Localization of Quantum Fields in a modified Randall-Sundrum scenario

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Mathematical Physical and Natural Sciences

Master's Degree in Physics

Localization of Quantum Fields in a

modied Randall-Sundrum scenario

Candidate:

Massimo Bedini

Prof. Roberto Contino

Supervisor:

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Chapter 1 Introduction

The Standard Model of Particle Physics (SM) is a theory of elementary par-ticles and their fundamental interactions formulated during the 1960s and the 1970s. The SM has been widely tested over the past decades and it has been extraordinarily accurate in its predictions [1]. Although it succesfully describes the strong and electroweak interactions, it fails to integrate grav-ity in its formulation. It contains two major theories: quantum chromody-namics (QCD), which describes the strong force, and the electroweak theory, which is the unied description of the electromagnetic and the weak interac-tions [2]. Specically, it's a non-abelian gauge theory with the symmetry group SU(3)C⊗SU(2)L⊗U(1)Y, where the mass of each particle is generated via the Higgs mechanism, by introducing a scalar eld known as the Higgs boson [3].

1.1 The Standard Model

The Standard Model is a non-abelian gauge theory, which is a quantum eld theory based on the principle of local gauge invariance. Let L(φ(x), ∂µφ(x)) be a Lagrangian invariant under the global transformations of a certain group G:

φ(x) → U φ(x), (1.1)

where U is a matrix representation of the group transformations. To make the theory invariant under the local transformations

φ(x) → U (x)φ(x), U = eiωa(x)Ta, (1.2) we must replace the ordinary derivatives ∂µby the covariant derivatives Dµ = ∂µ− igAaµTa, which have the property to transform as the eld itself:

Dµφ(x) → (Dµφ(x))0 = U (x)(Dµφ(x)). (1.3) We introduced a new set of vector (or gauge) elds Aa

µ. The constant g is the coupling constant of the group and the operators Ta _{are the generators of the} symmetry group, which obey to the commutation rules:

h Ta, Tb

i

= ifabcTc, (1.4)

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CHAPTER 1. INTRODUCTION 2 where the numbers fabc _{are the structure constants. Under the same group} G, in order to preserve the invariance, the newly added gauge elds must transform in a specic way:

Aa_µ→ Aa_µ0 = U (x)Aa_µ+ i g∂µ

U†(x). (1.5)

To complete the transition to a locally invariant gauge theory we need a term that depends on the gauge eld Aµand its derivatives that is locally invariant, to serve as a kinetic term. Thus, we dene the eld strength tensor Fµν as:

F_µνa = ∂µAaν− ∂νAaµ+ gf abc_Ab

µA c

ν, (1.6)

and see that the term Fa

µνFa,µν satises these conditions. Therefore, the new gauge invariant Lagrangian is:

L = L(φ(x), Dµφ(x)) − 1 4FµνF

µν_. _(1.7)

The introduction of the gauge elds not only makes the theory locally invariant, but denes the interaction between the matter elds and the vector elds and the interaction of the vector elds with themselves (in the non-abelian case).

The Standard Model is also a renormalizable theory [4]. Renormalizable means that it's a teory in which the innities, that arise from loop calculations, can be removed by redening a nite number of physical parameters. The Lagrangian must obey to certain rules if we want it to describe a renormalizable theory, for example we consider:

L ⊃ λ · O. (1.8)

λ is a parameter with positive mass dimension and O a certain operator built out of the elds. Since the mass dimension of a 4D Lagrangian is 4, only operators with mass dimension d ≤ 4 are allowed in the Lagrangian; operators with higher mass dimension lead to non-renormalizable interactions.

It's important to note that, nowadays, non-renormalizable theories are ac-cepted as theories that can describe nature correctly [5] [6]. There can be operators with mass dimension greater than 4 and they can contribute, but their eects are numerically suppresed by powers of the fundamental scale, which can be as large as the Planck scale MP l, much larger than the energies achievable experimentally today.

1.2 Structure of the Standard Model

As mentioned previously, the Standard Model is a gauge theory with gauge group SU(3)C⊗SU(2)L⊗U(1)Y. Let's now examine the gauge and the fermion sectors of the model.

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CHAPTER 1. INTRODUCTION 3

1.2.1 Gauge sector

The total number of generators of the gauge symmetry group is 1 + 3 + 8 = 12. For this reason the gauge sector contains 12 gauge elds, which mediate the interactions between the fermion elds. They appear in the Lagrangian through the covariant derivatives:

U (1)Y : Bµν = ∂µBν − ∂νBµ, SU (2)L : Wµνa = ∂µWνa− ∂νWµa+ g2abcWµbW c ν (1.9) SU (3)C : Gaµν = ∂µGaν − ∂νGaµ+ g3fabcGbµG c ν, (1.10)

where Bµ, Wµa (a = 1, 2, 3) and Gaµ (a = 1, ..., 8) are respectively the elds belonging to the weak-hypercharge U(1)Y, the weak-isospin SU(2)L and the color symmetry group SU(3)C. The Lagrangian then reads

L_Gauge = −1 4BµνB µν ₋1 4W a µνW a,µν ₋1 4G a µνG a,µν . (1.11)

1.2.2 Fermion sector

The fermions are organized in families. Each family consists of a charged lepton and a massless neutrino or a up and down quark. Since the gauge group of the SM contains the group SU(2)L, fermions are usually divided into left-handed fermions, which are doublets with respect to SU(2)L, and right-handed fermions, which are singlets with respect to SU(2)L. The former are given by: E_Li = νe e ! L , νµ µ ! L , ντ τ ! L ! , Qi_L= u d ! L , c s ! L , t b ! L ! . (1.12)

The latter by:

ei_R =(eR, µR, τR),

ui_R =(uR, cR, tR), (1.13) di_R =(dR, sR, bR).

With these elements we write the Lagrangian of the fermion sector:

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CHAPTER 1. INTRODUCTION 4 where we used the standard Feynman notation /D = Dµγµ. The covariant derivatives generate the interactions between the fermions and the gauge elds:

Dµ_E = ∂µ− ig1YLBµ− ig2 σa 2 W a,µ_, Dµ_Q = ∂µ− ig1YQBµ− ig2 σa 2 W a,µ_{− ig} 3taGa,µ, Dµ_e = ∂µ− ig1YeBµ, Dµ_q = ∂µ− ig1YqBµ− ig3taGa,µ, (q = u, d). (1.15) We introduced the generators of the symmetry group: σa_{are the Pauli matrices} for SU(2)L, taare the Gell-Mann matrices for SU(3)C and Y is the hypercharge for U(1)U.

1.3 Higgs mechanism

Introducing mass terms for the chiral fermions and gauge bosons is not straight-forward, because gauge invariance does not allow such terms. The solution is provided by the Higgs mechanism, which is based on the concept of sponta-neous symmetry breaking. We add a complex scalar eld Φ (the only one in the SM) to the theory, which is a doublet with respect to SU(2)L and has hypercharge Y = 1/2. This can be done by adding the following term to the Lagrangian:

L_Higgs= (DµΦ)†(DµΦ) − V (Φ), (1.16) where the potential has the form:

V (Φ) = −µ2Φ†Φ + λ 2

Φ†Φ2. (1.17)

If λ and µ2 _{are positive, the potential develops a non-zero vacuum expectation} value, linked to a set of phisically equivalent minima satisfying the equation:

Φ†Φ = µ 2 2λ ≡

v2

2. (1.18)

Choosing one of these minima (spontaneously) breaks the symmetry: even though the Lagrangian is symmetric under the group, the ground state is not. With a SU(2) rotation we can make the lower component of the Higgs doublet the only one to have a VEV dierent from zero:

hΦi = √1 2 0 v ! . (1.19)

With this choice, the gauge group SU(2)L⊗U(1)Y is said to be broken to the remaining U(1)EM.

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CHAPTER 1. INTRODUCTION 5

1.3.1 Fermion masses

To produce mass terms for the fermions, we add additional interactions be-tween the eld Φ and the fermion doublets. These new terms are called the Yukawa interactions and are allowed by the symmetry group:

L_Yukawa= − ¯E_LiΦ Y_eijej_R− ¯Qi_LΦ Y_dijdj_R− ¯Qi_LΦe Y_uijuj_R+h.c., (1.20) where the Y are the Yukawa matrices, 3 × 3 complex matrices, andΦ = iσe ₂Φ. With the spontaneous symmetry breaking this Lagrangian becomes

L_Yukawa = −√v 2¯eLYeeR− v √ 2 ¯ dLYddR− v √ 2u¯LYuuR+ .. (1.21) Diagonalizing the Yukawa matrices and nding their eigenvalues we nd that the mass terms for the fermions are proportional to the Higgs vacuum expec-tation value v: mf = λfv, where the λf are the so-called Yukawa couplings.

1.3.2 Gauge bosons masses

The kinetic term for the Higgs eld gives mass terms to the gauge bosons. The Higgs Lagrangian, evaluated at the vacuum expectation value of Φ, contains:

L_Higgs ⊃ 1 2 0 v g2Wµa σa 2 + 1 2g1Bµ ! g2Wµb σb 2 + 1 2g1Bµ ! 0 v ! . (1.22) Rotating and combining the four elds we obtain a massive complex vector eld, a massive real vector eld and a massless real vector eld:

W_µ± =√1 2(W 1 µ ∓ iW 2 µ), with m 2 W = 1 4g 2 2v 2_,

Z_µ0 = W_µ3cos θW − Bµsin θW, with m2Z = 1 4(g 2 1+ g 2 2)v 2 , Aµ= Wµ3sin θW + Bµcos θW, with m2A= 0, (1.23) where θW is the so-called Weinberg angle:

cos θW = g2 pg2 1 + g22 , sin θW = g1 pg2 1 + g22 . (1.24)

The fact that the eld Aµ, the one associated with the photon, is massless is a consequence of the particular breaking pattern of the gauge theory: SU(2)L⊗ U(1)Y →U(1)EM.

