerators play a crucial role in probing BSM physics [37].
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