1.4 Beyond the Standard Model

The fact that the SM is not able to describe gravity is not the only theoretical deciency of the theory. There are few shortcomings that lead us to believe

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CHAPTER 1. INTRODUCTION 6 that there must be a larger theory, a theory beyond the Standard Model. For example neutrinos are assumed to be massless, but this was shown to be not true by recent experiments [7]. Also, the theory fails to explain why fermions have the masses they do and to unify all the known interactions in nature [8]. Moreover, in the SM, there is the hierarchy problem: there is a very large dierence between the electroweak symmetry breaking scale v (∼ 246 GeV) and the fundamental quantum gravity scale MP l (∼ 1019 GeV) [9]. This problem can be seen as a lack of technical naturalness, because by computing the quantum corrections to the mass of the Higgs boson one can easily see that they are of order of the cuto scale of the theory: δm2 _{∼ Λ}2 _{(which can be as} high as the Planck scale). Since the mass of the Higgs boson is approximately 125 GeV, this implies that there is a ne tuning in the theory, where the quantum corrections balance in a way that the physical mass is much lower of MP l.

Supersimmetry (SUSY) is the most studied and possibly the most appealing option for solving the hierarchy problem: SUSY is a spacetime symmetry that mixes bosons with fermions (and vice versa) and it removes the power-law divergences of the quantum corrections by introducing superpartners for all the SM particles, making the corrections only logarithmically divergent [10]. In an exact supersymmetry each pair of superpartners would have exactly the same mass. Since none of these new particles has been found yet, it means that, if this symmetry exists, it must be spontaneously broken, allowing superpartners to dier in mass. Unfortunately, at this time there's still no evidence to show whether or not supersymmetry is correct [11].

Another way to approach this problem is by using the so called anthropic principle, which states that observations of the universe must be compatible with the conscious and sapient life that observes it [12]. For example, if the value of the vacuum expectation value of the Higgs boson were too large, the formation of the structure of our universe could be altered in a way that life wouldn't be possible. The same kind of arguments can be used to explain many other features of the SM, such as the value of the cosmological constant Λ, or the values of the parameters me, mu and md. From a phenomenological point of view, this isn't very satisfying, as it doesn't tell us what we should expect from experiments.

An alternative way to solve the hierarchy problem is to postulate the existence of additional spacetime dimensions. One of these extra dimensional models is the Randall-Sundrum model [13], which is the subject of the next chapter.

1.5 Contents

In the second chapter of this thesis, The Randall-Sundrum model, we intro-duce the Randall-Sundrum model, showing how its warped metric can solve the Hierarchy Problem through an exponential suppression of the energy scales. We show in detail the Kaluza-Klein decomposition for the scalar, gauge and

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CHAPTER 1. INTRODUCTION 7 fermionic elds and solve the equations for their wave functions along the extra dimension.

The third chapter, The RS model with an intermediate brane, concerns a variation on the RS scenario in which one introduces a (3 + 1)-dimensional imperfection in the 5D spacetime, a 3-brane. This allows us to take advantage of the fact that dierent kind of elds can propagate dierent amounts into the warped extra dimension. This new feature makes it possible to have a Higgs compositeness scenario at an energy scale higher than 1 TeV; in particular, if the energy scale relative to the intermediate brane is ∼ O(10) TeV, this exten-sion makes this modied model consistent with all the preciexten-sion experiments to this day. We show the mathematical implications that this extension has on the wave functions of the elds. Furthermore, we introduce localized brane terms (mass and kinetic) and study their eects on the spectrum of the KK modes and the shape of their proles.

In the fourth chapter, Localization of the elds, we show how one can achieve the localization of the elds in one region of the 5D bulk by tweaking the localized mass terms on the branes. It is shown that it is possible to localize the massless modes as well, if certain conditions are satised. We present a study of the spectrum and show the wave functions of the lightest KK modes. In the last chapter we oer a brief summary of the results and some con-cluding remarks about this extension of the Randall-Sundrum model.

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Chapter 2 The Randall-Sundrum model

The Randall-Sundrum model (RS), proposed by Lisa Randall and Raman Sun-drum in 1999 [14], is a 5-dimensional warped geometry theory where the extra dimension is spatial and compact. This compact extra dimension is not the Kaluza-Klein circle, but a circle of radius R with identity between upper and lower half (Fig. 2.1): (xµ_{, φ) = (x}µ_{, −φ)}, where xµ is the 4-dimensional coor-dinate and φ is the angular coorcoor-dinate for the fth dimension, with values in [0, π].

Figure 2.1: The S1_/Z

2 orbifold. In other words, we are working in the space S1_/Z

2, where S1 is the circle group and Z2 is the multiplicative group {−1, 1}. The Z2 group identies the point φ with the point −φ, eectively making the circle a line with two xed points, one at φ = 0 and the other one at φ = π. A circle with coordinate identication is called an orbifold. The functions f(φ) dened on this set are then required to be periodic, i.e. f(φ) = f(φ + 2π), and to be even, f (π) = f (−π). Two 3-branes, one sitting at φ = 0, the other at φ = π, enclose a 5-dimensional bulk (2.2). The former is commonly referred to as the Planck brane (or UV brane), the latter is called the TeV brane (or IR brane).

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 9

Figure 2.2: The 5-dimensional bulk.

2.1 Warped metric

Unlike the large extra dimensions model [15] (also known as the ADD model), the RS model considers a warped 5-dimensional metric. It's mandatory, if we want to describe actual physics, that the model preserves the Poincaré invariance in the standard 4-dimensional space. For this reason the following metric is considered:

ds2

= e−2σ(φ)ηµνdxµdxν − R2dφ2, (2.1) where ηµν =diag(1, −1, −1, −1) is the 4-dimensional Minkowski metric tensor and e−2σ(φ) _{is the so called exponential warping factor. The classical action for} this set-up is given by [16]:

S = Z d4 x Z dφ√g h 2M₅3R − ΛCC −X i Ti √ −giδ(φ − φi) i , (2.2) where M3

5 is the fundamental 5-dimensional mass scale, R is the 5-dimensional Ricci scalar, g is the determinant of the 5-dimensional metric and ΛCC _{is the} cosmological constant. The localized terms Tiand gi are respectively the brane tensions and the induced metric determinants in φ = 0 and φ = π. The warp factor can be determined by solving the 5D Einstein equations for the above action. Plugging the metric (2.1) into the equations leads to:

σ02= − Λ CC 24M3

5

≡ k2, (2.3)

which means that σ(φ) = kR|φ|. It's important to note that k is real only if the cosmological constant is negative; in other words, the 5D bulk is a slice of 5D Anti-de-Sitter space (AdS5). For xed φ, the metrics respects the 4D Poincaré

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 10 invariance as requested. The 4-dimensional part of the Einstein equations yields instead: 3σ00 = TU V 4M3 5 δ(φ) + TIR 4M3 5 δ(φ − π), (2.4)

where TU V and TIR are the Planck brane and TeV brane tensions respectively. By integrating this last equation in the intervals [−, ] and [π − , π + ], and by taking the limit → 0, we can obtain a relation between the two brane tensions and k:

TU V = 24M53k

TIR= −24M53k, (2.5)

so the solution holds only if TU V = −TIR is satised. The RS warped metric is then:

ds2 _{= e}−2kR|φ|

ηµνdxµdxν − R2dφ2. (2.6)

2.2 The Hierarchy Problem

The main goal of the RS model is to solve the Hierarchy Problem. Let's consider a low energy scale µ, with µ 1/(πR). The eective 4D theory can be obatined by intergrating over the extra dimension. This can be done by inserting the solution (2.7) into the classic 5D action (2.2):

S4D ⊃ Z d4_x Z dφp e g 2M₅3R ee −2kRφ , (2.7)

whereeg is the determinant ofegµν, the 4-dimensional metric describing the local gravitational uctuations hµν(x)around the at metric ηµν:

e

gµν = ηµν+ hµν. (2.8)

hµν(x) is none other than the physical 4D graviton eld; Re is the 4D Ricci scalar constructed with egµν. It's safe to suppose that at low energies the eld is independent of φ, so an explicit integration over the fth dimension can be performed. By making a comparison between the results and the known classical 4D Einstein action [17], we can obtain for the eective 4D Planck scale: M_{P l}2 = M 3 5 k h 1 − e−2kRπi. (2.9)

It's immediately clear that the energy scale of gravity depends weakly on the size of the extra dimension R, provided that kRπ is fairly large.

To address the Hierarchy Problem let's introduce a scalar eld H living on the TeV brane, with a vacuum expectation value hHi = v0. The eective 4D action for this eld is:

S4D ⊃ 1 R Z d4 x√gIR gµν_IR∂µH†∂νH − λ(H†H − v02)2 , (2.10)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 11 where gIR = g(φ = π) = R2e−8kRπ and gµν_IR = e2kRπηµν. We then dene the canonically normalized eld H ≡ ee −kRπH and obtain:

S4D ⊃ Z d4 xhηµν∂µHe†∂_νH − λe e H†H − (ee −kRπv₀)2 2i . (2.11)

Thus, while the 4D eective gravity scale MP l depends weakly on the length of the extra dimension, the eective symmetry breaking scale v ≡ e−kRπ_v

0 is exponentially suppressed, as depicted in Fig 2.3.

Figure 2.3: The exponential hierarchy.

This result allows us to obtain the scale of the electroweak symmetry break-ing with a small hierarchy. For example, if we assume that v0 ∼ MP l, with kRπ ∼ 30 we obtain:

v ∼ e−30v0 ∼ 10−16MP l∼TeV. (2.12) The scalar eld H can be identied with the Higgs boson, thus the Randall-Sundrum model provides a novel solution to the hierarchy problem: with a small hierarchy of ∼ 30 we are able to explain the large hierarchy of 16 orders of magnitude between the Planck scale and the electroweak symmetry breaking scale.

2.3 Bulk elds

Let's now consider a model with all the SM gauge bosons and fermions living in the extra dimension 1 _{. The SM particles will be identied with the zero}

1_{The Higgs boson must be conned to the TeV brane in order to generate the spontaneous}

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 12 modes of the ve-dimensional elds living in the bulk. Besides solving the hierarchy problem, this model will take care of the fermion mass hierarchy and allow the gauge coupling unication. More detailed analyses of the results will be shown in the next chapter, where a more complex and general model is considered.

2.3.1 Scalar elds

The action for a free scalar eld ϕ(x, φ), with mass m, in our model simply reads2 _[18]: S = 1 2 Z d4_x Z dφ√ggM N∂Mϕ∂Nϕ − m2ϕ2 = 1 2 Z d4_xZ _dφRh_e−2kRφ_ηµν_∂ µϕ∂νϕ − 1 R2ϕ∂5 e−4kRφ∂5ϕ − e−4kRφ_m2_ϕ2i_, (2.13)

where and integration by parts was made.

2.3.2 Kaluza-Klein decomposition of the scalar eld

By doing the Kaluza-Klein decomposition of our eld as 3_:

ϕ(x, φ) =X n

ϕ(n)(x)y√n(φ)

R , (2.14)

we are able to obtain the following action:

S = 1 2 X n,m Z d4 x Z dφ " e−2kRφηµν∂µϕ(n)(x)∂νϕ(m)(x)yn(φ)ym(φ) −ϕ(n)(x)ϕ(m)(x)yn(φ) 1 R2∂5 e−4kRφ∂5y(m)(φ) + e−4kRφm2ym(φ) !# . (2.15)

The functions yn(φ) are commonly known as proles. Now we require these functions to satisfy the following equation of motion:

− 1 R2∂5 " e−4kRφ∂5yn(φ) # + e−4kRφm2yn(φ) = m2ne −2kRφ yn(φ). (2.16)

If the equation of motion is satised, the action (2.3.2) reduces to:

S = 1 2 X n,m Z d4 x Z dφe−2kRφ " ηµν∂µϕ(n)(x)∂νϕ(m)(x) − m2nϕ(n)(x)ϕ(m)(x) # yn(φ)ym(φ). (2.17)

2_{From now on, we will use the notation} d dφ ≡ ∂5. 3_{With this notation we have [ϕ}(n)_{] = 1}_{and [y}

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 13 Finally, we require that the proles satisfy the following orthonormality

con-ditions: Z

dφe−2kRφ

yn(φ)ym(φ) = δnm, (2.18) this way we are able to obtain the 4-dimensional Kaluza-Klein action:

S = 1 2 X n Z d4x " ηµν∂µϕ(n)(x)∂νϕ(n)(x) − m2nϕ (n)_(x)2 # . (2.19)

This action describes a tower of 4D Kaluza-Klein modes ϕ(n)_{, scalar elds with} mass mn.

2.3.3 Solutions for the scalar eld

Before solving the equation of motion (2.16) it's important to note that when we made the integration by parts in (2.13) we implicitly required the elds ϕ to satisfy certain boundary conditions. More precisely, to make sure that the variation of the action is zero, the following condition must be satised:

e−4kRφyn(π)∂5yn(π) − yn(0)∂5yn(0) = 0. (2.20) This condition is satised either if yn(0) = yn(π) = 0(usually called Dirichlet-Dirichlet condition) or y0_{(0) = y}0_{(π) = 0} _{(Neumann-Neumann condition). In} this section we will impose the latter boundary conditions.

Let's now derive the form of the proles by solving the equation of motion. To do so, we make a change of variable: zn= m_knekRφand then the substitution fn(zn) = e−2kRφyn(φ). The derivative assumes the form: _dφd = kRzn_dzd_n. After this substitution the equation becomes:

z_n2d 2_f n dz2 n + zn dfn dzn + " z_n2 − 4 + m 2 k2 !# fn= 0. (2.21)

This equation is well known; the solutions to this equation dene the Bessel functions Jn and Yn. The equation has a regular singularity at 0 and an irregular singularity at innity. Going back from fn(zn) to the proles yn(φ), we can write the solution as a linear combination of the Bessel Functions:

yn(φ) = e2kRφ Nn " Jν m_n k e kRφ_{+ b} nYν m_n k e kRφ # , (2.22) where: • ν ≡ q 4 + m_k22.

• Nn are the normalization constants for each mode. • bn are constants that depend on n.

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 14 The spectrum of the KK modes can be found by imposing the boundary con-ditions. The condition y0

n(0) = 0gives us: bn = − Jν−1 mn k + (2 − ν)_mk nJν mn k Yν−1 mn k + (2 − ν)_mk nYν mn k , (2.23)

where we used the recurrence relations of the Bessel Functions:

νJν(x) + xJν0(x) = xJν−1(x), (2.24) νYν(x) + xYν0(x) = xYν−1(x). (2.25) The condition y0

n(π) = 0 gives us another equation for bn:

bn = − Jν−1 mn k e kRπ_{+ (2 − ν)} k mne −kRπ_J ν mn k e kRπ Yν−1 mn k e kRπ_{+ (2 − ν)} k mne −kRπ_Y ν mn k e kRπ . (2.26) The spectrum equation is then simply given by (2.23) = (2.26), which has to be solved numerically for mn. For each n one can then compute the numerical value of bn and, nally, calculate the value of the normalization constant using (2.18).

Also, we can compute the eective self-coupling for the Kaluza-Klein modes; the relevant action has the form:

S_int = Z d4_x Z dφ√g λ M₅3m−5 ϕ(x, φ) 2m , (2.27)

where the multiplication factor has been written in such a way that λ is a adimensional and ∼ O(1). After making the KK decomposition, we obtain for the 2nth mode: S_int(2n)= Z d4 x Z Rdφ e−4kRφ λ M₅3m−5 ϕ(2n)(x) 2m_y n(φ) √ R 2m . (2.28)

This means that the 4D eective action and the eective coupling for the 2nth mode are respectively:

S_int(2n) = Z d4_{x λ} e· ϕ(2n)(x) 2m , (2.29) λ_e = λ M₅3m−5kRm Z kRπ 0 dr e−4rh_y 2n r kR i2n . (2.30)

2.3.4 Gauge elds

Let's now consider a free abelian gauge eld AM(x, φ)in our warped bulk. The action for such eld reads:

S = − 1 4g2 5 Z d4 xdφ√g FM NFM N, (2.31)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 15 where:

FM N = ∂MAN − ∂NAM. (2.32) The gauge coupling g5is included in the gauge elds, so that [AM] = 1. Thanks to the special orbifold symmetry, it's always possible to nd a unitary gauge where the fth component A5 is zero [19]. Using this gauge we nd:

FM NFM N = gµαgνβFµνFαβ + 2gµνg55Fµ5Fα5= = e4kRφηµαηνβFµνFαβ− 2 R2e 2kRφ ηµν∂5Aµ∂5Aν. (2.33) We insert this into the action and, similarly to the scalar eld case, we make an integration by parts: S = − R 4g2 5 Z d4 xdφhηµαηνβFµνFαβ+ 2 R2η µν_A µ∂5 e−2kRφ∂5Aν i . (2.34)

2.3.5 Kaluza-Klein decomposition of the guage eld

Following the same process shown for the scalar eld, we write the gauge eld as Aµ(x, φ) =

P nA

(n)

µ (x)yn√(φ)_R . By inserting this into the action we obtain:

S = 1 g2 5 X n,m Z d4 xdφ " ηµαηνβ− 1 4 ∂µA(n)ν ∂αA (m) β − ∂µA(n)ν ∂βA(m)α −∂νA(n)µ ∂αA (m) β + ∂νA(n)µ ∂βA(m)α ynym− 1 2R2η µν_A(n) µ A(m)ν yn∂5 e−2kRφ∂5ym # . (2.35) In this case the equation of motion for the eld is:

− 1 R2∂5 e−2kRφ∂5yn = m2_nyn, (2.36)

and the orthonormality relations are: 1

g2 5

Z

dφ ynym = δnm. (2.37)

If these two equations are satised, the action becomes the one of a tower of free 4D Kaluza-Klein elds with mass mn:

S = X n Z d4 xh−1 4η µα_ηνβ_F(n) µν F (n) αβ + 1 2m 2 nA(n)µ A(n),µ i . (2.38)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 16

2.3.6 Solutions for the gauge eld

The equation of motion is slightly simpler to solve than the one of the scalar case, because the bulk mass m of the gauge eld is zero, which means that in this case ν = 2. We make the change of variable zn = m_knekRφ and the substitution fn(zn) = e−2kRφyn(φ), obtaining the Bessel dierential equation, which is solved by:

yn(φ) = 1 Nn ekRφ " J1 m_n k e kRφ_{+ b} nY1 m_n k e kRφ # . (2.39)

Imposing the Neumann-Neumann boundary conditions we can obtain two equations for bn, one given by y0n(0) = 0 and one by y

0

n(π) = 0, which leads us to the spectrum equation (to be solve numerically) that follows:

J0 mn k Y0 mn k = J0 mn k e kRπ Y0 mn k e kRπ . (2.40)

It's interesting to study the zero mode for the gauge eld. To do so we must solve the equation of motion with m0 = 0:

− 1 R2∂5 e−2kRφ∂5y0 = 0. (2.41)

Solving this equation is trivial and leads to the result: y0(φ) = 1 N0 h 1 + b0e2kRφ i (2.42) Contrary to the previous cases, the boundary condition in φ = 0 gives us:

b0 = 0, (2.43)

making the wave function constant. By imposing the orthonormalization con-dition of the prole we can compute the normalization constant:

1 = 1 g2 5 Z dφ y2 0 = 2 N2 0 g25 π 2. (2.44)

Thus, we've obtained for the zero mode the constant wave function: y0(φ) =

g5 √

π. (2.45)

2.3.7 Fermionic elds

The action for a free fermionic eld ψ(x, φ) in a warped bulk reads:

S = Z d4 xdφ√g " i 2 ¯ ψE_aAΓa∂Aψ − i 2 ∂Aψ¯ E_aAΓa− M ¯ψψ # , (2.46)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 17 where EA

a are the inverse vielbein elds, Γa are a representation of the 5D Cliord algebra (we choose the representation where Γa _{= (γ}µ_{, iγ}

5)) and M is the bulk mass of the eld. We furthermore dene our 5D Dirac spinor as the combination:

ψ = 1

2(1 − γ5)ψ + 1

2(1 + γ5)ψ ≡ ψL+ ψR, (2.47) where each eld has its own Kaluza-Klein decomposition

ψL= X n ψ_Ln(x)f n L(φ) √ R ; ψR = X n ψ_Rn(x)f n R(φ) √ R . (2.48)

We now integrate the action by parts and rearrange the terms using EA a = diag(ekRφ_{, e}kRφ_{, e}kRφ_{, e}kRφ_{, 1/R)}_{and (2.47) to obtain the form:}

S = R Z d4_x_{dφ e}−3kRφ " i ¯ψLγµ∂µψL+ i ¯ψRγµ∂µψR − e −kRφ R ¯_ψ L∂5ψR− ¯ψR∂5ψL +2ke−kRφ ¯ψLψR− ¯ψRψL − e−kRφM ¯ψLψR+ ¯ψRψL # . (2.49)

2.3.8 Kaluza-Klein decomposition of the fermionic eld

We now proceed with the combined Kaluza-Klein decomposition of the left and right fermionic elds. The decomposition leads to the action

S = X n,m Z d4 xdφ e−3kRφ " i ¯ψ_Lnγµ∂µψLmf n Lf m L + i ¯ψ n Rγ µ ∂µψRmf n Rf m R −e −kRφ R ∂5f m R − 2ke −kRφ f_Rm+ e−kRφM f_Rm ¯ψ_Lnψ_Rmf_Ln +e −kRφ R ∂5f m L − 2ke −kRφ f_Lm− e−kRφM f_Lm ¯ψn_Rψ_Lmf_Rn # . (2.50)

Contrary to the cases studied in the previous sections, in this particular case we have to require the KK modes to solve a coupled system of dierential equations:    e−kRφ R ∂5f n R− 2ke −kRφ_fn R+ e −kRφ_{M f}n R = mnfLn e−kRφ R ∂5f n L − 2ke−kRφfLn− e−kRφM fLn = −mnfRn

Furthermore, if we require the KK modes to satisfy the following orthonormal-ity relations Z dφ e−3kRφ f_Lnf_Lm = Z dφ e−3kRφ f_Rnf_Rm = δnm (2.51)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 18 we can rewrite the action in the following form:

S =X n Z d4 x " i ¯ψn_Lγµ∂µψnL+ i ¯ψRnγµ∂µψRn− mn ¯ψLnψRn + ¯ψnRψLn # = =X n Z d4 x " i ¯ψnγµ∂µψn− mnψ¯nψn # , (2.52)

where we dened the Dirac spinor for the nth mode as: ψn _{= ψ}n

L + ψRn. Therefore we obtained the action of a tower of Kaluza-Klein modes, each of mass mn.

2.3.9 Solutions for the fermionic eld

Following the same process of the previous sections leads us to these solutions of the coupled system:

f_(L,R)n (φ) = e 5 2kRφ Nn " Jν(L,R) m_n k e kRφ_{+ b}n (L,R)Yν(L,R) m_n k e kRφ # , (2.53) where we dened: ν(L,R) = 1 2 ± M k . (2.54)

It's signicant to note that if M > 1

2 k, νRbecomes negative; also, M = 0 gives νL = νR. We now focus on the left case, keeping in mind that obtaining the results for the right case only requires us to make the substitution M → −M. In order to soften the notation we will not explicitly write L and R in the next paragraph. The boundary conditions for the rst derivative in 0 and π give us two expressions for bn. Matching the two expressions we get the spectrum equation, to be solved numerically for mn:

Jν−1(z0) + 5 2 − ν z₀−1Jν(z0) Yν−1(z0) + 5 2 − ν z₀−1Yν(z0) = Jν−1(zπ) + 5 2 − ν z_π−1Jν(zπ) Yν−1(zπ) + 5 2 − ν z−1 π Yν(zπ) , (2.55)

where to further soften the notation we dened z0 = m_kn and zπ = m_knekRπ. The massless zero mode solution can be obtained by setting m0 = 0. In this case the system of equations decouples, making it a lot easier to solve (as above we focus on the left case):

e−kRφ R ∂5f 0 L− 2ke −kRφ f_L0+ e−kRφM f_L0 = 0. (2.56) The trivial solution is:

¯ f_L0(φ) = 1 N0 e(12− M k)kRφ, (2.57)

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CHAPTER 2. THE RANDALL-SUNDRUM MODEL 19 where we have introduced ¯f = e−32kRφf as the canonically normalized density

wave function. Using (2.51) we can easily compute the normalization constant, obtaining the nal result:

¯ f_L0(φ) = s kR(1 − 2M_k ) e(1−2M_k )kRπ_{− 1}e (1 2− M k)kRφ. (2.58)

Contrary to the gauge boson zero-mode prole, the fermion zero mode is not at. Instead, from (2.57), we can see that the prole depends exponentially on the bulk mass parameter M, which determines on which brane the wave function is localized. There are three dierent cases:

• For M > 1/2 k, the prole is localized close to the UV brane; • For M < 1/2 k the prole is localized on the TeV brane;

• For M = 1/2 k the prole is at, it is delocalized in the 5-dimensional bulk.

It's important to note that the variation of the action along the extra dimension impose opposite boundary conditions for each chirality. For example, if the Neumann-Neumann (or (++)) boundary conditions are imposed to the left components of the 5D Dirac fermion eld, then the boundary conditions for the right components must be Dirichlet-Dirichlet (or (−−)). It's crucial to understand that, for this reason, only one of the two zero modes can be a physical solution, resulting in one eective 4D chiral zero mode, as in the SM.

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Chapter 3 The RS model with an

intermediate brane

In the previous chapter we have seen that in the warped framework there can exist a regime of weakly coupled 5-dimensional eective theory, which allows a more denite phenomenological modeling, including a possible prototype for a Planck scale completion. In this framework, the Standard Model be-comes 5-dimensional, with its lightest modes being the known 4-dimensional SM particles, but with new phenomenological properties coming from their extra-dimensional wave function (proles). Specically, fermions naturally get diverse wave functions, which may as well lead to a new mechanism for the origin of the avor structure. Since the quantum corrections to the Higgs mass and the electroweak symmetry breaking are eectively cut o by the Kaluza-Klein excitations of the SM (which are on top of the lightest modes), the principle of naturalness then implies that these KK states must have masses of the order of the TeV scale. Electroweak precision tests strongly constrain the KK spectrum [1], but are still consistent with what observed at the LHC. On the other hand, the constraint from tests of CP violation and avor are very strict: CP and avor constraints imply mKK & O(10) TeV for the KK threshold.

In this thesis we study an extension of the standard Randall-Sundrum model that takes advantage of the fact that dierent kind of elds can prop-agate dierent amounts into the warped extra dimension [20], as illustrated in 3.1. We divide the elds in three categories: SM matter, gauge elds and gravity. Gravity must propagate in the entire length of the extra dimension in the form of a 5D General Relativity, since it is the dynamics of all space-time. However SM matter and the gauge elds can live in a smaller region1_. The two dierent regions of the fth dimension are separated by a 3-brane, a (3 + 1)-dimensional imperfection in the 5D spacetime. The new physics deriving from this framework is AdS/CFT dual to that of Vectorlike

Conne-1_{Matter elds can reside in an even smaller region than the gauge elds, but not viceversa,}

because charged matter always radiate gauge elds, but this case won't be considered in this work.

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 21

Figure 3.1: Model with an intermediate brane.

ment, described in [23], a structure safe from precision tests and a possibly a candidate for a more general dynamics that solves the hierarchy problem. As depicted in Fig 3.1, vectorlike connement represents the extension of the standard RS model to the one with an intermediate brane, resulting in dierent KK thresholds for the dierent kind of elds. In the next section we will show that the Goldberger-Wise mechanism for the radion stabilization will generate little hierarchies mKK_Matter,Gauge ≥ mKK_Grav.

From a 4-dimensional prespective, this sequence of KK thresholds is dual to a list of strong connement scales: ΛU V ≥ ΛHiggs ≥ ΛIR. The strong dynamics slowly evolves from the distant UV scale (the Planck scale) ΛU V to ΛHiggs, where the strong dynamics connes the preons (elementary point particles) into composites, among which there is the Higgs boson. In other words, if ΛHiggs ∼ O(10) TeV, we are in a scenario of Higgs compositeness at this scale [21]. This suppresses all the virutal KK-mediated electroweak, avor and CP violating eects enough to be consistent with all the precision experiments to this day. This is similar to QCD connement, where pions appear as composites of quarks and gluons.

3.1 The setup

We consider an orbifold with φ ∈ [−π, π]. Gravity lives in the bulk starting at the UV brane, with scale ΛU V ∼ MP l, and ending at the IR brane, with scale ΛIR ∼ O(1) TeV. The matter and gauge elds propagate from the UV brane to the intermediate brane (called the Higgs brane), which is taken to be around ∼ O(10) TeV, to be consistent with the avor bounds. The action in the bulk is similar to (2.2):

S = Z d4 x Z dφ√g h 2M₅3R − ΛCC(φ) −X i Ti √ −giδ(φ − φi) i , (3.1)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 22 with two key dierences:

• there is an additional tension term THiggs, relative to the intermediate brane, positioned in φ = φ0

• there can be two dierent eective cosmological constants, ΛCC

U V in [−φ0, φ0] and ΛCC

IR in [−π, −φ0] ∪ [φ0, π].

We solve the 5D Einstein equations by using the warped metric of the standard Randall-Sundrum model ds2 _{= e}−2σ(φ)_η

µνdxµdxν − R2dφ2, where σ(φ), the warp factor, has two dierent values in the bulk. We obtain two equations:

   σ02(φ) = −_24MΛCC3 5 ≡ k 2 3σ00(φ) = TU V 4M3 5δ Rφ +THiggs 4M3 5 δ R(φ − φ0) + TIR 4M3 5δ R(φ − π) The fact that there are now two dierent cosmological constatns leads us to two dierent values of k (the AdS curvature scale):

k_IR2 ≡ − Λ CC IR 24M3 5 , k_{U V}2 ≡ − Λ CC U V 24M3 5 . (3.2)

We can solve the rst equation by imposing the continuity of σ(φ) in φ0, obtaining: σ(φ) =        −kIRRφ + (kU V − kIR)Rφ0, −π < φ < −φ0 −kU VRφ, −φ0 < φ < 0 kU VRφ, 0 < φ < φ0 kIRRφ + (kU V − kIR)Rφ0, φ0 < φ < π

The plot of σ(φ) in the presence of an intermediate brane is shown in 3.2. Exactly like we have done in the case of the standard RS model, we integrate the second equation to obtain the values of the tensions Ti:

   TU V = 24M53kU V TIR = −24M53kIR THiggs= 12M53(kIR− kU V)

It's important to note that to avoid the presence of a branon2 _{(degree of} freedom associated to the bending of the intermediate brane) ghost, we must impose THiggs > 0, which means that kIR > kU V must be satised. If we integrate out the extra dimension from the action we obtain a relation between the eective 4-dimensional Planck scale and the fundamental scale M5:

M_{P l}2 = M₅3 " 1 kU V 1 − e−2kU Vφ0 ! + 1 kIR e2(kIR−kU V)φ0 _e−2kIRφ0 _{− e}−2kIRπ !# . (3.3)

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Figure 3.2: Values of σ(φ) on the orbifold with an intermediate brane.

Therefore, in this model, the presence of an intermediate brane, which is a (gravitating) physical object located in φ = φ0, with a non-zero tension, results in:

• k being dierent on the two sides of the brane

• an additional perturbativity constraint associated with the branon degree of freedom (that will not be considered in this thesis)

3.1.1 Parameters

We now make some assumptions to dene the set of parameters that will be used throughout this work. We require that the value of the energy scale relative of the intermediate brane ΛHiggsis 10 TeV, that the one relative to the IR brane ΛIR is 1 TeV and that the UV scale ΛU V is simply the Planck mass:

ΛHiggs= kU Ve−kU VRφ0 ' 10TeV,

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 24 By requiring kIR = r · kU V = r · MP l, we can write:

                 kU V = Mpl kIR= r · Mpl R = _rπM1 pl · log h _M pl ΛHiggs r · ΛHiggs ΛIR i φ0 = rπ · log Mpl ΛHiggs log h Mpl ΛHiggs r ·ΛHiggs ΛIR i

As done in [20], we choose r = 1.1, thus: R = 9.77·10−16_TeV−1 _{and φ}

0 = 2.96.

3.1.2 Radius stabilization

In the standard RS model, the radius R is stabilized (the IR scale is xed) using the so called Goldberger-Wise mechanism, which introduces a scalar eld, called radion, that propagates throughout the ve-dimensional bulk. On the UV and IR brane there is a potential for this scalar eld, which has two dierent minima on each brane, causing the vacuum expectation value of the eld to change along the extra dimension. This conguration generates a potential for the radion causing it to have a vacuum expectation value and a mass. In this extended model we expect to have two radions, the elds corresponding to the uctuations of the IR brane relative to the Higgs brane and to the ones of the Higgs brane relative the UV brane. As shown in [20], the stabilization of the two radii can be done sequentially: one stabilizes the ΛIR/ΛHiggs hierarchy, the other the ΛHiggs/ΛU V hierarchy.

We now summarize how this mechanism works in the CFT (Conformal Field Theory) language, considering the lighter radion, the one relative to the uctuations between the IR brane and the Higgs brane, that we take as xed and ∼ ΛU V. We consider the theory at ΛHiggs:

L(ΛHiggs∼ ΛU V) ⊃ LCF T + λΛU VOGW, (3.5) where OGW is Goldberger-Wise operator, a scalar operator that explicitly breaks the conformal symmetry. The operator has both energy and scaling dimension (4 − ), so that the coupling constant λ is dimensionless. We assume that OGW acquires a vacuum expectation value in the IR, breaking, this time sponta-neously, the symmetry. We interpret this scale as a VEV of the radion eld, that we denote as Φ. If we evolve the theory to the IR brane, we get for the radion potential: L(ΛIR) ⊃ (∂µΦ)2+ λ0Φ4+ cλΦ4 hΦi ΛU V !− , (3.6)

where c is a O(1) number. The last term sets the vacuum expectation value of Φ, with hΦi ∼ ΛIR. Minimizing the potential in the IR we see that the radius

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 25 is stabilized: ΛIR ∼ ΛU V − c λ λ0 !1 ΛU V, (3.7)

assuming 0 < < 1. Choosing the appriopriate values of (−cλ/λ0₎ _{and one} can generate the hierarchy between the two scales. Let's now denote by ϕ the phisical radion, which corresponds to the uctuations of Φ around its VEV: Φ ∼ ΛIR+ ag∗ϕ, where g∗ is the self-coupling of the graviton KK modes. By inserting this into the potential we can obtain the radion mass:

m2_ϕ ∼ λ0Λ2_IR, (3.8)

which is then slightly lighter than the IR energy scale.

3.2 Scalar eld

In this section we will show in detail how the calculations for the dynamics of a free scalar eld in 5-dimensions with an intermediate brane are made. As shown in the previous sections, the presence of an intermediate brane gives us two dierent curvature scales: kU V for the UV region of the bulk and kIR for the IR region. Consequently, we will have two dierent warp factors in the two regions. Furthermore, we will add localized lagrangian terms on the three branes and study how the dynamics change. First we will study the case with localized mass terms, then the case with localized kinetic terms.

3.2.1 Localized mass terms

Let's introduce a localized lagrangian mass term LM to the free scalar eld Lagrangian:

L_M= M2(φ)ϕ2 =m_{U V}2 δ(φ) + m2_Hδ(φ − φ0) + m2IRδ(φ − π)

ϕ2, (3.9) where mU V, mH and mIR are respectively the mass terms localized on the UV, Higgs and IR brane. The action then becomes:

S = Z d4_x_dφ√_g 1 2 h gM N∂Mϕ∂Nϕ − m2ϕ2− 2M2(φ)ϕ2 i . (3.10)

Like in the two-brane case, we perform a Kaluza-Klein decomposition: ϕ(x, φ) = P

nϕ(n)(x) y_√n(φ)

R . We nd that, if the following equation for the proles is sat-ised: − 1 R2∂5 " e−4σ(φ)∂5ym(φ) # + e−4σ(φ) " m2+ 2M2(φ) # ym(φ) = m2me −2σ(φ)_y m(φ), (3.11)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 26 and if they have the following orthonormalization rules:

Z

dφ e−2σ(φ)

yn(φ)ym(φ) = δnm, (3.12) then the action can be written as a tower of Kaluza-Klein modes of mass mn:

S = 1 2 X n Z d4 x " ηµν∂µϕ(n)(x)∂νϕ(n)(x) − m2nϕ(n)2(x) # . (3.13)

We now focus on equation (3.11). In the UV region the warp factor is e−2kU VRφ,

while in the IR region is e−2kIRRφ+2(kIR−kU V)Rφ0. By plugging the IR warp factor

in (3.11), it's easy to see that the equation is exactly the same of the UV region, a part from an exponential rescaling of the mode mass:

mn→ mne−(kIR−kU V)Rφ0 ≡ ¯mn. (3.14) Thus, if we impose the periodicity of yn on the orbifold, the solution of the equation (far from 0, φ0 and π) is:

yn(φ) =            1 NU V n e 2kU VR|φ| " JνU V mn kU Ve kU VR|φ| + bU V_n YνU V mn kU Ve kU VR|φ| # , −φ0 < φ < φ0 1 NIR n e 2kIRR|φ| " JνIR ¯ mn kIRe kIRR|φ| + bIR_n YνIR ¯ mn kIRe kIRR|φ| # , φ0<φ<π∪ −π<φ<−φ0 where we dened: νIR = s 4 + m 2 k2 IR , νU V = s 4 + m 2 k2 U V . (3.15)

For each n we have four free parameters: NU V

n , NnIR, bU Vn and bIRn . The periodicity implies the continuity in φ = 0 and in φ = π. So the four conditions that will x the parameters are: the normalization of the function and the three conditions for the rst derivative in 0, φ0 and π. The continuity of the function in φ0 will also dene the spectrum.

The three conditions for the rst derivative can be obtained by integrating the equation of motion in [−, ], [φ0 − , φ0 + ] and [π − , π + ], and then by taking the limit → 0. We nd:

dyn dφ(0 +_{) −}dyn dφ(0 − ) = 2m2_{U V}R2yn(0), dyn dφ(φ + 0) − dyn dφ(φ − 0) = 2m 2 HR 2_y n(φ0), (3.16) dyn dφ(π + ) − dyn dφ(π − ) = 2m2_IRR2yn(π).

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 27 We can calculate the value of bU V

n from the condition of the rst derivative in φ = 0: bU V_n = −JνU V−1(z0) + (2 − νU V)z −1 0 JνU V(z0) − m2 U VR kU V z −1 0 JνU V(z0) YνU V−1(z0) + (2 − νU V)z −1 0 YνU V(z0) − m2 U VR kU V z −1 0 YνU V(z0) , (3.17)

where we used the Bessel function recurrence relations and dened z0 = _km_{U V}n. In the same way, we can obtain bIR

n from the condition in φ = π: bIR_n = −JνIR−1(zπ) + (2 − νIR)z −1 π JνIR(zπ) + m2 IRR kIR z −1 π JνIR(zπ) YνIR−1(zπ) + (2 − νIR)z −1 π YνIR(zπ) + m2 IRR kIR z −1 π YνIR(zπ) , (3.18)

where we analogously dened zπ = _km¯_IRnekIRRπ = _km_IRnekIRRπ−(kIR−kU V)Rφ0. Continuity in φ0 simply gives:

1 NU V n e2kU VRφ0 h JνU V(zφ0) + b U V n YνU V(zφ0) i = = 1 NIR n e2kIRRφ0h_J νIR(¯zφ0) + b IR n YνIR(¯zφ0) i . (3.19) Where zφ0 = mn kU Ve kU VRφ0 and ¯z φ0 = ¯ mn kIRe kIRRφ0 ₌ mn kIRe kU VRφ0. To soften the

notation (and for later use), we dene the two constants B1 and B2: B1 =JνU V(zφ0) + b U V n YνU V(zφ0), B2 =JνIR(¯zφ0) + b IR n YνIR(¯zφ0). (3.20) So equation (3.19) becomes 1 NU V n e2kU VRφ0_B 1 = 1 NIR n e2kIRRφ0_B 2. (3.21)

The condition for the derivative in φ0 gives us the last equation we need to obtain the equation for the spectrum:

e2(kIR−kU V)Rφ0 h JνIR−1(¯zφ0)+(2 − νIR)¯z −1 φ0 JνIR(¯zφ0)+ bIR_n YνIR−1(¯zφ0) + (2−νIR)¯z −1 φ0YνIR(¯zφ0) i = NIR n NU V n h JνU V−1(zφ0) + (2 − νU V) + 2m2 HR kU V z−1_φ 0JνU V(zφ0)+ bU V_n YνU V−1(zφ0) + (2 − νU V) + 2m2 HR kU V z−1_φ 0YνU V(zφ0) i . (3.22)

Finally, combining equations (3.21) and (3.22), we obtain an equation that depends only on mn, that we can solve numerically:

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 28 mH = mIR= 0 mH = 0.5 kU V mH = kU V mH= 100 kU V mIR= 0.5 kU V mIR= kU V m1 4.21 4.22 4.22 4.22 4.73 5.24 m2 7.71 7.73 7.74 7.74 8.07 8.62 m3 11.18 11.24 11.26 11.27 11.45 11.96 m4 14.63 14.78 14.83 14.85 14.84 15.30

Table 3.1: Masses of the four lightest modes of a scalar KK, varying the localized mass terms mH and mIR in TeV; νU V = νIR= 2.

JνIR(¯zφ0) + b IR n YνIR(¯zφ0) JνU V(zφ0) + b U V n YνU V(zφ0) = JνIR−1(¯zφ0) + b IR n YνIR−1(¯zφ0) + (2 − νIR)¯z −1 φ0 h JνIR(¯zφ0) + b IR n YνIR(¯zφ0) i JνU V−1(zφ0) + b U V n YνU V−1(zφ0) + h (2 − νU V) + 2m2 HR kU V i z_φ−1 0 h JνU V(zφ0) + b U V n YνU V(zφ0) i .

The values of mn of the three lightest modes varying the localized terms in shown in table (3.1). The normalization constants NU V

n and NnIR are given by the orthonormality relations ():

1 = Z φ0 0 dφ 1 (NU V n )2 e2kU VRφ h JνU V m_n kU V ekU VRφ + bU V_n YνU V m_n kU V ekU VRφ i2 + +e2(kIR−kU V)Rφ0 Z π φ0 dφ 1 (NIR n )2 e2kIRRφ h JνIR m¯_n kIR ekIRRφ + bIR_n YνIR m¯_n kIR ekIRRφ i2 . (3.23) Dening the two integrals I1 and I2 (to be solved numerically)

I1 = Z φ0 0 dφe2kU VRφ h JνU V m_n kU V ekU VRφ + bU V_n YνU V m_n kU V ekU VRφ i2 , (3.24) I2 = e2(kIR−kU V)Rφ0 Z π φ0 dφ e2kIRRφ h JνIR m¯_n kIR ekIRRφ + bIR_n YνIR m¯_n kIR ekIRRφ i2 , we can rewrite the equation:

1 (NU V n )2 I1+ 1 (NIR n )2 I2 = 1. (3.25)

This equation, together with the continuity of the wave function in φ0, allows us to nd the two normalization constants:

N_nU V = r I1+ e−4(kIR−kU V)Rφ0 B₁ B2 2 I2, N_nIR = r e4(kIR−kU V)Rφ0 B₂ B1 2 I1+ I2. (3.26)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 29 0.5 1.0 1.5 2.0 2.5 3.0 ϕ 1× 1016 2× 1016 3× 1016 4× 1016 5× 1016 6× 1016 y1[ϕ], mIR= 0 0.5 1.0 1.5 2.0 2.5 3.0 ϕ 1× 1016 2× 1016 3× 1016 4× 1016 y1[ϕ], mIR= kUV 0.5 1.0 1.5 2.0 2.5 3.0 ϕ 1× 1016 2× 1016 3× 1016 4× 1016 y1[ϕ], mIR= 100 kUV

Figure 3.3: Eects of mIR on the wave functions.

3.2.2 Standard RS model limit

In the standard Randall-Sundrum model there is no intermediate brane. To study the limit of our extended model to get back to the standard one, we must remove the eects of the brane in φ0, by setting to zero its tension THiggs, which leads to kIR = kU V ≡ k. This also means that νIR = νU V ≡ ν. Setting M2_{(φ) = 0} for the sake of simplicity and manipulating the spectrum equation obtained for this model, we get:

bU V_n − bIR n h Jν(zφ0)Yν−1(zφ0) − Jν−1(zφ0)Yν(zφ0) i = 0. (3.27)

A set of solutions is given by the zeroes of the rst term, that is bIR

n = bU Vn : Jν−1(z0) + (2 − ν)z−10 Jν(z0) Yν−1(z0) + (2 − ν)z−10 Yν(z0) = Jν−1(zπ) + (2 − ν)z −1 π Jν(zπ) Yν−1(zπ) + (2 − ν)zπ−1Yν(zπ) (3.28) which is exactly the spectrum equation for a scalar eld in the standard RS model. It's important to note that there are no solutions that depend on φ0; to make sure this is true, we see that the second term in (3.27) has no zeros, it's always positive and tends towards ΛHiggs/mn for mn → 0 and mn → ∞. A plot of the second term is shown in Fig 3.4.

3.2.3 KK masses dependence on r and φ

0

In this subsection we see how the masses of the lightest three KK modes vary changing the values of r, the parameter that expresses the ratio between the

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 30 0 50 100 150 200 m(TeV) 0.5 1.0 1.5 2.0 ΛHiggs= 10 TeV, ν = 2

Figure 3.4: Second term vs mn.

two curvature scales (kIR = kU V, for r = 1), and φ0, which sets the position of the intermediate brane in the 5-dimensional bulk.

As seen before, if r = 1 the masses don't depend on φ0, because the eects of the intermediate brane are removed. Setting r = 1.1, varying φ0 between 2.94 and π, which correspond to ΛHiggs = 15TeV and ΛHiggs = 1 TeV respectively, we see that the KK modes become slightly lighter, as shown in Fig 3.5. Instead, by setting ΛHiggs = 10 TeV (which corresponds to φ0 ∼ 2.96) and varying r between r = 1 (standard RS scenario) and r = 5, it can be seen that the KK modes get heavier as r gets larger, as shown in Fig 3.6.

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Figure 3.5: Scalar KK masses varying φ0 between 2.94 and π, with r = 1.1.

m(TeV)

r

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3.2.4 Localized kinetic terms

In this case we introduce a Lagrangian term LK = A(φ)∂µϕ∂µϕin the action, where A(φ) = AU Vδ(φ) + AHδ(φ − φ0) + AIRδ(φ − π)determines the strength of the localized kinetic terms. Thus, we obtain the following action:

S = Z d4_x_dφ√_g1 2 h gM N∂Mϕ∂Nϕ − m2ϕ2+ 2A(φ)∂µϕ∂µϕ i . (3.29) Similarly to the previous case, we obtain the equation of motion for the proles yn(φ): − 1 R2∂φ " e−4σ(φ)∂φym(φ) # + e−4σ(φ)m2ym(φ) = m2me −2σ(φ) " 1 + 2A(φ) # ym(φ), (3.30) and their normalization rules3_:

Z

dφ e−2σ(φ)_{1 + 2A(φ)}_y

n(φ)ym(φ) = δnm. (3.31) If these equations are satised, the action becomes the usual 4D eective action for a tower of Kaluza-Klein modes:

S = 1 2 X n Z d4_x " ηµν∂µϕ(n)(x)∂νϕ(n)(x) − m2nϕ (n)2_(x) # . (3.32)

By integrating the equation of motion around φ = 0, φ = φ0 and φ = π, we obtain the conditions for the rst derivative:

dyn dφ(0 +_{) −} dyn dφ(0 − ) = −2m2_nR2AU Vyn(0), dyn dφ(φ + 0) − dyn dφ(φ − 0) = −2m 2 nR 2 AHe2kU VRφ0yn(φ0), (3.33) dyn dφ(π +_{) −}dyn dφ(π − ) = −2m2_nR2AIRe2kIRRπ−2(kIR−kU V)Rφ0yn(π).

Please note that, contrary to the localized mass terms case, now the mode masses mn appear in the conditions for yn0(φ). By imposing the conditions in φ = 0 and φ = π, we can nd the expressions for bU V_n and bIR_n :

bU V_n = − JνU V−1(z0) + (2 − νU V)z −1 0 JνU V(z0) + AU VkU VR z0JνU V(z0) YνU V−1(z0) + (2 − νU V)z −1 0 YνU V(z0) + AU VkU VR z0YνU V(z0) , (3.34) bIR_n = − JνIR−1(zπ) + (2 − νIR)z −1 π JνIR(zπ) − AIRkIRR zπJνIR(zπ) YνIR−1(zπ) + (2 − νIR)z −1 π YνIR(zπ) − AIRkIRR zπYνIR(zπ) . (3.35)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 33 AH = AIR= 0 AH = 1 AH = 3 AH = 5 AIR = 0.5 AIR= 1 m1 4.21 4.05 2.37 2.37 0.86 0.61 m2 7.71 4.25 4.22 4.22 5.69 5.67 m3 11.18 7.74 7.74 7.74 9.27 9.26 m4 14.63 11.29 11.28 14.86 12.79 12.77

Table 3.2: Values of mnfor a scalar KK, varying the localized kinetic terms, in TeV.

νU V = νIR= 2.

We se that in this case the localized terms are multiplied by z0 and zπ, instead of z−1

0 and z −1

π as in the localized mass terms case. Manipulating the equations for yn(φ) and y0n(φ) in φ = φ0 we obtain the spectrum equation, to be solved numerically: JνIR(¯zφ0) + b IR n YνIR(¯zφ0) JνU V(zφ0) + bU Vn YνU V(zφ0) = JνIR−1(¯zφ0) + b IR n YνIR−1(¯zφ0) + (2 − νIR)¯z −1 φ0 JνIR(¯zφ0) + b IR n YνIR(¯zφ0) JνU V−1(zφ0) + b U V n YνU V−1(zφ0) + h (2 − νU V)zφ−10 − 2AHkU VRzφ0 i JνU V(zφ0) + b U V n YνU V(zφ0) .

A list of values of mn, varying AIR and AU V, is shown in table 3.2. We write an expression for the normalization constants NU V

n and NnIR by dening the two integrals: I1 =2AU V h JνU V(z0) + b U V n YνU V(z0) i2 + 2AHe2kU VRφ0 h JνU V(zφ0) + b U V n YνU V(zφ0) i2 + Z φ0 0 dφ e2kU VRφ h JνU V m_n kU V ekU VRφ + bU V_n YνU V m_n kU V ekU VRφ i2 , I2 =2AIRe2(kIR−kU V)Rφ0e2kIRRπ h JνIR(zπ) + b IR n YνIR(zπ) i2 + + e2(kIR−kU V)Rφ0 Z π φ0 dφ e2kIRRφ h JνIR m¯_n kIR ekIRRφ + bIR_n YνIR m¯_n kIR ekIRRφ i2 . (3.36) Thus, N_nU V = r I1+ e−4(kIR−kU V)Rφ0 B₁ B2 2 I2, (3.37) N_nIR = r e4(kIR−kU V)Rφ0 B₂ B1 2 I1+ I2. (3.38)

Now we vary the localized parameters AH and AIR (we don't consider AU V because its eects are suppressed by MP l) and see how the mass of the three lightest KK modes change. As illustrated in Fig 3.8a and Fig 3.8b, we see that, as AH and AIR get larger, the modes get lighter, tending towards a non-zero constant value. In Fig. 3.7 we show how the wave functions are aected by the AH parameter.

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 34 0.5 1.0 1.5 2.0 2.5 3.0 ϕ 1× 1016 2× 1016 3× 1016 4× 1016 5× 1016 y1[ϕ], AH= 1 0.5 1.0 1.5 2.0 2.5 3.0 ϕ 2× 1015 4× 1015 6× 1015 8× 1015 1× 1016 y1[ϕ], AH= 3

Figure 3.7: Eects of AH on the wave functions.

(a) Scalar KK masses as a function of AIR. m(TeV)

A_IR

(b) Scalar KK masses as a function of AH. m(TeV)

A_H

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3.2.5 Dierent kinetic terms in the bulk

In this section we will show that, in the localized mass terms case, it's possible to force the rst derivative of the wave function to be continous, even if the brane terms are dierent from zero, by assuming that the eld has dierent kinetic terms in the two regions of the bulk. Let's consider the case where we want the wave function to have the rst derivative continous in φ = π. The general lagrangian for a scalar eld with bulk mass m, localized brane terms on the three branes and two dierent kinetic terms is:

S = 1 2 Z d4 xdφ√g h 1 g2 5(φ) gM N∂Mϕ∂Nϕ − m2ϕ2− 2M2(φ)ϕ2 i = = 1 2 Z d4_x Z φ0 0 dφ√gh 1 g2 U V gM N∂Mϕ∂Nϕ − m2ϕ2− 2M2(φ)ϕ2 i +1 2 Z d4_x Z π φ0 dφ√gh 1 g2 IR gM N∂Mϕ∂Nϕ − m2ϕ2− 2M2(φ)ϕ2 i . (3.39) After the eld redenition ϕ → gU V ϕ we obtain:

S = 1 2 Z d4 x Z φ0 0 dφ√g h gM N∂Mϕ∂Nϕ − gU V2 (m 2 + 2M2(φ))ϕ2 i +1 2 Z d4_x Z π φ0 dφ√ghg 2 U V g2 IR gM N∂Mϕ∂Nϕ − gU V2 (m 2_{+ 2M}2_(φ))ϕ2i_. _(3.40) Following the same steps of the previous sections, we get the following equa-tions for the proles yn(φ) in the two regions of the bulk:

− 1 R2∂5 e−4kU VRφ_∂ 5yn +e−4kU VRφ_g2 U V m2+ 2M2(φ) yn = e−2kU VRφm2nyn (UV) − 1 R2∂5 e−4kIRRφ_∂ 5yn +e−4kIRRφ_g2 IR m2+ 2M2(φ) yn= e−2kIRRφm¯2nyn (IR) (3.41) with the normalization condition:

Z φ0 0 dφ e−2kU VRφ_y n(φ)ym(φ)+ Z π φ0 dφg2U V g2_IRe −2kIRRφ+2(kIR−kU V)Rφ0_y n(φ)ym(φ) = δnm. (3.42) The solution to these equations is given by the usual combination of Bessel functions: yn(φ) =            1 NU V n e 2kU VR|φ| " JνU V mn kU Ve kU VR|φ| + bU V_n YνU V mn kU Ve kU VR|φ| # , 0 < φ < φ0 1 NIR n e 2kIRR|φ| " JνIR ¯ mn kIRe kIRR|φ| + bIR_n YνIR ¯ mn kIRe kIRR|φ| # , φ0 < φ < π

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 36 where in this case the parameters νU V and νIR are dened as:

νU V = s 4 + m 2_g2 U V k2 U V , νIR = s 4 + m 2_g2 IR k2 IR . (3.43)

The normalizations constants NU V

n and NnIR assume the value: N_nU V = s I1+ e−4(kIR−kU V)Rφ0 B₁ B2 2g2 U V g2 IR I2, (3.44) N_nIR= s e4(kIR−kU V)Rφ0 B₂ B1 2 I1+ g2_{U V} g2 IR I2, (3.45)

where I1 and I2 are the same dened in 3.24. By sending one of the two kinetic terms to innity (gU V,IR→ 0), we can modify one of the two ν parameters while leaving the other one equal to two.

More noteworthy is the eect that these two parameters have on the junc-tion equajunc-tions for the derivative of the wave funcjunc-tion:

dyn dφ(0 +_{) −}dyn dφ(0 − ) = 2m2_{U V}g_{U V}2 R2yn(0), dyn dφ(φ + 0) − dyn dφ(φ − 0) = m 2 H(g 2 U V + g 2 IR)R 2 yn(φ0), (3.46) dyn dφ(π +_{) −}dyn dφ(π − ) = 2m2_IRg_IR2 R2yn(π).

By looking at the junction equation for φ = π, we see that, even if m2

IR 6= 0, we can obtain ∆yn(π) = 0by sending the kinetic term in the IR region to innity (gIR → 0). As we will show in the next chapter, this result assures us that even a nite value of m2

IR is compatible with the connement of the massless mode, if we assume that the scalar eld has two dierent kinetic terms in the two regions of the bulk.

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3.3 Gauge eld

In this section we study the Kaluza-Klein modes for a gauge eld, introducing localized kinetic terms on the three branes4_{. The scenario is described by the} action S = − 1 4g₅2 Z d4_x_dφ√_gh_FM N_F M N + 2A(φ)FµνFµν i , (3.47)

where g is the determinant of the metric. In the above we have rescaled the bulk kinetic term by absorbing g5into AM, so that AM has mass dimension one, the canonical dimension for a gauge boson propagating in four dimensions. Then g₅−2 has dimensions of mass. We introduced a localized kinetic term A(φ) as done previously for the scalar eld. FM N is the usual eld-strength functional of the gauge elds,

F_{M N}a = ∂MAaN − ∂NAaM + f abc_Ab

MA c

N. (3.48)

for a non-Abelian Yang-Mills theory. Thanks to the orbifold symmetry, we choose a unitary gauge where A5 = 0, thus

FM NFM N = gµαgνβFµνFαβ+ 2gµνg55Fµ5Fα5= =e4σ(φ)ηµαηνβFµνFαβ − 2 R2e 2σ(φ)_ηµν_∂ 5Aµ∂5Aν. (3.49) After an integration by parts, we obtain the action:

S = − R 4g2 5 Z d4_x_dφh_{1 + 2A(φ)}_ηµα_ηνβ_F µνFαβ+ 2 R2η µν_A µ∂5 e−2σ(φ)∂5Aν i . (3.50) Now we can do the Kaluza-Klein decomposition of the elds:

Aµ(x, φ) = X

n

A(n)_µ (x)y√n(φ)

R . (3.51)

The action becomes:

S = 1 g₅2 X n,m Z d4_x_dφh_{1 + 2A(φ)}_ηµα_ηνβ₋ 1 4 ∂µA(n)ν ∂αA(m)_β − ∂µA(n)ν ∂βA(m)α − ∂νA(n)µ ∂αA(m)_β + ∂νA(n)µ ∂βA(m)α ynym− 1 2R2η µν_A(n) µ A(m)ν yn∂5 e−2σ(φ)∂5ym i . (3.52)

In this case the orthonormality conditions for the KK modes are: 1

g2 5

Z

dφ1 + 2A(φ)ynym = δnm. (3.53) The wave function equation instead is:

− 1 R2∂5 e−2σ(φ)∂5yn = m2_nyn 1 + 2A(φ). (3.54)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 38 If these conditions are satised, the action is reduced to the standard 4D action for a free massive gauge eld:

S =X n Z d4 x h − 1 4η µα ηνβF_µν(n)F_αβ(n)+1 2m 2 nA (n) µ A (n) ν i . (3.55)

The introduction of the localized kinetic terms doesn't aect the bulk solution for the KK modes, only the boundary conditions at φ = 0, φ = φ0 and φ = π. So the form of the proles is the same of (2.39), but in this case we will have dierent bIR

n and bU Vn . The solution of (3.54) is:

yn(φ) =            1 NU V n e kU VRφ " J1 mn kU Ve kU VRφ + bU V n Y1 mn kU Ve kU VRφ # , 0 < φ < φ0 1 NIR n e kIRRφ " J1 ¯ mn kIRe kIRRφ + bIR_n Y1 ¯ mn kIRe kIRRφ # , φ0 < φ < π

and the conditions on the branes for y0

n(φ)give: dyn dφ(0 + ) −dyn dφ(0 − ) = −2m2_nR2AU Vyn(0), (3.56) dyn dφ(φ + 0) − dyn dφ(φ − 0) = −2m 2 nR 2_A He2kU VRφ0yn(φ0), (3.57) dyn dφ(π +_{) −} dyn dφ(π − ) = −2m2_nR2AIRe2kIRRπ−2(kIR−kU V)Rφ0yn(π). (3.58) As done previously, the rst and the third conditions give us an expression for bU V_n and bIR_n : bU V_n = − J0(z0) + AU VkU VR z0J1(z0) Y0(z0) + AU VkU VR z0Y1(z0) , bIR_n = − J0(zπ) − AIRkIRR zπJ1(zπ) Y0(zπ) − AIRkIRR zπY1(zπ) . (3.59)

The conditions for yn(φ) and y0n(φ) in φ = φ0 give us the spectrum equation: J1(¯zφ0) + b IR n Y1(¯zφ0) J 1(zφ0) + bU Vn Y1(zφ0) = J0(¯zφ0) + b IR n Y0(¯zφ0) J0(zφ0) + b U V n Y0(zφ0) − 2AHkU VR zφ0 J1(zφ0) + b U V n Y1(zφ0) . (3.60)

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CHAPTER 3. THE RS MODEL WITH AN INTERMEDIATE BRANE 39 AH = AIR= 0 AH = 1 AH = 3 AH = 5 AIR = 0.5 AIR= 1 m1 2.68 2.68 2.68 2.66 0.65 0.35 m2 6.09 3.79 3.21 2.80 4.29 4.26 m3 9.50 6.36 6.33 6.31 7.75 7.73 m4 12.92 10.05 10.03 10.01 11.18 11.16

Table 3.3: Values of mn for a gauge KK varying the localized kinetic terms, in TeV.

normalization constants can be found using (3.53). We dene the integrals:

I1 =2AU V h J1(z0) + bU Vn Y1(z0) i2 + 2AHe2kU VRφ0 h J1(zφ0) + b U V n Y1(zφ0) i2 + Z φ0 0 dφ e2kU VRφ h J1 m_n kU V ekU VRφ + bU V_n Y1 m_n kU V ekU VRφ i2 , I2 =2AIRe2kIRRπ h J1(zπ) + bIRn Y1(zπ) i2 + (3.61) + Z π φ0 dφ e2kIRRφ h J1 m¯_n kIR ekIRRφ + bIR_n Y1 m¯_n kIR ekIRRφ i2 . Thus, the normalization constants are:

N_nU V = 1 g5 r I1+ e−2(kIR−kU V)Rφ0 B₁ B2 2 I2, (3.62) N_nIR = 1 g5 r e2(kIR−kU V)Rφ0 B₂ B1 2 I1+ I2, (3.63) where B1 and B2 are dened as in previous section.

Now we replicate what done for the scalar case and we vary the localized parameters AH and AIR (again, not considering AU V because its eects are suppressed by MP l) and see how the mass of the three lightest KK modes change. As illustrated in Fig 3.9 and Fig 3.10 we see that, as AH and AIR get larger, the modes get lighter, tending towards a non-zero constant value.

3.3.1 Zero mode

Equation (3.54) always has a solution with zero mass and constant wave func-tion. − 1 R2∂5 e−2σ∂5y0 = 0. (3.64)

Solving this equation is trivial and leads us to the following solution:

y0(φ) =        1 NU V 0 h 1 + bU V 0 e2kU VRφ i , φ < φ0 1 NIR 0 h 1 + bIR 0 e2kIRRφ i . φ > φ0

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m(TeV)

A_IR

Figure 3.9: Gauge KK masses as a function of AIR.

Contrary to the non-zero mode case, the conditions for y0

n(φ) in 0 and π lead to bU V

0 = bIR0 = 0, making the wave function constant along the 5D bulk. Furthermore, the continuity condition for y0(φ) in φ0 gives us N0U V = N0IR ≡ N−1. Finally, the normalization of the zero mode in terms of g5, AU V, AH and AIR is determined by the normalization condition:

1 = 1 g2 5 Z π 0 dφ1 + 2A(φ)y2₀ = 2N 2 g2 5 AU V + AH + AIR+ π 2 . (3.65) So the constant value of the prole of the zero mode is:

y0 =

g5

p2(AU V + AH + AIR+π₂)

. (3.66)

3.4 Fermion elds

As in the standard Randall-Sundrum model, in this model the Kaluza-Klein decomposition of the fermion elds relates the ve dimensional action to a sum over four-dimensional particle actions mass mn. The 5D action for fermions can be written as:

S = Z d4_x_dφ√_g " i 2 ¯ ψE_aAΓa∂Aψ − i 2 ∂Aψ¯ E_aAΓa− M ¯ψψ+ 2A(φ) i 2 ¯ ψE_aµΓa∂µψ − i 2 ∂µψ¯ E_aµΓaψ !# , (3.67)

Localization of Quantum Fields in a modified Randall-Sundrum scenario

Mathematical Physical and Natural Sciences

Master's Degree in Physics